Species-Mixed Chiral Phase

• Species-mixed chiral phase is a collective state where mixed species remain spatially integrated while exhibiting emergent chiral order.
• It arises from models like the multi-species Vicsek and Pisegna–Saha models that employ nonreciprocal and chiral couplings to induce collective rotation.
• The phase is characterized by quasi-long-range order with critical behaviors such as BKT transitions, providing insights for experimental design in active and soft matter systems.

A species-mixed chiral phase refers to a collective state in multicomponent systems—often composed of active or passive particles of distinct species—where chiral symmetry is manifest at the mesoscopic or macroscopic scale, while the different species remain spatially mixed rather than demixed. Such phases arise in diverse contexts across classical active matter, molecular systems, soft condensed phases, and quantum or statistical field theories, with the unifying feature that the interplay between multiple microscopic constituents and chirality-enforcing interactions leads to emergent chiral order with mixed-species statistics. The properties, order parameters, mechanisms of emergence, and phase boundaries of these phases are highly system-dependent but share commonalities in topology, symmetry, and critical behavior.

1. Theoretical Models and Defining Phenomenology

Species-mixed chiral phases can be rigorously defined within several theoretical frameworks. A prominent example is provided by nonreciprocal active matter models in which two or more density fields, $\rho_1(\mathbf r,t)$ and $\rho_2(\mathbf r,t)$, evolve according to coupled hydrodynamic or kinetic equations with both nonreciprocal and chiral couplings. In the minimal Pisegna–Saha model, the chemical potentials driving diffusive dynamics receive three contributions: an equilibrium (reciprocal) part, a linear nonreciprocal (off-diagonal) coupling of strength $\alpha$, and a nonlocal chiral term controlled by a pseudoscalar $\beta$ (Pisegna et al., 9 Sep 2025). The dynamical equations are

$$\partial_t \rho_a = \nabla^2 M_a - \nabla \cdot \mathbf{j}_a^c$$

where the chiral current $\mathbf{j}_a^c$ derives from the cross product of gradients of the two species densities.

A parallel, agent-based approach arises in the multi-species Vicsek model with nonreciprocal phase-shifted velocity alignment: $$\theta_n(t+\Delta t) = \operatorname{Arg} \sum_{m \in \mathcal N_n} e^{i[\theta_m(t) + \alpha_{nm}]} + \zeta_n$$ with $\alpha_{nm}=0$ for same-species and $\alpha$ for different species, and noise $\zeta_n$ (Woo et al., 21 Dec 2025, Woo et al., 21 Dec 2025). Here, collective motion and chiral rotation emerge when the phase shift $\alpha$ is nonzero and noise is subcritical.

The key phenomenological fingerprints of the species-mixed chiral phase are:

• The system remains spatially and compositionally mixed (no macroscopic demixing).
• Global or mesoscale chiral order arises: spontaneous collective rotation, nonzero angular momentum, or circulating currents.
• Signature quasi–long-range order (QLRO) in polarization or orientational correlations, reflected in algebraic decay of correlation functions, distinguishing the phase from fully disordered regimes.

2. Microscopic and Mesoscopic Order Parameters

Detection and quantification of the species-mixed chiral phase rely on appropriate order parameters:

Order Parameter	Definition / Observable	Phase Signature
Polarization, $m$	$|\frac{1}{N}\sum_n e^{i\theta_n}|$	$m > 0$ (coherent flocking)
Chirality, $\gamma$	Time-averaged $$\langle \sin(\theta_n(t+\Delta t) - \theta_n(t))\rangle_n$$	$\gamma \neq 0$ (collective rotation)
Species-mixing, $e$	Local neighbor-based index, $e \simeq 0$ for full mixing	$e \approx 0$ in mixed chiral phase
Correlation, $C_m(r)$	$$\langle \mathbf m(\mathbf r) \cdot \mathbf m(0) \rangle$$	Algebraic decay $\sim r^{-\tilde\eta}$
Vorticity, $\omega_c$	For hydrodynamic models: $$\omega_c = \hat z \cdot (\nabla \rho_1 \times \nabla \rho_2)$$	Nonzero, alternating, or disordered

In chiral-mixed phases, $m$ and $\gamma$ are both nonzero, manifesting flocking and persistent rotation, while the mixing parameter $e$ remains low, contrasting with demixed or vortex-cell phases where $e \gg 0$ (Woo et al., 21 Dec 2025, Woo et al., 21 Dec 2025).

In hydrodynamic settings, a macroscopic chiral order appears as circulating edge currents or nonzero mean vorticity (Pisegna et al., 9 Sep 2025).

3. Mechanisms for Chiral Order and Species Mixing

Chirality in mixed-species systems is typically imparted through one or more of:

• Nonreciprocal interactions (e.g., phase-shifted alignment, asymmetric couplings),
• Intrinsic particle chirality (self-rotation / handedness),
• Chiral stresses or nonlocal coupling (as nonlocal chemical potential terms).

In models with permutation (Potts) symmetry, a symmetric phase shift $\alpha$ prevents spontaneous species segregation for $\alpha$ below a critical value, enforcing a mixed but rotating state. The phase-shifted alignment rule produces a net torque, giving rise to a Hopf bifurcation at the hydrodynamic level and universally driving the flock into counterclockwise or chiral motion (Woo et al., 21 Dec 2025, Woo et al., 21 Dec 2025).

