superconductivity:aMonteCarlostudy
XiaoHu
NationalResearchInstituteforMetals,Tsukuba305-0047,Japan
(submittedtoPhys.Rev.BonFebruary1,2008)
ThermodynamicpropertiesoftheSO(5)theoryunifyingtheantiferromagnetism(AF)andthed-wavesuperconductivity(SC)areexploredbymeansofMonteCarlosimulationsonaclassicalmodelhamiltonian.ThepresentapproachtakesintoaccountthermalfluctuationsbothintherotationofSO(5)superspinsbetweentheAFandSCsubspaces,andinthephasevariablesofSCorderparameters.Temperaturevs.g-fieldphasediagramsfornullexternalmagneticfieldarepresented,wherethegfieldisconjugatewiththequadraticorderparametersandbreakstheSO(5)symmetry.Thenormal(N)/AFandN/SCphaseboundaries,bothassociatedwithsecond-orderphasetransitions,mergetangentiallyatthebicriticalpointintothefirst-orderAF/SCphaseboundary.HysteresisphenomenonisobservedattheAF/SCphasetransition,andthereforethepresentstudysuggeststheexistenceofaphase-separationregioninthephasediagram.EnhancementofAFcorrelationsisobservedabovetheSCcriticaltemperaturenearthebicriticalpointinsystemswithAFcouplingsstrongerthanSCones.Itsrelationwiththespin-gapphenomenonisaddressed.TheSO(5)theoryinanexternalmagneticfieldisalsoinvestigated,andthefollowingpropertiesareclarified:AtsufficientlylargegfieldstheSCorderisestablishedthroughafirst-orderfreezingtransitionfromtheflux-lineliquidintotheflux-linelattice.Short-rangeAFfluctuationsarelargeratcoresoffluxlinesthanelsewhere,anddecreasecontinuouslytozerowithincreasinggfield.Atintermediategfields,theflux-linelatticeoflong-rangeSCorderandthelong-rangeAFordercoexist.SuperlatticespotssurroundingthestrongAFBraggpeaksatQ=(±π,±π)areobservedinthesimulatedstructurefactor,andareidentifiedwiththemodulationbythetriangularflux-linelatticeofSC.TheAFphaseboundaryassociatedwiththecontinuousonsetoflong-rangeAForderdropssharplytothegaxisfromafinitetemperatureinthetemperaturevs.g-fieldphasediagram.
PACSnumbers:74.25.Dw,05.50.+q,74.20.-z,74.25.Ha
1
I.INTRODUCTION
High-Tcsuperconductivity(SC)incuprates[1]isachievedbyholedopingfromtheinsulatingstateofantiferromag-netism(AF).TheAFandSCphasesareproximateeachotherintemperaturevs.hole-doping-ratephasediagrams.EnhancementofAFcorrelationsisobservedabovetheSCtransitiontemperatureintheunderdopedregion[2].Aclearsignalfromtheseexperimentalfactsistheimportanceoftheinter-relationshipbetweenthesetwoverydifferentandevenexpelling,atafirstglance,properties.Toexplaintheoreticallythecomplexphasediagramsofthehigh-TcSCisstillverychallengingforthecondensedmatterphysics.Thisproblemhasbeenapproachedusingmicroscopicmodels,suchasHubbardhamiltonianandthesimplifiedt−Jhamiltonian[3,4].IthasbeentriedtoderivemicroscopicallytheattractiveforcenecessaryforCooper-pairformationfromthemagneticinteractions,andtoconstructthephasediagramwiththeAFandSCphasessidebyside.Uptodate,however,thereisnowellacceptedmicroscopictheorywhichcancountforthemostimportantfeaturesofthehigh-TcSCbothqualitativelyandquantitatively.
