Two-dimensional long-range uniaxial order in three-dimensional active fluids

Elongated active units cannot spontaneously break rotation symmetry in bulk fluids to form nematic or polar phases. This has led to the image of active suspensions as spontaneously evolving, spatiotemporally chaotic fluids. Here, in contrast, I show that bulk active fluids have stable active nematic and polar states at fluid–fluid or fluid–air interfaces. The active flow-mediated long-range interactions that destroy the ordered phase in bulk lead to long-range order at the interface. Thus, active fluids have a surface ordering transition and form states with quiescent, ordered surfaces and chaotic bulk. I further consider active units that are constrained to live at an interface to examine the minimal conditions for the existence of two-dimensional order in bulk three-dimensional fluids. In this case, immotile units do not order, but motile particles still form a long-range-ordered polar phase. This prediction of stable, uniaxial, active phases in bulk fluids may have functional consequences for active transport. Bulk active fluids are unstable because activity destroys long-range ordering. Now, a model of 3D active liquids shows that stable states can form at fluid–fluid surfaces.

Elongated active units cannot spontaneously break rotation symmetry in bulk fluids to form nematic or polar phases. This has led to the image of active suspensions as spontaneously evolving, spatiotemporally chaotic fluids. Here, in contrast, I show that bulk active fluids have stable active nematic and polar states at fluid-fluid or fluid-air interfaces. The active flow-mediated long-range interactions that destroy the ordered phase in bulk lead to long-range order at the interface. Thus, active fluids have a surface ordering transition and form states with quiescent, ordered surfaces and chaotic bulk. I further consider active units that are constrained to live at an interface to examine the minimal conditions for the existence of two-dimensional order in bulk three-dimensional fluids. In this case, immotile units do not order, but motile particles still form a long-range-ordered polar phase. This prediction of stable, uniaxial, active phases in bulk fluids may have functional consequences for active transport.
Active matter theories describe systems whose constituents convert a sustained supply of energy into work [1][2][3][4][5][6] . The microscopic energy input leads to macroscopic forces and currents, dramatically modifying the mechanics and statistical mechanics of active systems relative to their dead or equilibrium counterparts. This has spectacular effects on the phases and phase behaviours displayed by active materials.
One of the most notable consequences of microscopic drive on the phase behaviour of active systems is the threshold-free instability of uniaxial phases-both nematic and polar-in incompressible, bulk Stokesian fluids 3,7,8 as first discussed by Simha and Ramaswamy 7 . This bulk Simha-Ramaswamy instability, via which an oriented active fluid is driven to a spatiotemporally chaotic state 9,10 , has become the defining image of active suspensions. The impossibility of a bulk uniaxial state implies that active suspensions do not spontaneously break rotation symmetry; that is, there is no bulk isotropic-nematic or isotropic-polar transition in active Stokesian fluids. Instead, activity constantly stirs a bulk Stokesian fluid via random stresses, with a correlation length vanishing as the inverse of the square root of activity. This has led to the expectation that uniaxial phases cannot exist in momentum-conserved active systems, in contrast to fluids on substrates, where such order can not only exist at an arbitrarily high active drive, but can also be strengthened by it [11][12][13][14] .
In this article, I show that although bulk uniaxial ordering is forbidden in active suspensions, two-dimensional (2D) nematic and polar phases that spontaneously break continuous rotation symmetry exist at interfaces or boundaries of momentum-conserved fluids. In other words, bulk suspensions of elongated active units can have stable, long-range-ordered nematic or polar wetting layers at fluid interfaces or boundaries. Interface-associated ordering transitions and phases have been examined in detail in equilibrium systems [15][16][17][18][19] . However, in equilibrium magnets [17][18][19] and nematic fluids 15,20 , while such transitions precede bulk ordering, as the bulk critical point is approached from above, the thickness of the interfacial ordered phase diverges. In contrast, the thickness values of ordered wetting layers in active fluids do not change appreciably with noise strength since the bulk always remains disordered.
