By introducing a basis for a novel realization of the SU(4) Lie algebra, we exactly solve a spin-orbital chain with one-ion $L$-$S$ coupling (OILSC) via the Bethe ansatz (BA) approach. In the context of different Land\'e $g$ factors of the spin and orbital

E ectofone-ionL-Scouplingonmagneticpropertiesofacorrelatedspin-orbitsystem

Zu-JianYing1,2,3,Xi-WenGuan4,AngelaFoerster1,ItzhakRoditi3,BinChen2

1.InstitutodeF´ sicadaUFRGS,Av.BentoGon¸calves,9500,PortoAlegre,91501-970,Brasil

2.HangzhouTeachersCollege,Hangzhou310012,China

3.CentroBrasileirodePesquisasF´ sicas,RuaDr.XavierSigaud150,22290-180RiodeJaneiro,RJ,Brasil

4.DepartmentofTheoreticalPhysics,ResearchSchoolofPhysicalSciencesandEngineering,andCentreforMathematicsand

itsApplications,MathematicalSciencesInstitute,AustralianNationalUniversity,CanberraACT0200,Australia

()

ByintroducingabasisforanovelrealizationoftheSU(4)Liealgebra,weexactlysolveaspin-orbitalchainwithone-ionL-Scoupling(OILSC)viatheBetheansatz(BA)approach.Inthecontextofdi erentLand´egfactorsofthespinandorbitalsectors,theOILSCresultsinrichandnovelquantumphasetransitions.Someaccurateanalyticalexpressionsforthecritical eldsareobtained.Bothspinorderingandorbitalorderingarefoundinagappedsingletphase.Thesystemexhibitsmanyinterestingphenomenasuchasnonvanishingmagnetizationofthesingletphase,multi-entranceofthesingletinthegroundstatewhenthe eldvaries,unsymmetricmagnetizationintwo-componentphaseandmagnetizationcrossoverfordi erentOILSC.

arXiv:cond-mat/0309718v1 [cond-mat.str-el] 30 Sep 2003

Strongly-correlatedelectronsystemsinthepresenceoforbitaldegeneracyhaveattractedmuchinterestduetoexperimentaladvancesintransition-metalcompounds[1],andtheorbitalorderingaswellastheorbitaldensitywavehasbeenobservedinafamilyofmanganites[2].Amongtheexperimental ndingsarematerialsrelatedtospin-orbitalsystemsinonedimension,suchastetrahis(dimethylamino)ethylene-C60[3],arti cialquantumdotarrays[4],Na2Ti2Sb2OandNaV2O5degeneratechains[5],andsoforth.Awell-knownspin-orbitalmodelistheSU(4)model[6]whichisexactlysolvableinonedimen-sion[7,8].Thecompetitionoforbitaldegreeoffreedomwiththespinresultsinthreequantumphasetransitions(QPT’s)inamagnetic eld,accordingtobothnumerical[9]andanalytic[10]analysis.DeviationsfromtheSU(4)symmetrycanbecausedbyvariationsofdi erent-siteinteraction[3,5,11,12].AnotherpossibilityfordeviationwasconsideredforanSU(2) SU(2)Ising-typesame-siteanisotropyinRef.[13,10],whichmayoccurwhentheexternal eldturnsthespinorientedinparalleltotheorbitalangularmomentumfrozeninsomedirectionbyacrystalline eld.Amoregenerally-existingsame-sitein-teractionisSU(3) U(1)one-ionL-Scoupling(OILSC),thestrengthofwhichvariessubstantiallyinarangeof0～104cm 1(1cm 1～1K～10 4eV)fordi erentelements[14].Asitiswellunderstoodnow,L-Scouplingcancausemagneticanisotropy[15],asfarasasingleionisconcerned.Aninterestingproblemistoinvestigatethein uenceofsuchanOILSConthecollectiveproperties,especiallytheQPT’sofstrongly-correlatedspin-orbitalsystemsinamagnetic eld.ItcanbeexpectedthattheOILSCwillbringaboutnovelphysicsinthecompetitionofspinandorbitaldegreesoffreedom.

