The Field Theory Limit of Integrable Lattice Model

arXiv:hep-th/9406135v121Jun1994THEFIELDTHEORYLIMITOFINTEGRABLELATTICEMODELSH.J.deVega††LaboratoiredePhysiqueTh´eoriqueetHautesEnergies,Universit´ePierreetMarieCurie(ParisVI)etUniversit´eDenisDiderot(ParisVII),Tour16,1er.´etage,4,PlaceJussieu75252Paris,cedex05,France.LaboratoireAssoci´eauCNRSURA280.June1994AbstractThelight-coneapproachisreviewed.Thismethodallowstofindtheunderlyingquan-tumfieldtheoryforanyintegrablelatticemodelinitsgaplessregime.TherelativisticspectrumandS-matrixfollowsstraightforwardlyinthiswaythroughtheBetheAnsatz.Weshowherehowtoderivetheinfinitenumberoflocalcommutingandnon-localandnon-commutingconservedchargesinintegrableQFT,takingthemassiveThirringmodel(sine-Gordon)asanexample.Theyaregeneratedbyquantummonodromyoperatorsandprovidearepresentationofq−deformedaffineLiealgebrasUq(ˆG).LPTHE–PAR94/26June199401Yang-BaxterequationsandtheLight-coneApproachHowtotakethecontinuumlimitofintegrablelatticemodelshasalwaysbeenamajorproblem.Inthisshortreviewweshalltrytoconvincethereaderthatthelight-coneapproach[1,2]isthebestwaytoperformsuchcontinuumlimit.WeconsideraNxMtwodimensionalsquarelatticewhoselinksarelabeledbyanindexa=1,...,n.ThestatisticalweightofeachvertexwherethefourlinksmeetisdefinedbytheR−matrixRabcd(θ)where1≤a,b,c,d≤n(seefig.1).Hereθisacomplexvariablecalledspectralvariable.Inthepresentcontextθcanbeconsideredasasortofcouplingconstant.Itmustbenoticedthatuniversalmagnitudesareθ-independentinintegrablemodels[5].ItisconvenienttointroduceanoperatorTab(θ,˜ω)associatedtohorizontallines(seefig.2)Tab(θ,˜ω)=Xa1,...,aN−1ta1b(θ+ω1)⊗ta2a1(θ+ω2)⊗.....⊗taaN−1(θ+ωN)(1)Forfixeda,b,Tab(θ,˜ω)actsontheverticalspaceV=N1≤i≤NVi,Vi≡|Cn,andthelocalvertexoperatorsaredefinedas[tab(θ)]cd≡Rbdca(θ).Ineq.(1)weintroducedarbitraryinhomogeneityparametersω1,ω2,....,ωNassociatedtoeachsiteonahorizontalline.Whenperiodicboundaryconditions(PBC)areconsidered,itisusefultodefinetherow-to-rowtransfermatrixast(θ,˜ω)=XaTaa(θ,˜ω)(2)Forothertypesofboundaryconditions,seerefs.[5,6,7].ThefirstrelevantphysicalproblemistocomputethepartitionfunctionZ.Itisdefinedasthesumoverallpossibleconfigurationsofthestatisticalweightsforthewholelattice.ForaNxMlatticewithPBCinbothdirections,ZcanbewrittenasZ=Trht(θ,˜ω)Mi(3)whereTrstandsforthetraceontheverticalspaceV.Thefreeenergyisthengivenbyf(θ,˜ω)=−lim(N,M)→∞1NMlogZ(4)Allconsiderationsuptonowarevalidwhetherthemodelisintegrableornot.WeshallcallintegrablethosemodelswheretheR-matrixR(θ)obeystheYang-Baxterequations(YBE):X1≤k,l,m≤nRklba(θ−θ′)Rdmck