Superfluid-Mott Insulator Transition of Spin-2 Cold Bosons in an Optical Lattice in a Magne

The superfluid-Mott insulator transition of spin-2 boson atoms with repulsive interaction in an optical lattice in a magnetic field is presented. By using the mean field theory, Mott ground states and phase diagrams of superfluid-Mott insulator transition

Super uid-MottInsulatorTransitionofSpin-2ColdBosonsinan

OpticalLatticeinaMagneticField

arXiv:cond-mat/0312456v1 [cond-mat.soft] 18 Dec 2003ShuoJin1 ,Jing-MinHou1 ,Bing-HaoXie2,Li-JunTian1,andMo-LinGe11TheoreticalPhysicsDivision,NankaiInstituteofMathematics,NankaiUniversity,Tianjin,300071,China,andLiuhuiCenterforAppliedMathematics,Tianjin,300071,China2LaboratoryforComputationalPhysics,InstituteofAppliedPhysicsandComputationalMathematics,Beijing,100088,ChinaAbstractThesuper uid-Mottinsulatortransitionofspin-2bosonatomswithrepulsiveinteractioninanopticallatticeinamagnetic eldispresented.Byusingthemean eldtheory,Mottgroundstatesandphasediagramsofsuper uid-Mottinsulatortransitionatzerotemperaturearerevealed.Appliedmagnetic eldleadstosomephaseboundariessplitting.ForalltheinitialMottgroundstatescon-tainingmultiplespincomponents,di erentspincomponentstakeondi erentphaseboundaries.Itisfoundthatinthissystemthephaseboundarieswithdi erentmagnetizationcanbemovedindi erentwaysbyonlychangingtheintensityoftheappliedmagnetic eld.PACSnumber(s):03.75.Kk,03.75.Lm,03.75.Mn,32.80.Pj

The superfluid-Mott insulator transition of spin-2 boson atoms with repulsive interaction in an optical lattice in a magnetic field is presented. By using the mean field theory, Mott ground states and phase diagrams of superfluid-Mott insulator transition

I.INTRODUCTION

Recentremarkableexperiments[1,2,3,4]onthesuper uid(SF)toMottinsulator(MI)transitioninasystemofultracoldatomsinanopticallatticeopenintriguingprospectsforstudyingmany-bodyphenomena,associatedwithstronglycorrelatedsystemsinahighlycon-trollableenvironment.Theopticallattices[5,6]—arraysofmicroscopicpotentialsinducedbytheacstarke ectofinterferinglaserbeams—provideidealconditionsforthestudyofthelasercoolingandthequantumphasetransitionsofthecon nedcoldatoms.Thedynamicsofthecon nedatomsinopticallatticesisadequatelydescribedbytheBose-Hubbardmodel

[7,8],whichpredictsSF-MItransitionatlowtemperaturewithincreasingtheratiooftheon-siteinteractiontothehoppingmatrixelement.Besidesmanyexperimentale ortsmadetorealizeSF-MItransition,alargenumberoftheoreticalstudieshaveappeared[7,8,9,10].Inreference[9]anappropriatemean- eldapproximationwasdevelopedfortheHamiltonianofspinlessorpolarizedbosonsinanopticallattice,andinordertodescribethezero-temperaturephasetransitionfromthesuper uidtotheMott-insulatingphase,thephasediagramswerecalculated.

Sinceopticaltraps[11,12,13]liberatethespindegreesoffreedomandmakepossiblecondensationofspinorbosons,extensiveinterestshavebeenstimulatedinthestudyofmulti-componentspinorBEC.ThequantumphasetransitioninspinorBEC,aswellasavarietyofothernovelphenomena[14,15,16,17,18]werewellstudied.Subsequently,inspiredbytheseworks,thetheoreticalresearchesabouttheSF-MItransitionofthespinorbosonstrappedinanopticallatticearise.Mostrecently,DemlerandZhou[19]havestudiedspin-1Boseatomsinanopticallatticeandobtainedseveraluniqueproperties.Tsuchiyaetal.[20],HouandGe[21]haveinvestigatedthespin-1andspin-2bosonsinanopticallatticewiththemean- eldapproximationmethodandobtainedthephasediagramsshowingatransitionfromMottinsulatortosuper uid,respectively.

