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I formulate the theory of three dimensional "Malthusian flocks" -- i.e., coherently moving collections of self-propelled entities (such as living creatures) which are being "born" and "dying" during their motion -- whose constituents all have a preference for having their velocity vectors lie parallel to the same two-dimensional plane. I determine the universal scaling exponents characterizing such systems exactly, finding that the dynamical exponent $z=3/2$, the "anisotropy" exponent $\zeta=3/4$, and the "roughness" exponent $\chi=-1/2$. I also give the scaling laws implied by these exponents., Comment: 6 pages, no figures

We show that phase separation in ordered polar active fluids belongs to a new universality class. This describes large collections of self-propelled entities (``flocks"), all spontaneously moving in the same direction, in which attractive interactions (which can be caused by, e.g., autochemotaxis) cause phase separation: the system spontaneously separates into a high density band and a low density band, moving parallel to each other, and to the direction of mean flock motion, at different speeds. The upper critical dimension for this transition is $d_c=5$, in contrast to the well-known $d_c=4$ of equilibrium phase separation. We obtain the large-distance, long-time scaling laws of the velocity and density fluctuations, which are characterized by universal critical correlation length and order parameter exponents $\nu_\perp$, $\nu_\parallel$ and $\beta$ respectively. We calculate these to $\mathcal{O} (\epsilon)$ in a $d=5-\epsilon$ expansion., Comment: 21 pages, 9 figures

Motivated by a number of realizations of long-range interacting systems, including ultra-cold atomic and molecular gases, we study a neutral plasma with power-law interactions longer-ranged than Coulombic. We find that beyond a crossover length, such interactions are universally screened down to a standard Coulomb form in all spatial dimensions. This implies, counter-intuitively, that in two dimensions and below, such a "super-Coulombic" gas is asymptotically Coulombically confining at low temperatures. At higher temperatures, the plasma undergoes a deconfining transition that in two dimensions is the same Kosterlitz-Thouless transition that occurs in a conventional Coulomb gas, but at an elevated temperature that we calculate. We also predict that in contrast, above two dimensions, even when naively the bare potential is confining, there is no confined phase of the plasma at any nonzero temperature. In addition, the super-Coulomb to Coulomb crossover is followed at longer length scales by an unconventional "Debye-Huckel" screening, which leads to faster-than-Coulombic, power-law decay of the screened potential, in contrast to the usual exponentially decaying Yukawa potential. Furthermore, we show that power-law potentials, that fall off more rapidly than Coulomb, are screened down to a shorter-ranged power-law, rather than an exponential Debye-Huckel Yukawa form. We expect these prediction to be testable in simulations, and hope they will inspire experimental studies in various platforms., Comment: 17 pages, 2 figures

We develop the hydrodynamic theory of dry, polar ordered, active matter (``flocking") with autochemotaxis; i.e., self-propelled entities moving in the same direction, each emitting a substance which attracts the others (e.g., ants). We find that sufficiently strong autochemotaxis leads to an instability to phase separation into one high and one low density band. This is very analogous to both equilibrium phase separation, and ``motility induced phase separation" (``MIPS") and can occur in flocks due to any microscopic mechanism (e.g., sufficiently strong attractive interactions) that makes the entities cohere., Comment: 6 pages, 5 figures. arXiv admin note: substantial text overlap with arXiv:2401.09461

We analytically calculate the scaling exponents of a two-dimensional KPZ-like system: coherently moving incompressible polar active fluids. Using three different renormalization group approximation schemes, we obtain values for the ``roughness" exponent $\chi$ and anisotropy exponent $\zeta$ that are extremely near the known exact results. This implies our prediction for the previously completely unknown dynamic exponent $z$ is quantitatively accurate.

I show that non-equilibrium two-dimensional interfaces between three dimensional phase separated fluids exhibit a peculiar "sub-logarithmic" roughness. Specifically, an interface of lateral extent $L$ will fluctuate vertically (i.e., normal to the mean surface orientation) a typical RMS distance $w\equiv\sqrt{\langle |h(\br,t)|^2\rangle} \propto [\ln{(L/a)}]^{1/3}$ (where $a$ is a microscopic length, and $ h(\br,t)$ is the height of the interface at two dimensional position $\br$ at time $t$). In contrast, the roughness of equilibrium two-dimensional interfaces between three dimensional fluids, obeys $w \propto [\ln{(L/a)}]^{1/2}$. The exponent $1/3$ for the active case is exact. In addition, the characteristic time scales $\tau(L)$ in the active case scale according to $\tau(L)\propto L^3 [\ln{(L/a)}]^{1/3}$, in contrast to the simple $\tau(L)\propto L^3$ scaling found in equilibrium systems with conserved densities and no fluid flow., Comment: 4 pages, no figures

