dimensional Abelian Higgs model

35 Pages·2003·0.29 MB·English

NuclearPhysicsB666[FS](2003)361–395
www.elsevier.com/locate/npe
Kosterlitz–Thouless-like deconfinement mechanism
+
in the (2 1)-dimensional Abelian Higgs model
Hagen Kleinerta, Flavio S. Nogueiraa, Asle Sudbøb
aInstitutfürTheoretischePhysik,FreieUniversitätBerlin,Arnimallee14,D-14195Berlin,Germany
bDepartmentofPhysics,NorwegianUniversityofScienceandTechnology,N-7491Trondheim,Norway
Received17September2002;accepted16May2003
Abstract
We point out that the permanent confinement in a compact (2+1)-dimensional U(1) Abelian
Higgs model is destroyed by matter fields in the fundamental representation. The deconfinement
transitionisKosterlitz–Thouless-like.Thedualtheoryisshowntodescribeathree-dimensionalgas
of point charges with logarithmic interactions which arises from an anomalous dimension of the
gaugefieldcausedbycriticalmatterfieldfluctuations.Thetheoryisequivalenttoasine-Gordon-like
theory in(2+1)-dimensions withananomalous gradient energy proportional tok3.TheCallan–
Symanzikequationisusedtodemonstratethatthistheoryhasamasslessandamassivephase.The
renormalizationgroupequationsforthefugacityy(l)andstiffnessparameterK(l)ofthetheoryshow
thattherenormalizationofK(l)inducesananomalousscalingdimensionηy ofy(l).Thestiffness
parameterofthetheoryhasauniversaljumpatthetransitiondeterminedbythedimensionalityand
ηy. Asa byproduct of our analysis, we relate the critical coupling of thesine-Gordon-like theory
toanaprioriarbitraryconstantthatentersintothecomputationofcriticalexponentsintheAbelian
Higgs model at the charged infrared-stable fixed point of the theory, enabling a determination of
thisparameter.Thisfacilitatesthecomputationofthecriticalexponentν atthechargedfixedpoint
inexcellentagreementwithone-looprenormalizationgroupcalculationsforthethree-dimensional
XY model,thusconfirmingexpectationsbasedondualitytransformations.
2003ElsevierB.V.Allrightsreserved.
PACS:11.15.Ha;74.20.Mn;64.70.-p
Keywords:Gaugetheories;Confinement;Kosterlitz–Thoulesstransition
E-mailaddresses:[email protected](H.Kleinert),[email protected]
(F.S.Nogueira),[email protected](A.Sudbø).
0550-3213/$–seefrontmatter 2003ElsevierB.V.Allrightsreserved.
doi:10.1016/S0550-3213(03)00453-X
362 H.Kleinertetal./NuclearPhysicsB666[FS](2003)361–395
1. Introduction
Gauge theories in d =2+1 dimensions are often considered as effective theories of
stronglycorrelatedsystemsintwospatialdimensionsatzerotemperature[1–3].Prominent
examples of systems to which such theories are hoped to be applicable are the high-T
c
cuprates in the underdoped or undoped regime. In the undoped regime it is known that
spinorQED isaneffectivelowenergytheoryforthequantumHeisenbergantiferromagnet
3
(QHA)[1].Itishopedthatoneeffectivelycanaccountfordopingbycouplingthegauge
theorytoascalar bosonrepresentingtheholonpart(chargepart)ofcompositeHubbard-
operators describing projected electrons, which however do not satisfy simple fermion
commutationrelations.Similareffectivetheorieshavealonghistoryasusefultoy-models
inhigh-energyphysics[4–6],andhaverecentlybeensuggestedtodescribeneuralnetworks
[7].
Ofparticularinterestinthephysicsofstronglycorrelatedsystemsisthecompactversion
of the (2+1)-dimensional Abelian Higgs model with matter fields in the fundamental
representation.Thisisthe modelwe shallbe concernedwith inthispaperandforwhich
weshallfindtheresultssummarizedintheabstract.
