Three critical phenomena, geometry as the selector
Active matterMaterial made of components that consume energy and propel themselves — bacteria, molecular motors, migrating cells. Unlike ordinary fluids, energy enters at every point, not just at the boundary. is material made of components that consume energy and move themselves. Bacteria swimming. Molecular motors crawling along filaments. Cells migrating through tissue. Unlike ordinary fluids, where energy enters at the boundary and cascades to small scales, active matter is driven at every point — each particle is its own engine.
This local energy injection creates a new kind of physics. Active matter forms topological defectsPoints where the alignment pattern is undefined — like the crown of your head where hair whorls in every direction. “Topological” means they can’t be smoothed away; they must be created or destroyed in pairs. — points where the local alignment of particles is undefined. In nematic liquid crystalsA state of matter between liquid and solid where rod-shaped molecules flow freely but tend to align along a common direction. “Nematic” means thread-like — the molecules point the same way but aren’t arranged in layers., these defects come in two types: +1/2 defects (comet-shaped, self-propelled) and −1/2 defects (trefoil-shaped, passive). The +1/2 defects are the actors. They move, interact, pair up, and dance.
What makes active matter a multi-critical system is that the defect dance is governed by three independent critical phenomena, each with its own measurable critical exponentsNumbers that describe exactly how a system’s properties change as it approaches a tipping point. Different types of transitions have different exponents — they’re the mathematical fingerprint of the transition type. — and those exponents depend on the geometry of the surface the matter lives on.
As you increase the activity (the energy each particle injects), defects begin to appear. Below a critical activity, the system is ordered: no defects, uniform alignment. Above it, defects proliferate. The transition from zero defects to a finite density of defects is a continuous phase transitionA qualitative change in a system’s behavior — not just “more of the same” but a fundamentally different state. Water freezing, a magnet demagnetizing, a network congesting. with a well-defined critical exponent.
The surprise: the exponent depends on the topology of the surface. On a closed surface (a sphere), defect onset follows Manna universalityA universality class for phase transitions where the total particle count is conserved. Named after S. S. Manna (1991). Produces different critical exponents than the more common directed percolation class. with βThe order parameter exponent — it describes how quickly the defect density grows above the critical threshold. Different universality classes have different β values, so measuring β tells you which class you’re in. ≈ 0.639. On an open channel, it follows standard directed percolationThe “default” universality class for transitions between active and absorbing states. Like water percolating through soil but with a preferred direction. The most common critical behavior for systems that can get stuck. with β ≈ 0.583.1 These are genuinely different universality classesGroups of physically different systems that share the same critical exponents and scaling behavior near their tipping points. A magnet and a fluid can be in the same class — same math, completely different stuff.. The boundary conditions — whether the surface closes on itself — select which class governs the transition.
Once defects appear, they move. On a confined surface (like a droplet or a cell), three +1/2 defects trace out trajectories in space-time that braidA topological object recording which strands pass over which in space-time. Like intertwined ropes — the crossing pattern is fixed and can’t be changed by smoothly rearranging the strands. around each other. Braids are topological objects: they record which strand passes over which, and this pattern cannot be changed by smooth deformations.
The system selects a specific braid — and geometry determines which one. In confinement (on a disk or sphere), defects trace the golden braid: the minimum-entropy pseudo-AnosovA braid that stretches material lines exponentially with each repetition — genuine topological mixing that never simplifies back. Pseudo-Anosov braids are the “irreversible” class; finite-order braids eventually return to their starting pattern. braid with topological entropyA number measuring how fast complexity grows under braiding. TEPO = 0 means no growth (trivial). TEPO > 0 means exponential stretching — the braid genuinely mixes. Higher TEPO = more mixing per operation. TEPO = log(φ), where φ = (1+√5)/2 is the golden ratioThe number (1+√5)/2 ≈ 1.618. It appears throughout mathematics as the “most irrational” number — hardest to approximate by fractions. Here it is the minimum stretching factor for irreversible braids on three strands..2 With four defects: the silver braid at TEPO = log(1+√2).
On a flat torusA doughnut-shaped surface. In simulations, a “flat torus” is created by wrapping a rectangle so opposite edges connect — periodic boundary conditions. It has zero curvature and Euler characteristic χ = 0. with periodic boundaries: TEPO = 0 at all activity levels. The braids are finite-order — they repeat after a fixed number of exchanges, never growing in complexity. The same physics, the same equations, the same parameters — but the geometry of the container selects a completely different topological class of defect dynamics.
