Edge Currents Shape Condensates in Chiral Active Matter

Abstract

Chiral active matter, which breaks both parity symmetry and time-reversal symmetry, is ubiquitous in living systems. Here, we introduce a minimal two-dimensional chiral active lattice gas by incorporating stochastic, biased local rotations. At low temperatures, the system coarsens into condensates with chiral orientations and faceted, crystal-like shapes. The interfaces align at characteristic angles with respect to the lattice axes and exhibit edge currents that are persistent, unidirectional, and angle-dependent. To generalise these findings, we propose a continuum theory by adding an active chiral edge current term to Model B, which reveals the essential role of active chiral transport in the interfacial dynamics of phase separation. Edge currents with $n$-fold symmetry produce condensates whose shapes resemble regular $n$-sided polygons. In the thin-interface limit, we construct an effective interface potential governing edge currents, from which the steady-state condensate geometry can be obtained, both in the lattice model and the continuum description.

Figures (7)

• Figure 1: Elementary move of the chiral active lattice gas. Starting from the present configuration (middle), a square of $2\times 2$ lattice sites is randomly selected for rotation in counterclockwise direction at probability $1-p_c$ (left) or in clockwise direction at probability $p_c$ (right). Acceptance probabilities are determined by the change of Ising energy.
• Figure 2: Simulations of the chiral active lattice gas. Panels (a)-(c) show snapshots of lattice configurations at $k_BT=1.2J$ on a $512\times512$ lattice after $10^6$ Monte Carlo sweeps, starting from random initial conditions with $40\%$ type-A (white) and $60\%$ type-B (black) particles. (a) $p_c=1$, clockwise-only rotations; (b) $p_c=1/2$, unbiased rotations; (c) $p_c=0$, counter-clockwise-only rotations. Insets show the corresponding two-dimensional pattern Fourier spectra. (d) Zoom-in of a rectangular condensate illustrating the direction of the edge current at a straight interface with the tilt angle $\theta$. Panel (e) shows a microscopic interface between type A (yellow) and type B (blue) particles. The transport of a particle of type A along the interface, via repeated rotations, contributes to the edge current. Panel (f) shows the dependence of the edge-current magnitude $j$ on the interface angle $\theta$, measured from simulations of linear interfaces. Error bars indicate the standard deviation over 12 independent runs. Insets illustrate the stability (left) and instability (right) of interfaces depending on the sign of $\mathrm{d}j/\mathrm{d}\theta$. Orange arrows indicate the current direction and magnitude of the yellow phase, which as a result evolves in the direction indicated by the pink arrows and dashed lines.
• Figure 3: Illustration and numerical solutions of the chiral active Model B, with the two-dimensional scalar field $\phi$ indicated by the colourmap. (a) The active force is defined such that it drives an edge current $\mathbf{J}_A$ (red) in a direction perpendicular to the gradient $\nabla\phi$, i.e., tangentially along the interface. The strength of the active force depends on the magnitude of $\nabla \phi$ and is modulated by the angle $\theta$ between the interface and the horizontal direction. (b) Snapshot of a field at time 8000 after a random start, with 4-fold symmetry and activity parameter $\zeta=0.1$. Panels (c) and (d) illustrate the chirality of the system, leading to mirrored orientations of stationary condensates for $\zeta=+0.1$ and $\zeta=-0.1$. Panels (e) and (f) show stationary condensates for $n=3$ and $n=5$ fold symmetry. Other parameters are $a=-1/4$, $b=1/4$, $M=1$, $K=1$, $j_0=3/2$ and $|\zeta|=0.1$.
• Figure 4: Construction of condensate shapes and orientations in the thin interface limit. (a) The interface between the inside (yellow) and outside (white) of the condensate is parameterised by the coordinates $x(s)$ and $y(s)$, with an edge current $j$ flowing along the tangential direction at angle $\theta$. (b) Effective potential $V_\text{eff}$ corresponding to Eq. \ref{['eq:jA_4fold']} (with $j_0=1$, $\zeta=0.03$, $\beta=1$) leading to the dynamics of a particle in the Newton mapping (grey ball). The overall tilt of the potential is determined by $j$ (yellow: 0.02, pink: 0.03, blue: 0.04) in relation to the steady current $j^*=0.03$. Arrows indicate the particle's velocity in our examples (considering two different velocities $v_1$ and $v_2$ for the scenario with $j=j^*$). The locus $\theta^*$ of a potential maximum, where the particle moves most slowly, determines a stable slope of the condensate contour. (c) Trajectories $\theta(s)$ of a particle in the Newton mapping for the scenarios above. (d) Condensate contours corresponding to the trajectories above. Only the potential with $j=j^*$ leads to closed contours, otherwise one obtains an inspiral (yellow), or an extended wavy interface (blue). (e) Effective potential (blue) constructed from the current-angle relation (red) obtained from simulations of the lattice model. The maximum of the potential at $\theta^* \approx 0.384$ matches with the stationary angle $\theta^* = 0.3854 \pm 0.0007$ of interfaces observed in simulations.
• Figure A1: Illustration for the proof of ergodicity, showing how the states of sites labelled 4 and 5 at the top of column (a) can be exchanged through local rotations (clockwise: red, counterclockwise: blue), without affecting other sites. The strategy of column (a) leads to the exchange of sites 8 and 9, which only matters if they are in different states. The remaining columns (b-d) show alternative strategies for the case where both sites pair 2-3 and 8-9 are composed of different states, labelled as empty and "X" (which can stand for type A and B particles, or vice versa). For those strategies, also the states of some other neighbouring sites need to be specified.