Furthermore, in equilibrium materials, interface-associated phases are more susceptible to random fluctuations than bulk order. For instance, a 2D boundary-associated aligned phase displays only quasi-long-range order 15 , whereas the bulk, 3D orientationally ordered state has true long-range order because the effect of Goldstone fluctuations is more dominant in two dimensions than in three. In contrast, I show that both active nematic and polar phases at interfaces or boundaries of bulk fluids display true long-range order (LRO). This stabilization of interfacial order in momentum-conserved systems is due to active fluid flows; although such flows destabilize bulk uniaxial order, they anomalously stabilize interfacial order. In other words, Article https://doi.org/10.1038/s41567-023-01937-4 this motile phase display giant number fluctuations (GNFs) with the root-mean-square (r.m.s.) number fluctuations in a region containing on average 〈N〉 swimmers scaling as 〈N〉 3/4 instead of 〈N〉 1/2 as it would in equilibrium systems. The results for the different geometries are summarized in Table 1.

Model description
In this article, I consider the ordering of elongated active units at fluidfluid or fluid-air interfaces. The flat interface is assumed to be clean, that is, not coated by surfactants, and is taken to be situated at z = 0. Although the 3D rotation symmetry of the bulk fluid is broken at z = 0 by the interface, the interfacial plane has 1) symmetry, that is, 2D rotation symmetry. In the models considered here, the elongated active units at the interface are taken to be preferentially predisposed to lie parallel to it. That is, any out-of-plane component of the orientation of active units at the interface is taken to relax to zero, fast. This may be due to planar anchoring conditions at the interface or active torques 37,38 . However, because the in-plane axes 15 are degenerate, i.e. there is no preferred direction in the plane, aligned states at the interfacial plane must spontaneously break 1) symmetry. I consider such spontaneous breaking of 2D rotation invariance at fluid interfaces.

Interfaces in bulk active fluids
In the first class, I consider bulk fluids containing either active orientable units or components that form such objects. In these cases, the number of active units at the interface is not conserved, because active units diffuse into the bulk. Both nematic and polar orders are considered, using the nematic, non-conserved (NNC) and polar, non-conserved (PNC) models, respectively. These models apply to two distinct physical situations: (1) when monomers associate to form active units at the interface as well as dissociate and diffuse in the bulk and (2) when the active units themselves diffuse in the bulk fluid. These two situations are depicted in Fig. 1a,b (respectively at a two-fluid interface and a fluid-air interface). The first is relevant when monomers associate or dissociate at a fluid-fluid or fluid-air interface to form polymers, which then become active due to the action of motors. This is the case, for example, in the cellular cortex 5,39 . The second models systems in which elongated active swimmers, such as bacteria, form an ordered phase at an interface but can move into the bulk fluid.
In systems described by this class of models, the number of active units at the interface is not conserved, even though their total number (or the total number of monomers that compose the active units), both in the bulk and at the interface, is conserved. The dynamical equations of the in-plane order parameters only directly couple with the in-plane density of active particles and the in-plane velocity. Because, in this case, an excess in-plane concentration of active particles does not relax by diffusing through the interfacial layer, but instead by diffusing into the bulk fluid, the interfacial concentration of active units is not a slow variable-it relaxes within a microscopic time to a value determined by the bulk concentration of active units. The bulk fluid acts as a bath of particles that holds the interfacial particle concentration locally fixed 40,41 . Thus, the interfacial concentration is not a hydrodynamic variable in NNC and PNC models. Although the total concentration active flows 'anti-screen' fluctuations of bulk orientational order, but they screen fluctuations of interfacial uniaxial states in the same system. This is akin to observing a Jeans-like instability in the bulk, but a Coulomb-like screening at the interface. In bulk fluids, the ratio of active stress and viscosity introduces an inverse timescale 6 . However, this need not be proportional to the growth rate of an instability-it could be proportional to the relaxation rate of a massive mode. It turns out to be proportional to the growth rate of the instability of an aligned phase in bulk fluids ultimately because both total mass and total momentum are conserved. The crucial distinction that allows active flows to stabilize boundary-associated nematic order while destabilizing bulk order is that neither the number of active units nor the total mass at the interface is conserved. That is, both active units and the fluid can escape in the third dimension, with the 2D interfacial flow field not being incompressible even though the bulk 3D fluid is 16,21,22 . Interfacial polar order is more robust.
