THERMODYNAMIC BETHE ANSATZ IN RELATIVISTIC MODELS: …cftconf.itp.ac.ru/seminars/AlZamolodchikov/NPB_1990_342_00695.pdf · Nuclear Physics B342 (1990) 695—720 North-Holland THERMODYNAMIC

NuclearPhysicsB342(1990)695—720North-Holland

THERMODYNAMIC BETHE ANSATZ IN RELATIVISTIC MODELS:SCALING 3-STATE POTFS AND LEE-YANG MODELS

Al.B. ZAMOLODCHIKOV

Instituteof Theoreticaland ExperimentalPhysics,117259,B. Cheremushkinskaya25,Moscow, USSR

Received8 January1990

Two integrable2D relativistic field theorymodelsarestudied by the thermodynamicBetheansatzmethod.One of them describesthe scalinglimit T —~ 1~of the 3-statePottsmodel andthe other correspondsto the scaling region nearthe Lee—Yang singularity of the 2D Isingmodel. The finite volume groundstateenergyof these two theories is calculatednumericallyusing the integral equationsof the temperatureBetheansatzapproach.Numericalresultsarecomparedwith the perturbationsnearthe correspondingconformaltheories.This allows us torelatethe massscalesof the theoriesto their dimensionalcoupling constants.

1. Introduction

The critical point of the 3-statePottsmodel (for definition seee.g. ref. [11)isdescribedby the minimal model ~.%‘(5/6)of conformal field theory(CFT) [2,31.This conformal model correspondsto the Virasorocentralcharge

c=4/5 (1.1)

andthe following tableof primary field dimensions

3 13/8 2/3 1/8 0

7/5 21/40 1/15 1/40 2/5 1 22/5 1/40 1/15 21/40 7/5~ ( . )0 1/8 2/3 13/8 3

The scalar field Z = ‘~(2~) with dimensions(2/5,2/5) is relevantandcorrespondsto the temperaturedeformationof the critical theory [3]. This meansthat the

perturbationof thecritical fixed pointactionresultsin a field theorythat describesthe scaling limit T—s ~ of the Potts model. It seemsnatural to denotethis fieldtheory as [~á~(5/6)]~2,~, where the content of the squarebracketsrefers to theshort-distanceCFT and the index indicates the relevantscalarperturbation.To

0550-3213/90/$03.50©1990 — Elsevier SciencePublishersB.V. (North-Holland)

696 Al.B. Zamolodchikov/ ThermodynamicBetheansatz

avoid this somewhatoverequippedsymbol in this paperwe shallcall this theorythescaling Potts model (SPM). ConventionallySPM may be defined through theaction

AspM=A~(S/6)+Af~(x)d2x. (1.3)

The coupling constantA here hasdimensionA (mass)6”5.

It wasshownin ref. [4] that SPM is an integrablefield theory, i.e. it possessesaninfinite seriesof commutingintegralsof motion. SPM developsa finite correlationlength R~and therefore its spectrum is massive. In a theory with massiveexcitationsthe integrability impliesthe factorizationof their scattering(see e.g. ref.[5]). The structureof integralsof motion discoveredin ref. [41stronglysupportsthefollowing scatteringtheory, suggestedearlier in [6]. The particlespectrumconsistsof a Z

3 doubletof massiveparticlesA, A of the samemassmA.The antiparticleAcanbe viewed as a bound stateof two particlesA (since it shows up as a pole inthe correspondingscatteringamplitude)and vice versa.All the scatteringampli-tudesareexpressedin termsof two-particleamplitudes

A(J31)A(f32))~=S.~..,~(f31—p2)~A(I31)A(f32)>0~~,

A(Pl)A(f.2))Ifl=SA~(f3l—f32)~A(/3l)A(P2))OUt, (1.4)

where /3~and /~2 are rapidities of colliding particles. Note the absenceofreflection in AA scattering.The factorizedscatteringtheorywithout reflection iscalled purely elastic. Multiparticle amplitudesin this caseare simply productsofpairwisetransitionones [4,61

sinh(13/2+ i’i~-/3) sinh(/3/2+ in-/6)S~(/3)= sinh(/3/2—i~/3)’ S~(/3)= sinh(/3/2—i~/6) (1.5)

The scatteringtheorydescribedcontainsall the on-mass-shellinformation aboutSPM. It seemsinterestingto studythis relativistic field theory off-mass-shell,inparticularto makea connectionbetweenthe mass-shellregion andthe ultraviolet(UV) limit governedby the CFT ~%‘(5/6). In this paper the finite size effectsin SPM are consideredby meansof the thermodynamicBethe ansatz(TBA)method[7—121.

The relativistic versionof this approachgives information aboutthegroundstateenergyof the relativistic integrablefield theoryplaced in a box of finite length Rwith periodicboundaryconditions,i.e. on a circle.The ground stateenergyE(R) isa function of the circle circumferenceR. From dimensionalargumentsit is clearthat RE(R) is a function of the scalinglength r = R/RC. We define the groundstatescaling function F(r) as E(R)= 2v-F(R/R~)/R.Off-mass-shelleffectsbe-

A1.B. Zamolodchikov/ ThermodynamicBetheansatz 697

comeimportant if R is comparableor less than the correlationlength R~‘~ 1/rn,wherern is the massscaleof the theory. In the following considerationswe shall

always specifythis quantityas the inversemassm0 of the lightestparticle in thetheory. Thus the scalinglength is r = rn0R and

2~rE(R)=~j~~~F(rnoR). (1.6)

The limit R —s 0 correspondsto the UV limit and is described by CFT. InparticularE(R) behavesas E(R) ~rc/6R in this limit [13], where c is the CFTcentral charge.