IntheSO(5)theorythisproblemisapproachedinanotherway[5,6]:Thelong-rangeSCandAFordersarepresumedasthetwopossiblelong-rangeordersinpuresystems.ThethreecomponentsoftheAForderparameterandtherealandimaginarypartsoftheSCorderparametercomposeafive-componentsuperspinofSO(5)symmetry.AtlowtemperaturestheSO(5)symmetryisbrokenintotwosubspaces,theSO(3)oneassociatedwithAF,andtheU(1)oneassociatedwithSC.Thedestinationofthebrokensymmetryiscontrolledbythedopingrate,orthechemicalpotentialofholes.Therefore,thedopingrateplaystheroleofSO(5)-symmetrybreakingfield.MuchinteresthasbeenstimulatedbytheproposaloftheSO(5)theory,andconsiderableprogressesinexploringthistheoryhavebeenachieved[7–15].SincethesuperspinvectorintheSO(5)theoryisoffivedimensions,threeforAFandtwoforSC,entropyeffectsonthecompetitionbetweenthesetwolong-rangeordersarehighlynontrivial.Thermalfluctuationsareverycrucialindeterminingphasediagrams[13].Therefore,investigationoftheSO(5)theoryatfinitetemperaturesisessentialforacomprehensiveunderstandingofthetheory.Ultimatelyoneshouldcomparethepredictionsbythetheorywithphasediagramsobservedexperimentally.TheSO(5)theoryalsoraisesinterestsinthepointofviewofphasetransitionsandcriticalphenomena.TorevealthethermodynamicpropertiesoftheSO(5)theoryistheobjectiveofthepresentstudy.Anotherimportantissueisthecompetitionbetweenthelong-rangeSCorderinthepresenceofanexternalmagneticfield,realizedintheflux-linelattice(FLL),andthelong-rangeAForder.Sincemanyinterestingmagnetic-fieldresponseshavebeenclarifiedinthelong-rangeAFandSCordersseparately,theproximityoftheminhigh-Tccupratesisverylikelytoproducemoresophisticatedphenomena.Actually,itissuggestedthatvorticesinducedbyanexternalmagneticfieldinhigh-TcsuperconductorsmaypossessAFcores[5].Toexplorethevortexstatesinhigh-TccupratesintheschemeofSO(5)theoryisalsoveryimportant.
Inordertoachievetheabovepurposes,MonteCarlosimulationsonaclassicalmodelhamiltonianinthree-dimensional(3D)spaceareperformed[13].ThepresentapproachtakesintoaccountthermalfluctuationsbothintherotationofSO(5)superspinsbetweentheAFandSCsubspaces,andinthephasevariablesofSCorderparameters.Theremainderofthispaperisorganizedasfollows:ThehamiltonianispresentedinSec.II,withdescriptionson
2
technicaldetailsofsimulation.InSec.III,simulationresultsforthenullexternalmagneticfieldarepresented.TherearetheAF,AFandSCphase-separation,andSCphasesinthephasediagrams.Thespin-gapphenomenonisalsoaddressed.SectionIVisdevotedtorevealtheeffectsofanexternalmagneticfieldintheSO(5)theory.Coexistencebetweenthelong-rangeAForderandtheFLLoflong-rangeSCorderisobserved.VortexcoresarefoundoflargerAFcomponentsthanelsewhere.SummaryisgiveninSec.V.
II.HAMILTONIANANDSIMULATIONTECHNIQUES
Thehamiltonianinthepresentstudyisgivenby
H=−
SCJi,jti
i,j
·tj+
i,j
AF
Ji,jsi
·sj+g
i
s2i,
(1)
definedonthesimplecubiclattice.Thevectort,oftwocomponentsandcouplingferromagneticallywithnearestneighbors,isforthed-waveSCorderparameter;thevectors,ofthreecomponentsandwithAFcouplingbetweennearestneighbors,isfortheAForderparameter.TheinterplaybetweentheSCandAForderparametersisintroducedbytheSO(5)constraintonthesuperspin:
2
s2i+ti=1.
(2)
ThegfactorisafieldbreakingtheSO(5)symmetryintotheU(1)andSO(3)subgroups,andisproportionaltothedopingrateinaloosesense[5].
