Active fluids are a strange class of matter. Their microscopic constituents — motor proteins dragging filaments, bacteria propelling themselves through liquid — continuously convert chemical energy into motion, keeping the system permanently out of equilibrium. At low driving, the flows are weak and disorganized. Crank up the activity, and the fluid erupts into a self-sustained storm of vortices and jets known as active turbulence. The problem, the authors argue, is that unlike ordinary inertial turbulence, where well-established control parameters and asymptotic regimes exist, active turbulence has never had a clear dividing line separating a weakly disordered flow from a genuinely collective, system-scale state. Equilibrium statistical physics classifies phases through free energies, symmetry breaking and universal scaling, all rooted in detailed balance — tools that driven systems, lacking an underlying free energy, do not obviously inherit.

The team’s solution borrows from one of the most elegant developments in modern mathematical physics: Schramm–Loewner evolution, or SLE. SLE describes a one-parameter family of random, conformally invariant curves, labeled by a diffusivity κ, that arise as the scaling limits of interfaces in critical two-dimensional models. Specific values of κ correspond to famous systems — loop-erased random walks at κ = 2, Ising model interfaces at κ = 3, and critical percolation cluster boundaries at κ = 6. Because the parameter fully determines the curve statistics, measuring κ in a complex system provides a stringent test of whether that system shares the universality class of a known critical model. Demonstrating SLE behavior also implies conformal invariance — invariance under translations, rotations, rescalings and angle-preserving transformations — an exceptionally restrictive symmetry that tightly constrains the geometry of fluctuations across all scales.

Previous work had hinted that living matter might possess such hidden symmetry. In epithelial cell layers and bacterial colonies, the nodal lines of the vorticity field — the zero-vorticity contours separating regions of clockwise and counterclockwise rotation — had been found to follow the statistics of critical percolation, corresponding to SLE with κ = 6. But it remained unclear whether these statistics reflected a robust transition controlled by activity or were incidental features of particular biological systems. The new study settles the question by combining two independent experimental platforms with large-scale simulations. The first experiment used the canonical microtubule–kinesin active nematic, where activity is tuned through ATP concentration; the second used dense suspensions of swimming Bacillus subtilis, where oxygen availability controls self-propulsion.

The results were strikingly consistent. In the microtubule–kinesin system, at ATP concentrations above roughly 18 micromolar, the extracted vorticity nodal lines obeyed the analytical left-passage probability prediction for κ = 6, and the Loewner driving function behaved as a Brownian process with variance growing linearly in time at exactly the rate expected for SLE6. Below about 8 micromolar ATP, both signatures vanished. In the bacterial suspensions, high-oxygen conditions produced the same critical percolation statistics, while oxygen depletion — with the transition estimated to fall between 0.07 and 0.125 millimolar dissolved oxygen — destroyed them. Crucially, the velocity and vorticity fields look visually indistinguishable on either side of the threshold; the difference is not in how the flow appears, but in the symmetry class of its geometry.

Simulations revealed why. In the hydrodynamic active nematic model, where a nematic order parameter couples to flow through an active stress proportional to the activity parameter ζ, the same SLE6 signatures emerge only above a critical activity ζc. Below that threshold, the left-passage probability deviates from the analytical form and the driving function loses its Brownian character. Within the SLE framework, this breakdown corresponds directly to the loss of conformal invariance. The activity-controlled crossover is therefore a genuine symmetry-breaking transition — but of an unusual kind, in which the symmetry gained or lost is conformal, imposing exceptionally strong constraints on the flow geometry across scales. Even more remarkably, when the team replaced explicit active stress with stochastic forcing in a fluctuating nematohydrodynamic model, the conformally invariant state still appeared whenever detailed balance was violated, and disappeared when the fluctuations were constrained to satisfy the fluctuation–dissipation relation. Conformal symmetry, it turns out, is a generic consequence of non-equilibrium driving, not a property of any particular microscopic mechanism.

The transition is not merely geometric. Vortices are carriers of stress and momentum, so the team went on to ask how the vortex population reorganizes at the same threshold. Detecting vortex cores from the winding of the velocity field and connecting nearby vortices into geometric graphs, they found that below the transition, a spanning cluster forms only at connection distances far larger than the natural vortex spacing, while above ζc the percolation threshold snaps to roughly one characteristic length. More tellingly, a rigidity percolation analysis using the pebble game algorithm showed that precisely at ζc, a giant mechanically rigid cluster spanning the entire system emerges — the flow switches from a floppy collection of independent vortices to a self-supporting mechanical network. Both experimental systems showed the same trend, with high-activity states exhibiting larger geometric clusters and rigid clusters than their low-activity counterparts.

A third, fully independent diagnostic came from topology. Using persistent homology — a method that tracks how connected components and loops in the vorticity landscape are born and die as the field is systematically ‘flooded’ — the researchers converted persistence diagrams into persistence images and applied principal component analysis. The low-activity and high-activity regimes separated cleanly along the dominant principal component, with the transition occurring at the same critical activity identified by the SLE diagnostics and the vortex network analyses. Below the threshold, topological features had similar lifetimes, reflecting smooth, weakly structured flows; above it, the lifetime distributions broadened into long tails characteristic of heterogeneous, intermittent fields. Three independent lenses — nodal line geometry, vortex mechanics and field topology — all converge on the same critical point, establishing the transition as a robust, system-wide reorganization.

The implications reach well beyond soft matter laboratories. Because changes in flow organization in biological systems are tied to transport, mixing and force transmission, the identified threshold marks the point at which an active flow becomes coordinated across the entire system, both geometrically and mechanically. The work also connects active turbulence to a growing family of non-equilibrium systems in which conformal invariance has recently been spotted, including driven amorphous solids, water-wave turbulence and liquid-crystal interfaces, and it dovetails with theoretical efforts linking conformal symmetry, topology and non-equilibrium field theory. Open questions remain — most strikingly why the critical percolation of nodal lines and the rigidity percolation of vortex centres, distinct universality classes in equilibrium settings, coincide here. But the message is clear: fully developed active turbulence is not merely a disordered regime. It is a non-equilibrium critical state, and the boundary of that state can now be measured, in a microscope, with nothing more than the zeros of a vorticity field.

Subject of Research: The onset of fully developed active turbulence as an activity-driven non-equilibrium phase transition with critical percolation and rigidity statistics

Article Title: Fully developed active turbulence defined through a non-equilibrium phase transition

Article References: Bonn, L., Ma, T., Bantysh, O., Feng, W., Pedersen, M. C., Jing, G., Ignés-Mullol, J., Sagués, F., Araujo, N. A. M., & Doostmohammadi, A. (2026). Fully developed active turbulence defined through a non-equilibrium phase transition. Nature Physics. https://doi.org/10.1038/s41567-026-03408-y

Image Credits: AI Generated

DOI: 10.1038/s41567-026-03408-y

Keywords: active matter, active turbulence, phase transition, percolation, Schramm-Loewner evolution, conformal invariance, active nematics, bacterial suspensions, rigidity percolation, persistent homology, non-equilibrium physics, vorticity