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Macroscopic Quantum Coherence in Antiferromagnetic Molecular Magnets

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Macroscopic Quantum Coherence in Antiferromagnetic Molecular Magnets

    Macroscopic Quantum Coherence in

    Antiferromagnetic Molecular Magnets

Commun.TheorPhys.(Beijing,China)36(2001)PP245250

    ;@InternationalAcademicPublishersVol36,No2August15,2001

    ;MacroscopicQuantumCoherenceinAntiferromagneticMolecularMagnets ;HUHuiL0Rong.ZHUJiaLin,andXIONGJiaJiong

    ;DepartmentofPhysicsTsinghuaUniversity,Beijing100084,China

    ;.CenterforAdvancedStudy,TsinghuaUniversity,Beijing100084,China ;(ReceivedDecember18,2000)

    ;AbstractThemacroscopicquantumcoherenceinabiaxialantlferromagnetlcmolecularmagnetinthepresenceof

    ;magneticfieldactingparMldtoitshardanlso~opyaxisisstudiedwithinthetwosubtatticemodelOnthe

    basisof

    ;instantontechniqueinthespincoherent-statepathintegralrepresentation,boththerigorousWentzetKramersBriflouin

    ;exponentandpre-exponentialfactor[ortheground-statetunnelsplittingareobtainedWefindthattheQuantuln

    ;~uctuationsaroundtheclassicalpathscannotonlyinduceanewquantumphasepreviouslyreportedbyChioteroand

    ;Loss‟PhysRev.Lett80n9981169),butalsohavegreatin~uenceontheintensityoftheground-statetunnel

    ;spNttlngThosefeaturesclearlyhavenoanalogueintheferromagneticmolecularmagnetsHiesuggestthattheymaybe

    ;theuniversalbehaviorsinallantiferromagneticmolecularmagnets.Theanalyticalresultsarecomplementedbyexact

    ;diagonatlzatloncalculation.

    ;PACSnumbers:75.45+i03.65Bz,75.50Ee,7550.Xx

    ;Keywords:macroscopicquantumcoherence,two?sublatticemodel}antiferromagneticmolecularmagnets

    ;1Introduction

    ;Inrecentyears.owingmainlyt0therapidadvances

    ;bothinnewtecbnoloesofminiaturizationandinhighly

    ;sensitiveSQUIDmagnetometry}therehavebeenconsid

    ;erabletheoreticalandexperimentalstudiescarriedouton

    ;thenanometerscalemagnets[1,2whichhavebeeniden-

    ;tiffedascandidatesfortheobservationofmacroscopic

    ;quantumphenomena(MQP)[3,4suchasthetunnelingof

    ;thespinoutofnletnstablepotentialminimumthrorgh

    ;theclassicallyimpenetrablebarriert0astableone,je,

    ;macroscopicquantumtunneling(MQT),or1morestrik- ;ingly,macroscopicquantumcoherence(MQC)1wherethe ;spincoherentlyoscillatesbetweenenergeticallydegener? ;at.eeasydirectionsovermanyperiodsInthesemiclas? ;sicalspin?coherent?statepath?integraltheory,l5JMQCis ;connectedwiththepresenceofatopologicalterminthe ;Euclideanactionsz(e,)arisingfromthenonorthogonal? ;ityofspincoherentstates,wMchiscalledBerryphaseor ;thefssZuminoChernSimonsterm:

    ;f+T/2.

    ;iS/(1c.s)(r)dr,,T/2

    ;whereSisthewholespinofthesystem,0andarepolar ;andazimuthalspinangles,respectively.

    ;OneofthemanifestationsofMQCistheground-

    ;statetunnelsplittingofmagneticsystems.Intheab- ;senceofanexternalmagneticfield.jthasbeentheoreti. ;callydemonstratedthattheground-statetunnelsplitting ;iscompletelysuppressedtozeroforthehalf-integertotal ;spinferromagnetsorantiferromagnetswithbiaxialcrystal ;symmetry,[6‟7~resultingfromthedestructi~interference

    ;oftheBerryphaseintheEuclideanactionbetweenthe ;symmetry-relatedtunnelingpathsconnectingtwoclassi

    ;callydegenerateminimaSuchdestructive;nterference ;effectforhalSintegerspinsisknownasthetopological ;quenching.Butfortheintegerspins,thequantuminter

    ;ferencebetweentopologicallydifferenttunnelingpathsis ;constructive.andthereforetheground.statetunnelsplit. ;tingisnonzero.