1.1. Preliminaryremarks
OurstartingpointisthefollowingAbelianeuclideanfieldtheoryofascalarmatterfield
coupledtoamasslessgaugefield
(cid:1)(cid:2) (cid:3) (cid:1)
L =(cid:1) ∂ −iA0 φ (cid:1)2+m2|φ |2+ u0|φ |4, (1)
b µ µ 0 0 0 2 0
wherethesubscriptzerodenotesbarequantities.ItcorrespondstoatheorywithaMaxwell
term
1
L = F0 2, (2)
M 4e2 µν
0
where F0 =∂ A0 −∂ A0, in which the gauge coupling e goes to infinity. This limit
µν µ ν ν µ 0
↔
impliestheconstraintjbµ=0,wherejbµ=φ0∗ ∂ µφ0 isthebosoncurrent.
When deriving effective theories for the t–J model we arrive naturally at a compact
U(1) lattice gauge field [2]. For QHA, the gauge symmetry is larger and given by the
gaugegroupSU(2)[3].However,inthiscaseareducedU(1)formulationisalsopossible
[1]. Since this U(1) is a subgroupof SU(2), which is a compactgroup, the U(1) gauge
theoryofQHAisnecessarilyacompactAbeliangaugetheory.
It is well known that a compact U(1) theory of the pure Maxwell type in three
dimensionsconfineselectricchargespermanently[8].Intheliterature[9]itisalsoargued
that this permanent confinement should be present if an additional fermionic field ψ
coupledtothegaugefieldbyaLagrangian
(cid:4)N (cid:2) (cid:3)
L = ψ¯ ∂ −iA0 ψ . (3)
f i µ µ i
i=1
H.Kleinertetal./NuclearPhysicsB666[FS](2003)361–395 363
Thismeansthattheparticlesrepresentedbythefieldsψ andφ neverhaveanindependent
0
dynamics. In the context of many-body theory, the Dirac fermion ψ could represent a
spinon, while φ represents a holon. If electric test charges were permanently confined
in the model, then the spinon and the holon would only appear as composite particles.
In this case it would be impossible to fractionalize the electron, i.e. spin and charge
wouldalwaysremainattachedtoeachother.Spin-chargeseparationisknowntooccurin
1+1 dimensions [10]. There fermions can be transmuted into bosons via the so-called
Jordan–Wigner transformation. In 2+1 dimensions the situation is less clear, but for
matterfieldsinthefundamentalrepresentationthereisonecircumstancewherespin-charge
separationisknownrigorouslytooccur,namelythechiralspinliquidstate[11].However,
thestatisticsofparticlescanbechangedasin1+1dimensions.Inthechiralspinliquid,
spinons have anyonic statistics described by a Chern–Simons term [12] in the effective
gaugetheory,whichreflectsthebreakingofparityandtimereversalsymmetry.
The lack of consensus about spin-charge separation in (2+1)-dimensional compact
U(1) matter-coupledgaugetheorieswith matter fields in the fundamentalrepresentation
initiated investigations of other gauge theories for strongly correlated electron systems.
One of the most promising candidates is a Z gauge field coupled to matter fields
2
[13]. Similar ideas leading to electron fractionalizationhad earlier been presented in the
condensedmatterliterature[14,15].In2+1dimensionstheZ theoryhasadeconfinement
2
transition[5].Thus,Z gaugetheoriesarepotentiallygoodcandidatesfordescribingspin-
2
chargeseparationwithoutbreakingparityandtimereversalsymmetries.
The confinement properties of U(1) gauge theories for the cuprates and the relation
tospin-chargeseparationwererecentlydiscussedfromvariouspointsofview[9,16–19].
