A.E.Feiguin,J.A.Riera,A.DobryandH.A.Ceccatto
InstitutodeF´ısicaRosario,Boulevard27deFebrero210bis,2000Rosario,ArgentinaWeanalyzeseveralpropertiesofthelatticesolitonsintheincommensuratephaseofspin-PeierlssystemsusingexactdiagonalizationandquantumMonteCarlo.ThesesystemsaremodelledbyanantiferromagneticHeisenbergchainwithnearestandnext-nearestneighborinteractionscoupledtothelatticeintheadiabaticapproximation.Severalrelationsamongfeaturesofthesolitonsandmagneticpropertiesofthesystemhavebeendeterminedandcomparedwithanalyticalpredictions.Wehavestudiedinparticulartherelationbetweenthesolitonwidthandthespin-Peierlsgap.Althoughthisrelationhastheformpredictedbybosonizedfieldtheories,wehavefoundsomeimportantquantitativedifferenceswhichcouldberelevanttodescribeexperimentalstudiesofspin-Peierlsmaterials.
PACSnumbers:74.20.-z,74.20.Mn,74.25.Dw
arXiv:cond-mat/9705205v1 20 May 1997I.INTRODUCTION
One-dimensionalorquasi-one-dimensionalmagneticsystemsshowmanyfascinatingpropertieswhichcon-tinuetoattractanintensetheoreticalactivity.Oneofthesepropertiesisthepresenceofaspingapinantifer-romagneticHeisenbergchainswithintegerspin1andinladders.2Another,particularlycomplex,systemwhichpresentsaspingapisthespin-Peierls(SP)system.InthissystemaHeisenbergchaincoupledtothelatticepresentsaninstabilityatacriticaltemperature,TSP,belowwhichadimerizedlatticepatternappearsandaspingapopensintheexcitationspectrum.3
Theinterestinthespin-Peierlsphenomenawasre-centlyrevivedafterthefirstinorganicSPcompound,CuGeO3,wasfound.4ThisinorganicmaterialallowsthepreparationofbettersamplesthantheorganicSPcom-poundsandhenceseveralexperimentaltechniquescanbeappliedtocharacterizethepropertiesofthissystem.5Besides,thiscompoundcanbeeasilydopedwithmag-neticandnon-magneticimpurities,leadingtoabetterunderstandingofitsgroundstateandexcitations.6
Spin-Peierlssystemspresentalsoaveryrichandinter-estingbehaviorinthepresenceofanexternalmagneticfield.Belowthespin-Peierlstransitiontemperature,andformagneticfieldsHsmallerthanacriticalvalueHcr(T),thesystemisinitsspin-Peierlsphase,characterizedbyagappednonmagnetic(Sz=0)groundstatewithadimerizedpatternoralternatingnearest-neighbor(NN)interactions.ForT Asimplepictureofthedimerized-ICtransitioncanbe 1 obtainedbymappingtheHeisenbergspinchaintoaspin-lessfermionsystembyaJordan-Wignertransformation.TheeffectofthemagneticfieldfavoringanonzeroSzduetotheZeemanenergycanbeinterpretedasachangeinthebandfillingoftheequivalentspinlessfermionsystem.Asaresult,themomentumofthelatticedistortionmovesawayfromπasq˜=(1−Sz/N)π,whereNisthenumberofsitesonthechain.However,sinceumklappprocessespinthemomentumatπuptoacriticalfieldHcr(T),thelatticedistortionwillremainasimpledimerizationandthemagneticgroundstatewillremainasinglet.11Theoretical10,12,13andexperimental14,15studiesindicatethatthelatticedistortionpatternintheICphasecorre-spondstoanarrayofsolitons.Acomplementarypictureindicatinghowasolitonlatticecouldappearasaconse-quenceofthefinitemagnetizationintheICphaseisthefollowing.Let’sassumethatthedominantcontributiontothemagneticgroundstatecomesfromastateofNNsingletsordimers.Anupspinreplacingadownspindestroysasingletandgivesrisetotwodomain-wallsorsolitonsseparatingregionsofdimerizedorderwhichareshiftedinonelatticespacingwithrespecttoeachother.Eachsolitoncarriesaspin-1/2.Duetothespin-latticecouplingitisexpectedthatthelatticesolitonsaredrivenbythesemagneticsolitons. Thesolitonformationinspin-Peierlssystemshasbeenstudiedanalyticallybybosonizationtechniquesappliedtothespinlessfermionmodel.16Thecouplingtothelat-ticeistreatedusuallyintheadiabaticapproximation.Theresultingfield-theoryformalismhasleadtoimpor-tantresults,themostremarkablebeingtherelationbe-tweenthesolitonwidthandthespin-Peierlsgap,ξ∼∆−1.12AlthoughthisformalismhasbeenextendedtoaHeisenbergmodelwithcompetingNNandnext-nearest-neighbor(NNN)antiferromagneticinteractions17,18,itpresentssomeunsatisfactoryfeatures. Inthefirstplace,therearesomerecentexperimen-talresults14forthesolitonwidthintheICphaseinCuGeO3indicatingadisagreementwiththetheoreticalprediction.Althoughtheremightbeacontributionto thesolitonwidthcomingfrommagnetic17orelastic18in-terchaincouplingswhichwouldexplainatleastpartiallythisdisagreement,itisalsopossiblethatthedifferencescouldbeduetoseveralapproximationsinvolvedinthebosonizedfieldtheory.Oneshouldtakeintoaccountthatthesetheoriesarevalidinprincipleinthelongwave-lengthlimit,andtheapplicabilityoftheirresultstorealmaterialscannotbeinternallyassessed.Then,ourfirstmotivationtostartanumericalstudyoftheICphaseinspin-Peierlssystemsistomeasuretheimportanceoftheseapproximationsintheanalyticalapproach. Inthesecondplace,thefieldtheoryapproachdoesnotprovideadetaileddependenceofthemagnitudesinvolvedintermsoftheoriginalparametersofthemicroscopicalmodels.Forexample,evenforthesimplestcase12theexpressionobtainedforthespin-wavevelocitymustbereplacedbytheexactoneknownfromBethe’sexactso-lutionoftheHeisenbergchain.Inthissense,numericalstudiescouldgiveinformationabouthowtherelevantmagnitudesdependontheoriginalparameterswithoutfurtherapproximations. Withthesemotivations,inthisarticlewewanttoiniti-atethestudyoftheincommensuratephaseinSPsystemsusingnumericalmethods.Thesemethodsgiveessentiallyexactresultsforfiniteclusters,andtheycanbeusedtocheckvariousapproximationsrequiredbytheanalyticalapproachesandthevalidityoftheirpredictions.Besides,thenumericalsimulationsprovideadetailedinformationofthedominantmagneticandlatticestates.InSectionIIwepresentthemodelconsideredandwestudyseveralfeaturesofthesolitonformationintheICphaseusingtheLanczosalgorithm.InparticularweanalyzetheeffectofNNNinteractionsonthesolitonwidth.InSectionIIIweperformMonteCarlosimulationsusingtheworldlinealgorithm–whichallowsustostudylargerchainsthantheonesaccessibletotheLanczosalgorithm–inordertoreducefinitesizeeffects. II.EXACTDIAGONALIZATIONSTUDY Theone-dimensionalmodelwhichcontainsboththeantiferromagneticHeisenberginteractionsandthecou-plingtothelatticeis: H=J N(1+(ui+1−ui))Si·Si+1i=1 +NJ2 Si·Si+2+ K i=1 ∂δ+λ=0i leadtothesetofequations: JSi·Si+1+Kδi− J (3) obtained.Werepeatthisiterationuntilconvergence.Es-sentiallythesameprocedureisfollowedinthequantumMonteCarloalgorithm,asitisdiscussedinSectionIII.Wehaveappliedthisexactdiagonalizationproceduretodeterminethedistortionpatternsinthe20sitechainatT=0.InthefirstplaceweconsiderthecaseofSz=0.Asmentionedabove,thiscorrespondstoadimerizedlattice,i.e.