Circling crystals in chiral active matter with self-alignment

TL;DR

This work addresses how chirality and self-alignment compete in dense active crystals by modeling a two-dimensional solid of chiral active Brownian particles with a self-alignment torque and performing large-scale simulations alongside coarse-grained theory. The authors identify four phases—DS, CDS, CC, and VP—showing that weak chirality with strong self-alignment yields a circling crystal with global circular motion, while strong chirality suppresses global circling in favor of vortex-like velocity structures. Analyses of spatial velocity correlations and kinetic-energy spectra reveal a progression from exponential real-space correlations and $E(k)\sim k^{-1}$ behavior toward scale-free and sharper spectral features (e.g., $E(k)\sim k^{-4}$) as chirality strengthens, and velocity autocorrelations exhibit oscillations at the chiral frequency $\Omega$. The results provide experimentally testable predictions for epithelial tissues and swarming robots and motivate development of a field-theoretic description that incorporates the Lorentz-like chiral term and its coupling to self-alignment.

Abstract

We study a crystal composed of active units governed by self-alignment and chirality. The first mechanism acts as an effective torque that aligns the particle orientation with its velocity, while the second drives individual particles along circular orbits. We find that even a weak degree of chirality, when coupled with self-alignment, induces collective motion of the entire crystal along circular trajectories in space. We refer to this phase as a circling crystal. When chirality outweigh self-alignment, the circular global motion is suppressed in favor of vortex-like regions of coordinated motion. This state is characterized by oscillating spatial velocity correlations, a power law decay of the energy spectrum, and oscillatory temporal correlations. Our findings can be tested experimentally in systems ranging from epithelial tissues to swarming robots, governed by chirality and self-alignment.

Figures (4)

• Figure 1: Schematic and dynamics of chiral self-aligning active solids. (a) Schematic illustration of an active solid composed of active particles characterized by self-alignment and chirality. Here, the particle orientation, indicated by the black cap on the particle, is represented by $\hat{\mathbf{n}}$, while $\mathbf{v}$ denotes its velocity. The rotation direction, i.e. the particle chirality, is indicated through a purple circular arrow, while the self-alignment torque $\mathbf{T}_{sa}$ by a blue circular arrow. (b-e) Trajectories of a target particle within the active solid as the strengths of self-alignment and chirality are varied. The color along the trajectory encodes time, with darker points representing earlier positions and lighter points indicating later positions. The points along each trajectory are connected by gray lines to highlight the particle’s path. The simulation results presented here were obtained for reduced self-alignment values of $B = 0.1$ and $B = 0.5$, and reduced chirality values of $\omega = 0.5$ and $\omega = 40$. The other dimensionless parameters of the simulations are: $\text{Pe} = 10$, $M = 10^{-4}$, $\sqrt{\epsilon/m}/(D_r \sigma)=10^2$, and $\Phi = N \pi \sigma^2 / 4L^2 = 1.1$.
• Figure 2: Phase diagram. (a) Phase diagram in the plane of reduced self-alignment strength $B$ and reduced chirality $\omega$. Colors denote different phases which are visualized by the four snapshots (b)–(e), where vectors represent the particle velocities. (b) Circling crystal (CC) phase (light brown), where all particle velocities are aligned, and all the system performs synchronized circular orbits in space. (c) Vortex phase (VP), where the system develops vortex structures and the global circling is suppressed. (d) Disordered system (DS), where particle velocities form finite-size domains where particles are aligned. (e) Chiral disordered system (CDS), where the particles velocity are aligned in finite-size domains, but the particles show oscillatory dynamics. The orange region that separate CC and VP phases identify metastable configurations, showing circling crystal or vortex structures depending on the initial conditions. Colored rectangles in the phase diagram outline the values of parameters analyzed in successive figures. The remaining dimensionless parameters of the simulations are $\text{Pe} = 10$, $M = 10^{-4}$, $\sqrt{\epsilon/m}/(D_r \sigma)=10^2$, and $\phi = N \pi \sigma^2 / 4L^2 = 1.1$.
• Figure 3: Correlation Functions and Energy Spectra.(a), (c), and (e) Spatial velocity correlations as a function of the distance $r$ for reduced self-alignment values of $B = 0.05$, $0.2$, and $0.5$, respectively. (b), (d), and (f) Kinetic energy spectra of the system, $E(k)$, defined in Eq. \ref{['eq:energy_spectrum']}, as a function of $k$ for $B = 0.05$, $0.2$, and $0.5$. The simulations at $B=0.5$ are performed in a larger box ($180\times180$) in order to avoid finite-size effects. The other dimensionless parameters of the simulations are: $\text{Pe} = 10$, $M = 10^{-4}$, $\sqrt{\epsilon/m}/(D_r \sigma)=10^2$, and $\phi = N \pi \sigma^2 / 4L^2 = 1.1$.
• Figure 4: Velocity auto-correlation functions. (a)–(c) Velocity auto-correlation functions $C_t(t)$ for $B = 0.05$ and $\omega$ = 0.5, 50, 100. (d–f) $C_t(t)$ for $B = 0.2$ and $\omega$ = 0.5, 50, 100. (g–i) $C_t(t)$ for $B = 0.5$ and $\omega$ = 0.5, 50, 100. The simulations at $B=0.5$ are performed in a larger box (180×180) in order to avoid finite-size effects. The other dimensionless parameters of the simulations are: $\text{Pe} = 10$, $M = 10^{-4}$, $\sqrt{\epsilon/m}/(D_r \sigma)=10^2$, and $\phi = N \pi \sigma^2 / 4L^2 = 1.1$.