Thefollowingnotesontheabovehamiltonianseemappropriateatthisstage.First,theabovehamiltoniancanbeconsideredastheGinzburg-LandaudescriptionoftheSO(5)theory.BothoftheAFandSCorderparameters,sandt,aredefinedinascalelargerthantheatomicone,butmuchsmallerthanthemacroscopicone.Inthissensetheyshouldbecalledasthelocalorderparameters.Theconstraint(2)doesnotimplytheexistenceoflong-rangeorderinthemacroscopicscale.Thelong-rangeorderparameterfortheAFcomponentisthestaggeredmagnetization,andthatfortheSCcomponentisthehelicitymodulus[16].Second,althoughnoquantumeffectisincludedexplicitlyinhamiltonian(1),thecompetitionbetweenthetwodifferentlong-rangeordersistakenintoaccountsufficiently.Therefore,theprofound,nontrivialthermodynamicpropertiesoftheSO(5)theorycanbecaptured.Third,thermalfluctuationsinphasevariablesofSCorderparameters,whichareespeciallyimportantforunderdopedhigh-Tccuprates[17],aretakenintoaccountbythefirsttermintheabovehamiltonian,andtreatedusingtheMonteCarlotechnique.Furthermore,thishamiltonianiseasilydevelopedsoastoincorporateanexternalmagneticfieldforthestudyofvortexstates.Fourth,thesuperspinamplitudeisfixedtounityintheabovehamiltonian.Theonsetofsuperspinamplitudeitselfuponcoolingcanalsobetakenintoaccountinthemean-fieldfashion,andisexpectedtocorrespondtotheso-calledpseudo-gapphenomenon[5].Finally,onlythesimplestsymmetry-breakingfieldgassociatedwiththequadratictermsoforderparametersisincludedinthehamiltonian.Othersymmetry-breakingfieldsappearwhenhighordersoftheorderparametersareconsidered[5,12].Althoughanargumentonmagnitudesofthesefieldsis
3
absentrightnow,itisreasonablyexpectedthatthemostimportantfeaturesofthebreakingofSO(5)symmetryintotheAFandSCsubspacesarecapturedbythegfieldin(1).
Atypicalsimulationprocessstartsfromarandomconfigurationofsuperspinsatasufficientlyhightemperature.Thesystemisthencooledgradually.Theequilibriumstateatagiventemperatureisgeneratedusingtypically50,000MCsweepsofupdatefromthestateofaslightlyhighertemperature.Ineachsweepofupdate,candidatevectorsaregeneratedrandomlyonthefive-dimensionalunitsphereforsuperspinsonallsitesinthesystem,andaresubjecttothestandardMetropolisalgorithmtodetermineiftheyareacceptedforthenextconfiguration[18].Afterthisequilibriationprocess,statisticsonphysicalquantitiesisperformedover100,000MCsweeps.Aroundtransitiontemperatures,morethan106MCsweepsarespentinordertomakesureofsufficientequilibriationandstatistics.ThesystemsizeforsimulationsinnullexternalmagneticfieldisL3=403,withperiodicboundaryconditionsinallcrystaldirections.AstheSO(5)superspinsarecontinuousinfivedimensions,andthesystemisofthreedimensionsincrystalspace,athoroughanalysisoffinite-sizeeffectsonsimulationresults,whichisimportantfordeterminingtherelevantcriticalandbicriticalexponentsinhighprecisions,isextremelytimeconsuming.Onlyforseveralchosenparametersets,largersystemshavebeensimulatedinordertomakesurethatthemainpropertiesderivedfromthepresentsimulationsdonotsufferfromfinite-sizeeffects.Systematicerrors(finite-sizeeffects)arethereforenotestimatedfordatapresentedinthispaper.Statisticalerrorsarecomparabletosizesofmarksinfiguresasfarasnot
AF
specified.TheAFcouplingintheabplaneJab≡Jistakenastheenergyunit,andtemperatureismeasuredby
J/kBthroughoutthepresentpaper.
III.PHASEDIAGRAMSANDCORRELATIONFUNCTIONSFORH=0
SCSCAFAF
A.Isotropicsystem:Jab=Jc=Jc=Jab≡J
Figure1isthetemperaturevs.g-fieldphasediagramofthesystemwiththesameAFandSCcouplinginallcrystaldirections.BoththeN/AFandN/SCphasetransitionsareofsecondorder,inthe3DHeisenbergandXYuniversalityclass,respectively.Thetwophaseboundariesmergetangentiallyatthebicriticalpoint[gb,Tb]=[0,0.85J/kB][19].Forg=gbandT>Tb,theAFandSCcorrelationlengthsforthetwo-pointcorrelationfunctionsareequaltoeachother,andisotropicinallcrystaldirections;theweightsofAFandSCcomponentsare3/5and2/5,proportionaltothenumberofdegreesoffreedom.AwayfromtheSO(5)-symmetricline,positive(negative)gfieldssuppressAF(SC)correlationsatalltemperatures.
