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Braiding a flock: winding statistics of interacting flying spins Jean-Baptiste Caussin1 and Denis Bartolo1 1Laboratoire de Physique de l’Ećole Normale Supérieure de Lyon, Université de Lyon, 46, allée d’Italie, 69007 Lyon, France Whenanimalgroupsmovecoherentlyintheformofaflock,theirtrajectoriesarenotallparallel, the individuals exchange their position in the group. In this Letter we introduce a measure of this mixing dynamics, which we quantify as the winding of the braid formed from the particle trajectories. Building on a paradigmatic flocking model we numerically and theoretically explain the winding statistics, and show that it is predominantly set by the global twist of the trajectories 5 as a consequence of a spontaneous symmetry breaking. 1 0 PACSnumbers: 5.65.+b,87.23.Cc,5.40.Fb 2 n u The collective behaviors observed in animal groups of F . The projection operator (I−pˆ pˆ ) ensures that i i i J have attracted much attention in the biology, the math- pˆ lives on the unit sphere. We use a standard (metric) i 6 ematics and the physics communities over the last 20 form of F to account for the flocking dynamics. Noting i 2 years. Quantitative data analysis have established that r ≡r −r it reads: ij i j the salient traits of collective motion are very well cap- ] h turedbythedynamicsofflyingspins: persistentrandom 1 (cid:88) 1 (cid:88) F = pˆ + f(r )ˆr . (2) c walkers endowed with interactions akin to ferromagnetic i τNA j τNB ij ij e i j∈Ai i j∈Bi couplings between their velocities [1–6]. This framework m has been extensively exploited to rationalize structural τ is a relaxation time, which we henceforth set to τ =1. - t and dynamical properties starting from the emergence The first term in Eq. (2) is a ferromagnetic term which a of directed motion, to rapid (orientational) information promotes alignment with the mean direction of the NA t i s transfer, see [1, 6, 7, 9–11] and references therein. How- neighbors lying in the sphere A of radius R = 1. The . i A t ever, beyond these spectacular results, the internal dy- second term corresponds to attractive and repulsive in- a m namics of a flock, the relative motion of the individuals, teractions within the sphere B of radius R = 5, it is i B remains scarcely investigated both experimentally and introducedtoyieldcompactflocks[15]. Following[1,16], - d theoreticallyyetitisknowntodisplaynon-trivialanoma- we assume that these interactions are attractive above n lous behaviors [12, 13]. a distance 2r , and repulsive when r ≤ r < 2r : c c ij c o In this Letter, we theoretically describe the mixing f(r ) = 1 − [r /(r − r )]5 with r = 0.4. Eqs. (1) c ij c ij c c [ statistics of an archetypal flying-spin model. We first and (2) are solved numerically using an explicit Euler stresstheintrinsicgeometricalnatureofthedynamicsof scheme. In all that follows, our only control parameter 2 particlesinaflockandmapthisproblemtothebraiding isthenoiseamplitude,D (see[1,15]foracomprehensive v statisticsoftheirtrajectories. Weevidencethenontrivial investigationofthephasebehaviorofthismodel). Below 9 7 statistics of the winding between pairs of motile-particle a critical amplitude Dc ∼ 0.3, the rotational symmetry 8 trajectories, which is a robust measure of their entangle- of the particle orientation is spontaneously broken and 7 ment. This quantity displays spatial correlations at the collectivemotionemergesintheformofaflockingtransi- 0 population scale. We single out the reason for the non- tion(seetheSupplementaryDocument[35]). Acompact . 