Beyond the fundamental physical interest of realizing boundary-associated ordered phases in momentum-conserved active fluids, when no corresponding bulk phase exists, the fact that orientational order may exist even in highly active fluids has important experimental and biological consequences. Some of the more widely used experimental systems for studying pattern formation in active uniaxial systems are composed of motor-microtubule filaments either at a two-fluid interface 23,24 or forming a self-assembled layer immersed in a bulk fluid 25,26 . Although it had been assumed that uniaxial phases are forbidden in these geometries 27,28 , this article demonstrates that this conclusion is contingent on experimental details. Swimmers, such as bacteria [29][30][31][32] , generally aggregate at interfaces 22 and may form effectively 2D uniaxial phases. In cellular systems, uniaxial ordering is likely to be associated with interfaces or membranes-such as in the cortex-and is relevant for active transport [33][34][35][36] . Finally, there has recently been a great deal of interest in the possibility of forming protocells by microphase separation in active fluids. If such protocells contain elongated particles-for example, if the phase separation leads to droplets that are rich in elongated active units in a background that is poor in them-they can form an ordered boundary layer of orientable filaments in the droplet or a protocortex.
I now describe the key question and results of this article.

Summary of results
This article shows that active uniaxial particles in momentum-conserved fluids can spontaneously break rotation symmetry and form LRO polar or nematic phases at fluid-fluid or fluid-air interfaces. The hydrodynamic properties of these LRO uniaxial phases belong to a new universality class that I characterize exactly. How do these interfacial ordered phases escape the Simha-Ramaswamy instability that does not allow bulk ordering? Is it simply because the 2D velocity field at the fluid interface is compressible 16,21,22 ? To examine this, I consider active particles that are confined only to a fluid interface and cannot diffuse freely into the bulk fluid. The 2D interfacial velocity is not divergence-free in this system either, but, importantly, the number of active particles at the interface is conserved. This reveals that the minimal conditions for the existence of interfacial order differ for polar and apolar systems-although a homogeneous nematic phase is generically unstable in this case, the polar phase is more robust. The form of the nematic instability is, however, distinct from that in incompressible bulk fluids, and the lengthscale of the fluid structures formed at large active drive becomes independent of activity. I show that this is similar to an extended version of the Simha-Ramaswamy instability that generically destabilises bulk nematic states in all mass and momentum-conserved active fluids irrespective of whether they are incompressible or not. While conserved quantities at the interface destroy active nematic order, motile particles escape this instability and form an LRO state for large-enough propulsion speeds even when the number of active units at the interface is conserved. The concentration fluctuations in Article https://doi.org/10.1038/s41567-023-01937-4 of active units (or monomers) in the bulk and the interface is a hydrodynamic variable, this does not affect in-plane interfacial ordering. This is shown explicitly in Supplementary Section A. I now construct the dynamical equations for the NNC and PNC models.

NNC model-interfacial active nematic composed of nematogens diffusing in the bulk.