Nayak[9]states thatin gaugetheoriesof the t–J modelfermionsandbosonsinteractat
infinite(bare)gaugecouplingand,forthisreason,itisnecessarilyatheorywithpermanent
confinementofslaveparticles.Incontrast,IchinoseandMatsui[18]havearguedthatthe
couplingtomatterfieldsstronglyinfluencesthephasestructureofthesystem.InRef.[19],
it is correctly pointed out that if spin-charge separation occurs, it is not necessarily tied
to the notion of confinement–deconfinementof slave particles. The picture proposed in
Ref.[9]in2+1dimensionsisreminiscentof1+1dimensionswherespinonsandholons
are solitons and cannot be identified with the slave particles, which are not part of the
spectrum [10]. Nagaosa and Lee [17] discuss a compact U(1) gauge theory coupled to
bosonic matter field in the fundamentalrepresentation.They conclude that in d =2+1
this theory permanently confines electric charges, in contrast to the analysis by Einhorn
andSavitonthesamemodel[4].
In a recent letter [20], we have studied the confining properties of the Lagrangian
(1), as well as the case of a fermionic field ψ coupled to a gauge field, but with an
added Maxwell term. The Lagrangian (1) with a Maxwell term corresponds essentially
tothemodelconsideredbyNagaosaandLee[17],thoughtheseauthorshaveconsidereda
frozen-amplitudeversionofthemodel.InRef.[20],itwasemphasizedthatananomalous
scaling dimension of the gauge field, arising from matter-field fluctuations, changes the
interaction between monopolesfrom 1/r to lnr in three dimensions. It was then argued
that a monopole–antimonopole unbinding transition similar to the Kosterlitz–Thouless
(KT) transition takes place, but now in three dimensions. From this, we concluded that
testchargesundergoadeconfinementtransition.
364 H.Kleinertetal./NuclearPhysicsB666[FS](2003)361–395
It must be pointed out that the authors of Refs. [5,17], were looking for a transition
similar to those encountered in d =3+1, namely ordinary first- or second-order phase
transitions [5]. In Ref. [17], a duality transformation was performed showing that the
disorderparameter (cid:8)φ (cid:9) is always differentfromzero, implyingthat (cid:8)φ(cid:9) is always zero.
V
ThisresultisessentiallycorrectandisperfectlyconsistentwiththescenarioinRef.[20]
andexplainedfurtherinthepresentpaper.
A main result in our letter [20] is that there exists a non-trivial infrared stable fixed
point in the theory in d =2+1 which drives the deconfinement transition. There the
anomalousdimensionofthegaugefieldisgivenbyη =1ind=2+1[21,22].Thisresult
A
isexactasaconsequenceofgaugeinvariance.Itimpliesthatthenon-trivialinfraredfixed
point arises at an infinite bare gauge coupling. To see this, consider the boson–fermion
LagrangianL=L +L +L .Duetogaugeinvariance,thegaugecouplingrenormalizes
f b M
toe2=Z e2,whereZ isthewavefunctionrenormalizationconstantofthegaugefield.
A 0 A
The renormalization group (RG) β function for the renormalized dimensionless gauge
couplingα=e2/µhasthefollowingexactformin2+1dimensions
(cid:5) (cid:6)
∂α
β (α,g)=µ = γ (α,g)−1 α, (4)
α A
∂µ
where g is the renormalized dimensionless |φ|4 coupling and γ = µ∂lnZ /∂µ. Let
A A
us assume that there exist non-trivial infrared stable fixed points α∗ and g∗, where the
β functions β and β vanish. We have explained in Ref. [20] why such fixed points
α g
must exist. (For similar arguments, see Ref. [23].) Moreover, large-scale Monte Carlo
simulations have demonstrated explicitly the existence of such a non-trivial fixed point
[22,24] (see also Ref. [25]). Its existence has long been assured theoretically by duality
arguments [26,27] (see also Section 2.2). We shall not repeat the arguments and details
here.Instead,wefocusonthephysicalconsequencesofthenon-trivialfixedpoint.