From the space-timepoint of view this finite box theory lives on an infinitecylinderbasedon a circle of circumferenceR. In euclideanrelativistic field theoryonecanalternativelytreatthis geometryas the theory in an infinite spacevolumebut at finite temperature1/R. The infinite volume thermodynamiceffectscanbetreated completely in terms of on-mass-shelldata. In the case of factorizedscatteringtheory the finite-temperaturestatesare describedby the Bethe wavefunction.This permitsoneto reducethe problemto a systemof nonlinearintegralequations(the TBA equations),the contentof the scatteringtheorybeing encodedin their form. Equationsof this type were first consideredby Yang and Yang [7]andsuccessfullyused later in thermodynamicsof integrablelattice systems[8—121.

In sect. 3 the TBA equationsare specified for the caseof scatteringtheoryofSPM. It turns out that in the chargesymmetriccasetheseequationsareexactlythesameas the TBA equations,correspondingto the S-matrix of the field theorydescribingthe scalingregion nearthe Lee—Yangsingularityof the 2D Ising model[14,15].

Critical behaviorin the Lee—Yangsingularityis describedby the minimal model4’(2/5) [161.This model is nonunitaryandcorrespondsto the central charge

c= —22/5. (1.7)

The table of primary field dimensions

0 —1/5 —1/5 0 (1.8)

containsonly two primary fields: the unity operatorI and the scalar field ~ withdimensions(— 1/5, — 1/5). According to the conventionssuggestedabove thefield theorycorrespondingto perturbationof the conformal theoryby operator~is denotedas [.%~2/5)](1 2) or [~~2/5)](’,3). In thepresentcasethis generalnotationisredundantbecause~ is the uniquerelevantperturbationof .%‘(2/5). We shallcall


this field theory the scalingLee—Yangmodel (SLYM). The correspondingconven-tional action is

A —A +SLYM .4’(2/5) gjça~~x,uX,

wherethe couplingconstantg has dimensiong (mass)’2”5.In ref. [17] it was shown that this field theory is integrable and its particle

spectrumconsistsof a single massiveparticle B with mass mB. The two-particleamplitudeexhibitsa pole correspondingto fusion BB —s B —s BB andhasthe form

sinh(/3) + i sin(~/3)SBB(/3) = . . . . (1.10)

sinh(j3) — z sin(~-/3)

The scatteringtheory is nonunitaryandthe pole hasthe wrong residue.In sect. 2 we presentbriefly the TBA method for relativistic systems in the

simplest case of purely elastic scattering. In sect. 3 the TBA equationsarespecifiedfor the scatteringtheoriesof SPM andSLYM. In the chargesymmetriccasethis leadsto the samesingleintegralequation.This equationis studiedin thelow- and high-temperaturelimits. In particular, the nonperturbativebulk vacuumenergyis extractedexactly from this equation.In sect. 4 the equationis investi-gated numerically. The ground state scaling function is calculatedwith highaccuracy.Also we estimateseveralcoefficientsin its high-temperatureperturbativeexpansion.In sect. 5 thesecoefficientsare comparedwith perturbationsof SPMandSLYM in coupling constantsA and g, respectively.This providesuswith thenumericalrelation betweenthe dimensionalcouplingconstantsandmassscalesinthesetwo theories.With the conventionalCFT normalizationof fields in (1.3) and(1.9) we obtain

mA— (4.504307863...)A5”6,

m~=(2.642944662...)(—ig)5”2. (1.11)

2. TBA for relativistic purely elastic S-matrix

We start by consideringa relativistic field theory in toroidal geometry,having inmind to pass to a cylinder as a limiting caseof the torus. We take a flat torusgeneratedby two orthogonalgeodesiccirclesC andB of circumferenceR and L,respectively(fig. 1) andfollow a cartesiancoordinatesystemwith the x-axis alongthe contour C and the y-direction parallel to B. There are two topologicallydifferentwaysto developthe hamiltonianapproachto this situation.On the one

Al.B. Zamolodchikov/ ThermodynamicBetheansatz 699

Fig. 1. Flat torus generatedby two orthogonalgeodesiccircles C and B of circumferenceR and L,respectively.

handonecanconsiderthe field theorystateson circle C. Denotethe correspond-ing spaceof statesas -c’. The y axis plays the role of time directionand statesare

evolvedby the hamiltonian

Hcp~f~~~dx, (2.1)2’~rc

whereT~is the stresstensorof the theory. The momentum

Pc1~~ ~-fT~~dx (2.2)

is quantizedand its eigenvaluesare 2irn/R with n integer. Alternatively, thecontour B can be chosen as a quantizationsurface and the evolution of thecorrespondingspaceof states~ in the —x direction (we take —x as the timedirection to savethe frameorientation)is describedby hamiltonian

HB= p(B) ~—f1~~dy. (2.3)

The spacemomentum

1~ ~fTdy (2.4)

haseigenvalues2~,-n/L,n E Z in space ~.