Because, in both NNC and PNC models, the interfacial concentration of active units is not a hydrodynamic variable, the long-time, large-scale evolution of an in-plane ordered state is determined by the coupled dynamics for an in-plane order parameter that measures the degree of interfacial ordering of filaments and the in-plane velocity field. In nematic systems, the relevant order parameter is a rank-2 tensor: where the ordering direction is taken to be along x , θ is the devia tion of the local nematic order from x and S is the magnitude of the nematic order, whose steady-state value is S 0 = 〈S〉. This couples to the 2D, interfacial flow field v aṡ where the overdot denotes the convected derivative is the vorticity tensor at the  2 , which supports orientational order for α < 0. Note that the dynamical equation for the nematic order parameter at this order in gradients and fields is equivalent to that of an equilibrium system. There is no active term in this equation, because any such term involving just Q can be absorbed by redefining f Q . The lowest-order non-integrable term has two powers of gradient and two powers of Q and, as I will show post facto, is irrelevant. Furthermore, note that the entry of active particles into the interfacial layer at arbitrary in-plane angles, and their exit from it, imply a local, stochastic change of the orientation field leading to an additional source of noise 40,41 (that is, leading to a modification of Δ Q ). The in-plane velocity field v is obtained from the 3D bulk velocity field V as V ⊥ | z = 0 = v, where ⊥ represents the x and y components of a vector. For slow flows, V satisfies the incompressible Stokes equation forced by a surface force density f s , which, in momentum-conserved systems, can be written as a divergence of an in-plane stress ∇ ⊥ ⋅ σ s . The hydrodynamic behaviour of the in-plane ordered state is controlled by an interfacial active stress σ s = −ζQ, where ζ > 0 denotes an extensile suspension, and one with ζ < 0 signifies contractility 42 (see Supplementary Section D for a discussion of subdominant terms in σ s ). The equations for Q and V have to be solved simultaneously to examine the stability of interfacial order. Supplementary Section B shows that v appearing in equation (2) is obtained by solving the Stokes equation for V. It is related to f s via v = M ⋅ f s , where the mobility written in Fourier space, to the lowest order in wavenumbers, is with q ⊥ ≡ (q x , q y ) being the in-plane wavevector of a perturbation, and η the arithmetic mean of the viscosities of the fluid above and below the interface. See Supplementary Section B for higher order in q ⊥ corrections to this, arising from a finite Saffman-Delbrück length. Note that even though active units are present in the bulk fluid, because they do not order 7 , bulk active forcing leads only to a conserving noise, correlated over finite spatial and temporal scales 10,43 , which does not affect the hydrodynamic behaviour of the interfacial ordered phase (a demonstration is presented in Supplementary Section C). Article https://doi.org/10.1038/s41567-023-01937-4 p = p cos θ, sin θ) (here, p is the magnitude of the polar order and θ is the local deviation of the polarization from the ordering direction, which is taken to be x ) and the in-plane interfacial velocity field v. Just as in the NNC model, the interfacial concentration of polar active units is not a hydrodynamic variable. The dynamics of p is where v p denotes active self-advection due to the motility of the polar particles, F = ∫dr ⊥ [(α/2)p 2 + (β/4)p 4 + (K/2)(∇ ⊥ p) 2 ] = ∫dr ⊥ f p is the standard free energy for polar liquid crystals, h = δF/δp, and ξ p is a spatiotemporally white noise with variance 2Δ p . Other nonlinear gradient terms do not affect the hydrodynamics of the interfacial phase (Supplementary Section D) and have been suppressed here. As in the NNC model, the interfacial velocity is obtained by solving the interfacially forced bulk Stokes equation as v = M ⋅ f s . Here, M is given by equation (3) and f s = −ζ∇ ⟂ ⋅ (pp − p 2 I/2), where I is the rank-two identity tensor. This is equivalent to the active force in the NNC model.

Active interfacial layer in fluids
In the second class of models that I consider, active units are constrained to live at fluid-fluid or fluid-air interfaces. This is particularly relevant for experiments on motor-microtubule layers at two-fluid interfaces 23,24 , in which the numbers of microtubule filaments and motors are conserved. I will use these models to better understand the conditions that are required for stable active interfacial order. Because the active units are constrained to live at the interface, their number at the interface is conserved, unlike in the NNC and PNC models, and their concentration c is an additional hydrodynamic variable that couples to the in-plane order parameter and velocity fields. As earlier, I consider both nematic and polar orders, accounted for by the NC (nematic, conserved) and PC (polar, conserved) models, which I now describe.