SCSCAFB.Anisotropicsystem:Jab=10Jc=Jc=J
SCAFSC
Thetemperaturevs.g-fieldphasediagramofthesystemofcouplingsJab=Jc=JandJc=0.1Jispresented
inFig.2.Thebicriticalpointisat[gb,Tb]=[1.18J,0.64J/kB].Theequal-weightpartitionofthesuperspinatg=gb
4
observedintheisotropicsystemisbroken.Nevertheless,asindicatedintheinsetofFig.2,intheabplanetheAFcorrelationlengthisequaltotheSCcorrelationlengthwhenthegfieldisfixedatthebicriticalvalue.Thisagreementisnottrivialincontrastwiththeisotropicsystem.TheSCcorrelationlengthinthecaxisismuchsmallerthantheothercorrelationlengths.
SCSCAF
C.Stronglyanisotropicsystem:10Jab=100Jc=100Jc=J
Inordertosimulaterealhigh-Tccuprates,theAFexchangecouplingshouldbetakenmuchstrongerthantheeffectiveSCcoupling,andbothAFandSCcouplingsaremuchweakerinthecaxis.ThetemperaturedependenceoftheAFstaggeredmagnetizationandthehelicitymodulusoftheSCcomponents[20]areshowninFig.3forthe
SCSCAF
systemofcouplingsJab=0.1JandJc=Jc=0.01Jatthesymmetrybreakingfieldg=1.96J.Sincethehelicity
modulusisproportionaltothesuperfluiddensity[16],itisclearthatthelong-rangeSCorderisestablishedbelowthecriticaltemperatureTc≃0.115J/kB.Asshowninthesamefigure,theAFcorrelationlengthintheabplane,
SCξabistakenatthetemperatureTsg≃0.15J/kB.TheweightofAFcomponents,s2,decreasesmonotonicallyinthe
AF
,increasesatfirstastemperatureisreduced,andthenissuppressedastemperatureapproachesTc.Themaximalξab
AFwholecoolingprocessandshowsasharpdeclineamongTsgandTc.Therefore,theenhancementofξababoveTsgis
AF
belowTsgisbecauseofthelossoftheclearlytheresultofreductionofthermalfluctuations;thesuppressionofξab
AForderinitscompetitionwiththeSCorder.ThispeculiarbehavioroccursbecausetheAFcouplingintheabplaneoverwhelmsovertheSCone,whiletheSCgroundstateisestablishedbythelargegfield.TemperaturedependenceoftheinternalenergyandthespecificheatforthissystemaredepictedinFig.4.NofeaturecanbefoundaroundTsginthesetwothermodynamicquantities.Therefore,theonlyphasetransitiontakesplaceatTc,andTsgcorresponds
SC
merelytoacrossover.TheSCcorrelationlengthintheabplane,ξab,divergeswhentemperatureapproachesTcin
Fig.3,asusuallyinathermodynamicsecond-orderphasetransition.
Itisfoundexperimentallythatthespin-latticerelaxationrateassumesitsmaximumatatemperaturewellabovetheSCcriticalpoint[2].Thepresentsimulationresultsindicatethatthisspin-gapphenomenoncanbeexplainedbythecompetitionamongthelong-rangeSCandAForders,andthermalfluctuations.SincetheenhancementofAFcorrelationsabovetheSCcriticalpointisobservedinthestronglyanisotropicsystemofFig.3,butnotinisotropicandslightlyanisotropicsystemsofFigs.1,and2,itbecomesclearthatinordertoobservethespin-gapbehavior,thesystemshouldhaveSCcouplingsmuchweakerthanAFones,asinrealhigh-Tccuprates.