    ;Whilesuchspin.parityeffectsaresometimesrelated ;t0Kramersdegeneracy,theytypicallygobeyondthisthe? ;oreminratherunexpectedways.Inthepresenceofan ;externalmagneticfield,aspointedoutbyGarg,l8IgJthe ;gromld-statetunnelsplittingcanoscillateasafunctionof ;thefieldwhichisappliedalongitshardanisotropyaxis ;inferronlagnetswithbiaxialcrystalsymmetryandvan? ;ishesatcertainvaluesThispredictionisconfirmedin ;onerecentexperimentcarriedoutbyWernsdorferand ;Sessoli.Theydeveloped&newtechniquetomeaslMfethe ;verysmalltunnelsplittingontheorderof10Kinfer.

    ;ronlagneticmolecularFesclusters.Indeed}theyobserved ;aclearoscillationofthetmlnelsplittingasafunctionof ;themagneticfieldappliedalongthehardanisotropyaxis, ;whichisadirectevidenceoftheroleofthetopological ;spinphase(Berryphase)inthespindynamicsofthese

    ;moleculesAlthoughthisfield.indueedoscillationbehav- ;iorisinvestigatedingreatdetailinferromagneticsystems ;now.[8itisstilllessunderstoodinantiferromagnetic ;molecularmagnetssuchasFel9,Feg,Vs1andantiferrc- ;magneticferritin.[11,12JGolyshevandPopkovfirststudied ;„TheprojectsupportedbyNationalNaturalScienceFoundationofChinafGra~tNo199740191andChin

    ~‟”973”Program

    ;

    ;HUHu[,L0Rong.ZHUdiaLinandXIONGJiaJiong,l36

    ;MQCinauniaxialantiferromagneticfineparticleinthe ;presenceofmagneticfield.【】andfoundsimilaroscillation

    ;behaviorButtheyonlycalculatedtheWentzelKramers ;Brillouin(WKB)exponentinweak-fieldapproximation, ;andpaidnoattentiontoitspre-exponentialfactor.Later, ;in1998ChioleroandLossconsideredtheoscillationprop

    ;ertiesofaringlikemolecularmagnetusingananisotropic ;n0nlinearmodelfNLsM1.[121Inadditiont0theusual ;topologicalspinphase(Berryphase)term;theyfound ;anewquantumphasearisingfromfluctuationswhichis ;neverseeninferromagneticmolecularmagnetsItisreally ;astrikingquantumpropertyinantiferromagneticmolec- ;ularmagnetsUnfortunately,thefundamentalphysicsof ;thisnovelquantumphaseislessexploredsofar. ;Inthispaper.wewouldliketostudytheMQCof

    ;abiaxialsymmetryantiferromagneticmolecularmagnet ;basedonthetwo-sublatticemodel31Byapplying

    ;Lheinstantontechniqueinthespincoherentstatepath-

    ;integralrepresentation.qweobtaintherigorousinstan

    ;tonsolutionsandcalculateboththeWKBexponentand ;preexponentialfactorintheground-statetunnelsplitting ;%Vewillshowthatthequantumfluctuationsaroundthe ;classicalpathscannotonlyinduceanewquantumphase ;previouslyreportedbyChioleroandLoss,:1butalsohave

    ;greatinfluenceontheintensityofthegroundstatetun-

    ;nelsplitting.Thosefeaturesclearlyhavenoanalogalein ;theferromagneticmolecularmagnetsWesuggestthat ;the~,maybeuniversalbehaviorsinallantiferromagnetic ;raolecularmagnetsDuetotheinstantonmethodsbeing ;semiclassicalinnature.i.s..validi?largespinsandin

    ;continuumlimit,weperformexactdigonalizatj0ncalcu

    ;lationsandfindthattheyagreewellwiththeanalytical ;results.

    ;2InstantonCalculations

    ;Weconsiderthering-likeantiferromagneticmolecular

    ;magnets(i.e.Felo,FesandVs)composedofN=2n ;spinssregularlyspacedonacirclelyinginthe ;planewithanantiferromagneticexchangeinteractionbe- ;tweenthem.[n,lz]Ingeneral

    ;1

    ;thecrystallineanisotropy

    ;ateachsitehasbiaxialsymmetry.Asusuallyfor ;antiferromagnets351wedecomposethelocalspinsinto ;thetwomagneticsublattices:S1andwiththesame

    ;spinvalueS=ns.Thenthemolecularmagnetinanex

    ;ternalmagneticfieldHactingalongitshardanisotropy ;axiscDI1bedescribedbyaspinHamiltonianofthetype ;=

    ;J曼曼+(l+2雪一g.)

