Emergence of dynamic vortex glasses in disordered polar active fluids

Amelie Chardac,1 Suraj Shankar,2 M. Cristina Marchetti,3 and Denis Bartolo1

1Univ. Lyon, ENS de Lyon, Univ. Claude Bernard,CNRS, Laboratoire de Physique, F-69342, Lyon, France

2Department of Physics, Harvard University, Cambridge, MA 02318, USA3Department of Physics, University of California Santa Barbara, Santa Barbara, CA 93106, USA

(Dated: September 8, 2020)

In equilibrium, disorder conspires with topological defects to redefine the ordered states of matterin systems as diverse as crystals, superconductors and liquid crystals. Far from equilibrium, however,the consequences of quenched disorder on active condensed matter remain virtually uncharted.Here, we reveal a state of strongly disordered active matter with no counterparts in equilibrium: adynamical vortex glass. Combining high-content microfluidic experiments and theory, we show howcolloidal flocks collectively cruise through disorder without relaxing the topological singularities oftheir flows. The resulting state is highly dynamical but the flow patterns, shaped by a finite densityof frozen vortices, are stationary and exponentially degenerated. Quenched isotropic disorder actsas a random gauge field turning active liquids into dynamical vortex glasses. We argue that thisrobust mechanism should shape the collective dynamics of a broad class of disordered active matter,from synthetic active nematics to collections of living cells exploring heterogeneous media.

From a physicist perspective, flocks are ensembles ofliving or synthetic motile units collectively moving alonga common emerging direction [1–4]. They realize one ofthe most robust ordered states of matter observed overfive orders of magnitude in scale and in systems as di-verse as motility assays, self-propelled colloids, shakengrains and actual flocks of birds [3, 5–10]. The quietflows of flocks are in stark contrast with the spatio-temporal chaos consistently reported and predicted inactive nematic liquid crystals, another abundant formof ordered active matter realized in biological tissues,swimming cells, cellular extracts and shaken rods [2, 11].Active nematics do not support any form of long rangeorder [4, 12]. Their structure is continuously bent anddestroyed by the proliferation and annihilation of sin-gularities in their local orientation: topological defects[11, 13–15]. Unlike in active nematics, topological de-fects in flocking matter are merely transient excitationswhich annihilate rapidly and allow uniaxial order to ex-tend over system-spanning scales [4].

This idyllic view of the ordered phases of active liq-uids is limited, however, to pure systems. Disorder isknown to profoundly alter the stability of topological de-fects and the corresponding ordered states in equilibriumcondensed matter [16–18], but its role in active fluids re-mains virtually uncharted territory. All previous stud-ies [19–26], including our own early experiments [22],have been limited to weak disorder and smooth per-turbations around topologically trivial states. Unlikein equilibrium, disorder-induced topological excitationshave been out of reach of any active-matter experiment,theory and simulations.

In this article we show how isotropic disorder generi-cally challenges the extreme robustness of flocking mat-ter to topological defects. We map the full phase be-havior of colloidal flocks in disordered environments andreveal an unanticipated state of active matter: a dynam-ical vortex glass. In dynamical vortex glasses, millions

of self-propelled particles can steadily cruise though dis-order maintaining local orientational order and withoutrelaxing the topological singularities of their flows. Theassociated flow patterns are exponentially degeneratedand shaped by amorphous ensembles of frozen topologi-cal defects, yielding a dynamical state akin to the staticvortex-glass phase of dirty superconductors and random-gauge magnets [27–29]. Building a theory of flock hy-drodynamics beyond the spin-wave approximation, weelucidate the emergence and stabilization of topologicalvortices by quenched disorder. Finally, we discuss theuniversality of the dynamical vortex glass phase beyondthe specifics of polar active matter and colloidal flocks.

DISORDERED FLOCKS ON A CHIP

Our experiments are based on the model system we in-troduced in [6]. We use Quincke rotation to power inani-mate polystyrene beads of radius a = 2.4µm [30, 31], andturn them into colloidal rollers self-propelling at constantspeed ∼ 1 mm/s in circular microfluidic chambers of ra-dius R = 1.5 mm, see Fig. 1A and Movie S1. Beforeaddressing the impact of disorder on their collective dy-namics, it is worth recalling the phenomenology of thepure ordered phase [32, 33].