1 trivialstatisticsandshowthatthespontaneousbreaking polar flock forms and moves along Π(t) = (cid:104)pˆ (t)(cid:105) as i i 0 of a rotational symmetry causes the global twist of the exemplified in Fig. 1(a) and in the supplementary video 5 flock to chiefly rule the long-time winding fluctuations. S1 [34]. We restrict ourselves to this symmetry-broken 1 : Numerical flocking model. We build on a standard phase and investigate the relative motion of the individ- v flocking model used to model compact groups akin to uals within the flock. i X those observed in the wild [1, 1, 6, 14]. N persistent Quantifying the entanglement of the trajectories. r random walkers, ri(t), i = 1...N, propel at a constant We quantify the mixing dynamics of the flock by ex- a speed v0 = 1. The dynamics of their orientation pî(t) ploiting a powerful toolbox that has been introduced in generically takes the form: thecontextofLagrangianmixinginfluids. Theideaisto relatethemixingofanensembleofmovingparticletothe dpˆ i =(I−pˆ pˆ )·F ({r ,pˆ } )+ξ (t). (1) entanglement of their wordlines [17–20]. A convenient dt i i i j j j i measureoftheentanglementofabundleoftrajectoriesis Rotational diffusion is accounted for by the uncorrelated providedbyabraidrepresentationthatweintroducebe- Gaussianwhitenoisesξ ofvariance2D. Theinteractions low. Letusfirstdefineaconvenientrepresentationwhich i between self-propelled particles amounts to an effective disentangles the internal dynamics of the flock from the torque which aligns the orientation pˆ in the direction global turns of its mean direction of motion. We con- i 2 FIG. 1: (a) Instantaneous positions and orientations of the particles in a compact polar flock (250 particles), and definition of theparallel-transportedframe(ηˆ ,ηˆ ,Πˆ). (b)Worldlinesof5particlesinthesamepolarflock. D=2.6×10−2. (c)Definition 1 2 of the crossing index. (d) Braid diagram associated to the world lines drawn in (b). sideranorthogonalframe(G(t),ηˆ (t),ηˆ (t),Πˆ(t))shown all the crossings o this simplified representation. Prac- 1 2 in Fig. 1(a). The origin is the center of mass G(t) of tically we choose the Artin representation of the braid the flock. ηˆ (t = 0) is chosen arbitrarily; the ηˆ s are word [18, 19, 21]. The braidlab library [22] is used to 1 i then parallel-transported along the trajectory of G in compute both the pair and the total winding numbers. the course of the dynamics. If the flock were undergo- The topological nature of W(T) makes it a very robust ing a rigid-body motion, the particle positions in this measureoftheflockmixing. Wenowcarefullyinvestigate frame would be stationary. Conversely, mixing in the its statistics. flock translates into the winding of the particle positions Winding statistics. in(ηˆ1,ηˆ2)plane, seeFig.1(b)andsupplementarymovie The normalized distributions of W(T) are plotted in S2 [34]. In order to quantify this winding dynamics, we Fig. 2(a) for different values of the noise amplitude. For observetherelativepositionsoftheparticlesalongaref- all the trajectory lengths, the total winding follows a erence axis, say ηˆ1. Particle winding is measured by Gaussianstatisticswithzeromeansincetheflockhasno observing the particle exchanges along ηˆ1, and by as- intrinsicchirality: clockwiseandcounter-clockwisewind- signing an index (cid:15)=±1 to each crossing, as depicted in ings are equally probable. The winding distribution is Figs. 1(b), 1(c) and supplementary movie S3 [34]. The 1 fully characterized by its standard deviation (cid:104)W2(T)(cid:105)2, signofthisindexreflectstherelativepositionsofthepar- where the brackets denote the average over different ini- ticleintheorthogonaldirectionηˆ astheycrossalongηˆ . 