Active Matter Class with Second-Order Transition to Quasi-Long-Range Polar Order

We introduce and study in two dimensions a new class of dry, aligning, active matter that exhibits a direct transition to orientational order, without the phase-separation phenomenology usually observed in this context. Characterized by self-propelled particles with velocity reversals and ferromagnetic alignment of polarities, systems in this class display quasi-long-range polar order with continuously-varying scaling exponents and yet a numerical study of the transition leads to conclude that it does not belong to the Berezinskii-Kosterlitz-Thouless universality class, but is best described as a standard critical point with algebraic divergence of correlations. We rationalize these findings by showing that the interplay between order and density changes the role of defects.

We now have a satisfactory theoretical understanding of dry, aligning, active matter in the dilute limit, thanks to a series of works at particle, hydrodynamic, and kinetic levels . In these systems, the dominating interaction, local alignment, is in competition with noise, and can lead to orientationally-ordered phases that are endowed with generic long-range correlations and anomalous fluctuations [14-16, 20, 22, 25, 26, 36]. The emergence of order is not a direct, continuous phase transition, but occurs via phase-separation between a disordered gas and an ordered liquid separated by a coexistence phase [34,38,40]. Genuine critical behavior has only been found when long-range interactions are present, brought by non-metric, "topological" neighbors [43,44] or by imposing some incompressibility condition [45].
Vicsek-style models, which consist of constant-speed point particles that locally align their velocities in competition with some noise, have been instrumental in this success. Their simplicity allows for both in-depth numerical study and the controlled derivation of hydrodynamic theories [19,21,23,[27][28][29][30][31]37]. The main classes of dry aligning active matter studied so far each possess a Vicsek-style representative. Polar particles with ferromagnetic alignment -the case of the original Vicsek model [13]-give rise to true long-range polar order [14,15,32] and a coexistence phase made of quantized traveling bands [17,22,35,39,40,46]. Nematic alignment leads to global nematic order and a chaotic coexistence phase mediated by unstable nematic bands [20,33,36,41]. For polar particles nematic order seems to be long-ranged [7,25,42], whereas for finite velociyreversal rate it is only quasi-long-ranged.
In this Letter we show that self-propelled particles with velocity reversals and local ferromagnetic alignment exhibit novel collective properties and in particular a continuous transition to order. In the restricted Vicsek setting where a particle's polarity is simply given by its velocity, no order can emerge in this case. Here we relax this "Vicsek constraint" by conferring particles a polarity that they align with that of neighbors while they move either along or against it. Using kinetic and hydrodynamic-level descriptions derived from the microscopic model, we show that the analytic structure of this problem is qualitatively different from that of the three other classes. In particular, it is deprived from the generic linear instability at the root of the liquid/gas phase separation scenario. Particle-level simulations confirm this: the emerging polar order is only quasi-long-ranged with continuously-varying scaling exponents, while showing giant number fluctuations. Thus, this case possesses many of the properties of the (equilibrium) XY model. Yet, surprisingly, a numerical study of the continuous ordering transition leads to conclude that it does not belong to the Berezinskii-Kosterlitz-Thouless universality class [47][48][49] characteristic of the XY model, but is best described as a standard critical point with algebraic divergence of correlations. We rationalize these findings by showing that the coupling between order and density deprives defects from their usual role.
We first define our "Vicsek-shake" model. We restrict ourselves to two space dimensions and square domains of linear size L with periodic boundary conditions. The velocity v i of particle i is given by v i = ±v 0 p i where the unit vector p i is the particle's intrinsic polarity, and the sign is changed with probability α at each unit timestep. Positions and polarities are governed by: where Π normalizes vectors to unit length, range (−πη; πη], and the average is over the particles j present in ∂ i , the disk of radius r 0 = 1 centered on r i . We checked that the results presented in the following are not sensitive to the value of α, provided 0 < α < 1. Therefore, in the following, we only use the numericallyconvenient value α = 1 2 . With α fixed, the main parameters remain those of classic Vicsek-style models, the mean density of particles ρ 0 , and the noise amplitude η. We numerically determined the phase diagram of our model in the (ρ 0 , η) plane (Fig.1a). We find a single transition line from the disordered gas observed at strong noise and/or low density to a phase with global ordering of polarities, characterized, at finite system size, by a finite average value of the magnetization M (t) = | p i (t) i |. Contrary to the other known classes mentioned in the introduction, we do not see any sign of phase separation. The transition seems continuous, with only quasi-long-range order: the magnetization decreases algebraically with system size, M t ∼ L −κ(η) with κ increasing continuously with η ( Fig.1c). At strong-enough noise, a crossover to a fully disordered phase characterized by M t ∼ 1/L is observed at large-enough  (14) and (15) varying the exponent ν. The dashed lines represent the confidence intervals on ηc given by the fits. sizes (Fig.1d). Like in all known orientationally-ordered dry active matter phases, giant number fluctuations are present (Fig.1b): the variance ∆N 2 of the number of particles in a sub-system containing on average N particles scales faster than N . We find that ∆N 2 ∼ N ζ with ζ = 1.73(3), a value similar to those reported for the other classes [22,25,36].
