Surprising mappings of 2D polar active fluids to 2D soap and 1D sandblasting

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by  Leiming Chen

Surprising mappings of 2D polar active fluids to 2D soap and 1D sandblasting
Leiming Chen
College of Science, China University of Mining and Technology, Xuzhou Jiangsu, 221116, P. R. China
Chiu Fan Lee
Department of Bioengineering, Imperial College London,
South Kensington Campus, London SW7 2AZ, U.K.
John Toner
Department of Physics and Institute of Theoretical Science, University of Oregon, Eugene, OR 97403 and
Max Planck Institute for the Physics of Complex Systems, N¨othnitzer Str. 38, 01187 Dresden, Germany
6
1 Active fluids and growing interfaces are two well-studied but very different non-equilibrium sys-
0 tems. Each exhibits non-equilibrium behavior quite different from that of their equilibrium coun-
2 terparts. Here we demonstrate a surprising connection between these two: the ordered phase of
incompressible polar active fluids in two spatial dimensions without momentum conservation, and
n
growingone-dimensionalinterfaces(thatis,the1+1-dimensionalKardar-Parisi-Zhangequation),in
u
factbelongtothesameuniversalityclass. Thisuniversalityclassalsoincludestwoequilibriumsys-
J
tems : two-dimensional smectic liquid crystals, and a peculiar kind of constrained two-dimensional
4
ferromagnet. We use these connections to show that two-dimensional incompressible flocks are ro-
1
bustagainstfluctuations,andexhibituniversallong-ranged,anisotropicspatio-temporalcorrelations

ofthosefluctuations. Wealsotherebydeterminetheexactvaluesoftheanisotropyexponentζ and
]
t the roughness exponents χ that characterize these correlations.
f x,y
o
s PACSnumbers: 05.65.+b,64.60.Ht,87.18Gh
.
t
a
m Introduction . twocenturiesonequilibriumfluidshasfocusedonincom-
- Non-equilibriumsystemscanbehaveradicallydifferently pressiblefluids[23,24]. Inthispaper,weconsider2Dac-
d
from their equilibrium counterparts. Two of the most tive incompressible fluids; more specifically, we consider
n
striking examples of such exotic non-equilibrium behav- them in rotation invariant, but non-Galilean-invariant
o
c ior are moving interfaces (e.g., the surface of a grow- situationsinwhichmomentumisnotconserved(e.g.,ac-
[ ing crystal) [1], and “flocks” (i.e., coherently moving tive fluids moving over an isotropic frictional substrate

states of polar active fluids) [2–7]. The former is de- such as cells crawling on a substrate). Such an active
2
v scribed by the Kardar-Parisi-Zhang (KPZ) equation [8], system contains rich physics: it has recently been shown
4 which is also a model for erosion (i.e., sandblasting). that their static-moving transition belongs to a new uni-
2 Thisequationpredictsthatatwo-dimensional(2D)mov- versality class [25]. Here, we focus on the long-range
9
ing interface (i.e., the surface of a three-dimensional properties of the system in the moving phase.
1
crystal) is far rougher than the surface of a crystal in
0 We note that the incompressibility condition is not
1. equilibrium. In contrast, hydrodynamic theories of po- merelyatheoreticalcontrivance; notonlycanitberead-
lar active fluids [9–13] predict that a large collection of
0 ily simulated [26, 27] but it can arise in a variety of real
“active” (i.e., non-equilibrium) moving particles (which
6 experimental situations, including systems with long-
1 could be anything from motile organisms to molecular
ranged repulsive interactions [28], and dense systems of
: motor propelled biological macromolecules [2–7, 9–17])
v active particles with strong repulsive short-ranged inter-
can develop long-ranged orientational order in 2D, while
i actions, such as bacteria [26]. In addition, incompress-
X their equilibrium counterparts (e.g., ferromagnets), by
ibilityplaysanimportantroleinthemotilecolloidalsys-
r the Mermin-Wagner [18, 19] theorem, cannot. At the
a tems in fluid-filled microfluidic channels recently studied
same time, many non-equilibrium systems can also be
[29], although these systems differ in detail from those
mapped onto equilibrium systems [20]; an example of
westudyhereinbeingtwocomponent(backgroundfluid
this that proves very relevant is the connection between
plus colloids).
the 1+1-dimension KPZ model and the defect-free 2D
smectic (i.e., soap) model [21, 22]. Here, we add a living Inthispaper,weformulateahydrodynamic(i.e.,long-
systemtothislistbyshowingthatgenericincompressible wavelengthandlong-time)theoryoftheordered,moving
active polar fluids, e.g., an incompressible bird flock, all phase of a 2D incompressible polar active fluid. We find
belong to the same universality class. that the equal-time velocity correlation functions of the
type of incompressible polar active fluids we study here
Since many fluids flow much slower than the speed can be mapped exactly onto those of two equilibrium
of sound, a great deal of the work done over the past problems: a divergence-free 2D XY model (a peculiar
2
Ourresultsimplyinparticularthatincompressiblepo-
laractivefluidscandeveloplong-rangedorientationalor-
der (by developing a non-zero mean velocity (cid:104)v(cid:105)) in two
dimensions,justasfoundpreviouslyforcompressiblepo-
laractivefluids,butincompletecontrasttotheirequilib-
rium counterparts (i.e., ordinary divergenceful ferromag-
nets) with underlying rotation invariance, which cannot
so order. However, the scaling behavior of the veloc-
ity correlation functions is very different from those for
compressible polar active fluids studied in Ref. [11, 12].
