VOLUME 80, NUMBER 5 P H Y S I C A L R E V I E W L E T T E R S 2 FEBRUARY 1998

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Thermodynamics of Cosmic String Densities in U(1) Scalar Field Theory

Nuno D. Antunes,1 Luı́s M. A. Bettencourt,2 and Mark Hindmarsh11Centre for Theoretical Physics, University of Sussex, Brighton BN1 9QH, United Kingdom

2Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germ(Received 5 August 1997)

We present a full characterization of the phase transition in U(1) scalar field theory and of the asciated vortex string thermodynamics in 3D. We show that phase transitions in the string densities eand measure their critical exponents, both for the long string and the short loops. Evidence for a natseparation between these two string populations is presented. In particular, our results strongly indthat an infinite string population will only exist above the critical temperature. Canonical initial condtions for cosmic string evolution are shown to correspond to the infinite temperature limit of the theo[S0031-9007(97)05189-2]

PACS numbers: 11.27.+d, 05.70.Fh, 11.10.Wx, 98.80.Cq

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Topological defects appear in a great variety of systemfrom condensed matter laboratory experiments to the eaUniverse. Their importance in phase transitions in thlaboratory is known to be fundamental, and their presenin the early Universe may be the key to many of thunsolved questions in standard cosmology.

However, in spite of the universal relevance of topological defects, much about the fundamental descriptioof their formation and evolution remains qualitativeThis is a reflection of the complexities involved in firsprinciple studies, owing to their nature as nonperturbativexcitations of quantum field theories.

Particularly interesting for cosmology are stringliketopological defects [1]. Motivated by the study of theicreation in the early Universe [2], a variety of experimenthas been recently developed with the aim of studyinvortex string formation in liquid crystals [3], superfluid4He [4] and 3He [5] systems. Their new results permius to test with unprecedented precision theoretical ideabout defect formation and evolution.

From a theoretical standpoint, cosmic strings and othtopological defects have been traditionally thought to bproduced at phase transitions in the early Universe asresult of the formation of correlated domains [6]. Morerecently a refinement of this scenario [7] has been gainisupport, that most topological defects existing below thphase transition are, in fact, survivors of an equilibriumpopulation of unstable defects existing above the phatransition as nonlinear excitations of the fields. In ordeto exist at low energies these field configurations neeto be frozen in by some nonequilibrium process, whicis context dependent. In a continuous phase transitiolike the superfluid transition in4He, this process mayin addition be helped by the critical slowing down inthe response of the fields close to the critical point. Aestimate of the defect density hence produced is usuaachieved given the correlation length of the field athe time of freeze-out. By assigning random phasesdifferent domains and connecting by the path of minimumphase gradient, a network of defects can in principle b

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constructed. This picture generates order of magnitudestimates that successfully predicted the defect densitiobserved in the helium experiments.

In practice higher correlations in the phase structure othe fields exist and can change the picture considerabAs an illustration of the shortcomings of simply usingthe correlation length to predict produced defect densitienote that similar densities would be estimated if the defenetwork were frozen in above or below the critical pointprovided that the correlation length was chosen to be thsame in both cases. This is clearly not what is observeexperimentally [4], suggesting that the string networkmust be very different in two circumstances where thdomain structure can be expected to coincide.

In order to match theory to future, more accurate measurements, as well as for the sake of theoretical undestanding, it is highly desirable to generate more detailepredictions for the string density formed at phase transtions. In cosmology and in the laboratory, equilibriumis the natural starting point. If we know the statistics ostrings at any given temperature the reference to the dmain structure in the fields becomes obsolete, while throle of nonequilibrium in freezing in the string network, atleast in some scales, can be expected to hold as usual.

In this Letter we present the full characterization othe behavior of the string densities with temperaturin the U(1) scalar field theory. In doing so we gobeyond existing studies in theXY (see, e.g., [8]) model,which belongs to the same universality class, especialin determining the behavior of infinite strings, measuringcritical exponents associated with string densities anconfronting our findings with cosmological scenarios odefect formation.

