DOI: 10.1126/science.1236462, 365 (2013);341 Science

Kent Yagi and Nicolás YunesQuark StarsI-Love-Q: Unexpected Universal Relations for Neutron Stars and

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I-Love-Q: Unexpected UniversalRelations for Neutron Starsand Quark StarsKent Yagi* and Nicolás Yunes

Neutron stars and quark stars are not only characterized by their mass and radius but also byhow fast they spin, through their moment of inertia, and how much they can be deformed, throughtheir Love number and quadrupole moment. These depend sensitively on the star’s internalstructure and thus on unknown nuclear physics. We find universal relations between the momentof inertia, the Love number, and the quadrupole moment that are independent of the neutronand quark star’s internal structure. These can be used to learn about neutron star deformabilitythrough observations of the moment of inertia, break degeneracies in gravitational wavedetection to measure spin in binary inspirals, distinguish neutron stars from quark stars, andtest general relativity in a nuclear structure–independent fashion.

One of the largest uncertainties in nuclearphysics is the relation between energydensity (r) and pressure (p) at very high

densities, the so-called equation of state (EoS).The interior structure of very compact stars, likeneutron stars (NSs) and quark stars (QSs), de-pends sensitively on their EoS. This, in turn,determines their exterior properties, such as theirmass and radius; their rotation rate, character-ized by their moment of inertia; and their deform-

ability, characterized by their quadrupole momentand tidal Love number (1, 2).

Some astrophysical observations allow us toinfer properties of the EoS of compact stars (3–5).None of these observations, however, is current-ly accurate enough to select between the manydifferent EoSs that have been proposed or todistinguish between NSs and QSs, which thenleads to degeneracies in the extraction of infor-mation from observations. For example, gravi-tational wave (GW) observations of NS binaryinspirals may have difficulty in extracting the in-dividual spins, because these are degenerate withthe quadrupole moment for nonprecessing bina-

ries (6). Similarly, GWs from NS binary inspiralscannot be easily used to test general relativity (GR)because of EoS degeneracies (7, 8). We here findaway to uniquely break these degeneracies throughuniversal I-Love-Q relations between the reducedmoment of inertia, I ; tidal Love number, l

ðtidÞ;

and quadrupole moment, Q, that are essentiallyinsensitive to the star’s EoS (9).

Consider an isolated, slowly rotating NS orQS described by its mass,M*; the magnitude ofits spin angular momentum, J, and angular ve-locity,W; its (spin-induced) quadrupole moment,Q, and its moment of inertia, I ≡ J/W. Let usintroduce dimensionless quantities I ≡ I=M 3

* andQ ≡ −Q=ðM 3

*c2Þ, where c ≡ J=M 2

* is the di-mensionless spin parameter (10). I determineshow fast a body can spin given a fixed J, whereasQ encodes the amount of stellar quadrupolar de-formation. They are determined by numericallysolving the perturbedEinstein equations for realisticEoSs in a slow-rotation expansion (c « 1) to firstand second order in spin, respectively (9, 11).

The slow-rotation approximation requires thatc be small enough such that all equations canbe expanded in c « 1. In this approximation, theneglected corrections to I and Q are of O(c2)smaller than the leading-order contributions. Thus,demanding that any subleading terms be less than10% of the leading-order ones means that c « 0.3or equivalently spin frequencies « 600 Hz orspin periods » 1.7 ms. “True”millisecond pulsars(with periods of ~1 ms) cannot be modeled in aslow-rotation expansion. However, double NS bi-nary pulsars are expected to be spinning muchmore slowly, and the slow-rotation approxima-tion should be allowed.

REPORTS

Department of Physics, Montana State University, Bozeman,MT59717, USA.

*Corresponding author. E-mail: kyagi@physics.montana.edu

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Fig. 1. I-Love andQ-Love relations. (A and B) (Top) The neutron star (NS) andquark star (QS) universal I-Love and Love-Q relations for various EoSs, together withfitting curves (solid and dashed curves). On the top axis, we show the correspondingNS mass with an APR EoS. The thick vertical lines show the stability boundary for

the APS, SLy, LS220, and Shen EoSs from left to right. The parameter varied alongeach curve is the NS central density, or equivalently the NS compactness, with thelatter increasing to the left of the plots. (Bottom) Fractional errors between thefitting curves and numerical results. M◉ indicates the mass of the Sun.