2 1 tial conditions. The winding fluctuations increase lin- Morequantitatively,consideringtwoparticlesiandj,we early with the curvilinear length of the trajectories at introducetheirpair winding number,w (T),asthelink- ingnumberbetweentheirworldlines: wij(T)= 1(cid:80) (cid:15)ij, short times: (cid:104)W2(T)(cid:105)12 ∝ T, and crosses over to a where(cid:15)ij istheindexoftheath crossingi.jThesum2 isapear- diffusive regime where (cid:104)W2(T)(cid:105)21 ∝ T12 at long times, a formed over the crossings involving the particles i and j Fig. 2(b). These first results would naively suggest a only, over a time interval T. This quantity has a clear simple scenario. If the crossing events were uncorrelated meaning: it counts the number of turns of particle i the Gaussian statistics would readily stem from the cen- around j (or, equivalently, of j around i) over a time trallimittheoremasW(T)istheaverageofthecrossing T. The total winding in the flock between t and t +T signs. Inaddition,thevarianceofW(T)wouldobviously 0 0 is a measure of the entanglement of the world lines, and grow in a diffusive manner. However, this appealing ex- hence of the mixing: planation is inconsistent with a deeper analysis of the data. Letuscarefullystudythecorrelationsbetweenthe 1 (cid:88) W(T)= w (T), (3) pair windings, which are quantified by: N ij p (i,j) where Np = N(N −1)/2 is the number of pairs. Im- Cww(T)= N (N1 −1) (cid:88) (cid:88) (cid:104)wij(cid:104)w(t2)w(Tkl)((cid:105)T)(cid:105), portantly, W(T) does not depend on the distance be- p p (i,j)(k,l)(cid:54)=(i,j) tween the particles, only the index of the crossings and (4) the times at which they occur matter. Therefore the where (cid:104)w2(t)(cid:105) = N−1(cid:80) (cid:104)w2(T)(cid:105) is the variance of p (i,j) ij world lines define a braid that can be drawn in the form the pair windings. Given this definition, C (T)=0 for ww of a normalized braid diagram, Fig. 1(d) [19]. The to- uncorrelated w s, and C (T)=1 when the pair wind- ij ww tal winding corresponds to a topological invariant of the ings are fully correlated. Unexpectedly, we find that the (cid:80) braid: W(T) = (cid:15) /(2N ), where we now sum over correlation between the pair windings does not vanish at a a p 3 A twist in the statistics. We now elucidate the physical mechanism responsible for the winding correlations. We first note that a global instantaneous rotation of the flock around Πˆ would re- sultinafullycorrelatedpairwinding. Wethereforesepa- rate the associated global trajectory twist from the total winding. Denoting by η the position of particle i pro- i jected in the observation plane (G,ηˆ ,ηˆ ), the instanta- 1 2 neous rotation rate of the flock is: Ω= 1 (cid:88) 1 (η ×η˙ )·Πˆ. (5) N η2 i i i i Integrating over time, we define the global twist that is the number of turns of the flock around Πˆ: 1 (cid:90) t0+T Tw(T)= dtΩ(t). (6) 2π t0 We finally define the winding of the untwisted trajectories in the frame (G,ηˆ(cid:63),ηˆ(cid:63)), obtained by rotating 1 2 FIG. 2: (a) Probability distribution of the total winding the parallel-transported frame (G,ηˆ ,ηˆ ) by an angle W(T), normalized by its standard deviation (T =103). The 1 