As the rollers are set in motion, they interact and self-assemble into a spontaneously flowing liquid equivalentlyreferred to as a flock, a Toner–Tu fluid, or a polar liq-uid [2, 34] shown in Movie S1. The resulting flow pat-terns are initially isotropic and marred by a number of±1 topological defects clearly visible in the Schlieren pat-terns of Fig. 1A and Movie S2. However, the velocity-alignment interactions responsible for flocking motion pe-nalize flow distortions thereby causing the attraction andannihilation of defects of opposite charge. The resultinglively coarsening dynamics lasts few tens of seconds andultimately yields pristine azimuthal flows with long-range

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FIG. 1. Emergence and destruction of polar flows. A, Experimental picture of a Quincke-roller fluid at the onset ofself-propulsion and subsequent Schlieren patterns of the flow field. The colormap indicates the angle of the local velocity fieldwith the x-axis. The many topological defects formed at the onset of collective motion coarsen yielding a pristine azimuthalflow. Scale bar: 1 mm. B, Schematics of the microfluidic device. 32 circular chambers are patterned with random lattices ofcircular obstacles. Each chamber correspond to a different value of Φo. A T-junction makes it possible to continuously vary thepacking fraction of Quincke rollers in all chambers at once. Same color coding as in F. Scale bar: 3 mm. C, Closeup pictureshowing Quincke rollers collectively moving through a disordered lattice of micro-fabricated obstacles. A roller trajectory showsthat the obstacles repel the active colloids at a finite distance. Scale bar: 10µm. Color bar: time, from the oldest positions(bright colors) to the latest positions (dark colors). D, Polarization fields in the vortex, meander and gas states. The dotsindicate the location of the topological defects (Orange: +1, Blue: -1). Scale bar: 0.5 mm. E, Evolution of the number oftopological defects N (t) during the coarsening process for Φo = 4 % (Polar liquid), Φo = 14 % (Meander), Φo = 34 % (Gas).The error in the number of defect measurement is smaller than the symbol size, see SI. F, Variations of the average number oftopological defects in the steady state with the obstacle fraction. The meander phase emerges at Φc = 9± 1 %.

order in two dimensions, Fig. 1A.

To comprehensively investigate how disorder altersthe phases of polar active matter, we perform high-content experiments on the microfluidic device sketchedin Fig. 1B. On a single chip, we quantify the flows ofdisordered roller fluids in 32 different disordered geome-tries. Disorder is implemented in 3 mm wide circularchambers decorated by random arrays of isotropic ob-stacles of diameter 10µm that repel the rollers at a dis-tance, while leaving their speed unaltered, see Fig. 1Cand Movie S3. We replicate the experiments increas-

ing the obstacle fraction Φo from 0 % to 38 % keepingthe system below the Lorentz localization transition ofindividual rollers [35, 36]. All experiments are system-atically performed for several disorder realizations andinitial conditions, see also Methods.

Let us first discuss the impact of disorder on a polarliquid deep in the flocking phase, for a colloid numberdensity ρ0 = 7±0.5×10−2. For small disorder, Φo < 9 %,the obstacles hardly disrupt the collective flow, as ±1defects formed at the onset of collective motion quicklyannihilate. The system coarsens to an ordered flocking

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state which, in the disk geometry, takes the form of amacroscopic vortex centered around a single +1 defect,see Fig. 1 and Movie S4. In stark contrast, increasingΦo above 9 %, a very distinct type of organizationemerges. Disorder essentially arrests the coarsening bypinning ±1 defects. While the global flow vanishes,we find locally correlated flows through meanderingpatterns bent by disorder (Fig. 1D and Movies S5 andS6). Particles stream coherently through the meanders,resulting in a finite correlation length for orientationalorder, but the structure of the flow network itself isstatic and fixed in space, as shown in Fig. 1D and MovieS6. Further increasing the obstacle fraction (above26 %), we recover the liquid-gas transition reportedin [22]. In the gas phase, particles are motile, but thereis no orientational order as defect pairs continuouslyunbind and annihilate as seen in Figs. 1D, 1E. Thetwo transitions between the three dynamical states areunambiguously determined by the variations of N (Φo),the mean number of topological defects in the velocityfield, plotted in Fig. 1F (the detection of the topologicaldefects is detailed in Supplementary Note 3). Theemergence of meandering flows is signaled by the linearincrease of N as Φo exceeds the critical value Φc = 9 %,whereas the loss of local orientational order saturatesN (Φo) above Φo = 26 %.