δi=(−1)iδ0.Noticethatforthissimplecase,theequilibriumdistortionamplitudeδ0couldbedeterminedinaneasierwaybycomputingtheenergiesofthespinpartofHamiltonianforasetofvaluesofδ0.Then,addingtheelasticenergyandinterpolatingoneobtainstheminimumtotalenergy.Wehaveperformedthiscalculationinordertocheckouriterativealgorithm.Theresultsforδ0vs.K,forSz=0,areshownin z Fig.1forα=0.0,0.2and0.4,andJ2=0.2,and0.4.Itcanbeseenthat,asexpected,forα>0thedimerizedstateismorefavorableandthisleadstoalargerδ0foragivenK.ToalesserextentthistrendisalsopresentforzJ2>0. Thedependenceofδ0withKcanbeinferredfromthescalingrelationbetweentheenergyandthedimerization, 2ν E0(δ0)−E0(0)∼δ0(pluslogarithmiccorrections)withν=2/3,inprinciplevalidforα<αcandsmallδ0.11,25,26Then,itiseasytoobtainδ0∼K−3/2,arelationwhichisapproximatelysatisfiedbyournumericaldata.The ˆoftheelasticfactthatδ0vanishesatafinitevalueK constant,isjustafinitesizeeffect.BydiagonalizingchainsofN=12,16and20sites,forα=0,wehave ˆincreaseswiththelatticesize,asitcanverifiedthatK beseeninFig.2,anditshouldeventuallydivergeinthebulklimit. 0.4energyofthesystemforSz=1withthedimerizationobtainedforSz=0andthesamesetofparameters.TheresultsofthiscalculationareshowninFig.3.Con-sistentlywiththelargerδ0showninFig.1,thegapin-z creaseswithα.TheeffectofJ2ismuchweakerthanthatoftheisotropicsecondneighborinteractionwhichisnotsurprisingsincethe1Dgroundstatemagneticstructure,withadominantdimerizedstate,hasessentiallyaquan-tum(off-diagonal)origin.Thissmallincreasein∆foragivenKisconsistentwiththesmallincreaseinδ0showninFig.1.Thecorrespondingscalingrelation,∆∼K−1,obtainedfromtherelationbetweenthesinglet-tripletgap 2/3 andthedimerization,∆∼δ0,isagainreasonablysat-isfiedbyournumericaldata. 0.40.30.30.2δ00.1δ00.20.00.00.10.20.4K-3/20.60.81.00.01.02.03.04.0Kα = 0.00.3α = 0.2α = 0.4zzFIG.2.Dimerizationamplitudevs.elasticconstantob-tainedbyexactdiagonalizationforN=12,16,20(solidsquares,diamondandtriangles,respectively)andMonteCarlosimulationsforN=(opendots),withα=0.Theinsetshowstheexpectedscalingbehaviorδ0∼K−3/2forN=. δ00.2J2 = 0.2J2 = 0.41.50.11.00.01.02.03.04.05.06.07.08.09.010.0∆0.5KFIG.1.Dimerizationamplitudevs.elasticconstantob-tainedbyexactdiagonalizationinthe20sitechain,Sz=0, z forvariousvaluesofα=J2/JandJ2. OncewehavedeterminedtheequilibriumdistortionasafunctionofK,weareabletocomputethesinglet-tripletspingap,definedasthefollowingdifferenceofgroundstateenergies: ∆=E0,dim(Sz=1)−E0(Sz=0) (5) 0.00.00.51.01/KFIG.3.Singlet-tripletgapvs.elasticconstantobtainedbyexactdiagonalizationinthe20sitechain,forvariousvalues z ofαandJ2.ThesymbolshavethesamemeaningasinFig.1 ItisworthtoemphasizethatE0,dimisthegroundstate 3 . 0.4α = 0.00.3α = 0.2α = 0.4Jz2 = 0.2δ00.2Jz2 = 0.40.10.01.02.03.04.05.06.07.08.09.010.0KFIG.4.Dimerizationamplitudevs.elasticconstantob-tainedbyexactdiagonalizationinthe20sitechain,Sz=1, forvariousvaluesofαandJ2z . WenowconsiderthecaseofSz=1,whichcorrespondstotheincommensurateregionjustabovethedimerized-incommensuratetransition.Wehavedeterminedthedis-tortionpatternfora20sitechainusingtheiterativepro-ceduredescribedabove.Asdiscussedatthebeginningofthissection,thetwosolitonsordomainwallsseparatingdimerizedregionsareclearlydistinguishable.