Wewouldliketostressanimportantpoint,pertinenttod=2+1dimensions,andquite
differentfromthesituationford =3+1.Asα→α∗,thebarecouplinge2 musttendto
0
infinity.Bydefinition,theaboveβ functionisgivenatfixedΛ,α ,andg .Here,Λisthe
0 0
ultravioletcutoffwhile α =e2/Λ and g =u /Λ arethe dimensionlessbarecouplings.
0 0 0 0
Thefixedpointisreachedforµ→0.Alternatively,thefixedpointisreachedforΛ→∞
ifµisheldfixed.However,sinceα isfixeditfollowsthate2→∞asΛ→∞.Thus,in
0 0
d=2+1,thefixedpointtheoryisatinfinitebaregaugecoupling.Onemightobjectthat
thisinfinitegaugecouplingcannotberelevantforthecuprateswhichhaveaninfinitevalue
ofe2 atallscales,notonlyinthescaleinvariantregime.Thisistrue,butirrelevantasfar
0
asthedeconfinementtransitionisconcerned,whichisdeterminedbythenon-trivialfixed
point structure. The situation is analogous to the O(N) non-linear σ model as opposed
totheO(N) φ4 model.Thesemodelsarequitedifferent,butagreewitheachotheratthe
criticalpoint[28,29],thusbelongingtothesameuniversalityclass.Inourcase,themodel
withtheMaxwelltermatthefixedpointhasthesamecorrelationfunctionsasthemodel
withoutitalsoatthefixedpoint.
To summarize the discussion in the above paragraph, the non-compact action with
no Maxwell term has the same critical behavior as the compactone at the critical point
correspondingtoanon-trivialfixedpoint,characterizedbyaninfinitebarecoupling.Had
H.Kleinertetal./NuclearPhysicsB666[FS](2003)361–395 365
westartedfromaninfinitelyweakbarecoupling,theonlyfixedpointwewouldhaveany
hopeofreachingford=2+1wouldbetheGaussianfixedpoint.
In Ref. [20] we have pointed out that chiral symmetry breaking can destroy the
deconfinementinthefermioniccase.We wanttopointoutthatforthe combinedboson–
fermion model, L = L + L + L , chiral symmetry breaking does not spoil the
f b M
deconfinement transition. Chiral symmetry breaking occurs at a lower value of number
offermionflavoursN ,whenalsobosonsarepresent.KimandLee[30]claimedthatthe
f
criticalvalueofN isdecreasedbyafactortwo.SincewehavetypicallyNc ∼3andthe
f f
physicalnumberof fermioncomponentsin the cupratesis N =2, Kim andLee argued
f
thatspin-chargeseparationwouldoccuratfinitedoping[30].
1.2. Anomalousscalingandthepotentialbetweentestcharges
The high-energy physics literature is usually concerned with d = 4 and use low-
dimensionsonlyintoymodels.Incondensedmatterphysics,however,(2+1)-dimensional
gaugetheoriesare supposedto describereal physicalphenomenasuch as the anomalous
propertiesof high-T superconductors[31],or the physicsofQHA [1,32].For d ∈(2,4]
c
thegaugecouplingβ-functionmaybewrittenas
(cid:5) (cid:6)
β (α,g)= γ (α,g)+d−4 α. (5)
α A
Non-trivial fixed points induce an anomalous scaling behavior in the gauge field
propagator.IntheLandaugaugewehavethat
(cid:7) (cid:8)
p p
D (p)=D(p) δ − µ ν , (6)
µν µν p2
withthelargedistancebehaviorgivenby
1
D(p)∼ . (7)
|p|2−ηA
Theanomalousscalingdimensionisgivenexactlyby[21,22]
ηA≡γA(α∗,g∗)=4−d. (8)
Duetotheaboveresult,thepropagator(7)inconfigurationspacebecomes
1 1
D(x)∼ ∼ , (9)
|x|d−2+ηA |x|2
for all d ∈(2,4]. The potential between effective electric charges q(R), separated by a