NC model-interfacial active nematic composed of nematogenic species living at the interface. The dynamics of the Q tensor in this model is still described by equation (2). The interfacial velocity field is again obtained using the mobility in equation (3)

Long-range-ordered nematic and polar phase in NNC and PNC models
where Γ θ = Γ/4, and ξ is a non-conserving noise with variance Δ Q /2, from equation (2). The linear active force density is f s = −iζ q y θx + q x θŷ). This, in conjunction with equation (3), yields, to the lowest order in wavenumbers: where ϕ is the angle between q ⊥ and the mean ordering direction x . Equation (6) implies that the relaxation rate for angular fluctuations is positive for all ϕ when |λ| > 1 and ζλ > 0 (Supplementary Section E provides a detailed demonstration). This directly demonstrates that a 2D planar nematic phase is realized in the fully momentum-conserved NNC model (Fig. 2).
The 2D active nematic phases, both in the presence 11 and absence of fluids 48-51 were thought to be possible only in contact with substrates, which act as momentum sinks. Although nematic suspensions in contact with a substrate can retain order at arbitrarily high active drive 11 , this is a direct result of momentum exchange with the substrate. This allows for an active force that is not a divergence of a stress, and has a distinct angular symmetry, that tends to stabilize nematic ordering. In momentum-conserved systems, Simha-Ramaswamy instability was thought to forbid order. What allows interfacial states to evade this and order due to activity, even though here the active force is a divergence of a stress? It was believed that one of the key requirements for the generic instability of aligned states is fluid incompressibility. The interfacial fluid velocity is not incompressible 21,52 . It only becomes effectively incompressible when the surface is treated with surfactants 22,53,54 , as in refs. 24,27,28 . Moreover, all 2D films in a 3D medium, whether surfactant-coated or not, are compressible at large scales due to interfacial fluctuations 55 , even when they are incompressible at small scales. Because I consider a clean interface, a non-vanishing dilational or compressive flow in the plane is compensated by a non-vanishing z-gradient of the 3D velocity field (see equations (B12) and (B13) in Supplementary Section B for a direct demonstration). Indeed, an in-plane director distortion leads to a compressive or dilational flow iq ⟂ ⋅ v = ζθ/2η) q x q y /|q ⟂ |) = ζθ/4η)|q ⟂ | sin 2ϕ that leads to the final term in the square brackets in equation (6), which is essential for ensuring the stability of nematic order. Is the compressibility of the interfacial flow sufficient to ensure the stability of an ordered phase? Notice that the NNC model differs in two distinct ways from the conditions under which Simha-Ramaswamy instability is generally discussed: (1) the interfacial fluid layer is compressible and (2) the number of active units at the interface is not conserved. To check whether the former by itself ensures the stability of the ordered phase, I will examine in the next section whether an ordered state in which the active units are constrained to live at the interface-such that the number of particles at the interface is conserved (that is, (2) above does not hold)remains stable. Now I examine the properties of the interfacial ordered state. Because equation (6) implies that the relaxation rate of angular fluctuations scales as |q ⊥ | along all directions of the wavevector space-unlike the ∼q ⊥ 2 scaling of relaxation rate of equilibrium nematics or active nematics on substrates 3,11,48 -the static structure factor of angular fluctuations 〈|θ(q ⊥ , t)| 2 〉 ∝ Δ/|q ⊥ |. Using this to calculate the depression of the order parameter from its perfectly ordered value S 0 due to fluctuations, I get ⟨S⟩ Because W remains finite, 〈S〉 does not vanish due to fluctuations even in infinite systems (for small enough Δ). Thus, the linear theory predicts that an interfacial active nematic phase displays LRO (Supplementary Section E provides an expanded version of this discussion). The LRO phase here is a result of long-range interactions due to the bulk fluid. If this interaction is cut off at some (large) scale, for example, if instead of an interface in a bulk medium, one considers an interface in a thick layer of fluid resting on a substrate that acts as a momentum sink 56 , then the order is quasi-long-ranged instead of long-ranged-just as in active nematics in contact with a sub strate 11,48-51,57 -as I show in Supplementary Section F. Importantly, the ordered state is not destabilized; that is, the mechanism via which the aligned state is stabilized does not require the fluid interaction to be strictly infinite-ranged. To see whether the conclusion about the interfacial phase having LRO holds upon including the effect of nonlinearities, I use the standard renormalization group logic: I rescale lengths, time and the angle field as x → bx, y → b μ y, t → b z t and θ → b χ θ, where μ is the anisotropy exponent, z is the dynamical exponent and χ is the roughness exponent and examine whether any nonlinearity grows under this rescaling. Because ∂ t θ ~ −|q ⊥ |θ for all directions of the wavevector space, z = ζ = 1, within linear theory. Furthermore, as 〈|θ(q ⊥ , t)| 2 〉 ~ Δ/|q ⊥ |, χ = −1/2. I now use these exponents to check the relevance of nonlinearities. The most relevant nonlinearities that appear in equation (6) are from terms that have one power of the velocity field v and one power of θ, along with a gradient operator, such as the one due to advection v ⋅ ∇ ⊥ θ. This scales as ∼ q ⟂ θ 2 ) q because v ~ Φ(ϕ)θ where Φ(ϕ) is a vector function that depends on ϕ, but not on |q ⊥ |. This scales as b z − 1 + χ = b χ , which is irrelevant at the Gaussian fixed point because χ < 0. All other nonlinearites are even more irrelevant, implying that the linear exponents describe the exact hydrodynamic properties of the LRO active nematic phase.
Here, I have demonstrated that the interfacial nematic phase is stable to spin-wave fluctuations. In Supplementary Section G, I further argue that it should also be stable against unbinding of topo logical defects, confirming a 2D LRO nematic phase in 3D active fluids. In Supplementary Section H, I further demonstrate that polar flocks in the PNC model also display LRO with the same exponents as the ones obtained here.

Ordering when active units are confined to an interface
Although the last section demonstrated that active nematic and polar boundary layers exist at interfaces in bulk fluids, the minimal requirements for escaping the Simha-Ramaswamy instability were not clear. Are ordered phases possible in any compressible system, including momentum-conserved ones? To examine this, I consider systems in which active units are constrained to live only at the interface. The interfacial fluid velocity is still compressible, but the number of active particles at the interface is conserved. I first consider nematic ordering, then polar order.