We now derive hydrodynamic equations for the Vicsekshake class from our microscopic model. Encouraged by its overall success in the other cases, we adopt the Boltzmann-Ginzburg-Landau approach [19,23,37]. We write two coupled Boltzmann equations for the singlebody distributions f ± (x, θ, t) of "+" and "−" particles, i.e. those which, respectively, currently move along or against their polarity: where e(θ) in the material derivatives of f ± is the unit vector along θ [50], a is the exchange rate between the two subpopulations (akin to the microscopic reversal probability α), and the self-diffusion and collisional integrals read: where P η (σ) is the noise distribution of variance η 2 , δ 2π is the Dirac comb distribution of period 2π, and Ψ(θ 1 , θ 2 ) = Arg(e(θ 1 ) + e(θ 2 )) is the ferromagnetic alignment rule of polarities. The kernels K ± (θ 1 − θ 2 ) = |e(θ 1 ) ∓ e(θ 2 )| are different and are used depending on whether the two colliding particles belong to the same population or not.
Hydrodynamic equations are derived from (3) by expanding the distributions in angular Fourier modes: , and truncating and closing the resulting hierarchies in a controlled way. In the classic Vicsek model the remaining hydrodynamic (or slow) fields correspond to the first two angular modes, i.e. density and polarity/velocity. Here these fields are the zeroth and first modes of the sum f = f + + f − , i.e. the density and polarity of the total population, while the velocity field, now distinct from polarity, is the first mode of the difference g = f + − f − . Rewriting the Boltzmann equa-tions in terms of f and g modes, we obtain where the complex gradient ∇ = ∂ x + i∂ y , P k = dσP η (σ) exp(ikσ), and all other coefficients are listed in [51]. Setting f 1 ∼ ε near onset of polar order, Eqs. (6) and (7) impose the following scaling ansatz [37]: At the first non trivial order, ε 3 , we get equations for ρ, g 0 , f 1 , g 1 , f 2 and g 2 . The last two fields can then be enslaved to the four remaining ones, yielding: where all coefficients, expressed as functions of the microscopic parameters ρ 0 , σ and a are listed in [51] while their dependence on local density and g 0 has been made explicit. Eqs. (10) to (13) can be seen as two coupled Toner-Tu equations [14] [52]. Note that density is not advected by the order field f 1 , but by g 1 , in strong contrast to the classic polar case. Since ν 1 [ρ] < 0 and µ 1 [ρ] can change sign, the transition, as expected, is given by µ 1 [ρ 0 ] = 0, defining a line in the (ρ 0 , η) plane that goes to the origin as √ ρ 0 . Furthermore, since µ 1 [ρ] does not depend on a, this line is insensitive to the reversal rate, in agreement with the microscopic model. When µ 1 [ρ 0 ] < 0 the homogeneous disordered solution ρ = ρ 0 , f 1 = g 0 = g 1 = 0 is linearly stable, and becomes unstable when µ 1 [ρ 0 ] > 0. It is then replaced by the homogeneous ordered solution ρ = ρ 0 , g 0 = g 1 = 0, f 1 = µ 1 [ρ 0 ]/ξ. We studied its linear stability semi-numerically (see [51] for details) and analytically in the long wavelength limit (not shown). Apart from a pocket of weak, spurious instability at small a and low noises, it is essentially stable as soon as µ 1 [ρ 0 ] > 0 [53]. The analysis above confirms, at the mean-field level, the absence of the generic instability leading to the phase separation scenario in other classes of dry aligning active matter. Here we have a single transition line separating polar order from the disordered phase. Order (field f 1 ) and density are advected by the auxiliary field g 1 , and thus the mechanism proven by Toner and Tu to be responsible for the possibility of true long-range order is absent [14,15]. With fluctuations, polar order is only quasi-long range, as in equilibrium. Our problem thus possesses many of the hallmarks of the XY model. We now investigate whether this extends to the nature of the transition, i.e. whether it is in the well-known Berezinskii-Kosterlitz-Thouless (BKT) universality class. The BKT transition is characterized by an essential divergence of the correlation length ξ when approaching the critical point η c from the disorder side, together with the scaling of the susceptibility χ with ξ [48,49,54] log ξ ∼ (η−η c ) −ν ; χ ∼ ξγ with χ = L 2 ( M 2 − M 2 ) (14) with ν = 1 2 andγ = γ/ν = 7 4 . At finite size L, χ exhibits a maximum χ max (L) located at η χ (L). Increasing L, χ max diverges, and η χ converges to η c like: We have measured the dependence of M and χ on η for various systems sizes, all this at various global density values, but focussing most of our numerical effort on ρ 0 = 1. As shown in Fig. 2a, the susceptibility peak χ max does diverge algebraically with an exponentγ = 1.755 (6) in full agreement with the BKT/Ising value 7 4 . The peak location η χ is reasonably well fitted by Eq. (15) with ν = 1 2 , yielding an estimate of the asymptotic threshold η c = 0.247(2) (Fig. 2b).