In the following we considerthe limit L —s ~, L >> R. In this limit the partitionfunction of the field theoryZ(R,L) is dominatedby the groundstateof I~I~withthe ground stateenergyE(R)

Z(R,L) ~ (2.5)

700 ALB. Zamolodchikov/ ThermodynamicBetheansalz

On the otherhand

Z(R,L) = tr[e~h’B]. (2.6)

In the thermodynamiclimit L —s onehas

log Z(R,L) ‘~ —LRf(R), (2.7)

where f(R) is the bulk free energy of the systemon B at temperature1/R. Thisgives the relation

E(R)=Rf(R). (2.8)

Due to the translational invariance of the torus the one-point functions arecoordinateindependent.In particular

E(R) dE(R)KTVY) = 2~- R KTXX~= 2~- dR (2.9)

andthe traceof the stresstensor is

dKT~’)=KT~+T~)=21TRdR[RE(R)]. (2.10)

In the spaceparity invariant theorywith nondegenerategroundstatewe also have

= 0. (2.11)

Consider now more closely the structure of the space ~ in a theory withfactorizedpurely elasticscattering.The TBA equationsarise in the limit L >> R~,where the off-mass-shelleffectscan be neglected.To avoid irrelevant complica-tions we considerfirst the simplestscatteringtheorywith a singleneutralparticleof mass rn anda pair scatteringamplitude S(j3, — /32)~The rapidities /3, and I~2ofparticlesparameterizetheir on-shell energiesandmomenta

e/13) =mcoshf3,, p,(p) =msinh/3,. (2.12)

Amplitude S(/3) satisfiesunitarity

S(/3)S(—/3) = 1 (2.13)

andcrossingsymmetry

S(f3) =S(i7r—f3). (2.14)


The Bethewavefunction. In relativistic theory the wave function formalism isinappropriateto describea systemof relativisticparticles(this is dueto virtual and

real particle creation). In the configuration space,however, there are regionswhere we have a set of relativistic particles strongly separatedin their spacepositions x1. More specifically we take Ix~—x31 >> R~.With a short range interac-tion, in theseregionsthe particlesmove as free onesandoff-mass-shelleffectscanbe neglected.In theseregions,which we call free regions,we cantalk about thespacecoordinatesx, andmomentap, of particlesandintroducethe wave function

In the factorized picture the numberof particlesN is the samein all the freeregions.The set of their momentap,, i = 1,2,.. ., N, is also the sameandthe wave

function is the Bethewave function. We denotea free region as {i,,i2,.. .,iN) if

The transitionbetweentwo adjacentfree regionspassesthroughconfigurationswhere two or more particlesare close to eachother. In theseconfigurations,of

course,the relativistic effectsare essentialandwe cannotuse the wave function.The scatteringtheory, however,providesconditionsto match wave functions inadjacentfree regions. In the purely elastic caseevery transition, say {i~,...,~

i.,,+,,.. ~ —s{i,,. ~ ii,,. ..,iN}, results in multiplication of the wave func-tion by the correspondingscatteringamplitude, ~(1~,— f3~) in this case.Notethat for /3 real this amplitudeis anunimodularnumber

S(j3) = ~ (2.15)

with realphase~(f3).Thesematchingconditionslead in particularto the quantizationequationsfor

the momentap., i = 1, 2, . . . , N, of N particlesin a periodicbox of length L >> R~

e’~~i’flS(/3~_f31)=1; i=1,...,N (2.16)j~i

or

mLsinhf3,+ ~,6(f3~—f31)=2’n-n1 (2.17)j#i

with N integer numbers n,. This system of transcendentalequations selectsadmissiblesetsof rapidities(/3, f~N)in free regions{i1,. . . , ~N}• The energyandmomentumof the state(/3,,. . . , /~N) are

N N

HB= Erncoshi31, ‘~B Ernsinhl3,. (2.18)i=1 1=1


From eqs.(2.17) and(2.13) it is readilyseenthat 2’n-PBL is an integernumber,as

it shouldbe in a periodicbox.Additional selection rules on rapidity sets (/3,,. .., /3k) are to be taken into

accountif the particles are identical. The Bethe wave function should be sym-metrizedor antisymmetrizeddependingon their statistics.The unitarity condition(2.13) specifiesthat S2(0)= 1. Two different casesarepossible:

(a) 5(0) = —1. (2.19)

If thereare two identical particleswith the samerapidities,the wave function isantisymmetric in their coordinates.This is incompatible with Bose statistics.Thereforein the bosoniccasesuchstatesshouldbe excluded.In this sensebosonsbehave like fermions: each value of rapidity can be occupied by at most oneparticle. It implies in particularthat all the integersn

1 in (2.17) aredifferent.Weshall denotethis situation in the Bethe ansatzequationsas “fermionic”. On theotherhand,if identicalparticlesare fermions, thestateswith coinciding rapiditiesare allowedandfermionscanoccupyeachrapidity valuein any number.This casewill be referredto as “bosonic”.

(b) S(0)=1. (2.20)

In this casethe situation is inverted.Bose particlesoccupyeachrapidity value inany numberandfermionsbehavelike fermions.

Eq. (2.17) is a complicatedsystemof transcendentalequations.The situationyields to analysesin the thermodynamiclimit L —÷ ~. The numberof particlesinthermodynamicstatesgrows L andas L —s becomesvery large.The spectrumof rapidities, determinedby eq. (2.17), condensesand the distancebetween

adjacent levels behavesas ~ — 13~±, 1/rnL. It makes sensein this limit tointroducea continuousrapidity densityof particlesp,(/

3). Taking a small rapidityinterval ~/3with n particlesinside, onedefines

p,(f3) =n/~/3. (2.21)

This smooth function of /3 is independentof a choice of interval L1f3 while1/mL <<~lf3<<1. The phasesum in eq. (2.17) is nearly constantwhen varyingfrom one /3~to the next f3,~,and can be estimatedas an integral. Eq. (2.17)acquiresthe form

mLsinhf3~+ f~(/3~—13’)p,(/3’) d/3’ = 2i~-n1

andcan be consideredas the equationfor rapidity levels, definedas solutions tothis equationfor all integernumbersn on the r.h.s.butnot only n1 corresponding

