Long-Range Nematic Order in Two-Dimensional Active Matter

Working in two space dimensions, we show that the orientational order emerging from self-propelled polar particles aligning nematically is quasi-long-ranged beyond $\ell_{\rm r}$, the scale associated to induced velocity reversals, which is typically extremely large and often cannot even be measured. Below $\ell_{\rm r}$, nematic order is long-range. We construct and study a hydrodynamic theory for this de facto phase and show that its structure and symmetries differ from conventional descriptions of active nematics. We check numerically our theoretical predictions, in particular the presence of $\pi$-symmetric propagative sound modes, and provide estimates of all scaling exponents governing long-range space-time correlations.

Studies of active matter continue to flourish, exploring more and more complex situations in an increasingly quantitative manner [1]. Evidence accumulates showing that active matter exhibits collective properties impossible in thermal equilibrium or even in driven systems [2]. In spite of all this progress, important fundamental questions remain open. A long-standing such issue is whether true long-range nematic order can emerge in two space dimensions (2D).
In this Letter, we study 2D dry dilute active nematics -the framework in which the question of the asymptotic nature of nematic order was mostly discussed-using numerical simulations and theory. We show that the homogeneous ordered phase of a Vicsek-style model of polar self-propelled particles aligning nematically actually displays true long-range nematic order only up to r , the scale associated to typical time between velocity reversals induced by collisions and noise. Beyond r , global nematic order decays algebraically with system size, in agreement with general theoretical arguments. However r can easily take astronomically large values such that there exists a region of parameter space in which only true long-range nematic order can be observed. We derive a hydrodynamic theory for this regime and show that it possesses a structure and symmetries different from those of standard active nematics. Our analysis of this field theory predicts π-symmetric sound modes and the scaling form of space-time fluctuations. Finally, numerical results confirm the theory and allow us to estimate all scaling exponents.
We use the Vicsek-style model of polar particles with nematic alignment first introduced in [20]. Particles i = 1, . . . , N evolve at discrete timesteps with constant speed v 0 in square domains of linear size L with periodic boundary conditions, interacting with neighbors within unit distance. Their positions r i and unit-length orientations e i = e(θ i ) obey: where ϑ normalizes vectors (ϑ(u) = u/ u ), and R η rotates them by a random angle drawn from a uniform distribution in [−πη, πη], independently for every particle at every timestep. The two main parameters are the global densityρ = N/L 2 and the noise strength η. The phase diagram in the (ρ, η) plane is typical of Vicsek-style models [11]. All results presented below were obtained with v 0 = 0.5 andρ = 2.
We focus on the homogeneous nematic liquid that exists for η 0.21, where the global nematic order parameter S = | e i2θ t k k | t takes O(1) values. In this state, particles can be split into two 'polar' subpopulations according to which of the two opposite directions defined by the nematic order their orientation is closest. The nematic interaction in Eq. (1b) aligns particles belonging to the same population and anti-aligns particles belonging to opposite populations, so that particles mostly stay in the same population. Nevertheless, under the action of interactions and noise, they can eventually turn enough that they join the other population. It was shown in [20] that the distance traveled between such reversals is distributed exponentially with a characteristic length r independent of system size. In Fig. 1(a), we show that r grows very fast when the noise strength η decreases. A good fit of our data is that r ∼ η −8 .
In [20], the global nematic order parameter S was found to decrease slower than a power of L and consistent with an algebraic decay to a finite asymptotic value (S(L) − S(∞) ∼ L − ). These results led to conclude to true long-range nematic order, but they were obtained on a range of system sizes barely encompassing r . Here, choosing a noise strength such that r is not too large, we find that for L > r , S decays like a small power of L, in departure from the L < r behavior ( Fig. 1(b)). Asymptotically, nematic order is only quasi-long-range, in agreement with standard theories [18]. Nevertheless, in most of the homogeneous nematic phase, r is so large that only the L < r regime is accessible and it is thus important to study it per se. Working in this regime, we confirm that nematic order is fully long-range; moreover, the scaling of the local slope σ(L) ≡ −d ln(S)/d ln(L) ∼ L − allows to identify an internal crossover scale c separating two regimes with different values of ( Fig. 1(c)). We now present a theory of the long-range-ordered nematics present on scales much smaller than r . Full details of calculations are given in [22]. Our approach is not a perturbative version of active nematics: We directly consider two populations, R and L, of polar active particles with speed v 0 aligning their velocity with neighbors if those belong to the same population, and anti-align it otherwise. This is not equivalent to usual nematic alignment: two particles of the same population will align even if their relative angle is obtuse, and they will antialign if they belong to different populations, irrespective of their angle. We further assume that the populations exchange members randomly at rate 1/τ r r /v 0 . We first write Boltzmann equations ruling the evolution of the one-body probability density functions f L (r, θ, t) and f R (r, θ, t): (2) and the equation governing f R is given by swapping the L and R subscripts. In (2), v(θ) = v 0 e(θ) is the velocity of particles with orientation θ, whereas the integrals I sd and I co , given in [22], describe the effects of angular selfdiffusion and collisions.
