Integrable quantum field theory with boundaries th

arXiv:hep-th/0404014v311Nov2004DCPT-04/11SPhT-T04/037hep-th/0404014Integrablequantumfieldtheorywithboundaries:theexactg-functionPatrickDorey1,2,DavideFioravanti3,ChaihoRim4andRobertoTateo1,51Dept.ofMathematicalSciences,UniversityofDurham,DurhamDH13LE,UnitedKingdom2ServicedePhysiqueTh´eorique,CEA-Saclay,F-91191Gif-sur-YvetteCedex,France3Dept.ofMathematics,UniversityofYork,YorkYO105DD,UnitedKingdom4Dept.ofPhysics,ChonbukNationalUniversity,Chonju561-756,Korea5Dip.diFisicaTeoricaandINFN,Universit`adiTorino,ViaP.Giuria1,10125Torino,ItalyE-mails:[e-mail],[e-mail],[e-mail],[e-mail]-functionwasintroducedbyAffleckandLudwiginthecontextofcriticalquantumsystemswithboundaries.IntheframeworkofthethermodynamicBetheansatz(TBA)methodforrelativisticscatteringtheories,allattemptstowriteanexactintegralequationfortheoff-criticalversionofthisquantityhave,uptonow,beenunsuccesful.Wetacklethisproblembyusingann-particleclusterexpansion,closeinspirittoform-factorcalculationsofcorrelatorsontheplane.Theleadingcontributionalreadydisagreeswithallpreviousproposals,butastudyofthisandsubsequenttermsallowsustodeduceanexactinfraredexpansionforg,writtenpurelyintermsofTBApseudoenergies.Althoughweonlytreatthethermally-perturbedIsingandthescalingLee-Yangmodelsindetail,weproposeageneralformulaforgwhichshouldbevalidforanymodelwithentirelydiagonalscattering.Keywords:Boundaryproblems;Conformalfieldtheory;Integrability;ThermodynamicBetheansatzPACS:05.20.-y;11.25.Hf;11.55.Ds;68.35.Rh1IntroductionThestudyoftwo-dimensionalconformalfieldtheorieswithboundaries[1]andtheirintegrableperturbations[2,3,4]isofinterestbothincondensedmatterphysics[5]andinstringtheory[6].Animportantquantityemergingfromthedefinitionofthecylinderpartitionfunctionforthesefieldtheoriesistheg-function,the‘ground-statedegeneracy’or‘boundaryentropy’,whichformodelscriticalinthebulkwasintroducedsomeyearsagobyAffleckandLudwig[7].Whilemanyinterestingquestionsremaininthesecases[8,9],inthispaperweshalldealwiththefurtherissueswhichariseforoff-critical,massive,systems.Theg-functionformassivefieldtheoriescanbedefinedasfollows[10,11,12,13].TherearetwopossibleHamiltoniandescriptionsofthecylinderpartitionfunction.Intheso-calledL-channelrepresentationtherˆoleoftimeistakenbyL,thecircumferenceofthecircle:Zαβ=TrH(α,β)e−LHstripαβ(M,R)=∞Xn=0e−LEstripn(M,R).(1.1)Inthisformula,HstripαβpropagatesstatesinH(α,β),theHilbertspaceforanintervaloflengthRwithboundaryconditionsαandβimposedatthetwoends,Estripn∈spec(Hstripαβ),andMisthemassofthelightestparticleinthetheory.IntheR-channelrepresentationtherˆoleoftimeisinsteadtakenbyR,thelengthofthecylinder:Zαβ=hα|e−RHcirc(M,L)|βi=∞Xn=0G(n)α(l)G(n)β(l)e−REcircn(M,L),(l=ML)(1.2)whereEcircn∈spec(Hcirc)andG(n)α(l)=hα|ψnihψn|ψni1/2.(1.3)Inequation(1.2),theboundarystates|αi,|βiandtheeigenbasis{|ψni}oftheHamil-tonianHcirchavebeenused.ThesearedefinedonacircleofcircumferenceLandpropagatealongthe‘time’directionR.Atlargel,thefunctionlnG(0)α(l)growslinearly:lnG(0)α(l)∼−fαL,(1.4)wheretheconstantfαcontributestotheconstant(boundary)partoftheground-stateenergyonthestrip(seeeq.(A.5)).Thestandardg-functionisthendefinedaslngα(l)=lnG(0)α(l)+fαL.(1.5)Intheorieswithonlymassiveexcitationsinthebulk,lngα(l)tendsexponentiallytozeroatlargel.1RL(space)(time)αβFigure1:TheL-channeldecomposition;states|χniliveonthedottedlinesegmentalongthecylinder.R(time)L(space)αβFigure2:TheR-channeldecomposition;states|ψniliveonthedottedcirclearoundthecylinder.Thetwodecompositionsareillustratedinfigures1and2.Theequalityof(1.1)and(1.2)resultsinthefollowingimportantidentity:∞Xn=0e−LEstripn(M,R)=∞Xn=0G(n)α(l)G(n)β(l)e−REcircn(M,L).(1.6)Thepurposeofthispaperistodevelopanexactexpressionfortheground-statefunctionlnG(0)α(l)throughthelarge-Rlimitof(1.6),withboundaryconditionsβ=α.Asitstands,thefactthatEcirc0(M,L)isnegativemakestheRHSof(1.6)divergeasR→∞;however,rearranginggives2lnG(0)α(l)=REcirc0(M,L)−LEstrip0(M,R)(1.7)+ln1+∞Xn=1e−L(Estripn(M,R)−Estrip0(M,R))!+O(e−R(Ecirc1−Ecirc0)).Weshallrestrictourattentiontomassivetheorieswithnon-degenerategroundstateontheplane.Forthesemodelsthenon-zeromassgapgivesthefinaltermtheleadingbehaviourO(e−R(Ecirc1(M,L)−Ecirc0(M,L)))∼O(e−RM)(1.8)2inthedomainR≫L≫0.Inthissamedomain,Estrip0(M,R)tendstoitslimitingformasEstrip0(M,R)=EM2R+2fα+O(e−RM)(1.9)whereEandfαaretheextensivebulkandboundaryfreeenergies,asin(A.5).Theseconstraintsarecrucialforthevalidityoftheperturbativetreatmenttobeintroducedinthefollowingsections:thehighercorrectionshaveacleardependenceonRanddonotcontributetotheg-function.Discardingtheseexponentially-suppressedtermsandusingthedefinition(1.5),wefinallyobtain2lng(0)α(l)∼REcirc0(M,L)−EM2L+ln1+∞Xn=1e−L(Estripn(M,R)−Estrip0(M,R))!.(1.10)HavingtakenRtobelarge,theclusterexpansioninvolveslettingLtendtoinfinityaswell,sothatanexpansionoftheRHSof(1.10)canbedevelopedintermsofone-,two-andsoonparticlecontributions,whichthemselvescanbeconsistentlyestimatedusingtheBethe-ansatzapproximatedlevels(A.6),(A.7).Notethatthisdiffersfromthestrategyadoptedin[10],whereasaddle-poin