Ontheotherhand,theresponsetoexternalmagnetic eldofBECisalsoatopicwithinterests[17,18,22,23,24,25].Theexperimentalistshaveconcentratedoninvestigatingthesystemsinanappliedmagnetic eldbecausethephasetransitioncanbetunedbyadjustingthemagnetic eldratherthanchangingthesamplesmeasured[26,27,28,29].Thusthetheoreticalstudyinthisaspectisnecessary.Sofar,UedaandKoashi[17,18]havediscussedthemagneticresponseofspin-1andspin-2BECinamesoscopicregime;astothesysteminanopticallattice,SvicainskyandChui[30]havestudiedthespin-1bosonsinamagnetic eldandshown

The superfluid-Mott insulator transition of spin-2 boson atoms with repulsive interaction in an optical lattice in a magnetic field is presented. By using the mean field theory, Mott ground states and phase diagrams of superfluid-Mott insulator transition

somee ectsinducedbythemagnetic eld.However,thespin-2caseinamagnetic eldhasnotbeendiscussedyet.Whataboutthegroundstates,quantumphasetransitionandthein uenceofthemagnetic eldonthissystem?Thisisourmissioninthepresentletter.First,ignoringthehoppingtermoftheHamiltonian,wegetthesite-independentHamiltonian,itsenergyeigenvaluesandtheMottgroundstatesfordi erentcases.Then,applyingmean- eldapproximationandregardingthehoppingtermasaperturbation,weperformthecalculationsinsecond-orderanddrawthephasediagrams.Theresponseofthephasediagramstotheappliedmagnetic eldisqualitativelyanalyzedsubsequently.Finally,wegivesomeremarksandtheconclusion.

II.THEMODEL

Weconsideradilutegasofbosonatomswithhyper nespinF=2,suchas23Na,87Rbor85Rbsubjecttoanexternalmagnetic eldinanopticallattice.BasedonRef.[21],theHamiltonianofspin-2bosonswithrepulsiveinteractioninanopticallatticeincludingamagnetic eldtermcanbewritteninthesecond-quantizedform:

H=HA+HB,

(1)

Ψ αΨβΨβΨαHA=

+dr[h¯2c¯1

The superfluid-Mott insulator transition of spin-2 boson atoms with repulsive interaction in an optical lattice in a magnetic field is presented. By using the mean field theory, Mott ground states and phase diagrams of superfluid-Mott insulator transition

Forsimplicity,weassumeauniformmagnetic eldappliedalongthez-direction,anditisweakenoughtoignorethequadraticZeemane ect.Expandingthe eldoperatorsintheWannierbasisandkeepingonlythelowestvibrationalstates,Ψα=

reducestothegeneralizedBose-HubbardHamiltonian

H= t ibiαw(r ri),Eq.(1)b ibj

<i,j> µ in i+c02 i 2 6 (Fni)+i2c2

2n ( n 1)+c1

5 +S pF z.S(5)

Forthesakeofstudyingthequantumtransition,themean- eldapproximation[9]isusedandthehoppingtermisconsideredasaperturbation.Introducingthesuper uidorderpa-

√rameterφα=<biα>=

The superfluid-Mott insulator transition of spin-2 boson atoms with repulsive interaction in an optical lattice in a magnetic field is presented. By using the mean field theory, Mott ground states and phase diagrams of superfluid-Mott insulator transition

III.THEENERGYEIGENVALUESANDTHEMOTTGROUNDSTATESBeforeperturbativecalculations,wesolvetheequation

h0ψ=ε(0)ψ(8)

introducedandtheoperatorS+raisesnsbyonewhiletherelationn=2ns+n0holds,wheren

andthemagneticquantumisthetotalnumberofbosonsinasinglesite.ThespinoperatorF

zcommutewiththeoperatorsS ±,andthen,theeigenstatesψaredenotednumberoperatorF

as|n0,ns,F,Fz;λ>whereλlabelsorthonormaldegeneratestates.Theenergyeigenvalueisgivenby