In continuum models for scalar densities, the interplay of nonreciprocity ($\alpha$) and chirality ($\beta$) produces a phase diagram with three main regimes:

• A phase-separated regime with chiral edge currents (large $\beta$),
• A spatiotemporally disordered, species-mixed regime with locally fluctuating vorticity (comparable $\alpha$ and $\beta$),
• A traveling wave or homogeneous flocking phase at high nonreciprocity, low chirality (Pisegna et al., 9 Sep 2025).

In circle swimmer mixtures, alignment (local polar or nematic) competes with intrinsic chirality; only when the alignment timescale is short compared to the chiral precession timescale (i.e., when orbit radius $r_0$ is much larger than interaction range $R$) does a mixed, flocking phase persist (Kushwaha et al., 22 Oct 2024, Levis et al., 2019).

4. Phase Diagrams and Critical Behavior

Species-mixed chiral phases are found in sharply demarcated regions of parameter space, as revealed by extensive numerical integration and linear stability/bifurcation analysis. For the nonreciprocal multi-species Vicsek and active matter models, key boundaries include:

• The threshold in phase shift $\alpha$ (or nonreciprocity $\alpha$ and chirality $\beta$) marking transition from mixed chiral phase to species-separated or disordered states.
• The Berezinskii–Kosterlitz–Thouless (BKT)–type critical line where the finite-size scaling exponent crosses the universal value $1/8$, beyond which QLRO of polarization is lost (Woo et al., 21 Dec 2025, Woo et al., 21 Dec 2025).
• In hydrodynamic models, the conserved–Hopf bifurcation in $\alpha$ and the chiral-destabilization threshold in $\beta$ (Pisegna et al., 9 Sep 2025).

A typical schematic phase diagram for the Pisegna–Saha model in the $(\alpha,\ \beta)$ plane is:

Region	Physical State
High $\alpha$, low $\beta$	Traveling waves, macroscopic flocking
Intermediate $\alpha \sim \beta$	Spatiotemporally disordered species-mixed chiral phase
High $\beta$, low $\alpha$	Phase separation with chiral edge currents

Species-mixed chiral phases occupy a region with moderate nonreciprocity/phase shift and sufficiently low noise, below the onset of Potts-symmetry-breaking and above the flocking threshold (Woo et al., 21 Dec 2025, Pisegna et al., 9 Sep 2025).

5. Experimental Realizations and Physical Examples

Numerous experimental and simulation studies have realized or characterized species-mixed chiral phases. Distinct platforms include:

• Binary mixtures of air-fluidized, oppositely spinning disks, where at intermediate active driving and near-equal mixture ratio, the system forms a "complex chiral" phase with mixed composition and a vortex-dominated flow topology, evidenced by nonzero mean vorticity and spatial mixing of species (López-Castaño et al., 2022).
• Synthetic Janus colloids, bacterial swimmers, and chiral molecules in solution can realize both intrinsic chirality and engineered nonreciprocity via chemotactic or quorum sensing interactions (Pisegna et al., 9 Sep 2025).
• Mixtures of circle swimmers (biological or synthetic) display species-mixed chiral flocking at small intrinsic rotational velocities, demixing at threshold set by the ratio $v_0/\Omega$ (velocity to chirality), and complex superstructures when species differ in frequency (Kushwaha et al., 22 Oct 2024, Levis et al., 2019).
• Chiral molecular fluids can show racemic mixed phases above the critical temperature, with spontaneous symmetry breaking into enantiomer-enriched states below $T_c$ (Piaggi et al., 2023); mixing different chiral gases yields adjusted transition points but does not destroy the species-mixed critical regime (Presilla et al., 2015).
• In soft matter, mixtures of helical rods at two temperatures exhibit phase separation with a chiral-crystal ordering, where structural and orientational correlation functions confirm species mixing and chiral order (Chattopadhyay et al., 2023).

6. Broader Frameworks and Related Phenomena

Species-mixed chiral phases appear as generic features in systems with multiple interacting microscopic degrees of freedom and symmetry breaking, spanning active/classical, molecular, and quantum/statistical regimes:

• In quantum field theory, mixing between scalar quarkonium and tetraquark fields in extended sigma models introduces chiral symmetry breaking with mixed quark content, tuning phase transition order and critical properties (Mukherjee et al., 2013).
• In binary and multimodal chiral circle-swimmer mixtures, pattern formation can yield simultaneous macro- and microphase separation, multiscale flocking, or coexistence of racemic and chiral domains (Levis et al., 2019).
• Chiral stress balances and generalized (active) Laplace laws at interfaces provide a unifying mechanical perspective on the emergence of interfacial chiral currents in phase-separated mixtures (Pisegna et al., 9 Sep 2025).

The underlying symmetry principles—such as permutation (Potts) symmetry, nonreciprocity, and spontaneous breaking of continuous rotation symmetry—play crucial roles in allowing or forbidding species demixing, in setting phase boundaries, and in controlling the emergence of collective chiral order.

---

Principal references: (Pisegna et al., 9 Sep 2025, Woo et al., 21 Dec 2025, Woo et al., 21 Dec 2025, Kushwaha et al., 22 Oct 2024, Levis et al., 2019, Presilla et al., 2015, Piaggi et al., 2023, Chattopadhyay et al., 2023, López-Castaño et al., 2022, Mukherjee et al., 2013).