ALB. Zamolodchikov/ ThermodynamicBetheansatz 703

to actualstate.The situation is analogousto that for a systemof free particles,where the set of allowed levels is determinedby the one-particlequantizationcondition andone talksaboutoccupiedand free levels. The only differencefromthe free case is that now the set of levels is organizedself-consistentlywith theparticle distribution. Introducing the level densityp(f3) we arrive at the integralequation

2n-p(f3) = rnL cosh /3 + fq(/3 — f3’)p,(f3’) d/3’, (2.22)

where

q~(/3)=~6(/3)/~13. (2.23)

The energy(2.18) of the systemnow reads

HB=fmcosh/3p1(/3)d/3. (2.24)

It should be realized that in the thermodynamiclimit a large number ofquantumstatescorrespondto everyconsistentpair of densitiesp and p,. Considera partition of the rapidity axis in small intervals Z113a ~ 1. If also ~f3~~>>1/rnL,there is a large numberNa ~‘p(/

3a)~11~a of levels (note that p,p, ~L) in eachinterval and about na pt(f3a) ~f~a particlesare distributedbetweenthem. Theaverageddensities are not sensitive to local redistributionsof particles amonglevels.The numberof different distributionsin the interval ~lf3a amountsto

(N)!a (2.25)

(na)!(Na—na)!

in the “fermionic” caseand

(N +n —1)

!

a a (2.26)(Na~1)!(fla)!

in the “bosonic” case.The limiting behavior as L ._* ~ of the numberof states.i17(p, p,), correspondingto given consistentdensitiesp andp,, is estimatedby theentropy ,~/(p,p

1) = log .~‘K(p,p,). While 1/rnL ~ zlf3a ~ 1 the number ,.Y(p,p,)is correctlyestimatedas the productof numbers(2.25) or (2.26)over intervals ~Pa

andthe entropyis

~Fermifd/3[Pl0gPP,b0gPl(PP,)l0g(PPl)] (2.27)

704 A/B. Zamolodchikov/ ThermodynamicBetheansatz

in the “fermionic” caseand

~BosefdP[(P+P,)l0g(P+Pl) —plogp—p1logp,] (2.28)

in the “bosonic” one.With the entropytaken into accountthe summationoverstatesin (2.6) reduces

to minimization of the free energy

—RLf(p,p,) = —RHB(,o,) +~/(p,p,) (2.29)

in the macroscopiccharacteristicsp and p,, constrainedby the dynamicalrelation(2.22). It is convenientto introduce“pseudoenergy”E(/3) as

p, e~ p,= ; e” = (fermionic case), (2.30a)

p 1+e p—p,

p, e~ —. m= ; e = (bosoniccase). (2.30b)

p 1—e p+pi

In this notationthe extremumconditiontakesthe form

df3’—Rmcosh/3 + s(/3) + fw(/3 — /3’)log(l + e~”

3’~)—= 0 (fermioniccase),2~r

(2.31a)

df3’— Rmcosh/3 + ~(/3) — Jq~(p— /3’)log(l — e”~) -i--— = 0 (bosoniccase).

(2.3ib)

It turns out that densities p and p, enterthe expressionfor the extremalfreeenergyf in the ratio p,/p only for

dJ3Rf(R) = ~rnf cosh/3log(1±e~Y~—, (2.32)

wherethe uppersignrefersto the fermioniccaseandthe lower sign to the bosonic

case.In the moregeneralcasethereareseveraltypes of particlesAa, a = 1,2 M

with massesma. The purely elastic scatteringtheory is describedby a symmetricM xM matrix of pair transitionamplitudes5ab(f~)’eachsatisfyingthe unitarity


condition (2.13) (if there are charged particles, the crossing symmetry relationsmay havea slightly morecomplicatedform thaneq.(2.24)). In the TBA approach

oneconsidersM level densitiesp(a)(p) and M particle densitiesp(,a)• Eq. (2.22)turns into a systemof integral equations

maL bp(a)= —coshf3 +4~ah*P1~, (2.33)2~-

where ‘Pab is the symmetricmatrix kernel

d‘Pab(/3) = —j~j-log Sab(13) (2.34)

and * in (2.33) denotesthe convolution

df3’* p, = fq,(/3 - f3’)p,(/3’) —. (2.35)

The extremumequation(2.31) becomesa systemof nonlinearintegral equationsfor M pseudo-energies~a~I~)

ep(ja)( /3) = 1 ±~ p(a)( ~3), (2.36)

wherethe uppersignin the denominatoris chosenfor particlesAa of “fermionic”type (this dependsbothof their statisticsandthesign of the amplitude ~aa(0)) andthe lower sign refersto particlesof “bosonic” type. Introducingquantities

La(13) = ±log(1±e~”~), (2.37)

where again the upperand lower signs correspondto particlesof fermionic andbosonictypes respectively,we canwrite down the TBA equationsin unified form

M

—maRcosh/3+Ea+~co~6*L6_0. (2.38)

The bulk free energyf, and therefore the ground state energy in space ~ aregiven by the formula

M d/3E(R) =Rf(R) = — ~ majLa(13)coshl3—. (2.39)

a=I


Note that comparisonof eqs. (2.33) and(2.39) leadsto the useful relation

L 3e (/3)p(a)(f3) = a (2.40)2~- aR

The TBA expressionsfor the stresstensorexpectationvalueshavethe form

2~-M

K1’XX) = 7 a~imafp(,a)( /3)cosh/3 d/3 (2.41)

and

M ~ 9e (/3) 1 ôa~(/3)

(Tx) = a~,ma1 ±~ ( 3R cosh /3 — a sinh d/3. (2.42)

Introducingthe solutions ~ and ~1i~±~of the linear integral equations

M e~h

= mae~ + b=I ~Pat,* 1 ±e~h (2.43)

we get expression(2.42) in the form

1 M e”~~= ~ a>~i~fl~~1 ±~ ~~(/3)e~ + ±~(/3)e~)d/3. (2.44)

Theseformulas areuseful in the high-temperaturelimit of TBA equations.