Figure5isthetemperaturevs.g-fieldphasediagramofthesamecouplingsforFigs.3and4.Thebicriticalpointisat[gb,Tb]=[1.93J,0.12J/kB].Thelatentheatassociatedwiththefirst-ordertransitionbetweentheAFandSCphasesisapproximatelyQ≃0.05J,anddecreasestozeroasthebicriticalpointisapproached.Thespin-gaplikephenomenonisobservedintheregiongb 5 theSCcriticalpointisTsg/Tc≃1.6atg=2J,whichcountswelltheexperimentalobservation[2]. InFig.5,thespin-gaptemperatureTsgdecreasesasthebicriticalpointisapproached.Thismightseemcuriousatafirstglance,sinceitisclearfromhamiltonian(1)thatthelargerthegfieldthesmallertheAFcomponents.ShowninFigs.6(A)and(B)arethetemperaturedependenceoftheAFcorrelationlengthandthestaggeredsusceptibilityatseveralgfields.Althoughbothofthemaremonotonicallysuppressedbyincreasinggfieldwhentemperatureisfixed,thetemperaturewheretheytakemaxima,Tsg,increaseswiththegfield,asclearlyseeninFigs.6.Itisnotedthatthespin-gaptemperatureincreaseswithdecreasingdopingrateinexperiments.Thepresenttheorythereforeconflictswithexperimentalobservationsinthisaspect. TheSCcorrelationsaresuppressedinthenormalstateabovetheAFphaseboundaryinthepresentsystem.Inthissense,thereisnocounterpartofthespin-gaptemperatureaboveN´eelpoints.However,itisinterestingtoobserveinFig.7thatforthegfieldinacertainregionbelowthebicriticalvalue,theSCweight,t2,takesmaximumatatemperatureabovethecorrespondingN´eelpoint.ThetemperatureassociatedwiththemaximalSCweight,denotedbyTpinFig.5,maybeidentifiedwiththepairingtemperature[5].Thereisnofeatureintheinternalenergyandthespecificheataroundthiscrossovertemperature. IV.PHASEDIAGRAMANDVORTEXSTATESFORH>HC1 A.Modelhamiltonianandphasediagram Anexternalmagneticfieldpenetratesintoatype-IIsuperconductorviathinfluxlinesassociatedwithfluxquantaforHlargerthanthelowercriticalfieldHc1.SCisbrokenalongthefluxlines.High-Tcsuperconductorsareextremelytype-IIwithverylargeGinzburg-Landaunumbersκ∼100.Researchofthevortexstatesinhigh-TcSChasbeengrowingintoavividfieldofcondensedmatterphysicsandstatistics.ThemostimportantfeatureofthevortexstatesisthattheAbrikosovFLLmeltsintoFLliquidviaafirst-orderphasetransition[21,22,20]. IntheschemeoftheSO(5)theory,thefreeenergyofavortexstatecanbereducedbyrotatingthesuperspinsfromtheSCsubspaceintotheAFsubspaceattheflux-linecores[5].ThepossibilityofAFcoresoffluxlinesintheSO(5)theorywasfirstaddressedbyArovasetal.[8].Recently,Alamaetal.discussedtheκdependenceofthecorestate[15].Inthesestudies,theAbrikosovmean-fieldtheorywasdevelopedsoastoincorporatetheAFcomponents.However,theAbrikosovmean-fieldtheoryforthevortexstatesisnotappropriateforthehigh-TcSCsinceitonlytakesintoaccounttheamplitudeofSCorderparameter,andcannottreatthermalfluctuationsinthephasevariables,whichareessentiallyimportantfordeterminingthephasediagramofthevortexstatesinhigh-TcSC[20]. ThehamiltonianfortheSO(5)theoryinthepresenceofanexternalmagneticfieldmaybegivenasfollowing[13]:H=− SC Jij|ti||tj|cos(ϕi−ϕj−Aij)+ i,ji,j AF Jijsi·sj− i H·si+g i s2i, Aij= 2π