We show that dislocations in active 2d smectics with underlying rotational symmetry are always unbound in the presence of noise, meaning the active smectic phase does not exist for non-zero noise in $d=2$. The active smectic phase can, like equilibrium smectics in 2d, be stabilized by applying rotational symmetry breaking fields; however, even in the presence of such fields, active smectics are still much less stable against noise than equilibrium ones, when the symmetry breaking field(s) are weak., Comment: 23 pages, 3 figures

We study the moving phase of two-dimensional (2D) incompressible polar active fluids in the presence of both quenched and annealed disorder. We show that long-range polar order persists even in this defect-ridden two-dimensional system. We obtain the large-distance, long-time scaling laws of the velocity fluctuations using three distinct dynamic renormalization group schemes. These are an uncontrolled one-loop calculation in exactly two dimensions, and two $d=(d_c-\epsilon)$-expansions to $O(\epsilon)$, obtained by two different analytic continuations of our 2D model to higher spatial dimensions: a ``hard" continuation which has $d_c={7\over 3}$, and a ``soft" continuation with $d_c={5\over 2}$. Surprisingly, the quenched and annealed parts of the velocity correlation function have the same anisotropy exponent and the relaxational and propagating parts of the dispersion relation have the same dynamic exponent in the nonlinear theory even though they are distinct in the linearized theory. This is due to anomalous hydrodynamics. Furthermore, all three renormalization schemes yield very similar values for the universal exponents, and, therefore, we expect the numerical values we predict for them to be highly accurate., Comment: 39 pages, 11 figures

We present a hydrodynamic theory of incompressible polar active fluids with quenched disorder. This theory shows that such fluids can overcome the disruption caused by the quenched disorder and move coherently, in the sense of having a non-zero mean velocity in the hydrodynamic limit. However, the scaling behavior of this class of active systems cannot be described by linearized hydrodynamics in spatial dimensions between 2 and 5. Nonetheless, we obtain the exact dimension-dependent scaling exponents in these dimensions., Comment: 10 pages, 1 figure

We show that incompressible polar active fluids can exhibit an ordered, coherently moving phase even in the presence of quenched disorder in two dimensions. Unlike such active fluids with annealed (i.e., time-dependent) disorder only, which behave like equilibrium ferromagnets with long-range interactions, this robustness against quenched disorder is a fundamentally non-equilibrium phenomenon. The ordered state belongs to a new universality class, whose scaling laws we calculate using three different renormalization group schemes, which all give scaling exponents within 0.02 of each other, indicating that our results are quite accurate. Our predictions can be quantitatively tested in readily available artificial active systems, and imply that biological systems such as cell layers can move coherently in vivo, where disorder is inevitable., Comment: 6 pages

We consider the effect of spatial anisotropy on polar flocks by investigating active $q$-state clock models in two dimensions. In contrast to what happens in equilibrium, we find that, in the large-size limit, any amount of anisotropy changes drastically the phenomenology of the rotationally-invariant case, destroying long-range correlations, pinning the direction of global order, and transforming the traveling bands of the coexistence phase into a single moving domain. All this happens beyond a lengthscale that diverges in the $q\to\infty$ limit. A phenomenology akin to that of the Vicsek model can thus be observed in a finite system for large enough values of $q$. We provide a scaling argument which rationalizes why anisotropy has so different effects in the passive and active cases., Comment: 6 pages with 4 figures. 4 pages of supplementary information (as ancillary file)

We formulate a hydrodynamic theory of $p-$atic liquid crystals, namely two-dimensional anisotropic fluids endowed with generic $p-$fold rotational symmetry. Our approach, based on an order parameter tensor that directly embodies the discrete rotational symmetry of $p-$atic phases, allows us to unveil several unknown aspects of flowing $p-$atics, that previous theories, characterized by ${\rm O(2)}$ rotational symmetry, could not account for. This includes the onset of long-ranged orientational order in the presence of a simple shear flow of arbitrary shear rate, as opposed to the standard quasi-long-ranged order of two-dimensional liquid crystals, and the possibility of flow alignment at large shear rates., Comment: 6 pages, 5 figures. arXiv admin note: text overlap with arXiv:2106.11957