3. TBA for SPM and SLYM scattering theories

Considerfirst the SPM scatteringtheorywith the doublet of chargedbosonsAand A. We introduce the correspondingpseudoenergiesSA and t~A. The matrix

kernelin the integral (2.38) in thiscasehasthe following entries

13) = 4°AA(~ = — 2 cosh /3 + 1

‘P,~(/3) ~ = — 2coshf3—1 (3.1)

Note that

df3 1 df3 2J4OAA(13)~ = ~ fIPAA(13)-~—= ~-. (3.2)


For the scattering theory under considerationthe Bethe ansatz is of thefermionic type andTBA equationsread as

mARcosh/3+EA+’P~* LA+’P,~*LA=O,

_mARcoshP+eA+’P,~*LA+’PAA*LA=O (3.3)

where

L~=log(1+e~A); L~=log(1+e~~). (3.4)

The groundstateenergyis

d/3E(R)= —mAfcosh/3(LA+LA)~-----. (3.5)

Here we shall consideronly the chargesymmetric thermodynamicstate and setEA(13) = /3). It makessensehowever to study thermodynamicstateswith thecharge symmetry broken by, say, insertion of a chargeasymmetric operator,

commutingwith HB into the trace in eq. (2.6). An exampleis the Z3 charge11:= e

2’~~3A,12A = e2’~~”3A.From the point of view of the theoryquantizedonC this correspondsto the study of the ground state in sectorswith nonperiodic(twisted)boundaryconditions.At presentwe arenot concernedwith this interest-ing problem.

In the chargesymmetric casethe two equations(3.3) are the sameand we areleft with the singleintegral equationfor one pseudoenergye = =

_Rm~cosh/3+e+’P*log(1+e”)=0 (3.6)

where

2~sinh(2/3)

— sinh(3f3) (3.7)

Eq. (3.6) hasexactlythe sameform as the TBA equationfor the SLYM scatteringtheorywith pair amplitude(1.10).Thisobservationcanbe readily tracedto the factthat

5BB(/3) =S,~(/3)SAA(/3). (3.8)

It follows that the ground statescaling functions (1.6) in thesetwo field theoriesare the same except for factor of 2 which is due to two speciesof particles

contributingto (3.5)

FSPM(r) = 2FSLYM(r) = 2F(r) , (3.9)

708 A1.B. Zamolodchikov/ ThermodynamicBetheansatz

where the scaling length r = mAR in SPM and r = mBR in SLYM. By definition

df3F(r) = I coshl3L(13)—

2n-J 2ir

L(f3) = log(1 + e”~) (3.10)

where e(/3) is the solution to the integralequation

—rcoshf3+E+’P*L=0. (3.11)

Before turning to a numericalstudyof eq. (3.11) considera few limiting cases.Iterationsof eq.(3.11)producethe asymptoticlow-temperatureexpansion

rF(r) = —(C,(r) +C2(r) +...), (3.12)

2~-

where C~(r) exp(—nr)as r —‘ ~ up to a power-like factor. Separateterms C~(r)correspondto n-particleclustersin the trace(2.6). We have

—1C1(r) = ~

C2(r) = _~_-fcoshf3 e2~°~~3d/3— f—~-~--~-~-~e_~05b0~/32)’P(p, — /32).

(3.13)

Theseintegralscanbe expressedin termsof modified Besselfunctions

1C,(r) = — —K,(r)

IT

C2(r) = ~K1(2r) + ~2 [K,(r)f~Io(t)K~(t)t dt + Ii(r)fK~(t)tdtj

+ ~ [Ki(~~r)f°°Io(~t)K~(t)tdt + I,(V~r)fKo(~t)K~(t)tdt].

(3.14)

A1.B. Zamolodchikov/ ThermodynamicBetheansatz 709

Forany r andfor large 1131 the solution e(f3) to eq.(3.11) follows asymptotically

the “free” energy r cosh/3. Correctionsaregiven by the following /3 —s ~ asymp-totic expansion

— ~re~3= _2~/~((2IT/r)s,(r)e~— (2IT/r)5s5(r)e~~

— (2IT/r)tts,,(r)etl$ +

(3.15)

where

~r2s,(r) =F(r) —

24IT

df3(2IT/r)”s~(r) = — fL(/3)e’~P~-__, n = 5,7,11,13 (3.16)

Note that exponentsin the r.h.s. of expansion(3.15) are exactly the numbersofintegrals of motion in SLYM [17] and neutral integralsof motion in SPM [4].Functionss~(r)in (3.16) arechosenas to havea finite limit as r —s 0.

In the high-temperature limit r —~ 0 the solution e(/3) flattens in the centralregion — log(2/r) ~z/3 ~zlog(2/r), tendingto the limiting value there

V~+ie

0=log 2 =0.4812118251.... (3.17)

ThereforeL(j3) looks like a plateauin the central region with the samelimiting

height e0 anddoubleexponentialfalloff outsidethis region(see fig. 2, wheresomenumericalexamplesare plotted).As r —~0 the plateauwidensandthe form of itsright and left edgestends to some universalpattern. The r-dependenceof thefunction L(f3) reducesto shifts of the edges.The limiting form of, say, the right

L(~)

Fig. 2. Fewpatternsof L(f3) for differentvaluesof r.

