Universality Classes in Active Matter

• Active matter universality classes are defined by invariant emergent critical properties regardless of microscopic details, encompassing both equilibrium-like and novel non-equilibrium behaviors.
• They span models from scalar active matter and flocking (Toner–Tu) to chiral, Lévy, and interfacial systems, each exhibiting distinct scaling laws and critical exponents.
• Comprehensive studies using simulations, hydrodynamic field theories, and renormalization group analyses validate these classes, enhancing our understanding of collective non-equilibrium phenomena.

Active matter displays a remarkable diversity of large-scale behaviors, but central to its theoretical understanding is the concept of universality classes—categories that group systems according to the invariance of their emergent critical properties under changes of microscopic details. In active systems, non-equilibrium driving, broken detailed balance, and new conserved or slow modes lead to a proliferation of universality classes, many of which have no analog in equilibrium statistical mechanics. This article surveys the primary universality classes currently established or conjectured in active matter, drawing from simulations, hydrodynamic field theories, renormalization group analyses, and experimental connections.

1. Scalar Active Matter: Ising and Beyond

A considerable body of work indicates that scalar active matter without additional order (e.g., flocking) often shares its universality class with equilibrium scalar models. For Active Ornstein–Uhlenbeck Particles (AOUPs), quorum-sensing active particles, and models of Motility-Induced Phase Separation (MIPS), finite-size scaling analyses of order parameter, susceptibility, correlation length, and structure factor all confirm Ising universality in $d=2$—both in static and dynamic exponents (Maggi et al., 2020, Gnan et al., 2022, Paoluzzi et al., 2019). The key requirements are isotropic interactions, scalar order parameter symmetry, and absence of nontrivial coupling to other conserved or slow fields.

However, subtle crossovers can emerge: adding anisotropic interactions induces a cubic term in the field theory, generically replacing the critical point by a first-order transition (Paoluzzi et al., 2019). In models where activity modulates coupling to density through more complex dynamical channels (e.g., Active Model B$^+$), two distinct universality classes can be realized: the equilibrium Wilson–Fisher class (controlling bulk phase separation) and a non-equilibrium microphase separation class, related to a strong-coupling RG fixed point (Caballero et al., 2018). The transition between these regimes is often controlled by tuning activity parameters past a threshold, with the lower critical dimension and exponents governed by the RG flow.

2. Flocking, Ordering, and the Toner–Tu Class

For systems with spontaneous breaking of rotational symmetry and the advent of large-scale collective motion ("flocking"), active matter escapes the symmetry constraints of equilibrium, and the critical behavior is governed by distinct universality classes. The standard Toner–Tu field theory—formulated in terms of a conserved density and a vector/momentum density—predicts true long-range order and anomalous scaling exponents in two dimensions, in defiance of the Mermin–Wagner theorem (Jentsch et al., 2 Feb 2024, Cavagna et al., 2020). Non-perturbative functional RG studies resolve and correct earlier discrepancies between theory and simulation, yielding predictions for the roughness exponent $\chi$, dynamical exponent $z$, and anisotropy exponent $\zeta$ that quantitatively match Vicsek-model numerics: $$\chi = \frac{13(1-d)}{40}, \quad z = \frac{27+13d}{40}, \quad \zeta = \frac{41-d}{40} \quad [2402.01316].$$ In three dimensions, increasing activity drives a crossover from equilibrium ferromagnetic fixed-point behavior ($z=2$) to active (Vicsek/Toner–Tu) universality with $z\approx1.7$, controlled by a crossover length that depends on the strength of activity (Cavagna et al., 2020).

Modifications of the canonical active field theory—for example, the inclusion of non-local interactions or Lévy-walk shear stresses—yield new classes such as a "long-range Model A" regime, in which advective nonlinearities become irrelevant at strong non-locality, reducing critical scaling to a different set of exponents and suppressing usual active self-advection-driven phenomena (Skultety et al., 2020).

3. Order in Vector and Nematic Active Matter

Two-dimensional active matter with polar or nematic symmetry demonstrates a range of critical behaviors, including QLRO with varying exponents and a variety of transition types:

• The standard Vicsek model yields true long-range order (for ferromagnetic alignment), with a first-order transition (band formation).
• Modified models with velocity reversals produce a "Vicsek–shake" class, where the phase separation is suppressed, and the system undergoes a continuous transition to quasi-long-range polar order with exponents $\beta/\nu=1/8$, $\gamma/\nu=7/4$, $\nu\approx2.4$, but with a different (algebraic, not essential) divergence of the correlation length, and without BKT defect physics (Mahault et al., 2018). Defect proliferation is suppressed due to strong density–order coupling.
• Active nematics and self-propelled rods, depending on parameters, interpolate between LRO and QLRO but have intricate defect and coarsening dynamics.

4. New Classes from Anomalous Motility and Chiral Effects

Active Lévy Matter (ALM) introduces heavy-tailed step-size distributions (superdiffusion) while preserving local polar alignment (Cairoli et al., 2019). The resulting hydrodynamics involves fractional derivatives, dramatically altering large-scale collective motion. Unlike classic Vicsek-type models (which generically display a first-order transition via banding), ALM may exhibit a continuous disorder–order transition, whose critical exponents in 2D agree with those of equilibrium long-range models only for specific values (e.g., for Lévy index $\alpha=1/2$, $\beta=1/2$, $\gamma=1$, $\nu=1$), with different, $\alpha$-dependent non-classical exponents elsewhere. This constitutes a genuinely new universality class for active matter, determined by the interplay of anomalous single-particle dynamics and local alignment (Cairoli et al., 2019).