We first determine the field thermodynamics and locatthe critical point. In order to drive the system toequilibrium we evolved it stochastically according to

s≠2t 2 =2dfi 2 m2fi 1 lfi

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(1)

© 1998 The American Physical Society

VOLUME 80, NUMBER 5 P H Y S I C A L R E V I E W L E T T E R S 2 FEBRUARY 1998

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where hi, jj [ 1, 2. The stochastic variablejisx, td istaken to be Gaussian, characterized by

kjisx, tdl ­ 0 , (2)

kjisx, tdjjsx0, t0dl ­2h

bdsx 2 x0 ddst 2 t0ddij . (3)

These relations ensure that detailed balance appliesequilibrium, which will result, independently of the choiceof initial conditions, for large times at temperature1yb.We evolved the system Eqs. (1)–(3) using a staggerleapfrog method, on a lattice of sizeN3 ­ 1003. Wechose the lattice spacing to bedl ­ 0.5 and setl ­ 1,m2 ­ 1. We verified that the correlation length waalways resolved by at least four points in each linedimension, apart from the cases whenb ø 1. Aroundthe critical point all physical scales become much largthan the lattice spacing. Universal quantities can thbe measured and shown explicitly to be independentthis ultraviolet cutoff. Away from the critical point, athigh or low temperature, universality is lost in generaThere, the results lose some of their physical meanibut retain model properties that are interesting fromtheoretical point of view. This will be the case of thinfinite temperature limit of the theory.

An equivalent equilibrium state could have been otained using lattice Monte Carlo methods [9]. From thequilibrium point of view, the second order time derivative is redundant. However, this work is intended to bthe precursor to nonequilibrium studies of the relativistfield theory, where it is more natural to use a Langevmethod such as ours, and the field equations are natursecond order in time derivatives.

We decide when the system has reached equilibriuby monitoring several quantities associated with differelength scales. We measure the kinetic energy of tsystem and check for equipartition. This characterizthe smallest scales in the sample. We monitor the stridensities, associated with intermediate length scales,we measure the phase distribution functions acrosswhole volume. By making sure that these three quantithave reached asymptotic average values we concludethe system is in equilibrium. The temperature of the bacoincides with that measured using equipartition.

The transition between a system displaying a preferrrandom direction in its manifold of field minima at lowtemperature and a flat phase distribution, characteristictheUs1d symmetry, is shown in Fig. 1.

Although appealing the phase probability plots of Fig.do not permit a precise determination of the criticatemperature or any of its associated critical exponenThe critical point can more easily be found by observinthe temperature variation of

kjfV jl ­

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µ1V

ZV

dx fisxd∂2

+, (4)

where the angular brackets denote ensemble averagobtained in practice by averaging over many independe

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FIG. 1. Field phase probability distribution betweenb ­ 1.1(flattest) andb ­ 2.9 (sharpest). Atb . bc the distributionpeaks around a random phase and theUs1d symmetry arespontaneously broken.

measurements,200 1000, corresponding to well sepa-rated times. The variation of Eq. (4) and its derivativewith temperature is shown in Fig. 2. In particular, we sethat the derivative diverges at the critical point.

The critical temperature can be measured by assumithat Eq. (4) behaves like

kjfV sbdjl ­ Asb 2 bcda , (5)

which should hold immediately below the critical temperature. By fitting our data at several temperatures to Eq. (5for different inverse critical temperaturesbc, we can deter-mine its minimalx2 value. Simultaneously we obtain thecritical exponenta. The critical values hence determinedarebc ­ 1.906 6 0.008 anda ­ 0.43 6 0.07. The uni-versal value ofa is in reasonable agreement with renormalization group calculations [10].

Having measured the critical behavior in the fields wwant to analyze how vortex string densities vary with th

FIG. 2. The variation ofkjfV jl and its derivative, withb.Squares denote measured points.

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temperature. We assume ergodicity of the field evolutioonce equilibrium is reached. Using this fact we analyzat given time intervals the phase of the complex field, anassociate a vortex to each 2D lattice cell where theUs1dphase winds through2p. We then proceed to connect thevortices and construct the string network. Given a strinnetwork, we measure the string density in loops and lostring, for different values of the bath temperature arounthe phase transition.