vorticesislargerthantheSCcorrelationlengthandmuchsmallerthanthepenetrationdepthintheabplane,aconditionsatisfiedinlargeportionofH−Tphasediagramsofhigh-Tcsuperconductors.TheJosephsoncouplingshouldalsobedominantovertheelectromagneticcoupling.Thefirsttermintheabovehamiltonian,knownasthefullyfrustrated3DXYmodel,hasbeenusedsuccessfullyforexplainingmanyimportantthermodynamicpropertiesofthevortexstatesinhigh-TcSC[20,23]. Inordertosimplifythesituation,thecaseofanexternalmagneticfieldparalleltothecaxisisaddressedin 2 thepresentpaper[23].ThevectorpotentialisgivenbyA=(−yB/2,xB/2,0)withB=fφ0/lab.Here,φ0is thefluxquantum,labtheunitlengthintheabplane,andftheaveragenumberoffluxineachsquareunitcellintheabplane.Thedatashowninthefollowingareforf=1/25,correspondingtotheinter-vortexdistanceof √ 3lab/dv= almostverticalpartofAFphaseboundaryisinvestigatedbytuningthegfieldatafixedtemperature,inadditiontothecoolingprocessmentionedinSec.II.TheturningpointontheN/AFphaseboundaryinFig.8andthetricriticalpointinRef.[13]areatthesametemperature. B.AFvortexcores ThevortexcoresintheSCphaseoftheSO(5)theoryaredifferentfromthosewithoutAFcompetitionstudieduptodate.ThestructurefactorsS(qab,z=0)forvortices,s2,sab,andscforg=1.5JintheFLLphasearedisplayedinFigs.10.TheBraggpeaksinthestructurefactorFig.10(A)forthevortexcorrelationsarefromthetriangularFLL.OnealsofindsBraggpeaksinstructurefactorFig.10(B)fortheAFamplitudesatthesamewavenumbersofFig.10(A).ThiscoincidenceindicatesclearlythatcoresoffluxlinesareoflargerAFweightsthanelsewhere.Theg-fielddependenceoftheBragg-spotheightforAFweightsintheFLLphase,suchasthoseinFig.10(B),isinvestigatedwhentemperatureisfixed.AsshowninFig.11forT=0.3J/kB,S(qab=qmax,z=0)decayswithincreasinggfieldinapowerlawS≃p/gqwithp=0.8±0.05andq=3±0.1.ThehaloesatthewavenumbersQ=(±π,±π)inthestructurefactorsforsabandscinFigs.10(C)and(D)correspondtoshort-rangeAFfluctuations.TheweakspotatQ=(0,0)inFig.10(D)isfromthesmallferromagneticcomponentscinducedbytheexternalmagneticfield.ThestructurefactorsS(qab,z=0)intheAFandFLLcoexistencephaseforvortices,s2,sab,andscaredisplayedinFigs.12.FromthestructurefactorsFigs.12(A)and(B)forvorticesandAFamplitudes,itisclearthatAFcomponentsareenhancedincoresoffluxlines,asintheFLLphase.InstructurefactorsFigs.12(C)and(D)forsabandsc,therearestrongBraggpeaksatQ=(±π,±π)associatedwiththelong-rangeAForder.SatellitespotsareobservedaroundthemainBraggpeaksinFigs.12(C)and(D).ThesesatellitespotsareeasilyidentifiedwiththoseinFigs.12(A)and(B).Therefore,intheAFandFLLcoexistencephase,thephaseofthelong-rangeAForderispreservedincoresoffluxlines. V.SUMMARY ThermodynamicpropertiesoftheSO(5)theoryareinvestigatedusingMonteCarlosimulationsonamodelhamil-tonianwhichcountsthermalfluctuationsbothintherotationsofsuperspinsbetweentheSCandAFsubspaces,andinthephasevariablesoftheSCorderparameters.Thelatterfactorisessentiallyimportantforexplainingthermo-dynamicphasetransitionsassociatedwiththeonsetoflong-rangeSCorderinhigh-Tccupratesinnullandfiniteexternalmagneticfields.Therefore,thepresentapproachissuperiortoAbrikosov-typemean-fieldtreatmentsoftheSO(5)theory,inwhichthermalfluctuationsinthephasesofSCorderparametersareneglected. Fornullexternalmagneticfield,thereisabicriticalpointinthetemperaturevs.g-fieldphasediagram,atwhichthesecond-orderN/AFandN/SCphaseboundariesmergetangentiallyintothefirst-orderAF/SCphaseboundary.Hysteresisphenomenonisobservedatthefirst-orderAF/SCphasetransition,whichmaysuggestaphase-separation 