Specifically, we find that the equal-time velocity corre-
lation function in the ordered phase has the following
limiting behaviours:
(cid:10)|v(r,t)−v(r(cid:48),t)|2(cid:11)
FIG. 1: | Visual representation of the mappings. The

flow lines of the ordered phase of a 2D incompressible polar C0−AY−2/3 , κ(cid:28)1
active fluid, the magnetization lines of the ordered phase of = (cid:20) (cid:16) (cid:17)2(cid:21) (1)
C − 9c2Ae−Φ(κ) 1+ 4 x−x(cid:48) , κ(cid:29)1
divergence-free 2D XY magnets, dislocation-free 2D smectic  0 2 X 9 y−y(cid:48)
layers, and the surfaces of a growing one-dimensional crystal
(which can be obtained by taking equal-time-interval snap where X ≡ |x−x(cid:48)|/ξ and Y ≡ |y−y(cid:48)|/ξ are rescaled
x y
shots), undulate in exactly the same way over space; their
lengths in the x and y directions, and we define the scal-
fluctuations share exactly the same asymptotic scaling be-
ing ratio κ ≡ X . Here the function Φ(κ (cid:29) 1) ≈ cκ3
havior at large length scales. Note that the vertical axis is Y2/3
and the constant c ≈ 0.295 are both universal (i.e.,
time for KPZ surface growth and the y Cartesian coordinate
for the other three systems. system-independent), while C0 and A are non-universal
(i.e., system-dependent), positive, finite constants, and
ξ are non-universal lengths. Note that the fact that
x,y
typeofferromagnetdifferentfromordinaryferromagnets, (cid:10)|v(r,t)−v(r(cid:48),t)|2(cid:11) goes to a finite value in the large
which are divergenceful) and a dislocation-free 2D smec- separation limit |r − r(cid:48)| → ∞ implies long-ranged
ticAliquidcrystal[21,22,30–32],aswellasontothetime orientational order.
dependent correlation functions of the non-equilibrium
1+1-dimensional KPZ equation [8]. The mapping of the Results
2Dsmecticontothe1+1-dimensionalKPZequationwas Model. Westartwiththehydrodynamicmodelforcom-
discoveredbyGolubovicandWang[21,22];theothertwo pressible polar active fluids without momentum conser-
mappings are new (although 2D ferromagnets with 2D vation [9, 11, 12]:
dipolar interactions, which are similar but not identical
systems, have also been mapped onto 2D smectics [30]). ∂ ρ+∇·(vρ)=0 (2)
t
This series of mappings is illustrated in Fig. 1.
∂ v+λ (v·∇)v+λ (∇·v)v+λ ∇(|v|2)=Uv−∇P −v(v·∇P )+µ ∇(∇·v)+µ ∇2v+µ (v·∇)2v+f
t 1 2 3 2 B T 2
(3)
where v(r,t), and ρ(r,t) are respectively the coarse stability of the ordered phase.
grained continuous velocity and density fields. All of The U term makes the local v have a nonzero magni-
the parameters λ (i = 1 → 3), U, the “damping coef- tude v in the ordered phase, by the simple expedient of
i 0
ficients” µ , the “isotropic pressure” P(ρ,v) and the having U > 0 for v < v , U = 0 for v = v , and U < 0
B,T,2 0 0
“anisotropic Pressure” P (ρ,v) are, in general, functions for v > v . The f term is a random driving force. It is
2 0
of the density ρ and the magnitude v ≡ |v| of the local assumed to be Gaussian with white noise correlations:
velocity. Note that we omit higher order damping terms
because, as our analysis will show later, they are irrele- (cid:104)fi(r,t)fj(r(cid:48),t(cid:48))(cid:105)=2Dδijδd(r−r(cid:48))δ(t−t(cid:48)) (4)
vant. In addition, because we focus here on the ordered
where the “noise strength” D is a constant parameter of
phase, µ is taken to be positive, as required for the
T,B,2 the system, and i,j denote Cartesian components. Note
3
that in contrast to thermal fluids (e.g., Model A in [24]), tively, changes in the density are too small to affect
we are concerned with active systems that are not mo- U(ρ,v), λ (ρ,v), µ (ρ,v), and P (ρ,v). As a re-
1,2,3 B,T,2 2
mentumconserving. Asaresult,theleadingcontribution sult, in the incompressible limit taken this way, all of
to the noise correlations is of the form depicted in (4). them effectively become functions only of the speed v;
We now take the incompressible limit by taking the their ρ-dependence drops out since ρ is essentially con-
isotropic pressure P only to be extremely sensitive to stant.
departures from the mean density ρ . One could alter- Anotherconsequenceofthesuppressionofdensityfluc-
0
natively consider making U(ρ,v) and P (ρ,v) extremely tuationsbytheisotropicpressureP isthatthecontinuity
2
sensitive to changes in ρ as well. This would be appro- equation (2) reduces to the familiar condition for incom-
priate for an active fluid near its “active jamming” [33] pressible flow,
transition, since in that regime a small change in the lo-
cal density can change the speed from a non-zero value ∇·v=0, (5)
for ρ < ρ to zero for ρ > ρ . We will discuss this
jam jam
case in a future publication. which can, as in simple fluid mechanics, be used to de-
Focusing here on the case in which only the isotropic termine the isotropic pressure P.
pressureP becomesextremelysensitivetochangesinthe All of the above discussion taken together leads to the
density, we see that, in this limit, in which the isotropic followingequationofmotionintensornotationforanin-
pressure suppresses density fluctuations extremely effec- compressiblepolaractivefluid,ignoringirrelevantterms:
∂ v =−∂ P +U(v)v −λ (v)v (∂ v )−λ (v)v v v (∂ v )+µ (v)∂ ∂ v +µ (v)v v ∂ ∂ v +f , (6)
t m m m 1 n n m 4 m n (cid:96) n (cid:96) T n n m 2 (cid:96) n (cid:96) n m m
where λ (v)≡ 1dP2(v), and the λ and µ terms vanish all frequencies ω, and dividing by 2π, gives the equal
4 v dv 2 B
due to the incompressibility constraint ∇·v = 0 on v. time, spatially Fourier transformed velocity autocorrela-
In writing (6), we absorb a term W(v) into the pressure tion (cid:104)|u (q,t)|2(cid:105). Details of this straightforward calcula-
y
P, whereW(v)isderivedfromλ (v)bysolving 1 dW = tion are given in “Methods”; the result is
3 2v dv
λ (v).
3
We now analyze the implications of equation (6) for (cid:104)|u (q,t)|2(cid:105)= Dqx2 ≈ Dqx2 , (8)
the ordered state. y 2αq2+Γ(q)q2 2αq2+µq4
y y x
Linear theory. In the ordered phase, the system spon- where Γ(q) ≡ µq2 +µ0q2 with µ ≡ µ0 +µ0v2, where
x T y T 2 0
taneously breaks rotational symmetry by moving on av- µ0T,2 are µT,2(v) evaluated at v = v0, and the second,
erage along some spontaneously chosen direction which approximate equality applies for all q→0. This can be
we call xˆ; we call the direction orthogonal to this yˆ. In seenbynotingthat,forqy (cid:29)qx2andq→0,qy2 (cid:29)Γ(q)q2,
the absence of fluctuations (i.e., if we set the noise f in while for qy <∼ qx2 and q → 0, Γ(q)q2 ≈ µqx4. Hence,
(6) to zero), the velocity will be the same everywhere in in both cases, (which together cover all possible ranges
spaceandtime,andhavemagnitudev ,whichweremind of q for q → 0), the approximation 2αq2 + Γ(q)q2 ≈
0 y
thereaderisdefinedbyU(v )=0. Wetreatfluctuations 2αq2+µq4 is valid.