largedistanceRin(d−1)-dimensionalspaceisgivenby
q2(R)
V(R)∼ , (10)
Rd−3
where
1−(ΛR)−ηA (ΛR)d−4−1
q2(R)∼ ∼ , (11)
η d−4
A
366 H.Kleinertetal./NuclearPhysicsB666[FS](2003)361–395
andwhereΛisashortdistancecutoff.TheanomalousscalinginEq.(11)isaconsequence
ofthecouplingtomatterfields.Duetoit,thepotentialV(R)behaveseffectivelylike1/R
ford =3.Ford=4,itgoeslike ln(ΛR)/R, whileford=2,ithasaconfiningbehavior
proportionalto R. The regime governed by the Gaussian fixed point has q2(R)=q2 =
0
const,andcorrespondstotheso-calledCoulombphase.Inthisphase,thefour-dimensional
theoryhasV(R)=q2/R,whereasV(R)=q2lnR ford =3.We seethatthenon-trivial
0 0
infrared behavior induces an effective electric potential between test charges similar to
that which characterizes the Coulomb phase in d =4. If we extrapolate to d =2, we
obtain V(R)=q2R. Note that in d =2, we obtain a confining potential irrespective of
0
whetheranomalousscalingistakenintoaccountornot.
In compact Abelian gauge theories a confined phase is realized by the formation of
electric flux tubes connecting electric charges. These flux tubes are the dual analogs of
themagneticfluxtubesconnectingmagneticmonopoles[33,34].ThereisaDiracrelation
betweentheeffectiveelectricandmagneticcharges
q(R)q (R)∼1. (12)
m
Letusconsidernowthepotentialbetweenthemagneticcharges
q2(R) 1
V (R)∼ m ∼ . (13)
m Rd−3 q2(R)Rd−3
From Eq. (11) we see that for d =4 the magnetic potential behaves like 1/[Rln(ΛR)].
However,ford=3wehave
1
V (R)∼ , (14)
m
R
whichisself-dualwithrespecttothepotentialbetweenelectrictestcharges.
TheHiggsphaseforthe electric chargescorrespondsto V(R)∼constbecauseofthe
gauge field mass gap. The Higgs phase for magnetic test charges, on the other hand, is
givenbyV(R)∼R.Intheelectric–magneticdualitypicture[33,34]thisHiggsphasefor
magnetic charges is exchangedby the confined phase for electric charges. This scenario
should be valid for matter fields in the adjoint representation. In the absence of matter
fields, a compact (2+1)-dimensional gauge theory is definitely confined permanently
[8]. The above result shows that if matter fields are present, a deconfined phase is also
possible.However,ifthematterfieldsareinthefundamentalrepresentation,thesituation
is controversial[4,9,17–20].Our recent results in Ref. [20] seem to be confirmedby the
MonteCarlo workin Ref. [6].Themain purposeof thispaperis to givemoredetails on
thescenarioproposedinRef.[20]andtodescribeatheoryforadeconfinementtransition
inAbeliangaugetheoriescoupledtomatterfieldsinthefundamentalrepresentation.
1.3. Outlineofthepaper
InSection2,weconsiderthelatticedualitytransformationstothe(2+1)-dimensional
AbelianHiggslattice(AHL)model,firstthenon-compactcaseandlaterthecompactcase.
Wethendiscussthepossibleordinaryfirst-orsecond-orderphasetransitionsthesemodels
canhave,withmatterfieldsinthefundamentalrepresentationforthecompactcase.
H.Kleinertetal./NuclearPhysicsB666[FS](2003)361–395 367
InSection3,weconstructthecontinuumeffectiveLagrangiananditsdualcounterpart
forthecompact(2+1)-dimensionalAHLmodelwhenmatter-fieldshavebeenintegrated
out.Becausethesearecentralresultsofthepaper,itbehoovesustoannouncethemhere.