8 regioninthephasediagram. InsystemswithmuchstrongerAFcouplingsthanSConeswhiletheSCgroundstateisachievedbygfieldslargerthanthebicriticalvalue,AFcorrelationsareenhancedatacrossovertemperatureabovetheSCcriticalpoint.TheoriginofthisenhancementinAFcorrelationsisclarifiedtobethecompetitionamongthelong-rangeAFandSCordersandthermalfluctuations.Whenthegfieldbecomestoolarge,thiscrossoverfadesawaysinceAFcorrelationsaresuppressedatalltemperaturesbythelargeSCcomponent.Theseresultsareconsistentwiththespin-gapphenomenonobservedexperimentallyinthefollowingaspects:First,realcupratesareveryanisotropicintheAFandSCcouplingsJAF∼0.1eVandJSC∼0.01eV;Second,thespin-gapphenomenonhasbeenobservedexperimentallyonlyintheunderdopedregion.Incontrastwithexperimentalobservations,however,thespin-gaptemperaturedecreasesasthebicriticalpointisapproachedfromtheSCside.Nearthebicriticalpoint,thereisacrossovertemperatureabovetheN´eeltemperaturewheretheweightofSCcomponentstakesmaximum. TheSO(5)theoryinanexternalmagneticfieldisalsoinvestigated.Thelong-rangeSCorderisestablishedthroughafirst-orderfreezingtransitionfromtheFLliquidintotheFLL,whiletheonsetofthelong-rangeAForderisassociatedwithasecond-orderphasetransition.Thesetwophaseboundariescrosseachother,andthusproducearegioninthephasediagramwherethetwolong-rangeorderscoexist.IntheFLLphase,onlyshort-rangeAFfluctuationsareenhancedatcoresoffluxlines.1Dlong-rangeAForderalongthefluxlinecannotberealizedbecauseofstrongthermalfluctuations.Inthecoexistencephase,superlatticespotssurroundingthestrongAFBraggpeaksatQ=(±π,±π)areobservedinthesimulatedstructurefactor,andareidentifiedwiththemodulationbythetriangularflux-linelatticeofSC.Thissimulationresultcanbecheckedbytheneutronscatteringtechnique. Acknowledgements TheauthorwouldliketothankS.-C.Zhang,T.Koyama,andY.K.BangforstimulatingconversationsontheSO(5)theory.M.Tachikiisverygratefulfordrawingauthor’sattentiontovortexstatesinhigh-Tcsuperconductivityandcontinuousencouragement.HeappreciatesS.Miyashita,N.Akaiwa,M.Itakura,andY.NonomuraforhelpfuldiscussionsontechnicalpointsofMCsimulation.ThepresentsimulationsareperformedontheNumericalMaterialsSimulator(SX-4)ofNationalResearchInstituteforMetals(NRIM),Japan. 9 [20]X.Hu,S.Miyashita,andM.Tachiki,Phys.Rev.Lett.79,3498(1997);Phys.Rev.B58,3438(1998);referencestherein.[21]G.Blatteretal,Rev.Mod.Phys.66,1125(1994). [22]G.W.CrabtreeandD.R.Nelson,PhysicsToday45,39(1997).[23]X.HuandM.Tachiki,Phys.Rev.Lett.80,4044(1998). [24]ThisobservationisdifferentfromthatreportedinRef.