ε(0)= µn+c0

withS=(2n0+5)/4(n0=0,1,2,...)andSz=S+ns(ns=0,1,2,...),whichguarantees +S =S 2 S z S 2ispositivesemide nite.Thenewquantumnumbersnsandn0arethatSz ±andS z[17,18].TheeigenvaluesofthemutualeigenstatesforS 2andS zare{S(S 1),Sz}S +andS satisfythetogettheeigenvaluesandeigenstatesofh0.InEq.(5),theoperatersS z,S ±]=±S ±,[S +,S ]= 2S ztogetherwithS z≡SU(1,1)commutationrelations,namely[S 2≡ S +S +S 2 S zcommuteswith(2 n+5)/4,andasaconsequence,theCasimiroperatorSz2[F(F+1) 6n]+

i2c22 ) pFz,(9)SoMottstatescanbeexpressedas

numberoflatticesites.thehomogenouscase,thezerothordertotalenergyisE(0)=|n0,ns,F,Fz;λ>i,whereiislatticesiteindex.Foriε(0)=Nlε(0),whereNlisthe

Theenergyeigenstates|n0,ns,F,Fz;λ>canberepresentedas[18]

n20 n12 (2) n22 (3) n30 (3) n33 ) F(A (2)(FP(ns=0)(b (A2)(A0)(A3)|vac>,0)2)(10)

where

(2)A=0

(2)A2

(3)A01 2[(b 0) 2b1b 1+2b2b 2],10√123(b [2=1)],14√1√ 2b1b0b 1+3=[210√ 3[(b) 201(11)(12)23b 2(b 1) 6√

The superfluid-Mott insulator transition of spin-2 boson atoms with repulsive interaction in an optical lattice in a magnetic field is presented. By using the mean field theory, Mott ground states and phase diagrams of superfluid-Mott insulator transition

ns=n20,

F=2n12+2n22+3n33,

Fz=F F.(16)(17)(18)

FromEq.(9),weseethattheminimumenergystatesalwayssatisfyFz=Fwhenp>0.Thus,theproblemof ndingthegroundstatesreducestominimizingthefunction:

ε(0)(F,ns)= µn+

+2c2

2c02).[(F 2p c14c31](19)

Thegroundstatesofh0dependontherelationamongc1,c2,pandµ.Whenn=1,thegroundstateis|1,0,2,2;λ>;whenn≥2,therearefoursortsofgroundstatesclassi edbydi erentsigncombinationsofc1andc2.Heretheclassi cationoftheMottgroundstatesdi ersfromthatofRef.[18],whichislabelledbyferromagnetic,antiferromagneticandcyclicphases.Ingeneral,forp>0,whenc1<0,c2>0(ferromagneticcase),thethirdandthefourthterminEq.(19)possesstheminimalvaluessimultaneously,butwhenc1<0,c2<0(ferromagneticorantiferromagneticcase),c1>0,c2>0(ferromagneticorcycliccase)andc1>0,c2>0(ferromagneticorantiferromagneticcase),thesituationbecomescomplicated,sincesometimesthereexiststhecompetitionbetweencontributionsoftotalspinandthesinglet“pairs”toeigenenergy.WelistallMottgroundstatesindetailinappendix.

IV.PHASEDIAGRAMSOFSUPERFLUID-MOTTINSULATORTRANSITIONFromSectionIII,weknowthatF=Fzalwaysholdsinthegroundstates,whileintheexcitedstatesthisisnotthecase.Asaresultoftheappliedmagnetic eld,thestatedegeneracyfromthedi erentmagneticquantumnumbersFz[21]islifted.Morerichphasediagramsareexpectedthanthoseintheabsenceofthemagnetic eld.