2 2π×Tw: different noise amplitudes correspond to different polariza- tions of the flock [35]. Solid line: Gaussian distribution. W(cid:63) =W −Tw. (7) (b) Standard deviation, (cid:104)W2(T)(cid:105)1/2, as a function of time T, for different noise amplitudes. (c) Correlation function, HenceW(cid:63) =N−1(cid:80) w(cid:63),wherew(cid:63) =w −Twisthe defined by Eq. (4), as a function of T. (d) Variance of the p (i,j) ij ij ij winding between particles i and j in this rotating frame. totalwindingnormalizedbythevarianceofthepairwinding, plotted versus the particle number. Open circles: T =10−1. Within the braid picture, W(cid:63) is found by factorizing out Filled circles: T = 103. Black squares: random walkers con- the global twist of the braid word [24, 25]. finedinacircularboxofradius20,diffusivity: 10. Timestep: In Fig. 3(a), we plot the standard deviation of W, Tw 0.1. Dashed lines: slope −1. and W(cid:63) versus T.The twist contribution dominates the totalwindingatlongtimesand(cid:104)W2(cid:105)∼(cid:104)Tw2(cid:105). Thisnu- mericalfactisbetterunderstoodbynotingthatthewind- ingintherotatingframequalitativelyfollowsthebehav- long times. Conversely, it increases and plateaus at a ior displayed by confined random walkers. Above the re- finite value, Fig. 2(c), thereby ruling out the simple sce- laxationtimeτ =1introducedinEq.(2),W(cid:63)(T)follows nario sketched above. In order to check that this unex- the same diffusive evolution: (cid:104)W(cid:63)2(T)(cid:105) ∝ T [26]. More pected behavior is not a finite-size artifact, we first note importantly, we also find the same 1/N asymptotic scal- that C (T) ∼ (cid:104)W2(T)(cid:105)/(cid:104)w2(T)(cid:105) in the large-N limit ing for C ∼ (cid:104)W(cid:63)2(T)(cid:105)/(cid:104)w(cid:63)2(T)(cid:105) showing that the ww p w(cid:63)w(cid:63) and plot the ratio (cid:104)W2(T)(cid:105)/(cid:104)w2(T)(cid:105) for flocks of differ- spatialcorrelationsofthedisplacementsareshort-ranged entsizesN,inFig.2(d). Whereasshorttrajectorieshave at all times, Fig. 3(b). This result contrasts with the be- indeed winding correlations C (T) that decay with the havior of the total winding in the parallel-transported ww system size, as 1/N, the winding correlations of the long frame where C ∼ (cid:104)W2(T)(cid:105)/(cid:104)w2(T)(cid:105) hardly depends ww trajectoriesdonotdisplayanysignificantvariationswith on N at long times, Fig. 2(d). We can therefore propose N whenincreasingtheparticlenumberbyafactorof∼8. thefollowingscenario: thelong-rangespatialcorrelations In order to gain more insight into these tho opposite be- in the flock arise from the global rotation of the flock, haviors,wecomputedthesamequantityforthewordlines hence from the global twist of the trajectories as clearly of independent 2D random walkers confined in a circular exemplified in the supplementary video S4 [34]. box. C follows the same 1/N nontrivial scaling ob- Hydrodynamic description of the trajectories’ twist. ww served for short flocking trajectories. We shall note that Theveryoriginoftheglobalrotationoftheflockroots the finite time step of our numerical scheme regularizes from the spontaneous breaking of the rotational symme- the winding statistics of the random walkers and makes try of particle velocities. This symmetry breaking gives it possible to define its variance [23]. This second set risetoasoftorientationalmode[27]whichtwiststhetra- of observations confirms that the saturation of C (T) jectories in the parallel-transported