[13],wherethereisatricriticalpointontheN/AFphaseboundary at[gt,Tt]=[0.40J,0.59J/kB];thereisaregionatg>gtwheretheN/AFphasetransitionisoffirstorder.Thisdiscrepancycomesfromthedifferentwaysofgenerationofsuperspins[18]. 11 FigureCaptions Fig.1:Temperaturevs.g-fieldphasediagramofthesystemwithisotropicAFandSCcouplings.Thebicriticalpointisat[gb,Tb]=[0,0.85J/kB]. SCAFSC Fig.2:Temperaturevs.g-fieldphasediagramofthesystemwiththecouplingsJab=Jc=J,andJc=0.1J. Thebicriticalpointisat[gb,Tb]=[1.18J,0.64J/kB].Inset:temperaturedependenceoftheAFandSCcorrelationlengthsintheabplaneatg=gb. Fig.3:TemperaturedependenceoftheAFandSCorderparameters,correlationlengths,andtheweightofAF SCSCAFcomponentsatg=1.96JinthesystemwithcouplingsJab=0.1JandJc=Jc=0.01J.Here,TcistheSC transitionpoint,andTsgisthespin-gaptemperature. Fig.4:TemperaturedependenceoftheinternalenergyandthespecificheatpersiteforthesamesysteminFig.3.Fig.5:Temperaturevs.g-fieldphasediagramforthesamecouplingsinFig.3.Thebicriticalpointisat[gb,Tb]=[1.93J,0.12J/kB].Thespin-gaptemperatureTsgandpairingtemperatureTpfadeawayaroundg=2.2Jandg=1.5Jrespectively. Fig.6:TemperaturedependenceoftheAFcorrelationlengthintheabplane(A)andthestaggeredsusceptibility(B)atseveraltypicalgfields.Maximaareassumedatthespin-gaptemperaturesTsgforg>gb=1.93J. Fig.7:TemperaturedependenceoftheSCweightataseriesofgfields.MaximaareassumedatthepairingtemperaturesTpfor1.5J =0.1J.=JandJc=JcFig.8:Temperaturevs.g-fieldphasediagramofthesystemwiththecouplingsJab Thefluxdensityisgivenbyf=1/25,andtheZeemanfieldisH=0.1J. Fig.9:Temperaturedependenceofthehelicitymodulusalongthecaxis,thestaggeredmagnetization,andthespecificheatpersiteinthesystemofthesamecouplingsofFig.8atg=1.1J. Fig.10:StructurefactorsS(qab,z=0)forvortices(A),s2(B),sab(C),andsc(D)intheFLLphaseinFig.8.Fig.11:g-fielddependenceoftheBragg-spotheightfortheAFweightsatT=0.3J/kB. Fig.12:StructurefactorsS(qab,z=0)forvortices(A),s2(B),sab(C),andsc(D)intheAFandFLLcoexistencephaseinFig.8. 12 1.15 TcTN1.1 1.05 NT[J/kB]1 0.95 bAFSC0.9 0.85 0.8 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 g[J] Fig.1 by Xiao HU 1 N0.8 ξab10bAFξSCabξAFabg=gb0.6 T[J/kB]86SCTcTNTd0.4 420.2 00.6Tb0.91.2TuT[J/kB]00 0.5 1 1.5 2 g[J] 1 ξSCab0.8 ξAFab10 8 Υab[J], mstag, Υabmstag ξab0.44 0.22 Tc00 0.1 0.2 0.3 0.4 00.5 T[J/kB] Fig.3 by Xiao HU 0.65 4 0.4 e3 C[kB]e[J]0.2 Tc0 C2 Tsg-0.2 0 0.1 0.2 0.3 0.4 1 00.5 T[J/kB] Fig.4 by Xiao HU 0.5 TTNducsgp0.4 N0.3 TTTTT[J/kB]0.2 AFb0.1 SC01 1.5 2 2.5 3 g[J] Fig.5 by Xiao HU 12 Tsg9 g=1.92Jg=1.94Jg=1.96Jg=2.00JξAFab6 3 00 0.1 0.2 0.3 0.4 T[J/kB] Fig.6(A) by Xiao HU 120 Tsg100 g=1.92Jg=1.94Jg=1.96Jg=2.00J80 χstag60 40 20 00 0.1 0.2 0.3 0.4 T[J/kB] Fig.6(B) by Xiao HU 1 30.8 1.942.20.6 1.921.9 1.81.6 Tp0.2 g=1 J00 0.1 0.2 0.3 0.4 0.5 T[J/kB] Fig.7 by Xiao HU 1 TNTm0.8 0.6 NAFT[J/kB]0.4 coexistence0.2 FLL00.5 11.52 g[J] Fig.8 by Xiao HU 1 Υcmstag0.8 c3.5 3 2.5 Υc[J/Γ2], mstag0.6 2 C[kB]0.4 1.5 1 0.2 TmTN0.5 00 0.2 0.4 0.6 0.8 1 0 T[J/kB] Fig.9 by Xiao HU 0.12 0.1 S(qab=qmax,z=0)0.08 0.06 0.04 0.02 01 2 3 4 5 6 7 8 g[J] Fig.11 by Xiao HU 因篇幅问题不能全部显示，请点此查看更多更全内容