Datasets:

ZWG817
/

Full_Content

Dataset card Viewer Files Files and versions Community

Split (1)

train · 15.4k rows

title stringlengths 31 247	content stringlengths 15 7.12M
2311.14719v1.Thermal_Spin_Orbit_Torque_with_Dresselhaus_Spin_Orbit_Coupling.pdf	ThermalSpin-OrbitTorquewithDresselhaus Spin-OrbitCoupling Chun-YiXue,Ya-RuWang,Zheng-ChuanWang* SchoolofPhysicalSciences， UniversityofChineseAcademyofSciences,Beijing100049,China. *wangzc@ucas.ac.cn Abstract BasedonthespinorBoltzmannequation,weobtainatemperature dependentthermalspin-orbittorqueintermsofthelocalequilibrium distributionfunctioninatwo-dimensionalferromagnetwith Dresselhausspin-orbitcoupling.Wealsoderivethecontinuity equationofspinaccumulationandspincurrent—thespindiffusion equationinDresselhausferromagnet,whichcontainsthethermal spin-orbittorqueunderlocalequilibriumassumption.This temperaturedependentthermalspin-orbittorqueoriginatesfromthe temperaturegradientappliedtothesystem,itisalsosensitiveto temperatureduetothelocalequilibriumdistributionfunctiontherein. Inthespindiffusionequation,wecansingleouttheusualspin-orbit torqueaswellasthespintransfertorque,whichisconcededtoour previousresults.Finally,weillustratethembyanexampleof spin-polarizedtransportthroughaferromagnetwithDresselhaus spin-orbitcouplingdrivenbytemperaturegradient,thosetorquesincludingthermalspin-orbittorquearedemonstratednumerically. PACS:75.60.Jk,72.15Jf,75.70.Tj,85.70.-w I.Introduction Asanewbranchofspintronics,spin-orbitronicshasbeen exploredwhateverintheoriesorexperiments[1-2],becauseitcan provideanefficientwaytomanipulatethelocalmagneticmomentin thedevicesofspintronicsviaspin-orbittorques[3-5].Sincethe discoveryofgiantmagnetoresistance[6]andspintransfertorque[7], nowadaysmagnetoresistancerandomaccessmemory(MRAM) drivenbyspin-polarizedcurrenthasbeendesignedandrealized industrially[8-10].ThefirstgenerationMRAMistoggleMRAM [11],whichconsistofatransistorandamagnetictunneljunction (MTJ),butthisstructurebringsobviousdisadvantagestothetoggle MRAMduetothebigmagneticfield[12].Thesecondgeneration MRAMarespintransfertorqueMRAM(STT-MRAM)[13]and perpendicularspintransfertorqueMRAM(pSTT-MRAM)[14],in whichmagnetizationreversalinSTT-MRAMandpSTT-MRAMrely onthespin-polarizedelectricalcurrentratherthanbigmagneticfield, sotheyhavefasterwritingspeed.However,STT-MRAMdepends onthermalactivationtostartswitching[15],soithasaninitial latencywhichrestrictsitsmaximumcachespeed.Thus,oneproposedthethirdgenerationMRAMtosolvethisproblem,which isdrivenbyspinorbittorque(SOT).SOTcanbeemployedtoswitch themagnetizationinasystemwithabrokeninversionsymmetry. SinceSOTneedalowercriticalcurrent,ithasbetterthermal stability[16].Recently,ThermalSOThasbeenobservedin experiments[17-18].Tillnow,SOT-MRAMisagoodcandidateof magneticstoragedeviceforbetterperformance. In2007,HatamiproposedSTTwhichcanbedrivenbythermal spincurrent[19].Similarly,SOTcanalsobedrivenbythermalspin currentintheprocessofspin-polarizedelectrontransport,whichis calledthermalSOT(TSOT).TSOTwasfirstlyproposedby Freimuthin2014intermsofBerryphase[20-21]whichisexpressed bythequantumstatesofelectrons,whilethestatesusuallyshouldbe calculatedbytheDensityFunctionTheory(DFT),itissomewhat cumbersome.Thus,inthismanuscriptwewillgiveanother expressionofTSOTbyuseofdistributionfunctioninthespinor Boltzmannequation(SBE). TheSBEwasfirstlyproposedbyShengetalin1996atsteady state[22],theyderivedthisequationfromKadanoffnonequilibrium Greenfunction(NEGF)formalismbasedonthegradient approximation[23].ThenSBEwasaccomplishedbyLevyetalin 2004,whichcouldillustratethetimedependentprocessofspintransporteffectively[24].SBEwasalsoextendedtothecasebeyond gradientapproximationin2013[25].In2019,Wangetal. successfullyincludedtheRashbaandDresslhausspin-orbitcoupling intotheSBE[26],whichishelpfulforustoinvestigatetheSOT fromthespindiffusionequation–thecontinuityequationforthe spinaccumulationandspincurrent.Inthismanuscript,ourpurpose istoinvestigateTSOTina2-dimensionalferromagnetbymeansof SBE.WewillfindthatanunusualSOTdrivenbythetemperature gradientinthesystemwithDresselhausspinorbitcoupling,itisjust theTSOTwewant. II.TheoreticalFormalism Considerthespin-polarizedelectrontransportinatwo- dimensionalferromagnetwithDresselhausSOC.It’sHamiltonian canbewrittenas=−ℏ2 220+⋅+,whereis electroniceffectivemass,Jisthes-dexchangecouplingconstant, istheunitvectorforthelocalmagneticmomentofferromagnet, 0istheunitmatrix,isthePaulimatrixvector,describes theDresselhausSOCintwodimensionalferromagnet.In1955, DresselhausgavetheHamiltonianofDresselhausSOCinthree dimensionalsystemwithabulkinversionasymmetry(BIA)[29]. Whensuchathreedimensionalsystemissubjectedtostrainatthe interfaceorathinlayer,theHamiltonianwillreducetothefollowingsimplerform:=(−)[26],whereisthe couplingconstantofDresselhausSOC,andarethexandy componentsofgradientoperator,andtheandarethexand ycomponentsofthePaulioperator.ByKadanoffnonequilibrium Greenfunction(NEGF)formalism,wecanobtaintheSBEforthe spinordistributionoftransportelectron[24]: +⋅ −⋅ + ℏ,=− d#1 whereisthespinordistributionfunctionoftransportelectron, whichcanbeexpandedtoa2×2matrix=↑↑↑↓ ↓↑↓↓,isthe spinorenergy,it’sdefinedas()=()0+1 2()(,). IntheSBEframework,wecandecomposethespinordistribution functionintotwopartsbyusingthecompletebasisofmatrices0 andthe,(,)=(,)0+(,),where,is scalardistributionfunctionand,isvectordistribution function.Underthelocalequilibriumassumption,thespinor distributionfunctioncanalsobewrittenasfollow: ,,=0,+1,0+1,⋅2 where0(,)isthelocalequilibriumdistributionfunction. 1(,)and1(,)arethenonequilibriumpartsofspinor distribution.Intheferromagnet,wetakethelocalequilibrium distributionfunctionasadiagonalmatrix0(,)=↑↑0 0↓↓,herethediagonalcomponentsaretakenasFermidistribution functions↑↑(↓↓)=()±1 2−() ()+1−1,whereis Boltzmannconstant,()isthechemicalpotentialand≈ (FermiEnergy)atfinitetemperature.Wecanexpandthelocal equilibriumdistributionbasedonthecompletebasis 0,,,,whichis0(,)=1 2(↑↑+↓↓)0+1 2(↑↑− ↓↓),sothescalardistributionandvectordistributioncanbe rewrittenas=1 2(↑↑+↓↓)+1and=1,1,1+1 2(↑↑− ↓↓). Inspintronics,theSBEwithDresselhausspin-orbitcouplingina two-dimensionalmagneto-electricsystemunderanexternalelectric fieldhadbeengivenbyChaoYangetal.in2019[26] + ⋅−⋅,+ ℏ − =− #3 and + ⋅−⋅,− ℏ×,+ ℏ − + ℏ2−×,=− #4 where−( )and−( )representthecollisionterms. Undertherelaxationtimeapproximationassumption,wecan derivetheequationsforthescalardistributionfunctionandthe vectordistributionfunctionbasedonEqs.(3)and(4),whichcontainthelocalequilibriumdistributionfunctionas: + ⋅−⋅1 2↑↑+↓↓+ + ⋅−⋅1 + ℏ1 −1 =−− #5 + ⋅−⋅−×+ ℏ1 2↑↑+↓↓ −1 2↑↑+↓↓ + ℏ1 −1 ℏ− ℏ2−×=−− #6 whereandaretherelaxationtimesofelectronandspinflip, respectively.Eq.(5)andEq.(6)arecoupledtogether,weshould solvethemsimultaneously.Thephysicalobservablesinthe spin-polarizedtransportcanbeexpressedbythesolutionsofscalar andvectordistributionfunctions.Thechargedensityandcharge currentaredefinedasfollow, ,=1 ,,#7 and ,=1,, #8 whicharethemomentumintegralsoverthescalardistributions. Similarly,thespinaccumulationandspincurrentdensityaredefine as,=,,#9 and ,= ,,#10 whicharethemomentumintegralsoverthevectordistributions.It shouldbenotedthatthespincurrent (,)isatensor.Moreover, wecanalsodefinethethermalcurrentdensityas ,=1 ,,#11 whereisthescalarenergyofelectron.Sowhenwegetthe solutionsofscalarandvectordistributionfunctions,wecanobtain theseabovephysicalobservablesaccordingly. Ontheotherhand,wecanalsoobtainthecontinuityequations satisfiedbythesephysicalobservables.Ifweintegratethe momentumovertheFermisurfaceonthebothsidesofEq.(5)and Eq.(6),wehave +⋅=− + ⋅−⋅1 2↑↑+↓↓ − ℏ − −− #12 and +⋅= ℏ× − ℏ1 2↑↑+↓↓ −1 2↑↑+↓↓ − ℏ − + ℏ2−× =−− #13 Eq.(12)isjustthecontinuityequationforchargedensityandcharge current,whileEq.(13)isthecontinuityequationforthespin accumulationandspincurrent,thelatterisalsocalledspindiffusion equation.Whenthetime≫,theequationwillarriveatasteady state,thenthespindiffusionequationwillreduceto ℏ×=⋅ + ℏ1 2↑↑+↓↓ −1 2↑↑+↓↓ + ℏ − − ℏ2−× #14 Fromtheabovesteadystateequation,wecanreadoutallthetorques existinginthisspin-polarizedtransportprocess.Ontheleftsideof thisequation,theterm ℏ×isjustthespintransfertorque givenbyLevyetal.[27].Ontherighthandsideside,⋅ isthe divergenceofspincurrent,whichalsocanmakeacontributiontothe usualSTTasshownbyZhangetal[28].Besides,theterm ℏ( − )− ℏ2(−)× correspondstothe usualspin-orbittorquepresentedbyWangetal.[26],whilethe temperaturedependentterm ℏ1 2↑↑+↓↓ − 1 2↑↑+↓↓ isanewterm,itisinducedbythegradientof localequilibriumdistributionfunction,werefertothisasthe thermalSOT.Whenthegradientoftemperatureisappliedonly alongx-direction,itcanbeexpressedas: =− ℏ. 1 2(−+1 2 ) 1+(−−1 2 )2⋅−+1 2 + (−−1 2 ) 1+(−−1 2 )2⋅−−1 2 1 2 (15) wecanseethatitisproportionaltothegradientoftemperature, whichisconcededtothedefinitionofTSOTgivenbyFreimuthetal [20-21],sothistermisjusttheTSOTwesearchfor,itisthecentral resultinthismanuscript.Inthenext,wewillevaluatethesetorques numericallyinaferromagnetwithDresselhausSOC. III.NumericalResults Weconsideratwo-dimensionalferromagnetwithDresselhaus SOC,wherethesystemischosenasarectangularferromagnetwith ageometryof25×25nm².Thetemperaturedistributionissimply chosenas()=0+,whichislinearlydependentonthe positionofxcomponent,where0isaconstant,kisthetemperaturegradient.FromEq.(15),wecanseethatthetemperature gradientwillinducethermalspin-orbittorque. Inordertoquantifythesetorquesandcurrents,weneedtosolve Eq.(3)combiningwith(4)simultaneously,becausethescalar distributionfunctionandvectordistributionfunctionarecoupled togetherintheseequations.Tosimplifycalculation,wechosethe unitvectorofmagnetizationasafixedvector=(0,0,1),the equilibriumscalardistributionfunctionischosenas=1 2↑↑+ ↓↓,andtheequilibriumvectordistributionfunctionisadoptedas =(( + ),( + ),( + ))[22].Thedifferentialequations(3)and(4)aresolvedby differencemethod.Thephysicalconstantandparametersarelisted inTableⅠ,whereweadoptthematerialsparametersofferromagnet. TableⅠ.Thephysicalconstantsandparameters Physicalconstants/parametersSymbolValueUnit Momentumrelaxationtime 10−13s Spin-fliprelaxationtime 10−12s Fermienergy 4 eV Fermiwavevector 1.02×1010−1 s-dexchangecouplingstrengthJ 0.1 eV Electricalfield E −5×104.−1Temperaturegradient 5×109.−1 InFig.1,weplotthechargecurrentdensityasafunctionof positionxandy.Thevariationofchargecurrentwithrespectto positionandtimeisgovernedbythecontinuityequationofcharge densityandchargecurrentdensity(12).Forsimplicity,weonly studythechargecurrentatsteadystate.Wecanseethatthecharge currentdensitydecreasesgraduallyalongboththexandydirections, whichisduetotheresistanceintheferromagnet,inourcalculationit isconcernedwiththemomentumrelaxationtimeofelectrons. Sincetheexternalelectricfieldisappliedonlyalongthex-direction, thevariationofchargecurrentalongy-directionismainlycausedby theDresselhausSOC. Fig.1ThechargecurrentdensityFig.2Thethermalcurrentdensity vsposition vsposition Wealsoshowthecurveofthermalcurrentdensityasafunction ofpositioninFig.2,whichissimilartothechargecurrentdensity becauseoftheirdefinitions,italsodecreaseswithpositiongradually. Hereweonlyconsiderthethermalcurrentdensitycarriedbythetransportelectrons.Besidestheexternalelectricfield,thethermal currentdensitycanalsobedrivenbythetemperaturegradient.Since theelectricfieldandgradientoftemperatureareallalongthex-axis, thevariationofthermalcurrentdensityalongy-directionisprimarily inducedbytheDresselhausSOC. Becausewechoosethemagnetizationofferromagnet= (0,0,1),sothezcomponentofSTTis0.InFig.3,weshowthexand ycomponentsofSTTdensityasafunctionofposition.Theusual STTisthespaceintegralofthisdensityovertherectangular ferromagnet.ItisshownthatthemagnitudeofSTTdensityis differentatdifferentpositionforboththexandycomponents.The magnetizationofferromagnetat=10iseasiesttobe switchedbythebiggerSTT,andishardesttofliparoundtheline =becauseofthesmallerSTT.Theswitchingofmagnetization willproducethespinwavewithinthetwo-dimensionalferromagnet. Fig.3(a)Thex-componentofSTTdensity.(b)They-componentofSTTdensity ThespincurrentdensityvspositionisshowninFig.4.Sincethespincurrentisatensor,weonlydrawthexx-,xy-andxz- componentsofspincurrentdensityasafunctionofposition,they varyobviouslyaroundtheline=.Thevariationofspincurrent withpositionandtimesatisfiesthecontinuityequation(13)forthe spinaccumulationandspincurrent.AccordingtoEq.(14),the divergenceofspincurrentwillmakeacontributiontotheZhang-like STT[28]. Fig.4(a)Thexx-componentofspincurrent(b)Thexy-componentofspin current(c)Thexz-componentofspincurrent InFig.5,wedrawtheSOTasafunctionofposition.Theusual SOTisexpressedas ℏ( − )− ℏ2(−)× , soitisnotsensitivetothetemperature.Itdecreasesalongx- directionobviously,whilethereissmallvariationalongy-direction, becausetheexternalelectricfieldisappliedalongx-direction.For comparison,wealsoplottheTSOTatdifferenttemperature300K, 200Kand100KinFig.6,respectively,whichdependonthe temperatureandit’sgradientobviously.Thehigheroftemperature, thebiggerofTSOT,becausetherearemorepolarizedelectronFig.5TheSOTvsposition.Fig.6TheTSOTvspositionatdifferent temperatureT=300K,200K,100K. participatingintransportathighertemperature.ComparedFig.6 withFig.5,wecanfindthattheTSOTissmallerthantheusualSOT, whiletheTSOTcanbecomebiggerwhenweincreasethe temperature,sotheTSOTcannotbenegligibleathigher temperature.Certainly,TSOTisalsoproportionaltothetemperature gradient,itwillplayanimportantroleathighertemperaturegradient. ItshouldbepointedoutthattheTSOTiscalculatedbyEq.(15),we onlyneedtheexpressionoflocalequilibriumdistributionfunction,it isverysimplethanFreimuth’sexpressionofBerryphase[20-21], becausethelatterneedtheelectronicwavefunctionobtainedusually byDFT.BymeansofEq.(15),wecancalculatetheTSOTeasier, thisistheadvantageofSBEmethod. IV.SummaryandDiscussions Inthispaper,wehavederivedtheTSOTinatwo-dimensional ferromagnetwithDresselhausSOCbySBEunderthelocal equilibriumassumption.TheusualSOTisinducedbytheexternal electricfieldappliedtothesystem,whiletheTSOTisdrivenbythegradientoftemperature.WealsofindthatTSOTisverysensitiveto thetemperature,thehighertemperature,thebiggerTSOT.Our resultsshowthattheTSOTissmallerthanSOT,butitcannotbe negligibleathighertemperature.Certainly,TSOTisalso proportionaltothetemperaturegradient,accordingtoitsexpression Eq.(15).BecausethedirectexperimenttoobserveTSOTin two-dimensionalferromagnetswithDresselhausspin-orbitcoupling haven’tbeencarriedoutnow,weonlypredicttheoreticallythatone canobservetheeffectsofTSOTinthecaseofbiggradientof temperatureandhighertemperature. Tosimplifyourcalculation,weonlychooseasimpleuniform magnetizationina2-dimensionalferromagnet,whileinrealitythe magnetizationusuallyvarieswithtimeandposition.Thevariationof magnetizationwouldhaveinfluenceonthetransportpropertiesof thespin-polarizedelectrons.Ifweconsiderthevariationof magnetization,thecalculationwillbecomemuchmorecomplicated, itisleftforfutureexploration. Acknowledgments ThisstudyissupportedbytheNationalKeyR&DProgramof China(GrantNo.2022YFA1402703),theStrategicPriority ResearchProgramoftheChineseAcademyofSciences(GrantNo. XDB28000000).WealsothankProf.GangSu,Zhen-Gang.Zhu,Bo.GuandQing-Bo.Yanfortheirhelpfuldiscussions. AUTHORCONTRIBUTIONSTATEMENT Inthiswork,Zheng-ChuanWangproposedtheidea,Chun-Yi Xueperformedthecalculation,analyzedthenumericalresults,and wrotethemanuscript.Ya-RuWangassistedwiththecalculationand analysis. DataAvailabilityStatement Datasetsgeneratedduringthecurrentstudyareavailablefromthe correspondingauthoronreasonablerequest. References [1]A.ManchonandS.Zhang,Phys.Rev.B78,212405(2008). [2]T.KuschelandG.Reiss,Nat.Nanotechnol.10,22(2015). [3]A.Matos-AbiagueandR.L.Rodrıguez-Suarez,Phys.Rev.B80, 094424(2009). [4]K.M.D.HalsandA.Brataas,Phys.Rev.B88,085423(2013). [5]A.Brataas,A.D.KentandH.Ohno,Nat.Mater.11:372-81 (2012). [6]M.N.Baibichetal.,Phys.Rev.Lett.61,2472(1988). [7]J.C.Slonczewski,J.Magn.Magn.Mater.159,L1(1996). [8]W.J.GallagherandS.S.P.Parkin,IBMJ.Res.Dev.50,5 (2006). [9]X.Hanetal.,Appl.Phys.Lett.118,120502(2021).[10]S.Aggarwaletal.,2019IEEEInt.ElectronDevicesMeet. (IEDM)pp.2.1.1–2.1.4(2019). [11]L.Savtchenkoetal.,USPatent6546906B1(2003). [12]M.Motoyoshietal.,Dig.Tech.Pap.-Symp.VlSITechnol.p.22 (2004). [13]C.Chappert,A.FertandF.N.Van-Dau,Nat.Mater.6:813–23 (2007). [14]T.Kishietal.,Tech.Dig.-Int.ElectronDevicesMeet.309 (2008). [15]R.Patel,E.Ipek,E.G.Friedman,MicroelectronicsJournal, 45(2),133-143(2014). [16]C.Bi,N.SatoandS.X.Wang,AdvancesinNon-Volatile MemoryandStorageTechnology,203-235(2019). [17]J.M.Kimetal.,NanoLett.20(11):7803-7810(2020). [18]D.J.Kimetal.,Nat.Commun.8,1400(2017). [19]M.HatamiandG.E.W.Bauer,Phys.Rev.Lett.99,066603 (2007). [20]F.Freimuth,S.BlugelandY.Mokrousov,J.Phys:Condens. Matt.26,104202(2014). [21]G.Geranton,F.Freimuth,S.BlugelandY.Mokrousov,Phys. Rev.B91,014417(2015). [22]L.Sheng,D.Y.Xing,Z.D.Wang,andJ.Dong,Phys.Rev.B55,5908(1997). [23]L.P.Kadanoff,G.Baym,andJ.D.Trimmer,Quantum StatisticalMechanics(W.A.Benjamin,1962). [24]J.Zhang,P.M.Levy,S.ZhangandV.Antropov,Phys.Rev.Lett. 93,256602(2004). [25]Z.C.Wang,TheEuro.Phys.Jour.B86,206(2013). [26]C.Yang,Z.C.Wang,Q.R.ZhengandG.Su,TheEuro.Phys. Jour.B92,136(2019). [27]A.S.Shpiro,P.M.Levy,S.Zhang,Phys.Rev.B67,104430 (2003). [28]S.ZhangandZ.Li,Phys.Rev.Lett.93,27204(2004). [29]G.Dresselhaus,A.F.Kip,andC.Kittel,Phys.Rev. 100,618(1955).
1203.4079v2.Spin_orbit_couplings_between_distant_electrons_trapped_individually_on_liquid_helium.pdf	arXiv:1203.4079v2 [quant-ph] 13 Nov 2012Spin-orbit couplings between distant electrons trapped in dividually on liquid Helium M. Zhang1and L. F. Wei∗1,2 1Quantum Optoelectronics Laboratory, School of Physics, Southwest Jiaotong University, Chengdu 610031, China 2State Key Laboratory of Optoelectronic Materials and Techn ologies, School of Physics and Engineering, Sun Yat-sen University, Guangzhou 510275, China (Dated: November 10, 2018) Abstract We propose an approach to entangle spins of electrons ﬂoatin g on the liquid Helium by coherently ma- nipulating their spin-orbit interactions. Theconﬁgurati on consists ofsingleelectrons, conﬁnedindividually on liquid Helium by the micro-electrodes, moving along the s urface as the harmonic oscillators. It has been known that the spin of an electron could be coupled to its orbit (i.e., the vibrational motion) by prop- erly applying a magnetic ﬁeld. Based on this single electron spin-orbit coupling, here we show that a Jaynes-Cummings (JC) type interaction between the spin of a n electron and the orbit of another electron at a distance could be realized via the strong Coulomb interact ion between the electrons. Consequently, the proposed JC interaction could be utilized to realize a stron g orbit-mediated spin-spin coupling and imple- ment the desirable quantum information processing between the distant electrons trapped individually on liquid Helium. PACSnumbers: 73.20.-r, 03.67.Lx, 33.35.+r ∗weilianfu@gmail.com 1I. INTRODUCTION The interactions between the microscopic particles, e.g., the ions in Paul trap [1], the neutral atoms conﬁned in optical lattice [2], and the electrons in Pe nning trap [3], etc., relate usually to their masses and the inter-particle forces. Due to the small mass and the strong Coulomb interac- tion,theinteractingelectronscouldbeused toimplementq uantuminformationprocessing(QIP). Theideaofquantumcomputingwithstrongly-interactingel ectronsonliquidHeliumwasﬁrstpro- posedbyPlatzmanandDykmanin1999[4]. Intheirproposal,t hetwolowerhydrogen-likelevels of the surface-state electron are encoded as a qubit, and the effectively interbit couplings can be realizedbytheelectricdipole-dipoleinteraction. Whent heliquidheliumiscooledontheorderof mK temperature the qubit possesses long coherent time (e.g. , up to the order of ms) [5, 6]. Inter- estingly,Lyonsuggested[7]thatthequbitscouldalsobeen coded by thespinsoftheelectrons on liquidHelium,andestimatedthatthequbitcoherenttimeco uldreach 100s[7]. Heshowedfurther that the magneticdipole-dipoleinteractions between the s pinscould be used to couplethe qubits, if the electrons are conﬁned closed enough. For example, the coupling strength can reach to the order of kHz for the distance d= 0.1µm between the electrons [7]. Remarkably, recent experi- ments[8–10]demonstratedthemanipulationsofelectrons( conﬁning,transporting,anddetecting) onliquidHeliuminthesingle-electronregime. Thisprovid esreallytheexperimentalplatformsto realizetherelevantQIPwithelectrons onliquidHelium[11 –14]. Here, we propose an alternative approach to implement QIP wi th electronic spins on liquid Heliumbycoherentlymanipulatingthespin-orbitinteract ionsoftheelectrons. Inourproposal,the virtuesoflong-livedspinstates(toencodethequbit)ands trongCoulombinteraction(forrealizing theexpectably-fastinterbitoperations)arebothutilize d. Theelectronsaretrappedindividuallyon the surface of liquid Helium by the micro-electrodes. In the plane of liquid Helium surface each electron moves as a harmonic oscillator. It has been showed t hat such an external orbit-vibration could be effectively coupled to the internal spin of a single electron by applying a magnetic ﬁeld with a gradient along the vibrational axis [13]. Interestin gly, we show that the spin of an electron could be coupled to the vibrational motion of another distan t electron [as a Jaynes-Cummings (JC) type interaction], by designing a proper virtual excit ation of the electronic vibration. The present JC interaction could be utilized to signiﬁcantly en hance the spin-spin coupling between the distant electrons, and implement the desirable quantum computation with the spin qubits on liquidHelium. 2Qe HLiquid Helium xz y +I BzBsPotential h FIG. 1: (Color online) Sketch for a single electron trapped o n the surface of liquid Helium. The liquid Helium provides z-directional conﬁnement, and the micro-electrode Q (below the Helium surface at depth H) traps the electron in x-yplane. The desirable spin qubit is generated by an applied un iform magnetic ﬁeldBs, and the spin-orbit coupling of the trapped electron is obta ined by applying a current to another micro-electrode I (upon the liquid Helium surface at the hei ghth). Thepaperisorganizedasfollows: InSec. IIwediscusstheme chanismforspin-orbitcoupling withasingleelectrontrappedonliquidHelium[13],andthe nshowhowtoutilizesuchacoupling to realize the desirable quantum gate with the single electr on. By using the electron-electron Coulomb interaction, in Sec. III, we propose an approach to i mplement the JC coupling between the spin of an electron and the orbital motion of another elec tron. Based on such a distant spin- orbit interaction, we show that a two-qubit controlled-NOT (CNOT) gate and an orbit-enhanced coupling between the distant spins could be implemented. Fi nally, we give a conclusion in Sec. IV. II. SPIN-ORBITCOUPLINGWITHA SINGLETRAPPEDELECTRON We consider ﬁrst a single electron trap shown in Fig. 1 [13], w herein an electron (with mass meand charge e) on liquid Helium is weakly attracted by its dielectric imag e potential V(z) = −Λe2/z(withΛ = (ε−1)/4(ε+ 1)andεbeing the dielectric constant of liquid Helium). Due tothePauliexclusionprinciple,thereisanbarrier(about 1eV)topreventtheelectronpenetrating into the liquid Helium. As a consequence, z-directional conﬁnement of the electron is realized, yielding an one-dimensional (1D) hydrogenlike atom with th e spectrum En=−/planckover2pi1R/n2[15]. Here,R= Λ2e4me/(2/planckover2pi12)≈170GHz and rb=/planckover2pi12/(mee2Λ)≈7.6nm are theeffectiveRydberg 3energy and Bohr radius, respectively. In x-yplane, the electron can be conﬁned by the micro- electrode Q located at Hbeneath the liquid Helium surface. Typically, x,y,z≪H, and thus the potentialoftheelectron can bedescribed by[5] U(x,y,z)≈ −Λe2 z+E⊥z+me 2(ν2 xx2+ν2 yy2) (1) withE⊥=eQ/H2,νx=νy=/radicalbig eQ/(meH3), andQbeing the effective charge of the micro- electrode. This potential indicates that the motions of the trapped electron are a 1D Stark-shifted hydrogen along the z-direction, and a 2D harmonic oscillator in the plane parall el to the liquid Heliumsurface. TheHamiltonianfortheorbitalmotionsoft hetrapped electron can bewrittenas ˆHo=/summationdisplay nEn\|na/an}bracketri}ht/an}bracketle{tna\|+/summationdisplay k=x,y/planckover2pi1νk(ˆa† kˆak+1 2). (2) Here,\|na/an}bracketri}htis thenth boundstateofthehydrogenlikeatom, ˆa† kandˆakare thebosonicoperatorsof thevibrationalquantaoftheelectron alongthe k-direction. A spin qubit is generated by applying an uniform magnetic ﬁel dBsalongxdirection, and its Hamiltonian reads ˆHq= (gµBBs)ˆσx/2. Here, the Pauli operator is deﬁned as ˆσx=\| ↑/an}bracketri}ht/an}bracketle{t↑ \| − \| ↓/an}bracketri}ht/an}bracketle{t↓ \| with\| ↓/an}bracketri}htand\| ↑/an}bracketri}htbeing the two spin states. g= 2is the electronic g-factor, and µB= 9.3×10−24J/T is the Bohr magneton. The spin-orbit coupling of the trap ped electron can be realized by applying a dc current Ito the electrode I (located upon the liquid Helium surface with a height h) [13]. Typically, x,z≪hand the magnetic ﬁeld generated by the current I reads/vectorB= (Bx,0,Bz)withBx≈µ0I(1−z/h)/(2πh)andBz≈µ0Ix/(2πh2). Here,µ0is the permeability of free space. Therefore, the Hamiltonian describing theinteraction between the magnetic ﬁeld and spin can be expressed as: ˆHsb=gµB(Bzˆσz+B′ xˆσx)/2withB′ x=Bs+Bx, ˆσz= ˆσ−+ ˆσ+,ˆσ−=\| ↓/an}bracketri}ht/an}bracketle{t↑ \|andˆσ+=\| ↑/an}bracketri}ht/an}bracketle{t↓ \|. Consequently, the total Hamiltonian of the trapped electronin theappliedmagneticﬁelds reads ˆH=/planckover2pi1νs 2ˆσx+ˆHo+ˆHsx, (3) with ˆHsx=gµBµ0I 4πh2/radicalbigg /planckover2pi1 2meνx(ˆax+ˆa† x)ˆσz. (4) TheﬁrstandsecondtermsintherighthandofEq.(3)describe thefreeHamiltonianofthetrapped electron,with νs= (gµB//planckover2pi1)[Bs+(µ0I/2πh)]beingthetransitionfrequencybetweenitstwospin states, and ˆHsxdescribes the coupling between the spin and the orbital moti on along x-direction. 4Note that the coupling between the spin and z-directional orbital motion is neglected, due to the large-detuning. Also, theappliedstrongﬁeld Bs(e.g.,0.06T)does notaffect theinteraction ˆHsx, althoughitwillchangeslightlytheelectron’s motionsint hey-zplane[16]. Obviously,theHamiltonianin Eq.(3)can besimpliﬁedas ˆHe=/planckover2pi1Ω/parenleftbig eiδtˆσ+ˆa+e−iδtˆσ−ˆa†/parenrightbig (5) intheinteractionpicture. Here, δ=νs−νxisthedetuning, Ω =gµBµ0I 4πh2√2/planckover2pi1meνx(6) is the coupling strength, and ˆa= ˆax,ˆa†= ˆa† x. Note that, the Hamiltonian in Eq. (5) can also be obtained by applying an ac current I(t) =Icos(ωt)with frequency ω=νx−νs+δto the electrode. Specially, when δ= 0, this Hamiltonian describes a JC-type interaction between the spin and orbit of the single electron. In fact, Ref. [13] has a rranged this spin-orbit coupling of a single electron to increase the interaction between the sp in and a quantized microwave ﬁeld. Alternatively, we will utilize this spin-orbit coupling (t ogether with the electron-electron strong Coulomb interaction) to realize a strong interaction betwe en two electronic spins and generate certain typicalquantumgates. For the typical parameters: I= 1mA,h= 0.5µm, andνx= 10GHz [5, 13], we have Ω≈5.2MHz. Thisissigniﬁcantlylargerthanthedecoherencerate( whichistypicallyontheorder of10kHz [5, 13]) of thevibrational states of thetrapped electro n. Thus, the aboveJC interaction providesapossibleapproachtoimplementQIPbetweenthesp inandorbitstatesofasingletrapped electron. FortheJCinteraction,thestate-evolutionscan belimitedintheinvariant-subspaces {\| ↓ ,0/an}bracketri}ht}and{\| ↓,1/an}bracketri}ht,\| ↑,0/an}bracketri}ht},with\|0/an}bracketri}htand\|1/an}bracketri}htbeingthegroundandﬁrstexcitedstatesoftheharmonic oscillator. Thus,aphasegate ˆP=\|0,↓/an}bracketri}ht/an}bracketle{t0,↓ \|+\|0,↑/an}bracketri}ht/an}bracketle{t0,↑ \|+\|1,↓/an}bracketri}ht/an}bracketle{t1,↓ \|−\|1,↑/an}bracketri}ht/an}bracketle{t1,↑ \|couldbe implementedbyapplyingacurrentpulsetotheelectrodeI. T herelevantduration tissettosatisfy theconditions: sin(Ωt)≈0andcos(√ 2Ωt)≈ −1(e.g.,Ωt≈37.7numerically). Consequently,a CNOTgatewiththesingleelectroncouldberealizedas ˆS=ˆR(π/2,−π/2)ˆPˆR(π/2,π/2),where ˆR(α,β) = (\| ↑/an}bracketri}ht/an}bracketle{t↑ \|+\| ↓/an}bracketri}ht/an}bracketle{t↓ \|)cos(α)−i[exp(iβ)\| ↑/an}bracketri}ht/an}bracketle{t↓ \|+exp(−iβ)\| ↓/an}bracketri}ht/an}bracketle{t↑ \|]sin(α)isanarbitrary single-bit rotation [17]. This CNOT gate operation, betwee n the spin states and the two selected vibrational states of a single electron [18], is an intermed iate step for the later CNOT operation between twodistantspinqubits. 5Q1 Q2e1 e2 d Liquid Helium +I1 +I2 H Hh h Potential Bsz x y FIG. 2: (Color online) Two electrons (denoted by e1ande2) are conﬁned individually in two potential wells with the distance d, which is sufﬁciently large (e.g., d= 10µm) such that the magnetic dipole- dipole coupling between theelectronic spins isnegligible . Theorbital motionsof thetwoelectrons arealso decoupled from each other, since they are trapped in large-d etuning regime. By applying a current to the electrode I 1the spin of the electron e1could be coupled to the vibrational motions of electron e2, via a virtual excitation of the vibrational motion of electron e1. III. SPIN-ORBITJC COUPLINGBETWEENTHEDISTANTELECTRONS Without loss of generality, we consider here two electrons ( denoted by e1ande2) trapped individually in two potential wells, see Fig. 2. Suppose tha t the distance dbetween the potential wells is sufﬁciently large (e.g., d= 10µm), such that the directly magnetic interaction between the two spins could be neglected. Thus, the interaction betw een the two electrons leaves only the Coulomb one. Specially, the Coulomb interaction along t hex-direction can be approximately writtenas V(x)≈e2 2πǫ0d3x1x2 (7) withxjbeing the displacement of electron ejfrom its potential minima. By controlling the volt- ages applied on the electrodes Q1andQ2, the vibrational frequencies of the electrons are set as thelarge-detuning(and thustheelectrons aredecoupled fr omeach other). To couple the initially-decoupled electrons, we apply a cur rentIto the electrode I 1. As dis- cussed previously,such acurrent induces aspin-orbitcoup ling[i.e., ˆHein Eq. (5)]of theelectron e1. Therefore, thepresenttwo-electronssystemcan bedescri bed bythefollowingHamiltonian ˆHee=ˆHe+/planckover2pi1˜Ω/parenleftBig ei∆tˆaˆb†+e−i∆tˆa†ˆb/parenrightBig (8) in the interaction picture. Where, ˆbandˆb†are the bosonic operators of the vibrational motion of electrone2alongx-direction, ∆ =ν2x−ν1xisthedetuningbetweenthetwoelectronicvibrations 6alongx-direction, and ˜Ω =e2 4πǫ0med3√ν1xν2x, (9) the coupling strength. Numerically, for d= 10µm andνjx= 10GHz we have ˜Ω≈25MHz. Above,thespinofelectron e2wasdropped,asthedriving(inducedbyelectrode I1)onthisspinis negligible(dueto d≫h). The dynamical evolution ruled by the Hamiltonian in Eq. (8) i s given by the following time- evolutionoperator ˆU(t) = 1+/parenleftbig−i /planckover2pi1/parenrightbig/integraltextt 0ˆHee(t1)dt1 +/parenleftbig−i /planckover2pi1/parenrightbig2/integraltextt 0ˆHee(t1)/integraltextt1 0ˆHee(t2)dt2dt1+···.(10) Weassume δ= ∆forsimplicity,thentheabovetime-evolutionoperatorcan beapproximatedas ˆU(t)≈exp/parenleftbigg −it /planckover2pi1ˆHeﬀ/parenrightbigg , (11) withtheeffectiveHamiltonian ˆHeﬀ=/planckover2pi1Ω2 δ/bracketleftbig ˆa†ˆa(ˆσ+ˆσ−−ˆσ−ˆσ+)+ ˆσ+ˆσ−/bracketrightbig +/planckover2pi1˜Ω2 δ/parenleftBig ˆb†ˆb−ˆa†ˆa/parenrightBig +/planckover2pi1Ω˜Ω δ/parenleftBig ˆσ+ˆb+ ˆσ−ˆb†/parenrightBig .(12) Thesecondterm intherighthand ofEq.(10)andthetermsrela tingto thehighorders of Ω/δand ˜Ω/δwere neglected,since Ω,˜Ω≪δ. Furthermore, at theexperimentaltemperature(e.g., 20mK) the electrons are frozen well into their vibrational ground states (about 40mK for the vibrational frequency ∼10GHz). Thismeansthattheexcitationofthevibrationofelec trone1isvirtual,and thus the terms in Eq. (12) related to ˆa†ˆacan be adiabatically eliminated. As a consequence, the HamiltonianinEq. (12)reduces to ˆHeﬀ=/planckover2pi1Ω2 δˆσ+ˆσ−+/planckover2pi1˜Ω2 δˆb†ˆb+/planckover2pi1Ω˜Ω δ/parenleftBig ˆσ+ˆb+ ˆσ−ˆb†/parenrightBig (13) and furtherreads (for Ω =˜Ω) ˆHJC=/planckover2pi1Ω2 δ/parenleftBig ˆσ+ˆb+ ˆσ−ˆb†/parenrightBig (14) in the interaction picture. Obviously, this Hamiltonian de scribes a JC-type coupling between the spinofelectron e1and theorbitalmotionofelectron e2. 70 2 4 6 810 1200.20.40.60.81 t (s)Occupancies \|↑1,01,02〉 \|↓1,01,12〉 ×10−7 FIG. 3: (Color online) Numerical solutions for the Hamilton ian in Eq. (8): the occupancy evolutions of states\| ↑1,01,02/an}bracketri}ht(bluecurve)and \| ↓1,01,12/an}bracketri}ht(redcurve),with ˜Ω = Ω = 25 MHzand δ= ∆ = 250 MHz. Typically, the effective coupling strength can reach Ω′= Ω2/δ≈2.5MHz for d= 10µm, ν1x= 10GHz, and δ= 250MHz. With these parameters and the Hamiltonian in E.q (8), Fi g. 3 showsnumerically theoccupancy evolutionsof thestates \| ↑1,01,02/an}bracketri}htand\| ↓1,01,12/an}bracketri}ht. Here,\| ↓j/an}bracketri}ht and\| ↑j/an}bracketri}htare the two spin states of electron ej, and\|0j/an}bracketri}htand\|1j/an}bracketri}htare the two lower vibrational states of the electron. Obviously, the results are well agre ement with the solutions (i.e., the time- dependentoccupanciesof \| ↑1,02/an}bracketri}htand\| ↓1,12/an}bracketri}ht)fromtheHamiltonian ˆHJC. Thisveriﬁesthevalid- ityofˆHJC. Thespin-orbitJCcoupling(14)couldbeusedtoimplementQ IPbetweentheseparately trapped electrons. For example, by applying a current pulse with the duration t=π/(2Ω′)to an electrode, e.g., I 1, a two-qubit operation ˆV1,2(π/2) =\| ↓1,02/an}bracketri}ht/an}bracketle{t↓1,02\|−i\| ↓1,12/an}bracketri}ht/an}bracketle{t↑1,02\|between the electrons could be implemented. Consequently, a CNOT ga te between the qubits encoded by theelectronicspinscouldbeimplementedbytheoperationa lsequence ˆC=ˆV1,2(π/2)ˆS2ˆV1,2(π/2), withˆS2being the single-electron CNOT gate operated on the electro ne2. After this two-spin CNOT operation, the vibrational motions of the trapped elec trons return to their initial ground states. Furthermore, the mechanism used above for the distant spin- orbit coupling can be utilized to implement an orbit-mediated spin-spin interaction, whe rein the degrees freedom of the orbits of the two electrons are adiabatically eliminated. Indeed, by applying the current pulses to the electrodes simultaneously,theHamiltonianoftheindivid ually-drivenelectrons reads: ˆH′ ee=/planckover2pi1Ω/parenleftbig eiδtˆσ+ˆa+e−iδtˆσ−ˆa†/parenrightbig +/planckover2pi1˜Ω/parenleftBig eiδtˆaˆb†+e−iδtˆa†ˆb/parenrightBig +/planckover2pi1G/parenleftBig eiηtˆτ+ˆb+e−iηtˆτ−ˆb†/parenrightBig .(15) 80 20 40 60 80 100 12000.20.40.60.81 t (s)Occupancies × 10−6\|↓1,01,02,↑2〉 \|↑1,01,02,↓2〉 FIG. 4: (Color online) Numerical solutions for the Hamilton ian in Eq. (15): the occupancy evolutions of the states \| ↓1,01,02,↑2/an}bracketri}ht(blue curve) and \| ↑1,01,02,↓2/an}bracketri}ht(red curve), with ˜Ω = 25MHz,Ω = 2.6MHz, δ= 250MHz,and η= Ω2/δ. Here, the ﬁrst and third terms describe respectively the spi n-orbit couplings of the electrons e1 ande2, and the second term describes the Coulomb interaction betw een the electrons. Gandη are the coupling strength and the detuning between the spin a nd orbital motions of electron e2, respectively. ˆτ−=\| ↓2/an}bracketri}ht/an}bracketle{t↑2\|andˆτ+=\| ↑2/an}bracketri}ht/an}bracketle{t↓2\|are the corresponding spin operators of electron e2. The spin-orbit couplings, i.e., the ﬁrst and third terms in the Hamiltonian, can be realized by applying the ac currents I1(t) =I1cos(ω1t)andI2(t) =I2cos(ω2t)to the electrodes I 1and I2 respectively,with thefrequencies ω1=ν1x−νs+δandω2=ν2x−νs+η. Here the ac currents are appliedtorelatively-easilysatisfytheaboverequire ments forthedetunings. Withthehelp ofEq.(13), Eq.(15)can beeffectivelysimpliﬁ edas ˆH′ ee=ˆHeﬀ+/planckover2pi1G/parenleftBig eiηtˆτ+ˆb+e−iηtˆτ−ˆb†/parenrightBig , (16) i.e., ˆH′ ee=/planckover2pi1Ω˜Ω δ/parenleftBig eiγtˆσ+ˆb+e−itγˆσ−ˆb†/parenrightBig +/planckover2pi1G/parenleftBig ei(η−˜Ω2/δ)tˆτ+ˆb+e−i(η−˜Ω2/δ)tˆτ−ˆb†/parenrightBig (17) in the interaction picture, with γ= (Ω2−˜Ω2)/δ. We select G= Ω˜Ω/δandη= Ω2/δfor simplicity,suchthat ˆH′ ee=/planckover2pi1G/parenleftBig eiγtˆσ+ˆb+e−itγˆσ−ˆb†/parenrightBig +/planckover2pi1G/parenleftBig eiγtˆτ+ˆb+e−iγtˆτ−ˆb†/parenrightBig . (18) ByrepeatingthesamemethodforderivingtheeffectiveHami ltonianˆHeﬀ,i.e.,neglectingtheterms relatingto thehighorders of G/γinthetime-evolutionoperatorand eliminatingadiabatica llythe 9termsrelating to ˆb†ˆb, wehave ˆH′ eﬀ=/planckover2pi1G2 γ(ˆσ+ˆτ−+ ˆσ−ˆτ+). (19) This is an effectively interaction between the two spins, me diated by their no-excited orbital mo- tions[19]. Numerically, for ˜Ω≈25MHz,Ω≈2.6MHz, and δ≈250MHz, we have \|γ\| ≈2.5MHz, G≈0.26MHz, and Ω′′=\|G2/γ\| ≈27kHz. With these parameters, Fig. 4 shows numerically thetime-dependentoccupanciesof \| ↓1,01,02,↑2/an}bracketri}htand\| ↑1,01,02,↓2/an}bracketri}htfromtheHamiltonianinEq. (15). This provides the validity of the simpliﬁed Hamiltoni an in Eq. (19). Obviously,the present orbit-mediated spin-spin coupling is signiﬁcantly weaker than the above spin-orbit JC coupling (14) between the electrons, but still stronger than the dire ctly magnetic dipole-dipole coupling (which is estimated as ∼10−3Hz for the same distance) between the spins. Since the cohere nce time of the spin qubit is very long (e.g., could be up to minute s [7]), the orbit-mediated spin- spin coupling demonstrated above could be utilized to gener ate the spins entanglement and thus implementthedesirableQIP. Finally, we would like to emphasize that, the considered dou ble-trap conﬁguration shown in Fig. 2 seems similarly to that of the recent ion-trap experim ents [20, 21]. There, two ions are conﬁnedintwopotentialwellsseparatedby 40µm[20](or 54µm[21]),andtheion-ionvibrational coupling ˆHii=/planckover2pi1˜Ω[exp(i∆t)ˆaˆb†+ exp(−i∆t)ˆa†ˆb]is achieved up to ˜Ω≈10kHz [20] (or ˜Ω≈ 7kHz [21]). The coupling between theions was manipulated tun ablyby controllingthe potential wells(viasweepingthevoltagesontherelevantelectrodes )toadiabaticallytunetheoscillatorsinto or out of resonance, i.e., ∆ = 0or∆≫˜Ω[22], respectively. Instead, in the present proposal we suggested a JC-type coupling (and consequently an orbit-me diated spin-spin coupling) between the two separated electrons. Therefore, the operational st eps for implementing the QIP should be relativelysimple. Moreinterestingly,heretheelectron- electroncouplingstrength ˜Ωissigniﬁcantly stronger (about 103times) than that between the trapped ions (e.g,9Be+[20]), since the mass of electron ismuchsmallerthanthatoftheions. IV. CONCLUSION We have suggested an approach to implement the QIP with elect ronic spins on liquid helium. Twolong-livedspinstatesofthetrappedelectronwereenco dedasaqubit,andthestrongCoulomb 10interactionbetweentheelectronswasutilizedasthedatab us. Thespin-orbitJCcouplingbetween the spin of an electron and the vibrational motion of another distant electron is generated by designing a virtual excitation of the electronic vibration . Such a distant spin-orbit interaction is further utilized to realize an orbit-mediated spin-spin coupling and implement the desirable quantumgates. Compared with the ions in the Paul traps, here a feature is tha t the mass of the electron is much smaller than that of ions, and thus a strong Coulomb coup ling up to 25 MHz between the electrons could reached for a distance of d= 10µm. Finally, the construction suggested here for implementing quantum computation with trapped electro ns on the liquid helium should be scalable, andhopefullybefeasiblewithcurrent micro-sca letechnique. Acknowledgements : This work was partly supported by the National Natural Scie nce FoundationofChinaGrantsNo. 11204249,11147116,1117437 3,and90921010,theMajorState Basic Research Development Program of China Grant No. 2010C B923104, and the open project ofStateKeyLaboratoryofFunctionalMaterials forInforma tics. [1] W.Paul,Rev.Mod.Phys. 62,531(1990); D.J.Wineland, C.Monroe, W.M.Itano, D.Leibfr ied, B.E. King,andD.M.Meekhof, J.Res.Natl.Inst.Stand.Technol. 103, 259(1998); J.I.CiracandP.Zoller, Phys. Rev. Lett. 74, 4091 (1995). [2] O.Morsch and M.Oberthaler, Rev.Mod. Phys. 78, 179 (2006). [3] L. S. Brownand G. Gabrielse, Rev. Mod. Phys. 58, 233 (1986); L. Lamata, D. Porras, and J. I. Cirac, Phys. Rev. A 81, 022301 (2010). [4] P.M.Platzman and M.I. Dykman, Science 284, 1967 (1999). [5] M. I.Dykman, P.M.Platzman, and P.Seddighrad, Phys. Rev . B67, 155402 (2003). [6] E.Collin,W.Bailey,P.Fozooni,P.G.Frayne,P.Glasson ,K.Harrabi,M.J.Lea,andG.Papageorgiou, Phys. Rev. Lett. 89, 245301 (2002). [7] S.A.Lyon, Phys. Rev.A 74, 052338 (2006). [8] F. R. Bradbury, M. Takita, T. M. Gurrieri, K. J. Wilkel, K. Eng, M. S. Carroll, and S. A. Lyon, Phys. Rev. Lett. 107, 266803 (2011); K.Kono, Physics, 4, 110 (2011). [9] G. Papageorgiou, P. Glasson, K. Harrabi, V. Antonov, E. C ollin, P. Fozooni, P. G. Frayne, M. J. Lea, and D. G. Rees, Appl. Phys. Lett. 86, 153106 (2005); G. Sabouret, F. R. Bradbury, S. Shankar, J. A . 11Bert, and S.A.Lyon, Appl. Phys. Lett. 92, 082104 (2008). [10] M. Koch, G.Aub ¨ ock, C.Callegari, and W.E.Ernst, Phys. Rev. Lett. 103, 035302 (2009). [11] M. Zhang, H.Y. Jiaand L.F.Wei, Phys. Rev.A 80, 055801 (2009). [12] M. Zhang, H.Y. Jiaand L.F.Wei, Opt. Lett. 351686 (2010). [13] D. I. Schuster, A. Fragner, M. I. Dykman, S. A. Lyon, and R . J. Schoelkopf, Phys. Rev. Lett. 105, 040503 (2010). [14] S.Mostame and R.Sch ¨utzhold, Phys. Rev. Lett. 101, 220501 (2008). [15] C. C. Grimes and T. R. Brown, Phys. Rev. Lett. 32, 280 (1974); D. Konstantinov, M. I. Dykman, M. J. Lea, Y.Monarkha, and K.Kono, Phys.Rev. Lett. 103, 096801 (2009). [16] S.S.Sokolov, Phys.Rev. B 51, 2640 (1995). [17] Alternatively, the single-bit operations could be imp lemented by applying an ac current I(t)(with frequency ω=νsand phase θon the electrode I2, see, Fig. 2), which generates a magnetic ﬁeld Bz≈µ0I(t)d/[2π(d2+h2)]along the z-direction toexcite resonantly the spin of electron e1. [18] C. Monroe, D. M. Meekhof, B. E. King, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 754714 (1995). [19] K.Mølmer and A.Sørensen, Phys. Rev.Lett. 821835 (1999). [20] K. R. Brown, C. Ospelkaus, Y. Colombe, A. C. Wilson, D. Le ibfried, and D. J. Wineland, Nature (London) 471, 196 (2011). [21] M. Harlander, R.Lechner, M.Brownnutt, R.Blatt, and W. H¨ ansel, Nature (London) 471, 200 (2011). [22] M. Zhang and L.F.Wei, Phys. Rev.A, 83064301 (2011). 12
2012.02810v1.Effects_of_hybridization_and_spin_orbit_coupling_to_induce_odd_frequency_pairing_in_two_band_superconductors.pdf	arXiv:2012.02810v1 [cond-mat.supr-con] 4 Dec 2020✐✐ “paper” — 2020/12/8 — 1:58 — page 1 — #1 ✐✐ ✐ ✐✐ ✐Eﬀects of hybridization and spin-orbit coupling to induce odd frequency pairing in two-band superconductors Moloud Tamadonpour and Heshmatollah Yavari The eﬀects of spin independent hybridization potential and spin- orbit coupling on two-band superconductor with equal time s -wave interband pairing order parameter is investigated theoret ically. To study symmetry classes in two-band superconductors the Gor ’kov equations are solved analytically. By deﬁning spin singlet and spin triplet s-wave order parameter due to two-band degree of fre e- dom the symmetry classes of Cooper pair are studied. For spin singlet case it is shown that spin independent hybridizatio n gen- erates Cooper pair belongs to even-frequency spin singlet e ven- momentum even-band parity (ESEE) symmetry class for both in - traband and interband pairing correlations. For spin tripl et order parameter, intraband pairing correlation generates odd-f requency spin triplet even-momentum even-band parity (OTEE) symme- try class whereas, interband pairing correlation generate s even- frequency spin triplet even-momentum odd-band parity (ETE O) class. For the spin singlet, spin-orbit coupling generates pairing cor- relation that belongs to odd-frequency spin singlet odd-mo mentum even-band parity (OSOE) symmetry class and even-frequency spin singlet even-momentum even-band parity (ESEE) for intraba nd and interband pairing correlation respectively. In the spi n triplet case for itraband and interband correlation, spin-orbit co upling generates even-frequency spin triplet odd-momentum even- band parity (ETOE) and even-frequency spin triplet even-moment um odd-band parity (ETEO) respectively. 1. Introduction and summary Symmetries of order parameter in superconductors aﬀect the ir physical prop- erties. The total wave function of a pair of fermions, in acco rdance with the Pauli principle, should be asymmetric under the permutatio n of orbital, spin and time (or equivalently Matsubara frequency) coordinate s [1]. This leads 1✐✐ “paper” — 2020/12/8 — 1:58 — page 2 — #2 ✐✐ ✐ ✐✐ ✐2 Moloud Tamadonpour and Heshmatollah Yavari to four classes allowed combinations for the symmetries of t he wave function. This would imply that if the pairing is even in time, spin sing let pairs have even parity (ESE) and spin triplet pairs have odd parity (ETO ). While if the pairing is odd in time, spin singlet pairs have odd parity (OSO) and spin triplet pairs have even parity (OTE). Black-Schaﬀer and Balatsky [2] have shown that the multiband superconducting order parame ter has an ex- tra symmetry classiﬁcation that originates from the band de gree of freedom, so called even-band-parity and odd band-parity. As a conseq uence, Cooper pairs can be classiﬁed into eight symmetry classes [3]. Transport properties of multi-band superconductor are qua litatively dif- ferent from those of the one-band superconductor. For insta nce, two-band system with the non-magnetic impurity violates Anderson th eorem [4]. As a result, lots of eﬀorts have been devoted to understanding t he properties of such systems both theoretically and experimentally. For these materials band symmetry plays important role. A main hypothesis of the model is the formation of the Cooper pairs inside one energy band and tran sition of this pair from one band to another which leads to intra and inter ba nd electronic interactions. Multi-band model explained lots of strange p hysical properties of superconductive systems and were consistent with experi mental data. Fa- mous multiband superconductors are MgB2 [5, 6] and the iron-b ased super- conductors [7–9]. The nature of their two bands requires tha t the multiband approach be used to describe their properties. On the contra ry for cuprates despite their multiband nature a single-band approach is mo re appropriate. From a general symmetry analysis of even and odd-frequency p airing states, it was shown that odd-frequency pairing always exis ts in the form of odd-interband (orbital) pairing if there is any even-frequ ency even-interband pairing present consistent with the general symmetry requi rements [10]. The appearance of odd-frequency Cooper pairs in two-band super conductors by solving the Gor’kov equation was discussed analytically [1 1]. They considered the equal-time s-wave pair potential and introduced two typ es of hybridiza- tion potentials between the two conduction bands. One is a sp in-independent hybridization potential and the other is a spin-dependent h ybridization po- tential derived from the spin-orbit interaction. The eﬀect of random nonmagnetic impurities on the supercond ucting transition temperature in a two-band superconductor, by as suming the equal- time spin-singlet s-wave pair potential in each conduction band and the hy- bridization between the two bands as well as the band asymmet ry was stud- ied theoretically [11, 12]. The eﬀect of single-quasiparti cle hybridization or scattering in a two-band superconductor by performing pert urbation theory✐✐ “paper” — 2020/12/8 — 1:58 — page 3 — #3 ✐✐ ✐ ✐✐ ✐Eﬀects of hybridization and spin-orbit coupling 3 to inﬁnite order in the hybridization term, in a multiband su perconductor was investigated [13]. The superconducting state of multi-orbital spin-orbit cou pled systems in the presence of an orbitally driven inversion asymmetry, by assuming that the interorbital attraction is the dominant pairing channe l, was studied [14]. They have shown that in the absence of the inversion symmetry , supercon- ducting states that avoid mixing of spin-triplet and spin-s inglet conﬁgura- tions are allowed, and remarkably, spin-triplet states tha t are topologically nontrivial can be stabilized in a large portion of the phase d iagram. The impact of strong spin-orbit coupling (SOC) on the propertie s of new class superconductors has attracted much attentions. It has been the subject of great theoretical and experimental interest [15, 16]. The f ormation of unex- pected multi-component superconductors states allows for superconductors with magnetism and SOC. It was shown that for multi-orbital s ystems such as the Fe-pnictides SOC coupling, is much smaller than the or bit Hund’s coupling [17–19], In contrast for multiband systems such as Ir-based oxide materials it was found that the SOC interaction is comparabl e to the on-site Coulomb interaction [20]. The combined eﬀect of Hund’s and S OC coupling on superconductivity in multi-orbital systems was investi gated and it was shown that Hund’s interaction leads to orbital-singlet spi n-triplet supercon- ductivity, where the Cooper pair wave function is antisymme tric under the exchange of two orbitals [21]. Combined eﬀect of the spin-or bit coupling and scattering on the nonmagnetic disorder on the formation of t he spin reso- nance peak in iron-based superconductors was also studied [ 22]. In this paper by using Gor’kov equation the eﬀects of spin-or bit coupling and hybridization on the possibility of odd frequency pairi ng of a two-band superconductor with an equal time s-wave interband pairing order parameter are investigated theoretically. 2. Formalism 2.1. Two-band model The basic physics of multiband superconductors can be obtai ned by intro- ducing a two-band model. We start with a normal two-band Hami ltonian as✐✐ “paper” — 2020/12/8 — 1:58 — page 4 — #4 ✐✐ ✐ ✐✐ ✐4 Moloud Tamadonpour and Heshmatollah Yavari [12] (2.1) ˇHN=/integraldisplay dr/bracketleftig ψ† 1,↑(r), ψ† 1,↓(r), ψ† 2,↑(r), ψ† 2,↓(r)/bracketrightig ˇHN(r) ψ1,↑(r) ψ1,↓(r) ψ2,↑(r) ψ2,↓(r) , where (2.2) ˇHN=/parenleftbiggξ1kˆσ0/parenleftbig υeiθ+V/parenrightbig ˆσ0 /parenleftbig υe−iθ+V∗/parenrightbig ˆσ0ξ2kˆσ0/parenrightbigg . Hereψα,σ(r)is the annihilation ( ψ† α,σ(r))creation) operator of an electron with spin (σ=↑,↓) at theαth conduction band, ξαk=/planckover2pi12k2/2me−µFis the dispersion energy of band α,meis the mass of an electron, µFis the chemical potential. The spin independent hybridization potential i s a complex number characterized by a phase θ.υeiθdenotes the hybridization between the two bands, which is much smaller than the Fermi energy in the two c onduction bands. In the absence of spin ﬂip hybridization the spin-orb it coupling poten- tial isV(k) =ηˆz.(σ×/vectork) =η(kyσx−kxσy), whereηis the parameter that describes the strength of the Rashba spin-orbit coupling an dˆzis the unit vector perpendicular to the superconducting surface. This potential is odd- momentum-parity functions satisfying V(k) =−V(−k). Throughout this pa- per, Pauli matrices in spin, two-band, particle- hole space s are respectively denoted by ˆσj,ˆρjandˆτjforj= 1−3. Superconducting order parameter in bandαis: (2.3) ˆ∆αα′(k) =/parenleftbigg∆11(k) ∆12(k) ∆21(k) ∆22(k)/parenrightbigg . We focus only on interband superconducting order parameter (∆11(k) = ∆22(k) = 0) . The interband s-wave pair potential, is deﬁned by [12] (2.4) ∆12,σσ′(r) =g/an}bracketle{tψ1,σ(r)ψ2,σ′(r)/an}bracketri}ht. heregis interband attractive interaction between two electrons . By assuming the spatially uniform order parameter the Fourier transfor mation of the pair potential becomes (2.5) ∆12,↑↓=g Vvol/summationdisplay k/an}bracketle{tψ1,↑(k)ψ2,↓(−k)/an}bracketri}ht.✐✐ “paper” — 2020/12/8 — 1:58 — page 5 — #5 ✐✐ ✐ ✐✐ ✐Eﬀects of hybridization and spin-orbit coupling 5 In the two-band model, for spin singlet the order parameter i s symmetric (antisymmetric) under the permutation of band (spin) indic es. (2.6) ∆12,↑↓= ∆21;↑↓=−∆1,2;↓↑, But for spin triplet the order parameter is antisymmetric (sy mmetric) under the permutation of band (spin) indices. (2.7) ∆12,↑↓=−∆21;↑↓= ∆1,2;↓↑. For simplicity we omit the indices of ∆αα′. The Hamiltonian describing su- perconductor in the Nambu space, can be written as [11] (2.8)⌣HS(T)=1 2/summationdisplay kψ† k,σ/parenleftiggˇHN(k)ˇ∆S(T) ˇ∆† S(T)−ˇH∗ N(−k)/parenrightigg ψk,σ, where the spin-singlet and spin triplet pair potentials ( ˇ∆Sandˇ∆T) are respectively given by (2.9) ˇ∆S= ∆ˆρ1iˆσ2, (2.10) ˇ∆T= ∆iˆρ2ˆσ1. For a two-band system, the Bogoliubov- de Gennes Hamiltonian can be de- scribed by 8×8matrix reﬂecting spin, particle- hole and two band degrees of freedom. In particle-hole space N1, by considering the spin of electron as ↑and for hole as ↓, while in particle-hole space N2, we consider the spin of electron as ↓and for hole as ↑, we can describe the Hamiltonian ˇHS(T)by a 4×4matrix [3, 12] (2.11) ˇH0= ξkυeiθ+V(k) 0 ∆ υe−iθ+V∗(k)ξk −sspin∆ 0 0 −sspin∆ −ξk −υe−iθ−V∗(−k) ∆ 0 −υeiθ−V(−k) −ξk heresspin=−1for spin singlet and sspin= 1for spin triplet. To discuss the eﬀects of hybridizations and spin-orbit interaction on the properties of superconductors, we calculate the Green’s functions by sol ving the Gor’kov✐✐ “paper” — 2020/12/8 — 1:58 — page 6 — #6 ✐✐ ✐ ✐✐ ✐6 Moloud Tamadonpour and Heshmatollah Yavari equation [23] (2.12)/parenleftbig iωn−ˇH0/parenrightbigˇG0(k,iωn) =ˇ1, (2.13) ˇG0(k,iωn) =/parenleftigg ˆG0(k,iωn) ˆF0(k,iωn) −sspinˆF† 0(−k,iωn)−ˆG∗ 0(−k,iωn)/parenrightigg . whereωn= (2n+1)πkBTis the Matsubara frequency ( kBis the Boltz- mann constant), and ˇ1is the identity matrix in spin×band×particle− holespace.ˇG0is a4×4matrix where the diagonal components are nor- mal Green’s function and non-diagonal components are anoma lous Green’s function. 2.2. Spin Singlet Pairing Order According to Equation (2.11), the Hamiltonian of a two-band superconductor with spin singlet conﬁguration in the presence of spin-orbi t coupling is (2.14) ˇH0= ξk υeiθ+η(ky+ikx) 0 ∆ υe−iθ+η(ky−ikx) ξk ∆ 0 0 ∆ −ξk −υe−iθ+η(ky−ikx) ∆ 0 −υeiθ+η(ky+ikx) −ξk . By using Equation (2.12) and (2.13) , for spin singlet the solu tion of the normal Green’s function within the ﬁrst order of ∆is calculated as (2.15) ˆG0(k,iωn) =∆ Z0{[(ξ−iωn)(ν2+η2k2−2νη(kxsinθ+kycosθ)+(ξ+iωn) ×/parenleftbig −ξ2−ω2 n/parenrightbig ]ˆρ0+[(−νcosθ−ηky)(−(ξ+iωn)2+ν2+η2k2 −2νη(kxsinθ+kycosθ))]ˆρ1+[(νsinθ+ηkx)(−(ξ+iωn)2 +ν2+η2k2−2νη(kxsinθ+kycosθ))]ˆρ2} here (2.16) Z0=ξ4+2ξ2/parenleftbig ω2 n−ν2/parenrightbig +/parenleftbig ω2 n+ν2/parenrightbig2−8iηνξωn(kxsinθ+kycosθ) +2cos2θη2ν2(k2 x−k2 y)−4sin2θη2ν2kxky+2η2k2/parenleftbig ω2 n−ξ2/parenrightbig +η4k4. thatkx=kcosφandky=ksinφ, whereφis the angle between momentum and thexaxis. The matrix form of the normal Green’s function ( Eq. (2. 15))✐✐ “paper” — 2020/12/8 — 1:58 — page 7 — #7 ✐✐ ✐ ✐✐ ✐Eﬀects of hybridization and spin-orbit coupling 7 can be written as (2.17) ˆG0(k,iωn) =/parenleftbiggG11(k,iωn)G12(k,iωn) G21(k,iωn)G22(k,iωn)/parenrightbigg . where (2.18) G11(k,iωn) =∆ Z0[(ξ−iωn)(ν2+η2(k2 x+k2 y)−2νη(kxsinθ+kycosθ) −(ξ+iωn)(ξ2+ω2 n)] (2.19) G12(k,iωn) =∆ Z0[{−νeiθ−η(ikx+ky)}{−(ξ+iωn)2+ν2+η2k2 −2νη(kxsinθ+kycosθ)}] (2.20) G21(k,iωn) =∆ Z0[{−νe−iθ−η(−ikx+ky)}{−(ξ+iωn)2+ν2+η2k2 −2νη(kxsinθ+kycosθ)}] (2.21) G22(k,iωn) =∆ Z0[(ξ−iωn)(νe−iθ−η(−ikx+ky))(νeiθ−η(ikx+ky) −(ξ+iωn)/parenleftbig ξ2+ω2 n/parenrightbig ] By using Equation (2.12) and (2.13), the anomalous Green’s fu nction can be obtained as (2.22) ˆF0(k,iωn) =∆ Z0[(2νξcosθ+2ηωniky) ˆρ0+/parenleftbig −(ν2+ξ2+ω2 n)+η2k2/parenrightbig ˆρ1 +(2νη(kxcosθ−kysinθ)) ˆρ2+(2νξisinθ−2ηωnkx) ˆρ3]. In particle- hole space N1, the matrix form of the anomalous Green’s func- tion (Eq. (2.22)) is (2.23) ˆFN1 0(k,iωn) =/parenleftbiggF11,↑↓(k,iωn)F12,↑↓(k,iωn) F21,↑↓(k,iωn)F22,↑↓(k,iωn)/parenrightbigg , where (2.24) F11,↑↓(k,iωn) =∆ Z0[2νξeiθ−2ηωnkx+2ηωniky], (2.25) F12,↑↓(k,iωn) =∆ Z0[−(ν2+ξ2+ω2 n)+η2k2−2iνη(kxcosθ−kysinθ)],✐✐ “paper” — 2020/12/8 — 1:58 — page 8 — #8 ✐✐ ✐ ✐✐ ✐8 Moloud Tamadonpour and Heshmatollah Yavari (2.26) F21,↑↓(k,iωn) =∆ Z0[−(ν2+ξ2+ω2 n)+η2k2+2iνη(kxcosθ−kysinθ)], (2.27) F22,↑↓(k,iωn) =∆ Z0[2νξeiθ−2ηωnkx+2ηωniky]. In particle- hole space N2, the matrix form of the anomalous Green’s function is (2.28)ˆFN2 0(k,iωn) =/parenleftbigg F11,↓↑(k,iωn)F12,↓↑(k,iωn) F21,↓↑(k,iωn)F22,↓↑(k,iωn)/parenrightbigg =−ˆFN1 0(k,iωn). In the absence of spin-orbit coupling ( η) the anomalous Green’s function (Eq.(2.22)) becomes (2.29) ˆF0(k,iωn) =∆ Z0[2νξcosθˆρ0−(ν2+ξ2+ω2 n)ˆρ1+2νξisinθˆρ3], here (2.30) Z0=ξ4+2ξ2/parenleftbig ω2 n−ν2/parenrightbig +/parenleftbig ω2 n+ν2/parenrightbig2. The matrix form of the anomalous Green’s function in Equatio n (2.29) in particle-hole spaces N1andN2, are (2.31) ˆFN1 0(k,iωn) =/parenleftbigg F11,↑↓(k,iωn)F12,↑↓(k,iωn) F21,↑↓(k,iωn)F22,↑↓(k,iωn)/parenrightbigg =∆ Z0/parenleftbigg 2νξcosθ+2iνξsinθ−(ν2+ξ2+ω2 n) −(ν2+ξ2+ω2 n) 2νξcosθ−2iνξsinθ/parenrightbigg =∆ Z0/parenleftbigg 2ξνeiθ−(ν2+ξ2+ω2 n) −(ν2+ξ2+ω2 n) 2ξνe−iθ/parenrightbigg , and (2.32) ˆFN2 0(k,iωn) =/parenleftbiggF11,↓↑(k,iωn)F12,↓↑(k,iωn) F21,↓↑(k,iωn)F22,↓↑(k,iωn)/parenrightbigg =−ˆFN1 0(k,iωn) =∆ Z0/parenleftbigg −2νξcosθ−2iνξsinθ(ν2+ξ2+ω2 n) (ν2+ξ2+ω2 n)−2νξcosθ+2iνξsinθ/parenrightbigg =∆ Z0/parenleftbigg−2ξνeiθ(ν2+ξ2+ω2 n) (ν2+ξ2+ω2 n)−2ξνe−iθ/parenrightbigg .✐✐ “paper” — 2020/12/8 — 1:58 — page 9 — #9 ✐✐ ✐ ✐✐ ✐Eﬀects of hybridization and spin-orbit coupling 9 The intraband pairing correlations become (2.33) F11,↑↓(k,iωn)−F11,↓↑(k,iωn) =4∆ Z0ξνeiθ, (2.34) F22,↑↓(k,iωn)−F22,↓↑(k,iωn) =4∆ Z0ξνe−iθ. Hybridization generates ρ0andρ3components which belongs to even fre- quency symmetry class. It means that in the presence of inter band cou- pling, hybridization generates even frequency intra- subl attice pairing in the system. These components belong to even-frequency spin -singlet even- momentum even-band parity (ESEE) symmetry class. This resu lt is in agree- ment with the equation (20) presented in Ref [12]. Equation ( 2.33) and (2.34) are in agreement with the Equation (62) and (63) reported in R ef [3] in the ﬁrst order of ∆(\|∆\|2= 0) and equal energy bands ( ξ−= 0) and both belong to the (ESEE) symmetry class. The band symmetry generates in terband pairing correlation: (2.35)[F12,↑↓(k,iωn)−F12,↓↑(k,iωn)]+[F21,↑↓(k,iωn)−F21,↓↑(k,iωn)] =−4∆ Z0(ν2+ξ2+ω2 n). which belongs to (ESEE). This result is in agreement with the Equation (65) presented in Ref [3]. (2.36)[F12,↑↓(k,iωn)−F12,↓↑(k,iωn)]−[F21,↑↓(k,iωn)−F21,↓↑(k,iωn)] =2∆ Z3iωnξ−. which belongs to the symmetry (OSEO) class. We considered a t wo-band superconductor with an equal dispersion energy in each band (ξ+=ξ−). In this case the interband pairing correlation due to band asym metry is (2.37) [F12,↑↓(k,iωn)−F12,↓↑(k,iωn)]−[F21,↑↓(k,iωn)−F21,↓↑(k,iωn)] = 0. In the absence of hybridization within the second order of th e spin-orbit coupling constant ( η), we obtain (2.38) ˆG0(k,iωn) =∆ Z0[(η2k2−(ξ+iωn)2)(ξ−iωn) ˆρ0+η(ky+ikx)(ξ+iωn)2ˆρ1], where (2.39) Z0= (ξ2+ω2 n)2+2η2k2/parenleftbig ω2 n−ξ2/parenrightbig .✐✐ “paper” — 2020/12/8 — 1:58 — page 10 — #10 ✐✐ ✐ ✐✐ ✐10 Moloud Tamadonpour and Heshmatollah Yavari Equation (2.22) can be rewritten as (2.40) ˆF0(k,iω) =∆ Z0[2iηωnkyˆρ0−2ηωnkxˆρ3+/parenleftbig −ξ2−ω2 n+η2k2/parenrightbig ˆρ1]. In particle- hole space N1andN2,the matrix form of the anomalous Green’s function (Eq. (2.40)) is (2.41)ˆFN1 0(k,iωn) =/parenleftbigg F11,↑↓(k,iωn)F12,↑↓(k,iωn) F21,↑↓(k,iωn)F22,↑↓(k,iωn)/parenrightbigg =∆ Z0/parenleftbigg2iηωnky−2ηωnkx−ξ2−ω2 n+η2k2 −ξ2−ω2 n+η2k22iηωnky+2ηωnkx/parenrightbigg , (2.42) ˆFN2 0(k,iωn) =/parenleftbiggF11,↓↑(k,iωn)F12,↓↑(k,iωn) F21,↓↑(k,iωn)F22,↓↑(k,iωn)/parenrightbigg =−ˆFN1 0(k,iωn) =∆ Z0/parenleftbigg−2iηωnky+2ηωnkxξ2+ω2 n−η2k2 ξ2+ω2 n−η2k2−2iηωnky−2ηωnkx/parenrightbigg . The intraband pairing correlations are (2.43) F11,↑↓(k,iωn)−F11,↓↑(k,iωn) =4∆ Z0iωnη(ky+ikx), (2.44) F22,↑↓(k,iωn)−F22,↓↑(k,iωn) =4∆ Z0iωnη(ky−ikx). Spin-orbit coupling generates ρ0andρ3components which belong to odd frequency symmetry class. It means that in the presence of in ter-band cou- pling, spin-orbit coupling generates odd-frequency intra sublattice pairing in the system. These components belong to odd-frequency spin- singlet odd- momentum even-band parity (OSOE) symmetry class. In Ref [3] the intra- band pairing correlation is written as (2.45) F11,↑↓(k,iωn)+F11,↓↑(k,iωn) =∆ Z3(ξ+−ξ−)V3, (2.46) F22,↑↓(k,iωn)+F22,↓↑(k,iωn) =∆ Z3(ξ++ξ−)V3.✐✐ “paper” — 2020/12/8 — 1:58 — page 11 — #11 ✐✐ ✐ ✐✐ ✐Eﬀects of hybridization and spin-orbit coupling 11 The hybridization generates pairing correlations that bel ong to the (ETOE) class. The band asymmetry generates interband pairing corr elation as (2.47) [F12,↑↓(k,iωn)−F12,↓↑(k,iωn)]−[F21,↑↓(k,iωn)−F21,↓↑(k,iωn)] = 0. In Ref [3] for spin orbit hybridization the band asymmetry ge nerates the interband pair correlation as (2.48) [F12,↑↓(k,iω)−F12,↓↑(k,iω)]−[F21,↑↓(k,iω)−F21,↓↑(k,iω)] =2∆ Z3iωnξ−. which belongs to the odd-frequency spin-singlet even-mome ntum odd-band parity symmetry (OSEO). The interband pairing correlation due to band symmetry is (2.49)[F12,↑↓(k,iωn)−F12,↓↑(k,iωn)]+[F21,↑↓(k,iωn)−F21,↓↑(k,iωn)] =−4∆ Z0(ξ2+ω2 n−η2k2). This component belongs to even-frequency spin-singlet eve n-momentum even- band parity (ESEE) symmetry class. For spin singlet, hybrid ization potential generates ESEE symmetry class due to both intra and interban d correlation, whereas the spin dependent hybridization potential genera tes this class only for interband pairing correlation due to band symmetry. In t his case the odd frequency pairing arises only due to intraband pairing corr elations for spin dependent hybridization potential. 2.3. Spin Triplet Pairing Order By considering Equation (2.11), the Hamiltonian of a two-ban d supercon- ductor with spin triplet conﬁguration in the presence of spi n-orbit coupling is (2.50) ˇH0= ξk υeiθ+η(ky+ikx) 0 ∆ υe−iθ+η(ky−ikx) ξk −∆ 0 0 −∆ −ξk −υe−iθ+η(ky−ikx) ∆ 0 −υeiθ+η(ky+ikx) −ξk . The solution of the anomalous Green’s function within the ﬁr st order of ∆ is calculated as (2.51) ˆF0(k,iωn) =∆ Z0[(−2iηξkx+2νωnsinθ)ˆρ0+(2iνη(kxcosθ−kysinθ)) ˆρ1 +(i(ν2−ξ2−ω2 n)−iη2(k2 x+k2 y))ˆρ2+(−2ηξky−2iνωncosθ)ˆρ3].✐✐ “paper” — 2020/12/8 — 1:58 — page 12 — #12 ✐✐ ✐ ✐✐ ✐12 Moloud Tamadonpour and Heshmatollah Yavari The matrix form of the anomalous Green’s function (Eq. (2.51 ) ) can be written as (2.52) ˆF11,↑↓(k,iωn) =∆ Z0/parenleftig −2iνωneiθ−2iηξkx−2ηξky/parenrightig , (2.53) ˆF12,↑↓(k,iωn) =∆ Z0[(ν2−ξ2−ω2 n)−η2k2+2iνη(kxcosθ−kysinθ)], (2.54) ˆF21,↑↓(k,iωn) =∆ Z0[(−ν2+ξ2+ω2 n)+η2k2+2iνη(kxcosθ−kysinθ)], (2.55) ˆF22,↑↓(k,iωn) =∆ Z0/parenleftig 2iνωne−iθ−2iηξkx+2ηξky/parenrightig . In the absence of spin-orbit coupling ( η= 0) the anomalous Green’s function Equation (2.51) becomes (2.56)ˆF0(k,iωn) =∆ Z0[2νωnsinθˆρ0+i(ν2−ξ2−ω2 n)ˆρ2−2iνωncosθˆρ3]. here (2.57) Z0=ξ4+2ξ2/parenleftbig ω2 n−ν2/parenrightbig +/parenleftbig ω2 n+ν2/parenrightbig2. In particle-hole space N1andN2, the matrix form of the anomalous Green’s function (Eq. (2.56)) is (2.58) ˆFN1 0(k,iωn) =/parenleftbiggF11,↑↓(k,iωn)F12,↑↓(k,iωn) F21,↑↓(k,iωn)F22,↑↓(k,iωn)/parenrightbigg =∆ Z0/parenleftbigg 2νωn(sinθ−icosθ) (ν2−ξ2−ω2 n) −(ν2−ξ2−ω2 n) 2νωn(sinθ+icosθ)/parenrightbigg , (2.59)ˆFN2 0(k,iωn) =/parenleftbiggF11,↓↑(k,iωn)F12,↓↑(k,iωn) F21,↓↑(k,iωn)F22,↓↑(k,iωn)/parenrightbigg =ˆFN1 0(k,iωn) =∆ Z0/parenleftbigg 2νωn(sinθ−icosθ) (ν2−ξ2−ω2 n) −(ν2−ξ2−ω2 n) 2νωn(sinθ+icosθ)/parenrightbigg . The intraband pairing correlations becomes (2.60) F11,↑↓(k,iωn)+F11,↓↑(k,iωn) =−4∆ Z0iωnνe−iθ,✐✐ “paper” — 2020/12/8 — 1:58 — page 13 — #13 ✐✐ ✐ ✐✐ ✐Eﬀects of hybridization and spin-orbit coupling 13 (2.61) F22,↑↓(k,iωn)+F22,↓↑(k,iωn) =4∆ Z0iωnνeiθ. Hybridization generates ρ0andρ3which belongs to odd frequency sym- metry class. These components belong to odd-frequency spin -triplet even- momentum even-band parity (OTEE) symmetry class. This resu lt is in agree- ment with the Equation (24) presents in Ref [12] In the ﬁrst or der of∆ (\|∆\|2= 0) and equal energy bands ( ξ−= 0) Equation (2.60) and (2.61) are coincide with the Equation (83) and (84) presented in Ref [3] and both belong to the (OTEE) symmetry class. The band symmetry gener ates inter- band pairing correlation as (2.62)[F12,↑↓(k,iωn)+F12,↓↑(k,iωn)]−[F21,↑↓(k,iωn)+F21,↓↑(k,iωn)] =4∆ Z0(ν2−ξ2−ω2 n). which belongs to even-frequency spin triplet even-momentu m odd-band par- ity (ETEO) symmetry class. In Ref [3] the interband pairing c orrelation due to band asymmetry is (2.63)[F12,↑↓(k,iωn)+F12,↓↑(k,iωn)]+[F21,↑↓(k,iωn)+F21,↓↑(k,iωn)] =2∆ Z5iωnξ−. Thus the band hybridization generates pairing correlation s that belong to the odd-frequency spin triplet even-momentum even-band pa rity (OTEE) class. Since we considered a two-band superconductor with a n equal energy bands, the interband pairing correlation due to band asymme try is (2.64) [F12,↑↓(k,iωn)+F12,↓↑(k,iωn)]+[F21,↑↓(k,iωn)+F21,↓↑(k,iωn)] = 0. In the absence of hybridization, we obtain (2.65) ˆF0(k,iωn) =∆ Z0[−2iηξkxˆρ0−2ηξkyˆρ3−i/parenleftbig ξ2+ω2 n+η2k2/parenrightbig ˆρ2]. The matrix form of the anomalous Green’s function in Equatio n (2.65) in particle- hole spaces N1andN2, are (2.66)ˆFN1 0(k,iωn) =/parenleftbiggF11,↑↓(k,iωn)F12,↑↓(k,iωn) F21,↑↓(k,iωn)F22,↑↓(k,iωn)/parenrightbigg =∆ Z0/parenleftbigg−2iηξkx−2ηξky−(ξ2+ω2 n+η2k2) (ξ2+ω2 n+η2k2)−2iηξkx+2ηξky/parenrightbigg ,✐✐ “paper” — 2020/12/8 — 1:58 — page 14 — #14 ✐✐ ✐ ✐✐ ✐14 Moloud Tamadonpour and Heshmatollah Yavari (2.67)ˆFN2 0(k,iωn) =/parenleftbigg F11,↓↑(k,iωn)F12,↓↑(k,iωn) F21,↓↑(k,iωn)F22,↓↑(k,iωn)/parenrightbigg =ˆFN1 0(k,iωn) =∆ Z0/parenleftbigg −2iηξkx−2ηξky−(ξ2+ω2 n+η2k2) (ξ2+ω2 n+η2k2)−2iηξkx+2ηξky/parenrightbigg . The intraband pairing correlations are (2.68) F11,↑↓(k,iωn)+F11,↓↑(k,iωn) =−4∆ Z0ξη(ky+ikx), (2.69) F22,↑↓(k,iωn)+F22,↓↑(k,iωn) =4∆ Z0ξ(ky−ikx). Spin-orbit coupling generates ˆρ0andˆρ3which belong to even frequency sym- metry class. These components belong to even-frequency spi n-triplet odd- momentum even-band parity (ETOE) symmetry class. In Ref [3] the intra- band pairing correlation is calculated as (2.70) F11,↑↓(k,iωn)−F11,↓↑(k,iωn) =−∆ Z5iωnV3, (2.71) F22,↑↓(k,iωn)−F22,↓↑(k,iωn) =∆ Z0iωnV3. The hybridization generates pairing correlations that bel ong to the odd- frequency spin singlet odd-momentum even-band parity (OSO E) class. As mentioned in Ref [3] the interband pair correlation can be wr itten as (2.72)[F12,↑↓(k,iωn)+F12,↓↑(k,iωn)]+[F21,↑↓(k,iωn)+F21,↓↑(k,iωn)] =2∆ Z5iωnξ−. Thus the spin-orbit coupling generates pairing correlatio ns that belong to the odd-frequency spin triplet even-momentum even-band pa rity (OTEE) class. In contrast in our formalism the band asymmetry gener ates interband pairing correlation as (2.73) [F12,↑↓(k,iωn)+F12,↓↑(k,iωn)]+[F21,↑↓(k,iωn)+F21,↓↑(k,iωn)] = 0. The interband pairing correlation due to band symmetry is (2.74)[F12,↑↓(k,iωn)+F12,↓↑(k,iωn)]−[F21,↑↓(k,iωn)+F21,↓↑(k,iωn)] =−4∆ Z0(ξ2+ω2 n+η2k2).✐✐ “paper” — 2020/12/8 — 1:58 — page 15 — #15 ✐✐ ✐ ✐✐ ✐Eﬀects of hybridization and spin-orbit coupling 15 These components belong to even-frequency spin-triplet ev en-momentum odd-band parity (ETEO) symmetry class. Thus, for spin tripl et, the spin dependent and spin independent hybridization both generat e the same sym- metry class ETEO due to interband pairing correlation. The o dd frequency pairing arises in the presence of spin independent hybridiz ation due to in- traband pairing correlations. 3. Conclusion Within the theoretical model the existence of odd frequency pairs in two band superconductors by incorporating both spin independe nt hybridization and spin dependent spin-orbit interaction is investigated . This model also includes both the one-particle hybridization term and all p ossible intraband and interband superconducting pairing interaction terms i n a two-band sys- tem. The normal and anomalous thermal Green’s functions have bee n calcu- lated in the Nambu formalism as elements of the Fourier trans formed4×4 matrix Green’s function by taking into account of all possib le intraband and interband superconducting interaction terms coupling both bands in the mean ﬁeld approximation. By assuming that the attractive interaction acts on two electrons with diﬀerent spins in diﬀerent conduc tion bands dif- ferent symmetry classes were demonstrated in the presence o f hybridization and spin-orbit coupling. The role of intraband and interband pairing correlations to emerge the odd frequency in a two-band superconductor was examined. Fo r spin singlet, the odd-frequency is generated by spin dependent hybridiza tion potential owing to intraband pairing correlations in agreement with t he odd frequency generated by the interband pair correlation due to band asym metry in Ref [3]. On the other hand, for spin triplet the spin independent hybridization potential generates the odd-frequency pairing due to intra band correlations in agreement with the result of Ref [12]. References [1] M. Sigrist and K. Ueda, Phenomenological theory of unconventional su- perconductivity . Rev. Mod. Phys., 63, 239, (1991). [2] A. M. Black-Schaﬀer and A. V. Balasky, Proximity-induced unconven- tional superconductivity in topological insulators . Phys. Rev. B, 87, 220506(R), (2013).✐✐ “paper” — 2020/12/8 — 1:58 — page 16 — #16 ✐✐ ✐ ✐✐ ✐16 Moloud Tamadonpour and Heshmatollah Yavari [3] Y. Asano and A. Sasaki, Odd-frequency Cooper pairs in two-band su- perconductors and their magnetic response . Phys. Rev. B, 92, 224508, (2015). [4] V. A. Moskalenko and M. E. Palistrant, Two-band model determination of the critical temperature of a superconductor with an impu rity. Sov. Phys. JETP, 22, 526, (1966). [5] X. X. Xi, Two-band superconductor magnesium diboride . Rep. Prog. Phys., 71, 116501, (2008). [6] J. Nagamatsu, N. Nakagawa, T. Muranaka, Y. Zenitani, J. A kimutsu, Superconductivity at 39 K in magnesium diboride . Nature, 410, 63, (2001). [7] Y. Shun-Li and L. Jian-Xin, Spin ﬂuctuations and unconventional su- perconducting pairing in iron-based superconductors . Chin. Phys. B, 22, 087411, (2013). [8] Y. Tanaka, P. M. Shirage and A. Iyo, Disappearance of Meissner Eﬀect and Speciﬁc Heat Jump in a Multiband Superconductor . J Supercond Nov Magn, 23, 253–256, (2010). [9] P. O. Sprau et al, Discovery of orbital-selective Cooper pairing in FeSe . science, 357, 75-80, (2013). [10] A. M. Black-Schaﬀer and A. V. Balasky, Odd-frequency superconducting pairing in multiband superconductors . Phys. Rev. B, 88, 104514, (2013). [11] Y. Asano and A. A. Golubov, Green’s-function theory of dirty two-band superconductivity . Phys. Rev. B, 97, 214508, (2018). [12] Y. Asano, A Sasaki and A. A. Golubov, Dirty two-band superconductivity with interband pairing order . New J. Phys., 20, 043020, (2018). [13] L. Komendov’a, A. V. Balatsky, and A. M. Black-Schaﬀer, Band hy- bridization induced odd-frequency pairing in multiband su perconductors . Phys. Rev. B, 92, 094517, (2015). [14] Y. Fukaya, S. Tamura, K. Yada K, Y. Tanaka, P. Gentile, an d M. Cuoco, Interorbital topological superconductivity in spin-orbi t coupled supercon- ductors with inversion symmetry breaking . Phys. Rev. B , 97, 174522, (2018). [15] V. P. Mineev and M. Sigrist, Basic theory of superconductivity in metals without inversion center . Lect. Notes Phys., 847, 129–154, (2012).✐✐ “paper” — 2020/12/8 — 1:58 — page 17 — #17 ✐✐ ✐ ✐✐ ✐Eﬀects of hybridization and spin-orbit coupling 17 [16] M. Smidman, M. B Salamon, H. Q. Yuan and D. F. Agterberg, Spin- triplet p-wave pairing in a three-orbital model for iron pni ctide super- conductors . Rep. Prog. Phys, 80, 036501, (2017). [17] P. Lee and X. G. Wen, S, Superconductivity and spin-orbit coupling in non-centrosymmetric materials: A review . EPhys. Rev. B, 78, 2, (2008). [18] W. L. Yang et al, Evidence for weak electronic correlations in iron pnic- tidesPhys. Rev. B, 80, 014508, (2009). [19] P. Fazekas, Lecture Notes on Electron Correlation and Magnetism . World Scientiﬁc, Singapore, (1999). [20] H. Kuriyama et al, Epitaxially stabilized iridium espinel oxide without cations in the tetrahedral site . Appl. Phys. Lett., 96, 27010, (2010). [21] M. P. Christoph and H. Y. Kee, Identifying spin-triplet pairing in spin- orbit coupled multi-band superconductors . EPL, 98, 2, (2011). [22] M. M. Korshunov and Y. N. Togushova, Spin-orbit coupling and im- purity scattering on the spin resonance peak in three orbita l model for Fe-based superconductors . Journal of Siberian Federal University Math- ematics and Physics, 11, 998, (2018). [23] L. P. Gor’kov, Theory of superconducting alloys in a strong magnetic ﬁeld near the critical tepmerature . JETP, 10, 998, (1960). Department of Physics, University of Isfahan, Isfahan 81746, Iran E-mail address :h.yavary@sci.ui.ac.ir✐✐ “paper” — 2020/12/8 — 1:58 — page 18 — #18 ✐✐ ✐ ✐✐ ✐
1205.2162v3.3D_quaternionic_condensations__Hopf_invariants__and_skyrmion_lattices_with_synthetic_spin_orbit_coupling.pdf	arXiv:1205.2162v3 [cond-mat.quant-gas] 10 Feb 20163D quaternionic condensations, Hopf invariants, and skyrm ion lattices with synthetic spin-orbit coupling Yi Li,1,2Xiangfa Zhou,3and Congjun Wu1 1Department of Physics, University of California, San Diego , La Jolla, California 92093, USA 2Princeton Center for Theoretical Science, Princeton Unive rsity, Princeton, NJ 08544 3Key Laboratory of Quantum Information, University of Scien ce and Technology of China, CAS, Hefei, Anhui 230026, China We study the topological conﬁgurations of the two-componen t condensates of bosons with the 3D/vector σ·/vector pWeyl-type spin-orbit coupling subject to a harmonic trappi ng potential. The topology of the condensate wavefunctions manifests in the quaternio nic representation. In comparison to theU(1) complex phase, the quaternionic phase manifold is S3and the spin orientations form theS2Bloch sphere through the 1st Hopf mapping. The spatial distr ibutions of the quaternionic phases exhibit the 3D skyrmion conﬁgurations, and the spin d istributions possess non-trivial Hopf invariants. Spin textures evolve from the concentric distr ibutions at the weak spin-orbit coupling regime to the rotation symmetry breaking patterns at the int ermediate spin-orbit coupling regime. In the strong spin-orbit coupling regime, the single-parti cle spectra exhibit the Landau-level type quantization. In this regime, the three-dimensional skyrm ion lattice structures are formed when interactions are below the energy scale of Landau level mixi ngs. Suﬃciently strong interactions can change condensates into spin-polarized plane-wave sta tes, or, superpositions of two plane-waves exhibiting helical spin spirals. PACS numbers: 03.75.Mn, 03.75.Lm, 03.75.Nt, 67.85.Fg I. INTRODUCTION Quantum mechanical wavefunctions generally speak- ing are complex-valued. However, for the single com- ponent boson systems, their ground state many-body wavefunctions are highly constrained, which are usu- ally positive-deﬁnite [1], as a consequence of the Perron- Frobeniustheoreminthemathematicalcontextofmatrix analysis [2]. This is a generalization of the “no-node” theorem of the single-particle quantum mechanics, for example, both the ground state wavefunctions of har- monic oscillators and hydrogen atoms are nodeless. Al- though the positive-deﬁniteness does not apply to the many-body fermion wavefunctions because Fermi statis- tics necessarilyleads to nodal structures, it remains valid for many-body boson systems. It applies under the fol- lowing conditions: the Laplacian type kinetic energy, the arbitrary single-particle potential, and the coordinate- dependent interactions. The positive-deﬁniteness of the ground state wavefunctions implies that time-reversal (TR) symmetry cannot be spontaneously broken in con- ventional Bose-Einstein condensates (BEC), such as the superﬂuid4He and most ground state BECs of ultra-cold alkali bosons [3]. It would be interesting to seek unconventional BECs beyond the constraint of positive-deﬁnite condensate wavefunctions [4]. The spin-orbit coupled boson systems areanideal platformto studythis classofexoticstatesof bosons, which can spontaneously breaking the TR sym- metry. In addition to a simple Laplacian, the kinetic energy contains the spin-orbit coupling term linearly de- pendent on momentum. If the bare interaction is spin- independent, the condensate wavefunctions are heavily degenerate. An “order-from-disorder” calculation based on the zero-point energy of the Bogoliubov spectra wasperform to select the condensate conﬁguration[4]. Inside the harmonic trap, it is predicted that the condensates spontaneously develop the half-quantum vortex coexist- ing with 2D skyrmion-type spin textures [5]. Experi- mentally, spin-orbit coupled bosons have been realized in exciton systems in semi-conducting quantum wells. Spin texture conﬁgurations similar to those predicted in Ref. [5] have been observed [6]. On the other hand, the progress of synthetic artiﬁcial gauge ﬁelds in ultra- cold atomic gases greatly stimulates the investigation of the above exotic states of bosons [7, 8]. Extensive stud- ies have been performed for bosons with the 2D Rashba spin-orbit coupling, which exhibit various spin structures arising from the competitions among the spin-orbit cou- pling, interaction,andtheconﬁningtrapenergy[5,9–16]. Most studies so far have been on the two-dimensional spin-orbit coupled bosons. It would be interesting to fur- ther consider the unconventional condensates of bosons with the three-dimensional Weyl-type spin-orbit cou- pling, whose experimental realization has been proposed by the authors through atom-light interactions in a com- bined tripod and tetrapod level system [20] and also by Anderson et al.[21]. As will be shown below, the qua- terinon representationprovides a natural and most beau- tiful description of the topological condensation conﬁgu- rations. Quaternions are an extension of complex num- bers as the ﬁrst discovered non-commutative division al- gebra, which has provided a new formulation of quantum mechanics [17–19]. Similarly to complex numbers whose phasesspanaunitcircle S1, thequaternionicphasesspan a three dimensional unit sphere S3. The spin distribu- tions associated with quaternionic wavefunctions are ob- tained through the 1st Hopf map S3→S2as will be explained below. It would be interesting to search for BECswith non-trivialtopologicaldefects associatedwith2 the quaternionic phase structure. It will be a new class of unconventional BECs beyond the “no-node” theorem breaking TR symmetry spontaneously. In this article, we consider the unconventional conden- satewavefunctionswiththe3DWeyl-typespin-orbitcou- pling/vector σ·/vector p. The condensation wavefunctions exhibit topo- logically non-trivial conﬁgurations as 3D skyrmions, and spin density distributions are also non-trivial with non- zero Hopf invariants. These topological conﬁgurations can be best represented as defects of quaternion phase distributions. Spatial distributions of the quaternionic phase textures and spin textures are concentric at weak spin-orbit couplings. As increasing spin-orbit coupling, these textures evolve to lattice structures which are the 3D quaternionic analogy of the 2D Abrikosov lattice of the usual complex condensate. The rest part of this article is organized as follows. In Sect. II, we deﬁne the model Hamiltonian. In Sect. III, the condensate wavefunctions in the weak spin-orbit regimearestudied. Topologicalanalysesonthe skyrmion conﬁgurations and Hopf invariants are performed by us- ing the quaternion representation. In Sect. IV, the skyrmion lattice conﬁguration of the spin textures is studied in the intermediate and strong spin-orbit cou- pling regimes. In Sect. V, superpositions of plane-wave condensate conﬁgurations are studied. Conclusions are made in Sect. VI. II. THE MODEL HAMILTONIAN We consider a two-component boson system with the 3D spin-orbit coupling of the /vector σ·/vector p-type conﬁned in a har- monic trap. The free part of the Hamiltonian is deﬁned as H0=/integraldisplay d3/vector r ψ† γ(/vector r)/braceleftBig −/planckover2pi12/vector∇2 2m+i/planckover2pi1λ/vector σγδ·(/vector∇) +1 2mω2/vector r2/bracerightBig ψδ(/vector r), (1) whereγandδequal↑and↓referring to two internal states of bosons; /vector σare Pauli matrices; mis the bo- son mass;λis the spin-orbit coupling strength with the unit of velocity; ωis the trap frequency. At the single- particle level, Eq. (1) satisﬁes the Kramer-type time- reversal symmetry of T= (−iσ2)Cwith the property of T2=−1. However, parity is broken by spin-orbit cou- pling. In the absence ofthe trap, good quantum numbers for the single-particle states are the eigenvalues ±1 of he- licity/vector σ·/vector p/\|p\|, wherepis the momentum. This results in two branches of dispersions ǫ±(/vectork) =/planckover2pi12 2m(k∓kso)2, (2) where/planckover2pi1kso=mλ. The lowest single-particle energy states lie in the sphere with the radius ksodenoted as the spin-orbit sphere. It corresponds to a spin-orbit lengthscalelso= 1/ksoin real space. The harmonic trap has a natural length scale lT=/radicalBig /planckover2pi1 mω, and thus the dimension- less parameter α=lTksodescribes the relative spin-orbit coupling strength. As for the interaction Hamiltonian, we use the contact s-wave scattering interaction deﬁned as Hint=gγδ 2/integraldisplay d3/vector r ψ† γ(/vector r)ψ† δ(/vector r)ψδ(/vector r)ψγ(/vector r).(3) Two diﬀerent interaction parameters are allowed, in- cluding the intra and inter-component ones deﬁned as g↑↑=g↓↓=g, andg↑↓=cg, wherecis a constant. In the previous study of the 2D Rashba spin-orbit cou- pling with harmonic potentials [5, 15], the single-particle eigenstates are intuitively expressed in the momentum representation: the low energy state lies around a ring in momentum space, and the harmonic potential becomes the planar rotor operator on this ring subject to a π- ﬂux, which quantizes the angular momentum jzto half integers. Similar picture also applies in 3D [5, 22]. The low energy states are around the spin-orbit sphere. In the projected low energy Hilbert space, the eigenvectors read ψ+(/vectork) = (cosθk 2,sinθk 2eiφk)T. (4) The harmonic potential is again a rotor Hamiltonian on the spin-orbit sphere subject to the Berry gauge connec- tion as Vtp=1 2m(i∇k−/vectorAk)2(5) with the moment of inertial I=Mkk2 soandMk= /planckover2pi12/(mω2)./vectorAk=i/angb∇acketleftψ+(/vectork)\|∇k\|ψ+(/vectork)/angb∇acket∇ightis the vector po- tential of a U(1) magnetic monopole, which quantizes the angular momentum jto half-integers. While the ra- dial energy is still quantized in terms of /planckover2pi1ω, the angular energydispersion with respect to jis stronglysuppressed at large values of αas Enr,j,jz≈/parenleftBig nr+j(j+1) 2α2/parenrightBig /planckover2pi1ω+const,(6) wherenris the radial quantum number. As further shown in Ref. [20], in the case α≫1, all the states with the same nrbut diﬀerent jandjzare nearly degen- erate, thus can be viewed as one 3D Landau level with spherical symmetry but the broken parity. If ﬁlled with fermions, the system belongs to the Z2-class of 3D strong topological insulators. Now we load the system with bosons. The interaction energy scale is deﬁned as Eint=gN0/l3 T, whereN0is the total particle number in the condensate. The corre- sponding dimensionless parameter is β=Eint//planckover2pi1ω. At the Hartree-Fock level, the Gross-Pitaevskii energy func- tional is deﬁned in terms of the condensate wavefunction3 Ψ = (Ψ ↑,Ψ↓)Tas E=/integraldisplay d3/vector r(Ψ† ↑,Ψ† ↓)/braceleftBig −/planckover2pi12∇2 2m−iλ/planckover2pi1/vector∇·/vector σ+1 2mω2r2 +g/parenleftbigg n↑+cn↓0 0cn↑+n↓/parenrightbigg/bracerightBig/parenleftbigg Ψ↑ Ψ↓/parenrightbigg , (7) wheren↑,↓(/vector r) =N0\|Ψ↑,↓(/vector r)\|2are the particle densities of two components, respectively, and Ψ( /vector r) is normalized as/integraltext d3/vector rΨ†(/vector r)Ψ(/vector r) = 1. The condensate wavefunction Ψ( /vector r) is solved numerically by using the standard method of imaginary time evolution. The dimensionless form of the Gross-Pitaevskii equation is E′=/integraldisplay d3/vector r′(˜Ψ† ↑,˜Ψ† ↓)/braceleftBig −/vector∇′2 2−iα/vector∇′·/vector σ+r′2 2 +β/parenleftbigg ˜n↑+c˜n↓0 0c˜n↑+ ˜n↓/parenrightbigg/bracerightBig/parenleftbigg˜Ψ↑ ˜Ψ↓/parenrightbigg ,(8) whereE′=E/(/planckover2pi1ω),/vector∇′=lT/vector∇;/vector r′=/vector r/lT;˜Ψ↑and˜Ψ↓ arethe renormalizedcondensate wavefunctionssatisfying/integraltext d3r′\|˜Ψ↑\|2+\|˜Ψ↓\|2= 1; ˜n↑=\|˜Ψ↑\|2and ˜n↓=\|˜Ψ↓\|2. III. THE WEAK SPIN-ORBIT COUPLING REGIME In this section, we consider the condensate conﬁgura- tion in the limit of weak spin-orbit coupling, say, α∼1. In this regime, the single-particle spectra still resemble those of the harmonictrap. We study the case that inter- actions arenot strongenough to mix stateswith diﬀerent angular momenta. A. The spin-orbit coupled condensate In this regime, the condensate wavefunction Ψ remains the same symmetry structure as the single-particle wave- function over a wide range of interaction parameter β, i.e., Ψ remains the eigenstates of j=1 2as conﬁrmed numerically below. Ψ can be represented as Ψj=jz=1 2(r,ˆΩ) =f(r)Y+ j,jz(ˆΩ)+ig(r)Y− j,jz(ˆΩ),(9) wheref(r) andg(r) are real radial functions. Y± j,jz(ˆΩ) are the spin-orbit coupled spherical harmonic functions with even and odd parities, respectively. For example, for the case of j=jz=1 2, they are Y+ 1 2,1 2(r,ˆΩ) =/parenleftbigg 1 0/parenrightbigg , Y− 1 2,1 2(r,ˆΩ) =/parenleftbiggcosθ sinθeiφ/parenrightbigg ,(10) whose orbital partial-wavecomponents are sandp-wave, respectively. The TR partner of Eq. (9) is ψjz=−1 2= ˆTψj=jz=1 2=iσ2ψ∗ j=jz=1 2. The two terms in Eq. (9) are of opposite parity eigenvalues, mixed by the paritybreaking spin-orbit coupling /vector σ·/vector p. The coeﬃcient iof the Y− jjzterm is because the matrix element /angb∇acketleftY+ jjz\|/vector σ·/vector p\|Y− jjz/angb∇acket∇ight is purely imaginary. For the non-interacting case, the radial wavefunctions uptoaGaussianfactorcanbeapproximatedbyspherical Bessel functions as f(r)≈j0(ksor)e−r2/2l2 T, g(r)≈j1(ksor)e−r2/2l2 T,(11) which correspond to the sandp-partial waves, respec- tively. Both of them oscillate along the radial direction and the pitch values are around kso. Atr= 0,f(r) reaches the maximum and g(r) is 0. As rincreases, roughly speaking, the zero points of f(r) corresponds to the extrema of g(r) and vise versa. Repulsive interac- tions expand the spatial distributions of f(r) andg(r), but the above picture still holds qualitatively. In other words, there is aπ 2-phase shift between the oscillations off(r) andg(r). B. The quaternion representation Can we have unconventional BECs with non-trivial quaternionic condensate wavefunctions? Actually, the topological structure of condensate wavefunction Eq. (9) manifests clearly in the quaternion representation as shown below. We deﬁne the following mapping from the complex two-component vector Ψ = (Ψ ↑,Ψ↓)Tto a quaternion variable through ξ=ξ0+ξ1i+ξ2j+ξ3k, (12) where ξ0= ReΨ ↑,ξ1= ImΨ ↓,ξ2=−ReΨ↓,ξ3= ImΨ ↑.(13) i,j,kare the imaginary units satisfying i2=j2=k2= −1, and the anti-commutation relation ij=−ji=k. The TR transformation on ξis just−jξ. Eq. (9) can be expressed in the quaternionic exponen- tial form as ξj=jz=1 2(r,ˆΩ) =\|ξ(r)\|e/vector ω(ˆΩ)γ(r)=\|ξ\|(cosγ+/vector ωsinγ),(14) where \|ξ(r)\|= [f2(r)+g2(r)]1 2, /vector ω(ˆΩ) = sinθcosφ i+sinθsinφ j+cosθ k, cosγ(r) =f(r)/\|ξ(r)\|,sinγ(r) =g(r)/\|ξ(r)\|.(15) ω(ˆΩ) is the imaginary unit along the direction of ˆΩ sat- isfying/vector ω2(ˆΩ) =−1. According to the oscillating proper- ties off(r) andg(r),γ(r) spirals as rincreases. At the n-th zero point of g(r) denotedrn,γ(rn) =nπwhere n≥0 and we deﬁne r0= 0, while at the n-th zero point off(r) denotedr′ n,γ(r′ n) = (n−1 2)πwheren≥1. In 3D, the condensate wavefunctions can be topolog- ically non-trivial because the homotopy group of the4 quaternionic phase is π3(S3) =Z[23, 24]. The corre- sponding winding number, i.e. the Pontryagin index, of the mapping S3→S3is the 3D skyrmion number. The spatial distribution of the quaternionic phase e/vector ω(ˆΩ)γ(r) deﬁned in Eq. 14, which lies on S3, exhibits a topo- logically nontrivial mapping from R3toS3, i.e., a 3D multiple skyrmion conﬁguration. This type of topolog- ical defects are non-singular which is diﬀerent from the usual vortex in single component BEC. In realistic trap- ping systems, the coordinate space is the open R3. At large distance r≫lT,\|ξ(r)\|decays exponentially, where the quaternionic phase and the mapping are not well-deﬁned. Nevertheless, in each concentric spherical shell withrn<r<r n+1,γ(r) winds from nπto (n+1)π, and ω(ˆΩ) covers all the directions, thus this shell contributes 1 to the winding number of e/vector ω(ˆΩ)γ(r)onS3. If the system size is truncated at the order of lT, the skyrmion number can be approximated at the order of lTkso=α. There exists an interesting diﬀerence from the previ- ously studied 2D case: Although the spin density dis- tribution exhibit the 2D skyrmion conﬁguration due to π2(S2) [5, 15, 16], the 2D condensation wavefunctions have no well-deﬁned topology due to π2(S3) = 0. FIG. 1: The distribution of /vectorS(/vector r) in a) the xz-plane and in the horizontal planes with b)z= 0 andc)z/lT=1 2. The unit length is set as lT= 1 in all the ﬁgures in this article. The color scale shows the magnitude of out-plane component Syina) andSzinb) andc). The parameter values are α= 1.5,c= 1, andβ= 30, and the length unit in these and all the ﬁgures below is lT. C. The Hopf mapping and Hopf invariant Exotic spin textures in spinor condensates have been extensively investigated [25–27]. In our case, the 3D spin density distributions /vectorS(/vector r) exhibit a novel conﬁguration with non-trivial Hopf invariants due to the non-trivial homotopy group π3(S2) =Z[23, 24]./vectorS(/vector r) can be ob- tained from ξ(r) through the 1st Hopf map deﬁned as /vectorS(/vector r) =1 2ψ† γ/vector σγβψδ, or, in the quaternionicrepresentation, 1 2¯ξkξ=Sxi+Syj+Szk, (16) where¯ξ=ξ0−ξ1i−ξ2j−ξ3kisthequaternionicconjugate ofξ. The Hopf invariant of the 1st Hopf map is just 1 [24]. The real space concentric spherical shell rn< r < rn+1maps to the quaternionic phase S3, and the latter furthermapstothe S2Blochspherethroughthe1stHopf map. The winding number of the ﬁrst map is 1, and the Hopf invariant of the second map is also 1, thus the Hopf invariantoftheshell rn<r<r n+1toS2is1. Rigorously speaking, the magnitude of /vectorS(/vector r) decays exponentially at r≫lT, and thus the total Hopf invariant is not well- deﬁned in the open R3space. Again, if we truncate thesystem size at lT, the Hopf invariant is approximately at the order of α. FIG. 2: The Hopf ﬁbration of the spin texture conﬁgura- tion in Fig. 1. Every circle represents a spin orientation, and every two circles are linked with the linking number 1. Next we present numeric results for the spin textures associated with the condensation wavefunction Eq. 9 as5 plotted in Fig. 1. Explicitly, /vectorS(/vector r) is expressed as /bracketleftbigg Sx(/vector r) Sy(/vector r)/bracketrightbigg =g(r)sinθ/bracketleftbigg cosφ−sinφ sinφcosφ/bracketrightbigg/bracketleftbigg g(r)cosθ f(r)/bracketrightbigg , Sz(/vector r) =f2(r)+g2(r)cos2θ, (17) In thexz-plane, the in-plane components SxandSzform a vortex in the half plane of x >0 andSyis prominent in the core. The contribution at large distance is ne- glected, where /vectorS(/vector r) decays exponentially. Due to the axial symmetry of /vectorS(/vector r) in Eq. 17, the 3D distribution is just a rotation of that in Fig. 1 a) around the z-axis. In thexy-plane, spin distribution exhibits a 2D skyrmion pattern, whose in-plane components are along the tan- gential direction. As the horizontal cross-section shifted along thez-axis,/vectorS(/vector r) remains 2D skyrmion-like, but its in-plane components are twisted around the z-axis. The spin conﬁguration at z=−z0can be obtained by a com-bined operation of TR and rotation around the y-axis 180◦, thus its in-plane components are twisted in an op- posite way compared to those at z=z0. Combining the conﬁgurations on the vertical and horizontal cross sections, we complete the 3D distribution of /vectorS(/vector r) with non-zero Hopf invariant. The non-trivial structure of the Hopf invariant of the above spin conﬁguration can be revealed by plotting its Hopf ﬁbration in terms of the linked non-crossing circles in real space, as shown in Fig. 2. For all the points on each circle, their normalized spin polarizations /angb∇acketleft/vector σ/angb∇acket∇ight/\|/angb∇acketleft/vector σ/angb∇acket∇ight\| are the same, corresponding to a single point on the S2 sphere. Inaddition, everytwocirclesarelinkedwith each other with the linking number 1, which is the standard Hopf bundle structure describing a many-to-one map fromS3toS2. Ultracoldbosonswithsyntheticspin-orbit coupling providea novel platform to study such beautiful mathematical ideas in realistic physics systems. FIG. 3: The distribution of /vectorS(/vector r) in horizontal cross-sections with a) z/lT=−0.5, b)z/lT= 0, c)z/lT= 0.5, respectively. The color scale shows the value of Sz, and parameter values are α= 4,β= 2, andc= 1. IV. THE INTERMEDIATE AND STRONG SPIN-ORBIT COUPLING REGIME A. The intermediate spin-orbit coupling strength Next we consider the case of the intermediate spin- orbit coupling strength, i.e., 1 < α <10, at which the single-particle spectra evolve from the case of the har- monic potential to Landau level-like as shown in Eq. 6. Interactions are suﬃciently strong to mix a few lowest energy states with diﬀerent angular momenta j. As a re- sult, rotationalsymmetryisbrokenandcomplexpatterns appear. In this case, the topology of condensate wavefunctions isstill 3Dskyrmion-likemapping from R3toS3, andspin textures with the non-trivial Hopf invariant are obtained through the 1st Hopf map. Compared to the weak spin- orbit coupling case, the quaternionic phase skyrmions and spin textures are no longer concentric, but split to a multi-centered pattern. The numeric results of /vectorS(/vector r) areplotted in Fig. 3 for diﬀerent horizontal cross-sections. In thexy-plane,/vectorSexhibits the 2D skyrmion pattern as shown in Fig. 3 ( b): The in-plane components form two vortices and one anti-vortex, while Sz’s inside the vortex and anti-vortex cores are opposite in direction, thus they contribute to the skyrmion number with the same sign. The spin conﬁguration at z=z0>0 is shown in Fig. 3 (a), which is twisted around the z-axis clock-wise. After performing the combined TR and rotation around the y- axis 180◦, we arrive at the conﬁguration at z=−z0in Fig. 3(b). B. The strong spin-orbit coupling regime We next consider the case of strong spin-orbit cou- pling, i.e., α≫1. The single-particle spectra already exhibit the Landau-level type quantization in this regime as shown in Eq. 6. The single-particle eigenstates with nr= 0 are nearly degenerate i.e., they form the low-6 est Landau level states. We assume that the interaction strength is enough to mix states inside the lowest Lan- dau level but is still relatively weak not to induce inter- Landau level mixing. In this regime, the length scale of each skyrmion is shortened as enlarging the spin-orbit coupling strength. As we can imagine, more and more skyrmions appear and will form a 3D lattice structure, which is the SU(2) generalization of the 2D Abrikosov lattice of the usual U(1) superﬂuid. We have numerically solved the Gross- PitaevskiiequationEq. 7andfound thelattice structure: Each lattice site is a single skyrmion of the condensate wavefunction ξ(/vector r), whose spin conﬁguration exhibits the texture conﬁguration approximately with a unit Hopf in- variant. The numeric results for the spin texture conﬁg-uration are depicted in Fig. 4 a) andb) for two diﬀerent horizontal cross sections parallel to the xy-plane. In each cross section, spin textures form a square lattice, and the lattice constant dis estimated approximately the spin- orbit length scale as d≃2πlso= 2πlT/α. (18) For two horizontal cross sections with a distance of ∆z≃d/2, their square lattice conﬁgurations are dis- placedalongthediagonaldirection: Thesitesatonelayer sit abovethe plaquette centersofthe adjacentlayer. As a result, the overall three-dimensional conﬁguration of the topological defects is a body-centered cubic ( bcc) lattice, and its size is ﬁnite conﬁned by the trap. FIG. 4: The distribution of /vectorS(/vector r) in horizontal cross-sections with (a) z/lT= 0, (b)z/lT= 0.2, respectively. The color scale shows the value of Szand parameter values are α= 22,β= 1, andc= 1. The overall lattice exhibits the bcc structure. V. THE EFFECT OF STRONG INTERACTIONS In this section, we present the condensate conﬁgura- tions in the case that both spin-orbit coupling and inter- actions are strong, such that diﬀerent Landau levels are mixed by interactions. In this case, the eﬀect of the harmonic trapping poten- tial becomes weak compared with interaction energies, thus we can approximate the condensate wavefunctions as superpositions of plane-wave states. The plane-wave components are located on the spin-orbit sphere and the condensate wavefunctions are no longer topological. At c= 1, the interaction is spin-independent, and bosons select a superposition of a pair of states ±/vectorkon the spin- orbit sphere, say, ±ksoˆz. The condensate wavefunction is written as ψ(/vector r) =/radicalbigg Na N0eiksoz\| ↑/angb∇acket∇ight+/radicalbigg Nb N0e−iksoz\| ↓/angb∇acket∇ight,(19)withNa+Nb=N0. The density of Eq. 19 in real space is uniform to minimize the interaction energy at the Hartree-Fock level. However, all the diﬀerent parti- tions ofNa,byield the same Hartree-Fock energy. The quantum zero point energy from the Bogoliubov modes removes this accidental degeneracy through the “order- from-disorder” mechanism, which selects the equal par- titionNa=Nb. The calculation is in parallel to that of the 2D Rashba case performed in Ref. [5], thus will not be presented here. In this case, the condensate is a spin helix propagates along z-axis and spin spirals in the xy-plane. Atc/negationslash= 1, the spin-dependent part of the interaction can be written as Hsp=1−c 2g/integraldisplay d3r(ψ† ↑ψ↑−ψ† ↓ψ↓)2.(20) Atc >1, the interaction energy at Hartree-Fock level is minimized for the condensate wavefunction of a plane wave state eiksoz\| ↑/angb∇acket∇ight, or, its TR partner.7 Forc<1,/angb∇acketleftHsp/angb∇acket∇ightis minimized if /angb∇acketleftSz/angb∇acket∇ight= 0 in space. At the Hartree-Fock level, the condensate can either be a plane-wave state with momentum lying in the equator of the spin-orbit sphere and spin polarizing in the xy-plane, or, the spin spiral state described by Eq. 19 with Na= Nb. An“order-from-disorder”analysisontheBogoliubov zero-point energies indicates that the spin spiral stateis selected. We also present the numerical results for Eq. (4) in the main text with a harmonic trap in Fig. 5 for the case of c <1. The condensate momenta of two spin components have opposite signs, thus the trap inhomogeneity already prefers the spin spiral state Eq. 19 at the Hartree-Fock level. FIG. 5: The density proﬁle (a) for ↑-component, and that for ↓-component is the same. Phase proﬁles for (b) ↑and (c)↓- components, respectively. Parameter values are a= 10,β= 50, andc= 0.5. VI. CONCLUSION In summary, we have investigated the two-component unconventional BECs driven by the 3D spin-orbit coupling. In the quaternionic representation, the quaternionic phase distributions exhibit non-trivial 3D skyrmionconﬁgurationsfrom R3toS3. Thespinorienta- tion distributions exhibit texture conﬁgurations charac- terizedby non-zeroHopfinvariantsfrom R3toS2. These two topological structures are connected through the 1st Hopf map from S3toS2. At large spin-orbit coupling strength, the crystalline order of spin textures, or, wave- function skyrmions, are formed, which can be viewed as a generalization of the Abrikosov lattice in 3D. Note added.— Near the completion of this manuscript, we became aware of a related work by Kawakami et al.[28], in which the condensate wavefunction in the weak spin-orbit coupling case was studied. Acknowledgments.— Y.L. thanks the Princeton Cen- ter for Theoretical Science at Princeton University for support. X. F. Z. acknowledges the support of NFRP (2011CB921204, 2011CBA00200), the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB01030000),NSFC (11004186,11474266), and the Major Research plan of the National Natural Science Foundation of China (91536219). C. W. is sup- ported by the NSF DMR-1410375 and AFOSR FA9550- 14-1-0168. C. W. acknowledges the support from the Presidents Research Catalyst Awards of University of California, and National Natural Science Foundation of China (11328403). [1] R. P. Feynman, Statistical Mechanics, A Set of Lectures (Berlin: Addison-Wesley, 1972). [2] R. B. Bapat and T. Raghavan, Non-Negative Matri- ces and Applications (Cambridge University Press, Cam- bridge, United Kingdom, 1997). [3] A. J. Leggett, Rev. Mod. Phys. 73, 307 (2001). [4] C. Wu, Mod. Phys. Lett. 23, 1 (2009). [5] C. Wu , I. Mondragon-Shem, arXiv:0809.3532; C. Wu , I. Mondragon-Shem, and X. F. Zhou, Chin. Phys. Lett., 28, 097102 (2011). [6] A.A. High et al., Nature 483, 584 (2012). A.A. High et al., arXiv:1103.0321. [7] Y.-J. Lin et al., Nature 462, 628 (2009).[8] Y.-J. Lin et al., Nature 471, 83 (2011). [9] T. Stanescu et al., Phys. Rev. A 78, 023616 (2008). [10] T.-L. Ho et al., Phys. Rev. Lett. 107, 150403 (2011). [11] C. Wang et al., Phys. Rev. Lett. 105, 160403 (2010). [12] S.-K. Yip, Phys. Rev. A 83, 043616 (2011). [13] Y. Zhang et al., Phys. Rev. Lett. 108, 035302 (2012). [14] X.-F. Zhou et al., Phys. Rev. A 84, 063624 (2011). [15] H. Hu et al., Phys. Rev. Lett. 108, 010402(2012). [16] S. Sinha, R. Nath, and L. Santos, arXiv:1109.2045. [17] A. V. Balatsky, cond-mat/9205006. [18] S. L. Adler, Quaternionic Quantum Mechanics and Quantum Fields (OxfordUniversityPress, Oxford, 1995). [19] D.Finkelstein et al., J.Math.Phys.(N.Y.) 3, 207(1962).8 [20] Y. Li et al., Phys. Rev. B 85, 125122 (2012). [21] B. M. Anderson et al., arXiv:1112.6022. [22] S. K. Ghosh et al., Phys. Rev. A, 84, 053629 (2011). [23] F. Wilczek and A. Zee, Phys. Rev. Lett. 51, 2250 (1983). [24] M. Nakahara, Geometry, topology, and physics , (Taylor & Francis, 2003) [25] F. Zhou, Int. J. Mod. Phys. B 17, 2643-2698 (2003); E.Demler and F. Zhou, Phys. Rev. Lett. 88, 163001 (2002). G. W.SemenoﬀandF.Zhou, Phys.Rev.Lett. 98, 100401 (2007). [26] J. Zhang and T. L. Ho, arXiv:0908.1593. [27] D. M. Stamper-Kurn, and M. Ueda, arXiv:1205.1888. [28] T. Kawakami et al., arXiv:1204.3177.