Todepictthephasediagrams,weconsiderthehoppingtermastheperturbativeoneandcalculatethe rst-andsecond-ordercorrectionstothegroundenergy,whichareexpressedas

ε(1)g=<g|h1|g>=zt

ε(2)g= αφ αφα,α= 2,...,2,

,α= 2,...,2.(20)(21)|<g|h1|m>|2

n=g(0)εg (0)εm

The superfluid-Mott insulator transition of spin-2 boson atoms with repulsive interaction in an optical lattice in a magnetic field is presented. By using the mean field theory, Mott ground states and phase diagrams of superfluid-Mott insulator transition

Here|g>denotesthegroundstatediscussedintheappendix,and{|m>}representexcitedstatesexpressedasaclusterofquantumnumbersincludingn0,ns,F,Fz,λ.Wecancalculateallgivestheformofthemodi edgroundenergyas

(1)(2)(0)εg=ε(0)g+εg+εg=εg+ztthenonzeromatrixelementsof<g|bα+b α|m>.Therefore,second-orderperturbationtheory αAα(n,µ ,c 0,c 1,c 2,p )φ αφα,α= 2,...,2,(22)

whereAα(n,µ ,c 0,c 1,c 2,p )isrelatedtothe rst-andsecond-ordercorrectionsofthespincompo-nentwithmagneticquantumnumberαtothezeroth-ordergroundenergy.Itdependsonsystemparametersn,µ ,c 0,c 1,c 2andp ,whereµ =µ/zt,c 0=c0/zt,c 1=c1/zt,c 2=c2/zt,p =p/ztaredimensionless.MinimizingthegroundenergyfunctionEq.(22),we ndthatφα=0Aα(n,µ ,c 0,c 1,c 2,p )=0signi estheboundarybetweenthesuper uidandtheMottinsulatorphasesofthespincomponentwithmagneticquantumnumberα.

Usingtheperturbationtheory,wecananalyticallydeterminedthephasediagramsFig.1-5fordi erentcases.ThephasediagramsindicatethatthereexistsaphasetransitionfromMottinsulatorwithintegernumberbosonstosuper uidwhentheratioc0/tisdecreasedtoacriticalvalue.Inthezeroth-order,i.e.,neglectingthehoppingterm,thegroundstateisMottstateinwhichtheoccupationnumberpersiteispinnedatintegern=1,2,...,correspondingtoacom-mensurate llingofthelattice.Di erentgroundstatesmaycontaindi erentspincomponents.Forexample,thereisonlyspincomponentwithZeemanlevelm=2whenoccupationnumber

(2)persiten=1;spincomponentswithm=0,±1,±2forMottstatei(A0|0>)i,andspin

componentswithm=0,1,2for whenAα(n,µ ,c 0,c 1,c 2,p )>0andφα=0whenAα(n,µ ,c 0,c 1,c 2,p )<0.Thismeansthatonespincomponent,onesuper uidcomponentoccurswhenloweringtheratioc0/t,suchasthestatescontainingmultiplespincomponents,whenloweringtheratioc0/t,multiplesuper uid (2)i(A2|0>)i.FortheinitialMottgroundstateincludingonlycasen=1,2,3inFig.1,n=1,2inFig.2,andn=1inFig.3-5;foralltheinitialMottgroundcomponentsappear,andthephaseboundariesbetweensuper uidandMottinsulatorphasefordi erentspincomponentsaredistinct,forinstance,n=3inFig.2,n=2,3inFig.3-5.Afteranalyzingthephasediagrams,we ndthatthepositionofphaseboundaryisrelatedtoaverageoccupationnumberofspincomponentintheinitialMottgroundstate,i.e.,thelargertheaverageoccupationnumberofspincomponentpersiteis,theeasierthetransitionfromMottinsulatortosuper uidphase.Wealso ndthatsomeboundariesbetweensuper uidandMottinsulatorphaseswithmulti-spincomponents,suchasn=2inFig.3,5andn=3inFig.3,4,willturntobeidenticalwhenthemagnetic eldvanishes.Wecandrawtheconclusion

The superfluid-Mott insulator transition of spin-2 boson atoms with repulsive interaction in an optical lattice in a magnetic field is presented. By using the mean field theory, Mott ground states and phase diagrams of superfluid-Mott insulator transition

thattheappliedmagnetic eldresultsinsomephaseboundariessplitting.