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In the presence of a companion, a NS or a QSwill also be quadrupolarly deformed. The quad-rupole moment tensor determines the magnitudeof this deformation: Qij = −l(tid)eij, where l(tid) isthe tidal Love number and eij is the quadrupole(gravitoelectric) tidal tensor that characterizes thesource of the perturbation (12, 13). The dimension-less tidal Love number, lðtidÞ ¼ lðtidÞ=M 5

*, char-acterizes the tidal deformability of a star in thepresence of a companion, and it can be calculatedby treating the tidal effects as a perturbation to anisolated (nonrotating) NS or QS solution (9, 13).

One might have expected universal relationsbetween I ,Q, and lðtidÞ because I º C−2,Q ºC−1, and lðtidÞ º C−5 for polytropic EoSs in theNewtonian limit (9), where C ¼ M*=R* is thecompactness parameter with R* the stellar radius.In the slow-rotation and small-deformation ap-proximations, these barred quantities depend onspin only quadratically, and thus the relations areessentially spin-independent.

To find these relations, consider the follow-ing realistic EoSs for NSs: APR (14), SLy (15),Lattimer-Swesty with nuclear incompressibilityof 220 MeV (LS220) (16), Shen (17), PS (18),PCL2 (19), and a simple n = 1 polytropic EoSwith p = Kr1+1/n. For the LS220 and Shen EoSs,we adopt a temperature of 0.1 MeV and assumethey are neutrino-less and in b-equilibrium. ForQSs, we consider the EoSs: SQM1, SQM2, andSQM3 (19). We assume the stars are uniformlyrotating, with isotropic pressure.

As shown in Fig. 1, the I-Love-Q relations holduniversally for each NS and QS sequence, essen-tially independently of their EoSs for each class.Such relations can be numerically fitted with a poly-nomial on a log-log scale (9), shown in Fig. 1withsolid and dashed black curves:

lnyi = ai + bilnxi + ci(lnxi)2 + di(lnxi)

3+ ei(lnxi)4 (1)

where the coefficients are summarized in Table 1.The data in Fig. 1 were obtained by numericallysolving the perturbed Einstein equations, whichis unavoidable for realistic EoSs. For very simplepolytropic EoSs, the equations can be solvedanalytically in the Newtonian limit, and we ob-tain similar universal and analytic relations (9).

Two possible reasons may explain the I-Love-Q relations. First, the mathematical relations thatdefine I, l(tid), and Q seem to depend mostly onthe star’s internal structure near its outer layer,where nuclear physics constrains realistic EoSs.The integral that defines I andQ in the Newtonianlimit accumulates the most near the NS surface(9). This suggests that the relations should losetheir universality for unrealistic EoSs that modifythe star’s internal structure near its surface. Wehave verified this explicitly by computing theserelations for NSs with n = 2, 2.5, and 3 polytropicEoSs: the I-Love-Q curves deviate away fromthose in Fig. 1 as n increases. The NS and QSI-Love-Q relations present different universal be-havior because their EoSs are drastically differentin the low-density stellar region (20).

The second reason is related to the no-hair theoremsof GR. Figure 1 shows that the NS and the QSI-Love-Q relations approach each other as com-pactness is increased and approach the expectedI-Love-Q relations for black holes (BHs), that is,I → 4, lðtidÞ → 0, and Q → 1 (9). For BHs, allmultipole moments of the exterior spacetime arerelated to the BH mass and spin (21, 22) becauseof the no-hair theorems (23, 24). But for NSs andQSs, such relations do not hold because of thelack of no-hair theorems for nonvacuum space-times. Our results suggest the existence of NSuniversal relations that are similar to the BH onesand perhaps hint at the existence of no-hair–likerelations for nonvacuum spacetimes.

The I-Love-Q relations suggest an effacingof internal structure. This is not a consequenceof the effacement principle (25) in GR, becausethe latter applies only to the motion of BHs. TheI-Love-Q relations relate different multipole com-ponents of the exterior gravitational field of isolated,nonvacuum compact objects and say nothingabout their relative motion.

Double NS binary pulsars may allowmeasur-ing I to 10% accuracy in the near future (26, 27).This is because I induces additional periastron

precession, as well as precession of the angularmomentum vector and the NS spin vectors. Theformer translates into a time-dependent inclina-tion angle, whereas the latter may force the pulsarbeams to sweep in and out of Earth’s line of sight.Alternatively, this precession may only cause achange in the observed average pulse shape, as inthe Hulse-Taylor binary pulsar, in which casedirect measurement may be more difficult.