    ;+(1雪乞+2雪一gp日岛)(1)

    ;wheregistheLand6factor,andistheBohrmagneton

    ;1>2>0arethecrystallineanisotropycoefficients,and ;wetaketheeasy,medium1andhardaxesaNz,andz ;respectivelyforeachsublatticeJistheexchangeenergy. ;InaccordancewithexperimentalresultsjtwillbeaNsunled ;thatJ1k2forthestrongantiferromagneticcoupling. ;Notethatourtwo-sublatticeconfigurationisonlyvalid ;forthemagneticfieldHHere.=2js/g~is

    ;thecriticalfieldatwhichthestrongantiferromagneticex

    ;changeinteraction3S1S2iscomparabletotheZeeman ;termg日日(sl+)

    ;I?thesemiclassicalapproach.l6inordertoobtain

    ;thegroundstatetunnelsplitting1oneshouldcomputethe ;imaginarytimepropagatorinthespincoherentstaterep-

    ;resentation

    ;(,Iexp【一)=l厂口np(s)

    ;=

    ;.厂口{-}{).{-){)exp(ldrc)(2)

    ;overalltrajectorieswhichconnecttheinitialstatel)

    ;thefinalstatel,).Here0iejb=1,2)arethepolar ;andazimuthalanglesofeachsublatticespinvectorthe ;LagrangimlCincludestwopar~[“1

    ;cn=?iS(1c.soj)4/)J=1.2

    ;cl=all+COS01cos02+sin01sin02cos(~12)

    ;+?(C0S20j+K2sinsin.)J=1.2

    ;?gpSHcos0j

    ;i=1.2

    ;(3)

    ;(4)

;correspondingb)theBerryphasetermandthetotalEu

    ;clideanenergytermE(01,l1022)Herewehaveintro- ;ducedKa=k1S,K2=k2SandJ=jSAllterms

    ;inEq(4)areofapparentphysicalmeaningThefirst ;termistheexchangeinteractionenergy,thesecondterm ;ismagneticanisotropyenergyandthethirdtermisZee- ;nlanenergy.Thedominantcontributiontotheimaginary- ;timepropagatorcomesfromfiniteactionsolutionsofthe ;EulerLagrangeequationsofmotion(instantons)which ;canbeexpressedas

    ;_0-f5)

    ;:o,(6)0

    ;whereOi.=12)denotetheclassicalpaths.

    ;Accordingtotheinstantontechniqueinthespin- ;coherent.statepathintegralrepresentation..1theinstan-

    ;toncontributiontothetunnelsplitting?rnotincluding

    ;

    ;N02MacroscopicQuorumCoherenceinAntiferromagneticMolecularMagnets247

    ;thegeometricphasefactorgeneratedbytheBerryphase ;termintheEuclideanaction)isvenby

    ;?()e,(7]

    ;whereh;pisthesmall-angleprocessionoroscillationfie- ;quencyinthewell,andSclistheclassicalactionorthe ;WKBexponentdeterminedbyEqs5)alld6)Thepre-

    ;exponentialfactorP0orinaresfromthequantumfluctu- ;ationsaroundtheclassicalpaths,whichCallbeevaluated ;byexpandingtheEuclideanactiontothesecondorderin ;thesmalfluctuations

    ;2.1Wentzeb-Kramers-BrillouinE~vponent

    ;Ino121”case1onlylow-energytrajectorieswithalmost

    ;antiparallelS1ands2contributethepathintegralItis. ;therefore,safetosaythattunnelingofSafollowstunnel

    ;ingofS1i”?Forthatreasonwecanreplace02and2by

    ;T-01+e0and+1respectively(withl1,ll1)

    ;inC.Inthenewcoordinatestheimaginary-timepropa- ;gatorofthesystemcanberepresentedas

    ;/})/口慨})

    

    ;(dTc(,))(8)

    ;Bysimplealgebra,uptothesecond-orderapproximation ;aboutandd,weobtain

    ;c=i2S6+2(KICOS0+K2sin0sin.