Chiral active matter with topological features (e.g., phase-coherent agents performing chiral random walks) can organize via an inverse energy cascade into negative temperature Onsager dipoles, following a new "topological gas dynamics" universality class characterized by clustering of topological charge, vortex condensation, and critical transition thresholds set by disorder amplitude (Ivarsen, 1 Dec 2025).

5. Wetting, Interfaces, and Interfacial Universality Classes

Contact between active matter and boundaries introduces new types of critical phenomena. Active wetting transitions—for example, the sequence from total to partial wetting to dewetting in persistent active particles at a wall—are continuous and defined by universal exponents:

• Fraction of "dry sites" vanishes with exponent $\beta_1=1$
• Droplet-excess mass order parameter vanishes with exponent $\beta_2=3$

These exponents are robust across on- and off-lattice models, confirming the universality of the non-equilibrium active wetting class, distinct from equilibrium wetting transitions (Sepúlveda et al., 2018).

For fluctuations of phase-separated active interfaces, the class depends crucially on bulk conservation laws and slow modes:

• The $|\boldsymbol{q}|$KPZ class appears with only density conservation (roughness exponent $\chi\approx0.25$)
• The wet-$|\boldsymbol{q}|$KPZ class is realized with additional momentum conservation (exact $\chi=1$)
• Hyperuniform interfaces with center-of-mass-conserving noise yield $\chi=0$
• Slow bulk dynamics in crystalline or glassy active materials can produce novel crossovers or universality classes, with exponents distinguished by the particular slow modes present (Maire et al., 24 Nov 2025).

6. Multicriticality and New RG Fixed Points in Active Systems

Recent renormalization group analyses of active Ising models with coupled density and order-parameter fields reveal that incorporating active motility and density as a soft mode can render the standard Wilson–Fisher fixed point unstable. Instead, additional non-linear couplings generate new large-scale universality classes, one of which can robustly supersede the equilibrium class and dominate the critical scaling of the active Ising model (Wong et al., 8 Jul 2025). Similarly, compressible polar active fluids exhibit a rich fixed-point structure at multicriticality, with universality classes distinguished by the degree of nonequilibrium FDT violation, multiple dynamic exponents, and the presence of long-range interactions or additional conservation laws (Jentsch et al., 2022).

7. Turbulence and Scaling in Active Fluids

Active turbulence presents further examples of universality classes. In stratified driven active fluids, two distinct regimes are identified:

• The classical Kolmogorov $k^{-5/3}$ spectrum (for $R\ll 1$)
• A new active-stress-driven universality ($k^{-7/5}$ spectrum) for $R\gg 1$, where $R$ is a Richardson-like active disorder parameter. The crossover between cascades is determined by active stress and viscosity, and is directly amenable to experimental verification (Bhattacharjee et al., 2021).

Table: Selected Universality Classes in Active Matter

Physical System	Class Name / Main Reference	Key Exponents / Features
Scalar MIPS, AOUP, ALJ (isotropic)	Ising	$\beta=1/8$, $\gamma=7/4$, $\nu=1$
Active Model B$^+$: bulk separation	Wilson–Fisher	As above
Active Model B$^+$: microphase	Strong-coupling (non-eq.)	Unknown, non-WF, new non-eq. class
Vicsek / Flocking (polar order)	Toner–Tu (TT)	$\chi\approx-0.33$, $z\approx1.33$ (2D)
Vicsek–shake (velocity reversals)	New QLRO class (Mahault et al., 2018)	QLRO, $\beta/\nu=1/8$, $\nu\approx2.4$
Active Lévy Matter (fat-tailed steps)	ALM (Cairoli et al., 2019)	$\beta=1/2$ (for $\alpha=1/2$), $\nu=1$, $\gamma=1$
Interfacial roughening (density only)	$|\boldsymbol{q}|$KPZ	$\chi\approx0.25$, $z\approx2.78$
Interfacial roughening (mom. cons.)	wet-$|\boldsymbol{q}|$KPZ	$\chi=1$, $z=1$
Chiral active (topological)	Onsager/top. gas (Ivarsen, 1 Dec 2025)	Inverse energy cascade, $E(k)\sim k^{-5/3}$
Compressible polar fluids at MCP	Novel UCs (Jentsch et al., 2022)	$z$ and $\chi$ deviate from mean-field, FDT-violation

• Ising class, scalar active: (Maggi et al., 2020, Paoluzzi et al., 2019, Gnan et al., 2022)
• Active Model B$^+$ and microphase separation: (Caballero et al., 2018)
• Toner–Tu flocking: (Jentsch et al., 2 Feb 2024, Cavagna et al., 2020)
• Long-range Model A crossover: (Skultety et al., 2020)
• Vicsek–shake QLRO class: (Mahault et al., 2018)
• Active Lévy Matter: (Cairoli et al., 2019)
• Interfacial universality: (Maire et al., 24 Nov 2025)
• Active wetting: (Sepúlveda et al., 2018)
• Turbulence universality: (Bhattacharjee et al., 2021)
• Active Ising and multicritical polar fluids: (Wong et al., 8 Jul 2025, Jentsch et al., 2022)
• Chiral active, topological gas dynamics: (Ivarsen, 1 Dec 2025)

Active matter thus provides a landscape where universality emerges not only from symmetry and dimensionality, but is continuously reshaped by non-equilibrium driving, slow modes, conservation laws, and topological or anomalous transport, opening new avenues for the classification of collective phenomena.