Inthepresentpaperweshallexactlysolveaspin-orbitalchainwithsuchanOILSCviatheBetheansatz(BA)approach.Inthecontextofthedi erentgfac-torsofspinandorbitsectors,theOILSCleadstoarichvarietyofnovelQPT’s,withtheanalyticcritical elds

1

explicitlyobtained.Bothspinorderingandorbitalor-deringexistinagappedsingletphase,accompaniedbyanonvanishingmagnetization.Theenergyofthesingletisnolongerinvariantinthe eldduetothedi erentgfac-tors,whichbringsabout veconsecutiveQPT’s.Italsocontributestounsymmetricchangesofthetotalmagne-tizationwithinasamephaseinweakandinstrong elds.WeconsideranN-sitechainwiththeHamiltonian

H=H0+HL S+M,H0=HL S=λ

i

N i=1

Pi,i+1,

(1)

li· si,M= gsH

i

z(gsszi+gtli),

where sand larespin-1/2operatorsforspinandor-bitandgsandgtarethecorrespondingLand´egfactors,

1

Pi,i+1=(2 si· si+1+2)exhangestheneigher-sitestates.Histhemagnetic eldandperiodicboundaryconditionsareappliedthroughout.HL-SistheOILSCfromtheNsitesandnotethatelectronshavepositiveλwhereasholeshavenegativeλ[15].Intheabsenceofthe eld,HL-SbreaksthesystemsymmetryintoSU(3) U(1)whenthebulkpartH0isSU(4)invariant[6].ThebasisconsistsofsingletandtripletoftheSU(2)Liealgebra{s++l+,s +l ,sz+lz},whichisvalidinthepres-enceofthe eldonlyifgs=gtasintheintegrablespinladder[16,17].However,HL-SandMdonotcommuteduetothedi erentgfactorsofspinandorbit,thecon-ventionalsingletandtripletarenolongertheircommoneigenstatesandfailtobethesolutioninthespin-orbitalsystem.Nevertheless,wecandiagonalizeHL-S+Masawhole,whichrequiresanewbasisforallsites 0=

φ1 y 1φ2

1+

y 2

, 1=φ3, 2=

φ1+yφ2

1+

y2

, 3=φ4,

whereφ1=|↑↓ ,φ2=|↓↑ ,φ3=|↑↑ ,φ4=|↓↓ de-notesitestate|szlz andy±1=±g H/λ+γ(H/λ),

By introducing a basis for a novel realization of the SU(4) Lie algebra, we exactly solve a spin-orbital chain with one-ion $L$-$S$ coupling (OILSC) via the Bethe ansatz (BA) approach. In the context of different Land\'e $g$ factors of the spin and orbital

221/2

γ(h)=(1+g h),g±=gs±gt.Thenewrealiza-tionoftheSU(4)LiealgebratakesSmn i=δni m,

S10=p(y)r(y)[yl+(1/2+sz) s+(lz+1/2)],S30=p(y)r(y)[ys (1/2 lz) l (1/2 sz)],S12=r(y)[l+(1/2+sz)+ys+(1/2+lz)],

S32=r(y)[s (1/2 lz)+yl (1/2 sz)],S13=s+l+,S02=p(y)r2(y)[y2s+l s l++y(sz lz)],(2)

3π 3ln3)/2.Buta

strongpositiveOILSCλ>λc+keepsthetripletgapful,theGSonlyconsistsofsingletandthesystemexhibitsspinandorbitalordering,asshownlateron.The rstQPTwilloccurwhenthe eldbringsdownthetriplet 1andclosesthegap =g(1)(0)atacritical eld

wherer(y)=1/

2r(y)[(y 1)

s±lz l±sz +

y+1M

(k 1)

2

(µk,k+1j,m)Ξ1

m=1

2π

n4

1

λ

4

λ

1

4

+

4

+

H=λ 8+

c0

g(4)

+(1 2g)

where g=g /g+.Furtherincreaseofthe eldwill

exhaustallthesingletandsofullypolarizetheGSat

Hλ+8+

f=

g+(1 2g)

,(5)

whichisvalidforallvaluesofλ.ForaweakOILSC0<λ<λc+,the eldbringsoutthecomponents 3, 2, 0inturnatthreedi erentcriticalpoints.