The dual field theory is given by Eq. (51). It represents a description of a three
dimensional gas of point charges interacting with a logarithmic pair-potential, given
by Eq. (49). We emphasize that the 3d ln-plasma action of Eq. (49) emerges from an
underlyingmatter-coupledgaugetheory,Eq.(38),byintegratingoutthefluctuatingmatter
fields and considering the influence of critical matter fluctuations on the gauge-field
propagator. The result of this procedure is the effective theory Eq. (46). Such matter-
fieldfluctuationsendowthegauge-fieldpropagatorwithananomalousscalingdimension
η =4−d[21,22]whichinthree-dimensionsalterstheinteractionbetweenthemonopole
A
configurationsofthegauge-fieldfromaCoulomb-interaction1/RtoalnR interaction.
Recallthatincontrasttothis,intheclassictreatmentbyPolyakov[8]ofcompactthree-
dimensionalQEDwithnomatterfields,thestandardthree-dimensionalsine-Gordonfield
theory with a quadratic gradientterm, describing the three-dimensionalCoulomb gas, is
obtained.Thisactionisgivenby,inthenotationofEq.(51)
(cid:9)
(cid:5) (cid:2) (cid:3) (cid:6)
1
S = d3x ϕ −∂2 ϕ−2z cosϕ . (15)
SG 0
2t
Polyakov has demonstrated [8] that Eq. (15) has no phase transition, i.e., it is always
massive.OurEq.(51)differsdrasticallyfromEq.(15),duethepresenceofananomalous
gradientterm.
In Section 4.1,we show using the Callan–Symanzikequations,thatthe effectivedual
LagrangianEq. (51) has a massless and a massive phase separated at a critical coupling
t . Hence a phase transition must exist. This does not by itself suffice to show precisely
c
whatsort ofphase transition the system undergoes,nor doesit allow usto constructthe
correctflowdiagramofthecouplingconstantsoftheproblem.Itdoes,however,sufficeto
showthattwodifferentphasesexist.Sincethepropagatoroftheproblemislogarithmicin
d=2+1,aHohenberg–Mermin–Wagnertheorem[36]holds.Undersuchcircumstances,
it is very natural to conjecture that any phase transition in the system, if it exists, must
beofatopologicalcharacter.InSection4.2,weconstructtherenormalizationgroupflow
equationsfortheproblemandshowthatthephasetransitionisofaKT-liketype.
InSection5,weconsidertheconnectionbetweentherenormalizationgroupfunctions
obtaineddirectly from the Abelian Higgs model, and the KT phase transition we find in
Section4.Themainpointhereisthatwecanusethevalueofthecriticalcouplingofthe
dual effective Lagrangian for the topologicaldefects of the gauge field to fix an a priori
arbitrary constant which enters into evaluating critical exponents for the non-compact
AbelianHiggsmodel.
InSection6,weconcludewithasummaryandoutlook.AppendixAdiscussesanother
type of sine-Gordon theory also exhibiting a KT-like transition in three dimensions. In
AppendixB,wederivetheflowequationsforthestiffnessparameterandthefugacityofthe
systemdefinedbyEq.(49),andofwhichEq.(51)isafieldtheoryformulation.InAppendix
C,wecomputethescreenedeffectivepotentialbetweenchargesintheinsulatingphaseof
the3dln-plasma.InAppendixD,forcompleteness,wederivetheexactequationofstate
forad-dimensionalln-plasmawithnoshort-distancecutoff andrelatethesingularitiesin
368 H.Kleinertetal./NuclearPhysicsB666[FS](2003)361–395
thisplasma to theCallan–SymanzikapproachofSection4. InAppendixE, we consider,
alsoforcompleteness,thedualitytransformationoftheAHLmodelwithaChern–Simons
term added. This case is of interest in the fractional quantumHall effect [37] and chiral
spinliquids[11].