Furthermore,thephasediagramsFig.6(a)andFig.6(b)fordi erentintensitiesoftheapplied eldaredrawn.Thepositionofsomephaseboundariesisrelatedtotheintensityoftheappliedmagnetic eld.InFig.6(a),whentheappliedmagnetic eldincreases,forthesameMottgroundstatecontainingonlyonespincomponentwithZeemanlevelm=2,thephasediagramswillshiftalongthedirectionwithchemicalpotentialdecreasing.InFig.6(b),forthesameMottgroundstatecontainingspincomponentscorrespondingtoZeemanlevelsm=0,±1,±2,whenthemagnetic eldincreases,thephaseboundariesofspincomponentswithZeemanlevelsm=±1,±2move,butthephaseboundaryofspincomponentwithZeemanlevelm=0keepsinvariant.Moreover,onecanseethatforpositiveandnegativeZeemanlevels,thephaseboundariesbetweenSFandMIwillmoveinoppositedirections,andforthespincomponentwithpositiveZeemanlevel,thetransitionfromMItoSFbecomeseasierwhentheappliedmagnetic eldincreases.

V.REMARKSANDCONCLUSION

Forsimplicity,inthispaperweassumetheuniformmagnetic eldisappliedalongthez-direction,andpispositive.Infact,theanalysisoftheMottgroundstatesforp>0isenoughsincethesignofpdoesnotalterthephysics.Whenp<0,theMottgroundstatessatisfyFz= F,andhavethesameformasthoseinthecaseofp>0.We ndthatthein uenceofmagnetic eldonthephasediagramsismanifold,andtheboundariesbetweenSFandMIareessentiallydependentonthemagneticpropertiesofthegroundstates.Thisworkonlydiscussthephasediagramscorrespondingtodi erentmagnetic eldswiththesameMottgroundstate,i.e.,thecontinuouschangeofthephaseboundaries.ThecasethatMottgroundstateschangewhenthemagnetic eldincreases,i.e.,thesuddenjumpofthephaseboundaries,isbeyondthescopeofthepresentpaperandisnotshown.Forinstance,whenc1>0,c2>0,ifpincreasesfrom0tothevaluelargeenoughtosatisfyp/(2n+1)>c1,themagnetizationFcanjumpfromtheminimumtothemaximumone.Theyarethesubjectoffuturestudy.Inaddition,itisworthtonotethatphasediagramsofthezeromagnetic eldcannotbederivedbytakingp=0simply,sinceourderivationisbasedonthedegeneracylifting.

Inconclusion,wehaveinvestigatedthequantumphasetransitionfromMottinsulatortosuper uidphaseofspin-2coldbosonswithrepulsiveinteractioninopticallatticesunderthein uenceofauniformmagnetic eldatzerotemperature.Thephasediagramsshowthat

The superfluid-Mott insulator transition of spin-2 boson atoms with repulsive interaction in an optical lattice in a magnetic field is presented. By using the mean field theory, Mott ground states and phase diagrams of superfluid-Mott insulator transition