Introducing the more convenient f = f R + f L and g = f R − f L , expanding f and g in Fourier series of θ (e.g. f (r, θ, t) = 1 2π +∞ k=−∞ f k (r, t)e −ikθ ), the Boltzmann equations are de-dimensionalized and transformed into a hierarchy of partial differential equations for the f k and g k fields. As shown in [22], a linear stability analysis of the disordered solution ρ ≡ f 0 =ρ (the total density), f k>0 = g k = 0 reveals that it is unstable to g 1 perturbations at large density and/or weak noise. The field g 1 is thus responsible for the onset of orientational order. Note that g 1 measures polar order within each population, i.e. is a proxy for global nematic order. The equations for ρ and g 0 read where ≡ ∂ x + i∂ y denotes the complex gradient.
Following the Boltzmann-Ginzburg-Landau approach [11,[23][24][25], one can build step by step a scaling ansatz using a small parameter ε marking the magnitude of order near onset (|g 1 | ∼ ε). As detailed in [22], this leads to: |g k≥1 | ∼ ε k , |f k>1 | ∼ ε k , and ∂ t ∼ ∼ ε [26]. In addition, considering Eqs. (3a,3b), one completes the scaling ansatz by |g 0 | ∼ ε , |δρ| ∼ |f 1 | ∼ ε 2 . Truncating and closing the Boltzmann hierarchy at order ε 4 yields hydrodynamic equations for f 1 and g 1 : where all coefficients depend on the particle-level parametersρ, η and τ r . (see [22] for their explicit expressions), and local dependencies on ρ and g 0 are indicated. Eqs. (3), are structurally different from hydrodynamic theories written for active nematics. The 2π-symmetry of the interaction between our polar particles makes the pairs of equations for (ρ, f 1 ) and (g 0 , g 1 ) resemble two coupled Toner-Tu (TT) systems. Both ρ and g 0 are advected by the corresponding order fields f 1 and g 1 , which are not π-symmetric. Discarding the couplings to ρ and f 1 , Eqs. (3b) and (3d) are almost like the TT equations in the limit τ r → ∞. They however miss terms ∼ g 0 g 1 and ∼ g 1 g 1 that are forbidden by the R ↔ L symmetry of the problem, which imposes the equations to be invariant under g ↔ −g. Eqs.
We first note that in the small τ r limit, such that particles reverse their orientation many times on the scale at which we observe fluctuations, δg 0 is non-hydrodynamic (cf. Eq. (4b)). Eqs. (4) then reduce to those of an homogeneous active nematic (with δg ⊥ playing the role of the transverse fluctuations of nematic order, see [18,22]).
In the τ r → ∞ limit of main interest here, on the other hand, we neglect the term 2τ −1 r δg 0 in Eq. (4b). To compute space and time correlation functions of the three hydrodynamic fields δρ, δg 0 and δg ⊥ , we equip Eqs. (4) with additive, uncorrelated, zero-mean noise terms. For Eq. (4a), governing density fluctuations, this noise is conserved and we write it ∂ h ρ + ∂ ⊥ h ρ⊥ . Writing the (stochastic) Eqs. (4) in Fourier space, taking the long wavelength, low frequency limit q, ω → 0, rather tedious but standard calculations detailed in [22] lead to: where ∆ ρ and ∆ ρ⊥ are the amplitudes of the conserved ρ noise, ∆ 0 and ∆ ⊥ those of the g 0 and g ⊥ noises, and As shown in [22], where their explicit forms are given, ε d,p (q) ∼ q 2 , whereas the anisotropic speed is where θ q denotes the angle between q and the mean order. Eqs. (5,6,7) are fundamentally different from their counterparts in both active nematics and the TT class: at most orientations θ q correlations have a diffusive peak and two symmetric propagative peaks at ω = ±c(θ q )q.
