Effect of one-ion L-S coupling on magnetic properties of a c(3)

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

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