(θ)Refml(θ′)=X1≤k,l,m≤nRlkcb(θ′)Rmfka(θ)Rdelm(θ−θ′)(5)orintensorproductnotation[1⊗R(θ−θ′)][R(θ)⊗1][1⊗R(θ′)]=[R(θ′)⊗1][1⊗R(θ)][R(θ−θ′)⊗1](6)1ItmustbestressedthattheYBEareaheavilyoverdeterminedsetoffunctionalalgebraicequations.Theycontainapriorin4unknowns[theelementsofR(θ)]andn6equations.Despitethisfactalargesetofsolutionsisknown.Allofthempossesssymmetriesthatreducethenumberofindependentequationsandmakepossibletheexistenceofsolutions.ThesymmetriesmaybediscreteascyclicZnsymmetries,continuousabeliansymmetriesasU(1)nandnon-abelianasGL(n)(see[5]).TheYBE(5-6)admitthenaturalgraphicalrepresentationgiveninfig.3.Graphically,theYBEexpressthefreedomtopushlinesthroughintersectionsofpairoflines.Thispossibilityofrigidlineshiftingcanbeinterpretedasazerocurvatureconditiononthelattice.TheYBEenjoyapowerfulcoproductproperty.Namely,eqs.(5-6)impliesthattheoperatorsTab(θ,˜ω)fulfilltheYBalgebraR(λ−μ)[T(λ,˜ω)⊗T(μ,˜ω)]=[T(μ,˜ω)⊗T(λ,˜ω)]R(λ−μ)(7)Forone-site(N=1),Tab(θ,˜ω)reducestoR(θ)andeq.(7)becomeseq.(5).ForN-sites,eq.(7)canbeeasilyprovedbyrepeatedlypushinglinesthroughvertices(see[5]).Thatis,eq.(7)istheexpressionoftheYBEforN-sites.Thecoproductruleisheredefinedbyeq.(1).Eq.(7)impliesthecommutativityoftransfermatrices[t(θ,˜ω),t(θ′,˜ω)]=0(8)Thatis,thetransfermatricesforfixedω1,ω2,....,ωNformacommutingfamily.Hence,onecanexpecttodiagonalizeitwithθ-independenteigenvectors:t(θ,˜ω)Ψ(˜ω)=Λ(θ,˜ω)Ψ(˜ω)(9)TheBetheAnsatz(BA)actuallydoesthisjob[5].Then,thefreeenergyinthethermodynamiclimitturnstobegivenbythelargesteigenvalueΛ(θ,˜ω)maxoft(θ,˜ω).Wefindfromeqs.(4)and(9)f(θ,˜ω)=−limN→∞1NlogΛ(θ,˜ω)max(10)Whenθ=θ′,eq.(7)naturallysuggestthatR(0)isamultipleoftheunitmatrix.Thisisusuallythecase.Moreprecisely,asolutionoftheYBE(5)iscalledregularifR(0)=c1thatisRabcd(0)=cδacδbd(11)wherecisanon-zeroconstant.Settingθ=0ineqs.(5)yieldswiththehelpofeq.(11)Mefba(θ′)δdc=δfaMdecb(θ′)whereMabcb(θ)≡X1≤k,l≤nRabkl(−θ)Rklcd(θ)2WethusseethatMabcb(θ)musthavetheindexstructureMabcb(θ)=δacδbdρ(θ),whereρ(θ)isac-numberfunction.ThiscanbewrittenasR(θ)R(−θ)=ρ(θ),thatisX1≤c,d≤nRabcd(θ)Rcdef(−θ)=δaeδbfρ(θ)Itfollowsthatρ(θ)isanevenfunction.Thispropertyisusuallycalled‘unitarity’althoughthismaynotbealwaystheappropriatename.Fromeqs.(1)-(2)atzeroinhomogeneityω1=ω2=....=ωN=0andeq.(11),itfollowsthatt(0,{ωk=0})=cNΠswhereΠsistheunitshiftoperatorinthehorizontaldirection.ThemomentumoperatoristhengivenbyP≡−iloghc−Nt(0,{ωk=0})iMoreover,itcanbeshown[8,5]thattheoperatorsCm≡∂m∂θmlogt(θ,{ωk=0})θ=0couplem+1neighborsitesonthehorizontalline.UsuallyC1isaquantumspinchainhamil-tonian.Thecom