frame at the entire- ww at long time originates from extended correlations of the flock scale. We now lay out a more quantitative expla- crossing events. nation by switching to an hydrodynamic description of 4 momentum fluctuations are Gaussian. After space and time integration, Eqs. (9) and (6) imply that the twist alsofollowsanormaldistribution,inagreementwithour numerical findings reported in Fig. 3(c). Secondly, the damping of velocity fluctuations is set by the diffusive term Γ∇2(ρv), in Eq. (8) (see [2, 30] for more details). Hence the Fourier mode with wave-vector q decays in a time ∼ (Γq2)−1. While small-wavelength perturbations are quickly damped, the large-scale fluctuations that occur at the size of the flock, q ∼ 1/L , remain corre- flock latedoveratimeT ∼L2 /Γ. Thisobservationexplains flock the time behavior of the trajectories’ twist fluctuations. At short times, T < T, the small-q fluctuations result in finite-time correlations in the rotation rate Ω. Conse- quently,thetwistfluctuationspersistandundergoa“bal- istic” growth: (cid:104)Tw2(T)(cid:105) ∝ T2. At long times, T > T, alltheFouriermodeshavebeenrelaxed, thecorrelations vanish, and one recovers the observed diffusive behavior for the trajectory twist: (cid:104)Tw2(T)(cid:105) ∼ D T, Fig. (3)a. Tw The scaling of the effective diffusivity with the size of the flock is computed in [35]: D ∝ D˜T2L N−2. FIG. 3: (a) Standard deviations as a function of T: Tw flock This prediction agrees again with our numerical obser- (cid:104)W2(T)(cid:105)1/2 (open circles), (cid:104)W(cid:63)2(T)(cid:105)1/2 (diamonds) and vations, Fig. 3(d). Now that we have elucidated the (cid:104)Tw2(T)(cid:105)1/2 (filled circles), (cid:104)W2(T)(cid:105)1/2 for confined random twist statistics, we finally explain why it dominates the walkers (squares), same parameters as in Fig. 2. (b) Vari- winding in large flocks. Assuming that the flock density ance of the total winding normalized by the variance of the weakly depends on N, we find that (cid:104)Tw2(T)(cid:105) ∼ N−1/3. pair winding computed in the twisting frame for different particle numbers. Open circles: T = 10−1. Filled circles: As (cid:104)W(cid:63)2(cid:105)/(cid:104)w(cid:63)2(cid:105)∼N−1, Fig. 3(b), we conclude that the T = 103. Black squares: confined random walkers. Dashed ratiobetweenthelocalwindingfluctuationsandthetwist lines: slope −1. (c) Probability distribution of the twist nor- vanishes as N →∞. malized by its standard deviation (T = 103). Solid line: In order to unambiguously prove that the soft rota- Gaussian distribution. (d) The normalized twist diffusiv- tionalmodechieflyrulesthewindingstatistics, westress ity: D /(T2L ) decays quadratically with N. Solid line: Tw flock that all this phenomenology is lost in isotropic swarms. slope −2. WeshowintheSupplementaryDocument[35]thatwhen thenoiseamplitudeistoostrongtoobservedirectedmo- tion, (cid:104)W2(cid:105) deviates from (cid:104)Tw2(cid:105): increasing the noise re- the flock viewed as a active-fluid drop. We use the con- sults in the decorrelation of the winding and the twist ventional hydrodynamic framework first introduced phe- fluctuations [35]. Altogether, our numerical and analyti- nomenologically by Toner and Tu [12] and later derived cal results suggest a strong robustness of our main find- from microscopic theories [2, 28, 29]. The fluid density ings. Theprominenceofthetwistfluctuations,leadingto and velocity fields are ρ(r,t) and v(r,t). We focus on