Effect of one-ion L-S coupling on magnetic properties of a c(2)

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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.

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