710 ALB. Zamolodchikov/ ThermodynamicBetheansatz

edgeis determinedby the solution Lk~flk(I3)= log(1 + e~k1nk~) of the equation

eP+eklflk+’P*Lklflk=0 (3.18)

shifted to the right in /3 by log(2/r). We call the r-independentfunction LkIflk(13)the kink solution as it interpolatesbetweenlimiting valuese

0 and 0.In particular,thekink solution governsthe r —s 0 asymptoticsof the groundstate

scalingfunction F(r)

1F(0) = —~—-~fLkIflk(f3)e~df3. (3.19)

A known [8—10],but somewhatmysterious fact is that this integral can becalculatedexplicitly in terms of the Rogers dilogarithm function. Differentiating(3.18)w.r.t. /3 onesubstitutesfor e

13 in (3.19)

0Ekink eklflk ~kinke~= —ço* . (3.20)

1 + e~kink 9/3

This gives

2F(0) = - ~fde[1 + log(1 + e~)] (3.21)

and

F(0) = —1/30. (3.22)

This value matcheswell with the ground stateenergy asymptoticspredicted byconformal invariance.In the UV limit one has[13]

— C\E(R)=— ~+~1——I (3.23)

R 12)’

where ~l and ~ are right and left dimensionsof the ground state. In SPM thegroundstatecorrespondsto the vacuumstateof .A(5/6) with ii = = 0 and

ESPM= —2IT/15R. (3.24)

This correspondspreciselyto (3.22). In SLYM it is thenegativedimensionstate ‘Pof ~i’(2/5) with zl = = — 1/5 that plays the role of the ground state.Therefore

EsLyM’~ IT/15R, (3.25)

again in accordancewith (3.22).


Turn now to next-to-leadinghigh-temperaturecorrections.To get rid of thedominant“conformal” contribution(3.22) to the scalingfunction it is convenientto consider the stresstensortrace expectationvalue (2.42), which is zero in theconformallimit. Usingeq. (2.42)we have

(2IT)2 dF(r) ~mBKT~)5LYM = ~rn~2(T~)

5PM= r dr = 11+ e_)(e~,

(3.26)

where ~‘+(I3) is the solution to the linear equation

e~~i~= e~+ ‘P * 1 + e~~ (3.27)

In the limit r —s 0 the integrandin eq.(3.26) is localizednearthe right edgeof thecentral region. Its shapethereis determinedby the kink function Lklflk(/3). Itfollows that the limiting value is

T0= (2~)2 dF(r) r~O 1dL~~~(p)~ (3.28)

Note that this is just the quantity that governsthe /3 —s — ~ asymptoticsof theconvolution

‘P * Lklflk = —~ + —T0e~~+ ... . (3.29)IT

On the other hand,one can argue that Ek~flk(I3)at t = 0 is a regular function of= exp(613/5).The cancellationof e~termsin eq. (3.18)thereforerequires

T0=IT/V~. (3.30)

The numberdeterminesthe next-to-leadingterm in the scalingfunction

~r2F(r)=—~j+-~-—+.... (3.31)

It seemstrue that the remainderpart of F(r) nearG = 0 is a regularfunction of

G = rt2”5

~1~r2F(r)————=—~+~ (3.32)

24ir n=1


with a finite radiusof convergence.We shall seein sect. 5 that the serieson ther.h.s.of eq.(3.32)correspondsto the perturbativeexpansionsin couplingconstantsA in SPM and g in SLYM. This UV structureof the scaling function is also

supportedby numericalanalysesof sect.4, whereseveralfirst coefficientsF,~andthe radius of convergenceare estimated.Note in this connectionthat the coeffi-

cient s1(r) in the asymptoticexpansion(3.15) is regular in G. Remaindercoeffi-cients s~(r),n = 5,7,11,. . . arealso regular.

4. Numerical work

Eq. (3.11)wassolvedby iterativenumericalintegrationof the convolution ‘P * L

e~+,=rcoshI3+’P*L~ (4.1)

starting from the initial value e,~,(/3)= r cosh/3. The integrationwas replacedbysummationwith the /3-step~/3 = 0.1. The iterative processis well convergentandafter 3—70 iterations(dependenton the valueof r) onefinisheswith an s(/3) with14 significant digits accuracy.It wasverified that the final ~(j3) is unchangedunderthe integrationstepdecrease.Severalpatternsof L(f3) areplotted in fig. 2.

0 ~1 2 3 4 5

-aol

-aoz c(r)

s~(r)

- 0.04

Fig. 3. Scalingfunction F(r) andexpansioncoefficientss5(r)and s7(r)versusscalinglength r.

A/B. Zamo/odchikov/ ThermodynamicBetheansatz 713

~o G

-0.01 ~ ~

-0.02

-0.03 7 ~.

Fig. 4. Perturbativepartof scalingfunctionss,(r)versusG= r’2”5.