a coherent braiding of the flock trajectories, is expected strongly polarized flocks in which all particles follow the to be qualitatively robust to the very details of the in- same average direction. The momentum equation lin- teractions. It solely relies on the spontaneous breaking earized around the homogeneously polarized state takes of the rotational symmetry, and on its associated soft the simple form [2, 30]: mode. Itisthereforeadirectconsequenceoftheflocking transition and should be observed in all models yielding ∂t(ρv)+λ(Πˆ ·∇)ρv=−∇P(ρ)+Γ∇2(ρv)+f, (8) polarized flocks. We shall close with Letter from an experimental per- where P is the local pressure, and f(r,t) is a Gaus- spective. In most of the situations, in the wild, external sian white noise with correlation (cid:104)f (r,t)f (r(cid:48),t(cid:48))(cid:105) = α β perturbations and fields explicitly break the rotational 2D˜δ(r − r(cid:48))δ(t − t(cid:48))δ . In this continuous limit, the αβ symmetry (e.g. gravity, predators, obstacles), or cause global rotation rate, Eq. (5), is given by: suddencollectiveturnsasanalyzedin[13,31–33]forstar- 1 (cid:90) 1 ling flocks. The winding statistics should be an effective Ω= d3r (η×ρv)·Πˆ. (9) probe of the flock response to external bias without any N η2 a priori knowledge about the individual propulsion and Two comments are in order. Firstly, deep in the po- interaction mechanisms. larized phase, the linearity of Eq. (8) implies that the WeacknowledgesupportfromInstitutUniversitairede 5 France and ANR project MiTra. 69 (2003). [19] J.-L. Thiffeault, Chaos 20, 017516 (2010). [20] J. G. Puckett, F. Lechenault, K. E. Daniels, and J.-L. Thiffeault,J.Stat.Mech.Theor.Exp.2012,06008(2012). [21] M. D. Finn, and J.-L. Thiffeault, SIAM Rev. 53, 723 [1] T.VicsekandA.Zafeiris.PhysicsReports517,71(2012). (2011). [2] M.C.Marchetti,J.F.Joanny,S.Ramaswamy,T.B.Liver- [22] J.-L.ThiffeaultandM.Budisic,arXiv:1410.0849(2014). pool,J.Prost,M.Rao,andR.A.Simha,Rev. Mod. Phys. [23] C.Bélisle,andJ.Faraway,J.Appl.Prob.28,717(1991). 85, 1143 (2013). [24] A. T. Skjeltorp, S. Clausen, and G. Helgesen, Physica A [3] A. Cavagna, I. Giardina, Annu. Rev. Condens. Matter 274, 267 (1999). Phys. 5, 183 (2014). [25] P. Dehornoy, Contemporary Mathematics 360, 5 (2004). [4] D.S. Calovi, U. Lopez, S. Ngo, C. Sire, H. Chaté, and G. P. Dehornoy, Discrete Appl. Math. 156, 3091 (2008). Theraulaz, New J. Phys. 16, 015026 (2014). [26] A. Grosberg, and H. Frisch, J. Phys. A: Math. Gen. 36, [5] D.J.G.Pearce,andM.S.Turner,NewJ.Phys.16,082002 8955 (2003). (2014). [27] W.Bialek,A.Cavagna,etal.,Proc.Natl.Acad.Sci.USA [6] U. Lopez, J. Gautrais, I. D. Couzin, and G. Theraulaz, 111, 7212 (2014). Interface Focus 2, 693 (2012). [28] E. Bertin, M. Droz, and G. Grégoire, Phys. Rev. E [7] T. Vicsek, A. Cziro´k, E. Ben-Jacob, I. Cohen, and O. 74, 022101 (2006); J. Phys A: Math. Theor. 42, 445001 Shochet, Phys. Rev. Lett. 75, 1226 (1995). (2009). [8] G. Grégoire, and H. Chaté, Phys. Rev. Lett. 92, 025702 [29] F. D. C. Farrell, M. C. Marchetti, D. Marenduzzo, and (2004). J. Tailleur Phys. Rev. Lett. 108, 248101 (2012). [9] A. Cavagna, A. Cimarelli, et al., Proc. Natl. Acad. Sci. [30] J. Toner, Y. Tu, and S. Ramaswamy, Annals of Physics, USA 107, 11865 (2010). 