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

Abstract

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.

Figures (5)

• Figure 1: Sketch of the effect of symmetric phase shifts. The black arrows indicate the polar angle $\theta$ of particles, while the colored arrows indicate the apparent polar angle perceived by particles of different species. Particles belonging to different species are distinguished by color. The shaded sectors represent the phase shift $\alpha$. (a) When particles are moving in parallel, they tend to bend their trajectories counter-clockwise by $\alpha/2$. (b) When particles are moving in anti-parallel, they tend to bend their trajectories clockwise by $(\alpha-\pi)/2$.
• Figure 2: Numerical phase diagram in the $\alpha$-$\eta$ plane of the three-species system with $\rho_0=2.0$. Insets show the representative snapshots of size $128\times 128$ at the locations marked with symbols SM Each particle is drawn with a dot whose color represents its species.
• Figure 3: Order parameters (a) $e_{\rm s}$, (b) $\gamma_{\rm s}$, and (c) $m_{\rm s}$, and (d) effective FSS exponent $\tilde{\beta}_{\rm eff} = -\ln[m_{\rm s}(2L)/m_{\rm s}(L)] / \ln 2$. These are evaluated as functions of $\alpha$ with fixed $\eta=0.4$ for different system sizes $L=32, 64, 128, 256$ for the three-species system with $\rho_0=2$. The dashed line in (d) is drawn at the universal value, $\tilde{\beta}_{\rm BKT}=1/8$, for the BKT transition.
• Figure 4: Correlation functions for the three-species system with $\rho_0=2$ at representative $(\alpha,\eta)$ values corresponding to markers in Fig. \ref{['fig:PDQ3']}. (a) Power-law decay in the log-log scale for $r \gtrsim 10.0$. The dashed straight line is a guide for the eye. (b) Oscillatory behavior with rapidly decaying amplitude. The first zeros (vertical dashed lines) converge to a finite value as $L$ increases.
• Figure 5: (a-c) Successive snapshots of two flocks of species 1 (red) and 2 (blue), colliding at $t=0$SM with a collision front at $x=0$ (dashed line). The horizontal arrows represent the polarization field far from the collision front, while the vertical arrows represent the transverse component of the polarization field as a function of $x$. (d) Trajectories of tracer particles of each species, sitting at position $(x,y)=(0,0)$ at time $t=0$, obtained from $1000$ simulations. The mean positions at every $10$th steps are shown with symbols. Parameters: $L=128$, $Q=2$, $\rho_0=2$, $\eta=0.4$, and $\alpha=0.85\pi$.