thesystemundergoesaphasetransitionfromMottinsulatorwithintegernumberbosonsateachsitetosuper uidphasewhentheratioc0/tisdecreasedtoacriticalvalue.Di erentMottgroundstatesmaycontaindi erentspincomponents.ThepositionofphaseboundaryisrelatedtoaverageoccupationnumberofspincomponentintheinitialMottgroundstate.FortheinitialMottgroundstateincludingonlyonespincomponent,onesuper uidcomponentappearswhenloweringtheratioc0/t.ForsomeMottgroundstateswithmultiplespincomponents,theappliedmagnetic eldleadstothesplittingofthephaseboundaries,sothatthephaseboundariesbetweensuper uidandMott-insulatorphasefordi erentspincomponentsaredistinctinallgroundstates.Inparticular,wedrawthephasediagramscorrespondingtodi erentintensitiesmagnetic eldfortheinitialMottgroundstatecontainingone-spinandmulti-spincomponents.Theyqualitativelyshowthewayofthephaseboundaries’movingwiththeintensityoftheappliedmagnetic eld.Itisfoundthatthephaseboundariescanbemovedbyonlychangingtheintensityoftheappliedmagnetic eld.ForthespincomponentwithpositiveZeemanlevel,thelargerintensityofthemagnetic eldis,theeasierthetransitionfromMItoSFhappens.Thesetheoreticalresultsareexpectedtobepracticallyhelpfultotheexperimentalstudyofthe eld-tunedSF-MItransitionofboseatomswithhyper nespininanopticallattice.Acknowledgments

ThisworkisinpartsupportedbyNSFofChinaGrantNo.A0124015.

APPENDIX:LISTOFTHEMOTTGROUNDSTATES

1.c1<0,c2>0.Because(2p c1)/(2c1)≤0,thethirdtermandthefourthterminEq.(19)havetheminimalvaluesatthesametimewhenF=2nandns=0.Hence,thegroundstateis|n,0,2n,2n;λ>.

2.c1<0,c2<0.Therelation(2p c1)/(2c1)≤0stands,andthecompetitionbetweencontributionsoftotalspinandthesinglet“pair”happensinthepremiseofF

80c1 4c2

.(A.1)

The superfluid-Mott insulator transition of spin-2 boson atoms with repulsive interaction in an optical lattice in a magnetic field is presented. By using the mean field theory, Mott ground states and phase diagrams of superfluid-Mott insulator transition

Becausethesinglet“pair”numbermustbeaninteger,wewritensintermsoftheclosest

0integernumbern0sandthedecimalpart,i.e.ns=ns+α,wherethenumberαsatis es

1/2<α<1/2,whichcanberewrittenas,

n0s 1

80c1 4c2<n0s+1

2,theeigenenergyislowerwhenthesinglet“pair”numberisn0sthan

000anyotherinteger.Sothegroundstateis|n 2n0s,ns,2n 4ns,2n 4ns;λ>.

(c)Forn0s≥n

2c1=F0+α,(A.3)

whereF0istheintegernumberwhichistheclosestintegertoF1.InEq.(19),whenns=0thefourthtermhastheminimalvalue,andiftotalspinFtakestheappropriateintegerthethirdtermpossessestheminimalone.

(a)WhenF0≤0,i.e.,c1>p,theenergyeigenvaluehasitsminimumifF=0andns=

0.However,wemustnoticesomespecialcasesbecausetherehavesomeforbiddenandF=0,1,3,2n0 1areforbiddenwhenn0=3k±1(k∈Z).values[17,18],thatis,F=1,2,5,2n0 1arenotallowedwhenn0=3k(k∈Z),

i.Forn=3k(k∈Z),|n,0,0,0;λ>withns=0andF=0isthegroundstate.

ii.Forn=3k 1(k∈Z),thestatewithns=0andF=0isnotallowed

simultaneously.Ifns=0,thelowestallowedvalueoftotalspinFis2.On

theotherhand,F=0isnotforbiddenwhenn0is3k(k∈Z).ns=1isthe

lowestvaluesatisfyingtheconditiondueton=2ns+n0.Sothestatewith

ns=1andF=paringbotheigenenergies

forthetwocasesF=0,ns=1andF=2,ns=0,wegetthegroundstate,

(i)|n 2,1,0,0;λ>forc2<(15c1 10p)/(2n+1)and(ii)|n,0,2,2;λ>forc2>(15c1 10p)/(2n+1).