1004.3066v1.Spin_Orbit_Coupling_and_Spin_Waves_in_Ultrathin_Ferromagnets__The_Spin_Wave_Rashba_Effect.pdf	arXiv:1004.3066v1 [cond-mat.mes-hall] 18 Apr 2010Spin Orbit Coupling and Spin Waves in Ultrathin Ferromagnet s: The Spin Wave Rashba Eﬀect A. T. Costa,1R. B. Muniz,1S. Lounis,2A. B. Klautau,3and D. L. Mills2 1Instituto de F´ ısica, Universidade Federal Fluminense, 24 210-340 Niter´ oi, RJ, Brasil. 2Department of Physics and Astronomy, University of California Irvine, California, 92697, U. S. A . 3Departamento de Fisica, Universidade Federal do Par´ a, Bel ´ em, PA, Brazil. Abstract We present theoretical studies of the inﬂuence of spin orbit coupling on the spin wave excitations of the Fe monolayer and bilayer on the W(110) surface. The Dzy aloshinskii-Moriya interaction is active in such ﬁlms, by virtue of the absence of reﬂection sym metry in the plane of the ﬁlm. When the magnetization is in plane, this leads to a linear term in t he spin wave dispersion relation for propagation across the magnetization. The dispersion rela tion thus assumes a form similar to that of an energy band of an electron trapped on a semiconduct or surfaces with Rashba coupling active. We also show SPEELS response functions that illustr ate the role of spin orbit coupling in such measurements. In addition to the modiﬁcations of the di spersion relations for spin waves, the presence of spin orbit coupling in the W substrate leads to a s ubstantial increase in the linewidth of the spin wave modes. The formalism we have developed appli es to a wide range of systems, and the particular system explored in the numerical calcula tions provides us with an illustration of phenomena which will be present in other ultrathin ferrom agnet/substrate combinations. 1I. INTRODUCTION Thestudyofspindynamicsinultrathinferromagnetsisoffundamen talinterest, sincenew physics arises in these materials that has no counterpart in bulk mag netism. Examples are provided by relaxation mechanisms evident in ferromagnetic resona nce and Brillouin light scattering studies,1–3and also for the large wave vectors probed by spin polarized electro n loss spectroscopy (SPEELS).4Of course, by now the remarkable impact of ultrathin ﬁlm structures on magnetic data storage is very well known, and othe r applications that exploit spin dynamics in such materials are envisioned. Thus these issues are important from a practical point of view as well as from that of fundamental physics . Theoretical studies of the nature of spin waves in ultrathin ﬁlms ads orbed on metal sub- strates have been carried out for some years now, along with comp arison with descriptions provided with the Heisenberg model.5In this paper, we extend the earlier theoretical treat- ments to include the inﬂuence of spin orbit coupling on the spin wave sp ectrum of ultrathin ﬁlms. This extension is motivated by a most interesting discussion of t he ground state of the Mn monolayer on the W(110) surface. A nonrelativistic theoretical study of this system pre- dicted that the ground state would be antiferromagnetic in charac ter.7This prediction was conﬁrmed by spin polarized scanning tunneling microscope studies of the system.8However, recent experimental STM data with a more sensitive instrument sho wed a more complex ground state, wherein the ground state is in fact a spin density wav e.9One can construct the new state by beginning with the antiferromagnet, and then sup erimposing on this a long wavelength modulation on the direction of the moments on the lat tice. The authors of ref. 9 argued that the lack of reﬂection symmetry of the syste m in the plane of the ﬁlm activates theDzyaloshinskii Moriya (DM) interaction, andthe news tate hasitsorigininthis interaction. They also presented relativistic and ab initio calculations that gave an excellent account of the new data. The reﬂection symmetry is broken simply b y the presence of the substrate upon which the ﬁlm is grown. This argument to us is most int riguing, since one can then conclude that the DM interaction must be active in any ultra thin ferromagnet; the substrate is surely always present. The DM interaction has its o rigin in the spin orbit interaction, which of course is generally very weak in magnets that in corporate the 3d tran- sition elements as the moment bearing entities. However, in the case of the Mn monolayer on W(110) hybridization between the Mn 3d and the W 5d orbitals activ ates the very large 2W spin orbit coupling, with the consequence that the strength of th e DM interaction can be substantial, as illustrated by the calculations presented in ref. 9 . One may expect to see substantial impact of the DMinteraction in other ultrathin magnets grown on 5d substrates, and possibly 4d substrates as well. We have here another example of new physics present in ultrathin ma gnets that is not encountered inthebulk formofthematerial fromwhich theultrath instructure isfabricated. The purpose of this paper is to present our theoretical studies of spin orbit eﬀects on spin waves and also on the dynamic susceptibility of a much studied ultrath in ﬁlm/substrate combination, theFemonolayer andbilayer onW(110). Weﬁndstriking eﬀects. Forinstance, when the magnetization is in plane, as we shall see the DM interaction in troduces a term linear in wave vector in the dispersion relation of spin waves. Thus the uniform spin wave mode at zero wave vector acquires a ﬁnite group velocity. We ﬁnd th is to be in the range of 2×105cm/sec for the Fe monolayer on W(110). Furthermore, left/right asymmetries appear in the SPEELS response functions. Thus, we shall see that spin orbit coupling has clear eﬀects on the spin excitations of transition metal ultrathin fe rromagnets grown on 5d substrates. We comment brieﬂy on the philosophy of the approach used here, an d in various earlier publications.5Numerous authors proceed as follows. One may generate a descrip tion of the magnetic ground state of the adsorbed ﬁlms by means of an electro nic structure calculation based on density functional theory. It is then possible to calculate , within the framework of anadiabaticapproximation, eﬀectiveHeisenbergexchangeintegra lsJijbetweenthemagnetic moments in unit cell i and unit cell j. These may be entered into a Heise nberg Hamilto- nian, and then spin wave dispersion relations may be calculated throu gh use of spin wave theory. It has been known for decades10that in the itinerant 3d magnets, eﬀective exchange interactions calculated in such a manner have very long range in real space. Thus, one must include a very large number of distant neighbors in order to obtain co nverged results. This is very demanding to do with high accuracy for the very numerous dis tant neighbors, since the exchange interactions become very small as one moves out into distant neighbor shells. At a more fundamental level, as noted brieﬂy above, discussions in e arlier publications show that in systems such as we study here, the adiabatic approxim ation breaks down badly, with qualitative consequences.5First, spin wave modes of ﬁnite wave vector have very short lifetimes, by virtue of decay into the continuum of particle hole pairs ( Stoner excitations) 3even at the absolute zero of temperature5,11whereas in Heisenberg model descriptions their lifetime is inﬁnite. In multi layer ﬁlms, the earlier calculations show that as a consequence of the short lifetime, the spectrum of spin ﬂuctuations at large wave v ectors contains a single broad feature which disperses with wave vector in a manner similar to that of a spin wave; this is consistent with SPEELS data on an eight layer ﬁlm of Co on Cu(10 0).4This picture stands in contrast to that oﬀered by the Heisenberg model, in which a ﬁlm of N layers has N spin wave modes for each wave vector, and each mode has inﬁnite lif etime. The method developed earlier, and extended here to incorporate s pin orbit coupling, takes due account of the breakdown of the adiabatic approximatio n and also circumvents the need to calculate eﬀective exchange interactions in real space out to distant neighbor shells. Weworkdirectlyinwave vectorspacethroughstudyofthew ave vectorandfrequency dependent susceptibility discussed below, denotedas χ+,−(/vectorQ/bardbl,Ω;l⊥,l′ ⊥). Theimaginarypart of this object, evaluated for l⊥=l′ ⊥and considered as a function of frequency Ω for ﬁxed wave vector /vectorQ/bardblprovides us with the frequency spectrum of spin ﬂuctuations on lay erl⊥for the wave vector chosen. Spin waves appear as peaks in this functio n, very much as they do in SPEELS data, and in a manner very similar to that used by experimen talist we extract a dispersion relation for spin waves by following the wave vector depe ndence of the peak frequency. We never need to resort to a real space summation pr ocedure over large number of neighbors, coupled by very tiny exchange couplings. The spin wav e exchange stiﬀness can be extracted either by ﬁtting the small wave vector limit of the d ispersion relation so determined, or alternatively by utilizing an expression derived earlier5which once againdoes not require a summation in real space. Wecommentonanotherfeatureofthepresentstudy. Inearlierc alculations,5,11,14asinthe present paper, an empirical tight binding description forms the bas is for our description of the electronic structure. Within this approach, referred to as a m ulti band Hubbard model, we can generate the wave vector and frequency dependent susc eptibility for large systems. In the earlier papers, eﬀective tight binding parameters were extr acted from bulk electronic structure calculations. The present studies are based on tight bin ding parameters obtained directly from a RS-LMTO-ASA calculation for the Fe/W(110) system . We also obtain tight binding parameters by ﬁtting KKR based electronic structure calcu lations for the ultrathin ﬁlm/substrate combinations of interest. We ﬁnd that spin waves in t he Fe/W(110) system are quite sensitive to the empirical tight binding parameters which ar e employed, though as 4we shall see the various descriptions provide very similar pictures of the one electron local density of states. We note that Udvardi and Szunyogh12have also discussed the inﬂuence of spin orbit coupling on the dispersion relation of spin waves in the Fe monolayer on W(110) within the framework of the adiabatic approach discussed above, where exchange interactions and other magnetic parameters are calculated in real space. We shall d iscuss a comparison with our results and theirs below. There are diﬀerences. Most particula rly, we note that in Fig. 3, the authors of ref. 12 provide two dispersion curves for pr opagation perpendicular to the magnetization, whereas in a ﬁlm such as this with one spin per un it cell there can be only one magnon branch. Additionally and very recently, Bergman n and coworkers45 investigatedwithinanadiabaticapproachﬁnitetemperatureeﬀect sonthemagnonspectrum of Fe/W(110). Insection II, we comment onourmeans of introducing spin orbit cou pling into the theory. The results of our calculations are summarized in Section III and con cluding remarks are found in section IV. II. CALCULATION OF THE DYNAMIC SUSCEPTIBILITY IN THE PRESENCE OF SPIN ORBIT COUPLING Theformalism forincluding spin orbitcoupling eﬀects in our description of spin dynamics is quite involved, so in this section we conﬁne our attention to an outlin e of the key steps, and an exposition of the overall structure of the theory. Our sta rting point is the multi band Hubbard model of the system that was employed in our earlier study of spin dynamics in ultrathin ferromagnets. The starting Hamiltonian is written as5 H=/summationdisplay ij/summationdisplay µνσTµν ijc† iµσcjνσ+1 2/summationdisplay µνµ′ν′/summationdisplay iσσ′Ui;µν,µ′ν′c† iµσc† iνσ′ciν′σ′ciµ′σ (1) whereiandjaresite indices, σ,σ′refer to spin, and µ,νtothe tight binding orbitals, nine in numberforeachsite, whichareincludedinourtreatment. TheCoulo mbinteractionsoperate only within the 3d orbitals on a given lattice site. The ﬁlm, within which fer romagnetism is driven by the Coulomb interactions, sits on a semi-inﬁnite substrat e within which the Coulomb interaction is ignored. 5In our empirical tight binding picture, the spin orbit interaction adds a term we write as HSO=/summationdisplay i/summationdisplay µνλi 2/bracketleftBig Lz µν(c† iµ↑ciν↑−c† iµciν↓)+L+ µνc† iµ↓ciν↑+L− µνc† iµ↑ciν↓/bracketrightBig (2) where/vectorLis the angular momentum operator, λiis the local spin-orbit coupling constant, L±=Lx±iLyandLα µν=/an}bracketle{tµ\|Lα\|ν/an}bracketri}ht. We assume that the spin orbit interaction, present both within the ferromagnetic ﬁlm and the substrate, operates only with in the 3d atomic orbitals. A convenient tabulation of matrix elements of the orbital angular mo mentum operators is found in ref. 13. Information on the spin waves follows from the study of the spectr al density of the trans- verse dynamic susceptibility χ+,−(/vectorQ/bardbl,Ω;l⊥,l′ ⊥) as discussed above. From the text around Eq. (1) of ref. 14, we see that this function describes the amplitud e of the transverse spin motion (the expectation value of the spin operator S+in the layer labeled l⊥) to a ﬁctitious transverse magnetic ﬁeld of frequency Ω and wave vector /vectorQ/bardblparallel to the ﬁlm surface that is applied to layer l′ ⊥of the sample. The spectral density, given by Im {χ+,−(/vectorQ/bardbl,Ω;l⊥,l′ ⊥)}, when multiplied by the Bose Einstein function n(Ω) = [exp( βΩ)−1]−1is also the amplitude of thermal spin ﬂuctuations of wave vector /vectorQ/bardbland frequency Ω in layer l⊥. We obtain in- formation regarding the character (frequency, linewidth, and am plitude in layer l⊥) of spin waves from the study of this function, as discussed earlier.5 Our previous analyses are based on the study of the dynamic susce ptibility just described through use of the random phase approximation (RPA) of many bod y theory. The Feynman diagrams included in this method are the same as those incorporated into time dependent density functional theory, though use of our Hubbard model allow s us to solve the result- ing equation easily once the very large array of irreducible particle ho le propagators are generated numerically. Our task in the present paper is to extend the RPA treatment to inc orporate spin orbit coupling. The extension is non trivial. The quantity of interest, refe rred to in abbreviated notation as χ+,−, may be expressed as a commutator of the spin operators S+andS−whose precise deﬁnition is given earlier.5,12With spin orbit coupling ignored, the RPA decoupling procedure leads to a closed equation for χ+,−. When the RPA decoupling is carried out in its presence, we are led to a sequence of four coupled equations wh ich include new objects we may refer to as χ−,−,χ↑,−andχ↓,−. The number of irreducible particle hole propagators that must be computed likewise is increased by a factor of four. For a very simple version 6of a one band Hubbard model, and for a very diﬀerent purpose, Fuld e and Luther carried out an equivalent procedure many years ago15. In what follows, we provide a summary of key steps along with expressions for the ﬁnal set of equations. To generate the equation of motion, we need the commutator of th e operator S+ µν(l,l′) = c† lµ↑cl′ν↓with the Hamiltonian. One ﬁnds [S+ µν(l,l′),HSO] =1 2/summationdisplay η{λl′L+ νηc† lµ↑cl′η↑−λlL+ ηµc† lη↓cl′ν↓+λl′Lz νηc† lµ↑cl′η↓−λlLz ηµc† lη↑cl′ν↓}.(3) The last two terms on the right hand side of Eq. 3 lead to terms in the e quation of motion which involve χ+,−whereas the ﬁrst two terms couple us to the entities χ↑,−andχ↓,−. When we write down the commutator of these new correlation funct ions with the spin orbit Hamiltonian, we are led to terms which couple into the function χ−,−which is formed from the commutator of two S−operators. In the absence of spin orbit coupling, a consequence o f spin rotation invariance of the Hamiltonian is that the three new func tions just encountered vanish. But they do not in its presence, and they must be incorpora ted into the analysis. Onethen introduces theinﬂuence of theCoulomb interaction into th eequation ofmotion, and carries out an RPA decoupling of the resulting terms. The analys is is very lengthy, so here we summarize only the structure that results from this proce dure. Deﬁnitions of the various quantities that enter are given in the Appendix. We express the equations of motion in terms of a 4 ×4 matrix structure, where in schematic notation we let χ(1)=χ+,−, χ(2)=χ↑,−,χ(3)=χ↓,−andχ(4)=χ−,−. The four coupled equations then have the form Ωχ(s)=A(s)+/summationdisplay s′(Bss′+˜Bss′)χ(s′)(4) Each quantity in Eq. 4 has attached to it four orbital indices, and fo ur site indices. To be explicit, χ(2)=χ↑,−which enters Eq. (4) is formed from the commutator of the operat or c† lµ↑cl′ν↑withc† mµ′↓cm′ν′↑and in full we denote this quantity as χ(2) µν;µ′ν′(ll′;mm′). The site indices label the planes in the ﬁlm, and we suppress reference to Ω an d/vectorQ/bardbl. The products on the right hand side of Eq. 4 are matrix multiplications that involve th ese various indices. For instance, the object/summationtext s′Bss′χ(s′)is labeled by four orbital and four site indices so [Bss′χ(s′)]µν,µ′ν′(ll′;mm′) =/summationdisplay γδ/summationdisplay nn′Bss′ µν,γδ(ll′;nn′)χ(s′) γδ,µ′ν′(nn′;mm′). (5) One proceeds by writing Eq. 4 in terms of the dynamic susceptibilities t hat characterize the non-interacting system. These, referred to also as the irred ucible particle hole propaga- tors, are generated by evaluating the commutators which enter in to the deﬁnition of χ(s)in 7the non interacting ground state. These objects, denoted by χ(0s)obey a structure similar to Eq. 4, Ωχ(0s)=A(s)+/summationdisplay s′Bss′χ(s′). (6) It is then possible to relate χ(s)toχ(0s)through the relation, using four vector notation, /vector χ(Ω) =/vector χ(0)(Ω)+(Ω −B)−1˜B/vector χ(Ω). (7) The matrix structure Γ ≡(Ω−B)−1may be generated from the deﬁnition of B, which may be obtained from the equation of motion of the non-interacting sus ceptibility, Eq. 6. Then ˜Bfollows from the equation of motion of the full susceptibility, as gene rated in the RPA. One may solve Eq. 7 /vector χ(Ω) = [I−(Ω−B)−1¯B]−1/vector χ(0)(Ω), (8) so our basic task is to compute the non interacting susceptibility mat rix/vector χ(0)and then carry out the matrix inversion operation displayed in Eq. 8. For this we requ ire the single particle Greens functions (SPGFs) associated with our approach. To generate the SPGFs, we set up an eﬀective single particle Hamilton ianHspby intro- ducing a mean ﬁeld approximation for the Coulomb interaction. The ge neral structure of the single particle Hamiltonian is Hsp=/summationdisplay ij/summationdisplay µνσ˜Tµνσ ijc† iµσcjνσ+/summationdisplay i/summationdisplay µν{α∗ i;µνc† iµ↓ciν↑+αi;µνc† iµ↑ciν↓} (9) where the eﬀective hopping integral ˜Tµνσ ijcontains the spin diagonal portion of the spin orbit interaction, along withthemeanﬁeld contributions fromtheCoulomb interaction. The form we use for the latter is stated below. The coeﬃcients in the spin ﬂip te rms are given by αi;µν=λiL− µν−/summationdisplay ηγUi;ηµ,νγ/an}bracketle{tc† iη↓ciγ↑/an}bracketri}ht. (10) We then have the eigenvalue equation that generates the single par ticle eigenvalues and eigenfunctions in the form Hsp\|φs/an}bracketri}ht=Es\|φs/an}bracketri}ht; we can write this in the explicit form /summationdisplay l/summationdisplay ησ′/bracketleftBig δσσ′˜Tµησ′ il+δil(δσ′↓δσ↑α∗ l;µη+δσ′↑δσ↓αl;µη)/bracketrightBig /an}bracketle{tlησ′\|φs/an}bracketri}ht=Es/an}bracketle{tiµσ\|φs/an}bracketri}ht.(11) The single particle Greens function may be expressed in terms of the quantities that enter Eq. 11. We have for this object the deﬁnition Giµσ;jνσ′(t) =−iθ(t)/an}bracketle{t{ciµσ(t),c† jνσ′(0)}/an}bracketri}ht (12) 8and one has the representation Giµσ;jνσ′(Ω) =/summationdisplay s/an}bracketle{tiµσ\|φs/an}bracketri}ht/an}bracketle{tφs\|jνσ′/an}bracketri}ht Ω−Es+iη. (13) These functions may be constructed directly from their equations of motion, which read, after Fourier transforming with respect to time, −/summationdisplay l/summationdisplay ησ′′/bracketleftBig δσσ′′˜Tµησ′′ il+δil(δσ′′↓δσ↑α∗ l;µη+δσ′′↑δσ↓αl;µη)/bracketrightBig Glησ′′;jνσ′+ΩGiµσ;jνσ′=δσσ′δµνδij. (14) Forthecasewherethesubstrateissemi inﬁnite, ourmeansofgen eratinganumerical solution to the hierarchy of equations stated in Eq. 14 has been discussed e arlier. What remains is to describe how the Coulomb interaction enters the eﬀective hoppin g integrals ˜Tµνσ ijthat appear in Eq. 9, Eq. 11 and Eq. 14. There are, of course, a large number of Coulomb matrix elements in t he original Hamil- tonian, even if the Coulomb interactions are conﬁned to within the 3d shell. Through the use of group theory,17the complete set of Coulomb matrix elements may be expressed in terms of three parameters. These are given in Table I of the ﬁrst c ited paper in ref. 5. In subsequent work, we have found that a much simpler structure18nicely reproduces re- sults obtained with the full three parameter form. We use the simple r one parameter form here, for which Ui;µν,µ′ν′=Uiδµν′δµ′ν. Then in the mean ﬁeld approximation, the Coulomb contribution to the single particle Hamiltonian assumes the form H(C) sp=−/summationdisplay iUimi 2/summationdisplay µ(c† iµ↑ciµ↑−c† iµ↓ciµ↓) (15) Heremiis the magnitude of the moment on site i. The Coulomb interactions Uiare non zero only within the ultrathin ferromagnet, and the moments mi, determined self consistently, vary from layer to layer when we consider multi layer ferromagnetic ﬁlms. It should be noted that when the Ansatz just described is employed in Eq. 10, the term from the Coulomb interaction on the right hand side becomes propor tional to the transverse component of the moment located on site iand this vanishes identically. Thus, despite the complexity introduced by the spin orbit coupling, when the simple one p arameter Ansatz for the Coulomb matrix elements is employed, one needs no parameters b eyond the moment on each layer in the self consistent loops that describe the ground sta te. In the present context, this is an extraordinarily large savings in computational labor, and th is will allow us to 9address very large systems in the future. It is the case that cert ain oﬀ diagonal elements such as/an}bracketle{tc† mµ′↓cl′ν↑/an}bracketri}htappear in the quantities deﬁned in the Appendix. Notice, for example , the expressions in Eqs. A.1. Once the ground state single particle Gr eens functions are determined, such expectation values are readily computed. III. RESULTS AND DISCUSSION In earlier studies of Fe layers on W(110),19,20as noted above, the electronic structure was generated through use of tight binding parameters obtained f rom bulk electronic struc- ture calculations. These calculations generate eﬀective exchange interactions comparable in magnitude to those found in the bulk transition metals,20with the consequence that for both monolayer Fe and bilayer Fe on W(110) the large wave vector sp in waves generated by theory are very much stiﬀer than found experimentally21,22though it should be noted that for the bilayer, the calculated value of the spin wave exchange stiﬀness is in excellent accord with the data.23Subsequent calculations which construct the spin wave dispersion relation from adiabatic theory based on calculations of eﬀective exc hange integrals also gen- erate spin waves for the monolayer substantially stiﬀer than found experimentally,12though they are softer than in our earlier work by a factor of two or so. We remark that it has been suggested that the remarkably soft spin waves found exper imentally may have origin in carbon contamination of the monolayer and bilayer.20We remark here that this can be introduced during the SPEELS measurement. We note that the mag netic properties of Fe monolayers grown on carbon free W(110)24diﬀer dramatically from those grown on surfaces now known to be contaminated by carbon.25In the former case, the domain walls have a thickness of 2.15 nm,24whereas in the latter circumstance very narrow walls with thickness bounded from above by 0.6 nm are found.25This suggests that the strength of the eﬀective exchange is very diﬀerent in the two cases, with stiﬀer exchange in t he carbon free samples. The considerations of the previous paragraph have motivated us t o carry out a series of studies of the eﬀective exchange in the Fe monolayer on W(110) with in the framework of three diﬀerent electronic structure calculations. We ﬁnd that alth ough all three give local density of states that are very similar, along with very similar energy bands when these are examined, theintersiteexchangeinteractionsvarysubstantially. First, wehaveemployed the parameter set used earlier that is based on bulk electronic structu res19,20in new calculations 10we call case A. In case B, we have employed an approach very similar t o that used in ref. 12, though in what follows our calculation of eﬀective exchange integrals is non relativistic. This is the Korringa Kohn Rostoker Greens Function (KKR-GF) meth od,26which employs the atomic sphere approximation and makes use of the Dyson equat ionG=g+gVGas given in matrix notation. This allows us to calculate the Greens functio nGof an arbitrary complex system given the perturbing potential Vand the Greens function gof a reference unperturbed system. Within the Local Spin Density Approximation ( LSDA),27We consider a slab of ﬁve monolayers of W with the experimental lattice constant on top of which an Fe monolayer is deposited and relaxed by -12.9%12with respect to the W interlayer distance. Angular momenta up to lmax= 3 were included in the Greens functions with a kmesh of 6400 points in the full two dimensional Brillouin zone. The eﬀective exc hange interactions were calculated within the approximation of inﬁnitesmal rotations28that allows one to use the magnetic force theorem. This states that the energy change due to inﬁnitesmal rotations in the moment directions can be calculated through the Kohn Sham eig envalues. Method C is the Real Space Linear-Muﬃn-Tin-Orbital approach as im plemented, also, in the atomic sphere approximation (RS-LMTO-ASA).29–33Due to its linear scaling, this method allows one to address the electronic structure of systems with a large number of atoms for which the basic eigenvalue problem is solved in real space us ing the Haydock recursion method. The Fe overlayer on the W(110) substrate was simulated by a large bcc slab which contained ∼6800 atoms, arranged in 12 atomic planes parallel to the (110) surface, with the experimental lattice parameter of bulk W. One em pty sphere overlayer is included, and self consistent potential parameters were obtaine d for the empty sphere overlayer, the Fe monolayer, and the three W layers underneath u sing LSDA.34For deeper W layers we use bulk potential. Nine orbitals per site (the ﬁve 3d and 4 s p complex) were used to describe the Fe valence band and the empty sphere overlay er, and for W the fully occupied 4f orbitals were also included in the core. To evaluate the or bital moments we use a scalar relativistic (SR) approach and include a spin orbit coupling ter mλ/vectorL·/vectorSat each variational step.35In the recursion method the continued fraction has been terminat ed after 30 recursion levels with the Beer Pettifor terminator.36The TB parameters so obtained are inserted into our semi-empirical scheme and this allows us to generat e the non interacting susceptibilities which enter our full RPA description of the response of the structure. In order to compare the electronic structures generated by the approaches just described, 11we turn our attention to the local density of states for the major ity and minority spins in the adsorbed Fe monolayer. These are summarized in Fig. 1. The local densities of states (LDOS) generated by the three sets of TB parameters have approximately the same overall features, as we see from Fig. 1. Th e main diﬀerences appear in the majority spin band, which overlaps the 5d states in the W subst rate over a larger energy range than the minority band. This is also true if we compare t he LDOS generated by the tight binding parameters extracted from the KKR electronic structure to the LDOS obtaineddirectly fromtheKKR calculations (reddashed linein Fig.1b) . The Fe-Whopping parameters are indeed the least accurate portion of our paramet rization scheme. In case A wejustusedtheFe-FebulkparameterstodescribetheFe-Whopp ing. IncaseBweextracted TB parameters for Fe by ﬁtting a KKR calculation of an unsupported Fe monolayer with a lattice parameter matching that of the W substrate. For the Fe-W hopping we used the Fe parameters obtained from the ﬁtting, scaled to mimic the Fe-W dista nce relaxation. The relaxation parameter was chosen to give the correct spin magnetic moment for the adsorbed Fe monolayer. In case C all parameters were directly provided by th e RS-LMTO-ASA code, but in the DFT calculations the Fe-W distance was assumed to be equa l to the distance between W layers. Thus, the main diﬀerence between cases B and C is the treatment of the mixing between Fe and W states and this is expected to aﬀect more st rongly the states that occupy the same energy range. As noted above, while the local density of states provided by the th ree approaches to the electronic structure are quite similar as we see from Fig. 1 (and t he same is true of the electronic energy bands themselves if these are examined), the ex change interactions diﬀer substantially for the three cases. For the ﬁrst, second and third neighbors we have (in meV) 42.5, 3.72 and 0.46 for model A, 28.7, -7.87 and 0.31, for model B and 1 1.23, -7.31 and 0.22 for model C. The authors of ref. 12 ﬁnd 10.84, -3.34 and 3.64 for th ese exchange integrals. We now turn to our studies of spin excitations in the Fe monolayer and Fe bilayer on W(110) within the framework of the electronic structure generat ed through use of the ap- proach in case C. We will discuss the inﬂuence of spin orbit coupling on b oth the trans- verse wave vector dependent susceptibility though study of the s pectral density function A(/vectorQ/bardbl,Ω;l⊥) =−Im{χ+,−(/vectorQ/bardbl,Ω;l⊥)}discussed in section I. This function, for ﬁxed wave vector/vectorQ/bardbl, when considered as a function of frequency Ω, describes the fre quency spectrum of the ﬂuctuations of wave vector /vectorQ/bardblof the transverse magnetic moment in layer l⊥as noted 12-6 -4 -2 0 2 E-EF (eV)-2024LDOS (states/eV)-2024LDOS (states/eV)-2024LDOS (states/eV)a) b) c) FIG. 1: (color online) For the Fe monolayer adsorbed on W(110 ), we show the local density of states in the Fe monolayer. The majority spin density of stat es is shown positive and the minority spin density of states is negative. The zero of energy is at th e Fermi energy. In (a), bulk electronic structure parameters are used as in the second of the two pape rs cited in 5 (caseA). In (b), we have the density of states generated by method B. The black cu rve is found by ﬁtting the KKR electronic energy band structure to tight binding paramete rs as described in the text, and the red curve is calculated directly from the KKR calculation. In (c ) we have the local density of states generated by method C. 13above. In the frequency regime where spin waves are encountere d, this function is closely related to (but not identical to) the response function probed in a SPEELS measurement. In Fig. 2, for the Fe monolayer on W(110), we show the spectral de nsity function cal- culated for three values of \|/vectorQ/bardbl\|, for propagation across the magnetization. Thus, the wave vector is directed along the short axis in the surface. This is the dire ction probedin SPEELS studies ofthe Femonolayer onthissurface.22Ineach ﬁgure, we show three curves. The green dashed curve is calculated with spin orbit coupling set to zero. We sho w only a single curve for this case, because the spectral density is identical for the tw o directions of propagation across the magnetization, + /vectorQ/bardbland−/vectorQ/bardbl. When spin orbit coupling is switched on, for the two directions just mentioned the response function is very diﬀere nt, as we see from the red andblack curve inthevariouspanels. These spin wave frequencies, deduced fromthepeakin the response functions as discussed in section I, diﬀer for the two directions of propagation, and also note that the peak intensities and linewidths diﬀer as well. It is the absence of both time reversal symmetry and reﬂection symmetry which renders + /vectorQ/bardbland−/vectorQ/bardblinequivalent for this direction of propagation. The system senses this breakdo wn of symmetry through the spin orbit interaction. If one considers propagation parallel to the magnetization, the asymmetries displayed in Fig. 2 are absent. The reason is that for th is direction of propaga- tion, reﬂection in the plane that is perpendicular to both the magnet ization and the surface is a goodsymmetry operation of the system, but takes + /vectorQ/bardblinto−/vectorQ/bardblthus rendering the two directions equivalent. Recall, of course, that the magnetization is a pseudo vector in regard to reﬂections. Notice how very broad the curves are for large wav e vectors; the lifetime of the spin waves is very short indeed. As discussed in section I, we may construct a spin wave dispersion cu rve by plotting the maxima in spectral density plots such as those illustrated in Fig. 2 as a function of wave vector. We show dispersion relations constructed in this manner in F ig. 3, with spin orbit coupling both present and absent. In Fig. 3a, and for propagation perpendicular to the magnetization we show the dispersion curve so obtained for wave ve ctors throughout the surface Brillouin zone, and in Fig. 3b we show its behavior for small wav e vectors. Let is ﬁrst consider Fig, 3(a). Here the dispersion curve extends t hroughout the two dimensional Brillouin zone. At the zone boundary, quite clearly the slo pe of the dispersion curve does not vanish. In this direction of propagation, the natur e of the point at the zone boundary does not require the slope to vanish. What is most striking , clearly, is the anomaly 140 10 20 30 40 50 Ω (meV)05×1031×104-Imχ+,−(a) 0 50 100 150 200 Ω (meV)050010001500-Imχ+,-(b) 0 100 200 300 Ω (meV)0200400600800-Imχ+,-(c) FIG. 2: (color online) Thespectral density functions A(/vectorQ/bardbl,Ω;l⊥) evaluated in theFe monolayer for three values of the wave vector in the direction perpendicul ar to the magnetization. We have (a) \|/vectorQ/bardbl\|= 0.4˚A−1, (b)\|/vectorQ/bardbl\|= 1.0˚A−1and (c)\|/vectorQ/bardbl\|= 1.4˚A−1. The green curve (dashed) is calculated with spin orbit coupling set to zero; the spectral density he re is independent of the sign of /vectorQ/bardbl. The red and black curves are calculated with spin orbit coupling turned on. Now we see asymmetries for propagation across the magnetization, with the red curv e/vectorQ/bardbldirected from left to right and the black curve from right to left. in the vicinity of 1 ˚A−1. This feature is evident in the calculation with spin orbit coupling absent, and for positive values of the wave vector the feature be comes much more dramatic when spin orbit coupling is switched on. Anomalies rather similar to thos e in the black curve in Fig. 3(a) appear in the green dispersion curve found in Fig. 3 of ref. 12, though these authors did not continue their calculation much beyond the 1 ˚A−1regime. Our spin waves are very much softer than theirs in this spectral region, no tice. In Fig. 3 of ref. 12, one ﬁnds two dispersion curves, one a mirror image of the second. T hus, these authors 15-1.5 -1-0.5 00.5 11.5 Q (A-1)050100150200Ω (meV)(a) -0.4 -0.2 0 0.2 0.4 Q (A-1)010203040Ω (meV)(b) FIG. 3: (color online) Spin wave dispersion relations const ructed from peaks in the spectral den- sity, for the Fe monolayer on W(110). The wave vector is in the direction perpendicular to the magnetization. The red curve is constructed in the absence o f spin orbit coupling, it is included in the black curve. display two spin wave frequencies for each wave vector. This surely is not correct. For a structure with one atom per unit cell, there is one and only one spin wa ve mode for each wave vector, though as discussed above for the structure explo red here symmetry allow the left/right asymmetry in the dispersion curve illustrated in our Fig. 3. In Fig. 3b, again with spin orbit coupling switched on and oﬀ, we show an expanded view of the dispersion curve for small wave vectors. With spin orbit in teraction switched oﬀ, at zero wave vector we see a zero frequency spin wave mode, as re quired by the Goldstone theorem when the underlying Hamiltonian is form invariant under spin r otation. The curve is also symmetrical, and is accurately ﬁtted by the form Ω( /vectorQ/bardbl) = 149Q2 /bardbl(meV), with the wave vector in ˚A−1, whereas with spin orbit coupling turned on the dispersion relation is ﬁtted by Ω( /vectorQ/bardbl) = 3.4−11.8Q/bardbl+143Q2 /bardbl(meV). Spin orbit coupling introduces an anisotropy gap atQ/bardbl= 0, and most striking is the term linear in wave vector. This has its orig in in the Dzyaloshinskii Moriya interaction whose presence, as argued by th e authors of ref. 9, has its origin in the absence of both time reversal and inversion symmetry, for the adsorbed layer. At long wavelengths, one may describe spin waves by classical long wa velength phe- nomenology. The linear term in the dispersion curve has its origin in a te rm in the energy density of the spin system of the form VDM=−Γ/integraldisplay dxdzS y(x,z)∂Sx(x,z) ∂x(16) 16HereSα(x,z) is a spin density, the xzplane is parallel to the surface, and the magnetization is parallel to the zdirection. One interesting feature of the spin wave mode whose dispersion rela tion is illustrated in Fig. 3b is that at Q/bardbl= 0, the mode has a ﬁnite group velocity. The ﬁt to the dispersion curve gives this group velocity to be∂Ω(/vectorQ/bardbl) ∂Q/bardbl≈2×105cm/s , which is in the range of acoustic phonon group velocities. Weturnnowtoourcalculationsofspinwavesandtheresponsefunc tionsfortheFebilayer on W(110). Let us ﬁrst note that experimentally the orientation of the magnetization in the bilayer appears to be dependent on the surface upon which the bilayer is grown. For instance, when the bilayer is on the stepped W(110) surface, it is ma gnetized perpendicular to the surface,25a result inagreement with abinitio calculations of the anisotropy realiz ed in the epitaxial bilayer.38However, in the SPEELS studies of spin excitations in the bilayer22,39 the magnetization is in plane. In our calculations, we ﬁnd for model B t he magnetization is perpendicular to the surface, whereas in model C it lies in plane, along the long axis very much as in the SPEELS experiments. The anisotropy in the bilayer is no t particularly large, on the order of 0.5 meV/Fe atom, and one sees from these results t hat it is a property quite sensitive to the details of the electronic structure. The fact that model B and model C give the two diﬀerent stable orientation of the magnetization allows us to explore spin excitations for the two diﬀerent orientations of the magnetization. We ﬁrst turn our attention to the case where the magnetization lies in plane. The bilayer has two spin wave modes, an acoustic mode for which the magnetizat ion in the two planes precesses in phase, and an optical mode for which they precess 18 0 degrees out of phase. In Fig. 4, we show calculations of the dynamic susceptibility in the freq uency range of the acoustic mode for two values of the wave vector, Q/bardbl= +0.5˚A−1 andQ/bardbl=−0.5˚A−1. A spin orbit induced left right asymmetry is clearly evident both in the pe ak frequency and the height of the feature. Very recently, beautiful measuremen ts of spin orbit asymmetries in the Fe bilayer have appeared,39and the results of our Fig. 4 are to be compared with Fig. 3 of ref. 39. Theory and experiment are very similar, both in reg ard to the intensity asymmetry and also the spin orbit induced frequency shift, though our calculated spin wave frequencies are a little stiﬀer than those found experimentally. As remarked above, in Fig. 4 we show only the acoustical spin wave mo de frequency regime. In Fig. 5, for the spectral densities in the innermost layer ( upper panel) and the 170 50 100 Ω (meV)050010001500200025003000-Imχ+,− FIG. 4: (color online) The spectral density in the innermost layer, in the acoustic spin wave regime, for wave vectors of Q/bardbl= +0.5˚A−1(black curve) and Q/bardbl=−0.5˚A−1(red curve). Model C has been used for the calculation. In the ground state, the magne tization lies in plane along the long axis. outermost layer (lower panel) we show the spectral densities for t he entire spin wave regime, including the region where the optical spin wave is found. It is clear th at the spin orbit inducedfrequency shiftsarelargestfortheopticalmodewhich, u nfortunatelyisnotobserved in the experiments.39 In Fig. 6 for a sequence of wave vectors, all chosen positive, we sh ow a sequence of spectra calculated for the entire frequency range so both the acoustic an d optical spin wave feature are displayed. The black curves show the spectral density of the in nermost Fe layer, and the red curves are for the outer layer. The optical spin wave mode , not evident in the data, shows clearly in these ﬁgures. Notice that for wave vectors great er than 1 ˚A−1the acoustical mode is localized in the outer layer and the optical mode is localized on th e inner layer. The optical mode is very much broader than the acoustical mode at large wave vectors, by virtue of the strong coupling to the electron hole pairs in the W 5d ban ds. Aninteresting issue is the absence of the optical mode fromthe SPE ELS spectra reported 18-1500 -1000 -500 0-Imχ+,− 0 50 100 150 200 250 Ω (meV)-2000 -1000 0-Imχ+,− FIG. 5: (color online) For the wave vector Q/bardbl= 0.5˚A−1we show the spectral densities in the innermost Fe layer (upper panel) and in the outermost layer ( lower panel) for the Fe bilayer on W(110). Theﬁgureincludestheoptical spinwave feature. As inFig. 4, theblack curveiscalculated forQ/bardblpositive, and the red curves are for Q/bardblnegative. The calculations employ model C. in refs. 22 and 39. We note that these spectra are taken with only t wo beam energies, 4 eV and 6.75 eV. At such very low energies, the beam electron will sample b oth Fe layers, so the SPEELS signal will be a coherent superposition of electron wa ves backscattered from each layer; the excitation process involves coherent excitat ion of both layers by the incident electron. As a consequence of the 180 degree phase diﬀer ence in spin motions associated with the two modes it is quite possible, indeed even probab le, that for energies where the acoustical mode is strong the intensity of the optical mo de is weak, by virtue of quantum interference eﬀects in the excitation scattering amplit ude. In earlier studies of surface phonons, it is well documented that on surfaces where two surface phonons of diﬀerent polarization exist for the same wave vector, one can be sile nt and one active in electron loss spectroscopy.40It would require a full multiple scattering analysis of the spin waveexcitationprocesstoexplorethistheoretically. Whileearlier41calculationsthataddress SPEELS excitation of spin waves described by the Heisenberg model could be adapted for 190200040006000Imχ+,− 010002000Imχ+,− 05001000Imχ+,− 050010001500Imχ+,− 050010001500Imχ+,− 0 100 200 300 400 500 Ω (meV)05001000Imχ+,−Qy=0.4 A-1 Qy=0.6 A-1 Qy=0.8 A-1 Qy=1.0 A-1 Qy=1.2 A-1 Qy=1.4 A-1 FIG. 6: (color online) For the Fe bilayer and for several valu es of the wave vector (all positive), we show the spectral density functions for the innermost layer adjacent to the substrate (black curve) and those for the outer layer of the ﬁlm. The calculations emp loy model C. 20-1.5 -1-0.5 00.5 11.5 Q (A-1)0100200300Ω (meV) FIG. 7: (color online) For the Fe bilayer with magnetization in plane, we show the spin wave dispersion curves calculated with spin orbit coupling (bla ck points) and without spin orbit coupling (red curves). Model C has been employed for these calculatio ns. this purpose, in principle, a problem is that at such low beam energies it is necessary to take due account of image potential eﬀects to obtain meaningful result s.42This is very diﬃcult to do without considerable information on the electron reﬂectivity o f the surface.42It would be of great interest to see experimental SPEELS studies of the Fe bilayer with a wider range of beam energies to search for the optical mode, if this were possib le. In Fig. 7, we show dispersion curves for the optical and acoustic sp in wave branches for the bilayer. The magnetization lies in plane, and one can see that on th e scale of this ﬁgure, the spin orbit eﬀects on the dispersion curve are rather modest co mpared to those in the monolayer. For small wave vectors, with spin orbit coupling present , the dispersion curve of the acoustic spin wave branch is ﬁtted by the form Ω( Q/bardbl) = 0.49−0.85Q/bardbl+243Q2 /bardbl(meV) so at long wavelengths the inﬂuence of the Dzyaloshinskii Moriya inte raction is more than one order of magnitude smaller than it is in the monolayer. If the magnetization is perpendicular to the surface, then symmet ry considerations show that there are no left/right asymmetries in the spin wave propagat ion characteristics. One may see this as follows. Consider a wave vector /vectorQ/bardblin the plane of the surface, which also is perpendicular to the magnetization, and thus perpendicular t o the long axis. The reﬂection Rin the plane perpendicular to the surface and which contains the mag netization 210 100 200 300 400 Ω (meV)050010001500200025003000Imχ+,− FIG. 8: (color online) For the bilayer and the case where the m agnetization is perpendicular to the surface (model B), and for Q/bardbl= 0.6˚A−1, we show spectral density function calculated for positive values of Q/bardbl(continuous lines) and negative values of Q/bardbl(symbols). The black curve is the spectral density for the inner layer, and the red curve is the outermos t Fe layer. simultaneously changes the sign of wave vector and the magnetizat ion. If this is followed by the time reversal operation T, then/vectorQ/bardblremains reversed in sign but the magnetization changes back to its original orientation. Thus the product RTleaves the system invariant but transforms /vectorQ/bardblinto−/vectorQ/bardbl. The two propagation directions are then equivalent. We illustrate this in Fig. 8 where, for Q/bardbl= 0.6˚A−1, where it is shown that the spectral densities calculated for the two directions of propagation are ident ical, with spin orbit cou- pling switched on. Model B, in which the magnetization is perpendicular to the surface, has been used in these calculations. The spectral densities calculated f or the two signs of Q/bardbl cannot be distinguished to within the numerical precision we use. IV. CONCLUDING REMARKS We have developed the formalism which allows one to include the inﬂuenc e of spin orbit couplingonthespinexcitationsofultrathinferromagnetsonsemiin ﬁnitemetallicsubstrates. Our approach allows us to calculate the full dynamic susceptibility of t he system, so as illustrated by the calculations presented in section III we can examin e the inﬂuence of spin orbit coupling on the linewidth (or lifetime) of spin excitations, along wit h their oscillator 22strength. As in previous work, we can then construct eﬀective dis persion curves by following peaks in the spectral density as a function of wave vector, withou t resort to calculations of large numbers of very small distant neighbor exchange interaction s. The results presented in Fig. 4 are very similar to the experimental data reported in ref. 39 , as discussed above, though we see that in the bilayer the inﬂuence of the Dzyaloshinskii M oriya interaction is considerably more modest than in the monolayer. We will be exploring other issues in the near future. One interest in ou r minds is the inﬂuence of spin orbit coupling on the spin pumping contributions to th e ferromagnetic resonance linewidth, as observed in ferromagnetic resonance (FM R) studies of ultrathin ﬁlms.43It has been shown earlier44that the methodology employed in the present paper (without spin orbit coupling included) can be applied to the description of the spin pumping contribution to the FMR linewidth, and in fact an excellent quantitativ e account of the data on the Fe/Au(100) system was obtained. It is possible that fo r ﬁlms grown on 4d and 5d substrates that spin orbit coupling can inﬂuence the spin pum ping relaxation rate substantially. This willrequirecalculationsdirected towardmuchthic ker ﬁlmsthanexplored here. The formalism we have developed and described in the present paper will allow such studies in the future. Acknowledgments This research was supported by the U. S. Department of Energy, through grant No. DE-FG03-84ER-45083. S. L. wishes to thank the Alexander von Hu mboldt Foundation for a Feodor Lynen Fellowship. A.T.C. and R.B.M. acknowledge support from CNPq and FAPERJ and A.B.K. was supported also by the CNPq, Brazil. Appendix In this Appendix we provide explicit expressions for the various quan tities which enter the equations displayed in Section II. While these expressions are un fortunately lengthy, it will be useful for them to be given in full. A(1) µν,µ′ν′(ll′;mm′) =δl′mδνµ′/an}bracketle{tc† lµ↑cm′ν′↑/an}bracketri}ht−δlm′δµν′/an}bracketle{tc† mµ′↓cl′ν↓/an}bracketri}ht 23A(2) µν,µ′ν′(ll′;mm′) =−δlm′δµν′/an}bracketle{tc† mµ↓cl′ν↑/an}bracketri}ht A(3) µν,µ′ν′(ll′;mm′) =δl′mδνµ′/an}bracketle{tc† lµ↓cm′ν′↑/an}bracketri}ht A(4) µν,µ′ν′(ll′;mm′) = 0 (A.1) The various expectation values in the equations above and those dis played below are calculated from the single particle Greens functions once the self co nsistent ground state parameters are determined. Then ˜B11 µν,µ′ν′(ll′;mm′) =/summationdisplay η/parenleftBig Ul;µ′η,µν′/an}bracketle{tc† lη↓cl′ν↓/an}bracketri}htδlmδlm′−Ul′;µ′ν,ην′/an}bracketle{tc† lµ↑cl′η↑/an}bracketri}htδl′mδl′m′/parenrightBig ˜B12 µν,µ′ν′(ll′;mm′) =/summationdisplay η/parenleftBig Ul′;µ′ν,ν′η/an}bracketle{tc† lµ↑cl′η↓/an}bracketri}htδl′mδl′m′−Ul;ηµ′,µν′/an}bracketle{tc† lη↑cl′ν↓/an}bracketri}htδlmδlm′+ +Ul;µ′η,µν′/an}bracketle{tc† lη↑cl′ν↓/an}bracketri}htδlmδlm′/parenrightBig ˜B13 µν,µ′ν′(ll′;mm′) =/summationdisplay η/parenleftBig Ul′;µ′ν,ν′η/an}bracketle{tc† lµ↑cl′η↓/an}bracketri}htδl′mδl′m′−Ul;ηµ′,µν′/an}bracketle{tc† lη↑cl′ν↓/an}bracketri}htδlmδlm′− −Ul′;µ′ν,ην′/an}bracketle{tc† lµ↑cl′η↓/an}bracketri}htδl′mδl′m′/parenrightBig ˜B14 µν,µ′ν′(ll′;mm′) = 0 (A.2) ˜B21 µν,µ′ν′(ll′;mm′) =/summationdisplay ηUl;µ′η,µν′/an}bracketle{tc† lη↓cl′ν↑/an}bracketri}htδlmδlm′ ˜B22 µν,µ′ν′(ll′;mm′) =/summationdisplay η/bracketleftBig (Ul′;µ′ν,ν′η−Ul′;µ′ν,ην′)/an}bracketle{tc† lµ↑cl′η↑/an}bracketri}htδl′mδl′m′− −(Ul;ηµ′,µν′−Ul;µ′η,µν′)/an}bracketle{tc† lη↑cl′ν↑/an}bracketri}htδlmδlm′/bracketrightBig ˜B23 µν,µ′ν′(ll′;mm′) =/summationdisplay η/parenleftBig Ul;µ′ν,ν′η/an}bracketle{tc† lµ↑cl′η↑/an}bracketri}htδl′mδl′m′−Ul;ηµ′,µν′/an}bracketle{tc† lη↑cl′ν↑/an}bracketri}htδlmδl′m′/parenrightBig ˜B24 µν,µ′ν′(ll′;mm′) =−/summationdisplay ηUl′;µ′ν,ην′/an}bracketle{tc† lµ↑cl′η↓/an}bracketri}htδl′mδl′m′ (A.3) ˜B31 µν,µ′ν′(ll′;mm′) =−/summationdisplay ηUl′;µ′ν,ην′/an}bracketle{tc† lµ↓cl′η↑/an}bracketri}htδl′mδl′m′ ˜B32 µν,µ′ν′(ll′;mm′) =/summationdisplay η/parenleftBig Ul;µ′ν,ν′η/an}bracketle{tc† lµ↓cl′η↓/an}bracketri}htδl′mδl′m′−Ul;ηµ′,µν′/an}bracketle{tc† lη↓cl′ν↓/an}bracketri}htδlmδl′m′/parenrightBig 24˜B33 µν,µ′ν′(ll′;mm′) =/summationdisplay η/bracketleftBig (Ul′;µ′ν,ν′η−Ul′;µ′ν,ην′)/an}bracketle{tc† lµ↓cl′η↓/an}bracketri}htδl′mδl′m′− −(Ul;ηµ′,µν′−Ul;µ′η,µν′)/an}bracketle{tc† lη↓cl′ν↓/an}bracketri}htδlmδlm′/bracketrightBig ˜B34 µν,µ′ν′(ll′;mm′) =/summationdisplay ηUl;µ′η,µν′/an}bracketle{tc† lη↑cl′ν↓/an}bracketri}htδlmδlm′ (A.4) ˜B41 µν,µ′ν′(ll′;mm′) = 0 ˜B42 µν,µ′ν′(ll′;mm′) =/summationdisplay η/parenleftBig Ul′;µ′ν,ν′η/an}bracketle{tc† lµ↓cl′η↑/an}bracketri}htδl′mδl′m′−Ul;ηµ′,µν′/an}bracketle{tc† lη↓cl′ν↑/an}bracketri}htδlmδlm′− −Ul′;µ′ν,ην′/an}bracketle{tc† lµ↓cl′η↑/an}bracketri}htδl′mδl′m′/parenrightBig ˜B43 µν,µ′ν′(ll′;mm′) =/summationdisplay η/parenleftBig Ul′;µ′ν,ν′η/an}bracketle{tc† lµ↓cl′η↑/an}bracketri}htδl′mδl′m′−Ul;ηµ′,µν′/an}bracketle{tc† lη↓cl′ν↑/an}bracketri}htδlmδlm′+ +Ul;µ′η,µν′/an}bracketle{tc† lη↓cl′ν↑/an}bracketri}htδlmδlm′/parenrightBig ˜B44 µν,µ′ν′(ll′;mm′) =/summationdisplay η/parenleftBig Ul;µ′η,µν′/an}bracketle{tc† lη↑cl′ν↑/an}bracketri}htδlmδl′m−Ul′;µ′ν,ην′/an}bracketle{tc† lµ↓cl′η↓/an}bracketri}htδl′mδl′m′/parenrightBig (A.5) B11 µν,µ′ν′(ll′;mm′) =˜Tνν′↓ l′m′δlmδµµ′−(˜Tµµ′↑ lm)∗δl′m′δνν′ B12 µν,µ′ν′(ll′;mm′) =α∗ l′;ν′νδlmδl′m′δµµ′ B13 µν,µ′ν′(ll′;mm′) =−α∗ l;µµ′δlmδl′m′δνν′ B14 µν,µ′ν′(ll′;mm′) = 0 (A.6) B21 µν,µ′ν′(ll′;mm′) =αl′;νν′δlmδl′m′δµµ′ B22 µν,µ′ν′(ll′;mm′) =˜Tνν′↑ l′m′δlmδµµ′−(˜Tµµ′↑ lm)∗δl′m′δνν′ B23 µν,µ′ν′(ll′;mm′) = 0 B24 µν,µ′ν′(ll′;mm′) =−α∗ l;µµ′δlmδl′m′δνν′ (A.7) B31 µν,µ′ν′(ll′;mm′) =−αl;µ′µδlmδl′m′δνν′ 25B32 µν,µ′ν′(ll′;mm′) = 0 B33 µν,µ′ν′(ll′;mm′) =˜Tνν′↓ l′m′δlmδµµ′−(˜Tµµ′↓ lm)∗δl′m′δνν′ B34 µν,µ′ν′(ll′;mm′) =α∗ l′;ν′νδlmδl′m′δµµ′ (A.8) B41 µν,µ′ν′(ll′;mm′) = 0 B42 µν,µ′ν′(ll′;mm′) =−αl;µ′µδlmδl′m′δνν′ B43 µν,µ′ν′(ll′;mm′) =αl′;νν′δlmδl′m′δµµ′ B11 µν,µ′ν′(ll′;mm′) =˜Tνν′↑ l′m′δlmδµµ′−(˜Tµµ′↓ lm)∗δl′m′δνν′ (A.9) 1Rodrigo Arias and D. L. Mills, Phys. Rev. B 60, 7395 (1999); A. Azevedo, A. B. Oliveira, F. M. de Aguiar, and S. M. Rezende, Phys. Rev. B 62, 5331 (2000); D. L. Mills and S. M. Rezende in Spin Dynamics in Conﬁned Structures II, (Springer Verlag , Heidelberg, 2002); P. Landeros, Rodrigo E. Arias and D. L. Mills, Phys. Rev. B 77, 214405, (2008); J. Lindner, I. Barsukov, C. Raeder, C. Hassel, O. Posth, R. Meckenstock, P. Landeros and D. L. Mills, Phys. Rev. B 80, 224421 (2009). 2R.Urban, G. Woltersdorf andG. Heinrich, Phys.Rev. Lett. 87, 217204 (2001), Y.Tserkovnyak, A. Brataas and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002); Phys. Rev. B 66, 224403 (2002); D. L. Mills, Phys. Rev. B 68, 014419 (2003); M. Zwierzycki, Y. Tserkovnyak, P. J. Kelly, A. Brataas, and G. E. W. Bauer, Phys. Rev. B 71, 064420 (2005); A. T. Costa, R. B. Muniz and D. L. Mills, Phys. Rev. B 73, 054426 (2006). 3S. M. Rezende, A. Azevedo, M. A. Lucena and F. M. Aguiar, Phys. Rev. B63, 214416 (2001). 4R. Vollmer, M. Etzkorn, P. S. Anil Kumar, H. Ibach and J. R. Kir schner, Phys. Rev. Lett. 91, 14720 (2003); for a theoretical discussion of the linewidt h and dispersion observed in the experiment, see A. T. Costa, R. B. Muniz Phys. Rev. B 70, 0544406 (2004). 5Early papers are H. Tang, M. Plihal and D. L. Mills, Journal of Magnetism and Magnetic Materials 187, 23 (1998); R. B. Muniz and D. L. Mills, Phys. Rev. B 66, 174417 (2002); A. T. Costa, R. B. Muniz and D. L. Mills, Phys. Rev. B 68, 224435 (2003). 266For an explicit comparison between the predictions of the He isenberg model and that of an itinerant electron description of the spin excitations, se e Fig. 4 and Fig. 5 of the theoretical paper cited in ref. 4. 7S. Blgel, M. Weinert and P. H. Dederichs, Phys. Rev. Lett. 60, 1077 (1988). 8S. Heinze, M. Bode, A. Kubetzka, O. Pietzsch, X. Nie, S. Blgel , and R. Wiesendanger, Science 311, 1805 (2000). 9M. Bode, M. Heide, K. von Bergmann, P. Ferriani, S. Heinze, G. Bihlmayer, A. Kubetzka, O. Pietzsch, S. Blgel and R. Wiesendanger, Nature 447, 190 (2007). 10P. Lederer and A.Blandin, Phil. Mag. 14, 363 (1966). 11See also A. T. Costa, R. B. Muniz and D. L. Mills, Phys. Rev. B 73, 214403 (2006). 12L. Udvardi and L. Szunyogh, Phys. Rev. Lett. 102, 207204 (2009). 13E. Abate and M. Asdente, Phys. Rev. 140, A1303 (1965). 14A. T. Costa, Roberto Bechara Muniz and D. L. Mills, Phys. Rev. B73, 054426 (2006). 15Peter Fulde and Alan Luther, Phys. Rev. 175, 337 (1968). 16See the discussion in the second paper cited in footnote 5. 17See Chapter 6 and Chapter 7 and Appendix C of Group Theory and Q uantum Mechanics, M. Tinkham, (McGraw Hill, New York, 1964). 18See D. M. Edwards, page 114 of Moment Formation in Solids, Vol . 117 of NATO Advanced Study Institute, Series B: Physics (Plenum, New York, 1984) , edited by W. J. L. Buyers. 19See the second and third papers cited in footnote 5. 20A. T. Costa, R. B. Muniz, J. X. Cao, R. Q. Wu and D. L. Mills, Phys . Rev. B 78, 054439 (2008). 21J. Prokop, W. X. Tang, Y. Zhang, I Tudosa, T. R. F. Peixoto, Kh. Zakeri, and J. Kirschner, Phys. Rev. Lett. 102, 177206 (2009). 22W. X. Tang, Y. Zhang, I. Tudosa, J. Prokop, M. Etzkorn and J. Ki rschner, Phys. Rev. Lett. 99, 087202 (2007). 23Seethe discussion in A. T. Costa, R. B. Munizand D. L. Mills, I EEE Transactions on Magnetics 44, 1974 (2008). 24S. Krause, G. Herzog, T. Stapelfeldt, L. Berbil-Bautista, M . Bode E. Y. Vedmedenko, and R. Wiesendanger, Phys. Rev. Lett. 103, 127202 (2009). 25M. Pratzer, H. J. Elmers, M. Bode, O. Pietzsch, A. Kubetzka an d R. Wiesendanger, Phys. Rev. 27Lett.88, 057201 (2002). 26N. Papanikolaou, R. Zeller and P. H. Dederichs, J. Phys. Cond ens. Matter 14, 2799 (2002). 27S. H. Vosko, L. Wilk and M. Nusair, Can. J. Phys. 58, 1200 (1980). 28A. I. Liechtenstein, M. I. Katsnelson, V. P. Antropov, and V. A. Gubanov, J. Magnetism and Magnetic Materials 67, 65 (1987). 29P. R. Peduto, S. Frota-Pessa and M. S. Methfessel, Phys. Rev. B44, 13283 (1991). 30S. Frota-Pessˆ oa, Phys. Rev. B 46, 14570 (1992). 31R. Haydock, Solid State Physics, edited by H. Ehrenreich, F. Seitz and D. Turnbull, (Academic Press, New York, 1980), vol. 35. 32O. K. Andersen, Phys. Rev. B 12, 3060 (1975). 33O. K. Andersen, O. Jepsen and D. Gltzel, Highlights of Conden sed Matter Theory, edited by F. Bassani, F. Fumi, and M. P. Tosi, (North Holland, Amsterda m, 1985). 34U. von Barth and L. A. Hedin, J. Phys. C. 5, 1629 (1972). 35O. Eriksson, B. Johansson, R. C. Albers, A. M. Boring, and M. S . S. Brooks, Phys. Rev. B 42, 2707 (1990). 36N. Beer and D. Pettifor, The Electronic Structure of Complex Systems, (Plenum Press, New York, 1984). 37M. Plihal and D. L. Mills, Phys. Rev. B 58, 14407 (1998). 38M. Heide, G. Bihlmayer and S. Blgel, Phys. Rev. B 78, 140403 (2008). 39Kh. Zakeri, Y. Zhang, J. Prokop, T. H. Chuang, N. Sakr, W. X. Ta ng and J. Kirschner, Phys. Rev. Lett. 104, 137203 (2010). 40B. M. Hall andD. L. Mills, Phys. Rev. B 34, 8318 (1986); BurlM. Hall, D. L. Mills, Mohammed H. Mohammed and L. L. Kesmodel, Phys. Rev. B 38, 5856 (1988). 41M. P. Gokhale, A. Ormeci and D. L. Mills, Phys. Rev. B 46, 8978 (1992). 42B. M. Hall, S. Y. Tong and D. L. Mills, Phys. Rev. Lett. 50, 1277 (1983). 43R. Urban, G. Woltersdorf, and B. Heinrich, Phys. Rev. Lett. 87, 217204 (2001). 44A. T. Costa, R. B. Muniz and D. L. Mills, Phys. Rev. B 73, 054426 (2006). 45A. Bergman, A. Taroni, L. Bergqvist, J. Hellsvik, B. Hrvarss on, and O. Eriksson, Phys. Rev. B 81, 144416 (2010). 28
1408.1838v3.Spin_Orbital_Order_Modified_by_Orbital_Dilution_in_Transition_Metal_Oxides__From_Spin_Defects_to_Frustrated_Spins_Polarizing_Host_Orbitals.pdf	Spin-Orbital Order Modied by Orbital Dilution in Transition Metal Oxides: From Spin Defects to Frustrated Spins Polarizing Host Orbitals Wojciech Brzezicki,1, 2Andrzej M. Ole s,3, 1and Mario Cuoco2 1Marian Smoluchowski Institute of Physics, Jagiellonian University, prof. S. Lojasiewicza 11, PL-30348 Krak ow, Poland 2CNR-SPIN, IT-84084 Fisciano (SA), Italy, and Dipartimento di Fisica \E. R. Caianiello", Universit a di Salerno, IT-84084 Fisciano (SA), Italy 3Max-Planck-Institut f ur Festk orperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany (Dated: 24 December 2014) We investigate the changes in spin and orbital patterns induced by magnetic transition metal ions without an orbital degree of freedom doped in a strongly correlated insulator with spin-orbital order. In this context we study the 3 dion substitution in 4 dtransition metal oxides in the case of 3d3doping at either 3 d2or 4d4sites which realizes orbital dilution in a Mott insulator. Although we concentrate on this doping case as it is known experimentally and more challenging than other oxides due to nite spin-orbit coupling, the conclusions are more general. We derive the eective 3 d4d(or 3d3d) superexchange in a Mott insulator with dierent ionic valencies, underlining the emerging structure of the spin-orbital coupling between the impurity and the host sites and demonstrate that it is qualitatively dierent from that encountered in the host itself. This derivation shows that the interaction between the host and the impurity depends in a crucial way on the type of doubly occupied t2gorbital. One nds that in some cases, due to the quench of the orbital degree of freedom at the 3 dimpurity, the spin and orbital order within the host is drastically modied by doping. The impurity acts either as a spin defect accompanied by an orbital vacancy in the spin- orbital structure when the host-impurity coupling is weak, or it favors doubly occupied active orbitals (orbital polarons) along the 3 d4dbond leading to antiferromagnetic or ferromagnetic spin coupling. This competition between dierent magnetic couplings leads to quite dierent ground states. In particular, for the case of a nite and periodic 3 datom substitution, it leads to striped patterns either with alternating ferromagnetic/antiferromagnetic domains or with islands of saturated ferromagnetic order. We nd that magnetic frustration and spin degeneracy can be lifted by the quantum orbital ips of the host but they are robust in special regions of the incommensurate phase diagram. Orbital quantum uctuations modify quantitatively spin-orbital order imposed by superexchange. In contrast, the spin-orbit coupling can lead to anisotropic spin and orbital patterns along the symmetry directions and cause a radical modication of the order imposed by the spin-orbital superexchange. Our ndings are expected to be of importance for future theoretical understanding of experimental results for 4 dtransition metal oxides doped with 3 d3ions. We suggest how the local or global changes of the spin-orbital order induced by such impurities could be detected experimentally. PACS numbers: 75.25.Dk, 03.65.Ud, 64.70.Tg, 75.30.Et I. INTRODUCTION The studies of strongly correlated electrons in transi- tion metal oxides (TMOs) focus traditionally on 3 dma- terials [1], mainly because of high-temperature super- conductivity discovered in cuprates and more recently in iron-pnictides, and because of colossal magnetoresis- tance manganites. The competition of dierent and com- plex types of order is ubiquitous in strongly correlated TMOs mainly due to coupled spin-charge-orbital where frustrated exchange competes with the kinetic energy of charge carriers. The best known example is spin-charge competition in cuprates, where spin, charge and super- conducting orders intertwine [2] and stripe order emerges in the normal phase as a compromise between the mag- netic and kinetic energy [3, 4]. Remarkable evolution of the stripe order under increasing doping is observed [5] and could be reproduced by the theory based on the extended Hubbard model [6]. Hole doping in cuprates corresponds to the removal of the spin degree of free-dom. Similarly, hole doping in a simplest system with the orbital order in d1conguration removes locally orbital degrees of freedom and generates stripe phases which in- volve orbital polarons [7]. It was predicted recently that orbital domain walls in bilayer manganites should be par- tially charged as a result of competition between orbital- induced strain and Coulomb repulsion [8], which opens a new route towards charge-orbital physics in TMOs. We will show below that the stripe-like order may also occur in doped spin-orbital systems. These systems are very challenging and their doping leads to very complex and yet unexplored spin-orbital-charge phenomena [9]. A prerequisite to the phenomena with spin-orbital- charge coupled degrees of freedom is the understanding of undoped systems [10], where the low-energy physics and spin-orbital order are dictated by eective spin- orbital superexchange [11{13] and compete with spin- orbital quantum uctuations [14{16]. Although ordered states occur in many cases, the most intriguing are quan- tum phases such as spin [17] or orbital [18] liquids. Re-arXiv:1408.1838v3 [cond-mat.str-el] 30 Jan 20152 cent experiments on a copper oxide Ba 3CuSb 2O9[19, 20] have triggered renewed eorts in a fundamental search for a quantum spin-orbital liquid [21{24], where spin-orbital order is absent and electron spins are randomly choosing orbitals which they occupy. A signature of strong quan- tum eects in a spin-orbital system is a disordered state which persists down to very low temperatures. A good example of such a disordered spin-orbital liquid state is as well FeSc 2S4which does not order in spite of nite Curie-Weiss temperature CW=45 K [25], but shows instead signatures of quantum criticality [26, 27]. Spin-orbital interactions may be even more challenging \| for instance previous attempts to nd a spin-orbital liquid in the Kugel-Khomskii model [14] or in LiNiO 2[28] turned out to be unsuccessful. In fact, in the former case certain types of exotic spin order arise as a consequence of frustrated and entangled spin-orbital interactions [29, 30], and a spin-orbital entangled resonating valence bond state was recently shown to be a quantum superposition of strped spin-singlet covering on a square lattice [31]. In contrast, spin and orbital superexchange have dierent energy scales and orbital interactions in LiNiO 2are much stronger and dominated by frustration [32]. Hence the reasons behind the absence of magnetic long range order are more subtle [33]. In all these cases orbital uctuations play a prominent role and spin-orbital entanglement [34] determines the ground state. The role of charge carriers in spin-orbital systems is under very active investigation at present. In doped La1x(Sr,Ca)xMnO 3manganites several dierent types of magnetic order compete with one another and occur at increasing hole doping [35{37]. Undoped LaMnO 3 is an antiferromagnetic (AF) Mott insulator, with large S= 2 spins for 3 d4ionic congurations of Mn3+ions stabilized by Hund's exchange, coupled via the spin- orbital superexchange due to egandt2gelectron exci- tations [38]. The orbital egdegree of freedom is re- moved by hole doping when Mn3+ions are generated, and this requires careful modeling in the theory that takes into account both 3 d4and 3d3electronic cong- urations of Mn3+and Mn4+ions [39{44]. In fact, the orbital order changes radically with increasing doping in La 1x(Sr,Ca)xMnO 3systems at the magnetic phase transitions between dierent types of magnetic order [37], as weel as at La 0:7Ca0:3MnO 3/BiFeO 3heterostructures, where it oers a new route to enhancing multiferroic func- tionality [45]. The double exchange mechanism [46] trig- gers ferromagnetic (FM) metallic phase at sucient dop- ing; in this phase the spin and orbital degrees of freedom decouple and spin excitations are explained by the or- bital liquid [47, 48]. Due to distinct magnetic and kinetic energy scales, even low doping may suce for a drastic change in the magnetic order, as observed in electron- doped manganites [49]. A rather unique example of a spin-orbital system with strongly uctuating orbitals, as predicted in the theory [50{52] and seen experimentally [53{55], are the per- ovskite vanadates with competing spin-orbital order [56].In theset2gsystemsxyorbitals are lled by one electron and orbital order of active fyz;zxgorbitals is strongly in uenced by doping with Ca (Sr) ions which replace Y (La) ones in YVO 3(LaVO 3). In this case nite spin- orbit coupling modies the spin-orbital phase diagram [57]. In addition, the AF order switches easily from the G-type AF (G-AF) toC-type AF (C-AF) order in the presence of charge defects in Y 1xCaxVO3. Already at lowx'0:02 doping the spin-orbital order changes and spectral weight is generated within the Mott-Hubbard gap [58]. Although one might imagine that the orbital degree of freedom is thereby removed, a closer inspec- tion shows that this is not the case as the orbitals are polarized by charge defects [59] and readjust near them [60]. Removing the orbital degree of freedom in vana- dates would be only possible by electron doping generat- ing instead d3ionic congurations, but such a doping by charge defects would be very dierent from the doping by transition metal ions of the same valence considered below. Also in 4dmaterials spin-orbital physics plays a role [61], as for instance in Ca 2xSrxRuO 4systems with Ru4+ ions in 4d4conguration [62{66]. Recently it has been shown that unconventional magnetism is possible for Ru4+and similar ions where spin-orbit coupling plaus a role [67, 68]. Surprisingly, these systems are not similar to manganites but to vanadates where one nds as well ions with active t2gorbitals. In the case of ruthenates thet4 2gRu4+ions have low S= 1 spin as the splitting between the t2gandeglevels is large. Thus the undoped Ca2RuO 4is a hole analogue of a vanadate [50, 51], with t2gorbital degree of freedom and S= 1 spin per site in both cases. This gives new opportunities to investigate spin-orbital entangled states in t2gsystem, observed re- cently by angle resolved photoemission [69]. Here we focus on a novel and very dierent doping from all those considered above, namely on a substitu- tional doping by other magnetic ions in a plane built by transition metal and oxygen ions, for instance in the (a;b) plane of a monolayer or in perovskite ruthenates or vanadates. In this study we are interested primarily in doping of a TMO with t2gorbital degrees of freedom, where doped magnetic ions have no orbital degree of free- dom and realize orbital dilution . In addition, we deal with the simpler case of 3 ddoped ions where we can neglect spin-orbit interaction which should not be ignored for 4 d ions. We emphasize that in contrast to manganites where holes within egorbitals participate in transport and are responsible for the colossal magnetoresitance, such doped hole are immobile due to the ionic potential at 3 dsites and form defects in spin-orbital order of a Mott insulator. We encounter here a dierent situation from the dilution eects in the 2D egorbital system considered so far [70] as we deel with magnetic ions at doped sites. It is chal- lenging to investigate how such impurities modify locally or globally spin-orbital order of the host. The doping which realizes this paradigm is by either Mn4+or Cr3+ions with large S= 3=2 spins stabilized by3 FIG. 1. (a) Schematic view of the orbital dilution when the 3d3ion with no orbital degree of freedom and spin S= 3=2 substitutes 4 d4one with spin S= 1 on a bond having specic spin and orbital character in the host (gray arrows). Spins are shown by red arrows and doubly occupied t2gorbitals (doublons) are shown by green symbols for aandcorbitals, respectively. (b) If an inactive orbital along the bond is re- moved by doping, the total spin exchange is AF. (c) On the contrary, active orbitals at the host site can lead to either FM (top) or AF (bottom) exchange coupling, depending on the energy levels mismatch and dierence in the Coulomb cou- plings between the impurity and the host. We show the case when the host site is unchanged in the doping process. Hund's exchange, and orbital dilution occurs either in a TMO with d2ionic conguration as in the vanadium per- ovskites, or in 4 dMott insulators as in ruthenates. It has been shown that dilute Cr doping for Ru reduces the tem- perature of the orthorhombic distortion and induces FM behavior in Ca 2Ru1xCrxO4(with 0<x< 0:13) [71]. It also induces surprising negative volume thermal expan- sion via spin-orbital order. Such defects, on one hand, can weaken the spin-orbital coupling in the host, but on the other hand, may open a new channel of interaction between the spin and orbital degree of freedom through the host-impurity exchange, see Fig. 1. Consequences of such doping are yet unexplored and are expected to open a new route in the research on strongly correlated oxides. The physical example for the present theory are the in- sulating phases of 3 d4dhybrid structures, where doping happens at d4transition metal sites, and the value of the spin is locally changed from S= 1 toS= 3=2. As a demonstration of the highly nontrivial physics emerging abFIG. 2. Schematic view of C-AF spin order coexisting with G-AO orbital order in the ( a;b) plane of an undoped Mott insulator with 4 d4ionic congurations. Spins are shown by arrows while doubly occupied xyandyzorbitals (canda doublons, see text) form a checkerboard pattern. Equivalent spin-orbital order is realized for V3+ions in (b;c) planes of LaVO 3[56], with orbitals standing for empty orbitals (holes). in 3d4doxides, remarkable eects have already been observed, for instance, when Ru ions are replaced by Mn, Ti, Cr or other 3 delements. The role of Mn doping in the SrRuO Ruddlesden-Popper series is strongly linked to the dimensionality through the number nof RuO 2 layers in the unit cell. The Mn doping of the SrRuO 3 cubic member drives the system from the itinerant FM state to an insulating AF conguration in a continuous way via a possible unconventional quantum phase tran- sition [72]. Doping by Mn ions in Sr 3Ru2O7leads to a metal-to-insulator transition and AF long-range order for more than 5% Mn concentration [73]. Subtle orbital rearrangement can occur at the Mn site, as for instance the inversion of the crystal eld in the egsector observed via x-ray absorption spectroscopy [74]. Neutron scatter- ing studies indicate the occurrence of an unusual E-type antiferromagnetism in doped systems (planar order with FM zigzag chains with AF order between them) with mo- ments aligned along the caxis within a single bilayer [75]. Furthermore, the more extended 4 dorbitals would a priori suggest a weaker correlation than in 3 dTMOs due to a reduced ratio between the intraatomic Coulomb in- teraction and the electron bandwidth. Nevertheless, the (eective)d-bandwidth is reduced by the changes in the 3d-2p-3dbond angles in distorted structures which typi- cally arise in these materials. This brings these systems on the verge of a metal-insulator transition [76], or even into the Mott insulating state with spin-orbital order, see Fig. 2. Hence, not only 4 dmaterials share common fea- tures with 3 dsystems, but are also richer due to their sen- sitivity to the lattice structure and to relativistic eects due to larger spin-orbit [77] or other magneto-crystalline couplings. To simplify the analysis we assume that onsite Coulomb interactions are so strong that charge degrees of freedom are projected out, and only virtual charge trans- fer can occur between 3 dand 4dions via the oxygen lig-4 ands. For convenience, we dene the orbital degree of freedom as a doublon (double occupancy) in the t4 2gcon- guration. The above 3 ddoping leads then eectively to the removal of a doublon in one of t2gorbitals which we label asfa;b;cg(this notation is introduced in Ref. [16] and explained below) and to replacing it by a t3 2gion. To our knowledge, this is the only example of remov- ing the orbital degree of freedom in t2gmanifold realized so far and below we investigate possible consequences of this phenomenon. Another possibility of orbital dilution which awaits experimental realization would occur when at2gdegree of freedom is removed by replacing a d2ion by ad3one, as for instance by Cr3+doping in a vanadate \| here a doublon is an empty t2gorbital, i.e., lled by two holes. Before presenting the details of the quantitative anal- ysis, let us concentrate of the main idea of the superex- change modied by doping in a spin-orbital system. The d3ions have singly occupied all three t2gorbitals and S= 3=2 spins due to Hund's exchange. While a pair ofd3ions, e.g. in SrMnO 3, is coupled by AF superex- change [48], the superexchange for the d3d4bond has a rather rich structure and may also be FM. The spin exchange depends then on whether the orbital degree of freedom is active and participates in charge excitations along a considered bond or electrons of the doublon can- not move along this bond due to the symmetry of t2g orbital, as explained in Fig. 1. This qualitative dier- ence to systems without active orbital degrees of freedom is investigated in detail in Sec. II. The main outcomes of our analysis are: (i) the de- termination of the eective spin-orbital exchange Hamil- tonian describing the low-energy sector for the 3 d4d hybrid structure, (ii) establishing that a 3 d3impurity without an orbital degree of freedom modies the orbital order in the 4 d4host, (iii) providing the detailed way how the microscopic spin-orbital order within the 4 d4host is modied around the 3 d3impurity, and (iv) suggesting possible spin-orbital patterns that arise due to periodic and nite substitution (doping) of 4 datoms in the host by 3dones. The emerging physical scenario is that the 3dimpurity acts as an orbital vacancy when the host- impurity coupling is weak and as an orbital polarizer of the bonds active t2gdoublon congurations when it is strongly coupled to the host. The tendency to polarize the host orbitals around the impurity turns out to be ro- bust and independent of spin conguration. Otherwise, it is the resulting orbital arrangement around the impu- rity and the strength of Hund's coupling at the impurity that set the character of the host-impurity magnetic ex- change. The remaining of the paper is organized as follows. In Sec. II we introduce the eective model describing the spin-orbital superexchange at the 3 d4dbonds which serves to investigate the changes of spin and orbital or- der around individual impurities and at nite doping. We arrive at a rather general formulation which empha- sizes the impurity orbital degree of freedom, being a dou-blon, and present some technical details of the deriva- tion in Appendix A. The strategy we adopt is to analyze rst the ground state properties of a single 3 d3impu- rity surrounded by 4 d4atoms by investigating how the spin-orbital pattern in the host may be modied at the nearest neighbor (NN) sites to the 3 datom. This study is performed for dierent spin-orbital patterns of the 4 d host with special emphasis on the alternating FM chains (C-AF order) which coexist with G-type alternating or- bital (G-AO) order, see Fig. 2. We address the impurity problem within the classical approximation in Sec. III A. As explained in Sec. III B, there are two nonequivalent cases which depend on the precise modication of the or- bital order by the 3 dimpurity, doped either to replace a doublon in aorbital (Sec. III C) or the one in corbital (Sec. III D). Starting from the single impurity solution we next ad- dress periodic arrangements of 3 datoms at dierent con- centrations. We demonstrate that the spin-orbital or- der in the host can be radically changed by the presence of impurities, leading to striped patterns with alternat- ing FM/AF domains and islands of fully FM states. In Sec. IV A we consider the modications of spin-orbital order which arise at periodic doping with macroscopic concentration. Here we limit ourselves to two represen- tative cases: (i) commensurate x= 1=8 doping in Sec. IV B, and (ii) two doping levels x= 1=5 andx= 1=9 be- ing incommensurate with underlying two-sublattice order (Fig. 2) which implies simultaneous doping at two sub- lattices, i.e., at both aandcdoublon sites, as presented in Secs. IV C and IV D. Finally, in Sec. V A we inves- tigate the modications of the classical phase diagram induced by quantum uctuations, and in Secs. V B and V C we discuss representative results obtained for nite spin-orbit coupling (calculation details of the treatment of spin-orbit interaction are presented in Appendix B). The paper is concluded by a general discussion of possi- ble emerging scenarios for the 3 d3impurities in 4 d4host, a summary of the main results and perspective of future experimental investigations of orbital dilution in Sec. VI. II. THE SPIN-ORBITAL MODEL In this Section we consider a 3 dimpurity in a strongly correlated 4 dTMO and derive the eective 3 d34d4 spin-orbital superexchange. It follows from the coupling between 3dand 4dorbitals via oxygen 2 porbitals due to thepdhybridization. In a strongly correlated system it suces to concentrate on a pair of atoms forming a bondhiji, as the eective interactions are generated by charge excitations d4 id4 jd5 id3 jalong a single bond [12]. In the reference 4 dhost both atoms on the bond hijiare equivalent and one considers, H(i;j) =Ht(i;j) +Hint(i) +Hint(j): (1) The Coulomb interaction Hint(i) is local at site iand we describe it by the degenerate Hubbard model [80], see5 below. We implement a strict rule that the hopping within thet2gsector is allowed in a TMO only between two neighboring orbitals of the same symmetry which are ac- tive along the bond direction [15, 78, 79], and neglect the interorbital processes originating from the octahe- dral distortions such as rotation or tilting. Indeed, in ideal undistorted (perovskite or square lattice) geometry the orbital avor is conserved as long as the spin-orbit coupling may be neglected. The interorbital hopping ele- ments are smaller by at least one order of magnitude and may be treated as corrections in cases where distortions play a role to the overall scenario established below. The kinetic energy for a representative 3 d-2p-4dbond, i.e., after projecting out the oxygen degrees of freedom, is given by the hopping in the host /thbetween sites i andj, Ht(i;j) =thX ( ); dy idj+dy jdi :(2) Heredy iare the electron creation operators at site iin the spin-orbital state ( ). The bondhijipoints along one of the two crystallographic directions, =a;b, in the two-dimensional (2D) square lattice. Without distor- tions, only two out of three t2gorbitals are active along each bondh12iand contribute to Ht(i;j), while the third orbital lies in the plane perpendicular to the axis and thus the hopping via oxygen is forbidden by symmetry. This motivates a convenient notation as follows [15], jaijyzi;jbijxzi;jcijxyi; (3) with thet2gorbital inactive along a given direction 2 fa;b;cglabeled by the index . We consider a 2D square lattice with transition metal ions connected via oxygen orbitals as in a RuO 2(a;b) plane of Ca 2RuO 4(SrRuO 3). In this casejai(jbi) orbitals are active along the b(a) axis, whilejciorbitals are active along both a;baxes. To derive the superexchange in a Mott insulator, it is sucient to consider a bond which connects nearest neighbor sites,hijih 12i. Below we consider a bond between an impurity site i= 1 occupied by a 3 dion and a neighboring host 4 dion at sitej= 2. The Hamiltonian for this bond can be then expressed in the following form, H(1;2) =Ht(1;2) +Hint(1) +Hint(2) +Hion(2):(4) The total Hamiltonian contains the kinetic energy term Ht(1;2) describing the electron charge transfer via oxy- gen orbitals, the onsite interaction terms Hint(m) for the 3d(4d) ion at site m= 1;2, and the local potential of the 4datom,Hion(2), which takes into account the mismatch of the energy level structure between the two (4 dand 3d) atomic species and prevents valence uctuations when the host is doped, even in the absence of local Coulomb interaction. The kinetic energy in Eq. (4) is given by, Ht(1;2) =tX ( ); dy 1d2+dy 2d1 ;(5)wheredy mis the electron creation operator at site m= 1;2 in the spin-orbital state ( ). The bondh12ipoints along one of the two crystallographic directions, =a;b, and again the orbital avor is conserved [15, 78, 79]. The Coulomb interaction on an atom at site m= 1;2 depends on two parameters [80]: (i) intraorbital Coulomb repulsionUm, and (ii) Hund's exchange JH m. The label m stands for the ion and distinguishes between these terms at the 3dand 4dion, respectively. The interaction is expressed in the form, Hint(m) =UmX nm"nm#2JH mX <~Sm~Sm + Um5 2JH mX < 0nmnm0 +JH mX 6=dy m"dy m#dm#dm": (6) The terms standing in the rst line of Eq. (6) con- tribute to the magnetic instabilities in degenerate Hub- bard model [80] and decide about spin order, both in an itinerant system and in a Mott insulator. The remaining terms contribute to the multiplet structure and are of im- portance for the correct derivation of the superexchange which follows from charge excitations, see below. Finally, we include a local potential on the 4 datom which encodes the energy mismatch between the host and the impurity orbitals close to the Fermi level and prevents valence uctuations on the 4 dion due to the 3 d doping. This term has the following general structure, Hion(2) =Ie 2 4X ;n2!2 ; (7) with=a;b;c . The eective Hamiltonian for the low energy processes is derived from H(1;2) (4) by a second order expansion for charge excitations generated by Ht(1;2), and treating the remaining part of H(1;2) as an unperturbed Hamil- tonian. We are basically interested in virtual charge ex- citations in the manifold of degenerate ground states of a pair of 3dand 4datoms on a bond, see Fig. 3. These quantum states are labeled as ek 1 withk= 1;:::; 4 and fep 2gwithp= 1;:::; 9 and their number follows from the solution of the onsite quantum problem for the Hamilto- nianHint(i). For the 3 datom the relevant states can be classied according to the four components of the total spinS1= 3=2 for the 3dimpurity atom at site m= 1, three components of S2= 1 spin for the 4 dhost atom at sitem= 2 and for the three dierent positions of the double occupied orbital (doublon). Thus, the eective Hamiltonian will contain spin products ( ~S1~S2) between spin operators dened as, ~Sm=1 2X dy m ~ dm ; (8)6 3d atom 4d atomabc eI2site 1 site 2 FIG. 3. Schematic representation of one conguration be- longing to the manifold of 36 degenerate ground states for a representative 3 d4dbondh12ias given by the local Coulomb Hamiltonian Hint(m) (6) withm= 1;2. The dominant ex- change processes considered here are those that move one of the four electrons on the 4 datom to the 3 dneighbor and back. The stability of the 3 d3-4d4charge congurations is provided by the local potential energy Ie 2, see Eq. (7). form= 1;2 sites and the operator of the doublon posi- tion at site m= 2, D( ) 2= dy 2 "d2 " dy 2 #d2 # : (9) The doublon operator identies the orbital within the t2gmanifold of the 4 dion with a double occupancy (oc- cupied by the doublon) and stands in what follows for the orbital degree of freedom. It is worth noting that the hopping (5) does not change the orbital avor thus we expect that the resulting Hamiltonian is diagonal in the orbital degrees of freedom with only D( ) 2operators. Following the standard second order perturbation ex- pansion for spin-orbital systems [12], we can write the matrix elements of the low energy exchange Hamiltonian, H( ) J(i;j), for a bondh12ik along the axis as follows, ek 1;el 2H( ) J(1;2)ek0 1;el0 2 =X n1;n21 "n1+"n2 ek 1;el 2Ht(1;2)n1;n2 n1;n2Ht(1;2)ek0 1;el0 2 ;(10) with"nm=En;mE0;mbeing the excitation energies for atoms at site m= 1;2 with respect to the unperturbed ground state. The superexchange Hamiltonian H( ) J(1;2) for a bond along can be expressed in a matrix form by a 3636 matrix, with dependence on Um,JH m, and Ieelements. There are two types of charge excitations: (i)d3 1d4 2d4 1d3 2one which creates a doublon at the 3 d impurity, and (ii) d3 1d4 2d2 1d5 2one which adds another doublon at the 4 dhost site in the intermediate state. The second type of excitations involves more doubly oc- cupied orbitals and has much larger excitation energy. It is therefore only a small correction to the leading term (i), as we discuss in Appendix A. Similar as in the case of doped manganites [48], the dominant contribution to the eective low-energy spin- orbital Hamiltonian for the 3 d4dbond stems from thed3 1d4 2d4 1d3 2charge excitations, as they do not involve an extra double occupancy and the Coulomb energy U2. The 3d3 14d4 23d4 14d3 2charge excitations can be analyzed in a similar way as the 3 d3 i3d4 j3d4 i3d3 jones for anhiji bond in doped manganites [48]. In both cases the total number of doubly occupied orbitals does not change, so the main contributions come due to Hund's exchange. In the present case, one more parameter plays a role, =Ie+ 3(U1U2)4(JH 1JH 2); (11) which stands for the mismatch potential energy (7) renor- malized by the onsite Coulomb interactions fUmgand by Hund's exchange fJH mg. On a general ground we expect to be a positive quantity, since the repulsion Umshould be larger for smaller 3 dshells than for the 4 dones and Umis the largest energy scale in the problem. Let us have a closer view on this dominant contribu- tion of the eective low-energy spin-orbital Hamiltonian for the 3d4dbond, given by Eq. (A2). For the anal- ysis performed below and the clarity of our presentation it is convenient to introduce some scaled parameters re- lated to the interactions within the host and between the host and the impurity. For this purpose we employ the exchange couplings JimpandJhost, Jimp=t2 4; (12) Jhost=4t2 h U2; (13) which follow from the virtual charge excitations gener- ated by the kinetic energy, see Eqs. (2) and (5). We use their ratio to investigate the in uence of the impurity on the spin-orbital order in the host. Here this the hopping amplitude between two t2gorbitals at NN 4 datoms,JH 2 andU2refer to the host, and (11) is the renormalized ionization energy of the 3 d4dbonds. The results de- pend as well on Hund's exchange element for the impurity and on the one at host atoms, imp=JH 1 ; (14) host=JH 2 U2; (15) Note that the ratio introduced for the impurity, imp(14), has here a dierent meaning from Hund's exchange used here for the host, host(15), which cannot be too large by construction, i.e., host<1=3. With the parametrization introduced above, the dom- inant term in the impurity-host Hamiltonian for the im- purity spin ~Siinteracting with the neighboring host spins f~Sjgatj2N(i), deduced fromH( ) 3d4d(1;2) Eq. (A2), can be written in a rather compact form as follows H3d4d(i)'X ;j2N(i)n JS(D( ) j)(~Si~Sj) +EDD( ) jo ; (16)7 0.0 0.2 0.4 0.6 0.8 1.0 ηimp-0.20.00.20.40.6JS/Jimp, ED/JimpED JS(Dγ= 1) JS(Dγ= 0)22 FIG. 4. Evolution of the spin exchange JS(D( ) 2) and the doublon energy ED, both given in Eq. (16) for increasing Hund's exchange impat the impurity. where the orbital (doublon) dependent spin couplings JS(D( ) j) and the doublon energy EDdepend on imp. The evolution of the exchange couplings are shown in Fig. 4. We note that the dominant energy scale is E D, so for a single 3 d4dbond the doublon will avoid occupy- ing the inactive ( ) orbital and the spins will couple with JS(D( ) j= 0) which can be either AF if imp.0:43 or FM ifimp>0:43. Thus the spins at imp=c imp'0:43 will decouple according to the H( ) J(i;j) exchange. Let us conclude this Section by writing the complete superexchange Hamiltonian, H=H3d4d+H4d4d+Hso; (17) whereH3d4dP iH3d4d(i) includes all the 3 d4d bonds around impurities, H4d4dstands for the the ef- fective spin-orbital Hamiltonian for the 4 dhost bonds, andHsois the spin-orbit interaction in the host. The former term we explain below, while the latter one is dened in Sec. V B, where we analyze the quantum cor- rections and the consequences of spin-orbit interaction. The superexchange in the host for the bonds hijialong the =a;baxes [81], H4d4d=JhostX hijik n J( ) ij(~Si~Sj+ 1) +K( ) ijo ;(18) depends on J( ) ijandK( ) ijoperators acting only in the orbital space. They are expressed in terms of the pseu- dospin operators dened in the orbital subspace spanned by the two orbital avors active along a given direction , i.e., J( ) ij=1 2(2r1+ 1) ~ i~ j( )1 2r2 z iz j( ) +1 8 ninj( )(2r1r2+1)1 4r1 ni+nj( );(19)K( ) ij=r1(~ i~ j)( )+r2 z iz j( )+1 4(r1+r2) ninj( ) 1 4(r1+ 1) ni+nj( ): (20) with r1=host 13host; r 2=host 1 + 2host; (21) standing for the multiplet structure in charge excitations, and the orbital operators f~ ( ) i;n( ) igthat for the =c axis take the form: ~ (c) i=1 2 ay iby i ~ aibi\|; (22) n(c) i=ay iai+by ibi: (23) For the directions =a;bin the considered ( a;b) plane one nds equivalent expressions by cyclic permutation of the axis labelsfa;b;cgin the above formulas. This prob- lem is isomorphic with the spin-orbital superexchange in the vanadium perovskites [50, 51], where a hole in the fa;bgdoublet plays an equivalent role to the doublon in the present case. The operators fay i;by i;cy igare the dou- blon (hard core boson) creation operators in the orbital =a;b;c , respectively, and they satisfy the local con- straint, ay iai+by ibi+cy ici= 1; (24) meaning that exactly onedoublon (9) occupies one of the threet2gorbitals at each site i. These bosonic occu- pation operators coincide with the previously used dou- blon occupation operators D( ) j, i.e.,D( ) j= y j jwith =a;b;c . Below we follow rst the classical procedure to determine the ground states of single impurities in Sec. III, and at macroscopic doping in Sec IV. III. SINGLE 3d IMPURITY IN 4d HOST A. Classical treatment of the impurity problem In this Section we describe the methodology that we applied for the determination of the phase diagrams for a single impurity reported below in Secs. III C, and next at macroscopic doping, as presented in Sec. IV. Let us con- sider rst the case of a single 3 dimpurity in the 4 dhost. Since the interactions in the model Hamiltonian are only eective ones between NN sites, it is sucient to study the modication of the spin-orbital order around the im- purity for a given spin-orbital conguration of the host by investigating a cluster of L= 13 sites shown in Fig. 5. We assume the C-AF spin order (FM chains coupled an- tiferromagnetically) accompanied by G-AO order within the host which is the spin-orbital order occurring for the realistic parameters of a RuO 2plane [81], see Fig. 2. Such a spin-orbital pattern turns out to be the most relevant one when considering the competition between the host8 FIG. 5. Schematic top view of the cluster used to obtain the phase diagrams of the 3 dimpurity within the 4 dhost in an (a;b) plane. The impurity is at the central site i= 0 which belongs to the corbital host sublattice. For the outer sites in this cluster the spin-orbital conguration is xed and determined by the undoped 4 dhost (with spins and cor- bitals shown here) having C-AF/G-AO order, see Fig. 2. For the central i= 0 site the spin state and for the host sites i= 1;:::; 4 the spin-orbital congurations are determined by minimizing the energy of the cluster. and the impurity as due to the AO order within the ( a;b) plane. Other possible congurations with uniform orbital order and AF spin pattern, e.g. G-AF order, will also be considered in the discussion throughout the manuscript. The sitesi= 1;2;3;4 inside the cluster in Fig. 5 have active spin and orbital degrees of freedom while the im- purity at site i= 0 has only spin degree of freedom. At the remaining sites the spin-orbital conguration is as- signed, following the order in the host, and it does not change along the computation. To determine the ground state we assume that the spin-orbital degrees of freedom are treated as classical variables. This implies that for the bonds between atoms in the host we use the Hamiltonian (18) and neglect quan- tum uctuations, i.e., in the spin sector we keep only the zth (Ising) spin components and in the orbital one only the terms which are proportional to the doublon occupa- tion numbers (9) and to the identity operators. Similarly, for the impurity-host bonds we use the Hamiltonian Eq. (16) by keeping only the zth projections of spin operators. Since we do neglect the uctuation of the spin amplitude it is enough to consider only the maximal and minimal values ofhSz iifor spinS= 3=2 at the impurity sites and S= 1 at the host atoms. With these assumptions we can construct all the possible congurations by varying the spin and orbital congurations at the sites from i= 1 toi= 4 in the cluster shown in Fig. 5. Note that the outer ions in the cluster belong all to the same sublattice, so two distinct cases have to be considered to probe all the congurations. Since physically it is unlikely that asingle impurity will change the orbital order of the host globally thus we will not compare the energies from these two cases and analyze two classes of solutions separately, see Sec. III B. Then, the lowest energy conguration in each class provides the optimal spin-orbital pattern for the NNs around the 3 dimpurity. In the case of degener- ate classical states, the spin-orbital order is established by including quantum uctuations. In the case of a periodic doping analyzed in Sec. IV, we use a similar strategy in the computation. Taking the most general formulation, we employ larger clusters having both size and shape that depend on the impurity distribution and on the spin-orbital order in the host. For this purpose, the most natural choice is to search for the minimum energy conguration in the elementary unit cell that can reproduce the full lattice by a suitable choice of the translation vectors. This is computation- ally expensive but doable for a periodic distribution of the impurities that is commensurate to the lattice be- cause it yields a unit cell of relatively small size for dop- ing around x= 0:1. Otherwise, for the incommensurate doping the size of the unit cell can lead to a conguration space of a dimension that impedes nding of the ground state. This problem is computationally more demanding and to avoid the comparison of all the energy congu- rations, we have employed the Metropolis algorithm at low temperature to achieve the optimal conguration it- eratively along the convergence process. Note that this approach is fully classical, meaning that the spins of the host and impurity are treated as Ising variables and the orbital uctuations in the host's Hamiltonian Eq. (18) are omitted. They will be addressed in Sec. V A. B. Two nonequivalent 3ddoping cases The single impurity problem is the key case to start with because it shows how the short-range spin-orbital correlations are modied around the 3 datom due to the host and host-impurity interactions in Eq. (17). The analysis is performed by xing the strength between Hund's exchange and Coulomb interaction within the host (6) at host= 0:1, and by allowing for a variation of the ratio between the host-impurity interaction (16) and the Coulomb coupling at the impurity site. The choice ofhost= 0:1 is made here because this value is within the physically relevant range for the case of the ruthe- nium materials. Small variations of hostdo not aect the obtained results qualitatively. As we have already discussed in the model derivation, the sign of the magnetic exchange between the impurity and the host depends on the orbital occupation of the 4ddoublon around the 3 dimpurity. The main aspect that controls the resulting magnetic conguration is then given by the character of the doublon orbitals around the impurity, depending on whether they are active or inactive along the considered 3 d4dbond. To explore such a competition quantitatively we investigate G-AO9 order for the host with alternation of aandcdoublon congurations accompanied by the C-AF spin pattern, see Fig. 2. Note that the aorbitals are active only along thebaxis, while the corbitals are active along the both axes:aandb[79]. This state has the lowest energy for the host in a wide range of parameters for Hund's exchange, Coulomb element and crystal-eld potential [81]. Due to the specic orbital pattern of Fig. 2, the 3 dim- purity can substitute one of two distinct 4 dsites which are considered separately below, either with aor withc orbital occupied by the doublon. Since the two 4 datoms have nonequivalent surrounding orbitals, not always ac- tive along the 3 d4dbond, we expect that the result- ing ground state will have a modied spin-orbital order. Indeed, if the 3 datom replaces the 4 done with the dou- blon in the aorbital, then all the 4 dneighboring sites have active doublons along the connecting 3 d4dbonds because they are in the corbitals. On the contrary, the substitution at the 4 dsite withcorbital doublon con- guration leads to an impurity state with its neighbors having both active and inactive doublons. Therefore, we do expect a more intricate competition for the latter case of an impurity occupying the 4 dsite withcorbital con- guration. Indeed, this leads to frustrated host-impurity interactions, as we show in Sec. III D. C. Doping removing a doublon in aorbital We start by considering the physical situation where the 3dimpurity replaces a 4 dion with the doublon within theaorbital. The ground state phase diagram and the schematic view of the spin-orbital pattern are reported in Fig. 6 in terms of the ratio Jimp=Jhost(14) and the strength of Hund's exchange coupling imp(12) at the 3 d site. There are three dierent ground states that appear in the phase diagram. Taking into account the struc- ture of the 3 d4dspin-orbital exchange (16) we expect that, in the regime where the host-impurity interaction is greater than that in the host, the 4 dneighbors to the impurity tend to favor the spin-orbital conguration set by the 3d4dexchange. In this case, since the orbitals surrounding the 3 dsite already minimize the 3 d4d Hamiltonian, we expect that the optimal spin congura- tion corresponds to the 4 dspins aligned either antifer- romagnetically or ferromagnetically with respect to the impurity 3dspin. The neighbor spins are AF to the 3 dspin impurity in the AFaphase, while the FM aphase is just obtained from AFaby reversing the spin at the impurity, and hav- ing all the 3 d4dbonds FM. It is interesting to note that due to the host-impurity interaction the C-AF spin pat- tern of the host is modied in both the AF aand the FMaground states. Another intermediate conguration which emerges when the host-impurity exchange is weak in the intermediate FS aphase where the impurity spin is undetermined and its conguration in the initial C- AF phase is degenerate with the one obtained after the 0.00.20.40.60.81.0ηimp 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Jimp/JhostAFaFMa AFa FSa abFSa ?FIG. 6. Top \| Phase diagram of the 3 dimpurity in the ( a;b) plane with the C-AF/G-AO order in 4 dhost for the impurity doped at the sublattice with an a-orbital doublon. Dierent colors refer to local spin order around the impurity, AF and FM, while FS indicates the intermediate regime of frustrated impurity spin. Bottom \| Schematic view of spin-orbital pat- terns for the ground state congurations shown in the top panel. The 3 datom is at the central site, the dotted frame highlights the 4 dsites where the impurity induces a a spin reversal. In the FS aphase the question mark stands for that the frustrated impurity spin within the classical approach but frustration is released by the quantum uctuations of the NN corbitals in the adirection resulting in small AF couplings along theaaxis, and spins obey the C-AF order (small ar- row). The labels FM aand AFarefer to the local spin order around the 3 dimpurity site with respect to the host \| these states dier by spin inversion at the 3 datom site. spin-inversion operation. This is a singular physical sit- uation because the impurity does not select a specic direction even if the surrounding host has a given spin- orbital conguration. Such a degeneracy is clearly veri- ed at the critical point c imp'0:43 where the amplitude of the 3d4dcoupling vanishes when the doublon occu- pies the active orbital. Interestingly, such a degenerate conguration is also obtained at Jimp=Jhost<1 when the host dominates and the spin conguration at the 4 dsites around the impurity are basically determined by Jhost. In this case, due to the C-AF spin order, always two bonds are FM and other two have AF order, independently of the spin orientation at the 3 dimpurity. This implies that10 orbital fliporbital flip0(a) (b) FIG. 7. Schematic view of the two types of orbital bonds found in the 4 dhost: (a) an active bond with respect to orbital ips, ( + i j+H:c:), and (b) an inactive bond, where orbital uctuations are blocked by the orbital symmetry \| here the orbitals are static and only Ising terms contribute to the ground state energy. both FM or AF couplings along the 3 d4dbonds per- fectly balance each other which results in the degenerate FSaphase. It is worth pointing out that there is a quite large re- gion of the phase diagram where the FS astate is stabi- lized and the spin-orbital order of the host is not aected by doping with the possibility of having large degeneracy in the spin conguration of the impurities. On the other hand, by inspecting the corbitals around the impurity (Fig. 6) from the point of view of the full host's Hamilto- nian Eq. (18) with orbital ips included, ( + i j+H:c:), one can easily nd out that the frustration of the impu- rity spin can be released by quantum orbital uctuations. Note that the corbitals around the impurity in the a (b) direction have quite dierent surroundings. The ones along theaaxis are connected by two active bonds along thebaxis with orbitals a, as in Fig. 7(a), while the ones alongbare connected with only one activeaorbital along the same baxis. This means that in the perturba- tive expansion the orbital ips will contribute only along thebbonds (for the present G-AO order) and admix the aorbital character to corbitals along them, while such processes will be blocked for the bonds along the aaxis, as also forborbitals along the baxis, see Fig. 7(b). This fundamental dierence can be easily included in the host-impurity bond in the mean-eld manner by set- tinghDib; i= 0 for the bonds along the baxis and 0<hDia; i<1 for the bonds along the aaxis. Then one can easily check that for the impurity spin point- ing downwards we get the energy contribution from the spin-spin bonds which is given by E#=(host)hDia; i, and for the impurity spin pointing upwards we have E"=(host)hDia; i, with(host)>0. Thus, it is clear that any admixture of the virtual orbital ips in the host's wave function polarize the impurity spin upwards so that the C-AF order of the host will be restored. D. Doping removing a doublon in corbital Let us move to the case of the 3 datom replacing the doublon at corbital. As anticipated above, this congu- ration is more intricate because the orbitals surroundingthe impurity, as originated by the C-AF/AO order within the host, lead to nonequivalent 3 d4dbonds. There are two bonds with the doublon occupying an inactive or- bital (and has no hybridization with the t2gorbitals at 3d atom), and two remaining bonds with doublons in activet2gorbitals. Since the 3d4dspin-orbital exchange depends on the orbital polarization of 4 dsites we do expect a competition which may modify signicantly the spin-orbital correla- tions in the host. Indeed, one observes that three cong- urations compete, denoted as AF1 c, AF2cand FMc, see Fig. 8. In the regime where the host-impurity exchange dominates the system tends to minimize the energy due to the 3d4dspin-orbital coupling and, thus, the orbitals become polarized in the active congurations compatible with theC-AF/G-AO pattern and the host-impurity spin coupling is AF for imp0:43, while it is FM otherwise. This region resembles orbital polarons in doped mangan- ites [39, 42]. Also in this case, the orbital polarons arise because they minimize the double exchange energy [46]. On the contrary, for weak spin-orbital coupling be- tween the impurity and the host there is an interesting cooperation between the 3 dand 4datoms. Since the strength of the impurity-host coupling is not sucient to polarize the orbitals at the 4 dsites, it is preferable to have an orbital rearrangement to the conguration with inactive orbitals on 3 d4dbonds and spin ips at 4 d sites. In this way the spin-orbital exchange is optimized in the host and also on the 3 d4dbonds. The resulting state has an AF coupling between the host and the im- purity as it should when all the orbitals surrounding the 3datoms are inactive with respect to the bond direction. This modication of the orbital conguration induces the change in spin orientation. The double exchange bonds (with inactive doublon orbitals) along the baxis are then blocked and the total energy is lowered, in spite of the frustrated spin-orbital exchange in the host. As a result, the AF1cstate the spins surrounding the impurity are aligned and antiparallel to the spin at the 3 dsite. Concerning the host C-AF/G-AO order we note that it is modied only along the direction where the FM cor- relations develop and spin defects occur within the chain doped by the 3 datom. The FM order is locally disturbed by the 3ddefect antiferromagnetically coupled spins sur- rounding it. Note that this phase is driven by the or- bital vacancy as the host develops more favorable orbital bonds to gain the energy in the absence of the orbital degrees of freedom at the impurity. At the same time the impurity-host bonds do not generate too big energy losses as: (i) either impis so small that the loss due to EDis compensated by the gain from the superexchange /JS(D( ) j= 1) (all these bonds are AF), see Fig. 4, or (ii)Jimp=Jhostis small meaning that the overall energy scale of the impurity-host exchange remains small. Inter- estingly, if we compare the AF1 cwith the AF2 cground states we observe that the disruption of the C-AF/G- AO order is anisotropic and occurs either along the FM chains in the AF1 cphase or perpendicular to the FM11 0.00.20.40.60.81.0ηimp 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Jimp/JhostAF1cFMc AF2c AF1c AF2c ab FIG. 8. Top \| Phase diagram of the 3 dimpurity in the 4dhost withC-AF/G-AO order and the impurity doped at thecdoublon sublattice. Dierent colors refer to local spin order around the impurity: AF c1, AFc2, and FM. Bottom \| Schematic view of spin-orbital patterns for the two AF ground state congurations shown in the top panel; the FM cphase diers from the AF c2 one only by spin inversion at the 3 d atom. The 3 datom is at the central site and has no doublon orbital, the frames highlight the spin-orbital defects caused by the impurity. As in Fig. 6, the labels AF and FM refer to the impurity spin orientation with respect to the neighboring 4dsites. chains in the AF2 cphase. No spin frustration is found here, in contrast to the FS aphase in the case of adoublon doping, see Fig. 6. Finally, we point out that a very similar phase diagram can be obtained assuming that the host has the FM/ G- AO order with aandborbitals alternating from site to site. Such conguration can be stabilized by a distortion that favors the out-of-plane orbitals. In this case there is no dierence in doping at one or the other sublattice. The main dierence is found in energy scales \| for the G-AF/C-AO order the diagram is similar to the one of Fig. 8 if we rescale Jimpby half, which means that the G-AF order is softer than the C-AF one. Note also that in the peculiar AF1 cphase the impurity does not induce any changes in the host for the FM/AO ordered host. Thus we can safely conclude that the observed change in the orbital order for the C-AF host in the AF1 cphase is due to the presence of the corbitals which are notdirectional in the ( a;b) plane. Summarizing, we have shown the complexity of local spin-orbital order around t3 2gimpurities in a 4 t4 2ghost. It is remarkable that such impurity spins not only modify the spin-orbital order around them in a broad regime of parameters, but also are frequently frustrated. This highlights the importance of quantum eects beyond the present classical approach which release frustration as we show in Sec. V A. IV. PERIODIC 3d DOPING IN 4d HOST A. General remarks on nite doping In this Section we analyze the spin-orbital patterns due to a nite concentration xof 3dimpurities within the 4 d host withC-AF/G-AO order, assuming that the 3 dim- purities are distributed in a periodic way. The study is performed for three representative doping distributions \| the rst one x= 1=8 is commensurate with the un- derlying spin-orbital order and the other two are incom- mensurate with respect to it, meaning that in such cases doping at both aandcdoublon sites is imposed simulta- neously. As the impurities lead to local energy gains due to 3d4dbonds surrounding them, we expect that the most favorable situation is when they are isolated and have maximal distances between one another. Therefore, we selected the largest possible distances for the three dop- ing levels used in our study: x= 1=8,x= 1=5, and x= 1=9. This choice allows us to cover dierent regimes of competition between the spin-orbital coupling within the host and the 3 d4dcoupling. While single impu- rities may only change spin-orbital order locally, we use here a high enough doping to investigate possible global changes in spin-orbital order, i.e., whether they can oc- cur in the respective parameter regime. The analysis is performed as for a single impurity, by assuming the clas- sical spin and orbital variables and by determining the conguration with the lowest energy. For this analysis we set the spatial distribution of the 3 datoms and we determine the spin and orbital prole that minimizes the energy. B.C-AF phase with x= 1=8doping We begin with the phase diagram obtained at x= 1=8 3ddoping, see Fig. 9. In the regime of strong impurity- host coupling the 3 d4dspin-orbital exchange deter- mines the orbital and spin conguration of the 4 datoms around the impurity. The most favorable state is when the doublon occupies corbitals at the NN sites to the im- purity. The spin correlations between the impurity and the host are AF (FM), if the amplitude of impis below (above)c imp, leading to the AF aand the FM astates, see Fig. 9. The AF astate has a striped-like prole with12 AF chains alternated by FM domains (consisting of three chains) along the diagonal of the square lattice. Even if the coupling between the impurity and the host is AF for all the bonds in the AF astate, the overall congura- tion has a residual magnetic moment originating by the uncompensated spins and by the cooperation between the spin-orbital exchange in the 4 dhost and that for the 3d4dbonds. Interestingly, at the point where the domi- nant 3d4dexchange tends to zero (i.e., for imp'c imp), one nds a region of the FS aphase which is analogous to the FSaphase found in Sec. III C for a single impu- rity, see Fig. 6. Again the impurity spin is frustrated in purely classical approach but this frustration is easily released by the orbital uctuations in the host so that theC-AF order of the host can be restored. This state is stable for the amplitude of impbeing close to c imp. The regime of small Jimp=Jhostratio is qualitatively dierent \| an orbital rearrangement around the impu- rity takes place, with a preference to move the doublons into the inactive orbitals along the 3 d4dbonds. Such orbital congurations favor the AF spin coupling at all the 3d4dbonds which is stabilized by the 4 d4dsu- perexchange [38]. This conguration is peculiar because it generally breaks inversion and does not have any plane of mirror symmetry. It is worth pointing out that the original order in the 4 dhost is completely modied by the small concentration of 3 dions and one nds that the AF coupling between the 3 dimpurity and the 4 dhost generally leads to patterns such as the AF cphase where FM chains alternate with AF ones in the ( a;b) plane. Another relevant issue is that the cooperation between the host and impurity can lead to a fully polarized FM a state. This implies that doping can release the orbital frustration which was present in the host with the C- AF/G-AO order. C. Phase diagram for periodic x= 1=5doping Next we consider doping x= 1=5 with a given periodic spatial prole which concerns both doublon sublattices. We investigate the 3 dspin impurities separated by the translation vectors ~ u= (i;j) and~ v= (2;1) (one can show that for general periodic doping x,j~ uj2=x1) so there is a mismatch between the impurity periodicity and the two-sublattice G-AO order in the host. One nds that the present case, see Fig. 10, has similar general structure of the phase diagram to the case of x= 1=8 (Fig. 9), with AF correlations dominating for implower thanc impand FM ones otherwise. Due to the specic doping distribution there are more phases appearing in the ground state phase diagram. For imp< c impthe most stable spin conguration is with the impurity cou- pled antiferromagnetically to the host. This happens both in the AF vacancy (AF v) and the AF polaronic (AFp) ground states. The dierence between the two AF states arises due to the orbital arrangement around the impurity. For weak ratio of the impurity to the host 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Jimp/Jhost0.00.20.40.60.81.0ηimp FMa FSa AFc AFaAFc FSa AFa ab ??? ?FIG. 9. Top panel \| Ground state diagram obtained for pe- riodic 3ddopingx= 1=8. Dierent colors refer to local spin order around the impurity: AF a, AFc, FSa, and FMa. Bot- tom panel \| Schematic view of the ground state congura- tions within the four 8-site unit cells (indicated by blue dashed lines) for the phases shown in the phase diagram. The ques- tion marks in FS aphase indicate frustrated impurity spins within the classical approach \| the spin direction (small ar- rows) is xed only by quantum uctuations. The 3 datoms are placed at the sites where orbitals are absent. spin-orbital exchange, Jimp=Jhost, the orbitals around the impurity are all inactive ones. On the contrary, in the strong impurity-host coupling regime all the orbitals are polarized to be in active (polaronic) states around the im- purity. Both states have been found as AF1 cand AF2c phase in the single impurity problem (Fig. 8). More generally, for all phases the boundary given by an approximate hyperbolic relation imp/J1 impsepa- rates the phases where the orbitals around impurities in13 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Jimp/Jhost0.00.20.40.60.81.0ηimp AFp FMp FMv AFv FSvFS1p FS2p ?? ?? ab?? FIG. 10. Ground state diagram for x= 1=5 periodic concentration of 3 dimpurities (sites where orbitals are absent) with schematic views of the ground state congurations obtained for the unit cell of 20 sites. Spin and orbital order are shown by arrows and orbitals occupied by doublons; magnetic phases (AF, FS, and FM) are highlighted by dierent color. The question marks in FS states (red circles) indicate frustrated impurity spins within the classical approach. ? ?