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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.