(AtypicalpatterncanbeseeninFig.7.)Themaximumdistortionδ0,showninFig.4,presentssimilarbehaviorastheoneshowninFig.1correspondingtoSz=0.Inparticular,thefactthatδ0vanishesatafiniteKisagainduetofinitesizeeffects. Inordertocomputethesolitonwidth,weusethefol-lowingformtofitthenumericallyobtaineddistortionpatterns: δi=(−1)i˜δtanhi−i0−di−i0+d ξtanhξ ,(6) whichcorrespondstomodelingeachsolitonasanhyper-bolictangent,asobtainedinthethisproblem.12Theamplitude˜analyticalapproachto δ ,thesolitonwidthξ,andthesoliton-antisolitondistanced,aretheparame-ters˜determinedbythenumericalfitting.Theamplitudeδ shouldbeequaltothemaximumdistortionδ0definedaboveforwellseparatedsolitons,i.e.d≫ξ.Themainlimitationofthiscalculationarisesintheregionwhere,foragivenα,Kissolargethatthesolitonshaveasub-stantialoverlapinthe20sitechain,andthefittingfunc-tion(6)isnolongerappropriate.Inthiscase,theellipticsineshouldbeusedtodescribethesolitonlattice.Thisistheregionwherefinitesizeeffectsareimportant,asitwasdiscussedabovewithrespecttoFigs.1and4.How-ever,thissituationisnotdirectlyrelevanttoexperimentsinceinrealmaterialsthesolitonsarewell-separated.14WeshowinFig.5thesolitonwidthasafunctionofthegap∆forthe20sitechain,forthesamevaluesof αandJ2z asbefore.Itcanbeseenthatthethereisalineardependenceofthesolitonwidthwiththeinverseofthegap.This12 behaviorisconsistentwiththetheoreticalprediction:ξ=vs/∆,(7) wherevsisthespin-wavevelocityforα<αc.Itwasrecentlyshownthattherelation(7),originallyobtainedfortheunfrustratedchain,12isalsovalidinthepresenceoffrustration.18Forα>αc,∆containsacontributionfromthefrustrationduetothepresenceofagapevenintheabsenceofdimerization. Alinearfittingofthesecurvesintheregionξ>2.5givestheslopes1.87,1.70and1.63,forα=0.0,0.2and0.4respectively.Recently,anumericalstudy27hasproposedthelaw:vs=π III.MONTECARLOSIMULATIONS whereJi=J(1+δi).ThesematrixelementsaretheBoltzmannweightsassociatedwithabond(i,i+1)inatimestep∆τ=1/mTintheTrotterdirection,whereTisthetemperatureandmistheTrotternumber.Sincetheexchangecouplingsdependonthelatticedisplacements,thesematrixelementsaresitedependent. Weimplementedthealgorithmwiththeadditionofadynamicminimizationofthefreeenergywithrespecttothelatticedisplacements.Startingfromagiveninitialconfiguration(randomdistributionofspinsandadimer-izedpatternforthelatticedisplacements)wetypicallyconsidered2×103sweepsforthermalization.Duringthenext4×103sweepswemeasuredthederivativeofthemagneticfreeenergy,which,inthelimitofT→0,isgivenby ∂FM δInordertotreatlongerchainsthanthoseconsid-eredintheLanczosdiagonalizationstudyoftheprevi-oussection,wehaveimplementedaworld-lineMonteCarloalgorithm28suitedtothisproblem.Theparti-tionfunctionisre-expressedasafunctionalintegraloverwordlineconfigurations,wherethecontributiononeachimaginary-timesliceisgivenbytheproductofthetwo-siteevolutionmatrixelements, −∆τJS·Sz zzziii+1Wi,i+1(τ)=Si,τSi+1,τeSi,τ+∆τSi+1,τ+∆τ foralongerchainwithN=128indicatesthatinthepa-rameterrangeofourcalculationstheMonteCarloresults havenosizeablefinite-sizeeffects. 