We formulate a comprehensive hydrodynamic theory of two-dimensional liquid crystals with generic $p-$fold rotational symmetry, also known as $p-$atics, of which mematics $(p=2)$ and hexatics $(p=6)$ are the two best known examples. Previous hydrodynamic theories of $p-$atics are characrerized by continuous ${\rm O}(2)$ rotational symmetry, which is higher than the discrete rotational symmetry of $p-$atic phases. By contrast, here we demonstrate that the discrete rotational symmetry allows the inclusion of additional terms in the hydrodynamic equations, which, in turn, lead to novel phenomena, such as the possibility of flow alignment at high shear rates, even for $p>2$. Furthermore, we show that any finite imposed shear will induce long-ranged orientational order in any $p-$atic liquid crystal, in contrast to the quasi-long-ranged order that occurs in the absence of shear. The induced order parameter scales like a non-universal power of the applied shear rate at small shear rates., Comment: 25 pages, 7 figures

We construct the hydrodynamic theory of coherent collective motion ("flocking") at a solid-liquid interface. The polar order parameter and concentration of a collection of "active" (self-propelled) particles at a planar interface between a passive, isotropic bulk fluid and a solid surface are dynamically coupled to the bulk fluid. We find that such systems are stable, and have long-range orientational order, over a wide range of parameters. When stable, these systems exhibit "giant number fluctuations", i.e., large fluctuations of the number of active particles in a fixed large area. Specifically, these number fluctuations grow as the $3/4$th power of the mean number within the area. Stable systems also exhibit anomalously rapid diffusion of tagged particles suspended in the passive fluid along any directions in a plane parallel to the solid-liquid interface, whereas the diffusivity along the direction perpendicular to the plane is non-anomalous. In other parameter regimes, the system becomes unstable., Comment: 52 pages, 3 figures

We present the hydrodynamic theory of coherent collective motion ("flocking") at a solid-liquid interface, and many of its predictions for experiment. We find that such systems are stable, and have long-range orientational order, over a wide range of parameters. When stable, these systems exhibit "giant number fluctuations", which grow as the 3/4th power of the mean number. Stable systems also exhibit anomalous rapid diffusion of tagged particles suspended in the passive fluid along any directions in a plane parallel to the solid-liquid interface, whereas the diffusivity along the direction perpendicular to the plane is not anomalous. In the remaining parameter space, the system becomes unstable., Comment: 6 pages, 2 figs

We show that "Malthusian flocks" -- i.e., coherently moving collections of self-propelled entities (such as living creatures) which are being "born" and "dying" during their motion -- belong to a new universality class in spatial dimensions $d>2$. We calculate the universal exponents and scaling laws of this new universality class to $O(\epsilon)$ in a $d=4-\epsilon$ expansion, and find these are different from the "canonical" exponents previously conjectured to hold for "immortal" flocks (i.e., those without birth and death) and shown to hold for incompressible flocks with spatial dimensions in the range of $2 < d \leq 4$. We also obtain a universal amplitude ratio relating the damping of transverse and longitudinal velocity and density fluctuations in these systems. Furthermore, we find a universal separatrix in real (${\bf r}$) space between two regions in which the equal time density correlation $\langle\delta\rho({\bf r}, t)\delta\rho(0, t)\rangle$ has opposite signs. Our expansion should be quite accurate in $d=3$, allowing precise quantitative comparisons between our theory, simulations, and experiments., Comment: 48 pages, 7 figures. This is the long paper accompanying the short paper "Moving, reproducing, and dying beyond Flatland: Malthusian flocks in dimensions $d> 2$" [arXiv:2001.01300]

We show that "Malthusian flocks" -- i.e., coherently moving collections of self-propelled entities (such as living creatures) which are being "born" and "dying" during their motion -- belong to a new universality class in spatial dimensions $d>2$. We calculate the universal exponents and scaling laws of this new universality class to $O(\epsilon)$ in an $\epsilon=4-d$ expansion, and find these are different from the "canonical" exponents previously conjectured to hold for "immortal" flocks (i.e., those without birth and death) and shown to hold for incompressible flocks in $d>2$. Our expansion should be quite accurate in $d=3$, allowing precise quantitative comparisons between our theory, simulations, and experiments., Comment: 5 pges, 2 figures. This new version has essentially the same content as the earlier one, but differs cosmetically, and is the version actually submitted to PRL. The accompanying long paper is titled "A novel nonequilibrium state of matter: a $d=4-\epsilon$ expansion study of Malthusian flocks"