Generic instability of a nematic phase in the NC model
I now show that when active units are constrained to live at the interface, active nematic order is impossible; that is, nematic order is generically unstable in the NC model. To see this, I expand the equations of motion for concentration and angular fluctuations about a homogeneous, perfectly ordered state with S 0 = 1 and a mean concentration c 0 . The linear fluctuations of the osmotic pressure about c 0 are taken to be Π c (c) ≈ A r (c 0 )δc and the coupled linearized equations for the angle and concentration fields are and Here all coefficients such as ζ, λ are evaluated at c 0 (this is a simplified limit of the full coupled equations of motion for δc and θ displayed in Supplementary Section J; here I assume that ζ is not a function of c). One of the two eigenfrequencies implied by equations (7) and (8) changes sign as the angle, ϕ, between the wavevector of perturbation and the ordering direction passes through π/4: implying an instability of the ordered state for perturbations with ϕ either just above or just below π/4. Interestingly, this eigenfrequency becomes independent of ζ, and therefore of its sign (that is, its contractile or extensile character), at small A r (that is, when ζλ ≫ A r c 0 ). Of course, ω − vanishes when ζ = 0. Therefore, ω − crosses over from being independent of ζ when ζλ ≫ A r c 0 to linearly depending on ζ when ζλ ≪ A r c 0 . That the character of the instability can be controlled by changing the concentration of the active particles (which controls A r ) has important consequences for experiments on motor-microtubule films 24,56 . Often, these experiments measure the 'activity lengthscale' ℓ a . This scale is associated with the fastest-growing mode obtained by retaining the O q 2 ⟂ ) terms in equations (7) and (8) that stabilize the dynamics at larger wavenumbers (this explicitly depends on the Saffman-Delbrück length; see Supplementary Section I). The form of equation (9) demonstrates that, for small enough A r , ℓ a ∝ (2A r c 0 + ζλ)/A r c 0 ζ saturates at a finite value as activity is increased, while for larger A r it scales as 1/ζ (Fig. 3). This is in contrast to current predictions that ℓ a decreases monotonically with activity 24,56 . This implies that the fastest-growing mode, or the typical vortex size in the spatiotemporally chaotic state, in motor-microtubule experiments 24 should become independent of activity as the density of the active particles is reduced; that is, at low microtubule concentration, ℓ a should become independent of motor concentration.
The generic instability of the nematic state in this model implies that the lack of 2D incompressibility of an interfacial layer is not enough Article https://doi.org/10.1038/s41567-023-01937-4 to stabilize nematic order. Instead, stability also requires the number of active units at the interface to not be conserved. Indeed, although Simha-Ramaswamy instability has, until now, been thought to be associated with incompressible bulk fluids, I show in Supplementary Section L that bulk nematic order is unstable-with the instability having a character similar to the one discussed in this section (albeit with a wavenumber-independent growth rate)-in any momentum and mass-conserved active fluid and not just incompressible ones. This provides a rationalization for nematic interfacial order additionally requiring an exchange of active particles between the interface and the bulk.

Stable polar phase PC model-outrunning the generic instability
Although apolar active particles constrained to live at surfaces of momentum-conserving fluids cannot order, surprisingly, polar swimmers can. In other words, although both bulk polar and apolar orders are generically unstable in Stokesian, momentum-conserved active fluids (Supplementary Section L), the minimal conditions for the existence of an interfacial ordered state differ for these two kinds of order. Apolar order only exists in the absence of any interfacial conserved quantity, but polar order is more robust. Highly motile units outrun the instability suffered by their immotile counterparts. Here I demonstrate this in a particularly simple case (Supplementary Section M presents a discussion of the full model) where the only polar term I retain is the self-advection of the angular fluctuations. The concentration dynamics is still described by equation (8) (that is, I artificially set the active motility to 0 for this illustration; see Supplementary Section M) and I add a term −iv p q x θ to the right-hand side of equation (7). This turns out to be enough to ensure the existence of a stable active polar phase. The eigenfrequencies, for fluctuations about a polarized state, written in terms of two dimensionless numbers ℛ 1 = A r c 0 /4v p η and ℛ 2 = ζ/4v p η are where . The stability boundary of the polar phase depends on the non-dimensional parameters ℛ 1 , ℛ 2 and λ. A representative plot of the stability region of a homogeneous flock in the ℛ 1 , ℛ 2 plane, for a specific choice of λ (λ = 2), is displayed in Fig. 4.