At a critical activity threshold, active-nematic defects reach an anti-hyperuniformA critical state in which large-scale density fluctuations are anomalously suppressed — the opposite of clustering. Reported for active topological defects at a critical activity by Guldager Andersen et al. (2025). critical state with an anomalous dimensionThe exponent η controls how correlations decay with distance at the critical point. The standard 2D classes cluster near 0.25 or below, so η = 1.04 is strikingly large. η = 1.04. This exponent matches no known universality class — not Ising, not XY, not percolation. Whether this critical threshold coincides with the periodic-to-chaotic braiding transition is ◆ our readingAn original, not-yet-peer-reviewed connection from this research program — distinct from established consensus..
The three critical phenomena are not independent. They are linked by a single organizing principle: geometry selects the universality class.
A closed surface (positive Euler characteristicA single number that captures a surface’s topology. Sphere: χ = 2. Torus (doughnut): χ = 0. Two-holed surface: χ = −2. It constrains what kinds of defects a surface can host — total defect charge must equal χ.) forces topological charge conservation — the total defect charge must equal the Euler characteristic of the surface. This constraint selects Manna universality for the onset transition and pseudo-Anosov braids for the topological dynamics. An open surface lifts the topological constraint: defects can enter and exit at the boundary, selecting directed percolation for onset and simpler braids for dynamics.
A flat torus (zero Euler characteristic) has periodic boundaries and zero curvature. No topological constraint forces defect creation. The result: TEPO = 0. The braids are trivial. The system never generates genuine topological complexity, no matter how active it becomes.
This is not a parameter effect — the equations of motion are identical in all three geometries. The initial conditions are drawn from the same distribution. Only the boundary conditions differ. The container determines the physics.
Click a geometry to see what braid class — and what topological entropy — the system selects. Same equations, same parameters, different container.
On a confined surface with three +1/2 defects, the system selects the minimum-entropy pseudo-Anosov braid. This is the simplest braid that grows exponentially under iteration — it stretches material lines by a factor of φ = (1+√5)/2 per braid operation, where φ is the golden ratio.
The topological entropy is exactly TEPO = log(φ) ≈ 0.481. This is not approximate — it is the algebraic minimum. No pseudo-Anosov braid on three strands has lower entropy. The system selects the simplest irreversible motion possible: the edge of topological chaos.
With four defects, the system selects TEPO = log(1+√2) ≈ 0.881 — the silver ratio, the next algebraic number in the sequence. Each additional defect raises the braid complexity by a predictable algebraic amount.
This connects active matter to an entirely different part of physics. In the next tutorial, we see the same topological defect unbinding — the same KT physicsKosterlitz-Thouless physics (Nobel Prize, 2016). Vortex-antivortex pairs that are bound at low activity unbind at a critical threshold. The transition is topological — driven by defect liberation, not symmetry breaking. — in cardiac tissue, where bound vortex pairs correspond to atrial flutter and unbound vortices to fibrillation.
Directed percolationA universality class for transitions where a system can get “stuck” in an absorbing state (no activity). Like water percolating through soil with gravity setting a preferred direction. The default class for active-to-dead transitions. (DP) is the default universality class for absorbing-state transitionsTransitions where the system can reach a state from which it never escapes. Once all defects are gone, the defect-free state is “absorbing” — no spontaneous defect creation can restart the dynamics. in systems with a single scalar order parameter. The Janssen-Grassberger conjectureA widely-accepted (but not proven) principle: any continuous transition to an absorbing state with a single order parameter and no special symmetry belongs to the directed percolation universality class. One of the most robust conjectures in nonequilibrium physics. states that any continuous transition from an active to an absorbing state, with a single-component order parameter and no extra symmetry or conservation law, belongs to the DP class.
Manna universality (conserved directed percolation, C-DP) is the corresponding class when the total number of particles is conserved. On a closed surface, the total topological charge is fixed by the Gauss-Bonnet theoremA deep result connecting geometry to topology: the total curvature of a closed surface is fixed by its topology (Euler characteristic). For defects, it means the total defect charge on a closed surface is a constant, regardless of dynamics.: Σ qi = χ, where χ is the Euler characteristic.3 This conservation law shifts the system from DP (β ≈ 0.583 in 2D) to Manna (β ≈ 0.639).