2. DualityintheAbelianHiggslatticemodel
In this section we review the duality approach to the AHL model. Although this is a
wellstudiedtopic[4,26,27,38,39],itisworthreviewingithereinordertoemphasizethe
differences and similarities between the non-compact and compact cases. In particular,
we shall discuss the extent to which these cases exhibit ordinary first- or second-order
phasetransitions.TheinterestingcaseincludingaChern–Simonstermwillbediscussedin
AppendixD.
Theessentialpointisthatstartingfromanon-compactorcompactAHLmodel,thedual
actionhasthegeneralform
(cid:4) (cid:4)
1
S = h M (r −r )h −i2π l ·h , (16)
dual iµ µν i j iν i i
2
i,j i
where h ∈(−∞,∞) and l are integer dual link variables. In the non-compactcase l
iµ i i
satisfytheconstraint
∇·l =0, (17)
i
whereasinthecompactcase,theright-handsideisnon-zero
∇·l =Q , (18)
i i
duetomonopolechargesQ ∈Z.Thesymbol∇ denotesthegradientvectoronasimple
i
cubiclatticeofunitspacingwithcomponents∇µfi ≡fi+µˆ −fi.
2.1. Thenon-compactcaseandthe“inverted”XY transition
Inthenon-compactcase,thepartitionfunctionoftheAHLmodelisgivenby
(cid:4) (cid:9)π(cid:10)(cid:11) (cid:12) (cid:9)∞(cid:10)(cid:11) (cid:12)
dθ
Z= i dA exp(−S), (19)
iµ
2π
{niµ}−π i −∞ i,µ
wheretheactionS isgivenbytheVillainapproximation
(cid:4) (cid:4)
β 1
S= (∇ θ −A −2πn )2+ (∇×A )2. (20)
2 µ i iµ iµ 2e2 i
i,µ i
Usingtheidentity
(cid:13)
(cid:4)∞ (cid:4)∞
2π
e(−t/2)m2+ixm= e(−1/2t)(x−2πn)2, (21)
t
m=−∞ n=−∞
H.Kleinertetal./NuclearPhysicsB666[FS](2003)361–395 369
followingdirectlyfromPoisson’sformula
(cid:9)∞
(cid:4)∞ (cid:4)∞
F(n)= dxF(x)e2πimx, (22)
n=−∞ m=−∞−∞
weobtain
(cid:9)∞(cid:10) (cid:12) (cid:14) (cid:10) (cid:12)(cid:15)
(cid:11) (cid:4) (cid:4)
1 1
Z= dAiµ δ∇·mi,0exp −2βm2ii+Ai·mi− 2e2(∇×Ai)2 .
−∞ i,µ {mi} i
(23)
The Kronecker delta in Eq. (23) is generated by the θ integrations. Now we should
i
integrateoutthe gaugefield A . The easiest way of performingthis integrationis by the
i
introductionofanauxiliaryfieldh suchthatthepartitionfunctioncanberewrittenas
i
(cid:9)∞ (cid:9)∞ (cid:9)∞(cid:10) (cid:12)
(cid:11) (cid:4)
Z= dA dh db δ(∇·b )
iµ iµ iµ i
−∞−∞−∞ i,µ {Mi}
(cid:14) (cid:10) (cid:12)(cid:15)
(cid:4) 1 e2
×exp − b2+iA ·(b −∇×h )− h2+2πiM ·b , (24)
2β i i i i 2 i i i
i
whereasummationbypartshasbeendonetoreplaceh ·(∇×A )byA ·(∇×h ),and
i i i i
we haveused the Poisson formula(22)to replacethe integervariablesm by continuum
i
variables b , at the cost of an additional sum over integer variables M . We may now
i i
integrate out A to obtain a delta function δ(b −∇×h ), after which also b can be
i i i i
integratedoutb ,yielding
i
(cid:9)∞(cid:10) (cid:12) (cid:14) (cid:10) (cid:12)(cid:15)
(cid:4) (cid:11) (cid:4) 1 e2
Z= dh exp − (∇×h )2+ h2−2πiM ·(∇×h ) .
iµ 2β i 2 i i i
{Mi}−∞ i,µ i
(25)
Summing the last term in the exponent by parts and going over to integer variables
l =∇×M ,weobtain
i i
(cid:9)∞(cid:10) (cid:12) (cid:14) (cid:10) (cid:12)(cid:15)
(cid:4) (cid:11) (cid:4) 1 e2
Z= dhiµ δ∇·li,0exp − 2β(∇×hi)2+ 2 h2i −2πili·hi .
