Predicting the critical density of topological defects in O(N) scalar field theories

4 Pages·0.19 MB·English

by  Nuno D. Antunes

Predicting the critical density of topological defects in O(N) scalar field theories.
Nuno D. Antunes1, Lu´ıs M. A. Bettencourt2 and Andrew Yates3
1D´ept. de Physique Th´eorique, Universit´e de Gen`eve, 24 quai E. Ansermet, CH 1211, Gen`eve 4, Switzerland
2T-6/T-11, Theoretical Division, Los Alamos National Laboratory, Los Alamos NM 87545,USA
3Centre for Nonlinear Dynamics and its Applications, University College, Gower Street, London WC1E 6BT, UK
(February 1, 2008)
O(N) symmetric λφ4 field theories describe many critical phenomenain thelaboratory and in the
earlyUniverse. GivenN andD≤3,thedimensionofspace,thesemodelsexhibittopological defect
classical solutions that in some cases fully determine their critical behavior. For N = 2, D = 3 it
has been observed that the defect density is seemingly a universal quantity at Tc. We prove this
conjecture and show how to predict its value based on the universal critical exponents of the field
theory. Analogously,forgeneralN andD wepredicttheuniversalcriticaldensitiesofdomainwalls
9 and monopoles, for which no detailed thermodynamic study exists. This procedure can also be
9 inverted,producingan algorithm forgenerating typicaldefect networksat criticality, incontrast to
9 the canonical procedure[1], which applies only in theunphysicallimit of infinitetemperature.
1
PACS Numbers: 05.70.Fh, 11.27.+d, 98.80.Cq UNIGE-TH-98-12-1023
n
a
J

9
O(N) symmetric scalar field theories are a class of defect densities at criticality. We will show that it leads
2
modelsdescribingthecriticalbehaviorofangreatvariety totheproofthatthecriticaldensityofvortexstrings,ob-

2 of important physical systems. For example, for N = 3 servedinrecentnon-perturbativethermodynamicstudies
v they describe ferromagnets, the liquid vapor transition of O(2), is a universalnumber. Among other insights [4]
1 and binary mixtures for N = 1 and superfluid 4He and thisshowsthatthephasetransitioninO(2)in3Doccurs
9
the statistical properties of polymers, for N = 2. In the when a critical density of defects is reached, connecting
3
early Universe N = 2 describes the phase transition as- directly the familiar picture of the Hagedorn transition
1
sociated with the breakdownof Peccei-Quinnsymmetry, in vortex densities to the more abstractcritical behavior
0
9 and models of high energy particle physics may belong ofthefields. WealsoextendourproceduretodifferentN
9 totheuniversalityclassofO(N)scalarmodels,whenever andD,makingpredictionsforthevaluesoftheuniversal
/ the mass of the Higgs bosons is larger that that of the densities of domain walls and monopoles, in 2 and 3D.
h
gauge bosons. O(N) scalar models are also invoked in Finallytheinversionofthisprocedureallowsustoeas-
p
- most implementations of cosmological inflation. ilygeneratetypicalfieldconfigurationsatcriticality. This
p One of the fundamental properties of O(N) λφ4 field is of fundamental practical importance. Recent experi-
e theories is the existence, for N D, of static no|n|-linear ments in 3He [5]andlargescale numericalstudies ofthe
h ≤
classical solutions, (domain walls, vortices, monopoles) theory[6]havelentquantitativesupporttotheideas,due
:
v that we will refer to henceforth as topological defects. toKibble[7]andZurek[8],thatdefectsformatasecond
i
X Atsufficientlyhightemperatures,topologicaldefectscan orderphasetransitionduetocriticalslowingdownofthe
be excited as non-perturbative fluctuations. Their dom- fields response over large length scales, in the vicinity of
r
a inance over the thermodynamics, due to their large con- the critical point. Defect networks hence formed have
figurational entropy, is known to trigger the phase tran- densities andlengthdistributions setby thermalequilib-
sitioninO(2)in3Dand2D,andtheir persistenceatlow rium at T =T+.
c
energies prevents the onset of long range order in O(2) In contrast most realizations of defect networks used,
D 2 and in O(1) in 1D. eg., in cosmological studies are generated using the
≤
It is therefore natural that the universalcritical expo- Vachaspati-Vilenkin[1] (VV) algorithm. It relies on lay-
nents characterizing the phase transition in terms of de- ing down random field phases on a lattice and searching
fects and through the behavior of field correlators must for their integer windings along closed paths. The abso-
beconnected. Thisconnectionismademorequantitative luterandomnessofthephasescorrespondstotheT
→∞
whenever one can construct dual models, field theories limit of the theory. More fundamentally it yields defect
which possess these collective solutions as their funda- networks that are quantitatively distinct from those in
mental excitations [2]. In the absence of supersymmetry equilibrium at criticality, i.e., at formation.
rigorousmappings between the fundamental models and Fig.1showsthe behaviorofasystemofvortexstrings
their dual counterparts exist only in very special cases at a second-order phase-transition, for O(2) in 3D. The
[2,3]. Duality has been suggested and empirically ob- datawasobtainedfromthestudyofthenon-perturbative
served to be a much more general phenomenon, though. thermodynamics of the field theory [9]. At T the total
c
In this letter we explore the duality between the criti- densityofstringρ displaysadiscontinuityinitsderiva-
tot
cal behavior of the 2-point field correlationfunction and tive, signaling a second order phase transition.