Forλ<0,thesingle-ionenergyE0ofthesinglet 0riseswhenthe eldbecomesstronger.Thetripletcom-ponent 3haslowerenergyinzero eldbutwillberaisedbythe eldinaquickerwaythan 0.Theenergycom-petitionof 3and 0resultsinnewkindsofQPT’s.Inthefour-componentGSaweaknegativeOILSCdoesnotexpelthesingletfarawayfromthetriplet.Inanincreas-ing eld 3risesinenergymorequicklyandsoongoesbeyond 0,so 3getsoutoftheGSbefore 0.Therearethreecritical eldsconsecutivelyexhausting 3, 0and 2fromtheGS.However,astrongernegativeOILSCleavesalongerdistancebetweenthetripletandsinglet.Before 3coversthisdistanceto 0,thesinglet 0hasalreadyrisencompletelyoutoftheGS,whichhappensinthe rstcriticalpoint.But 3soondrawsverynearto 0suchthat 0becomescloseenoughtotheGS,inwhich 3stilllies,andisdrawnbackintotheGS.ThisbringsaboutthesecondQPT.Afterwards,the eldover-whelmsthein uenceofOILSCandpumpout 3, 0and 2onebyone,whichgivesotherthreeQPT’s.Thereforethesystemundergoes veconsecutiveQPT’s(5-QPT)inthiscase,thevariationsofthecomponentsintheGSare:1230→123→1230(and1203)→120→12→1,wherewedenotethecomponent ibynumberiandeacharrowindicatesaQPT.Suchunusual5-QPToccurnearλc

(λ5c<λ<λc ,λ5c=.

0.712,forgs=2.0,gt=1.0).WhentheOILSCisnegativelystrongerthanJc ,thesingletisgapfulanddoesnotexistintheGSbeforethe eldisapplied.Butthequickerrisingof 3willgetthesingletinvolvedintheGS,aswhathappensinthesecondQPTofthe5-QPT.Insuchcase,fourconsecu-tiveQPT’s(4-QPT)willbeobserved:123→1230(and1203)→120→12→1.AnegativelystrongerOLSCthan2

By introducing a basis for a novel realization of the SU(4) Lie algebra, we exactly solve a spin-orbital chain with one-ion $L$-$S$ coupling (OILSC) via the Bethe ansatz (BA) approach. In the context of different Land\'e $g$ factors of the spin and orbital

λQwillexpelthesinglettoofarawayfromtheGScom-posedofthetriplet.Thetripletcomponent 3itselfwillgetoutoftheGSbeforeitbecomesadhesiveanddrawthesingletintotheGS.WethenhaveonlythetwoQPT’s.WepresentadetailedphasediagraminFig.1,thein-setillustratesthe5-and4-QPT.5-QPTexistifC QtendstoriseupatC ,withrequirementonthegfac-..

torsgt<(1 g5)gs/(1+ g5)=0.516gs,where g5=3[(1+w)a0]1/2/(4π)andw=2/3,satis edformosttyp-icalspin-orbitalsystems.Whenthetriplet 3attractsthesinglet 0intotheGS,somedensityof 3themselvesareattractedupwardsabitby 0,whichcausesaquickerincreaseinmagnetization.Thisattraction-and-counter-attractionismoresensitivenearthepointQwhere 3hassmallproportionintheGS.A4-QPTmagnetizationnearpointQisplottedinFig.2B.Themagnetizationinthe5-QPTcasehaslessapparentchangesinthe rsttwotransitionpointsduetoheavierdensityof 3.