The superfluid-Mott insulator transition of spin-2 boson atoms with repulsive interaction in an optical lattice in a magnetic field is presented. By using the mean field theory, Mott ground states and phase diagrams of superfluid-Mott insulator transition

iii.Forn=3k+1(k∈Z),thecompetitionexistsbetweencontributionsoftotalspin

andthesinglet“pair”toeigenenergy.ButifF=0,nsisatleast2.Therefore,

thegroundstateis(i)|n 4,2,0,0;λ>forc2<(15c1 10p)/2(2n 1);(ii)|n,0,2,2;λ>forc2>(15c1 10p)/2(2n 1).

(b)WhenF0≥2n,i.e.,c1<p/(2n+1),thegroundstateis|n,0,2n,2n;λ>with

F=2nandns=0.

(c)When0<F0<2n,i.e.,p/(2n+1)<c1<p,theeigenenergyisthuslowerwhenF

isclosertoF1andwhennsissmaller.ExceptforthecaseofF0takingtheforbiddenvaluesofF,thegroundstateis|n,0,F0,F0;λ>.WhenF0equalstotheforbiddenvaluesofF,Fmaytaketheallowedintegernext-nearesttoF1,i.e.,F0±1orF0±2,3isrealizedbysettingnsas0,1,or2owingtotherelationn=2ns+n0.Whetherorand|n 4,2,F0,F0;λ>canbethelowest-energystatedependsontheratioc2/c1.notthestates|n,0,F0±1,F0±1;λ>,|n,0,F0±2,F0±2;λ>,|n 2,1,F0,F0;λ>,oratthecostofincreasingnsto1or2,sinceanyofthethreevalues0,1,2ofn0mod

4.c1>0,c2<0.TheeigenenergyisthelowestifF=F0andnsatitshighestvalue.However,thesetwochoicesarenotalwayssatis edsimultaneously.

(a)F0≤0,i.e.,c1>p.

i.Whenniseven,|0,n/2,0,0;λ>isthegroundstate.

ii.Whennisodd,nshasthehighestvalue(n 1)/2.ButFisnotzerowhenns=

(n 1)/2.Alternatively,thereisanothercasethatF=0andns=(n 3)/2.

(ii)|3,(n 3)/2,0,0;λ>forc1>(7|c2|+10p)/15.

case2.,itisdividedintothreecases.

i.Forn0s≤0,thegroundstateis|n,0,2n,2n;λ>.

ii.For0<n0s<

2nHence,thegroundstateis(i)|1,(n 1)/2,2,2;λ>forc1<(7|c2|+10p)/15;(b)F0>0,i.e.,c1<p.Notethatn0sisshowninEq.(A.2),similartotheanalysisof,(i)whenniseven,|0,n/2,0,0;λ>isthegroundstate;(ii)whenn

isodd,|1,(n 1)/2,2,2;λ>does.

The superfluid-Mott insulator transition of spin-2 boson atoms with repulsive interaction in an optical lattice in a magnetic field is presented. By using the mean field theory, Mott ground states and phase diagrams of superfluid-Mott insulator transition

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The superfluid-Mott insulator transition of spin-2 boson atoms with repulsive interaction in an optical lattice in a magnetic field is presented. By using the mean field theory, Mott ground states and phase diagrams of superfluid-Mott insulator transition

FIG.1:ThephasediagramofBose-HubbardHamiltonianobtainedfromsecond-orderperturbationtheorywithsolidlinesforc1= 0.1zt,c2=0.1ztandp=0.2zt.Thedashedlinesindicatethezeroth-orderphasediagram.

The superfluid-Mott insulator transition of spin-2 boson atoms with repulsive interaction in an optical lattice in a magnetic field is presented. By using the mean field theory, Mott ground states and phase diagrams of superfluid-Mott insulator transition

FIG.2:ThesameasinFig.1,butforc1= 0.02zt,c2= 0.25ztandp=0.01ztwithsolidlines.Forn=3,theinteriorlineisthephaseboundaryofspincomponentwithZeemanlevelm=2;themiddlelinethephaseboundaryofspincomponentwithZeemanlevelm= 2,andtheexternaltriplelines,whicharetooclosetobedistinguished,thephaseboundariesofspincomponentswithZeemanlevelsm=0,±1.TheinsetshowsanexpansionoftheregionlabelledbyA.