Given a measurement of I, M, and W, theI-Love-Q relations automatically provide thevalue of l(tid) and Q, assuming the star is eithera NS or a QS. The latter would not be easilyobservable with binary pulsars directly; althoughQ and l(tid) do induce additional precession, theireffect is greatly suppressed relative to that of I.The I-Love-Q relations refer to reduced (barred)quantities, which must be appropriately normal-ized by the mass and the spin period. A smallerror in the latter could induce a large error inderived quantities. Such an error is smaller thanthe nonuniversality of the I-Love-Q relations ifthe NS spin period is much greater than 8.5 ms,which is the case for the double pulsar binary anda NS binary in the Laser Interferometer Gravita-tional Wave Observatory (LIGO) band.

Table 1. Fit parameters. Estimated numerical coefficients for the fitting formulas of the NS and QS I-Love,I-Q, and Love-Q relations.

yi xi ai bi ci di eiNS

I l(tid) 1.47 0.0817 0.0149 2.87 × 10−4 −3.64 × 10−5

I Q 1.35 0.697 –0.143 9.94 × 10−2 −1.24 × 10−2

Q l(tid) 0.194 0.0936 0.0474 −4.21 × 10−3 1.23 × 10−4

QSI l(tid) 1.52 0.0100 0.0418 −2.26 × 10−3 5.35 × 10−5

I Q 1.30 0.757 –0.139 7.87 × 10−2 −7.29 × 10−3

Q l(tid) 0.286 0.126 0.0900 −1.13 × 10−2 4.57 × 10−4

0.01 0.1

0.01

0.1

Mea

sure

men

t Acc

urac

y of

Spi

ns

∆β∆χ∆χ∆χ∆χ

Fig. 2. Breaking spindegeneracies.Measure-ment accuracy of spinparameters b, cs, and causing a Fisher analysiswithAdvanced LIGO,givenadetection at a distanceofDL = 100Mpc with signal-to-noise ratio ≈ 30. Thespin-dependent part ofthe waveform phase isparameterized with thedimensionless averagedspin, cs, and spin differ-ence, ca, when using theLove-Q relation and withthe effective spin param-eter b, constructed froma certain combination ofthe individual spins, when not using the relation. We consider three different NS binaries [(i), (ii), and(iii)], with parameters (m1,m2) = (1.45,1.35)M◉ and c1 = c2, (m1,m2) = (1.45,1.35)M◉ and c1 = 2 c2,and (m1,m2) = (1.4,1.35)M◉ and c1 = c2, respectively. Db is computed without use of the NS Love-Q relation,whereas Dcs,a are computed by using this relation. Db and Dca are almost identical for all systems, becausethey are dominated by their priors (|b| < 0.2 and |ca| < 0.1). Thanks to the Love-Q relation, cs can bemeasuredto ~0.01.

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Given independent measurements of any twomembers of the I-Love-Q trio, one could also dis-tinguish NSs from QSs, assuming GR is correct.These relations are different for NSs and QSs,even though they are universal within their classof EoSs, as shown in Fig. 1. Given two such mea-surements with sufficiently small uncertainties,one could determine whether the observed com-pact object is a NS or a QS.

Interferometric GW detectors are most sensi-tive to theGWphase. Forwaves emitted duringNSbinary inspirals, the phase contains a term pro-portional to the NSs’ spin-induced quadrupolemoments, Q1 and Q2, and another term propor-tional to their tidally induced quadrupolar de-formations, l1

(tid) and l2(tid). The former enters

with a factor proportional to v4/c4, where v is theorbital velocity of binary constituents (28), whereasthe latter is proportional to v10/c10 (12), relative tothe leading-order term. Because by Kepler’s thirdlaw v º f 1/3 (where f is the GW frequency), eachterm has a distinct GW frequency dependence thatmakes them nondegenerate.

TheNS quadrupolemoment is degeneratewiththe NSs’ individual spins, because there is a spin-spin interaction term in the GW phase that entersat the same order in v/c as the quadrupole term (28).Such a degeneracy may prevent us from simul-taneously extracting the quadrupole moment andthe individual spins from a GW detection. How-ever, the Love-Q relation can be used to break thisdegeneracy by rewriting Q as a function of lðtidÞ.If the Love number can be measured with a GWdetection, then one can also separately measurethe spins, as shown in Fig. 2.