We develop the elastic theory for inversion-asymmetric tethered membranes and use it to identify and study their possible phases. Asymmetry in a tethered membrane causes spontaneous curvature, which in general depends upon the local in-plane dilation of the tethered network. This in turn leads to long-ranged interactions between the local mean and Gaussian curvatures, which is not present in symmetric tethered membranes. This interplay between asymmetry and Gaussian curvature leads to a new {\em double-spiral} phase not found in symmetric tethered membranes. At temperature $T=0$, tethered membranes of arbitrarily large size are always rolled up tightly into a conjoined pair of Archimedes' spirals. At finite $T$ this spiral structure swells up significantly into algebraic spirals characterized by universal exponents which we calculate. These spirals have long range orientational order, and are the asymmetric analogs of statistically flat symmetric tethered membranes. We also find that sufficiently strong asymmetry can trigger a structural instability leading to crumpling of these membranes as well. This provides a new route to crumpling for asymmetric tethered membranes. We calculate the maximum linear extent $L_c$ beyond which the membrane crumples, and calculate the universal dependence of $L_c$ on the membrane parameters. By tuning the asymmetry parameter, $L_c$ can be continuously varied, implying a {\em scale-dependent} crumpling. Our theory can be tested on controlled experiments on lipids with artificial deposits of spectrin filaments, in-vitro experiments on %\sout{artificial deposition of spectrin filaments on} red blood cell membrane extracts, %\sout{after %depletion of adenosine-tri-phosphate molecules} and on graphene coated on one side., Comment: 28 pages, 12 captioned figures. This is an accompanying longer version of our already submitted work, arXiv:1811.12440, containing all the technical details required for a thorough reading of arXiv:1811.12440

The hydrodynamic equations of dry active polar fluids (i.e., moving flocks without momentum conservation) are shown to imply giant number fluctuations. Specifically, the rms fluctuations $\sqrt {<(\delta N)^2>}$ of the number $N$ of active particles in a region containing a mean number of active particles $$ scales according to the law $\sqrt {<(\delta N)^2>} = K'^{\phi(d)}$ with $\phi(d)=\frac{7}{10}+\frac{1}{5d}$ in $d\le4$ spatial dimensions. This is much larger the "law of large numbers" scaling $\sqrt {<(\delta N)^2>} = K\sqrt{}$ found in most equilibrium and non-equilibrium systems. In further contrast to most other systems, the coefficient $K'$ also depends singularly on the shape of the box in which one counts the particles, vanishing in the limit of very thin boxes. These fluctuations arise {\it not} from large density fluctuations - indeed, the density fluctuations in \dry s are not in general particularly large - but from long ranged spatial correlations between those fluctuations. These are shown to be closely related in two spatial dimensions to the electrostatic potential near a sharp upward pointing conducting wedge of opening angle ${3\pi\over8}=67.5^\circ$, and in three dimensions to the electrostatic potential near a sharp upward pointing charged cone of opening angle $37.16^\circ$. This very precise prediction can be stringently tested by alternative box counting experiments that directly measure this density-density correlation function., Comment: 23 pages; Invited contribution to Journal of Chemical Physics Special Issue on Active Matter

Although human beings have known about the phenomenon of "flocking"- that is, the coherent movement of large numbers of creatures (flocks of birds, schools of fish, herds of woolly mammoths, etc.)- since prehistoric times, it is only in the last two decades that we have begun to truly understand this phenomenon. In particular, the surprising fact that a very large collection of organisms in two dimensions cannot all {\it point} in the same direction, but can quite easily {\it move} in the same direction, can now be explained. In these lectures, I'll review one of the principle theoretical tools that made this possible: hydrodynamics. My intention is both to elucidate flocking- or, to use the specific technical mouthful, "polar ordered dry active fluids"-, and to use flocking as an illustration of how to use the hydrodynamic approach on new and unfamiliar systems., Comment: Notes for lectures given at the Les Houches summer school " Active matter and non-equilibrium statistical physics", August 27-September 21, 2018. arXiv admin note: text overlap with arXiv:1010.5017 by other authors

We show that inversion-asymmetric tethered membranes exhibit a new double-spiral phase with long range orientational order not present in symmetric membranes. We calculate the universal algebraic spiral shapes of these membranes in this phase. Asymmetry can trigger the crumpling of these membranes as well. In-vitro experiments on lipid, red blood cell membrane extracts, and on graphene coated on one side, could test these predictions., Comment: Significant modification on arXiv:1610.01992 and new results added

We study universal behavior in the moving phase of a generic system of motile particles with alignment interactions in the incompressible limit for spatial dimensions $d>2$. Using a dynamical renormalization group analysis, we obtain the exact dynamic, roughness, and anisotropy exponents that describe the scaling behavior of such incompressible systems. This is the first time a compelling argument has been given for the exact values of the anomalous scaling exponents of a flock moving through an isotropic medium in $d>2$.