Equal-time correlation functions are easily obtained by integrating Eqs. (5) over ω. They all diverge as q −2 for most θ q , which means that nematic order is only quasilong-range at this linear level, a situation similar to that of polar order in TT theory. To resolve this marginal situation, one needs to study nonlinear hydrodynamics. We first repeat the calculations leading to Eqs.(4) keeping the leading order nonlinearities (in fields and gradients). The structure of our theory shares similarities with the polar case. We thus limit ourselves to terms of order 3 in fields and gradients [6]. After lengthy but straightfor-ward manipulations (detailed in [22]), we obtain: where L is the linear part (Eqs. (4)). Introducing the scaling exponents via We thus have isotropic (ξ = 1) diffusive (z = 2) scaling with quasi-long-range order in d = 2 (χ = 0) at the linear level, as for both active nematics and TT theory. At the linear fixed point, 9 of the 10 nonlinear terms in Eqs. (8) scale like b (4−d)/2 , i.e. are relevant in d ≤ d c ≡ 4 (the exception is ω 5 ). This means that the linear theory breaks down in d ≤ 4, and that we should in principle embark on a complete renormalization group analysis to obtain exponent values. We leave this challenging task for future studies. Instead we rely on general considerations and formal similarities with TT theory to make predictions that we test numerically.
Given that the structure of Eqs. (8) is similar to that found in TT theory, we follow [6] and conjecture that the scaling of correlation functions in the nonlinear theory is obtained using renormalized noise coefficients , with functions f ∆ and f ε expected to be universal and x z/ξ . On the other hand the speeds c(θ q ) should not be renormalized. Under all these assumptions, it is possible to predict the asymptotic behavior of equal-time correlation functions. For instance: (For other functions, see [22].) We now come back to our Vicsek-style model at noise strength η = 0.1 and show data for the order correlations confirming the structure of the above theory and providing estimates of the scaling exponents. Additional results for the densities ρ and g 0 will be published elsewhere [29]. We actually measure the transverse nematic order δQ ⊥ , which, when aligned along the horizontal direction and assuming small angular deviations, is a good proxy of δg ⊥ . (δQ ⊥ ∼ cos(θ) sin(θ) ∼ δθ ∼ρ −1 δg ⊥ ).
The frequency spectra do have the qualitative structure predicted by Eqs. (5): two symmetric propagative peaks and a central diffusive one (Fig. 2(a)). As expected, peak locations, at a fixed angle θ q are proportional to q, allowing the easy measurement of the sound speed c(θ q ), which we find in perfect quantitative agreement with Eq. (7) (Fig. 2(b)). Peak widths provide estimates of z and z/ξ in the ⊥ and directions, as in TT theory. As shown in Fig. 2(c), we find a crossover at the same scale c as observed in Fig. 1(c). For scales below c we find z 1.75 and z /ξ 1.4, while we are only able to estimate z/ξ 1.1 in the asymptotic regime (we use primes to denote exponent values measured below c ). The equal-time order correlation function shown in Fig. 2(c) in the and ⊥ directions, also exhibits a crossover at c . From the pre-crossover scaling we estimate ζ 1.75 and ζ /ξ 1.4, while we find ζ/ξ 1.1 in the q < 2π/ c regime.  We thus have two sets of scaling exponents: for scales below c , the above estimates lead to z = ζ 1.4, ξ 1.25, and 2χ −0.5. Note that this yields −2χ /ξ 0.4, in agreement with our estimate of 0.45 in Fig. 1(c) [30]. For scales beyond c , we have z = ζ, but cannot estimate ξ from correlation functions. Using 0.8 ( Fig. 1(c)), yields ξ 1.1 and 2χ −0.9, and finally z = ζ 1.2. A few remarks are in order: (i) both below and above c , z = ζ = 1 + 2χ + ξ, a hyperscaling relation also verified by polar flocks that implies that the dominant noises are additive and their amplitude is not renormalized [8]; (ii) in our nematic phase the anisotropy exponent ξ 1.1 > 1, at odds with 2D polar flocks for which ξ 0.95 < 1 [8], but in both cases we cannot exclude that scaling is asymptotically isotropic.
To summarize, the orientational order emerging from self-propelled polar particles aligning nematically is always quasi-long-range asymptotically, but this regime is only observed beyond r , the scale associated to induced velocity reversals, which can easily take very large values and often cannot even be measured. Below r , nematic order is fully long-range. Constructing a hydrodynamic theory from microscopic grounds, we showed that this de facto phase has a structure and symmetries distinct from both conventional descriptions of active nematics and Toner and Tu theory. Consequently, systems in the corresponding class exhibit features never reported so far, such as long-range nematic order and the presence πsymmetric propagative sound modes.
Finally, we believe our findings can be observed experimentally, as long as the rate of velocity reversals, be they induced or spontaneous, is small. After all, nematic alignment resulting from inelastic collisions between elongated objects is quite generic. Confined bacteria and motility assays are promising systems in this regard.
We thank Xia-qing Shi and Alexandre Solon for a critical reading of this manuscript. We acknowledge generous allocations of cpu time on the Living Matter Department cluster in MPIDS, and on Beijing CSRC's Tianhe supercomputer.