0 y x
by expanding v around v xˆ, defining u(r,t) as the small We can now obtain the real space transverse fluctua-
0
fluctuation in the velocity field about this mean: tions
(cid:90) d2q
v=(v0+ux(r,t))xˆ+uy(r,t)yˆ. (7) (cid:104)u2(r,t)(cid:105)= (cid:104)|u (q,t)|2(cid:105), (9)
y qx>∼L1 (2π)2 y
Plugging Eq. (7) into Eq. (6) and expanding to lin-
ear order in u, leads to a linear stochastic partial dif- where L is the lateral extent of the system in the x-
ferential equation with constant coefficients. Like all direction (its extent in the y-direction is taken for the
such equations, this can be solved simply by spatio- purposes of this argument to be infinite). Note that the
temporally Fourier transforming, and solving the resul- longitudinal fluctuations (cid:104)u2(r,t)(cid:105) are negligibale com-
x
tant linear algebraic equations for the Fourier trans- pared to (cid:104)u2(r,t)(cid:105). Using (8), the integral in (9) is read-
y
formed field u(q,ω) in terms of the Fourier transformed ily seen to converge in the infra-red, and, hence, as sys-
noise f(q,ω). We can thereby relate the two point cor- tem size L → ∞. Since the integral is finite, and pro-
relation function (cid:104)|u (q,ω)|2(cid:105) to the known correlations portional to the noise strength D, it is clear that, for
y
(4) of the random force f. Integrating the result over sufficientlysmallD,thetransversefluctuations(cid:104)u2(r,t)(cid:105)
y
4
canbemadesmallenoughthatlong-rangedorientational remains valid when nonlinear effects are taken into
order-i.e., anon-zero(cid:104)v(r,t)(cid:105)-ispreservedinthepres- account (even though those nonlinearities change the
enceoffluctuations; therefore,theorderedstateisstable scaling laws from those predicted by the linear theory).
against fluctuations for sufficiently small noise strength
D. Nonlinear Theory. We begin by expanding the full
We show in the next section that this conclusion equation of motion (6) to higher order in u. This gives
∂ u = −∂ P −2αu δ −λ0v ∂ u +µ0∇2u +µ0v2∂2u +f
t m m x mx 1 0 x m T m 2 0 x m m
(cid:32) (cid:33)
α u3
− yδ +2u u δ +u2δ −λ0u ∂ u δ , (10)
v v my x y my y mx 1 y y y my
0 0
where the superscript “0” means that the v-dependent namely, λ0u ∂ u actually remains at this point.
1 y y y
coefficients are evaluated at v = v , and we define the
0
“longitudinal mass” α≡−v0 (cid:16)dU(v)(cid:17) . To proceed further, we must power count more care-
2 dv v=v0 fully.
Thefirstlineofequation(10)containsthelinearterms,
including the noise f; the first three terms on the second We only need to calculate one of the two fields u ,
x,y
linearetherelevantnon-linearities,whilethefourthterm since they are related by the incompressibility condition
proves to be irrelevant, as we’ll soon show. ∇·v = 0. We choose to solve for u ; its Fourier trans-
y
In writing (10), we have neglected “obviously irrele- formed equation of motion can be obtained by Fourier
vant” terms, by which we mean terms that differ from transforming (10) and acting on both sides of the re-
those explicitly displayed in (10) by having more powers sultant equation with the transverse projection operator
of the small fluctuations u, or more spatial derivatives P (q)=δ −q q /q2 whichprojectsorthogonaltothe
lm lm l m
of a given type. For more discussion of these “obviously spatial wavevector q. This eliminates the pressure term.
irrelevant” terms, see “Methods”. Note that only one Taking the l = y component of the resulting equation
of the non-linearities associated with the λ terms, gives:
1,2,3
(cid:34) (cid:32) (cid:33)(cid:35)
u2(r,t)
∂ u (q,t) = −iv q u (q,t)−Γ(q)u (q,t)+P (q)F −2α u (r,t)+ y
t y 1 x y y yx q x 2v
0
(cid:34) (cid:32) (cid:33) (cid:35)
α u3
+P (q)F − y +2u u −λ0u ∂ u +P (q)f (q,t), (11)
yy q v v x y 1 y y y ym m
0 0
where F represents the Fourier component at wavevec- the incompressibility condition. Note that our conven-
q
tor q, i.e., F [g(r)] ≡ (cid:82) d2rg(r)e−iq·r; the “bare” value tion for the anisotropy exponent here is exactly the op-
q
of the speed v , before rescaling and renormalization, is positeofthatusedinreferences[9–13]; thatis, wedefine
1
v =λ0v , and Γ(q) is given after equation (8). ζ by q ∼ qζ being the dominant regime of wavevector,
1 1 0 y x
We now rescale co-ordinates (x,y), time t, and the while [9–13] defines this regime as q ∼qζ.
x y
components of the real space velocity field ux,y(r,t) ac- Upon this rescaling, the form of Eq. (11) remains un-
cording to changed, but the various coefficients become dependent
on the rescaling parameter (cid:96).
x(cid:55)→e(cid:96)x, y (cid:55)→eζ(cid:96)y, t(cid:55)→ez(cid:96)t (12)
Details of this simple power counting (including the
uy(r,t)(cid:55)→eχy(cid:96)uy(r,t), (13) slightly subtle question of how to rescale the projection
u (r,t)(cid:55)→eχx(cid:96)u (r,t)=e(χy+1−ζ)(cid:96)u (r,t), (14) operators) are given in “Methods”. The results for the
x x x
three parameters (damping coefficient µ, “longitudinal
where the scalings of u (r,t) and u (r,t) are related by mass” α, and noise strength D) that control the size of
x y
5
the fluctuations in the linear theory are: µ (cid:55)→ e(z−2)(cid:96)µ, relevant terms, as [34]
α(cid:55)→e(z−2ζ+2)(cid:96)α, and D (cid:55)→e(z−2χy−ζ−1)(cid:96)D.