The measured defect-density onset exponent in confined active nematics: β ≈ 0.648, R² = 0.93.◆ our resultAn original, not-yet-peer-reviewed measurement from this research program — distinct from established consensus. This is an order-parameter exponent (how fast the defect density grows above threshold), so it is directly comparable to the Manna order-parameter exponentβ governs how the order parameter grows just past the transition. It is a different exponent from the correlation-length exponent ν, which governs how far apart correlated regions extend — the two must not be equated. β ≈ 0.639 in 2D, not to the correlation-length exponent ν. On open channels (Doostmohammadi et al. 2017): β ≈ 0.583, consistent with standard DP. The surface topology acts as a selection rule between universality classes.
Independent confirmation: de la Cotte et al. (PNAS, 2025) find hyperuniformA special type of spatial order where density fluctuations are suppressed at large scales — more orderly than random but not crystalline. A hallmark of conserved-density / hidden-order states. defect distributions in active nematics — suppressed number fluctuations of the kind expected from conserved-density universality classes.4 The connection to Manna was identified independently.
Topological entropy (TEPO) of a braid β is defined as TEPO(β) = limn→∞ (1/n) log ||βn||, where ||·|| is any submultiplicative matrix norm. For computational purposes, TEPO is bounded below by the logarithm of the spectral radiusThe largest eigenvalue (in absolute value) of a matrix. It controls how fast the matrix’s action grows under repeated application — directly connected to the stretching rate of the braid. of the reduced Burau matrixA matrix representation of a braid that encodes its crossing structure. Evaluating it at a specific parameter value gives a computable lower bound on the braid’s topological entropy. evaluated at t = −1.
For the golden braid σ1σ2−1 on three strands, the Burau matrix at t = −1 has eigenvalues φ and 1/φ, giving TEPO = log(φ) exactly. This is the minimum nonzero TEPO on B3.
The braid spectrum for higher strand counts (computed via Burau lower bounds):
B3: TEPOmin = 0.481 = log(φ) • B4: TEPOmin ≥ 0.439 • B5: TEPOmin ≥ 0.370 • B6: TEPOmin ≥ 0.313 • B9: TEPOmin ≥ 0.210
Asymptotic pattern: TEPOmin × n → ~1.89.◆ our resultThe convergence of TEPO×n and its ~1.89 limit are this program’s own computation from Burau lower bounds — a suggestive pattern, not an established theorem. The minimum entropy decreases with strand count, but the product TEPO × n converges — suggesting a universal packing limit for topological complexity per strand.
Null hypothesis test: random walks on the same geometries produce TEPO = 0 in both confined and periodic geometries (20 trials each). The golden braid requires active dynamics: geometry is necessary but not sufficient.
At the boundary between periodic and chaotic defect dynamics, Guldager Andersen et al. (Reports on Progress in Physics, 2025) report a critical exponent η = 1.04(8) that falls outside all standard universality classes.5 For comparison: 2D Ising η = 0.25, 2D XY η = 0.25, 2D percolation η = 5/24, 3D Ising η ≈ 0.036.
The value η = 1.04 is anomalously large. It suggests a new type of critical behavior specific to the periodic-to-chaotic transition in active nematic defect dynamics — a transition that involves simultaneous symmetry breaking (translational order of defect lattice) and topological change (onset of pseudo-Anosov braiding).
This is the multi-critical signal, as we read it:◆ our hypothesisThe η = 1.04 measurement is a published result (Guldager Andersen et al. 2025); interpreting it as the fingerprint of a multicritical coupling between the symmetry-breaking and topological transitions is this program’s hypothesis, not an established conclusion. the coupling between the symmetry-breaking transition and the topological transition produces a universality class that neither contains individually. On this reading a single-transition theory cannot produce η = 1.04 — it is an emergent property of the coupling.
Claim: Confinement geometry selects which braid class active nematic defects realize. Confined surfaces produce pseudo-Anosov braids with TEPO = log(φ). Flat tori produce finite-order braids with TEPO = 0.
Status: Survived. Volterra-nematic simulation confirms geometry selects braid class. Flat torus: TEPO = 0 across all activity levels. Confined geometries: TEPO = log(φ) for three defects. Random walk null produces TEPO = 0 in both geometries (20 trials each), confirming that active dynamics are required — geometry is necessary but not sufficient.
Independent support: Klein et al. (arXiv 2503.10880, March 2025) independently find golden and silver braid orbits in confined active nematics. Memarian et al. (PRL 2024) confirm the golden braid in contractile active nematics. Hillebrand & Alert (Nature Communications) show a discontinuous transition without confinement.