PHASE BEHAVIOUR OF DISORDERED POLARACTIVE MATTER

In order to quantitatively distinguish the dynamics inthe three regimes, we inspect the statistics of the time-averaged polarization defined as p(r) = 〈v(r, t)〉t, wherethe unit vector v(r, t) is the instantaneous orientationof the velocity field, see Methods. The distributions ofp(r) in Fig. 2A show that, for small disorder, the flow isuniformly polarized along the azimuthal direction, whichreflects nearly perfect polar order. Conversely, above Φc

global polar order is suppressed, and the polarization isisotropically distributed on the scale of the system. Thestrong localization of the distribution on the unit circledemonstrates, however, that meanders persistently dis-tort the streamlines without arresting the local flows.Above Φo = 26 %, in the active-gas phase, the typicalpolarization vanishes, polar order melts and flows aresuppressed at all scales.

We now demonstrate that the two flowing patterns cor-respond to two distinct dynamical phases. We first stressthat polar liquids with genuine long range orientationalorder survive at finite disorder (see also SupplementaryNote 6). Unlike active nematics, deconfining the systemdoes not suppress the stationary vortex patterns shownin Fig. 1D [37–39]. This crucial result follows from thefinite-size analysis of the polarization order parameterP(`) =

√〈pr(r)〉2r + 〈pθ(r)〉2r, where ` is the size of the

region where spatial averaging is performed. This defi-

nition of P is natural in a circular geometry where thespatial average of p vanishes, even in pure systems. Be-low Φc, Supplementary Fig. S1 clearly indicates thatP(`) converges to a finite value over a finite length scaleξP , signaling long range polar order. Conversely, deep inthe meandering regime P(`) vanishes over a finite scale.Polar liquids and meanders are therefore two genuinelydistinct phases of active matter.

The transition between these two dynamical phases iscaptured by the global polarization P ≡ P(` = 2R),where R is the chamber radius. We can see in Fig. 2Bthat P decays weakly and linearly with Φo for small disor-der, but drops sharply to 0 at the critical value Φc definedfrom the proliferation of quenched topological defects inFig 1F. The bifurcation of P in Fig. 2B suggests a crit-ical scenario. This hypothesis is further supported byFig. 2C which reveals a divergence, or at the very leasta drastic increase of the correlation length ξP at the on-set of meandering motion. The polarization P alone,however, does not distinguish the meander from the gasphase. We thus introduce the Edwards–Anderson pa-rameter QEA = 〈v(r, t) · v(r, t+ T )〉r,t,T→∞ that quanti-fies the temporal persistence of the emergent flows [40].The finite non-zero value of QEA for Φc < Φo < 26 %(Fig. 2B) confirms the persistence of polar order alongthe meanders. We also find that the continuous tran-sition, or crossover, between the meander and the gasphase, where QEA ∼ 0, coincides with that identifiedfrom the topological defect statistics in Fig. 1F.

Taking advantage of our high-content microfluidicexperiments, we measure P and QEA in more than 400experiments performed varying the roller fraction from10−3 to 1.5 × 10−1, and Φo from 0 to 38 % for multipledisorder realizations and initial conditions. The resultingphase diagram shown in Fig. 2D firmly establishes thatdisorder and topological defects do not merely offsetthe flocking transition as reported in [22], but conspireto bend the dynamics of polar liquids into amorphousand singular flows while preserving local flocking motion.