(a) FSv (b) FS1p, FS2p FIG. 11. Isotropic surrounding of the degenerate impurity spins in the FS vand FSpphases in the case of x= 1=5 pe- riodic doping (Fig. 10). Frames mark the clusters which are not connected with orbitally active bonds.thec-orbital sublattice are all inactive (small imp) from those where all the orbitals are active (large imp). The inactive orbital around the impurity stabilize always the AF coupling between the impurity spin and host spins whereas the active orbitals can give either AF or FM exchange depending on imp(hencec imp, see Fig. 4). Since the doping does not match the size of the elemen- tary unit cell, the resulting ground states do not exhibit specic symmetries in the spin-orbital pattern. They are generally FM due to the uncompensated magnetic mo- ments and the impurity feels screening by the presence of the surrounding it host spins being antiparallel to the impurity spin. By increasing Hund's exchange coupling at the 3 dion the system develops fully FM state in a large region of the ground state diagram due to the possibility of suit- able orbital polarization around the impurity. On the other hand, in the limit where the impurity-host bonds14 ab0.0 0.5 1.0 1.5 2.0 2.5 3.0 Jimp/Jhost0.00.20.40.60.81.0ηimp AFp FMp FMv AFv FSvFS1p FS2p ?? ?? ?? FIG. 12. Ground state diagram for x= 1=9 periodic concentration of 3 dimpurities with schematic views of the ground state congurations obtained for the unit cell of 36 sites. Spin order (AF, FS, and FM) is highlighted by dierent color. The question marks in FS states (red squares) indicate frustrated impurity spins within the classical approach \| the spin is xed here by quantum uctuations (small arrows). Doped 3 datoms are at the sites where orbitals are absent. are weak, so either for imp'c impand large enough Jimp=Jhostso that all orbitals around the impurity are active, or just for small Jimp=Jhostwe get the FS phases where the impurity spin at the a-orbital sublattice is un- determined in the present classical approach. This is a similar situation to the one found in the FS aphase of a single impurity problem and at x= 1=8 periodic doping, see Figs. 6 and 9, but there it was still possible to iden- tify the favored impurity spin polarization by considering the orbital ips in the host around the impurity. However, the situation here is dierent as the host's order is completely altered by doping and has became isotropic, in contrast to the initial C-AF order (Fig. 2) which breaks the planar symmetry between the aandb direction. It was precisely this symmetry breaking that favored one impurity spin polarization over the other one. Here this mechanism is absent \| one can easily checkthat the neighborhood of the corbitals surrounding im- purities is completely equivalent in both directions (see Fig. 11 for the view of these surroundings) so that the orbital ip argument is no longer applicable. This is a peculiar situation in the classical approach and we indi- cate frustration in spin direction by question marks in Fig. 10. In Fig. 11 we can see that both in FS vacancy (FS v) and FS polaronic (FS p) phase the orbitals are grouped in 33 clusters and 22 plaquettes, respectively, that en- circle the degenerate impurity spins. For the FS vphase we can distinguish between two kind of plaquettes with non-zero spin polarization diering by a global spin inver- sion. In the case of FS pphases we observe four plaquettes with zero spin polarization arranged in two pairs related by a point re ection with respect to the impurity site. It is worthwhile to realize that these plaquettes are com-15 pletely disconnected in the orbital sector, i.e., there are no orbitally active bonds connecting them (see Fig. 7 for the pictorial denition of orbitally active bonds). This means that quantum eects of purely orbital nature can appear only at the short range, i.e., inside the plaquettes. However, one can expect that if for some reason the two degenerate spins in a single elementary cell will polarize then they will also polarize in the same way in all the other cells to favor long-range quantum uctuations in the spin sector related to the translational invariance of the system. D. Phase diagram for periodic x= 1=9doping Finally we investigate low doping x= 1=9 with a given periodic spatial prole, see Fig. 12. Here the impurities are separated by the translation vectors ~ u= (0;3) and ~ v= (3;0). Once again there is a mismatch between the periodic distribution of impurities and the host's two- sublattice AO order, so we again call this doping incom- mensurate as it also imposes doping at both doublon sublattices. The ground state diagram presents gradu- ally increasing tendency towards FM 3 d4dbonds with increasingimp, see Fig. 12. These polaronic bonds po- larize as well the 4 d4dbonds and one nds an almost FM order in the FM pstate. Altogether, we have found the same phases as at the higher doping of x= 1=5, see Fig. 10, i.e., AF vand AFpat low values imp, FMv and FMpin the regime of high imp, separated by the regime of frustrated impurity spins which occur within the phases: FS v, FS1p, and FS2p. The dierence between the two AF (FM) states in Fig. 12 is due to the orbital arrangement around the impurity. As for the other doping levels considered so far, x= 1=8 andx= 1=5, we nd neutral (inactive) orbitals around 3dimpurities in the regime of low Jimp=Jhostin AFvand FMvphases which lead to spin defects within the 1D FM chains in the C-AF spin order. A similar behavior was reported for single impurities in the low doping regime in Sec. III. This changes radically above the orbital tran- sition for both types of local magnetic order, where the orbitals reorient into the active ones. One nds that spin orientations are then the same as those of their neigh- boring 4datoms, with some similarities to those found atx= 1=5, see Fig. 10. Frustrated impurity spins occur in the crossover regime between the AF and FM local order around impurities. This follows from the local congurations around them, which include two "-spins and two#-spins accompanied bycorbitals at the NN 4 dsites. This frustration is easily removed by quantum uctuations and we suggest that this happens again in the same way as for x= 1=8 doping, as indicated by small arrows in the respective FS phases shown in Fig. 12.V. QUANTUM EFFECTS BEYOND THE CLASSICAL APPROACH A. Spin-orbital quantum uctuations So far, we analyzed the ground states of 3 dimpurities in the (a;b) plane of a 4 dsystem using the classical ap- proach. Here we show that this classical picture may be used as a guideline and is only quantitatively changed by quantum uctuations if the spin-orbit coupling is weak. We start the analysis by considering the quantum prob- lem in the absence of spin-orbit coupling (at = 0). The orbital doublon densities, N X i2hosthni i; (25) with =a;b;c , and totalSzare conserved quantities and thus good quantum numbers for a numerical simulation. To determine the ground state congurations in the pa- rameters space and the relevant correlation functions we diagonalize exactly the Hamiltonian matrix (17) for the cluster ofL= 8 sites by means of the Lanczos algorithm. In Fig. 13(a) we report the resulting quantum phase diagram for an 8-site cluster having one impurity and assuming periodic boundary conditions, see Fig. 13(b). This appears to be an optimal cluster conguration be- cause it contains a number of sites and connectivities that allows us to analyze separately the interplay between the host-host and the host-impurity interactions and to sim- ulate a physical situation when the interactions within the host dominate over those between the host and the impurity. Such a problem is a quantum analogue of the single unit cell presented in Fig. 9 for x= 1=8 periodic doping. As a general feature that resembles the classical phase diagram, we observe that there is a prevalent tendency to have AF-like (FM-like) spin correlations between the impurity and the host sites in the region of impbelow (above) the critical point at c imp'0:43 which separates these two regimes, with intermediate congurations hav- ing frustrated magnetic exchange. As we shall discuss below it is the orbital degree of freedom that turns out to be more aected by the quantum eects. Following the notation used for the classical case, we distinguish various quantum AF (QAF) ground states, i.e., QAF cn (n= 1;2) and QAF an(n= 1;2), as well as a uni- form quantum FM (QFM) conguration, i.e., QFM a, and quantum frustrated one labeled as QFS a. In order to visualize the main spin-orbital patterns con- tributing to the quantum ground state it is convenient to adopt a representation with arrows for the spin and el- lipsoids for the orbital sector at any given host site. The arrows stand for the on-site spin projection hSz ii, with the length being proportional to the amplitude. The length scale for the arrows is the same for all the congurations. Moreover, in order to describe the orbital character of the ground state we employed a graphical representation that makes use of an ellipsoid whose semi-axes fa;b;cg16 0.0 0.3 0.6 0.9 1.2 1.5 Jimp/Jhost0.000.250.500.751.00 ηimpQFMa QFSa1 QAFc2 QAFc1QFSa2 QAFa1 QAFa2 12 8 34 5 673 2 6 41 71 71 7(a) (b) ab FIG. 13. (a) Phase diagram for the quantum problem at zero spin-orbit simulated on the 8-site cluster in the presence of one-impurity. Arrows and ellipsoids indicate the spin-orbital state at a given site i, while the shapes of ellipsoids re ect the orbital avarages: hay iaii,hby ibiiandhcy icii(i.e., a circle in the plane perpendicular to the axis implies 100% occupation of the orbital ). (b) The periodic cluster of L= 8 sites used, with the orbital dilution (3 d3impurity) at site i= 8. The dotted lines identify the basic unit cell adopted for the simulation with the same symmetries of the square lattice. length are given by the average amplitude of the squared angular momentum components f(Lx i)2;(Ly i)2;(Lz i)2g, or equivalently by the doublon occupation Eq. (9). For in- stance, for a completely at circle (degenerate ellipsoid) lying in the plane perpendicular to the axis only the corresponding orbital is occupied. On the other hand, if the ellipsoid develops in all three directions fa;b;cgit implies that more than one orbital is occupied and the distribution can be anisotropic in general. If all the or- bitals contribute equally, one nds an isotropic spherical ellipsoid. Due to the symmetry of the Hamiltonian, the phases shown in the phase diagram of Fig. 13(a) can be characterized by the quantum numbers for the z-th spin projection, Sz, and the doublon orbital occupa- tionN(25), (Sz;Na;Nb;Nc): QAFc1 (3:5;2;2;3), QFSa2 (1:5;3;1;3), QAFa1 (5:5;1;3;3), QAFa2 and QAFa2 (5:5;2;2;3), QFSa1 (0:5;3;0;4), and QFM a (8:5;2;1;4). Despite the irregular shape of the clus- ter [Fig. 13(b)] there is also symmetry between the aandbdirections. For this reason, the phases with Na6=Nbcan be equivalently described either by the set (Sz;Na;Nb;Nc) or (Sz;Nb;Na;Nc). The outcome of the quantum analysis indicates thatthe spin patterns are quite robust as the spin congu- rations of the phases QAF a, QAFc, QFSaand QFMa are the analogues of the classical ones. The eects of quantum uctuations are more evident in the orbital sec- tor where mixed orbital patterns occur if compared to the classical case. In particular, orbital inactive states around the impurity are softened by quantum uctua- tions and on some bonds we nd an orbital conguration with a superposition of active and inactive states. The unique AF states where the classical inactive scenario is recovered corresponds to the QAF c1 and QAF c2 ones in the regime of small imp. A small hybridization of active and inactive orbitals along both the AF and FM bonds is also observed around the impurity for the QFS aphases as one can note by the shape of the ellipsoid at host sites. Moreover, in the range of large impwhere the FM state is stabilized, the orbital pattern around the impurity is again like in the classical case. A signicant orbital rearrangement is also obtained within the host. We generally obtain an orbital pattern that is slightly modied from the pure AO conguration assumed in the classical case. The eect is dramatically dierent in the regime of strong impurity-host coupling (i.e., for large Jimp) with AF exchange (QAF a2) with the17 formation of an orbital liquid around the impurity and within the host, with doublon occupation represented by an almost isotropic shaped ellipsoid. Interestingly, though with a dierent orbital arrangement, the QFS a1 and the QFS a2 states are the only ones where the C-AF order of the host is recovered. For all the other phases shown in the diagram of Fig. 13 the coupling between the host and the impurity is generally leading to a uni- form spin polarization with FM or AF coupling between the host and the impurity depending on the strength of the host-impurity coupling. Altogether, we conclude that the classical spin patterns are only quantitatively modi- ed and are robust with respect to quantum uctuations. B. Finite spin-orbit coupling In this Section we analyze the quantum eects in the spin and orbital order around the impurity in the pres- ence of the spin-orbit coupling at the host d4sites. For thet4 2gconguration the strong spin-orbit regime has been considered recently by performing an expansion around the atomic limit where the angular ~Liand spin ~Simomenta form a spin-orbit singlet for the amplitude of the total angular momentum, ~Ji=~Li+~Si(i.e.,J= 0) [67]. The instability towards an AF state starting from theJ= 0 liquid has been obtained within the spin-wave theory [68] for the low energy excitations emerging from the spin-orbital exchange. In the analysis presented here we proceed from the limit of zero spin-orbit to investigate how the spin and orbital order are gradually suppressed when approaching theJ= 0 spin-orbit singlet state. This issue is addressed by solving the full quantum Hamiltonian (17) exactly on a cluster of L= 8 sites including the spin-orbital ex- change for the host and that one derived for the host- impurity coupling (17) as well as the spin-orbit term, Hso=X i2host~Li~Si: (26) where the sum includes the ions of the 4 dhost and we use the spin S= 1 and the angular momentum L= 1, as in the ionic 4 d4congurations. Here is the spin-orbit coupling constant at 4 dhost ions, and the components of the orbital momentum ~LifLx i;Ly i;Lz igare dened as follows: Lx i=iX (dy i;xydi;xzdy i;xzdi;xy); Ly i=iX (dy i;xydi;yzdy i;yzdi;xy); Lz i=iX (dy i;xzdi;yzdy i;yzdi;xz): (27) To determine the ground state and the relevant corre- lation functions we use again the Lanczos algorithm for the cluster of L= 8 sites. Such an approach allows us tostudy the competition between the spin-orbital exchange and the spin-orbit coupling on equal footing without any simplifying approximation. Moreover, the cluster calcu- lation permits to include the impurity in the host and deal with the numerous degrees of freedom without mak- ing approximations that would constrain the interplay of the impurity-host versus host-host interactions. Finite spin-orbit coupling signicantly modies the symmetry properties of the problem. Instead of the SU(2) spin invariance one has to deal with the rotational invariance related to the total angular momentum per site~Ji=~Li+~Si. Though the ~Li~Siterm in Eq. (26) commutes with both total ~J2andJz, the full Hamilto- nian for the host with impurities Eq. (17) has a reduced symmetry because the spin sector is now linearly coupled to the orbital which has only the cubic symmetry. Thus the remaining symmetry is a cyclic permutation of the fx;y;zgaxes. Moreover,Jzis not a conserved quantity due to the or- bital anisotropy of the spin-orbital exchange in the host and the orbital character of the impurity-host coupling. There one has a Z2symmetry associated with the parity operator (-1)Jz. Hence, the ground state can be clas- sied as even or odd with respect to the value of Jz. This symmetry aspect can introduce a constraint on the character of the ground state and on the impurity-host coupling since the Jzvalue for the impurity is only due to the spin projection while in the host it is due to the combination of the orbital and spin projection. A direct consequence is that the parity constraint together with the unbalance between the spin at the host and the im- purity sites leads to a nonvanishing total projection of the spin and angular momentum with respect to a sym- metry axis, e.g. the zthaxis. It is worth to note that a xed parity for the impurity spin means that it prefers to point in one direction rather than the other one which is not the case for the host's spin and angular momen- tum. Thus the presence on the impurity for a xed P will give a nonzero polarization along the quantization axis for every site of the system. Such a property holds for any single impurity with a half-integer spin. Another important consequence of the spin-orbit cou- pling is that it introduces local quantum uctuations in the orbital sector even at the sites close to the impurity where the orbital pattern is disturbed. The spin-orbit term makes the on-site problem around the impurity ef- fectively analogous to the Ising model in a transverse eld for the orbital sector, with nontrivial spin-orbital entan- glement [34] extending over the impurity neighborhood. In Figs. 14 and 15 we report the schematic evolu- tion of the ground state congurations for the cluster ofL= 8 sites, with one-impurity and periodic boundary conditions as a function of increasing spin-orbit coupling. These patterns have been determined by taking into ac- count the sign and the amplitude of the relevant spatial dependent spin and orbital correlation functions. The arrows associated to the spin degree of freedom can lie in xyplane or out-of-plane (along z, chosen to be parallel18 QAFc1 QAFa1 QAFc2 QFMa a ba ba ba b λ1 λ2 λ3 λ4 λ5 λ10λ1 λ2 λ3 λ4 λ7 λ10λ1 λ2 λ4 λ5 λ7 λ10λ1 λ2 λ3 λ4 λ7 λ10 FIG. 14. Evolution of the ground state congurations for the AF phases for selected increasing values of spin-orbit coupling m, see Eq. (28). Arrows and ellipsoids indicate the spin-orbital state at a given site i. Color map indicates the strength of the average spin-orbit, h~Li~Sii, i.e., red, yellow, green, blue, violet correspond to the growing amplitude of the above correlation function. Small arrows at 5and10indicate quenched magnetization at the impurity by large spin-orbit coupling at the neighboring host sites. to thecaxis) to indicate the anisotropic spin pattern. The out-of-plane arrow length is given by the on-site ex- pectation value of hSz iiwhile the in-plane arrow length is obtained by computing the square root of the second moment, i.e.,p h(Sx i)2iandp h(Sy i)2iof thexandyspin components corresponding to the arrows along aandb, respectively. Moreover, the in-plane arrow orientation for a given direction is determined by the sign of the correspond- ing spin-spin correlation function assuming as a refer-ence the orientation of the impurity spin. The ellipsoid is constructed in the same way as for the zero spin-orbit case above, with the addition of a color map that indi- cates the strength of the average ~Li~Si(i.e., red, yel- low, green, blue, violet correspond with a growing ampli- tude of the local spin-orbit correlation function). The scale for the spin-orbit amplitude is set to be in the interval 0 < < J host. The selected values for the ground state evolution are given by the relation (with19 m= 1;2;:::; 10), m= 0:04 + 0:96(m1) 9 Jhost: (28) The scale is set such that 1= 0:04Jhostand10=Jhost. This range of values allows us to explore the relevant physical regimes when moving from 3 dto 4dand 5dmate- rials with corresponding being much smaller that Jhost, Jhost=2 and > J host, respectively. For the per- formed analysis the selected values of (28) are also rep- resentative of the most interesting regimes of the ground state as induced by the spin-orbit coupling. Let us start with the quantum AF phases QAF c1, QAFc2, QFSa1, QFSa2, QAFa1, and QAF a2. As one can observe the switching on of the spin-orbit coupling (i.e.,1in Fig. 14) leads to anisotropic spin patterns with unequal moments for the in-plane and out-of-plane components. From weak to strong spin-orbit coupling, the character of the spin correlations keeps being AF be- tween the impurity and the neighboring host sites in all the spin directions. The main change for the spin sector occurs for the planar components. For weak spin-orbit coupling the in-plane spin pattern is generally AF for the whole system in all the spatial directions (i.e., G-AF or- der). Further increase of the spin-orbit does not modify qualitatively the character of the spin pattern for the out- of-plane components as long as we do not go to maximal values ofJhostwhere localhSz iimoments are strongly suppressed. In this limit the dominant tendency of the system is towards formation of the spin-orbital singlets and the spin patterns shown in Fig. 14 are the eect of the virtual singlet-triplet excitations [67]. Concerning the orbital sector, only for weak spin-orbit coupling around the impurity one can still observe a rem- iniscence of inactive orbitals as related to the orbital va- cancy role at the impurity site in the AF phase. Such an orbital conguration is quickly modied by increasing the spin-orbit interaction and it evolves into a uniform pattern with almost degenerate orbital occupations in all the directions, and with preferential superpositions of c and (a;b) states associated with dominating LxandLy orbital angular components ( attened ellipsoids along the cdirection). An exception is the QAF c2 phase with the orbital inactive polaron that is stable up to large spin- orbit coupling of the order of Jhost. When considering the quantum FM congurations QFMa1 in Fig. 14, we observe similar trends in the evo- lution of the spin correlation functions as obtained for the AF states. Indeed, the QFM aexhibits a tendency to form FM chains with AF coupling for the in-plane compo- nents at weak spin-orbit that evolve into more dominant AF correlations in all the spatial directions within the host. Interestingly, the spin exchange between the impu- rity and the neighboring host sites shows a changeover from AF to FM for the range of intermediate-to-strong spin-orbit amplitudes. A peculiar response to the spin-orbit coupling is ob- tained for the QFS a1 phase, see Fig. 15, which showed QFSa2 QFSa1 λ1 λ2 λ3λ4 λ8 λ10λ1 λ2 λ3 λ5 λ6 λ10a ba bFIG. 15. Evolution of the ground state congurations for the QFSa1 and QFSa2 phases for selected increasing values of spin-orbit coupling m, see Eq. (28). Arrows and ellip- soids indicate the spin-orbital state at a given site i. Color map indicates the strength of the average spin-orbit, h~Li~Sii, i.e., red, yellow, green, blue, violet correspond to the growing amplitude of the above local correlation function. a frustrated spin pattern around the impurity already in the classical regime, with FM and AF bonds. It is re- markable that due to the close proximity with uniform FM and the AF states, the spin-orbit interaction can lead to a dramatic rearrangement of the spin and orbital cor- relations for such a conguration. At weak spin-orbit coupling (i.e., '1) the spin-pattern is C-AF and the increased coupling ( '2)) keeps the C-AF order only20 for the in-plane components with the exception of the im- purity site. It also modulates the spin moment distribu- tion around the impurity along the zdirection. Further increase of leads to complete spin polarization along thezdirection in the host, with antiparallel orientation with respect to the impurity spin. This pattern is guided by the proximity to the FM phase. The in-plane compo- nents develop a mixed FM-AF pattern with a strong xy anisotropy most probably related to the dierent bond exchange between the impurity and the host. When approaching the regime of a spin-orbit coupling that is comparable to Jhost, the out-of-plane spin compo- nents dominate and the only out-of-plane spin polariza- tion is observed at the impurity site. Such a behavior is unique and occurs only in the QFS aphases. The coop- eration between the strong spin-orbit coupling and the frustrated host-impurity spin-orbital exchange leads to an eective decoupling in the spin sector at the impu- rity with a resulting maximal polarization. On the other hand, as for the AF states, the most favorable congu- ration for strong spin-orbit has AF in-plane spin corre- lations. The orbital pattern for the QFS astates evolves similarly to the AF cases with a suppression of the active- inactive interplay around the impurity and the setting of a uniform-like orbital conguration with unquenched an- gular momentum on site and predominant in-plane com- ponents. The response of the FM state is dierent in this respect as the orbital active states around the impu- rity are hardly aected by the spin-orbit while the host sites far from the impurity the local spin-orbit coupling is more pronounced. Finally, to understand the peculiar evolution of the spin conguration it is useful to consider the lowest order terms in the spin-orbital exchange that couple directly the orbital angular momentum with the spin. Taking into account the expression of the spin-orbital exchange in the host (26) and the expression of ~Lione can show that the low energy terms on a bond that get more relevant in the Hamiltonian when the spin-orbit coupling makes a non-vanishing local angular momentum. As a result, the corresponding expressions are: Ha(b) host(i;j)Jhostn a1~Si~Sj+b1Sz iSz jLy(x) iLy(x) jo +n ~Li~Si+~Lj~Sjo ; (29) with positive coecients a1andb1that depend on r1and r2(21). A denite sign for the spin exchange in the limit of vanishing spin-orbit coupling is given by the terms which go beyond Eq. (29). Then, if the ground state has isotropic FM correlations (e.g. QFM a) at= 0, the termSz iSz jLy(x) iLy(x) jwould tend to favor AF-like con- gurations for the in-plane orbital angular components when the spin-orbit interaction is switched on. This op- posite tendency between the zandfx;ygcomponents is counteracted by the local spin-orbit coupling that pre- vents to have coexisting FM and AF spin-orbital corre- lations. Such patterns would not allow to optimize the~Li~Siamplitudes. One way out is to reduce the zthspin projection and to get planar AF correlations in the spin and in the host. A similar reasoning applies to the AF states where the negative sign of the Sz iSz jcorrelations favors FO alignment of the angular momentum compo- nents. As for the previous case, the opposite trend of in- and out-of-plane spin-orbital components is suppressed by the spin-orbit coupling and the in-plane FO correla- tions for thefLx;Lygcomponents leads to FM patterns for the in-plane spin part as well. Summarizing, by close inspection of Figs. 14 and 15 one nds an interesting evolution of the spin patterns in the quantum phases: (i) For the QAF states (Fig. 14), a spin canting devel- ops at the host sites (i.e., the relative angle is between 0 and) while the spins on impurity-host bonds are al- ways AF. The canting in the host evolves, sometime in an inhomogeneous way, to become reduced in the strong spin-orbit coupling regime where ferro-like correlations tend to dominate. In this respect, when the impurity is coupled antiferromagnetically to the host it does not fol- low the tendency to form spin canting. (ii) In the QFM states (Fig. 15), at weak spin-orbit one observes spin-canting in the host and for the host- impurity coupling that persists only in the host whereas the spin-orbit interaction is increasing. C. Spin-orbit coupling versus Hund's exchange To probe the phase diagram of the system in presence of the spin-orbit coupling ( > 0) we solved the same cluster ofL= 8 sites as before along three dierent cuts in the phase diagram of Fig. 13(a) for three values of , i.e., small= 0:1Jhost, intermediate = 0:5Jhost, and large=Jhost. Each cut contained ten points, the cuts were parameterized as follows: (i) Jimp= 0:7Jhostand 0imp0:7, (ii)Jimp= 1:3Jhostand 0imp0:7, and (iii)imp=c imp'0:43 and 0Jimp1:5Jhost. In Fig. 16(a) we show the representative spin-orbital congurations obtained for = 0:5Jhostalong the rst cut shown in Fig. 16(b). Values of impare chosen as imp=m0:7(m1) 9; (30) withm= 1;:::; 10 but not all the points are shown in Fig. 16(a) \| only the ones for which the spin-orbital conguration changes substantially. The cut starts in the QAF c2 phase, according to the phase diagram of Fig. 16(b), and indeed we nd a simi- lar conguration to the one shown in Fig. 14 for QAF c2 phase at=5. Moving up in the phase diagram from 1to2we see that the conguration evolves smoothly to the one which we have found in the QAF a1 phase at =5(not shown in Fig. 14). The evolution of spins is such that the out-of-plane moments are suppressed while in-plane ones are slightly enhanced. The orbitals become more spherical and the local spin-orbit average, h~Li~Sii,21 FIG. 16. (a) Evolution of the ground state congurations as for increasing impand for a xed value of spin-orbit coupling = 0:5Jhostalong a cut in the phase diagram shown in panel (b), i.e., for Jimp= 0:7Jhostand 0imp0:7. Arrows and ellipsoids indicate the spin-orbital state at a given site i. Color map indicates the strength of the average spin-orbit, h~Li~Sii, i.e., red, yellow, green, blue, violet correspond to the growing amplitude of the above correlation function. becomes larger and more uniform, however for the apical sitei= 7 in the cluster [Fig. 13(b)] the trend is oppo- site \| initially large value of spin-orbit coupling drops towards the uniform value. The points between 3and 7we skip as the evolution is smooth and the trend is clear, however the impurity out-of-plane moment begins to grow above 5, indicating proximity to the QFS a1 phase. For this phase at intermediate and high the impurity moment is much larger than all the others (see Fig. 15). Forimp=7the orbital pattern clearly shows that we are in the QFS a1 phase at=5which agrees with the position of the 7point in the phase diagram, see Fig. 16(b). On the other hand, moving to the next imp point upward along the cut Eq. (30) we already observe a conguration which is very typical for the QFM aphase at intermediate (here=7shown in Fig. 14 but also 6, not shown). This indicates that the QFS a1 phase can be still distinguished at = 0:5Jhostand its position in the phase diagram is similar as in the = 0 case, i.e., as an intermediate phase between the QAF a1(2) and QFM a one. Finally, we have found that also the two other cuts which were not shown here, i.e., for Jimp= 1:3Jhostand increasingimpand forimp=c imp'0:43 and increas- ingJimpconrm that the overall character of the phase diagram of Fig. 13(a) is preserved at this value of spin-orbit coupling, however rstly, the transitions between the phases are smooth and secondly, the subtle dier- ences between the two QFS a, QAFaand QAFcphases are no longer present. This also refers to the smaller value of, i.e.,= 0:1Jhost, but already for =Jhost the out-of-plane moments are so strongly suppressed (ex- cept for the impurity moment in the QFS a1 phase) and the orbital polarization is so weak (i.e., almost spheri- cal ellipsoids) that typically the only distinction between the phases can be made by looking at the in-plane spin correlations and the average spin-orbit, h~Li~Sii. In this limit we conclude that the phase diagram is (partially) melted by large spin-orbit coupling but for lower values ofit is still valid. VI. SUMMARY AND CONCLUSIONS We have derived the spin-orbital superexchange model for 3d3impurities replacing 4 d4(or 3d2) ions in the 4 d (3d) host in the regime of Mott insulating phase. Al- though the impurity has no orbital degree of freedom, we have shown that it contributes to the spin-orbital physics and in uences strongly the orbital order. In fact, it tends to project out the inactive orbitals at the impurity-host bonds to maximize the energy gain from virtual charge uctuations. In this case the interaction along the su- perexchange bond can be either antiferromagnetic or fer- romagnetic, depending on the ratio of Hund's exchange coupling at impurity ( JH 1) and host ( JH 2) ions and on the mismatch between the 3 dand 4datomic energies, modied by the dierence in Hubbard U's and Hund's exchangeJH's at both atoms. This ratio, denoted imp (14), replaces here the conventional parameter =JH=U often found in the spin-orbital superexchange models of undoped compounds (e.g., in the Kugel-Khomskii model for KCuF 3[14]) where it quanties the proximity to fer- romagnetism. On the other hand, if the overall coupling between the host and impurity is weak in the sense of the total superexchange, Jimp, with respect to the host value,Jhost, the orbitals being next to the impurity may be forced to stay inactive which modies the magnetic properties \| in such cases the impurity-host bond is al- ways antiferromagnetic. As we have seen in the case of a single impurity, the above two mechanisms can have a nontrivial eect on the host, especially if the host itself is characterized by frustrated interactions, as it happens in the parameter regime where the C-AF phase is stable. For this rea- son we have focused mostly on the latter phase of the host and we have presented the phase diagrams of a sin- gle impurity conguration in the case when the impu- rity is doped on the sublattice where the orbitals form a checkerboard pattern with alternating candaorbitals occupied by doublons. The diagram for the c-sublattice doping shows that in some sense the impurity is never weak, because even for a very small value of Jimp=Jhost it can release the host's frustration around the impurity22 site acting as an orbital vacancy. On the other hand, for thea-sublattice doping when the impurity-host coupling is weak, i.e., either Jimp=Jhostis weak orimpis close to c imp, we have identied an interesting quantum mech- anism releasing frustration of the impurity spin (that cannot be avoided in the purely classical approach). It turned out that in such situations the orbital ips in the host make the impurity spin polarize in such a way that theC-AF order of the host is completely restored. The cases of the periodic doping studied in this pa- per show that the host's order can be completely altered already for rather low doping ofx= 1=8, even if the Jimp=Jhostis small. In this case we can stabilize a ferri- magnetic type of phase with a four-site unit cell having magnetizationhSz ii= 3=2, reduced further by quantum uctuations. We have established that the only param- eter range where the host's order remains unchanged is whenimpis close toc impandJimp=Jhost&1. The latter value is very surprising as it means that the impurity-host coupling must be large enough to keep the host's order unchanged \| this is another manifestation of the orbital vacancy mechanism that we have already observed for a single impurity. Also in this case the impurity spins are xed with the help of orbital ips in the host that lift the degeneracy which arises in the classical approach. We would like to point out that the quantum mechanism that lifts the ground state degeneracy mentioned above and the role of quantum uctuations are of particular interest for the periodically doped checkerboard systems withx= 1=2 doping which is a challenging problem for future research. From the point of view of generic, i.e., non-periodic doping, the most representative cases are those of a doping which is incommensurate with the two-sublattice spin-orbital pattern. To uncover the generic rules in such cases, we have studied periodic x= 1=5 andx= 1=9 dop- ing. One nds that when the period of the impurity posi- tions does not match the period of 2 for both the spin and orbital order of the host, interesting novel types of order emerge. In such cases the elementary cell must be dou- bled in both lattice directions which clearly gives a chance of realizing more phases than in the case of commensu- rate doping. Our results show that indeed, the number of phases increases from 4 to 7 and the host's order is altered in each of them. Quite surprisingly, the overall character of the phase diagram remained unchanged with respect to the one for x= 1=8 doping and, if we ignore the dif- ferences in conguration, it seems that only some of the phases got divided into two versions diering either by the spin bond's polarizations around impurities (phases aroundc imp), or by the character of the orbitals around the impurities (phases with inactive orbitals in the limit of small enough product impJimp, versus phases with ac- tive orbitals in the opposite limit). Orbital polarization in this latter region resembles orbital polarons in doped manganites [42, 43] \| also here such states are stabilized by the double exchange [46]. A closer inspection of underlying phases reveals how-ever a very interesting degeneracy of the impurity spins atx= 1=5 that arises again from the classical approach but this time it cannot be released by short-range orbital ips. This happens because the host's order is already so strongly altered that it is no longer anisotropic (as it was the case of the C-AF phase) and there is no way to restore the orbital anisotropy around the impurities that could lead to spin-bonds imbalance and polarize the spin. In the case of lower x= 1=9 doping such an eect is absent and the impurity spins are always polarized, as it happens for x= 1=8. It shows that this is rather a peculiarity of the x= 1=5 periodic doping. Indeed, one can easily notice that for x= 1=5 every atom of the host is a nearest neighbor of some impurity. In contrast, for x= 1=8 we can nd three host's atoms per unit cell which do not neighbor any impurity and for x= 1=9 there are sixteen of them. For this reason the impurity eects are amplied for x= 1=5 which is not unexpected although one may nd somewhat surprising that the ground state diagrams for the lowest and the highest doping considered here are very similar. This suggests that the cooperative eects of multiple impuri- ties are indeed not very strong in the low-doping regime, so the diagram obtained for x= 1=9 can be regarded as generic for the dilute doping regime with uniform spatial prole. For the representative case of x= 1=8 doping, we have presented the consequences of quantum eects beyond the classical approach. Spin uctuations are rather weak for the considered case of large S= 1 andS= 3=2 spins, and we have shown that orbital uctuations on superex- change bonds are more important. They are strongest in the regime of antiferromagnetic impurity-host coupling (which suggests importance of entangled states [34]) and enhance the tendency towards frustrated impurity spin congurations but do not destroy other generic trends observed when the parameters impandJimp=Jhostin- crease. Increasing spin-orbit coupling leads to qualitative changes in the spin-orbital order. When Hund's ex- change is small at the impurity sites, the antiferromag- netic bonds around it have reduced values of spin-orbit coupling term, but the magnetic moments reorient and survive in the ( a;b) planes, with some similarity to the phenomena occurring in the perovskite vanadates [57]. This quenches the magnetic moments at 3 dimpurities and leads to almost uniform orbital occupancies at the host sites. In contrast, frustration of impurity spins is re- moved and the impurity magnetization along the caxis survives for large spin-orbit coupling. We would like to emphasize that the orbital dilution considered here in uences directly the orbital degrees of freedom in the host around the impurities. The synthesis of hybrid compounds having both 3 dand 4dtransition metal ions will likely open a novel route for unconven- tional eects in complex materials. There are several reasons for expecting new scenarios in mixed 3 d4d spin-orbital-lattice materials, and we pointed out only23 some of them. On the experimental side, the changes of local order could be captured using inelastic neutron scattering or resonant inelastic x-ray scattering (RIXS). In fact, using RIXS can also bring an additional advan- tage: RIXS, besides being a perfect probe of both spin and orbital excitations, can also (indirectly) detect the nature of orbital ground state (supposedly also including the nature of impurities in the crystal) [82]. Unfortu- nately, there are no such experiments yet but we believe that they will be available soon. Short range order around impurities could be inves- tigated by the excitation spectra at the resonant edges of the substituting atoms. Taking them both at nite energy and momentum can dive insights into the nature of the short range order around the impurity and then unveil information of the order within the host as well. Even if there are no elastic superlattice extra peaks one can expect that the spin-orbital correlations will emerge in the integrated RIXS spectra providing information of the impurity-host coupling and of the short range order around the impurity. Even more interesting is the case where the substituting atom forms a periodic array with small deviation from the perfect superlattice when one expects the emergence of extra elastic peaks which will clearly indicate the spin-orbital reconstruction. In our case an active orbital diluted site cannot participate co- herently in the host spin-orbital order but rather may to restructure the host ordering [83]. At dilute impurity concentration we may expect broad peaks emerging at nite momenta in the Brillouin zone, indicating the for- mation of coherent islands with short range order around impurities. We also note that local susceptibility can be suit- ably measured by making use of resonant spectroscopies (e.g. nuclear magnetic resonance (NMR), electron spin resonance (ESR), nuclear quadrupole resonance (NQR), muon spin resonance ( SR), etcetera ) that exploit the dierent magnetic or electric character of the atomic nu- clei for the impurity and the host in the hybrid system. Finally, the random implantation of the muons in the sample can provide information of the relaxation time in dierent domains with unequal dopant concentration which may be nonuniform. For the given problem the dierences in the resonant response can give relevant in- formation about the distribution of the local elds, the occurrence of local order and provide access to the dy- namical response within doped domains. The use of local spectroscopic resonance methods has been widely demon- strated to be successful when probing the nature and the evolution of the ground state in the presence of spin va- cancies both for ordered and disordered magnetic cong- urations [84{87]. In summary, this study highlights the role of spin de- fects which lead to orbital dilution in spin-orbital sys- tems. Using an example of 3 d3impurities in a 4 d4(or 3d2) host we have shown that impurities change radi- cally the spin-orbital order around them, independently of the parameter regime. As a general feature we havefound that doped 3 d3ions within the host with spin- orbital order have frustrated spins and polarize the or- bitals of the host when the impurity-host exchange as well as Hund's exchange at the impurity are both suf- ciently large. This remarkable trend is independent of doping and is expected to lead to global changes of spin- orbital order in doped materials. While the latter eect is robust, we argue that the long-range spin uctuations resulting from the translational invariance of the system will likely prevent the ground state from being macro- scopically degenerate, so if the impurity spins in one unit cell happens to choose its polarization then the others will follow. On the contrary, in the regime of weak Hund's exchange 3d3ions act not only as spin defects which or- der antiferromagnetically with respect to their neighbors, but also induce doublons in inactive orbitals. Finally, we remark that this behavior with switching between inactive and active orbitals by an orbitally neu- tral impurity may lead to multiple interesting phenomena at macroscopic doping when global modications of the spin-orbital order are expected to occur. Most of the re- sults were obtained in the classical approximation but we have shown that modications due to spin-orbit coupling do not change the main conclusion. We note that this generic treatment and the general questions addressed here, such as the release of frustration for competing spin structures due to periodic impurities, are relevant to double perovskites [88]. While the local orbital polar- ization should be similar, it is challenging to investigate disordered impurities, both theoretically and in experi- ment, to nd out whether their in uence on the global spin-orbital order in the host is equally strong. ACKNOWLEDGMENTS We thank Maria Daghofer and Krzysztof Wohlfeld for insightful discussions. W. B. and A. M. O. kindly ac- knowledge support by the Polish National Science Cen- ter (NCN) under Project No. 2012/04/A/ST3/00331. W. B. was also supported by the Foundation for Pol- ish Science (FNP) within the START program. M. C. acknowledges funding from the EU \| FP7/2007-2013 under Grant Agreement No. 264098 \| MAMA. Appendix A: Derivation of 3d4dsuperexchange Here we present the details of the derivation of the low energy spin-orbital Hamiltonian for the 3 d34d4bonds around the impurity at site i.H3d4d(i), which follows from the perfurbation theory, as given in Eq. (10). Here we consider a single 3 d34d4bondhiji. Two contri- butions to the eective Hamiltonian follow from charge excitations: (i)H( ) J;43(i;j) due tod3 id4 jd4 id3 j, and (ii) H( ) J;25(i;j) due tod3 id4 jd2 id5 j. Therefore the low energy24 Hamiltonian is, H( ) J(i;j) =H( ) J;43(i;j) +H( ) J;25(i;j): (A1) Consider rst the processes which conserve the num- ber of doubly occupied orbitals, d3 id4 jd4 id3 j. Then by means of spin and orbital projectors, it is possible to ex- pressH( ) J;43(i;j) fori= 1 andj= 2 as H( ) J;43(1;2) = ~S1~S2t2 184 7 + 3JH 23 + 5JH 2 +D( ) 2 ~S1~S2t2 184 1 + 3JH 2+3 + 5JH 2 + D( ) 21t2 128 +1 + 3JH 23 + 5JH 2 ;(A2) with the excitation energy dened in Eq. (11). The resulting eective 3 d4dexchange in Eq. (A2) consists of three terms: (i) The rst one does not depend on the orbital conguration of the 4 datom and it can be FM or AF depending on the values and the Hund's exchange on the 3dion. In particular, if is the largest or the smallest energy scale, the coupling will be either AF or FM, respectively. (ii) The second term has an explicit dependence on the occupation of the doublon on the 4 d atom via the projecting operator D( ) 2. This implies that a magnetic exchange is possible only if the doublon occu- pies the inactive orbital for a bond along a given direction . Unlike in the rst term, the sign of this interaction is always positive favoring an AF conguration at any strength of and JH 1. (iii) Finally, the last term de- scribes the eective processes which do not depend on the spin states on the 3 dand 4datoms. This contri- bution is of pure orbital nature, as it originates from the hopping between 3 dand 4datoms without aecting their spin conguration, and for this reason favors the occupa- tion of active t2gorbitals along the bond by the doublon. Within the same scheme, we have derived the ef- fective spin-orbital exchange that originates from the charge transfer processes of the type 3 d3 14d4 23d2 i4d5 j, H( ) J;25(1;2). The eective low-energy contribution to the Hamiltonian for i= 1 andj= 2 reads H( ) J;25(1;2) =t2 U1+U2 + 3JH 22JH 1 1 3D( ) 2 ~S1~S2 +1 3 ~S1~S2 1 2 D( ) 2+ 1 :(A3) By inspection of the spin structure involved in the ele- mental processes that generate H( ) J;25(1;2), one can note that it is always AF independently of the orbital con- guration on the 4 datom exhibiting with a larger spin- exchange and an orbital energy gain if the doublon is occupying the inactive orbital along a given bond. We have veried that the amplitude of the exchange termsinH( ) J;25(1;2) is much smaller than the ones which enter inH( ) J;43(1;2) which justies that one may simplify Eq. (A1) fori= 1 andj= 2 to H( ) J(1;2)'H( ) J;43(1;2); (A4) and neglectH( ) J;25(1;2) terms altogether. This approxi- mation is used in Sec. II. Appendix B: Orbital operators in the L-basis The starting point to express the orbital operators ap- pearing in the spin-orbital superexchange model (17) is the relation between quenched jaii,jbii, andjciiorbitals at siteiand the eigenvectors j1ii,j0ii, andj1iiof the angular momentum operator Lz i. These are known to be jaii=1p 2(j1ii+j1ii); jbii=ip 2(j1iij1ii); jcii=j0ii: (B1) From this we can immediately get the occupation number operators for the doublon, D(a) i=ay iai=jaiihaji= 1(Lx i)2; D(b) i=by ibi=jbiihbji= 1(Ly i)2; D(c) i=cy ici=jciihcji= 1(Lz i)2; (B2) and the relatedfn( ) igoperators, n(a) i=by ibi+cy ici= (Lx i)2; n(b) i=cy ici+ay iai= (Ly i)2; n(c) i=ay iai+by ibi= (Lz i)2: (B3) The doublon hopping operators have a slightly dierent structure that re ects their noncommutivity, i.e., ay ibi=jaiihbji=iLy iLx i; by ici=jbiihcji=iLz iLy i; cy iai=jciihaji=iLx iLz i: (B4) These relations are sucient to write the superexchange Hamiltonian for the host-host and impurity-host bonds in thefLx i;Ly i;Lz igoperator basis for the orbital part. However, in practice it is more convenient to work with real operators L+ i;L i;Lz i rather than with the origi- nal ones,fLx i;Ly i;Lz ig. Thus we write the nal relations which we used for the numerical calculations in terms of these operators, D(a) i=1 4h L+ i2+ L i2i +1 2(Lz i)2; D(b) i=1 4h L+ i2+ L i2i +1 2(Lz i)2;25 D(c) i= 1(Lz i)2; (B5) for the doublon occupation numbers and going directly to the orbital ~ ioperators we nd that, +(a) i =1 2 L iL+ i Lz i; +(b) i=i 2Lz i L+ i+L i ; +(c) i=i 4h L+ i2 L i2i i 2Lz i; (B6) for the o-diagonal part and z(a) i=1 8h L+ i2+ L i2i +3 4(Lz i)21 2; z(b) i=1 8h L+ i2+ L i2i 3 4(Lz i)2+1 2;z(c) i=1 4h L+ i2+ L i2i ; (B7) for the diagonal one. Note that the complex phase in +(b) i and+(c) i is irrelevant and can be omitted here as +( ) i is always accompanied by ( ) j on a neighboring site. This is a consequence of the cubic symmetry in the orbital part of the superexchange Hamiltonian and it can be altered by a presence of a distortion, e.g., octahedral rotation. For completeness we also give the backward relation between angular momentum components, fL ig with=x;y;z , and the orbital operators f( ) ig; these are: Lx i= 2x(a) i; Ly i= 2x(b) i; Lz i= 2y(c) i: (B8) [1] M. Imada, A. Fujimori, and Y. Tokura, Metal-Insulator Transitions , Rev. Mod. Phys. 70, 1039 (1998). [2] E. Berg, E. Fradkin, S. A. Kivelson, and J. M. Tran- quada, Striped Superconductors: how Spin, Charge and Superconducting Orders Intertwine in the Cuprates , New J. Phys. 11, 115004 (2011). [3] P. A. Lee, N. Nagaosa, and X.-G. Wen, Doping a Mott Insulator: Physics of High-Temperature Superconductiv- ity, Rev. Mod. Phys. 78, 17 (2006). [4] M. Vojta, Lattice Symmetry Breaking in Cuprate Super- conductors: Stripes, Nematics, and Superconductivity , Adv. Phys. 58, 699 (2009). [5] K. Yamada, C. H. Lee, K. Kurahashi, J. Wada, S. Waki- moto, S. Ueki, H. Kimura, Y. Endoh, S. Hosoya, G. Shi- rane, R. J. Birgeneau, M. Greven, M. A. Kastner, and Y. J. Kim, Doping Dependence of the Spatially Modulated Dynamical Spin Correlations and the Superconducting- Transition Temperature in La2xSrxCuO 4, Phys. Rev. B 57, 6165 (1998). [6] M. Fleck, A. I. Lichtenstein, and A. M. Ole s, Spectral Properties and Pseudogap in the Stripe Phases of Cuprate Superconductors , Phys. Rev. B 64, 134528 (2001). [7] P. Wr obel and A. M. Ole s, Ferro-Orbitally Ordered Stripes in Systems with Alternating Orbital Order , Phys. Rev. Lett. 104, 206401 (2010). [8] Q. Li, K. E. Gray, S. B. Wilkins, M. Garcia Fernan- dez, S. Rosenkranz, H. Zheng, and J.F. Mitchell, Predic- tion and Experimental Evidence for Thermodynamically Stable Charged Orbital Domain Walls , Phys. Rev. X 4, 031028 (2014). [9] J. Zaanen and A. M. Ole s, Carriers Binding to Excitons: Crystal-Field Excitations in Doped Mott-Hubbard Insula- tors, Phys. Rev. B 48, 7197 (1993). [10] Y. Tokura and N. Nagaosa, Orbital Physics in Transition-Metal Oxides , Science 288, 462 (2000). [11] K. I. Kugel and D. I. Khomskii, The Jahn-Teller Ef- fect and Magnetism: Transition Metal Compounds , Sov. Phys. Usp. 25, 231 (1982).[12] A. M. Ole s, G. Khaliullin, P. Horsch, and L. F. Feiner, Fingerprints of Spin-Orbital Physics in Cubic Mott In- sulators: Magnetic Exchange Interactions and Optical Spectral Weights , Phys. Rev. B 72, 214431 (2005). [13] K. Wohlfeld, M. Daghofer, S. Nishimoto, G. Khaliullin, and J. van den Brink. Intrinsic Coupling of Orbital Ex- citations to Spin Fluctuations in Mott Insulators , Phys. Rev. Lett. 107, 147201 (2011). [14] L. F. Feiner, A. M. Ole s, and J. Zaanen, Quantum Melt- ing of Magnetic Order due to Orbital Fluctuations , Phys. Rev. Lett. 78, 2799 (1997). [15] G. Khaliullin and S. Maekawa, Orbital Liquid in Three- Dimensional Mott Insulator: LaTiO 3, Phys. Rev. Lett. 85, 3950 (2000). [16] G. Khaliullin, Orbital Order and Fluctuations in Mott Insulators , Prog. Theor. Phys. Suppl. 160, 155 (2005). [17] Leon Balents, Spin Liquids in Frustrated Magnets , Na- ture (London) 464, 199 (2010). [18] L. F. Feiner and A. M. Ole s, Orbital Liquid in Ferro- magnetic Manganites: The Orbital Hubbard Model for eg Electrons , Phys. Rev. B 71, 144422 (2005). [19] S. Nakatsuji, K. Kuga, K. Kimura, R. Satake, N. Katayama, E. Nishibori, H. Sawa, R. Ishii, M. Hagi- wara, F. Bridges, T. U. Ito, W. Higemoto, Y. Karaki, M. Halim, A. A. Nugroho, J. A. Rodriguez-Rivera, M. A. Green, and C. Broholm, Spin-Orbital Short-Range Order on a Honeycomb-Based Lattice , Science 336, 559 (2012). [20] J. A. Quilliam, F. Bert, E. Kermarrec, C. Payen, C. Guillot-Deudon, P. Bonville, C. Baines, H. Luetkens, and P. Mendels, Singlet Ground State of the Quantum Antiferromagnet Ba3CuSb 2O9, Phys. Rev. Lett. 109, 117203 (2012). [21] B. Normand and A. M. Ole s, Frustration and Entangle- ment in the t2gSpin-Orbital Model on a Triangular Lat- tice: Valence-bond and Generalized Liquid States , Phys. Rev. B 78, 094427 (2008). [22] P. Corboz, M. Lajk o, A. M. La uchli, K. Penc, and F. Mila, Spin-Orbital Quantum Liquid on the Honeycomb Lattice , Phys. Rev. X 2, 041013 (2012).26 [23] J. Nasu and S. Ishihara, Dynamical Jahn-Teller Eect in a Spin-Orbital Coupled System , Phys. Rev. B 88, 094408 (2012). [24] A. Smerald and F. Mila, Exploring the Spin-Orbital Ground State of Ba3CuSb 2O9, Phys. Rev. B 90, 094422 (2014). [25] V. Fritsch, J. Hemberger, N. B uttgen, E.-W. Scheidt, H.- A. Krug von Nidda, A. Loidl, and V. Tsurkan, Spin and Orbital Frustration in MnSc 2S2andFeSc 2S4, Phys. Rev. Lett. 92, 116401 (2004). [26] G. Chen, L. Balents, and A. P. Schnyder, Spin-Orbital Singlet and Quantum Critical Point on the Diamond Lat- tice: FeSc 2S4, Phys. Rev. Lett. 102, 096406 (2009). [27] L. Mittelst adt, M. Schmidt, Z. Wang, F. Mayr, V. Tsurkan, P. Lunkenheimer, D. Ish, L. Balents, J. Deisenhofer, and A. Loidl, Spin-Orbiton and Quantum Criticality in FeSc 2S4, Phys. Rev. B 91, in press (2015). [28] F. Vernay, K. Penc, P. Fazekas, and F. Mila, Orbital De- generacy as a Source of Frustration in LiNiO 2, Phys. Rev. B 70, 014428 (2004). [29] W. Brzezicki, J. Dziarmaga, and A. M. Ole s, Non- collinear Magnetic Order Stabilized by Entangled Spin- Orbital Fluctuations , Phys. Rev. Lett. 109, 237201 (2012). [30] W. Brzezicki, J. Dziarmaga, and A. M. Ole s, Exotic Spin Orders Driven by Orbital Fluctuations in the Kugel- Khomskii Model , Phys. Rev. B 87, 064407 (2013). [31] P. Czarnik and J. Dziarmaga, Striped Critical Spin Liquid in a Spin-Orbital Entangled RVB State in a Projected Entangled-Pair State Representation , Phys. Rev. B 91, 045101 (2015). [32] F. Reynaud, D. Mertz, F. Celestini, J. Debierre, A. M. Ghorayeb, P. Simon, A. Stepanov, J. Voiron, and C. Delmas, Orbital Frustration at the Origin of the Mag- netic Behavior in LiNiO 2, Phys. Rev. Lett. 86, 3638 (2001). [33] A. Reitsma, L. F. Feiner, and A. M. Ole s, Orbital and Spin Physics in LiNiO 2and NaNiO 2, New J. Phys. 7, 121 (2005). [34] A. M. Ole s, Fingerprints of Spin-Orbital Entanglement in Transition Metal Oxides , J. Phys.: Condens. Matter 24, 313201 (2012). [35] E. Dagotto, T. Hotta, and A. Moreo, Colossal Magne- toresistant Materials: The Key Role of Phase Separation , Phys. Rep. 344, 1 (2001). [36] E. Dagotto, Open Questions in CMR Manganites, Rel- evance of Clustered States and Analogies with Other Compounds Including the Cuprates , New J. Phys. 7, 67 (2005). [37] Y. Tokura, Critical Features of Colossal Magnetoresistive Manganites , Rep. Prog. Phys. 69, 797 (2006). [38] L. F. Feiner and A. M. Ole s, Electronic Origin of Mag- netic and Orbital Ordering in Insulating LaMnO 3, Phys. Rev. B 59, 3295 (1999). [39] R. Kilian and G. Khaliullin, Orbital Polarons in the Metal-Insulator Transition of Manganites , Phys. Rev. B 60, 13458 (1999). [40] M. Cuoco, C. Noce, and A. M. Ole s, Origin of the Optical Gap in Half-Doped Manganites , Phys. Rev. B 66, 094427 (2002). [41] A. Weisse and H. Fehske, Microscopic Modelling of Doped Manganites , New J. Phys. 6, 158 (2004). [42] M. Daghofer, A. M. Ole s, and W. von der Linden, Orbital Polarons versus Itinerant egElectrons in Doped Mangan-ites, Phys. Rev. B 70, 184430 (2004). [43] J. Geck, P. Wochner, S. Kiele, R. Klingeler, P. Reutler, A. Revcolevschi, and B. B uchner, Orbital Polaron Lattice Formation in Lightly Doped La1xSrxMnO 3, Phys. Rev. Lett. 95, 236401 (2005). [44] A. M. Ole s and G. Khaliullin, Dimensional Crossover and the Magnetic Transition in Electron Doped Manganites , Phys. Rev. B 84, 214414 (2011). [45] Y. M. Sheu, S. A. Trugman, L. Yan, J. Qi, Q. X. Jia, A. J. Taylor, and R. P. Prasankumar, Polaronic Trans- port Induced by Competing Interfacial Magnetic Order in aLa0:7Ca0:3MnO 3/BiFeO 3Heterostructure , Phys. Rev. X4, 021001 (2014). [46] P.-G. de Gennes, Eects of Double Exchange in Magnetic Crystals , Phys. Rev. 118, 141 (1960). [47] G. Khaliullin and R. Kilian, Theory of Anomalous Magnon Softening in Ferromagnetic Manganites , Phys. Rev. B 61, 3494 (2000). [48] A. M. Ole s and L. F. Feiner, Why Spin Excitations in Metallic Ferromagnetic Manganites are Isotropic , Phys. Rev. B 65, 052414 (2002). [49] H. Sakai, S. Ishiwata, D. Okuyama, A. Nakao, H. Nakao, Y. Murakami, Y. Taguchi, and Y. Tokura, Electron Dop- ing in the Cubic Perovskite SrMnO 3:Isotropic Metal ver- sus Chainlike Ordering of Jahn-Teller Polarons , Phys. Rev. B 82, 180409 (2010). [50] G. Khaliullin, P. Horsch, and A. M. Ole s, Spin Order due to Orbital Fluctuations: Cubic Vanadates , Phys. Rev. Lett. 86, 3879 (2001). [51] G. Khaliullin, P. Horsch, and A. M. Ole s, Theory of Op- tical Spectral Weights in Mott Insulators with Orbital De- grees of Freedom , Phys. Rev. B 70, 195103 (2004). [52] P. Horsch, A. M. Ole s, L. F. Feiner, and G. Khaliullin, Evolution of Spin-Orbital-Lattice Coupling in the RVO3 Perovskites , Phys. Rev. Lett. 100, 167205 (2008). [53] C. Ulrich, G. Khaliullin, J. Sirker, M. Reehuis, M. Ohl, S. Miyasaka, Y. Tokura, and B. Keimer, Magnetic Neu- tron Scattering Study of YVO 3:Evidence for an Orbital Peierls State , Phys. Rev. Lett. 91, 257202 (2003). [54] J.-S. Zhou, J. B. Goodenough, J.-Q. Yan, and Y. Ren, Superexchange Interaction in Orbitally Fluctu- ating RVO 3, Phys. Rev. Lett. 99, 156401 (2007). [55] M. Reehuis, C. Ulrich, K. Proke s, S. Mat'a s, J. Fu- jioka, S. Miyasaka, Y. Tokura, and B. Keimer, Structural and Magnetic Phase Transitions of the Orthovanadates RVO3(R=Dy,Ho,Er) as seen via Neutron Diraction , Phys. Rev. B 83, 064404 (2011). [56] J. Fujioka, T. Yasue, S. Miyasaka, Y. Yamasaki, T. Arima, H. Sagayama, T. Inami, K. Ishii, and Y. Tokura, Critical Competition between two Distinct Orbital-Spin Ordered States in Perovskite Vanadates , Phys. Rev. B 82, 144425 (2010). [57] P. Horsch, G. Khaliullin, and A. M. Ole s, Dimeriza- tion versus Orbital-Moment Ordering in a Mott Insulator YVO 3, Phys. Rev. Lett. 91, 257203 (2003). [58] J. Fujioka, S. Miyasaka, and Y. Tokura, Doping Vari- ation of Anisotropic Charge and Orbital Dynamics in Y1xCaxVO3:Comparison with La1xSrxVO3, Phys. Rev. B 77, 144402 (2008). [59] P. Horsch and A. M. Ole s, Defect States and Spin-Orbital Physics in Doped Vanadates Y1xCaxVO3, Phys. Rev. B 84, 064429 (2011). [60] A. Avella, P. Horsch, and A. M. Ole s, Defect States and Excitations in a Mott Insulator with Orbital Degrees of27 Freedom: Mott-Hubbard Gap versus Optical and Trans- port Gaps in Doped Systems , Phys. Rev. B 87, 045132 (2013). [61] T. Hotta, Orbital Ordering Phenomena in d- andf- Electron Systems , Rep. Prog. Phys. 69, 2061 (2006). [62] T. Mizokawa, L. H. Tjeng, G. A. Sawatzky, G. Ghir- inghelli, O. Tjernberg, N. B. Brookes, H. Fukazawa, S. Nakatsuji, and Y. Maeno, Spin-Orbit Coupling in the Mott Insulator Ca2RuO 4, Phys. Rev. Lett. 87, 077202 (2001). [63] J. S. Lee, Y. S. Lee, T. W. Noh, S.-J. Oh, J. Yu, S. Nakat- suji, H. Fukazawa, and Y. Maeno, Electron and Orbital Correlations in Ca2xSrxRuO 4Probed by Optical Spec- troscopy , Phys. Rev. Lett. 89, 257402 (2002). [64] A. Koga, N. Kawakami, T. M. Rice, and M. Sigrist, Orbital-Selective Mott Transitions in the Degenerate Hubbard Model , Phys. Rev. Lett. 92, 216402 (2004). [65] Z. Fang, K. Terakura, and N. Nagaosa, Orbital Physics in Ruthenates: First-Principles Study , New J. Phys. 7, 66 (2005). [66] T. Sugimoto, D. Ootsuki, and T. Mizokawa, Impact of Local Lattice Disorder on Spin and Orbital Orders in Ca2xSrxRuO 4, Phys. Rev. B 84, 064429 (2011). [67] G. Khaliullin, Excitonic Magnetism in Van Vleck-type d4 Mott Insulators , Phys. Rev. Lett. 111, 197201 (2013). [68] A. Akbari and G. Khaliullin, Magnetic Excitations in a Spin-Orbit-Coupled d4Mott Insulator on the Square Lat- tice, Phys. Rev. B 90, 035137 (2014). [69] C. N. Veenstra, Z.-H. Zhu, M. Raichle, B. M. Lud- brook, A. Nicolaou, B. Slomski, G. Landolt, S. Kittaka, Y. Maeno, J. H. Dil, I. S. Elmov, M. W. Haverkort, and A. Damascelli, Spin-Orbital Entanglement and the Breakdown of Singlets and Triplets in Sr2RuO 4Revealed by Spin- and Angle-Resolved Photoemission Spectroscopy , Phys. Rev. Lett. 112, 127002 (2014). [70] T. Tanaka and S. Ishihara, Dilution Eects in Two- Dimensional Quantum Orbital Systems , Phys. Rev. Lett. 98, 256402 (2007). [71] T. F. Qi, O. B. Korneta, S. Parkin, L. E. De Long, P. Schlottmann, and G. Cao, Negative Volume Ther- mal Expansion via Orbital and Magnetic Orders in Ca2Ru1xCrxO4(0< x < 0:13), Phys. Rev. Lett. 105, 177203 (2010). [72] G. Cao, S. Chikara, X. N. Lin, E. Elhami, V. Durairaj, and P. Schlottmann, Itinerant Ferromagnetism to Insu- lating Antiferromagnetism: A Magnetic and Transport Study of Single Crystal SrRu 1xMnxO3(0x<0:60), Phys. Rev. B 71, 035104 (2005). [73] M. A. Hossain, B. Bohnenbuck, Y. D. Chuang, M. W. Haverkort, I. S. Elmov, A. Tanaka, A. G. Cruz Gonzalez, Z. Hu, H.-J. Lin, C. T. Chen, R. Math- ieu, Y. Tokura, Y. Yoshida, L. H. Tjeng, Z. Hus- sain, B. Keimer, G. A. Sawatzky, and A. Damascelli, Mott versus Slater-type Metal-Insulator Iransition in Mn- Substituted Sr3Ru2O7, Phys. Rev. B 86, 041102(R) (2012). [74] M. A. Hossain, Z. Hu, M. W. Haverkort, T. Bur- nus, C. F. Chang, S. Klein, J. D. Denlinger, H.- J. Lin, C. T. Chen, R. Mathieu, Y. Kaneko, Y. Tokura, S. Satow, Y. Yoshida, H. Takagi, A. Tanaka, I. S. Elmov, G. A. Sawatzky, L. H. Tjeng, and A. Damascelli, Crystal- Field Level Inversion in Lightly Mn-Doped Sr3Ru2O7,Phys. Rev. Lett. 101, 016404 (2008). [75] D. Mesa, F. Ye, S. Chi, J. A. Fernandez-Baca, W. Tian, B. Hu, R. Jin, E. W. Plummer, and J. Zhang, Single- Bilayer E-type Antiferromagnetism in Mn-Substituted Sr3Ru2O7:Neutron Scattering Study , Phys. Rev. B 85, 180410(R) (2012). [76] S. A. J. Kimber, J. A. Rodgers, H. Wu, C. A. Murray, D. N. Argyriou, A. N. Fitch, D. I. Khomskii, and J. P. Atteld, Metal-Insulator Transition and Orbital Order in PbRuO 3, Phys. Rev. Lett. 102, 046409 (2009). [77] J. F. Annett, G. Litak, B. L. Gy ory, and K. I. Wysoki nski, Spin-Orbit Coupling and Symmetry of the Order Parameter in Strontium Ruthenate , Phys. Rev. B73, 134501 (2006). [78] A. B. Harris, T. Yildirim, A. Aharony, O. Entin- Wohlman, and I. Ya. Korenblit, Unusual Symmetries in the Kugel-Khomskii Hamiltonian , Phys. Rev. Lett. 91, 087206 (2003). [79] M. Daghofer, K. Wohlfeld, A. M. Ole s, E. Arrigoni, and P. Horsch, Absence of Hole Connement in Transition- Metal Oxides with Orbital Degeneracy , Phys. Rev. Lett. 100, 066403 (2008). [80] A. M. Ole s, Antiferromagnetism and Correlation of Elec- trons in Transition Metals , Phys. Rev. B 28, 327 (1983). [81] M. Cuoco, F. Forte, and C. Noce, Interplay of Coulomb Interactions and c-Axis Octahedra Distortions in Single- Layer Ruthenates , Phys. Rev. B 74, 195124 (2006). [82] P. Marra, K. Wohlfeld, and J. van den Brink, Unraveling Orbital Correlations with Magnetic Resonant Inelastic X- Ray Scattering , Phys. Rev. Lett. 109, 117401 (2012). [83] M. A. Hossain, I. Zegkinoglou, Y.-D. Chuang, J. Geck, B. Bohnenbuck, A. G. Cruz Gonzalez, H.-H. Wu, C. Sch uler-Langeheine, D. G. Hawthorn, J. D. Den- linger, R. Mathieu, Y. Tokura, S. Satow, H. Takagi, Y. Yoshida, Z. Hussain, B. Keimer, G. A. Sawatzky, and A. Damascelli, Electronic Superlattice Revealed by Reso- nant Scattering from Random Impurities in Sr3Ru2O7, Sci. Rep. 3, 2299 (2013). [84] L. Limot, P. Mendels, G. Collin, C. Mondelli, B. Oulad- diaf, H. Mutka, N. Blanchard, and M. Mekata, Suscep- tibility and Dilution Eects of the kagom e Bilayer Ge- ometrically Frustrated Network: A Ga NMR study of SrCr 9pGa129pO19, Phys. Rev. B 65, 144447 (2002). [85] J. Bobro, N. La orencie, L. K. Alexander, A. V. Maha- jan, B. Koteswararao, and P. Mendels, Impurity-Induced Magnetic Order in Low-Dimensional Spin-Gapped Mate- rials, Phys. Rev. Lett. 103, 047201 (2009). [86] A. Sen, K. Damle, and R. Moessner, Fractional Spin Tex- tures in the Frustrated Magnet SrCr 9pGa129pO19, Phys. Rev. Lett. 106, 127203 (2011). [87] P. Bonf a, P. Carretta, S. Sanna, G. Lamura, G. Prando, A. Martinelli, A. Palenzona, M. Tropeano, M. Putti, and R. De Renzi, Magnetic Properties of Spin-diluted Iron Pnictides from SR and NMR in LaFe 1xRuxAsO, Phys. Rev. B 85, 054518 (2012). [88] A. K. Paul, M. Reehuis, V. Ksenofontov, B. Yan, A. Hoser, D. M. T obbens, P. M. Abdala, P. Adler, M. Jansen, and C. Felser, Lattice Instability and Com- peting Spin Structures in the Double Perovskite Insulator Sr2FeOsO 6, Phys. Rev. Lett. 111, 167205 (2013).
2102.01400v2.Coupled_spin_orbital_fluctuations_in_a_three_orbital_model_for__4d__and__5d__oxides_with_electron_fillings__n_3_4_5_____Application_to___rm_NaOsO_3_____rm_Ca_2RuO_4___and___rm_Sr_2IrO_4_.pdf	arXiv:2102.01400v2 [cond-mat.str-el] 20 Mar 2021Coupled spin-orbital ﬂuctuations in a three orbital model f or4d and5doxides with electron ﬁllings n= 3,4,5— Application to NaOsO 3, Ca2RuO4, and Sr 2IrO4 Shubhajyoti Mohapatra and Avinash Singh∗ Department of Physics, Indian Institute of Technology, Kanpu r - 208016, India (Dated: March 23, 2021) A uniﬁed approach is presented for investigating coupled sp in-orbital ﬂuctuations within a realistic three-orbital model for strongly spin-o rbit coupled systems with electron ﬁllings n= 3,4,5 in the t2gsector of dyz,dxz,dxyorbitals. A generalized ﬂuctuation propagator is constructed which is consistent w ith the generalized self- consistent Hartree-Fock approximation where all Coulomb i nteraction contributions involving orbital diagonal and oﬀ-diagonal spin and charge c ondensates are included. Besides the low-energy magnon, intermediate-energy orbit on and spin-orbiton, and high-energy spin-orbit exciton modes, the generalized spe ctral function also shows other high-energy excitations such as the Hund’s coupling i nduced gapped magnon modes. We relate the characteristic features of the coupled spin-orbital excitations to the complex magnetic behavior resulting from the interpl ay between electronic bands, spin-orbit coupling, Coulomb interactions, and str uctural distortion eﬀects, as realized in the compounds NaOsO 3, Ca2RuO4, and Sr 2IrO4.2 I. INTRODUCTION The 4dand 5dtransition metal (TM) oxides exhibit an unprecedented coupling bet ween spin, charge, orbital, and structural degrees of freedom. The c omplex interplay between the diﬀerent physical elements such as strong spin-orbit coupling ( SOC), Coulomb interac- tions, and structural distortions results in novel magnetic state s and unconventional collec- tive excitations.1–6In particular, the cubic structured NaOsO 3and perovskite structured Ca2RuO4and Sr 2IrO4compounds, corresponding to dnelectronic conﬁguration of the TM ion with electron ﬁllings n=3,4,5 in the t 2gsector, respectively, are at the emerging research frontier as they provide versatile platform for the exploration of S OC-driven phenomena involving collective electronic and magnetic behavior including coupled s pin-orbital excita- tions. Thediﬀerent physical elements giverisetoarichvarietyofnontrivia l microscopic features which contribute to the complex interplay. These include spin-orbita l-entangled states, bandnarrowing, spin-orbit gap, andexplicit spin-rotation-symmet ry breaking (dueto SOC), electronic band narrowing due to reduced eﬀective hopping (octah edral tilting and rotation), crystal ﬁeld induced tetragonal splitting (octahedral compress ion), orbital mixing (SOC and octahedral tilting, rotation) which self consistently generate s induced SOC terms and orbitalmomentinteractionfromtheCoulombinteractionterms, sig niﬁcantlyweakerelectron correlation term Ucompared to 3 dorbitals and therefore critical contribution of Hund’s coupling to local magnetic moment. These microscopic features con tribute to the complex interplay in diﬀerent ways for electron ﬁllings n=3,4,5, resulting in signiﬁcantly diﬀerent macroscopic properties of the three compounds, which are brieﬂy reviewed below along with experimental observations about the collective and coupled sp in-orbital excitations as obtained from recent resonant inelastic X-ray scattering (RIXS) studies. The nominally orbitally quenched d3compound NaOsO 3undergoes a metal-insulator transition (MIT) ( TMI=TN= 410 K) that is closely related to the onset of long-range antiferromagnetic (AFM) order.7–10Various mechanisms, such as Slater-like, magnetic Lif- shitz transition, and AFM band insulator have been proposed to exp lain this unusual and intriguing nature of the MIT.8,11–14Interplay of electronic correlations, Hund’s coupling, and octahedral tilting and rotation induced band narrowing near th e Fermi level in this weakly correlated compound results in the weakly insulating state wit h G-type AFM or-3 der, with magnetic anisotropy and large magnon gap resulting from in terplay of SOC, band structure, and the tetragonal splitting.14,15The OsL3resonant edge RIXS measurements at room temperature show four inelastic peak features below 1.5 eV , which have been in- terpreted to correspond to the strongly gapped ( ∼58 meV) dispersive magnon excitations with bandwidth ∼100 meV, excitations (centered at ∼1 eV) within the t2gmanifold, and excitations from t2gtoegstates and ligand-to-metal charge transfer for the remaining tw o higher-energypeaks.13,16,17Theintensity andpositionsofthethreehigh-energypeaksappear to be essentially temperature independent. The nominally spin S=1d4compound Ca 2RuO4undergoes a MIT at TMI=357 K and magnetic transition at TN=110 K ( ≪TMI) via a structural phase transition involving a compressive tetragonal distortion, tilt, and rotation of the RuO 6octahedra.18–21The low- temperature AFM insulating phase is thus characterized by highly dis torted octahedra with nominally ﬁlled xyorbital and half-ﬁlled yz,xzorbitals.22–24This transition has also been identiﬁed in pressure,25–27chemical substitution,28–30strain,31and electrical current studies,32,33and highlights the complex interplay between SOC, Coulomb interactio ns, and structural distortions. Inelastic neutron scattering (INS)34–36and Raman37studies on Ca 2RuO4have revealed unconventional low-energy ( ∼50 and 80 meV) excitations interpreted as gapped trans- verse magnon modes and possibly soft longitudinal (“Higgs-like”) or two-magnon excitation modes. From both Ru L3-edge and oxygen K-edge RIXS studies, multiple nontrivial exci- tations within the t2gmanifold were observed recently below 1 eV.38–40Two low-energy ( ∼ 80 and 350 meV) and two high-energy ( ∼750 meV and 1 eV) excitations were identiﬁed within the limited energy resolution of RIXS. From the incident angle an d polarisation de- pendence of the RIXS spectra, the orbital character of the 80 m eV peak was inferred to be mixture of xyandxz/yzstates, whereas the 0.4 eV peak was linked to unoccupied xz/yz states. Guided by phenomenological spin models, the low-energy ex citations (consisting of multiple branches) were interpreted as composite spin-orbital exc itations (also termed as “spin orbitons”). Finally, SOC induced novel Mott insulating state is realized in the d5compound Sr2IrO4,41,42where band narrowing of the spin-orbital-entangled electronic sta tes near the Fermi level plays a critical role in the insulating behavior. The AFM insu lating ground state is characterized by the correlation induced insulating gap with in the nominally J=1/24 bandsemerging fromtheKramersdoublet, which areseparatedfr omthebands ofthe J=3/2 quartet by energy 3 λ/2, whereλis the SOC strength. The RIXS spectra show low-energy dispersive magnon excitations (up to 200 meV), further resolved in to two gapped magnon modes with energy gaps ∼40 meV and 3 meV at the Γ point corresponding to out-of- plane and in-plane ﬂuctuation modes, respectively.43–46Weak electron correlation eﬀect and mixing between the J=1/2 and 3/2 sectors were identiﬁed as contributing signiﬁcantly to the strong zone-boundary magnon dispersion as measured in RIXS studies.47In addition, high-energy dispersive spin-orbit exciton modes have also been rev ealed in RIXS studies in the energy range 0.4-0.8 eV.48This distinctive mode is also referred to as the spin-orbiton mode,49,50and has been attributed to the correlated motion of electron-hole pair excitations across the renormalized spin-orbit gap between the J=1/2 and 3/2 bands.51 Most of the theoretical studies involving magnetic anisotropy eﬀec ts and excitations in abovesystems havemainlyfocusedonphenomenological spinmodels withdiﬀerent exchange interactions obtained as ﬁtting parameters to the experimental s pectra. However, the in- terpretation of experimental data remains incomplete since the ch aracter of the eﬀective spins, the microscopic origin of their interactions, and the microsco pic nature of the mag- netic excitations are still debated.2–6Realistic information about the spin-orbital character of both low and high-energy collective excitations, as inferred from the study of coupled spin-orbital excitations, is clearly important since the spin and orbit al degrees of freedom are explicitly coupled, and both are controlled by the diﬀerent physic al elements such as SOC, Coulomb interaction terms, tetragonal compression induced crystal-ﬁeld splitting be- tweenxyandyz,xzorbitals, octahedral tilting and rotation induced orbital mixing hopp ing terms, and band physics. Due to the intimately intertwined roles of the diﬀerent physical eleme nts, a uniﬁed ap- proach is therefore required for the realistic modeling of these sys tems in which all physical elements are treated on an equal footing. The generalized self-co nsistent approximation ap- pliedrecentlytothe n= 4compoundCa 2RuO4providessuch auniﬁedapproach.52Involving theself-consistent determinationofmagneticorderwithinathree -orbitalinteractingelectron model including all orbital-diagonal and oﬀ-diagonal spin and charge condensates generated by the diﬀerent Coulomb interaction terms, this approach explicitly in corporates the com- plex interplay and accounts for the observed behavior including the tetragonal distortion induced magnetic reorientation transition, orbital moment interac tion induced orbital gap,5 SOC and octahedral tilting induced easy-axis anisotropy, and Coulo mb interaction induced anisotropic SOC renormalization. Extension to the n= 5 compound Sr 2IrO4,53provides conﬁrmation of the Hund’s coupling induced easy-plane magnetic anis otropy, which is re- ponsible for the ∼40 meV magnon gap measured for the out-of-plane ﬂuctuation mod e.46 Towards a generalized non-perturbative formalism unifying the mag netic order and anisotropy eﬀects on one hand and collective excitations on the oth er, the natural exten- sion of the above generalized condensate approach is therefore t o consider the generalized ﬂuctuation propagator in terms of the generalized spin ( ψ† µ[σα]ψν) and charge ( ψ† µ[1]ψν) op- erators in the pure spin-orbital basis of the t 2gorbitalsµ,ν=yz,xz,xy and spin components α=x,y,z. The generalized operators include the normal ( µ=ν) spin and charge opera- tors as well as the orbital oﬀ-diagonal ( µ/negationslash=ν) cases which are related to the generalized spin-orbit coupling terms ( LαSβ, whereα,β=x,y,z) and the orbital angular momentum operatorsLα. Constructing the generalized ﬂuctuation propagator as above w ill ensure that this scheme is fully consistent with the generalized self-consistent a pproach involving the generalized condensates. The diﬀerent components of the generalized ﬂuctuation propagat or will therefore natu- rallyincludespin-orbitonsandorbitons, corresponding tothespin- orbital(LαSβ)andorbital (Lα) moment ﬂuctuations, besides the normal spin andchargeﬂuctua tions. The normal spin ﬂuctuations will include in-phase and out-of phase ﬂuctuations with respect to diﬀerent or- bitals, the latter being strongly gapped due to Hund’s coupling. The s pin-orbitons will include the spin-orbit excitons measured in RIXS studies of Sr 2IrO4. The structure of this paper is as below. The three-orbital model w ithin the t 2gsector (including SOC, hopping, Coulomb interaction, and structural disto rtion terms), and the generalized self-consistent formalism including orbital diagonal and oﬀ-diagonal condensates are reviewed in Sec. II and III. After introducing the generalized ﬂ uctuation propagator in Sec. IV, results of the calculated ﬂuctuation spectral functions are presented for the cases n= 3,4,5 (corresponding to the three compounds NaOsO 3, Ca2RuO4, Sr2IrO4) in Sections V, VI, VII. Finally, conclusions are presented in Sec. VIII. The bas is-resolved contributions to the total spectral function showing the detailed spin-orbital c haracter of the collective excitations are presented in the Appendix.6 II. THREE ORBITAL MODEL WITH SOC AND COULOMB INTERACTIONS In the three-orbital ( µ=yz,xz,xy ), two-spin ( σ=↑,↓) basis deﬁned with respect to a common spin-orbital coordinate axes (Fig. 1), we consider the Ham iltonianH=Hband+ Hcf+Hint+HSOCwithin the t2gmanifold. For the band and crystal ﬁeld terms together, we consider: Hband+cf=/summationdisplay kσsψ† kσs ǫyz k′0 0 0ǫxz k′0 0 0ǫxy k′+ǫxy δss′+ ǫyz kǫyz\|xz kǫyz\|xy k −ǫyz\|xz kǫxz kǫxz\|xy k −ǫyz\|xy k−ǫxz\|xy kǫxy k δ¯ss′ ψkσs′ (1) in the composite three-orbital, two-sublattice ( s,s′= A,B) basis. Here the energy oﬀset ǫxy(relative to the degenerate yz/xzorbitals) represents the tetragonal distortion induced crystal ﬁeld eﬀect. The band dispersion terms in the two groups co rrespond to hopping terms connecting the same and opposite sublattice(s), and are giv en by: ǫxy k=−2t1(coskx+cosky) ǫxy k′=−4t2coskxcosky−2t3(cos2kx+cos2ky) ǫyz k=−2t5coskx−2t4cosky ǫxz k=−2t4coskx−2t5cosky ǫyz\|xz k=−2tm1(coskx+cosky) ǫxz\|xy k=−2tm2(2coskx+cosky) ǫyz\|xy k=−2tm3(coskx+2cosky). (2) Heret1,t2,t3are respectively the ﬁrst, second, and third neighbor hopping ter ms for thexyorbital. For the yz(xz) orbital,t4andt5are the nearest-neighbor (NN) hopping terms iny(x) andx(y) directions, respectively, corresponding to πandδorbital overlaps. Octahedral rotation and tilting induced orbital mixings are represe nted by the NN hopping termstm1(betweenyzandxz) andtm2,tm3(betweenxyandxz,yz). In then= 4 case corresponding to the Ca 2RuO4compound, we have taken hopping parameter values: ( t1,t2, t3,t4,t5)=(−1.0,0.5,0,−1.0,0.2),orbital mixing hopping terms: tm1=0.2andtm2=tm3=0.15 (≈0.2/√ 2), andǫxy=−0.8, all in units of the realistic hopping energy scale \|t1\|=150 meV.54–56The choice tm2=tm3corresponds to the octahedral tilting axis oriented along7 (a)(b) FIG. 1: (a) The common spin-orbital coordinate axes ( x−y) along the Ru-O-Ru directions, shown along with the crystal axes a,b. (b) Octahedral tilting about the crystal aaxis is resolved along thex,yaxes, resulting in orbital mixing hopping terms between the xyandyz,xzorbitals. the±(−ˆx+ ˆy) direction, which is equivalent to the crystal ∓adirection (Fig. 1). The tm1 andtm2,m3values taken above approximately correspond to octahedral rot ation and tilting angles of about 12◦(≈0.2 rad) as reported in experimental studies.26 For the on-site Coulomb interaction terms in the t2gbasis (µ,ν=yz,xz,xy ), we consider: Hint=U/summationdisplay i,µniµ↑niµ↓+U′/summationdisplay i,µ<ν,σniµσniνσ+(U′−JH)/summationdisplay i,µ<ν,σniµσniνσ +JH/summationdisplay i,µ/negationslash=νa† iµ↑a† iν↓aiµ↓aiν↑+JP/summationdisplay i,µ/negationslash=νa† iµ↑a† iµ↓aiν↓aiν↑ =U/summationdisplay i,µniµ↑niµ↓+U′′/summationdisplay i,µ<νniµniν−2JH/summationdisplay i,µ<νSiµ.Siν+JP/summationdisplay i,µ/negationslash=νa† iµ↑a† iµ↓aiν↓aiν↑(3) including the intra-orbital ( U) and inter-orbital ( U′) density interaction terms, the Hund’s coupling term ( JH), and the pair hopping interaction term ( JP), withU′′≡U′−JH/2 = U−5JH/2 from the spherical symmetry condition U′=U−2JH. Herea† iµσandaiµσare the electron creation and annihilation operators for site i, orbitalµ, spinσ=↑,↓. The density operator niµσ=a† iµσaiµσ, total density operator niµ=niµ↑+niµ↓=ψ† iµψiµ, and spin density operator Siµ=ψ† iµσψiµin terms of the electron ﬁeld operator ψ† iµ= (a† iµ↑a† iµ↓). All interaction terms above are SU(2) invariant and thus possess spin rotation symmetry. Finally, for the bare spin-orbit coupling term (for site i), we consider the spin-space8 representation: HSOC(i) =−λL.S=−λ(LzSz+LxSx+LySy) = /parenleftig ψ† yz↑ψ† yz↓/parenrightig/parenleftig iσzλ/2/parenrightig ψxz↑ ψxz↓ +/parenleftig ψ† xz↑ψ† xz↓/parenrightig/parenleftig iσxλ/2/parenrightig ψxy↑ ψxy↓ +/parenleftig ψ† xy↑ψ† xy↓/parenrightig/parenleftig iσyλ/2/parenrightig ψyz↑ ψyz↓ +H.c. (4) which explicitly breaks SU(2) spin rotation symmetry and therefore generates anisotropic magnetic interactions from its interplay with other Hamiltonian terms . Here we have used the matrix representation: Lz= 0−i0 i0 0 0 0 0 , Lx= 0 0 0 0 0−i 0i0 , Ly= 0 0i 0 0 0 −i0 0 , (5) for the orbital angular momentum operators in the three-orbital (yz,xz,xy ) basis. As the orbital “hopping” terms in Eq. (4) have the same form as spin -dependent hopping termsiσ.t′ ij, carrying out the strong-coupling expansion57for the−λLzSzterm to second order inλyields the anisotropic diagonal (AD) intra-site interactions: [H(2) eﬀ](z) AD(i) =4(λ/2)2 U/bracketleftbig Sz yzSz xz−(Sx yzSx xz+Sy yzSy xz)/bracketrightbig (6) betweenyz,xzmoments if these orbitals arenominally half-ﬁlled, as in thecase ofCa 2RuO4. This term explicitly yields preferential x−yplane ordering (easy-plane anisotropy) for parallelyz,xzmoments, as enforced by the relatively stronger Hund’s coupling. Similarly, from the strong coupling expansion for the other two SOC t erms, we obtain additional anisotropic interaction terms which are shown below to yie ldC4symmetric easy- axis anisotropy within the easy plane. From the −λLxSxand−λLySyterms, we obtain: [H(2) eﬀ](x,y) AD(i) =4(λ/2)2 U/bracketleftbig Sx xzSx xy−(Sy xzSy xy+Sz xzSz xy)/bracketrightbig +4(λ/2)2 U/bracketleftbig Sy xySy yz−(Sx xySx yz+Sz xySz yz)/bracketrightbig (7) Neglecting the terms involving the Szcomponents which are suppressed by the easy-plane anisotropy discussed above, we obtain: [H(2) eﬀ](x,y) AD(i) =−4(λ/2)2 U/bracketleftbig Sx xy(Sx yz−Sx xz)+Sy xy(Sy xz−Sy yz)/bracketrightbig =−4(λ/2)2 UfxyS2[sin2φsinφc] (8)9 where the spin components are expressed as: Sx xy=fxyScosφ,Sx yz=Scos(φ−φc),Sx xz= Scos(φ+φc) (and similarly for the ycomponents) in terms of the overall orientation angle φ of the magnetic order and the relative canting angle 2 φcbetween the yz,xzmoments. Here the factorfxy<1 represents the reduced moment for the xyorbital. The above expression shows the composite orientation and canting angle dependence of the anisotropic interaction energy having the C4symmetry. Minimum energy is obtained at orientations φ=nπ/4 (wheren= 1,3,5,7) since the canting angle has the approximate functional form φc≈φmax csin2φin terms of the orientation φ. Thus, while the easy-plane anisotropy involves only the yz,xzmoments, the xymoment plays a crucial role in the easy-axis anisotropy, which is directly relevant for NaOsO 3(xyorbital is also nominally half-ﬁlled), but also for Ca 2RuO4with the factor fxyas incorporated above. For later reference, we note here that condensates of the orbit al oﬀ-diagonal (OOD) one- body operators as in Eq. (4) directly yield physical quantities such a s orbital magnetic moments and spin-orbital correlations: /angbracketleftLα/angbracketright=−i/bracketleftbig /angbracketleftψ† µψν/angbracketright−/angbracketleftψ† µψν/angbracketright∗/bracketrightbig = 2 Im/angbracketleftψ† µψν/angbracketright /angbracketleftLαSα/angbracketright=−i/bracketleftbig /angbracketleftψ† µσαψν/angbracketright−/angbracketleftψ† µσαψν/angbracketright∗/bracketrightbig /2 = Im/angbracketleftψ† µσαψν/angbracketright λint α= (U′′−JH/2)/angbracketleftLαSα/angbracketright= (U′′−JH/2)Im/angbracketleftψ† µσαψν/angbracketright (9) where the orbital pair ( µ,ν) corresponds to the component α=x,y,z, and the last equation yields the interaction induced SOC renormalization, as discussed in th e next section. III. SELF-CONSISTENT DETERMINATION OF MAGNETIC ORDER We consider the various contributions from the Coulomb interaction terms (Eq. 3) in the HF approximation, focussing ﬁrst on terms with normal (orbital dia gonal) spin and charge condensates. The resulting local spin and charge terms can be writ ten as: [HHF int]normal=/summationdisplay iµψ† iµ[−σ.∆iµ+Eiµ1]ψiµ (10) where the spin and charge ﬁelds are self-consistently determined f rom: 2∆α iµ=U/angbracketleftσα iµ/angbracketright+JH/summationdisplay ν<µ/angbracketleftσα iν/angbracketright(α=x,y,z) Eiµ=U/angbracketleftniµ/angbracketright 2+U′′/summationdisplay ν<µ/angbracketleftniν/angbracketright (11)10 in terms of the local charge density /angbracketleftniµ/angbracketrightand the spin density components /angbracketleftσα iµ/angbracketright. There are additional contributions resulting from orbital oﬀ-diago nal (OOD) spin and charge condensates which are ﬁnite due to orbital mixing induced by SOC and structural distortions (octahedral tilting and rotation). The contributions c orresponding to diﬀerent Coulomb interaction terms are summarized in Appendix A, and can be g rouped in analogy with Eq. (10) as: [HHF int]OOD=/summationdisplay i,µ<νψ† iµ[−σ.∆iµν+Eiµν1]ψiν+H.c. (12) where the orbital oﬀ-diagonal spin and charge ﬁelds are self-cons istently determined from: ∆iµν=/parenleftbiggU′′ 2+JH 4/parenrightbigg /angbracketleftσiνµ/angbracketright+/parenleftbiggJP 2/parenrightbigg /angbracketleftσiµν/angbracketright Eiµν=/parenleftbigg −U′′ 2+3JH 4/parenrightbigg /angbracketleftniνµ/angbracketright+/parenleftbiggJP 2/parenrightbigg /angbracketleftniµν/angbracketright (13) in terms of the corresponding condensates /angbracketleftσiµν/angbracketright ≡ /angbracketleftψ† iµσψiν/angbracketrightand/angbracketleftniµν/angbracketright ≡ /angbracketleftψ† iµ1ψiν/angbracketright. The spin andcharge condensates inEqs. 11and 13 areevaluated us ing the eigenfunctions (φk) and eigenvalues ( Ek) of the full Hamiltonian in the given basis including the interaction contributions [ HHF int] (Eqs. 10 and 12) using: /angbracketleftσα iµν/angbracketright ≡ /angbracketleftψ† iµσαψiν/angbracketright=Ek<EF/summationdisplay k(φ∗ kµs↑φ∗ kµs↓)[σα] φkνs↑ φkνs↓ (14) for siteion thes=A/Bsublattice, and similarly for the charge condensates /angbracketleftniµν/angbracketright ≡ /angbracketleftψ† iµ1ψiν/angbracketrightwith the Pauli matrices [ σα] replaced by the unit matrix [ 1]. The normal spin and charge condensates correspond to ν=µ. For each orbital pair ( µ,ν) = (yz,xz), (xz,xy), (xy,yz), there are three components ( α=x,y,z) for the spin condensates /angbracketleftψ† µσαψν/angbracketrightand one charge condensate /angbracketleftψ† µ1ψν/angbracketright. This is analogous to the three-plus-one normal spin and charge condensates for each of the three orbitals µ=yz,xz,xy . The above additional terms involving orbital oﬀ-diagonal condensa tes contribute to or- bital physics. Thus, the charge terms lead to coupling of orbital an gular momentum opera- tors to weak orbital ﬁelds, the spin terms result in interaction-indu ced SOC renormalization as given in Eq. (9), and the self consistently determined renormalize d SOC values are obtained as: λα=λ+λint α (15)11 forthethreecomponents α=x,y,z. Resultsoftheselfconsistentdeterminationofmagnetic order including all orbital diagonal and oﬀ-diagonal spin and charge condensates have been presented for the 4d4compound Ca 2RuO4recently,52illustrating the rich interplay between diﬀerent physical elements. IV. GENERALIZED FLUCTUATION PROPAGATOR Sinceallgeneralizedspin /angbracketleftψ† µσψν/angbracketrightandcharge /angbracketleftψ† µψν/angbracketrightcondensateswereincludedintheself consistent determination of magnetic order, the ﬂuctuation prop agator must also be deﬁned in terms of the generalized operators. We therefore consider the time-ordered generalized ﬂuctuation propagator: [χ(q,ω)] =/integraldisplay dt/summationdisplay ieiω(t−t′)e−iq.