1
A disorder parameter can be constructed in terms of formulaallowsustocomputethedensityofvortexstrings
string quantities by dividing the string population into crossinganarbitraryplane in three dimensionalspace, a
long string (typically string longer than L2, where L quantity that is clearly proportional to ρ .
tot
∼
isthe sizeofthecomputationaldomain)andloops,com- The last key observation is that in the critical domain
prising of shorter strings. The corresponding densities of a second order transition, all O(N) theories are ef-
are denoted by ρ and ρ . In Fig. 1 we can observe fectively approximately Gaussian, but with non-trivial
long loop
that ρ consistently vanishes below T , except for a critical exponents. In particular renormalization group
long c
small range of β where it increases rapidly to a finite analysisshowsthatthemassandquarticcouplingvanish
critical value. In [9] we conjectured that in the infinite at T [12,13]. Higher order polynomial terms (eg. φ6)
c
∝
volume limit ρ exhibits a discontinuous transition. may be generated but are small. Hence in the critical
inf
domain the field two-point function can be written as
eik·x
G(x) dDk , (2)
∝ k−2+η
Z | |
where η << 1. is the universal critical exponent taking
into account deviations from the mean-field result.
Thus the effective Gaussianity of the theory allows us
to use Halperin’s result to compute the critical value of
ρ (β ). Notethat,modulorenormalization,thefinalre-
tot c
sult depends only on η establishing, as conjectured, that
ρ (β ) is a universal quantity.
tot c
Substituting, Eq. (2) into (1) we obtain:
η+1 k3+η k3+η
ρ max− min , (3)
tot ∝ η+3 k1+η k1+η
(cid:18) (cid:19) max− min
where we have introduced upper and lower momentum
cut-offs k and k . In the case of a lattice of size L
max min
and lattice spacing a we take k = 2π/L and k =
min max
2π/a which leads to
η+1 1 (a/L)3+η
FIG.1. Thestringdensities,total,loopsandlongstring,as ρ − . (4)
a function of inverse temperatureβ. Atβc,thedensities dis- tot ∝(cid:18)η+3(cid:19)1−(a/L)1+η
playderivativediscontinuities,signalingasecondorderphase
For large enough lattices, a/L<<1, and we obtain
transition. ρtot(βc) coincides for different studies, leading to
theconjecture that it is a universal number. η+1
ρ . (5)
tot
∝ η+3
The value of the total string density at β , ρ (β ) (cid:18) (cid:19)
c tot c
≃
0.20coincideswith resultsfromstudies ofdifferent mod- In order to generate quantitative predictions we need
elsinthe sameuniversalityclass[9,10]. This factleadus to determine the exact proportionality factor in Eq.(5).
to the conjecture [9] that ρ (β ) is universal. Thiscanbeachievedbyinvokingtheotherinstancewhen
tot c
In order to prove this conjecture we appeal to a the interacting theory becomes Gaussian. In the high
well known result, due to Halperin and Mazenko [11]. temperature limit β 0, the effective interaction be-
→
Halperin’s formula expresses ρ , the density of zeros of comes irrelevant. In terms of renormalization group the
0
a Gaussian field distribution in terms of its two-point theorydisplaysa(trivial)Gaussianfixedpoint,withvan-
function. For an O(N) theory the relevant quantity ishingeffectivecoupling. Onthelatticefieldsatdifferent
is the O(N) symmetric correlation function G(x) = points will be completely uncorrelated. Note that this
φ(0)φ(x)† , resulting in situation corresponds to the usual VV algorithm where
h i
a field is thrown randomly on the lattice and a network
G′′(x=0) N2 of strings is built by identifying phase windings. Fig. 1
ρ . (1)
0 ∝ G(x=0) shows the agreement of the densities observed at β = 0.