MEANDER FLOWS ARE DYNAMICAL VORTEXGLASSES

We now establish that the meanders realize a dynam-ical vortex glass. To do so, we compare the flow ori-entations of fifty replicas of the very same experiment(Φo = 15 %). We solely vary the initial conditions, keep-ing the colloid fraction and disorder realization iden-tical. We then quantify the resemblance between theflow patterns by introducing a measure of the local over-lap qαβ(r) = pα(r) · pβ(r) between replicas α and β.Examples of overlap maps are shown in Fig. 3A forfour different pairs of experiments, see also Supplemen-tary Fig. S5. The replicated flows are identical whenqαβ(r) = +1, opposite when qαβ(r) = −1 and orthogo-nal when qαβ(r) = 0. A simple inspection of the maps

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FIG. 2. Suppression of long-range orientational order. a, PDF of the polarization fields P(pr, p✓) in the three dynamicalphases state. Note that the distribution is isotropic but peaked along the unit circle in the meander phase: spontaneous flowsare locally preserved but global polar order is suppressed by disorder. b, Finite-size analysis of the polarization order parameterP(`) measured in square boxes of width `. The polarization decays exponentially towards a finite value in the polar-liquid phase(�o < �c) demonstrating the existence of genuine long-range orientational order. No polar order is observed above �c. Thechamber diameter 2R is 3 mm. c, Variation of the global polarization P and Edwards–Anderson QEA order parameters. Thepolar liquid phase is defined by P > 0, the meander phase by P = 0, QEA > 0 and the gas phase by QEA = 0. d, Correlationlength of the polarization field defined from the exponential decay of P(`) as sketched in (b). The cusp of the ⇠P(�o) curve at�c hints towards a critical dynamical transition.

distinct phases of active matter.

The transition between these two dynamical phases iscaptured by P ⌘ P(` = 2R), where R is the chamberradius. We show in Fig. 2c that P decays weakly andlinearly with �o for small disorder, but drops sharplyto 0 at the critical value �c defined from the prolifer-ation of quenched topological defects in Fig 1e. Thebifurcation of P in Fig. 2c suggests a critical scenario.This hypothesis is further supported by Fig. 2d whichreveals a divergence, or at the very least a drastic in-crease of the correlation length ⇠P at the onset of me-andering motion. The polarization P alone, however,does not distinguish the meander from the gas phase.We thus introduce the Edwards–Anderson parameterQEA = hv(r, t) · v(r, t + T )ir,t,T!1 that quantifies thetemporal persistence of the emergent flows [24]. The fi-nite non-zero value of QEA for �c < �o < 26 % (Fig. 2c)confirms the persistence of polar order along the mean-ders. We also find that the continuous transition, orcrossover, between the meander and the gas phase, whereQEA ⇠ 0, coincides with that identified from the topo-logical defect statistics in Fig. 1e. Taking advantage of

our high-content microfludic experiments, we measure Pand QEA in 400 experiments performed at XX di↵erentroller densities, and YY obstacle fractions for multipledisorder realizations and initial conditions. The resultingphase diagram shown in Fig. 4a firmly establishes thatdisorder and topological defects do not merely o↵set theflocking transition as reported in [14] but generically con-spire to bend the polar liquid into an amorphous meanderphase where flocking motion is locally preserved. Wenow establish that the meanders realize a dynamical vor-tex glass. To do so, we compare the flow orientations offifty replicas of the very same experiment (�o = 15%).We solely vary the initial conditions, keeping the col-loid fraction and disorder realization identical. We thenquantify the resemblance between the flow patterns byintroducing the local overlap q↵�(r) = p↵(r) · p�(r) be-tween the replicas ↵ and �. Examples of overlap mapsare shown in Fig. 4a for four di↵erent pairs of experi-ments, see also Supplementary Fig. S5. The replicatedflows are identical when q↵�(r) = +1, opposite whenq↵�(r) = �1 and orthogonal when q↵�(r) = 0. A simpleinspection of the maps readily indicates that all exper-

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FIG. 2. Suppression of long-range orientational order. A, PDF of the polarization fields P(pr, pθ) in the three dynamicalphases. Note that the distribution is isotropic but peaked along the unit circle in the meander phase: spontaneous flows arelocally preserved but global polar order is suppressed by disorder. B, Variation of the global polarization P and Edwards–Anderson QEA order parameters. In polar liquids P > 0, in meanders P = 0 but QEA > 0, and in a gas both P and QEA

vanish. C, Correlation length of the polarization field defined from the exponential decay of P(`) plotted in SupplementaryFigure S1. The cusp of the ξP(Φo) curve at Φc hints towards a critical dynamical transition. The accuracy in the measurementof ξ/2R is of the order of 10−2. D, Phase diagram of polar active matter in isotropic disorder. The phase boundaries areequivalently determined from the number of topological defects (Fig. 1F), or from the variations of P and QEA. In practice, wedefine the boundary separating the polar-liquid from the meander phase as P > 0.45 and the boundary between the meanderand gas phases as P < 0.45 and QEA > 0.2. The geometry of the phase boundaries does not crucially depend on thesecriteria. The extent of the meander phase becomes vanishingly small only at the onset of the flocking transition. The dashedrectangle indicates the region of the phase diagram explored in [22] where the meander phase was missed. Orientational orderis generically lost due to the emergence of meander patterns. The gray rectangle corresponds to the series of experimentsdiscussed in the main text.