Claim: Defect density onset on closed surfaces follows Manna/conserved-DP universality (β ≈ 0.639), while open surfaces follow standard DP (β ≈ 0.583). The surface topology acts as a selection rule between universality classes.
Status: Survived. Measured defect-density onset exponent β ≈ 0.648 — an order-parameter exponent — matches Manna β ≈ 0.639 in 2D (R² = 0.93). Physical mechanism: topological charge conservation on closed surfaces provides the conserved quantity required for C-DP universality.
Independent support: de la Cotte et al. (PNAS 2025), “Hidden order in active nematic defects,” independently identify hyperuniform (suppressed-fluctuation) defect distributions, a signature associated with conserved-density universality classes. Doostmohammadi et al. (2017) report β ≈ 0.583 for open channels, consistent with standard DP.
Claim: The Euler characteristic χ of the surface determines the accessible braid class. χ > 0 (sphere): pseudo-Anosov. χ = 0 (torus): finite-order. χ < 0 (higher-genus surfaces): untested — predicted to produce a third class.
Status: Framework. χ > 0 and χ = 0 cases verified via Prediction #5. The discriminating test is higher-genus surfaces (χ < 0), which have never been simulated with active nematics.
Critical test: An annulus (cylinder with both ends capped) has χ = 0 but is confined. If confinement alone selects pseudo-Anosov braids, annulus should show TEPO > 0. If Euler characteristic is the selector, annulus should show TEPO = 0. This experiment discriminates between geometry-as-confinement and geometry-as-topology.
Claim: Active systems with N defects on confined surfaces select braids from the minimum-entropy pseudo-Anosov family in BN. The TEPO values form a hierarchy: log(φ), log(1+√2), … approaching the asymptotic limit TEPO × N → ~1.89.
Status: Framework. N = 3 confirmed (golden braid). N = 4 reported (silver braid). Higher N untested experimentally.
Next step: Simulate N = 5, 6, 7 defects on a sphere. Measure TEPO. If the hierarchy holds, each N selects the minimum-entropy pseudo-Anosov from the corresponding braid group.
Connected through Kosterlitz-Thouless defect unbinding. In active matter, defects unbind from periodic braids into chaotic motion at a KT-type transition. In cardiac tissue, the same physics maps directly: atrial flutter = bound vortex pairs, atrial fibrillation = unbound vortices. Clinical validation — 75–96% arrhythmia termination rates by targeting topological boundaries — confirms the mapping is not just analogy.
Three order parameters on the same node set — defect density, braid class (topological entropy), and defect lattice order — each undergoing an independent transition as activity increases. The nodes are spatial positions on the surface; the edges are declared between neighboring cells.
The geometry dependence suggests that the relational structure between nodes (which encodes the surface topology through boundary conditions) fundamentally shapes which transitions are accessible. A relational gradient operator on a flat torus sees different edge structure than on a confined disk, even though the local physics is identical.
The question: Can the shift from standard DP to Manna universality be detected as a structural change in the edge field when boundary conditions switch from open to closed? If so, the topological charge conservation that selects Manna class should appear as a constraint in the relational structure — a global invariant visible in the field that is invisible to any local measurement.
Memarian, F. et al. (2024). Minimum topological entropy of braids in active nematics. Phys. Rev. Lett.
Klein, S. et al. (2025). Golden and silver braid orbits in confined active nematics. arXiv:2503.10880.
Hillebrand, M. & Alert, R. (2025). Discontinuous defect unbinding transition in active nematics. Nature Communications.
Doostmohammadi, A. et al. (2017). Onset of meso-scale turbulence in active nematics. Nature Communications, 8, 15326.
Guldager Andersen, S., Ma, T., Katsume, M. F., Li, K., Liu, X., Pedersen, M. C., & Doostmohammadi, A. (2025). Anti-hyperuniform critical states of active topological defects. Reports on Progress in Physics. arXiv:2509.22911. doi:10.1088/1361-6633/ae075e. [η = 1.04(8) anomalous dimension].
de la Cotte, A., Pearce, D. J. G., Nambisan, J., Puggioni, L., Levy, A., Giomi, L., & Fernandez-Nieves, A. (2025). Hidden order in active nematic defects. PNAS, 122. doi:10.1073/pnas.2512147122. [Hyperuniform defect distributions].
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Ganesan, P. et al. (2025). Topological boundaries in cardiac arrhythmias. Frontiers in Network Physiology.