(cid:90) (cid:20) 1 (cid:21)
We now use the standard renormalization group logic H = d2r V(|M|)+ µ|∇(cid:126)M|2 , (16)
to assess the importance of the non-linear terms in (11). XY 2
This logic is to choose the rescaling exponents z, ζ, and
whereµisthe“spinwavestiffness”. Intheorderedphase,
χ so as to keep the size of the fluctuations in the field
y the “potential” V(|M|) has a circle of global minima at
u fixed upon rescaling. This is clearly accomplished by
a non-zero value of |M|, which we will take to be v .
keeping α, µ, and D fixed. From the rescalings just 0
Expanding in small fluctuations about this minimum
found, this leads to three simple linear equations in the
by writing M = (v +u )xˆ+u yˆ, we obtain, keeping
three unknown exponents z, ζ, and χ ; solving these, we 0 x y
y
only “relevant” terms,
find the values of these exponents in the linearized the-
oilnirnyeh:aarnζtldien,rm=wesziclniann(1=1n)o2wa,tχayloslisnnegs=sletn−hg1et.himWscpaiotlerhtsa,tnshicemesepolfeyxtbphyoenlnoeoonnkts-- HXY = 12(cid:90) d2r2α(cid:32)ux+ 2uv2y (cid:33)2+µ|∇(cid:126)u|2 , (17)
0
ing at how their coefficients rescale. (We don’t have to
worryaboutthesizeoftheactualnon-lineartermsthem- where we define the “longitudinal mass” 2α ≡
selves changing upon rescaling, because we have chosen (cid:16) ∂2V (cid:17)(cid:12)(cid:12) .
the rescalings to keep them constant in the linear the- ∂|M|2 (cid:12)|M|=v0
ory.) We find that all of the non-linearities whose coef- We now add to this model the divergence-free con-
ficients are proportional to α are “relevant” (i.e., grow straint ∇·M = 0, which obviously implies ∇·u = 0.
upon rescaling), while those associated with the last re- To enforce this constraint, we introduce to the Hamilto-
maining non-linearity, λ0, associated with the λ terms nian a Lagrange multiplier P(r):
1
get smaller upon rescaling: λ01 (cid:55)→ e−2(cid:96)λ01. Hence, this (cid:90)
term will not affect the long-distance behavior, and can H(cid:48) =H − d2r P(r)(∇·u). (18)
XY
bedroppedfromtheproblem. Thisisverydifferentfrom
the compressible problem, in which the α non-linearities The simplest dynamical model that relaxes back to
are unimportant, while the λ ones dominate; the reasons the equilibrium Boltzmann distribution e−βH(cid:48)(u) for
for this difference are discussed in “Methods”. the Hamiltonian H(cid:48) is [34, 35] the “time-dependent-
Dropping the λ0 term in (10), and making a Galilean Ginsburg-Landau”(TDGL)model∂ u =−δH(cid:48)/δu +f ,
1 t l l l
transformationtoa“pseudo-co-moving”co-ordinatesys- where f is the thermal noise whose statistics can also
tem moving in the direction xˆ of mean flock motion be described by Eq. (4) with D = k T = 1/β. This
B
at speed v1 ≡ λ01v0 to eliminate the “convective term” TDGL equation is readily seen to be exactly Eq. (15)
v1∂xum from the right hand side of (10), leaves us with with µ0T = µ. Therefore, we conclude that the ordered
our final simplified form for the equation of motion: phase of 2D incompressible polar active fluids has the
same static (i.e., equal-time) scaling behaviors as the or-
(cid:32) (cid:33)
u2 dered phase of the 2D XY model subject to the con-
∂ u = −∂ P −2α u + y δ
t m m x 2v xm straint ∇·M=0.
0
This mapping between a nonequilibrium active fluid
(cid:32) (cid:33)
−2α u + u2y u δ model and a “divergence-free” XY model allows us to
v x 2v y ym investigate the fluctuations in our original active fluid
0 0
model by studying the partition function of the equilib-
+µ∂2u +µ0∂2u +f . (15)
x m T y m m rium model.
Todealwiththeexactidentity∇·u=0,weuseatrick
We now show that Eq. (15) also describes an equilib-
familiarfromthestudyofincompressiblefluidmechanics:
rium system: the ordered phase of the 2D XY model
we introduce a “streaming function”; i.e., a new scalar
subject to the divergence-free constraint ∇ · M = 0,
field h(r) such that
where M is the magnetization. This connection enables
us to use purely equilibrium statistical mechanics (in u =−v ∂ h , u =v ∂ h . (19)
x 0 y y 0 x
particular, the Boltzmann distribution) to determine
the equal-time correlations of 2D incompressible polar Because this construction guarantees that the incom-
active fluids. pressibilitycondition∇·u=0isautomaticallysatisfied,
there is no constraint on the field h(r).
Divergence-free 2D XY model. The 2D XY model The field h(r) has a simple interpretation as the dis-
describes a 2D ferromagnet whose magnetization field placement of the fluid flow lines from set of parallel lines
M(r) and position r both have two components. The along xˆ that would occur in the absence of fluctuations,
Hamiltonian for this model can be written, ignoring ir- as illustrated in Fig. 2. (We thank Pawel Romanczuk
6
time correlation functions of the 2D smectic to the KPZ
equation,they-coordinateinthesmecticismappedonto
timetintheKPZequationwithh(x,t)theheightofthe
“surface” at position x and time t above some reference
height. As a result, the dynamical exponent z of the
KPZ
1+1-dimensional KPZ equation becomes the anisotropy
exponent ζ of the 2D smectic. Since the scaling laws
ofthe1+1-dimensionalKPZequationareknownexactly
[8],thoseoftheequal-timecorrelationsofthe2Dsmectic
can be obtained as well.