Besidestheexactsingletphasecriticalpoint(4)andfull-polarizedpoint(5),othercriticalpointsoftheQPTcanbeobtainedfromWiener-Hopfmethod[21]

C+NQ.Hc=H++τk+G2 1,1, 1(λ,H+),QD.Hc=H +τk G2 1,1, 1(λ,H ),C+M.Hc=H0+τk0[wG231

2,

MQ.2Hc=H0+τk0[wG2 1,1,0(λ,H0)+G 1

sandorbitalangularmomentum l,themagnetization

zzz

M=gss+gtlcanbeobtainedforasingletsiteManytransition-metalelementsinvolveastrongOILSCandtheGSofthesystemcanbelocatedinthesingletphase,inwhichallsitesareoccupiedbythesinglet.Con-sequently,thespinswillbealignedferromagneticallyinthezdirectionwhilealltheorbitalsareferromagneticallyorderedintheoppositedirection.Therefore,bothorbitalorderingandspinorderingcanbeobserved.Inaddition,unliketheconventionalsinglet,themagnetization Mz 0isnolongerzerointhe eld,asshownbythedashlinesofcurveseandfinFig.2A. Mz 0increasesalmostinalinearwayduetoasmallvalueofH/λbeforethesingletphaseistransitedtoanotherphaseatHc0.Butthein-creasingrated Mz 0/dHbecomessmallerwhenthe eldgetsstronger.Thiswillleadtounsymmetricincreasingofthemagnetizationinweakandinstrong elds.Thedot-tedregionsofcurvesc,dandeinFig.2Ashowaquickerclimbingthantheregionsthereafter,whichhappensinasamephasewithonlytwocomponents 0and 1intheGS.Curvedonlyinvolvesthesetwocomponentsfromthebeginningandprovidesafullviewofthisunsymmetry.Besidestheslowdownofd Mz 0/dH,anothercontribu-tiontotheunsymmetrycomesfromthevariationintheenergy-di erenceincreasingrate.Theenergydi erence|E1 E0|betweentheGScomponents 1and 0variesinaslowerrated|E1 E0|/dHwhenthe eldreacheshigher.

Wealsopresenttwotypicalmagnetizationfornega-tiveOILSC,shownbycurvesaandbinFig.2A.ThecorrespondingQPT’sare:(a)123→12→1;(b)1230(and1203)→120→12→1.Onecannoticethecrossoverofthetwocurvesinphase12beforefullpolarization,whereonlytwocomponents 1and 2areleftintheGS.Curvebliesinfour-componentphase1230intheabsenceofthe eld,whileastrongerOILSCexpelsthesingleto theGSandcurveastartsfromtripletphase123.Thenegativeλmakes 0risinginenergylevelwhile 2sinkswhenthe eldisapplied.Buttheloweststate 1sinksfastandsoalltheothercomponentsaremovingoutoftheGS.Thusthereisonemorecomponent 0gettingoutoftheGSforcasebthancasea.Therefore,theproportionofthemostenergeticallyfavorablecomponent 1,withaposi-tivemagnetizationg+/2,increasesmorequicklyincasebthanincasea,whichgivesaquickerriseofmagneti-zationincurvebbeforethe rstQPT.Thecomponent 3incasebhassmallerproportioninthebeginningofafour-componentphase1230,whileithaslargerpro-portionincaseastartingfromathree-componentphase123.Consequently, 3getsoutoftheGSearlierincasebandthe rstQPToccursaheadofcasea.Buttheriseofcurvebsoonslowsdownafteritsearlier rstQPT,sincethe 0hasweakernegativemagnetizationthan 3incasea.After 3getsoutincasea,curvebsoonrises3

s 0= l 0=g H[2λγ(H/λ)] 1, Mz 0=g s 0.