Pulsar observations allow GR tests (8) whenthe gravitational field is much stronger than thatin the Solar System (7). Unfortunately, these testsare not effective most of the time because of de-generacies between modified gravity effects andthe EoS. The I-Love-Q relations can be used tobreak this degeneracy.

A robust GR test would require at least twoindependent measurements of any two quanti-ties in the I-Love-Q trio. Given a single measure-ment, the I-Love-Q relations give us the othertwo for a NS or a QS in GR. A second indepen-dent measurement can then be used as a redun-dancy test.

Depending on where the two observed valueslie in the I-Love-Q plane, different tests are pos-sible. If the two observables lie on the GR NS(QS) I-Love-Q line, then

1) Case a. The object is a NS (QS), and anymodified gravity effect must be small enough tofit this observation.

2) Case b. The object is a QS (NS) in amodified gravity theory with the right couplingparameters to still fit this observation.

If the twoobservables lie on neitherGR I-Love-Qline, then the observations would indicate a GRdeviation, provided the assumptions made hereare correct. An example of such a GR test is givenin Fig. 3. One can carry out a similar test by usingthe Love-C relation (shown in Fig. 3), which iseffectively universal.

References and Notes1. A. E. H. Love, Proc. R. Soc. London Ser. A 82, 73–88

(1909).

2. S. Postnikov, M. Prakash, J. M. Lattimer, Phys. Rev. D 82,024016 (2010).

3. J. M. Lattimer, M. Prakash, Phys. Rep. 442, 109–165(2007).

4. F. Ozel, Rep. Prog. Phys. 76, 016901 (2013).5. For example, the observation of x-ray bursters and low-

mass x-ray binaries has allowed for the simultaneousdetermination of the star’s mass and radius to O(10)%accuracy (4). Observations of double NS pulsars, such asJ0737-3039 (29), may allow for the measurement of themoment of inertia to the same accuracy (26, 27).Gravitational wave (GW) observations from binary NSinspirals with second-generation ground-based detectors,such as Advanced LIGO, Advanced Virgo, and KAGRA,may allow for the measurement of the tidal Love number(12, 30, 31).

6. If the NS binary has misaligned spins, precession maybreak these degeneracies.

7. C. M. Will, Living Rev. Relativ. 9, 3 (2006).8. I. H. Stairs, Living Rev. Relativ. 6, 5 (2003).9. K. Yagi, N. Yunes (2013), http://arxiv.org/abs/1303.1528.

10. Throughout the paper, we use geometric units with G(Newton’s gravitational constant) and c (the speed oflight) set to unity.

11. J. B. Hartle, Astrophys. J. 150, 1005 (1967).12. E. E. Flanagan, T. Hinderer, Phys. Rev. D 77, 021502

(2008).13. T. Hinderer, Astrophys. J. 677, 1216–1220 (2008).14. A. Akmal, V. Pandharipande, D. Ravenhall, Phys. Rev. C

58, 1804–1828 (1998).15. F. Douchin, P. Haensel, Astron. Astrophys. 380, 151–167

(2001).16. J. M. Lattimer, F. Douglas Swesty, Nucl. Phys. A 535,

331–376 (1991).17. H. Shen, H. Toki, K. Oyamatsu, K. Sumiyoshi, Nucl. Phys. A

637, 435–450 (1998).18. V. R. Pandharipande, R. A. Smith, Nucl. Phys. A 237,

507–532 (1975).19. M. Prakash, J. R. Cooke, J. M. Lattimer, Phys. Rev. D 52,

661–665 (1995).20. J. Lattimer, M. Prakash, Astrophys. J. 550, 426–442

(2001).21. R. P. Geroch, J. Math. Phys. 11, 2580 (1970).

