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

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