Figure 3 shows how the total equilibrium density pelattice link of strings,rtot, as well as the density of longand short strings (rinf and rloop, respectively) changeswith temperature. These quantities allow us a direcomparison to algorithms of cosmic string formation. Wobserve a dramatic change in the behavior of the strisystem across the phase transition. The curves of Figsuggest that we can decompose strings into two distinpopulations, one of loops, say, strings smaller thancertain cutoff lengthL` , N2 and another populationconsisting of much longer strings. We will refer to theformer as the string loops and to the latter, owing to thterminology in cosmology, as infinite strings.

At bc all three densities display abrupt changestheir behavior. These changes are phase transitionsin all three cases the derivatives of the densities presdiscontinuities at the critical point. Even more remarkabis the fact that the appearance of infinite string seemsbe linked to the criticality in the fields. The very sharprise in the infinite string density at the critical point maysuggest that the phase transition associated with it mactually be discontinuous in the infinite volume limit.

In order to clarify this question we measured the infinitstring density variation withb for various values ofL`. The results are shown in Fig. 4. The temperatuat which infinite strings first appear clearly depends othe choice ofL`, getting closer tobc as it increases.We also verified that for smaller lattice sizes infinitestring first appeared at lower temperatures; eg., in a203

FIG. 3. The dependence ofrtot, rloop , rinf on b.

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computational domain there was infinite string as sooas b ­ 2.3. Both these variations supply evidence thaas the volume is increased the temperature at whicinfinite strings first appear migrates towards the criticapoint implying a discontinuous phase transition inrinf.At present it is, however, impossible to perform suchextrapolation with full confidence.

We can establish that the phase transition in the fieldand in the string densities coincide by finding the criticapoint and exponents for the latter.

At the critical temperature the density of strings dropsuddenly and keeps falling for lower temperatures. Thdecay ofrtot, which coincides withrloop , is excellentlydescribed by the exponential law

rtotsbd ­ rtotsbcde2sb2bcdE , (6)

where rtotsbcd . 0.2 is a universal number (see, e.g.,[8] for a very different study yielding the same result)and E ­ 1.836, implying that at temperatures below thecritical point strings are Boltzmann suppressed. At lowtemperatures (in practice forb * 6) there is long-rangeorder and no strings survive.

Just above the critical temperature the variation of thstring densities can be best characterized by defining tquantitydr ; frsT d 2 rsTcdgyrsTcd. The behavior ofthe infinite string densityrinf is well described by anansatz similar to Eq. (5),drinf ~ sT 2 Tcdginf . Note thechange fromb in Eq. (5) toT . We find ginf ­ 0.25 6

0.01 and bc ­ 1.911 6 0.001, in agreement with allprevious estimates. Forrtot we finddrtot ~ sT 2 Tcdgtot ,with bc ­ 1.912 6 0.002 andgtot ­ 0.39 6 0.01.

For temperatures well above the phase transition thloop density can be well described by a quadratic form

rloopsbd . rloops0d

"1 1

b

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13

√b

2

!2#, (7)

whererloops0d . 1y12. This corresponds to a fraction of25 6 1% of all strings in loops forb ­ 0.

FIG. 4. The dependence ofrinf on b, for several values ofL`, measured in lattice units.

VOLUME 80, NUMBER 5 P H Y S I C A L R E V I E W L E T T E R S 2 FEBRUARY 1998

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The variations are more complicated forrtot andrinf.The large size of the derivatives close tob ­ 0 makes ithard to approximate the curves, and a good fit can onbe achieved by a polynomial of large degree, typical9 or 10, with coefficients of alternating signs. Indeedthe behavior looks nonpolynomial instead, which woulmean that there is no standard high-temperature expansfor rinf. This is explicable since there is no locaoperator which can distinguish an infinite string, and thuwe would not necessarily expect a polynomial behaviaroundb ­ 0.