Fig. 3. TestingGR. (A) Error box (shaded region) in the I-Love plane, given twoindependent observations of I and l(tid) consistent with GR (black star) to 10%accuracy with a binary pulsar observation (26, 27) and to 60% with a GW ob-servation (9, 12, 30, 31), respectively. The black solid and dotted-dashed linesshow theGRNS andQS I-Love relations, respectively, whereas other lines show therelations in dynamical Chern-Simons (CS) gravity, which are still EoS-insensitive.This theory modifies Einstein’s by introducing a gravitational parity-violatinginteraction through a dynamical scalar field, and it is currently constrained onlyvery weakly by known tests and experiments (32). Given this observation, anymodified theory would be constrained to predict an I-Love relation that runs

through this shaded region. The CS I-Love relation lies above the GR oneirrespective of the CS coupling constant; thus, this is an example of case a thatwould place a constraint on the theory that is 6 orders of magnitude stronger thancurrent solar system ones (33). (B) Error box (shaded region) in the Love-C plane,given two independent observations of l(tid) and C consistent with GR (black star)to 60% accuracy with a GW observation (9, 12, 30, 31) and to 5% with a futurelow-mass x-ray binary observation (4), respectively. The Love-C relation is depen-dent on the EoS, as shown by the spread in the curves. However, the differencebetween these curves is smaller than the error box, making the Love-C relationeffectively EoS-independent and allowing for a generic test of GR.

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22. R. O. Hansen, J. Math. Phys. 15, 46 (1974).23. S. W. Hawking, Commun. Math. Phys. 25, 152–166

(1972).24. B. Carter, Phys. Rev. Lett. 26, 331–333 (1971).25. T. Damour, Gravitational Radiation, N. Deruelle,

T. Piran, Eds. (North-Holland, Amsterdam,1983).

26. J. M. Lattimer, B. F. Schutz, Astrophys. J. 629, 979–984(2005).

27. M. Kramer, N. Wex, Class. Quantum Gravity 26, 073001(2009).

28. E. Poisson, Phys. Rev. D 57, 5287–5290 (1998).

29. M. Burgay et al., Nature 426, 531–533 (2003).30. T. Hinderer, B. D. Lackey, R. N. Lang, J. S. Read,

Phys. Rev. D 81, 123016 (2010).31. T. Damour, A. Nagar, L. Villain, Phys. Rev. D 85, 123007

(2012).32. S. Alexander, N. Yunes, Phys. Rep. 480, 1–55

(2009).33. Y. Ali-Haimoud, Y. Chen, Phys. Rev. D 84, 124033

(2011).

Acknowledgments: We thank E. O’Connor and B. Lackey forproviding tabulated EoSs, as well as E. Berti, L. Blanchet,

V. Cardoso, T. Hinderer, K. Hotokezaka, M. Kramer, L. Lindblom,F. Nakano, F. Özel, P. Pani, E. Poisson, S. Ransom, L. Rezzolla,M. Shibata, T. Tanaka, and four anonymous reviewers for theircomments. We also thank the Yukawa Institute for TheoreticalPhysics at Kyoto University, where this work was initiatedduring the Long-Term Workshop YITP-T-12-03 on “Gravity andCosmology 2012.” N.Y. acknowledges support from NSFgrant PHY-1114374, as well as support provided by NASAgrant NNX11AI49G.

12 February 2013; accepted 20 June 201310.1126/science.1236462

Holographic Vortex Liquidsand Superfluid TurbulencePaul M. Chesler,* Hong Liu,* Allan Adams*

Superfluid turbulence is a fascinating phenomenon for which a satisfactory theoretical framework islacking. Holographic duality provides a systematic approach to studying such quantum turbulenceby mapping the dynamics of a strongly interacting quantum liquid into the dynamics of classicalgravity. We use this gravitational description to numerically construct turbulent flows in aholographic superfluid in two spatial dimensions. We find that the superfluid kinetic energyspectrum obeys the Kolmogorov −5=3 scaling law, with energy injected at long wavelengthsundergoing a direct cascade to short wavelengths where dissipation by vortex annihilation andvortex drag becomes efficient. This dissipation has a simple gravitational interpretation as energyflux across a black hole event horizon.

Superfluid turbulence is a non-equilibriumphenomenon dominated by the dynam-ics of quantized vortices (1–7), which drive

the system outside the hydrodynamic regimeof normal turbulent fluids. Powerful numeri-cal simulations of phenomenological models ofvortex dynamics have produced considerableinsight. However, a complete theoretical frame-work remains lacking, and an ab initio study isdesirable.

Here, we use holographic duality to studysuperfluid turbulence in two spatial dimensions.Holographic duality is a precise equivalence be-tween certain systems of quantum matter with-out gravity and classical gravitational systems ina curved spacetime with one additional spatialdimension (8–16). This allows a first-principlesstudy in these systems of superfluid dynamics, in-cluding turbulent flows, by using the correspond-ing gravity description of the superfluid phase. Inthis framework, dissipation in the gravitationaldescription can be understood in terms of excita-tions falling through a black hole event horizon.This provides a direct measure of the rate ofenergy dissipation and its spectrum.