readily indicates that, although all experiments corre-spond to overall different patterns, a few macroscopic re-gions are virtually identical from one replica to another.More quantitatively, we plot in Fig. 3C the distributionof the global overlap Qαβ = 〈qαβ(r)〉r, and compare it tothat of the macroscopic vortex obtained in the flockingstate. The two distributions are markedly different. Inthe flocking state, P(Qαβ) is composed of two symmet-ric peaks at Qαβ = ±1, as global circulation of eitherhandedness are equally probable. Conversely, in the me-ander phase, P(Qαβ) is not symmetric. It is a broadGaussian distribution centered around a positive meanvalue. In other words, even though disorder is isotropicand homogeneous, the obstacles bias the orientation ofthe flow field over macroscopic regions of space, as a re-sult reversing the orientation of the flow does not yieldan equally probable pattern. The bias in the flow struc-ture is clearly visible in the replica-averaged velocity fieldshown in Fig 3B.

Crucially, the large width of the overlap distributiondemonstrates a broad range of meandering patterns that

do not merely differ from each other by continuous dis-tortions. The replicas are actually topologically inequiva-lent, as demonstrated by the distribution of the differencein the number of topological-defect between each pair ofreplicas (Nα − Nβ). The regions of the flow patternsthat are robust to variations in the initial conditions aretherefore shaped by pinned defects of identical charge andorientation which persist in all replicas. In other words,defects are strongly correlated not only in their spatiallocation, but also in their flow direction. This becomesevident upon inspecting the statistics of local overlap.

Unlike the global overlap distribution, P(qαβ(r))is bimodal and sharply peaks at qαβ(r) = ±1, seeFig. 3E. Simply put, the meanders in each pair ofreplicas differ by the reversal of the flow orientationover a finite fraction of space. This area fraction isgiven by 1 − [P(qαβ = +1)− P(qαβ = −1)] ∼ 0.5, seealso Supplementary Note III. We can then estimate thetypical extent of the compact regions where the flowcan flip sign from one replica to another by measuringthe correlation length of the local overlap, or of the

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FIG. 3. Meanders as dynamical vortex glasses. A, Maps of the overlap field qαβ(r) for 4 out of 1225 different pairsof replicas (α, β). Scalebar: 1 mm. B, Magnitude of the replica-averaged polarization field. Disorder robustly sets the floworientation in macroscopic regions (light colors). Scalebar: 1 mm. C, Overlap distributions in the polar liquid and meanderphases. In the polar liquid phase (light orange) the distribution is symmetric and bimodal. Clockwise and counterclockwisevortices are equiprobable. In the meander phase, the distribution is a wide Gaussian centered on 0.46. D, Distribution of thedifference between the number of topological defects in each pair of replicas. The distribution is a Gaussian with a standarddeviation σ = 18 ± 3 much larger than the uncertainty defined by the standard deviation of the topological charge (whichshould be constrained to equal 1 in a circular geometry). Inset: distribution of the number of defects in each replica. Thereplicated flows are topologically distinct. E, Probability density of the local overlap qαβ(r). The distribution is asymmetricbut sharply peaked at ±1. Locally the active fluid either flows along identical or opposite directions in each replica of the sameexperiment. F, Decay of the spatial correlation of qαβ(r) averaged over all replicas and of the replica-averaged polarization fieldplotted in (B). Both decorrelations occur over the same finite distance ξQ = 100±10µm. All panels correspond to experimentsperformed deep in the meander phase Φo = 15 %.

replica-averaged flow shown in Fig. 3F. We find that theoverlap decorrelates exponentially over a finite distanceξQ = 100± 10µm. This finite correlation length impliesthat the meanders explore a conformational landscapeincluding a number of steady states that scales as

∼ 2(R/ξQ)2 . The exponential degeneracy of the flowsdemonstrates that the meander phase is the dynamicalanalogue of a topological defect glass. It is stronglyreminiscent of the vortex glass phases found in disor-dered flux lines in superconductors and in random-gaugeHeisenberg magnets [28, 29, 41].