(ri−rj)×/angbracketleftΨ0\|T[σα µν(i,t)σα′ µ′ν′(j,t′)]\|Ψ0/angbracketright (16) in the self-consistent AFM ground state \|Ψ0/angbracketright, where the generalized spin-charge operators at lattice sites i,jare deﬁned as σα µν=ψ† µσαψν, which include both the orbital diagonal (µ=ν) and oﬀ-diagonal ( µ/negationslash=ν) cases, as well as the spin ( α=x,y,z) and charge ( α=c) operators, with σαdeﬁned as Pauli matrices for α=x,y,zand unit matrix for α=c. In the random phase approximation (RPA), the generalized ﬂuctua tion propagator is obtained as: [χ(q,ω)]RPA=2[χ0(q,ω)] 1−[U][χ0(q,ω)](17) in terms of the bare particle-hole propagator [ χ0(q,ω)] which is evaluated by integrating out the electronic degrees of freedom: [χ0(q,ω)]µ′ν′α′s′ µναs=1 2/summationdisplay k/bracketleftigg /angbracketleftk\|σα µν\|k−q/angbracketrights/angbracketleftk\|σα′ µ′ν′\|k−q/angbracketright∗ s′ E⊕ k−q−E⊖ k+ω−iη+/angbracketleftk\|σα µν\|k−q/angbracketrights/angbracketleftk\|σα′ µ′ν′\|k−q/angbracketright∗ s′ E⊕ k−E⊖ k−q−ω−iη/bracketrightigg (18) The matrix elements in the above expression are evaluated using the eigenvectors of the HF Hamiltonian in the self-consistent AFM state: /angbracketleftk\|σα µν\|k−q/angbracketrights= (φ∗ kµ↑sφ∗ kµ↓s)[σα] φk−qν↑s φk−qν↓s (19)12 and the superscripts ⊕(⊖) refer to particle (hole) states above (below) the Fermi energy. The subscripts s,s′indicate thetwo (A/B) sublattices. Inthe compositespin-charge- orbital- sublattice ( µναs) basis, the [ χ0(q,ω)] matrix is of order 72 ×72, and the form of the [ U] matrix in the RPA expression (Eq. 17) is given in Appendix B. The spectral function of the excitations will be determined from: Aq(ω) =1 πIm Tr[χ(q,ω)]RPA (20) using the RPA expression for [ χ(q,ω)]. When the collective excitation energies lie within the AFM band gap, it is convenient to consider the symmetric form of the denominator in the RPA expression (Eq. 17): [U][χ0(q,ω)][U]−[U] (21) and in terms of the real eigenvalues λq(ω) of this Hermitian matrix, the magnon energies ωqfor momentum qare determined by solving for the zeroes: λq(ω=ωq) = 0 (22) corresponding to the poles in the propagator. Results of the calculated spectral function will be discussed in the s ubsequent sections for diﬀerent electron ﬁlling cases ( n= 3,4,5) with applications to corresponding 4 dand 5dtransition metal compounds. Broadly, our investigation of the gen eralized ﬂuctuation propagator will provide information about the dominantly spin, orbit al, and spin-orbital excitations, as the generalized spin and charge operators ψ† µσαψνinclude spin ( µ=ν, α=x,y,z), orbital (µ/negationslash=ν,α=c), and spin-orbital ( µ/negationslash=ν,α=x,y,z) cases. Also included will be the high-energy spin-orbit exciton modes involving par ticle-hole excitations across the renormalized spin-orbit gap between spin-orbital enta ngled states of diﬀerent J sectors, as in the n= 5 case relevant for the Sr 2IrO4compound. V.n= 3— APPLICATION TO NaOsO 3 The strongly spin-orbit coupled orthorhomic structured 5 d3osmium compound NaOsO 3, with nominally three electrons in the Os t2gsector, exhibits several novel electronic and magnetic properties. These include a G-type antiferromagnetic (A FM) structure with spins13 oriented along the caxis, a signiﬁcantly reduced magnetic moment ∼1µBas measured from neutron scattering, a continuous metal-insulator transition (MIT ) that coincides with the AFM transition ( TN=TMIT= 410 K) as seen in neutron and X-ray scattering, and a large magnon gap of 58 meV as seen in resonant inelastic X-ray scattering (RIXS) measurements indicating strong magnetic anisotropy.8–10,16 Two diﬀerent mechanisms contributing to SOC-induced easy-plane a nisotropy and large magnon gap for out-of-plane ﬂuctuation modes were identiﬁed for the weakly correlated 5d3compound NaOsO 3in terms of a simpliﬁed picture involving only the normal spin and charge densities.14,15Both essential ingredients — (i) small moment disparity between yz,xz andxyorbitals and (ii) spin-charge coupling eﬀect in presence of tetragon al splitting — are intrinsically present in the considered three-orbital model on the s quare lattice. A realistic representation of magnetic anisotropy in NaOsO 3is therefore provided by the considered model, while maintaining uniformity of lattice structure across the n= 3,4,5 cases consid- ered in order to keep the focus on coupled spin-orbital ﬂuctuation s. The ﬁrst mechanism involves the SOC-induced anisotropic interactio n terms as in Eq. (6) resulting from the three SOC terms −λLαSαforα=x,y,z. Due to the small moment disparitymyz,xz>mxyresulting from the broader xyband, the interaction term in Eq. (6) dominates over the other two terms, leading to the easy-plane anis otropy for parallel yz,xz moments enforced by the Hund’s coupling. With increasing U, this eﬀect weakens as the moments saturate myz,xz,xy≈1 in the large Ulimit. In the second mechanism, the SOC induced decreasing xyorbital density nxywith spin rotation from zdirection to x−yplane couples to the tetragonal distortion term, and for positive ǫxythe energy is minimized for spin orientation in the x−yplane. We will consider the parameter set values U= 4,JH=U/5,U′′=U−5JH/2, bare SOC value λ=1.0, andǫxy= 0.5 unless otherwise indicated, with the hopping energy scale \|t1\|=300 meV. Thus, U= 1.2 eV,λ=0.3 eV,ǫxy= 0.15 eV, which are realistic values for the NaOsO 3compound. Initially, we will also set tm1,m2,m3= 0 for simplicity, and focus on the easy-plane anisotropy and large magnon gap for out-of-plane ﬂuc tuations. Self consistent determination of magnetic order using the generaliz ed approach discussed in Sec. III conﬁrms the easy-plane anisotropy. Starting in nearly zdirection, the AFM order direction self consistently approaches the x−yplane in a few hundred iterations. Initially, we will discuss magnetic excitations in the self consistent state with A FM order along the14 (π/2,π/2) (π,0) ( π,π) (0,0) 0 50 100 150 200 250 300ω (meV) 0.01 0.1 1 10 100 (a) 0 50 100 150 200 250 300 (π/2,π/2) (π,0) ( π,π) (π/2,π/2)(b)ωq (meV) FIG. 2: (a) Low energy part of the calculated spectral functi on from the generalized ﬂuctuation propagator shows the magnon excitations in the self-consis tent state with planar AFM order, and (b) magnon dispersion showing the gapless and gapped modes c orresponding to in-plane and out- of-plane ﬂuctuations. ˆxor ˆydirections. Although these orientations correspond to metastab le states as discussed later, they provide convenient test cases for explicitly conﬁrming t he gapless in-plane and gapped out-of-plane magnon modes in the generalized ﬂuctuation p ropagator calculation. The low-energy part of the calculated spectral function using Eq. (20) is shown in the Fig. 2(a) as an intensity plot for qalong symmetry directions of the Brillouin zone. The gapless and gapped modes corresponding to in-plane and out-of-p lane ﬂuctuations reﬂect the easy-plane magnetic anisotropy. The calculated gap energy 60 meV is close to the measured spin wave gap of 58 meV in NaOsO 3. Also shown for comparison in Fig. 2(b) is the magnon dispersion calculated from the poles of the RPA propag ator as described in Sec. IV. Focussing on the magnon gap in Fig. 2(b), which provides a m easure of the SOC induced easy-plane anisotropy, eﬀects of various physical quant ities are shown in Fig. 3. ThegaplessGoldstonemodecorrespondingtoin-planerotationofA FMorderingdirection in thex−yplane involves only small changes in spin densities /angbracketleftψ† µσαψµ/angbracketrightforα=x,yand µ=yz,xz, and also in generalized spin densities /angbracketleftψ† µσαψν/angbracketrightforα=x,yandµ=yz,xzwith ﬁxedν=xy. For example, the magnetization values mx yz= 0.82 andmx xz= 0.84 change tomy yz= 0.84 andmy xz= 0.82 when the ordering is rotated from xtoydirection. Thus, the Goldstone mode is nearly pure spin mode and the small orbital cha racter reﬂects the eﬀectively suppressed spin-orbital entangement in the n= 3 AFM state. In contrast, the15 0 20 40 60 80 100 0 0.06 0.12 0.18 0.24 0.30 0.36(a)magnon gap (meV) SOC (eV) 0 20 40 60 80 100 0.9 1.2 1.5 1.8 2.1 2.4(b)magnon gap (meV) U (eV) 0 20 40 60 80 100 -0.15-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25(c)magnon gap (meV) εxy (eV) FIG. 3: Variation of the calculated magnon gap showing eﬀects of (a) SOC, (b) Hubbard U, and (c) tetragonal distortion ǫxy, on the easy-plane magnetic anisotropy. n= 5 case corresponding to Sr 2IrO4shows strongly coupled spin-orbital character of the Goldstone mode (Appendix C) due to the extreme spin-orbital enta nglement. We now consider the easy-axis anisotropy eﬀects in our self consist ent determination of magnetic order. With respect to the AFM order orientation (azimut hal angleφ) within the easy (x−y) plane, we ﬁnd an easy-axis anisotropy along the diagonal orientat ionsφ=nπ/4 (n= 1,3,5,7) even for no octahedral tilting. This anisotropy is due to the orien tation and canting angle dependent anisotropic interaction (Eq. 8) as disc ussed in Sec. II. The anisotropic interaction energy vanishes for φalong thex,yaxes (hence the gapless in-plane mode in Fig. 2), and is signiﬁcant near the diagonal orientations, res ulting in easy-axis anisotropy and small relative canting between yz,xzmoments which is explicitly conﬁrmed in our self-consistent calculation. The resulting C4symmetry of the easy-axis ±(ˆx±ˆy) is reduced to C2symmetry ±(ˆx−ˆy) in the presence of octahedral tilting. The important anisotropy eﬀ ects of the octahedral tilting induced inter-site DM interactions are discussed below. We ﬁnd that the DM axis lies along the crystal baxis, leading to easy axis direction along the crystal aaxis. Both these directions are interchanged in comparison to the Ca 2RuO4case, which follows from a subtle diﬀerence in the present n= 3 case as explained below. Following the analysis carried out for the Ca 2RuO4compound,52within the usual strong- coupling expansion in terms of the normal ( t) and spin-dependent ( t′ x,t′ y) hopping terms induced by the combination of SOC and orbital mixing hopping terms tm2,m3due to octa- hedral tilting, the DM interaction terms generated in the eﬀective s pin model are obtained16 TABLE I: Self consistently determined magnetization and de nsity values for the three orbitals ( µ) on the two sublattices ( s), showing easy-axis anisotropy along the crystal aaxis due to octahedral tilting induced DM interaction. Here tm2,m3= 0.15. µ(s)mx µmy µmz µnµ yz(A) 0.598 −0.557 0.006 1.012 xz(A) 0.557 −0.598−0.006 1.012 xy(A) 0.541 −0.541 0.0 0.977µ(s)mx µmy µmz µnµ yz(B)−0.598 0.557 0.006 1.012 xz(B)−0.557 0.598 −0.006 1.012 xy(B)−0.541 0.541 0.0 0.977 TABLE II: Self consistently determined renormalized SOC va luesλα=λ+λint αand the orbital magnetic moments /angbracketleftLα/angbracketrightforα=x,y,zon the two sublattices. Bare SOC value λ=1.0. s λ xλyλz/angbracketleftLx/angbracketright /angbracketleftLy/angbracketright /angbracketleftLz/angbracketright A 1.179 1.179 1.364 0.032 −0.032 0.0 B 1.179 1.179 1.364 −0.032 0.032 0.0 as: [H(2) eﬀ](x,y) DM=8tt′ x U/summationdisplay /angbracketlefti,j/angbracketrightxˆx.(Si,xz×Sj,xz)+8tt′ y U/summationdisplay /angbracketlefti,j/angbracketrightyˆy.(Si,yz×Sj,yz) ≈8t\|t′ x\| U/summationdisplay /angbracketlefti,j/angbracketright(ˆx+ ˆy).(Si,yz×Sj,yz) (23) where we have taken t′ x=−t′ y=−ive andSx i,xz=Sx i,yz(due to Hund’s coupling) as earlier, but withSz i,xz=−Sz i,yzfor then= 3 case as obtained in our self consistent calculation which is discussed below. The eﬀective DM axis (ˆ x+ ˆy) is thus along the crystal baxis (Fig. 1) for theyzorbital, resulting in easy-axis anisotropy along the crystal adirection, as well as spin canting about the DM axis in the zdirection. Results for various physical quantities are shown in Tables I and II. Starting with initial orientation along the ˆ xor ˆydirections, the AFM order direction self consistently approaches the easy-axis direction in a few hundred iterations, explicitly exhibitin g the strong easy-axis anisotropywithintheeasy( x−y)planeduetotheoctahedraltiltinginducedDMinteraction,17 -3-2-1 0 1 2 (0,0) ( π,0) (π,π) (0,0) (0, π) (π,0)yz xz xy Ek - EF (eV) 0 50 100 150 200 250 300ωq (meV) (π/2,π/2) (π,0) ( π,π) (π/2,π/2) FIG. 4: (a) Calculated orbital resolved electronic band str ucture in the self-consistent state with AFMorderalongthecrystal aaxisduetooctahedraltiltinginducedDMinteraction. Here tm2,m3= 0.15. Colors indicate dominant orbital weight: red ( yz), green ( xz), blue ( xy). (b) Magnon dispersion for the magnetic order as given in Table I, showin g that both in-plane and out-of-plane modes are appreciably gapped due to the easy-axis and easy-p lane anisotropies. along with small spin canting in the zdirection about the DM axis. The small moment disparitymyz,xz>mxyand the negligible orbital moments can also be seen here explicitly. The renormalized SOC strength λzis enhanced relative to the other two components, which further reduces the SOC induced frustration in this system with no minally one electron in each of the three orbitals. Withoctahedral tilting included, theorbital resolved electronic ban dstructure intheself- consistent AFM state (Fig. 4(a)) shows the AFM band gap between valence and conduction bands, SOC induced orbital mixing and band splittings, the ﬁne splittin g due to octahedral tilting, andtheasymmetricbandwidthfor xyorbitalbandscharacteristicofthe2ndneighbor hopping term t2which connects the same magnetic sublattice. The calculated magno n dispersion evaluated using Eq. 22 is shown in Fig. 4(b). As expected, both in-plane and out-of-plane magnon modes are gapped due to the easy-axis and e asy-plane anisotropies discussed above. The high energy part of the spectral function is shown in the series of panels in Fig. 5 for diﬀerent SOC strengths. The two groups of modes here corre spond to: (i) the Hund’s coupling induced gapped magnon modes for out-of-phase spin ﬂuct uations (the two disper- sive modes starting at energies 0.7 and 0.8 eV from the left edge in pan el (a)), and (ii)18 (π/2,π/2) (π,0) ( π,π) (0,0) 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4ω (eV) 0.01 0.1 1 10 100 λ = 0(a) (π/2,π/2) (π,0) ( π,π) (0,0) 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4ω (eV) 0.01 0.1 1 10 100 λ = 0.5(b) (π/2,π/2) (π,0) ( π,π) (0,0) 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4ω (eV) 0.01 0.1 1 10 100 λ = 1.0(c) (π/2,π/2) (π,0) ( π,π) (0,0) 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4ω (eV) 0.01 0.1 1 10 100 λ = εxy = 0 (d) FIG. 5: Gapped magnon modes and dominantly magnetic exciton modes for the n= 3 case seen in the high-energy part of the spectral function calculated in the self consistent AFM state including octahedral tilting, for diﬀerent SOC ( λ) values shown in the panels. the spin-orbiton modes (starting at energy below 0.6 eV) which are in ter-orbital magnetic excitons corresponding to the lowest-energy particle-hole excita tions across the AFM band gap involving yz/xzorbitals (particle) and xyorbital (hole) states (Fig. 4(a)). Through the usual resonant scattering mechanism, these modes are pulled down in energy below the continuum by the U′′interaction term, and form well deﬁned propagating modes. The spin-orbiton mode involving xyorbital shifts to higher energy when ǫxydecreases to zero (panel (d)) which lowers the dominantly xyvalence band (Fig. 4(a)) and thus increases the particle-hole excitation energy. The splitting of the exciton mod es in panel (c) is due to the SOC induced splitting of electronic bands as seen in Fig. 4(a), whic h is then reﬂected in the particle-hole excitation energies. The combination of orbitals f or these exciton modes indicates that LxandLycomponents of the orbital angular momentum are involved in these coupled spin-orbital ﬂuctuations. There is an additional spin-orbit on mode involving only yz,xzorbitals (and Lzcomponent) which is formed at higher energy near 0.8 eV (ﬂat band near the left edge iin panel (a)). With increasing SOC, the high-ener gy modes involving19 yz,xzorbitals acquire signiﬁcant spin-orbit exciton character. VI.n= 5— APPLICATION TO Sr2IrO4 The perovskite structured 5 d5compound Sr 2IrO4exhibits an AFM insulating state due to strong SOC induced splitting of the t2gstates, with four electrons in the nominally ﬁlled and non-magnetic J=3/2 sector and one electron in the nominally half ﬁlled and magnetically active J=1/2 sector. The SOC induced splitting of 3 λ/2 between states of the two total angular momentum sectors, strong spin-orbital entan glement, and band narrowing of states in the J=1/2 sector, all of these play a crucial role in the stabilization of the AFM insulator state. Both low-energy magnon excitations and high-ene rgy spin-orbit excitons acrosstherenormalizedspin-orbit gaphave beenintensively studie dusing RIXSexperiments and variety of theoretical approaches.43,46,47,51,53 In this case, we have taken realistic parameter values U= 3,JH=U/7, bare SOC valueλ= 1.35, andǫxy=−0.5 for simplicity, along with hopping terms: ( t1,t2,t3,t4, t5,tm1)=(-1.0, 0.5, 0.25, -1.0, 0.0, 0.2), all in units of the realistic hopping en ergy scale \|t1\|=290 meV. The self consistently determined results for various phy sical quantities are given in Table III for magnetic order in the xdirection. All ordering directions within the x−yplane are nearly equivalent. Besides the dominant Hund’s coupling indu ced easy-plane anisotropy,53there is an extremely weak easy-axis anisotropy which will be discuss ed at the end of this section. The octahedral rotation induces small in-plane canting of spins but the canting axis is free to orient in any direction. The strong Coulomb inte raction induced SOC renormalization by nearly 2/3 (Table IV) agrees with the pseudo-or bital based approach.51 TABLE III: Self consistently determined magnetization and density values, showing small spin canting about the zaxis due to octahedral rotation induced DM interaction. Her etm1= 0.2. µ(s)mx µmy µmz µnµ yz(A) 0.186 −0.052 0 1.653 xz(A)−0.185 0.049 0 1.654 xy(A)−0.172−0.047 0 1.693µ(s)mx µmy µmz µnµ yz(B)−0.186−0.052 0 1.653 xz(B) 0.185 0.049 0 1.654 xy(B) 0.172 −0.047 0 1.69320 TABLE IV: Self consistently determined renormalized SOC va luesλα=λ+λint αand the orbital magnetic moments /angbracketleftLα/angbracketrightforα=x,y,zon the two sublattices. Bare SOC value λ=1.35. s λ xλyλz/angbracketleftLx/angbracketright /angbracketleftLy/angbracketright /angbracketleftLz/angbracketright A 1.882 1.882 1.871 0.367 0.091 0 B 1.882 1.882 1.871 −0.367 0.091 0 The strong orbital moments and their correlation with the magnetic order direction (Table IV) reﬂect the strong SOC induced spin-orbital entanglement. The low energy part of the spectral function (Fig. 6(a)) clearly sh ows the gapless and gapped modes corresponding to in-plane and out-of-plane ﬂuctua tions, consistent with the easy-plane anisotropy. The magnon gap ≈45 meV is close to the result obtained using the pseudo-orbital based approach,53and in agreement with recent experiments.46It should be noted that along with the full generalized spin sector, the orbital o ﬀ-diagonal charge sector (ψ† µ1ψν)relatedtotheorbitalmomentoperators Lx,y,zwasincludedintheabovecalculations which allows for the accompanying transverse ﬂuctuations of orbit al moments. Indeed, the exactly gapless Goldstone mode seen in Fig. 6(a) is obtained only if the ψ† µ1ψνsector is included, indicating the coupled spin-orbital nature of the Goldston e mode, as illustrated in (π/2,π/2) (π,0) ( π,π) (0,0) 0 50 100 150 200 250 300ω (meV) 0.01 0.1 1 10 100 (a) (π/2,π/2) (π,0) ( π,π) (0,0) 300 400 500 600 700 800ω (meV) 0.01 0.1 1 10 100 (b) FIG. 6: The spectral function in the self-consistent state f or then= 5 case with planar AFM order including octahedral rotation, showing the (a) gaple ss and gapped modes corresponding to in-plane and out-of-plane ﬂuctuations and (b) the spin-orb it exciton modes near 500 meV and 300 meV in the high-energy part.21 (π/2,π/2) (π,0) ( π,π) (0,0) 0 200 400 600 800 1000 1200ω (meV) 0.01 0.1 1 10 100 (a)JH = 0 (π/2,π/2) (π,0) ( π,π) (0,0) 0 0.4 0.8 1.2 1.6 2 2.4ω (eV) 0.01 0.1 1 10 100 (b) λ = 5, JH = U/7 (π/2,π/2) (π,0) ( π,π) (0,0) 1.9 2 2.1 2.2 2.3ω (eV) 0.01 0.1 1 10 100 (c) λ = 5, JH = U/7 FIG. 7: The spin-orbit exciton modes ( n= 5) for special cases showing (a) no splitting in the weak branch (∼400 meV) for Hund’s coupling JH= 0, (b) disappearance of the weak branch for large SOC value λ= 5, and (c) expanded view of the multiple exciton modes ( ∼2 eV) in case (b). Here octahedral rotation is included in all three cases. Appendix C showing the detailed spin-orbital composition. Fig. 6(b) shows the spin-orbit exciton modes ( ∼500 meV) involving particle-hole excita- tions between the J=1/2 and 3/2 sectors, which matches closely with results obtained u sing the pseudo-orbital based approach.51As discussed in the previous ( n= 3) case, collective modes arise from particle-hole excitations which are converted to w ell deﬁned propagating modes split oﬀ from the continuum by the Coulomb interaction induced resonant scattering mechanism. The signiﬁcantly weaker modes ( ∼300 meV) just below the particle-hole con- tinuum for the nominally J= 1/2 sector are also spin-orbit exciton modes. The splitting seen beyond ( π,0) vanishes for JH= 0, as seen in Fig. 7(a). The weak intensity corresponds to the small J= 3/2 character (mainly mJ=±3/2) in the nominally J= 1/2 bands due to strong mixing between the two sectors induced by the band (hop ping) terms. For large SOC strength λ, the weak exciton modes disappear (Fig. 7(b)), conﬁrming the abo ve pic- ture. Thus, the (low) intensity of the weak exciton modes provides a direct measure of the mixing between the J= 1/2 and 3/2 sectors. The ﬁne splitting of exciton bands in Fig. 7(c) corresponds to four possible mJvalues (±3/2,±1/2) for the hole in the J= 3/2 sector and the exciton hopping terms connecting the two sublattices. We now discuss the extremely weak easy-axis anisotropy which leads to preferred isospin (J= 1/2) orientation along the diagonal directions ±(ˆx±ˆy) within the easy plane. Fig. 8(a) shows the small magnon gap ( ≈3 meV) for the in-plane magnon mode induced by the Hund’s coupling JHdue to the extremely weak spin twisting as shown in Fig. 8(b) which results in an easy-axis anisotropy with C4symmetry. Here the parameter set is same as22 0 2 4 6 8 10 12 14 0 0.02 0.04 0.06 0.08in-plane mode(a)ωq (meV) qx = qy (b) isospin FIG. 8: (a) Magnon energies for isospin order along ˆ x+ˆydirection showing the small magnon gap (≈3 meV) for the in-plane ﬂuctuation mode. (b) The isospin and yz,xz,xy moment orientations fortheideal spin-orbital entangled state, whichis extrem ely weakly perturbedbyﬁnite JHresulting in slight twisting of the yz,xzmoments as indicated, leading to the easy-axis anisotropy w ithC4 symmetry. The isospin easy axes are along φ=nπ/4 wheren= 1,3,5,7. earlier including the octahedral rotation which only weakly enhances the magnon gap. The above weak perturbative eﬀect of JHonthestrongly spin-orbital entangled statecorresponds to the opposite end of the competition between SOC and JHas compared to the n= 3 case discussed in Sec. V. VII.n= 4— APPLICATION TO Ca2RuO4 For moderate tetragonal distortion ( ǫxy≈ −1), thexyorbital in the 4 d4compound Ca2RuO4is nominally doubly occupied and magnetically inactive, while the nominally h alf- ﬁlledandmagnetically active yz,xzorbitalsyieldaneﬀectively two-orbitalmagneticsystem. Hund’s coupling between the two S= 1/2 spins results in low-lying (in-phase) and apprecia- bly gapped (out-of-phase) spin ﬂuctuation modes. The in-phase m odes of the yz,xzorbital S= 1/2 spins correspond to an eﬀective S= 1 spin system. However, the rich interplay be- tween SOC, Coulomb interaction, octahedral rotations, and tetr agonal distortion results in complex magnetic behaviour which crucially involves the xyorbital and is therefore beyond the above simplistic picture. Treating all the diﬀerent physical elements on the same footing with in the uniﬁed frame- work of the generalized self-consistent approach explicitly shows t he variety of physical23 (π/2,π/2) (π,0) ( π,π) (0,0) 0 20 40 60 80 100 120 140 160 180ω (meV) 0.01 0.1 1 10 100 (a) (π/2,π/2) (π,0) ( π,π) (0,0) 250 300 350 400 450 500ω (meV) 0.01 0.1 1 10 100 (b) FIG. 9: The generalized ﬂuctuation spectral function for th en= 4 case, showing coupled spin- orbital excitations including low-energy magnon modes (be low∼60 meV), intermediate-energy orbiton (100 and 140 meV) and spin-orbiton (300 and 350 meV) m odes, and high-energy spin- orbit exciton (425 meV) modes. eﬀects arising from the rich interplay in Ca 2RuO4. These include: SOC induced easy-plane and easy-axis anisotropies similar to the n=3 case, octahedral tilting induced reduction of easy-axis anisotropyfrom C4toC2symmetry, spin-orbital coupling induced orbital magnetic moments, Coulomb interaction induced strongly anisotropic SOC ren ormalization, decreas- ing tetragonal distortion induced magnetic reorientation transitio n from planar AFM order to FM (z) order, and orbital moment interaction induced orbital gap.52Stable FM and AFM metallic states were also obtained near the magnetic phase boun dary separating the two magnetic orders. The self-consistent determination of magne tic order has also explicitly shown the coupled nature of spin and orbital ﬂuctuations, as reﬂe cted in the ferro and an- tiferro orbital ﬂuctuations associated with in-phase and out-of- phase spin twisting modes, highlighting the strong deviation from conventional Heisenberg beh aviour in eﬀective spin models, as discussed recently to account for the magnetic excitat ion measurements in INS experiments on Ca 2RuO4.35 In the following, we will take the same parameter set as considered in the self-consistent study,52along with U= 8 andJH=U/5 in the energy scale unit (150 meV), so that U= 1.2 eV,U′′=U/2 = 0.6 eV, andJH= 0.24 eV. These are comparable to reported values extracted from RIXS ( JH= 0.34 eV) and ARPES ( JH= 0.4 eV) studies.24,40The hopping parameter values considered are as given in Sec. II, and th e bare SOC value λ= 1.24 Fig. 9 shows the calculated generalized ﬂuctuation spectral funct ion. Several well de- ﬁned propagating modes are seen here including: (i) the low-energy (below∼60 meV) dominantly spin (magnon) excitations involving the magnetically active yz,xzorbitals and corresponding to in-plane and out-of-plane ﬂuctuations which are gapped due to the mag- netic anisotropies, (ii) the intermediate-energy (100 and 140 meV) dominantly orbital exci- tations (orbitons) involving particle-hole excitations between xy(hole) andyz,xz(particle) states, (iii) the intermediate-energy (300 and 350 meV) dominantly spin-orbital excitations (spin-orbitons) involving xy(hole) and yz,xz(particle) states, and (iv) the high-energy (425 meV) dominantly spin-orbital excitations (spin-orbit excitons ) involving particle-hole excitations between yz,xz(hole) and yz,xz(particle) states of nominally diﬀerent Jsec- tors. The SOC-induced spin-orbital entangled Jstates are strongly renormalized by the tetragonal splitting and the electronic correlation induced stagge red ﬁeld. The spin-orbital characterization of the various collective excitat ions mentioned above is inferred from the basis-resolved contributions to the total spec tral functions which explicitly show the relative spin-orbital composition of the various excitation s (Appendix C). The presence of sharply deﬁned collective excitations for the magnon, orbiton, and spin-orbiton modes which are clearly separated from the particle-hole continuum highlights the rich spin- orbital physics in the n= 4 case corresponding to the Ca 2RuO4compound. Many of our calculated magnon spectra features such as the magnon gaps for in-plane and out-of-plane modes, weak dispersive nature along the magnetic zone boundary, as well as the overall magnon energy scale are in excellent agreement with the INS study.34,35The orbiton mode energy scale is also qualitatively comparable to the composite excitat ion peaks obtained around 80 meV in Raman and RIXS studies.37–40The calculated spin-orbiton and spin-orbit exciton energies are also in agreement with the excitation peaks obt ained around 300-350 meV energy range and 400 meV in RIXS studies. We also obtained excit ations in the high- energy range 750-800 meV and 900 meV (not shown), which are com parable to the peaks obtained around 750 meV and 1000 meV in RIXS studies.25 VIII. CONCLUSIONS Following up on the generalized self-consistent approach including or bital oﬀ-diagonal spin and charge condensates, investigation of the generalized ﬂuc tuation propagator reveals thecomposite spin-orbital character ofthe diﬀerent types ofco llective excitations instrongly spin-orbit coupled systems. A realistic representation of magnetic anisotropy eﬀects due to the interplay of SOC, Coulomb interaction, and structural distort ion terms was included in the three-orbital model, while maintaining uniformity of lattice stru cture in order to focus on the coupled spin-orbital excitations. Our uniﬁed investiga tion of the three electron ﬁlling cases n= 3,4,5 corresponding to the three compounds NaOsO 3, Ca2RuO4, Sr2IrO4 provides deep insight into how the spin-orbital physics in the magnet ic ground state is reﬂected in the collective excitations. The calculated spectral fun ctions show well deﬁned propagating modes corresponding to dominantly spin (magnon), or bital (orbiton), and spin- orbital (spin-orbiton) excitations, along with the spin-orbit excito n modes involving spin- orbital excitations between states of diﬀerent Jsectors induced by the spin-orbit coupling. Appendix A: Orbital oﬀ-diagonal condensates in the HF approximation TheadditionalcontributionsintheHFapproximationarisingfromthe orbitaloﬀ-diagonal spin and charge condensates are given below. For the density, Hun d’s coupling, and pair hopping interaction terms in Eq. 3, we obtain (for site i): U′′/summationdisplay µ<νnµnν→ −U′′ 2/summationdisplay µ<ν[nµν/angbracketleftnνµ/angbracketright+σµν./angbracketleftσνµ/angbracketright]+H.c. −2JH/summationdisplay µ<νSµ.Sν→JH 4/summationdisplay µ<ν[3nµν/angbracketleftnνµ/angbracketright−σµν./angbracketleftσνµ/angbracketright]+H.c. JP/summationdisplay µ/negationslash=νa† µ↑a† µ↓aν↓aν↑→JP 2/summationdisplay µ<ν[nµν/angbracketleftnµν/angbracketright−σµν./angbracketleftσµν/angbracketright]+H.c. (A1) in terms of the orbital oﬀ-diagonal spin ( σµν=ψ† µσψν) and charge ( nµν=ψ† µ1ψν) oper- ators. The orbital oﬀ-diagonal condensates are ﬁnite due to the SOC-induced spin-orbital correlations. These additional terms in the HF theory explicitly pres erve the SU(2) spin rotation symmetry of the various Coulomb interaction terms. Collecting all the spin and charge terms together, we obtain the orb ital oﬀ-diagonal26 (OOD) contributions of the Coulomb interaction terms: [HHF int]OOD=/summationdisplay µ<ν/bracketleftbigg/parenleftbigg −U′′ 2+3JH 4/parenrightbigg nµν/angbracketleftnνµ/angbracketright+/parenleftbiggJP 2/parenrightbigg nµν/angbracketleftnµν/angbracketright −/parenleftbiggU′′ 2+JH 4/parenrightbigg σµν./angbracketleftσνµ/angbracketright−/parenleftbiggJP 2/parenrightbigg σµν./angbracketleftσµν/angbracketright/bracketrightbigg +H.c. (A2) Appendix B: Coulomb interaction matrix elements in the orbital-pair basis CorrespondingtotheaboveHFcontributionsintheorbitaloﬀ-diag onalsector, weexpress the Coulomb interactions in terms of the generalized spin and charge operators (for site i): [Hint]OOD=/summationdisplay µ<ν/bracketleftbigg/parenleftbigg −U′′ 2+3JH 4/parenrightbigg nµνn† µν−/parenleftbiggU′′ 2+JH 4/parenrightbigg σµν.σ† µν/bracketrightbigg +/summationdisplay µ<ν/bracketleftbigg/parenleftbiggJP 4/parenrightbigg nµνn† νµ−/parenleftbiggJP 4/parenrightbigg σµν.σ† νµ+H.c./bracketrightbigg (B1) wheren† µν=nνµandσ† µν=σνµ. The above form shows that only the pair-hopping interaction terms ( JP) are oﬀ-diagonal in the orbital-pair ( µν) basis. We will use the above Coulomb interaction terms in the orbital oﬀ-diagonal sector in the R PA series in order to ensure consistency with the self-consistent determination of mag netic order including the orbital oﬀ-diagonal condensates. The Coulomb interaction terms in the orbital diagonal sector can be cast in a similar form: [Hint]OD=/summationdisplay µ/bracketleftbigg/parenleftbigg −U 4/parenrightbigg σµ.σµ+/parenleftbiggU 4/parenrightbigg nµnµ/bracketrightbigg +/summationdisplay µ<ν/bracketleftbigg/parenleftbigg −2JH 4/parenrightbigg σµ.σν+U′′nµnν/bracketrightbigg (B2) which include the Hubbard, Hund’s coupling, and density interaction t erms. The form of the [ U] matrix used in the RPA series Eq. (17) is now discussed below. In the composite spin-charge-orbital-sublattice ( µναs) basis, the [ U] matrix is diagonal in spin, charge, and sublattice sectors. There are two possible cases invo lving the orbital-pair ( µν) basis. In the case µ=ν, the [U] matrices in the spin ( α=x,y,z) and charge ( α=c) sectors are obtained as: [U]µ′µ′α′=α µµα=x,y,z= U JHJH JHU JH JHJHU [U]µ′µ′α′=α µµα=c= −U−2U′′−2U′′ −2U′′−U−2U′′ −2U′′−2U′′−U (B3)27 FIG. 10: The basis-resolved contributions to the total spec tral function for the low-energy magnon (left panel) and intermediate-energy orbiton (center and r ight panels) modes, showing dominantly spin (µ=ν,α=x,y,z) and orbital ( µ/negationslash=ν,α=c) character of the ﬂuctuation modes, respectively. corresponding to the interaction terms (Eq. B2) for the normal s pin and charge density operators. Similarly, for the six orbital-pair cases ( µ,ν) corresponding to µ/negationslash=ν, the [U] matrix elements in the spin ( α=x,y,z) and charge ( α=c) sectors are obtained as: [U]µνα µνα=x,y,z=U′′+JH/2 [U]µνα µνα=c=U′′−3JH/2 [U]νµα µνα=x,y,z=JP [U]νµα µνα=c=−JP (B4) corresponding to the interaction terms (Eq. B1) involving the orbit al oﬀ-diagonal spin and charge operators. Appendix C: Basis-resolved contributions to the total spectral function The detailed spin-orbital character of the collective excitations ca n be identiﬁed from the basis-resolved contributions to the total spectral functions. T his is illustrated here for the28 FIG. 11: The basis-resolved contributions to the total spec tral function for the intermediate- energy spin-orbiton (left and center panels) and high-ener gy spin-orbit exciton (right panel) modes, showing dominantly spin-orbital character ( µ/negationslash=ν,α=x,y,z) involving xyandyz,xzorbitals (left and center panels) and yz,xzorbitals (right panel). excitations shown in Fig. 9 for the n= 4 case corresponding to the Ca 2RuO4compound. Fig. 10 shows dominantly spin excitations involving yz,xzorbitals for the magnon modes (below 60 meV) and dominantly orbital excitations involving xyandyz,xzorbitals for the orbiton modes (100 and 140 meV). Similarly, Fig. 11 shows dominan tly spin-orbital excitations involving xyandyz,xzorbitals for the spin-orbiton modes (300 and 350 meV), and dominantly spin-orbital excitations involving yz,xzorbitals for the spin-orbit exciton modes (425 meV).29 FIG. 12: The extreme spin-orbital-entanglement induced co rrespondence between (a) magnetic ordering directions, (b) sign of magnetic moments for the th ree orbitals, and (c) orbital current induced orbital moments for the three orbitals, for the n= 5 case corresponding to Sr 2IrO4. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x y z cω = 0 meV (a)π-1Im[χ(q,ω)]µναµναµν=yz yz xz xz xy xy yz xz xz xyxy yz xz yz xy xz yz xyq =(0,0) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x y z cω = 46 meV (b)π-1Im[χ(q,ω)]µναµναq =(0,0) FIG. 13: The basis-resolved contributions to the total spec tral function for the (a) gapless in-plane magnon mode and (b) gapped out-of-plane magnon mode for the n= 5 case corresponding to Sr2IrO4with extreme spin-orbital entanglement. Similarly, for the n= 5 case corresponding to Sr 2IrO4, the detailed spin-orbital character of the Goldstone mode and gapped mode at q= (0,0) seen in Fig. 6 is shown in Fig. 13, explicitly illustrating the eﬀect of extreme spin-orbital entangle ment and the resulting correspondence (Fig. 12) between magnetic ordering directions, spin moments, and orbital moments.30 ∗Electronic address: avinas@iitk.ac.in 1W. Witczak-Krempa, G. Chen, Y. B. Kim, and L. Balents, Correlated Quantum Phenomena in the Strong Spin-Orbit Regime , Annu. Rev. Condens. Matter Phys. 5, 57 (2014). 2J. G. Rau, E. Kin-Ho Lee, and H.-Y. Kee, Spin-Orbit Physics Giving Rise to Novel Phases in Correlated Systems: Iridates and Related Materials , Annu. Rev. Condens. Matter Phys. 7, 195 (2016). 3G. Cao and P. Schlottmann, The Challenge of Spin-Orbit-Tuned Ground States in Iridates: A Key Issues Review , Rep. Prog. Phys. 81, 042502 (2018). 4G. Zhang and E. Pavarini, Spin-Orbit and Coulomb Eﬀects in Single-Layered Ruthenates , Phys. Status Solidi RRL 12, 1800211 (2018). 5J. Bertinshaw, Y. K. Kim, G. Khaliullin, and B. J. Kim, Square Lattice Iridates , Annu. Rev. Condens. Matter Phys. 10, 315 (2019). 6G. Zhang and E. Pavarini, Higgs Mode and Stability of xy-orbital Ordering in Ca2RuO4, Phys. Rev. B101, 205128 (2020). 7Y. G. Shi, Y. F. Guo, S. Yu, M. Arai, A. A. Belik, A. Sato, K. Yama ura, E. Takayama- Muromachi, H. F. Tian, H. X. Yang, J. Q. Li, T. Varga, J. F. Mitc hell, and S. Okamoto, Continuous Metal-Insulator Transition of the Antiferromagn etic Perovskite NaOsO 3, Phys.Rev. B80, 161104(R) (2009). 8S. Calder, V. O. Garlea, D. F. McMorrow, M. D. Lumsden, M. B. St one, J. C. Lang, J.-W. Kim, J. A. Schlueter, Y. G. Shi, K. Yamaura, Y. S. Sun, Y. Tsuji moto, and A. D. Christian- son,Magnetically Driven Metal-Insulator Transition in NaOsO 3, Phys. Rev. Lett. 108, 257209 (2012). 9Y. Du, X. Wan, L. Sheng, J. Dong, and S. Y. Savrasov, Electronic Structure and Magnetic Properties of NaOsO 3, Phys. Rev. B 85, 174424 (2012). 10M.-C. Jung, Y.-J. Song, K.-W. Lee, and W. E. Pickett, Structural and Correlation eﬀects in the Itinerant Insulating Antiferromagnetic Perovskite NaOsO 3, Phys. Rev. B 87, 115119 (2013). 11I. Lo Vecchio, A. Perucchi, P. Di Pietro, O. Limaj, U. Schade, Y. Sun, M. Arai, K. Yamaura, and S. Lupi, Infrared evidence of a Slater Metal-Insulator Transition in NaOsO 3, Sci. Rep. 3, 2990 (2013).31 12B. Kim, P. Liu, Z. Erg¨ onenc, A. Toschi, S. Khmelevskyi, and C . Franchini, Lifshitz Transition Driven by Spin Fluctuations and Spin-Orbit Renormalizatio n inNaOsO 3, Phys. Rev. B 94, 241113(R) (2016). 13J.G. Vale, S.Calder, C.Donnerer, D. Pincini, Y. G. Shi, Y. Ts ujimoto, K. Yamaura, M. Moretti Sala, J. van den Brink, A. D. Christianson, and D. F. McMorrow ,Evolution of the Magnetic Excitations in NaOsO 3through its Metal-Insulator Transition , Phys. Rev. Lett. 120, 227203 (2018). 14S. Mohapatra, C. Bhandari, S. Satpathy, and A. Singh, Eﬀect of Structural Distortion on the Electronic Band Structure of NaOsO 3studied within Density Functional Theory and a Three- Orbital Model , Phys. Rev. B 97, 155154 (2018). 15A. Singh, S. Mohapatra, C. Bhandari, and S. Satpathy, Spin-Orbit Coupling Induced Magnetic Anisotropy and Large Spin Wave Gap in NaOsO 3, J. Phys. Commun. 2, 115016 (2018). 16S. Calder, J. G. Vale, N. Bogdanov, C. Donnerer, D. Pincini, M . Moretti Sala, X. Liu, M. H. Upton, D. Casa, Y. G. Shi, Y. Tsujimoto, K. Yamaura, J. P. Hill , J. van den Brink, D. F. McMorrow, and A. D. Christianson, Strongly Gapped Spin-Wave Excitation in the Insulating Phase of NaOsO 3, Phys. Rev. B 95, 020413(R) (2017). 17J.G. Vale, S.Calder, C.Donnerer, D. Pincini, Y. G. Shi, Y. Ts ujimoto, K. Yamaura, M. Moretti Sala, J. van den Brink, A. D. Christianson, and D. F. McMorrow ,Crossover from Itinerant to Localized Magnetic Excitations Through the Metal-Insulator Transition in NaOsO 3, Phys. Rev. B97, 184429 (2018). 18S. Nakatsuji, S.-i. Ikeda, and Y. Maeno, Ca 2RuO4: New Mott Insulators of Layered Ruthenate , J. Phys. Soc. Jpn 66(7), 1868 (1997). 19M.Braden, G.Andr´ e, S.Nakatsuji, andY. Maeno, Crystal and Magnetic Structure of Ca2RuO4: Magnetoelastic Coupling and the Metal-Insulator Transition , Phys. Rev. B 58, 847 (1998). 20C. S. Alexander, G. Cao, V. Dobrosavljevic, S. McCall, J. E. C row, E. Lochner, and R. P. Guertin, Destruction of the Mott Insulating Ground State of Ca2RuO4by a Structural Transi- tion, Phys. Rev. B 60, R8422 (1999). 21O. Friedt, M. Braden, G. Andr´ e, P. Adelmann, S. Nakatsuji, a nd Y. Maeno, Structural and Magnetic Aspects of the Metal-Insulator Transition in Ca2−xSrxRuO4, Phys. Rev. B 63, 174432 (2001). 22E. Gorelov, M. Karolak, T. O. Wehling, F. Lechermann, A. I. Li chtenstein, and E. Pavarini,32 Nature of the Mott Transition in Ca2RuO4, Phys. Rev. Lett. 104, 226401 (2010). 23G. Zhang and E. Pavarini, Mott transition, Spin-Orbit Eﬀects, and Magnetism in Ca2RuO4, Phys. Rev. B 95, 075145 (2017). 24D. Sutter, C. G. Fatuzzo, S. Moser, M. Kim, R. Fittipaldi, A. V ecchione, V. Granata, Y. Sassa, F. Cossalter, G. Gatti, M. Grioni, H. M. Rønnow, N. C. Plumb, C . E. Matt, M. Shi, M. Hoesch, T. K. Kim, T.-R. Chang, H.-T. Jeng, C. Jozwiak, A. Bostwick, E . Rotenberg, A. Georges, T. Neupert, and J. Chang, Hallmarks of Hund’s Coupling in the Mott Insulator Ca2RuO4, Nat. Comm.8, 15176 (2017). 25F. Nakamura, T. Goko, M. Ito, T. Fujita, S. Nakatsuji, H. Fuka zawa, Y. Maeno, P. Alireza, D. Forsythe, and S. R. Julian, From Mott Insulator to Ferromagnetic Metal: A Pressure Study of Ca2RuO4, Phys. Rev. B 65, 220402(R) (2002). 26P. Steﬀens, O. Friedt, P. Alireza, W. G. Marshall, W. Schmidt, F. Nakamura, S. Nakatsuji, Y. Maeno, R. Lengsdorf, M. M. Abd-Elmeguid, and M. Braden, High-pressure Diﬀraction Studies onCa2RuO4, Phys. Rev. B 72, 094104 (2005). 27H. Taniguchi, K. Nishimura, R. Ishikawa, S. Yonezawa, S. K. G oh, F. Nakamura, and Y. Maeno,Anisotropic Uniaxial Pressure Response of the Mott Insulato rCa2RuO4, Phys. Rev. B 88, 205111 (2013). 28S. Nakatsuji and Y. Maeno, Quasi-Two-Dimensional Mott Transition System Ca2−xSrxRuO4, Phys. Rev. Lett. 84, 2666 (2000). 29Z. Fang and K. Terakura. Magnetic Phase Diagram of Ca2−xSrxRuO4Governed by Structural Distortions , Phys. Rev. B 64, 020509 (2001). 30P. Steﬀens, O. Friedt, Y. Sidis, P. Link, J. Kulda, K. Schmalzl , S. Nakatsuji, and M. Braden, Magnetic Excitations in the Metallic Single-Layer Ruthena tesCa2−xSrxRuO4Studied by Inelas- tic Neutron Scattering , Phys. Rev. B 83, 054429 (2011). 31C. Dietl, S. K. Sinha, G. Christiani, Y. Khaydukov, T. Keller , D. Putzky, S. Ibrahimkutty, P. Wochner, G. Logvenov, P. A. van Aken, B. J. Kim, and B. Keimer, Tailoring the Electronic Properties of Ca2RuO4via Epitaxial Strain , Appl. Phys. Lett. 112, 031902 (2018). 32F. Nakamura, M. Sakaki, Y. Yamanaka, S. Tamaru, T. Suzuki, an d Y. Maeno, Electric-Field- Induced Metal Maintained by Current of the Mott Insulator Ca2RuO4, Sci. Rep. 3, 2536 (2013). 33R. Okazaki, Y. Nishina, Y. Yasui, F. Nakamura, T. Suzuki, and I. Terasaki, Current-Induced Gap Suppression in the Mott Insulator Ca2RuO4, J. Phys. Soc. Jpn. 82, 103702 (2013).33 34S. Kunkem¨ oller, D. Khomskii, P. Steﬀens, A. Piovano, A. A. Nu groho, and M. Braden, Highly Anisotropic Magnon Dispersion in Ca2RuO4: Evidence for Strong Spin-Orbit Coupling , Phys. Rev. Lett. 115, 247201 (2015). 35A. Jain, M. Krautloher, J. Porras, G. H. Ryu, D. P. Chen, D. L. A bernathy, J. T. Park, A. Ivanov, J. Chaloupka, G. Khaliullin, B. Keimer, and B. J. Kim ,Higgs Mode and Its Decay in a Two-Dimensional Antiferromagnet , Nat. Phys. 13, 633 (2017). 36S. Kunkem¨ oller, E. Komleva, S. V. Streltsov, S. Hoﬀmann, D. I . Khomskii, P. Steﬀens, Y. Sidis, K. Schmalzl, and M. Braden, Magnon dispersion in Ca2Ru1−xTixO4: Impact of Spin-Orbit Coupling and Oxygen Moments , Phys. Rev. B 95, 214408 (2017). 37S.-M. Souliou, J. Chaloupka, G. Khaliullin, G. Ryu, A. Jain, B. J. Kim, M. Le Tacon, and B. Keimer, Raman Scattering from Higgs Mode Oscillations in the Two-Dim ensional Antiferro- magnetCa2RuO4, Phys. Rev. Lett. 119, 067201 (2017). 38C. G. Fatuzzo, M. Dantz, S. Fatale, P. Olalde-Velasco, N. E. S haik, B. Dalla Piazza, S. Toth, J. Pelliciari, R. Fittipaldi, A. Vecchione, N. Kikugawa, J. S. Brooks, H. M. Rønnow, M. Grioni, Ch. R¨ uegg, T. Schmitt, and J. Chang, Spin-Orbit-Induced Orbital Excitations in Sr2RuO4and Ca2RuO4: A resonant Inelastic X-Ray Scattering Study , Phys. Rev. B 91, 155104 (2015). 39L. Das, F. Forte, R. Fittipaldi, C. G. Fatuzzo, V. Granata, O. Ivashko, M. Horio, F. Schindler, M. Dantz, Yi Tseng, D. E. McNally, H. M. Rønnow, W. Wan, N. B. Ch ristensen, J. Pelliciari, P. Olalde-Velasco, N. Kikugawa, T. Neupert, A. Vecchione, T. S chmitt, M. Cuoco, and J. Chang, Spin-Orbital Excitations in Ca2RuO4Revealed by Resonant Inelastic X-Ray Scattering , Phys. Rev. X8, 011048 (2018). 40H. Gretarsson, H. Suzuki, Hoon Kim, K. Ueda, M. Krautloher, B . J. Kim, H. Yava¸ s, G. Khali- ullin, and B. Keimer, Observation of Spin-Orbit Excitations and Hund’s Multiple ts inCa2RuO4, Phys. Rev. B 100, 045123 (2019). 41B. J. Kim, H. Jin, S. J. Moon, J.-Y. Kim, B.-G. Park, C. S. Leem, J. Yu, T. W. Noh, C. Kim, S.-J. Oh, J.-H. Park, V. Durairaj, G. Cao and E. Rotenber g,NovelJeﬀ= 1/2Mott State Induced by Relativistic Spin-Orbit Coupling in Sr2IrO4, Phys. Rev. Lett. 101, 076402 (2008). 42B.J.Kim, H.Ohsumi,T.Komesu, S.Sakai, T.Morita, H. Takagi andT.Arima, Phase-Sensitive Observation of a Spin-Orbital Mott State in Sr2IrO4, Science 323, 1329–1332 (2009). 43J. Kim, D. Casa, M. H. Upton, T. Gog, Y.-J. Kim, J. F. Mitchell, M. van Veenendaal, M. Daghofer, J.vandenBrink, G.Khaliullin, andB.J.Kim, Magnetic Excitation Spectra of Sr2IrO434 Probed by Resonant Inelastic X-Ray Scattering: Establishin g Links to Cuprate Superconductors , Phys. Rev. Lett. 108, 177003 (2012). 44J-i Igarashi and T. Nagao, Interplay between Hund’s Coupling and Spin-Orbit Interaction on Elementary Excitations in Sr2IrO4, J. Phys. Soc. Jpn. 83, 053709 (2014). 45D. Pincini, J. G. Vale, C. Donnerer, A. de la Torre, E. C. Hunte r, R. Perry, M. Moretti Sala, F. Baumberger, and D. F. McMorrow, Anisotropic Exchange and Spin-Wave Damping in Pure and Electron-Doped Sr2IrO4, Phys. Rev. B 96, 075162 (2017). 46J. Porras, J. Bertinshaw, H. Liu, G. Khaliullin, N. H. Sung, J .-W. Kim, S. Francoual, P. Steﬀens, G. Deng, M. Moretti Sala, A. Eﬃmenko, A. Said, D. Casa , X. Huang, T. Gog, J. Kim, B. Keimer, and B. J. Kim, Pseudospin-Lattice Coupling in the Spin-Orbit Mott Insulato r Sr2IrO4, Phys. Rev. B 99, 085125 (2019). 47S. Mohapatra, J. van den Brink, and A. Singh, Magnetic Excitations in a Three-Orbital Model for the Strongly Spin-Orbit Coupled Iridates: Eﬀect of Mixing Between the J= 1/2and3/2 Sectors, Phys. Rev. B 95, 094435 (2017). 48J. Kim, M. Daghofer, A. H. Said, T. Gog, J. van den Brink, G. Kha liullin, and B. J. Kim, Excitonic Quasiparticles in a Spin-Orbit Mott Insulator , Nat. Commun. 5, 4453 (2014). 49O. Krupin, G. L. Dakovski, B. J. Kim, J. Kim, S. Mishra, Yi-De C huang, C. R. Serrao, W.-S. Lee, W. F. Schlotter, M. P. Minitti, D. Zhu, D. Fritz, M. Choll et, R. Ramesh, S. L. Molodtsov, and J. J. Turner, Ultrafast Dynamics of Localized Magnetic Moments in the Unc onventional Mott Insulator Sr2IrO4, J. Phys.: Condens. Matter 28, 32LT01 (2016). 50M. Souri, B. H. Kim, J. H. Gruenewald, J. G. Connell, J. Thomps on, J. Nichols, J. Terzic, B. I. Min, G. Cao, J. W. Brill, and A. Seo, Optical Signatures of Spin-Orbit Exciton in Bandwidth- Controlled Sr2IrO4Epitaxial Films via High-Concentration Ca and Ba Doping , Phys. Rev. B 95, 235125 (2017). 51S. Mohapatra and A. Singh, Correlated Motion of Particle-Hole Excitations Across the R enor- malized Spin-Orbit Gap in Sr2IrO4, J. Magn. Magn. Mater 512, 166997 (2020). 52S. Mohapatra and A. Singh, Magnetic Reorientation Transition in a Three Orbital Model f or Ca2RuO4— Interplay of Spin-Orbit Coupling, Tetragonal Distortion, a nd Coulomb Interac- tions, J. Phys.: Condens. Matter 32, 485805 (2020). 53S. Mohapatra and A. Singh, Pseudo-Spin Rotation Symmetry Breaking by Coulomb Interacti on terms in Spin-Orbit Coupled Systems , J. Phys.: Condens. Matter 33, 065802 (2021).35 54G. Khaliullin, Excitonic Magnetism in Van Vleck–type d4Mott Insulators , Phys. Rev. Lett. 111, 197201 (2013). 55A. Akbari and G. Khaliullin, Magnetic Excitations in a Spin-Orbit-Coupled d4Mott Insulator on the Square Lattice , Phys. Rev. B 90, 035137 (2014). 56T. Feldmaier, P. Strobel, M. Schmid, P. Hansmann, and M. Dagh ofer,Excitonic Magnetism at the Intersection of Spin-Orbit Coupling and Crystal-Field S plitting, Phys. Rev. Research 2, 033201 (2020). 57S.MohapatraandA.Singh, Spin Waves and Stability of Zigzag Order in the Hubbard Model with Spin-Dependent Hopping Terms: Application to the Honeycom b Lattice Compounds Na2IrO3 andα-RuCl3, J. Magn. Magn. Mater 479, 229 (2019).(π/2,π/2) (π,0) ( π,π) (0,0) 600 650 700 750 800 850 900 950 1000ω (meV) 0.01 0.1 1 10 100 (c)
1205.6629v1.Conservation_law_in_noncommutative_geometry____Application_to_spin_orbit_coupled_systems.pdf	arXiv:1205.6629v1 [math-ph] 30 May 2012Conservation law in noncommutative geometry – Application to spin-orbit coupled systems Naoyuki Sugimoto1and Naoto Nagaosa1,2 1Cross-correlated Materials Research Group (CMRG) and Corr elated Electron Research Group (CERG), RIKEN, Saitama 351-0198, J apan 2Department of Applied Physics University of Tokyo, Tokyo 11 3-8656, Japan The quantization scheme by noncommutative geometry develo ped in string theory is applied to establish the conservation law of twisted spin and spin curr ent densities in the spin-orbit coupled systems. Starting from the pedagogical introduction to Hop f algebra and deformation quantization, the detailed derivation of the conservation law is given.2 CONTENTS I. Introduction 3 II. Noether’s theorem in ﬁeld theory 4 A. Conventional formulation of Noether’s theorem 4 B. Generalization of Noether’s theorem 7 III. Hopf algebra 8 1. Algebra 8 2. Coalgebra 10 3. Dual-algebra and Hopf algebra 11 IV. Deformation quantization 12 A. Wigner representation 13 B. Star product 14 1. Cohomology equation 16 2.L∞algebra 18 C. Topological string theory 20 1. Ghost ﬁelds and anti-ﬁelds 21 2. Condition of gauge invariance of classical action 22 3. Gauge invariance in path integral 23 D. Equivalence between deformation quantization and topological s tring theory 25 1. Path integral as L∞map 25 2. Perturbation theory 26 E. Diagram rules of deformation quantization 27 F. Gauge invariant star product 29 V. Twisted spin 31 A. Derivation of a twisted spin in Wigner space 31 B. Rashba-Dresselhaus model 33 VI. Conclusions 35 References 363 I. INTRODUCTION Electrons are described by the Dirac equation where the U(1) Maxw ell electromagnetic ﬁeld (emf) Aµis coupled to the charge current jµas described by the Lagrangian (in the natural unit where /planckover2pi1=c= 1;µ= 0,1,2,3) [1] L=¯ψ[iγµˆDµ−m]ψ. (1) whereˆDµ=∂µ−ieAµis the covariant derivative, mis the electron mass. Note that the spin is encoded by 4 component nature of the spinors ψand¯ψ=ψ†γ0and the 4×4 gamma matrices γµ, but the charge and charge current alone determine the electromagnetic properties of the ele ctrons, which are given by jµ=−∂L ∂Aµ=−e¯ψγµψ. (2) In condensed matter physics, on the other hand, the low energy p henomena compared with the mass gap 2 mc2∼ 106eVare considered, and only the positive energy states described by t he two-component spinor are relevant. Then, the relativistic spin-orbit interaction originates when the negative e nergy states (positron stats) are projected out to derive the eﬀective Hamiltonian or Lagrangian. The projection to a subspace of the Hilbert space leads to the nontrivial geometrical structure which is often described by the g auge theory. This is also the case for the Dirac equation, and the resultant gauge ﬁeld is SU(2) non-Abelian gauge ﬁ eld corresponding to the Zeeman eﬀect (time- component) and the spin-orbit interaction (spatial components) as described below. The eﬀective Lagrangianforthe positiveenergystatescan be der ivedby the expansionwith respect to 1 /(mc2) [2–4] L= iψ†D0ψ+ψ†D2 2mψ+1 2mψ†/bracketleftbigg eqσaA·Aa+q2 4Aa·Aa/bracketrightbigg ψ, (3) whereψis now the two-component spinor and D0=∂0+ieA0+iqAa 0σa 2, andDi=∂i−ieAi−iqAa iσa 2(i= 1,2,3) are the gauge covariant derivatives with qbeing the quantity proportional to the Bohr magneton [2, 4]. Aµis the Maxwell emf, and the SU(2) gauge potential are deﬁned as Aa 0=Ba Aa i=ǫiaℓEℓ, (4) andσx,y,zrepresent the Pauli matrices. The SU(2) gauge ﬁeld is coupled to th e 4-component spin current ja 0=ψσaψ, ja i=1 2mi[ψ†σaDiψ−Diψ†σaψ]. (5) Namely, the Zeeman coupling and the spin-orbit interaction can be re garded as the gauge coupling between the 4-spin current and the SU(2) gauge potential. (The spin current is the te nsor quantity with one suﬃx for the direction of the spin polarization while the other for the direction of the ﬂow.) N ote that the system has no SU(2) gauge symmetry since the “vector potential” Aa µis given by the physical ﬁeld strength BandE, i.e., the relation ∂µAa µ= 0 automatically holds. This fact is connected to the absence of the co nservation law for the spin density and spin current density in the presence of the relativistic spin-orbit intera ction. In the spherically symmetric systems, the total angular momentum, i.e., the sum of the orbital and spin angular momenta, is conserved, but the rotational symmetry is usually broken by the periodic or disorder potential A0in condensed matter systems. Therefore, it is usually assumed that the conservation law of spin is lost by the spin-o rbit interaction. However, it is noted that the spin and spin current densities are “co variantly” conserved as described by the “continuity equation” [2–4] D0Ja 0+D·Ja= 0. (6) replacing the usual derivative ∂µby the covariant derivative Dµ. This suggest that the conservation law holds in the co-moving frame, but the crucial issue is how to translate this la w to the laboratory frame, which is the issue addressed in this paper. Note again that the SU(2) gauge symmetr y is absent in the present problem, and hence the Lagrangian like tr( FµνFµν), which usually leads to the generalized Maxwell equation and also to t he conservation law of 4 spin current including both the matter ﬁeld and gauge ﬁeld [1], is missing. Instead, we will regard Aa µas the frozen background gauge ﬁeld, and focus on the quantum dynamic s of noninteracting electrons only.4 In this paper, we derive the hidden conservation law by deﬁning the “ twisted” spin and spin current densities which satisfy the continuity equation with the usual derivative ∂µ. The description is intended to be pedagogical and self- contained. For this purpose, the theoretical techniques develop ed in high energy physics is useful. The essential idea is to take into account the eﬀect of the background gauge ﬁeld in te rms of the noncommutative geometry generalizing the concept of “product”. This is achieved by extending the usual Lie algebra to Hopf algebra. Usually, a conservation law is derived from symmetry of an action, i.e., Noether’s theorem. The symmetry in the noncommutative geometry is called as a “twisted” symmetry, and th is symmetry and the corresponding generalized Noether’s theorem have been studied in the high energy physics. Se iberg and Witten proposed that an equivalence of a certain string theory and a certain ﬁeld theory in noncommutative geometry [5]. Since then, the noncommutative geometry have been attracted many researchers. On the other hand, it is known that the Poincaresymmetry is broken in a ﬁeld theory on a noncommutative geometry. It is a serious proble m because the energy and momentum cannot be deﬁned. M. Chaichian, et al. proposed the twisted symmetry in the Minkowski spacetime, and a lleged that the twisted Poincare symmetry is substituted for the Poincare symmet ry [6, 7]. Moreover, G. Amelino-Camelia, et al. discussed Noether’s theorem in the noncommutative geometry [8, 9 ]. As we will discuss in detail later, a certain type ofa noncommutative g eometry space is equal to a spin-orbit coupled system. Therefore, a global SU(2) gauge symmetry in the noncom mutative geometry space gives a Noether current corresponding to the “twisted” spin and spin current in the spin-or bit coupling system. This enables us to derive the generalized Noether’s theorem for the twisted spin and spin curren t densities. Now some remarks about the application is in order. Spintronics is an e merging ﬁeld of electronics where the role of charge and charge current are replaced by the spin and spin cur rent aiming at the low energy cost functions [10]. The relativistic spin-orbit interaction plays the key role there since it enables the manipulation of spins by the electric ﬁeld. However, this very spin-orbit interaction introduces the spin relaxation which destroys the spin information in sharp contrast to the case of charge where the information is pro tected by the conservation law. Therefore, it has been believed that the spintronics is possible in a short time-scale or t he small size devices. The discovery of the conservation law of twisted spin and spin current densities means th at the quantum information of spin is preserved by this hidden conservation law, and could be recovered. Actually, it has been recently predicted that the adiabatic change in the spin-orbit interaction leads to the recovery of the sp in moment called spin-orbit echo [11]. Therefore, the conservation law of the twisted spin and spin current densities is directly related to the applications in spintronics. The plan ofthis paper follows (see Fig. 1). In section II, we reviewth e conventionalNoether’s theorem, and describe brieﬂy its generalization to motivate the use of Hopf algebra and def ormation quantization. In section III, the Hopf algebra is introduced, and section IV gives the explanation of the de formation quantization with the star product. The gauge interaction is compactly taken into account in the deﬁnitio n of the star product. These two sections are sort of short review for the self-containedness and do not conta in any original results except the derivation of the star product with gauge interaction. Section V is the main body of this pap er. By combining the Hopf algebra and the deformation quantization, we present the derivation of the conse rved twisted spin and spin current densities. Section VI is a brief summary of the paper and contains the possible new direc tions for future studies. The readers familiar with the noncommutative geometry and deformation quantization c an skip sections III, IV, and directly go to section V. II. NOETHER’S THEOREM IN FIELD THEORY In this section, we discuss Noether’s theorem [12], and its generaliza tion as a motivation to introduce the Hopf algebra and deformation quantization. In section IIA, we will recall Noether theorem, and rewrite it using the so- called “coproduct”, which is an element of the Hopf algebra. In sect ion IIB, we will sketch a derivation of generalized Noether theorem. A. Conventional formulation of Noether’s theorem We start with the action Igiven by I=/integraldisplay ΩdDimxL(x) =/integraldisplay dDimxhΩ(x)L(x), (7)5 FIG. 1. Flows of derivation of generalized Noether’s theore m. Roman numerals and capital letters in boxes represent section and subsection numbers, respectively. A generaliz ation of the Noether’s theorem is achieved through Hopf alge bra and deformation quantization (section V). Hopf algebra appear to characterize feature of an inﬁnitesimal transformed var iation operator (sections II and III). The SU(2) gauge structure is embedded in the star product (section IV). where Ω represents a range of the spacetime coordinate x(≡(x0,xi)≡(ct,x)) with a dimension Dim, i.e., (Dim −1) is the dimension of the space, Ldescribes a Lagrangian density, and hΩ(x) =/braceleftbigg 1 for{x\|x∈Ω} 0 for{x\|x /∈Ω}.; (8) crepresents light speed. We introduce a ﬁeld φrwith internal degree of freedom r, and inﬁnitesimal transformations: xµ/ma√sto→(x′)µ:=xµ+δζxµ, (9) φr(x)/ma√sto→φ′ r(x′) :=φr(x)+δζφr(x), (10) where we characterize the transformations by the subscript; sp eciﬁcally,ζrepresents a general inﬁnitesimal transfor- mation. Hereafter, we will employ Einstein summation convention, i.e., aµbµ≡aµbµ≡/summationtextDim−1 µ=0ηµνaµbµwith vectors aµandbµ(µ= 0,1,...,(Dim−1)), and the Minkowski metric: ηµν:= diag(−1,1,1,...,1/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright Dim−1). We deﬁne the variation operator of the action as follow: δζI:=/integraldisplay Ω′dDimx′L′(x′)−/integraldisplay ΩdDimxL(x) =/integraldisplay dDimx′hΩ′(x′)L′(x′)−/integraldisplay dDimxhΩ(x)L(x), (11) where we characterize this variation by ζ, because this variation is derived from the inﬁnitesimal transforma tions Eqs. (9) and (10). Since the integration variable x′can be replaced by x, Eq. (11) is δζI=/integraldisplay dDimxhΩ′(x)L′(x)−/integraldisplay dDimxhΩ(x)L(x) =/integraldisplay dDimx(hΩ′(x)−hΩ(x))L′(x)+/integraldisplay dDimxhΩ(x)[L′(x)−L(x)] =/integraldisplay dDimxhδΩ(x)L′(x)+/integraldisplay dDimxhΩ(x)[L′(x)−L(x)], (12) whereδΩ := Ω′−Ω andhδΩ=−(∂µhΩ)δζxµ+O((δζx)2). Therefore, we obtain the following equation through partial integration: δζI=/integraldisplay dDimxhΩ(x)/bracketleftbig ∂µ(L(x)δζxµ)+δL ζL(x)/bracketrightbig +O((δζx)2), (13)6 where we have introduced the so-called Lie derivative: δL ζφr(x) :=φ′ r(x)−φr(x) =δζφr(x)−(∂µφr)δζxµ+O(δζx2), (14) and we replacedL′byLdue toL′δζxµ=Lδζxµ+O(δζx2). Hereafter, we assume that the action is invariant under the inﬁnite simal transformations Eqs. (9) and (10). In the case where the Lagrangian density is a function of φrand∂µφr, i.e.,L(x) =L[φr(x),∂µφr(x)], the Lie derivative of the Lagrangian is given by δL ζL:=L′(x)−L(x) =L[φ′ r(x),∂µφ′ r(x)]−L[φr(x),∂µφr(x)] =/braceleftbigg∂L ∂φrδL ζφr+∂L ∂(∂µφr)∂µδL ζφr/bracerightbigg =/parenleftbigg∂L ∂φr−∂µ∂L ∂(∂µφr)/parenrightbigg δL ζφr+∂µ/parenleftbigg∂L ∂(∂µφr)δL ζφr/parenrightbigg , (15) and the variation of the action is calculated by δζI=/integraldisplay ΩdDimx/braceleftbigg/parenleftbigg∂L ∂φr−∂µ∂L ∂(∂µφr)/parenrightbigg δL ζφr+∂µ/parenleftbigg Lδζxµ+∂L ∂(∂µφr)δL ζφr/parenrightbigg/bracerightbigg . (16) If we require that δζxandδL ζφrvanish on the surface ∂Ω, we obtain the Euler-Lagrange equation. On the other hand, if we require that ﬁelds φrsatisfy the Euler-Lagrange equation, we obtain continuity equatio n∂µjµ= 0 with a Noether current jµ:=/parenleftbigg Lδζxµ+∂L ∂(∂µφr)δL ζφr/parenrightbigg . (17) Hereafter let us discuss an inﬁnitesimal global U(1) ×SU(2) gauge transformation and inﬁnitesimal translation and rotation transformations, which are denoted by χin this paper. Variations in terms of χare deﬁned by δχxµ:= Γµ νxν, (18) δχφr:= iϑµν(ξµν)r′ rφr′ (19) with an inﬁnitesimal parameter ϑµν, and symmetry generators Γµ νand (ξµν)r′ r. 1. Forthe globalU(1) ×SU(2) gaugetransformation, Γµ ν≡0,ϑµν≡ϑµδµν, and (ξµν)r′ r≡δµν(ˆsµ)r′ r(µ,ν= 0,1,2,3; r,r′= 1,2), where ˆs0:=/planckover2pi1/2, and ˆs1,2,3:=/planckover2pi1ˆσx,y,z/2 with the Planck constant h= 2π/planckover2pi1and Pauli matrices ˆσx,y,z. 2. For the translation, Γµ ν≡εµδµ ν,ϑµν≡εµδµνand (ξµν)r′ r≡ˆpµδµνδr′ rwith an inﬁnitesimal parameter εµand the momentum operator ˆ pµ=−i/planckover2pi1∂µ(µ,ν= 1,2,3;r,r′= 1,2). 3. For the rotation, Γµ ν≡ωµ ν,ϑµν≡ωµν, and (ξµν)r′ r≡δr′ rxµˆpν, which corresponds to the angular momentum tensor (µ,ν= 1,2,3;r,r′= 1,2). For these transformations, equation ∂µδχxµ= 0 is satisﬁed. This can be seen explicitly as follows. The variations of space coordinates of the global U(1) ×SU(2) and the translation transformations are given by δχxµ= 0 orδχxµ= constant, respectively, and thus ∂µδχxµ= 0 is trivial. The variation of the rotation transformation is given by δχxµ=ωµ νxν, therefore∂µδχxµ=∂µωµ νxν=ωµ µ= 0. We consider a variation of the Lagrangian density; δζL:=L′(x′)−L(x) =L′(x′)−L(x′)+L(x′)−L(x) =δL ζL(x′)+δζxµ∂µL+O((δζx)2). (20) Note that Eq. (20) is correct for any inﬁnitesimal transformation . Here we consider the global U(1) ×SU(2) gauge transformation and/or the translation and rotation transforma tionsδχ. Because∂µδχxµ= 0, we obtain the following equation: δχL=δL χL(x)+∂µ(L(x)δχxµ)+O((δx)2). (21)7 From Eqs.(13) and (21), one can see δ(ζ=χ)I=/integraldisplay dDimxδχL(x), (22) where (ζ=χ) denotes that the type of the variation in Eq. (13) is restricted to the global U(1)×SU(2) or Poincare transformations. (For simplicity we omitted the subscript Ω in the int egral). Finally, for ζ=χ, the variation of the action is equal to the variation of Lagrangian. This fact will be used la ter in section V where the variation of the Lagrangian density instead of the action will be considered. B. Generalization of Noether’s theorem Now, we wouldlike to introduce a Hopfalgebraforthe purposeofgen eralizingNoether’s theorem [8, 9, 13]. At ﬁrst, werewriteNoether’stheoreminsectionIIbyusingtheHopfalgebra ,andnext, weintroduceatwistedsymmetry [6,7]. For simplicity, we only consider the global U(1) ×SU(2) gauge symmetry and the Poincare symmetry. We assume tha t the Lagrangian density is written as L(x) =ψ†(x)ˆL(x)ψ(x) (23) with a ﬁeld ψ:=/parenleftbigg ψ1 ψ2/parenrightbigg , a Hermitian conjugate ψ†≡/parenleftbig ψ1,ψ2/parenrightbig , and an single-particle Lagrangian density operator ˆL, which is a 2×2 matrix; the overline represents the complex conjugate. The act ion can be rewritten as I=/integraldisplay dDimx1dDimx2ψ†(x2)δ(x2−x1)ˆL(x1)ψ(x1) = tr/integraldisplay dDimx1dDimx2δ(x2−x1)ˆL(x1)ψ(x1)ψ†(x2) = tr/integraldisplay dDimx1/braceleftbigg/integraldisplay dDimx2˜L(x1,x2)G(x2,x1)/bracerightbigg = tr/integraldisplay dDimx1/braceleftbigg lim x3→x1(˜L∗CG)(x1,x3)/bracerightbigg , (24) where “tr” represents the trace in the spin space, G(x1,x2) :=ψ(x1)ψ†(x2),˜L(x1,x2) :=δ(x1−x2)ˆL(x2), and∗C represents the convolution integral: (f∗Cg) :=/integraldisplay dDimx3f(x1,x3)g(x3,x2) (25) with smooth two-variable functions fandg. The variation operator δχof the action can be also rewritten as δχI= tr/integraldisplay dDimx1dDimx2˜L(x2,x1)/bracketleftbig iϑξψ(x1)ψ†(x2)−ψ(x1)ψ†(x2)iϑξ/bracketrightbig = tr/integraldisplay dDimx1dDimx2/bracketleftig ˜L(x2,x1)iϑξG(x1,x2)−iϑξ˜L(x2,x1)G(x1,x2)/bracketrightig (26) withϑξ≡ϑµν(ξµν); in addition, we assumed that the single-particle Lagrangian densit y operator is invariant under the inﬁnitesimal transformation δχ. Here, we introduce Grassmann numbers θ1andθ2; an integral is deﬁned by/integraltext dθi(θj) =δij. The variation of the8 right-hand side of Eq.(26) can be rewritten as follow: δχI=−itr/integraldisplay dθ1dθ2dDimx1dDimx2/bracketleftig θ1˜L(x2,x1)ϑθ2ξG(x1,x2)+ϑθ2ξθ1˜L(x2,x1)G(x1,x2)/bracketrightig =−itr/integraldisplay dθ1dθ2dDimx1dDimx2µ◦(µ⊗id) ◦/bracketleftig/parenleftig θ1˜L(x2,x1)⊗ϑθ2ξ+ϑθ2ξ⊗θ1˜L(x2,x1)/parenrightig ⊗G(x1,x2)/bracketrightig =−itr/integraldisplay dθ1dθ2dDimx1dDimx2µ◦(µ⊗id)/bracketleftig △(ϑθ2ξ)◦/parenleftig θ1˜L(x2,x1)⊗G(x1,x2)/parenrightig/bracketrightig ≡ˆTr/bracketleftig △(ϑθ2ξ)◦((θ1˜L)⊗G)/bracketrightig , (27) where⊗and◦represent a tensor product and a product of operators, respe ctively. The operator µdenotes the transformation of the tensor product to the usual product µ:x⊗y/ma√sto→xy, and△represents a coproduct: △(ζ) :=ζ⊗id+id⊗ζ, (28) whereζand id represent a certain operator and the identity map, respect ively. These operators constitutes the Hopf algebraaswillbeexplainedinthenextsection. Moreover,wehavede ﬁnedˆTr :=−itr/integraltext dθ1dθ2dDimx1dDimx2µ◦(µ⊗id). We emphasize here that the variation is written by the coproduct △, which is important to formulate the generalized Noether theorem in the presence of the gauge potential. The copr oduct determines an operation rule of a variation operator; for example, the coproduct (28) represents the Leib niz rule. A twisted symmetry transformation is given by deformation of the coproduct. We now sketch the concept of the twisted symmetry in deformation quantization [6, 7]. First, we assume that the variation of action δζI0is zero, i.e., ζrepresents the symmetry transformation of the system corres ponding to the actionI0. Next, we consider the action IAwith external gauge ﬁelds A. Usually, external gauge ﬁelds breaks symmetries of I0, i.e.,δζIA/ne}ationslash= 0. Here we introduce a map: F(0/mapsto→A):I0/ma√sto→IA, which will be deﬁned in section IVF. The basic idea is to generalize the ”product” taking into accoun t the gauge interaction. Using this map, the variation is rewritten as δζF(0/mapsto→A)I0/ne}ationslash= 0. On the other hand, when the twisted symmetry δt ζ:=F(0/mapsto→A)δζF−1 (0/mapsto→A)can be deﬁned, we obtain the following equation: δt χIA=F(0/mapsto→A)δχF−1 (0/mapsto→A)IA =F(0/mapsto→A)δχF−1 (0/mapsto→A)F(0/mapsto→A)I0 =F(0/mapsto→A)δχI0 = 0. (29) Namely,δt χcorresponds to a symmetry with external gauge ﬁelds. In the exp ression for the variation of action in terms of the Hopf algebra Eq.(27), we can replace ∆ by ∆tcorresponding to the change from δχtoδt χas shown in section V. This is achieved by using the Hopf algebra and the deforma tion quantization, which will be explained in sections III and IV, respectively. Therefore, we can generalize t he Noether’s theorem and derive the conservation law even in the presence of the gauge ﬁeld A. III. HOPF ALGEBRA Here we introduce a Hopf algebra. First, we rewrite the algebra usin g tensor and linear maps. Secondly, a coalgebra is deﬁned using diagrams corresponding to the algebra. Finally, we de ﬁne a dual-algebra and Hopf algebra. 1. Algebra We deﬁne the algebra as a k-vector space Vhaving product µand unitε. Here,krepresents a ﬁeld such as the complex number or real number. In this paper, we consider Vas the space of functions or operators. A space of linear maps from a vector space V1to a vector space V2is written as Hom( V1,V2). A productµis a bilinear map: µ∈Hom(V/circlemultiplytextV,V), i.e., µ:V/circlemultiplydisplay V→V,(x,y)/ma√sto→xy, (30)9 and a unit is a linear map: ε∈Hom(k,V), i.e., ε:k→V, α/ma√sto→α·1 (31) withx,y,xy∈Vandα∈k. Hereµandεsatisﬁes µ((x+y)⊗z) =µ(x⊗z)+µ(y⊗z), µ(x⊗(y+z)) =µ(x⊗y)+µ(x⊗z), (32) µ(αx⊗y) =αµ(x⊗y), µ(x⊗αy) =αµ(x⊗y), (33) ε(α+β) =ε(α)+ε(β) (34) withx,y,z∈Vandα,β∈k. The product µhas the association property, which is written as µ◦(id⊗µ) =µ◦(µ⊗id). Because the left-hand side and the right-hand side of the previous equation give the followin g equations: µ◦(id⊗µ)(x⊗y⊗z) =µ(x⊗(yz)) =x(yz) (35) and µ◦(µ⊗id)(x⊗y⊗z) =µ((xy⊗z)) = (xy)z, (36) for allx,y,z,xy,yz,xyz ∈V, thenµ◦(id⊗µ) =µ◦(µ⊗id) is equal to the association property x(yz) = (xy)z. This property is illustrated as the following diagram: V/circlemultiplytextV/circlemultiplytextV V/circlemultiplytextV V/circlemultiplytextV V❄✲ ✲❄µ⊗id id⊗µ µ µ/clockwise Here/clockwisedenotes that this graph is the commutative diagram. The unitεsatisﬁes the following equation: µ◦(ε⊗id) =µ◦(id⊗ε). Since the left-hand side and the right-hand side of the previous equation give the following equations µ◦(ε⊗id)(α⊗x) =µ⊗(α1V⊗x) =α1Vx=αx (37) and µ◦(id⊗ε)(x⊗α) =µ◦(x⊗α1V) =αx1V=αx (38) for allx∈Vandα∈k, and∃1V∈V, then the unit can be written as µ◦(id⊗ε) =µ◦(ε⊗id). Note that V∼k/circlemultiplytextV∼V/circlemultiplytextk, where∼represents the equivalence relation, i.e., a∼bdenotes that aandbare identiﬁed. This property is illustrated as: k/circlemultiplytextV V/circlemultiplytextk V/circlemultiplytextV V✲ ❄✛ ❍❍❍❍❍❍❍ ❥✟✟✟✟✟✟✟ ✙ε⊗id id ⊗ε µ ∼ ∼ Algebra is deﬁned as a set ( V,µ,ε).10 2. Coalgebra A coalgebra is deﬁned by reversing the direction of the arrows in the diagrams corresponding to the algebra. Thus, we will deﬁne a coproduct △∈Hom(V,V/circlemultiplytextV) and counit η∈Hom(V,V) with ak-vector space V. A coproduct is a bilinear map from VtoV/circlemultiplytextV: △:V→V/circlemultiplydisplay V, (39) and satisﬁes co-association property: V/circlemultiplytextV/circlemultiplytextV V/circlemultiplytextV V/circlemultiplytextV V✻✛ ✛✻△⊗id id⊗△ △ △/clockwise Namely, (id⊗△)◦△= (△⊗id)◦△ (40) (Compare the diagram corresponding to the association property and that corresponding to the co-association prop- erty). A counitηis a linear map from Vto ﬁeldk: η:V→k, (41) and satisﬁes the following diagram: k/circlemultiplytextV V/circlemultiplytextk V/circlemultiplytextV V✛ ✻✲ ❍❍❍❍❍❍❍ ❨ ✟✟✟✟✟✟✟ ✯η⊗id id ⊗η △∼ ∼ Namely, (η⊗id)◦△= (id⊗η)◦△, (42) whereV∼k/circlemultiplytextV∼V/circlemultiplytextk. Since△andηare linear maps,△andηsatisfy △(x+y) =△(x)+△(y),△(αx) =α△(x), (43) η(x+y) =η(x)+η(y), η(αx) =αη(x) (44) withx,y∈Vandα∈k. Note that V∼k/circlemultiplytextV∼V/circlemultiplytextkandV/circlemultiplytextV∼k/circlemultiplytextV/circlemultiplytextV∼V/circlemultiplytextk/circlemultiplytextV∼V/circlemultiplytextV/circlemultiplytextk. A coalgebra is deﬁned as a set ( V,△,η). For example, in the vector space D≡k/circleplustextk∂:={a0+a1∂\|a0,a1∈k}, we deﬁne a coproduct △D(∂) =∂⊗1 + 1⊗∂and△D(1) = 1⊗1, and a counit ηD(∂) = 0 and ηD(1) = 1. The set (D,△D,ηD) is coalgebra, because this set satisﬁes the equations: ( △D⊗id)◦△D= (id⊗△D)◦△Dand (ηD⊗id)◦△D= (id⊗ηD)◦△D. Because the coproduct and counit are linear map, we only check th e above equations with respect to x= 1 and∂. Forx= 1, (△D⊗id)◦△D(1) =△D(1)⊗1 = 1⊗1⊗1, (45)11 and (id⊗△D)◦△D(1) = 1⊗△D(1) = 1⊗1⊗1. (46) Therefore, (△D⊗id)◦△D(1) = (id⊗△D)◦△D(1). Moreover, (ηD⊗id)◦△D(1) =ηD(1)⊗1 = 1⊗1, (47) and (id⊗ηD)◦△D(1) = 1⊗ηD(1) = 1⊗1. (48) Therefore ( ηD⊗id)◦△D(1) = (id⊗ηD)◦△D(1). Forx=∂, (△D⊗id)◦△D(∂) =△D(∂)⊗1+△D(1)⊗∂=∂⊗1⊗1+1⊗∂⊗1+1⊗1⊗∂, (49) and (id⊗△D)◦△D(∂) =∂⊗△D(1)+1⊗△D(∂) =∂⊗1⊗1+1⊗∂⊗1+1⊗1⊗∂. (50) Therefore, (id⊗△D)◦△D(∂) = (△D⊗id)◦△D(∂). Finally, (ηD⊗id)◦△D(∂) =ηD(∂)⊗1+ηD(1)⊗∂= 1⊗∂=∂, (51) and (id⊗ηD)◦△D(∂) =∂⊗ηD(1)+1⊗ηD(∂) =∂⊗1 =∂. (52) Therefore, ( ηD⊗id)◦△D(∂) = (id⊗ηD)◦△D(∂). Namely, the set ( D,△D,ηD) is the coalgebra. Note that △D(1) corresponds to the product with a constant: a(fg) =a1(fg) =a(1f1g) =aµ◦△D(1)(f⊗g), where we have used the coproduct△D(1) = 1⊗1attheﬁnalequalsign. Here f,andgaresmoothfunctions, 1isincludedinthefunctionspace, anda∈k.△D(∂) represents the Leibniz rule: ∂(fg) = (∂f)g+f∂(g) =µ◦(∂⊗1+1⊗∂)◦(f⊗g) =µ◦△D(∂)(f⊗g), where we have used the coproduct △D(∂) = 1⊗∂+∂⊗1 at the last equal sign. ηD(1) andηD(∂) represent the ﬁltering action to a constant function: 1 a=a=ηD(1)aand∂(a) = 0 =ηD(∂)a, respectively. 3. Dual-algebra and Hopf algebra A dual-algebra is the set of an algebra and a coalgebra, i.e., the set of (V,µ,ε,△,η). On a dual-algebra, we deﬁne a∗-product as f∗g=µ◦(f⊗g)◦△ (53) withf,g∈Hom(V,V). We deﬁne an antipode S∈Hom(V,V) which satisﬁes the following equation: µ◦(id⊗S)◦△=µ◦(S⊗id)◦△=ε◦η, (54) whereε◦ηcorresponds to the identity mapping, i.e., Sis an inverse of unit. For example, SDin the set (D,µD,εD,△D,ηD) is deﬁned as SD(1) = 1 and SD(∂) =−∂. Forx= 1, µD◦(id⊗SD)◦△D(1) =µD◦(1⊗1) = 1, (55) and µD◦(SD⊗id)◦△D(1) =µD◦(1⊗1) = 1. (56) Therefore, we obtain µD◦(id⊗SD)◦△D(1) =µD◦(SD⊗id)◦△D(1) =εD◦ηD. For∂, µD◦(id⊗SD)◦△D(∂) =µD◦(∂⊗1−1⊗∂) = 0, (57)12 and µD◦(SD⊗id)◦△D(∂) =µD◦(−∂⊗1+1⊗∂) = 0. (58) Therefore, we obtain µD◦(id⊗SD)◦△D(∂) =µD◦(SD⊗id)◦△D(∂) =εD◦ηD(∂). Namely, ( D,µD,εD,△D,ηD) is the Hopf algebra. A dual-algebra with an antipode S, i.e., (V,µ,ε,△,η,S), is called a Hopf algebra. By using the approach similar to a coproduct and counit, we can deﬁn e a codiﬀerential operator Q∈Hom(V,V) from a diagram of the diﬀerential ∂∈Hom(V,V). The diﬀerential ∂is the linear map: ∂:V→V, (59) and satisﬁes Leibniz rule ∂◦µ=µ◦(id⊗∂+∂⊗id), (60) which is illustrated as V V V/circlemultiplytextV V/circlemultiplytextV✻✛ ✛✻∂ µ µ (id⊗∂+∂⊗id)/clockwise A codiﬀerential operator Qis a linear map; Q:V→V, and satisﬁes the following diagrams: V V V/circlemultiplytextV V/circlemultiplytextV❄✲ ✲❄Q △ △ (id⊗Q+Q⊗id)/clockwise Namely, a codiﬀerential operator Qsatisﬁes△◦Q= (id⊗Q+Q⊗id)◦△. In section IVB2, the codiﬀerential operator will be introduced. IV. DEFORMATION QUANTIZATION In this section, we explain the deformation quantization using the no ncommutative product encoding the commu- tation relationships. At ﬁrst, in section IVA, we introduce the so-c alled Wigner representation and Wigner space, and show that a product in the Wigner space is noncommutative. This product is called Moyal product and it guarantees the commutation relationship of the coordinate and ca nonical momentum. Next, we add spin functions and background gauge ﬁelds to the Wigner space, and rewrite the c oordinates of Wigner space as a set of spacetime coordinates X, mechanical momenta p, and spins s:= (sx,sy,sz) (pincludes the background gauge ﬁelds). To gen- eralize the Moyal product for the deformed Wigner space, which is a set of function deﬁned on ( X,p,s), we explain the general constructing method of the noncommutative produc t in section IVB; the noncommutative product is the generalized Moyal product, which is called “star product”. This constructing method is given as a map from a Poisson bracket in the Wigner space to the noncommutative produc t (see section IVB), and we see the condition of this deformation quantization map in section IVB. This map is describe d by the path integral of a two-dimensional ﬁeld theory, which is called the topological string theory. In section IVC, we explain this topological string theory, and in section IVD, we discuss the perturbative treatment of this t heory. In section IVE, we summarize the diagram technique. Finally, in section IVF, we construct the star product in (X,p,s) space. We note that the star product guarantees the background gauge structure.13 A. Wigner representation We start with the introduction of the Wigner representation. From Equation (24), a natural product is the convolution integral: (f∗Cg)(x1,x2) :=/integraldisplay dDimx3f(x1,x3)g(x3,x2), (61) wheref,g∈Gwith a two spacetime arguments function space G. Here we introduce the center of mass coordinate X and the relative coordinate ξas follows: X≡(T,X) := ((t1+t2)/2,(x1+x2)/2), (62) ξ≡(ξt,ξ) := (t1−t2,x1−x2). (63) Moreover we employ the following Fourier transformation: FT:f(x1,x2)/ma√sto→f(X,p) =/integraldisplay dDimξe−ipµξµ//planckover2pi1f(X+ξ/2,X−ξ/2). (64) Now we deﬁne the Wigner space: W:={FT[f]\|f∈G}[14]. In this space, the convolution is transformed to the so-called Moyal product [15, 16]: (f ⋆Mg)(X,p) :=f(X,p)ei/planckover2pi1 2/parenleftBig← −∂X− →∂pν−← −∂p− →∂X/parenrightBig g(X,p), (65) because F−1 T[f ⋆Mg] =/integraldisplaydDimp (2π/planckover2pi1)Dimeipνξν//planckover2pi1/braceleftbigg f(X,p)ei/planckover2pi1 2/parenleftBig← −∂Xν− →∂pν−← −∂pν− →∂Xν/parenrightBig g(X,p)/bracerightbigg =/integraldisplaydDimp (2π/planckover2pi1)DimdDimξ1dDimξ2eipνξν//planckover2pi1/braceleftig e−ipνξ1//planckover2pi1f(X+ξ1/2,X−ξ1/2) ×ei/planckover2pi1 2/parenleftBig← −∂Xν− →∂pν−← −∂pν− →∂Xν/parenrightBig e−ipνξν 2//planckover2pi1g(X+ξ2/2,X−ξ2/2)/bracerightig =/integraldisplaydDimp (2π/planckover2pi1)DimdDimξ1dDimξ2eipν(ξν−ξν 1−ξν 2)//planckover2pi1 ×f(X+ξ1/2,X−ξ1/2)e1 2/parenleftBig← −∂Xνξν 2−ξν 1− →∂Xν/parenrightBig g(X+ξ2/2,X−ξ2/2) =/integraldisplay dDimξ1dDimξ2δ(ξ−ξ1−ξ2) ×f/parenleftbigg X+ξ1+ξ2 2,X−ξ1−ξ2 2/parenrightbigg g/parenleftbigg X−ξ1−ξ2 2,X−ξ1+ξ2 2/parenrightbigg =/integraldisplay dDimx+dDimx−δ(ξ−x+)f/parenleftig X+x+ 2,x−/parenrightig g/parenleftig x−,X−x+ 2/parenrightig =/integraldisplay dDimx−f(X+ξ/2,x−)g(x−,X−ξ/2) =f∗Cg (66) withξ1+ξ2≡x+andξ1−ξ2≡2(X−x−). In the Wigner space, the position operator ˆ xµ=xµand the momentum operator ˆ pµ=−i/planckover2pi1∂µbecomesXµ⋆Mand pµ⋆Mbecause FT[ˆxµ 1g(x1,x2)] =/integraldisplay dDimξe−i /planckover2pi1pνξν(Xµ+ξµ/2)g(X+ξ/2,X−ξ/2)=/parenleftbigg Xµ+i/planckover2pi1 2∂pµ/parenrightbigg g(X,p) =Xµ⋆Mg(X,p),(67) FT[ˆxµ 2g(x1,x2)] =/integraldisplay dDimξe−i /planckover2pi1pνξνg(X+ξ/2,X−ξ/2)(Xµ−ξµ/2) =g(X,p)⋆MXµ, (68) FT[(ˆp1)µg(x1,x2)] =/integraldisplay dDimξe−i /planckover2pi1pνξν/planckover2pi1 i∂xµ 1g(x1,x2) =pµ⋆g(X,p), (69)14 and FT[(ˆp2)µg(x1,x2)] =/integraldisplay dDimξe−i /planckover2pi1pνξν/planckover2pi1 i∂xµ 2g(x1,x2) =g(X,p)⋆pµ. (70) The commutation relationship of operators is [ Xµ,pν]⋆M:=Xµ⋆Mpν−pν⋆MXµ= i/planckover2pi1δµ ν, which corresponds to the canonical commutation relationship of operators: [ˆ xµ,ˆpν] = i/planckover2pi1δµ ν. To add the spin arguments in W, we will employ the following bilinear map: FM/ma√sto→FA=0:=ei/planckover2pi1 2(∂Xµ⊗∂pµ−∂pµ⊗∂Xµ)+i 2ǫabcsa∂sb⊗∂sc(71) with∂saf:=faandf≡f0+/summationtext a=x,y,zsafa. Note that the spin operator ˆs:= (ˆsx,ˆsy,ˆsz) is characterized by the commutation relation [ˆ sa,ˆsb] = iǫabcˆsc(a,b,c=x,y,z) with the Levi-Civita tensor ǫabc, and the star product (71) reproduces the relation, i.e., the operator ( sa⋆) satisﬁes [sa,sb]⋆= iǫabcsc. To obtain the map F(0/mapsto→A):I0/ma√sto→IA, we introduce the variables transformation ( Xµ,pµ,s)/ma√sto→(Xµ,ˆpµ,s) where ˆpµ=pµ−qAa µ(Xν)sa+eAµ (72) withq=\|e\|/mc2, the electric charge −e=−\|e\|, a U(1) gauge ﬁeld Aµ, and a SU(2) gauge ﬁeld Aa µ. Their ﬁelds are treated as real numbers, and the integral over pµcan be replaced by an integral over ˆ pµ. This transformation induces the following transformations of diﬀerential operators: ∂Xµ⊗∂pµ−∂pµ⊗∂Xµ/ma√sto→∂Xµ⊗∂ˆpµ−∂ˆpµ⊗∂Xµ+q/parenleftig ∂XµˆAν−∂XνˆAµ/parenrightig ∂ˆpµ⊗∂ˆpν, (73) ǫabcsa∂sb⊗∂sc/ma√sto→ǫabcsa∂sb⊗∂sc−qǫabcAb µsa∂ˆpµ⊗∂sc−qǫabcAc µsa∂sb⊗∂ˆpµ +q2ǫabcsaAb µAc ν∂ˆpµ⊗∂ˆpν, (74) whereˆAµ:=Aa µsa−(e/q)Aµ. We expandFA=0in terms of /planckover2pi1as FA=0=∞/summationdisplay n=0/parenleftbiggi/planckover2pi1 2/parenrightbiggn Fn A=0. (75) We deﬁne the bilinear map FAcorresponding to the commutation relation in terms of the phase sp ace (Xµ,ˆpµ,s), and expand it in terms of /planckover2pi1as FA=∞/summationdisplay n=0/parenleftbiggi/planckover2pi1 2/parenrightbiggn Fn A. (76) From Eqs. (73) and (74), F1 Ais given as follows: F1 A=∂Xµ⊗∂ˆpµ−∂ˆpµ⊗∂Xµ+qˆFµν∂ˆpµ⊗∂ˆpν+ǫabcsa∂sb⊗∂sc −qǫabcsaAb µ∂ˆpµ⊗∂sc+qǫabcsaAb µ∂sc⊗∂ˆpµ (77) withˆFµν:=∂XµˆAν−∂XνˆAµ+(q//planckover2pi1)εabcsaˆAb µˆAb ν. Note that µ◦F1 Ais the Poisson bracket. A constitution method of higher order terms Fn Awithn>1 is called a deformation quantization, which is given by Kontsevich [17], as will be described in the next subsection. B. Star product In this subsection, we explain the Kontsevich’s deformation quantiz ation method [17]. We deﬁne a star product as f ⋆g≡µ◦FA(f⊗g) =f·g+∞/summationdisplay n=1νnβn(f⊗g) (78) withν= i/planckover2pi1/2 [18, 19]. Here βn∈Hom(Vf⊗Vf,Vf) is called the two-cochain ( Vfrepresents the function space). We require that the star product satisﬁes the association property (f ⋆g)⋆h=f ⋆(g⋆h), which limits forms of Fn≥1 A15 L∞L∞ d.g.L.aαβαβ /g54 /g54 /g37/g372/g5422/g372/g40 /g40(a) (b) (c) FIG. 2. Steps of the derivation of the deformation quantizat ion. (a): The image of the deformation quantization, which i s the map from T2with the Jacobi identity to C2with the association property. (b): Enlargement of algebra s. The two vector space T2and two cochain space C2generalize to multi-vector space Tand cochain space C, respectively. These spaces are compiled in the d.g.L.a; ﬁnally, L∞algebra is introduced by using the d.g.L.a. (c): The deforma tion quantization is redeﬁned as the map on the L∞algebra. andβn≥1. We note that the association property is necessary for the exist ence of the inverse with respect to the star product. For example, the inverse of the Lagrangian is a Green fun ction, which always exists as ψψ†with a wave functionψ. Now, we deﬁne a p-cochain spaceCp:= Hom(V⊗p f,Vf) withV⊗p f≡Vf⊗Vf⊗···⊗Vf/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright p, whereVfrepresents a function space such as the Wigner space W; we deﬁne a multi-vector space Tk:= Γ(M,/logicalandtextkTM), whereMrepresents a manifold such as a classical phase space (dimension d),TM:=/uniontext p∈MTpMdenotes a tangent vector bundle with a tangent vector space TpM≡{/summationtextd iai(x)∂xi}atp∈M(xis a coordinate at p;airepresents a certain coeﬃcient),/logicalandtextk denotes ak-th completely antisymmetric tensor product, (for example, ∂i∧∂j=1 2!(∂i⊗∂j−∂j⊗∂i)∈/logicalandtext2TM), and Γ represents the section; for example, Γ( M,TM) is deﬁned as a set of tangent vector at each position p∈M. The Poisson bracket{f,g}≡α(f⊗g) :=αij(x)(∂i∧∂j)(f⊗g) is element ofT1, whereαij=−αjiis called the Poisson structure (i,j= 1,2,···,d). The deformation quantization is the constitution method of higher o rder cochains βn≥2∈C2from the Poisson bracketα∈T2. In other words, the deformation quantization is the following map F: F:T2→C2 α/ma√sto→β≡/summationdisplay n≥1νnβn, (79) whereαsatisﬁes the Jacobi identity and βsatisﬁes the association property, as shown in Fig. 2(a). In the following sections, we will generalize the two-cochain C2and the second order diﬀerential operator T2to the so-calledL∞algebra(the deﬁnition is given in section IVB2). In the section IVB1, we will introduce the two-cochain C2and second order diﬀerential operator T2, and thep-cochainCpandk-th order diﬀerential operator Tk. We will show that these operators satisfy certain conditions, and CpandTkare embedded in a diﬀerential graded Lie algebra (d.g.L.a) (the deﬁnition is shown in section IVB1). Moreover, in sectio n IVB2, the d.g.L.a will be embedded in the L∞algebra (see Fig. 2(b)). In the L∞algebra, the Jacobi identity and the association property are com piled in the following equation Q(eγ) = 0, (80) whereγ=αorβ, andQis called the codiﬀerential operator, which will be introduced in sectio n IVB2. Namely, in theL∞algebra, the deformation quantization is a map from α∈T2toβ∈C2holding the solution of Eq. (80) (Figure 2(c)). Such a map is uniquely determined in the L∞algebra. In this paper, we will identify the tensor product ⊗with the direct product ×, i.e.,V1/circlemultiplytextV2∼V1×V2:f⊗g∼(f,g) withf∈V1andg∈V2(a∼bdenotes that aandbare identiﬁed; ( f,g) represents the ordered pair, i.e., it is a set of fandg, and (a,b)/ne}ationslash= (b,a)).16 1. Cohomology equation From Eq. (78), the association property is given by the following equ ation: /summationdisplay i+j=m i,j≥0βi(βj(f,g),h)) =/summationdisplay i+j=m i,j≥0βi(f,βj(g,h)) (81) withβ0(f,g)≡f·g. (The symbol “·” represents the usual commutative product, and βj∈C2, j= 0,1,···.) Because β1is the Poisson bracket, which is bi-linear diﬀerential operator, we de ﬁneβj(∈C2, j= 2,3,···) as a diﬀerential operator on a manifold M; moreover, we also assume that p-cochains are diﬀerential operators and products of functions. Here,AandCk(A;A) represent a space of smooth functions on a manifold Mand a space of multilinear diﬀerential maps fromA⊗ktoA, respectively. Degree of βk∈Ck(A;A) is deﬁned by deg(βk) :=kfork≥2. (82) Now, we introduce a coboundary operator ∂C:Ck(A;A)→Ck+1(A;A) [20, 21]; (∂Cβk)(f0,···,fk/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright k+1) :=f0βk(f1,···,fk/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright k)+k/summationdisplay r=1(−1)rβk(f0,···,fr−1·fr,···fk/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright k) +(−1)k−1βk(f0,···,fk−1/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright k)fk (83) withβk∈Ck(A;A); note that ∂2 C= 0, and thus, ∂Cis the boundary operator. The Gerstenhaber bracket is deﬁned as [,]C:Ck(A;A)⊗Ck′(A;A)→Ck+k′−1(A;A) [22]: [βk,βk′]C(f0,f1,···,fk+k′−2/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright k+k′−1) :=k−1/summationdisplay r=0(−1)r(k′−1)βk(f0,···,fr−1,βk′(fr,···,fr+k′−1),fr+k′,···,fk+k′−2/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright k) −k′−1/summationdisplay r=0(−1)(k−1)(r+k′−1)βk′(f0,···,fr−1,βk(fr,···,fr+k−1),fr+k,···,fk+k′−2/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright k′), (84) whereβk∈Ck(A;A) andβk′∈Ck′(A;A). Note that ∂2 C= 0, and thus, ∂Cis the boundary operator. By using the coboundary operator and the Gerstenhaber bracke t, Eq. (81) is rewritten as ∂Cβm=−1 2/summationdisplay i+j=m i,j≥0[βi,βj]C (85) withβj∈C2(A;A) (j= 1,2,···). For example, Eq. (81) for m= 0,1,2 is given as: (f·g)·h=f·(h·g) for m= 0, {f·g,h}+{f,g}·h={f,g·h}+f·{g,h} form= 1, β2(f·g,h)+{{f,g},h}+β2(f,g)·h=β2(f,g·h)+{f,{g,h}}+f·β2(g,h) form= 2,(86) where we have used β0(f,g) :=f·gandβ1(f,g)≡{f,g}. The coboundary operator for β∈C2(A;A) is given by: (∂Cβ)(f,g,h) =f·β(g,h)−β(f·g,h)+β(f,g·h)−β(f,g)·h; (87) moreover, the Gerstenhaber bracket in terms of βi,βj∈C2(A;A) is given by [βi,βj]C(f,g,h) =βi(βj(f,g),h)−βi(f,βj(g,h))+βj(βi(f,g),h)−βj(f,βi(g,h)). (88) Using the above Eqs. (86-88), we can check the equivalence betwe en Eq. (81) and Eq. (85).17 Equation (85) is called the cohomology equation, and the star produ ct is constructed by using solutions of the cohomology equation. If we add Eq. (85) with respect to m= 0,1,2,···, we obtain the following equation: ∂Cβ+1 2[β,β]C= 0 (89) withβ≡/summationtext∞ j=0βj;β,βj∈C2(A;A),j= 0,1,2,.... Here, we identify the vector ﬁelds ∂i,∂j∈TMwith anti-commuting numbers ˜ ηi,˜ηj(˜ηi˜ηj=−˜ηj˜ηi),i,j= 1,2,...,d; thus the Poisson bracket αij(∂i∧∂j)/2 is rewritten by α=αij˜ηi˜ηj/2. Now, we deﬁne the Batalin-Vilkovisky (BV) bracket: [α1,α2]BV:=−d/summationdisplay i=1/parenleftigg α1← −∂ ∂xi− →∂α2 ∂˜ηi−α1← −∂ ∂˜ηi− →∂α2 ∂xi/parenrightigg (90) withα1,α2∈T2. By using the BV bracket, the Jacobi identity is rewritten as ∂BVα+1 2[α,α]BV= 0, (91) withα1,α2∈T2; forα=αij˜ηi˜ηj,α← −∂/∂xl=− →∂α/∂xl:= (∂xlαij)˜ηi˜ηjand− →∂α/∂˜ηl=−α← −∂/∂˜ηl:=αij(δil˜ηj−˜ηiδjl). By using the BV bracket, the Jacobi identity is rewritten as ∂BVα+1 2[α,α]BV= 0, (92) where∂BV≡0, i.e.,∂2 BV= 0. Now, we generalize the diﬀerential ∂BVand BV-bracket [ ,]BVforαk∈Tkandαk′∈Tk′as follows (Tk≡ Γ(M,/logicalandtextkTM)): ∂BV:Tk→Tk+1 ∂BV:= 0, (93) [,]BV:Tk/circlemultiplydisplay Tk′→Tk+k′−1 [αk,αk′]BV:=−d/summationdisplay i=1/parenleftigg αk← −∂ ∂˜ηi− →∂αk′ ∂xi−αk← −∂ ∂xi− →∂αk′ ∂˜ηi/parenrightigg (94) withαk= (αk)i1,···,ik(x)ηi1∧···∧ηik∼(αk)i1,···,ik(x)˜ηi1···˜ηik, andαk′= (αk′)i0,···,ik′(x)ηi0∧···∧ηik′∼ (αk′)i0,···,ik′(x)˜ηi0···˜ηik′; degree ofα∈Tkis deﬁned by deg(α) =k−1, α∈Tk. (95) The cochain algebra is deﬁned by the set of the diﬀerential operato r∂C, the Gerstenhaber bracket [ ,]Cand C:=/circleplustext∞ k=2Ck, i.e., (∂C,[,]C,C); in addition, the multi-vector algebra is deﬁned by the set of the diﬀ erential operator ∂BV:= 0, BV bracket [ ,]BVandT:=/circleplustext∞ k=1Tk, i.e., (∂BV,[,]BV,T). The cochain algebra and the multi-vector algebra satisfy the following common relations: ∂2= 0, (96) ∂[γ1,γ2] = [∂γ1,γ2]+(−1)deg(γ1)[γ1,∂γ2], (97) [γ1,γ2] =−(−1)deg(γ1)deg(γ2)[γ2,γ1] (98) [γ1,[γ2,γ3]]+(−1)deg(γ3)(deg(γ1)+deg(γ2))[γ3,[γ1,γ2]]+(−1)deg(γ1)(deg(γ2)+deg(γ3))[γ2,[γ3,γ1]] = 0 (99) withγ1,γ2,γ3∈G≡(CorT),∂≡∂(CorBV), and [,]≡[,](CorBV). Therefore, the two algebra can be compiled in the so-called the diﬀerential graded Lie algebra (d.g.L.a) ( ∂,[,],G), whereG:=/circleplustext∞ k=1Gkis a graded k-vector18 space withGkhas a degree deg( x)∈Z(x∈Gk;Zis the set of integers), and d.g.L.a. has the linear operator ∂and the bi-linear operator [ ,]: ∂:Gk→Gl, xk∈Gk, xl∈Gl, deg(∂xk) = deg(xk)+1 = deg( xl), (100) [,] :Gk/circlemultiplydisplay Gl→Gm, xk∈Gk, xl∈Gl, xm∈Gm, deg([xk,xl]) = deg(xk)+deg(xl) = deg(xm), (101) where∂and [,] satisfy Eqs. (96), (97), (98) and (99). In d.g.L.a., Eqs. (89) and (92) are compiled in the so-called Maurer-Cartan equation [23]: ∂γ+1 2[γ,γ] = 0 (102) withγ∈G. Therefore, the deformation quantization Fis a map: F:G→G, γ1/ma√sto→γ2, ∂γi+1 2[γi,γi] = 0, i= 1,2. (103) Namely, the deformation quantization is a map holding a solution of the Maurer-Cartan equation (102). In the section IVB2, we will introduce a L∞algebra, and will redeﬁne the deformation quantization; in the L∞algebra, the Maurer-Cartan equation (102) is rewritten as Q(eγ) = 0 (Qandeγwill be deﬁned in IVB2). 2.L∞algebra Now we deﬁne a commutative graded coalgebra C(V). First, wedeﬁneaset( V,△,τ,Q), whereV:=/circleplustext n=1,2,···V⊗nwithagraded k-vectorspace V⊗n(n= 1,2,...),△and Qrepresent the coproduct and codiﬀerential operator, respect ively; moreover, τdenotes cocommutation (deﬁnition is given later). The coproduct, cocommutation and codiﬀerential o perator satisfy the following equations: (△⊗id)◦△= (id⊗△)◦△, (104) τ△=△, (105) △◦Q= (id⊗Q+Q⊗id)◦△, (106) τ(x⊗y) := (−1)degco(x)degco(y)y⊗x, (107) with degco(x) := deg(x)−1, wherex∈V⊗deg(x)andy∈V⊗deg(y).Qrepresents a codiﬀerential operator adding one degree:Q∈Hom(V⊗m,V⊗(m+1)) with degco(Q(x)) = degco(x)+1 forx∈V⊗m,∃m∈Z+(the explicit form of Qis given later; Z+:={i\|i>0, i∈Z}). By usingτ, we deﬁne the commutative graded coalgebra C(V) from (V,△,τ,Q); the identify relation ∼is deﬁned as x⊗y∼(−1)degco(x)degco(y)y⊗x, i.e.,x⊗yand (−1)degco(x)degco(y)y⊗xare identiﬁed. Now, we deﬁne the commutative graded tensor algebra: C(V) :=V/∼ ≡{[x]\|x∈V}, (108) where [x] ={y\|y∈V, x∼y}, and degco(x1⊗x2⊗···⊗xn) = degco(x1) + degco(x2) +···+ degco(xn) with x1⊗x2⊗···⊗xn∈V⊗degco(x1)⊗V⊗degco(x2)⊗···⊗V⊗degco(xn); a product inC(V) is deﬁned by xy:= [x⊗y]. Namely, inC(V), x1x2···xixi+1···xn= (−1)degco(xi)degco(xi+1)x1x2···xi+1xi···xn (109) withn≥2. (Let us recall that the derivation of the exterior algebra from t he tensor space;VandC(V) correspond to the tensor space and the exterior algebra, respectively.) Moreover, in the case that Q2= 0, the commutative graded coalgebra C(V) is called the L∞algebra. For the L∞ algebra, the coproduct and codiﬀerential operator are uniquely d etermined by using multilinear operators: lk: (V⊗k∈C(V)))→V∈C(V) (110) degco(lk(x1···xk)) = degco(x1)+···+degco(xk)+1 (111)19 as follows: △(x1···xn) =/summationdisplay σn−1/summationdisplay k=1ε(σ) k!(n−k)!(xσ(1)···xσ(k))⊗(xσ(k+1)···xσ(n)), (112) Q=∞/summationdisplay k=1Qk, (113) Qk(x1···xn) =/summationdisplay σε(σ) k!(n−k)!lk(xσ(1)···xσ(k))⊗xσ(k+1)⊗···⊗xσ(n), (114) whereε(σ) represents a sign with a replacement σ:x1x2···xn/ma√sto→xσ(1)xσ(2)···xσ(n). From the condition Q2= 0, we can identify ( l1, l2) with (∂,[,]) in d.g.L.a. If we put l3=l4=···= 0,Q(eα) = 0 forα∈Vis equal to the Maurer-Cartan equation Eq. (102) in d.g.L.a [17], where eα≡1+α+1 2!α⊗α+··· (115) withα⊗n⊗1≡1⊗α⊗n≡α⊗nforn= 1,2,···. Therefore, the deformation quantization is a map: F:C(V)→C(V), γ1/ma√sto→γ2 (116) with Q(eγi) = 0, i= 1,2. (117) To constitute such a map F, we introduce the L∞mapF, which is deﬁned as the following map holding degrees of coalgebra: F:C(V)→C(V), v1,v2∈C(V), v1/ma√sto→v2, degco(v1) = degco(v2); (118) moreover, the L∞map satisﬁes the following equations: △◦F= (F⊗F)◦△, (119) Q◦F=F◦Q. (120) A form of such a map is limited as [17]: F=F1+1 2!F2+1 3!F3+···, (121) Fl:C(V)→V⊗l(⊂C(V)) Fl(x1···xn) =/summationdisplay σ/summationdisplay n1,···,nl≥1 n1+···+nl=nε(σ) n1!···nl! ·Fn1(xσ(1)···xσ(n1))⊗···⊗ Fnl(xσ(n−nl+1)···xσ(n)), (122) where Fnis a map fromC(V) toV(⊂C(V)) holding degrees; Fn: V⊗n(⊂C(V))→V(⊂C(V)) x1⊗···⊗xn/ma√sto→x′, degco(x1)+···+degco(xn) = degco(x′). (123) Here we deﬁne β:=/summationtext∞ n=11 n!Fn(α···α), which satisﬁes F(eα) =eβ. The map Fholds solutions ofMaurer-Cartan equationsQ(eα) = 0 andQ/parenleftbig eβ/parenrightbig = 0; from Q(eβ)≡Q◦F(eα) (124)20 and the deﬁnition of the L∞map:Q◦F=F◦Q, we obtain the following equation: Q(eβ) =F◦Q(eα) = 0, (125) which means that the L∞map transfers a solution of the Maurer-Cartan equation from ano ther solution. Now, we return to the deformation quantization. The multi-vector spaceT, is embedded in C(V);C(V) = (T,△T,τ,QT), where△T(x1x2) :=x1∧x2forx1,x2∈T, (QT)1:=∂BV≡0, (QT)2:= [,]BV, and (QT)l:= 0 for l= 3,4,...;τreplaces the wedge product “ ∧” with the product “ ·”. For the cochain space C, it is also embedded in C(V);C(V) = (C,△C,τ,QC), where△C(x1x2) :=x1∧x2forx1,x2∈C, (QC)1:=∂C, (QC)2:= [,]C, and (QC)l:= 0 forl= 3,4,.... The star product is given by f ⋆g=f·g+β(f⊗g), which is identiﬁed as the map F0+F1withF0:=µ◦. Here we summarize the main results of the succeeding sections witho ut explaining their derivations. The map F1is given by a path integral of a topological ﬁeld theory having super ﬁ elds:X:= (X1,...,XN); and scalar ﬁelds:ψ:= (ψ1,...,ψN),λ:= (λ1,...,λN), andγ:= (γ1,...,γN); and one-form ﬁelds: θ:= (θ1,θ2,...,θm), A:= (A1,...,A N),A+:= (A+1,...,A+N), andη:= (η1,...,ηN); and Grassmann ﬁelds ci:= (c1,...,cN); on a disk Σ ={z\|z=u+iv, u,v∈R, v≥0}[17, 24, 25]. These ﬁelds are deﬁned in section IVC. Using these ﬁeld s, the mapFn:V⊗n→Vis given as follows: Fn(α1,···,αn)(f1⊗···⊗fm)(x) =/integraldisplay ei /planckover2pi1S0 ghi /planckover2pi1Sα1···i /planckover2pi1SαnOx(f1,...,f m) (126) for any function f1,...,f m, which depend on x; in this paper, xrepresents the coordinate in the classical phase space. Hereα1,α2,...,α n∈V, andmis deﬁned by degco(αi)+2, which is common and independent of i(i= 1,2,...,n). The operatorOxis deﬁned as Ox(f1,...,f m) :=/integraldisplay [f1(X(t1,θ1))···fm(X(tm,θm))]δx(ψ(∞)) (127) ≡/integraldisplay 1=t1>t2>···>tm=0f1(ψ(t1))m−1/productdisplay k=2∂ik/bracketleftbig f(ψ(tk))A+ik(tk)/bracketrightbig fm(ψ(0))δx(ψ(∞)) (128) form,δx(ψ(t)) :=/producttextd i=1δ(ψi−xi)γi(t), andt∈∂Σ, where S0 gh:=/integraldisplay Σ/bracketleftbig Ai∧dψi−∗Hdγi∧dci−λid∗HAi/bracketrightbig (129) with a Hodge operator ∗H:∧k→∧2−k, (k= 0,1,2); we will introduce the explicit deﬁnition in section IVD2. Moreover, for αr:=αi1,···,inrr(X)∂i1∧···∧∂inr(nr>1 is an integer number; degco=nr−2), Sαr:=/parenleftbigg/integraldisplay Σ/integraldisplay d2θ1 nrαi1···inrr(X)ηi1···ηinr/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle Φ∗=∂ϕ, (130) where the subscript Φ∗=∂ϕmeans that the ﬁelds ( X,η,A+) go to (ψ,A,0). These results lead to the diagram technique in section IVE and the explicit expression of the star prod uct in section IVF. C. Topological string theory In this section, we expound the ﬁelds: A,ψ,c,γ,λ,θ,η,A+, andX. The simplest topological string theory is deﬁned the following action: S0:=/integraldisplay Σd2σǫµνAµ,i∂νψi=−1 2/integraldisplay Σd2σǫµνFµν,iψi(131) with local coordinates σ= (σ1,σ2) on a disk Σ (we consider that the disk is the upper-half plain in the com plex one, i.e., Σ :={z\|z=u+iw;w≥0;u,w∈R}), whereAµ,i(σ) andψi(σ) are U(1) gauge ﬁelds and scalar ﬁelds, respectively; Fµν,i(σ) is a gauge strength ( µ= 1,2 andi= 1,...,N). The other ﬁelds c,γ,λ,θ,η, andA+are introduced in section IVC1; we discuss the gauge ﬁxing method using the so-called BV-BRST formalism [26, 27] (where the BV refers to Batalin and Vilkovisky; BEST refers to Becc hi, Rouet, Stora and Tyutin). In section IVC2, we discuss the gauge invariance of the path integral, and introduce the SD operator. In section IVC3, we see that correspondence of the deformation quantization and topological string theory.21 1. Ghost ﬁelds and anti-ﬁelds Here, we quantize the action (131) using the path integral. Roughly speaking, the path integralis the Gaussintegral around a solution of an equation of motion. In many cases, a genera l actionShas no inverse. Therefore, we will add some extra ﬁelds, and obtain the action Sghhaving inverse, which is called as the quantized action. Now, we discuss a general ﬁeld theory. We assume that a general a ctionSis a function of ﬁelds φi, i.e.,S=S[φi]; each ﬁeldφiis labeled by a certain integer number, which is called as a ghost number gh(φi) (it is deﬁned below). φi Cdenotes that the ﬁelds ﬁxed on the solution of the classical kinetic e quation:δS0/δφi= 0, and the subscript of the ﬁelds represents a number of ﬁelds. Because the Gauss integr al is an inverse of a Hessian, a rank of the Hessian should be equal to the number of the ﬁelds. Here a Hessian is deﬁned by: K[φi,φj] :=− →δ δφiS← −δ δφj, (132) where− →δ δφiφj1φj2···φin:=δj1 iφj2···φj2+(−1)j1iφj1δj2 i···φjn i+···+(−1)i(j1+j2+···+j(n−1))φj1φj2···φj(n−1)δjn i, and φi1φi2···φin← −δ δφj:=φi1φi2···φi(n−1)δin j+(−1)injφi1φi2···δi(n−1) jφin+(−1)(i2+···+in)jδi1 jφi2···φin; for a boson φi,i is a ghost number gh( φi); for a fermion φi,iis gh(φi)+1. We deﬁne the rank of the Hessian Kand the number of the ﬁelds φiby♯Kand♯φi, respectively. Generally speaking,♯K <♯φi, because an action has some symmetries δRφi:=Ri jφjwith nontrivial symmetry generators Ri j, where is satisﬁes the following equation: S← −δ δφiRi j= 0 (133) withRi j\|φk=φkc/ne}ationslash= 0. The nontrivial symmetry generator decrease the rank of Hes sian from the number of ﬁelds. To deﬁne the path integral, we should add ( ♯φi−♯K) virtual ﬁelds [26–28]. The additional ﬁelds are called as ghost ﬁelds Φα1and antiﬁelds Φ∗ αl(l= 0,1), and these ﬁelds are labeled by ghost numbers. For Φα1, the ghost number is deﬁned by gh(Φα1) := 1. The ﬁelds and ghost ﬁelds have antiﬁelds Φ∗ αl. The antiﬁelds corresponding to φ≡Φα0and Φα1 are described as Φ∗ α0and Φ∗ α1, respectively. The ghost number of Φ∗ αlis deﬁned by gh(Φ∗ αl) =−l−1. Statistics of the anti-ﬁelds is opposite of ﬁelds, i.e., if the ﬁelds are fermions(boso ns), the anti-ﬁelds are bosons(fermions). (Here we only consider the so-called irreducible theory. For a general the ory, see references [26–28].) Using these ﬁelds, we will transform the action S[Φα0]/ma√sto→Sgh[Ψ], where Ψ := (Φαl,Φ∗ αl) withl= 0,1 and αl∈Z+:={i\|i>0, i∈Z}(Zrepresents the set of integers), Φα0:=φiare ﬁelds, Φα1represents ghost ﬁelds, and Φ∗ αldenotes anti-ﬁelds of the ﬁelds Φαl. Hereafter we write a function space created by the ﬁelds and ant i-ﬁelds as C(Ψ). It is known that Sghis given by Sgh=S+Φ∗ α0Rα0α1Φα1+O(Ψ3). (134) Note that the anti-ﬁelds will be ﬁxed, and ♯Ψ =/summationtext l♯Φαl(see section IVC3). For the topological string theory, the ﬁelds φαare U(1) gauge ﬁelds Ai,µand scalar ﬁelds ψiwithi= 1,...,Nand µ= 1,2; namely, Φα0≡φα:= (Ai,µ,ψi). Since♯Ai,µ= 2Nand♯ψi=N, the ﬁelds number ♯φαis 3N. The action (131) has the U(1) gauge invariance: δ0Aµ,i=∂µδj iχj, (135) δ0ψi= 0, (136) δ0χi= 0 (137) withχirepresentsa scalarfunction ( i= 1,...,N). Therefore, the topologicalstring theory has 2 Nlinear-independent nontrivial symmetry generators. Here we replace the scalar ﬁelds chiiwith ghost ﬁelds ci(BRST transformation). Moreover, we add antiﬁelds A∗ i,µ; since the gauge transformation does not connect to ψand the other ﬁelds, we does not add ψ∗(the space of ﬁelds and ghost ﬁelds has 2 Nsymmetry generators, and the space of anti-ﬁelds and the anti-ghosts also have 2Nsymmetry generators corresponding to U(1) gauge symmetry; see Figure 3): R(µ,i) β=∂µδi β,(β= 1,2,...,N). (138)22 FIG. 3. The Hessian matrix: K[Ψα(σ1),Ψβ(σ2)] :=δ δΨα(σ2)Sδ δΨβ(σ1)The ﬁrst column and ﬁrst raw represent the right-hand side and the left-hand side of variation functions, respect ively.ˆ∂σj:=δ(σ−σ1)∂σjδ(σ−σ2) represents a non-trivia Noether current ( j= 1,2). The Hessian is block diagonal matrix; the ranks of the upp er left and the lower right parts are 2 = 3 −1. Therefore, the total rank of the Hessian is 2+2 = 4 ( iis ﬁxed). In this case, ♯K(φα,φβ) = 3N−2N; on the other hand, the action is a function of 3 Nﬁelds (Aµ,i,ψi),Nghost ﬁelds ciand 2Nanti-ﬁeldsA∗ i,µ. Therefore, a rank of the Hessian corresponding to ( S0)ghis calculated by rankK(Ψ,Ψ)\|Ψc= rankK(φ,φ)\|Ψc+♯ci+♯A∗ i,µ =N+N+2N = 4N. (139) Since♯Φ = 4N(antiﬁelds will be ﬁxed), the ﬁeld number of the path-integral of ( S0)ghis equal to the rank of the Hessian of ( S0)gh; hence, the path-integral of the action ( S0)ghbecome well-deﬁned. Finally, the gauge invariance action is written by (S0)gh=S0+/integraldisplay Σ(Ai)+∧δ0Ai (140) withAi:=Aµ,idσµandδ0Aµ,i=∂µδj icj, where we deﬁne Φ+ αusing a Hodge operator ∗H: Φ+ α≡ ∗HΦ∗ α(the deﬁnition of the Hodge operator is depend on the geometry of the d isk Σ; we will introduce the explicit deﬁnition in section IVD2), which is also called as the anti-ﬁeld. 2. Condition of gauge invariance of classical action In this section, we will add interaction terms: Sgh:= (S0)gh+g(S1)gh+···, wheregrepresents an expansion parameter, and we will see that Sghis uniquely ﬁxed except a certain two form αby a gauge invariance condition. Note thatαsatisﬁes the Jacobi’s identity. Therefore, we can identify αwith the Poisson bracket. First we discuss the gauge invariant condition. If we identify the ﬁeld s and anti-ﬁelds with coordinates qand canonical momentum p, i.e., (Φαi,Φ∗ αi)↔(qαi,pαi), and we also identify the action Sand the Hamiltonian H: S↔H. In the analytical mechanics, δam:={H,}represents a transform along the surface H(q,p) = constant, i.e.,δamholds the Hamiltonian. Similarly, we can deﬁne a gauge transformation , which holds the action S, using the Poisson bracket in the two-dimension ﬁeld theory. It is known as the Batalin-Vilkovisky (BV) bracket [26, 27]; the deﬁnitions of the bracket are {f,g}BV:=/summationdisplay αi i=0,1,.../parenleftbiggδf δΦαiδg δΦ∗αi−δf δΦ∗αiδg δΦαi/parenrightbigg (141) withf, g∈C(Ψ). The BV bracket has the ghost number 1, then a BV-BRST operator δBV:={S,}BVadds one ghost number. The BV bracket satisﬁes the following equations: {f,g}BV=−(−1)(gh(f)+1)(gh( g)+1){g,f}BV, (142) (−1)(gh(f)+1)(gh( h)+1){f,{g,h}BV}BV+cyclic = 0 , (143) {f,gh}BV={f,g}BVh+(−1)(gh(f)−1)gh(g)g{f,h}BV (144)23 withf, g, h∈C(Ψ). Using the BV-BRST operator, the gauge invariance of action Sis written as δBVS= 0, i.e., {S,S}BV= 0, (145) which is called the classical master equation. We use this equation and Eqs. (142) and (143); we obtain δ2 BV= 0, which corresponds to the condition of the BRST operator: δ2 BRST= 0 (δBRSTis the BRST operator). Therefore, the DV-BRST operator is the generalized BRST one. Next, we discuss generalization of the topological ﬁeld theory. Let us write a generalized action Sghas Sgh= (S0)gh+g(S1)gh+g2(S2)gh+···, (146) wheregis an expansion parameter. Using gauge invariance condition (145), (Sn)gh(n= 1,2,···) is given by a solution of the following equation: ∂n ∂gn{Sgh,Sgh}BV/vextendsingle/vextendsingle/vextendsingle/vextendsingle g=0= 0. (147) The general solution is given by [25] (S1)gh=/integraldisplay Σd2σ/bracketleftigg 1 2αij(AiAj−2ψ+ icj)+∂αij ∂ψk/parenleftbigg1 2(c+)kcicj−(A+)kAicj/parenrightbigg +1 4∂2αij ∂ψk∂ψl(A+)k(A+)lcicj/bracketrightigg , (148) (Sn>1)gh= 0 (149) with (A+)i≡ ∗HA∗ µ,i=dσµεµνA∗ ν,i,ψ+ i≡∗Hψ∗ i=εµνdσµ∧dσνψ∗ iand (c+)i≡∗H(c∗)i=εµνdσµ∧dσν(c∗)i (εµν=−ενµ, ε12= 1), where αijis a function of ψ, and satisﬁes the following equation: ∂αij ∂ψmαmk+∂αjk ∂ψmαmi+∂αkl ∂ψmαmj= 0. (150) Here, if we identify ψiwithxi, this equation is the Jacobi identity of Poisson bracket. Therefor e, we can identify the Poisson bracket with the topological string theory. 3. Gauge invariance in path integral Now we discuss the path integral of the topological string theory/integraltext DΦV(Ψ) withV(Ψ) =Oei /planckover2pi1S, and an observable quantity operator O. Note that this path-integral does not include integrals in terms of the anti-ﬁelds. Therefore, we must ﬁx the anti-ﬁelds; then, we consider that the anti-ﬁeld Φ∗is a function of the ﬁeld Φ, i.e., Φ∗= Ω(Φ) and Ω∈C(Φ = Ψ) Namely, the path integral is deﬁned by /integraldisplay DΦV(Ψ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle Φ∗=Ω. (151) A choice of Ω(Φ) is corresponding to the gauge ﬁxing in the gauge the ory. The path integral must be independent to the gauge choice (gauge invariance). To obtain a gauge invariant condition, we take the variation of the path integral in terms of anti-ﬁelds, and obtain the following gauge invaria nt condition [29]: △SDV(Ψ) = 0, (152) where we have introduced the Schwinger-Dyson (SD) operator: △SD:=/summationdisplay αl(−1)αlδ δΦαlδ δΦ∗αl, (153)24 where (−1)αlis deﬁned as follows: if Φαlrepresents a boson, ( −1)αl= (−1)gh(Φαl); if Φαlrepresents a fermion, (−1)αl= (−1)(gh(Φαl)+1). Equation (152) is called the quantum master equation. It is known t hat the following two conditions are equivalence: △SDV(Ψ) = 0⇐⇒Ω =− →δϕ δΦa,∃ϕ, (154) whereϕis called the gauge-ﬁxing fermion (an example will be shown later). To performthe path integral, we generalizethe classicalaction Sghtoa quantum action W=Sgh+i/planckover2pi1W1+(i/planckover2pi1)2W2+ ···. The correction terms Wn(n= 1,2,...) are calculated from the master equation: △SDei /planckover2pi1W= 0, (155) or {Sgh,Sgh}BV= 0, (156) {W1,Sgh}BV+i/planckover2pi1△SDSgh= 0, (157) {W2,Sgh}BV+i/planckover2pi1△SDW1+1 2{W1,W1}BV= 0, (158) ··· In the case where △SDSgh= 0, we can put W1=W2=···= 0. Fortunately, the topological string theory satisﬁes △SDSgh= 0. Therefore, we do not have to be concerned about the quantu m correction of the action. Finally, we consider the gauge ﬁxing. Here we employ the Lorentz gau ge: d∗HAi= 0, (159) and we add the integral of the Lorentz gauge to Sgh. However, the path integral should hold gauge invariance, i.e., the path integral should be independent of gauge ﬁxing term. Then, th e gauge ﬁxing can be written gauge-ﬁxed fermion: ϕ:=/integraldisplay Σγi(d∗HAi) =−/integraldisplay Σdγi∗HAi, (160) where we introduced Nﬁeldsγi(i= 1,2,...,N), and anti-ﬁelds γ+ iare given by γ+ i=− →∂ϕ ∂γi=d∗HAi. (161) Now, we employ the Lagrange multiplier method, and introduce Nscalar ﬁelds λi. The gauge-ﬁxed action is written by Sgf=Sgh−/integraldisplay Σγid∗HAi (162) =Sgh−/integraldisplay Σλiγ+ i. (163) The other anti-ﬁelds are also ﬁxed by this gauge-ﬁxing fermion: ψ+ i=c+ i=λ+ i= 0, (164) A+ i=∗Hdγi. (165) Gauge ﬁxed action Sgfis written by Sgf=/integraldisplay Σ/bracketleftigg Ai∧dψi+1 2αijAi∧Aj−∗Hdγi∧/parenleftbigg dci+∂αkl ∂ψiAkcl/parenrightbigg −1 4∗Hdγi∧∗Hdγj∂2αkl ∂ψi∂ψjckcl−λid∗HAi/bracketrightigg . (166)25 Here we perform the following variable transformations: Xi:=ψi+θµA∗ µ−1 2θµθνc+i µν, (167) ηi:=ci+θµAi,µ+1 2θµθνψ+ i,µν, (168) whereθµϑν=−θνθµ; gh(θµ) = 1. For any scalar ﬁeld f(u) (u∈Σ),˜f(u,θ) :=f(u)+θµf(1) µ(u)+1 2θµθνf(2) µνis called as the super ﬁeld, where f(1)andf(2)represent a one-form ﬁeld and a two-form ﬁeld, respectively. By using the super ﬁelds, the gauge ﬁxed action Sgfcan be rewritten as Sgf=/integraldisplay Σ/integraldisplay d2θ/bracketleftbigg ηiDXi−λid∗HAi+1 2αij(X)ηiηj/bracketrightbigg , (169) whereD:=θµ∂ ∂uµ. This is the ﬁnal result in this section. Hereafter, we write S0 gf:=/integraltext Σ/bracketleftbig ηiDXi−λid∗HAi/bracketrightbig and S1 gf:=/integraltext Σαijηiηj/2. D. Equivalence between deformation quantization and topol ogical string theory Wereturntothediscussionaboutthedeformationquantization. H ereweseethatthe equivalenceofthedeformation quantization and the topological string theory, and introduce the perturbation theory of the topological string theory, which is equal to Kontsevich’s deformation quantization [17]. 