Multi-Species Nonreciprocal Active Matter Model

• The paper introduces a minimal framework of multi-species nonreciprocal active matter that uses a constant inter-species phase shift to break action–reaction symmetry.
• It employs Vicsek-type dynamics and hydrodynamic continuum modeling to derive order parameters, phase boundaries, and scaling laws, providing clear analytic tractability.
• The work reveals emergent phenomena such as chiral flocking and vortex-cell species separation, emphasizing the role of permutation symmetry in non-equilibrium collective behavior.

A multi-species nonreciprocal active matter model describes an ensemble of $N$ self-propelled agents, each belonging to one of $Q$ distinguishable species, with interaction rules breaking action–reaction symmetry while retaining full permutation symmetry over species (Potts symmetry). The fundamental microscopic symmetry is that action–reaction is violated solely by a constant phase shift in inter-species velocity alignment, yet all species are otherwise dynamically equivalent, giving rise to a unique set of emergent phenomena including chiral flocking, vortex-cell species separation, and rich phase coexistence (Woo et al., 21 Dec 2025). This paradigm establishes a minimal, analytically tractable framework for nonreciprocal active mixtures with maximal symmetry.

1. Vicsek-Type Microscopic Dynamics with Permutational Nonreciprocity

The system consists of self-propelled particles in two spatial dimensions indexed by species label $s_n \in \{1, ..., Q\}$, with position $\mathbf{r}_n$ and orientation $\theta_n$. The evolution is governed by discrete or continuous-time generalizations of the Vicsek alignment protocol: $\begin{aligned} \theta_n(t+\Delta t) &= \Arg\bigg[\sum_{m \in \mathcal{N}_n} \exp\big(i(\theta_m(t)+\alpha_{nm})\big)\bigg] + \zeta_n(t),\ \mathbf{r}_n(t+\Delta t) &= \mathbf{r}_n(t) + v_0\,\hat{\mathbf{e}}(\theta_n(t))\,\Delta t, \end{aligned}$ where $\mathcal{N}_n$ is the set of neighbors within a radius $r_0$, $v_0$ is the self-propulsion speed, and $\zeta_n$ is angular noise of magnitude $\eta$.

Nonreciprocity is encoded via the phase shift

$$\alpha_{nm} = \alpha\,\left[1-\delta_{s_n,s_m}\right], \quad 0 \le \alpha \le \pi,$$

so intra-species alignment is standard ($\alpha_{nm}=0$), but all inter-species alignments carry the same constant shift $\alpha$. This ensures \emph{permutation invariance (Potts symmetry)}: the dynamics is invariant under species relabeling, $\mathcal{S}_Q$.

The continuous-time Langevin representation is

$$\dot\theta_n = -J \sum_{m \in \mathcal{N}_n} \sin(\theta_n - \theta_m - \alpha_{nm}) + \xi_n,$$

with $J$ the alignment strength and $\xi_n$ angular white noise.

2. Hydrodynamic and Boltzmann Continuum Description

At the continuum level, fields are resolved by species. The local angular Fourier modes are introduced: $$f_k^\mu(\mathbf{r}, t) = \left\langle\sum_{n=1}^N \delta_{s_n,\mu}\,\delta(\mathbf{r} - \mathbf{r}_n)\,e^{ik\theta_n}\right\rangle,$$ with $\rho^\mu = f_0^\mu$ (density) and $w^\mu = f_1^\mu$ (polarization field). The Boltzmann moment expansion and Ginzburg–Landau truncation yield, near onset,

$$\begin{aligned} \partial_t w^\mu &= \mu_0\,w^\mu - \xi_0\,|w^\mu|^2 w^\mu - c_1 \nabla \rho^\mu + c_2 \nabla^2 w^\mu + \dots\ \partial_t \rho^\mu &= -\nabla\cdot w^\mu + D_\rho \nabla^2 \rho^\mu + \dots \end{aligned}$$

with the complex coefficients $\mu_0, \xi_0$ depending on $\alpha$.

A further reduction in the mixed-chiral regime gives an effective $XY$-Langevin model in a co-rotating frame,

$$\dot{\phi}_n = -J_{\rm eff} \sum_{m\in \mathcal{N}_n} \sin(\phi_n - \phi_m) + \zeta_n, \qquad J_{\rm eff} = J \frac{Q-1}{Q} \cos\alpha,$$

where $\phi_n = \theta_n - \Omega_0 t$.