DYNAMICAL VORTEX GLASS AS A GAUGEGLASS

We now make a precise connection between the dy-namical vortex glass we observe here in active fluids andits static counterpart in superconductors.

To do so, we construct a hydrodynamic theory of disor-dered polar flows, building on a final set of experiments

performed in periodic lattices of obstacles, see Supple-mentary Fig. S7 and Movie S7. A systematic investi-gation reveals that no meander forms in the absence ofdisorder, see Supplementary Note 5. The spatial hetero-geneities in the local obstacle fraction φo(r) are essentialto generate the meandering flow patterns. At lowest or-der in gradients, disorder thus minimally alters Toner–Tuhydrodynamics for the conserved roller density ρ and theflow velocity v [1, 34] ,

∂tρ = −∇ · (ρv), (1)

∂tv + λv ·∇v = (a2 − a4|v|2)v +K∇2v − β∇ρ− βo∇φo ,(2)

All the transport coefficients are taken to be constantpositive parameters. When ∇φo = 0, Eq. 2 reducesto the standard Toner–Tu equations, which quantita-tively predicts the vortical flows of Fig. 1A [32]. Theterms a2, a4 set the mean value of the flow velocity(v0 =

√a2/a4) deep in the homogeneous ordered phase,

while λ controls self-advection and the elastic constant Kpenalizes flow distortions. The βo term reflects the cou-pling to disorder, and provides a direct qualitative expla-

6

nation for the bias of the overlap distribution in Fig. 3C.Although the obstacles are isotropic and, on average, ho-mogeneously distributed, φo(r) acts as a random pres-sure field that locally drives the flows towards the sameobstacle-depleted regions in all replicas. For weak disor-der (Φo < Φc), the ordered flows are only marginally per-turbed by the obstacles. As detailed in SupplementaryNote 6, we can then perform a simple linearized analysisabout the polarized state, deep in the order phase wherev = v0(cos θ, sin θ). We readily obtain the steady stateorientational correlator in Fourier space

θ2q =β2oq

2xq

2y

[(c‖v0q2x − c2⊥q2y)2 + v20K2q2xq

4]Φo , (3)

where the overbar denotes a disorder average. c‖ = λv0and c⊥ =

√βρ0 are the longitudinal and transverse sound

speeds respectively, and ρ0 is the mean colloidal rollerdensity. Velocity fluctuations hence remain finite on largescales signaling true long-range order. The mean po-larization merely decays linearly with Φo, in agreementwith our experimental measurements in the small disor-der limit (Φo < Φc), see Fig. 2C and Supplementary Note6.

For strong disorder, we have to go beyond this spin-wave theory and account for both nonlinearities and thepresence of topological defects. Unlike in equilibrium fer-romagnets and active-nematic films, 2D colloidal flockssupport genuine long range order, thereby ruling out theclassical Kosterlitz-Thouless picture of defect unbinding.Topological defects in clean polar flocks only participatein a rapid coarsening process by annihilating quickly toyield pristine ordered flow, as evidenced in Fig. 1A. Dis-order changes this picture in important ways. Quenchedfluctuations due to the random obstacles provides an ad-ditional contribution that competes with orientationalelasticity to uncover a new active defect-unbinding sce-nario. For simplicity, we fix the colloidal roller density tobe a constant (ρ = ρ0); the sole hydrodynamic mode, istherefore the orientation θ. As detailed in SupplementaryNotes 7 and 8, retaining the full nonlinear dependence onθ in Toner-Tu equations, we find that at long-wavelengthsthe convective nonlinearities solely compete with the ran-dom pressure term in Eq. 2. To gain some insight onthis competition, let us first consider a local minimumof φo(r) at the origin, the inward disorder pressure mustbe balanced by a centrifugal kinematic pressure ∼ λv2/rassociated with an azimuthal flow. In a more generalsetting, the competition between convection and randompressure constrain the angular fluctuations to obey,

∇θ = − βoλv20

z× P ·∇φo ≡ A(r) , (4)

where P = 1 − vv. Remarkably, this quenched distri-bution of phase distortions is reminiscent of random-gauge XY models, and flux-line glasses in type II su-perconductors, see e.g. [27–29]. Eq. (4) reveals that ac-tive self-advection nonlinearly transforms the quenched

disorder potential φo into an effective random gaugevector field A. The local circulation of A then de-fines a quenched topological charge in the backgroundQ(r) = z · [∇×A(r)].