This gives [21, 22] ζ = 3/2 and χ = 1/2 as the ex-
h
FIG.2: |Analogybetweendisplacmentfieldoftheflow ponents for the 2D smectic, where χ gives the scaling
linesin2Dincompressiblepolaractivefluidsandthat of the smectic layer displacement fieldh h(r) with spatial
of2Dsmecticlayers. Inthecaseof2Dincompressiblepolar
coordinatex. Giventhestreamingfunctionrelation(19)
activefluids, thefieldh(r)istheverticaldisplacementofthe
between h(r) and u(r), we see that the scaling exponent
flow lines (i.e., the solid lines) from the set of parallel lines
χ for u is just χ =χ −1=−1/2 and that the scal-
(i.e.,thedottedlines)alongxˆthatwouldoccurintheabsence y y y h
offluctuations. Foradefect-free2Dsmectic, itlikewisegives ing exponent χx for ux is just χx = χy +1−ζ = −1.
the vertical displacement of the smectic layers (i.e., the solid Note that these exponents are different from those for
lines) from their reference positions (i.e., the dotted lines) at compressible polar active fluids[9–13] where ζ =5/3 and
zero temperature. χ = −1/5. (Note that our convention here (q ∼ qζ) is
y y x
the inverse of that (q ∼qζ) used in references [9–13].)
x y
The fact that both of the scaling exponents χ and
y
for pointing out this pictorial interpretation to us.) This
χ are less than zero implies that both u and u fluc-
x y x
fact,whichisexplainedinmoredetailin“Methods”,isa
tuations remain finite as system size L → ∞; this, in
consequence of the fact that, as in conventional 2D fluid
turn, implies that the system has long-ranged orienta-
mechanics, contours of the streaming function h(r) are
tional order since (cid:104)|v(r,t)−v(r(cid:48),t)|2(cid:105) remains finite as
flow lines.
|r−r(cid:48)|→∞. That is, the ordered state is stable against
This picture of a set of lines that “wants” to be
fluctuations, at least for sufficiently small noise D.
parallel being displaced by a fluctuation h(r) looks very
The velocity correlation function can be calculated
much like a 2D smectic liquid crystal (i.e., “soap”), for
through the connection between u and h. Using the
which the layers are actually 1D fluid stripes.
aforementioned connection between 2D smectics and the
1+1-dimensionalKPZequation,theequaltimelayerdis-
2D smectic and KPZ models. This resemblance
placement correlation function takes the form [21, 22]:
between our system and a 2D smectic is not purely
visual. Indeed, making the substitution (19), the C (r−r(cid:48)) ≡ (cid:10)[h(r,t)−h(r(cid:48),t)]2(cid:11)
h
Hamiltonian(17)becomes(ignoringirrelevanttermslike = B|x−x(cid:48)|Ψ(κ). (21)
(∂ ∂ h)2,whichisirrelevantcomparedto(∂2h)2 because
x y x
y-derivatives are less relevant than x-derivatives): where we define the scaling variable κ≡ X , with X ≡
Y2/3
|x−x(cid:48)|/ξ , and Y ≡ |y−y(cid:48)|/ξ , and the non-universal
x y
1(cid:90) (cid:34) (cid:18) (∂ h)2(cid:19)2 (cid:35) constant B is an overall multiplicative factor; estimates
Hs = 2 d2r B ∂yh− x2 +K(∂x2h)2 , of the non-universal nonlinear lengths ξx,y are given in
“Methods”.
(20)
The limiting behaviors of the universal scaling func-
where B = 2αv2 and K = µv2. This Hamiltonian is ex-
0 0 tionΨhavebeenstudiednumericallypreviously[37,38].
actlytheHamiltonianforthedislocation-free2Dsmectic
Here, we use the most accurate version currently known
model with h(r) in Eq. (20) interpreted as the displace-
(www-m5.ma.tum.de/KPZ) [39, 40]:
mentfieldofthesmecticlayers,asalsoillustratedinFig.

2. Ψ(κ)≈c1+e−Φh(κ) , , κ(cid:29)1
Thescalingbehavioursofthedislocation-free2Dsmec- Ψ(κ)≈ (22)
ticmodelareextremelynon-trivial,sincethe“criticaldi- κ+ c2 , κ(cid:28)1 ,
κ
mension” d below which a purely harmonic description
c
where for κ(cid:29)1,
of these systems breaks down is d =3[36]. Fortunately,
c
these non-trivial scaling behaviours are known, thanks Φ (κ)=cκ3+O(κ) . (23)
h
to an ingenious further mapping [21, 22] of this prob-
lem onto the 1+1-dimensional KPZ equation [8], which Here, the constants c and c are all universal and are
1,2
is a model for interface growth or erosion (e.g., “sand- given by c ≈ 0.295, c ≈ 1.843465, and c ≈ 1.060...
1 2
blasting”). In this mapping, which connects the equal- [39, 40].
7
Rewritingthevelocitycorrelationfunction(1)interms to identify the relevant non-linearities of this model,
of the fluctuation u using (7) gives we perform a series of mathematical transformation
which map our model to three other interesting, but
(cid:104)|v(r,t)−v(r(cid:48),t)|2(cid:105) seemingly unrelated, models. Specifically, we make
= C −2(cid:104)u (r,t)u (r(cid:48),t)(cid:105)−2(cid:104)u (r,t)u (r(cid:48),t)(cid:105),(24) heretofore unanticipated connections between four
0 y y x x
seemingly unrelated systems: the ordered phase of 2D
where C0 = 2(cid:104)u2(r,t)(cid:105) is finite, and the two correlation incompressible polar active fluids, the ordered phase
functions on the right hand side of the equality are just of the divergence-free 2D XY model, dislocation-free
thederivativesofthelayerdisplacementcorrelationfunc- 2D smectics, and growing one-dimensional interfaces.
tion: Through this connection, we show that 2D incompress-
ible polar active fluids spontaneously break continuous
(cid:104)uy(r,t)uy(r(cid:48),t)(cid:105) =−v202∂x∂x(cid:48)Ch(r−r(cid:48)), (25) rotational invariance (which their equilibrium coun-
(cid:104)ux(r,t)ux(r(cid:48),t)(cid:105) =−v202∂y∂y(cid:48)Ch(r−r(cid:48)). (26) tcearnpnaorttsdo)(,i.ea.n,d oobrdtaininartyhedeixvaecrtgesncacleifnugl bfeehrarvoimoragonfetthse)
To derive (25, 26) we use (19) and the definition of C equal-time velocity correlation function of the original
h
(i.e., the first equality of formula (21)). model. Because this mapping only involves equal-time
Inserting (25, 26) into (24) and using the asymptotic correlations, the dynamical scaling of the original model
forms (21, 22) for C , we obtain (as explained in more iscurrentlyunknown. Wehopetodeterminethisscaling
h
detailin“Methods”)theasymptoticformofthevelocity in further work.