1. Path integral as L∞map Here we summarize correspondence between Path integral with L∞map. First we note that the map: α:=α(x)µνηµην/2/ma√sto→Sα:=S1 gf=/integraltext Σα(x)µνηµην/2 is isomorphic, because {Sα1,Sα2}BV=S{α1,α2}BV. SD operator satisﬁes the conditions of codiﬀerential operator QinL∞algebra, where the vector space and the degree of the space correspond to C(Ψ) and the ghost number, respectively. The path integral/integraltext ei /planckover2pi1S0 gfgives the deformation quantization F0+F1. The master equation Qei /planckover2pi1Sα= 0 (170) withQ=△SDis corresponding to the L∞map’s condition QF= 0 Forαr:=αi1,i2,···,imηi1ηi2···ηim/m! with a positive integer m,Fn:V⊗n 1⊗→V2is given by Fn(α1,...,α n)(f1⊗···⊗fm)(x) :=/integraldisplay ei /planckover2pi1S0i /planckover2pi1Sα1···i /planckover2pi1SαnO(f1,...,f m), (171) whereSαris the expansion of Sα, and is deﬁned as Sαr:=/parenleftbigg/integraldisplay Σ1 m!αi1···im(X)ηi1···ηim/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle Φ∗=∂ϕ, (172) andOis chosen to satisfy Q◦F=F◦Q, (173) where F:=F0+F1+F2+···. We putOas follow: O(f1,...,f m) =/integraldisplay Bm[X(t1,θ1))···fm(X(tm,θm))](m−2)δx(X(∞)), (174) ≡/integraldisplay 1=t1>t2>···>tm=0f1(ψ(1))m−1/productdisplay k=2∂ik/bracketleftbig f(ψ(tk))A+ik(tk)/bracketrightbig fm(ψ(0))δx(ψ(∞)) (175)26 where the subscript ( m−2) denotes that ( m−2) forms are picked up from the products of super ﬁelds, and Bm represents the surface of the disk Σ, i.e., tis the parameter specifying the position on the boundary ∂Σ (1 =t1> t2>···>tm−1>tm= 0). To be exact, the action and ﬁelds include gauge ﬁxing terms, ghost ﬁ elds and anti-ﬁeld. Finally, the deformation quantization is given as follow: (f ⋆g)(x) =/integraldisplay DΦf(ψ(1))g(ψ(0))δ(xi−ψi(∞))ei /planckover2pi1Sgf. (176) 2. Perturbation theory Nowweseethattheperturbationtheoryofthetopologicalstring theory. First, wewritetheactionas Sgf=S0 gf+S1 gf. The ﬁrst term is deﬁned as S0 gf=/integraldisplay Σ/bracketleftbig Ai∧(dξi+∗Hdλi)+cid∗Hdγi/bracketrightbig , (177) whereξi≡ψi−xi, and we have expanded ψiaroundxi. The path integral of an observable quantity /an}bracketle{tO/an}bracketri}htis given by /integraldisplay ei /planckover2pi1SgfO=∞/summationdisplay n=0in /planckover2pi1nn!/integraldisplay ei /planckover2pi1S0 gf(S1 gf)nO, (178) where/integraltext :=/integraltext DξDADcDγDλ. This expansion corresponds to the summation of all diagrams by th e contractions of all pairs in terms of ﬁelds and ghost ﬁelds. From equation (177), pro pagators are inverses of d⊕∗Hd, d∗Hd. (179) Here we assume that the disk is the upper complex plane: Σ = {z\|z=u+iv, u,v∈R, v≥0}withi2=−1, and the boundary is ∂Σ ={z\|z=u, u∈R}. (Rrepresents the real number space, and zdenotes a complex number.) The Hodge operator ∗His deﬁned by /braceleftbigg ∗Hdu=dv ∗Hdv=−du/ma√sto−→/braceleftbigg ∗Hdz=−idz ∗Hdz=idz, (180) wherezrepresents the complex conjugate of z. Moreover, dz=du∂ ∂u+dv∂ ∂v=dz∂ ∂z+dz∂ ∂z, (181) δz(w) :=δ(w−z)duw∧dvw,/integraldisplay δz(w) = 1, (182) wherew∈Cwith the complex number plane C, andw≡uw+ivw. Now, we calculate Green functions of d⊕∗Hdandd∗Hd, because the Green functions are inversesof these operators: DwG(z,w) = i/planckover2pi1δz(w), (183) whereDw=dw⊕∗Hdwordw∗Hdw. The solution depends on the boundary condition. In the case that zandw satisfy the Neumann boundary condition, a solution is a function of φh(z,w) :=1 2ilog(z−w)(z−w) (z−w)(z−w). (184) On the other hand, zandwsatisfy the Dirichlet boundary condition, a solution is a function of ψh(z,w) := log/vextendsingle/vextendsingle/vextendsingle/vextendsinglez−w z−w/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (185) TheNeumannboundaryconditionis0 = ∂u1G(z,w)\|u2=0,andtheDirichletboundaryconditionis0 = ∂u2G(z,w)\|u1=0.27 The propagators are given by /an}bracketle{tγk(w)cj(z)/an}bracketri}ht=i/planckover2pi1 2πδk jψh(z,w), (186) /an}bracketle{tξk(w)Aj(z)/an}bracketri}ht=i/planckover2pi1 2πδk jdzφh(z,w), (187) /an}bracketle{t(∗Hdγk)(w)cj(z)/an}bracketri}ht=i/planckover2pi1 2πδk jδwφh(z,w), (188) and so on. From these propagators, we can obtain diagram rules co rresponding to the deformation quantization. In section IVE, we will introduce exact diagram rules. To obtain the star product, we choice Ox=f(X(1))g(X(0))δx(ψ(∞)) (189) E. Diagram rules of deformation quantization From the perturbation theory of the topological string theory, w e can obtain the following diagram rules of the star product, which is ﬁrst given by Kontsevich [17, 30]: (f ⋆g)(x) =f(x)g(x)+∞/summationdisplay n=1/parenleftbiggi/planckover2pi1 2/parenrightbiggn/summationdisplay Γ∈GnwΓBΓ,α(f,g). (190) where Γ,BΓ,α(f,g) andwΓare deﬁned as follows: Deﬁnition. 1 Gnis a set of the graphs Γwhich have n+ 2vertices and 2nedges. Vertices are labeled by symbols “1”, “2”,..., “n”, “L”, and “R”. Edges are labeled by symbol (k,v), wherek= 1,2,...,n,v= 1,2,...,n,L,R , and k/ne}ationslash=v.(k,v)represents the edge which starts at “ k” and ends at “ v”. There are two edges starting from each vertex withk= 1,2,...,n;LandRare the exception, i.e., they act only as the end points of the edges. Hereafter, VΓand EΓrepresent the set of the vertices and the edges, respectivel y. Deﬁnition. 2 BΓ,α(f,g)is the operator deﬁned by: BΓ,α(f,g) :=/summationdisplay I:EΓ→{i1,i2,···,i2n} n/productdisplay k=1 /productdisplay e∈EΓ,e=(k,∗)∂I(e) αI((k,v1 k),(k,v2 k)) × /productdisplay e∈EΓ,e=(∗,L)∂I(e) f × /productdisplay e∈EΓ,e=(∗,R)∂I(e) g , (191) where,Iis a map from the list of edges ((k,v1,2 k)),k= 1,2,...,nto integer numbers {i1,i2,···,i2n}. Here1≤in≤d; drepresents a dimension of the manifold M.BΓ,α(f,g)corresponds to the graph Γin the following way: The vertices “1”, “2”,..., “n”, correspond to the Poisson structure αij.RandLcorrespond to the functions fandg, respectively. The edgee= (k,v)represents the diﬀerential operator ∂(iorj)acting on the vertex v. The simplest diagram for n= 1is shown in Fig. 4(a), which corresponds to the Poisson bracket: {f,g}=/summationtext i1,i2αi1i2(∂xi1f)(∂xi2g). The higher order terms are the generalizations of this Pois son bracket. Figure4(b)shows a graph Γex.2withn= 2corresponding to the list of edges ((1,L),(1,R),(2,R),(2,3)); (192) in addition, the operator BΓex.2,αis given by (f,g)/ma√sto→/summationdisplay i1,···,i4(∂xi3αi1i2)αi3i4(∂xi1f)(∂xi2∂xi4g). (193) Deﬁnition. 3 We put the coordinates for the vertices in the upper-half com plex planeH+:={z∈C\|Im(z)>0} (Crepresents the complex plain; Im(z)denotes the imaginary part of z). Therefore, RandLare put at 0and1, respectively. We associate a weight wΓwith each graph Γ∈Gnas wΓ:=1 n!(2π)2n/integraldisplay Hnn/logicalanddisplay k=1/parenleftig dφh (k,v1 k)∧dφh (k,v2 k)/parenrightig , (194)28 FIG. 4. (a): The graph Γ ex.1∈G1corresponding to Poisson bracket. (b): A graph Γ ∈G2correspond to the list of edges: ((1,L),(1,R),(2,R),(2,1))/mapsto→ {i1,i2,i3,i4}. whereφis deﬁned by φh (k,v):=1 2iLog/parenleftbigg(q−p)(¯q−p) (q−¯p)(¯q−¯p)/parenrightbigg . (195) pandqare the coordinates of the vertexes “ k” and “v”, respectively. ¯prepresents the complex conjugate of p∈C. Hndenotes the space of conﬁgurations of nnumbered pair-wise distinct points on H+: Hn:={(p1,···,pn)\|pk∈H+, pk/ne}ationslash=plfork/ne}ationslash=l}. (196) Here we assume that H+has the metric: ds2= (d(Re(p))2+d(Im(p))2)/(Im(p))2, (197) withp∈H+;φh(p,q)is the angle which is deﬁned by (p,q)and(∞,p), i.e.,φh(p,q) =∠pq∞with the metric (197). For example, wΓex.1corresponding to Fig. 4(a)is calculated as: wΓex.1=2 1!(2π)2/integraldisplay H1d1 2iLog/parenleftbiggp2 p2/parenrightbigg ∧d1 2iLog/parenleftbigg(1−p)2 (1−p)2/parenrightbigg = 1, (198) where we have included the factor “ 2” arising from the interchange between two edges in Γ.wΓcorresponding to the Fig.4(b)is wΓ(b)=1 2!(2π)4/integraldisplay H2d1 iLog/parenleftbiggp1 p1/parenrightbigg ∧d1 iLog/parenleftbigg1−p1 1−p1/parenrightbigg ∧d1 iLog/parenleftbiggp2 p2/parenrightbigg ∧d1 2iLog/parenleftbigg(p1−p2)(p1−p2) (p1−p2)(p1−p2)/parenrightbigg =1 2!(2π)4/integraldisplay H2d1 iLog/parenleftbiggp1 p1/parenrightbigg ∧d1 iLog/parenleftbigg1−p1 1−p1/parenrightbigg ∧d(2arg(p2))∧d\|p2\|∂ ∂\|p2\|1 2iLog/parenleftbigg(p1−p2)(p1−p2) (p1−p2)(p1−p2)/parenrightbigg =1 2!(2π)4/integraldisplay H2d1 iLog/parenleftbiggp1 p1/parenrightbigg ∧d1 iLog/parenleftbigg1−p1 1−p1/parenrightbigg ∧d1 iLog/parenleftbiggp2 p2/parenrightbigg ∧d1 iLog/parenleftbigg1−p2 1−p2/parenrightbigg =w2 1 2! =1 2, (199) wherep1andp2are the coordinates of vertexes “ 1” and “2”, respectively. Here, we have used the following facts: /integraldisplay∞ 0d\|p2\|∂\|p2\|Log/parenleftbigg(p1−p2)(p1−p2) (p1−p2)(p1−p2)/parenrightbigg = lim Λ→∞Log/parenleftbigg(p1−Λeiarg(p2))(p1−Λeiarg(p2)) (p1−Λe−iarg(p2))(p1−Λe−iarg(p2))/parenrightbigg = lim Λ→∞Log/parenleftbigg(1−Λeiarg(p2))(1−Λeiarg(p2)) (1−Λe−iarg(p2))(1−Λe−iarg(p2))/parenrightbigg , /integraldisplay \|p1\|>ΛdLog/parenleftbiggp1 p1/parenrightbigg ∧dLog/parenleftbigg1−p1 1−p1/parenrightbigg Λ→∞−→/integraldisplay \|p1\|>ΛdLog/parenleftbiggp1 p1/parenrightbigg ∧dLog/parenleftbiggp1 p1/parenrightbigg = 0. (200) Generally speaking, the integrals are entangled for n≥3graphs, and the weight of these are not so easy to evaluate as Eq.(199). Note that the above diagram rules also deﬁne the twisted element as the following relation: ( f⋆g)≡µ◦F(f⊗g).29 FIG. 5. A four vertexes graph, where the white circle and the w hite square represent αAandαF, respectively; the dotted arrow, waved arrow, and real arrow represent ∂p,∂s, and∂X, respectively. F. Gauge invariant star product From Eq. (77), the Poisson structure corresponding to our mode l is αij= 0ηµν0 −ηµν−qˆFµν−qǫabcsaAb µ 0qǫabcsaAb µǫabcsc , (201) where the symbols iandjrepresent indexes of the phase space ( TX,ωp,s). We separate the Poisson structure as follows: α:= 0ηµν0 −ηµν0 0 0 0 0 + 0 0 0 0 0−qǫabcAb µsa 0qǫabcsaAb µǫabcsa + 0 0 0 0qˆFµν0 0 0 0 ≡α0+αA+αF. (202) Here, forf=f0+faσa,∂saf:=fa(a=x,y,z), wheref0,x,y,zare functions Xandp. Becauseα0is constant and αAandαFare functions of Xµands, and any function fis written as f=f0+/summationtext a=x,y,zfasa(f0,aonly depends on Xandp), then we obtain additional diagram rules: A1. Two edges starting from αFconnect with both vertices “ L” and “R”. A2. At least one edge from vertices α0orαFconnect with vertices “ L” or “R”. A3. A number of the edges entering αAis one or zero. We also separate the graph Γ into Γ α0, ΓαAand Γ αF. Here, we deﬁne the numbers of vertices α0,αA, andαF asnα0,nαA, andnαF, respectively. Γ αFis the graph consisted by vertices corresponding to αF, and “L” and “R”, and edges starting from these vertices. We consider Γ αFas a cluster, and deﬁne Γ αAas the graph consisted by the vertices corresponding to αA, which acts on the cluster corresponding to Γ αF. Γα0is the rest of the graph Γ without ΓαAand Γ αF. Here, we label vertexes Γ αF, ΓαAand Γ α0by “k= 1−nαF”, “k= (nαF+ 1)−(nαF+nαA)” and “k= (nαF+nαA+1)−(nαF+nαA+nα0)”, respectively. The edge starting from “ k” and ending to “ v1,2 k” represents (k,v1,2 k). Next, we calculate weight wnαFand the operator BΓαF,αFcorresponding to Γ αF, and later those for Γ αAor0. Separation of graph Γ We now sketch the proof of wΓBΓ,α=wnα0BΓα0,α0·wnαABΓαA,αA·wnαFBΓαF,αF, wherewnαa=wnαa 1 nαa!fora= 0,A,F, andw1is given by Eq. (198). From the additional rule A1, each operator corresponding to vert exesαFand edges ( αF,LorR) acts onfandg independently. Thus wαF∼wnαF 1. Secondly we consider the graph which consists of four vertexes c orresponding to αA,αF, and “L”and“R” as shownin Fig. 5. We alsoassume that one edge ofthe vertex corr espondingto αAconnects with a vertex corresponding to αF. In this case, from additional diagram rule A3, another edge of the vertex has to connect with “ L” or “R”. Since we can exchange the role “ R” and “L” by the variable transformation p/ma√sto→1−p, (p∈H+), we assume that one edge of the vertex corresponding to αAconnect with “ L”. The weight wΓin this case is30 FIG. 6. This ﬁgure shows the calculation method of the graph ( a), where the dotted arrow and real arrow represent the derivative with respect to pandX, respectively, and the white circle and the white triangle r epresent α0andαF, respectively. We rewrite the graph ( a) as the graph ( c) which is given by the cluster represented by the big circle a nd the operators into it, where the big circle represents the graph ( b). given by Eq. (199), i.e., the integrals for the weight is given by replacin g coordinate of the vertex corresponding to αF with coordinate of “ R” inH+. This result can be expanded to every graph though a graph include s the vertices α0. For example, we illustrate the calculation of a six vertices graph, whic h only includes α0andαF, in Fig. 6. At ﬁrst we make the cluster having only vertices αF,fandg(ﬁg. 6(b)), which is corresponding to the following operator: wnαF nαF!αi1i2 Fαi3i4 F(∂pi1∂pi3f)(∂pi2∂pi4g). The edges from the vertices act on the cluster independently (ﬁ g. 6(c)); we obtain the following operator:wnα0 nα0!wnαF nαF!αj1j2 0αj3j4 0(∂Xj1αi1i2 F)(∂Xj3αi3i4 F)(∂Xj2∂pi1∂pi3f)(∂Xj4∂pi2∂pi4g). The position of each vertex corresponding to αAandαFcan be move independently in integrals, and the entangled integral does not appear. Therefore the weight wnαAof a graph Γ αA∼wnαA 1only depends on the number of vertexes correspondingto αAandαF, andwΓ=wnαA·wnαFholds generally. From additional rule A2, we can similarly discuss aboutagraphΓ α0, andobtain wnα0∼wnα0 1. Finally,wecancountthecombinationof nα0,nαAandnαF, anditisgiven by(nα0+nαA+nαF)! (nαA+nαF)!nα0!·(nαA+nαF)! nαA!nαF!. Therefore we obtain the Eq: wΓBΓ,α=wnα0BΓα0,α0·wnαABΓαA,αA·wnαFBΓαF,αF. The summation of each graph is easy, and we can derive the star pro duct:f ⋆g=µ◦FA(f⊗g), where twisted elementFAis written as follow: FA= exp/braceleftbiggi/planckover2pi1 2/parenleftbig ∂Xµ⊗∂pµ−∂pµ⊗∂Xµ/parenrightbig/bracerightbigg ◦exp/braceleftbiggi/planckover2pi1 2εijksk/parenleftbig Ai µ∂pµ⊗∂sj+∂si⊗∂sj−∂sj⊗Ai µ⊗∂pµ/parenrightbig/bracerightbigg ◦exp/braceleftbiggi/planckover2pi1 2/parenleftbig Fa µνsa+Fµν/parenrightbig ∂pµ⊗∂pν/bracerightbigg . (203) Because the action IAincluding a global U(1) ×SU(2) gauge ﬁeld, the action IAis written as IA=FA◦F−1 0I0, thus, the mapF0/mapsto→A:I0/ma√sto→IAis given byF0/mapsto→A=FA◦F−1 0, i.e., F(0/mapsto→A)= exp/braceleftbiggi/planckover2pi1 2/parenleftbig ∂Xµ⊗∂pµ−∂pµ⊗∂Xµ/parenrightbig/bracerightbigg ◦exp/braceleftbiggi/planckover2pi1 2εijksk/parenleftbig Ai µ∂pµ⊗∂sj+∂si⊗∂sj−∂sj⊗Ai µ⊗∂pµ/parenrightbig/bracerightbigg ◦exp/braceleftbiggi/planckover2pi1 2/parenleftbig Fa µνsa+Fµν/parenrightbig ∂pµ⊗∂pν/bracerightbigg ◦exp/braceleftbigg −i/planckover2pi1 2εijksk∂si⊗∂sj/bracerightbigg ◦exp/braceleftbigg −i/planckover2pi1 2/parenleftbig ∂Xµ⊗∂pµ−∂Xµ⊗∂pµ/parenrightbig/bracerightbigg . (204) The inverse map is given by the replacement of i by −i in the map (204).31 V. TWISTED SPIN In this section, we will derive the twisted spin density, which corresp onds to the spin density in commutative spacetime without the background SU(2) gauge ﬁeld. First, we will derive a general form of the twisted spin current in sec tion VA, which is written by using the twisted variation operator. This operator is constituted of the coproduc t and twisted element; the coproduct reﬂects the action rule of the global SU(2) gauge symmetry generator, and th e twisted element represents gauge structure of the background gauge ﬁelds. In section VB, at ﬁrst, we will calculate the twisted spin density of th e so-called Rashba-Dresselhaus model in the Wigner representation using the general form of the twisted s pin density, and next, we will ﬁnd the twisted spin operator in real spacetime using correspondence between opera tors in commutative spacetime and noncommutative phasespace. A. Derivation of a twisted spin in Wigner space The Lagrangian density in the Wigner space is given by L(X,p) =/parenleftbigg p0−p2 2m/parenrightbigg ⋆/parenleftbig ψψ†/parenrightbig , (205) wheremis the electric mass, f ⋆g:=µ◦FA(f⊗g) for any functions fandg. The variation corresponding to the inﬁnitesimal global SU(2) gauge transformation is deﬁned as δsaψ= iϑsaψ, (206) δsaψ†=−iψ†sa, (207) δsaxµ= 0, (208) whereθrepresents an inﬁnitesimal parameter. Therefore, the variation of the Lagrangian density L(X,p) :=ˆL⋆ψψ† is given by δsa(L(X,p)) :=ˆL⋆iθsa⋆ψψ†−ˆL⋆ψψ†⋆iθsa(209) Here we introduce the Grassmann numbers λ1,2,3(λ3:=λ1λ2), the product µ, and the coproduct △η, where the coproduct satisﬁes △η(f) :=f⊗η+η⊗f (210) for any functions fand operator η. The equation (209) can be rewritten as δsa(L) =/integraldisplay dλ3µ◦FA/parenleftig ˆL⊗/parenleftbig iλ1θsa⋆λ2ψψ†+λ2ψψ†⋆iλ1θsa/parenrightbig/parenrightig =/integraldisplay dλ3µ◦FA/parenleftig ˆL⊗µ◦FA△iλ1θsa/parenrightig =/integraldisplay dλ3µ◦FA(id⊗µ)◦(id⊗FA)◦(id⊗△iλ1θsa)◦(ˆL⊗λ2ψψ†) = i/integraldisplay dλ3µ◦(id⊗µ)◦(id⊗△)FA◦(id⊗FA) ◦(id◦θ1/2⊗θ1/2)◦(id⊗△λ1sa)◦(ˆL⊗λ2ψψ†), (211) where we used/integraltextdλiλj=δij(i,j= 1,2,3) with the Kronecker delta δijHere we introduce the following symbols: ˆµ:=µ◦(id⊗µ), (212) ˆFA:= (id⊗△)FA◦(id⊗FA), (213) ˆθ:=θ1/2⊗θ1/2, (214) ˆ△λ1sa:= (id⊗△λ1sa), (215)32 where the coproduct in the diﬀerential operator space is deﬁned a s △(dn) :=/summationdisplay i+j=n i≥0, j≥0di⊗dj. (216) Here vectors{d0,d1,...}corresponding to following operators: d0:= id anddn:= (1/n!)(∂n/∂pn), ord0:= id and dn:= (1/n!)(∂n/∂xn) (n= 1,2,...), anddlare bases of a vector space B(k) :=/circleplustext∞ l=0kdl(krepresents a scalar). The coproduct△in the vector space B(k) satisﬁes the coassociation law: ( △⊗id)⊗△= (id⊗△)◦△because (△⊗id)◦△(dn) =/summationdisplay i+j=n i≥0, j≥0△(di)⊗dj =/summationdisplay i+j=n i≥0, j≥0 /summationdisplay k+l=i k≥0, l≥0dk⊗dl⊗dj =/summationdisplay k+l+j=n k≥0, l≥0, j≥0dk⊗dl⊗dj (217) and (id⊗△)◦△(dn) =/summationdisplay i+j=n i≥0, j≥0di⊗△(dj) =/summationdisplay i+k+l=n i≥0, k≥0, l≥0di⊗dk⊗dl. (218) It represents the Leibniz rule with respect to the diﬀerential oper ator∂µ. For instance, (id ⊗△)(∂µ⊗∂ν) := ∂µ⊗△(∂ν) =∂µ⊗∂ν⊗id+∂µ⊗id⊗∂ν; it corresponds to the following calculation: ∂µf·∂ν(g·h) =∂µf·∂νg·h+∂µf·g·∂νh. (219) The variation (211) is rewritten as δsaL(X,p) = i/integraldisplay dλ3ˆµ◦ˆFA◦ˆθ◦ˆ△λ1sa◦(ˆL⊗λ2ψψ†). (220) If we replace ˆ△t λ1sa:=ˆF−1 Aˆ△λ1saF0withˆ△λ1sain equation (220), the integrals in terms of xandpof the right-hand side of Eq. (220) become zero because ˆFA◦ˆ△t λ1sadoes not include the SU(2) ﬁeld, which breaks the global SU(2) gauge symmetry, in the case that the parameter θis constant. Therefore, for the action S:=/integraltext dDimXdDimpL(X,p)/(2π/planckover2pi1)Dim, δt saS:= tr/integraldisplay/integraldisplay/integraldisplay dλ3dDimXdDimp (2π/planckover2pi1)Dimˆµ◦ˆFA◦ˆθ◦ˆ△t λ1sa(ˆL⊗λ2ψψ†) (221) is the inﬁnitesimal SU(2) gauge transformation with background SU (2) gauge ﬁelds. Becauseδt saS= 0, we can write δt saS=/integraldisplay dDimXθ/parenleftbig ∂µjt µ/parenrightbig . (222) In the case that the inﬁnitesimal parameter depends on the space time coordinate, this equation can be written as δt saS=/integraldisplay dDimXθ(X)/parenleftbig ∂µjt µ/parenrightbig =−/integraldisplay dDimX/parenleftbigg∂θ(X′) ∂X′µ/parenrightbigg jt µ(X′). (223)33 Therefore, we obtain the twisted Noether current jt µ=−δt saS δ(∂µθ(X)). (224) In particular, the twisted spin St a=/integraldisplay dXjt 0 =/integraldisplay dXδt saS δ(∂Tθ)(225) is conserved quantity. Here, we assumed that the SU(2) gauge is s tatic one. However, we do not use this condition in the derivation of the twisted Noether charge and current density . Then, we can derive the virtual twisted spin with a time-dependent SU(2) gauge: ˜St a. In this case, we only use the time-dependent SU(2) gauge ﬁeld str engthFa µν, which has non-zero space-time components Fa 0i(i= 1,2,...,Dim−1). Here, we discuss the adiabaticity of the twisted spin. In the case th at SU(2) gauge ﬁelds have time dependence, the twisted spin is not conserved. Now, we assume that Aa µ=λ(t)Ca µ(a=x,y,z) with constant ﬁelds Ca µ= (0,Ca);λ(t) is an adequate slowly function dependent on time. Because ˜St aincludesF−1 µν∼1 (1+(˙λ)2C·C)/parenleftbigg 0 ˙λC −˙λCλ−2[Ci,Cj]−1/parenrightbigg with˙λ≡dλ/dt, the diﬀerence between ˜St aandSt acomes from only that between inverse of ﬁeld strength: ∆ F−1∼ 1 1+(˙λ)2λ−2[Ci,Cj]−1−λ−2[Ci,Cj]−1∼(˙λ/λ)2[Ci,Cj]−1. Therefore we obtain dSt a dt=O(˙λ2). (226) This means that St ais the adiabatic invariance. Namely, for the inﬁnitely slow change in λ(t) during the time period T(→∞),St aremains constant while ∆ λ=λ(T)−λ(0) is ﬁnite. This fact is essential for the spin-orbit echo proposed in [11]. B. Rashba-Dresselhaus model Here we apply the formalism developed so far to an explicit model, i.e., th e so-called Rashba-Dresselhaus model given by H=ˆp2 2m+α(ˆpxˆσy−ˆpyˆσx)+β(ˆpxˆσx−ˆpyˆσy)+V(ˆx) (227) with a potential V(ˆx), whereαandβare the Rashba and Dresselhaus parameters, respectively. Comp leting square in terms of ˆ p, we obtain Ax x=−2mβ/(/planckover2pi1q),Ay x=−2mα/(/planckover2pi1q),Ax y= 2mα/(/planckover2pi1q),Ay y= 2mβ/(/planckover2pi1q),A0=m(α2+β2)/e, andAz x,y=Ax,y z=Az z=Ax,y,z 0=Ax,y,z= 0, where q=\|e\|/(mc2). To calculate the twisted symmetry generator ˆ△t(λ2sa), we ﬁrst consider the ˆF0. (id⊗△)F0is given by (id⊗△)F0= exp/braceleftbiggi/planckover2pi1 2/parenleftig ∂Xµ⊗1⊗∂pµ+∂Xµ⊗∂pµ⊗1−∂pµ⊗1⊗∂Xµ−∂pµ⊗∂Xµ⊗1/parenrightig +i 2εijk/parenleftig sk∂si⊗1⊗∂sj+sk∂si⊗∂sj⊗1/parenrightig/bracerightbigg , (228)34 andˆF0is given by ˆF0= exp/braceleftbiggi/planckover2pi1 2/parenleftig ∂Xµ⊗1⊗∂pµ+∂Xµ⊗∂pµ⊗1−∂pµ⊗1⊗∂Xµ−∂pµ⊗∂Xµ⊗1/parenrightig +i 2εijk/parenleftig sk∂si⊗1⊗∂sj+sk∂si⊗∂sj⊗1/parenrightig/bracerightbigg ◦exp/braceleftbiggi/planckover2pi1 2/parenleftig 1⊗∂Xµ⊗∂pµ−1⊗∂pµ⊗∂Xµ/parenrightig +i 2εijk/parenleftig 1⊗sk∂si⊗∂sj/parenrightig/bracerightbigg = exp/braceleftbiggi/planckover2pi1 2/parenleftig 1⊗∂Xµ⊗∂pµ+∂Xµ⊗1⊗∂pµ+∂Xµ⊗∂pµ⊗1 −1⊗∂pµ⊗∂Xµ−∂pµ⊗1⊗∂Xµ−∂pµ⊗∂Xµ⊗1/parenrightig +i 2εijk/parenleftig 1⊗sk∂si⊗∂sj+sk∂si⊗1⊗∂sj+sk∂si⊗∂sj⊗1/parenrightig/bracerightbigg ≡G0. (229) Similarly, ˆFAis given by ˆFA= exp/braceleftbiggi/planckover2pi1 2/parenleftig 1⊗∂Xµ⊗∂pµ+∂Xµ⊗1⊗∂pµ+∂Xµ⊗∂pµ⊗1 −1⊗∂pµ⊗∂Xµ−∂pµ⊗1⊗∂Xµ−∂pµ⊗∂Xµ⊗1/parenrightig/bracerightig ◦exp/braceleftbiggi 2εijk/parenleftig 1⊗sk∂si⊗∂sj+sk∂si⊗1⊗∂sj+sk∂si⊗∂sj⊗1 +1⊗skAi µ∂pµ⊗∂sj+skAi µ∂pµ⊗1⊗∂sj+skAi µ∂pµ⊗∂sj⊗1 −1⊗∂sj⊗skAi µ∂pµ−∂sj⊗1⊗skAi µ∂pµ−∂sj⊗skAi µ∂pµ⊗1/parenrightig/bracerightig ◦exp/braceleftbiggi/planckover2pi1 2/parenleftig 1⊗ˆFµν∂pµ⊗∂pν+ˆFµν∂pµ⊗1⊗∂pν+ˆFµν∂pµ⊗∂pν⊗1/parenrightig/bracerightbigg ≡GA Xp◦GA sp◦GA pp. (230) We note that the operators G0andGA Xp,sp,pphave each inverse operator, which are denoted by G0andGA Xp,sp,pp, respectively. Here, the overline −represents the complex conjugate. Because the twisted variation is ˆ µ◦ˆFA◦ˆθ◦ˆF−1 A◦ˆ△λ1sa◦F0(L⊗λ2ψψ†), the inﬁnitesimal parameter ˆθbecomes an operator ˆFA◦ˆθ◦ˆF−1 A. It is calculated by using the operator formula eBCe−B=∞/summationdisplay n=01 n![B,[B,···[B,/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright nC]···]] (231) for any operators BandC. In the calculation, one will use the following formula in midstream: /summationdisplay l=0Dl (l+1)!=/integraldisplay1 0dλeλD, (232) /summationdisplay l=0Dl (l+2)!=/integraldisplay1 0dλ/integraldisplayλ 0dλ′eλ′D, (233) and so on, for any operator D. From these results of the calculations, we obtain the twisted spin as follow st a=µ◦FA◦(sa⊗id)◦˜Υ◦(id⊗G)◦˜Υ†, (234)35 where˜Υ is deﬁned as ˜Υ :=1 2/bracketleftbigg ei 2(α+β)σ−/parenleftbig ∂px+∂py/parenrightbig ⊗ei 2(α−β)σ+/parenleftbig ∂px−∂py/parenrightbig/bracketrightbigg ◦/bracketleftigg e−i 2/parenleftbig ∂X⊗∂Y+∂Y⊗∂X/parenrightig ◦/parenleftbig 1 2m(α2−β2)⊗σz/parenrightbig +e−i 2/parenleftbig ∂X⊗∂Y+∂Y⊗∂X/parenrightig ◦/parenleftbig σz⊗1 2m(α2−β2)/parenrightbig/bracketrightigg +ei 2(α+β)σ−/parenleftbig ∂px+∂py/parenrightbig ⊗sin2/parenleftigg i/planckover2pi1/parenleftbig ∂X−∂Y/parenrightbig 8√ 2m(α+β)/parenrightigg ,(235) whereσ±:=σx±σy. Finally, we will rewrite the twisted spin as an operator form in commuta tive spacetime. Roughly speaking, the operator in the commutative spacetime. and the one in the noncomm utative Wigner space have the following rela- tions (the left-hand side represents operators on the Wigner spa ce; the right-hand side represents operators on the commutative spacetime): Xµ⋆⇔ˆxµ, (236) pµ⋆⇔ˆpµ, (237) sa⋆⇔ˆsa, (238) i/planckover2pi1∂pµ⋆⇔ˆxµ, (239) −i/planckover2pi1∂Xµ⋆⇔ˆpµ, (240) because [Xµ,pν]⋆:=Xµ⋆pν−pν⋆Xµ= i/planckover2pi1δµ νis equal to the commutation of the operator form: [ˆ xµ,ˆpν] = i/planckover2pi1δµ ν. The equivalence of st aand the twisted spin in the operator form on the commutative space time ˆst acan be conﬁrmed using the Wigner transformation in terms of ˆ st a. The operator form of the twisted spin is given by st a=/planckover2pi1 2ψ†Υ†σaΥψ, (241) where Υ = lim x′→x1 2ei 2m(α+β)σ−x+/bracketleftbigg e−i 2m(α−β)σ+x′ −e−i 2σz 2m2(α2−β2)/parenleftbig← −∂x′∂y+← −∂y′∂x/parenrightbig +e−i 2σz 2m2(α2−β2)/parenleftbig ∂x′∂y+∂y′∂x/parenrightbig e−i 2m(α−β)σ+x′ −+2sin2/parenleftbiggi/planckover2pi1(∂x−∂y) 8√ 2m(α+β)/parenrightbigg/bracketrightbigg (242) withx±:=x±y. This operator in Eq. (241), when integrated over X, is the conserved quantity for any potential conﬁguration V(ˆx) as long asα,βare static and the electron-electron interaction is neglected. VI. CONCLUSIONS In this paper, we have derived the conservation of the twisted spin and spin current densities. Also the adiabatic invariant nature of the total twisted spin integrated over the spa ce is shown. Here we remark about the limit of validity of this conservation law. First, we neglected the dynamics of the electromagnetic ﬁeld Aµwhich leads to the electron-electron interaction. This leads to the inelastic electr on scattering, which is not included in the present analysis, and most likely gives rise to the spin relaxation. This inelastic s cattering causes the energy relaxation and hence the memory of the spin will be totally lost after the inelastic lifet ime. This situation is analogous to the two relaxation times T1andT2in spin echo in NMR and ESR. Namely, the phase relaxation time T2is usually much shorter than the energy relaxationtime T1, and the spin echo is possible for T <T 1. Similar story applies to spin-orbit echo [11] where the recoveryof the spin moment is possible only within the inelastic lifetime of the spins. However, the generalization of the present study to the dynamical Aµis a diﬃcult but important issue left for future investigations. Also the eﬀect of the higher order terms in 1 /(mc2) in the derivation of the eﬀective Lagrangian from Dirac theory requires to be scrutinized. Another direction is to explore the twisted conserved quantities in t he non-equilibrium states. Under the static electric ﬁeld, the system is usually in the current ﬂowing steady stat e. Usually this situation is described by the linear response theory, but the far from equilibrium states can in pr inciple be described by the non-commutative geometry [16, 30]. The nonperturbative eﬀects in this non-equilibriu m states are the challenge for theories, and deserve the further investigations.36 The author thanks Y.S. Wu, F.C. Zhang, K. Richter, V. Krueckl, J. N itta, and S. Onoda for useful discussions. This workwassupportedbyPriorityAreaGrants, Grant-in-Aidsunder the Grantnumber21244054,StrategicInternational Cooperative Program (Joint Research Type) from Japan Science a nd Technology Agency, and by Funding Program for World-Leading Innovative R and D on Science and Technology (FI RST Program). [1] M. E. Peshkin and D. V. Schroeder. Introduction to Quantum Field Theory . Addison-Wesley, New York (1995). [2] J. Fr¨ ohlich and U. M. Studer. Gauge invariance and curre nt algebra in nonrelativistic many-body theory. Rev. Mod. P hys., 65, 733–802 (1993). [3] G. Volovik. The Universe in a Helium Droplet . Oxford University Press, Oxford, U.K. (2003). [4] B. Leurs, Z. Nazario, D. Santiago and J. Zaanen. Non-Abel ian hydrodynamics and the ﬂow of spin in spinorbit coupled substances. Annals of Physics, 323(4), 907 – 945 (2008). [5] N. Seiberg and E. Witten. String theory and noncommutati ve geometry. Journal of High Energy Physics, 09, 032 (1999). [6] M. Chaichian, P. Kulish, K. Nishijima and A. Tureanu. On a Lorentz-invariant interpretation of noncommutative spac e- time and its implications on noncommutative QFT. Physics Le tters B, 604(1-2), 98–102 (2004). [7] M. Chaichian, P. Preˇ snajder and A. Tureanu. New Concept of Relativistic Invariance in Noncommutative Space-Time: Twisted Poincar´ e Symmetry and Its Implications. Phys. Rev . Lett.,94, 151602 (2005). [8] G. Amelino-Camelia, G. Gubitosi, A. Marcian` o, P. Marti netti, F. Mercati, D. Pranzetti and R. A. Tacchi. First Resul ts of the Noether Theorem for Hopf-Algebra Spacetime Symmetries. Progress of Theore tical Physics Supplement, 171, 65–78 (2007). [9] G. Amelino-Camelia, G. Gubitosi, A. Marcian` o, P. Marti netti and F. Mercati. A no-pure-boost uncertainty principl e from spacetime noncommutativity. Physics Letters B, 671(2), 298–302 (2009). [10] S. Maekawa. Concepts in Spin Electronics . OXFORD SCIENCE PUBLICATIONS (2006). [11] N. Sugimoto and N. Nagaosa. Spin-orbit echo. To apper in Science. [12] E. L. Hill. Hamilton’s Principle and the Conservation T heorems of Mathematical Physics. Rev. Mod. Phys., 23, 253–260 (1951). [13] C. Gonera, P. Kosi´ nski, P. Ma´ slanka and S. Giller. Spa ce-time symmetry of noncommutative ﬁeld theory. Physics Le tters B,622(1-2), 192–197 (2005). [14] E. Wigner. On the Quantum Correction For Thermodynamic Equilibrium. Phys. Rev., 40, 749–759 (1932). [15] J. E. Moyal. Mathematical Proceedings of the Cambridge Philosophical Society. Proc. Cambridge Phil. Soc., 45, 99–124 (1949). [16] S. Onoda, N. Sugimoto and N. Nagaosa. Theory of Non-Equi libirum States Driven by Constant Electromagnetic Fields. Progress of The oretical Physics, 116(1), 61–86 (2006). [17] M. Kontsevich. Deformation Quantization of Poisson Ma nifolds. Letters in Mathematical Physics, 66, 157–216 (2003). [18] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. S ternheimer. Deformation theory and quantization. I. Defor ma- tions of symplectic structures. Annals of Physics, 111(1), 61 – 110 (1978). [19] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. S ternheimer. Deformation theory and quantization. II. Phys ical applications. Annals of Physics, 111(1), 111 – 151 (1978). [20] G. Hochschild. On the Cohomology Groups of an Associati ve Algebra. The Annals of Mathematics, 46(1), pp. 58–67 (1945). [21] D. Happel. Hochschild cohomology of ﬁnitedimensional algebras. In M.-P. Malliavin, editor, S´ eminaire d’Alg` ebre Paul Dubreil et Marie-Paul Malliavin , vol. 1404 of Lecture Notes in Mathematics , pages 108–126. Springer Berlin / Heidelberg. ISBN 978-3-540-51812-9 (1989). [22] M. Gerstenhaber. The Cohomology Structure of an Associ ative Ring. The Annals of Mathematics, 78(2), 267–288 (1963). [23] E. Cartan. Sur la structure des groupes inﬁnis de transf ormation. Annales Scientiﬁques de l’ ´Ecole Normale Sup´ erieure, 22, 219–308 (0012-9593). [24] A. S. Cattaneo and G. Felder. A Path Integral Approach to the Kontsevich Quantization Formula. Communications in Mathematical Physics, 212, 591–611 (2000). [25] N. Ikeda and I. Ken-iti. Dimensional reduction of nonli near gauge theories. Journal of High Energy Physics, 09, 030 (2004). [26] I. A. Batalin and G. A. Vilkovisky. Gauge algebra and qua ntization. Physics Letters B, 102(1), 27–31 (1981). [27] I. A. Batalin and G. A. Vilkovisky. Quantization of gaug e theories with linearly dependent generators. Phys. Rev. D ,28, 2567–2582 (1983). [28] J. Alfaro and P. Damgaard. Origin of antiﬁelds in the Bat alin-Vilkovisky lagrangian formalism. Nuclear Physics B, 404(3), 751–793 (1993). [29] I. Batalin and G. Vilkovisky. Relativistic S-matrix of dynamical systems with boson and fermion constraints. Phys ics Letters B, 69(3), 309–312 (1977). [30] N. Sugimoto, S. Onoda and N. Nagaosa. Gauge Covariant Fo rmulation of the Wigner Representation through Deformatio n Quantization. Progress of Theoretical Physics, 117(3), 415–429 (2007).
1304.2766v1.Spin_orbital_liquids_in_non_Kramers_magnet_on_Kagome_lattice.pdf	Spin-orbital liquids in non-Kramers magnet on Kagome lattice Robert Schaffer1, Subhro Bhattacharjee1;2, and Yong Baek Kim1;3 1Department of Physics and Center for Quantum Materials, University of Toronto, Toronto, Ontario M5S 1A7, Canada. 2Department of Physics and Astronomy, McMaster University, Hamilton, Ontario L8S 4M1, Canada. 3School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Korea. (Dated: June 21, 2018) Localized magnetic moments with crystal-ﬁeld doublet or pseudo-spin 1/2 may arise in correlated insulators with even number of electrons and strong spin-orbit coupling. Such a non-Kramers pseudo-spin 1/2 is the consequence of crystalline symmetries as opposed to the Kramers doublet arising from time-reversal invariance, and is necessarily a composite of spin and orbital degrees of freedom. We investigate possible spin-orbital liquids with fermionic spinons for such non-Kramers pseudo-spin 1/2 systems on the Kagome lattice. Using the projective symmetry group analysis, we ﬁnd tennew phases that are not allowed in the corresponding Kramers systems. These new phases are allowed due to unusual action of the time reversal operation on non-Kramers pseudo-spins. We compute the spin-spin dynamic structure factor that shows characteristic features of these non-Kramers spin-orbital liquids arising from their unusual coupling to neutrons, which is therefore relevant for neutron scattering experiments. We also point out possible anomalous broadening of Raman scattering intensity that may serve as a signature experimental feature for gapless non-Kramers spin-orbital liquids. PACS numbers: I. INTRODUCTION The low energy magnetic degrees of freedom of a Mott insulator, in the presence of strong spin-orbit coupling, are described by states with entangled spin and orbital wave functions.1,2In certain crystalline materials, for ions with even numbers of electrons, a low energy spin-orbit entan- gled “pseudo-spin”-1/2 may emerge, which is not protected by time-reversal symmetry (Kramers degeneracy)12but rather by the crystal symmetries.3,4Various phases of such non- Kramers pseudo-spin systems on geometrically frustrated lat- tices, particularly various quantum paramagnetic phases, are of much recent theoretical and experimental interest in the context of a number of rare earth materials including frus- trated pyrochlores5–9and heavy fermion systems.10,11 In this paper, we explore novel spin-orbital liquids that may emerge in these systems due to the unusual transformation of the non-Kramers pseudo-spins under the time reversal trans- formation. Contrary to Kramers spin-1/2, where the spins transform as S!Sunder time reversal,12here only one component of the pseudo-spin operators changes sign under time reversal:f1;2;3g!f1;2;3g.3,4This is be- cause, due to the nature of the wave-function content, the 3component of the pseudo-spin carries a dipolar magnetic moment while the other two components carry quadrupolar moments of the underlying electrons. Hence the time rever- sal operator for the non-Kramers pseudo-spins is given by T=1K(whereKis the complex conjugation operator), which allows for new spin-orbital liquid phases. Since the magnetic degrees of freedom are composed out of wave func- tions with entangled spin and orbital components, we prefer to refer the above quantum paramagnetic states as spin-orbital liquids, rather than spin liquids. Since the degeneracy of the non-Kramers doublet is pro- tected by crystal symmetries, the transformation properties of the pseudo-spin under various lattice symmetries intimatelydepend on the content of the wave-functions that make up the doublet. To this end, we focus our attention on the ex- ample of Praseodymium ions (Pr3+) in a localD3denviron- ment, which is a well known non-Kramers ion that occurs in a number of materials with interesting properties.5–7Such an environment typically occurs in Praseodymium pyrochlores given by the generic formulae Pr 2TM2O7, where TM(= Zr, Sn, Hf, or Ir) is a transition metal. In these compounds, the Pr3+ions host a pair of 4 felectrons which form a J= 4 ground state manifold with S= 1andL= 5, as expected due to Hund’s rules. In terms of this local environment we have a nine fold degeneracy of the electronic states.4This degener- acy is broken by the crystalline electric ﬁeld. The oxygen and TM ions form a D3dlocal symmetry environment around the Pr3+ions, splitting the nine fold degeneracy. A standard anal- ysis of the symmetries of this system (see appendix A) shows that theJ= 4 manifold splits into three doublets and three singlets ( j=4= 3Eg+ 2A1g+A2g) out of which one of the doublets is found to have the lowest energy, usually well sepa- rated from the other crystal ﬁeld states.4This doublet (details in Appendix A), formed out of a linear combination of the Jz=4withJz=1andJz=2states, is given by ji=jm=4ijm=1i jm=2i:(1) The non-Kramers nature of this doublet is evident from the nature of the “spin” raising and lowering operators within the doublet manifold; the projection of the angular momen- tum raising and lowering operators to the space of doublets is zero (PJPji= 0 wherePprojects into the doublet manifold). However, the projection of the Jzoperator to this manifold is non-zero, and describes the z component of the pseudo-spin ( 3). In addition, there is a non-trivial projection of the quadrupole operators fJ;Jzgin this manifold. These have off-diagonal matrix elements, and are identiﬁed with the pseudo-spin raising and lowering operators ( =1i2). In a pyrochlore lattice the local D3daxes point to the centre of the tetrahedra.4On looking at the pyrochlore lattice alongarXiv:1304.2766v1 [cond-mat.str-el] 9 Apr 20132 the [111] direction, it is found to be made out of alternate layers of Kagome and triangular lattices. For each Kagome layer (shown in Fig. 1) the local D3daxes make an angle of cos1(p 2=3)with the plane of the Kagome layer. We imag- ine replacing the Pr3+ions from the triangular lattice layer with non-magnetic ions so as to obtain decoupled Kagome layers with Pr3+ions on the sites. The resulting structure is obtained in the same spirit as the now well-known Kagome compound Herbertsmithite was envisioned. As long as the lo- cal crystal ﬁeld has D3dsymmetry, the doublet remains well deﬁned. A suitable candidate non-magnetic ion may be iso- valent but non-magnetic La3+. Notice that the most extended orbitals in both cases are the ﬁfth shell orbitals and the crystal ﬁeld at each Pr3+site is mainly determined by the surrounding oxygens and the transition metal element. Hence, we expect that the splitting of the non-Kramers doublet due to the above substitution would be very small and the doublet will remain well deﬁned. In this work we shall consider such a Kagome lattice layer and analyze possible Z2spin-orbital liquids, with gapped or gapless fermionic spinons . The rest of the paper is organized as follows. In Sec. II, we begin with a discussion of the symmetries of the non-Kramers system on a Kagome lattice and write down the most general pseudo-spin model with pseudo-spin exchange interactions up to second nearest neighbours. In Sec. III formulate the pro- jective symmetry group (PSG) analysis for singlet and triplet decouplings. Using this we demonstrate that the non-Kramers transformation of our pseudo-spin degrees of freedom under time reversal leads to a set of ten spin-orbital liquids which cannot be realized in the Kramers case. In Sec. IV we de- rive the dynamic spin-spin structure factor for a representa- tive spin liquid for the case of both Kramers and non-Kramers doublets, demonstrating that experimentally measurable prop- erties of these two types of spin-orbital liquids differ qualita- tively. Finally, in Sec. V, we discuss our results, and propose an experimental test which can detect a non-Kramers spin- orbital liquid. The details of various calculations are discussed in different appendices. II. SYMMETRIES AND THE PSEUDO-SPIN HAMILTONIAN Since the local D3daxes of the three sites in the Kagome unit cell differ from each other a general pseudo-spin Hamil- tonian is not symmetric under continuous global pseudo-spin rotations. However, it is symmetric under various symmetry transformations of the Kagome lattice as well as time rever- sal symmetry. Such symmetry transformations play a major role in the remainder of our analysis. We start by describing the effect of various lattice symmetry transformations on the non-Kramers doublet. We consider the symmetry operations that generate the space group of the above Kagome lattice. These are (as shown in Fig. 2(a)) T1,T2: generate the two lattice translations. =C0 2I: (not to be confused with the pseudo-spin FIG. 1: A Kagome layer, in the pyrochlore lattice environment. We consider sites labelled z and z’ replaced by non-magnetic ions, de- coupling the Kagome layers. The local axis at the u,v and w sites point towards the center of the tetrahedron on which these lie. u vw FIG. 2: (color online) (a) The symmetries of the Kagome lattice. Also shown are the labels for the sublattices and the orientation of the local z-axis. (b) Nearest and next nearest neighbour bonds. Colors refer to the phases r;r0and0 r;r0, with these being 0 on blue bonds, 1 on green bonds and 2 on red bonds. operators which come with a superscript) where Iis the three dimensional inversion operator about a plaquette center andC0 2refers to a two-fold rotation about a line joining two opposite sites on the plaquette. S6=C2 3I: whereC3is the threefold rotation op- erator about the center of a hexagonal plaquette of the Kagome lattice. T=1K: Time reversal. Here, we consider a three dimensional inversion operator since the local D3daxes point out of the Kagome plane. The above symmetries act non-trivially on the pseudo-spin degrees of freedom, as well as the lattice degrees of freedom. The ac- tion of the symmetry transformations on the pseudo-spin op- erators is given by, S6:f3;+;g!f3;!+;!g; T:f3;+;g!f3;;+g; C0 2:f3;+;g!f3;;+g; T1:f3;+;g!f3;+;g; T2:f3;+;g!f3;+;g; (2) (!= !1=ei2 3). Operationally their action on the doublet (j+i ji )can be written in form of 22matrices. The translationsT1;T2act trivially on the pseudo-spin degrees of3 freedom, and the remaining operators act as T=1K; =1; S 6= !0 0! ; (3) whereKrefers to complex conjugation. The above expres- sions can be derived by examining the effect of these operators on the wave-function describing the doublet (Eq. 1). We can now write down the most generic pseudo-spin Hamiltonian allowed by the above lattice symmetries that is bilinear in pseudo-spin operators. The form of the time- reversal symmetry restricts our attention to those products which are formed by a pair of 3operators or those which mix the pseudo-spin raising and lowering operators. Any term which mixes 3andchanges sign under the sym- metry, and can thus be excluded. Under the C3transforma- tion about a site, the terms C3:3 r3 r0!3 C3(r)3 C3(r0)and C3:+ r r0!+ C3(r) C3(r0). However, the term + r+ r0(and its Hermitian conjugate) gain additional phase factors when transformed; under the C3symmetry transformation, this term becomesC3:+ r+ r0!!+ C3(r)+ C3(r0). In addition, un- der thesymmetry, this term transforms as :+ r+ r0! (r) (r0). Thus the Hamiltonian with spin-spin exchange interactions up to next-nearest neighbour is given by Heff=JnnX hr;r0i[3 r3 r0+ 2(+ r r0+h:c:) +2q(e2ir;r 0 3+ r+ r0+h:c:)] +JnnnX hhr;r0ii[3 r3 r0+ 2(0+ r r0+h:c:) +2q0(e2i0 r;r 0 3+ r+ r0+h:c:)]; (4) whereand0take values 0, 1 and 2 depending on the bonds on which they are deﬁned (Fig. 2(b)). III. SPINON REPRESENTATION OF THE PSEUDO-SPINS AND PSG ANALYSIS Having written down the pseudo-spin Hamiltonian, we now discuss the possible spin-orbital liquid phases. We do this in two stages in the following sub-sections. A. Slave fermion representation and spinon decoupling In order to understand these phases, we will use the fermionic slave-particle decomposition of the pseudo-spin op- erators. At this point, we note that the pseudo-spins satisfy S= 1=2representations of a “SU(2)” algebra among their generators (not to be confused with the regular spin rotation symmetry). We represent the pseudo-spin degrees of freedom in terms of a fermion bilinear. This is very similar to usual slave fermion construction for spin liquids13,14. We take j=1 2fy j[]fj; (5)where;=";#is deﬁned along the local zaxis andfy(f) is anS= 1=2fermionic creation (annihilation) operator. Fol- lowing standard nomenclature, we refer to the f(fy)as the spinon annihilation (creation) operator, and note that these satisfy standard fermionic anti-commutation relations. The above spinon representation, along with the single occupancy constraint fy i"fi"+fy i#fi#= 1; (6) form a faithful representation of the pseudo-spin-1/2 Hilbert space. The above representation of the pseudo-spins, when used in Eq. 4, leads to a quartic spinon Hamiltonian. Fol- lowing standard procedure,13,14this is then decomposed us- ing auxiliary ﬁelds into a quadratic spinon Hamiltonian (af- ter writing down the corresponding Eucledian action). The mean ﬁeld description of the phases is then characterized by the possible saddle point values of the auxiliary ﬁelds. There are eight such auxiliary ﬁelds per bond, corresponding to ij=hfy ifji;ij=hfi i2 fji; (7a) Ea ij=hfy i[a]fji;Da ij=hfi i2a fii; (7b) wherea(a= 1;2;3) are the Pauli matrices. While Eq. 7a represents the usual singlet spinon hopping (particle-hole) and pairing (particle-particle) channels, Eq. 7b represents the cor- responding triplet decoupling channels. Since the Hamilto- nian (Eq. 4) does not have pseudo-spin rotation symmetry, both the singlet and the triplet decouplings are necessary.16,17 From this decoupling, we obtain a mean-ﬁeld Hamiltonian which is quadratic in the spinon operators. We write this com- pactly in the following form17(subject to the constraint Eq.6) H0=X ijJij~fy iUij~fj; (8) ~fy i=h fy i"fi#fy i#fi"i ; (9) Uij= ij; (10) = I; =I ; (11) whereare the Identity (for = 0) and Pauli matrices (= 1;2;3) acting on pseudo-spin degrees of freedom, and represents the same in the gauge space. We immediately note that ; = 08;: (12) The requirement that our H0be Hermitian restricts the coefﬁ- cientsijto satisfy 00 ij;ab ij2=;a0 ij;0b ij2<: (13) fora;b2 f 1;2;3g. The relations between ijs and fij;ij;Eij;Dijgare given in Appendix C.17As a straight forward extension of the SU(2)gauge theory formulation for4 spin liquids,13,18we ﬁnd that H0is invariant under the gauge transformation ~fj!Wj~fj; (14) Uij!WiUijWy j; (15) where theWimatrices are SU(2) matrices of the form Wi= ei~~ ai(~(1;2;3)). Noting that the physical pseudo- spin operators are given by ~ i=1 4~fy i~~fi; (16) Eq. 12 shows that the spin operators, as expected, are gauge invariant. It is useful to deﬁne the “ -components” of the Uij matrices as follows: Uij=V ij; (17) where V ij= ij=J ij0 0J ij ; (18) and J ij=0 ij+3 ij1 iji2 ij 1 ij+i2 ij0 ij3 ij : (19) Under global spin rotations the fermions transform as ~fi!V~fi; (20) where V is an SU(2) matrix of the form V=ei~~b(~ f1;2;3g). So whileV0 ij(the singlet hopping and pairing) is invariant under spin rotation, fV1 ij;V2 ij;V3 ijgtransforms as a vector as expected since they represent triplet hopping and pairing amplitudes. B. PSG Classiﬁcation We now classify the non-Kramers spin-orbital liquids based on projective representation similar to that of the conven- tional quantum spin liquids.13Each spin-orbital liquid ground state of the quadratic Hamiltonian (Eq 11) is character- ized by the mean ﬁeld parameters (eight on each bond, ;;E1;E2;E3;D2;D2;D3, or equivalently Uij). However, due to the gauge redundancy of the spinon parametrization (as shown in Eq. 15), a general mean-ﬁeld ansatz need not be invariant under the symmetry transformations on their own but may be transformed to a gauge equivalent form without breaking the symmetry. Therefore, we must consider its trans- formation properties under a projective representation of the symmetry group.13For this, we need to know the various pro- jective representations of the lattice symmetries of the Hamil- tonian (Eq. 4) in order to classify different spin-orbital liquid states. Operationally, we need to ﬁnd different possible sets of gauge transformations fGGgwhich act in combination withthe symmetry transformations fSGgsuch that the mean-ﬁeld ansatzUijis invariant under such a combined transformation. In the case of spin rotation invariant spin-liquids (where only the singlet channels andare present), the above statement is equivalent to demanding the following invariance: Uij= [GSS]Uij[GSS]y=GS(i)US(i)S(j)Gy S(j);(21) whereS2SGis a symmetry transformation and GS2GG is the corresponding gauge transformation. The different pos- siblefGSj8S2SGggive the possible algebraic PSGs that can characterize the different spin-orbital liquid phases. To obtain the different PSGs, we start with various lattice sym- metries of the Hamiltonian. The action of various lattice transformations15is given by T1:(x;y;s )!(x+ 1;y;s); T2:(x;y;s )!(x;y+ 1;s); :(x;y;u )!(y;x;u ); (x;y;v )!(y;x;w ); (x;y;w )!(y;x;v ); S6:(x;y;u )!(y1;x+y+ 1;v); (x;y;v )!(y;x+y;w); (x;y;w )!(y1;x+y;u); (22) where (x;y)denotes the lattice coordinates and s2fu;v;wg denotes the sub-lattice index (see ﬁgure 2). In terms of the symmetries of the Kagome lattice, these op- erators obey the following conditions T2=2= (S6)6=e; g1T1gT=e8g2SG; T1 2T1 1T2T1=e; 1T1 1T2=e; 1T1 2T1=e; S1 6T1 2S6T1=e; S1 6T1 2T1S6T2=e; 1S6S6=e: (23) In addition, these commutation relations are valid in terms of the operations on the pseudo-spin degrees of freedom, as can be veriﬁed from Eq. 3. In addition to the conditions in Eq. 23, the Hamiltonian is trivially invariant under the identity transformation. The in- variant gauge group (IGG) of an ansatz is deﬁned as the set of all pure gauge transformations GIsuch thatGI:Uij!Uij. The nature of such pure gauge transformations immediately dictates the nature of the low energy ﬂuctuations about the mean ﬁeld state. If these ﬂuctuations do not destabilize the mean-ﬁeld state, we get stable spin liquid phases whose low energy properties are controlled by the IGG. Accordingly, spin liquids obtained within projective classiﬁcation are pri- marily labelled by their IGGs and we have Z2;U(1)and SU(2)spin liquids corresponding to IGGs of Z2;U(1)and5 SU(2)respectively. In this work we concentrate on the set of Z2“spin liquids” (spin-orbital liquids with a Z2IGG). We now focus on the PSG classiﬁcation. As shown in Eq. 2, in the present case, the pseudo-spins transform non-trivially under different lattice symmetry transformations. Due to the presence of the triplet decoupling channels the non-Kramers doublet transforms non-trivially under lattice symmetries (Eq. 3). Thus, the invariance condition on the Uijs is not given by Eq. 21, but by a more general condition Uij= [GSS]Uij[GSS]y=GS(i)S US(i)S(j) Gy S(j): (24) Here S US(i)S(j) =DSUS(i)S(j)Dy S; (25) andDSgenerates the pseudo-spin rotation associated with the symmetry transformation ( S) on the doublet. The matrices DShave the form DS6=1 20ip 3 23; (26) D=DT=i1;DT1=DT2= 0: (27) Under these constraints, we must determine the relations between the gauge transformation matrices GS(i)for our set of ansatz. The additional spin transformation (Eq. 25) does not affect the structure of the gauge transformations, as the gauge and spin portions of our ansatz are naturally separate (Eq. 12). In particular, we can choose to deﬁne our gauge transformations such that GS:Uij=GS: ij! ijGy S(i)GS(j); (28) S:Uij=S: ij! S(i)S(j)DSDy S;(29) where we have used the notation GS:UijGy S(i)UijGS(j) and so forth. As a result, we can build on the general con- struction of Lu et al.15to derive the form of the gauge trans- formation matrices. The details are given in Appendix B. A major difference arises when examining the set of alge- braic PSGs for Z2spin liquids found on the Kagome lattice due to the difference between the structure of the time re- versal symmetry operation on the Kramers and non-Kramers pseudo-spin- 1=2s. In the present case, we ﬁnd there are 30 invariant PSGs leading to thirty possible spin-orbital liquids. This is in contrast with the Kramers case analysed by Lu et al.,15where tenof the algebraic PSGs cannot be realized as invariant PSGs, as all bonds in these ansatz are predicted to vanish identically due to the form of the time reversal oper- ator, and hence there are only twenty possible spin liquids. However, with the inclusion of spin triplet terms and the non- Kramers form of our time reversal operator, these ansatz are now realizable as invariant PSGs as well. The time reversal operator, as deﬁned in Appendix B, acts as T: ij!~ ij; (30)where ~=if2f1;2gand~=if2f0;3g. The projective implementation of the time-reversal symmetry condition (Eq. 23) takes the form (see Appendix B) [GT(i)]2=TI8i; (31) whereGT(i)is the gauge transformation associated with time reversal operation and T=1for aZ2IGG. Therefore, the terms allowed by the time reversal symmetry to be non zero are, for T= 1, 10;11;12;13;20;21;22;23; (32) and forT=1, with the choice GT(i) =i1(see appendix B), 02;03;10;11;20;21;32;33: (33) This contrasts with the case of Kramers doublets, in which no terms are allowed for T=1, and forT=1the allowed terms are 02;03;12;13;22;23;32;33: (34) Further restrictions on the allowed terms on each link arise from the form of the gauge transformations deﬁned for the symmetry transformations. All nearest neighbour bonds can then be generated from Uijdeﬁned on a single bond, by per- forming appropriate symmetry operations. Using the methods outlined in earlier works (Ref. 13, 15) we ﬁnd the minimum set of parameters required to stabilize Z2 spin-orbital liquids. We take into consideration up to second neighbour hopping and pairing amplitudes (both singlet and triplet channels). The results are listed in Table I. The spin-orbital liquids listed from 2130are not allowed in the case of Kramers doublets and, as pointed out before, their existence is solely due to the unusual action of the time- reversal symmetry operator on the non-Kramers spins. Hence these tenspin-orbital liquids are qualitatively new phases that may appear in these systems. Of these ten phases, only two (labelled as 21and22in Table I) require next nearest neigh- bour amplitudes to obtain a Z2spin-orbital liquid. For the other eight , nearest neighbour amplitudes are already sufﬁ- cient to stabilize a Z2spin-orbital liquid. It is interesting to note (see below) that bond-pseudo-spin- nematic order (Eq. 35 and Eq. 36) can signal spontaneous time-reversal symmetry breaking. Generally, since the triplet decouplings are present, the bond nematic order parameter for the pseudo-spins21,22 Q ij=h S iS j+S iS j =2(~Si~Sj)=3i;(35) as well as vector chirality order ~Jij=h~Si~Sji; (36) are non zero. Since the underlying Hamiltonian Eq. 4) gener- ally does not have pseudo-spin rotation symmetry, the above non-zero expectation values do not spontaneously break any6 TABLE I: Symmetry allowed terms: We list the terms allowed to be non-zero by symmetry, for the 30 PSGs determined by Yuan-Ming Lu et al15. The PSGs listed together are those with 12=1and all other factors equal. Included are terms allowed on nearest and next-nearest neighbour bonds, as well as chemical potential terms which can be non zero on all sites for certain spin-orbital liquids. Also included is the distance of bond up to which we must include in order to gap out the gauge ﬂuctuations to Z2via the Anderson-Higgs mechanism13. Only PSGs 9 and 10 can not host Z2spin-orbital liquids with up to second nearest neighbour bonds. No. s n.n. n.n.n. Z2 1-2 2;310;21;02;03;32;3310;21;02;03;32;33n.n. 3-4 010;21;02;03;32;3310;21n.n. 5-6 310;21;02;03;32;3310;21;03;33n.n. 7-8 0 11;2011;20;02;03;32;33n.n.n. 9-10 0 11;2011;20- 11-12 0 11;2010;11;02;32n.n.n. 13-14 310;11;03;3310;21;02;03;32;33n.n. 15-16 310;11;03;3310;21;03;33n.n. 17-18 010;11;03;3310;11;02;32n.n. 19-20 010;11;03;3310;21n.n. 21-22 010;21;22;2310;21;22;23n.n.n. 23-24 010;21;22;2310;11;12;23n.n. 25-26 011;12;13;2013;20;21;22n.n. 27-28 011;12;13;2010;11;13;22n.n. 29-30 011;12;13;2011;12;13;20n.n. pseudo-spin rotation symmetry. However, because of the un- usual transformation property of the non-Kramers pseudo- spins under time reversal, the operators corresponding to Q13 ij;Q23 ij;J1 ij;J2 ijare odd under time reversal, a symmetry of the pseudo-spin Hamiltonian. Hence if any of the above op- erators gain a non-zero expectation value in the ground state, then the corresponding spin-orbital liquid breaks time rever- sal symmetry. While this can occur in principle, we check explicitly (see Appendix C) that in all the spin-orbital liquids discussed above, the expectation values of these operators are identically zero. This provides a non-trivial consistency check on our PSG calculations. We now brieﬂy dicuss the effect of the ﬂuctuations about the mean-ﬁeld states. In the absence of pairing channels (both singlet and triplet) the gauge group is U(1). In this case, the ﬂuctuations of the gauge ﬁeld about the mean ﬁeld (Eq. 15) are related to the scalar pseudo-spin chirality ~S1~S2~S3, where the three sites form a triangle.19Such ﬂuctuations are gapless in a U(1)spin liquid. It is interesting to note that the scalar spin-chirality is odd under time-reversal symmetry and it has been proposed that such ﬂuctuations can be detected in neutron scattering experiments in presence of spin rotation symmetry breaking.20In the present case, however, due to the presence of spinon pairing, the gauge group is broken down toZ2and the above gauge ﬂuctuations are rendered gapped through Anderson-Higg’s mechanism.13 In addition to the above gauge ﬂuctuations, because of thetriplet decouplings which break pseudo-spin rotational sym- metry, there are bond quadrupolar ﬂuctuations of the pseudo- spinsQ ij(Eq. 35), as well as vector chirality ﬂuctuations ~Jij (Eq. 36)21,22on the bonds. These nematic and vector chirality ﬂuctuations are gapped because the underlying pseudo-spin Hamiltonian (Eq. 4) breaks pseudo-spin-rotation symmetry. However, we note that because of the unusual transformation of the non-Kramers pseudo-spins under time reversal (only thezcomponent of pseudo-spins being odd under time re- versal),Q13 ij;Q23 ij;J1 ijandJ2 ijare odd under time reversal. Hence, while their mean ﬁeld expectation values are zero (see above), the ﬂuctuations of these quantities can in principle lin- early couple to the neutrons in addition to the zcomponent of the pseudo-spins. Having identiﬁed the possible Z2spin-orbital liquids, we can now study typical dynamic structure factors for these spin-orbital liquids. In the next section we examine the typi- cal spinon band structure for different spin-orbital liquids ob- tained above and ﬁnd their dynamic spin structure factor.7 FIG. 3: The spin structure factor for an ansatz in spin liquid 17, with the spin variables transforming as a Kramers doublet. FIG. 4: The spin structure factor for an ansatz in spin liquid 17, with the spin variables transforming as a non-Kramers doublet. IV . DYNAMIC SPIN STRUCTURE FACTOR We compute the dynamic spin structure factor S(q;!) =Zdt 2ei!tX ijeiq(rirj)X a=1;2;3ha i(t)a j(0)i; (37) for an example ansatz of our spin liquid candidates, in order to demonstrate the qualitative differences between the Kramers and non-Kramers spin-orbital liquids. In the above equation, the pseudo-spin variables are deﬁned in a global basis (with the z-axis perpendicular to the Kagome plane). In computing the structure factor for the non-Kramers example, we include only the3components of the pseudo-spin operator in the lo- cal basis, since only the z-components carry magnetic dipole moment (see discussion before). Hence, only this component couples linearly to neutrons in a neutron scattering experi- ment. Eq. 37 fails to be periodic in the ﬁrst Brillouin zone of the Kagome lattice16, as the term rirjin eq. 37 is a half- integer multiple of the primitive lattice vectors when the sub- lattices of sites i and j are not equal. As such, we examine the structure factor in the extended brillouin zone, which consists of those momenta of length up to double that of those in theﬁrst brillouin zone. We plot the structure factor along the cut !M0!K0!, whereM0=2MandK0=2K. We examine the structure factors for two ansatz of spin liquid # 17 which has both Kramers and non-Kramers analogues. As expected, we ﬁnd that the structure factor has greater weight in the case of a Kramers spin liquid. This is par- tially due to the fact that the moment of the scattering par- ticle couples with all components of the spin, rather than sim- ply thez-component. In addition, we note that the presence of terms allowed in the non-Kramers spin-orbital liquid in- duce the formation of a gap, which is absent for the Kramers case with up to second nearest neighbour singlet and triplet terms in this particular spin-orbital liquid. Qualitative and quantitative differences such as these, which can be observed in these structure factors between Kramers and non-Kramers spin-orbital liquids, provides one possible distinguishing ex- perimental signature of these states. We shall not pursue this in detail in the present work. V . DISCUSSION AND POSSIBLE EXPERIMENTAL SIGNATURE OF NON-KRAMERS SPIN-ORBITAL LIQUIDS In this work, we have outlined the possible Z2spin-orbital liquids, with gapped or gapless fermionic spinons, that can be obtained in a system of non-Kramers pseudo-spin-1/2s on a Kagome lattice of Pr+3ions. We ﬁnd a total of thirty , 10 more than in the case of corresponding Kramers system, allowed within PSG analysis in presence of time reversal symmetry. The larger number of spin-orbital liquids is a result of the dif- ference in the action of the time-reversal operator, when real- ized projectively. We note that the spin-spin dynamic struc- ture factor can bear important signatures of a non-Kramers spin-orbital liquid when compared to their Kramers counter- parts. Our analysis of the number of invariant PSGs leading to possibly different spin-orbital liquids that may be realizable in other lattice geometries will form interesting future directions. We now brieﬂy discuss an experiment that can play an im- portant role in determining non-Kramers spin-orbital liquids. Since the non-Kramers doublets are protected by crystalline symmetries, lattice strains can linearly couple to the pseudo- spins. As we discussed, the transverse ( xandy) components of the pseudo-spins f1;2gcarry quadrupolar moments and hence are even under the time reversal transformation. Fur- ther, they transform under an Egirreducible representation of the local D3dcrystal ﬁeld. Hence any lattice strain which has this symmetry can linearly couple to the above two trans- verse components. It turns out that in the crystal type that we are concerned, there is indeed such a mode related to the distortion of the oxygen octahedra. Symmetry considerations show that the linear coupling is of the form Eg11+Eg22 (fEg1;Eg2gbeing the two components of the distortion in the local basis). The above mode is Raman active. For a spin-liquid, we expect that as the temperature is lowered, the spinons become more prominent as deconﬁned quasipar- ticles. So the Raman active phonon can efﬁciently decay into the spinons due to the above coupling channel. If the spin liquid is gapless, then this will lead to anomalous broaden-8 ing of the above Raman mode as the temperature is lowered, which, if observed, can be an experimental signature of the non-Kramers spin-orbital liquid. The above coupling is for- bidden in Kramers doublets by time-reversal symmetry and hence no such anomalous broadening is expected. Acknowledgments We thank T. Dodds, SungBin Lee, A. Paramekanti and J. Rau for insightful discussions. This research was supported by the NSERC, CIFAR, and Centre for Quantum Materials at the University of Toronto. Appendix A: Crystal Field Effects In this appendix, we explore the breaking of the J= 4spin degeneracy by the crystalline electric ﬁeld. The oxygen and TM ions form a D3dlocal symmetry environment around the Pr3+ions, splitting the ground state degeneracy of the elec- trons. This symmetry group contains 6 classes of elements: E, 2C3,3C0 2,i,2S6, and 3d, where theC3are rotations by 2=3 about the local z axis, the C0 2are rotations by about axis per- pendicular to the local z axis, iis inversion, S6is a rotation by4=3combined with inversion and dis a reﬂection about the plane connecting one corner and the opposing plane, run- ning through the Prmolecule about which this is measured (or, equivalently, a rotation about the x axis combined with inversion). For our J=4 manifold, these have characters given by (4)(E) = 24 + 1 = 9 = (4)(i) (A1) (4)(C3) =(4)(2 3) =sin(3) sin(=3)= 0 =(4)(S6)(A2) (4)(d) =(4)() =sin(9=2) sin(=2)= 1 =(4)(C0 2)(A3) where the latter equalities are given by the fact that our J=4 manifold is inversion symmetric. Thus, decomposing this in terms ofD3dirreps, our l=4 manifold splits into a sum of doublet and singlet manifolds as l=4= 3Eg+ 2A1g+A2g: (A4) To examine this further, we need to consider the matrix ele- ments of the crystal ﬁeld potential between the states of differ- ent angular momenta. We know that this potential must be in- variant under all group operations of D3d, so we can examine the transformation properties of individual matrix elements, hmjVjm0i. Under the C3operation, these states of ﬁxed m transform as C3jmi=e2im 3jmi=!mjmi (!=e2i 3) (A5) and thus the matrix elements transform as C3:hmjVjm0i!hmj(C3)1VC3jm0i=!m0mhmjVjm0i: (A6)By requiring that this matrix be invariant under this transfor- mation, we can see that this potential only contains matrix elements for mixing of states which have the z-component of angular momentum which differ by 3. Thus, our eigenstates are mixtures of the jm= 4i,jm= 1i, andjm=2istates, of thejm= 3i,jm= 0i, andjm=3istates, and of the jm=4i,jm=1i, andjm= 2istates. In addition to this, we have the transformation properties Tjmi= (1)mjmi (A7) and jmi= (1)mjmi (A8) (where the operators for time reversal and reﬂection are bolded for future clarity). Inversion acts trivially on these states, as we have total angular momentum even. Thus our time-reversal and lattice reﬂection (about one axis) symme- tries give us doublet states of eigenstates jm= 4i+jm= 1i jm=2iandjm=4ijm=1i jm= 2i (with,, 2< in order to respect the time reversal symme- try) for the three eigenstates of V in these sectors. The eigen- states of thejm= 3i,jm= 0i, andjm=3iportion of V must therefore split into three singlet states, by our represen- tation theory argument A4. Due to the expected strong Ising term in our potential, we expect the eigenstate with maximal J to be the ground state, meaning that to analyze the properties of this ground state we are interested in a single doublet state, one with large (close to one). We will restrict ourselves to this manifold from this point forward, and deﬁne the two states in this doublet as j+i=jm= 4i+jm= 1i jm=2i (A9) ji=jm=4ijm=1i jm= 2i:(A10) We shall also refer to states of angular momentum jm=ni asjnifor simplicity of notation. Appendix B: Gauge transformations We begin by describing the action of time reversal on our ansatz. The operation is antiunitary, and must be combined with a spin transformation 1in the case of non-Kramers dou- blets. As a result, the operation acts as T: ij! ij11. However, we can simplify this consid- erably by performing a gauge transformation in addition to the above transformation, which yields the same transforma- tion on any physical variables. The gauge transformation we perform isi2, which changes the form of the time reversal operation to T: ij! ij1122= ~ ij, where ~=if2f1;2gand~=if 2f0;3g. On the Kagome lattice, the allowed form of the gauge trans- formations has been determined by Yuan-Ming Lu et al.15For completeness, we will reproduce that calculation, valid also9 for our spin triplet ansatz, here. The relations between the gauge transformation matrices, [GT(i)]2=TI; (B1) G((i))G(i) =I; (B2) Gy T1(i)Gy T(i)GT1(i)GT(T1 1(i)) =T1TI; (B3) Gy T2(i)Gy T(i)GT2(i)GT(T1 2(i)) =T2TI; (B4) Gy (i)Gy T(i)G(i)GT(1(i)) =TI; (B5) Gy S6(i)Gy T(i)GS6(i)GT(S1 6(i)) =S6TI; (B6) Gy T2(T1 1(i))Gy T1(i)GT2(i)GT1(T1 2(i)) =12I; (B7) GS6(S1 6(i))GS6(S2 6(i))GS6(S3 6(i)) GS6(S2 6(i))GS6(S6(i))GS6(i) =S6I; (B8) Gy (T1 2(i))Gy T2(i)G(i)GT1((i)) =T1I; (B9) Gy (T1 1(i))Gy T1(i)G(i)GT2((i)) =T2I; (B10) Gy (S6(i))GS6(S6(i))G(i)GS6((i)) =S6I; (B11) Gy S6(T1 2(i))Gy T2(i)GS6(i)GT1(S1 6(i)) =S6T1I;(B12) Gy S6(T1 2T1(i))Gy T2(T1(i))GT1(T1(i)) GS6(i)GT2(S1 6(i)) =S6T2I; (B13) are valid for our case as well, due to the decoupling of spin and gauge portions of our ansatz. In the above, the relations are valid for all lattice sites i= (x;y;s ), I is the 4x4 identity matrix, and the GSmatrices are gauge transformation matri- ces generated by exponentiation of the matrices. The ’s are1, the choice of which characterize different spin liquid states. In deriving this form of the commutation relations, we have included a gauge transformation i2in our deﬁnition of the time reversal operator, as this simpliﬁes the effect of the operator on the mean ﬁeld ansatz. We turn next to the calculation of the gauge transforma- tions. We look ﬁrst at the gauge transformations associated with the translations. We can perform a site dependent gauge transformation W(i), under which the gauge transformations associated with the translational symmetries transform as GT1(i)!W(i)GT1(i)Wy(i^x) (B14) GT2(i)!W(i)GT2(i)Wy(i^y): (B15) As such, we can choose a gauge transformation W(i) to sim- plify the form of GT1andGT2. Using such a transformation, along with condition B7, we can restrict the form of these gauge transformations to be GT1(i) =iy 12I GT2(i) =I: (B16) To preserve this choice, we can now only perform gauge transformations which are equivalent on all lattice positions (W(x;y;s ) =W(s)) or transformations which change the shown matrices by an IGG transformation. Next, we look at adding the reﬂection symmetry . Given our formulae for GT1andGT2, along with the relations be-tween the gauge transformations, we have that Gy (T1 2(i))G(i)x 12=T1I (B17) Gy (T1 1(i))G(i)y 12=T2I: (B18) DeﬁningG(0,0,s) =g(s), we have, by repeated application of the above, G(0;y;s) =y T1g(s) (B19) G(x;y;s ) =y T1xy 12x T2g(s): (B20) Next, using G((i))G(i) =I (B21) we ﬁnd that I=G(y;x; (s))G(x;y;s ) (B22) = (T1T2)x+yg((s))g(s): (B23) Since this is true for all x and y, T1T2= 1 and thus T1=T2andg((s))g(s) =I(where(u) = u;(v) =wand(w) =v). Our ﬁnal form for the gauge transformation is G(x;y;s ) =x+y T1xy 12g(s): (B24) Next we look at adding the S6symmetry to our calcula- tion. We can do an IGG transformation, taking GT1(T1(i)) toS6T2GT1(T1(i)), with the net effect being that S6T2be- comes one (previous calculations are unaffected). We now have that Gy S6(T1 2T1(i))GS6(i)y 12=I (B25) Gy S6(T1 2(i))GS6(i)x1 12 =S6T1I (s=u;v)(B26) Gy S6(T1 2(i))GS6(i)x 12=S6T1I (s=w):(B27) DeﬁningGS6(0;0;s) =gS6(s), we ﬁnd that GS6(n;n;s) =n(n1)=2 12gS6(s) (B28) GS6(x;y;s ) =x(x1)=2+y+xy 12 x+y S6T1gS6(s) (s=u;v) (B29) GS6(x;y;s ) =x(x1)=2+xy 12 x+y S6T1gS6(s) (s=w): (B30) Using the commutation relation between the andS6gauge transformations, we ﬁnd that S6I=y T1y 12y S6T1gy (v)gS6(v)g(u)gS6(u) (B31) =y T1y 12y S6T1g(w)gS6(v)g(u)gS6(u)(B32) giving us that T112S6T1 = 1 and g(u)gS6(u)g(w)gS6(v) =S6I. A similar calcu- lation on a different sublattice gives us S6I=y T1y 12y S6T1gy (w)gS6(w)g(v)gS6(w) (B33) =y T1y 12y S6T1g(v)gS6(w)g(v)gS6(w)(B34)10 TABLE II: We list the solutions of Eq. B43 - B54, along with a set of gauge transformations which realize these solutions. No.TTS6TS6S612g(u)g(v)g(w)gS6(u)gS6(v)gS6(w) 1,2 -1 1 1 1 1 11 000000 3,4 -1 1 1 1 -1 11 0000-0i1 5,6 -1 1 -1 1 -1 11 000i3i3i3 7,8 -1 1 1 -1 -1 11i10-00i10 9,10 -1 1 1 -1 1 11i10-00-i1i1 11,12 -1 1 -1 -1 1 11i10-0i3-i2i3 13,14 -1 -1 -1 -1 -1 11i3i3i3i3i3i3 15,16 -1 -1 1 -1 1 11i3i3i3000 17,18 -1 -1 1 -1 1 11i3i3i300i1 19,20 -1 -1 -1 -1 1 11i3i3i3i3-i3i3 21,22 1 1 1 1 1 11 000000 23,24 1 1 1 1 -1 11 0000-0i3 25,26 1 1 1 -1 -1 11i30-00i30 27,28 1 1 1 -1 1 11i30-00-i3i1 29,30 1 1 1 -1 1 11i30-00-i3i3 giving us (g(v)gS6(w))2=S6I. AZ2(IGG) gauge transformation of the form W(x;y;s ) =y T1changesT1 to 1. Using the cyclic relation of the gauge transformations related to the S6operators, we ﬁnd S6I=12(gS6(w)gS6(v)gS6(u))2(B35) giving us that [gS6(w)gS6(v)gS6(u)]2=S612I: (B36) Next we turn to the time reversal symmetry. Similar meth- ods to the above give us that [GT(i)]2=TI (B37) Gy T(i)GT(i+ ^x) =T1TI (B38) Gy T(i)GT(i+ ^y) =T2TI: (B39) The ﬁrst of these relations tells us that GT(i)is either the identity (for T= 1) ori~ a~ (forT=1, wherej~ aj= 1. DeﬁningGT(0;0;s) =gT(s), GT(x;y;s ) =x T1Ty T2TgT(s) (B40) and further, using the commutation relations between the andTgauge transformations and the S6andTgauge trans- formations, gy (s)gy T(s)g(s)gT((s))x+y T1Tx+y T2T=TI(B41) gy S6(s)gy T(s)gS6(s)gT(S1 6(s))f1(i) T1Tf2(i) T2T=S6TI:(B42)Because this is true for all x and y, and f1(i)is not equal to f2(i),T1T=T2T= 1. IfGT(i) =i~ a~ , we perform a gauge transformation W on GT(i)such thatWyGT(i)W= i1(as this is the same on all sites, it does not affect our gauge ﬁxing for the translation gauge transformations). Collecting the necessary results for further use, GT1(x;y;s ) =y 12I (B43) GT2(x;y;s ) =I (B44) G(x;y;s ) =xy 12g(s) (B45) GS6(x;y;s ) =xy+(x+1)x=2 12 gS6(s)s=u;v (B46) GS6(x;y;s ) =xy+x+y+(x+1)x=2 12 gS6(s)s=w(B47) GT(s) =I=gT(s)T= 1 (B48) GT(s) =i1=gT(s)T=1 (B49) g((s))g(s) =I (B50) g(u)gS6(u)g(w)gS6(v) = (g(v)gS6(w))2=S6I (B51) (gS6(w)gS6(v)gS6(u))2=S612I (B52) g(s)gT((s)) =TgT(s)g(s) (B53) gS6(s)gT(S1 6(s)) =S6TgT(s)gS6(s): (B54) We also have the gauge freedom left to perform a gauge ro- tation arbitrarily at all positions for T= 1 or an arbitrary gauge rotation about the x axis for T=1. The solution to the above equations is derived in detail by Luet al.15and as such we simply list the results in table II. The basic method of obtaining these solutions is as follows:11 for each choice of Z2parameter set, we determine whether there is a choice of gauge matrices fgSgwhich satisfy the equations B43 - B54. In order to do so, we determine the allowed forms of the gSmatrices from the equations, then use the gauge freedom on each site to ﬁx the form of these. Of particular not is the fact that in the consistency equations for thegmatrices, the terms 12andS6only appear multiplied together, meaning that for any choice of the gauge matrices gSwe can choose 12=1, which ﬁxes the form of S6. Appendix C: Relation among the mean-ﬁeld paramters The relation among the different singlet and triplet param- eters in terms of ijis given by ij=00 ij+03 ij;ij=01 ij+i02 ij; E1 ij=10 ij+13 ij;E2 ij=20 ij+23 ij;E3 ij=30 ij+33 ij D1 ij=11 ij+i12 ij;D2 ij=21 ij+i22 ij;D3 ij=31 ij+i32 ij (C1) Using these, we can derive the form of the bond nematic order parameter and vector chirality order parameters, which are given in terms of the mean ﬁeld parameters21as Q; ij=1 2 E ijE ij1 3;j~Eijj2 +h:c: 1 2 D ijD ij1 3;j~Dijj2 +h:c: J ij=i 2 ijE ij ijE ij +i 2 ijD ij ijD ij (C2)where our deﬁnition of ijdiffers by a factor of (-1) from that of the cited work. We rewrite this in terms of our variables, ﬁnding Q ij=0 ij0 ij+X aa ija ij + 3X b (b0 ij)2X a(ba ij)2) J ij=i(00 ij0 ijX a0a ija ij) (C3) In particular, we ﬁnd that J1,J2,Q13andQ23must be zero for all non-Kramers spin liquids, as the terms allowed by sym- metry in Eq. 32 and 33 do not allow non-zero values for these order parameters. 1P. Fazekas, Lecture Notes on Electron Correlation and Magnetism (Series in Modern Condensed Matter Physics) , World Scientiﬁc Pub Co Inc (1999). 2K. Yosida, Theory of Magnetism , Springer (2001). 3B. Bleaney and H. E. D. Scovil, Phil. Mag. 43, 999 (1952). 4Shigeki Onoda and Yoichi Tanaka, Phys. Rev. B 83, 094411(R) (2011). 5K. Matsuhira, Y . Hinatsu, K. Tenya1, H. Amitsuka and T. Sakak- ibara, J. Phys. Soc. Jpn. 71, 1576 (2002). 6K. Matsuhira, C. Sekine, C. Paulsen, M. Wakeshima, Y . Hinatsu, T. Kitazawa, Y . Kiuchi, Z. Hiroi, and S. Takagi, J. Phys. Conf. Ser. 145, 012031 (2009). 7S. Nakatsuji, Y . Machida, Y . Maeno, T. Tayama, T. Sakakibara, J. van Duijn, L. Balicas, J. N. Millican, R. T. Macaluso, and J. Y . Chan, Phys. Rev. Lett. 96, 087204 (2006). 8SB. Lee, S. Onoda, and L. Balents, Phys. Rev. B 86, 104412 (2012). 9R. Flint, and T. Senthil, arXiv:1301.0815 (2013). 10P. Chandra, P. Coleman, and R. Flint, Nature 493, 621 (2013). 11O. Suzuki, H. S. Suzuki, H. Kitazawa, G. Kido, T. Ueno, T. Ya-maguchi, Y . Nemoto, T. Goto, J. Phys. Soc. Japan 75, 013704 (2006). 12H. A. Kramers, Proc. Amsterdam Acad. 33, 959 (1930). 13X. G. Wen, Phys. Rev. B 65, 165113 (2002); X.-G. Wen, Quantum Field Theory of Many-Body Systems. Oxford University Press, Oxford (2004). 14P. W. Anderson, G. Baskaran, Z. Zou, and T. Hsu, Phys. Rev. Lett. 58, 2790 (1987). 15Yuan-Ming Lu, Ying Ran, and Patrick A. Lee, Phys. Rev. B 83, 224412 (2011). 16T. Dodds, S. Bhattacharjee, and Y . B. Kim, arXiv:1303.1154 17R. Schaffer, S. Bhattacharjee, and Y . B. Kim, Phys. Rev. B 86, 224417 (2012). 18I. Afﬂeck, and J. B. Marston, Phys. Rev. B 37, 3774 (1988). 19P. A. Lee, and N. Nagaosa, Phys. Rev. B 46, 5621 (1992). 20P. A. Lee, and N. Nagaosa, Phys. Rev. B 87, 064423 (2013). 21R. Shindou, and T. Momoi, Phys. Rev. B 80, 064410 (2009). 22S. Bhattacharjee, Y . B. Kim, S.-S. Lee, and D.-H. Lee, Phys. Rev. B 85, 224428 (2012).
1609.03141v1.Spin_Orbit_Coupling_Induced_Spin_Squeezing_in_Three_Component_Bose_Gases.pdf	Spin-Orbit Coupling Induced Spin Squeezing in Three-Component Bose Gases X. Y. Huang,1F. X. Sun,1W. Zhang,2, 3,Q. Y. He,1, 4,yand C. P. Sun5,z 1State Key Laboratory of Mesoscopic Physics, School of Physics, Peking University, Collaborative Innovation Center of Quantum Matter, Beijing 100871, China 2Department of Physics, Renmin University of China, Beijing 100872, China 3Beijing Key Laboratory of Opto-electronic Functional Materials and Micro-nano Devices, Renmin University of China, Beijing 100872, China 4Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, Shanxi 030006, China 5Beijing Computational Science Research Center, Beijing 100084, China Weobservespinsqueezinginthree-componentBosegaseswhereallthreehyperﬁnestatesarecoupled by synthetic spin-orbit coupling. This phenomenon is a direct consequence of spin-orbit coupling, as can be seen clearly from an eﬀective spin Hamiltonian. By solving this eﬀective model analytically with the aid of a Holstein-Primakoﬀ transformation for spin-1 system in the low excitation limit, we conclude that the spin-nematic squeezing, a novel category of spin squeezing existing exclusively in large spin systems, is enhanced with increasing spin-orbit intensity and eﬀective Zeeman ﬁeld, which correspond to Rabi frequency Rand two-photon detuning within the Raman scheme for synthetic spin-orbit coupling, respectively. These trends of dependence are in clear contrast to spin-orbit coupling induced spin squeezing in spin-1/2 systems. We also analyze the eﬀects of harmonic trap and interaction with realistic experimental parameters numerically, and ﬁnd that a strong harmonic trap favors spin-nematic squeezing. We further show spin-nematic squeezing can be interpreted as two-mode entanglement or two-spin squeezing at low excitation. Our ﬁndings can be observed in87Rb gases with existing techniques of synthetic spin-orbit coupling and spin-selectively imaging. I. INTRODUCTION Spin squeezing is an important resource which has many potential applications not only in quantum metrol- ogy and atom interferometers [1–5], but also in many aspects of quantum information due to its close rela- tion with quantum entanglement [6–10]. In conventional experiments, squeezing is usually achieved via the non- linearity induced by the inter-particle interaction [3–5]. As an example, spin squeezing has been obtained ex- perimentally in a Bose-Einstein condensate (BEC) of a three-component Bose gas [11]. However, the intensity of spin squeezing in these experiments crucially depends on the interaction between atoms. In cold atom exper- iments, the background interaction is usually very weak such that the observation of squeezing is relatively hard. Although there are some techniques to enhance the in- teraction, e.g., by tuning the state-dependent microwave potentials [4], or through a magnetic Feshbach resonance in alkali atoms [5, 12], the side eﬀects of decoherence, severe atom loss and dynamical instability induced by strong interaction still hinder the achievement of strong spin squeezing. The experimental realization of synthetic spin-orbit coupling (SOC) in ultracold atomic gases [13–15] has at- tracted much attention, partly due to its close relation wzhangl@ruc.edu.cn yqiongyihe@pku.edu.cn zcpsun@csrc.ac.cnto exotic many-particle states and novel excitations [16– 18]. Recently, theoreticalstudieshaveproposedtorealize spin squeezing in two-component BEC by synthetic spin- orbit coupling (SOC) [19, 20]. It has been shown that the presence of SOC will induce an eﬀective spin-spin inter- action which can lead to spin squeezing. However, there are two disadvantages of these proposals. First, the syn- thetic SOC requires a Raman transition between two hy- perﬁne states. The Rabi frequency of this Raman transi- tion is detrimental to spin squeezing, i.e., a stronger SOC leads to a weaker squeezing. Besides, the two-photon de- tuning of this Raman transition is also unfavorable such that best squeezing will be achieved when the detuning is zero. Nonetheless, in realistic experiments one would en- counter severe heating eﬀect when the detuning is tuned on resonance. In this paper, we study spin squeezing in a three- component Bose gases where all three hyperﬁne states are coupled by spin-orbit coupling induced by Raman transitions. As a result, this system has pseudo-spin-1, andthespinoperatorshereinmustbedescribedbySU(3) spin matrices, i.e., the Gell-Mann matrices. These Gell- Mann matrices span an eight-dimensional spin hyper- space, with three of them are usually refereed as spin vectors, and the other ﬁve as nematic tensors [21]. The squeezed spin operators hence can be categorized into three types, including the spin-spin squeezing, nematic- nematic squeezing, and spin-nematic squeezing. Here, we focus on the spin-nematic squeezing, as it is a novel typeofsqueezingwhichexistsexclusivelyinsystemswith large spins. We ﬁnd that the presence of SOC can induce spin-nematic squeezing, which can be further enhancedarXiv:1609.03141v1 [quant-ph] 11 Sep 20162 byincreasingtheSOCintensityorreducingthequadratic Zeeman splitting. These trends of dependence can be understood from an eﬀective Hamiltonian, in which the Rabi frequency and quadratic Zeeman splitting corre- spond to eﬀective Zeeman ﬁelds in the spin and nematic sectors, respectively, hence causing opposite eﬀects on various types of spin squeezing. More importantly, we ﬁnd that the squeezing is favored by two-photon detun- ing of the Raman transition within a fairly large param- eter regime, which is beneﬁcial for experimental realiza- tions to avoid severe heating eﬀect. When the system ex- hibits spin-nematic squeezing in the low excitation limit, we also ﬁnd two-mode entanglement [22] and two-spin squeezing [23]inthesystem. Wefurtherstudytheeﬀects of an external trapping potential and inter-atomic inter- action which are present in realistic experimental situa- tionsbynumericallyanalysis, andconcludethatthespin- nematic squeezing is favored by stronger trapping poten- tials. Finally, we discuss possible detection scheme via a spin-selective imaging technique and a radio-frequency (RF) rotation of the spin axes [11]. The remainder of this paper is organized as follows. In Sec. II, we introduce the system under investigation and discuss the single-particle spectra. We then derive an ef- fectivespinHamiltonianfromwhichitcanbeseenclearly that SOC induces an eﬀective spin-spin interaction. We then analyze the spin-nematic squeezing and its depen- dence of various factors in Sec. III. Finally, we discuss possible experimental detection scheme and summarize in Sec. IV. II. SINGLE-PARTICLE SPECTRA AND EFFECTIVE HAMILTONIAN Spin-orbit coupled three-component Bose gas can be generalized by counter-propagating Raman lasers along ^xto couple the three hyperﬁne states with momentum transfer of the Raman process 2kr. The non-interacting Hamiltonian can be written in the matrix form as [24] H=0 BBBBB@(kx+2kr)2 2 R=2 0 R=2k2 x 2 R=2 0 R=2(kx2kr)2 2+1 CCCCCA+k2 ? 2; (1) wherek?=q k2y+k2zis the transverse momentum, is the two-photon detuning from the Raman resonance, is the quadratic Zeeman shift induced by the magnetic ﬁeld along ^y, and RrepresentstheRabifrequencyoftheRa- man transition. Notice that throughout the manuscript, we use the natural units of }=m= 1, and deﬁne krand the recoil energy Er=k2 r=2as the units of momentum and energy, respectively. The single-particle dispersion can be obtained by di- agonalizing th non-interacting Hamiltonian of Eq. (1). 1minimum 1minimum2minima 2minima3minimaæcritical point ÑdEr=1HaL 0 1 2 3 4 5 6-6-4-20246 ÑWRErÑeEr 1minimum1minimum 1minimum2minima 2minimaæ æcritical point critical point ÑeEr=6HbL 0 2 4 6 8-10-50510 ÑWRErÑdEr HcL ÑWREr=76543210 -4-2024-4-3-2-1012 kxkrEEr ÑeEr=6543210-1-2-3HdL -4-2024-6-4-202 kxkrEErFigure 1. (Color online) (a-b) Single-particle phase diagrams of a three-component Bose gas with one-dimensional SOC in the (a) R–plane with = 1and (b) R–plane with = 6. The lowest branch of the single-particle dispersion spectrum acquires either one, two, or three local minima in diﬀerent parameter regimes separated by solid lines. On the dashed lines within regions of multiple minima, two of the local minima are degenerate. Typical examples for the lowest branch of dispersion curves by changing (c) Rabi frequency Rwith= 1and= 0and (d) quadratic Zeeman energy with= 1and R= 2. The resulting spectra has three branches, among which thelowestonecanhavethreeminima,twominima,orone minimum depending on the combination of parameters. In Figs. 1(a) and 1(b), we show the parameter regions ex- hibitingdiﬀerentstructuresforthecaseof ~=Er= 1and ~=Er= 6respectively. From Fig. 1(a), we can identify various regions where the lowest branch of single-particle dispersion acquires 1, 2, or 3 minima. Speciﬁcally, for the case of a large positive quadratic Zeeman splitting , thej0istate is far detuned from the other two high- lying hyperﬁne states, so that the spectrum has only one minimum. On the other hand, if is large negative, the j0istate becomes the high-lying state and the system essentially turn into a spin-1/2 Bose gas where the two j1ispin components are spin-orbit coupled via virtual processes involving the j0istate. As a result, the single- particle dispersion can have either two or one minima, depending on the SOC intensity Rand two-photon de- tuning. For the case of intermediate jj, all three hyper- ﬁne states are spin-orbit coupled and the shape of spec- trum is sensitively dependent on all parameters. Typical examples of dispersion spectra along the kxaxis showing one minimum, two minima, and three minima, as well as the trends of evolution depending on Randare illustrated in Figs. 1(c) and 1(d), respectively. As the spin operators in spin-1/2 systems all belong to the SU(2) group, those in spin-1 systems discussed3 here are elements in the SU(3) group. The SU(3) group is locally isomorphic to the O(8) group, which has eight linearly independent observables as generators. These generatorscanbegroupedintotwotypes, includingthree spin vectors (or angular momentum operators) and ﬁve nematic tensors. The irreducible matrix representations of these observables are given by [21] Jx=1p 20 @0 1 0 1 0 1 0 1 01 A,Jy=ip 20 @01 0 1 01 0 1 01 A, Jz=0 @1 0 0 0 0 0 0 011 A,Qxy=i0 @0 01 0 0 0 1 0 01 A, Qyz=ip 20 @01 0 1 0 1 01 01 A,Qzx=1p 20 @0 1 0 1 01 01 01 A, D=0 @0 0 1 0 0 0 1 0 01 A,Y=1p 30 @1 0 0 02 0 0 0 11 A: The commutators between these spin operators can then be classiﬁed into three categories: [Jy;Jz] =iJxas spin-spin group, [Qxy;Qxz] =iJx,[Qyz;D] =iJx, and [Qyz;Y] =p 3iJxas nematic-nematic group, and [Jx;Qyz] =i(p 3Y+D),[Jy;Qzx] =i(p 3Y+D)as spin-nematic group. To study the eﬀective spin-spin interaction induced by SOC,aswellastheinducedspinsqueezingeﬀect, nextwe derive an eﬀective spin model. To facilitate the deriva- tion, we impose a weak harmonic trap V(x) =!2 xx2=2 + !2 yy2=2 +!2 zz2=2. We will ﬁnd that the resulting form of the eﬀective model does not depend on the absolute value of trapping frequency, hence incorporate solely the eﬀect of SOC. In the presence of such an auxiliary trap- ping potential, we can quantize the motional degrees of freedom along the trapping direction to a discrete energy spectrum. In particular, by introducing the bosonic op- eratorsap !x=2(x+ikx=!x),bp !y=2(y+iky=!y), cp !z=2(z+ikz=!z)and the collective spin operators Fs=x;y;z =PN i=1Ji;s,FY=PN i=1Yi, the Hamiltonian of Eq. (1) for N-particle can be rewritten as ~H=!xNaya+N4k2 r 3+ Rp 2Fx +ikrp 2!x(aya)FzFz+2k2 r+p 3FY:(2) Here we ignore !yNbyb+!zNcycsince the boson modes iny,zdirection do not interact with the ultracold atoms. Employing the unitary transformation U= exp[iG(ay+a)Fz]withG=p 2=!xkr=N, the Hamiltonian thus can be transformed as ~H0=!xNayaqF2 zFz+2k2 r+p 3FY + Rp 2fFxcos[G(ay+a)]Fysin[G(ay+a)]g;(3) whereq= 4k2 r=N= 8Er=N. Notice that the term of N(4k2 r)=3has been dropped out as the zero-point energy. For a BEC, the expectation value of hayaiis in the order ofNfor the ground state, and about unity for excited states. Considering the prefactor of 1=Nin the deﬁnition of G, the leading order of the arguments in the cosine and sine functions in Eq. (3) are 1=p N, which is negligible for systems of large particle number. As a result, we can approximate the cosine and sine functions tothezerothorder,andtheHamiltonianEq. (3)becomes separable in spatial and spin degrees of freedom, leading to an eﬀective spin Hamiltonian He=qF2 z+ Rp 2FxFz+4Er+p 3FY:(4) One can see clearly that an eﬀective spin-spin interac- tion emerges as a result of SOC, and the Rabi frequency R, two-photon detuning , and the quadratic Zeeman splittingact as eﬀective Zeeman ﬁelds along diﬀerent directions in the eight-dimensional spin hyperspace. III. SPIN-NEMATIC SQUEEZING With the aid of the eﬀective spin model of Eq. (4), we can study the spin squeezing in the underlying system. As the commutators relation between spin and nematic operators are not present in the spin-1/2 case, next we focus on spin squeezing of this type. The method can be straightforwardly applied to the spin-spin and nematic- nematic commutators, and the results are qualitatively consistent with the ﬁndings for the spin-spin case in spin- 1/2 system with SOC [19, 20]. The spin model of Eq. (4) can not be solved analyt- ically due to the presence of nonlinear interaction. In the low excitation limit, however, we can introduce the Holstein-Primakoﬀ transformation for spin-1 systems Fx1p 2 by 1N0 0+N0 0b1+ h:c: ; Fy1p 2i by 1N0 0+N0 0b1h:c: ; (5) whereN0 0q Nby 1b1by 1b1, and the operators b1 andb1representing spin ﬂipping processes between the internal levelsj1iandj0i, represented by the bosonic modesa1anda0. For the case that most of the par- ticles remain in the mode a0, i.e.,hay 0a0i 'Nand4 hby 1b1iN, the operators b1=a1ay 0=p Nare eﬀec- tive bosonic modes satisfying the bosonic commutation relationsh b;by i =with; =1. Within the assumption that the majority of the particles are resid- ing in thej0istate, or equivalently the excitations to thej1istates are rare, we can rewrite the bosonic operators as a mean-ﬁeld value plus some ﬂuctuations b1=p N1+b1. The ground state energy can the be obtained by minimizing the energy functional E(1;1). As in this the low excitation limit, nearly all the spins are polarized in FYdirection, which means p 3FY+FD2N, the squeezing parameter is then given by [25] xmin 42Jn? J=242Fx=N; (6) Here,Jis the expectation value of mean spin, Jn?is a spin component along the direction perpendicular to the mean spin direction. So in our case, it is clear that x can be obtained by calculated the variance of Fx, one has spin squeezing in the spin-nematic channel as x<1. We ﬁrst discuss the case of zero two-photon detuning = 0, and show in Fig. 2 the spin-nematic squeezing pa- rameter as functions of Rabi frequency Rand quadratic Zeeman splitting . One can see clearly that the ground stateis aspinsqueezed stateundertheeﬀect ofSOC.Im- portantly, as shown in Fig. 2(a), spin-nematic squeezing can be enhanced with increasing R. This behavior is in stark contrast to the case of spin-1/2 systems, where the spin-spin squeezing is favored by decreasing R[19, 20]. We then extend the discussion to the more general case of a nonzero two-photon detuning 6= 0. This scenario is experimentally relevant because a severe heating eﬀect is usually present as the Raman transition is on-resonance. As shown in Fig. 3(a), a ﬁnite favors spin-nematic squeezingwithinafairlylargeregionof j~=Erj<5. This result can be understood by analyzing the single-particle Hamiltonian of Eq. 1, where andare energy oﬀsets of the diagonal elements. As Raman transitions will be enhanced when diﬀerence states are near resonance, spin squeezing will be favored when the absolute value of is close to. To further clarify this argument, we ana- lyze the atom populations of diﬀerent ground states with changing. As shown in Fig. 3(b), the presence of a ﬁ- nitewill enhance the transition between the j0istate and one of thej1istates, while the transition to the otherj1istate is reduced. Notice that this behavior is very diﬀerent from the spin-1/2 case, where the two spin components are moved away from each other with increasing, leading to an eﬀectively weaker SOC. The dependences of spin-nematic squeezing on the var- ious parameters of R,andcan also be interpreted from the eﬀective spin model of Eq. (4), within which the three parameters correspond to eﬀective Zeeman ﬁelds along theFx,FY, andFydirections, respectively. Con- sidering that in the low excitation limit nearly all spins are polarized along the FYdirection, a stronger Zeeman HaL 012340.70.80.91.0 ÑWRErxx HbL 2468100.70.80.91.0 ÑeErxxFigure 2. (Color online) Spin-nematic squeezing parameter x as a function of (a) Rabi frequency Rwith= 0and= 6 and (b) quadratic Zeeman splitting with= 0and R= 2. In both ﬁgures, results obtained from the eﬀective spin model Eq.(4) are illustrated by blue solid lines, in comparison to the numerical solutions of the GP equation for a pancake-shaped trapwith!x=!y= 50Hz,!z= 1500Hz(blackdashed), and for a cigar-shaped trap with !x=!y= 5000Hz,!z= 1500 Hz (red dotted). Here, we consider a gas of87Rb atoms in theF= 1manifold with background interaction and total particle number N= 105. ﬁeld along the same direction, i.e., a larger value of , will further intensify the polarization so that the eﬀec- tive spin-spin interaction becomes relatively weak, lead- ing to a less spin-nematic squeezing eﬀect. On the other hand, eﬀective Zeeman ﬁelds along the perpendicular di- rections, either FxorFz, will tilt the spin polarization from theFYaxis slightly but the eﬀect spin-spin interac- tion is enhanced obviously, resulting an increased squeez- ing parameter as in Eq. (6). In addition to the spin-nematic squeezing, we notice that in the low excitation limit with the majority of par- ticles residing in the j0istate, the two eﬀective bosonic modesb1andb1can be entangled, which is referred as two-mode entanglement. A suﬃcient criterion for en- tanglement between the modes b1andb1from the spin squeezing parameters is then given by [22] DCZ= ( +++=2 )=2<1; (7) where h2F i=Nrepresentsthevarianceofquadra- ture phase amplitudes which depends on the parameter5 HaL -5.0-2.50.02.55.00.750.800.850.900.951.00 ÑdErxx HbL -10-505100.000.200.400.600.801.00 ÑdErr Figure 3. (Color online)(a) Variations of spin-nematic squeez- ing parameter as a function of two-photon detuning with R= 2and= 6. Analytic result obtained from the eﬀective spin model Eq. (4) within low-density excitation approxima- tion (blue solid) is compared with numerical solutions of the GP equation for a pancake-shaped trap with !x=!y= 50 Hz,!z= 1500Hz (black dashed), and for a cigar-shaped trap with!x=!y= 5000Hz,!z= 1500Hz (red dotted). (b) Atom number fractions of the j1i(black dotted),j0i(blue dashed), andj+ 1i(red solid) states. , andwehaveusethedeﬁnitions F += cosFx+sinFyz andF = cosFzx+ sinFyin this system. Here, the collective nematic operators are Fyz=PN i=1Qyzand Fzx=PN i=1Qzx. Figure 4(a) shows that DCZreaches its minimum for =nwithnan integer, and the en- tanglement is enhanced by Raman transition. Another representation of two-mode entanglement is called two-spin squeezing, which is deﬁned by dividing the spin-1 space into three subspaces pseudospins (each of spin-1/2) U,VandTassociated the three relative number diﬀerences of particles N+1N0,N1N0, and N+1N1in three-component labeled by f+1;1;0g [23]. Two-spin squeezing parameter is given to describe the correlation between the spin subspace U(the spin ﬂipping process between internal levels j+ 1iandj0i) andV(the spin ﬂipping process between internal levels j1iandj0i) [23] UV=2F ++ 2F+=2 p 3jhFYij<1; (8) In Fig. 4(b), one can see clearly that the optimal correla- 0p 2p3p 22p0.920.961.00 qxDCZqHaL 0 1 2 3 40.700.800.901.00 ÑWRErxDCZ0 0p 2p3p 22p0.961.001.04 qxUVqHbL 0 1 2 3 40.850.900.951.00 ÑWRErxuv0Figure 4. (Color online) (a) Two-mode entanglement param- eter0 DCZand (b) two-spin squeezing parameter 0 UVver- sus Rabi frequency Rwith other parameters being = 0 and= 6. Analytic result obtained from the eﬀective spin model Eq. (4) within low-density excitation approximation (blue solid) is compared with numerical solutions of the GP equation for a pancake-shaped trap with !x=!y= 50Hz, !z= 1500Hz (black dashed), and for a cigar-shaped trap with!x=!y= 5000Hz,!z= 1500Hz (red dotted). The insets show the squeezing parameters as functions of . No- tice that the optimal squeezing in both criteria are obtained when=nwithnan integer. tion is obtained when =nwithnan integer, and in- creases with the the Raman transition. When comparing spin-nematic squeezing parameter (Fig. 2(a)) with these two criterions (Fig. 4), we ﬁnd that the eﬀect of squeez- ing in spin-nematic channel is another representation of the correlation between two spin subspaces and entangle- ment between two eﬀective modes in the low excitation limit. Finally, we notice that in realistic experiments, one also needs to take the eﬀects of inter-atomic interaction and a global harmonic trap into consideration. Taking 87Rb as a particular example, the interaction among the three hyperﬁne states of the ground state manifold can be categorized into two groups, depending on the total angularmomentumofthetwocollidingatoms. Theback- ground scattering lengths are taken as as0= 101:8a0for F= 0, andas2= 100:4a0forF= 2, wherea0denotes the Bohr radius [26]. For the eﬀects of trapping potentials, we consider two types of global harmonic traps includ- ing a pancake-shaped quasi-two-dimensional trap with6 !x=!y= 50Hz and!z= 1500Hz, and a cigar-shaped three-dimensional trap with !x=!y= 5000 Hz and !z= 1500Hz. BynumericallysolvingtheGross-Pitaevski(GP)equa- tion for a total number of N= 105atoms, we obtain the ground state of the system, and calculate the spin- nematic squeezing parameter x, the two-mode entangle- ment parameter DCA, and the two-spin squeezing pa- rameter UV. The corresponding results are shown in Figs.2, 3and4. Bycomparingthenumericalresultswith the outcome from the eﬀective spin model, we conclude that the eﬀective model Eq. (4) is qualitatively valid in thelowexcitationlimit. Ontheotherhand, astronghar- monic trap can cause sizable increment on spin-nematic squeezing and two-mode entanglement. This observation can be understood by noticing that in the presence of a strong harmonic trap, the particles will be more con- densed with a higher number density at the trap center. As a result, the inter-particle interaction has stronger eﬀect and causes better spin-nematic squeezing and two- mode entanglement. IV. EXPERIMENTAL DETECTION AND CONCLUSION We have shown that an eﬀective spin-spin interac- tion can be induced in spin-orbit coupled spin-1 BEC, whichcanproduceaspecialkindofsqueezingcalledspin- nematicsqueezing. Thistypeofspinsqueezingcanbeen- hanced by increasing Raman transition intensity and de-creasing quadratic Zeeman splitting. More importantly, the squeezing is favored by a ﬁnite two-photon detuning in a fairly large parameter regime, which could be bene- ﬁcial for experiments to reduce heating eﬀect. These be- haviors are in clear contrast to the spin squeezing within spin-orbit coupled spin-1/2 systems, where the trends of dependence on Raman transition intensity and two- photon detuning are opposite. We also observe SOC induced two-mode entanglement and two-spin squeezing in such a system, and investigate their dependence on Raman transition intensity. We further analyze the ef- fects of inter-particle interaction and external harmonic trap by numerically solving the GP equation, and ﬁnd good agreement with approximate solutions of the eﬀec- tive spin model. In order to detect such an exotic type of spin squeezing in this system, one may need to rotate Jxinto the easily measuredJzdirection by applying a =2radio-frequency (RF) rotation about the Jyaxis. This operation can be accomplished with a two-turn coil on the experimental y-axis driven at the frequency splitting of the mFstates. Then, we can measure the variance of spin via a spin- selective imaging technique. ACKNOWLEDGMENTS This work is supported by NSFC (11274009, 11622428, 11274025,11434011,11522436,61475006,and61675007), NKBRP (2013CB922000) and the Research Funds of Renmin University of China (10XNL016, 16XNLQ03). [1] D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heinzen, Phys. Rev. A 50, 67 (1994). [2] M.KitagawaandM.Ueda, Phys.Rev.A 47, 5138(1993). [3] J. Estève, C. Gross, A. Weller, S. Giovanazzi, and M. K. Oberthaler, Nature 455, 1216 (2008). [4] M. F. Riedel, P. Böhi, Y. Li, T. W. Hänsch, A. Sinatra, and P. Treutlein, Nature 464, 1170 (2010). [5] C. Gross, T. Zibold, E. Nicklas, J. Estève, and M. K. Oberthaler, Nature 464, 1165 (2010). [6] H.F.HofmannandS.Takeuchi, Phys.Rev.A 68, 032103 (2003). [7] G. Tóth, C. Knapp, O. Gühne, and H. J. Briegel, Phys. Rev. A 79, 042334 (2009). [8] E. G. Cavalcanti, P. D. Drummond, H. A. Bachor, and M. D. Reid, Opt. Express 17, 18693 (2009). [9] M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cav- alcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, Rev. Mod. Phys. 81, 1727 (2009). [10] E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, Phys. Rev. A 80, 032112 (2009). [11] C. D. Hamley, C. S. Gerving, T. M. Hoang, E. M. Book- jans, and M. S. Chapman, Nat. Phys. 8, 305 (2012). [12] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Rev. Mod. Phys. 82, 1225 (2010).[13] Y. J. Lin, K. Jiménez-García, and I. B. Spielman, Nature 471, 83 (2011). [14] P. Wang, Z. Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H. Zhai, and J. Zhang, Phys. Rev. Lett. 109, 095301 (2012). [15] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah, W. S. Bakr, and M. W. Zwierlein, Phys. Rev. Lett. 109, 095302 (2012). [16] H. Zhai, Rep. Prog. Phys. 78, 026001 (2015). [17] W. Yi, W. Zhang, and X. Cui, Sci. China: Phys. Mech. Astron. 58, 014201 (2015). [18] J. Zhang, H. Hu, X. J. Liu, and H. Pu, Annu. Rev. Cold At. Mol. 2, 81 (2014). [19] J. Lian, L. Yu, J.-Q. Liang, G. Chen, and S. Jia, Sci. Rep.3, 3166 (2013). [20] Y. Huang and Z.-D. Hu, Sci. Rep. 5, 8006 (2015). [21] E. Yukawa, M. Ueda, and K. Nemoto, Phys. Rev. A 88, 033629 (2013). [22] L. M. Duan, J. I. Cirac, and P. Zoller, Phys. Rev. A 65, 033619 (2002). [23] Özgür E. Müstecaphoˇ glu, M. Zhang, and L. You, Phys. Rev. A 66, 033611 (2002). [24] Z. Lan and P. Ohberg, Phys. Rev. A 89, 023630 (2014). [25] J. Ma, X. Wang, C. P. Sun, and F. Nori, Phys. Rep. 509, 89 (2011). [26] W. Zhang, D. L. Zhou, M. S. Chang, M. S. Chapman, and L. You, Phys. Rev. A 72, 013602 (2005).