2,2,

1

2

(λ,H0)],

3

1±1λ2) 2+λ2(λ2± 3g

2H2/2,λ2+g

k±=

4

λ±

3(1 1)/2g

+H±

) 1,

andλ±=8ln2±λ.H±andH0arethecontribu-tionsofin nityFermipoints(FP),theothertermscome

fromtherevisionofthe niteFP.Forgs=2gt,H+

2

canbesimpli edasH+=[λ2+ 9λ]/[2g+λ+].Theaboveanalyticresultsarecomparedwiththenumeri-calonesinFig.1,whichshowsanexcellentaccuracy.ForC+NQDandC+MQ,deviationsonlyoccurnearC+duetosmallFQinvalidforWiener-Hopfmethod,butanexpansionbasedonsmallFPcanbecarriedout

1

(λc+ λ)3/2,c=1forC+NandandHc～=(2c/π)g+

CNQMQ

,Hc+c=(ln2)/3forC+M.Settingλ=0inHc

andHfwillrecovertheveryaccurateexpressionsoftheSU(4)modelobtainedinRef.[10].

Aremarkableprogressinexperimentsofspin-orbitalsystemsistheobservationoforbitalorderinginsometransition-metalcompounds[2],whichcanbealsofoundinthepresentmodel.Theexpectationsofthespin

By introducing a basis for a novel realization of the SU(4) Lie algebra, we exactly solve a spin-orbital chain with one-ion $L$-$S$ coupling (OILSC) via the Bethe ansatz (BA) approach. In the context of different Land\'e $g$ factors of the spin and orbital

beyondcurveaagainasithastwocomponents 2and 0movingoutwhileonlyonecomponent 2incasea.Asaresult,casebhashighermagnetizationthancaseawhentheybothcometophase12.Butinthisphase, 2risesawayfromthelowerstate 1alwaysmorequicklyincaseawithastrongerλ,asshownbyd|E2 E1|/dHinFig.2B.Therefore, 2getsoutfasterincasea,anditslowermagnetizationrisesinadvanceofcasebandreachesthesaturationpoint rst.Thisresultsinthecrossover.Finallywestressonlyinthecontextofthedi erentgfactorsofspinandorbitthattheOILSCleadstotheseproperties.Ifgs=gt[17],thenovelphenomenaincludingthe veconsecutiveQPT’s,thespinandorbtalordering,nonzeromagnetizationofthesinglet,theunsymmetricmagnetizationandthemagnetizationcrossover,willalldisappear.

WethankHuan-QiangZhou,You-QuanLiandMur-rayT.Batchelorforhelpfuldiscussions.ZJYthanksFAPERGSandFAPERJforsupport.XWGthanksAus-tralianResearchCouncil.AFthanksFAPERGSandCNPq.IRthanksPRONEXandCNPq.BCthanksZhejiangNaturalScienceFoundationRC02068.

By introducing a basis for a novel realization of the SU(4) Lie algebra, we exactly solve a spin-orbital chain with one-ion $L$-$S$ coupling (OILSC) via the Bethe ansatz (BA) approach. In the context of different Land\'e $g$ factors of the spin and orbital

1.5(gs+gt)/2

(C)

10.8

(B)

Hc1

HHd(E E1 )/dH2

λ= 1.5

1.1

λ= 0.5

0.9

1

H2

3

4

0.60.40.2

0.45

123

120

12

0.55

H

0.65

d(λ=4)

0.75

1.5Magnetization M

z

a(λ= 1.5)b(λ= 0.5)c(λ=2)

λ=6)

1

f(λ=10)

0.5

(A)

00

2

46Magnetic field H

8

10

FIG.2.(A)Typicalmagnetizationbehaviorsindi erentQPT’s,gs=2.0,gt=1.0.ThedashedlinesaregappedsingletphasewithanonzeromagnetizationMz,bothspinandorbitalorderingcanbeobservedinthisphase.CurvedprovidesafullviewofunsymmetricgrowthofMz,thedotted-lineregion(phase01)inweak eldclimbsfasterthanthesolid-lineregion(phase10)instrong eld.Alsonotethecrossoverbetweencurveaandbbeforefullpolarization.(B)MagnetizationoffourconsecutiveQPT’s,λ= 0.76,Hc4=3.65,thenumbersdenotethephasesasinFig.1.(C)Comparisononincreasingspeedofenergydi erenceE2 E1inthe eldforλ= 1.5andλ= 0.5,theformeralwayshasalargerspeed.

M

1.3

5