For a given disorder configuration, the flow stream-lines on large scales must satisfy the nonlinear constraintin Eq. 4, which can be used to obtain the steady statedistribution of θ to be P[θ(r)] ∝ limT→0 exp(−E/T ) withan auxiliary temperature T and an effective energy (seeSupplementary Note 7 for details)

E[θ(r)] =K

2

∫d2r |∇θ −A(r)|2 , . (5)

The zero temperature ground state controls the flow con-figurations of the disordered flock (T → 0). This com-pletes our mapping of the nonlinear meandering flows ina polar fluid on a disordered substrate to the well knownproblem of finding the ground states of a random-gaugeXY model [27, 29, 42].

The effective energy in Eq. 5 captures the competitionbetween disorder generating meanders dictated by A andelastic restoring forces penalizing these distortions. Thisallows a Kosterlitz-Thouless style argument for prolifer-ating free vortices detailed in Supplementary Note 8. Westress, however, on the crucial role of activity in our anal-ysis. Although we obtain an equilibrium-like mapping inEq. 5, activity is crucial in generating the gauge fieldA via self-advection. Noting a the vortex-core size, theelastic penalty of an isolated defect ∼ K ln(R/a) can beoffset by the energy gain ∼ KΦo(βo/λv

20) ln(R/a) from

optimally screening out the background charge. Theirbalance hence predicts a critical threshold Φc ∝ λv20/βobeyond which pinned vortices proliferate, in agreementwith our measurements of the number of frozen defectsat the onset of of the vortex glass formation, Fig. 1F.

A more sophisticated analysis including renormaliza-tion corrections to the elastic constant is presented inSupplementary Note 8. This rationalizes the existenceof the dynamical vortex glass phase along with its phasetransition as being driven by disorder driven unbindingof ±1 vortices.

CONCLUDING REMARKS

In conclusion, combining high-content experimentsand theory we have shown how dynamical vortex glassesgenerically emerge when the polar order of flocking mat-ter competes with isotropic disorder. We emphasize thegenerality of our predictions. Both the topological-defectstabilisation scenario and the extensive complexity ofthe flow patterns solely rely on stable uniaxial order in afluid assembled from motile units. Beyond the specifics,of collodial active matter, we therefore expect dynamicalvortex glasses to emerge in a host of active materialsranging from confined microtubule nematics [11], to con-centrated bacteria suspensions [37], and cell tissues [43]cruising through disorder. Establishing a quantitative

7

theory of their amorphous patterns remains a formidablechallenge.

Acknowledgements. We acknowledge supportfrom ANR program WTF and Idex program ToRe.M. C. M. was supported by the US National ScienceFoundation through award DMR-1938187. S. S. is sup-ported by the Harvard Society of Fellows. We thankD. Carpentier, O. Dauchot, W. T. M. Irvine, andA. Morin for insightful comments and suggestions.Data availability The data that support the plotswithin this paper and other findings of this study areavailable from the corresponding author upon request.Author Contributions D. B. conceived the project.A. C. and D. B. designed the experiments. A. C.performed the experiments and analyzed data. A. C.and D. B. interpreted and discussed the experiments.M. C. M. and S. S. performed the theory. All authorsdiscussed the results and wrote the manuscript. A. C.and S. S. contributed equally.Author Information Correspondence and requestsfor materials should be addressed to D. B. (email:denis.bartolo@ens-lyon.fr).Competing interests The authors declare no compet-ing interests.