correlationfunctiongivenby(1). Wecanalsoobtainthe
Fourier transformed equal time correlation functions; Methods
these are given in “Methods”. Linear theory. In this section we give the details of
the derivation of the linearized theory of incompressible
Discussion polaractivefluids. Webeginwiththelinearizedequation
We formulate a universal equation of motion describing of motion, obtained by expanding Eq. (6) of the main
the ordered phase of 2D incompressible polar active text to linear order in the fluctuation u of the velocity
fluids. After using renormalization group analysis around its mean value v xˆ:
0
∂ u =−∂ P −2αu δ −λ0v (∂ u )−λ0v3δ (∂ u )+µ0∇2u +µ0v2∂2u +f , (27)
t m m x mx 1 0 x m 4 0 xm x x T m 2 0 x m m
where the superscript “0” means that the v-dependent x. Knowing this scaling (in particular, χ ) allows us to
x,y
coefficients are evaluated at v = v , and we define the answer the most important question about this system:
0
(cid:16) (cid:17)
“longitudinal mass” α≡−v0 dU(v) . is the ordered state actually stable against fluctuations?
2 dv
v=v0
Our goal now is to determine the scaling of the fluc- Toobtainthisscalinginthelineartheory,webeginby
tuations u of the velocity with length and time scales, calculating the fluctuations of u predicted by that the-
and to determine the relative scaling of the two Carte- ory. Sincethetwocomponentsofuarenotindependent,
siancomponentsxandy ofpositionwitheachother,and but, rather, locked to each other by the incompressibil-
with time t. That is, in the language of hydrodynamics, ity condition ∇·v = 0, it is only necessary to calculate
we seek the “roughness exponents” χ , the anisotropy one of them. We choose to focus on the y-component,
x,y
exponent ζ, and the dynamical exponent z character- whichcanbecalculatedbyfirstspatio-temporallyFourier
izing respectively the scaling of: velocity fluctuations transforming(27),andthenactingonbothsideswiththe
u , “transverse” (i.e., perpendicular to the direction of transverse projection operator P (q) = δ −q q /q2
x,y lm lm l m
flock motion) position y, and time t with “longitudinal” which projects orthogonal to the spatial wavevector q.
(i.e., parallel to the direction of flock motion) position Thecomponent(cid:96)=yoftheresultantequationthengives
q q
−i(ω−λ0v q )u (q,ω)=(2α+iλ0v3q ) x yu (q,ω)−Γ(q)u (q,ω)+P f (q,ω) , (28)
1 0 x y 4 0 x q2 x y ym m
where we define with µ≡µ0 +µ0v2.
T 2 0
Γ(q)≡µ0q2+µ0v2q2 =µq2+µ0q2, (29)
T 2 0 x x T y
8
We can eliminate u from (28) using the incompress- istakenforthepurposesofthisargumenttobeinfinite).
x
ibilitycondition∇·v=0,whichimplies,inFourierspace, Using(32),thisintegralisreadilyseentoconvergeinthe
q u = −q u . Solving the resultant linear algebraic infra-red, and, hence, as system size L → ∞. Since the
x x y y
equation for u (q,ω) in terms of f (q,ω) gives integral is finite, and proportional to the noise strength
y m
D,itisclearthat,forsufficientlysmallD,thetransverse
u (q,ω)= Pym(q)fm(q,ω) , (30) uy fluctuations in real space can be made small enough
y −i[ω−c(qˆ)q]+Γ(q)+2α(cid:16)qy(cid:17)2 that long-ranged orientational order, and, hence, a non-
q zero(cid:104)v(r,t)(cid:105),ispreservedinthepresenceoffluctuations;
the ordered state is stable against fluctuations for suffi-
where we define the direction-dependent “sound speed”
ciently small noise strength D.
c(qˆ)≡λ01v0qqx +λ04v03qyq2q3x . (31) deTpahretuerxepoδnue2yntofχythceanuybefluocbttuaaintieodnsbyfrloomokitnhgeiartinthfie-
nite system limit: δu2 ≡ (cid:104)u2(r,t)(cid:105)| −(cid:104)u2(r,t)(cid:105)| =
y y L=∞ y L
Using Eq. (28), we can obtain (cid:104)|uy(q,ω)|2(cid:105) from the (cid:82) d2q (cid:104)|u (q,t)|2(cid:105); we define the “roughness expo-
known correlations of the random force f (i.e., formula qx<∼L1 (2π)2 y
nent”χ bythewaythisquantityscaleswithsystemsize
(4) in the main text). Integrating the result over all fre- y
quencies ω, and dividing by 2π, gives the equal time, L: δu2y ∝ L2χy. Note that this definition of χy requires
χ < 0, since it depends on the existence of an ordered
spatially Fourier transformed velocity autocorrelation: y
state, which necessarily implies that the velocity fluctu-
Dq2 Dq2 ations δu2y do not diverge as L → ∞. If (cid:104)u2y(r,t)(cid:105)|L=∞
(cid:104)|uy(q,t)|2(cid:105)= 2αq2+Γx(q)q2 ≈ 2αq2+xµq4 . (32) is not finite, one can obtain χy by performing exactly
y y x the type of scaling argument outlined here directly on
where the second, approximate equality applies for all (cid:104)u2y(r,t)(cid:105)|L itself.
q → 0. This can be seen by noting that, for q (cid:29) q2 Approximating (32) for the dominant regime of
y x
and q → 0, qy2 (cid:29) Γ(q)q2, while for qy <∼ qx2 and q → 0, wavevector qy ∼ qx2, and changing variables in the in-
Γ(q)q2 ≈ µq4. Hence, in both cases, (which together tegral from q to Q according to q ≡ Qx, q ≡ Qy
x x,y x,y x L y L2
cover all possible ranges of q for q→0), the approxima- shows that δu2 ∝L−1, and hence χ =−1.