2201.11823v2.Superfluid_transition_temperature_and_fluctuation_theory_of_spin_orbit_and_Rabi_coupled_fermions_with_tunable_interactions.pdf	Super uid transition temperature and uctuation theory of spin-orbit and Rabi coupled fermions with tunable interactions Philip D. Powell,1, 2Gordon Baym,2and C. A. R. S a de Melo3 1Lawrence Livermore National Laboratory, 7000 East Avenue, Livermore, California 94550, USA 2Department of Physics, University of Illinois at Urbana-Champaign, 1110 W. Green Street, Urbana, Illinois 61801, USA 3School of Physics, Georgia Institute of Technology, 837 State Street, Atlanta, Georgia 30332, USA (Dated: June 6, 2022) We obtain the super uid transition temperature of equal Rashba-Dresselhaus spin-orbit- and Rabi-coupled Fermi super uids, from the Bardeen-Cooper-Schrieer (BCS) to Bose-Einstein con- densate (BEC) regimes in three dimensions for tunable s-wave interactions. In the presence of Rabi coupling, we nd that spin-orbit coupling enhances (reduces) the critical temperature in the BEC (BCS) limit. For xed interactions, we show that spin-orbit coupling can convert a rst-order (dis- continuous) phase transition into a second-order (continuous) phase transition, as a function of Rabi coupling. We derive the Ginzburg-Landau free energy to sixth power in the super uid order pa- rameter to describe both continuous and discontinuous phase transitions as a function of spin-orbit and Rabi couplings. Lastly, we develop a time-dependent Ginzburg-Landau uctuation theory for an arbitrary mixture of Rashba and Dresselhaus spin-orbit couplings at any interaction strength. I. INTRODUCTION The ability to simulate magnetic and other external elds [1{12] in cold atomic gases has created the oppor- tunity to explore a wide variety of new interactions and complex phase structures otherwise inaccessible in the laboratory. Moreover, the capacity to generate these syn- thetic elds in both bosonic and fermionic systems, and to continuously tune two-body interactions by means of a Feshbach resonance, has opened up a wonderland of tun- able systems, previously restricted to theorists' dreams. For example, the possibility of simulating quantum chro- modynamics (QCD) on an optical lattice [13{16] is a tan- talizing prospect for researchers whose current theoreti- cal tools remain limited by QCD's non-perturbative char- acter and the restriction of lattice techniques to near-zero chemical potential. Previous theoretical analyses of three-dimensional spin-orbit-coupled Fermi gases (e.g.,6Li,40K) have fo- cused mainly on the zero-temperature limit, in which sev- eral exotic phases characterized by unconventional pair- ing are expected to emerge [17{22]. However, the Ra- man laser platforms currently employed to produce syn- thetic spin-orbit elds also induce heating that prevents the realization of temperatures suciently low to observe the super uid transition in either the weakly coupled Bardeen-Cooper-Schrieer (BCS) or the strongly coupled Bose-Einstein condensate (BEC) regimes [5, 7]. Thus, while two-body bound states (Feshbach molecules) have been observed in the BEC limit of40K [7, 23], the obser- vation of super uid states remain elusive. Future exper- iments, however, may break this impasse by employing a new platform currently under development\|the radio- frequency atom chip\|which avoids heating of the atom cloud entirely [24]. While rf atom chips are somewhat more restricted than the Raman scheme in the maximum obtainable spin-orbit coupling, its potential to reach su-per uid temperatures is leading to its adoption in the next generation of experiments probing the topological super uid phases of spin-orbit-coupled fermions [25]. One class of systems of particular interest in the con- text of quantum simulation is that of Rashba-Dresselhaus spin-orbit-coupled gases [17{22, 26, 27]. These systems are intriguing both because they re ect physics studied extensively in the context of semiconductors [28, 29], and because they provide a platform for realizing tunable non-Abelian elds in the laboratory. Thus, while the holy grail of a full optical simulation of QCD remains years in the future, there do exist notable analogies be- tween quark matter and cold atomic systems (e.g., non- Abelian elds, evolution between strongly and weakly coupled limits) within near-term experimental reach [30{ 33]. Investigations of spin-orbit-coupled ultracold gases have also included optical lattices [34{43], thus enlarg- ing the number of possible physical systems that can be accessible experimentally. To date, most experimental realizations of these systems have adopted equal Rashba-Dresselhaus cou- plings [4{6, 44], but systems exhibiting Rashba-only cou- plings have also been created [11, 45, 47]. Other ex- periments have generated spin-orbit coupling dynami- cally [48] or even created three-dimensional spin-orbit coupling [49]. Due to the versatility of Rashba- Dresselhaus coupled systems, the ability to realize these systems in the laboratory, and the myriad technical challenges inherent in reaching arbitrarily low temper- atures, it is increasingly important to provide a theoret- ical framework for guiding and testing these simulators against experimental probes at realistic (nonzero) tem- peratures. This problem bears a close relation to spin-orbit cou- pling in solids, where the role of the Rabi frequency is played by an external Zeeman magnetic eld. While a mean-eld treatment describes well the evolution from the BCS to the BEC regime at zero temperature [50, 51],arXiv:2201.11823v2 [cond-mat.quant-gas] 2 Jun 20222 this order of approximation fails to describe the correct critical temperature of the system in the BEC regime because the physics of two-body bound states, i.e., Fes- hbach molecules, is not captured when the pairing order parameter goes to zero [52]. To remedy this problem, we include the eects of order-parameter uctuations in the thermodynamic potential. In this paper, we investigate the impact of a specic class of spin-orbit coupling, namely, an equal mixture of Rasha and Dresselhaus terms, on the super uid tran- sition temperature of a three-dimensional Rabi-coupled Fermi gas, but also give general results for an arbitrary mixture of Rashba and Dresselhaus components. This paper is the longer version of our preliminary work [53]. We stress that the present results are applicable to both neutral cold atomic and charged condensed-matter sys- tems. We show that spin-orbit coupling, in the presence of a Rabi eld (or Zeeman eld, in solids), enhances the critical temperature of the super uid in the BEC regime and converts a discontinuous rst-order phase transition into a continuous second-order transition, as a function of the Rabi frequency for given two-body interactions. We analyze the nature of the phase transition in terms of the Ginzburg-Landau free energy, calculating it to the sixth power of the super uid order parameter, as required to describe both discontinuous transitions as a function of the spin-orbit coupling, Rabi frequency, and two-body interactions. This paper is organized as follows. In Sec. II, we de- scribe the Hamiltonian and action for three-dimensional Fermi gases in the presence of a general Rashba- Dresselhaus spin-orbit coupling, Rabi eld, and tunable s-wave interactions. We also obtain the inverse Green operator that is used in the calculation of the thermo- dynamic potential and Ginzburg-Landau theory of sub- sequent sections. In Sec. III, we analyze the thermo- dynamic potential across the entire BCS-to-BEC evolu- tion, including contributions from both the mean-eld and Gaussian uctuations, and obtain the order parame- ter and number equations. In Sec. IV, we study the com- bined eects of Rabi elds and spin-orbit coupling on the super uid critical temperature, constructing the nite- temperature phase diagram versus Rabi elds and scat- tering parameter. In Sec. V, we present the Ginzburg- Landau (GL) theory for the super uid order parameter and investigate further corrections to the critical tem- perature in the BEC limit by including interactions be- tween bosonic bound states. The GL action is obtained to sixth order in the order parameter to allow for the exis- tence of discontinuous (rst-order) phase transitions. In Sec. VI, we compare our work on the experimentally rel- evant equal Rashba-Dresselhaus spin-orbit coupling with earlier work that has considered dierent forms of the- oretically motivated spin-orbit couplings. In Sec. VII, we conclude and look toward the future of experimental work in this eld. In the interest of readability, we relegate a number of detailed calculations to appendices. In Appendix A, wediscuss the Hamiltonian and eective Lagrangian for a general Rashba-Dresselhaus spin-orbit coupling. In Ap- pendix B, we analyze the saddle-point approximation for general Rashba-Dresselhaus spin-orbit coupling. In Ap- pendix C, we derive the modied number equation, in- cluding the contribution arising from Gaussian uctua- tions, which renormalizes the chemical potential obtained at the saddle-point level. In Appendix D, using a gen- eral Rashba-Dresselhaus spin-orbit coupling, we obtain expressions for the coecients of the Ginzburg-Landau theory up to sixth order in order parameter. II. HAMILTONIAN AND ACTION Throughout this paper, we adopt units in which ~= kB= 1. The Hamiltonian density of a three-dimensional Fermi gas in the presence of Rashba-Dresselhaus spin- orbit coupling and Rabi eld is H(r) =Hk(r) +Hso(r) +HI(r)n(r): (1) The rst term in Eq. (1) is the kinetic energy, Hk(r) =X s y s(r)^k2 2m s(r); (2) where ^k=iris the momentum operator, s(r) is the fermion eld at position rwith (real or pseudo-) spin s and massm. The second term is the spin-orbit interac- tion, Hso(r) =X ss0 y s(r)h Hso(^k)i ss0 s0(r); (3) with the spin-orbit coupling matrix in momentum ( k) space being Hso(^k) = m(^kxx+^kyy) R 2z; (4) where (x;y;z) are the Pauli matrices in spin space, is the momentum transfer to the atoms in a two-photon Raman process [7] or on a radio frequency atom chip [24], is the anisotropy of the Rashba-Dresselhaus eld, and Ris the Rabi frequency. The third term is the two-body s-wave contact interaction, HI(r) =g y "(r) y #(r) #(r) "(r); (5) whereg>0 corresponds to a constant attraction between opposite spins. Finally, is the chemical potential and n(r) =P s y s(r) s(r) is the local density. While the gen- eral Rashba-Dresselhaus spin-orbit coupling is discussed in Appendix A, in what follows we focus on the more experimentally relevant situation of equal Rashba and Dresselhaus couplings ( = 0). Standard manipulations (see Appendix A) lead to the Lagrangian density, L(r;) =1 2 y(r;)G1(^k;) (r;) +1 gj(r;)j2 +K(^k)(rr0); (6)3 where=itis the imaginary time, = ( " # y " y #)T is the Nambu spinor, K(^k) =^k2=2mis the kinetic energy operator with respect to the chemical potential, and ( r;) =gh #(r;) "(r;)iis the pairing eld describing the formation of pairs of two fermions with opposite spins. Note that includes the overall positive shift2=2min the single-particle kinetic energies due to spin-orbit coupling. The inverse Green's operator ap- pearing in Eq. (6) is G1(^k;) =0 BB@@K"i^kx=m 0 i^kx=m @K# 0 0 @+K"i^kx=m 0i^kx=m @+K#1 CCA; (7) whereK";#=K(^k) R=2;are the kinetic energy terms shifted by the Rabi coupling. As noted above, a mean-eld treatment of this La- grangian fails to correctly describe the super uid critical temperature in the BEC regime. However, the inclu- sion of Gaussian uctuations of captures the eects of two-body bound states and leads to a physical super uid transition temperature. It is to this task that we now turn. III. THERMODYNAMIC POTENTIAL The system's partition function may be expressed in terms of the functional integral, Z=Z DDD D yeS; (8) where the Euclidean action is S=Z 0dZ d3rL(r;); (9) = 1=Tis the inverse temperature, and the Lagrangian density is given by Eq. (6). Integrating over the fermion elds yields the thermodynamic potential, =TlnZ= 0+ F; (10) where 0=TlnZ0=TS0is the mean-eld (saddle- point) contribution, for which ( r;) = 0, and F=TlnZFis the contribution arising from order- parameter uctuations. Detailed derivations of the ther- modynamic potential for a general Rashba-Dresselhaus spin-orbit coupling, as well as the associated order pa- rameter and number equations, are given in Appen- dices B and C. The contributions to the thermody- namic potential for the experimentally relevant situa- tion of equal Rashba-Dresselhaus spin-orbit coupling are discussed below in Sec. III A at the mean-eld and in Sec. III B at the Gaussian uctuation level.A. Mean-Field Approximation The mean-eld, or saddle-point, term in the thermo- dynamic potential is 0=Vj0j2 gT 2X k;jlnh 1 +eEj(k)i +X kk;(11) wherek="k,"k=k2=2m, and theEj(k), with j=f1;2;3;4g, are the eigenvalues of the momentum space Nambu Hamiltonian matrix, H0(k) =@G1(k;)j= 0; (12) where the operator @=I@, andIis the identity matrix. The rst set of eigenvalues, E1;2(k) =2 42 k2s E2 0;kh2 kkx m2 j0j23 51=2 (13) describe quasiparticle excitations, with the plus (+) as- sociated with E1and the minus ( ) withE2. The second set of eigenvalues, E3;4(k) =E2;1(k);corre- sponds to quasiholes. Further, 2 k=E2 0;k+h2 k, where E0;k=p 2 k+j0j2;andhk=p (kx=m)2+ 2 R=4 is the magnitude of the combined spin-orbit and Rabi cou- plings. We express the two-body interaction parameter gin terms of the renormalized s-wave scattering length as via the relation [52] 1 g=m 4as+1 VX k1 2"k: (14) Note thatasis thes-wave scattering length in the absence of spin-orbit and Rabi elds. It is, of course, possible to expressg, and all subsequent relations, in terms of a scat- tering length which is renormalized by the presence of the spin-orbit and Rabi elds [54, 55], but for both simplic- ity and the sake of referring to the more experimentally accessible quantity, we do not do so here. The order-parameter equation is obtained from the saddle-point condition 0= 0jT;V; = 0, leading to m 4as=1 2VX k1 "kA+(k) 2 R 4khkA(k) ;(15) where we introduced the notation A(k) =12n1(k) 2E1(k)12n2(k) 2E2(k); (16) withnj(k) = 1= eEj(k)+ 1 being the Fermi function. In addition, the particle number at the saddle point N0= @ 0=@jT;V;is given by N0=X k 1k A+(k) +(kx=m)2 khkA(k) :(17)4 The mean-eld temperature T0is determined by solv- ing Eq. (15) for the given . The corresponding number of particles is given by Eq. (17). This mean-eld treat- ment leads to a transition temperature e1=kFas, where kFis the Fermi momentum. This result gives the correct transition temperature on the BCS limit; however, it is unphysical on the BEC regime for kFas!0. In order to nd a physical result, we need to include order-parameter uctuations, which we now do. B. Gaussian Fluctuations In discussing Gaussian uctuations, we concentrate on equal Rasha-Dresselhaus couplings, leaving details for general Rashba-Dresselhaus coupling to Appendix C. To obtain the correct super uid transition temperature in the BEC limit we must include the physics of two-body bound states near the transition, as described by the two- particleT-matrix [56, 57]. Accounting for all two-particle channels, the T-matrix calculation leads to a two-particle scattering amplitude , where 1(q;z) =m 4as1 2VX k1 "k+2X i;j=1ijWij ; (18) zis the complex frequency and Wij=1ni(k)nj(k+q) zEi(k)Ej(k+q): (19) At the super uid phase boundary 0!0, the eigenval- ues appearing in Eq. (19) reduce to E1;2(k) =jjkjhkj, but it is straightforward to show that ignoring the abso- lute values does not result in any change in either the mean-eld order parameter or number equation. Mean- while, the coecients 11=22=jukuk+qvkv k+qj2; (20) 12=21=jukvk+q+uk+qvkj2; (21) are the coherence factors associated with the quasi- particle amplitudes for 0= 0: uk=s 1 2 1 + R 2hk ; v k=is 1 2 1 R 2hk :(22) The Gaussian uctuation correction to the thermody- namic potential is F=TX q;iqnln [(q;iqn)=V]: (23) over the entire BCS-to-BEC evolution. The uctuation contribution to the particle number is therefore NF= @ F=@jT;V, where NF=X qZ1 1d! nB(!)@(q;!) @@(q;0) @ T;V; (24)with the phase shift (q;!) dened via the relation (q;!i) =j(q;!)jei(q;!): (25) When two-body states are present, the uctuation con- tribution can be written as NF=Nsc+Nb, where Nsc=X qZ1 !tp(q)d! nB(!)@(q;!) @@(q;0) @ T;V (26) is the number of particles in scattering states, and !tp(q) is the two-particle continuum threshold corresponding to the branch point of 1(q;z) [56, 58], Nb= 2X qnB(Ebs(q)2); (27) is the number of fermions in bound states, where nB(!) = 1=(e!1) is the Bose distribution function, andEbs(q) is the energy of the bound states obtained from 1(q;z=E2) = 0, corresponding to a pole in the scattering amplitude ( q;z). In the limit of large and negative fermion chemical potential, the system be- comes non-degenerate and 1(q;z) = 0 becomes the exact eigenvalue equation for the two-body bound state in the presence of spin-orbit and Rabi coupling [23]. The total number of fermions, as a function of , thus be- comes N=N0+NF; (28) whereN0is given in Eq. (17) and NFis the sum of Nsc andNb, as discussed above [52, 56]. IV. CRITICAL TEMPERATURE We calculate numerically the transition temperature Tcbetween the normal and uniform super uid states, as a function of the scattering parameter 1 =kFas, by simul- taneously solving the order parameter and number equa- tions (15) and (28). The solutions correspond to the min- ima of the free energy, F= +N. We do not discuss the cases of Fulde-Ferrell [59] or Larkin-Ovchinnikov [60] nonuniform super uid phases since they only exist over a very narrow region of the phase diagram deep in the BCS regime [59, 60], which is not experimentally accessible for ultracold fermions. Figure 1, in which we scale temperatures by the Fermi temperature TF=k2 F=2m, shows the eects of spin- orbit and Rabi couplings on Tc. The solid (black) line in Fig. 1(a) shows Tcversus 1=kFasfor zero Rabi cou- pling ( R= 0) and zero spin-orbit coupling . If R= 0, the spin-orbit coupling can be removed by a simple gauge transformation, and thus plays no role. In this situation, the pairing is purely s-wave. The dashed (blue) line shows Tcfor R6= 0, with vanishing equal Rashba-Dresselhaus spin-orbit coupling. We see that for xed interaction strength, the pair-breaking eect of the5 FIG. 1. (Color online) (a) The super uid transition tem- peratureTc=TF, whereTFis the Fermi temperature, vs the scattering parameter 1 =kFasfor equal Rashba-Dresselhaus spin-orbit coupling and two dierent Rabi coupling strengths, R= 0 and"F. For R= 0 [solid (black) curve], Tcis the same as for zero spin-orbit coupling since the equal spin-orbit eld can be gauged away. The dashed (blue) line shows Tc for zero spin-orbit coupling, with R="F, while the dot- ted (green) line shows Tcfor R="Fand ~==kF= 0:5. (b)Tcis drawn at unitarity, 1 =kFas= 0, and in the inset at 1=kFas=2:0, as a function of e R= R="F. The solid (red) curves represent ~ = 0 and the dotted (blue) curves represent ~= 0:5. Across the dotted (red) curves, the phase transition is rst order. Rabi coupling suppresses super uidity, compared with R= 0; the Rabi eld here plays the pair-breaking role of the Zeeman eld in an superconductor. With both spin-orbit and Rabi couplings present, the two-particle pairing is no longer purely singlet s-wave, but obtains a triplet p-wave component; the admixture stabilizes the super uid phase, as shown by the dotted (green) line. The latter curve shows that in the BEC regime with large positive 1 =kFas, the super uid tran- sition temperature is enhanced by the presence of spin- orbit and Rabi couplings, a consequence of the reduction FIG. 2. (Color online) Chemical potential at the super uid critical temperature ( Tc) for ~==kF= 0:5 and various Rabi elds, e R= R="F. of the bosonic eective mass in the xdirection below 2 m. However, for suciently large R, the geometric mean bosonic mass MBincreases above 2 mandTcdecreases. This renormalization of the mass of the bosons can be traced back to a change in the energy dispersion of the fermions when both spin-orbit coupling and Rabi elds are present. Figure 1(b) shows Tcversus Rfor xed 1=kFas, both with and without equal Rashba-Dresselhaus spin-orbit coupling at = 0:5kF. When both andTare zero, super uidity is destroyed at a critical value of Rcorre- sponding to the Clogston limit [61]. At low temperature, the phase transition to the normal state is rst order because the Rabi coupling is suciently large to break singlet Cooper pairs. However, at higher temperatures the singlet s-wave super uid starts to become polarized by thermally excited quasiparticles that produce a para- magnetic response. Thus, above the characteristic tem- perature indicated by the large (red) dots, the transition becomes second order, as pointed out by Sarma [62]. The change in the transition order occurs not only for = 0, but also for nonzero values of both in the BCS regime and near unitarity, depending on the choice of parame- ters, as illustrated in Fig. 1(b). The critical temperature for 6= 0 vanishes only asymptotically in the limit of large R. We note that for R=EFand= 0, the transition from the super- uid to the normal state is continuous at unitarity, but very close to a discontinuous transition. In the range 1:05. R=EF.1:10, numerical uncertainties as !0 prevent us from predicting exactly whether the transition at unitarity is continuous or discontinuous. Figure 2 shows (Tc) for xed spin-orbit coupling and several Rabi couplings. The solid (black) curve, which represents the situation in which no Rabi eld is present, is equivalent to the situation in which spin-orbit coupling is also absent, as noted in the discussion of Fig. 1. It is6 FIG. 3. (Color online) Phase diagram of critical temperature Tc=TFvs 1=kFasand R="Ffor equal Rashba-Dresselhaus coupling=kF= 0:5. The nite-temperature uniform super- uid phases re ect those at T= 0 shown in the background. These phases are distinguished by the number of rings (line nodes) in the quasiparticle excitation spectrum [i.e., where E2(k) = 0] and type of gap: (1) direct gapped super uid with zero rings (magenta diamonds), (2) indirect gapped super uid with zero rings (red circles), (3) gapless super uid with two rings (blue square), and (4) gapless one-ring super uid (green stars). evident that while the Rabi eld reduces the chemical potential in the BCS limit, it also shifts the onset of the system's evolution to the BEC limit to larger inverse scat- tering lengths, and produces a non-monotonic behavior of(Tc) near unitarity. Figure 3 shows Tcfor equal Rashba-Dresselhaus cou- pling= 0:5kF, as a function of Rabi eld and scattering parameter. We also superpose the zero- temperature phase diagram to illustrate the dierent su- per uid ground states of this system. According to the zeros of the lowest quasiparticle energy E2(k), the uni- form super uid phases that emerge are [21] direct gapped with zero rings (line nodes), indirectly gapped with zero rings, gapless with one ring, and gapless with two rings. Figure 4 shows the fractional number Nb=Nof bound fermions at Tcas a function of 1 =kFasfor two sets of external elds. In the BCS (BEC) regime, the rela- tive contribution to Nis dominated by unbound (bound) fermions. The main eect of spin-orbit and Rabi elds on Nb=Nis to shift the location where the two-body bound states emerge. For xed spin-orbit coupling (Rabi eld) and increasing Rabi eld (spin-orbit coupling), two-body bound states emerge at larger (smaller) scattering pa- rameters. These shifts are in agreement with the cal- culated shifts in binding energies of Feshbach molecules in the presence of equal Rashba-Dresselhaus spin-orbit coupling and Rabi elds [23]. FIG. 4. (Color online) Fractional number Nb=Nof bound fermions as a function of the interaction parameter 1 =kFas, for equal Rashba-Dresselhaus coupling =kF= 0:5 and Rabi frequencies e R= R="F= 0 (black solid line) and e R= R="F= 2 (red dot-dashed line). V. GINZBURG-LANDAU THEORY To further elucidate the eects of uctuations on the order of the super uid transition, as well as to assess the impact of spin-orbit and Rabi couplings near the crit- ical temperature, we now derive the Ginzburg-Landau description of the free energy near the transition. In the limit of small order parameter, the uctuation action SF can be expanded in powers of the order parameter ( q) beyond Gaussian order. The expansion of SFto quar- tic order is sucient to describe the continuous (second- order) transition in Tcversus 1=kFasin the absence of a Rabi eld [52]. However, to correctly describe the rst- order transition [61, 62] at low temperature (Fig. 1), it is necessary to expand the free energy to sixth order in . The quadratic (Gaussian-order) term in the action is SG=VX qjqj2 (q;z): (29) For an order parameter varying slowly in space and time, we may expand 1as 1(q;z) =a+X `c`q2 ` 2md0z+; (30) with the sum over `=fx;y;zg. The full result, as a functional of ( r;), has the form SF=Z 0dZ d3r d0@ @ +ajj2 +X `c`jr`j2 2m+b 2jj4+f 3jj6 :(31) The full time-dependent Ginzburg-Landau action de- scribes systems in and near equilibrium (e.g., with col-7 lective modes). The imaginary part of d0measures the non-conservation of jj2in time (i.e., the Cooper pair lifetime). Details of the derivation of SFare found in Appendix D. We are interested in systems at thermodynamic equi- librium, where the order parameter is independent of time, that is, ( r;) = ( r). In this situation, mini- mizing the free energy TSFwith respect to yields the Ginzburg-Landau equation, X `c`r2 ` 2m+bj(r)j2+fj(r)j4+a! (r) = 0: (32) Forb > 0, the system undergoes a continuous phase transition when achanges sign. However, when b <0, the system is unstable in the absence of f. Forb <0 anda > 0, a rst-order phase transition occurs when 3b2= 16af. Positivefstabilizes the system even when b<0. In the BEC regime, where d0is purely real, we dene an eective bosonic wave function ( r) =pd0(r) to re- cast Eq. (32) in the form of the Gross-Pitaevskii equation for a dilute Bose gas, X `r2 ` 2M`+U2j (r)j2+U3j (r)j4B! (r) = 0: (33) Here,B=a=d0is the bosonic chemical potential, M`=m(d0=c`) are the anisotropic bosonic masses, and U2=b=d2 0andU3=f=d3 0represent contact interac- tions of two and three bosons. In the BEC regime, these terms are always positive, leading to a dilute gas of stable bosons. The boson chemical potential Bis 2+Eb<0, whereEb=Ebs(q=0) is the two- body binding energy in the presence of spin-orbit cou- pling and Rabi frequency, obtained from the condition 1(q;E2) = 0, discussed earlier. The anisotropy of the eective bosonic masses, Mx6= My=MzM?, stems from the anisotropy of the equal Rashba-Dresselhaus spin-orbit coupling, which together with the Rabi coupling modies the dispersion of the constituent fermions along the xdirection. In the limit kFas1, the many-body eective masses reduce to those obtained by expanding the two-body binding en- ergy,Ebs(q)Eb+P `q2 `=2M`;and agree with known results [23]. However, for 1 =kFas.2, many-body and thermal eects produce deviations from the two-body re- sult. In the absence of two- and three-body boson-boson interactions, U2andU3, we directly obtain an analytic expression for Tcin the Bose limit from Eq. (27), Tc=2 MBnB (3=2)2=3 ; (34) withMB= (MxM2 ?)1=3, by noting that B= 0 or Ebs(q=0)2= 0, and using the condition that nB'n=2 [with corrections exponentially small in (1 =kFas)2], wherenBis the density of bosons. In the BEC regime, the results shown in Fig. 1 include the eects of the mass anisotropy, but do not include the eects of boson-boson interactions. To account for boson-boson interactions, we adopt the Hamiltonian of Eq. (33) with U26= 0, but with U3= 0, and apply the method developed in Ref. [63] to show that these interactions further increase TBEC to Tc(aB) = (1 + )TBEC; (35) where =n1=3 BaB. Here,aBis thes-wave boson- boson scattering length, is a dimensionless constant 1, and we use the relation U2= 4aB=MB. Since nB=k3 F=62and the boson-boson scattering length is aB=U2MB=4, we have =~fMBeU2;wherefMB= MB=2m;eU2=U2k3 F="F;and~==4(65)1=3=50: For xed 1=kFas,Tcis enhanced by the spin-orbit eld, a R-dependent decrease in the eective boson mass MB (10-15%), as well as a stabilizing boson-boson repulsion U2(2-3%), for the parameters used in Fig. 1. In closing our discussion of the strongly bound BEC limit, we note that in the absence of spin-orbit coupling, a Gaussian-order calculation of the two-boson scattering length yields the erroneous Born approximation result aB= 2as. However, an analysis of the T-matrix beyond Gaussian order, which includes the eects of two-body bound states, obtains the correct result aB= 0:6asat very low densities [64] and agrees with four-body calcu- lations [65]. The same method can be used to estimate U2 oraBbeyond the Born approximation discussed above. Nevertheless, while the precise quantitative relation be- tweenaBandasin the presence of spin-orbit coupling is yet unknown, the trend of increasing Tcdue to spin-orbit coupling has been clearly shown. VI. COMPARISON TO EARLIER WORK In this section, we brie y compare our results with ear- lier investigations of dierent types of theoretically mo- tivated spin-orbit couplings, worked in dierent dimen- sions, or at zero temperature. Our results focus mainly on an analysis of the critical super uid temperature and the eects thereon of order-parameter uctuations for a three-dimensional Fermi gas in the presence of equal Rashba-Dresselhaus spin-orbit coupling and Rabi elds. The appendices consider the more general situation of arbitrary Rashba and Dresselhaus components. Several works have analyzed the eects of spin-orbit- coupled fermions in three dimensions at zero tempera- ture [17{22, 66{69]. While some authors have described the situation of Rashba-only couplings [17{19, 66], others have assessed the case of equal Rashba and Dresselhaus components [21, 22] or a general mixture of the two [20]. It has been demonstrated that in the absence of a Rabi eld, the zero-temperature evolution from BCS to BEC8 super uidity is a crossover for s-wave systems, not only for Rashba-only couplings [17{20, 66], but also for ar- bitrary Rashba and Dresselhaus components [20]. This result directly follows from the fact that the quasiparti- cle excitation spectrum remains fully gapped throughout the evolution. In contrast, the addition of a Rabi eld gives rise to topological phase transitions for Rashba-only cou- plings [17] and equal Rashba and Dresselhaus compo- nents [21, 22], a situation which certainly persists for general Rashba-Dresselhaus couplings. The simultane- ous presence of a general Rashba-Dresselhaus spin-orbit coupling and Rabi elds leads to a qualitative change in the quasiparticle excitation spectrum and to the emer- gence of topological super uid phases [17, 21, 22]. Two- dimensional systems have also been investigated at zero temperature, where topological phase transitions have been identied for Rashba-only [70] and equal Rashba- Dresselhaus [71] couplings, in the presence of a Rabi eld. While early papers in this eld focused mainly on the zero-temperature limit, progress toward nite- temperature theories was made rst in two dimen- sions [72, 73] and later in three dimensions [74{76]. The eects of a general Rashba-Dresselhaus spin-orbit coupling and Rabi eld on the Berezenskii-Kosterlitz- Thouless transition were thoroughly investigated for two-dimensional Fermi gases at nite temperatures [72, 73], including both Rashba-only and equal Rashba- Dresselhaus spin-orbit couplings as examples. The super uid critical temperature in three dimen- sions was investigated using a spherical (3D) spin-orbit couplingkin the absence of a Rabi eld [74, 75], and also for Rashba-only (2D) couplings in the pres- ence of a Rabi eld [76]. In a recent review article [77], the critical temperature throughout the BCS-BEC evo- lution was discussed both in the absence [52] and pres- ence [53] of Rashba-Dresselhaus spin-orbit coupling. In Secs. 5 and 6 of this review, the authors describe the same method and expressions we obtained in our earlier preliminary work [53] for the analytical relations required to obtain the critical temperature at the Gaussian order; they include, however, only the contribution of bound states discussed earlier in the literature for Rashba-only spin-orbit coupling without Rabi elds [18]. In contrast, here we develop a complete Gaussian theory to compute the super uid critical temperature of a three-dimensional Fermi gas in the presence of both a general Rashba- Dresselhaus (2D) spin-orbit coupling and Rabi elds. We focus our numerical calculations on the specic situation of equal Rashba-Dresselhaus components, which is eas- ier to achieve experimentally in the context of ultracold atoms. Our key results, already announced in our earlier work [53], include the contributions of bound and scat- tering states at the Gaussian level. As seen in Fig. 4 of this present paper, there is a wide region of interac- tion parameters for which the contribution of scattering states cannot be neglected. Furthermore, unlike previ- ous work [74{77], we provide a comprehensive analysisof the Ginzburg-Landau uctuation theory and include the eects of boson-boson interactions on the super uid critical temperature in the BEC regime. VII. CONCLUSION We have evaluated the super uid critical tempera- ture throughout the BCS-to-BEC evolution of three- dimensional Fermi gases in the presence of equal Rashba- Dresselhaus spin-orbit couplings, Rabi elds, and tun- ables-wave interactions. Furthermore, we have devel- oped the Ginzburg-Landau theory up to sixth power in the order parameter to elucidate the origin of rst-order phase transitions when the spin-orbit eld is absent and the Rabi eld is suciently large. Lastly, in the appen- dices, we have presented the nite-temperature theory of s-wave interacting fermions in the presence of a generic Rashba-Dresselhaus coupling and external Rabi elds, as well as the corresponding time-dependent Ginzburd- Landau theory near the super uid critical temperature. ACKNOWLEDGMENTS We thank I. B. Spielman for discussions. The re- search of P.D.P. was supported in part by NSF Grant No. PHY1305891 and that of G.B. by NSF Grants No. PHY1305891 and No. PHY1714042. Both G.B. and C.A.R. SdM. thank the Aspen Center for Physics, supported by NSF Grants No. PHY1066292 and No. PHY1607611, where part of this work was done. This work was performed under the auspices of the U.S. De- partment of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344. Appendix A: Hamiltonian and eective Lagrangian for general Rashba-Dresselhaus spin-orbit coupling In this appendix, we consider a larger class of spin- coupled fermions in three dimensions with a general Rashba-Dresselhaus (GRD) coupling. The Hamiltoninan density for equal Rashba-Dresselhaus (ERD) discussed in Sec. II is a particular case of the general Rashba- Dresselhaus Hamiltonian density, H(r) =H0(r) +Hso(r) +HI(r): (A1) Adopting units in which ~=kB= 1, the independent- particle Hamiltonian density without spin-orbit coupling is H0(r) =X jr (r)j2 2m y (r) (r) ;(A2) where ,m, andare the fermion eld operator, mass, and chemical potentials for internal state , re- spectively. The spin-orbit Hamiltonian can be written9 as Hso(r) =X i y (r)i;hi(r) (r); (A3) where theiare the Pauli matrices in isospin (internal state) space and h= (hx;hy;hz) includes both the spin- orbit coupling and Zeeman elds. Finally, we consider a two-bodys-wave contact interaction, HI(r) =g y "(r) y #(r) #(r) "(r); (A4) whereg>0 corresponds to an attractive interaction. By introducing the pairing eld ( r;) = gh #(r;) "(r;)i;we remove the quartic inter- action and obtain the Lagrangian density, L(r;) =1 2 y(r;)G1(^k;) (r;) +j(r;)j2 g +eK+(^k)(rr0); (A5) where we introduced the momentum operator ^k=ir, the Nambu spinor = ( " # y " y #)T, and dened eK= (eK"eK#)=2:Here,eK"=K"hz;andeK#= K#+hz;withK(^k) =^k2=(2m)being the kinetic energy operator of internal state with respect to its chemical potential. Lastly, the inverse Green's operator appearing in Eq. (A5) is G1(^k;) =0 BB@@eK"h ? 0 h?@eK# 0 0 @+eK"h? 0h ?@+eK#1 CCA; (A6) whereh?(^k) =hx(^k) +ihy(^k) plays the role of the spin- orbit coupling, and hzis the Zeeman eld along the z direction. To make progress, we expand the order parameter about its saddle-point (mean-eld) value 0by writ- ing ( r;) = 0+(r;):Next, we integrate over the fermionic elds and use the decomposition G1(^k;) = G1 0(^k;)+G1 F(^k;);where G1 0(^k;) is the mean-eld Green's operator, given by Eq. (A6) with ( r;) = 0, andG1 F(^k;) is the contribution to the inverse Green's operator arising from uctuations. These steps yield the saddle-point Lagrangian density, L0(r;) =T 2VTr ln(G1 0) +j0j2 g+eK+(^k)(rr0); (A7) and the uctuation contribution, LF(r;) =T 2VTr ln( I+G0G1 F) + ( r;) +j(r;)j2 g; (A8) resulting in the eective Lagrangian density Le(r;) = L0(r;) +LF(r;):In the expressions above, we workin a volume Vand take traces over both discrete and continuous indices. Notice that the term ( r;) = [0(r;) + 0(r;)]=gin the uctuation Lagrangian cancels out the linear terms in andwhen the loga- rithm is expanded, due to the saddle point condition S0 0= 0; (A9) whereS0=R 0dd3rL0(r;) is the saddle-point action. Appendix B: Saddle Point Approximation for general Rashba-Dresselhaus spin-orbit coupling We rst analyze the saddle-point contribution. The saddle-point thermodynamic potential 0=TlnZ0 can be obtained for the saddle-point partition function Z=eS0as 0=TS0. Transforming the saddle-point LagrangianL0from Eq. (A7) into momentum space and integrating over spatial coordinates and imaginary time leads to the saddle-point thermodynamic potential, 0=Vj0j2 gT 2X k;jln(1+eEk;j)+X keK+(k);(B1) whereK(k) =k2=2mand the eigenvalues Ek;j are the poles of G0(k;z), withj=f1;2;3;4g. Next, we restrict our analysis to mass balanced sys- tems (m"=m#) in diusive equilibrium ( "=#). We also consider the general Rashba-Dresselhaus (GRD) spin-orbit eld h?(k) =(kx+iky)=m;whereand are the magnitude and anisotropy of the spin-orbit cou- pling, respectively. Note that this form is equivalent to another common form of the Rashba-Dresselhaus cou- pling found in the literature [21, 22]: hso=hR+hD where hR=vR(kx^yky^x) and hD=vD(kx^y+ky^x). The two forms are related via a momentum-space ro- tation and the correspondences =m(vR+vD) and = (vRvD)=(vR+vD). The equal Rashba-Dresselhaus limit (ERD) corresponds to vR=vD=v, leading to = 0 and= 2mv. The specic case of equal Rashba- Dresselhaus spin-orbit coupling discussed in the main part of the paper corresponds to the case where = 0, that is,h?(k) =kx=m: For the general Rashba-Dresselhaus case, the four eigenvalues are E1;2(k) =h 2 k2q E2 0;kh2 kj0j2jh?(k)j2i1=2 ;(B2) E3;4(k) =E2;1(k); (B3) where the + () sign within the outermost square root corresponds to E1(E2), and the functions inside the square roots are 2 k=E2 0;k+h2 k, with contributions E0;k=q 2 k+j0j2; (B4) hk=p jh?(k)j2+h2z; (B5)10 wherek="k;and"k=k2=2m:The order- parameter equation is found from the saddle point con- dition 0= 0jT;V; = 0. At the phase boundary be- tween the super uid and normal phases, 0!0, and the order-parameter equation becomes m 4as=1 2VX k1 "ktanh(E1=2) 2E1tanh(E2=2) 2E2 h2 z khktanh(E1=2) 2E1tanh(E2=2) 2E2 ; (B6) after expressing the interaction parameter gin terms of thes-wave scattering length via the relation 1 g=m 4as+1 VX k1 2"k: (B7) We note that asis thes-wave scattering length in the absence of spin-orbit and Zeeman elds. It is, of course, possible to express all relations obtained in terms of a scattering length which is renormalized by the presence of the spin-orbit and Rabi elds [54, 55]. However, in ad- dition to complicating our already cumbersome expres- sions, it would make reference to a quantity that is more dicult to measure experimentally and that would hide the explicit dependence of the properties that we analyze in terms of the spin-orbit and Rabi elds, so we do not consider such complications here. Note that since 0= 0 at the phase boundary, the eigenvalues in Eq. (B2) re- duce toE1(k) =jkj+hk,E2(k) =jkjhk, which is the absolute value of the normal-state energy dispersions. However, it is straightforward to show that ignoring the absolute values does not result in any change in either the mean-eld order parameter given by Eq. (B6) or number equation shown in Eq. (B8), when 0!0. The saddle-point critical temperature T0is determined by solving Eq. (B6) subject to the thermodynamic con- straintN0=@ 0=@jT;V;which yields N0=X k 1k1 "k+tanh(E1=2) 2E1+tanh(E2=2) 2E2 +jh?(k)j2 khktanh(E1=2) 2E1tanh(E2=2) 2E2 : (B8) A mean-eld description of the system, which involves a simultaneous solution of Eqs. (B6) and (B8), yields the asymptotically correct description of the system in the BCS limit; however, such a description fails miserably in the BEC regime where it does not ac- count for the formation of two-body bound states. The general Rashba-Dresselhaus spin-orbit saddle-point equations (B6) and (B8) reduce to the equal Rashba- Dresselhaus equations (15) and (17) of the main part of the paper with the explicit use of hz= R=2 and h?(k) =kx=m, where Ris the Rabi coupling.Appendix C: Derivation of the modied number equation with Gaussian uctuations We begin by deriving the modied number equation arising from Gaussian uctuations of the order parame- ter near the super uid phase boundary. The uctuation thermodynamic potential Fresults from the Gaussian integration of the elds (r;) and(r;) in the uctua- tion partition function ZF=R ddeSF, where the ac- tionSF=R d 0R d3rLF(r;) is calculated to quadratic order. The contribution to the thermodynamic potential due to Gaussian uctuations is F=TX iqn;qln [(q;iqn)=V] (C1) whereqn= 2nT are the bosonic Matsubara frequencies and ( q;iqn) is directly related to the pair uctuation propagator pair(q;iqn) =V1(q;iqn): The Matsubara sum can be evaluated via contour in- tegration, F=TX qI Cdz 2inB(z) ln [(q;z)=V]; (C2) wherenB(z) = 1=(ez1) is the Bose function and the countourCencloses all of the Matsubara poles of the Bose function. Next, we deform the contour around the Matsubara frequencies towards innity, taking into ac- count the branch cut and the possibility of poles coming from the logarithmic term inside the countour integral. We take the branch cut to be along the real axis, then add and subtract the pole at iqn= 0 to obtain F=TX qZ1 1d! nB(!) [(q;!)(q;0)];(C3) where the phase shift (q;!) is dened via ( q;!i) = j(q;!)jei(q;!);and arises from the contour segments above and below the real axis. The thermodynamic identity N=@ =@jT;Vthen yields to the uctuation correction, NF=TX qZ1 1d! nB(!)@(q;!) @@(q;0) @ ; (C4) to the the saddle-point number equation, and has a sim- ilar analytical structure as in the case without spin-orbit and Zeeman elds [52, 56]. Thus, we can write the - nal number equation at the critical temperature Tcas N=N0+NF. Since the phase shift (q;z) vanishes ev- erywhere that ( q;z) is analytic, the only contributions to Eq. (C4) arise from a possible isolated pole at !p(q) and a branch cut extending from the two-particle contin- uum threshold !tp(q) = minfi;j;kg[Ei(k) +Ej(k+q)] to z!1 along the positive real axis. The explicit form of (q;z) can be extracted from Eq. (D15) of Appendix D. When there is a pole corresponding to the emergence of a two-body bound state, we can explicitly write ( q;z)11 R(q)=(z!p(q));from which we obtain @(q;!)=@= 2(z!p(q));leading to the bound state density Nb= 2X qnB(!p(q)); (C5) where the energy !p(q) must lie below the two-particle continuum threshold !tp(q). The factor of 2, which arises naturally, is due to the two fermions comprising a bosonic molecule. Naturally, the presence of this term in the uctuation-modied number equation is dependent upon the existence of such a pole, that is, a molecular bound state. These bound states correspond to the Feshbach molecules in the presence of spin-orbit coupling and Zee- man elds [7, 23]. Having extracted the pole contribution to Eq. (C4), when it exists, the remaining integral over the branch cut corresponds to scattering state fermions, Nsc=TX qZ1 !tp(q)d! nB(!)@(q;!) @@(q;0) @ ; (C6) whose energy is larger than the minimum energy !tp(q) of two free fermions. Thus, when bound states are present, we arrive at the modied number equation, N=N0+Nsc+Nb (C7) whereN0is the number of free fermions obtained from the saddle-point analysis in Eq. (B8), and NbandNsc are the bound state and scattering contributions given in Eqs. (C5) and (C6), respectively. These general results are particularized to the equal Rashba-Dresselhaus case in Sec. III B of this paper. The number of unbound states Nuis then easily seen to beNu=N0+Nsc, that is, the sum of the free-fermion (N0) and scattering ( Nsc) contributions. Naturally, the number of unbound states is also equal to the total num- ber of states, N, minus the number of bound states, Nb, that is,Nu=NNb. Appendix D: Derivation of Ginzburg-Landau coecients for general Rashba-Dresselhaus spin-orbit coupling Next, we derive explicit expressions for the coe- cients of the time-dependent Ginzburg-Landau theory valid near the critical temperature of the super uid. We start from the uctuation Lagrangian, LF(r;) =T 2VTr ln( I+G0G1 F) + ( r;) +j(r;)j2 g; (D1) in a volume V, and take the traces over both discrete and continuous indices. Notice that the term ( r;) = [0(r;) + 0(r;)]=gin the uctuation Lagrangian cancels out the linear terms in andwhen the log- arithm is expanded, due to the saddle-point condition.Since the expansion is performed near Tc, we take the saddle-point order parameter 0!0 and redene the uctuation eld as (r;) = ( r;) to obtain LF(r;) =jj2 gT 2VTr ln( I+G0[0]G1 F[]):(D2) Notice that the arguments in G0[0] and G1 F[] represent the values of 0= 0 and= , respectively. We expand the logarithm to sixth order in to obtain LF(r;) =jj2 g+T 2VTr1 2(G0G1 F)2+1 4(G0G1 F)4 +1 6(G0G1 F)6+::: ; (D3) where the higher-order odd (cubic and quintic) terms in the order-parameter amplitudes expansion can be shown to vanish due to conservation laws and energy or momen- tum considerations. The traces can be evaluated explicitly by using the momentum-space inverse single-particle Green's function G1 0(k;k0) =A1(k) 0 0 A1(k)T kk0;(D4) derived from Eq. (A6). Here, we use the shorthand no- tationk(i!;k), where!n= 2nT are bosonic Mat- subara frequencies and dene the 2 2 matrix, A1(k) = i!neK"(k)h ?(k) h?(k)i!neK#(k) ; (D5) whereeK"=khz,eK#=k+hz, withk=k2=2m the kinetic energy relative to the chemical potential, hz the external Zeeman eld, and h?(k) =hx(k) +ihy(k) the spin-orbit eld. We also dene the uctuation con- tribution to the inverse Green's function, G1 F(k;k0) =0iykk0 iyy k0k0 ; (D6) whereyis the second Pauli matrix in isospin (internal state) space and k= VZ 0dZ d3rei(kr!)(r) (D7) is the Fourier transform of ( r), withr(r;), and also has dimensions of energy. Recall that we set ~=kB= 1, such that energy, frequency and temperature have the same units. Inversion of Eq. (D4) yields G0(k;k0) =A(k) 0 0[A(k)]T kk0; (D8) where the matrix A(k) is A(k) =1 det[A1(k)] i!neK#(k)h ?(k) h?(k)i!neK"k) : (D9)12 with det[ A1(k)] =Q2 j=1[i!nEj(k)] and where the independent-particle eigenvalues Ej(k) are two of the poles of G0(k;k). These poles are exactly the gen- eral eigenvalues described in Eqs. (B2) in the limit of 0!0. Note that setting 0= 0 in the general eigen- value expressions yields E1;2(k) =jjkjhkj. The other set of poles of G0(k;k) corresponds to the eigenvalues E3;4(k) =E2;1(k) found from det A1(k)T= 0. Using Eq. (D3) to write the uctuation action as SF=R 0dR d3rLF(r;);results in SF=VX qjqj2 (q)+V 2X q1;q2;q3b1;2;31 23 12+3 +V 3X q1q5f151 23 45 12+34+5;(D10) where summation over q(iqn;q) indicates sums over both the bosonic Matsubara frequencies qn= 2nT and momentum q. Here, we used the shorthand notation jqjto represent the labels of qjor qj. The quadratic order appearing in Eq. (D10) arises from the termsj(r;)j2=gand (T=2V)Tr(G0G1 F)2=2 in Eq. (D3), and is directly related to the pair propaga- torpair(q) =V1(q), with 1(q) =1 gT 2VX kTr A(k)A1(qk) det[A1(qk)];(D11) where we use the identity yAy= det( A)(AT)1: The fourth-order contribution arises from1 4(G0G1 F)4and leads to b(q1;q2;q3) =T 2VX kTr A(k)A1(q1k)A(kq1+q2)A1(q1q2+q3k) det [A1(q1k)] det [ A1(q1q2+q3k)]; (D12) while the sixth order contribution emergences from1 6(G0G1 F)6, giving f(q1;;q5) =T 2VX kdet [A(q1k)] det [ A(q1q2+q3k)] det [ A(q1q2+q3q4+q5k)] Tr A(k)A1(q1k)A(kq1+q2)A1(q1q2+q3k) A(kq1+q2q3+q4)A1(q1q2+q3q4+q5k) : (D13) Evaluating the expressions given in Eqs. (D11) through (D13) requires us to perform summations over Matsubara frequencies of the type TX i!n1 i!nE(k)=( n(k) if \+" 1n(k) if \";(D14) wheren(k) = 1= eE(k)+ 1 is the Fermi function. For the quadratic term, we obtain the result 1(q;iqn) =m 4as+1 2VX k1 "k +2X i;j=1ij(k;q)Wij(k;q;iqn) ;(D15) where the functions in the last term are Wij(k;q;iqn) =1ni(k)nj(k+q) iqnEi(k)Ej(k+q); (D16)corresponding to the contribution of bubble diagrams to the pair susceptibility. The coherence factors are 11(k;q) =jukuk+qvkv k+qj2; (D17) 12(k;q) =jukvk+q+uk+qvkj2; (D18) with11(k;q) =22(k;q) and12(k;q) =21(k;q); where the quasiparticle amplitudes are uk=s 1 2 1 +hz hk ; (D19) vk=eiks 1 2 1hz hk : (D20) The anglekis the phase associated with the spin-orbit eldh?(k) =jh?(k)jeik;and we replaced the interac- tion parameter gby thes-wave scattering length asvia13 Eq. (B7), recalling that "k=k2=2m. The phase and modulus of h?(k) are k= arctanky kx ; (D21) jh?(k)j=jj mq k2x+k2y; (D22) and the total eective eld is hk=p h2z+jh?(k)j2: (D23)Since we are interested only in the long-wavelength and low-frequency regime, we perform an analytic con- tinuation to real frequencies iqn=!+iafter calculat- ing the Matsubara sums for all coecients appearing in Eq. (D10) and perform a small momentum qand low- frequency!expansion resulting in the Ginzburg-Landau action, SF=SGL=VX q a+X `c`q2 ` 2md0!! jqj2+V 2X q1;q2;q3b(q1;q2;q3)q1 q2q3 q1q2+q3 +V 3X q1q5f(q1;q2;q3;q4;q5)q1 q2q3 q4q5 q1q2+q3q4+q5: (D24) Here, the label `appearing explicitly in the termP `c`q2 `=(2m) represents the spatial directions fx;y;zg, while theqj's in the sums correspond to ( qj;!j) and the summationsP qjrepresent integrals VR d!jR d3qj, wherejlabels a fermion pair and can take values in the setf1;2;3;4;5g. In the expression above, we used the result 1(q;!) =a+X `c`q2 ` 2md0!+ (D25) for the analytically continued expression of 1(q;iqn) appearing in Eq. (D15). To write the coecients above in a more compact notation, we dene Xi=Xi(k) = tanh [Ei(k)=2]; (D26) Yi=Yi(k) = sech2[Ei(k)=2]: (D27) The frequency- and momentum-independent coecient is a=m 4as+1 VX k1 2"kX1 4E1+X2 4E2 h2 z khkX1 4E1X2 4E2 ;(D28)whereE1=E1(k) andE2=E2(k). The coecient d0=dR+idImultiplying the linear term in frequency has a real component given by dR=1 2VPX k2X i;j=1ij(k;0)1ni(k)nj(k) [Ei(k) +Ej(k)]2:(D29) Using the explicit forms of the coherence factors ukand vkthat dene ij(k;q=0), the above expression can be rewritten as dR=1 2VPX k 1 +h2 z 2 kX1 4E2 1+X2 4E2 2 +2h2 z khkX1 4E2 1X2 4E2 2 ;(D30) which denes the time scale for temporal oscillations of the order parameter. Here, the symbol Pdenotes the principal value, and the coecient dRis obtained from Re 1(q=0;!+i) =m 4as+1 2VX k2 41 "k+P2X i;j=1ij(k;q=0)1ni(k)nj(k) !Ei(k)Ej(k)3 5: (D31) The imaginary component of the coecient dhas the form dI= 2VX k2X i;j=1ij(k;0) [1ni(k)nj(k)]0(Ei(k) +Ej(k)); (D32)14 where the derivative of the delta function is 0() =@(x+)=@xjx=0:Using again the expressions of the coherence factorsukandvkleads to dI= 2VX k (X1+X2)0(2k) +jh?j2 h2 k X10(2E1) +X20(2E2)(X1+X2)0(2k) ; (D33) which determines the lifetime of fermion pairs. This result originates from Im 1(q=0;!+i) = 2VX k2X i;j=1ij(k;q=0) [1ni(k)nj(k)](!Ei(k)Ej(k)); (D34) which immediately reveals that below the two-particle threshold!tp(q=0) = minfi;j;kg[Ei(k) +Ej(k)] at center-of-mass momentum q=0, the lifetime of the pairs is innitely long due to the emergence of stable two-body bound states. Note that collisions between bound states are not yet included. The expressions for the c`coecients appearing in Eq. (D25) are quite long and complex. Since these coef- cients are responsible for the mass renormalization and anisotropy within the Ginzburg-Landau theory, we out- line below their derivation in detail. These coecients can be obtained from the last term in Eq. (D15), which we dene as F(q) =1 2VX k2X i;j=1ij(k;q)Wij(k;q;iqn= 0):(D35) The relation between c`and the function F(q) dened above is c`=m@2F(q) @q2 ` q=0: (D36) A more explicit form of c`is obtained by analyzing the symmetry properties of F(q) under inversion and re ec- tion symmetries. To make these properties clear, we rewrite the summand in Eq. (D35) by making use of the transformation k!kq=2. This procedure leads to the symmetric form, F(q) =1 2VX k2X i;j=1eij(k;k+)fWij[Ei(k);Ej(k+)]: (D37) Here, k+=k+q=2 and k=kq=2 are new momentum labels, and e11(k;k+) =jukuk+vkv k+j2; (D38) e12(k;k+) =jukvk+vkuk+j2(D39) are coherence factors, with e11(k;k+) =e22(k;k+) ande12(k;k+) =e21(k;k+) The functions ukand vkare dened in Eqs. (D19) and (D20). It is now very easy to show that eij(k;k+) =eij(k+;k), that is, eij(k;k+) is an even function of q, since taking q!q leads to k!k+andk+!kleavingeijinvariant.It is also clear, from its denition, that eijis symmetric in the band indices fi;jg. Furthermore, the function fWij[Ei(k);Ej(k+)] =Nij Dij; (D40) dened above, is the ratio between the numerator, Nij= tanh [Ei(k)=2] + tanh [Ej(k+)=2];(D41) representing the Fermi occupations and the denominator, Dij= 2 [Ei(k) +Ej(k+)]; (D42) representing the sum of the quasi-particle excitation energies. To elliminate the Fermi distributions ni(k) in the numerator, we used the relation 1 2ni(k) = tanh [Ei(k)=2]. Notice that fWij[Ei(k);Ej(k+)] is not generally symmetric under inversion q! q, that is, under the transformation k!k+and k+!k. This means that fWij[Ei(k);Ej(k+)]6= fWij[Ei(k+);Ej(k)], unless when i=j, where it is trivially an even function of q. However, fWij[Ei(k);Ej(k+)] is always symmetric under simul- taneous momentum inversion ( q!q) and band index exchange, that is, fWij[Ei(k);Ej(k+)] =fWji[Ej(k+);Ei(k)] (D43) for anyfi;jg. This property will be used later to write a nal expression for c`. Next, we write @2F(q) @q2 ` q=0=1 2VX k2X i;j=1Fij; (D44) where the function inside the summation is Fij=" @2eij @q2 `fWij+ij@2fWij @q2 `# q=0: (D45) Notice the absence of terms containing the product of the rst-order derivatives of eijandfWij. These terms vanish due to parity since eijis an even function of q, leading to [@eij=@q`]q=0= 0. The last expression can be15 further developed upon summation over the band indices, leading to @2F(q) @q2 ` q=0=A+B: (D46) The rst contribution is given by A=1 2VX k@2e11 @q2 `fWdi+@2e12 @q2 `fWod q=0;(D47) and contains the second derivatives of eijand the sym- metric terms fWdi= fW11+fW22 ; (D48) fWod= fW12+fW21 ; (D49)The second contribution is given by B=1 2VX k" e11@2fWdi @q2 `+e12@2fWod @q2 `# q=0:(D50) Next, we explicitly write eij,fWijand their second derivatives with respect to q`atq=0. We start with h fWiji q=0=Xi+Xj 2 [Ei+Ej](D51) and for the second derivative, we write " @2fWij @q2 `# q=0=1 Dij@2Nij @q2 ` q=0" 2 D2 ij@Dij @q`@Nij @q`# q=0+" 2Nij D3 ij@Dij @q`2# q=0" Nij D2 ij@2Dij @q2 `# q=0: Each one of the four terms in the above expression is evaluated at q=0and can be written in terms of specic expressions that are given below. The numerator is [Nij]q=0=Xi+Xj; (D52) the rst derivative of Nijis @Nij @q` q=0=Y2 j 4T@Ej @k`Y2 i 4T@Ei @k`; (D53) and the second derivative of Nijis @2Nij @q2 ` q=0=XjY2 j 8T2@Ej @k`2 +Yi 8T@2Ei @k2 ` XiY2 i 8T2@Ei @k`2 +Y2 i 8T@2Ei @k2 `:(D54) The denominator Dijand its rst derivative are [Dij]q=0= 2(Ei+Ej); (D55) @Dij @q` q=0=@Ej @k`@Ei @k`; (D56) while the second derivative of Dijis @2Dij @q2 ` q=0=1 2@2Ei @k2 `+@2Ej @k2 ` : (D57) When the order parameter is zero, that is, j0j= 0, the energiesE1(k) andE2(k) become E1(k) =jkj+hk (D58) E2(k) =jkjhk: (D59)The rst derivatives of these energies are @E1(k) @k`=S1(k)k` m+@hk @k`; (D60) @E2(k) @k`=S2(k)k` m@hk @k`; (D61) with the functions S1(k) = sgn [jkj+hk] sgn [k] and S2(k) = sgn [jkjhk] sgn [k]:The derivative of the ef- fective Zeeman eld is @hk @k`=1 hk2 m2(kx`x+ky`y): (D62) The second derivatives of the energies are @2E1(k) @k2 `=S1(k) m+@2hk @k2 `(D63) @2E2(k) @k2 `=S2(k) m@2hk @k2 `; (D64) where the second derivative of the eective eld is @2hk @k2 `=1 hk2 m2 (`x+`y)1 h2 k2 m2 k2 x`x+2k2 y`y : (D65) Since the diagonal elements fWiiare even functions of qand so areNiiandDii, their expressions are simpler than in the general case discussed above, because the rst order derivatives of NiiandDiivanish. The surviving terms involve only the second derivatives of NiiandDii leading to the expression " @2fWii @q2 `# q=0=1 Dii@2Nii @q2 ` q=0Nii D2 ii@2Dii @q2 ` q=0: (D66)16 Here, the numerator and denominator functions are [Nii]q=0= 2Xiand [Dii]q=0= 4Ei; (D67) while their second derivatives are @2Nii @q2 ` q=0=XiY2 i 4T2@Ei @k`2 +Y2 i 4T@2Ei @k2 `;(D68) @2Dii @q2 ` q=0=@2Ei @k2 `: (D69) The next step in obtaining the c`coecients is to an- alyze the functions eijand their second derivatives. We begin by writing e11atq=0: [e11]q=0=u2 kjvkj22=h2 z h2 k: (D70) To investigate the second derivative of e11, we write e11= 11 11; (D71) where the complex function is given by 11=ukuk+vkvk+: (D72) In this case, we write the rst derivative of e11as @e11 @q`=@ 11 @q` 11+ 11@ 11 @q`(D73) and the second derivative as @2e11 @q2 `=@2 11 @q2 ` 11+ 2@ 11 @q`@ 11 @q`+ 11@2 11 @q2 `:(D74) To explore the symmetry with respect to q, we express 11in terms of its odd and even components via the re- lation 11= 11;e+ 11;o, where the even component 11;e= [ 11(q) + 11(q)]=2 is 11;e=ukuk+jvkjjvk+jcos kk+ (D75) and the odd component 11;o= [ 11(q) 11(q)]=2 is 11;o=ijvk+jjvkjsin k+k : (D76) Expressed via the even 11;eand odd 11;ocomponents, the second derivative in Eq. (D74) is @2e11 @q2 `=@2 11;e @q2 ` 11:e+ 2@ 11;o @q`@ 11;o @q`+ 11;e@2 11;e @q2 `: (D77) Notice that the even component is purely real, that is, 11;e= 11;e, and that the odd component is purely imag- inary, 11;o= 11;o. Use of this property leads to @2e11 @q2 `= 2 11;e@2 11;e @q2 `2@ 11;o @q`2 : (D78)The contribution from the even term 11;eis [ 11;e]q=0=u2 kjvkj2=hz hk; (D79) and from its second derivative is @2 11;e @q2 ` q=0=1 2@jvkj @k`2 1 2jvkj@2jvkj @k2 `+jvkj2@k @k`2 ; (D80) while the contribution from the odd term 11;ois @ 11;o @q` q=0=ijvkj2@k @k`: (D81) Now, we turn our attention to e12and its second derivative. From Eq. (D39), we notice that 12is ex- plicitly odd in qbecause 12(q) = 12(q), since the operation q!qtakes k!k+and vice versa, leading to [e12]q=0= 0: (D82) To calculate the second derivative of e12, we write e12= 12 12; (D83) where the complex function 12=ukvk+vkuk+: (D84) We relate @2e12=@q2 `to 12and its rst and second derivatives via @2e12 @q2 `=@2 12 @q2 ` 12+ 2@ 12 @q`@ 12 @q`+ 12@2 12 @q2 `:(D85) Given that [ 12]q=0= 0 and [ 12]q=0= 0, the expression above simplies to @2e12 @q2 ` q=0= 2@ 12 @q`@ 12 @q` q=0= [`(q)]2;(D86) where we used the expressions @ 12 @q` q=0=eik`(k) (D87) for the derivatives of 12atq=0with the function `(k) =uk@jvkj @k`jvkj@uk @k`+ukjvkj@k @k`: (D88) The last information needed is the derivatives of uk, jvkj, andk, which are given by @uk @k`=1 2hz h3 k2 m2(kx`x+ky`y) (1 +hz=hk)1=2; (D89) @jvkj @k`=1 2hz h3 k2 m2(kx`x+ky`y) (1hz=hk)1=2; (D90) @k @k`=(kx`yky`x) k2x+2k2y: (D91)17 The long steps discussed above complete the derivation of all the functions needed to compute the c`coecients for an arbitrary spin-orbit coupling, expressed as a general linear combination of Rashba and Dresselhaus terms. As announced earlier, the calculation of c`, dened in Eq. (D36), is indeed very long and requires the use of all the expressions given from Eq. (D37) to Eq. (D91). Despite this complexity, that are a few important com- ments about the symmetries of the c`coecients that are worth mentioning. Given that c`determines the mass anisotropies in the Ginzburg-Landau (GL) theory, we discuss next the anisotropies of c`as a function of the spin-orbit coupling parameters and. First, in the limit of zero spin-orbit coupling, where andare equal to zero, all the c`coecients are identical re ect- ing the isotropy of the system, that is, cx=cy=cz and reduce to previously known results [52]. In this case, the GL eective masses m`=mdR=c`are isotropic: mx=my=mz. Second, in the limit of 6= 0 and=1, the spin-orbit coupling has the same strength along thexandydirections, and thus for the Rashba (= 1) or Dresselhaus ( =1) cases, the coecients obey the relation cx=cy6=cz. This leads to eective massesmx=my6=mz. Third, in the limit 6= 0, but = 0, corresponding to the ERD case, the coecients have the symmetry cx6=cy=cz. Now the eective masses obey the relation mx6=my=mz. Finally, in the case where 6= 0, and 06=jj<1, all thec`coe- cients are dierent, that is, cx6=cy6=cz. Therefore, the eective masses are also dierent in all three directions: mx6=my6=mz. Following an analogous procedure, we analyze the co- ecientsb(q1;q2;q3), ande(q1;q2;q3;q4;q5) with allqi= (0;0), and dene Zij=Xi+EiYj=2: (D92) Using the notation b(0;0;0) =b(0), we obtain b(0) =1 8VX k 1 +h4 z 2 kh2 kZ11 E3 1+Z22 E3 2 +2h2 z khkZ11 E3 1Z22 E3 2 +h4 z 3 kh3 kX1 E1X2 E2 ; (D93) which is a measure of the local interaction between two pairing elds. Using the notation f(0;0;0;0;0) =f(0), we obtain f(0) =3 32VX k 1 +3h4 z 2 kh2 kZ11 E5 1+Z22 E5 2 h2 z khk 3 +h4 z 2 kh2 kZ11 E5 1Z22 E5 2 h6 z 4 kh4 kZ11 E3 1+Z22 E3 2 h4 z 3 kh3 kZ11 E3 1Z22 E3 2 +2 6X1Y1 E3 1+X2Y2 E3 2 +2h2 z 6khkX1Y1 E3 1X2Y2 E3 2 h6 z 5 kh5 kX1 E1X2 E2 ; (D94) which is a measure of the local interaction between three pairing elds. It is important to mention that in the absence of spin-orbit and Zeeman elds, the Ginzburg-Landau coecients obtained above reduce to those reported in the literature [52]. As we proceed to explicitly write the Ginzburg-Landau action and Lagrangian density, we emphasize that in con- trast to the standard crossover that one observes in the absence of an external Zeeman eld [52], for xed hz6= 0 it is possible for the system to undergo a rst-order phase transition with increasing 1 =kFas. The same applies for xed 1=kFaswith increasing hz. Thus, while an expan- sion ofSFto quartic order is sucient when no Zeeman elds are present, when Zeeman elds are turned on, the fourth-order coecient b(0) =bmay become negative. Such a situation requires the analysis of the sixth-order coecientf(0) =fto describe this rst-order transition correctly and to stabilize the theory since f >0. The Ginzburg-Landau action in Euclidean space can be written asSGL=R dtR d3rLGL(r);wherer(r;t).Here, the Lagrangian density is LGL(r)=aj(r)j2+b 2j(r)j4+f 3j(r)j6 +X `c`jr`(r)j2 2mid0(r)@(r) @t;(D95) where`=fx;y;zg,b=b(0) andf=f(0). A variation ofSGLwith respect to (r) viaSGL=(r) = 0 yields the time-dependent Ginzburg-Landau (TDGL) equation, id0@ @tX `c`r2 ` 2m+bjj2+fjj4+a! (r) = 0 (D96) with cubic and quintic terms, where = ( r) are de- pendent on space and time. This equation describes the spatio-temporal behavior of the order parameter ( r;t) in the long-wavelength and long-time regime.18 In the static homogeneous case with b>0, Eq. (D96) reduces to either the trivial (normal-state) solution = 0 whena > 0 or to the nontrivial (super uid state) jj=p jaj=b, whena < 0. The coecient dprovides the timescale of the TDGL equation, and thereby deter- mines the lifetime associated with the pairing eld ( r). This can be seen directly by again considering the ho- mogeneous case to linear order in ( r), in which case the TDGL equation has the solution ( t)(0)eiat=d 0: This last expression can be rewritten more explicitly as (t)(0)ei!0tet=0;where!0=jajdR=jd0j2is the oscillation frequency of the pairing eld, and 0= jd0j2=(jajdI) is the lifetime of the pairs, where both dR anddIare positive denite, that is, dR>0 anddI>0. In the BEC regime, where stable two-body bound states exist, the imaginary part of d0vanishes (dI= 0), and the lifetime time of the pairs is innitely long. In this case,d0=dRand we can dene the eective bosonic wave function =pdR to recast Eq. (D96) in the form of the Gross-Pitaevskii equation, i@ @tX `r2 ` 2M`+U2j j2+U3j j4B! (r) = 0; (D97)with cubic and quintic nonlinearities, where = ( r), to describe a dilute Bose gas. Here, B=a=dRis the bosonic chemical potential, M`=m(dR=c`) are the anisotropic masses of the bosons, and U2=b=d2 Rand U3=f=d3 Rrepresent contact interactions of two and three bosons, respectively. In the Bose regime, the life- timeof the composite boson is /1=dI!1 and the interactions U2andU3are always repulsive, thus leading to a system consisting of a dilute gas of stable bosons. In this regime, the chemical potential of the bosons is B2+Eb<0, whereEbis the two-body bound state energy in the presence of spin-orbit coupling and Zeeman elds obtained from the condition 1(q;E2) = 0 discussed in the main text. Notice that when B!0, in the absence of boson-boson interactions, the bosons condense. [1] Y-J. Lin, R. L. Compton, K. Jimin ez-Garc a, J. V. Porto, and I. B. Spielman, Synthetic magnetic elds for ultra- cold neutral atoms, Nature (London) 462, 628 (2009). [2] Y-J. Lin, R. L. Compton, A. R. Perry, W. D. Phillips, J. V. Porto, and I. B. Spielman, Bose-Einstein condensate in a uniform light-induced vector potential, Phys. Rev. Lett. 102, 130401 (2009). [3] C. J. Kennedy, W. C. Burton, W. C. Chung and W. Ketterle, Observation of Bose{Einstein condensation in a strong synthetic magnetic eld, Nature Physics, 11, 859 (2015). [4] Y.-J. Lin, K. Jim enez-Garc a and I. B. Spielman, Spin{orbit-coupled Bose{Einstein condensates, Nature 471, 83 (2011). [5] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah, W. S. Bakr, and M. W. Zwierlein, Spin-injection spec- troscopy of a spin-orbit coupled Fermi gas, Phys. Rev. Lett. 109, 095302 (2012). [6] P. Wang, Z.-Q. Yu, Z, Fu, J, Miao, L, Huang, S. Chai, H. Zhai, and J. Zhang, Spin-orbit coupled degenerate Fermi gases, Phys. Rev. Lett. 109, 095301 (2012). [7] R. A. Williams, M. C. Beeler, L. J. LeBlanc, K. Jim enez- Garc a and I. B. Spielman, Raman-Induced Interactions in a Single-Component Fermi Gas Near an s-Wave Fesh- bach Resonance, Phys. Rev. Lett. 111, 095301 (2013). [8] V. Galitski and I. B. Spielman, Spin{orbit coupling in quantum gases, Nature 494, 49 (2013). [9] Z. Fu, L. Huang, Z. Meng, P. Wang, L. Zhang, S. Zhang, H. Zhai, P. Zhang, and J. Zhang, Production of Feshbach molecules induced by spin-orbit coupling in Fermi gases, Nat. Phys. 10, 110 (2014). [10] M. Mancini, G. Pagano, G. Cappellini, L. Livi, M. Rider,J. Catani, C. Sias, P. Zoller, M. Inguscio, M. Dalmonte, and L. Fallani, Observation of chiral edge states with neutral fermions in synthetic Hall ribbons, Science 349, 1510 (2015). [11] L. Huang, Z. Meng, P. Wang, P. Peng, S.-L. Zhang, L. Chen, D. Li, Q. Zhou, and J. Zhang, Experimental real- ization of two-dimensional synthetic spin{orbit coupling in ultracold Fermi gases, Nature Physics 12, 540 (2016). [12] Z. Wu, L.Zhang, W. Sun, X.-T. Xu, B.-Z. Wang, S.-C. Ji, Y. Deng, S. Chen, X.-J. Liu, and J.-W. Pan, Realization of two-dimensional spin-orbit coupling for Bose-Einstein condensates Science 354, 83 (2016). [13] J. I. Cirac and P. Zoller, Goals and opportunities in quan- tum simulation, Nature Phys. 8, 264 (2012). [14] U.-J. Wiese, Ultracold quantum gases and lattice sys- tems: Quantum simulation of lattice gauge theories, An- nalen der Physik 525, 777 (2013). [15] E. Zohar, J. I. Cirac, and B. Reznik, Quantum simula- tions of lattice gauge theories using ultracold atoms in optical lattices, Rep. Prog. Phys. 79, 1 (2015). [16] M. Dalmonte and S. Montangero, Lattice gauge theory simulations in the quantum information era, Contemp. Phys. 57, 388 (2016). [17] M. Gong, S. Tewari, and C. Zhang, BCS-BEC crossover and topological phase transition in 3D spin-orbit cou- pled degenerate Fermi gases Phys. Rev. Lett. 107, 195303 (2011). [18] Z.-Q. Yu and H. Zhai, Spin-orbit coupled Fermi gases across a Feshbach resonance, Phys. Rev. Lett. 107, 195305 (2011). [19] H. Hu, L. Jiang, X.-J. Liu, and H. Pu, Probing anisotropic super uidity in atomic Fermi gases with19 Rashba spin-orbit coupling, Phys. Rev. Lett. 107, 195304 (2011). [20] L. Han and C. A. R. S a de Melo, Evolution from BCS to BEC super uidity in the presence of spin-orbit coupling, Phys. Rev. A 85, 011606(R) (2012). [21] K. Seo, L. Han, and C. A. R. S a de Melo, Emergence of Majorana and Dirac particles in ultracold fermions via tunable interactions, spin-orbit eects, and Zeeman elds, Phys. Rev. Lett. 109, 105303 (2012). [22] K. Seo, L. Han, and C. A. R. S a de Melo, Topological phase transitions in ultracold Fermi super uids: The evo- lution from Bardeen-Cooper-Schrieer to Bose-Einstein- condensate super uids under articial spin-orbit elds, Phys. Rev. A 85, 033601 (2012). [23] D. M. Kurkcuoglu and C. A. R. S a de Melo, Formation of Feshbach molecules in the presence of articial spin-orbit coupling and Zeeman elds, Phys. Rev. A 93, 023611 (2016). [24] N. Goldman, I. Satija, P. Nikolic, A. Bermudez, M. A. Martin-Delgado, M. Lewenstein, and I. B. Spielman, Re- alistic time-reversal invariant topological Insulators with neutral atoms, Phys. Rev. Lett. 105, 255302 (2010). [25] I. B. Spielman, private communication. [26] G. Juzeli unas, J. Ruseckas, and J. Dalibard, Generalized Rashba-Dresselhaus spin-orbit coupling for cold atoms, Phys. Rev. A 81, 053403 (2010). [27] D. L. Campbell, G. Juzeli unas, and I. Spielman, Realistic Rashba and Dresselhaus spin-orbit coupling for neutral atoms Phys. Rev. A 84, 025602 (2011). [28] G. Dresselhaus, Spin-orbit coupling eects in zinc blende structures Phys. Rev. 100, 580 (1955). [29] E. I. Rashba, Properties of semiconductors with an ex- tremum loop: I. Cyclotron and combinational resonance in a magnetic eld perpendicular to the plane of the loop, Sov. Phys. Solid State 2, 1224 (1960) [30] C. A. R. S a de Melo, When fermions become bosons: Pairing in ultracold gases, Physics Today vol. 61issue 10, page 45 (2008). [31] T. Ozawa and G. Baym, Population imbalance and pair- ing in the BCS-BEC crossover of three-component ultra- cold fermions, Phys. Rev. A 82, 063615 (2010). [32] P. D. Powell, From Quarks to Cold Atoms: The Phases of Strongly-Interacting Systems, Ph.D. Thesis, University of Illinois at Urbana-Champaign (2013). [33] D. M. Kurkcuoglu and C. A. R. S a de Melo, Color super- uidity of neutral ultracold fermions in the presence of color- ip and color-orbit elds, Phys. Rev. A 97, 023632 (2018) [34] Daisuke Yamamoto, I. B. Spielman, and C. A. R. S a de Melo, Quantum phases of two-component bosons with spin-orbit coupling in optical lattices, Phys. Rev. A 96, 061603(R) (2017). [35] S. L. Bromley, S. Kolkowitz, T. Bothwell, D. Kedar, A. Safavi-Naini, M. L. Wall, C. Salomon, A. M. Rey, and J. Ye, Dynamics of interacting fermions under spin{orbit coupling in an optical lattice clock, Nature Physics 14, 399 (2018). [36] Yu Yi-Xiang, Fadi Sun, and Jinwu Ye, Class of topo- logical phase transitions of Rashba spin-orbit coupled fermions on a square lattice, Phys. Rev. B 98, 174506 (2018). [37] M. Mamaev, R. Blatt, J Ye, and A. M. Rey, Cluster state generation with spin-orbit coupled fermionic atoms in optical lattices, Phys. Rev. Lett. 122, 160402 (2019).[38] Baihua Gong, Shuai Li, Xin-Hui Zhang, Bo Liu, and Wei Yi, Bloch bound state of spin-orbit-coupled fermions in an optical lattice, Phys. Rev. A 99, 012703 (2019). [39] Man Hon Yau and C. A. R. S a de Melo Chern-number spectrum of ultracold fermions in optical lattices tuned independently via articial magnetic, Zeeman, and spin- orbit elds, Phys. Rev. A 99, 043625 (2019) [40] Wei Jia, Zhi-Hao Huang, Xian Wei, Qing Zhao, and Xiong-Jun Liu, Topological super uids for spin-orbit cou- pled ultracold Fermi gases, Phys. Rev. B 99, 094520 (2019). [41] Majid Kheirkhah, Zhongbo Yan, Yuki Nagai, and Frank Marsiglio, First- and second-order topological supercon- ductivity and temperature-driven topological phase tran- sitions in the extended hubbard model with spin-orbit coupling, Phys. Rev. Lett. 125, 017001 (2020). [42] Urs Gebert, Bernhard Irsigler, and Walter Hofstetter, Local Chern marker of smoothly conned Hofstadter fermions, Phys. Rev. A 101, 063606 (2020). [43] Irakli Titvinidze, Julian Legendre, Maarten Grothus, Bernhard Irsigler, Karyn Le Hur, and Walter Hofstet- ter, Spin-orbit coupling in the kagome lattice with ux and time-reversal symmetry, Phys. Rev. B 103, 195105 (2021). [44] R. A. Williams, L. J. LeBlanc, K. Jim enez-Garc a, M. C. Beeler, A. R. Perry, W. D. Phillips, and I. B. Spielman, Synthetic partial waves in ultracold atomic collisions, Sci- ence335, 314 (2012). [45] Z. Meng, L. Huang, P. Peng, D. Li, L. Chen, Y. Xu, C. Zhang, P. Wang, and J. Zhang Experimental obser- vation of a topological band gap opening in ultracold Fermi gases with two-dimensional spin-orbit coupling, Phys. Rev. Lett. 117, 235304 (2016). [46] L. Huang, Z. Meng, P. Wang, P. Peng, S.-L. Zhang, L. Chen, D. Li, Q. Zhou, and J. Zhang, Experimental real- ization of two-dimensional synthetic spin{orbit coupling in ultracold Fermi gases, Nature Physics 12, 540 (2016). [47] A. Vald es-Curiel, D. Trypogeorgos, Q.-Y. Liang, R. P. Anderson, and I. B. Spielman, Topological features with- out a lattice in Rashba spin-orbit coupled atoms, Nature Communications 12, 593 (2021). [48] Ronen M. Kroeze, Yudan Guo, and Benjamin L. Lev, Dynamical spin-orbit coupling of a quantum gas Phys. Rev. Lett. 123, 160404 (2019). [49] Zong-Yao Wang, Xiang-Can Cheng, Bao-Zong Wang, Jin-Yi Zhang, Yue-Hui Lu, Chang-Rui Yi, Sen Niu, You- jin Deng, Xiong-Jun Liu, Shuai Chen, and Jian-Wei Pan, Realization of an ideal Weyl semimetal band in a quan- tum gas with 3D spin-orbit coupling, Science 372, 271 (2021). [50] A. J. Leggett in Modern Trends in the Theory of Con- densed Matter , pp. 13{27, edited by A. Pekalski and R. Przystawa, Springer-Verlag, Berlin (1980). [51] J. R. Engelbrecht, M. Randeria, and C. A. R. S a de Melo, BCS to Bose crossover: Broken-symmetry state, Phys. Rev. B 55, 15153 (1997). [52] C. A. R. S a de Melo, M. Randeria, and J. Engelbrecht, Crossover from BCS to Bose superconductivity: Transi- tion temperature and time-dependent Ginzburg-Landau theory, Phys. Rev. Lett. 71, 3202 (1993). [53] P. D. Powell, G. Baym, C. A. R. S a de Melo, Super uid transition temperature of spin-orbit and Rabi coupled fermions with tunable interactions, arXiv:1709.07042v1 (2017).20 [54] S. Gopalakrishnan, A. Lamacraft, and P. M. Gold- bart, Universal phase structure of dilute Bose gases with Rashba spin-orbit coupling, Phys. Rev. A 84, 061604 (2011). [55] T. Ozawa, Topics in Multi-component Ultracold Gases and Gauge Fields, Ph.D. Thesis, University of Illinois at Urbana-Champaign (2012). [56] P. Nozi eres and S. Schmitt-Rink, Bose condensation in an attractive fermion gas: From weak to strong coupling superconductivity, J. Low Temp. Phys. 59, 195 (1985). [57] Z. Yu and G. Baym, Spin correlation functions in ul- tracold paired atomic-fermion systems: Sum rules, self- consistent approximations, and mean elds, Phys. Rev. A73, 063601 (2006). [58] Z. Yu, G. Baym, and C. J. Pethick, Calculating energy shifts in terms of phase shifts, J. Phys. B: At. Mol. Opt. Phys. 44, 195207 (2011). [59] P. Fulde and R. A. Ferrell, Superconductivity in a strong spin-exchange eld, Phys. Rev. 135, A550 (1964). [60] A. I. Larkin and Y. N. Ovchinnikov, Nonuniform state of superconductors, Sov. Phys. JETP 20, 762 (1965). [61] A. M. Clogston, Upper limit for the critical eld in hard superconductors, Phys. Rev. Lett. 9, 266 (1962). [62] G. Sarma, On the in uence of a uniform exchange eld acting on the spins of the conduction electrons in a su- perconductor, J. Phys. Chem. Sol. 24, 1029 (1963). [63] G. Baym, J.-P. Blaizot, M. Holzmann, F. Lalo e and D. Vautherin, The transition temperature of the dilute in- teracting Bose gas, Phys. Rev. Lett. 83, 1703 (1999). [64] M. Iskin and C. A. R. S a de Melo, Fermi-Fermi mixtures in the strong-attraction limit, Phys. Rev. A 77, 013625 (2008). [65] D. S. Petrov, C. Salomon, and G. V. Shlyapnikov, Di- atomic molecules in ultracold Fermi gases: Novel com- posite bosons, J. Phys. B 38, S645 (2005). [66] J. P. Vyasanakere, S. Zhang, and V. B. Shenoy, BCS- BEC crossover induced by a synthetic non-Abelian gauge eld, Phys, Rev. B 84, 014512 (2011) [67] X.-J. Feng and L. Yin, Phase diagram of a spin-orbit coupled dipolar Fermi gas at T= 0 K, Chin. Phys. Lett. 37, 020301 (2020).[68] L. Dell'Anna, G. Mazzarella, and L. Salasnich, Conden- sate fraction of a resonant Fermi gas with spin-orbit cou- pling in three and two dimensions, Physical Review A 84, 033633 (2011). [69] L. Dell'Anna, G. Mazzarella, and L. Salasnich, Tuning Rashba and Dresselhaus spin-orbit couplings: Eects on singlet and triplet condensation with Fermi atoms, Phys- ical Review A 86, 053632 (2012). [70] S. Tewari, T. D, Stanescu, J. D. Sau and S. Das Sarma, Topologically non-trivial superconductivity in spin{orbit-coupled systems: Bulk phases and quantum phase transitions, New Journal of Physics, 13, 065004 (2011). [71] Li Han and C. A. R. S a de Melo, Ultra-cold fermions in the atland: Evolution from BCS to Bose super uid- ity in two-dimensions with spin-orbit and Zeeman elds, arXiv:1206.4984v1 (2012) [72] J. P. A. Devreese, J. Tempere, and C. A. R. S a de Melo Eects of spin-orbit coupling on the Berezinskii- Kosterlitz-Thouless transition and the vortex-antivortex structure in two-dimensional Fermi gases, Phys. Rev. Lett. 113, 165304 (2014). [73] J. P. A. Devreese, J. Tempere, and C. A. R. S a de Melo Quantum phase transitions and Berezinskii-Kosterlitz- Thouless temperature in a two-dimensional spin-orbit- coupled Fermi gas, Phys. Rev. A 92, 043618 (2015) [74] J. P. Vyasanakere and V. B. Shenoy, Fluctuation theory of Rashba Fermi gases: Gaussian and beyond, Phys. Rev. B.92, 121111(R) (2015). [75] T. Yamaguchi, D. Inotani, and Y. Ohashi, Rashbon bound states associated with a spherical spin{orbit cou- pling in an ultracold Fermi gas with an s-wave interac- tion, J. Low Temp. Phys. 183, 161 (2016). [76] B. M. Anderson, C.-T. Wu, R. Boyack, and K. Levin Topological eects on transition temperatures and re- sponse functions in three-dimensional Fermi super uids, Phys. Rev. B 92, 134523 (2015). [77] L. Dell'Anna, and S. Grava, Critical temperature in the BCS-BEC crossover with spin-orbit coupling, Condens. Matter 6, 16 (2021).