We focus on “non-counterflow” superfluidturbulence (17–25). Experimentally, these sys-tems appear to obey Kolmogorov’s −5=3 scalinglaw in the kinetic energy spectrum, which sug-gests a remnant similarity between quantum andclassical turbulence. In classical turbulence, this

scaling behavior can be understood as a conse-quence of an energy cascade in which the injectedenergy is passed from one scale to another withoutsubstantial loss. Whether quantum turbulence ad-mits a similar cascade picture—and, if so, whetheror when the cascade drives energy to long or toshort wavelengths—remain important open ques-tions. Several recent numerical studies of thephenomenological Gross-Pitaevskii equation ina two-dimensional (2D) superfluid (with dissi-pation put in by hand) observed Kolmogorovscaling but came to conflicting conclusions regard-ing the direction of the cascade (26–29).

To set up our superfluid, consider a quantumfield theory in two spatial dimensions with acomplex scalar operator, y(x), carrying charge qunder a global U(1) symmetry. Let jm(x) denotethe conserved current operator of this globalU(1) symmetry. To induce a superfluid conden-sate for y, we will turn on a chemical potentialm for the U(1) charge. For sufficiently large m, ycan develop a nonzero expectation value ⟨y⟩ ≠ 0when the temperature falls below a critical tem-perature Tc, spontaneously breaking the globalU(1) symmetry and driving the system into asuperfluid phase.

We now construct a simple holographic modelof our 2D superfluid. We begin with a classicalfield theory in an asymptotically anti–de Sitterspacetime with three spatial dimensions (AdS4).Under the standard holographic dictionary, theconserved current jm(x) is mapped to a dynam-ical U(1) gauge field AM(x, z) in the gravitationalbulk, and the scalar operator y(x) is mapped toa bulk scalar field F(x, z) carrying charge qunder the gauge field AM (where z is the radial

coordinate of AdS4) (30). A nonzero temper-ature corresponds to adding to the bulk space-time a black hole whose horizon is a 2D planeextended in boundary spatial directions. Addinga chemical potential corresponds to imposinga boundary condition on the bulk gauge field At =m at the boundary of AdS4 (31). If the charge qand scaling dimension ∆ of y lie in certain range(32, 33), a sufficiently large m drives the bulkscalar field F to condense through the Higgsmechanism. Different values of the charge q andpotential for F define different holographic quan-tum theories, each with a low-temperature su-perfluid phase (34, 35). We choose a quadraticpotential with a mass for F corresponding to yhaving scaling dimension ∆ = 2, and we work inthe limit of large q [the probe limit of (33); seesupplementary text].

A superfluid state generically has gappedvortex excitations, which play an important rolein our discussion below. Around a vortex, the fluidcirculation is quantized. Introducing the (un-normalized) superfluid velocity

u ≡J

j⟨y⟩j2 , J ≡i

2½⟨y*⟩∇⟨y⟩ − ⟨y⟩∇⟨y⟩*�

ð1Þ

the winding number W of a vortex is deter-mined by

W ¼ 1

2p∮Gdx ⋅ u ð2Þ

where the path G encloses a single vortex and isoriented counterclockwise. Boldface symbolsdenote vectors along boundary spatial directions;x ≡ {x1, x2}, where xi denotes the two spatialdirections and∇ ¼ f∂=∂x1, ∂=∂x2g. Vortices mapinto the gravitational bulk as flux tubes extend-ing along the AdS radial direction from the bound-ary, where they have a characteristic size 1/m, tothe horizon. Inside the flux tube, the condensategoes to zero, effectively punching a hole throughthe bulk scalar condensate. Explicit gravity solu-tions corresponding to a static vortex of arbitrarywinding number were previously constructed nu-merically (36–38).

The gravity dual thus provides a first-principlesdescription of superfluid flows involving vortices,as well as tools to describe dissipation in the sys-tem. Consider turning on a perturbation of jm in theboundary theory, which on the gravity side cor-

Department of Physics, Massachusetts Institute of Technol-ogy, Cambridge, MA 02139, USA.

*Corresponding author. E-mail: awa@mit.edu (A.A.); pchesler@mit.edu (P.M.C.); hong_liu@mit.edu (H.L.)

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