318, 170 (2005). [10] W.Bialek,A.Cavagna,etal.,Proc.Natl.Acad.Sci.USA [31] A. Attanasi, A. Cavagna et al., Nature Phys., 10, 691 109, 4786 (2012). (2014). [11] A. Attanasi, A. Cavagna et al. Nature Phys. 10, 691 [32] A. Attanasi, A. Cavagna et al., preprint at (2014). arXiv:1410.3330 (2014). [12] J. Toner, and Y. Tu, Phys. Rev. Lett. 75, 4326 (1995). [33] A. Cavagna, L. Del Castello et al., J. Stat. Phys., 158, J. Toner, Y. Tu, and M. Ulm, Phys. Rev. Lett. 80, 4819 601 (2014). (1998). [34] See the Supplementary Movies. Movie S1– Time evolu- [13] A. Cavagna, S. M. D. Queiros, I. Giardina, F. Stefanini tion of the flock (the center of the frame has been kept and M. Viale, Proc. R. Soc. B 208, 0122484 (2013). at the center of the flock, D = 0.026). Movie S2– Posi- [14] W.Bialek,A.Cavagnaetal.,Proc.Natl.Acad.Sci.USA tion of the particles projected in the the plane (ηˆ ,ηˆ ) 1 2 109, 4786 (2012). (total time T = 14.4, D = 0.026). Movie S3– Growth of [15] I.D.Cousin,J.Krause,R.James,G.D.Ruxton,andN. the world lines defined from the particle fluctuations in R. Franks, J. Theor. Biol. 218, 1 (2002). the (ηˆ ,ηˆ ) plane. D = 0.026. Movie S4– Same as S2, 1 2 [16] H. Chaté, F. Ginelli, G. Grégoire, F. Peruani, and F. spedup60times: theglobalrotationoftheflockbecomes Raynaud, Eur. Phys. J. B. 64, 451 (2008). clearly apparent (total time T =900). [17] P. Pieranski, S. Clausen, G. Helgesen, and A. T. Skjel- [35] See the Supplementary Document: winding statistics of torp, Phys. Rev. Lett. 77, 1620 (1996). isotropic flocks, and scaling law for the twist diffusivity. [18] P. Boyland, M. Stremler, and H. Aref, Physica D 175, 6 Supplementary Information TRANSITION TO COLLECTIVE MOTION Upon decreasing the noise amplitude, the model defined by Eqs. (1) and (2) in the main text displays a transition to directed motion, upon decreasing the noise amplitude below a critical value D ∼ 0.2. The mean polarization, c Π = N−1|(cid:80) pˆ |, increases from 0 in isotropic flocks to 1 in coherenty-moving groups, as shown in Fig. 4. Both 0 i i polar and isotropic flocks are compact: due to the attractive interactions, they do not span the entire simulation box but keep a finite size. A comprehensive characterization of the nature of the transition is provided in [1]. FIG. 4: Mean polarization of the flock plotted versus the noise amplitude. LONG-TIME BEHAVIOR OF THE TWIST FLUCTUATIONS IN POLAR FLOCKS We detail the derivation of the scaling law for the twist diffusivity D from the hydrodynamic description of the Tw flock. For the sake of clarity, we introduce the momentum field V˜ = ρv. As we noted in the main text, its Fourier component V˜ , with wave-vector q, is damped over a typical time ∼ (Γq2)−1. From Eq. (8), the time correlations q therefore decay as: D˜ (cid:104)V˜ (t)V˜ (t(cid:48))(cid:105)∝ e−Γq2|t−t(cid:48)|δ δ(q+q(cid:48)), (10) q,α q(cid:48),β Γq2 αβ wheretheindicesα,β denotethespatialcomponentsofV˜ . Thesmallestwave-vectorbeingq ∼1/L ,thetime q min flock correlations vanish is the limit where |t−t(cid:48)|(cid:29)T ∼L2 /Γ. In this regime, the momentum field is delta-correlated: flock D˜ (cid:104)V˜ (t)V˜ (t(cid:48))(cid:105)∝ δ(t−t(cid:48))δ δ(q+q(cid:48)). (11) q,α q(cid:48),β Γ2q4 αβ From this result, we deduce the autocorrelation of the global rotation rate. Eq. (9) provides the following expression: 1 (cid:90) (cid:90) 1 (cid:104)Ω(t)Ω(t(cid:48))(cid:105)= d3r d3r(cid:48) Πˆ Πˆ (cid:15) (cid:15) η η(cid:48) (cid:104)V˜ (t)V˜ (t(cid:48))(cid:105), (12) N2 η2η(cid:48)2 α λ αβγ λµν β µ γ ν where we recall that η = (I−ΠˆΠˆ)·r and (cid:15) is the fully antisymmetric Levi-Civita symbol. Moving to Fourier αβγ space and using Eq. (11), we have equivalently: D˜ (cid:90) (cid:90) (cid:90) η·η(cid:48) 