Spinning mixtures: nonreciprocity transfers chirality across scales in scalar densities

TL;DR

The paper addresses how microscopic chirality can influence macroscopic dynamics in a two-species active mixture by formulating a minimal, conserved-density model with nonreciprocal interactions and a chiral current. Chirality enters through a nonlocal chiral chemical potential driven by the curl of the nonreciprocal current, linking micro-scale handedness to macro-scale flows and domain formation. The interplay between chirality and nonreciprocity yields novel states, including phase separation with edge currents and a spatio-temporally disordered chiral phase, as well as traveling waves when nonreciprocity dominates. These results illuminate how large-scale chirality can emerge across scales in active matter and offer concrete predictions for experiments with spinning or chemotactic chiral colloids, with potential implications for metamaterials and biological collectives.

Abstract

A mixture of spinning particles of two different types represents a system where both nonreciprocity and chirality determine the emergent dynamics. In this work we present a minimal model for a two-species mixture of chiral active particles, formulated solely in terms of the number densities of the species. Both nonreciprocity and chirality enter the bulk part of the chemical potential, taking the form of local and non-local contributions, respectively. The chiral term manifests as the curl of the nonreciprocal current, which is non-zero when chasing interactions produces a local phase shift between the number densities. Chiral domains are localised and, as a result of number conservation, they have either positive or negative sign. The chiral domains pull in or push out particles depending on their sign and strongly modify the nonreciprocal dynamics. Their interplay generates distinctive dynamical states, including phase separation with edge currents and a spatio-temporally disordered phase with both chiral and nonreciprocal signatures.

Figures (9)

• Figure 1: Multispecies chiral systems. (a) A broken mirror symmetry in flagellar rotation causes an opposite rotation of the bacterial cell body beppu2021edge. As a result, the motion of the cell is chiral, meaning it cannot be mapped onto its mirror image by any rotation or translation. (b) Two species of active particles with an intrinsic angular velocity: activity arises through effective nonreciprocal interactions between species, which couple with the particles’ internal chirality. (c) Binary mixtures of chemically active colloids with chiral chemotaxis: a non-axisymmetric particle (for example with two mobility patches separated by a fixed angle $\theta_0$) responds to the chemical fields produced by other colloids by drifting to the left or the right w.r.t. the local chemical gradient. (d) Chirality interplays with nonreciprocity to produce phase separated states with edge currents. Nonreciprocity and chirality act together to respectively generate and twist a local polar order parameter $\bm J$pisegna2024emergent, with finite vorticity $\omega_c$ at the interface of a chiral phase-separated state.
• Figure 2: Dynamics in the steady state. (a-c) Chiral disordered state: when chirality and nonreciprocity compete, the system remains disordered in the steady state; the fields lack any definite pattern and fluctuate across a broad spectrum of length and time scales. The polar order (depicted with streamlines) and the chiral vorticity (the magnitude of the circulation of $\bm{J}$) also appear featureless. (d-f) Phase separation with edge currents: increasing $\beta$ drives the system into a macroscopic bulk phase–separated state, with circulating currents localized at the interface. Both forms of activity produce effective repulsive interactions between the two fields for the chosen parameters, as seen in panels (d) and (e). In this phase, $\omega_c$ behaves strikingly differently: domains of opposite sign form along the interface, around which $\bm J$ circulates with alternating handedness. (g) The phase diagram shows the existence of four distinct phases at the steady state. Black markers: in the absence of stochastic terms, for $\alpha=0$ and $\beta>0$, we observe only bulk phase separation in the steady state. Blue markers: when $\alpha$ dominates over $\beta$, chiral effects are undetectable in the stable traveling waves (TW). Red markers: fully disordered chiral phase where we observe spatiotemporal chaos. Green markers: bulk phase separation with edge waves. Parameters for panels (a-c): $\alpha=6, \beta=6 ,K=1, dt=0.05, T=3\times 10^4$. Parameters for bottom panels (d-f): $\alpha=1, \beta=10,K=1, dt=0.05, T=3\times 10^4$. $\omega_c = (\bm \nabla \times \bm J)_{\hat{\bm z}}$.
• Figure 3: Characterising the dynamical steady states. We compare and contrast the properties of the two chiral steady states of Fig. \ref{['fig:phase_diagram']}: chaotic disorder (red rhombus) and phase separation (green square). (a) The probability distribution functions $P$ of the chiral vorticity $\omega_c$ and the two components $J_{x,y}$ are shown. In both states, $\omega_c, J_{x,y}$, $P$ are symmetric about zero meaning that there is no coherent travelling state. The tails assume a clear exponential form in the disordered state, while in the phase separated state, it is less pronounced. (b) The static correlation function $S_c$ [see Eq. \ref{['eq:StructureChirality']}] is isotropic in the disordered phase but anisotropic in the phase separated state, reflecting that the directed edge currents still involve multiple length scales. (c) Power spectra of the densities in the disordered state reveal broad frequency fluctuations; in phase separation, edge currents produce multiple characteristic peaks.
• Figure 4: Coarsening dynamics. (a) The coarsening dynamics does not show self-similarity as clear from the snapshots of the density fields. (b) The differences are clearer when we look at patterns of $\omega_c$. At early times, the $\omega_c$ is disordered. At later times, the sources of opposite signs cancel one another to produce linear arrangements of charges along the interface. (c) The length-scale associated with the coarsening dynamics shows different coarsening regimes. The curves represent the growing $L$ for the values of $\alpha$ indicated in the legend. (d) $\langle \omega^2_c \rangle$, the average of the squared chiral vorticity demonstrates two points: the the disordered state occurs for large $\langle \omega^2_c \rangle$, it decays at later times and is then accompanied by a ripening regime similar to Oswald-ripening in passive phase separation.
• Figure 5: Linear stability of the traveling wave. We solve for the largest real eigenvalue of the matrix with $n \in [-N,N]$ and $N=100$. Here $\psi$ is the angle of perturbation, such that $\bm q= q(\cos \psi, \sin \psi)$. (a) Largest eigenvalue at different $\beta$, at $q_0=0.5,\alpha=1$, $K=1$; inset: $q=0.5$, and varying angle $\psi$, $\pi/2$ is the direction of the most unstable mode; main panel, trend in $q$ varying $\beta$ at $\psi=\pi/2, q_0=0.5, \alpha=1$,$K=1$. (b) Largest eigenvalue as a function of $\alpha$ and $\beta$. Large nonreciprocity stabilizes a wave-like solution, while large $\beta$ makes the ordered pattern unstable (parameters: $q=0.5, \psi=\pi/2, q_0=0.5$.)