Particularly interesting is theT ! ` limit. Thenrtot ! 1y3 andrinfyrtot ! 75 6 1%. These results co-incide with those obtained from canonical VachaspaVilenkin (VV) algorithms for simulating string formationin the early Universe [11–13]. Their underlying motivation stems from the picture that strings result from randophase fluctuations within spatial domains of a given sizTypically, the complex field is given unit modulus, withphases assigned randomly to each site on a cubic lattThere are variations on this theme: e.g., the continuophase may be approximated by three values [11], whileads to slightly different values ofrtot and rinfyrtot.Nonetheless, assigning phases randomly to each latsite corresponds precisely to the infinite temperature limand thus VV algorithms generate ensembles of stringT ­ `. These ensembles are used as initial conditiofor the free evolution of the string network, following thenumerical solution to the classical dynamics of relativistic strings [1]. Thus, numerical simulations of this typemodel an instantaneous quench from infinite to zero temperature. While these initial conditions are ultimately unphysical, the network of strings is observed to approarapidly a self-similar evolution, which serves to disguisthe initial state. Alternatively, if one wished to describa network of strings evolving from an equilibrium statebelow Tc, the VV algorithm would be a completely inap-propriate starting point. In practice, to obtain the valuof rinfyrtot corresponding to finiteT , one could removea fraction of the infinite strings from a network generated using VV algorithms. This procedure was first implemented in [14] to study the scaling properties of strinnetworks. In practice it may miss important features othe actual equilibrium distribution, though.

Finally, our observation that the phase transition in thfields may coincide with the point where infinite stringstarts forming suggests that strings do indeed play a fudamental role in the critical behavior of the system, ahas been suggested so often in the literature [15–1Statistical studies of string networks [19], however, commonly identify the critical point in string densities withthe Ginzburg temperatureTG , Tc. Above we presentedevidence for the coincidence of the critical behavior in th

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fields and in the strings atTc. We hope to clarify thesequestions in a future publication, where we analyze thstring network’s scale dependence, through the propertof the string length distribution.

Our numerical studies were performed at the CraT3E of the Department of Computer Science, T. U. oBerlin. We thank A. Yates for his string tracing rou-tine. N. D. A. thanks E. Copeland for useful commentsL. M. A. B. is supported by theDeutsche Forschungsge-meinschaft. N. D. A. is supported by JNICT-ProgramaPraxis XXI, Contract No. BD/2794/93-RM. M. H. issupported by PPARC (U.K.), Grants No. B/93/AF/1642No. GR/L12899, and No. GR/K55967. This work waspartially supported by the European Science Foundation

[1] M. Hindmarsh and T. W. B. Kibble, Rep. Prog. Phys.58,477 (1995); A. Vilenkin and E. P. S. Shellard,CosmicStrings and Other Topological Defects(Cambridge Uni-versity Press, Cambridge, 1994).

[2] W. H. Zurek, Nature (London)317, 505 (1985); ActaPhys. Pol. B24, 1301 (1993).

[3] I. Chuanget al., Science251, 1336 (1991); M. J. Bowicket al., Science263, 943 (1994).

[4] P. C. Hendyet al., Nature (London)368, 315 (1994); inFormation and Interaction of Topological Defects,editedby A.-C. Davis and R. Brandenberger (Plenum, NewYork, 1995), p. 379.

[5] C. Bauerle et al., Nature (London)382, 332 (1996);V. M. H. Ruutuet al., Nature (London)382, 334 (1996).

[6] T. W. B. Kibble, J. Phys. A9, 1387 (1976).[7] W. H. Zurek, Phys. Rep.276, 177 (1996).[8] G. Kohring, R. E. Shrock, and P. Wills, Phys. Rev. Lett

57, 1358 (1986).[9] See, e.g., M. Creutz,Quarks, Gluons and Lattices(Cam-

bridge University Press, Cambridge, 1983).[10] Z. C. Le Guillou and J. Zinn-Justin, Phys. Rev. B21, 3976

(1980).[11] T. Vachaspati and A. Vilenkin, Phys. Rev. D30, 2036

(1984).[12] A. Yates and T. W. B. Kibble, Phys. Lett. B364, 149

(1995).[13] A. Achucarro, J. Borrill, and A. R. Liddle, Report

No. hep-ph/9702368.[14] P. G. Ferreira, Ph.D. thesis, University of London, 1995.[15] L. Onsager, Nuovo Cimento Suppl.6, 249 (1949).[16] R. P. Feynman, inProgress in Low Temperature Physics

edited by C. J. Gorter (North-Holland, Amsterdam, 1955Vol. 1, p. 17.

[17] V. Kotsubo and G. A. Williams, Phys. Rev. B33, 6106(1986).

[18] H. Kleinert, Gauge Fields in Condensed Matter Physic(World Scientific, Singapore, 1989).

[19] E. J. Copelandet al., Physica (Amsterdam)179A, 507(1991).

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