(cid:12) (cid:12)
(cid:12) (cid:12) with the well known VV result of ρtot = 1/3 with 75%
Eq. (1) measures the(cid:12)(cid:12)density of(cid:12)(cid:12)coincident zeros of all long string. Since the totally uncorrelated field corre-
N components of the field at a point. Coincident zeros spondstoaflatpowerspectrumG(k) k0 wenormalize
occur at the core of topological defects. Depending on Halperin’s expression by imposing ρ ∼=1/3 for η =2,
tot
N and D, coincident zeros can be interpreted as either
monopoles, strings or domain walls. In the particular 5 η+1
ρ = . (6)
case of a Gaussian O(2) theory in D = 3, Halperin’s tot 9 η+3
(cid:18) (cid:19)
2
η isalwaysmuchsmallerthan1. Settingη =0weobtain of size N3 with N = 16,32,64 and 128. All results
lat lat
ρ (T ) = 5/27 = 0.185, close to the value ρ 0.2 are averagesover 50 samples obtained from independent
tot c tot
≃
observed both in the λφ4 [9] and XY [10] studies in 3D. randomrealizations. Fig.2showsthestringdensitiesfor
A more accurate estimate can be obtained by replacing values of η between 0. and 0.1, including all reasonable
η by its precise value for the universality class to which values of η in 3D.
these theories belong. From [12] we have η 0.035 we
≃
get ρ =0.190, closer to the observed value.
tot
Asimilarexercisepermitsthecomputationofthecrit-
icaldensityofdomainwallsforO(1) andmonopolesina
O(3) theory at the critical temperature in 3D.
The density of domain walls per link is
1/2 1/2
1 5 η+1
ρ = . (7)
tot
2 3 η+3
(cid:18) (cid:19) (cid:18) (cid:19)
Note thevalue ofρ (β =0)=1/2correspondingtothe
tot
high-temperature limit. At β we get, with η = 0.034
c
[12], ρ (β ) 0.38. For monopoles we will take for the
tot c
≃
flat-spectrum case ρ (β = 0) 0.1. A better estimate
tot
≃
can be obtained from a tetrahedral discretization of the
sphere, resulting in ρ (β =0)=3/32 and
tot
3 5 3/2 η+1 3/2
ρ = . (8)
tot
32 3 η+3
(cid:18) (cid:19) (cid:18) (cid:19)
withη =0.038[12]leadingtothecriticalvalueρ (β )=
tot c
0.040. Finallyfordomainwallsin2D,thedensityperlink
at β is (using ρ (β =0)=1/2): FIG.2. The string densities from the 50 Gaussian realiza-
c tot
tions as a function of η for a lattice with Nlat = 64. Error
1 η 1/2 bars indicate standard deviation from themean.
ρ = . (9)
tot
√2 η+2
(cid:18) (cid:19) The values for the densities depend on the size of the
Taking η =0.26 [12] we obtain ρtot(βc)=0.24. lattice,convergingtofinitevaluesforlargeNlat. InFig. 3
The present procedure can be inverted to generate a wecanseethescalingofρtot withboxsizefortwochoices
typical defect network at criticality. The approximate of the critical exponent, the mean field value η =0. and
Gaussianity of the field theory at T implies that the the theoretical result for the O(2) universality class in
c
statistical distribution of fields, P[φ] is given by 3D,η 0.035. Wecanpredicttheformandthepowerof
∼
this scaling through Eq. (4). Writing a/L = 1/N , the
lat
P[φ]= e−β d3k12φ−kG(|k|)φk (10) numberofpointsinthelattice,andexpandingEq.(4)in
N powers of 1/N we see that Halperin’s result converges
R lat
This distribution can be sampled by generating fields as to its infinite volume limit according to:
φ =R(k) β−1G(k ) (11) 1
k | | ρtot(∞)−ρtot(Nlat)= N1+η +O(1/Nl2at) (13)
where R(k) is a randompnumber extracted from a Gaus- lat
siandistribution,withzeromeanandunitvariance. The To check these scalings we fitted the data of Fig. 3 to a
fieldcanthenbeFouriertransformedtocoordinatespace, power law of the form:
itsphasesidentifiedateachsite,andvorticesfoundinthe
A
standardway. Sincewewillbewillingtocompareresults
ρ (N )=ρ ( )+ (14)
from this algorithm with the ones measured in lattice tot lat tot ∞ Nα
lat
Langevinsimulations we chose to employ the exact form
For η =0 and η =0.035 we found:
of the field correlator on the lattice:
D 2−2η η =0.0, ρtot(∞)=0.1969, A=0.3259, α=1.060;
G(|k|)−1 =" 2(1−cos(ki))# ∼|k|→0 |k|2−η. (12) η =0.035, ρtot(∞)=0.2012, A=0.3422, α=1.124.
i=1
X These values of α are indeed close to 1, with a larger
We have performed several tests on the algorithm, by correction for η =0.035 as expected from Eq. (13).
comparing it to the results of the non-perturbative ther- In [9] for a lattice of size N = 100 we measured
lat
modynamics of the fields at criticality. We used lattices ρ (β ) = 0.198 0.004. For a Gaussian field with η =
tot c
±
3
0.035 we obtain ρ = 0.203 0.003. The agreement of performaprecisecomparisonhoweverHalperin’sformula
tot
±
the two results is very satisfactory. should be rederived for a field theory on the lattice. De-
spitetheseshortcomingsHalperin’sformulahasthemerit
of being the only analytic way of estimating the critical
densities of defects in theories where non-perturbative
thermodynamic results are scarce.