I show in Supplementary Section M that the linear theory describes the exact hydrodynamic behaviour of this flock, and the angular fluctuations have the same exponents that characterize nematic and polar flocks in the NNC and PNC models. Furthermore, considering the full PC model (that is, one in which the coefficient of the active motility has not been set to 0), I find (Supplementary Section M) that the roughness exponent for the concentration fluctuations is χ c = −1/2 as well. This implies that the equal-time concentration fluctuations 〈|δc(q ⊥ , t)| 2 〉 ~ 1/|q ⊥ | in the polar phase. This divergence at small wavenumbers implies that r.m.s. number fluctuations √⟨ N 2 ⟩ in a region containing 〈N〉 particles on average scales as 〈N〉 3/4 , instead of as 〈N〉 1/2 as it would in equilibrium systems not at a critical point, and thus violates the law of large numbers. Such GNFs-albeit with a different exponent-have been extensively discussed for active systems in contact with substrates 3,6,11,48,58 , but here appear in a fully momentum-conserved system.
The mechanism via which polar order is stabilized even when the active units are constrained to live at the interface bears some resemblance to how motility stabilizes bulk flocks in inertial fluids 59 . However, here, the interfacial flock is stabilized even in the strict Stokesian regime.

Conclusions
In this article, I have shown that active orientable particles form ordered phases at interfaces of momentum-conserved bulk fluids. While nematic order requires the exchange of particles between the bulk and the interface, polar order does not. In Supplementary Section N I further demonstrate that small fluctuations of the interface itself do not destroy this ordering. This is particularly relevant for order on membranes immersed in bulk fluids.
I now discuss some experimental systems and realistic situations in which the results obtained in this article should be observed and tested. Self-assembled motor-microtubule layers at oil-water interfaces 23,24,60 or in bulk fluids 25,26 are one of the mainstays for the study of active pattern formation. In current experiments, the number of capped microtubule filaments at the interface is essentially constant, and so the nematic state is generically unstable, as expected. Furthermore, For an essentially incompressible system, that is, A r → ∞, ℓ a scales as 1/ζ, but for any finite A r , ℓ a scales as 1/ζ only until ζ ~ 2A r c 0 /λ, saturating beyond that. due to the surfactants present in these systems 61,62 , the interfacial layer is essentially incompressible in 2D. However, the incompressibility constraint may be removed and the value of A r in these experiments may be varied by not using surfactants and changing the concentrations of microtubule filaments. As A r is varied, the activity lengthscaleor the scale of the vortices in the chaotic state-should display the behaviour described in the subsection Generic instability of a nematic phase in the NC model. More radically, an ordered nematic phase may be realized by allowing the microtubule filaments to associate and dissociate at the interface and the monomers to diffuse in the bulk. A further complication is that these experiments [23][24][25][26]56,60 are performed in confined channels. This cuts off the long-range fluid interactions, but I demonstrate in Supplementary Section F, that a stable uniaxial phase is still realized at arbitrarily high values of activity. However, the suppression of long-range interactions enhances orientational fluctuations leading to only quasi-long-range order. Beyond experiments on intracellular gels, the theory presented here naturally models boundary layers of bacteria that aggregate at air-water interfaces [29][30][31][32] . Such bacterial films may display the ordered nematic phases predicted in the section Long-range-ordered nematic and polar phase in NNC and PNC models. Interfacial active nematics can also be created by impregnating an equilibrium interfacial nematic with bacteria.
Active polar phases are often associated with membranes 63 in cellular systems, and have important consequences for signalling and transport. One particularly interesting example of this is the polar ordering of short actomyosin filaments at cell membranes, which have been argued to be crucial for nanoclustering of cell-surface molecules [33][34][35][36] . Cell membranes are in contact with the bulk, active cytoskeletal fluid, which may promote ordering and, thus, membrane-associated transport.
A route to the spontaneous generation of surfaces is via phase separation, which is common in active systems 64 , and this article demonstrates that such surfaces can naturally host a uniaxial wetting layer. Droplets formed due to the segregation of molecules in complex active mixtures 65 can grow and separate due to chemical activity, thus modelling protocells. If some of the components in these protocells are elongated, they may form a stable and aligned boundary layer on the surface of the protocell, leading to an aligned protocortex.
I look forward to quantitative and qualitative experimental examinations of the properties of ordered active wetting layers.

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