The effect of quenched (frozen) disorder on the collective motion of active particles is analyzed. We find that active polar systems are far more robust against quenched disorder than equilibrium ferromagnets. Long ranged order (a non-zero average velocity $\langle{\bf v}\rangle$) persists in the presence of quenched disorder even in spatial dimensions $d=3$; in $d=2$, quasi-long-ranged order (i.e., spatial velocity correlations that decay as a power law with distance) occurs. In equilibrium systems, only quasi-long-ranged order in $d=3$ and short ranged order in $d=2$ are possible. Our theoretical predictions for two dimensions are borne out by simulations., Comment: 5 pages, 1 figure. The short paper on flocking with quenched noise

The effect of quenched (frozen) orientational disorder on the collective motion of active particles is analyzed. We find that, as with annealed disorder (Langevin noise), active polar systems are far more robust against quenched disorder than their equilibrium counterparts. In particular, long ranged order (i.e., the existence of a non-zero average velocity $\langle {\bf v} \rangle$) persists in the presence of quenched disorder even in spatial dimensions $d=3$, while it is destroyed even by arbitrarily weak disorder in $d \le 4$ in equilibrium systems. Furthermore, in $d=2$, quasi-long-ranged order (i.e., spatial velocity correlations that decay as a power law with distance) occurs when quenched disorder is present, in contrast to the short-ranged order that is all that can survive in equilibrium. These predictions are borne out by simulations in both two and three dimensions., Comment: 22 pages, 6 figures. The long paper on flocking with quenched noise

We study the hydrodynamic behavior of three dimensional (3D) incompressible collections of self-propelled entities in contact with a momentum sink in a state with non-zero average velocity, hereafter called 3D easy-plane incompressible polar active fluids. We show that the hydrodynamic model for this system belongs to the same universality class as that of an equilibrium system, namely a special 3D anisotropic magnet. The latter can be further mapped onto yet another equilibrium system, a DNA-lipid mixture in the sliding columnar phase. Through these connections we find a divergent renormalization of the damping coefficients in 3D easy-plane incompressible polar active fluids, and obtain their equal-time velocity correlation functions.

The dynamics of self-locomotion of active particles in aligned or liquid crystalline fluids strongly deviates from that in simple isotropic media. We explore the long-time dynamics of a swimmer moving in a three-dimensional smectic liquid crystal and find that the mean-square displacement (MSD) transverse to the director exhibits a distinct logarithmic tail at long times. The scaling is distinctly different from that in an isotropic or nematic fluid and hints at the subtle but important role of the director fluctuation spectrum in governing the long-time motility of active particles. Our findings are based on a generic hydrodynamic theory and Brownian dynamics computer simulation of a three-dimensional soft mesogen model., Comment: 12 pages, 2 figures

It is shown that gravitational radiation can bind two initially unbound bodies; no third body is needed. Such captured bodies will almost always inspiral and merge due to further gravitational radiation on cosmologically negligible time scales (e.g., @ 5 years for GW150914). The capture cross-section $\sigma$ for such "capture and inspiraling" is far larger, for initial relative speed of the two objects $v_\infty\ll c$, than that $\sigma_{d}$ for "direct capture": $\sigma\propto\left(c / v_\infty\right)^{18/7}$, while $\sigma_{d}\propto\left(c / v_\infty\right)^2$. Implications of these results for black hole binary mergers, and giant black holes at galactic centers, are discussed., Comment: 4 pages of main text p[lus 9 pages of supplemental materials

Active (i.e., self-propelled or swimming) particles moving through an isotropic fluid exhibit conventional diffusive behavior. We report anomalous diffusion of an active particle moving in an anisotropic, nematic background. Whilst the translational motion parallel to the nematic director shows ballistic behavior, the long-time transverse motion is super-diffusive, with an anomalous scaling $\propto t \ln t$ of the mean squared displacement with time $t$. This behavior is predicted by an analytical theory that we present here, and is corroborated by numerical simulation of active particle diffusion in a simple lattice model for a nematic liquid crystal. It is universal for any collection of self-propelled elements (e.g., bacteria or active rods) moving in a nematic background, provided only that the swimmers are sufficiently dilute that their interactions with each other can be neglected, and that they do not perform "hairpin" turns., Comment: 12 pages, 3 figures