1 (cid:104)Ω(t)Ω(t(cid:48))(cid:105)= d3r d3r(cid:48) d3q e−iq·(r−r(cid:48)) δ(t−t(cid:48)). (13) Γ2N2 η2η(cid:48)2 q4 7 We therefore obtain (cid:104)Ω(t)Ω(t(cid:48))(cid:105)∼D δ(t−t(cid:48)). A simple dimensional estimate of the value of the integral in Eq. 13 Tw readily provides the scaling D ∝D˜Γ−2N−2L5 for the effective diffusivity. Recalling that the persistence time of Tw flock the twist is T ∼L2 /Γ, we recover the scaling given in the main text: flock D ∝D˜T2L N−2. (14) Tw flock WINDING STATISTICS ACROSS THE TRANSITION TO COLLECTIVE MOTION FIG. 5: (a) Variance of the total winding (cid:104)W2(T) (blue open circles), and of the twist (cid:104)Tw2(T) (red filled circles), plotted as a function of time T for a polarized flock (D = 0.02). (b) Same as (a) for an isotropic flock (D = 1.23). (c) Ratio between thevariancesofthetwistandofthetotalwindingatlongtimes,plottedversusthenoiseamplitude. (d)Long-timecorrelation between the twist and the total winding, C =(cid:104)WTw(cid:105)/[(cid:104)W2(cid:105)(cid:104)Tw2(cid:105)]1/2. WTw Inthemaintext,werestrictedourselvestopolarizedflocks. Here,weextendtheanalysistohighernoiseamplitudes, above the transition to collective motion. As the flock has no well-defined direction of motion, we project the particle positionsalongafixeddirection(ratherthaninaparallel-transportedframe). Wecomputethewindingandthetwist statistics as explained in the main text. We then compare, in Fig. 5, the results obtained for polarized and isotropic flocks. We plot the variances (cid:104)W2(T)(cid:105) and (cid:104)Tw2(T)(cid:105) in two extreme cases, see Figs. 5(a) and (b). We also quantify theratio(cid:104)Tw2(cid:105)/(cid:104)W2(cid:105)atlongtimes,aswellasthecorrelationsbetweenthewindingandthetwistfluctuations,C , WTw in Figs. 5(c) and (d). Upon varying the noise across the transition to collective motion, the emergence of polar order is associated with a clear change in the winding statistics. • In polar flocks (at low noise amplitude, D < D ∼ 0.2), we find that (cid:104)Tw2(cid:105) ∼ (cid:104)W2(cid:105). The total winding and c the twist are fully correlated, C ∼1. As they both follow a Gaussian distribution, these two quantities are WTw statistically equivalent. • In isotropic flocks (D > D ), the total winding is well distinct from the twist as exemplified in Fig. 5(b). In c contrast with polar flocks, the ratio (cid:104)Tw2(cid:105)/(cid:104)W2(cid:105) deviates from unity upon increasing the noise amplitude, see 8 Fig. 5(c). This behavior is associated with a clear decay of the correlation between the winding and the twist in an isotropic flock, Fig. 5(d). These results further confirm the scenario proposed in the main text to explain the winding statistics. In polar flocks, the total winding and the twist are both ruled by the same stochastic process: the spontaneous breaking of the rotational symmetry yields a spatially coherent rotation of the flock yet stochastic in time, i.e. a global braiding of the trajectories. This mechanism does not rely on the specific form of the interactions leading to polar order, it is thereforegenerictoallflockingmodels. Bycontrast,inisotropicflocks,thiscorrelatedrotationalmodeissuppressedas rotationalsymmetryisnotbroken. Theresidualwindingandtwistfluctuationssolelyarisefromtheweakly-correlated displacements of the individuals: in isotropic flocks, they correspond to different random processes. [1] G. Grégoire, and H. Chaté, Phys. Rev. Lett. 92, 025702 (2004).