Allowing now ν to vary in a range [0.1, 2], we find fits of the variations of η χ and ξ as convincing as for the BKT value ν = 1 2 . Interestingly, the two independent estimates of η c then become closer to each other as ν → 0 (Fig. 2d). This suggests an algebraic divergence for ξ at threshold. We therefore redefine the ν exponent as that of a standard second-order phase transition: Fitting our data accordingly, we obtain better fits for both ξ and η χ and, importantly, fully-compatible threshold values at which, moreover, κ(η c ) 1 8 . Imposing a common value for the asymptotic threshold, both datasets give the same estimate of ν, and we finally conclude that η c = 0.257(1) with ν = 2.4 (1). From these values, we compute β/ν using a collapse of the magnetization curves (shown in [51]) and find a value fully compatible with β/ν = 1 8 , which satisfies the hyperscaling relation 2β/ν + γ/ν = d with d = 2. Using data obtained at various global densities, we find the same estimates of γ/ν and β/ν, although our estimate of ν shows some variation due to its sensitivity to the estimated value of η c . The asymptotic threshold values thus obtained behave as η c ∼ ρ h 0 with h ∼ 0.66, in clear departure from the mean-field value 1 2 (Fig. 1a). Our numerical analysis leads us to conclude that the transition to polar order exhibited by our system is not of the BKT type. In the XY model, this transition is closely related to the (effective) Coulomb interaction between topological singularities that unbind and proliferate above the critical temperature [48,49]. Detecting these defects in simulations of our model is made very difficult, if not impossible, by the presence of strong density fluctuations. Indeed, the very existence of topologically constrained defects requires that order can be defined everywhere. Here, the local order is hard to measure in sparse regions, and even impossible to define if the local density is below the ordering threshold ρ c 0 (η), the transitional density found by varying ρ 0 keeping η fixed. One can nevertheless study the fate of defects from carefully prepared initial configurations containing a ±1 pair. Running the model deep in the ordered phase, we observe that the positive defect expels particles from its core and is quickly transformed into a sparse, almost void region whose diameter grows like √ t (Fig. 3c). After some time, this region has become sufficiently large so that it reaches the negative defect and the system eventually repairs itself (Fig. 3a, movie in [51]). This mean-field behavior is also observed in simulations of the (deterministic) hydrodynamic equations (Fig. 3b, movie in [51]). It is intrinsically related to the coupling between density and order, and indicates that the very idea of topologically-bound point defects is not relevant in our system. Closer to the transition, we of course expect fluctuations to play a major role, but this conclusion should still hold. This is the topic of ongoing work.
To summarize, we have shown that the collective behavior of active particles with velocity reversals that align ferromagnetically their polarities is different from that of other classes of dry, dilute, aligning active matter. This new class is characterized by the emergence of a phase with quasi-long-range polar order and anomalous number fluctuations. Like in the XY model, scaling exponents vary continuously in this phase, but the transition point to order shows algebraic divergences governed by ν = 2.4(1), not the essential singularity of the BKT class. Nevertheless, the exponent ratios β/ν and γ/ν take the BKT/Ising values 1 8 and 7 4 . These results constitute the first case where the phase separation scenario at play in most dry aligning active matter systems is prevented "structurally", not by imposing incompressibility [45] or by resorting to non-metric neighbors [43,44]. The estimated exponents seem to correspond to a new type of non-equilibrium critical point.