0.100.060.02-0.020.280.240.200.160.12050100150dF/dδ+λiterationFIG.6.Minimizationofthefreeenergyforauniformdimerizedchain.Weplotthederivativeofthefreeenergyandtheparameterδalongthesuccessiveiterations. InFig.2weshowtheMonteCarloresultsfortheho-mogeneousdimerizationofthesitechainintheSz=0subspaceasafunctionoftheelasticconstantK,togetherwiththeLanczosresultsforsmallerchains.Noticethatintheparameterrangeconsideredthesitechaindoesnothavethefinitesizeeffectspresentforsmallerchains,namely,thevanishingofδ0forfinitevaluesofK.Theinsetshowstheexpectedscalingbehaviorδ∝K−3/2discussedintheprevioussection.Asafurthercheck,wehavealsoreproducedthescalingbehavioroftheenergygainE0(δ0)−E0(0)andgapwithδ0withameasuredexponentν=2/3withinstatisticalerrors. 1.20.60.0-0.6-1.21.20.60.0-0.6-1.21.20.60.0-0.6-1.2K=1K=2.5∂δiandob-servables,andiii)correctionofthedisplacementpatternduetostatisticalfluctuations. InourcalculationsweconsideredchainsofsiteswithperiodicboundaryconditionsandatemperatureT=0.05J.Wecheckedthatthisvalueislowenoughtostudyground-statepropertiesbycomparisonwithmea-surementsatevenlowertemperatures.Ontheotherhand,athighertemperaturesthesolitonisnotobservedandthereisnodefinitepatternoflatticedisplacements.Wetookm=80fortheTrotternumber,whichislargeenoughtoreproducetheLanczosresultsonsmallerchains(seeFig.2).Forsomeparticularquantitiesliketheenergygap,whichrequiremoreprecision,weconsid-eredalsom=160.Inaddition,comparisonwithresults K=1.5K=3K=20102030405060K=3.5102030405060xx FIG.7.Latticedistortionpatternsandlocalmagnetiza-tionsofthesitechainobtainedbyMonteCarlo,forSz=1anddifferentvaluesofK.Ineverypanelthemaximumlat-ticedistortionisnormalizedtoone,sothattheycannotbedirectlycompared. 5 10.0J2 = 0.0 Lanczoszzzz8.06.0J2 = 0.4 LanczosJ2 = 0.0 MCJ2 = 0.3 MCα = 0.4ξ4.02.0valueofξ,andconsideredincreasingvaluesofd.ForsmallK(≤2J)wefoundthatthetotalenergybecomesacon-stant(withinstatisticalfluctuations)whend≥4ξ,whichimpliesthatthesoliton-antisolitonpairsshownintheleftpanelsofFig.4arenotinteracting.Thiswasconfirmedbyallowingthelatticedistortiontoevolvestartingfromapatternlike(6)withaninitialseparationlargerthand,whichproducesthesameresultforξandthetotalenergy. 8.06.00.00.01.02.03.0KFIG.8.Solitonwidthvs.elasticconstantKobtainedby z MonteCarlosimulationsinthesitechainforJ2=0.0and0.3,togetherwithLanczosresultsforthe20sitechain,zJ2=0.0and0.4,andα=0.4.Thedashedlinecorresponds z toalinearfittotheMonteCarloresultsforJ2=0. ξ4.0J2 = 0.0J2 = 0.3zz2.0ThesolitonstructureinthesubspacewithS=1isgiveninFig.7,whereweplotthedisplacementenvelopei=(−)iδiandthelocalmagnetizationSz,fordif-δi ferentvaluesoftheelasticconstantK.Noticethatthedisplacementsarenormalizedbytheirmaximumvalues(showninFig.2)andthelocalmagnetizationbytheclas-sicalvalueS=1/2.Consequently,thesizeoflatticedis-tortionsindifferentpanelscannotbedirectlycompared.ForsmallvaluesofKthereisawelldefinedsoliton-antisolitonstructureinthedistortionpattern,withtheassociatedlocalmagnetizationfollowingastaggeredor-der.Thereisanet1/2spindensityneareachdomainwall,whichmakestheexcessSz=1.Asinthepre-vioussection,wefittedatwo-solitonsolution(6),with˜0=1becauseofthenormalizationadopted.There-δ sultsforthesolitonwidthξareshowninFig.8.ForincreasingvaluesofKthesolitonwidthgrowsuntilthedisplacementprofileresemblesasinelaw(seeFig.7).Thissinusoidalpatternistypicalofthesolitonlattice,observedforlargevaluesofSz.Itcanbeseenthatthescalingξ∼Kobtainedin[12]iswellreproducedinthewholeparameterrangeconsidered,asindicatedbythelinearfittothedata(dashedline).Thisfigureshows z thatthesolitonwidthforJ2=0.3alsopresentsalin-eardependencewithK.Thesefeaturesobservedinthesitechainarequalitativelysimilartothosepresentinthe20sitechainasdeterminedbyexactdiagonalization.Besides,itcanseeninthisfigurethatthereductionofξismuchstrongerwhentheisotropicNNNistakingintoaccount. Wehaveperformedasimplestudyonthesoliton-antisolitoninteraction.Forthisstudywefixedthedis-tortionpatterntothelaw(6)withthepreviouslyfitted 6 z 0.00.01.02.0∆FIG.9.Solitonwidthvs.inverseofsinglet-tripletgapob-z tainedbyMonteCarlosimulationsforJ2=0.0and0.3.Thehorizontalerrorbarsgivetheestimatederrorinthedetermi-nationofthegap∆. -13.04.05.0Next,westudythebehaviorofthesolitonwidthξwiththespin-Peierlsgap∆.Thatis,wecomparethequantityξthatcharacterizestheSz=1solitonstate,withthesinglet-tripletexcitationgap∆abovethedimerizedSz=0groundstate.AsshowninFig.9,thesetwoquantitiesareinverselyrelatedtoeachother,asdiscussedintheprevioussection.Theslopeofthelinearfitis1.9,veryclosetothevalue1.87obtainedbyexactdiagonalizationofthe20sitechainintheprevioussection.ThisresultconfirmsthedisagreementbetweenthenumericalresultswiththeanalyticalpredictionpointedoutinSectionII. z AlsoshowninFig.9aretheresultsforJ2=0.3.Alinearfittotheseresultsleadstoaslope≈2.3,i.e.largerthan z thevaluecorrespondingtoJ2=0.0.Thisincreaseofthe −1 slopebetweenξand∆isconsistentwiththeresultobtainedforthe20sitelatticebyexactdiagonalization z andJ2=0.4.ThisbehaviorshouldbecontrastedwiththereductionoftheslopefoundfortheisotropicNNNinteraction.Apossibleexplanationofthisbehaviorcouldbethefollowing.Asdiscussedintheprevioussection,the zz leadstoasmallerincreaseofthespingapthantermH2 thefullyisotropicNNNinteraction.Ontheotherhand,theIsinginteractioncouldbemoreeffectiveinpunishingtheexcessSzwhichappearsaroundasolitonleadingtoasmallerreductionofthesolitonwidththantheonecausedbytheisotropicterm,asitcanbeseeninFig.8.AmoredetailedstudyoftheHamiltonianinthepresence ofthetermofHzz thisbehavior. 2isclearlynecessarytofullyunderstandFinally,itispossibletoestimatethecriticalvalueofthemagneticfieldatzerotemperature.ByaddingaZeemantermtotheHamiltonian(1),−gµBSzH(µB:Bohr’smagneton),Hcrmaybecalculatedas: Hcr=E0(Sz=1)−E0(Sz=0) (9) inunitsofgµB.E0(Sz=1)isthegroundstateenergyof(1),andthenHcr<∆,whichisthevalueexpectedofagappedsystemintheabsenceofmagneto-elasticcou-pling.ThebehaviorofHcrasafunctioninFig.10forthesitechain,α=J=0.4.Itisapparent2z of∆isshown =0.0,andfor the20sitechain,α=J2z alineardependenceoveralltherangestudied,11whichisinagree-mentwiththemean-fieldprediction3,,Hcr≈0.84∆.However,weobtainacoefficientconsiderablesmaller,Hcr/∆≈0.47,almostindependentofα.Thisvalueisalsosmallerthantwicethesolitonformationenergycal-culatedinRef.[12].Thefinitevaluethecurvescorrespondingtoα=J2z attheoriginof =0.4isafinitesizeeffect. 