Model Exact Low-Lying States and Spin Dynamics in

arXiv:cond-mat/0201330v1[cond-mat.mes-hall]18Jan2002ModelExactLow-LyingStatesandSpinDynamicsinFerricWheels;Fe6toFe12IndranilRudra1,S.Ramasesha1andDiptimanSen21SolidStateandStructuralChemistryUnitIndianInstituteofScience,Bangalore560012,India2CentreforTheoreticalStudiesIndianInstituteofScience,Bangalore560012,IndiaUsinganefficientnumericalschemethatexploitsspatialsymmetriesandspin-parity,wehaveobtainedtheexactlow-lyingeigenstatesofexchangeHamiltoniansforferricwheelsuptoFe12.ThelargestcalculationinvolvestheFe12ringwhichspansaHilbertspacedimensionofabout145millionforMs=0subspace.Ourcal-culatedgapsfromthesingletgroundstatetotheexcitedtripletstateagreeswellwiththeexperimentallymeasuredvalues.Studyofthestaticstructurefactorshowsthatthegroundstateisspontaneouslydimerizedforferricwheels.Spinstatesofferricwheelscanbeviewedasquantizedstatesofarigidrotorwiththegapbetweenthegroundandthefirstexcitedstatedefiningtheinverseofmomentofinertia.WehavestudiedthequantumdynamicsofFe10asarepresentativeofferricwheels.Weusethelow-lyingstatesofFe10tosolveexactlythetime-dependentSchr¨odingerequationandfindthemagnetizationofthemoleculeinthepresenceofanalter-natingmagneticfieldatzerotemperature.Weobserveanontrivialoscillationofmagnetizationwhichisdependentontheamplitudeoftheacfield.WehavealsostudiedthetorqueresponseofFe12asafunctionofmagneticfield,whichclearlyshowsspin-statecrossover.PACSnumbers:75.45.+j,75.50.Xx,75.60.EjI.INTRODUCTIONInrecentyearspolyoxometalateswhicharepracticalrealizationofnanomagnetshavebecometheareaofintenseresearchbecauseoftheirenormousvarietyofstructureandfascinatingmagneticproperties.Aparticularaestheticclassisthatofthering-shapediron(III)compoundsdenotedasmolecularferricwheels.ThedecanuclearwheelFe10synthesizedbyLippardet.al.1mayberegardedasaprototypeofthisclass.Meanwhilesynthesisofferricwheelswith6,8,12and18havealsobeenreported2.Thesematerialshavedominantanti-ferromagneticcouplingbetweenFe(III)spinsandasingletgroundstate.Themagneticpropertiesofsuchnanoscopicmoleculesresultfromtheinterplayofsuperexchangeinteractionsbetweentheatomicspins,dipolarcouplingofthelocalmomentsandon-siteanisotropiesarisingfromligandconfigurations.TheemergenceofferricwheelshasledtoarenewedinterestinthepropertiesoftheHeisenbergchain,especiallyforlargespinvalues.In1964Fisherusedaclassicaltreatment3,inwhichspinquantization1isabsent,tostudythepropertiesofHeisenbergHamiltonianandfoundanalyticalsolutionsforthethermodynamicpropertiesofthesystem.NumerousquantummechanicalcalculationshavebeenmadeforHeisenbergHamiltonian,mostlyforspin-1/2chains.CalculationsforlargerSbecamemoreinterestingafterHaldane’sconjecture4regardingthepresenceofanenergygapintheexcitationspectrumfromthegroundstateforintegerSchains.TofittheexperimentaltemperaturedependenceofthemagneticsusceptibilitydataforFe10Lip-pardet.al.adoptedaclassicalspintreatmenttoobtainthevalueofexchangeinteractionparameterJ.Butbelow50Kthistreatmentfails.FerricwheelswithS=5/2,andsystemsizeofupto8siteshavebeentreatedexactlyusingirreducibletensoroperatortechnique5withtheaidofpointgroupsymmetryaswellasbyusingtheinvarianceofspinHamiltonianwithregardtointerchangeofspinsites6.RecentlytherehavebeentheoreticalstudiestoexplainthelowtemperaturemagneticsusceptibilitydataofFe107.Themagnetizationofferricwheelsexhibitasteplikefielddependenceatlowtemperatureduetotheoccurrenceoffieldinducedgroundstatelevelcrossing,aspectacularmanifestationofquantumsizeeffectinthesenanomagnets1,8.TherehavebeenNMRandspecificheatstudiesoftheseferricwheelstoinvestigatetheenergylevelstructure9,10.Inappropriateparameterregimetheseringsystemsareconsideredtobethecandidatesfortheobservationofmacroscopicquantumphenomena,intheformofquantumcoherenttunnelingoftheNeelvector11.Tounderstandtheselowtemperaturepropertiesofferricwheelsindetailweneedtoknowthelowlyingeigenvaluespectrumforthesesystems.ItisalsoofinteresttocompareandcontrastthezerotemperaturedensityofstateofS=5/2ringswiththatoftheexactlyknownS=1/2chain.Inthispaperwehaveusedspinparitytogetherwithrotationalsymmetryoftheferricwheelstoobtainthelow-lyingeigenvaluespectrumforringswith6,8,10and12spin-5/2.Thedispersionspectrumrevealsinterestingfeatures.Therehavebeenrecentreportsofsolvingthelow-lyingeigen-spectrumofFe10usingtheDensityMatrixRenormalizationGroup12.Tothebestofourknowledgeoursisthefirststudyofofferricwheelsusingexactmethodsuptoaringsizeof12,S=5/2iron(III)ions.WehavealsostudiedthequantumspindynamicsofferricwheelsinpresenceofanalternatingfieldaftersettinguptheHamiltonianmatrixinthesubspaceoflow-lyingstates.ThisHamiltonianincludesmultipolarspin-spininteractiontermsbesidesatimevaryingmagneticfield(Zeemanterm).Wehavethenevolvedaninitialstate,whichistakentobethegroundstatewithaspecificvalueofMS(thez-componentofthetotalspin)intheabsenceofthemagneticfield,byusingthetime-dependentformulationoftheproblemintherestrictedsubspace.Weobserveanontrivialoscillationofthemagnetizationwhosefrequencydependsontheamplitudeofthealternatingfield.Thisphenomenaisverysimilartowhathasbeenobservedincaseofuniaxialmagnets13.2II.MODELHAMILT