METHODS

A. Quincke rollers experiments

The experimental setup is similar to that describedin [44]. We disperse polystyrene colloids of radius a =2.4µm (Thermo Scientific G0500) in a solution of hex-adecane including 5.5 × 10−2 wt % of dioctyl sulfosucci-nate sodium salt (AOT). We then inject the solution inmicrofluidic chambers made of two electrodes spaced by a25µm-thick scotch tape. The electrodes are glass slides,coated with indium tin oxide (Solems, ITOSOL30, thick-ness: 80 nm). We let the colloids sediment on the bottomelectrode and apply a DC voltage of 120 V. The result-ing electric field triggers the so-called Quincke electro-rotation and causes the colloids to roll at a constantspeed v0 = 0.8 mm/s [6, 45]. The microfluidic deviceis sketched in . 1b. We confine the colloidal rollers insidecircular chambers of radius R = 1.5 mm including ran-dom lattices of circular posts of radius 10µm. Both theobstacles and the confining disks are made of a 2µm-thicklayer of insulating photoresist resin (Microposit S1818)patterned by means of conventional UV-Lithography asexplained in [22]. The patterns are lithographed on thebottom electrode. Note that the distribution of the ob-stacle centers corresponds to a planar Poisson process,the circular posts can therefore overlap as see in Fig. 1band Movie S3. The experiments reported in the maintext correspond to thirty different microfluidic chambersincluding obstacle fractions ranging from 0 % to 38 %. Wefine tune the mean roller fraction ρ0 using a T-junction to

inject sequentially a colloidal suspension or a colloid-freesolvent.

If not specified otherwise, the stationary flows are mea-sured in the first chamber 120 s after the application ofthe DC field. This waiting time is more than one or-der of magnitude larger that the flows’ relaxation time,see Supplementary Note I. For the replica experimentswe proceed as follows: the chambers are filled with thecolloids at the desired area fraction. We motorize the col-loids, wait for 120 s and film their motion for 5 s. We thenswitch the voltage off, and switch it on again repeatingthe same procedure fifty times in a row.

B. From Lagrangian trajectories to Eulerian flowfields

In order to track of the trajectories of the rollers, weimage them with a Nikon AZ100 microscope with a 4.8Xmagnification and record videos with a CMOS camera(Basler Ace) at 190 fps. All measurements are systemat-ically repeated three times for different initial conditionsand same disorder configuration. If not specified other-wise, we measure all quantities reported in the main textafter the ensemble of rollers has reached its stationarystate.

1. Lagrangian trajectories

We detect the position of all the rollers with a sub-pixel accuracy using the algorithm introduced by Lu etal in [46]. We then reconstruct their trajectories over thewhole 3 mm wide circular chambers using the Crockerand Grier algorithm [47] with the MATLAB routineavailable at [48]. We define the individual roller veloci-ties from their displacements over two subsequent frames(time interval: δt = 5.3 ms): vi(t) = ri(t + δt) − ri(t),where ri(t) and vi(t) are respectively the position andvelocity of particule i at time t. The accuracy of the po-sition measurements is of the order of 0.1µm, inducingan accuracy of the order of 40µm/s for individual speedmeasurements. When powered with an electric field E ofmagnitude 120 V, all colloids roll at a constant speed:

v0 = 0.80± 0.04 mm/s. (6)

In addition, when isolated, their direction of motionfreely diffuses on the unit circle with a rotational dif-fusivity DR defined as the exponential decorrelation rateof the velocity orientation in an isotropic phase:

DR = 2.2± 0.1 s−1. (7)

2. Eulerian fields

Building on these Lagrangian measurements, we recon-struct the instantaneous Eulerian velocity fields v(r, t) as

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follows. We average the instantaneous roller velocities in76.4µm×76.4µm binning windows arranged on a squarelattice with a lattice spacing of 15.3µm. Given the rollerdensity, each PIV window typically averages the veloc-ity of 25 rollers. We systematically checked that none ofour results crucially depends on the specific choice of thesize of the binning windows. The polarization and over-lap fields are computed with the same spatial resolutionfrom v(r, t).

To compute the polarization order parameter from theinstantaneous velocity field, we first average the radialand azimuthal components of the polarization fieldp(r) ≡ 〈v(r, t)〉t over square boxes of size `. We thencompute the spatial average: pB ≡ (〈pr(r)〉r, 〈pθ(r)〉r),in each box B. Finally, P(`) corresponds to the norm ofpB averaged over all boxes B.

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