y y 2
tion 2αq2+Γ(q)q2 ≈2αq2+µq4 is valid. Note also that the fluctuations of u are much smaller
y y x x
Equation (32) implies that fluctuations diverge most than those of u . This can be seen by using the incom-
y
rapidly as q → 0 if q is taken to zero along a locus pressibility condition, which implies, in Fourier space,
iansytmheptqotpiclaalnlye,t(cid:104)h|uat(oqb,te)y|s2(cid:105)q∝y <∼1q.x2I;nalcoonngtrsaustc,haalonlogcuasll, ux =−qyquxy, which implies
y q2
oqxt2he(cid:28)r lo1ci.i, iI.ne.,ththisosseenfoser,wohniechcaqny (cid:29)sayqx2t,h(cid:104)a|tuyt(hqe,tr)e|g2i(cid:105)m∝e (cid:104)|ux(q,t)|2(cid:105)= 2αq2D+qΓy2(q)q2 ≈ 2αqD2+qy2µq4 , (33)
qy2 q2 y y x
qy <∼ qx2 shows the largest fluctuations at small q; this
implies the anisotropy exponent ζ =2. which is clearly finite as q → 0 along any locus; indeed,
Wecangetthedynamicalexponentz predictedbythe it is bounded above by D.
2α
lineartheorybyinspectionof(30),althoughsomecareis We can calculate a roughness exponent χ for u for
x x
required. The form of the first term in the denominator the linear theory from this result exactly as we calculate
mightsuggestω ∝q,whichwouldimplyz =1. However, the roughness exponent χ for u ; we find χ = 1 −
y y x
the propagating c(qˆ)q term in this expression does not ζ +χ = −3. We shall see in the next section that the
y 2
appear in our final expression (32) for the fluctuations; first line of this equality also holds in the full non-linear
rather,thesearecontrolledentirelybythedampingterm theory, even though the values of the exponents χ , ζ,
Γ(q)+2α(cid:16)qy(cid:17)2. Balancing ω against that term in the and χy all change. x
q The fact that u has much smaller fluctuations than
dominant regime of wavevector q ∼ q2 gives ω ∝ q2, x
y x x u means that we have to work to higher order in u
which implies z =2. y y
than in u when we treat the non-linear theory, as we
Now we seek χ , which determines whether or not x
y do in next section.
the ordered state is stable against fluctuations in an
arbitrarily large system. This can be obtained by
Mapping to an equilibrium “incompressible”
looking at the real space fluctuations (cid:104)u2(r,t)(cid:105) =
y magnet. We now go beyond the linear theory, and ex-
(cid:82) d2q (cid:104)|u (q,t)|2(cid:105), where L is the lateral extent of
qx>∼L1 (2π)2 y pand the full equation of motion (6) of the main text to
thesysteminthex-direction(itsextentinthey-direction higher order in u. We obtain
9
∂ u = −∂ P −2αu δ −λ0v ∂ u +µ0∇2u +µ0v2∂2u +f
t m m x mx 1 0 x m T m 2 0 x m m
(cid:34) (cid:35)
α u3
− yδ +2u u δ +u2δ −λ0u ∂ u δ . (34)
v v my x y my y mx 1 y y y my
0 0
We keep terms that might naively appear to be higher NotethattheΓ(q)termin(11)ofthemaintextinvolves
order in the small fluctuations (e.g., the u3δ term rel- two parameters (µ and µ0); hence, we get the rescalings
y my T
ative to the u u δ term) because, as we saw in the of both of these parameters from this term.
x y my
linearized theory, the two different components u of u Similarly, looking at the rescaling of the non-linear
x,y
scaledifferentlyatlonglengthscales. Hence,itisnotim- terms proportional to u2 and u3, respectively, we obtain
y y
mediatelyobvious,e.g.,whichofthetwotermsjustmen- the rescalings:
tioned is actually most important at long distances. We
α α α α
therefore, for now, keep them both. For essentially the (cid:55)→e(z+χy−ζ+1)(cid:96) , (cid:55)→e(z+2χy)(cid:96) . (38)
v v v2 v2
samereason,itisnotobviouswhetheru2δ oru3δ is 0 0 0 0
y mx y my
more important, so we shall for now keep both of these
We recover the first of these by looking at the rescaling
terms as well.
of the non-linear term proportional to u u as well.
x y
On the other hand, it is immediately obvious that a
We note that the two rescalings (38) are both consis-
termlike,e.g.,u u2δ islessrelevantthanu2δ ,since,
x y mx y mx tent with (37) if we rescale v0 according to
whatever the relative scaling of u and u , u u2δ is
x y x y mx
much smaller at large distances than u2yδmx, since ux is. v0 (cid:55)→e(1−ζ−χy)(cid:96)v0 . (39)
Likewise, we drop the term 21(cid:0)ddλv1(cid:1)v=v0u2y∂xuyδmy, By power counting on the u ∂ u term, we obtain the
since it is manifestly smaller, by one ∂ , than the u3δ y y y
x y my rescaling of λ0:
term already displayed explicitly in (34). 1
This sort of reasoning guides us very quickly to the
λ0 (cid:55)→e(z+χy−ζ)(cid:96)λ0 . (40)
reduced model (34). As explained in the main text, act- 1 1
ing on both sides of (34) with the transverse projection
Finally, by looking at the rescaling of the noise corre-
operator P (q)=δ −q q /q2 which projects orthog-
lm lm l m lations (i.e., (4) of the main text), we obtain the scaling
onal to the spatial wavevector q eliminates the pressure
of the noise strength D:
term. Then taking the l =y component of the resulting
equation gives (11) of the main text, which we now use D (cid:55)→e(z−2χy−ζ−1)(cid:96)D . (41)
to calculate the rescaled coefficients.