0.6α = 0.0 MCα = 0.4Jz0.42 = 0.4Hcr0.20.00.00.20.40.60.81.0FIG.10.H∆crvs.spingapobtainedbyMonteCarlointhe sitechainforα=J2z =0.0andbyexactdiagonalization inthe20sitechain,forα=J2z =0.4. IV.CONCLUSIONS Inthisarticlewehaveanalyzedthemagneticsolitonlatticeintheincommensuratephaseofspin-Peierlssys-temsusingnumericalmethods.Thereisaremarkableagreementbetweentheresultsobtainedbyexactdiag-onalizationusingtheLanczosalgorithmandthoseob-tainedbyquantumMonteCarlowiththeworld-lineal-gorithm.Therelationsamongvariousfeaturesofthesolitonsandmagneticpropertiesofthesystemhavebeendeterminedandcomparedwithanalyticalresults.OurstartingpointisamicroscopicalmodelproposedtodescribeseveralpropertiesofCuGeO3,consistingofa1DAFHeisenbergmodelwithnearestandnext-nearestneighborinteractions. Inthefirstplacewehavenotdetectedanycrossoverinthebehaviorofthequantitiesexaminedasα,theratioofNNNtoNNinteractions,becomesgreaterthanαcatleastforthesmallchainsconsidered.Thatis,thereareonlysmoothchangesasαvariesbetween0.0and0.4.ThemostimportanteffectofthecompetingNNNinteractionisareductionofthesolitonwidthξasafunctionoftheinverseofthesinglet-tripletspingap∆.Furthermore,theeffectofthediagonalterm(2)ismuchlessimportantandinsomecasesevenqualitativelydifferenttothatoftheisotropicNNNterm. Althoughseveralfunctionalformspredictedbycon-tinuumanalyticaltheorieshavebeenconfirmedbyournumericaldata,therearesomeimportantquantitativedifferences.Themostimportantdisagreementbetweenournumericalresultsandtheanalyticalpredictionsisrelatedtothecoefficientintherelationξ∼∆−1,i.e.wehaveobtainedasystematicallyhighervaluethanthetheoreticalvaluewhichisthespin-wavevelocity.Thees-timatedvalueofHcr/∆isalsonoticeablesmallerthanthemean-fieldresultandslighlysmallerthanthepre-dictionofbosonizedfieldtheory.TherelevanceofthesenumericalresultstorealSPmaterials,suchasCuGeO3andtherecentlydiscoveredNaV2O5,29hastobedeter-minedexperimentally. Thenumericalproceduresdevelopedinthisarticlecouldbeappliedtothestudyofseveralotherproper-tiesoftheincommensuratephaseofspin-Peierlssystems,suchasthestaticmagnetizationasafunctionofthemagneticfield(recentlymeasuredinCuGeO3byFagot-Revuratetal.15)andtheorderofthetransitionfromthedimerizedtotheincommensuratephases.9 16 I.Affleck,Fields,StringsandCriticalPhenomena,editedbyE.Br´ezinandJ.Zinn-Justin(North-Holland,Amster-dam,1990),pg.563.17 J.Zang,S.ChakravartyandA.R.Bishop,cond-mat/9702185.18 A.DobryandJ.Riera,(tobepublished).19 J.RieraandA.Dobry,Phys.Rev.B51,16098(1995).20 G.Castilla,S.ChakravartyandV.J.Emery,Phys.Rev.Lett.75,1823(1995).21 S.HaasandE.Dagotto,Phys.Rev.B52,14396(1995).22 J.RieraandS.Koval,Phys.Rev.B53,770(1996).23 D.Poilblancetal.,Phys.Rev.B,toappear(1997).24 K.OkamotoandK.Nomura,Phys.Lett.A169,433(1992).25 G.Spronken,B.Fourcade,andY.L´epine,Phys.Rev.33,1886(1986),andreferencestherein.26 Numericalcalculationsindicatethatthisrelationstillholdswithanexponentνcloseto2/3,forα>αc,atleastfornottoosmallδ0andα<1/2;M.LaukampandJ.Riera,(tobepublished).27 A.FledderjohannandC.Gros,cond-mat/9612013.28 J.E.Hirschetal.,Phys.Rev.B26,5033(1982).29 M.IsobeandY.Ueda,J.Phys.Soc.Jpn.65,1178(1996);M.Weiden,R.Hauptmann,C.Geibel,F.Steglich,M.Fis-cher,P.LemmensandG.G¨untherodt,preprintcond-mat/9703052. 8 因篇幅问题不能全部显示，请点此查看更多更全内容
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