The scalingfunction is plotted in fig. 3. The first two expansioncoefficientsin

eq. (3.15), s5(r) and s7(r), arealso presentedthere.The r —s 0 limiting valuesare

estimatedas

s5(0) = —1.13476515585...x ~

s7(0) = —3.56664913023...x 10~. (4.2)

In fig. 4 the “perturbativepart” of the scalingfunction s,(r) = F(r) — V~r2/24IT

is plottedagainstG = r12”5. Supposingthis to be expandablein power series(3.32)

in G onecan estimateseveralfirst expansioncoefficients F~from numericaldataon F(r). The result for the first ten F~,’sis presentedin table 1. The accuracyoftheseextractednumbersfall rapidly with the order numbern. A few partial sumsof theperturbativeseries(3.32) are plotted in fig. 4 andshow fast convergencefor

IGI ~ 14 (notethat this correspondsto 3 correlationlengths).To estimatethe convergenceregionof the series(3.32)thevaluesof F,, listed in

table 1 for n> 1 were fitted by the formula

= (1+ a/n + O(1/n2)) (4.3)

714 A/B. Zamo/odchikov/ ThermodynamicBetheansatz

TABIE ICoefficientsin theperturbativeexpansion(3.32)of thegroundstatescalingfunction

(numericalestimations)

n F,,

1 — 1.415365357153x 10~2 1.3587274489x iO~3 —4.75827523x 10~’4 2.130038x 1O~5 —1.07141 x io~6 5.7788±0.001x i0’°7 —3.264 ±0.003x10”8 1.90 ±0.01 X 10_12

9 —1.11 ±0.08xi0’~10 7 ±3x10’

5

7\ 400.03

0.02

all

G -,~4

Fig. 5. Truncatedseries(3.32) for negativeG. G0 is the estimatedpositionof singularity.

A/B. Zamo/odchikov/ ThermodynamicBetheansatz 715

correspondingto a singularityof the type

F=(G—G0y’~. (4.4)

Supposingthe coefficientsnearthe termsomittedin the 1/n expansion(4.3) to beof order 1 onefinds

G0= —14.3±0.4; a= —1.3±0.3. (4.5)

The value of G0 gives the position of singularityanddeterminesthe convergenceregion of the series(3.32). In fig. 5, the truncatedseries(3.32) is presentedfornegativeG.

5. Perturbative calculations

Herewe considerthe scalingfunction from the pointof view of perturbationsinthecoupling constantsA in SPM and g in SLYM. Theperturbationtheory for thefinite sizeeffectsin CFT perturbedby a relevantoperatorwasstudiedin ref. [181.In unperturbedCFT the circle hamiltonianand momentumare

H~=-~-(L0+L0_ h.), p —1~-(L0—L0). (5.1)

The perturbativecorrectionsto the ground stateenergyof SPM are given by theseries

E~= 15R -R~ 1~ (5.2)

where X, = (x,,y,) are points on the cylinder we consideredabove and theconnectedcorrelation functions (... >,~,definedvia the usual combinatorialpre-scription,arecalculatedin CFT (at A = 0). We adoptthe usualCFT normalizationof field cD (andhencethe coupling constantA) by the relation

IXI_&s, (5.3)

wherei~= 2/5 in SPM. Usingthe conformalmapping z= exp(— 2’n-i~/R),where= x + iy is the complex coordinate on the cylinder, one can expressintegrals in

eq.(5.2) in terms of connectedCFT correlationfunctions (<...))~ on the infinite


planez

~rc (—A)~21T 2(~1—t)n+2

E(Pe~t)=—~ —R ~6R n=1 n! R

xfK(v(0)~(z,,2,)...~(Zn,2n)V(~)))~fl(Zj2j)~ld2Z2...d2Zfl,

(5.4)

where V is the CFT field correspondingto the cylinderunperturbedground state(V=I in the caseof SPM).

In fact, in SPM due to CFT selectionrules the seriesin (5.2) is overevenpowersof the coupling A only. Sincetheperturbationdimension~i <1/2, all the integralsin eq.(5.2) areUV convergent.On the cylinder they arealso infraredconvergent.It follows from dimensionalargumentsthat the expansion is in fact in thedimensionless parameter A

2R’2”5, i.e.

~ = — 1/15 + ~,A2R’2~5+ ~2A

4R24~5+...). (5.5)

We supposethat this seriesconvergesin some finite region near A2R’2”5 = 0.

Thermodynamicargumentsrequire that as R .._*

~ ~0(A)R, (5.6)

the asymptoticcorrectionsbeing exponentiallysmall in the massive theory. Thisquantity

~5~’0(A)=f0A5~3 (5.7)

is the singular part of the infinite volume bulk free energy caused by thelong-range fluctuations of the Potts model near criticality. In field theorythe vacuumenergyis conventionallynormalizedto zero.Subtractingthis from theperturbativeseries(5.5) one has

ESPM(R)= ~ — 6~’0R. (5.8)

This leadsto the following short-distancestructure

ESPM(R) = —f0A5/3R+ (— 1/15 + ~

1A2R’2~5+ ~

2A4R24~5+ ...). (5.9)

This is exactlythe structurewe havefound from the analysesof the TBA equation.


Therefore, from the perturbative point of view the term V~r2/24IT in the scalingfunction is dueto infrareddivergentsubtractionsandshouldbe attributedto the

singularinfinite volumevacuumenergyc~

~(SPM) —~m~=~ (5.10)

The behavior of c~0—~R~

2is predicted by scaling. It is surprising that thenumericalfactor canbe so easilyextractedfrom the TBA approach.

The perturbativeseriesin SLYM is

E~= -~-R~ 1 K’P(xl)...’P(xn)~~d2x2...d

2xn. (5.11)

With the sameargumentsas usedabovewe find

‘~‘ (R\ = — ~f(SLYM)A5/3RSLYM\ / 0

+ — 1/30+ ~sLYM)gRl2/5 + ~SLYM)g2R24/S +...). (5.12)

As we know from the TBA analyses,when suitably normalized,this is the samefunction as ESPM. In particular,

~‘(SLYM) ~mB 12R~ (5.13)

Also, the perturbativecoefficientsare the sameup to the rescalingfactor

(~5)

2~~(SPM)=2(rn~/s )~~SLYM~2F (5.14)

where I~,are expansioncoefficients of the TBA ground state scaling function(3.32). This permits us to estimate the mass scales of thesetwo modelsin termsofcouplings by comparing the numerical results of sect. 4 with perturbative calcula-tions.