{li}−∞ i,µ i
(26)
Note that the Kronecker delta constraint above is a direct consequenceof our change to
integer-valuedvariables.Ifh isintegratedoutweobtain
i
(cid:10) (cid:12)
(cid:4) (cid:4)
Z=Z0 δ∇·li,0exp −2π2β liµD(ri−rj)ljµ , (27)
li i,j,µ
wheretheGreenfunctionGhasthelarge-distancebehavior
√
e− βe|ri−rj|
D(r −r )∼ . (28)
i j 4π|r −r |
i j
370 H.Kleinertetal./NuclearPhysicsB666[FS](2003)361–395
The factor Z in Eq. (27) corresponds to the partition function of a free massive gauge
0
bosontheory.
Eq.(27) is the dualrepresentationof the partitionfunctionforthe non-compactAHL
model.Duetotheconstraint∇·l =0,theintegerlinksl formclosedloops.
i i
Bytakingthelimite→0inEq.(23),weobtain
(cid:7) (cid:8)
(cid:4) (cid:4)
1
Z|e=0= δ∇·mi,0exp −2β m2i , (29)
{mi} i
which is the loop gas representation of the XY model. If, on the other hand, we take
the limit β → ∞ in Eq. (27), we obtain the loop gas representation of the “frozen
superconductor”[38]
(cid:7) (cid:8)
(cid:4) 2π2(cid:4)
Z|β=∞= δ∇·li,0exp − e2 l2i , (30)
{li} i
whichhaspreciselythesameformasinEq.(29).Therefore,theXY modelisequivalent
to thefrozensuperconductor,providedthe Dirac-likerelation e2=4π2β holds.Eq.(27)
isareformulationofEq.(19)intermsofthetopologicaldefectsofthemodel,whichare
identifiedasinteger-valuedvortexstringsformingclosedloops.
If we consider the phase diagram in the (e2–T)-plane (with T =1/β), we can use
Eqs. (29)and (30)to establish the criticalpointson the axese2 and T, correspondingto
T →0ande2→0limits,respectively.FromEq.(29)weseethatwhene2→0wehave
a XY critical point on the T-axis. Eq. (30) has exactly the same form as Eq. (29), but
correspondstothe T →0 limit. Thecriticalpointinthislimitisthereforee2=4π2/T ,
c c
with T being the critical temperature of the XY transition as described by the Villain
c
approximation. This is the so-called “inverted” XY transition (IXY) [26]. From the
existence of these two critical points we can establish a phase diagram where there is a
criticallineconnectingthem[26].Theorderedsuperconductingphasecorrespondstothe
region0<e2<e2.
c
2.2. Thecompactcaseandtheabsenceofanordinaryphasetransition
InthecompactAHLmodelthegaugefieldA ∈[−π,π].ThecorrespondingVillain
iµ
actionisnowgivenby
(cid:4) (cid:4)
(cid:16)S= β (∇ θ −A −2πn )2+ 1 (> ∇ A −2πN )2, (31)
2 µ i iµ iµ 2e2 µνλ ν iλ iµ
i i
and in the partition function we should sum over both integers n and N . Using the
iµ iµ
identity(21)weobtain
(cid:4) (cid:4) (cid:9)π(cid:10)(cid:11) (cid:12)(cid:9)π(cid:10)(cid:11) (cid:12)
dA dθ
Z= iµ i exp(S(cid:24)), (32)
2π 2π
{ni} {mi}−π i,µ −π i