3. Spontaneous Symmetry Breaking and Collective Phases

Key order parameters characterize macroscopic states:

• Global polarization: $$P = \langle \frac{1}{N}|\sum_{n}\hat{\mathbf{e}}(\theta_n)|\rangle$$
• Net chirality: $$\Gamma = \langle \frac{1}{N\Delta t}\sum_{n}\sin(\theta_n(t+\Delta t) - \theta_n(t))\rangle$$
• Species (Potts) order: $$E = \langle \frac{1}{2\pi r_0^2\rho_{\rm tot}N} \sum_{|\mathbf{r}_n - \mathbf{r}_m| < r_0} \frac{Q\delta_{s_n,s_m}-1}{Q-1} \rangle$$

Depending on $(\alpha, \eta)$ and density, the model displays (Woo et al., 21 Dec 2025):

• Chiral–mixed phase: $P > 0$, $\Gamma > 0$, $E \approx 0$, quasi-long-range order (QLRO): all species participate in a synchronized, rotating flock.
• Species–separated (“vortex cell”) phase: $P \to 0$, $E > 0$, $\Gamma < 0$, Potts symmetry is spontaneously broken, and each species occupies rotating vortex domains.
• Disordered phase: $P \approx 0$, $E \approx 0$, no global order.
• Coexistence: spatially heterogeneous coexistence of chiral clusters and vortex cells.

Phase boundaries are set by:

• Hopf bifurcation condition: $\Re[\mu_0(\alpha_c, \eta)] = 0$.
• Species-separation instability: for large $\alpha \lesssim \pi$, the antisymmetric mode destabilizes.

4. Linear Stability and Bifurcation Analysis

Linearizing the hydrodynamic equations near homogeneous states allows the identification of instabilities:

• The Stuart–Landau amplitude equation for the collective order parameter $w$,

$\dot w = \mu_0 w - \xi_0 |w|^2 w,$

with Hopf bifurcation at $\Re[\mu_0] = 0$.

• Near $\alpha \to \pi$, a two-species antisymmetric fluctuation $w^1 = -w^2$ acquires a positive Lyapunov exponent:

$$\delta w_A(t) \sim e^{\Lambda t} e^{i\Theta}, \quad \Lambda \propto (\pi - \alpha)^2 > 0,$$

corresponding to lateral repulsion and rotation of antiparallel flocks.

Finite-size scaling of QLRO is quantified by

$$m_s(L) \sim L^{-\tilde{\beta}}, \quad \tilde{\beta} \le 1/8,$$

indicating a BKT-type transition at the edge of the QLRO phase.

5. Spatio-Temporal Patterns and Scaling Laws

Direct simulations reveal:

• Mixed-chiral state: all particles (all species) traverse large-scale (usually counterclockwise) orbits; local clusters exhibit coherent rotation.
• Vortex-cell state: sharply demixed, each cell predominantly a single species, with clockwise circulation.
• Coexistence regime: spontaneous nucleation, growth, and dissolution of both cluster types, exhibiting lane-like and hybrid “bubble” patterns.

Polarization correlations in the QLRO regime decay algebraically: $$C_m(\mathbf{r}) \sim r^{-\tilde{\eta}}, \qquad m_s(L) \sim L^{-\tilde{\beta}}, \quad \tilde{\beta} = \tfrac{1}{2}\tilde{\eta},$$ with $\tilde{\beta}=1/8$ the BKT threshold.

6. Physical Mechanism: Interaction of Nonreciprocity and Permutational Symmetry

The model features a uniform inter-species phase shift $\alpha$ as the only source of nonreciprocity, yet treats all species equivalently (full $\mathcal{S}_Q$ symmetry). For small $\alpha,\eta$, global chiral flocking emerges through a Hopf bifurcation, with QLRO enforced by the underlying two-dimensional symmetry.

At large $\alpha \rightarrow \pi$, inter-species alignment becomes antagonistic (anti-alignment), yielding mutual repulsion between species and breaking Potts symmetry: the minimal mechanism for spontaneous species separation into vortex-cell mosaics.

Unlike generic nonreciprocal models, where interaction matrices are asymmetric and potentially hierarchy-forming or explicit chase–run–evade, this construction yields nonreciprocal but permutation-symmetric macrodynamics, with rich collective outcomes not observed in either symmetric-reciprocal or strongly asymmetric (e.g., predator–prey) settings.

7. General Significance and Links with Broader Active Matter Models

The multi-species nonreciprocal active matter model with Potts symmetry (Woo et al., 21 Dec 2025) is distinct from previously studied classes of nonreciprocal mixtures, such as Cahn-Hilliard-type (scalar) models (Saha et al., 2020), quorum-sensing mixtures (Duan et al., 2023, Duan et al., 8 Nov 2024), and asymmetric Vicsek-type models (Lardet et al., 22 Dec 2025), in which the interaction matrices lack full species-exchange symmetry.

The chiral phases, vortex-cell tiling, and dynamic bubble/lane coexistence observed here depend crucially on the coexistence of uniform nonreciprocity and permutation invariance, and provide a minimal route to symmetry-breaking phenomena with analytically tractable order parameters, phase diagrams, and scaling laws. These results underscore the importance of symmetry constraints, even in driven, non-equilibrium active mixtures, for determining the repertoire of emergent collective states.