The simplest is the first-order perturbation in SLYM. It is given by the groundstateexpectationof the field ‘p in .%‘(2/5)

R— 2/5~71(5LYM) K’P)g.-o (5.15)

As the ground statein this casecorrespondsto the primary state~p,this expecta-

718 A/B. Zamo/odchikov/ ThermodynamicBetheansatz

tion valueis determinedby the .%‘(2/5) triple-’p structureconstant[161

—2/5

(‘p)= (~)C~. (5.16)

The latter canbe extractedfrom the generalexpressionsfound in ref. [19]

= ~y(l/5)3/2y(2/5)t/2~ (1.911312699...)i (5.17)

with y(x) = F(x)/F(1 —x). Therefore

~7(SLYM) = (2IT) 7”5C~~(0.1458410994... )i.

Comparingthis with the valueof F, quotedin table 1 we find

g=(0.09704845636...)irn~2/’S. (5.18)

In particular, the coupling constant g in SLYM is imaginary with Im g> 0.Note that the singularity discussedat the end of sect. 4 is located at negativeimaginary g.

In SPM the first correction is givenby the following integral

R2~5— f(q~(0)~p(X))d2X (5.19)

41T

The groundstatenow correspondsto the identity operatorand

= [(~)2sin(IT~/R)’sin(IT~/R) ]4/5. (5.20)

The integral in (5.19) canbe calculatedexplicitly

y(2/5)2y(1/5)— —1.048590494.... (5.21)

4(2IT) 2/5

By comparingwith the relation(5.14)this leadsto

A=(0.1643033129...)m~(5. (5.22)

For confidence it is worth verifying relation (5.14) in the next order of theSLYM perturbation theory. To order g2 we find using the form (5.4) of the

Al.B. Zamo/odchikov/ ThermodynamicBetheansatz 719

perturbationexpansion

~SLYM) = — 2(2IT)’9~5f~(z, 2)(z2) -6/5 d2z, (5.23)

where ~1 is the connectedfour-point correlationfunction of fields cp on infiniteplane

~(z,2) = ((‘p(cc)’p(1)’P(z,2)’P(0)))~. (5.24)

It canbe found explicitly in termsof hypergeometricfunctions[2, 16]

236 236~(z, 2) = (z2)2”5[(1 —z)(1 —2)] t/5

2F( ~, ~, z) 2F1(~~~, ~,

124 124

+ C~Q~(z2)’~5[(1—z)(1_2)J’/5

2F1(~~—, _~~)2F1(_~—, _~2).

(5.25)

Integral (5.23)wasevaluatednumericallywith the result

f~(z,2)(z2yo~’Sd2z=2IT(4.955511876...). (5.26)

This gives

~SLYM) —1.442629784...x102 (5.27)

in good agreementwith the valueof F2 quotedin table1.

The author is grateful to A.B. Zamolodchikov for discussionson the subjectmatter and interest in the work and to V. Yurov for valuable advice about thenumericalanalyses.

References

[1] R. Baxter,Exactly solvedmodelsin statisticalmechanics(AcademicPress,London,1982)[2] A.A. Belavin, AM. Polyakovand A.B. Zamolodchikov,NucI. Phys.B241 (1984) 333[3] VI.S. Dotsenko,J. Stat. Phys.34 (1984)781[4] A.B. Zamolodchikov,mt. J. Mod. Phys.A3 (1988)743[5] A.B. ZamolodchikovandAl.B. Zamolodchikov,Ann. Phys. (N.Y.) 120 (1979)253[6] R. Koberleand J.A. Swieca,Phys.Lett. B86 (1979)209[71C.N. Yang andC.P. Yang, J. Math. Phys.10(1969)1115[8] G.E. Andrews,R.J. Baxter andP.J. Forrester,J. Stat. Phys.35 (1984)93[9] V.V. Bazhanovand N.Yu. Reshetikhin,SerpukhovpreprintN27 (1987)

[10] V.V. Bazhanov,A.N. Kirillov and NYu. Reshetikhin,PismaZhETF 46 (1987)500

720 A1.B. Zamolodchikov/ ThermodynamicBetheansatz

[111E. Ogievetsky,N. Reshetikhinand P. Wiegmann,NucI. Phys.B280 (1987)45[12] A.D. Tsvelic,NucI. Phys.B305 (1989)675[13] J.L. Cardy,J. Phys.A16 (1984) L385[14] T.D. Lee andC.N. Yang,Phys.Rev.87 (1952) 404, 410[15] M.E. Fisher, Phys.Rev. Lett. 40 (1978) 1610[16] J.L. Cardy,Phys.Rev. Lett. 54 (1985)1354[171J.L. CardyandG. Mussardo,SantaBarbarapreprint UCSBTH-11 (1989)[181H. Saleurand C. Itzykson, J. Stat.Phys. 48 (1987)449[19] VI.S. DotsenkoandV.A. Fateev,Nucl. Phys.B240[F512] (1984) 312; B251[F513] (1985)691

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THERMODYNAMIC BETHE ANSATZ IN RELATIVISTIC MODELS: …cftconf.itp.ac.ru/seminars/AlZamolodchikov/NPB_1990_342_00695.pdf · Nuclear Physics B342 (1990) 695—720 North-Holland THERMODYNAMIC