1110.6364v1.Artificial_spin_orbit_coupling_in_ultra_cold_Fermi_superfluids.pdf	arXiv:1110.6364v1 [cond-mat.quant-gas] 28 Oct 2011Artiﬁcial spin-orbit coupling in ultra-cold Fermi superﬂu ids Kangjun Seo, Li Han and C. A. R. S´ a de Melo School of Physics, Georgia Institute of Technology, Atlant a, Georgia 30332, USA (Dated: November 23, 2018) The control and understanding of interactions in many parti cle systems has been a major chal- lenge in contemporary science, from atomic to condensed mat ter and astrophysics. One of the most intriguing types of interactions is the so-called spin -orbit coupling - the coupling between the spin (rotation) of a particle and its momentum (orbital moti on), which is omnipresent both in the macroscopic and microscopic world. In astrophysics, the sp in-orbit coupling is responsible for the synchronization of the rotation (spinning) of the Moon and i ts orbit around Earth, such that we can only see one face of our natural satellite. In atomic phys ics, the spin-orbit coupling of electrons orbiting around the nucleus gives rise to the atom’s ﬁne stru cture (small shifts in its energy levels). In condensed matter physics, spin-orbit eﬀects are respons ible for exotic electronic phenomena in semiconductors (topological insulators) and in supercond uctors without inversion symmetry. Al- though spin-orbit coupling is ubiquitous in nature, it was n ot possible to control it in any area of physics, until it was demonstrated in a breakthrough experi ment [1] that the spin of an atom could be coupled to its center-of-mass motion by dressing twoatom ic spin states with apair of laser beams. This unprecedented engineered spin-orbit coupling was pro duced in ultra-cold bosonic atoms, but can also be created for ultra-cold fermionic atoms [1–3]. In anticipation of experiments, we develop a theory for interacting fermions in the presence of spin-or bit coupling and Zeeman ﬁelds, and show that many new superﬂuids phases, which are topological in na ture, emerge. Depending on values of spin-orbit coupling, Zeeman ﬁelds, and interactions, in itially gapped s-wave superﬂuids acquire p-wave,d-wave,f-wave and higher angular momentum components, which produc e zeros in the excitation spectrum, rendering the superﬂuid gapless. Sev eral multi-critical points, which separate topological superﬂuid phases from normal or non-uniform, a re accessible depending on spin-orbit coupling, Zeeman ﬁelds or interactions, setting the stage f or the study of tunable topological super- ﬂuids. PACS numbers: 03.75.Ss, 67.85.Lm, 67.85.-d The eﬀects of spin-orbit coupling in few body systems like the Earth-Moon complex in astrophysics or the elec- tron spin and its orbital motion around the nucleus in isolated atoms of atomic physics are reasonably well un- derstood due to the simpliﬁcity of these systems. How- ever, in the setting of many identical particles, spin-orbit eﬀects have revealed quite interesting surprises recently running from topological insulators in semiconductors [4] to exotic superconductivity [5] and non-equillibrium ef- fects [6] depending on the precise form of the spin-orbit coupling. In atomic physics the coupling arises from the interaction of the magnetic moment of the electron and a magnetic ﬁeld, present in the frame of electron, due to the electric ﬁeld of the nucleus. Similarly in con- densed matter physics, the coupling arises from the mag- netic moment mof electrons, which move in the back- ground of ions. In the electron’s reference frame, these ions are responsible for a magnetic ﬁeld B, which de- pends on the electron’s momentum kand couple to elec- tron’s spin. The resulting spin-orbit coupling has the formHSO=−m·B=−/summationtext jhj(k)σj,whereσjrep- resents the Pauli matrices and hj(k) describes the j-th component ( j=x,y,z) of the eﬀective magnetic ﬁeld vectorh. For some materials hcan take the Dressel- haus [7] form hD(k) =vD(kyˆx+kxˆy),the Rashba [8] formhR(k) =vR(−kyˆx+kxˆy),or more generally a lin- ear combination of the two h⊥(k) =hD(k)+hR(k).In all these situations the type of spin-orbit coupling cannot be changed arbitrarily and the magnitude can not be tuned from weak to strong, making the experimental control of spin-orbit eﬀects very diﬃcult. Recently, however, it has been demonstrated experi- mentally that spin-orbit coupling can be engineered in a ultra-cold gas of bosonic atoms in their Bose-Einstein condensatephase[1], whenapairofRamanlaserscreates a coupling between two internal spin states of the atoms and its center-of-mass motion (momentum). Thus far, the type of spin-orbit ﬁeld that has been created in the laboratory [1] has the equal-Rashba-Dresselhaus (ERD) formh⊥(k) =hERD(k) =vkxˆy, wherevR=vD=v/2. Other forms of spin-orbit ﬁelds require additional lasers and create further experimental diﬃculties [9]. In ultra- coldbosonsthemomentum-dependent ERDcouplinghas been created in conjunction with uniform Zeeman terms, which are independent of momentum, along the z axis (controlled by the Raman coupling Ω R), and along the y-axis (controled by the detuning δ). The simultaneous presence of hz,hyandhERD(k) leads to the Zeeman- spin-orbit (ZSO) Hamiltonian HZSO(k) =−hzσz−hyσy−hERD(k)σy for an atom with center-of-mass momentum kand spin basis\| ↑/an}b∇acket∇i}ht,\| ↓/an}b∇acket∇i}ht. The ﬁelds hz=−ΩR/2,hy=−δ/2 and hERD=vkxˆycan be controlled independently, and thus can be used as tunable parameters to explore the avail- able phase space and to investigate phase transitions, as2 achieved in the experiment involving a bosonic isotope of Rubidium (87Rb). Although current experiments have focused on Bose atoms, there is no fundamental reason that impeeds the realization of a similar set up for Fermi atoms [1–3] designed to study fermionic superﬂuidity [3]. Considering possible experiments with fermionic atoms such as6Li,40K, we discuss in this letter phase diagrams, topological phase transitions, spectroscopic and thermo- dynamic properties at zero and ﬁnite temperatures dur- ing the evolution from BCS to BEC superﬂuidity in the presence of controllable Zeeman and spin-orbit ﬁelds in three dimensions. To investigate artiﬁcial spin-orbit and Zeeman ﬁelds in ultra-cold Fermi superﬂuids, we start from the Hamilto- nian density H(r) =H0(r)+HI(r), (1) where the single-particle term is simply H0(r) =/summationdisplay αβψ† α(r) ˆKαδαβ−/summationdisplay jˆhj(r)σj,αβ ψβ(r). (2) Here,ˆKα=−∇2/(2m)−µαis the kinetic energyin refer- ence to the chemical potential µα,ˆhj(r) is the combined eﬀective ﬁeld including Zeeman and spin-orbit compo- nents along the j-direction ( j=x,y,z), andψ† α(r) are creation operators for fermions with spin αat position r. Notice that we allow the chemical potential µ↑to be diﬀerent from µ↓, such that the number of fermions N↑ with spin ↑may be diﬀerent from the number of fermions with spin ↓. The interaction term is HI(r) =−gψ† ↑(r)ψ† ↓(r)ψ↓(r)ψ↑(r), (3) wheregrepresents a contact interaction that can be expressed in terms of the scattering length via the Lippman-Schwinger relation V/g=−Vm/(4πas) +/summationtext k1/(2ǫk).The introduction of the averagepairing ﬁeld ∆(r)≡g/an}b∇acketle{tψ↓(r)ψ↑(r)/an}b∇acket∇i}ht ≈∆0and its spatio-temporal ﬂuc- tuationη(r,τ) produce a complete theory for superﬂuid- ity in this system. From now on, we focus on the experimental case where a) the Raman detuning is zero ( δ= 0) indicating that there is no component of the Zeeman ﬁeld along the ydi- rection; b) the Raman coupling Ω Ris non-zero meaning that a Zeeman component along the zdirection exists, that is,hz=−ΩR/2; and c) the spin-orbit ﬁeld has com- ponentshy(k) andhx(k) alongtheyandxdirections. To start our discussion, we neglect ﬂuctuations, and trans- formH0(r) into momentum space as H0(k). Using the basisψ† ↑(k)\|0/an}b∇acket∇i}ht ≡ \|k↑/an}b∇acket∇i}ht, ψ† ↓(k)\|0/an}b∇acket∇i}ht ≡ \|k↓/an}b∇acket∇i}ht,where\|0/an}b∇acket∇i}htis the vacuum state, the Fourier-transformed Hamiltonian H0(k) becomes the matrix H0(k) =K+(k)1+K−σz−hzσz−hy(k)σy−hx(k)σx, Such matrix can be diagonalized in the helicity basis Φ† ⇑(k)\|0/an}b∇acket∇i}ht ≡ \|k⇑/an}b∇acket∇i}ht,Φ† ⇓(k)\|0/an}b∇acket∇i}ht ≡ \|k⇓/an}b∇acket∇i}ht,where the spins⇑and⇓are aligned or antialigned with respect to the eﬀective magnetic ﬁeld heﬀ(k) =h/bardbl(k) +h⊥(k).Here, K+(k) = (K↑+K↓)/2 =ǫk−µ+,is a measure of the average kinetic energy ǫk=k2/2min relation to the average chemical potential µ+= (µ↑+µ↓)/2.While h⊥(k) =hx(k)ˆx+hy(k)ˆyis the spin-orbit ﬁeld and h/bardbl(k) = (hz−K−)ˆzis the eﬀective Zeeman ﬁeld, with K−= (K↑−K↓)/2 =−µ−whereµ−= (µ↑−µ↓)/2 is the internal Zeeman ﬁeld due to initial population im- balance, and hzis the external Zeeman ﬁeld. When there is no population imbalance the internal Zeeman ﬁeld isµ−= 0, and we have only hz. In general, the eigenvalues of the Hamiltonian matrix H0(k) are ξ⇑(k) =K+(k)−\|heﬀ(k)\|andξ⇓(k) =K+(k)+\|heﬀ(k)\|, where\|heﬀ(k)\|=/radicalbig (µ−+hz)2+\|h⊥(k)\|2is the magni- tude of the eﬀective magnetic ﬁeld, with the transverse component being expressedin termsof the complex func- tionh⊥(k) =hx(k) +ihy(k).In the limit where the in- ternalµ−and external hzZeeman ﬁelds vanish and the spin-orbit ﬁeld is null ( h⊥= 0), the energies of the helic- ity bands are identical ξ⇑(k) =ξ⇓(k) producing no eﬀect in the original energy dispersions [10]. When interactions are added to the problem, pair- ing can occur within the same helicity band (intra- helicity pairing) or between two diﬀerent helicity bands (inter-helicity pairing). This leads to a tensor order parameter for superﬂuidity that has four components ∆⇑⇑(k) =−∆T(k)e−iϕ,corresponding to the helicity projectionλ= +1; ∆ ⇑⇓(k) =−∆S(k),and ∆ ⇓⇑(k) = ∆S(k),corresponding to helicity projection λ= 0; and ∆⇓⇓(k) =−∆T(k)eiϕ,corresponding to helicity pro- jectionλ=−1. The phase ϕ(k) is deﬁned from the amplitude-phase representation of the complex spin- orbit ﬁeldh⊥(k) =\|h⊥(k)\|eiϕ(k),while the amplitude ∆T(k) = ∆0\|h⊥(k)\|/\|heﬀ(k)\|for helicities λ=±1aredi- rectly proportional to the scalar order parameter ∆ 0and to the relative magnitude of the spin-orbit ﬁeld \|h⊥(k)\| with respect to the magnitude of the eﬀective magnetic ﬁeld\|heﬀ(k)\|. Additionally, ∆ Thas the simple physi- cal interpretation of being the triplet component of the order parameter in the helicity basis, which is induced by the presence of non-zero spin-orbit ﬁeld h⊥, but van- ishes when h⊥= 0. Analogously the amplitude ∆ S(k) = ∆0h/bardbl(k)/\|heﬀ(k)\|for helicity λ= 0 are directly propor- tional to the scalar order parameter ∆ 0and to the rela- tive magnitude of the total Zeeman ﬁeld h/bardbl(k) =µ−+hz with respect to the magnitude of the eﬀective magnetic ﬁeld\|heﬀ(k)\|. Additionally, ∆ Shas the simple physical interpretationofbeingthesingletcomponentofthe order parameter in the helicity basis. It is interesting to note the relation \|∆T(k)\|2+\|∆S(k)\|2=\|∆0\|2,which, for ﬁxed \|∆0\|, shows that as \|∆S(k)\|increases, \|∆T(k)\|decreases and vice-versa. Such relation indicates that the singlet and triplet channels are not separable in the presence of spin-orbitcoupling. Furthermore, the orderparameterin the triplet sector ∆ ⇑⇑(k) and ∆ ⇓⇓(k) contains not only p-wave,but also f-waveand evenhigherodd angularmo- mentum contributions, as long as the total Zeeman ﬁeld3 µ−+hzis non-zero. Similarly, the orderparameterin the singlet sector ∆ ⇑⇓(k) and ∆ ⇓⇑(k) contains not only only s-wave, but also d-wave and even higher even angular momentum contributions, as long as the total Zeeman ﬁeldµ−+hzis non-zero. Higher angular momentum pairing in the helicity basis, occurs because the original local (zero-ranged) interaction in the original ( ↑,↓) spin basis is transformed into a ﬁnite-ranged interaction in the helicity basis ( ⇑,⇓). In the limiting case of zero total Zeemanﬁeld µ−+hz= 0, the singletcomponentvanishes (∆S(k) = 0), while the triplet component becomes inde- pendent of momentum (∆ T(k) = ∆ 0), leading to order parameter ∆ ⇑⇑(k) =−h∗ ⊥(k), and ∆ ⇓⇓(k) =−h⊥(k) which contains only p-wave contributions [11], since the components of h⊥(k) depend linearly on momentum k. The eigenvalues Ej(k) of the Hamiltonian including the order parameter contribution emerge from the diago- nalization ofa 4 ×4 matrix (see supplementary material). The two eigenvalues for quasiparticles are E1(k) =/radicalbigg/parenleftig ξh−−/radicalig ξ2 h++\|∆S(k)\|2/parenrightig2 +\|∆T(k)\|2,(4) corresponding to the highest-energy quasiparticle band, and E2(k) =/radicalbigg/parenleftig ξh−+/radicalig ξ2 h++\|∆S(k)\|2/parenrightig2 +\|∆T(k)\|2,(5) corresponding to the lowest-energy quasiparticle band, while the eigenvalues for quasiholes are E3(k) =−E2(k) for highest-energy quasihole band and E4(k) =−E1(k) for the lowest-energy quasihole band. The energy ξh−= [ξ⇑(k)−ξ⇓(k)]/2 is momentum-dependent, corresponds to the average energy diﬀerence between the helicity bands and can be written as ξh−=−\|heﬀ(k)\|,while the energy ξh+= [ξ⇑(k)+ξ⇓(k)]/2 is also momentum dependent, corresponds to the averaged energy sum of the helicity bands and can be written as ξh+=K+(k) = ǫk−µ+. There are a few important points to notice about the excitation spectrum of this system. First, notice that E1(k)> E2(k)≥0. Second, that the eigenergies are symmetric about zero, such that we can regard quasi- holes (negative energy solutions) as anti-quasiparticles. Third, that only E2(k) can have zeros (nodal regions) correspondingto the locus in momentum space satisfying the following conditions: a) ξh−=−/radicalig ξ2 h++\|∆S(k)\|2, which corresponds physically to the equality between the eﬀective magnetic ﬁeld energy \|heﬀ(k)\|and the excita- tion energy for the singlet component/radicalig ξ2 h++\|∆S(k)\|2; and b)\|∆T(k)\|= 0,corresponding to zeros of the triplet component of the order parameter in momentum space. SinceE2(k)< E1(k), and only E2(k) can have ze- ros, the low energy physics is dominated by this ein- genvalue. In the case of equal Rashba-Dresselhaus (ERD) where h⊥(k) =v\|kx\|, zeros ofE2(k) can oc- cur whenkx= 0, leading to the following cases: (a)two possible lines (rings) of nodes at ( k2 y+k2 z)/(2m) = µ++/radicalbig (µ−+hz)2−\|∆0\|2for the outer ring, and ( k2 y+ k2 z)/(2m) =µ+−/radicalbig (µ−+hz)2−\|∆0\|2forthe innerring, when (µ−+hz)2− \|∆0\|2>0; (b) doubly-degenerate line of nodes ( k2 y+k2 z)/(2m) =µ+forµ+>0, doubly- degenerate point nodes for µ+= 0, or no-line of nodes forµ+<0, when (µ−+hz)2− \|∆0\|2= 0; (c) no line of nodes when ( µ−+hz)2− \|∆0\|2<0. In ad- dition, case (a) can be reﬁned into cases (a2), (a1) and (a0). In case (a2), two rings indeed exist pro- vided that µ+>/radicalbig (µ−+hz)2−\|∆0\|2. However, the inner ring disappears when µ+=/radicalbig (µ−+hz)2−\|∆0\|2. In case (a1), there is only one ring when \|µ+\|</radicalbig (µ−+hz)2−\|∆0\|2,In case (a0), the outer ring dis- appears at µ+=−/radicalbig (µ−+hz)2−\|∆0\|2, and forµ+< −/radicalbig (µ−+hz)2−\|∆0\|2no rings exist. We choose our momentum, energy and velocity scales through the Fermi momentum kF+deﬁned from the to- tal density of fermions n+=n↑+n↓=k3 F+/(3π2).This choice leads to the Fermi energy ǫF+=k2 F+/2mand to the Fermi velocity vF+=kF+/m, as energy and veloc- ity scales respectively. In Fig. 1, we show the phase diagram of Zeeman ﬁeld hz/ǫF+versus chemical poten- tialµ+/ǫF+describing possible superﬂuid phases accord- ing to their quasiparticle excitation spectrum. We la- bel the uniform superﬂuid phases with zero, one or two rings of nodes as US-0, US-1, and US-2, respectively. Non-uniform (NU) phases also emerge in regions where uniform phases are thermodynamically unstable. The US-2/US-1 phase boundary is determined by the condi- tionµ+=/radicalbig (µ−+hz)2−\|∆0\|2, when\|µ−+hz\|>\|∆0\|; the US-0/US-2 boundary is determined by the Clogston- like condition \|(µ−+hz)\|=\|∆0\|whenµ+>0, where the gapped US-0 phase disappears leading to the gap- less US-2 phase; and the US-0/US-1 phase boundary is determined by µ+=−/radicalbig (µ−+hz)2−\|∆0\|2, when \|µ−+hz\|>\|∆0\|. Furthermore, with the US-0 bound- aries, a crossover line between an indirectly gapped and a directly gapped US-0 phase occurs at µ+= 0. Lastly, some important multi-critical points arise at the inter- sections of phase boundaries. First the point µ+= 0 and \|(µ−+hz)\|=\|∆0\|corresponds to a tri-critical point for phasesUS-0, US-1, and US-2. Second, the point \|∆0\|= 0 andµ+=\|(µ−+hz)\|corresponds to a tri-critical point for phases N, US-1 and US-2. In the limit where both µ−andhzvanish no phase transitions take place and the problem is reduced to a crossover [12–14]. In the US-1 and US-2 phases near the zeros of E2(k), quasiparticles have linear dispersion and behave as Dirac fermions. Such change in nodal structures is associated with bulk topological phase transitions of the Lifshitz classasnotedfor p-wave[15]and d-wave[16,17]superﬂu- ids. Such Lifshitz topological phase transitions are possi- ble here because the spin-orbit coupling ﬁeld induces the triplet component of the order parameter ∆ T(k). The loss of nodal regions correspond to annihilation of Dirac4 a N NUS-1 US-1 NU NU hz / !F+ "+ / !F+ -2 -1 0 10123 -3 -2 -1 b -2 -1 0 10123 -3 -2 -1 N NUS-1 US-1 Indirect US-0 Direct US-0 "+ / !F+ Indirect US-0 NU NU hz / !F+ Direct US-0 US-2 US-2 FIG. 1: Phase diagram of Zeeman ﬁeld hz/ǫF+versus chem- ical potential µ+/ǫF+for a)v/vF+= 0 and b) v/vF+= 0.28 identifying uniform superﬂuid phases US-0 (gapped), US-1 (gapless with one ring of nodes), and US-2 (gapless with two-rings of nodes). The NU region corresponds to unsta- ble uniform superﬂuids which may include phase separation and/or a modulated superﬂuid (supersolid). Solid lines rep - resent phase boundaries, while the dashed line represents t he crossover from the direct-gap to the indirect-gap US-0 phas e. quasiparticles with opposite momenta, which lead to the disappearance of rings. The transition from phase US- 2 to indirect gapped US-0 occurs through the merger of the two-rings at the phase boundary followed by the im- mediate opening of the indirect gap at ﬁnite momentum. However, the transition from phase US-2 to US-1 corre- sponds to the disappearance of the inner ring through the origin of momenta, similarly the transition from US- 1 to the directly gappped US-0 corresponds to the dis- appearance of the last ring also through the origin of momenta. In the case of Rashba-only coupling rings of nodes are absent and it is possible to have at most nodal points [18, 19]. The last two phase transitions are special because the zero-momentum quasiparticles at thesephaseboundariescorrespondtotrueMajoranazero energy modes if the phase ϕ(k) of the spin-orbit ﬁeld h⊥(k) =\|h⊥(k)\|eiϕ(k)and the phase θ(k) of the order parameter ∆ 0=\|∆0\|eiθ(k)have opposite phases at zero momentum: ϕ(0) =−θ(0) [mod(2π)]. This can be seen from an analysis of the quasiparticle eigenfunction Φ2(k) =u1(k)ψk↑+u2(k)ψk↓+u3(k)ψ† −k↑+u4(k)ψ† −k↓ corresponding to the eigenvalue E2(k). The emergence ofzero-energyMajoranafermionsrequiresthequasiparti- cle to be its own anti-quasiparticle: Φ† 2(k) = Φ2(k). This canonlyhappenatzeromomentum k=0,wheretheam- plitudesu1(0) =u∗ 3(0) andu2(0) =u∗ 4(0). Such require- ment leads to the conditions µ2 += (µ−+hz)2+\|∆0\|2, andϕ(0) =−θ(0) [mod(2π)], showing that Majorana fermions can exist only at the US-0/US-1 and US-2/US- 1 phase boundaries. It is important to emphasize that the Majorana fermions found here exist in the bulk, and thus their emergence or disappeareance aﬀect bulk ther-modynamic properties, unlike Majorana fermions found at the edge (surfaces) of topological insulators and some topological superﬂuids. The common ground between bulk and surface Majorana fermions is that both exist at boundaries: the bulk Majorana zero-energy modes may exist at the phase boundaries between two topologically distinct superﬂuid phases, while surface Majorana zero- energy modes may exist at the spatial boundaries of a topologically non-trivial superﬂuid. It is evident that the transition between diﬀerent su- perﬂuid phases occurs without a change in symmetry in the orderparameter∆ 0, and thus violatesthe symmetry- based Landau classiﬁcation of phase transitions. In the presentcase, the simultaneousexistenceofspin-orbitand Zeeman ﬁelds (internal or external) couple the singlet ∆S(k) and triplet ∆ T(k) channels and all the super- ﬂuid phases US-0, US-1 and US-2 just have diﬀerent weights from each order parameter component. How- ever a ﬁner classiﬁcation based on topological charges can be made via the construction of topological invari- ants. Since the superﬂuid phases US-0, US-1, US-2 are characterized by diﬀerent excitation spectra correspond- ing to the eigenvalues of the Hamiltonian matrix includ- ing interactions H(k), we can use the resolvent matrix R(ω,k) = [−ω1+H(k)]−1and the methods of algebraic topology [20] to construct the topological invariant ℓ=/integraldisplay DdSγ 24π2ǫµνλγTr/bracketleftbig ΛkµΛkνΛkλ/bracketrightbig , whereΛkµ=R∂kµR−1.The topological invariant is ℓ= 0 in the gapped US-0 phase, is ℓ= 1 in the gap- less US-1 phase and ℓ= 2 in the gapless US-2 phase, showing that, for ERD spin-orbit coupling, ℓcounts the number of rings of zero-energy excitations in each super- ﬂuid phase. The integral above has a hyper-surface mea- suredSγandadomain Dthat enclosesthe regionofzeros ofω=Ej(k) = 0. Here µ,ν,λ,γ run from 0 to 3, and kµ has components k0=ω,k1=kx,k2=ky, andk3=kz. The topological invariant measures the ﬂux of the four- dimensional vector Fγ=ǫµνλγTr/bracketleftbig ΛkµΛkνΛkλ/bracketrightbig /24π2, through a hypercube including the singular region of the resolvent matrix R(ω,k), much in the same way that the ﬂux of the electric ﬁeld Ein Gauss’ law of classical elec- tromagnetism measures the electric charge qenclosed by a Gaussian surface:/contintegraltext dS·(ǫ0E) =q. Thus, the topologi- cal invariant deﬁned above deﬁnes the topological charge of fermionic excitations, in the same sense as Gauss’ law for the electric ﬂux deﬁnes the electric charge. A full phase diagram can be constructed only upon veriﬁcation of thermodynamic stability of all the pro- posed phases. For this purpose it becomes imper- ative to investigate the maximum entropy condition (see supplementary material). Independent of any microscopic approximations, the necessary and suﬃ- cient conditions for thermodynamic stability of a given phase are: positive isovolumetric heat capacity CV= T(∂S/∂T)V,{Nα}≥0; positive chemical susceptibility matrixξαβ= (∂µα/∂Nβ)T,V,i.e, eigenvalues of the5 matrix [ξ] are both positive; and positive bulk mod- ulusB= 1/κTor isothermal compressibility κT= −V−1(∂V/∂P)T,{Nα}.Using these conditions, we con- struct the full phase diagramdescribed in Fig. 1 for equal Rasha-Dresselhaus (ERD) spin-orbit coupling. The re- gions, where the uniform superﬂuid phases are unsta- ble are labeled by the abbreviation NU to indicate that non-uniform phases such as phase separation or modu- lated superﬂuid (supersolid) may emerge. In Fig. 2, we showthephasediagramofZeemanﬁeld hz/ǫF+versusin- teraction parameter 1 /(kF+as), for population balanced fermions, where the number of spin-up fermions N↑is equal to the number of spin-down fermions N↓. v / v_F = 0.141 b N 0 1 2 3 4-1 -2 2468 0hz / !F+ US-2 Indirect US-0 Direct US-0 US-1 1/k F+ asNU v / v_F = 0.283 c N US-1 Direct US-0 US-2 Indirect US-0 0 1 2 3 4-1 -2 2468 0hz / !F+ 1/k F+ asNU v / v_F = 0.424 0 1 2 3 4-1 -2 2468 0d NDirect US-0 US-2 Indirect US-0 hz / !F+ 1/k F+ asUS-1 NU v / v_F = 0 a N 0 1 2 3 4-1 -2 2468 0hz / !F+ Indirect US-0 Direct US-0 1/k F+ asNU US-1 FIG.2: Phasediagram ofZeemanﬁeld hz/ǫF+versusinterac- tion 1/(kF+as) showing uniform superﬂuid phases US-0, US- 1, and US-2, and non-uniform (NU) region for a) v/vF+= 0; b)v/vF+= 0.14; c)v/vF+= 0.28; d)v/vF+= 0.56. Solid lines are phase boundaries, the dashed line indicates a crossover from the indirect- to direct-gapped US-0. Since these superﬂuid phases exhibit major changes in momentum-frequency space as evidenced by their single particle excitation spectrum, it is important to explore additional spectroscopic quantitities to characterize fur- ther the nature of these phases and the phase transitions between them. An important quantity is the 4 ×4 resol- vent matrix R(iω,k) =/parenleftbiggG(iω,k)F(iω,k) F†(iω,k)G(iω,k)/parenrightbigg ,(6) from where the spectral density Aα(ω,k) = −(1/π)ImGαα(iω=ω+iδ,k) for spin α=↑,↓can be extracted. The spectral function Aα(ω,k) in the plane of momenta ky-kzwithkx= 0 and frequency ω= 0 reveals the existence of rings of zero-energy excitations in the US-1 and US-2 phases. The density of statesDα(ω) =/summationtext kAα(ω,k) for spinαas a function of frequencyωis also an important spectroscopic quantitywhich is shown in Fig. 3 along with excitation spectra Ej(k) for phases US-1 and US-2 at ﬁxed ERD spin-orbit couplingv/vF+= 0.28. The parameters used for phase US-1 arehz/ǫF+= 0.5 and 1/(kF+as) =−0.4, while for phase US-2 they are hz/ǫF+= 2.0 and 1/(kF+as) = 1.0. Notice that, even though the excitation spectrum Ej(k) is symmetric, the coherence factors appearing in the matrixGare not, such that the density of states Dα(ω) is not an even function of ω, and thus it is not particle- hole symmetric. The main feature of Dα(ω) at low frequencies is the linear behavior due to the existence of Dirac quasiparticles and quasiholes in the US-1 and US-2 phases, which are absent in the direct-gap and the indirect-gap US-0 phases. The peaks and structures in Dα(ω) mostly emerge due to the maxima and minima of Ej(k). Notice that for ﬁnite Zeeman ﬁeld hz, the density of states D↑(ω)/ne}ationslash=D↓(ω) because the induced population imbalanceP= (N↑−N↓)/(N↑+N↓) is non-zero. For the US-2 case shown in Fig. 3b, the induced population imbalanceP≪1 sincehz/ǫF+is small, while for the US-1 case shown in Fig. 3e, P≈1 as the spins are almost fully polarized since hz/ǫF+is large. !!"# $$"# %!!"& !!"' !!"' !"& !!"# $$"# %!!"& !!"' !!"' !"& !"#$%!!&' !!&% !!&% !&' !!"# $$"# %!!"& !!"' !!"' !"& !!"# $$"# %!!"& !!"' !!"' !"& !!"# $$"# %!!"& !!"' !!"' !"& \|k x\| / k F+ Ej / !F+ \|k y\| / k F+ Ej / !F+ \|k x\| / k F+ Ej / !F+ \|k y\| / k F+ Ej / !F+ D( ") !F+ " / !F+ D( ") !F+ " / !F+ a b c f e d FIG. 3: Energy spectrum and density of states in phase US-2 areshown ina), b), c)for hz/ǫF+= 0.5and1/(kF+as) =−0.4 and in phase US-1 are shown in d), e), f) for hz/ǫF+= 2.0 and 1/(kF+as) = 1.0. Energies Ej(kx,0,0) versus \|kx\|in a) and d); frequency ωversus density of states D↑(ω) (dashed), D↓(ω) (dot-dashed), and their sum D(ω) (solid) in b) and e); energies Ej(0,ky,0) versus \|ky\|in c) and f). In summary, we have discussed the eﬀects of spin-orbit and Zeeman ﬁelds in ultra-cold Fermi superﬂuids, ob- tained the phase diagrams of Zeeman ﬁeld versus inter- action parameter or versus chemical potential, and iden- tiﬁed several bulk topological phase transitions between gapped and gapless superﬂuids as well as a variety of multi-critical points. We haveshown that the presenceof simultaneousZeeman and spin-orbitﬁelds induces higher6 angular momentum pairing, as manifested in the emer- gence of momentum dependence ofthe singlet and triplet components of order parameter expressed in the helicity basis. Finally, we have characterized topological phases and phase transitions between them through their exci- tation spectra (existence of Dirac quasiparticles or Majo- rana zero-energy modes), topological charges, and spec- troscopic and thermodynamic properties, such as densityof states and isothermal compressibility. Acknowledgments We thank ARO (W911NF-09-1-0220) for support. [1] Y. J. Lin, K. Jimenez-Garcia, and I. B. Spielman, Nature 471, 83-86 (2011). [2] X. J. Liu, M. F. Borunda, X. Liu, and J. Sinova, Phys. Rev. Lett. 102, 046402 (2009). [3] M. Chapman and C. S´ a de Melo, Nature 471, 41-42 (2011). [4] C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 146802 (2005). [5] L. P. Gor’kov and E. I. Rashba, Phys. Rev. Lett. 87, 037004 (2001). [6] T. D. Stanescu, C. Zhang, and V. Galitski, Phys. Rev. Lett.99110403 (2007). [7] G. Dresselhaus, Phys. Rev. 100, 580 (1955). [8] Y. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6039 (1984). [9] J. Dalibard, F. Gerbier, G. Juzeliunas, and P. Ohberg, arXiv:1008.5378 (2010). [10] J. P. Vyasanakere, S. Zhang, and V. B. Shenoy, arXiv:1104.5633 (2011). [11] C. Zhang, S. Tewari, R. M. Lutchyn, and S. Das Sarma, Phys. Rev. Lett. 101160401 (2008). [12] Z.-Q. Yu and H. Zhai, arXiv:1105.2250 (2011). [13] H. Hu, L. Jiang, X. Jiu, and H. Pu, arXiv:1105.2488 (2011). [14] L. Han and C. A. R. S´ a de Melo, arXiv:1106.3613 (2011). [15] G. E. Volovik, World Scientiﬁc, Singapore (1992). [16] R. D. Duncan and C. A. R. S´ a de Melo, Phys. Rev. B 62, 9675 (2000). [17] S. S. Botelho and C. A. R. S´ a de Melo, Phys. Rev. B 71, 134507 (2005). [18] M. Gong, S. Tewari, and C. Zhang, arXiv:1105.1796 (2011). [19] M. Iskin and A. L. Subasi, arXiv:1106.0473 (2011). [20] M. Nakahara, Adam Hilger, Bristol (1990). I. ARTIFICIAL SPIN-ORBIT COUPLING IN ULTRA-COLD FERMI SUPERFLUIDS: (SUPPLEMENTARY MATERIAL) The method used to study the spin-orbit and Zeeman eﬀects in ultra-cold Fermi superﬂuids is the functional integral method and its saddle-point approximation in conjunction with ﬂuctuation eﬀects. To describe the thermodynamic phases and the corresponding phase dia- gram in terms of the interactions, Zeeman and spin-orbit ﬁelds, we calculate partition function at temperature TZ=/integraltext D[ψ,ψ†]exp/parenleftbig −S[ψ,ψ†]/parenrightbig with action S[ψ,ψ†] =/integraldisplay dτdr/bracketleftigg/summationdisplay αψ† α(r,τ)∂ ∂τψα(r,τ)+H(r,τ)/bracketrightigg , where the Hamiltonian density is given in Eq. (1). Using the standard Hubbard-Stratanovich transfor- mation that introduces the pairing ﬁeld ∆( r,τ) = g/an}b∇acketle{tψ↓(r,τ)ψ↑(r,τ)/an}b∇acket∇i}htand integrating over the fermion vari- ables lead to the eﬀective action Seﬀ=/integraldisplay dτdr/bracketleftbigg\|∆(r,τ)\|2 g−T 2VlndetM T+/tildewideK+δ(r−r′)/bracketrightbigg , where/tildewideK+= (/tildewideK↑+/tildewideK↓)/2.The matrix Mis M= ∂τ+/tildewideK↑−h⊥0−∆ −h∗ ⊥∂τ+/tildewideK↓∆ 0 0 ∆†∂τ−/tildewideK↑h∗ ⊥ −∆†0h⊥∂τ−/tildewideK↓ ,(7) whereh⊥=hx−ihycorresponds to the transverse com- ponent of the spin-orbit ﬁeld, hzto the parallel com- ponent with respect to the quantization axis z,/tildewideK↑= ˆK↑−hz, and/tildewideK↓=ˆK↓+hz. To make progress, we use the saddle point approxi- mation ∆( r,τ) = ∆ 0+η(r,τ),and write M=Msp+ Mf. The matrix Mspis obtained via the saddle point ∆(r,τ)→∆0which takes M→Msp, and the ﬂuctua- tion matrix Mf=M−Mspdepends only on η(r,τ) and its Hermitian conjugate. Thus, we write the eﬀective ac- tion asSeﬀ=Ssp+Sf. The ﬁrst term is Ssp=V T\|∆0\|2 g−1 2/summationdisplay k,iωn,jln/bracketleftbigg−iωn+Ej(k) T/bracketrightbigg +/summationdisplay k/tildewideK+ T, in momentum-frequency coordinates ( k,iωn), where ωn= (2n+1)πT. Here,Ej(k) are the eigenvalues of Hsp= /tildewideK↑(k)−h⊥(k) 0 −∆0 −h∗ ⊥(k)/tildewideK↓(k) ∆ 0 0 0 ∆† 0−/tildewideK↑(−k)h∗ ⊥(−k) −∆† 00h⊥(−k)−/tildewideK↓(−k) , (8) which describes the Hamiltonian of elemen- tary excitations in the four-dimensional basis7 Ψ†=/braceleftig ψ† ↑(k),ψ† ↓(k),ψ↑(−k),ψ↓(−k)/bracerightig .The ﬂuctu- ation action is Sf=/integraldisplay dτdr/bracketleftbigg\|η(r,τ)\|2 g−T 2Vlndet/parenleftbig 1+M−1 spMf/parenrightbig/bracketrightbigg . The spin-orbit ﬁeld is h⊥(k) =hR(k) +hD(k), wherehR(k) =vR(−kyˆx+kxˆy) is of Rashba-type and hD(k) =vD(kyˆx+kxˆy) is of Dresselhaus-type, has magnitude \|h⊥(k)\|=/radicalig (vD−vR)2k2y+(vD+vR)2k2x. For Rashba-only (RO) ( vD= 0) and for equal Rashba- Dresselhaus(ERD) couplings( vR=vD=v/2), the mag- nitude of the transverse ﬁelds are \|h⊥(k)\|=vR/radicalig k2x+k2y (vR>0) andh⊥(k) =v\|kx\|(v>0), respectively. The Hamiltonian in the helicity basis Φ = UΨ, where Uis the unitary matrix that diagonalizes the Hamilto- nian in the normal state, is /tildewideHsp(k) = ξ⇑(k) 0 ∆ ⇑⇑(k) ∆⇑⇓(k) 0ξ⇓(k) ∆⇓⇑(k) ∆⇓⇓(k) ∆∗ ⇑⇑(k) ∆∗ ⇑⇓(k)−ξ⇑(k) 0 ∆∗ ⇑⇓(k) ∆∗ ⇓⇓(k) 0 −ξ⇓(k) . The components of the order parameter in the helic- ity basis are given by ∆ ⇑⇑(k) = ∆ T(k)e−iϕk,and ∆⇓⇓(k) =−∆T(k)eiϕkfor the triplet channel and by ∆⇑⇓(k) =−∆S(k) and ∆ ⇓⇑(k) = ∆ S(k) for the sin- glet channel. The eigenvalues of Hsp(k) for quasiparti- clesE1(k),E2(k) are listed in Eqs. (4) and (5), while the eigenvalues for quasiholes are E3(k) =−E2(k), and E4(k) =−E1(k). The thermodynamic potential is Ω = Ω sp+Ωf, where Ωsp=V\|∆0\|2 g−T 2/summationdisplay k,jln{1+exp[−Ej(k)/T]}+/summationdisplay k¯K+, with¯K+=/bracketleftig /tildewideK↑(−k)+/tildewideK↓(−k)/bracketrightig /2 is the saddle point contribution and Ω f=−TlnZf, withZf=/integraltext D[¯η,η]exp[−Sf(¯η,η)] is the ﬂuctuation contribution. The order parameter is determined via the minimization of Ωspwith respect to \|∆0\|2, leading to V g=−1 2/summationdisplay k,jnF[Ej(k)]∂Ej(k) ∂\|∆0\|2, (9) wherenF[Ej(k)] = 1/(exp[Ej(k)/T] + 1) is the Fermi function for energy Ej(k). The contact interaction gis expressed in terms of the scattering parameter asvia the Lippman-Schwinger relation discussed in the main text. The total number of particles N+=N↑+N↓is deﬁned fromthethermodynamicrelation N+=−(∂Ω/∂µ+)T,V, and can be written as N+=Nsp+Nf. (10) The saddle point contribution is Nsp=−/parenleftbigg∂Ωsp ∂µ+/parenrightbigg T,V=1 2/summationdisplay k 1−/summationdisplay jnF[Ej(k)]∂Ej(k) ∂µ+ ,and the ﬂuctuation contribution is Nf= −(∂Ωf/∂µ+,)T,Vleading to Nf=T Zf/integraldisplay D[¯η,η]exp[−Sf(¯η,η)]/parenleftbigg −∂Sf(¯η,η) ∂µ+/parenrightbigg , with the partial derivative being ∂SF(¯η,η) ∂µ+=−T 2VTr/bracketleftbigg/parenleftbig 1+M−1 spMf/parenrightbig−1∂ ∂µ+/parenleftbig M−1 spMf/parenrightbig/bracketrightbigg . Knowledge of the thermodynamic potential Ω, of the order parameter Eq. (9) and number Eq. (10) provides a complete theory for spectroscopic and thermodynamic properties of attractive ultra-cold fermions in the pres- ence of Zeeman and spin-orbit ﬁelds. Representative Saddle point solutions for chemical potential µ+and or- der parameter amplitude \|∆0\|as a function of 1 /(kF+as) in the equal Rashba-Dresselhaus (ERD) case ( v/vF+= 0.28) are shown in Fig. 4 for hz/ǫF+= 0,0.5,1.0,2.0. These parameters are used to obtain the phase diagrams described in Figs. 1 and 2 in combination with an anal- ysis of the excitation spectrum Ej(k) given in Eqs. (4) and (5) and the thermodynamic stability conditions for all the uniform superﬂuid phases: directly or indirectly gapped superﬂuid with zero nodal rings (US-0); gapless superﬂuid with one ring of nodes (US-1); and gapless superﬂuid with two rings of nodes (US-2). !!!"#"!$%#"!$ !!!"#"!$%!%!!#! 1/k F+ as 1/k F+ as!+ / "F+ \|#0\| / "F+ b a FIG. 4: a) Chemical potential µ+/ǫF+and b) order pa- rameter amplitude \|∆0\|/ǫF+versus interaction parameter 1/(kF+as) for spin-orbit parameter v/vF+= 0.28 and val- ues of the Zeeman ﬁeld hz/ǫF+= 0 (solid); hz/ǫF+= 0.5 (dashed); hz/ǫF+= 1.0 (dotted); and hz/ǫF+= 2.0 (dot- dashed). A thermodynamic stability analysis of all proposed phases can be performed by investigating the maximum entropy condition. The total change in entropy due to thermodynamic ﬂuctuations, irrespective to any approx- imations imposed on the microscopic Hamiltonian, can be written as ∆Stot=−1 2T(∆T∆S−∆P∆V+∆µα∆Nα), where the repeated αindex indicates summation, and the condition ∆ Stot≤0 guarantees that the entropy is maximum. Considering the entropy Sto be a function of temperature T, number of particles Nαand volume8 V, we can elliminate the ﬂuctuations ∆ S, ∆P, and ∆µα in favor of ﬂuctuations ∆ T, ∆Vand ∆Nα, and show that the ﬂuctuations ∆ Tare statistically independent of ∆Nαand ∆V, while ﬂuctuations ∆ Nαand ∆Vare not. The ﬁrst condition for thermodynamic stability leads to the requirement that the isovolumetric heat capacity CV=T(∂S/∂T)V,{Nα}≥0.Additional conditions are directly related to number ∆ Nαand volume ∆ Vﬂuctu- ations. They require the chemical susceptibility matrix ξαβ= (∂µα/∂Nβ)T,Vto be positive deﬁnite, i.e, that its eigenvalues are both positive. This is guaranteed by det[ξ] =ξ↑↑ξ↓↓−ξ↑↓ξ↓↑>0 andξ↑↑>0. The last con- dition for thermodynamic stability is that the bulk mod- ulusB= 1/κTor the isothermal compressibility κT= −V−1(∂V/∂P)T,{Nα},are positive. Since the number ∆Nαand volume ∆ Vﬂuctuations are not statistically independent, the bulk modulus is related to the matrix [ξ] viaV/κT=N2 ↑ξ↑↑+N↑N↓ξ↑↓+N↓N↑ξ↓↑+N2 ↓ξ↓↓. The positivity of the volumetric speciﬁc heat CV, chemi- cal susceptibility matrix [ ξ] and bulk modulus B= 1/κT are the necessary and suﬃcient conditions for thermody- namic stability, which must be satisﬁed irrespective of approximations used at the microscopic level. !!!"#"!$%#%&"! "' !!!"#"!$%#%&"! "' !!!"#"!$%#%&"! "' !!!"#"!$%#%&"! "' !T "F+ !T "F+ !T "F+ !T "F+ 1/k F+ as 1/k F+ as 1/k F+ as 1/k F+ asd cb a 0.5 11.5 025 50 !!"# $!#! !#! !!"# !!"$# !"% !#! !#! FIG. 5: Isothermal compressibility ¯ κT= (N2 +)V−1κT= (∂N+/∂µ+)T,Vin units of 3 N+/(4ǫF+) versus interaction 1/(kF+as) at spin-orbit coupling v/vF+= 0.28 for the val- ues of the Zeeman ﬁeld a) hz/ǫF+= 0; b) hz/ǫF+= 0.5; c)hz/ǫF+= 1.0; and d) hz/ǫF+= 2.0. Insets show regions where the compressibility is large. Further characterization of phases US-0, US-1 and US-2 is made via thermodynamic prop- erties such as the isothermal compressibility κT= (V/N2 +)(∂N+/∂µ+)T,V,which is shown in Fig. 5 versus 1 /(kF+as) for the values of the Zee- man ﬁeld hz/ǫF+= 0,0.5,1.0,2.0 and spin-orbit couplingv/vF+= 0.28. Notice the negative re- gions ofκTindicating that the uniform superﬂuid phases are unstable, and its discontinuities at phase boundaries. The normal state compressibility κTor ¯κT= (N2 +)V−1κT= (∂N+/∂µ+)T,Vcan be obtained analytically for arbitrary Zeeman hzand spin-orbit parametervin the BCS limit where 1 /kF+as→ −∞as ¯κT=3N+ 4ǫF+/summationdisplay j=±/bracketleftbigg Aj+/bracketleftbigg ˜µ+−A2 j+/radicalig ˜h2z+2˜vA2 j/bracketrightbigg∂Aj ∂˜µ+/bracketrightbigg , (11) where the auxiliary function Ajis A±=/radicaligg (˜µ++ ˜v)±/radicalbigg (˜µ++ ˜v)2−/parenleftig ˜µ2 +−˜h2z/parenrightig and its derivative is ∂A± ∂˜µ+=/bracketleftbigg 1±˜v//radicalig (˜µ++ ˜v)2−(˜µ2 +−˜h2z)/bracketrightbigg /(2A±) with ˜µ+=µ+/ǫF+,˜hz=hz/ǫF+, and ˜v=v/(2ǫF+). Notice that, as hz→0 and ˜v→0,A±→√˜µ+and ¯κT→(3N+)/(2ǫF+) is reduced to the standard result, since ˜µ+→1. In addition, κTor ¯κTcan be obtained analytically in the BEC limit where 1 /kF+as→+∞. Whenhzandvare zero, then ¯κT=3N+ 2ǫF+π kF+as(12) can also be written in terms of bosonic properties 1 V/parenleftbigg∂N+ ∂µ+/parenrightbigg T,V=1 π/parenleftbiggmB aB/parenrightbigg , (13) wheremB= 2mis the boson mass and aB= 2asin the boson-boson interaction. In the case where hz/ne}ationslash= 0 andv/ne}ationslash= 0, a similar expression can be derived for V−1(∂N+/∂µ+)T,Vbut the eﬀective boson mass mB= 2mf(hz,v), and the eﬀective boson-boson interaction aB= 2asg(hz,v) are now functions of hzandv. Notice that the ratio mB/aBin the BEC limit can be directly extracted from the behavior of ¯ κTfor large 1/(kF+as).
2201.06265v2.Spin_orbit_coupled_superconductivity_with_spin_singlet_non_unitary_pairing.pdf	Spin-orbit-coupled superconductivity with spin-singlet non-unitary pairing Meng Zeng,1Dong-Hui Xu,2, 3Zi-Ming Wang,2, 3and Lun-Hui Hu4, 5, 1Department of Physics, University of California, San Diego, California 92093, USA 2Department of Physics and Chongqing Key Laboratory for Strongly Coupled Physics, Chongqing University, Chongqing 400044, China 3Center of Quantum Materials and Devices, Chongqing University, Chongqing 400044, China 4Department of Physics, the Pennsylvania State University, University Park, PA, 16802, USA 5Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA The gap functions for a single-band model for unconventional superconductivity are distinguished by their unitary or non-unitary forms. Here we generalize this classication to a two-band super- conductor with two nearly degenerate orbitals. We focus on spin-singlet pairings and investigate the eects of the atomic spin-orbit coupling (SOC) on superconductivity which is a driving force behind the discovery of a new spin-orbit-coupled non-unitary superconductor. Multi-orbital eects like or- bital hybridization and strain induced anisotropy will also be considered. The spin-orbit-coupled non-unitary superconductor has three main features. First, the atomic SOC locks the electron spins to be out-of-plane, leading to a new Type II Ising superconductor with a large in-plane upper critical eld beyond the conventional Pauli limit. Second, it provides a promising platform to re- alize the topological chiral or helical Majorana edge state even without external magnetic elds or Zeeman elds. More surprisingly, a spin-polarized superconducting state could be generated by spin- singlet non-unitary pairings when time-reversal symmetry is spontaneously broken, which serves as a smoking gun to detect this exotic state by measuring the spin-resolved density of states. Our work indicates the essential roles of orbital-triplet pairings in both unconventional and topological superconductivity. I. INTRODUCTION In condensed matter physics, research on unconven- tional superconductivity [1, 2] remains a crucial topic and continues to uncover new questions and challenges in both theory and experiment, since the discovery of the heavy-fermion superconductors (SCs) [3] and the d-wave pairing states in high-temperature cuprate SCs [4{7]. In addition to the anisotropic gap functions (e.g., p;d;f;g - wave...), the sublattice or orbital-dependent pairings [8{ 10] are shown to be an alternative avenue to real- ize unconventional SCs. They might be realized in multi-orbital correlated electronic systems, whose candi- date materials include iron-based SCs [11{22], Cu-doped Bi2Se3[23, 24], half-Heusler compounds [25{34], and pos- sibly Sr 2RuO 4[35{40] etc. In particular, considering the atomic orbital degrees of freedom, the classication of unconventional pairing states could be signicantly enriched. Among them, SCs with spontaneous time- reversal symmetry (TRS) breaking is of special interest, in which two mutually exclusive quantum phenomena, spin magnetism, and superconductivity may coexist with each other peacefully[41{46]. On the other hand, the orbital multiplicity could also give rise to non-unitary pairings, which again include both time-reversal breaking (TRB) and time-reversal invariant (TRI) pairings. Very recently, prior studies have demonstrated the existence of spin-singlet non- unitary pairing states that break the inversion symme- try in Dirac materials [9]. One aim of this work is the hu.lunhui.zju@gmail.comgeneralization of unitary and non-unitary gap functions in a two-band SC while preserving inversion symme- try, which is possible exactly due to the multi-orbital degrees of freedom [47]. We focus on a system with two nearly degenerate orbitals and nd that the non- unitary pairing state is generally a mixed superconduct- ing state with both orbital-independent pairings and orbital-dependent pairings. Recently, the interplay be- tween orbital-independent pairings and spin-orbit cou- pling (SOC) has been shown to demonstrate the intrigu- ing phenomenon of a large in-plane upper critical eld compared with the Pauli paramagnetic eld for a two- dimensional SC. For example, the Type I Ising super- conductivity in monolayer MoS 2[48, 49] and NbSe 2[50] and the Type II Ising superconductivity in monolayer stanene [51]. Therefore, the interplay of atomic SOC and the multi-orbital pairing could potentially give rise to exciting physics. However, to the best of our knowl- edge, the in uence of the atomic SOC on the orbital- dependent pairings remains unsolved. Furthermore, the multi-orbital nature also gives rise to possible orbital hy- bridization eects and provides an experimentally con- trollable handle using lattice strains, both of which could lead to orbital anisotropy and could potentially change the pairing symmetry. In particular, lattice strain has been a useful experimental tool to study unconventional superconductors [52{54] and has even been proposed to induce the elusive charge-4e phase [55]. We will be do- ing an extensive investigation on all the aforementioned multi-orbital eects. Another topic of this work is concerned with the co- existence of TRB pairings and spin magnetism even in a spin-singlet SC. It is well-known that spin-polarizationarXiv:2201.06265v2 [cond-mat.supr-con] 1 Mar 20232 (SP) can be generated by nonunitary spin-triplet super- conductivity, which is believed to be the case for LaNiC 2 [56] and LaNiGa 2[57, 58]. More recently, the coexis- tence of magnetism and spin-singlet superconductivity is experimentally suggested in multi-orbital SCs, such as iron-based superconductors [59, 60] and LaPt 3P [61]. Therefore, in addition to the spin-triplet theory, it will be interesting to examine how SP develops in multi-orbital spin-singlet SCs as spontaneous TRS breaking in the ab- sence of external magnetic elds or Zeeman elds. In this work, we address the above two major issues by studying a two-band SC with two atomic orbitals (e.g., dxzanddyz). We start with the construction of a kp model Hamiltonian on a square lattice with applied lat- tice strain. The breaking of C4vdown toC2point group generally leads to the degeneracy lifting of dxzanddyz. Based on this model, we study the stability of supercon- ductivity and the realization of 2D topological supercon- ductors in both class D and DIII. First and foremost, the in uence of atomic SOC is studied, which gives birth to a new spin-orbit-coupled SC. This exotic state shows the following features: rstly, a large Pauli-limit violation is found for the orbital-independent pairing part, which belongs to the Type II Ising superconductivity. Further- more, the orbital-dependent pairing part also shows a weak Pauli-limit violation even though it does not belong to the family of Ising SCs. Secondly, topological super- conductivity can be realized with a physical set of pa- rameters even in the absence of external magnetic elds or Zeeman elds. In addition, a spin-polarized super- conducting state could be energetically favored with the spontaneous breaking of time-reversal symmetry. Our work implies a new mechanism for the establishment of spin magnetism in the spin-singlet SC. In the end, we also discuss how to detect this eect by spin-resolved scanning tunneling microscopy measurements. The paper is organized as follows: in section II, we discuss a two-orbital normal-state Hamiltonian on a 2D square lattice and also its variants caused by applied in- plane strain eects, then we show the spin-singlet unitary or non-unitary pairing states with or without TRS. The strain eect on pairing symmetries is also studied based on a weak-coupling theory. In section III, the eects of atomic SOC on such pairing states are extensively stud- ied, as well as the in-plane paramagnetic depairing eect. Besides, the topological superconductivity is studied in section IV even in the absence of external magnetic elds or Zeeman elds, after which we consider the spontaneous TRB eects in section V and show that spin-singlet SC- induced spin magnetism could emerge in the presence of orbital SOC. In the end, a brief discussion and conclu- sion are given in section VI. We will also brie y comment on a very recent experiment [62], demonstrating that a fully gapped superconductor becomes a nodal phase by substituting S into single-layer FeSe/SrTiO 3.II. MODEL HAMILTONIAN In this section, we rst discuss the normal-state Hamil- tonian that will be used throughout this work for an electronic system consisting of both spin and two locally degenerate atomic orbitals (e.g., dxzanddyz) on a 2D square lattice. We assume each unit cell contains only one atom, so there is no sublattice degree of freedom. The orbital degeneracy can be reduced by applying the in- plane lattice strain because the original C4vpoint group is reduced down to its subgroup C2vfor strain10;01or 11(A more generic strain would reduce the symmetry directly to C2). Heren1n2represents the strain tensor whose form will be given later. We will apply the sym- metry analysis to construct the strained Hamiltonian in the spirit of kptheory. Then, we discuss the pairing Hamiltonian and the corresponding classication of spin- singlet pairing symmetries including non-unitary pairing states. The strain eect is also investigated on the super- conducting pairing symmetries based on a weak-coupling scheme [10]. A. Normal-state Hamiltonian In this subsection, we construct the two-orbital normal-state Hamiltonian H0(k) with lattice strain- induced symmetry-breaking terms. Before that, We rst showH0(k) in the absence of external lattice strains. For a square lattice as illustrated in Fig. 1 (a), it owns theC4vpoint group that is generated by two symme- try operators: a fourfold rotation symmetry around the ^z-axisC4z: (x;y)!(y;x) and a mirror re ection about the ^y^zplaneMx: (x;y)!(x;y). Other symmetries can be generated by multiplications, such as the mirror re ection about the (^ x+ ^y)^zplane Mx+y: (x;y)!(y;x) is given by C4zMx. In the absence of Rashba spin-orbit coupling (SOC), the sys- tem also harbors inversion symmetry I, enlarging the symmetry group to D4h=C4v fE;Ig. In the spirit of kpexpansion around the point or the Mpoint, we consider a two-orbital system described by the inversion- symmetric Hamiltonian in two dimensions (2D), H0(k) =(k)00+soc23+o[go(k)]0;(1) where the basis is made of fdxz;dyzg-orbitals y k= (cy dxz;"(k);cy dxz;#(k);cy dyz;"(k);cy dyz;#(k)). Herecyis the creation operator of electrons, andare Pauli ma- trices acting on the orbital and spin subspace, respec- tively, and 0,0are 2-by-2 identity matrices. Besides, (k) =(k2 x+k2 y)=2mis the band energy measured relative to the chemical potential ,mis the eective mass,socis the atomic SOC [63{65] and ocharacter- izes the strength of orbital hybridization. This model could describe the two hole pockets of iron-based super- conductors [66, 67]. Moreover, the rst two components ofgo(k) are for the inter-orbital hopping term, while the3 𝑀𝑥 (a) 𝐶4𝑧 𝑥𝑦 𝑑𝑥𝑧orbital ++− −𝑥𝑧𝑑𝑦𝑧orbital ++− −𝑦𝑧𝑪𝟒𝒗𝑪𝟐𝒗 𝑪𝟐𝒗 𝑪𝟐𝑪𝟐𝒗𝑴𝒙𝑴𝒚 Basis Orbitals 𝐴1++𝑥2,𝑦2 𝐴2−−𝑥𝑦 𝐵1−+𝑥𝑧𝑑𝑥𝑧 𝐵2+−𝑦𝑧𝑑𝑦𝑧Strain𝜎10(b) 𝑀𝑥𝑀𝑦 (c) Strain𝜎11 (d)𝑪𝟐𝒗𝑴𝒙′𝑴𝒚′ Basis Orbitals 𝐴1++𝑥′2,𝑦′2 𝐴2−−𝑥′𝑦′ 𝐵1−+𝑥′𝑧𝑑𝑥′𝑧 𝐵2+−𝑦′𝑧𝑑𝑦′𝑧 Strain𝜎1𝛿𝐶2𝑧 𝐶2𝑧 𝐶2𝑧𝑪𝟐𝑪𝟐𝒛 Basis Orbitals 𝐴1+𝑥2,𝑦2,𝑥𝑦 𝐵1−𝑦𝑧,𝑥𝑧𝑑𝑥𝑧,𝑑𝑦𝑧Strain𝑥𝑦 𝑥𝑦 FIG. 1. The strain eect on a two-dimensional square lattice. In the absence of lattice strain, (a) shows the square lattice owing theC4vpoint group that is generated by C4zandMx. We consider the normal-state Hamiltonian with dxz;dyz-orbitals. Inversion symmetry ( I) is broken by growing crystal samples on an insulating substrate. The in-plane strain eects on the square lattice are illustrated in (b-d) for applied strain along dierent directions. (b) shows that the ^ xor ^y-axis strain breaks the square lattice into the rectangular lattice with two independent mirror re ection symmetries MxandMy, obeying the subgroupC2vofC4v. TheC2vpoint group contains four one-dimensional irreducible representations (irrep.) A1;A2;B1;B2. (c) shows that the strain along the ^ x+ ^y-direction also reduces the C4vdown toC2v. (d) represents a general case, where the subgroupC2is preserved that only has A1andB1irreps. third term is for the anisotropic eective mass, explained below in detail. TheC4v(orD4h) point group restricts go(k) = (aokxky;0;k2 xk2 y), whereao= 2 is a symmetric case that increases the C4zto a continues rotational sym- metry about the ^ z-axis. To be precise, the g1-term, 2okxky10, is attributed to the inter-orbital hopping integral along the ^x^ydirections, o 2(cy dxz;(ix;iy)cdyz;(ix+ 1;iy+ 1) +cy dxz;(ix;iy)cdyz;(ix1;iy1) cy dxz;(ix;iy)cdyz;(ix+ 1;iy1) cy dxz;(ix;iy)cdyz;(ix1;iy+ 1) + h.c.) ;(2) where (ix;iy) represents the lattice site. In addition, the g3-term,o(k2 xk2 y)30, causes the anisotropic eective masses. For example, the eective mass of the dxzorbital is1 1=m2oalong the ^x-axis while that is1 1=m+2oalong the ^y-axis. This means that the hopping integrals aredierent along ^ xand ^ydirections, (1 2mo)cy dxz;(ix;iy)cdxz;(ix+ 1;iy) +(1 2m+o)cy dxz;(ix;iy)cdxz;(ix;iy+ 1) +(1 2m+o)cy dyz;(ix;iy)cdyz;(ix+ 1;iy) +(1 2mo)cy dyz;(ix;iy)cdyz;(ix;iy+ 1) + h.c.:(3) In this work, we focus on a negative eective mass case by choosing 1 =m2o>0. However, using a posi- tive eective mass does not change our main conclusion. Moreover, our results can be generally applied to other systems with two orbitals px;py, once it satises the C4v point group. The time-reversal symmetry operator is presented as T=i02KwithKbeing the complex conjugate. And the inversion symmetry is presented as I=00. It is easy to show Eq. (1) is invariant under both TandI. However, inversion can be broken by growing the sample on insulating substrates, the asymmetric Rashba SOC is described by HR(k) =R0[gR(k)]; (4)4 whereRis the strength of the Rashba SOC with gR(k) = (ky;kx;0) as required by the C4vpoint group. Next, we consider the lattice strain eect on the two- dimensional crystal with a square lattice, as summarized in Fig. 1 (b-d). The in-plane strain eect is characterized by the 2-by-2 strain tensor whose elements are dened asij=1 2 @xiuj+@xjui , whereuiis the displacement atralong the ^eidirection. Even though it is an abuse of notation, it should be self-evident that the here does not represent the Pauli matrices. The strain tensor can be parametrized as the following =cos2cossin cossin sin2 ; (5) whereis the polar angle with respect to the ^ x-axis. For the= 0 (=2) case, the compressive or tensile strain applied along the ^ x-axis (^y-axis) makes the square lattice as a rectangular lattice, as illustrated in Fig. 1 (b). And the==4 case is for the shear strain along the (^ x+ ^y)- direction in Fig. 1 (c). All the above cases reduce the C4vpoint group into its subgroup C2vthat is generated by two independent mirror re ections. Otherwise, it is generally reduced to C2. The irreducible representations forC2vandC2are shown in Fig. 1 (b-d). Based on the standard symmetry analysis, to the leading order, the strained Hamiltonian is given by Hstr=tstr[sin(2)1+ cos(2)3]0; (6) where both tstrandcan be controlled in experi- ments [68]. AndHstrcan be absorbed into the go-vector in Eq. (1), renormalizing the orbital hybridization as ex- pected. Furthermore, one can check that Hstrpreserves bothTandI, but explicitly breaks the C4z=i2ei 43 because of [Hstr;C4z]6= 0. Interestingly, the orbital tex- ture on the Fermi surface can be engineered by strain, and its eect on superconducting pairing symmetries is brie y discussed in the Appendix C. Therefore, a strained normal-state Hamiltonian is HN(k) =H0(k) +HR(k) +Hstr; (7) which will be used throughout this work. The Rashba SOC induced spin-splitting bands are considered only when we discuss the topological superconducting phases in section IV and V, even though the normal-state Hamil- tonianHN(k) is topologically trivial. For the supercon- ducting states, we focus on the inversion symmetric pair- ings (i.e., spin-singlet s-wave pairing) and their response to applied strains or in-plane magnetic elds. In the absence of Rashba SOC, the band structures of HN(k) in Eq. (7) are given by E(k) =1 2m(k2 x+k2 y)q 2soc+ ~g2 1+ ~g2 3;(8) where we dene the strained orbital hybridization ~gvec- tor with ~g1=aookxky+tstrsin(2) and ~g3=o(k2 x k2 y)+tstrcos(2). Each band has two-fold degeneracy, en- forced by the presence of both TandI. At the point 𝑘𝑥𝑘𝑦 𝑘𝑥𝑘𝑦(a) (b)FIG. 2. The lattice strain eect on the Fermi surfaces of the normal-state Hamiltonian without Rashba SOC. (a) shows the two Fermi surfaces without lattice strain (i.e., tstr= 0), thusC4z-symmetric energy contours are formed. (b) shows the breaking of C4zby lattice strain with tstr= 0:4 and = 0, onlyC2z-symmetric energy contours appear. Other parameters used here are m= 0:5;a0= 1;o= 0:4,R= 0 and=0:5. (kx=ky= 0),E =p 2soc+t2 str. The two Fermi sur- faces with and without strain are numerically calculated and shown in Fig. 2, where we choose <p 2soc+t2 str. These are two hole pockets because of the negative eec- tive mass of both orbitals. The Fermi surfaces in Fig. 2 (a) areC4-symmetric ( tstr= 0), while those in Fig. 2 (b) are only C2-symmetric due to the symmetry break- ing of lattice strains. Please note that there is only one Fermi surface when jj<p 2soc+t2 str, which is a neces- sary condition to realize topological superconductors as we will discuss in Sec. IV. B. Review of singlet-triplet mixed pairings Before discussing the possible unconventional pairing symmetry forHNin Eq. (7), we brie y review both uni- tary and non-unitary gap functions for a single-band SC in the absence of inversion symmetry. In this case, a singlet-triplet mixed pairing potential is given by (k) = [s s(k)0+ t(ds(k))] (i2);(9) where are Pauli matrices acting in the spin subspace. Here s(k) represents even-parity spin-singlet pairings and the odd-parity ds(k) is for the spin-triplet ds-vector. Physically, the unitary SC has only one superconducting gap like in the conventional BCS theory, while a two- gap feature comes into being by the non-unitary pairing potential. More explicitly, the unitary or non-unitary is dened by whether the following is proportional to the identity matrix 0: (k)y(k) =jsj2 2 s+jtj2jdsj2 + 2Re[s t sd s]+ijtj2(dsd s);(10) Therefore, Eq. (10) gives rise to a possible classication by assuming a non-vanishing s2Rand a proper choice5 Pairings TRS Unitary Δ𝑠 Δ𝑜 𝒅𝑜 TRI unitary Yes YesReal ZeroZero RealReal TRI non -unitary Yes No Real Real Real TRB unitary No Yes RealPurely imaginaryReal TRB non -unitary No No Real Real Complex TABLE I. The four pairing states classied by time-reversal symmetry and unitary for a spin-singlet superconductor with both orbital-independent pairing sand orbital-dependent pairing oanddo. of a global phase. In principle, there are four possible phases, including the TRI non-unitary SCs ( t2R;ds2 R), the TRB unitary SCs ( ti;ds2R), and the TRB non-unitary SCs ( t2R;ds2C). On the other hand, the TRI unitary SCs are achieved only with s= 0 or t= 0 and real ds, meaning a purely spin-singlet SC or a purely spin-triplet SC. These states might be distin- guished in experiments, for example, the TRB unitary pairing state might induce a spontaneous magnetization with the help of Rashba spin-orbit coupling [69], which can be detected by SR [70]. As we know, the spin-singlet pairings do not coexist with the spin-triplet pairings in the presence of inversion symmetry (e.g. centrosymmetric SCs). Roughly speak- ing, it seems out of the question to realize non-unitary pairing states in purely spin-singlet SCs. However, this is a challenge but not an impossibility for an SC with multi-orbitals, which is one of the aims of this work. In the following, we will discuss how to generalize the classi- cation of TRI or TRB and unitary or non-unitary pair- ing states to a spin-singlet SC with two atomic orbitals in the presence of inversion symmetry. The four cases are summarized in Table I. C. Pairing Hamiltonian of a two-orbital model We now consider the pairing Hamiltonian for HNin Eq. (7). By ignoring the uctuations, the mean-eld pair- ing Hamiltonian is generally given by, H=X k;s1a;s2ba;b s1;s2(k)cy as1(k)cy bs2(k) + h.c.;(11) wheres1;s2are index for spin and a;bare for orbitals. As studied in Ref. [10], the orbital-triplet pairing is robust even against orbital hybridization and electron-electron interactions. Thus, we consider both orbital-independent and orbital-dependent pairings for generality. In analogy to spin-triplet SCs, we use an orbital do(k)-vector for the spin-singlet orbital-dependent pairing potential [12], which takes the generic form tot(k) = [s s(k)0+ o(do(k))] (i2);(12)where sand oare pairing strengths in orbital- independent and orbital-dependent channels, respec- tively. The Fermi statistics requires that s(k) = s(k), while the three components of dosatisfy d1;3 o(k) =d1;3 o(k) andd2 o(k) =d2 o(k). Namely, d2 o(k) represents odd-parity spin-singlet orbital-singlet pairings and the others are for even-parity spin-singlet orbital-triplet pairings. The pairing potential presented in this form is quite convenient, similar to the spin-triplet case [10, 12, 14, 71]. The benets of this form in Eq. (12) will be shown when we discuss the mixture of orbital- independent and orbital-dependent pairings. Combining Eq. (12) with Eq. (1), the Bogoliubov-de-Gennes (BdG) Hamiltonian is HBdG(k) =HN(k) tot(k) y tot(k)H N(k) ; (13) which is based on the Nambu basis ( y k; T k). Same with the spin case in Eq. (10), the non-unitarity of a spin- singlet pairing potential dened in Eq. (12) is determined by whether tot(k)y tot(k) is proportional to an identity matrix. More explicitly we have tot(k)y tot(k) =jsj2 2 s00+joj2jdoj200 + 2Rejs o sd oj0+ijoj2(dod o)0;(14) which could also exhibit four general possibilities: time- reversal-invariant (TRI) or time-reversal-breaking (TRB) and unitary or non-unitary SCs, with a simple replace- mentft;dsg!f o;dog. In the absence of band split- tings, i.e.,soc=o=R=tstr= 0 as an illustration, the superconducting excitation gaps on the Fermi sur- faces of a TRI unitary SC are E;(k) =p 2(k) + (s s(k) +ojdoj)2;(15) with;=. It is similar to the superconducting gaps for non-unitary spin-triplet SCs [1]. Moreover, the two- gap feature indicates the non-unitarity of the supercon- ducting states, which implies the possibility of a nodal SC as long as s s(k)ojdoj= 0 is satised on the Fermi surfaces. And, the nodal quasi-particle states can be experimentally detected by measuring specic heat, London penetration depths, SR, NMR, etc. As a re- sult, this provides possible evidence to get a sight of TRI non-unitary phases in real materials (e.g. centrosymmet- ric SCs). Furthermore, the above conclusion is still valid when we turn on soc,o, andtstr. III. THE PAULI LIMIT VIOLATION: A LARGE IN-PLANE UPPER CRITICAL FIELD In this section, we study the Pauli limit violation for the spin-singlet TRI non-unitary SC against an in-plane magnetic eld (e.g. Hc2;k> HP). For a 2D crystalline SC or a thin lm SC, the realization of superconducting states that are resilient to a strong external magnetic eld6 has remained a signicant pursuit, namely, the pairing mechanism can remarkably enlarge the in-plane upper critical eld. Along this crucial research direction, one recent breakthrough has been the identication of \Ising pairing" formed with the help of Ising-type spin-orbit coupling (SOC), which breaks the SU(2) spin rotation and pins the electron spins to the out-of-plane direction. Depending on whether the inversion symmetry is broken or not by the Ising-type SOC, the Ising pairing is clas- sied as Type I (broken) and Type II (preserved) Ising superconductivity, where the breaking of Cooper pairs is dicult under an in-plane magnetic eld. To demonstrate the underlying physics, in the fol- lowing, we consider the interplay between atomic SOC soc6= 0 and spin-singlet TRI non-unitary pairing state. Thus, we consider the pairing potential tot= s0+ o(d1 o1+d3 o3) (i2); (16) where s, o,d1 o, andd3 oare all real constant. This can be realized once we have on-site attractive interactions in both orbital channels. Another reason for studying the atomic SOC is that it is not negligible in many real ma- terials. It is interesting to note that the strength of SOC can be tuned in experiments, for example, by substituting S into single-layer FeSe/SrTiO 3[62] or growing a super- conductor/topological insulator heterostructure [72]. Without loss of generality, the direction of the mag- netic eld can be taken to be the x-direction, i.e., H= (Hx;0;0) withHx0. Therefore, the normal Hamilto- nian becomes HN(k) +h01; (17) where the rst part is given by Eq. (7) and h=1 2gBHx is the Zeeman energy with g= 2 the electron's g-factor. To explicitly investigate the violation of the Pauli limit for the spin-orbit coupled SCs, we calculate the in-plane upper critical magnetic eld normalized to the Pauli- limit paramagnetic eld Hc2;k=HPas a function of the normalized temperature Tc=T0, by solving the linearized gap equation. Here HP= 1:86T0represents the Pauli limit withT0the critical temperature in the absence of an external magnetic eld. Following the standard BCS decoupling scheme [10], we rst solve Tcfor the orbital-independent pairing chan- nel by solving the linearized gap equation, v0s(T)1 = 0, wherev0is eective attractive interaction and the su- perconductivity susceptibility s(T) is dened by s(T) =1 X k;!nTrh Ge(k;i!n)Gh(k;i!n)i ;(18) where the conventional s-wave pairing with s(k) = 1 is considered for Eq. (16). Here Ge(k;i!n) = [i!n H0(k)]1is the Matsubara Green's function for elec- trons and that for holes is dened as Gh(k;i!n) = 2G e(k;i!n)2. Here= 1=kBTand!n= (2n+ 1)=withninteger. Likewise, for the orbital-dependentpairing channels, the superconductivity susceptibility o(T) is dened as o(T) =1 X k;!nTrh (do(k))yGe(k;i!n) (do(k))Gh(k;i!n)i ;(19) where the orbital-dependent pairing ( Agrepresentation) with the vector-form as do= (d1 o;0;d3 o) for Eq. (16) is used for the Tccalculations. However, the momentum- dependent do-vector does not aect the formalism and main results, as we will discuss in the appendix C. The coupling between orbital-independent and orbital- dependent channels leads to a high-order correction ( 2k2 F=2, withbeing the coupling strength of the eec- tive~gin the Hamiltonian representing orbital hybridiza- tion and strain ), which can be ignored once =kF. A. Type II Ising superconductivity In this subsection, we rst consider the orbital- independent pairing state (i.e., s6= 0 and o= 0) and show it is a Type II Ising SC protected from the out- of-plane spin polarization by the atomic SOC soc23. To demonstrate that, one generally needs to investigate the eects of atomic SOC on the SC Tcas a function of the in-plane magnetic eld hbased on Eq. (18), in the presence of both orbital hybridization oand straintstr. As dened in Eq. (8), the eects of orbital hybridiza- tion and lattice strain on the system can be captured by an eective ~g(aookxky+tstrsin(2);o(k2 xk2 y) + tstrcos(2)). The case with tstr= 0 has been studied in Ref. [73], however, the strain eect on the type II Ising SC has not been explored yet. To reveal the pure role of lattice strains, we consider kFto be close to the -point so that the k-dependent hybridization part is dominated by the strain part for generic . Therefore, we focus on ~g=tstr(sin 2;cos 2) in the following discussions. After a straightforward calculation (see details in Ap- pendix B), the superconductivity susceptibility s(T) in Eq. (18) is calculated as s(T) =0(T) +N0fs(T;soc;tstr;h); (20) withN0is the DOS near the Fermi surface and the pair- breaking term is given by fs(T;soc;tstr;h) =1 2[C0(T;) +C0(T;+)] + [C0(T;)C0(T;+)]2 soc+t2 strh2 2E+E ;(21) whereEp 2soc+ (tstrh)2,=1 2(E+E), and 0(T) =N0ln 2e !D kBT is the superconducting suscep- tibility when soc;tstr;h= 0. Here = 0:57721is7 λsoc=0,tstr=0λsoc=1.5,tstr=20.00.20.40.60.81.00.00.20.40.60.81.0λsoc=0,tstr=0λsoc=1.5,tstr=0λsoc=1.5,tstr=10.00.20.40.60.81.00.00.20.40.60.81.0 Tc/T0Tc/T0Tc/T0(a)(b)(c)λsoc/tstrhhtstr=0.10tstr=0.12tstr=0.160.00.51.01.52.02.50.750.800.850.900.951.00 FIG. 3. The pair-breaking eects. (a) A signicant Pauli limit violation is due to the atomic SOC for the orbital-independent pairing with s= 1. However, the lattice strain might slightly suppress the Hc2by comparing the blue and red curves. (b) A weak Pauli limit violation due to the atomic SOC for the orbital-dependent pairing with o= 1. (c) The suppression of Tcby atomic SOC for orbital-dependent pairing at zero external magnetic elds with o= 1. For the three gures here, we have set the strain parameter = 8, i.e. ~g= (p 2 2;0;p 2 2). the Euler-Mascheroni constant. Furthermore, the kernel function of the pair-breaking term fsis given by C0(T;E) = Re (0)(1 2) (0)(1 2+iE 2kBT) ;(22) with (0)(z) being the digamma function. Note that C0(T;E)0 and it monotonically decreases as Ein- creases, indicating the reduction of Tc. Namely,C0(T;E) gets smaller for a larger E. We rst discuss the simplest case with soc=tstr= 0, where the pair-breaking function becomes fs(T;0;0;h) = C0(T;h), which just leads to the Pauli limit Hc2;k HP= 1:86Tc, as shown in Fig. 3(a). Furthermore, we turn onsocwhile take the tstr!0 limit, the pair- breaking term in Eq. (21) is reduced to fs(T;soc;0;h) =C0 T;p 2soc+h2h2 2soc+h2;(23) which reproduces the same results of Type II Ising super- conductors in Ref. [73]. Under a relatively weak magnetic eld (hsoc), the factor h2=(2 soc+h2)1 leads to fs(T;soc;0;h)!0, which in turn induces a large in- planeHc2;k=HP. Next, we investigate the eect of lattice strain tstron the in-plane upper critical eld Hc2;k. Interestingly, tstr would generally instead reduce Hc2;k. To see it explicitly, we expand the pair-breaking function fsin Eq. (21) up to the leading order of t2 str, fs(T;soc;tstr;h)fs(T;soc;0;h) +F(T;soc;h)t2 str+O(t4 str);(24) whereF(T;soc;h) is given in Appendix B and we nd it is always negative (i.e., F(T;soc;h)<0). In addition to the rst term fs(T;soc;0;h) discussed in Eq. (23), the second term F(T;soc;h)t2 stralso serves as a pair- breaking eect on Tcat non-zero eld. Therefore, the secondo-term further reduces Tc, leading to the reduc- tion of the in-plane upper critical eld.We then numerically conrm the above discussions. We solve the linearized gap equation v0s(T)1 = 0 and arrive at log( Tc=T0) =fs(Tc;soc;tstr;h), from which Tc=T0is numerically calculated in Fig. 3 (a). Here T0is the critical temperature at zero external magnetic elds. The Pauli limit corresponds to T0(soc= 0;tstr= 0;h= 0). The non-monotonic behavior of the curves at small Tc=T0(.0:5, i.e. dashed line) from solving the linearized gap equation calls for a comment. In the small temper- ature range, the transition by tuning the eld strength becomes the rst order supercooling transition [74]. Here we mainly focus on the solid line part, which is second order and gives the critical eld Hc2. We see that in general there is a Pauli limit violation for non-zero soc andtstr. Furthermore, by comparing the two cases with soc= 1:5;tstr= 0 andsoc= 1:5;tstr= 1, we conrm the above approximated analysis. We believe the strain eect on the type II Ising SC will be tested in experiments soon. B.Hc2;kfor orbital-dependent pairings In this subsection, we further study the in uence of the atomic SOC socon the paramagnetic pair-breaking eect for orbital-dependent pairings (i.e., s= 0 and o6= 0). We nd a weak enhancement of the in-plane upper critical eld Hc2;kcompared with the Pauli limit. Following the criteria of the orbital do-vector in Ref. [10] (also discussed in the Appendix C), we take doto be parallel to the vector ~gby assuming soctstr, which leads to the maximal condensation energy. This would be justied in the next subsection. After a straightforward calculation (see details in Appendix B), the superconduc- tivity susceptibility o(T) in Eq. (19) is calculated as, o(T) =0(T) +N0fo(T;soc;tstr;h); (25)8 where the pair-breaking term is given by fo(T;soc;tstr;h) =1 2[C0(T;) +C0(T;+)] + [C0(T;)C0(T;+)]t2 str2 soch2 2E+E ;(26) which diers from fs(T;soc;tstr;h) for orbital- independent pairings in Eq. (21). The only dierence between them lies in the factor ( t2 str2 soch2)=2E+E, compared with that of fs(T;soc;tstr;h) (i.e. (t2 str+2 soch2)=2E+E), which leads to a com- pletely distinct superconducting state, demonstrated as follows. To understand Eq. (26), we rst discuss the simplest case withsoc=tstr= 0, where the pair-breaking func- tion becomes f(T;0;0;h) =C0(T;h), which just leads to the Pauli limit Hc2;kHP= 1:86Tc, as shown in Fig. 3(b). Likewise, when soc= 0 andtstr6= 0, the pair- breaking function again simplies to C0(T;h). Therefore, the Pauli limit of the in-plane upper critical eld is not aected by tstritself. Physically, this is because spin and orbital degrees of freedom are completely decoupled in this case, and it has also been shown that a similar or- bital eect does not suppress Tcwhen dok~g[10], which is what we assumed here. On the other hand, if we turn on merely the atomic SOCsoc6= 0 while keeping tstr= 0, the pair-breaking function is given by fo(T;soc;0;h) =C0(T;p 2soc+h2); (27) which leads to the reduction of the upper critical eld, i.e., Hc2;k< HP, because of f(T;soc;0;h)< f(T;0;0;h)<0. Remarkably, we nd that the atomic SOC also plays a similar role of \magnetic eld" to sup- press the orbital-dependent pairing, as discussed in the next subsection. Thus, it does not belong to the family of Ising SCs, which makes the orbital-dependent pair- ing signicantly dierent from the orbital-independent pairings. Moreover, their dierent dependence on the in- plane magnetic eld might also be tested in experiments, which is beyond this work and left for future work. This also indicates the dierence between orbital-triplet SC and spin-triplet SC in responses to Zeeman elds. However, it is surprising to notice that there is a weak enhancement of the in-plane upper critical eld Hc2;kfor the case with both tstr6= 0 andsoc6= 0. Solving the gap equation v0o(T)1 = 0, we obtain lnTc T0 =fo(T;soc;tstr;h): (28) Fig. 3 (b) shows how Tc=T0changes with the applied in-plane magnetic eld, where the Pauli limit curve cor- responds to soc;tstr= 0. When both the atomic SOC and strain are included, the critical eld Hc2exceeds the Pauli limit by a small margin. Therefore, a spin-orbit- coupled SC with spin-singlet non-unitary pairing sym- metries does not belong to the reported family of Ising superconductivity.C. Atomic SOC induced zero-eld Pauli limit As mentioned above, the atomic SOC breaks the spin degeneracy, which generally suppresses the even parity orbital-dependent pairings, in the case with s= 0 and o6= 0. Thus, the robustness of such pairings in the presence of atomic SOC is the preliminary issue that we need to address. And we nd that the spin-singlet orbital-dependent pairing is also prevalent in solid-state systems when the energy scale of atomic SOC is smaller than that of the orbital hybridization or external strain. In this case, we focus on the zero magnetic eld limit. Using the general results from the calculations in the pre- vious section, we have lnTc T0 =fo(T;soc;tstr;h= 0) =C0 T;q t2 str+2soc2 soc t2 str+2soc;(29) whereC0(T;E) is dened in Eq. (22). In the case of soc= 0, it can be been that Tc(tstr) =T0(tstr= 0), i.e. the superconducting Tcis not suppressed by stain or the orbital hybridization when the orbital do-vector is parallel to ~g[10]. However, in the presence of non-zero atomic SOC soc, theTcwill be suppressed even when dok~gis satised. Fig. 3 (c) shows the behavior of Tcas a function of the soc=tstrfor two dierent values of tstr. We see the suppression of Tcas long assoc6= 0, and the suppression is more prominent when tstris larger. To understand the suppression of orbital-dependent pairings by the atomic SOC, we take the tstr= 0 limit. Eq. (29) leads to lnTc T0 =C0(T;soc); (30) which implies that socplays the same role of \magnetic eld" that suppresses the Tcof the orbital-dependent pairing states. And socHProughly measures the zero-eld \Pauli-limit" of the orbital-dependent pairing states. We dub this new eect as zero-eld Pauli limit for orbital-dependent pairings induced by the atomic SOC, which can serve as the preliminary analysis of whether orbital-dependent pairings exist or not in real materials by simply calculating soc=Tc. Motivated by this observation, we notice that the nor- mal Hamiltonian given in Eq. (7) satises [ HN(k);2] = 0 with botho!0 andtstr!0. It stands for the U(1) rotation in the orbital subspace. As a result, we can project the normal Hamiltonian HN(k) in Eq. (7) into block-diagonal form corresponding to the 1 eigenvalues of2by using the basis transformation U=0 1p 2 1i 1i : (31) The new basis is given by ~ y(k) = (cy +;";cy +;#;cy ;#;cy ;"); (32)9 wherecy ;s1p 2(cy dxz;sicy dyz;s). On this basis, the normal Hamiltonian is given by H0=H+ 0H 0; (33) whereH 0are given by H 0=(k)soc3: (34) Note that the time-reversal transforms H 0(k) to H 0(k). Explicitly, the atomic SOC is indeed a \mag- netic eld" in each subspace, while it switches signs in the two subspaces to conserve TRS. Next, we project the pairing Hamiltonian to the new basis, and we nd that it also decouples as H=H+ H ; (35) whereH are given by H = 2h cy ;"(k)cy ;#(k)("$# )i + h.c.;(36) where o(id1 o+d3 o) are the gap strengths in each subspace. In each subspace, it resembles an s-wave su- perconductor under an eective \magnetic eld" of the atomic SOC along the out-of-plane direction. It natu- rally explains the \zero-eld Pauli-limit" pair-breaking eect of atomic SOC on the orbital-dependent pairings with thetstr!0 limit. As a brief conclusion, our results demonstrate that the spin-singlet orbital-dependent pair- ings occur only in weak atomic SOC electronic systems. IV. 2D HELICAL SUPERCONDUCTIVITY In the above sections, the spin-orbit-coupled SCs concerning inversion symmetry have been comprehen- sively studied. In addition to that, it will be natural to ask if there exist more interesting superconducting states (e.g. topological phases) by including an inversion- symmetry breaking to the normal Hamiltonian in Eq. (7), namely,R6= 0. For this purpose, in this section, we focus on the Rashba SOC and explore its eect on the spin-orbit-coupled SCs, especially the orbital-dependent pairings. Even though the 2D bulk SC or thin lm SC preserves the inversion symmetry, a Rashba SOC appears near an interface between the superconducting layer and the insulating substrate. Remarkably, we nd a TRI topological SC (helical TSC) phase generated by the interplay between the two types of SOC (atomic and Rashba) and spin-singlet orbital-dependent pairings. Since TRS is preserved, it belongs to Class DIII accord- ing to the ten-fold classication. On the boundary of the interface, there exists a pair of helical Majorana edge states [75{84]. To explore the topological phases, we consider the normal-state Hamiltonian in Eq. (7), and the TRI spin- singlet non-unitary pairing symmetry in Eq. (16) for theBdG Hamiltonian (13), namely, a real orbital do-vector is assumed for the orbital-dependent pairings. In thetstr!0 and s!0 limit, the bulk band gap closes only at the -point for c=p 2soc4joj2 while no gap-closing happens at other TRI momenta, leading to a topological phase transition. Thus, we con- clude that the topological conditions are c< < + c and an arbitrary orbital do-vector. In Appendix D, we show theZ2topological invariant can be analytically mapped to a BdG-version spin Chern number, similar to the spin Chern number in the 2D topological insula- tors. As mentioned in Sec. III (c), the conservation of 2, the U(1) symmetry in the orbital subspace, leads to the decomposition of the BdG Hamiltonian into two blocks for dierent eigenvalues of 2. In each subspace, we can dene the BdG Chern number as C=1 2X lled bandsZ BZdkh n(k)jirkj n(k)i;(37) withj nibeing the energy eigenstate of H BdG(see the details in Appendix D). Then the Z2topological invari- ant, in this case, is then explicitly given by, C+C 2; (38) whereCare the Chern numbers of the channels. = 1 corresponds to the TSC phase, shown in Fig. 4 (a). Based on the analysis for the topological condition, we learn that oshould be smaller than soc. How- ever, as shown in Sec. III, the atomic SOC actually will reduce the Tcof orbital-dependent pairings, which set a guideline to a physically realizable set of parameters, T0soco, beyond the BCS theory ( o1:76T0). For example, the monolayer FeSe superconductor lms on dierent substrates achieve a very high critical tem- peratureT070 K [85]. As for a more general case with non-zero o,tstrand s, the BdG Hamiltonian can no longer be decomposed into two decoupled blocks, hence the Chern number ap- proach fails to characterize the Z2invariant. However, the more general Wilson-loop approach still works (see details in Appendix E). In general, the Z2-type topolog- ical invariant of helical superconductivity could be char- acterized by the Wilson loop spectrum [86, 87], shown in Fig. 4 (b), which demonstrates the non-trivial Z2index. To verify the helical topological nature, we calculate the edge spectrum in a semi-innite geometry with kybe- ing a good quantum number. Fig. 4 (c) conrms clearly that there is a pair of 1D helical Majorana edge modes (MEMs) propagating on the boundary of the 2D system. V. TRB NON-UNITARY SUPERCONDUCTOR So far, the TRI non-unitary pairing states are investi- gated, which exhibit the Pauli-limit violation for in-plane upper critical eld and topological phases. Furthermore,10 0.40.20-0.2-0.40.3-0.30(c)Eky/π-1-0.500.51-0.500.511.5-1-0.50110.5-0.51.500.5(a)μChern numberky−ππ0−π0πθ(b) -0kx-0.500.5vys=0 ; 0=0 ; FIG. 4. Topological helical superconductivity for spin-singlet orbital-dependent pairing in the presence of Rashba SOC. (a) TheZ2index is calculated by decoupling the BdG Hamiltonian into two chiral blocks when s= 0 andtstr= 0. The other parameters used: m= 0:5,= 0:2,soc= 0:4,R= 1, o= 0:1,do= (1;0;1). (b) The Wilson loop calculation of the Z2 invariant for s= 0:05,tstr= 0:1,= 8andgo= (1;0;1). The other parameters remain the same as those in (a). The spectrum of edge states in (c) shows two counter-propagating Majorana edge states of the helical TSC. in this section, we study the TRB non-unitary pairing states characterized by a complex do-vector when both sand oare real. As it is well known, the experiments by zero-eld muon-spin relaxation ( SR) and the polar Kerr eect (PKE) can provide strong evidence for the observation of spontaneous magnetization or spin polar- ization in the superconducting states, which indicates a TRB superconducting pairing symmetry. On the theory side, the non-unitary spin-triplet pairing potentials are always adopted to explain the experiments. However, for a spin-singlet SC, a theory with TRB pairing-induced spin-magnetization is in great demand. Addressing this crucial issue is one of the aims of this work, and we nd that a spin-singlet TRB non-unitary SCs supports a TRB atomic orbital polarization, which in turn would give rise to spin polarization in the presence of atomic SOC. A. 2D chiral TSC We rst explore the possible 2D chiral topological phases by considering the simplest case with o= tstr= s= 0 to demonstrate the essential physics. For the TRB non-unitary pairing, a complex orbital do-vector can be generally parameterized as do= (cos;0;eisin). And the relative phase ==2 is energetically favored by minimizing the free energy. At the point, the bulk gap closes at c;i= p 2soc4jij2, wherei= 1;2 and 1;2=io(sin cos). Due to TRB, c;16= c;2. Accordingly, we semi- qualitatively map out the phase diagram in Fig. 5 by tun- ingand, and label the dierent phase regions by the number of Majorana edge modes (MEMs), denoted as Q. Whenjj>maxfjc;1j;jc;2jg, the topologically trivial phase is achieved with Q= 0. As for minfjc;1j;jc;2jg< jj<maxfjc;1j;jc;2jg, there is only one MEM on the boundary, corresponding to the Q= 1 regions [88, 89]. Whenjj<minfjc;1j;jc;2jg, there areQ= 2 MEMs. 0.00.51.01.52.02.53.0-0.55-0.35-0.150.050.250.45 θμπ02π0-0.60.60.3-0.3μ/u1D4AC=2/u1D4AC=2/u1D4AC=2/u1D4AC=1/u1D4AC=1/u1D4AC=1/u1D4AC=1Trivial SC Trivial SCθFIG. 5. Topological chiral superconductivity. We plot the phase diagram in terms of the number of MEMs ( Q) of the TSC. Parameters used: o= 0:14,soc= 0:4,==2, o= 0,tstr= 0 and s= 0. The chiral TSC might be detected by anomalous thermal Hall conductivity Kxy=Q 2T 6[90]. B. Atomic orbital polarization and spin polarization Next, we show how spin-singlet TRB non-unitary pair- ing can induce spin polarization, and discuss how to identify such pairings by using spin-polarized scanning tunneling microscopy measurements. We assume a TRB complex orbital do-vector and nd that it can generate the orbital orderings as Mo=i 1=M(dod o); (39) of which the y-component breaks TRS shown in Fig. 6 (a). More precisely, we nd that My o/11 (a)𝑴𝑜∝𝑖𝒅𝑜×𝒅𝑜∗𝒅𝑜𝒅𝑜∗ (a)TRS-breaking OP(c)(b)𝐸𝑘𝑥𝑘𝑥𝜔𝐷−𝐷+ TRS-breaking OP(b)Two orbitals{dxz,dyz}{dxy,dyz}{dxz,dxy}SP direction-axisz-axisy-axisxλsocL⋅σ -0.5 0 0.50123456(c)Orbital DOS (a.u.)κ=+1κ=−1E-0.5 0 0.50123456Spin DOS (a.u.)σ=↓σ=↑(d) E FIG. 6. (a) Schematic diagram showing the TRB orbital po- larization (OP) induced by complex do-vector. (b) Spin could be polarized in dierent directions based on the two active orbitals involved in the pairing. (c) Orbital DOS projected into the chiral =1 basis, showing a two-gap feature due to TRB. (d) The corresponding spin DOS, shifted relative to the fermi level due to the non-zero eective Zeeman eld from the OP. Parameters used: m= 0:5;=2;R= 0;soc= 0:2;o= 0:4;o= 0;tstr= 0;do= (1;0;ei=10). P k;h^ndxz+idyz;(k)^ndxzidyz;(k)i6= 0 indicates the atomic orbital-polarization (OP) (see Appendix F for de- tails). Here, ^ nis the density operator of electrons. Once My odevelops a nite value, it leads to orbital-polarized DOS and two distinct superconducting gaps of the quasi- particle spectrum (Fig. 6 (c), more details below). There- fore, the orbital degree of freedom in spin-singlet SCs plays a similar role as the spin degree of freedom of spin- triplet SCs. Once the atomic SOC is present, spin-polarization (SP) could be induced indirectly. A possible Ginzburg- Landau term could be F=sjMsj2+ socMz sMy o; (40) withs>0 and soc6= 0. Here, Mz s/P k;h^n;" ^n;#i. Therefore, the complex orbital do-vector can be identied by the spin-resolved density of states (DOS) for spin-singlet superconductors.Minimizing Eq. (40)directly leads toMz s= socMy o=Mz s, which indicates the OP- induced spin magnetism. In addition, the direction of SP can be also aligned to xoryaxes, discussed later. To verify the above analysis, we numerically solve the BdG Hamiltonian (13), HBdGjEn(k)i=En(k)jEn(k)i, where the n-th eigenstate is given by jEn(k)i= (un dxz;";un dxz;#;vn dxz;";vn dxz;#;un dyz;";un dyz;#;vn dyz;";vn dyz;#)T. Thus, the atomic-orbital and spin-resolved DOS can becalculated as the following, D orbit(E) =X ;n;kjun ;j2(EEn(k)); D spin(E) =X ;n;kjun ;j2(EEn(k));(41) whereun ;=1p 2(un dxz;iun dyz;) and=1 for dxzidyzorbitals. In Fig. 6 (c), D+1 orbit6=D1 orbitin- dicates that the DOS is orbital-polarized. Remarkably, we also have D" spin6=D# spindue to coupling between elec- tron spin and atomic orbitals, shown in Fig. 6 (d). The dierence in orbital DOS acts as an eective Zeeman eld for the electron spins, hence shifting the spin DOS rela- tive to the fermi level in opposite directions for up spin and down spin. This interesting phenomenon is quite dierent from spin-triplet SCs. In TRB spin-triplet SCs, The spin-up channel and spin-down channel will form dierent symmetric gaps in spin DOS, similar to the two orbital channels in Fig. 6 (c) for our case. Therefore, the spin DOS proles are distinct in the two cases. As a result, the spin-resolved DOS, which can be probed by spin-resolved STM [91] and muon-spin relaxation [92, 93], can serve as a smoking gun evidence to identify TRB due to complex orbital do-vector in multi-orbital SCs. VI. DISCUSSIONS AND CONCLUSIONS In the end, we brie y discuss the direction of spin- polarization induced by atomic orbital-polarization, sum- marized in Fig. 6 (b). We consider the three-dimensional subspace of t2gorbitals spanned by fdyz;dxz;dxyg, where the matrix form of the angular momentum operators L reads [63], Lx=0 @0 0 0 0 0i 0i01 A; Ly=0 @0 0i 0 0 0 i0 01 A; Lz=0 @0i0 i0 0 0 0 01 A; (42) which satisfy the commutation relation [ Lm;Ln] = imnlLl. Therefore, the spin-orbit coupling for a system with thet2gorbitals is given by, Hsoc=socL: (43) Then, let us consider a two-orbital system, the above SOC Hamiltonian will be reduced to, 8 >< >:Forfdyz;dxzg:Hsoc=soc23; Forfdyz;dxyg:Hsoc=soc22; Forfdxz;dxyg:Hsoc=soc21:(44) Therefore, in the above three cases, the spin-polarization is pointed to z;y;x -axis, respectively. Because the atomic12 orbital polarization is induced by the complex orbital do- vector as (0 ;My o;0)/id odo. To summarize, we establish a phenomenological the- ory for spin-singlet two-band SCs and discuss the distinct features of both TRI non-unitary pairings and TRB non- unitary pairings by studying the eects of atomic spin- orbit coupling (SOC), lattice strain eect, and Rashba SOC. Practically, we demonstrate that the stability of orbital-dependent pairing states could give birth to the non-unitary pairing states in a purely spin-singlet SC. Remarkably, the interplay between atomic SOC and orbital-dependent pairings is also investigated and we nd a new spin-orbit-coupled SC with spin-singlet non- unitary pairing. For this exotic state, there are mainly three features. Firstly, the atomic SOC could enlarge the in-plane upper critical eld compared to the Pauli limit. A new eect dubbed as \zero-eld Pauli limit" for orbital-dependent pairings is discovered. Secondly, topological chiral or helical superconductivity could berealized even in the absence of external magnetic elds or Zeeman elds. Furthermore, a spontaneous TRB SC could even generate a spin-polarized superconducting state that can be detected by measuring the spin-resolved density of states. We hope our theory leads to a deeper understanding of spin-singlet non-unitary SCs. Our theory might have potential applications to the intriguing Sr 2SuO 4[94, 95], LaNiGa 2[58], iron-based SCs [59, 60] and ultra-cold atomic systems with large spin alkali and alkaline-earth fermions [96{100]. VII. ACKNOWLEDGMENTS We thank J.-L. Lado, R.-X. Zhang and C.-X. Liu for helpful discussions. We especially acknowledge J.- L. Lado's careful reading of the manuscript. D.-H.X. was supported by the NSFC (under Grant Nos. 12074108 and 11704106). [1] M. Sigrist and K. Ueda, Rev. Mod. Phys. 63, 239 (1991). [2] V. P. Mineev, K. Samokhin, and L. Landau, Introduc- tion to unconventional superconductivity (CRC Press, 1999). [3] F. Steglich, J. Aarts, C. D. Bredl, W. Lieke, D. Meschede, W. Franz, and H. Sch afer, Phys. Rev. Lett. 43, 1892 (1979). [4] J. G. Bednorz and K. A. M 'uller, Z. Physik B - Con- densed Matter 64, 189 (1986). [5] F. C. Zhang and T. M. Rice, Phys. Rev. B 37, 3759 (1988). [6] P. W. Anderson, P. Lee, M. Randeria, T. Rice, N. Trivedi, and F. Zhang, Journal of Physics: Con- densed Matter 16, R755 (2004). [7] D. J. Scalapino, Rev. Mod. Phys. 84, 1383 (2012). [8] P. M. R. Brydon, D. S. L. Abergel, D. F. Agterberg, and V. M. Yakovenko, Phys. Rev. X 9, 031025 (2019). [9] T. M. R. Wolf, M. F. Holst, M. Sigrist, and J. L. Lado, Phys. Rev. Res. 4, L012036 (2022). [10] M. Zeng, D.-H. Xu, Z.-M. Wang, L.-H. Hu, and F.- C. Zhang, arXiv e-prints , arXiv:2109.06039 (2021), arXiv:2109.06039 [cond-mat.supr-con]. [11] X. Dai, Z. Fang, Y. Zhou, and F.-C. Zhang, Phys. Rev. Lett. 101, 057008 (2008). [12] T. T. Ong and P. Coleman, Phys. Rev. Lett. 111, 217003 (2013). [13] Z. P. Yin, K. Haule, and G. Kotliar, Nature Physics 10, 845 (2014). [14] T. Ong, P. Coleman, and J. Schmalian, Proceedings of the National Academy of Sciences 113, 5486 (2016). [15] R. Nourafkan, G. Kotliar, and A.-M. S. Tremblay, Phys. Rev. Lett. 117, 137001 (2016). [16] A. V. Chubukov, O. Vafek, and R. M. Fernandes, Phys. Rev. B 94, 174518 (2016). [17] M. Yi, Y. Zhang, Z.-X. Shen, and D. Lu, npj Quantum Materials 2, 1 (2017). [18] E. M. Nica, R. Yu, and Q. Si, npj Quantum Materials 2, 1 (2017).[19] P. O. Sprau, A. Kostin, A. Kreisel, A. E. B ohmer, V. Taufour, P. C. Caneld, S. Mukherjee, P. J. Hirschfeld, B. M. Andersen, and J. S. Davis, Science 357, 75 (2017). [20] H. Hu, R. Yu, E. M. Nica, J.-X. Zhu, and Q. Si, Phys. Rev. B 98, 220503 (2018). [21] W. Chen and W. Huang, Phys. Rev. Research 3, L042018 (2021). [22] E. M. Nica and Q. Si, npj Quantum Materials 6, 1 (2021). [23] L. A. Wray, S.-Y. Xu, Y. Xia, Y. San Hor, D. Qian, A. V. Fedorov, H. Lin, A. Bansil, R. J. Cava, and M. Z. Hasan, Nature Physics 6, 855 (2010). [24] L. Fu and E. Berg, Phys. Rev. Lett. 105, 097001 (2010). [25] P. M. R. Brydon, L. Wang, M. Weinert, and D. F. Agter- berg, Phys. Rev. Lett. 116, 177001 (2016). [26] W. Yang, Y. Li, and C. Wu, Phys. Rev. Lett. 117, 075301 (2016). [27] D. F. Agterberg, P. M. R. Brydon, and C. Timm, Phys. Rev. Lett. 118, 127001 (2017). [28] L. Savary, J. Ruhman, J. W. F. Venderbos, L. Fu, and P. A. Lee, Phys. Rev. B 96, 214514 (2017). [29] W. Yang, T. Xiang, and C. Wu, Phys. Rev. B 96, 144514 (2017). [30] C. Timm, A. P. Schnyder, D. F. Agterberg, and P. M. R. Brydon, Phys. Rev. B 96, 094526 (2017). [31] J. Yu and C.-X. Liu, Phys. Rev. B 98, 104514 (2018). [32] I. Boettcher and I. F. Herbut, Phys. Rev. Lett. 120, 057002 (2018). [33] B. Roy, S. A. A. Ghorashi, M. S. Foster, and A. H. Nevidomskyy, Phys. Rev. B 99, 054505 (2019). [34] H. Kim, K. Wang, Y. Nakajima, R. Hu, S. Ziemak, P. Syers, L. Wang, H. Hodovanets, J. D. Denlinger, P. M. Brydon, et al. , Science advances 4, eaao4513 (2018). [35] D. F. Agterberg, T. M. Rice, and M. Sigrist, Phys. Rev. Lett. 78, 3374 (1997). [36] T. Takimoto, Phys. Rev. B 62, R14641 (2000).13 [37] W. Huang, Y. Zhou, and H. Yao, Phys. Rev. B 100, 134506 (2019). [38] Z.-M. Wang, M. Zeng, C. Lu, L.-H. Hu, and D.-H. Xu, In preparation (2022). [39] J. Clepkens, A. W. Lindquist, X. Liu, and H.-Y. Kee, Phys. Rev. B 104, 104512 (2021). [40] A. Ramires and M. Sigrist, Phys. Rev. B 100, 104501 (2019). [41] W. Yang, C. Xu, and C. Wu, Phys. Rev. Research 2, 042047 (2020). [42] N. F. Q. Yuan, W.-Y. He, and K. T. Law, Phys. Rev. B95, 201109 (2017). [43] L. Chirolli, F. de Juan, and F. Guinea, Phys. Rev. B 95, 201110 (2017). [44] A. Robins and P. Brydon, Journal of Physics: Con- densed Matter 30, 405602 (2018). [45] J. L. Lado and M. Sigrist, Phys. Rev. Research 1, 033107 (2019). [46] L.-H. Hu, P. D. Johnson, and C. Wu, Phys. Rev. Re- search 2, 022021 (2020). [47] A. Ramires, Journal of Physics: Condensed Matter 34, 304001 (2022). [48] Y. Saito, Y. Nakamura, M. S. Bahramy, Y. Kohama, J. Ye, Y. Kasahara, Y. Nakagawa, M. Onga, M. Toku- naga, T. Nojima, et al. , Nature Physics 12, 144 (2016). [49] J. Lu, O. Zheliuk, I. Leermakers, N. F. Yuan, U. Zeitler, K. T. Law, and J. Ye, Science 350, 1353 (2015). [50] X. Xi, Z. Wang, W. Zhao, J.-H. Park, K. T. Law, H. Berger, L. Forr o, J. Shan, and K. F. Mak, Nature Physics 12, 139 (2016). [51] J. Falson, Y. Xu, M. Liao, Y. Zang, K. Zhu, C. Wang, Z. Zhang, H. Liu, W. Duan, K. He, et al. , Science 367, 1454 (2020). [52] J. P. Ruf, H. Paik, N. J. Schreiber, H. P. Nair, L. Miao, J. K. Kawasaki, J. N. Nelson, B. D. Faeth, Y. Lee, B. H. Goodge, et al. , Nature Communications 12, 59 (2021). [53] S. Beck, A. Hampel, M. Zingl, C. Timm, and A. Ramires, Phys. Rev. Res. 4, 023060 (2022). [54] K. Ahadi, L. Galletti, Y. Li, S. Salmani-Rezaie, W. Wu, and S. Stemmer, Science advances 5, eaaw0120 (2019). [55] R. M. Fernandes and L. Fu, Phys. Rev. Lett. 127, 047001 (2021). [56] A. D. Hillier, J. Quintanilla, and R. Cywinski, Phys. Rev. Lett. 102, 117007 (2009). [57] A. D. Hillier, J. Quintanilla, B. Mazidian, J. F. Annett, and R. Cywinski, Phys. Rev. Lett. 109, 097001 (2012). [58] Z. F. Weng, J. L. Zhang, M. Smidman, T. Shang, J. Quintanilla, J. F. Annett, M. Nicklas, G. M. Pang, L. Jiao, W. B. Jiang, Y. Chen, F. Steglich, and H. Q. Yuan, Phys. Rev. Lett. 117, 027001 (2016). [59] V. Grinenko, R. Sarkar, K. Kihou, C. Lee, I. Morozov, S. Aswartham, B. B uchner, P. Chekhonin, W. Skrotzki, K. Nenkov, et al. , Nature Physics , 1 (2020). [60] N. Zaki, G. Gu, A. Tsvelik, C. Wu, and P. D. Johnson, Proceedings of the National Academy of Sciences 118 (2021). [61] P. K. Biswas, S. K. Ghosh, J. Z. Zhao, D. A. Mayoh, N. D. Zhigadlo, X. Xu, C. Baines, A. D. Hillier, G. Bal- akrishnan, and M. R. Lees, Nature communications 12, 2504 (2021). [62] Q. Zou, B. D. Oli, H. Zhang, T. Shishidou, D. Agterberg, M. Weinert, and L. Li, arXiv preprint arXiv:2212.13603 (2022).[63] W.-C. Lee, D. P. Arovas, and C. Wu, Phys. Rev. B 81, 184403 (2010). [64] J. Clepkens, A. W. Lindquist, and H.-Y. Kee, Phys. Rev. Research 3, 013001 (2021). [65] J. B oker, P. A. Volkov, P. J. Hirschfeld, and I. Eremin, New Journal of Physics 21, 083021 (2019). [66] S. Raghu, X.-L. Qi, C.-X. Liu, D. J. Scalapino, and S.-C. Zhang, Phys. Rev. B 77, 220503 (2008). [67] M. Yi, Y. Zhang, Z.-X. Shen, and D. Lu, npj Quantum Materials 2, 57 (2017). [68] C. Guo, L. Hu, C. Putzke, J. Diaz, X. Huang, K. Manna, F.-R. Fan, C. Shekhar, Y. Sun, C. Felser, et al. , Nature Physics , 1 (2022). [69] L.-H. Hu, X. Wang, and T. Shang, Phys. Rev. B 104, 054520 (2021). [70] T. Shang, J. Zhao, L.-H. Hu, J. Ma, D. J. Gawryluk, X. Zhu, H. Zhang, Z. Zhen, B. Yu, Y. Xu, et al. , Science advances 8, eabq6589 (2022). [71] P. A. Frigeri, D. F. Agterberg, A. Koga, and M. Sigrist, Phys. Rev. Lett. 92, 097001 (2004). [72] H. Yi, L.-H. Hu, Y. Wang, R. Xiao, J. Cai, D. R. Hickey, C. Dong, Y.-F. Zhao, L.-J. Zhou, R. Zhang, et al. , Na- ture Materials , 1 (2022). [73] C. Wang, B. Lian, X. Guo, J. Mao, Z. Zhang, D. Zhang, B.-L. Gu, Y. Xu, and W. Duan, Phys. Rev. Lett. 123, 126402 (2019). [74] K. Maki and T. Tsuneto, Progress of Theoretical Physics 31, 945 (1964). [75] X.-L. Qi, T. L. Hughes, S. Raghu, and S.-C. Zhang, Phys. Rev. Lett. 102, 187001 (2009). [76] C.-X. Liu and B. Trauzettel, Phys. Rev. B 83, 220510 (2011). [77] S. Nakosai, Y. Tanaka, and N. Nagaosa, Phys. Rev. Lett. 108, 147003 (2012). [78] S. Deng, L. Viola, and G. Ortiz, Phys. Rev. Lett. 108, 036803 (2012). [79] F. Zhang, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. 111, 056402 (2013). [80] J. Wang, Y. Xu, and S.-C. Zhang, Phys. Rev. B 90, 054503 (2014). [81] A. Haim and Y. Oreg, Physics Reports 825, 1 (2019). [82] O. E. Casas, L. Arrachea, W. J. Herrera, and A. L. Yeyati, Phys. Rev. B 99, 161301 (2019). [83] Y. Volpez, D. Loss, and J. Klinovaja, Phys. Rev. Re- search 2, 023415 (2020). [84] R.-X. Zhang and S. Das Sarma, Phys. Rev. Lett. 126, 137001 (2021). [85] J.-F. Ge, Z.-L. Liu, C. Liu, C.-L. Gao, D. Qian, Q.-K. Xue, Y. Liu, and J.-F. Jia, Nature Materials 14, 285 (2014). [86] R. Yu, X. L. Qi, A. Bernevig, Z. Fang, and X. Dai, Phys. Rev. B 84, 075119 (2011). [87] W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Phys. Rev. B 96, 245115 (2017). [88] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B 82, 184516 (2010). [89] J. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma, Phys. Rev. Lett. 104, 040502 (2010). [90] T. Meng and L. Balents, Phys. Rev. B 86, 054504 (2012). [91] R. Wiesendanger, Reviews of Modern Physics 81, 1495 (2009).14 [92] G. Csire, J. Annett, J. Quintanilla, and B. Ujfalussy, Magnetically-textured superconductivity in elemental rhenium (2021), arXiv:2005.05702 [cond-mat.supr-con]. [93] T. Shang, M. Smidman, S. K. Ghosh, C. Baines, L. J. Chang, D. J. Gawryluk, J. A. T. Barker, R. P. Singh, D. M. Paul, G. Balakrishnan, E. Pomjakushina, M. Shi, M. Medarde, A. D. Hillier, H. Q. Yuan, J. Quintanilla, J. Mesot, and T. Shiroka, Phys. Rev. Lett. 121, 257002 (2018). [94] G. M. Luke, Y. Fudamoto, K. Kojima, M. Larkin, J. Merrin, B. Nachumi, Y. Uemura, Y. Maeno, Z. Mao, Y. Mori, et al. , Nature 394, 558 (1998). [95] J. Xia, Y. Maeno, P. T. Beyersdorf, M. M. Fejer, and A. Kapitulnik, Phys. Rev. Lett. 97, 167002 (2006). [96] T.-L. Ho and S. Yip, Phys. Rev. Lett. 82, 247 (1999). [97] C. Wu, J.-p. Hu, and S.-c. Zhang, Phys. Rev. Lett. 91, 186402 (2003). [98] B. J. DeSalvo, M. Yan, P. G. Mickelson, Y. N. Mar- tinez de Escobar, and T. C. Killian, Phys. Rev. Lett. 105, 030402 (2010). [99] S. Taie, Y. Takasu, S. Sugawa, R. Yamazaki, T. Tsuji- moto, R. Murakami, and Y. Takahashi, Phys. Rev. Lett. 105, 190401 (2010). [100] A. V. Gorshkov, M. Hermele, V. Gurarie, C. Xu, P. S. Julienne, J. Ye, P. Zoller, E. Demler, M. D. Lukin, and A. Rey, Nature physics 6, 289 (2010). [101] Y. Wang and L. Fu, Phys. Rev. Lett. 119, 187003 (2017). [102] C.-X. Liu, Phys. Rev. Lett. 118, 087001 (2017). [103] L.-H. Hu, C.-X. Liu, and F.-C. Zhang, Communications Physics 2, 1 (2019). [104] S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. Ludwig, New Journal of Physics 12, 065010 (2010). [105] C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, Rev. Mod. Phys. 88, 035005 (2016).15 Appendix A: Toy model for two-band superconducting phase diagrams In this part of the appendix, we explore a possible su- perconducting phase diagram including the non-unitary pairing states in the GL framework. Here we assume a two-band SC with tot= s0+ o(d1 o1+d3 o3) (i2): (A1) In terms of the superconducting order parameters fs;o;do= (d1 o;0;d3 o)gand the order parameter for the orbital orderings Mo/P k;hcy a(k)abcb(k)i, the total GL free energy can be constructed to address the homogeneous superconducting phase without external magnetic elds, F[s;o;do;Mo] =F0+Fb+Fo; (A2) where F0=1 2(T)joj2+1 20(T)jsj2+1 2MjMoj2 +1 4joj4+1 40jsj4+00jsj2joj2 +ojd1 oj4+0 ojd3 oj4;(A3) wherejdoj= 1 is adopted, (T) =0(T=Tc11), 0(T) =0 0(T=Tc21) and the coecients 0,0 0,M,, 0,00,o,0 oare all positive. Tc1;Tc2are critical tem- peratures in orbital-dependent and orbital-independent channels respectively, which are in general dierent from each other. And M>0 means that there is no spon- taneous atomic orbital polarization. In the supercon- ducting state with both non-zero sand odeveloped already, additionally, there are two possible ways to pur- sue the spontaneous TRB, denoted as FbandFo. Firstly, we consider theFbterm Fb=b1 so+b2( so)2+ h.c.; (A4) where the sign of b2determines the breaking of TRS. Here we focus on the generic case where sand obelong to dierent symmetry representations so that there is no linear order coupling between them, i.e. b1= 0. Given b1= 0 andb2>0, we have a o==2 relative phase dierence between sand oeio[101], which gives to the achievement of the TRB unitary pairing state ( s2 R;oi;do2R). More generally, a TRB non-unitary SC arises from the non-zero bilinear b1-term, which is symmetry-allowedonly when sand obelong to the same symmetry rep- resentation of the crystalline symmetry group. Namely, the case with b16= 0 andb2>0 can pin the phase dierenceoto an arbitrary nonzero value, i.e., o2 (0;). Then, this case can also give rise to TRB non- unitary pairing with ( s2R;t2C;do2R) or (s2R;t2C;do2C). On the other hand, the b2<0 situation makes TRI non-unitary pairing states (s2R;o2R;do2R). However, even in the case with b2<0, we still have an alternative approach to reach TRB pairing states, driven by theFoterm Fo= 0jdod oj2+i 1Mo(dod o) + h.c.;(A5) where the sign of 0identies the TRB due to a complex do. In particular 0<0 results in a TRB non-unitary state (s2R;o2R;do2C). We summarize many of the possible interesting super- conducting phases in Fig. 7, which schematically shows a superconducting phase diagram as a function of b2and 0 by settingb1= 0, i.e. the generic case where o;sbe- long to dierent representations. Notice that this phase diagram characterized by b2and 0does not contain the TRI unitary pairing phase. b2γ0(TRB + NU)(TRB + NU)(TRB + U)(TRI + NU)Δo∝i,do∈ℂΔo∈ℝ,do∈ℂΔo∝i,do∈ℝΔo∈ℝ,do∈ℝ0 FIG. 7. Schematic superconducting phase diagrams on the b2- 0plane when b1= 0 and sis real and non-zero. Here, TRB and TRI are short for TR-breaking and TR-invariant, respectively; U and NU represent unitary and non-unitary, respectively. Appendix B: Derivation of Tcfrom linearized gap equation Starting from the generic Hamiltonian, containing atomic SOC, generic ~g= (~g1;0;~g2) withj~gj= 1 and in-plane magnetic eld, H0(k) =(k) +soc32+(~g11+ ~g33) +h1: (B1)16 The Matsubara Green's function for electrons is Ge(k;i!n) = [i!nH 0(k)]1 =P i!nk+E+P++ i!nk+E++P+ i!nkE+P+++ i!nkE+;(B2) where the projection operator P =1 4[1 +(~g111+ ~g313)][1 + E (soc32+o(~g11+ ~g33) +h1)]; (B3) with;; 2f+;gandE =p 2soc+ (+ h)2. The Green's function for hole is Gh(k;i!n) =G e(k;i!n). Here !n= (2n+ 1)kBT. The linearized gap equation is given by a;b s1;s2(k) =1 X !nX s0 1a0;s0 2b0Vs1a;s2b s0 1a0;s0 2b0(k;k0)[Ge(k0;i!n)(k0)Gh(k0;i!n)]s0 1a0;s0 2b0; (B4) where the generic attractive interaction can be expanded as Vs1a;s2b s0 1a0;s0 2b0(k;k0) =v0X ;m[d;m o(k)i2]s1a;s2b[d;m o(k0)i2]s0 1a0;s0 2b0; (B5) wherev0>0 and labels the irreducible representation with m-dimension of crystalline groups. The linearized gap equation is reduced to v0(T)1 = 0 where (T) is the superconductivity susceptibility. We have •For orbital-independent pairing: (T)s=1 X k;!nTr ( s(k)i2)yGe(k;i!n)( s(k)i2)Gh(k;i!n) : (B6) •For orbital-dependent pairing: (T)o=1 X k;!nTr (do(k)i2)yGe(k;i!n)(do(k)i2)Gh(k;i!n) : (B7) Then we take the standard replacement, X k;!n!N0 4Z+!D !DdZZ Sd X !n; (B8) whereN0is the density of states at Fermi surface, is the solid angle of kon Fermi surfaces and !Dthe Deybe frequency. We will also be making use of, N0 Z+!D !DX !ndG+ e(k;i!n)G+ h(k;i!n) =N0 Z+!D !DX !nG e(k;i!n)G h(k;i!n) =0(T); (B9) N0 Z+!D !DX !ndG e(k;i!n)G+ h(k;i!n) =N0 Z+!D !DX !nG+ e(k;i!n)G h(k;i!n) =0(T) +N0C0(T);(B10) where0(T) =N0ln 2e !D kBT , = 0:57721the Euler-Mascheroni constant and C0(T) = Re[ (0)(1 2) (0)(1 2+ iE(k) 2kBT)] with (0)(z) being the digamma function. For orbital-independent pairing considered in the main text s0i2, we have s(T) =0(T) +N0 2 C0 T;E+E 2 +C0 T;E++E 2 +N0 2 C0 T;E+E 2 C0 T;E++E 2 2+2 soch2 E+E 0(T) +N0fs(T;soc;;h):(B11)17 In order to look at the eect of on the Pauli limit, we could Taylor expand fs(T;soc;;h) for small: fs(T;soc;;h) =fs(T;soc;0;h) +F(T;soc;h)2+O(4); (B12) with F(T;soc;h) = (2)(1 2)2 soch2 4k2 BT2(2soc+h2)2 Ref (0)(1 2) (0)(1 2+ip 2soc+h2 2kBT)g42 soch2 (2soc+h2)3 + Imf (1)(1 2+ip 2soc+h2 2kBT)g2 soch2 2kBT(2soc+h2)5=2:(B13) This is used in the main text. For orbital-dependent pairing o(d11+d33)i2withdo=~g, we have o(T) =0(T) +N0 2 C0 T;E+E 2 +C0 T;E++E 2 +N0 2 C0 T;E+E 2 C0 T;E++E 2 22 soch2 E+E 0(T) +N0fo(T;soc;;h):(B14) Appendix C: Strain eect on Tcand pairing symmetry The strain eect characterized by Eq. (6) in the main text can be absorbed into the orbital hybridiza- tion vector goand gives rise to an eective ~ggo+ tstr=o(sin 2;0;cos 2). Then in the absence of SOC terms, the corrected critical temperature Tcdue to the strain and hybridization eects is perturbatively given by lnTc T0 =Z Z Sd C0(T0) jdoj2jdo^~gj2 ;(C1) whereT0is the critical temperature without strain or hy- bridization and the integration is over the solid angle of kover the Fermi surface. Similar to previous discussions, the strain generally suppresses the critical temperature when ~gis not exactly parallel to do, as shown in Fig. 8 (a). For non-zero strain, the Tcis not suppressed when dojj~g. Fig. 8 (b) shows the symmetry breaking pattern of thejdoj, which is proportional to the SC gap (the propor- tionality constant has been normalized to 1 in the gure), around the Fermi surface. The strain would reduce the symmetry from C4toC2, as expected. Appendix D: TSC with s= 0;o= 0 To demonstrate the topology, we also show a simple case with s= 0 ando= 0, where the Z2can be characterized analytically. In this section, we focus on the simplied case without orbital independent pairing or orbital hybridization. InFig. 4 (c), we calculate the edge spectrum with kxbe- ing a good quantum number in a semi-innite geometry, and it shows the corresponding bulk band structure to- gether with two counter-propagating MEMs. The bulk topology of the 2D helical TSC phase is characterized by theZ2topological invariant , which can be extracted by calculating the Wilson-loop spectrum. And, = 1 mod 2 characterizes the helical TSC. In Fig. 4 (b), we plot the evolution of as a function of ky, and the winding pattern indicates the topological Z2invariant= 1. On the other hand, with s= 0, which is the case if we only consider on-site attractive interactions be- tween electrons [102, 103], the BdG Hamiltonian (13) can be decomposed into two orbital subspaces that are related through time-reversal transformation. Each of these blocks has a well-dened Chern number because each block alone breaks TRS. The two Chern numbers can then be used to dene the Z2invariant of the whole BdG system. The detailed procedures are the following. For the normal Hamiltonian given in Eq. (1), we have [H0;2] = 0. As a result, we can project the normal HamiltonianH0in Eq. (1) into block-diagonal form cor- responding to the 1 eigenvalues of 2by using the basis transformationU=0 1p 2 1i 1i . The new basis is given by ~ y(k) = (cy +;";cy +;#;cy ;#;cy ;"); (D1) wherecy ;s1p 2(cy dxz;sicy dyz;s). On this basis, the normal Hamiltonian is given by H0=H+ 0H 0; (D2)18 Tc/T0t/tstr(a)tstr/λokF2=0tstr/λokF2=1tstr/λokF2=20.00.51.01.52.00.50.60.70.80.91.0 \|do(θ)\|θ(b)tstr/λokF2=0tstr/λokF2=0.30π2π0.00.51.01.52.0 FIG. 8. (a) shows the suppression of Tcfor dierent strain strengths. Here do=go+t ogstrwhereas ~g=go+tstr ogstr. (b) shows the symmetry breaking of the SC gap from C4toC2due to the existence of the external strain. We have chosen go= (3kxky;0;k2 xk2 y) and the strain parameter = 0 in gstr. whereH 0are given by H 0=(k) +R(kx2ky1)soc3: (D3) Note that the time-reversal transforms H 0(k) to H 0(k). In the new basis the pairing Hamiltonian also decouples asH=H+ H withH given by H = 2h cy ;"(k)cy ;#(k)("$# )i + h.c.;(D4) where o(id1 o+d3 o) are the gap strengths in each subspace. Therefore, the Bogoliubov de-Gennes (BDG) Hamiltonian takes the following block-diagonal form, HBdG=H+ BdGH BdG; (D5) where H BdG(k) = ((k)soc3) 3+R(kx2 3ky1 0) 2d12 12d32 2; (D6) with being the Pauli matrices in the particle- hole space. The Nambu basis is y (k) = (cy ;"(k);cy ;#(k);c;"(k);c;#(k)). Each subspace has its own particle-hole symmetry. By symmetry, the 2D BdG Hamiltonian in Eq. (D5) belongs to Class DIII of the A-Z classication[104, 105] for topological insulators and superconductors because both TRS and particle-hole symmetry are preserved. However, it is not the case for our model. The BdG Hamiltonian here could exhibit topological states with Z2 type topological invariant, which can be dened as the following. In each subspace, we dene the BdG Chern number as C=1 2X lled bandsZ BZdkh n(k)jirkj n(k)i;(D7) withj nibeing the energy eigenstate of H BdG. Then theZ2invariant, in this case, is then explicitly given by, C+C 2; (D8) whereCare the Chern numbers of the channels. This has been discussed in the main text.Appendix E: Wilson loop calculation for Z2TSC In the thermodynamics limit, the Wilson loop operator along a closed path pis expressed as Wp=Pexp iI pA(k)dk ; (E1) wherePmeans path ordering and A(k) is the non- Abelian Berry connection Anm(k) =ihn(k)jrkjm(k)i; (E2) withjm;n(k)ithe occupied eigenstates. The Wilson line element is dened as Gnm(k) =hn(k+ k)jm(k)i; (E3) where the k= (kx;ky), and k= (0;2=Ny) is the steps. In the discrete case, the Wilson loop operator on a path along kyfrom the initial point kto the nal point k+ (0;2) can be written as Wy;k=G(k+ (Ny 1)k)G(k+(Ny2)k):::G(k+k)G(k), which satises the eigenvalue equation Wy;kjj y;ki=ei2j y(kx)jj y;ki (E4) The phase of eigenvalue = 2j y(kx) is the Wannier function center. Appendix F: Spin and orbital magnetizations: M s and Mo In this section, we show the denition of spin and or- bital magnetization at the mean-eld level. The spin magnetization in orbital-inactive systems takes the form Ms/X k;s1;s2hcy s1(k)s1s2cs2(k)i; (F1)19 which tells us the magnetic moments generated by spin polarization. Similarly, the orbital magnetization in orbital-active system is given by Mo/X k;s;a;bhcy s;a(k)abcs;b(k)i: (F2) The dierent components of the orbital magnetization vector represent dierent orders in the SC ground state.More specically, we have Mx o=X k;shcy s;dxzcs;dyz+cy s;dyzcs;dxzi; (F3) My o=iX k;shcy s;dxzcs;dyzcy s;dyzcs;dxzi (F4) =1 2X k;sh^ns;dxz+idyz^ns;dxzidyzi; (F5) Mz o=X k;shcy s;dxzcs;dxzcy s;dyzcs;dyzi: (F6) We see that Mx;z obreaks theC4rotation symmetry and My obreaks TRS. In our work, we only consider the pos- sibility of spontaneous TRS breaking, thus the Mx;z o will not couple to the superconducting order parameters, which are required to be invariant under Cn. BecauseMy o breaks TRS so that it could be coupled to the supercon- ducting order parameters, which spontaneously breaks TRS. This is one of the main results of our work, (0;My o;0)/id odo; (F7) where the complex orbital do-vector breaks TRS.