To do this, we must also determine how the projec- We now use the standard renormalization group logic
tion operators P and P rescale upon the rescalings to assess the importance of the non-linear terms in (11)
yx yy
(i.e., (12) of the main text). Since in the linear theory of the main text. This logic is to choose the rescaling
(see, e.g., the u –u correlation function (32)) fluctua- exponents z, ζ, and χ so as to keep the size of the fluc-
y y y
tionsaredominatedbytheregimeqy <∼qx2,itfollowsthat tuations in the field u fixed upon rescaling. Since, as we
P (q)=−qxqy ≈−q /q (cid:28)1andP (q)=1−qy2 ≈1. sawinourtreatmentofthelinearizedtheory(inparticu-
yx q2 y x yy q2 lar,Eq.(32)),thatsizeiscontrolledbythreeparameters:
This implies that these rescale according to
the“longitudinalmass”α,thedampingcoefficientµ,and
Pyx(q)(cid:55)→e(1−ζ)(cid:96)Pyx(q) , Pyy(q)(cid:55)→Pyy(q) .(35) thenoisestrengthD,thechoiceofz,ζ,andχy thatkeeps
these fixed will clearly accomplish this. From the rescal-
Performingtherescalings(12-14)ofthemaintext,and ings(36),(37), and(41),thisleadstothreesimplelinear
(35) above on the equation of motion (11) of the main equations in the three unknown exponents z, ζ, and χ ;
y
text, we obtain, from the rescalings of first three (i.e., solving these, we find the values of these exponents in
the linear) terms on the right hand side the following the linearized theory:
rescalings of the parameters:
ζ =z =2, χ =−1/2, χ =−3/2 (42)
lin lin ylin xlin
v (cid:55)→e(z−1)(cid:96)v , µ(cid:55)→e(z−2)(cid:96)µ ,
1 1
which, unsurprisingly, are the linearized exponents we
µ0 (cid:55)→e(z−2ζ)(cid:96)µ0 , (36)
T T found earlier.
and With these exponents in hand, we can now assess the
importance of the non-linear terms in (11) of the main
α(cid:55)→e(z−2ζ+2)(cid:96)α . (37) textatlonglengthscales,simplybylookingathowtheir
10
coefficients rescale. (We don’t have to worry about the free to relax in such a way as to cancel out the α non-
size of the actual non-linear terms themselves changing linearities,which,becausetheyinvolvenospatialderiva-
upon rescaling, because we have chosen the rescalings to tives,windupdominatingtheλnon-linearities,whichdo
keepthemconstantinthelineartheory.) Themassα, of involve spatial derivatives. In addition, the suppression
course, is kept fixed. Inserting the linearized exponents of fluctuations by the incompressibility condition, which
(42) into the rescaling relation (39) for v , we see that as we’ve already seen in the linear theory, makes the λ
0
non-linearitiesnotonlylessrelevantthantheαones,but
v0 (cid:55)→e−2(cid:96)v0 . (43) actually irrelevant. Hence, we can drop them in this in-
compressible problem, leaving us with Eq. (45) as our
Since v0 appears in the denominator of all three of the equation of motion.
non-linear terms associated with α, and α itself is fixed, As one final simplification, we make a Galilean trans-
this implies that all three of those terms are “relevant”, formation to a “pseudo-co-moving” co-ordinate system
in the renormalization group sense of growing larger as moving in the direction xˆ of mean flock motion at speed
we go to longer wavelengths (i.e., as (cid:96) grows). As usual λ0v . Note that if the parameter λ0 had been equal to
1 0 1
in the RG, this implies that these terms ultimately alter 1, this would be precisely the frame co-moving with the
thescalingbehaviorofthesystematsufficientlylongdis- flock. The fact that it is not is a consequence of the lack
tances. Inparticular,theexponentsz,ζ,andχx,y change of Galilean invariance in our problem.
from their values (42) predicted by the linear theory. Thisboosteliminatesthe“convective”termλ0v ∂ u
1 0 x m
The same is not true of the λ01 non-linearity, however, fromtherighthandsideof(45),leavinguswithourfinal
becauseitisirrelevant;thatis,itgetssmalleruponrenor- simplified form for the equation of motion:
malization. Thisfollowsfrominsertingthelinearizedex-
(cid:32) (cid:33)
ponents(42)intotherescalingrelation(40)forλ0,which u2
1 ∂ u = −∂ P −2α u + y δ
gives t m m x 2v xm
0
(cid:32) (cid:33)
λ01 (cid:55)→e−2(cid:96)λ01 , (44) −2α u + u2y u δ
v x 2v y ym
0 0
which shows clearly that λ0 vanishes as (cid:96) → ∞; that is,
in the long-wavelength limi1t. +µ∂x2um+µ0T∂y2um+fm . (46)
Since λ0 was the only remaining non-linearity associ-
1 which is just equation (15) of the main text.
ated with the λ terms in our original equation of motion
(34), we can accurately treat the full, long distance be-
Mappingofequilibrium“incompressible”magnet
havior of this problem by leaving out all of those non-
to 2D smectic. We begin by demonstrating the picto-
linear terms.
rialinterpretationofthe“streamingfunction”introduced
Doing so reduces the equation of motion (34) to
in the main text via
(cid:32) (cid:33)
u2 u =−v ∂ h , u =v ∂ h . (47)
∂ u = −λ0v ∂ u −∂ P −2α u + y δ x 0 y y 0 x
t m 1 0 x m m x 2v xm
0 This implies that the streaming function φ for the full
(cid:32) (cid:33)
−2vα ux+ 2uv2y uyδym v−e∂loxcφi,tyisfigeilvdenv(bry)=v0xˆ+u, defined via vx =∂yφ, vy =
0 0
+µ∂2u +µ0∂2u +f . (45) φ=v0(y−h(r)). (48)
x m T y m m
As in conventional 2d fluid mechanics, contours of the
Before proceeding to analyze this equation, we note
streaming function φ are flow lines. When the system is
thedifferencesbetweenthestructureofthisproblemand
in its uniform steady state (i.e., v =v xˆ), these contour
thatforthecompressiblecase. Inthecompressibleprob- 0
lines, defined via
lem, there is no constraint analogous to the incompress-
ibility condition relating ux and uy. Hence, ux is free to φ=nC, n=0,1,2,3... (49)
relaxquickly(tobeprecise,onatimescale 1 )toitslocal
2α
u2 where C is some arbitrary constant, are a set of parallel,
“optimal” value, which is readily seen to be − y . Once
2v0 uniformly spaced lines given by yn =nC/v0.
this relaxation has occurred, all of the non-linearities
Now let’s ask what the flow lines are if there are fluc-
associated with α drop out of that compressible prob-
tuations in the velocity field: v = v xˆ+u. Combining
0
lem, leaving the λ non-linearities as the dominant ones.
our expression for φ (48) and the expression (49) for the
For a detailed discussion of the rather tricky analysis
flow lines, we see that the positions of the flow lines are
of the compressible problem that leads to this conclu-
now given by
sion, see Ref. [41]. Here, in the incompressible problem,
u is, because of the incompressibility constraint, not y =nC/v +h, n=0,1,2,3..., (50)
x n 0