Novel "smectic-P" behavior, in which self-propelled particles form rows and move on average along them, occurs generically within the orientationally-ordered phase of simple models that we simulate. Both apolar (head-tail symmetric) and polar (head-tail asymmetric) models with aligning and repulsive interactions exhibit slow algebraic decay of smectic order with system size up to some finite length scale, after which faster decay occurs. In the apolar case, this scale is that of an undulation instability of the rows. In the polar case, this instability is absent, but traveling fluctuations disrupt the rows in large systems and motion and smectic order may spontaneously globally rotate. These observations agree with a new hydrodynamic theory which we present here. Variants of our models also exhibit active smectic "A" and "C" order, with motion orthogonal and oblique to the layers respectively., Comment: revised version

"Active nematics" are orientationally ordered but apolar fluids composed of interacting constituents individually powered by an internal source of energy. When activity exceeds a system-size dependent threshold, spatially uniform active apolar fluids undergo a hydrodynamic instability leading to spontaneous macroscopic fluid flow. Here, we show that a special class of spatially non-uniform configurations of such active apolar fluids display laminar (i.e., time-independent) flow even for arbitrarily small activity. We also show that two-dimensional active nematics confined on a surface of non-vanishing Gaussian curvature must necessarily experience a non-vanishing active force. This general conclusion follows from a key result of differential geometry: geodesics must converge or diverge on surfaces with non-zero Gaussian curvature. We derive the conditions under which such curvature-induced active forces generate "thresholdless flow" for two-dimensional curved shells. We then extend our analysis to bulk systems and show how to induce thresholdless active flow by controlling the curvature of confining surfaces, external fields, or both. The resulting laminar flow fields are determined analytically in three experimentally realizable configurations that exemplify this general phenomenon: i) toroidal shells with planar alignment, ii) a cylinder with non-planar boundary conditions, and iii) a "Frederiks cell" that functions like a pump without moving parts. Our work suggests a robust design strategy for active microfluidic chips and could be tested with the recently discovered "living liquid crystals"., Comment: The rewritten paper has several changes, principally: 1. A separate section III for two-dimensional curved systems, illustrated with an new example. 2. Remarks about the relevance of the frozen director approximation in the case of weak nematic order; and 3. A separate Supplemental Material document, containing material previously in the Appendix, along with additional material

Active fluids and growing interfaces are two well-studied but very different non-equilibrium systems. Each exhibits non-equilibrium behavior quite different from that of their equilibrium counterparts. 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 growing one-dimensional interfaces (that is, the 1+1-dimensional Kardar-Parisi-Zhang equation), in fact belong to the same universality class. This universality class also includes two equilibrium systems : two-dimensional smectic liquid crystals, and a peculiar kind of constrained two-dimensional ferromagnet. We use these connections to show that two-dimensional incompressible flocks are robust against fluctuations, and exhibit universal long-ranged, anisotropic spatio-temporal correlations of those fluctuations. We also thereby determine the exact values of the anisotropy exponent $\zeta$ and the roughness exponents $\chi_{_{x,y}}$ that characterize these correlations., Comment: 14 pages, 3 figures, to appear in Nature Communications

We study the asymptotic response of polar ordered active fluids ("flocks") to small external aligning fields $h$. The longitudinal susceptibility $\chi_{_\parallel}$ diverges, in the thermodynamic limit, like $h^{-\nu}$ as $h \rightarrow 0$. In finite systems of linear size $L$, $\chi_{_\parallel}$ saturates to a value $\sim L^\gamma$. The universal exponents $\nu$ and $\gamma$ depend only on the spatial dimensionality $d$, and are related to the dynamical exponent $z$ and the "roughness exponent" $\alpha$ characterizing the unperturbed flock dynamics. Using a well supported conjecture for the values of these two exponents, we obtain $\nu = 2/3$, $\gamma = 4/5$ in $d = 2$ and $\nu = 1/4$, $\gamma = 2/5$ in $d = 3$. These values are confirmed by our simulations., Comment: 20 pages, 3 figures, accepted for publication in New Journal of Physics