The superfluid-Mott insulator transition of spin-2 boson atoms with repulsive interaction in an optical lattice in a magnetic field is presented. By using the mean field theory, Mott ground states and phase diagrams of superfluid-Mott insulator transition

FIG.3:ThesameasinFig.1,butforc1=0.22zt,c2=0.1ztandp=0.05ztwithsolidlines.Whenn=2,theregionlabelledbyBisenlargedintheleftinset;the velinesrepresentthephaseboundariesofspincomponentswithZeemanlevelsm= 2, 1,0,1,2respectively(frominteriortoexternal).Forn=3,the velinesaretooclosetobedistinguished,sotheregionlabelledbyCisenlargedintherightinset,inwhichthe velinesthephaseboundariesofspincomponentswithZeemanlevelsm=0,±1,±2.

The superfluid-Mott insulator transition of spin-2 boson atoms with repulsive interaction in an optical lattice in a magnetic field is presented. By using the mean field theory, Mott ground states and phase diagrams of superfluid-Mott insulator transition

FIG.4:ThesameasinFig.1,butforc1=0.18zt,c2=1.0ztandp=0.05ztwithsolidlines.Forn=2,thethreelinesrepresentthephaseboundariesofspincomponentswithZeemanlevelm=1,0,2respectively(frominteriortoexternal).Whenn=3,the velinesaretooclosetobedistinguished,sotheregionlabelledbyDisenlargedintheinset,inwhichtheexternallineisthephaseboundaryofspincomponentwithZeemanlevelm=2,andtheotherfourlinesthephaseboundariesofspincomponentswithZeemanlevelsm=0,±1, 2.

The superfluid-Mott insulator transition of spin-2 boson atoms with repulsive interaction in an optical lattice in a magnetic field is presented. By using the mean field theory, Mott ground states and phase diagrams of superfluid-Mott insulator transition

FIG.5:ThesameasinFig.1,butforc1=0.02zt,c2= 0.1ztandp=0.008ztwithsolidlines.Whenn=2,the velinesaretooclosetobedistinguished,sotheregionlabelledbyEisextendedintheleftinset,inwhichtheinteriorlineisthephaseboundaryofspincomponentwithZeemanlevelm= 1;theexternallinethephaseboundaryofspincomponentwithZeemanlevelm=1;thethreemiddlelinesthephaseboundariesofspincomponentswithZeemanlevelsm=0,±2.Forn=3,theexteriorlineisthephaseboundaryofspincomponentwithZeemanlevelm=2;themiddlelinethephaseboundaryofspincomponentwithZeemanlevelm= 2,andtheexternaltriplelines,whicharetooclosetobedistinguished,thephaseboundariesofspincomponentswithZeemanlevelsm=0,±1.TherightinsetshowsanexpansionoftheregionlabelledbyF.

The superfluid-Mott insulator transition of spin-2 boson atoms with repulsive interaction in an optical lattice in a magnetic field is presented. By using the mean field theory, Mott ground states and phase diagrams of superfluid-Mott insulator transition

FIG.6:ThephasediagramsofBose-HubbardHamiltonianobtainedfromsecond-orderperturbationtheoryfordi erentp.In(a),c1= 0.1zt,c2=0.1zt,withp=0.2zt(solidlines)andp=0.8zt(dashedlines)respectively.In(b),whenn=2,c1=0.22zt,c2=0.1zt,withp=0.05zt(solidlines)andp=0.08zt(dashedlines)respectively;themiddlesolidlineexpressesthephaseboundaryofspincomponentwithZeemanlevelm=0,andkeepsinvariantwhenp=0.08zt;the vesolidlinesrepresentthephaseboundariesofspincomponentswithZeemanlevelsm= 2, 1,0,1,2respectively(frominteriortoexternal);thefourdashedlinesrepresentthephaseboundariesofspincomponentswithZeemanlevelsm= 2, 1,1,2respectively(frominteriortoexternal).

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