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