    ;+fiS~sin0K1sin20

    ;+K2sin20sin9BSHsin)

;+(K2sin20sin24)e,

    ;+is[(1+cos01sinOeo]g~

    ;+(Aooe;+Aoeeoe+AEi),

    ;where

    ;A=ic.s+iJ

    ;c.s2

    ;+cos2sin2~cos,

    ;Aoe=K2sin20sin2.

    ;A=s+K2si?o~s2~.

    ;(9)

    ;f101

    ;InEqs(9)and(10)wehavedroppedthesubscriptsof0l ;andlforclarity.

    ;UponGaussianintegrating(8)over

    ;andoneCallobtainthefollowingeffectiveLagrm~gian ;f2eff=i2S~+2(K1COS0+Kasin0sin)

    ;+

    ;_sin20(_I-.]_…)

    ;Notethatmagneticfieldentersonlythroughthelastterm ;inEq(11Jandhasnoinfluenceonthetunnelingbarrier ;Becauseofthecondition1>,theequilibriumori

    ;entationsofs1are(0,=/2,0)and(/2,)which ;correspondtotwodegenerateclassicalminimaofthean

    ;ergyE=0Itisobviousfromsymmetrythatthereare ;twodifferenttypesofinstantontrajectoriesofopposite ;windingsaroundhardanisotropyaxisl{denotethem ;asjnstalltons

    ;=0=?/2--+=~mr(=1,3,5,)(12)

    ;Toexecutethefirst,weshouldseekfortileclassical ;pathorpaths)i2l(T)=((),())connectingthetwo ;minima,thatminimizestheaction

    ;%/dr

    ;ThispathsatisfiestheEuler-LagrallgeequationsoflnO- ;tion(SeealsoEqs(5)and(6)),

    ;()一研OZ:eff-o-(13)lJ一研(J

    ;SubstitutingtheeffectiveLagrangianintoEq(13),we ;obtain

    ;[]:(-2+zsirI2n.

    ;S2

    ;l/de-ig#B

    ;H),04)

    ;d[sin:(-ig#BH)]:sintnl5)

    ;Consequently,aquasiclassicaltunnelingofS1mayoccnr ;;n.plane0=/2,andthen,equation(15)reducesto

;sineGordonequation1

    ;orequivalently,

    ;d2_

    ;=2

    ;de7

    ;1

    ;f161

    ;(17)

    ;wherel=(4JKa/S.)/Undertheboundarycondi

    ;tionsinwhichtheclassicalpathapproachesthetwomin

    ;imaasr_?~oeweobtainanexactsolutionofthis ;equation(3]

    ;()=2arctall(exp(~,lr)).(18)

    ;Itiseasilyverifiedthat_?0,as~oc.Thecorre

    ;spondingclassicalaction,Le.,theWKBexponentinthe ;rateofquantumtunnelingatfinitemagneticfield,Callbe ;

    ;HUHui.LURongZHUJia-LinandXIONGJia-Jiong,,o1.36 ;evaluatedbyintegratingtheEuclideanactionwithabove ;classicaltrajectories1andtheresultisfoundtobe ;=

    ;ReScI?iImScl(19)

    ;f201

    ;f211

    ;wherethepositiveandnegativesignsinEq(19)areCOt- ;respondingtoinstaaltonsrespectively.

    ;ItisclearlyseenfromEqs(20)and(21)that,theclas

    ;sicalactionhastwounusualfeaturesinthepresenceof ;magneticfieldFirstof_therealpartofactionhasno

    ;dependenceonthemagneticfieldandisdeterminedby ;materialparametersofthesystemonlyThisfeatureis ;quitedifferentfromthatinferromagneticmolecularmug- ;nets,andcanbeunderstoodeasilyfromEq.(zz),sincethe ;tunnelingbarrierremainsunchangedunderthemagnetic ;field.Further,asshownfromEq.(21)ifweignorethe ;contributionfromquantumfluctuationsaroundtheclas- ;sicalpaths,thegroundstatetunnelsplittingwhichisprO- ;portionaltoexp(ReSc1)lCOS(Imsd)loscillatesasthe ;fieldHisincreased,andtheThesecondmajorstepistoevaluatethepre-

    ;exponentialfactorofsmallfluctuationsaroundtheclassb ;calinstantonpaths.Wewrite

    ;Off)=(r)+60(r)

    ;?(r)=(r)+6O(r)

    ;f231

;f24,

    ;andevaluatetheactiontothesecondorderin(60,) ;WritingS=&l+S,wehave

    ;=

    ;/drS2{+(sH)2+w

    ;uCOS2?i2g#SHwlsin~]60

    ;++(cos25)6~.),(25)

    ;extratermhasimportantconsequencesatthehighmag

    ;neticfield.