We have therefore established the connection between
the universalcritical exponent characterizing the behav-
ior of the O(N) field 2-point correlator and the critical
density of defects. This relation implies that defect den-
sitiesatT forasystemundergoingasecondorderphase
c
transitionareuniversalnumbers. We predicted them for
severalO(N)modelsin2and3D.Basedontheseinsights
we proposed a new algorithm for generating networks of
defects at the time of formation. In particular, we have
shown that this algorithm reproduces accurately all the
featuresofastringnetworkin3Datcriticality. Thispro-
cedure,instead of the more usualalgorithmof [1]should
be used to generate typical defect networks at the time
of their formation.
We thank R. Du¨rrer and W. Zurek for useful discus-
sions. ThisworkwaspartiallysupportedbytheESFand
by the DOE, under contract W-7405-ENG-36.
FIG. 3. The total string density for two values of η for
Nlat = 16,32,64 and 128 and respective fits to a power law.
Statistical errors are much larger than the deviation of the
points to thefits.
Theresultsforρ andρ usingthesetwodifferent
long loop [1] T. Vachaspati and A. Vilenkin, Phys. Rev. D 30, 2036
methodsarealsoingoodagreement. Inthiscasewewere
(1984).
not able to find a reasonable scaling expression though,
[2] H. Kleinert, Gauge Fields in Condensed Matter Physics
but this is to be expected given the arbitrariness of the
(World Scientific, Singapore, 1989).
long string definition. The results for N = 100, using
lat [3] R. J. Baxter, Exactly solvable models in statistical me-
η = 0.035 are ρ = 0.080 0.004, ρ = 0.121
long ± loop ± chanics (Academy Press, London, 1982).
0.004. These compare well with the non-perturbative
[4] G. A.Williams, cond-mat/9807338
resultsρlong =0.076 0.005,ρloop =0.120 0.004. Even [5] C.B¨auerleetal.,Nature382,332(1996);V.M.H.Ruutu
± ±
more impressive is that the string length distribution at et al.,Nature382, 334 (1996).
criticality can also be reproduced by our Gaussian field [6] N. D. Antunes, L. M. A. Bettencourt and W. H. Zurek,
algorithm. This distributioncanbe successfully fitted to hep-ph/9811426 and references therein.
an expression of the form [9], [7] T.W.B. Kibble, J. Phys. A 9, 1387 (1976).
[8] W. H. Zurek, Nature 317 505 (1985), Acta Physica
n(l)=Al−γe−βσl. (15) Polonica B 24 1301 (1993).
[9] N.D.Antunes,L.M.A.BettencourtandM.Hindmarsh,
The fit to the results of the Gaussian field algorithm Phys. Rev. Lett. 80, 908 (1998); N. D. Antunes and L.
shows a small variation of the parameters A,γ, and σ M. A.Bettencourt, Phys. Rev.Lett. 81, 3083 (1998).
for η 0. 0.1. σ is consistently zero,reflectingthe fact [10] G. Kohring,R.E.Shrockand P.Wills, Phys.Rev.Lett.
∼ −
that the spectrum is always scale invariant. The value 57, 1358 (1986).
of γ varies between 2.34 and 2.40. For the critical ex- [11] B.I.Halperin,inPhysics of defects,LesHouchesXXXV
ponent η = 0.035 we obtained γ 2.35. Once again NATOASI,Eds.Balian,KlemanandPoirier(NorthHol-
≃
this isingoodagreementwiththe resultfromthe lattice land) 816 (1981), F.Liu and G.F. Mazenko, Phys.Rev.
non-perturbative thermodynamics at T [9], γ 2.36. D 46, 5963 (1992).
c
≃
Finally the predictions for ρ from Halperin’s for- [12] J. Zinn-Justin, Quantum field theory and critical phe-
tot
mula, when compared to the accuracy of the Gaussian nomena, (Oxford Science Publications, U.K., 1996).
algorithm seem rather poor. The expression is meant to [13] C. Wetterich, in Electroweak Physics and the early Uni-
apply for continuum distributions, while all other values verse,Ed.J.C.Rom˜aoandF.Freire,Plenum(NewYork)
338 (1994) and references therein.
of ρ were obtained on the lattice. A straight substitu-
tot
tion of the lattice correlators (12) into (1) increases ρ
tot
to 0.21 from 0.19, covering our full range of results. To
4