    ;Now,theimaginarytimepropagatorisfoundtobe

    ;,lexp[一】l)=exp()DD(26)

    ;with

    ;.=/)ex{/a+((.+

    ;

    ;+c0s2i2g#SHwxsin$)6ol},(27)

    ;D?=/v{6O}

    ;…p

    ;{/drs2+(c.s2?]).(28)

    ;whereNeandNarethenormalizationfactors.The ;fluctuationdeterminantforisstandard.Following ;Eq(2.44)inRef.10],weobtain

    ;()ReS~t,(2g)

    ;Forthe0-fluctuationdeterminantwefindinthefirst-order ;perturbationtheory(ThedetailedcalculationofDwill ;bereportedelsewhere),forthehighmagneticfield, ;e

    ;[罢】,?印【_J.

    ;where=l/9s‟=(HJH)(K2/J)/isusedasa

    ;smallparameterinthehighmagneticfieldItisclearly ;showninEq(30jthattheexistenceofmagneticfieldcaJ1 ;bringaphaseshiftwhichapproachesappraximately/2 ;inthehigh-fieldregime.Notethatthevalueofphaseshift ;agreeswellwiththatfoundbyChioleroandLoss.]0n

    ;theotherhandasthefielddecreasesdowntozero,and ;thus”_†o..thephaseshivanishesdespitethebreak.

    ;downofourfirst-orderperturbationca]culations.?the

    ;low-fieldregime

    ;Combingtheclassicalactionandtwofluctuationde. ;termina~lts,wearriveatthedesiredground-statetunnel ;sphtting,

    ;where

    ;?x[Re.Sd,~.

    ;×p(ReSc1)Icos()l,(31)

    ;s(一甍)+.(32)

;where0=(4JKl/S.)/Notethat0and?fluctuations

    ;aredecoupledinEq(25),andinthe0-fluctuation,anun-3ResultsandDiscussions

    ;usualterm~i29#

    SHw1sindistinguishing+instantonsThesemiclassicalanalysispresentedsofarapplies,

    ;frominstaaltoasappears.Aswewillshowbelow,thisstrictlyspeaking,onlytoasizablenumberofspin

    swith

    ;,,

    ;,旦巩

    ;,,

    ;,=/,IS

    ;ch

    ;h

    ;w

    ;

    ;N02~acros~opicQuaneumCoherenceinAntiferromagnetlcMolecularMagnets

    ;S1H0wever,asisoftenthecasewithsuchmeth

    ;odstheresultsarevalid(atleastqualitatively1evendown ;toafewspinsofsmallsizeThisexpectationisindeed ;confirmedbyexactdiagonalizationcalculationswhichwe ;haveperformedo12Hamiltonianfl1ResultforS=5. ;andforSOOletypicalvaluesk1=003K,k2=0.0lK ;and=10KispresentedinFig.1,thecriticalfieldis ;foandtobe7.44T.Here,thetmitsfortheenergyand ;magneticfieldasetakentobeKelvinandTeslarespec- ;tively.Wecapriseethatthenumericandsemiclassical ;approachshowsreasonableagreementinthewholemag- ;neticfieldregilne.Sinceourperturbedcalculationforthe ;0-fluctuationdeterminantisonlyvalidinthehighmag- ;neticfield,theagreementinthelow?fieldregimeissur. ;prisinginsonicways.

    ;AsshowninFig.1,thegroundstatetunnelsplitting

    ;vanishesatthefieldHiHa=1.0Thisdisappearanceis ;evidentfortheextra/2phaseshift.sinceaccordingto ;Eq(21).thereshouldbepeakinusua1.Itisworth ;whilenotingthattheextra/2phaseshiftisnotlira- ;itedtoourbiaxialsymmetricantiferromagneticmolecu

    ;laxmagnetcaseIndeed,forthestrongantiferromagnetic ;coupling,uponGaussianintegrating(8)overthesmall ;displacementsandE,onecanobtaintheeffectiveL.d_ ;grangianingenera

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