1212.0420v5.Normal_state_properties_of_spin_orbit_coupled_Fermi_gases_in_the_upper_branch_of_energy_spectrum.pdf	arXiv:1212.0420v5 [cond-mat.quant-gas] 2 May 2013Normal state properties of spin-orbit coupled Fermi gases i n the upper branch of energy spectrum Xiao-Lu Yu, Shang-Shun Zhang, and Wu-Ming Liu Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China (Dated: June 17, 2021) We investigate normal state properties of spin-orbit coupl ed Fermi gases with repulsive s-wave interaction, in the absence of molecule formation, i.e., in the so-called “upper branch”. Within the framework of random phase approximation, we derive anal ytical expressions for the quasi- particle lifetime τs, the eﬀective mass m∗ s, and the Green’s function renormalization factor Zsin the presence of Rashba spin-orbit coupling. In contrast to spin -orbit coupled electron gas with Coulomb interaction, we show that the modiﬁcations are dependent on the Rashba band index s, and occur in the ﬁrst order of the spin-orbit coupling strength. We als o calculate experimental observable such as spectral weight, density of state and speciﬁc heat, which exhibit signiﬁcant diﬀerences from their counterparts without spin-orbit coupling. We expect our mi croscopic calculations of these Fermi liquid parameters would have the immediate applications to the spin-orbit coupled Fermi gases in the upper branch of the energy spectrum. PACS numbers: 03.75.Ss, 05.30.Fk, 67.85.Lm I. INTRODUCTION Motivated by the recent success on the evidence of Stoner ferromagnetism for the repulsive Fermi gas in the upper branch of the energy spectrum [1], there has been increasing interest in the nature of uncondensed Fermi gas (free of molecules) within the repulsively interacting regime [2], which naturedly becomes the well-controlled platform for simulating Landau Fermi liquid. Much of the interest in ultracold atomic gases comes from their amazing tunability. A wide range of atomic physics and quantum optics technology provides unprecedented ma- nipulationofavarietyofintriguingquantumphenomena. Based on the Berry phase eﬀect [3] and its non-Abelian generalization [4], Spielman’s group in NIST has suc- cessfully generated a synthetic external Abelian or non- Abelian gauge potential coupled to neutral atoms. Re- cent experiments have realized the atomic40K [5] or6Li [6] gases with spin-orbit coupling (SOC). These achieve- ments will open a whole new avenue in cold atom physics [7–10]. The eﬀect of SOC in fermionic systems has conse- quently become an important issue in recent years, and attracts a great deal of attentions in ultracold Fermi gases. Most of existing works are devoted to the eﬀect of SOC on the superﬂuid state with negative s-wave scat- tering length [11–13] and the Bardeen-Cooper-Schrieﬀer (BCS) to Bose-Einstein condensation (BEC) crossover [14–17]. Furthermore, the SOC give rise to a variety of topological phases, such as the quantum spin Hall state and the topological superﬂuid [18–21]. In addition to the study of SOC eﬀect on these symmetry-breakingor topo- logical phases, the normal state contains various poten- tial instabilities and deserves attention. The considera- tion of SOC systems in the framework of Fermi liquid theory is therefore desirable. As is well known, Lan- dau’s Fermi-liquid theory provides a phenomenologicalapproach to describe the properties of strongly interact- ing fermions. Fermi liquid parameters characterizing the renormalizedmany-bodyeﬀective interactionshaveto be determined through experimental results. The purpose of this paper is to study key normal-state propertiesoftwodimensional(2D)FermigaseswithSOC in the repulsive regime—their quasi-particle lifetime, ef- fective massand Green’s function renormalizationfactor. Following previous studies of Landau’s Fermi-liquid the- ory including SOC [22, 23], we are attempted to build a microscopic foundation of phenomenological parame- ters. Therefore within the framework of random phase approximation (RPA), we derive analytical expressions for the quasi-particle lifetime τs, the eﬀective mass m∗ s, and the Green’s function renormalization factor Zsfor a 2D Fermi gases with repulsive s-wave interaction in the presence of Rashba SOC. To make contact with cur- rent experiments directly, we also calculate experimental observable such as spectral weight, density of state and speciﬁc heat, and discuss their corresponding experimen- tal signatures. We shall show that the modiﬁcations are dependent on the Rashba band index sdenoting the two directionsofthe eigenspinorsofthe RashbaHamiltonian. The paper is organizedasfollows. The model Hamilto- nian and the renormalizations due to s-wave interaction in the presence of SOC is discussed in Sec. II. In Sec. III, Starting from the RPA self-energy of the SOC Fermi liquid, we derived all the analytical formula of the Fermi liquid parameters in presence of SOC. The experimen- tal observable quantities such as the spectral function, density of states and speciﬁc heat are calculated. Sec. IV is devoted to discuss the experimental measurements of these fundamental parameters and their correspond- ing experimental signatures. The comparisons with the ordinary Fermi liquid are also presented.2 II. PERTUBATIVE THEORY OF 2D FERMI GASES WITH s-WAVE INTERACTION IN THE PRESENCE OF RASHBA SOC A. Model Hamiltonian We consider a 2D spin-1/2 fermionic system with Rashba-type SOC and s-wave interaction, described by the model Hamiltonian H=H0+HI. (1) The non-interacting part H0reads as, H0=/summationdisplay pc† p[p2 2m+α(ˆ z×p)·σ−µ]cp,(2) wherecp= (cp,↑,cp,↓)T,µ=k2 F/2mis the chemical po- tential,αrepresents SOC strength and kFis the Fermi momentum in the absence of Rashba-type SOC. The re- duced Plank constant /planckover2pi1is taken as 1 in this paper. The non-interacting Hamiltonian H0can be diagonalized in the helicity bases \|k,s/angb∇acket∇ight=1√ 2/parenleftbigg1 iseiφ(k)/parenrightbigg ,s=±1, (3) whereφ(k) = arctan( ky/kx) andsis the helicity of Fermi surfaces, which represents that the in-plane spin is right-handed or left-handed to the momentum. The dispersion relations for two helical branches are ξk,s= (k2+2skR\|k\|−k2 F)/2m, wherekR=mαcorresponds to the recoil momentum in experiments [5, 6]. The Fermi surfaces are given by ξk,s= 0, which yields two Fermi momenta ks=κkF−skRwithκ=/radicalbig 1+γ2. We have deﬁned the dimensionless SOC strength γ=kR/kF. Re- cently, the experimental realization of the SOC degen- erated Fermi gases have been reported [5–7]. By apply- ing a pair of laser beams to the ultracold40K or6Li atoms trapped in a anisotropic harmony trap, the equal weight combination of the Rashba-type and Dresselhaus- type SOC is generated. Their elegant experiments are performed in the weakly repulsive regime, which could provide the possibilities to study the SOC degenerated Fermi gases in the normal state. The interacting part reads as HI= 2g/integraldisplayd2kd2pd2q (2π)6c† k+q,↑c† p−q,↓cp,↓ck,↑,(4) whereg= 2πNas/3√ 2πmζz, which is controlled by the s-wave scattering length as. HereNis the total atom number, ζz=/radicalbig 1/mωzis the conﬁnement scale of the atomic cloud in the ˆ zdirection perpendicular to the 2D plane, and mis the mass for the ultracold atoms. Notice that the universal properties of the low-energy interac- tion among ultra-cold atoms depend only on the scat- tering length as[24–32]. We focus on the normal stateregime of Fermi atomic gas in this work assuming posi- tive scattering length, which could be reached in the up- per branch of Feshbach resonance. We note that the gas of dimers and the repulsive gas of atoms represent two diﬀerent branches of the many-body system, both corre- sponding to positive values of the scattering length [33]. The atomic repulsive gas conﬁguration has been experi- mentally achieved by ramping up adiabatically the value of the scattering length, starting from the value a= 0 [34]. B. Renormalizations due to the s-wave interaction. Let’s consider the problem in the helicity bases \|k,s/angb∇acket∇ight. The non-interacting Green’s function is G0 s(k,ω) =1 ω−ξk,s+isgn(ω)0+. (5) The Dyson’s equation expresses the relation between the non-interacting and interacting Green’s functions in terms of the self-energy Σ sas Gs(k,ω) =1 ω−ξk,s−Σs(k,ω). (6) All the many-body physics is contained in the self-energy Σs. The poles of interacting Green’s function Gs(k,ω) give the quasi-particle excitations, of which lifetime τs= 1/Γscan be obtained from the imaginary part of self- energy as Γs(k) =−2ImΣs(k,ξk,s). (7) The real part of the self-energy gives a modiﬁcation of dispersion relations. At low temperature, the properties of the low energy excitation in the vicinity of the Fermi surface is essential. Thus we can expand the real part of self-energy to the ﬁrst order of ωand\|k\|−ksas ReΣs(k,ω) = ReΣ s(ks,0)+ω∂ωReΣs(ks,ω)\|0 +(k−ks)·∇kReΣs(k,0)\|ks.(8) The interacting Green’s function now becomes Gs(k,ω) =Zs ω−ξ∗ k,s+i(1/2)Γs(k),(9) whereξ∗ k,sisthemodiﬁed energydispersion. TheGreen’s function acquires a renormalized factor Zs=1 1−As, (10) whereAs=∂ωReΣs(ks,ω)\|0. The Fermi velocity vks= ∂ξ∗ k,s/∂k\|kscan be calculated as follows ∂ξ∗ k,s ∂k\|\|k\|=ks=Zs(κkF/m+∂kReΣs(k,0)\|ks).(11)3 k sq,q 0 k-q rk s p+q,p +q 0 0 p,p 0= + + + ...q,q 0,k -q 0 0 k-q r ,k -q 0 0 FIG. 1. Feynman diagrams for the self-energy of the SOC Fermi liquid in the presence of s-wave interaction. The Feyn- man rules are deﬁned under the helicity bases. The label s andrdenote the helicity index. The self-energy is calculated within the framework of RPA [35]. Fornon-interactingFermigas, the Fermivelocityis v0 ks= κkF/m. Therefore we introduce the eﬀective mass via ∂ξ∗ k,s/∂k\|\|k\|=ks=κkF/m∗ s. The eﬀective mass in terms of self-energy is m∗ s m=1 Zs/parenleftbigg 1+m κkF∂kReΣR s(ks,0)/parenrightbigg−1 .(12) The Eqs. (7), (10) and (12) are our starting points of microscopic calculations of normal states properties, which embody the main properties of a quasi-particle in the Landau theory of Fermi liquids. III. SOC FERMI LIQUID PARAMETERS WITH REPULSIVE s-WAVE INTERACTION A. RPA Self-energy To investigate the renormalization eﬀects in two Rashba energy bands separately, it is convenient to work in the helicity bases. The interacting part of the Hamil- tonian in the helicity bases is rewritten as HI=/summationdisplay k,p,qVss′;rr′(k,p,q)ϕ† k+q,s′ϕ† p−q,r′ϕp,rϕk,s,(13) where the interaction vertex Vss′;rr′(k,p,q) = gfss′(θk,θk+q)frr′(θp,θp−q). Due to the presence of SOC, the spin is locked to momentum. The interac- tion vertex acquires an overlap factor fss′(θk,θk+q) and frr′(θp,θp−q), which is deﬁned by fss′(θk,θp) =1 2(1+ss′ei[θk−θp]),(14) whereθkandθpare the azimythal angles of kandpre- spectively. WithintheframeworkofRPA,theinteractionPath C1Path CRe[ω] i Im[ ω] Re[ω] i Im[ ω] Path C2 FIG. 2. (Color online). (a): The contours of integration in the complex ωplane for the self-energy given byEq. (19). For intermediate states with the energy 0 < ξk−q,r< ω, the pole of the propagator falls in the ﬁrst quadrant. (b): Schematic of the deformation of the contour into the imaginary axis. The self-energy given by Eq. (19) is equal to the integration alo ng the path C1 with an additional contribution of the residue as shown by the integral path C2. vertex is modiﬁed as (see Fig. 1) VRPA ss′,rr′(k,p,q,ω) =g ǫ(q,ω)fss′(θk,θk+q)frr′(θp,θp−q), (15) where the dielectric function ǫ(q,ω) = 1+gχ(q,ω) and χ(q,ω) is the bare density-density susceptibility of non- interacting SOC Fermi gas. In the long wavelength and low frequency limit, the susceptibility can be carried out as Reχ(y) =m π/bracketleftBigg 1−\|y\|/radicalbig y2−1Θ(\|y\|−1)/bracketrightBigg , Imχ(y) =m πy/radicalbig 1−y2Θ(1−\|y\|), (16) wherey=mω/κk F\|q\|. It can be seen from Eq. (16) that the susceptibility satisﬁes the following relations χ∗(q,ω) =χ(−q,−ω) =χ(q,−ω).(17) It’simportanttonoticethat thebaresusceptibilityisreal for\|ω\|> vF/radicalbig 1+γ2\|q\|. For\|ω\|< vF/radicalbig 1+γ2\|q\|, the baresusceptibilityinEq. (16)containsanimaginarypart which represents the absorptive behavior of the medium. Thisimaginarypartisresponsiblefortheﬁnitelifetimeof the quasi-particle in the medium. Along the imaginary axis, the analytical formula of χ(q,ω) is much simpler (see Fig. 2) as χ(q,iω) =m π[1−\|y\|/radicalbig y2+1]. (18) The RPA self-energy (see Fig. 1) is Σs(k,ω) =i/integraldisplay Cd2qdq0 (2π)3/summationdisplay rgFsr ǫ(q,q0)G0 r(k−q,ω−q0), (19)4 where the Fsris the overlap factor Fsr=1+srcos(θk−θk−q) 2. (20) The integral path Cis shown in Fig. 2 (a). After defor- mation of contour path in Fig. 2 (b), the pole gives rise to a residue contribution Σpole s(k,ω) =−/summationdisplay r/integraldisplay Drd2q (2π)2g ǫFsr,(21) where the region of integration Driskr<\|k−q\|< \|k\|−(r−s)kRandǫ= 1+gχ0(q,ω−ξk−q,r). The line integral along the imaginary axis is Σline s(k,ω) =−/integraldisplay∞ −∞d2qdq0 (2π)3/summationdisplay rgFsr ǫ(q,iq0)1 ω−iq0−ξk−qr. (22) The total self-energy is given by Σs= Σpole s+Σline s. (23) The advantage of this decomposition is that the imagi- nary part is given by the contribution of residue, and the real part is mainly determined by the line integral. B. Quasi-particle lifetime The quasi-particle lifetime can be calculated by the imaginary part of self-energy, which comes from the con- tribution of residue in Eq. (21). At zero temperature and for quasi-particle ξk,s>0, the imaginary part of the self-energy reads Γs(k) =2/summationdisplay q,rΘ(ξk−q,r)Θ(ξk,s−ξk−q,r) ×ImVRPA sr;sr(k,k−q,q;ξk,s−ξk−q,r).(24) The imaginary part of the RPA vertex in the helicity bases is ImVRPA sr;rs(k,k−q,q,k;w) =gFsrIm1 ǫ(y),(25) wherew=ξk,s−ξk−q,randy=w/vF\|q\|. Sincethe main contribution of integral comes from the forward scatter- ing, i.e., y≪1, the density-density susceptibility can be expanded about y= 0. One ﬁnds that the susceptibility in this neighborhood is χ(y) =m π(1+iy)+O(y2). (26) The imaginary part of RPA vertex goes to ImVRPA sr;rs(k,k−q,q;w)≃ −m2g2 (mg+π)2π myFsr.(27)(k-ks)/k Fτs-1 (units of εF) γ=0.5γ=0 (ordinary Fermi liquid ) 0.00 0.02 0.04 0.06 0.08 0.100.000.050.100.150.20 FIG. 3. (Color online). The inverse of the lifetime τsfor 40K ultracold atoms as a function of the momentum kin the vicinity of the Fermi surface. The lifetime of quasi-partic le is enhanced due to the presence of SOC. The parameters taken here are: the number of atoms is about 104,kR=h/λwith λ= 773nm, γ= 0.5, trap frequency ωz= 2π×400Hz, and as= 32a0, where a0is the Bohr Radius. The unit ǫF= /planckover2pi1×0.21MHz. Substituting Eq. (27) into Eq. (24), the inverse lifetime Γs(k) can be evaluated as Γs(k) =−m2g2ǫF π(mg+π)2δ2/braceleftbigg lnδ 8−1 2−γ2lnγ 4/bracerightbigg ,(28) whereδ= (k−ks)/kF. The result for ordinary Fermi liquid in the presence of s-wave repulsive interaction can be obtained by taking the limit γ→0 in Eq. (28). The eﬀect of SOC is shown in Fig. 3 via the comparison with the ordinary Fermi liquid with the same strength of s- wave repulsive interaction, where we can conclude that the quasi-particleis much stablerin the presence ofSOC. C. Green’s function renormalization factor Our starting point is the real part of the self-energy which contains two parts: one is the residue contribution given by Eq. (21), the other is the integral along the imaginary axis given by Eq. (22). Thus we want to evaluate As=Apole s+Aline s =∂ξΣpole s(ks,ξ)\|ξ=0+∂ξΣline s(ks,ξ)\|ξ=0.(29) Given the rotation symmetry, we only need to consider ks=ksex. The ﬁrst term in Eq. (29) is Apole s=/summationdisplay r/integraldisplayd2q (2π)2δ(ξks−q,r)g ǫFsr.(30)5 50 100 150 200Analytical Numerical γ=0 FIG. 4. (Color online). Renormalization factor Zsas func- tions of scattering length aswith the same parameters in Fig. 3. The black thick line represents the renormalization fact or of ordinary Fermi gases ( γ= 0). The red (dashed) and blue (dotted) lines represent the analytical results for the SOC Fermi gas given by Eq. (34). The discrete points are evalu- ated numerically. The second term in Eq. (29) can be integrated by parts, which gives two parts: the ﬁrst one reads as Aline s=−/summationdisplay r/integraldisplayd2q (2π)2δ(ξks−q,r)g ǫFsr,(31) which cancels the residue contribution given by Eq. (30). Thus the ﬁnal result of Eq. (29) can be expressed as As=−mgk2 F (2π)3k2s/integraldisplay2π 0dφ/integraldisplay∞ 0d¯y/integraldisplay∞ 0dx/summationdisplay rImfsr(x,¯y,φ), (32) where we have deﬁned ¯ y=mw/qk s,x=q/2ksand fsr(x,¯y,φ) =Fsr i¯y−µ(x,φ)1 ǫ2∂ǫ ∂¯y. (33) Hereµs,r(x,φ) =mξk−q,r/\|q\|ks, and the overlap fac- torFsr(x,φ) = 1/2 +sr(1−2xcosφ)/2l, withl= \|k−q\|/ks=/radicalbig 1−4xcosφ+4x2. FromEqs. (10), (32)and(33), wecanobtaintherenor- malization factor straightforwardly Z−1 s= 1+m2g2 8π(mg+π)1 (κ−sγ)2 = 1+m2g2 8π(mg+π)(1+sγ+O(γ2)).(34) The renormalization factor turns out to be dependent on the helicity s, and the leading correction due to SOC is O(γ), which is diﬀerent from the results of two dimen- sional electron gas (2DEG) with SOC in semiconductors [36]. We show the analytical results along with the nu- merical calculation for Zsas a function of the s-wave scattering length in Fig. 4 and the strength of SOC in Fig. 5. The Green’s function renormalization factor for ordinaryFermiliquid isalsoshownforcomparisonin Fig.γAnalytical Numerical FIG. 5. (Color online). Renormalization factor Zsas func- tions of dimensionless SOC strength γwith the same param- eters in Fig. 3. The red (dashed) and blue (dotted) lines represent the analytical results for the SOC Fermi gas given by Eq. (34). The discrete points are evaluated numerically. 4. We can see from Fig. 5 that the renormalization fac- tor is reduced for the s= +1 branch while enhanced for thes=−1 branch with increasing strength of SOC. D. Eﬀective mass The eﬀective mass can be evaluated by the real part of the static self-energy in the vicinity of the Fermi sur- face from Eq.(12). In contrast to the calculation of the renormalization factor Zs, the contribution of residue is irrelevant now. The correction of the eﬀective mass is isotropic due to the rotation symmetry. Without loss of generalities, we assume ks=ksexin the following. We begin with ∂kReΣs(k,0)\|\|k\|=ks= Re/integraldisplay∞ −∞d2qdw (2π)3 ×∂k/summationdisplay r1 iw+ξk−q,rVRPA sr;sr(k,q,iw).(35) The interaction vertex is dependent on the external mo- mentum kbecause of the overlap of the helical eigen- states. It is instructive to consider some special cases. For weak SOC ( γ≪1), the integration in Eq. (35) can be expanded to ∂kReΣs(k,0)\|\|k\|=ks=kFg 4πsγ+O(γ2).(36) In contrast to the SOC Fermi liquid with Coulomb inter- action [36], our result is band dependent and has a ﬁrst order correction γ. The eﬀective mass reads as m∗ s m=Z−1 s(1+smg mg+πγ 4κ)−1. (37) For strong SOC ( γ∼1), we show the numerical results along with the analytical results of Eq. (37) in Fig. 6.6 Analytical Numerical 10 20 30 40 0γ=0 FIG. 6. (Color online). Eﬀective mass as functions of s-wave scattering length aswith the same parameters as in Fig. 3. The red (dashed) and blue (dotted) line denote the case for s= +1 and s=−1 respectively. The solid triangle and circle points are the corresponding numerical results. The black line represents the analytical results for ordinary Fermi l iquid (γ= 0). The many-body modiﬁcations of the eﬀective mass are dependent on the helical bands. The eﬀect of SOC is shown in Fig. 7, from which we can see that the eﬀective mass for the s= +1 branch is enhanced while the s=−1 branch is reduced with increasing strength of SOC. E. Spectral function, density of state and speciﬁc heat A close related quantity is the spectral function A(k,ω), which is the imaginary part of the single par- ticle Green’s function [39, 40] A(k,ω) =−1 πImGret(k,ω), (38) whereGret(k,ω) is the retarded Green’s function. It canbe straightforwardlyevaluatedfromthe time-ordered Green’s function G(k,ω) as ImGret(k,ω) = ImG(k,ω)sign(ω), ReGret(k,ω) = ReG(k,ω). (39) With the great improvements in the spectroscopic tech- nique, it has been possible to directly measure the low-energy spectral weight function of a 2D system in the momentum-resolvedradiofrequency(rf) experiments [42–44]. In Fig. 8, we show the spectral functions of the two Rashba bands separately. The spectral functions at theFermisurfacesasshowninFig. 8(a)and(b)(vertical arrow) have the form A(ks,ω) =Zsδ(ω). (40) Fig. 8 (c) and (d) are density plots of the spectral func- tions, which could be compared with the results of the momentum-resolved rf spectroscopy in current experi-γ0.25 0.5 0.75 1.0 0Analytical Numerical FIG. 7. (Color online). Eﬀective mass as functions of the strength of SOC γwith the same parameters as in Fig. 3. The red (dashed) and blue (dotted) line denote the case for s= +1 and s=−1 respectively. The solid triangle and circle points are the corresponding numerical results. ments. Given the spectralfunctions, we canobtain the density of states (DOS) of the fermionic system with SOC via [39, 40] ρ(ω) =/summationdisplay sρs(ω) =/summationdisplay s/integraldisplayd2k (2π)2As(k,ω).(41) For the non-interacting case, the DOS now becomes ρ0(ω) = 0, ω < −κ2k2 F 2m, m πγ√ 2mω/k2 F+κ2,−κ2k2 F 2m< ω <−k2 F 2m, m π, ω > −k2 F 2m.(42) The DOS is modiﬁed in the presence of s-wave in- teraction. On the Fermi surfaces, for non-interacting SOC Fermi gas, the DOS in units of m/πis given by: ρ0 +1(EF) = 0.28,ρ0 −1(EF) = 0.72, and ρ0(EF) = 1 forγ= 0.5. For the interacting SOC Fermi gas, the DOS is evaluated numerically as follow: ρ+1(EF) = 1.37, ρ−1(EF) = 0.83 andρ(EF) = 2.20, where all the param- eters are the same with Fig. 3. The quasi-particles from two helical bands both con- tribute to the speciﬁc heat. At low temperature, the speciﬁc heat is proportional to the DOS on the Fermi surfaces and the temperature T. Thus the ratio between the speciﬁc heats at low temperature is cv c0v=ρ(EF) ρ0(EF)= 2.20, (43) wherec0 vis the speciﬁc heat of SOC Fermi gases without interactions.7 (a) (b) (c) (d)ω/ εFs=+1 0.2 0.050.5 δ(ω) δ(ω) s=+1 s=-1 1.01.52.02.5 0.01.0 0.02.03.04.05.06.0 s=-1ω/εF ω/ εF k/kF 3 A(k, ω)/ εF 3 0.0 0.05 0.1 0.0 0.1 0.2 A(k, ω)/ εF ω/εF k/kF FIG. 8. (Color online). Zero temperature spectral func- tion at diﬀerent values of ( k−ks)/kFare shown at (a) and (b). All the parameters are the same with Fig. 3. (a) is fors= +1 and (b) is for s=−1 respectively. The eight peaks, from left to right, correspond to ( k−ks)/kF= −0.01,−0.075,−0.05,−0.025,0.025,0.05,0.075,0.01. The vertical arrows in (a) and (b) denote δfunctions at ω= 0 with weights 0 .23 and 0 .67 respectively. (c) and (d) are den- sity plots of the spectral function for the same parameters a s above, which correspond to s= +1 and s=−1 respectively. The white dashed line denotes the modiﬁed single particle dispersion. IV. DISCUSSIONS AND SUMMARIES Wehaveobtainedvariousnormalstatequantitiesof2D SOC Fermi gases in the presence of repulsive s-wave in- teraction. Ultimately, to makecontact with experiments, two practical considerations warrant mention. (i): The repulsive s-wave interaction can be achieved on the up- per branchof a Feshbach resonance. One problem should be considered is that the upper branch of a Feshbach resonance is an excited branch, and will decay to the BEC molecule state due to inelastic three-body collisions [37]. However, with small scattering length, the decay rate is well suppressed [1, 38] and the system may be metastable for observation. (ii): Recently, the SOC de- generate Fermi gases have been realized in the ultra-cold atom systems[5, 6]. The eﬀective SOC generatedin their experimental schemes is an equal weight combination of Rashba-type and Dresselhaus-type SOC. In this paper, we have investigated the case of Rashba-type SOC. The Dresselhaus-type SOC is presented in Appendix, which is demonstrated to give the same normal state properties as the Rashba case. In current experiments, we consider the following typ- ical experimental parameters for quasi-2D systems. One can trap about 104 40K atoms within a pancake-shaped harmonic potential with the trap frequency chosen as 2π×(10,10,400)Hz along the (ˆ x,ˆy,ˆz) direction. The system size can be estimated as (37 .8,37.8,5.98)µm.The other related parameters are taken as: as= 32a0, γ= 0.5,kR= 2π/λ= 8.128×106m. The Fermi liquid parameters are listed in Table. I under this typical ex- perimental setup. For comparison, we give the results of ordinary Fermi liquid and the 2DEG in semiconductors together. We should notice that the single particle spec- tral function captures the valuable information for low energy excitations of the Fermi liquid and can be mea- sured by means of mometnum-resolved rf experiments [41–44]. The rf spectroscopy is a technique used to probe atomic correlation by exciting atoms from occupied hy- perﬁne states to another (usually empty) reference hy- perﬁne state. The single-particle spectral function is ob- tained in experiment through the mometnum-resolvedrf- transfer strength. As a result, the inferences drawn from the spectral function in the vicinity of the Fermi surfaces can be used to determine the Fermi liquid parametersde- scribingthe lowenergybehaviorsofthenormalstate. Up to now, the rf experiment is the most promising method in ultracold atomic gases to measure these Fermi liquid parameters. TABLE I. Normal state properties for SOC Fermi liquid ( γ= 0.5), ordinary Fermi liquid ( γ= 0), and 2DEG ( γ= 0.051) in semiconductor. All other parameters used here are the same with Fig. 3. γ= 0.5 γ= 0 2DEG( γ= 0.051)† 1/τ†† s0.73kHz 0 .67kHz 55 .36GHz ZsZ+1= 0.23, Z−1= 0.67.0.96 0 .97 m∗ s/mm∗ +1/m= 4.88, m∗ −1/m= 1.16.1.04 0 .98 †The results for 2DEG in InGaAs are taken from Ref. [36]. Compared with the SOC Fermi liquid in the pres- ence ofs-wave interaction, the results for the 2DEG with Coulomb interaction are independent on the en- ergy band, and the leading order correction relative to the SOC strength is γ2. ††The values of the inverse lifetime are evaluated at ( k− k±1)/kF= 0.01. Compared with 2DEG in the typical semiconductor, the quasi-particle in ultracold atomic gases is much stabler. In summary, we studied the normal state properties of the SOC Fermi gas with repulsive s-wave interaction. The quasi-particle lifetime τs, the renormalization factor Zs, the eﬀective mass m∗ s/mare calculated, which em- body the main properties of a quasi-particle. To make contactwithexperimentsdirectly,wecalculatedthespec- tral function A(k,ω), density of state ρ(ω), and the spe- ciﬁcheat cvatlowtemperature. Thesequantitiesprovide a good description of the low energy physics with SOC ands-wave interaction, which are measurable in current experiments. The analytical and numerical results show8 that the normal state properties are distinct for the two energy bands, and the leading correction relative to the ordinary Fermi liquid with s-wave interaction is on the order of γ, which are strikingly diﬀerent from the SOC Fermi liquid with Coulomb interaction [36]. We expect our microscopic calculations of the Fermi liquid param- eters and related quantities would have the immediate applicability to the SOC Fermi gasesin the upper branch of the energy spectrum. ACKNOWLEDGMENTS We acknowledge helpful discussions with Jinwu Ye, Han Pu, Congjun Wu, and Hui Hu. This work was supported by the NKBRSFC under Grants No. 2009CB930701, No. 2010CB922904, No. 2011CB921502, and No. 2012CB821300, NSFC under GrantsNo. 10934010,and NSFC-RGC under Grants No. 11061160490 and No. 1386-N-HKU748/10. Appendix A: Relationships between the Rashba SOC and Dresselhaus SOC In this paper, the normal state properties of the Fermi liquid with Rashba-type SOC is studied. Now, we would like to demonstrate that another type of SOC, namely the Dresselhaus-type, gives exactly the same results for the normal state properties considered here. We start with the single particle Hamiltonian with Dresselhaus SOC [7] HD=p2 2m+α(−pyσx−pxσy)−µ.(A1) The helicity bases with Dresselhaus SOC are \|p,s/angb∇acket∇ightD=−1√ 2/parenleftBigg 1 ise−iφ(p)/parenrightBigg ,s=±1,(A2) where the subscript Drepresents the Dresselhaus-type SOC. The starting point of the microscopic calculation is the self-energy Σ sfor each band. To illustrate the relationships of the two types of SOC, it is essential to derive the relationships of the Feynman rules between the two cases. For the Dresselhaus-type SOC, the non- interacting Green’s function and the interaction vertex in the helicity bases reads GD s(k,ω) =1 ω−ξk,s+isgn(ω)0+,(A3) VD ss′;rr′(k,p,q) =gfD ss′(θk,θk+q)fD rr′(θp,θp−q).(A4) The energy spectrum of the Dresselhaus-type SOC is the same with the Rashba-type. Thus the single particle re- tarded Green’s function within the Dresselhaus represen- tation is the same with the result for Rashba-type SOCgiven in Eq. (5). The overlap factor fD ss′(θk,θk+q) for the Dresselhaus-type SOC is given by fD ss′(θk,θk+q) =1 2(1+ss′e−i[φ(k)−φ(k+q)]).(A5) Compared with Eq. (14), we ﬁnd the overlap factors are conjugated for the two cases fD ss′(θk,θk+q) =fR ss′(θk,θk+q)∗.(A6) The bare susceptibility in Matsubara formalism is given by χD(q,iωn)=kBT/summationdisplay k,iωms,rGD s(k,iωm)GD r(k−q,iωm−iωn)FD sr, (A7) where the factor FD sr=1+srcosθ 2(A8) is the same with the Rashba case in Eq. (20). So the bare susceptibility χDfor the Dresselhaus type SOC is equal to the case for the Rashba type SOC. The self- energy is given by Eq. (21) and (22). The integrand function includes the following factors: the single parti- cle’s Green’s function, the bare susceptibility χD(q,iωn) and the overlapfactor FD sr, which areall demonstratedto be the same for the two types of SOC. Therefore we con- clude that the normal state properties calculated here, such as the quasi-particle lifetime τs(k), the renormal- ization factor Zsand the eﬀective mass m∗ s/mare all the same exactly for the two types of SOC. Appendix B: spectral function in the spin representation In this work, we studied the microscopic parameters such asτk,s,Zsandm∗ s/mand associated experimental observable such as As(k,ω),ρ(EF) andcvin the helicity representation. Theoreticallyspeaking, the helicity bases provides a clear representation for the description of the microscopicpicture ofthe singleparticle excitation ofthe repulsiveSOC Fermi gas. In the normalstate regime, the quasi-particle in the weakly repulsive SOC Fermi gas can be adiabatically connected with the particle with helicity in the non-interacting theory. In this Appendix, we will also present the main results in the spin representation in our manuscript, since experimentalists ﬁnd it handier to work with spin representation. In the following, we will transform the results to the spin representation. In the spin representation, the Green’s functions of the SOC Fermi gas has the of 2 ×2 matrix form. The non- interacting form is given by G0(k,ω)α,β=/summationdisplay s1 ω−ξk,s+isgn(ω)0+Ps(k),(B1)9 wherePs(k) = [1+s(ˆz×ˆk)·σ]/2 is the projection opera- tor to the helicity bases. Considering the s-wave interac- tion, there is a modiﬁcation of the quasi-particle disper- sionξ∗ k,s, and a ﬁnite lifetime of the quasi-particle τk,s corresponding to the imaginary part of the self-energy. In the weakly repulsive regime, the polarization of the quasi-particle is hold because of the stability from its non-trivial topology structure, such that we obtain the many-body Green’s function as Gα,β(k,ω) =/summationdisplay s1 ω−ξ∗ k,s−isgn(ω)Γk,s 2Ps(k).(B2) Based on this form of Green’s function modiﬁed with s- wave interaction, we could obtain the spectral function in the spin representation, Aα,β(k,ω) =/summationdisplay sPs(k)α,βAs(k,ω),(B3)whereAsis the spectral function in the helicity bases representation. We could ﬁnd that A↑,↑(k,ω) =A↓,↓(k,ω) =1 2(A+(k,ω)+A−(k,ω)),(B4) A↑,↓(k,ω) =A↓,↑(k,ω)∗ =−ie−iθk 2(A+(k,ω)−A−(k,ω)).(B5) The elements A↑,↑,A↓,↓are real and equal to each other, while the oﬀ-diagonal elements A↑,↓,A↑,↓have an angle dependence e−iθk,eiθk, which is the very eﬀect of spin- orbit coupling. The other qualities such as the density of states at the Fermi energy, and the low temperature speciﬁc heat are both independent to the representation we use. There- fore they could be compared with experiment straight- forwardly. [1] G. B. Jo, Y. R. Lee, J. H. Choi, C. A. Christensen, T. H. Kim, J. H. Thywissen, D. E. Pritchard, and W. Ketterle, Science325, 1521 (2009). [2] V. B. Shenoy and T.-L. Ho, Phys. Rev. Lett. 107, 210401 (2011). [3] F. Wilczek and A. Zee, Proc. R. Soc. Lond. A 392, 45 (1984). [4] F. Wilczek and A. Zee, Phys. Rev. Lett. 52, 2111 (1984). [5] P. Wang, Z. Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H. Zhai, and J. Zhang, Phys. Rev. Lett. 109, 095301 (2012). [6] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah, W. S. Bakr, and M. W. Zwierlein, Phys. Rev. Lett. 109, 095302 (2012). [7] Y.-J. Lin, K. Jimenez-Garcia, and I. B. Spielman, Nature (London) 471, 83 (2011). [8] Y.-J. Lin, R. L. Compton, A. R. Perry, W. D. Phillips, J. V. Porto, and I. B. Spielman, Phys. Rev. Lett. 102, 130401 (2009). [9] Y. J. Lin, R. L. Compton, K. Jimnez-Garca, J. V. Porto, and I. B. Spielman, Nature 462, 628 (2009). [10] Y.-J. Lin, R. L. Compton, K. Jimenez-Garcia, W. D. Phillips, J. V. Porto, and I. B. Spielman, Nat. Phys. 7, 531 (2011). [11] Z. Q. Yu and H. Zhai, Phys. Rev. Lett. 107, 195305 (2011). [12] H. Hu, L. Jiang, X. J. Liu, and H. Pu, Phys. Rev. Lett. 107, 195304 (2011). [13] R. Liao, Y. Yi-Xiang, and W. M. Liu, Phys. Rev. Lett. 108, 080406 (2012). [14] J. P. Vyasanakere, S. Zhang, and V. B. Shenoy, Phys. Rev. B84, 014512 (2011). [15] L. He and X. G. Huang, Phys. Rev. Lett. 108, 145302 (2012). [16] L. Han and C. A. R. S´ ade Melo, Phys. Rev. A 85, 011606(R) (2012). [17] G. Chen, M. Gong, and C. Zhang, Phys. Rev. A 85, 013601 (2012). [18] A. Kubasiak, P. Massignan, and M. Lewenstein, Euro- phys. Lett. 92, 46004 (2010). [19] A. Bermudez, N. Goldman, A. Kubasiak, M. Lewenstein,and M. A. Martin-Delgado, New J. Phys. 12, 033041 (2010). [20] G. Liu, S. L. Zhu, S. Jiang, F. Sun, and W. M. Liu, Phys. Rev. A82, 053605 (2010). [21] J. D. Sau, R. Sensarma, S. Powell, I. B. Spielman, and S. Das Sarma, Phys. Rev. B 83, 140510 (2011). [22] T. Fujita and K. F. Quader , Phys. Rev. B 36, 5152 (1987). [23] Y. Li and C. Wu, Phys. Rev. B 85, 205126 (2012). [24] N. F. Mott and H. S. W. Massey, The Theory of Atomic Collisions , 3rd ed. (Oxford University Press, London, 1965). [25] A. Messiah, Quantum Mechanics (Wiley, New York, 1962). [26] E. Braaten and H. W. Hammer, Phys. Rep. 428, 259 (2006). [27] T. K¨ ohler and K. G´ oral, Rev. Mod. Phys. 78, 1311 (2006). [28] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Rev. Mod. Phys. 82, 1225 (2010). [29] S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D. M. Stamper-Kurn, and W. Ketterle, Nature (London) 392, 151 (1998). [30] P. Courteille, R. S. Freeland, and D. J. Heinzen, F. A. van Abeelen, and B. J. Verhaar, Phys. Rev. Lett. 81, 69 (1998). [31] J. L. Roberts, N. R. Claussen, J. P. Burke, C. H. Greene, E. A. Cornell, and C. E. Wieman, Phys. Rev. Lett. 81, 5109 (1998). [32] V. Vuleti´ c, A. J. Kerman, C. Chin, and S. Chu, Phys. Rev. Lett. 82, 1406 (1999). [33] L. Pricoupenko and Y. Castin, Phys. Rev. A 69, 051601(R) (2004). [34] T. Bourdel, J. Cubizolles, L. Khaykovich, K. M. F. Magalh˜ aes, S. J. J. M. F. Kokkelmans, G. V. Shlyap- nikov, and C. Salomon, Phys. Rev. Lett. 91, 020402 (2003). [35] J. W. Negele and H. Orland, Quantum Many-Particle Systems (Addison-Wesley, New York, 1988). [36] D. S. Saraga and Daniel Loss, Phys. Rev. B 72, 19531910 (2005). [37] D. S. Petrov, Phys. Rev. A 67, 010703 (2003). [38] J. P. DIncao and B. D. Esry, Phys. Rev. Lett. 94, 213201 (2005). [39] G. D. Mahan, Many-Particle Physics (Plenum, New York, 1990). [40] A.L.FetterandJ. D.Walecka, Quantum Theory of Many Particle Systems (McGraw Hill, New York, 1971). [41] H. Hu, H. Pu, J. Zhang, S. G. Peng, and X. J. Liu, arXiv:1208.5841.[42] C. Chin, M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, J. H. Denschlag and R. Grimm, Science 305, 1128 (2004). [43] C. H. Schunck, Y. Shin, A. Schirotzek, M. W. Zwierlein, and W. Ketterle, Science 316, 867 (2007). [44] M. Veillette, E. G. Moon, A. Lamacraft, L. Radzihovsky, S. Sachdev, and D. E. Sheehy, Phys. Rev. A 78, 033614 (2008).
0905.1389v1.Spin_state_transition_and_phase_separation_in_multi_orbital_Hubbard_model.pdf	arXiv:0905.1389v1 [cond-mat.str-el] 9 May 2009Spin-state transitionand phaseseparationinmulti-orbit al Hubbard model Ryo Suzuki1†, Tsutomu Watanabe2∗, and Sumio Ishihara1 1Department of Physics, Tohoku University, Sendai 980-8578 , Japan 2Institute of Multidisciplinary Research for Advanced Mate rials, Tohoku University, Sendai 980-8577, Japan (Dated: December 4, 2018) Westudyspin-statetransitionandphaseseparationinvolv ingthistransitionbasedonthemilti-orbitalHubbard model. Multiple spinstatesare realizedbychanging theene rgy separationbetweenthe twoorbitalsandthe on- siteHundcoupling. ByutilizingthevariationalMonte-Car losimulation,weanalyzetheelectronicandmagnetic structures in hole doped and undoped states. Electronic pha se separation occurs between the low-spin band insulating state and the high-spin ferromagnetic metallic one. Difference of the band widths inthe two orbitals isof prime importance for the spin-state transitionandthe phase separation. PACS numbers: 75.25.+z, 71.70.-d,71.30.+h Novel electric and magnetic phenomena observed in cor- related electron systems are responsible for competition a nd cooperationbetweenmulti-electronicphaseswithdelicat een- ergybalance. These areowingto the internaldegreesof free - dom of electrons, i.e. spin, charge and orbital, under stron g electron correlation,and their couplingwith crystal latt ice.1,2 Insome transition-metalions, thereis anadditionaldegre eof freedom, termed the spin-state degree of freedom, i.e. mul- tiple spin states due to the different electron conﬁguratio ns in a single ion. One prototypical example is the perovskite cobaltites R1−xAxCoO3(R: rare earth ion, A: alkaline earth ion) where transitions between the multiple spin states occ ur by changing carrier concentration, temperature and so on. I n Co3+with thed6conﬁguration, there are three possible spin states, the high-spin (HS) state (e2 gt4 2g)with an amplitude of S=2, the intermediate-spin (IS) one (e1 gt5 2g)withS=1, and thelow-spin(LS)one (t6 2g)withS=0. Severalmagnetic,electricandtransportmeasurementshav e been carried out in the insulating and metallic cobaltites. It is known that LaCoO 3is a non-magnetic LS band-insulator (BI) at low temperatures, although there is still controver sy in the spin-state transition and the IS state at ﬁnite temper - ature.3,4,5,6,7In high hole doping region of x>0.3−0.4 in La1−xSrxCoO3,theferromagnetic(FM)metallicstatewasex- perimentally conﬁrmed. In the lightly hole doped region be- tween the two, a number of inhomogeneousfeatures in mag- netic, electric and lattice structures have been reported e x- perimentally. Spatial segregation of hole-rich FM regions and hole-poor insulating ones have been suggested by the neutron diffraction, the electron microscopy, NMR and so on.8,9,10,11Magnetic/non-magnetic clusters have been found by the small-angle and inelastic neutron scattering experi - ments.12,13,14It is widely believed that the observed giant magneto-resistance effect in the lightly doped region resu lts fromtheelectronicandmagneticinhomogeneity.12 Electronic phase separation (PS) phenomena in transition- metal compounds have been studied extensively and inten- sively, in particular, in the high Tc superconducting cupra tes and the colossal magnetoresistive manganites.15,16,17,18In these materials, the long-rangespin/orbitalordersin the Mott insulating phases and their melting by carrier doping are of essence in the electronic PS. The exchangeenergyfor the lo-calizedspins/orbitalsandthekineticonefortheitineran telec- trons are gained in spatially separate regions. On the other hand, in the present case, the non-magnetic band insulator i s realizedintheinsulatingphase,andthespin-statetransi tionis brought about by carrier doping. Thus, the present phenom- ena belong to a new class of the electronic PS in correlated system, although only a little theoretical studies have bee n done until now. In this paper, we address the issues of the spin-state transitionand the PS associated with thistrans ition by analyzing the multi-orbital Hubbard model. We examine the electronic structures in hole doped and undoped systems by utilizing the variational Monte-Carlo (VMC) method. We ﬁndthat,betweenthenon-magneticBIandtheHSFMmetal, theelectronicPS isrealized. We claimthat the differentba nd widthsplayanessentialrolein thepresentelectronicPS. We set up a minimal model, the two-orbital Hubbard model,19,20,21,22where the spin-state degrees of freedom and a transition between them are able to be examined. In each site in a crystal lattice, we introduce two orbitals, termed A and B, which represent one of the egandt2gorbitals, respec- tively. Anisotropic shape of the orbital wave function is no t concerned. An energy difference between the two orbitals is denotedby Δ≡εA−εB>0 whereεA(εB) is the level energy forA(B).Whentheelectronnumberpersiteistwo,thelowest two electronic states in a single site are \|B2/angbracketrightand\|A1B1/angbracketrightwith tripletspinstate whicharetermedtheLSandHSstatesinthe present model, respectively. The explicit form of the model Hamiltonianisgivenby H=Δ∑ iσc† iAσciAσ−∑ /angbracketleftij/angbracketrightγσtγ/parenleftBig c† iγσcjγσ+H.c./parenrightBig +U∑ iγniγ↑niγ↓+U′∑ iσσ′niAσniBσ′ −J∑ iσσ′c† iAσciBσc† iBσ′ciAσ′−J′∑ iγc† iγ↑ci¯γ↑c† iγ↓ci¯γ↓,(1) whereciγσis the annihilation operator of an electron at site i with orbital γ(=A,B)andspinσ(=↑,↓), andniγσ≡c† iγσciγσ is the number operator. A subscript ¯γtakesA(B), whenγis B(A). Weassumethatthetransferintegralisdiagonalwithre- spectto theorbitalsand \|tA\|>\|tB\|, bothofwhichare justiﬁed inperovskitecobaltites. Inmostofthenumericalcalculat ions, a relation tB/tA=1/4is chosen. As the intra-site electronin-2 teractions, we introduce the intra- and inter-orbital Coul omb interactions, UandU′, respectively,the Hundcoupling Jand thepair-hopping J′. Therelations U=U′+2JandJ=J′sat- isﬁedinanisolatedionareassumed. Inaddition,weintrodu ce therelation U=4Jinthenumericalcalculation. We adopt the VMC method where the electron correlation is treated in an unbiased manner and simulations in a large clustersizearepossible. Forsimplicityandalimitationi nthe computerresource, we introducetwo-dimensionalsquare la t- tices with a system size of N≡L2(L≤6) and the periodic and anti-periodic boundary conditions. The number of elec- tron isNe, and the hole concentrationper site measured from Ne=2Nisdenotedas x≡(2N−Ne)/N. Thevariationalwave function is given as a product form of Ψ=G\|Φ/angbracketrightwhereGis the correlation factor and \|Φ/angbracketrightis the one-bodywave function. The two types of the wave function are considered in \|Φ/angbracketright: theSlaterdeterminantobtainedbythesecondterminEq.(1) , and that for the HS antiferromagnetic (AFM) order given by applying the Hartree-Fock approximationto the third term i n Eq. (1). In the latter, the AFM order parameteris treated as a variationalparameter. WeassumetheGutzwiller-typecorr ela- tion factor Πil(1−ξlPil)wherelindicatesthe local electron conﬁgurations, Pilis the projection operator at site ifor the conﬁguration l, andξlis the variational parameter. Here we introducethe10variationalparametersforthe10inequiva lent electron conﬁgurations in a single site.23The ﬁxed-sampling method is used to optimize the variational parameters.24In addition to the standard VMC method, we improve the vari- ational wave function by estimating analytically the weigh ts for the conﬁgurations which are sampled by the MC simu- lations. This method is valid for the LS state and reduces the CPUtimebymorethanoneorder. Inmostofthecalculations, 104−105MCsamplesareadoptedformeasurements. We start from the case at x=0 where the average electron number per site is two. The electronic states obtained by the simulation are monitored by the total spin amplitude deﬁned by S2= (1/N)∑i/angbracketleftS2 i/angbracketrightwhereSi= ∑γSiγ= (1/2)∑ss′γc† iγsσss′ciγs′is the spin operator with the Pauli matrices σ, the spin correlation function Sγ(q) = (4/N)∑ijeiq·(ri−rj)/angbracketleftSz iγSz jγ/angbracketright, and the momentum-distribution function nγ(k) = (1/2)∑σ/angbracketleftc† kγσckγσ/angbracketrightwhereckγσis the Fourier transform of ciγσ. Size dependences of S2andSγ(q) inL=4−8 are within a few percent. We obtain the three phases, the HS Mott insulator (MI), the LS BI and the metal- lic (ML) phase. In the HS-MI phase, S2is about 1.6 being about 80% of the maximumvalue for S=1. A sharp peak in Sγ(q)atq= (π,π)and no discontinuity in nγ(k)imply that this is the AFM MI. In the LS-BI phase, nA(k) [nB(k)]is al- mostzero(one)inallmomenta,and S2≃0. IntheMLphase, discontinuous jumps are observed in both nA(k)andnB(k). The electron(hole)fermi surface is located around k=(0,0) [(π,π)]in the A (B) band;this is a semi metal. A valueof S2 isabout0.3,andnoremarkablestructureis seenin Sγ(q). Thephasediagramat x=0ispresentedinFig.1. Theerror bars imply the upper and lower bounds of the phase bound- ary, and symbols are plotted at the middle of the bars. In the region of large Δ(J), the LS-BI (HS-MI) phase is realized, ! " # $ % & ' ( )* + ,- . / 0 12 34 56 7 8 9 : ; < =>? @ AB C D FIG.1: (coloronline)Phasediagramsat x=0. Aratiooftheelectron transfers is taken to be tB/tA=1/4 in (a) and tB/tA=1 in (b). In (b),ﬁlledsquares andopen circles are forthe results obtai ned bythe VMC method and the previous DMFT one in Ref. 19, respectively . Broken curves are guides for eyes. Stars represent the param eters where the carrier dopings are examined. and between the two with small ΔandJ, the ML phase ap- pears. To compare the present results with the previous ones calculatedbythedynamical-meanﬁeldtheory(DMFT),19we present,inFig.1(b),thephasediagramwherethetwotransf er integrals are chosen to be equal, i.e. tB/tA=1. Although the global features in the phase diagrams are the same with each other,theHS-MIphaseobtainedbytheVMCmethodappears in a broader parameter region than that in DMFT, in partic- ular, near the boundary of the HS-MI and ML phases. This is because the AFM long-range order in the HS-MI phase is treatedproperlyintheVMCmethod. We haveconﬁrmedthat thephaseboundariesobtainedbytheVMCmethodwherethe AFM orderisnotconsideredalmostreproducetheDMFT re- sults. Now we show the results at ﬁnite x. Holes are introduced into the LS-BI phase near the phase boundary with the pa- rametervaluesof (Δ/tA,J/tA)=(12.2,4)and(8.25,2.5)[see Fig. 1]. By changingthe initial conditionsin the VMC simu- lation,weobtainthefollowingfourstates: i)theLS-MLsta te wherenA(k)is almost zero in all k, and the fermi surface is located in the B band around k= (π,π), ii) the FM HS-ML state where nB(k)is about 1 /2 in allk, the fermi surface is in the A band, and Sγ(q)has a sharp peak at q= (0,0), iii) theAFMHS-MLstatewherethefermisurfaceexistsintheA band around k= (π,0), andSγ(q)has a peak at q= (π,π), and iv) the mixed state where the wave function is a linear- combinationoftheLS-MLandFM HS-MLstates. InFig.2(a),theenergyexpectationvalues E≡/angbracketleftH/angbracketrightforthe several states in (Δ/tA,J/tA) = (12.2,4)are plotted as func- tionsofx. Thetransferintegralsarechosentobe tB/tA=1/4. Toshowthenumericaldataclearly,weplot E′=(E/tA)+Cx with a numerical constant C, instead of E. This transforma- tiondoesnotaffecttheMaxwell’sconstructionintroduced be- low. The results in the AFM HS-ML are not plotted, because of their higher energy values than others. We also present, in Fig. 3, a ratio of the LS sites to the LS and HS ones in3E F GH I J K LMN OPQ R ST U VW X Y Z [ \] ^_` a b c d e f gh i j k l m no p q rs t u v w x y z { \| } ~ ¡ ¢FIG.2: (coloronline) Holeconcentrationdependences ofth eenergy expectations for several states at (Δ/tA,J/tA)=(12.2,4)in (a), and those at(Δ/tA,J/tA)=(8.25,2.5)in (b). Broken lines are given by the Maxwell’s construction. Aratioof the electrontransfe rs is taken to betB/tA=1/4. A constant parameter Cin the deﬁnition of E′is taken tobe 8.2in(a)and 5.25in(b). the mixedstates deﬁned by RLS=nLS/(nLS+nHS). HerenLS (nHS) is a number of the sites where the LS (HS) state is re- alized. As shown in Fig. 2(a), the LS state, where holes are dopedinto the B band,is destabilizedmonotonicallywith in - creasingx. On the other side, in a region of x>0.5, the FM HS-ML state is realized. In between the two regions, the mixed state is the lowest energy state. The mixed state is smoothlyconnectedtotheLSandHSonesinthelowandhigh xregions, respectively. As shown in Fig. 3, a discontinuous jumpinthemixedstate isseenaround x=0.25;thesystemis changedfromthe LS dominantmixedstate into the HS dom- inant one with x. It is noticeable that the E′versusxcurve in themixedstateisconvexintheregionof0 <x<0.33. Thatis, byfollowingtheMaxwell’sconstruction,thePSoftheLS-BI andtheFMHS dominantmixedstatesismorestabilizedthan the homogeneous phase in this region of x. In the Fig. 2(b), we show the results in (Δ/tA,J/tA) = (8.25,2.5)where the system at x=0 is close to the ML phase [see Fig. 1(a)]. The PS appears,butits regionisshrunken. The magnetization per site in the lowest energy state de- ﬁned by M(x) = (1/N)/angbracketleft∑iSz i/angbracketrightis plotted in Fig. 3. A zero magnetization at x=0 reﬂects the LS-BI ground state. In a highdopedregionof x>0.33,the magnetizationdata almost follow a relation M(x)≃(1+x)/2: the system is expected to consist of the N/2 HS sites, the (1/2−x)NLS ones, and £ ¤ ¥ ¦ § ¨ © ª« ¬ ® ¯ °± ² ³´ µ ¶· ¸ ¹ º» ¼ ½ ¾¿ À Á ÂÃ Ä Å Æ Ç È É Ê Ë Ì Í Î Ï Ð Ñ Ò Ó Ô Õ Ö × Ø Ù Ú ÛÜ Ý Þß à áâ ã äå æ çè é ê ëì í îï FIG. 3: (color online) A ratio of the LS sites to the LS and HS ones in the mixed state, RLS, and magnetization M(x)as func- tions of the hole concentration x. A broken line connecting data at M(x=0)andM(x=0.33)is drawn by the Maxwell’s rule. For comparison, we plot a M(x) =x/2 curve which is expected from the hole doping in the LS-BI phase. Parameters are chosen to b e (Δ/tA,J/tA)=(12.2,4)andtB/tA=1/4.ð ñ ò ó ô õ ö ÷ ø ù ú û ü ý þ ÿ FIG.4: (coloronline)Hole concentrationdependences ofth eenergy expectations for several states where the electron transfe r integrals arechosentobeequalas tB/tA=1. Otherparametersaretakentobe (Δ/tA,J/tA)=(12.2,4),andaconstantparameter Cinthedeﬁnition ofE′istaken tobe 8. thexNsingly electron occupiedones. In this scheme, we ob- tainRLS=(1−2x)/(2−2x)which is consistent with the nu- merical data of RLSinx>0.33. Between x=0 and 0.33, wherethe PS is realized, M(x=0)andM(x=0.33)are con- nectedbyastraightlineaccordingtothevolume-fractionr ule in the Maxwell’s construction. The slope of M(x)is about three times higher than M(x) =x/2 which is expected in the holedopingintotheLS-BIphase. Thisisqualitativelycons is- tent with the experimental observations in the magnetizati on wheredopedholesinducehighspin value.3,25 We now address an origin of the electronic PS where the spin-state degree of freedom is concerned. In Fig. 4, we present the hole concentration dependence of the energy ex- pectationswherethebandwidthsaresettobeequalwitheach other,tB/tA=1. AswellasthecalculationinFig.2(a),theen- ergyparametersaretakentobe (Δ/tA,J/tA)=(12.2,4)which4 ! "# $ % &' ( )*+ , - . / 0 1 2 3 4 5 6 789 :; <= > ? @A B CDE F G H I J K L M N O P Q FIG.5: (color online) Schematic density-of-states inthe L S-BIstate atx=0and that inthe HS-ML one ina highhole doped region. isclosetotheLS-HSphaseboundaryat x=0[see Fig.1(b)]. Themixedstateisnotobtainedinthesimulation. Inallregi on ofxuptox=0.45,theLSstateisthelowestgroundstate,and neither the spin-state transition nor the PS occur. The diff er- enceofthebandwidthsinthetwoorbitalsisofessenceinthe electronicPS phenomena. ToclarifythemechanismofPSfurthermore,schematicpic- turesof the density of states (DOS) in the LS-BI at x=0 and the FM HS-ML in a high hole doped region are presented in Fig. 5. For simplicity, detailed shapes of DOS are not taken into account. In LS-BI state at x=0, the fermi level is lo- catedinside ofthe bandgapbetweenthe A andB bands. The bandwidthintheAbandislargerthanthatinB.Ontheother hand, in the FM HS-ML state which is realized in x/greaterorsimilar0.5 in Fig. 2(a), the system is a doped MI with ferromagnetic spin polarization. The fermi level is located in the A band. Be- cause of the large band width in the A band, there is a large kineticenergygainincomparisonwiththedopedLS-BIstate wherethefermilevelislocatedintheBbandintherigidband scheme. Thiskineticenergygainistheoriginofthespinsta te transitionbydoping. ItisshowninFig.4that,whentheequa l band widths are assumed, the E′v.s.xcurvesfor the LS-ML and FM HS-ML states are almost parallel and do not crosswith each other. This data implies that there is no differenc e in the kinetic energy gains for the two states, when the band widths are assumed to be equal. The present PS phenomena arealso attributedto thisbandwidthdifferenceasfollows . In the rigid-band sense, by doping of holes in the LS-BI state, thefermilevelfallsintothetopoftheBbandfromthemiddle of the gap in Fig. 5(a). If we suppose that this state is real- ized in a low xregion and is transferred into the FM HS-ML state shown in Fig. 5(b) with increasing x, the fermi level is increased with increasing hole concentration because of th e differentbandwidths. Thisis nothingbutthe negativechar ge compressibility κ=(∂µ/∂x)<0withthechemicalpotential µ,i.e. appearanceoftheelectronicPS. Finally, we discuss implications of the perovskite cobaltites. The obtained PS between the insulating nonmag- netic state and the hole-rich FM one is qualitatively consis - tent with the inhomogeneity suggested by a number of ex- periments. The PS and the spin-state transition are attribu ted to the band-width difference of the two bands correspond- ing to the egandt2gbands in the perovskite cobaltites. This electronic PS is robust by changingthe model parameter val- ues, except for tB/tA, when the non-doped system is located near the phase boundarybetween the LS-BI and HS-MI. The present phenomena are different from the previous PS’s dis- cussed in the high-Tc cuprates and the manganiteswhere the long-range spin/orbital orders are realized in the MI’s; th e spatialsegregationsoccurbetweenthelong-rangeordered MI and the ML states where the superexchange interaction en- ergyand the kinetic one of dopedholes are separately gained in the different spatial regions. Our scenario of the PS base d on the band-widthdifferencemay be checkedexperimentally by adjusting the tolerance factor, i.e. the Co-O-Co bond an- gle; the smaller tolerance factor implies the smaller (larg er) bandwidthin the eg(t2g)orbitals,and suppressionof the PS. Detailed values of xwhere the PS is realized, and a typical size of the clusters remain as questions. Several factors no t considered here, the intermediate-spin state, the long-ra nge Coulombinteraction,thelatticevolumedependingonthesp in states, andso on,are requiredtoanswerthese questions. Authors would like to thank H. Yokoyama and H. Takashima for their valuable discussions. This work was supported by JSPS KAKENHI, TOKUTEI from MEXT, and Grand challenges in next-generation integrated nanoscience. 1S. Maekawa, et al. Physics of TransitionMetal Oxides , (Springer Verlag, Berlin,2004), andreferences therein. 2M. Imada, et al.Rev. Mod. Phys. 70, 1039 (1998). 3S.Yamaguchi et al.Phys.Rev. B 53, R2926 (1996). 4M. A. Korotin, etal.Phys.Rev. B 54, 5309 (1996). 5M. W.Haverkort, et al.Phys. Rev. Lett. 97, 176405 (2006). 6S.Noguchi, et al.Phys. Rev. B 66, 094404 (2002). 7Y. Kobayashi, et al.Phys.Rev. B 72, 174405 (2005). 8M. Itoh,et al.J. Phys.Soc.Jpn. 63, 1486 (1994). 9R. Caciuffo, etal.Phys.Rev. B 59, 1068 (1999). 10P.L.Kuhns, et al.Phys.Rev. Lett. 91, 127202 (2003).11A. Ghoshray, etal.Phys.Rev. B 69, 064424 (2004). 12J. Wu,et al.Phys. Rev. Lett. 94, 037201 (2005). 13D. Phelan, et al.Phys. Rev. Lett. 96, 027201 (2006). 14D. Phelan, et al.Phys. Rev. Lett. 97, 235501 (2006). 15E.L.Nagaev, Phys.State.Sol.(b) 186, 9(1994). 16E. Dagotto, The Physics of Manganites and RelatedCompounds , (Springer-Verlag,Berlin2003). 17S.Okamoto, etal.Phys.Rev. B 61, 451(2000). 18K. I.Kugel, etal.Phys.Rev. Lett.95, 267210 (2005) 19P.Werner,and A.J. Millis,Phys.Rev. Lett. 99, 126405 (2007). 20K. SanoandY. Ono, J.Phys. Soc.Jpn. 72, 1847 (2003).5 21K.Kobayashi,andH.Yokoyama,PhysicaC 445-448,162(2006). 22K. Kubo, Phys.Rev. B 79, 020407 (2009). 23The 10 inequivalent conﬁgurations considered here are \|0/angbracketright,\|Aσ/angbracketright, \|Bσ/angbracketright,\|AσB¯σ/angbracketright,\|AσBσ/angbracketright,\|B↑B↓/angbracketright,\|A↑A↓/angbracketright,\|AσB↑B↓/angbracketright,\|A↑A↓Bσ/angbracketright, \|A↑A↓B↑B↓/angbracketrightwithσ=↑and↓, whereAσ(Bσ)impies that the A (B) orbital isoccupied by the up(down) spinelectron. 24C. J.Umrigar, et al.Phys. Rev. Lett 60, 1719 (1988).25J. Okamoto, et al.Phys. Rev. B 62, 4455 (2000). †Presentaddress: ThebankofTokyo-MitsubishiUFJ,Tokyo, Japan. ∗Presentaddress: ChibaInstituteofTechnology,Tsudanuma , Chiba275-0016,Japan.
1807.04579v1.Spin_Phonon_coupling_parameters_from_maximally_localized_Wannier_functions_and_first_principles_electronic_structure__the_case_of_durene_single_crystal.pdf	Spin-phonon coupling parameters from maximally localized Wannier functions and rst principles electronic structure: the case of durene single crystal Subhayan Roychoudhury and Stefano Sanvito School of Physics and CRANN Institute, Trinity College, Dublin 2, Ireland Spin-orbit interaction is an important vehicle for spin relaxation. At nite temperature lattice vibrations modulate the spin-orbit interaction and thus generate a mechanism for spin-phonon coupling, which needs to be incorporated in any quantitative analysis of spin transport. Starting from a density functional theory ab initio electronic structure, we calculate spin-phonon matrix elements over the basis of maximally localized Wannier functions. Such coupling terms form an eective Hamiltonian to be used to extract thermodynamic quantities, within a multiscale approach particularly suitable for organic crystals. The symmetry of the various matrix elements are analyzed by using the -point phonon modes of a one-dimensional chain of Pb atoms. Then the method is employed to extract the spin-phonon coupling of solid durene, a high-mobility crystal organic semiconducting. Owing to the small masses of carbon and hydrogen spin-orbit is weak in durene and so is the spin-phonon coupling. Most importantly we demonstrate that the largest contribution to the spin-phonon interaction originates from Holstein-like phonons, namely from internal molecular vibrations. I. INTRODUCTION In a non-magnetic material the electrical resistance ex- perienced by a charge carrier is independent of its spin. In contrast, when the material is magnetic the resistance typically depends on the relative orientation of the car- rier spin and the local magnetization1. This observa- tion inspired the advent of the eld of spin-electronics or spintronics2, which concerns the injection, manipu- lation and detection of spins in a solid-state environ- ment. A prototype spintronics device, the spin-valve3, consists of two ferromagnetic layers sandwiching a non- magnetic spacer4, which can display a metallic5,6, insu- lating7or semiconducting8,9electronic structure. The carriers, which are spin-polarized by one ferromagnet, travel through the spacer to the other ferromagnet. If the spin direction is maintained during such transfer, then the total resistance of the device will depend on the mu- tual orientation of the magnetization vectors of the two ferromagnets. It is then crucial to understand how the spin direction evolves during the motion of the carriers through the spacer, and in particular to understand how this is preserved. There are multiple possible sources of spin relaxation in a material, such as the presence of impurities, hyper- ne interaction and spin-orbit (SO) coupling. A theoret- ical description of all such phenomena is needed for an accurate evaluation of the quantities related to spin re- laxation. The relative dominance of one interaction over the others is typically highly dependent on the specic material. In this work, we shall focus on SO interac- tion, more specically on the modulation of such inter- action due to lattice vibrations. The spin of an electron interacts with the magnetic eld generated by the rel- ative motion of the nucleus about the electron, giving rise to SO interaction. At nite temperature the atoms of a solid vibrate with respect to their equilibrium posi- tions with the amplitudes of such vibrations increasingwith temperature. Such vibrations, the phonons, change the potential felt by the electrons, including the compo- nent due to SO coupling10. This eectively generates a mechanism for spin-phonon coupling11, which is key for the calculation of quantities related to spin-relaxation in many systems. It must be noted that in current liter- ature the term `spin-phonon' coupling has been used to denote dierent eects. For instance in the study of mul- tiferroic compounds `spin-phonon coupling' indicates the modulation of the phonon frequencies due to changes in the magnetic ordering.12{16Here we are interested in the opposite, namely in the change of electronic structure brought by the vibrations, in particular for the case of organic crystals. Recent years have witnessed a growing interest in ex- ploring the possibility of using organic crystal semicon- ductors for electronic and spintronic applications17{21. This stems from the high degree of mechanical exibility, the light weight and the ease of synthesis and patterning that characterize organic compounds. In these systems covalently bonded organic molecules are held together by weak van der Waals interactions. Due to the weak bonds between the individual molecules, vibrational mo- tions are prominent in organic crystals and the coupling of the vibrations to the charge carriers plays a crucial role22in the transport properties of such materials. The presence of experimental evidence in support of dierent transport regimes23{27has generated a signif- icant debate on whether the transport in organic crys- tals is dominated by delocalized band-like transport, as in covalently bonded inorganic semiconductors, by local- ized hopping, or by a combination of both. This can very well depend on the specic crystal and the exper- imental conditions, such as the temperature. Typically, in organic crystals the vibrational degrees of freedom are thought to introduce signicant dynamical disorder28 and thereby have paramount in uence on the transport properties. Since the typical energies associated to lat- tice vibrations in organics are of the same order of magni-arXiv:1807.04579v1 [cond-mat.mtrl-sci] 12 Jul 20182 tude of the electronic bandwidth, the coupling between carriers and phonons can not be treated by perturba- tion theory. Thus, in general, formulating a complete theoretical framework for the description of transport in organic crystals is more challenging than that for cova- lently bonded inorganic semiconductors22,29. Even more complex is the situation concerning spin transport, for which the theoretical description often relies on param- eters extracted from experiments30, or on approximate spin Hamiltonians31 One viable option towards a complete ab initio de- scription of spin transport consists in constructing a mul- tiscale approach, where information about the electronic and vibrational properties calculated with rst-principles techniques are mapped onto an eective Hamiltonian re- taining only the relevant degrees of freedom. For instance this is the strategy for constructing eective giant-spin Hamiltonians with spin-phonon coupling for the study of spin relaxation in molecular magnets32,33. The approach presented here instead consists in projecting the elec- tronic structure over appropriately chosen maximally lo- calized Wannier functions (MLWFs)34, which eectively dene a tight-binding (TB) Hamiltonian. In a previous work35we have described a computationally convenient scheme for extracting the SO coupling matrix elements for MLWFs. Here we extend the method to the com- putation of the spin-phonon matrix elements. Our de- rived Hamiltonian can be readily used to compute spin- transport quantities, such as the spin relaxation length. The paper is organized as follows. In the next section we introduce our computational approach and describe the specic implementation used. Then we present our results. We analyze rst the symmetry of the various matrix elements by considering the simple case of a lin- ear atomic chain of Pb atoms. Then we move to the most complex case of the durene crystal, a popular high- mobility organic semiconductor. Finally we conclude. II. METHOD Wannier functions, which form the basis functions of the proposed TB Hamiltonian, are essentially the weighted Fourier transforms of the Bloch states of a crys- tal. From a set of N0isolated Bloch states, fj mkig, which for instance can be the Kohn-Sham (KS) eigen- states of a DFT calculation, one can obtain N0Wannier functions. The n-th Wannier ket centred at the lattice siteR,jwnRi, is found from the prescription, jwnRi=V (2)3Z BZ"NX m=1Uk mnj mki# eik:Rdk;(1) whereVis the volume of the primitive cell, j mkiis them-th bloch vector, and the integration is performed over the rst Brillouin zone (BZ). Here Ukis a unitary operator that mixes the Bloch states. In order to x the gauge choice brought by Ukone minimizes the spread ofa Wannier function, which is dened as =X n hwn0jr2jwn0ijhwn0jrjwn0ij2 :(2) Such choice denes the so-called MLWFs36. We use the codewannier9037for construction of such MLWFs. Since the TB Hamiltonian operator, ^H, depends on the ionic positions, the ionic motions give rise to changes in ^H. In addition, since the MLWFs are constructed from the Bloch states, which themselves depend on the ionic coordinates, lattice vibrations result in a change of the MLWFs as well. Therefore, the change in the Hamilto- nian matrix elements due to the ionic motion, namely the onsite energies and hopping integrals, originates from the combined action of 1) the change in ^Hand 2) the change in the MLWFs basis. Hence, in the MLWFs TB picture the variation of the matrix element, "nm, due to an ionic displacement, is given by "nm=hwf nj^Hfjwf mihwi nj^Hijwi mi; (3) wherewi m(wf m) and ^Hi(^Hf) are the initial (nal) MLWF and the Hamiltonian operator, respectively. Eq. (3) describes the variation of an onsite energy or a hopping integral depending on whether jwmiandjwni are located on the same site or at dierent sites. Since any general lattice vibration can be expanded as a linear combination of normal modes, one is typically interested in calculating "nmdue to vibrations along the normal mode coordinates. In order to quantify the rate of such change, we dene the electron-phonon coupling parame- ter,g mn, for the-th phonon mode as the rate of change, "mn, of"mnwith respect to a displacement Qalong such normal mode, namely g mn=@"mn @Q Q!Q+Q: (4) HereQdescribes the system's geometry, so that Q! Q+ Qindicates that the partial derivative is to be taken with respect to the atomic displacement along the phonon eigenvector corresponding to the mode . This coupling constant is fundamentally dierent from that dened in a conventional TB formulation. In that case the electron-phonon coupling is simply dened as nm=@ hi nj^Hf^Hiji mi @Q Q!Q+Q;(5) whereji niis then-th basis function before the motion. Note that, at variance with Eq. (3), which takes into ac- count both the changes in the operator and the basis set, in Eq. (5) only the Hamiltonian operator is modied and the matrix element is evaluated with respect to the basis set corresponding to the equilibrium structure. For the remaining of this paper, unless stated otherwise, electron- phonon coupling will always denote the rst description, i.e. theg mns of Eq. (4). The eect of such coupling on3 charge transport has been the subject of many previous investigations.38{41 As all matrix elements, also those associated to the SO coupling depend on the ionic coordinates. In a pre- vious paper35we have described a method to calculate the SO matrix elements associated to the MLWFs ba- sis,hws1 mRj^VSOjws2 nR0i, from those computed over the spin- polarized Bloch states, h s1 m;kj^VSOj s2 n;k0i(the superscript denotes the magnetic spin quantum number). Note that here the MLWFs computed in absence of SO coupling are used as basis functions, since they span the entire relevant Hilbert space. The term h s1 m;kj^VSOj s2 n;k0ican be, in principle, calculated from any DFT implementa- tion that incorporates SO coupling. Our choice is the siesta code42, which uses an on-site approximation43for the SO coupling and gives the SO elements in terms of a set of localized atomic orbitals fjs ;Rlig44. Hence, the basic owchart for such calculation follows the general prescription hs1 ;Rjj^VSOjs2 ;Rli!h s1 m;kj^VSOj s2 n;k0i!hws1 mRj^VSOjws2 nR0i; (6) namely from the SO matrix elements calculated for the siesta local orbitals one computes those over the Bloch functions and then the ones over the MLWFs. Once the matrix elements hws1 mRj^VSOjws2 nR0iare known, it is possible to determine the spin-phonon coupling by following a prescription similar to that used for comput- ing the electron-phonon coupling in Eq. (4), gs1s2() m;n =@"s1s2 (SO)mn @Q Q!Q+Q; (7) where"s1s2 (SO)mnis the SO matrix element between the ML- WFsjws1miandjws2ni,Qdenotes the atomic positions and Qrefers to an innitesimal displacement of the coordinates along the -th phonon mode. As noted ear- lier, a change in atomic coordinates results in a change in the MLWFs and such change must be taken into ac- count when calculating the dierence in the SO elements "s1s2 (SO)mn. We use the same symbol gto denote both the electron-phonon and the spin-phonon coupling, since they can be distinguished by the presence or absence of the spin indices. In practice, when calculating both the electron-phonon and the spin-phonon coupling each atom iin the unit cell is innitesimally displaced by Qei along the direc- tion of the corresponding phonon eigenvector, ei . Then the electron-phonon (spin-phonon) coupling is calculated as "mn=Q("s1s2 (SO)mn=Q), i.e. from nite dier- ences. If Qis too large, then the harmonic approxima- tion, which is the basis of this approach, breaks down. In contrast, if Qis too small, then the quantity will have a signicant numerical error. Hence, for any system stud- ied, one must evaluate the coupling term for a range of Qand, from a plot of coupling terms vs Q, choose Γ π /a-30-20-100E-EF (eV) Omitted bands Included bandsFIG. 1. Band structure of a diatomic Pb chain calculated with a minimal basis set in siesta . The black and the red lines correspond to bands omitted from and included in the construction of the MLWFs, respectively. the most suitable value of Q. It is important to note that the coupling terms so dened have the dimension of energy/length. This is consistent with the semiclassical TB Hamiltonian used, for example in Ref.45, for treat- ing transport in organic crystals with signicant dynamic disorder. However, various other denitions and dimen- sions for the electron-phonon coupling can be found in literature.41,46{48 III. RESULTS AND DISCUSSION A. One Dimensional Pb Chain A linear chain of Pb atoms with a diatomic unit cell has 6 phonon modes for each wave-vector, q. For simplicity we restrict our calculations to the -point, q=0, so that equivalent atoms in all unit cells have the same displace- ments with respect to their equilibrium positions. Since for the acoustic modes there is no relative displacement between the atoms of a unit cell, we are left with three optical modes of vibration as shown in the bottom panel of Fig. 2. The electronic band structure of a diatomic Pb chain calculated with a single-zeta basis functions is shown in Fig. 1. Note that two of the bands marked in red are composed mostly of porbitals-bonding and are doubly degenerate. Thus, as expected, the band struc- ture contains 8 bands in total. The MLWFs are con- structed by omitting the lowest two bands (mostly made ofs-orbitals) and retaining the remaining 6 bands. This gives us six MLWFs per unit cell, three centred on each atom. For each of the three modes, we evaluate the cou-4 FIG. 2. The unit cell of the Pb chain containing two atoms. The gures in the top panel show isovalue plots of the three MLWFs (from left to right: jw1;0i,jw2;0iandjw3;0i) cen- tred on the rst atom. The bottom panels indicate the direc- tions of the atomic motion corresponding to the three phonon modes (mode 1, mode 2 and mode 3, from left to right). pling matrix elements between the MLWFs of the same unit cell for a range of Q. By analysing these results we nd that Q= 0:03 is an acceptable value for such fractional displacement. The top panel of Fig. 2 shows the MLWFs correspond- ing to the rst atom of the unit cell at the equilibrium geometry. From this gure one can see that jw1;0i,jw2;0i andjw3;0iclosely resemble the porbitals of the rst atom, which we can denote arbitrarily (the denition of the axes is arbitary) as pz,pxandpy, respectively. By symmetry, jw4;0i,jw5;0i,jw6;0ican be associated with the pz,px andpyorbitals located on the second atom. However, it is important to note that such similarity between the ML- WFs and the orbital angular momentum eigenstates does not mean that they are equivalent . In order to appreciate this point, note that hwi;0jwj;0i= 0;8i6=jbut this is not necessarily true forhpm;1jpn;2i, wherejpm;1iandjpn;2iare or- bital angular momentum eigenkets centred on the rst and the second atom, respectively. When an atom is displaced from its equilibrium position, the porbitals (e.g. the basis orbitals of siesta ) experience a rigid shift only, but do not change in shape. In contrast, the MLWFs change in shape along with being displaced. Most importantly, in the on-site SO approximation used in siesta , the hopping term for SO coupling, i.e. the SO matrix element between two orbitals lo- cated on two dierent atoms, is always zero. As for the on-site term, the SO matrix element between two orbitals of the same atom is independent of the position of the other atom. Thus, the spin-phonon matrix elements are always zero, when calculatedMode Element value(meV/ A) Mode 1 [ w3jw4] -0.85 Mode 2 [ w1jw4] 4.03 [w2jw5] -1.51 [w3jw6] -1.51 Mode 3 [ w2jw4] -0.85 TABLE I. The non-vanishing electron-phonon coupling ma- trix elements for the -point phonon modes of the Pb chain with a diatomic unit cell. [ wjw] denotes the electron- phonon coupling matrix element between the MLWFs jwi andjwi. One must keep in mind that the matrix elements are real and the remaining non-vanishing ones not reported in the table can be found from the relation [ wjw] = [wjw]. See Fig. 2 for a diagram of the modes and the MLWFs. Mode Element Value(meV/ A) Mode 1 [ w" 1jw" 5] (0.0,-0.07) [w" 2jw" 4] (0.0,0.07) [w" 2jw# 6] (-0.19,0.0) [w" 3jw# 5] (0.19,0.0) Mode 2 [ w" 1jw# 5] (0.05,0.0) [w" 2jw# 4] (-0.05,0.0) [w" 1jw# 6] (0.0,-0.05) [w" 3jw# 4] (0.0,0.05) Mode 3 [ w" 1jw" 6] (0.0,0.07) [w" 3jw" 4] (0.0,-0.07) [w" 2jw# 6] (0.0,-0.19) [w" 3jw# 5] (0.00,0.19) TABLE II. Spin-phonon coupling matrix elements for the - point phonon modes of the Pb chain with diatomic unit cell. [ws1jws2] denotes the complex spin-phonon coupling matrix element between the MLWFs jws1iandjws2i. The remain- ing non-vanishing matrix elements can be found from the re- lations in Eq. (8). The phonon modes and the MLWFs are shown in Fig. 2 . with the on-site SO approximation over the siesta basis set. This is not the case for the MLWFs. Even when used in conjunction with an on-site SO ap- proximation, the spin-phonon coupling is typically non-zero for a MLWF basis owing to the change in the basis functions upon ionic displacement. Before calculating the spin-phonon coupling, let us take a brief look at the electron-phonon coupling ma- trix elements for the three phonon modes. The non-zero matrix elements are presented in Tab. I for each of the normal modes. It is interesting to note that the change in overlap between the associated ` p' orbitals due to the atomic displacements corresponding to the normal modes can be intuitively expected to have the same trend as the electron-phonon coupling matrix elements calculated with respect to the MLWFs (since the MLWFs closely resembleporbitals). For example, for an atomic motion along mode 3 (see Fig. 2), hpy;1jpz;2imust be zero, since jpz;2ihas always equal overlap with the positive and neg-5 ative lobe ofjpy;1i. Keeping in mind that modes 1, 2 and 3 correspond, respectively, to a motion in the y,zandx direction, one can easily show that hpz;1jpz;2imode:2>hpz;1jpx;2imode:3, hpz;1jpx;2imode:3 = hpx;1jpz;2imode:3 = hpy;1jpz;2imode:1, hpx;1jpy;2imode:1 = hpx;1jpz;2imode:2 = hpy;1jpz;2imode:3 = 0 where denotes a change in the overlap of the orbitals due to their corresponding atomic motion. Now we proceed to present our results for the spin- phonon coupling. At variance with the electron-phonon coupling matrix elements, the spin-phonon ones are not necessarily real valued. For each mode of the three modes, the inequivalent non-zero spin-phonon coupling matrix elements are tabulated in Tab. II. We denote the spin-phonon matrix element between jws1iandjws2ias [ws1jws2]. All other (equivalent) non-zero spin-phonon matrix elements can be found from those presented in Tab. II by using the following relations [w" jw# ] =[w# jw" ]; [w" jw# ] = [w# jw" ]; =[w" jw" ] ==[w# jw# ]: (8) Also, from the symmetry of the MLWFs, it is easy to show that [w" 1jw" 5]Mode1 =[w" 2jw" 4]Mode1; (9) [w" 1jw" 6]Mode3 =[w" 3jw" 4]Mode3: (10) We have noted that in the on-site approximation, the spin-phonon coupling (according to our denition) of the Pb chain should be zero, when calculated over the siesta basis set. However, if such on-site approximation is re- laxed, one will be able to determine a number of ana- lytical expressions for these coupling elements in terms of the change in orbital overlaps. It is interesting to note that the analytical expressions calculated in this way share many qualitative similarities with those pre- sented in Tab. II. We summarize the ndings of this sec- tion by noting that the spin-phonon couplings matrix ele- ments corresponding to the two equivalent normal modes show the expected symmetry. We have also seen that the non-zero spin-phonon coupling matrix elements for mode 2 are, in general smaller than those for the symmetry- equivalent modes 1 and 3. -15-10-505 ΓZ Z'Y ΓB B' Y A'Energy (eV) -7.5-7.0-6.5-6.0 ΓZ Z'Y ΓB B' Y A'Energy (eV) kPlot from SIESTA Plot from MLWFsFIG. 3. Band structure of the durene crystal. Panel (a) shows all the occupied and many unoccupied bands. MLWFs are constructed from the 4 highest occupied bands, which are plotted in black. Panel (b) shows the magnied structure of these 4 bands plotted with siesta (green line) and obtained from the MLWFs computed with wannier90 (red circle). B. Durene Crystal Finally we are in the position to discuss the spin- phonon coupling in a real organic crystal, namely in durene. In an electron-phonon or spin-phonon coupling calculation, one needs to make sure that the construc- tion of the MLWFs converges to a global minimum, oth- erwise the various displaced geometries may correspond to dierent local minima resulting in the description of a dierent energy landscape. Typically, a MLWF calcu- lation with dense k-mesh is likely to converge to a local minimum, while a calculation with coarse k-mesh has a higher probability of giving the global minimum (-point calculation always converges to the global minimum). However, a coarse k-mesh translates in a small period for the Born-Von-Karman boundary conditions, i.e. a poorer description of the crystal. In our calculation, we6 (b)(c)(d)(a) FIG. 4. Isovalue plots for MLWFs of the four topmost va- lence bands of a durene crystal. Panels (a), (b), (c) and (d) correspond to jw1;0i,jw2;0i,jw3;0iandjw4;0irespectively. use a 444k-grid and construct the MLWFs from the top four valence bands. This enables the calculation to converge to a global minimum, identied by vanish- ing or negligible imaginary elements in the Hamiltonian matrix. In Fig. 3(a) we show a plot of the durene band- structure (within a large energy window) and in Fig. 3(b), the bandstructure corresponding to the four bands used toconstruct MLWFs. These are plotted from the DFT siesta eigenvalues and by diagonalizing the tight-binding Hamiltonian constructed over the MLWFs. Since the unit cell of durene contains two molecules, the four valence bands give us four MLWFs per unit cell, so that each molecule has associated two MLWFs. In Fig. 4 we show an isovalue plot of the 4 MLWFs cor- responding to R=0. We see that unlike jw3;0iand jw4;0i, which are situated on the same molecule, jw1;0i andjw2;0iare on dierent but equivalent molecules dis- placed by a primitive lattice vector a2. Thus,jw1;0iand jw2;R0iare on the same molecule for R0=a2, where fa1;a2;a3gis the set of primitive vectors. This means that for our tight-binding picture hw1;0j^Hjw2;0icorre- sponds to a non-local (hopping) matrix element, whereas hw1;0j^Hjw2;R0iis a local (on-site) energy term. In the fol- lowing, we shall calculate the electron-phonon and spin- phonon coupling corresponding to various modes of the durene crystal and compare: 1) the relative contribution of the dierent modes, 2) for each mode, the relative con- tribution of the local and non-local terms. Since the unit cell contains two molecules, each with 24 atoms (48 atoms in the unit cell), a -point phonon calcu- lation will give us 144 modes, with 141 being non-trivial. Among these, 12 will be predominantly intermolecular modes (3 translational and 9 rotational modes, where the molecules move rigidly with respect to each other) and the remaining ones will be of predominantly intramolecu- lar nature. Here we shall consider only the phonon modes with an energy less than 75 meV, as the modes with higher energy are accessible only at high temperature41. FIG. 5. Histogram of the eective electron-phonon coupling as a function of the phonons energy. The local and the non- local contributions are denoted by green and red bars, respec- tively. Thus, we take into account 25 modes, of which the rst 12 are intermolecular (these are lower in energy) and the rest are symmetry inequivalent intramolecular ones49. In order to compare the contributions of the dier- ent phonon modes and of the local (Holstein-type) and non-local (Peierls-type) contributions, we calculate the following eective electron-phonon coupling parameters GL =X m;njg mnj2; (11) wheremandnare functions centred on same molecule, and GN =X m6=njg mnj2; (12) wheremandnare on dierent molecules. Here the superscripts L and N stand for Local and Non-local, respectively. A crucial point to be noted for treating bulk crystals is that in wannier90 , the direct lattice points, where the MLWFs are calculated, are the lattice points of the Wigner-Seitz cell about the cell ori- gin,R=0. Typically, one should expect the number of such lattice points to be the same as the number of k-points in reciprocal space. However, in a 3-D crystal it is possible to have lattice points, which are equidistant from the R=0cell and (say) nnumber of other cells. This means that such lattice point is shared by Wigner- Seitz cells of n+ 1 cells. In this case, this degenerate lattice point is taken into consideration by wannier90 , but a degeneracy weight of 1 =(n+ 1) is associated with it. Consequently in further calculations (such as the band structure interpolation), its contribution carries a factor of 1=(n+ 1). Keeping this in mind, we multiply the con- tributions from the MLWFs of degenerate direct lattice7 points by their corresponding weighting factors. Fig. 5 shows a histogram of the Gterms as function of the phonons energy. It must be kept in mind that the cou- pling matrix elements are strongly dependent on the ML- WFs. Therefore, constructing Wannier functions from a dierent set of Bloch states can in principle result in dif- ferent values of G. We see that in our case, most of the modes with high G(=GL +GN ) are located at high phonon energies. Also, the electron-phonon couplings for modes with lower Gare dominated by the non-local con- tributions, while those with higher Gare dominated by local contributions. Concerning the spin-phonon coupling, we can dene spin-dependent Gterms, namely the eective spin- phonon couplings, GL(s1s2) =X m;njgs1s2() mnj2; (13) wheremandnare on same molecule and GN(s1s2) =X m6=njgs1s2() mnj2; (14) wheremandnare on dierent molecules. In Fig. 6, we plot these eective spin-phonon coupling terms, and we break down the local and non-local con- tributions. The top and the bottom panels correspond to the (s1=";s2=") and (s1=";s2=#) case, respec- tively. As expected, the spin-phonon coupling terms are extremely small (about four orders of magnitude smaller than those of the Pb chain), owing to the small atomic masses in the crystal (the SOC is small). As in the case of the electron-phonon interaction, the eective spin- phonon coupling terms are dominated by non-local con- tributions for low G(s1s2) =GL(s1s2) +GN(s1s2) and by local contributions for high G. We also see that the spin-phonon coupling (for same spin, as well as for dif- ferent spins) is very small for the rst few modes, which represent intermolecular motions. This is fully consis- tent with the short-ranged nature of SO coupling. An important message emerging from these results is that phonon modes having high eective electron-phonon cou- pling do not necessarily have high eective spin-phonon coupling, and vice-versa. This means that the knowledge of the phonon spectrum says little a priori about the spin-phonon coupling, so that any quantitative theory of spin relaxation cannot proceed unless a detail analysis along the lines outlined here is performed. In conclusion, we have discovered that both the electron-phonon and the spin-phonon coupling constants are, in general, dominated by the local modes, as ex- pected by the short-range nature of the SOC. However, modes with very small eective coupling tend to have a larger relative contribution arising from non-local modes. No apparent correlation can be found between the ef- fective coupling constants pertaining to various phonon modes for the electron-phonon coupling and those for the spin-phonon coupling. FIG. 6. Histogram plot of the eective spin-phonon coupling parameters, G(s1s2) , as a function of the phonons energy. The top panel corresponds to the s 1= s2case (same spins), while the bottom one corresponds to s 16= s2(dierent spins). The local and the non-local contributions are denoted by green and red bars respectively. IV. CONCLUSION Based on our previous work concerning the calculation of the SO matrix elements with respect to MLWFs basis sets, we have presented calculations of the spin-phonon coupling matrix elements of periodic systems. We note that, in order to be useful in a multiscale approach based on an eective Hamiltonian, the electron-phonon and the spin-phonon coupling are not to be calculated in terms of a xed set of MLWFs. Instead, one must take into account the change in the MLWFs as a result of the ionic motions. The coupling matrix elements for a given phonon mode are calculated by displacing atoms from the ground state geometry along that phonon eigenvec- tor and by taking nite dierences. For phonon modes at the -point, we have calculated the electron-phonon and8 spin-phonon coupling elements of a 1D chain of Pb atoms with two atoms per unit cell and of a bulk durene crys- tal. This latter is a widely-studied and well-known or- ganic semiconductor. The spin-phonon coupling matrix elements of the Pb chain obey the expected symmetry relations. For durene we have observed that, in general, the spin-phonon coupling is dominated by local contribu- tions (Holstein-modes), although, for phonon modes with a small net eective coupling, the non-local part seems to dominate. Our calculations of spin-phonon coupling matrix elements are expected to be valuable in the con- struction of a eective Hamiltonians to be used for com- puting transport-related quantities. This is particularlywelcome in the case of organic crystals, where ab initio computation of transport properties is a challenging task. V. ACKNOWLEDGEMENTS This work is supported by the European Research Council, Quest project. Computational resources have been provided by the supercomputer facilities at the Trin- ity Center for High Performance Computing (TCHPC) and at the Irish Center for High End Computing (ICHEC). We thank Carlo Motta for providing the - point phonon eigenvalues and eigenvectors of durene crystal used in Ref.41. 1N. F. Mott, Proc. Royal Soc. A 153, 699 (1936). 2S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Moln ar, M. L. Roukes, A. Y. Chtchelka- nova, and D. M. Treger, Science 294, 1488 (2001). 3B. Dieny, V. S. Speriosu, S. S. P. Parkin, B. A. Gurney, D. R. Wilhoit, and D. Mauri, Phys. Rev. B 43, 1297 (1991). 4S. Sanvito, Nature Mater. 10, 484 (2011). 5M. N. Baibich, J. M. Broto, A. Fert, F. N. Van Dau, F. Petro, P. Etienne, G. Creuzet, A. Friederich, and J. Chazelas, Phys. Rev. Lett. 61, 2472 (1988). 6G. Binasch, P. Gr unberg, F. Saurenbach, and W. Zinn, Phys. Rev. B 39, 4828 (1989). 7J. S. Moodera, L. R. Kinder, T. M. Wong, and R. Meser- vey, Phys. Rev. Lett. 74, 3273 (1995). 8J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and I. Zutic, Acta Phys. Slovaca 57, 565 (2007). 9D. D. Awschalom and M. E. Flatt e, Nature Phys. 3, 153 (2007). 10F. P. Marn and H. Suhl, Phys. Rev. Lett. 63, 442 (1989). 11H. Ochoa, A. H. Castro Neto, V. I. Fal'ko, and F. Guinea, Phys. Rev. B 86, 245411 (2012). 12T. Birol and C. J. Fennie, Phys. Rev. B 88, 094103 (2013). 13J. H. Lee and K. M. Rabe, Phys. Rev. B 84, 104440 (2011). 14J. Hong, A. Stroppa, J. I~ niguez, S. Picozzi, and D. Van- derbilt, Phys. Rev. B 85, 054417 (2012). 15T. Birol, N. A. Benedek, H. Das, A. L. Wysocki, A. T. Mulder, B. M. Abbett, E. H. Smith, S. Ghosh, and C. J. Fennie, Curr. Opin. Solid State Mater. Sci. 16, 227 (2012). 16M. Mochizuki, N. Furukawa, and N. Nagaosa, Phys. Rev. B84, 144409 (2011). 17G. Malliaras and R. Friend, Phys. Today 58, 53 (2005). 18Nature Mater. 12, 591 (2013), editorial. 19S. R. Forrest, Nature 428, 911 (2004). 20S. Sanvito, Chem. Soc. Rev. 40, 3336 (2011). 21S. Wang, A. Wittmann, K. Kang, S. Schott, G. Schwe- icher, R. D. Pietro, J. Wunderlich, D. Venkateshvaran, M. Cubukcu, and H. Sirringhaus, in 2017 IEEE Inter- national Magnetics Conference (INTERMAG) (2017) pp. 1{1. 22A. Troisi, Chem. Soc. Rev. 40, 2347 (2011). 23C. Dimitrakopoulos and P. Malenfant, Adv. Mater. 14, 99 (2002).24O. D. Jurchescu, J. Baas, and T. T. M. Palstra, Appl. Phys. Lett. 84, 3061 (2004). 25O. Ostroverkhova, D. G. Cooke, F. A. Hegmann, J. E. Anthony, V. Podzorov, M. E. Gershenson, O. D. Jurchescu, and T. T. M. Palstra, Appl. Phys. Lett. 88, 162101 (2006). 26K. Marumoto, S.-i. Kuroda, T. Takenobu, and Y. Iwasa, Phys. Rev. Lett. 97, 256603 (2006). 27T. Sakanoue and H. Sirringhaus, Nature Mater. 9, 736 (2010). 28A. Troisi and G. Orlandi, J. Phys. Chem. A 110, 4065 (2006). 29K. Hannewald and P. A. Bobbert, Phys. Rev. B 69, 075212 (2004). 30P. A. Bobbert, W. Wagemans, F. W. A. van Oost, B. Koopmans, and M. Wohlgenannt, Phys. Rev. Lett. 102, 156604 (2009). 31S. Bhattacharya, A. Akande, and S. Sanvito, Chem. Com- mun. 50, 6626 (2014). 32A. Lunghi, F. Totti, R. Sessoli, and S. Sanvito, Nature Commun. 8, 14620 (2017). 33A. Lunghi, F. Totti, S. Sanvito, and R. Sessoli, Chem. Sci. 8, 6051 (2017). 34N. Marzari, A. A. Mosto, J. R. Yates, I. Souza, and D. Vanderbilt, Rev. Mod. Phys. 84, 1419 (2012). 35S. Roychoudhury and S. Sanvito, Phys. Rev. B 95, 085126 (2017). 36N. Marzari and D. Vanderbilt, Phys. Rev. B 56, 12847 (1997). 37A. A. Mosto, J. R. Yates, Y.-S. Lee, I. Souza, D. Vander- bilt, and N. Marzari, Comput. Phys. Commun. 178, 685 (2008). 38F. Ortmann, K. Hannewald, and F. Bechstedt, phys. sta- tus solidi B 245, 825 (2008). 39F. Ortmann, K. Hannewald, and F. Bechstedt, Appl. Phys. Lett. 93, 222105 (2008). 40H. Ishii, N. Kobayashi, and K. Hirose, Phys. Rev. B 95, 035433 (2017). 41C. Motta and S. Sanvito, J. Chem. Theo. Comp. 10, 4624 (2014). 42J. M. Soler, E. Artacho, J. D. Gale, A. Garc a, J. Junquera, P. Ordejon, and D. Sanchez-Portal, J. Phys. Condens. Matter 14, 2745 (2002). 43In the on-site approximation a SO coupling matrix element is neglected unless both orbitals and the SO pseudopoten-9 tial are on the same atom. 44L. Fern andez-Seivane, M. A. Oliveira, S. Sanvito, and J. Ferrer, J. Phys. Condens. Matter 18, 7999 (2006). 45A. Troisi and G. Orlandi, Phys. Rev. Lett. 96, 086601 (2006). 46K. Kaasbjerg, K. S. Thygesen, and K. W. Jacobsen, Phys. Rev. B 85, 115317 (2012). 47A. Girlando, L. Grisanti, M. Masino, I. Bilotti, A. Bril- lante, R. G. Della Valle, and E. Venuti, Phys. Rev. B 82,035208 (2010). 48V. Coropceanu, J. Cornil, D. A. da Silva Filho, Y. Olivier, R. Silbey, and J.-L. Br edas, Chem. Rev. 107, 926 (2007). 49The phonon spectrum is calculated with FHI-AIMS. Cal- culation courtesy: Dr. Carlo Motta.
1012.4757v1.Spin_dephasing_and_pumping_in_graphene_due_to_random_spin_orbit_interaction.pdf	arXiv:1012.4757v1 [cond-mat.mes-hall] 21 Dec 2010Spin dephasing and pumping in graphene due to random spin-or bit interaction V. K. Dugaev1,2, E. Ya. Sherman3,4, and J. Barna´ s5∗ 1Department of Physics, Rzesz´ ow University of Technology, al. Powsta´ nc´ ow Warszawy 6, 35-959 Rzesz´ ow, Poland 2Department of Physics and CFIF, Instituto Superior T´ ecnic o, TU Lisbon, Av. Rovisco Pais, 1049-001 Lisbon, Portugal 3Department of Physical Chemistry, Universidad del Pa´ ıs Va sco UPV-EHU, 48080 Bilbao, Spain 4IKERBASQUE, Basque Foundation for Science, 48011, Bilbao, Spain 5Institute of Molecular Physics, Polish Academy of Sciences , ul. Smoluchowskiego 17, 60-179 Pozna´ n, Poland (Dated: April 25, 2022) We consider spin eﬀects related to the random spin-orbit int eraction in graphene. Such a random interaction can result from the presence of ripples and/or o ther inhomogeneities at the graphene surface. We show that the random spin-orbit interaction gen erally reduces the spin dephasing (relaxation) time, even if the interaction vanishes on aver age. Moreover, the random spin-orbit coupling also allows for spin manipulation with an external electric ﬁeld. Due to the spin-ﬂip interband as well as intraband optical transitions, the spi n density can be eﬀectively generated by periodic electric ﬁeld in a relatively broad range of freque ncies. PACS numbers: 72.25.Hg,72.25.Rb,81.05.ue,85.75.-d I. INTRODUCTION Graphene is currently attracting much attention as a new excellent material for modern electronics1–3. The natural two-dimensionality of graphene matches per- fectly to the dominating planar technology of other semi- conducting materials, and correspondingly gives the way to creatingnewhybrid systems. However,the moststrik- ing properties of graphene are not directly related to its two-dimensionality. Due to the bandstructure eﬀects, electrons in pure graphene can be described by the rela- tivistic Dirac Hamiltonian, leading to the linear electron energy spectrum near the Dirac points. As a result, the electronic and transport properties of graphene are sig- niﬁcantly diﬀerent from those of any other metallic or semiconducting material,3–6except (to some extent) its parent material – the clean graphite.7 It has been also suggested that graphene may have good perspectives as a new material for applications in spintronics.8–11The intrinsic spin-orbit interaction in graphene is usually very small, and therefore one can expect extremely long spin dephasing (relaxation) time.12–16Thus, spin injected to graphene, for instance from ferromagnetic contacts, can maintain its coherence for a relatively long time. Experiments demonstrate spin relaxation times for various graphene-based systems spanned over several orders of magnitude17–21with some of them being much shorter than expected.17,18The rea- son of this contradiction is not quite clear, and several diﬀerent explanations of these observations have been already put forward.22–24In this paper we present an- ∗Also at Faculty of Physics, Adam Mickiewicz University, ul. Umultowska 85, 61-614 Pozna´ n, Polandother model based on the random Rashba spin-orbit interaction.25,26Physical origin of such random spin- orbit interaction can be related to the ripples existing at the surface of graphene27–29and/or to some impuri- ties adsorbedat the surface, which randomlyenhance the magnitude of spin-orbit coupling as compared to that in the clean graphene.23 One of the key issues in spintronics (including graphene-based spintronics) is the possibility of spin ma- nipulation with an external electric/optical ﬁeld. This includes spin generation, spin rotation, spin switching, etc. Here we consider the possibility of spin pumping in grapheneusingtheideaofcombinedresonanceinsystems with Rashba spin-orbit interaction.30,31The possibility of spin manipulation using optical excitation32is based on various mechanisms of spin-orbit interaction in semi- conductor systems. In particular, the spin polarization appears in systems with regular spin-orbit coupling, sub- ject to periodic electric ﬁeld.33–35It has been shown re- cently, that the randomspin-orbit interaction also can be applied to generate spin polarization in symmetric semi- conductor quantum wells.36In this paper we show that similar method can be used to generate spin polarization in graphene with random Rashba spin-orbit interaction. To do this we analyze the intensity of optically-induced spin-ﬂip transitions assuming two-dimensional massless Dirac model of electron energy spectrum in graphene, and calculate the magnitude of spin-polarization induced by the optical pumping. The paper is structured as follows. In section 2 we describe the model Hamiltonian assumed for graphene. Spindephasingduetotherandomspin-orbitRashbacou- pling is calculated in section 3. In turn, spin pumping by an external electric ﬁeld is considered in section 4. Final conclusions are presented in section 5.2 II. MODEL To describe electrons and holes in the vicinity of Dirac points we use the model Hamiltonian H0which is suﬃ- cient when considering the eﬀects related to low-energy electron and hole excitations. We also include the spin- dependent perturbation in the form of a spatially ﬂuc- tuating Rashba spin-orbit interaction, Hso. Thus, the system Hamiltonian can be written as (we use system of units with ¯h≡1) H=H0+Hso, (1) H0=vτ·k, (2) Hso=λ(r) v/parenleftbigg∂H0 ∂kxσy−∂H0 ∂kyσx/parenrightbigg =λ(r) (τxσy−τyσx), (3) wherevis the electron velocity, λ(r) is the random spin- orbit parameter, r= (x,y) is the two-dimensional coor- dinate, and τandσare the Pauli matrices acting in the sublattice and spin spaces, respectively.37Equations (2) and (3) show that spin-orbit coupling can be described by a conventional Rashba Hamiltonian, proportional to vxσy−vyσx, where the velocity components vx,vyare, in general, obtained with the unperturbed Hamiltonian H0. It is well known that there is an intrinsic (internal) spin- orbit coupling in graphene, which is related to relativistic corrections to the crystal ﬁeld of the corresponding lat- tice. In addition, a spatially uniform Rashba ﬁeld can be induced by the substrate on which the graphene sheet is located. The reported results suggest that these inter- actions, which can be considered as independent sources of spin relaxation, are either very weak15,38or do not contribute to the dephasing rate by symmetry reasons.24 Therefore, we will neglect them in our considerationsand will brieﬂy discuss their role in the following. The Schr¨ odinger equation, ( H0−ε)ψk= 0, for the pseudospinor components of the wavefunction ψkis /parenleftbigg −ε vk− vk+−ε/parenrightbigg/parenleftbigg ϕk χk/parenrightbigg = 0, (4) wherek±=kx±iky. The normalized solutions corre- sponding to the eigenstates εk=±vkof Hamiltonian H0 can be written in the form ψkσ±(r) =eik·r √ 2/parenleftbigg \|1σ/angbracketright±k+ k\|2σ/angbracketright/parenrightbigg ,(5) where the ±signs correspond to the states in upper and lower branches, respectively. We assume that the average value of spin-orbit inter- action vanishes, while the spatial ﬂuctuation of λ(r) can be described by the correlation function F(r−r′) of a certain form, /angbracketleftλ(r)/angbracketright= 0, (6) C(r−r′)≡ /angbracketleftλ(r)λ(r′)/angbracketright=/angbracketleftbig λ2/angbracketrightbig F(r−r′).(7)When calculating spin dephasing, one can consider only the electron states corresponding to the upper branch (conduction band) of the energy spectrum, εk= vk. The spin-ﬂip scattering from the random potential determines the spin relaxation in this particular band, while from symmetry of the system follows that spin de- phasing in the lower (valence) band is the same. The intraband matrix elements of the random spin-orbit in- teraction (3) in the basis of wavefunctions (5) for the conduction band form the following matrix in the spin subspace Vkk′=λkk′/parenleftbigg 0−ik−/k ik′ +/k′0/parenrightbigg , (8) whereλkk′is the Fourier component of the random spin- orbit coupling. Since scattering from the random spin- orbit potential is elastic, only the intraband transitions contribute to the spin relaxation. III. SPIN DEPHASING To demonstrate how the random spin-orbit coupling works in graphene and how its eﬀects can be observed in experiment, as well as to compare graphene and con- ventional two-dimensional semiconductor structures, we calculate in this Section the corresponding spin dephas- ing time. For this purpose we use the kinetic equation for the density matrix (Wigner distribution function),36,40 ∂ρk ∂t= Stρk. (9) The collision integral St ρkon the right-hand side of this equation is due to the spin-ﬂip scattering from the ran- dom spin-orbit interaction, Stρk=π/summationdisplay k′(2Vkk′ρk′Vk′k−Vkk′Vk′kρk−ρkVkk′Vk′k) ×δ(εk−εk′). (10) We assume the following form of the density matrix: ρk=ρ0k+skσz, (11) where the ﬁrst term ρ0kcorresponds to the spin- unpolarized equilibrium state. On substituting (8) and (11) into Eq. (10) we ﬁnd Stρk=−2πσz/summationdisplay k′C(q)(sk+sk′)δ(εk−εk′),(12) whereC(q)≡/angbracketleftbig λ2 kk′/angbracketrightbig andq=k′−k. Assuming that skdoes not depend on the point at the Fermi surface we obtain Stρk=−4kσzsk πv/integraldisplay2k 0C(q)/radicalbig 4k2−q2dq. (13)3 20 30 40 50 60 R (nm)0.30.40.50.60.70.80.91 k = 106 /cmτs /τs0 FIG. 1: Spindephasingtimeas afunctionofthecharacterist ic rangeRof the random spin-orbit ﬂuctuations. Fordeﬁniteness, we assumethat the characteristicspa- tial range of the random spin-orbit ﬂuctuations is R, and takeC(q) in the following form: C(q) = 2π/angbracketleftλ2/angbracketrightR2e−qR, (14) satisfying the normalization condition /integraldisplay C(q)d2q (2π)2=/angbracketleftλ2/angbracketright. (15) The resulting spin relaxation rate is not strongly sensi- tive to the shape of the correlator. However, the ap- plicability of the approach based on Eq.(12) depends on the ratio of electron mean free path ℓtoR, being valid only ifℓ/R≫1. In such a case, typically realized in graphene, the electron spin experiences indeed random, weakly correlated in time ﬂuctuations of the spin-orbit coupling. In addition, we can safely neglect the eﬀect of the random spin-orbit coupling on the momentum relax- ation rate. Finally, for the spin dephasing time we obtain the following expression: 1 τs k=8k v/angbracketleftbig λ2/angbracketrightbig R2/integraldisplay2kR 0e−xdx√ 4k2R2−x2 =4πk v/angbracketleftbig λ2/angbracketrightbig R2[I0(2kR)−L0(2kR)],(16) whereI0(x)andL0(x)arethemodiﬁedBesselandStruve functions of zeroth order, respectively. In the limiting semiclassical ( kR≫1) and quantum (kR≪1) cases we ﬁnd 1 τs k≃4R v/angbracketleftbig λ2/angbracketrightbig 1, kR≫1, πkR, kR ≪1.(17) ForkR≫1, the result in Eq.(17) can be interpreted as the special realization of the Dyakonov-Perel’ spin relax- ation mechanism. To see this, we note that the electron spin rotates at the rate Ω ∼/angbracketleftbig λ2/angbracketrightbig1/2, with the preces- sion direction changing randomly at the timescale of the order of time that electron needs to pass through one do- main of the size R, i.e.,τR∼R/v.The resulting spinrelaxation rate 1 /τs kis of the order of Ω2τR.It is worth mentioning that at given spatial and energy scale of the ﬂuctuating spin-orbit ﬁeld, the decrease in the electron free path and in the momentum relaxation time leads to the decrease in the spin dephasing rate: if ℓ≪R, elec- tron spin interacts with the local rather than with the rapidly changing random ﬁeld, and spin relaxation rate becomes of the order of/angbracketleftbig λ2/angbracketrightbig τ, whereτis the momen- tum relaxation time. This agrees qualitatively with the observations of Ref.[20], however, a quantitative compar- ison needs a more detailed analysis. Taking examples with typical values v= 108cm/s, R∼50 nm, and /angbracketleftλ2/angbracketright ∼500µeV2, similar to what can be expected from Ref.[12], we obtain τs kless than or of the order of 10 ns. As one can see from Eq.(17), the spin relaxation for small kR≪1 is suppressed, as can be understood in terms of the averaging of the random ﬁeld over a large 1 /k2≫R2area. The full k-dependence in Eq.(17) implies that the spin dephasing rate is pro- portional to n1/2at low carrier concentrations nand is independent of nat higher ones. Therefore, at the charge neutrality point, where n= 0, the spin relaxation vanishes, in agreement with the observations of Ref.[21]. Moreover, our approach qualitatively agrees with the in- crease in the spin relaxation time in the bilayer graphene comparedtothesinglelayerone21: thetransverserigidity of the bilayer can be larger, thus suppressing formation of the long-range ripples, and, as a result, the random spin-orbit coupling. Equation (17) shows that, as far as the spin dephasing is considered, the only diﬀerence between graphene and conventional semiconductors25,26,36is related to the fact that the electron velocity is constant for the former case and is proportional to the momentum in the latter one. Aswewill seein the nextSection, this diﬀerence becomes crucial for the spin pumping processes. The dependence of spin relaxation time, calculated from Eq. (16) as a function of the characteristic domain sizeRof the random spin-orbit interaction is presented in Fig. 1, were τs0is deﬁned as τ−1 s0≡8/angbracketleftbig λ2/angbracketrightbig /vk. The curves corresponding to diﬀerent values of k>106cm−1 would be practically indistinguishable in this ﬁgure. Here several comments on the numerical values of spin relaxation are in order. The values observed in experi- ments on spin injection from ferromagnetic contacts17,18 are of the order of 10−10s, two orders of magnitude less than our estimate which does not take into account ex- plicitly the role of the Si-based substrates. The eﬀect of the SiO 2substrate, including the contributions from im- purities and electron-phonon coupling, was thoroughly analyzed in Ref.[23]. However, the obtained dephas- ing rates were well below the experimental values and also below the estimate obtained here, leading the au- thors of Ref.[23] to the suggestion of an important role of heavy adatoms in the spin-orbit coupling.39On the other hand, it was shown that the spin dephasing rate canbestronglyinﬂuencedatrelativelyhightemperatures by the electron-electron collisions.24However, including4 these collisions does not bring theoretical values closer to the experimental ones. The discrepancies between theory and experiment and between experimental data obtained on diﬀerent systems call for a more detailed analysis of the experimental sit- uation, including the dependence of spin relaxation time on the device functional properties and the experimental techniques applied. IV. SPIN PUMPING Let us consider now spin pumping by an external elec- tromagnetic periodic ﬁeld corresponding to the vector potential A(t). We assume that the system described by Eqs. (1)-(3) is additionally in a constant magnetic ﬁeld B. For simplicity, we consider the Voigt geometry with the ﬁeld in the graphene plane, so that the eﬀects of Lan- dau quantization are absent. Thus, the Hamiltonian H0 includes now the constant ﬁeld and can be written as H0=vτ·k+∆σx, (18) where 2∆ = gµBBis the spin splitting and the magnetic ﬁeld is orientedalongthe x−axis (the electronLand´ efac- tor for graphene is g= 2). The induced spin polarization is opposite to the direction of the magnetic ﬁeld. Since we are interested in real transitions in which the energy is conserved, we consider interaction with a single component of the periodic electromagnetic ﬁeld, A(t) = Ae−iωt,which enters in the gauge-invariant form HA=−e c∂H ∂k·A(t) =−ev cτ·A(t),(19) and in the following we treat the term HAas a small perturbation. The absorption in a periodic ﬁeld (probability of ﬁeld- induced transitions in unit time) can be written as I(ω) = ReTr/summationdisplay k/integraldisplaydε 2πHAGk(ε+ω)HAGk(ε),(20) whereGk(ε) is the Green function. In the absence of spin-orbit interaction, the absorption (20) does not in- clude any spin-ﬂip transitions. We can account for the spin-orbit interaction (3) in the second order perturba- tion theory, including the corresponding matrix elements as shown in Fig. 2. This means that we do not con- siderperturbationtermsasthe self-energywithin asingle Green function assuming that they are already included in the electron relaxation rate. Thus, in the second order perturbation theory with re- spect to the random spin-orbit interaction Hsowe obtain I(ω) = ReTr/summationdisplay kk′/integraldisplaydε 2πHAG0 k(ε+ω)Hkk′ soG0 k′(ε+ω) ×HAG0 k′(ε)Hk′k soG0 k(ε), (21)FIG. 2: (Color online.) Feynman diagrams for the light ab- sorption in the second order perturbation theory with respe ct to the random spin-orbit interaction. The coupling and elec - tromagnetic vertices are shown as white and ﬁlled circles, r e- spectively. FIG. 3: (Color online.) The energy spectrum and indirect spin-ﬂip transitions in graphene in a uniform magnetic ﬁeld B. Long and short arrows correspond to the interband and intraband transitions, respectively. where Green’s function G0 k(ε) = diag/braceleftBig G0 k↑(ε), G0 k↓(ε)/bracerightBig corresponds to the Hamiltonian H0of Eq. (18), G0 kσ(ε) =ε+vτ·k+∆σ+µ (ε−ε1kσ+µ+iδsgnε)(ε−ε2kσ+µ+iδsgnε), (22) withσ=±1 corresponding to the spins oriented along and opposite to the x−axis, respectively, ε(1,2)kσ= ±vk+∆σ,δbeing the half of the momentum relaxation rate,δ= 1/2τ, andµdenoting the chemical potential. Diagrams in Fig. 2 show the qualitative diﬀerence between the graphene and semiconductor quantum well with respect to the eﬀects of random spin-orbit coupling. In semiconductors, the external electric ﬁeld is explicitly coupled to the anomalous spin-dependent term in the electron velocity, which is random, and therefore the di- agrams describing the corresponding transitions include only two Green functions. In graphene, due to the ab- sence of randomness-originated term in the Hamiltonian HA,four Green functions are required to take into ac- countthe randomcontributionofthe spin-orbitcoupling. This situation, in some sense, is more close to what is ob- served in the conventionalkinetic theory of normal metal5 conductivity, where the coupling to the external ﬁeld does not depend on the randomness explicitly, and the additional disorder eﬀects appear due to the self-energy and/or due to the vertex corrections, as in Fig. 2. Upon calculating contributions from the diagrams of Fig. 2, one ﬁnds the total rate of spin-ﬂip and spin- conserving transitions due to the random spin-orbit cou- pling in the form Irso(ω) =e2A2 c2ReTr/summationdisplay σσ′/summationdisplay kk′v2C(q)/integraldisplaydε 2π(τ·nA) ×G0 kσ(ε+ω)τ−G0 k′σ′(ε+ω)(τ·nA) ×G0 k′σ′(ε)τ+G0 kσ(ε), (23) whereτ±=τx±iτy,andnAdescribes the direction of A. In the following we assume linear polarization of light,A= (A,0). After calculating the trace and in- tegrating over energy εin Eq. (23) one ﬁnds a rather cumbersome expression (see Appendix) consisting of sev- eral terms, each of them corresponding to transitions be- tween certain branches of the spectrum (Fig. 3). For deﬁniteness, we locate the chemical potential in the va- lence band. Correspondingly, only the transitions from the bands (2 ↑) and (2 ↓) to the unoccupied states in the bands (1 ↑), (1↓), (2↑) and (2 ↓) are possible. We will concentrate on the optically induced spin-ﬂip transi- tions contributing to the optically-generated spin pump- ing. Hence, wedonotconsiderspin-conservedtransitions contributing to the usual Drude conductivity.41–43 A. Interband spin-ﬂip transitions Let us consider ﬁrst the spin-ﬂip transitions from the valence to conduction bands, such as k2→k′ 1.For con- venience we introduce the parameter kj= (k,σx)j,de- scribing the momentum and spin projection for an elec- tron in the subband j.The corresponding expression for the transition rate can be obtained from the equations presented in the Appendix, and considerably simpliﬁed by taking into account that: (i) in the nonsingular terms εk1−εk2can be substituted by ω,(ii) in the semiclassical limitkR≫1 the energy change due to the change in the momentumissmallcomparedto1 /τ, and(iii)thephoton energy is much larger than the characteristic low-energy scale parameters, i.e., ω≫1/τ,andω≫∆. We mention that the linear in ∆ terms have to be kept despite ∆ ≪ω since the resulting spin pumping rate, determined by the contributions of both initial spin states, is linear in ∆. The expressions for the parameters K2σ(ω), which deter- mine the spin-ﬂip rate as introduced in the Appendix, Eq.(A2), can be then simpliﬁed, and as a result one ob- tains the following formula for the spin-ﬂip rate in the relevant frequency domain: I2σ→1σ′(ω) = 4πσe2A2 c2v4 ω/summationdisplay kk′C(q)/bracketleftBig f(εk2)−f(εk′ 1)/bracketrightBig0.020.040.060.080.11010K [s-3eV-2] 5 10 15 20 photon energy [meV]00.020.040.061010 K [s-3eV-2] R=20 nmR=30 nm R=20 nm R=10 nmR=30 nmR=10 nm(a) (b)2σ 2σ FIG. 4: (Color online.) The parameter K2↓(ω) for the rate of the spin-ﬂip interband transitions in the high-frequenc y domain (a) and K2↑(b) for diﬀerent values of the range pa- rameter Rdescribing characteristic size of the ﬂuctuations in spin-orbit interaction and /angbracketleftλ2/angbracketright= 100µeV2. ×k2−kqcosϕ εk2+ω−εk1δ/parenleftBig εk2+ω−εk′ 1/parenrightBig /parenleftBig εk2+ω−εk′ 2/parenrightBig/parenleftBig εk2−εk′ 2/parenrightBig.(24) The transitions are constrained due to the δ-function in Eq. (24) corresponding to the energy conservation with the change in the momentum q(here we use ϕ= cos−1(k,q)) δ/parenleftBig εk2+ω−εk′ 1/parenrightBig =\|ω−vk−2σ∆\|δ(ϕ−ϕ0) v2kq\|sinϕ0\|,(25) whereϕ0is a solution of the equation vk+2σ∆−ω+v/radicalbig k2−2kqcosϕ+q2= 0,(26) which gives us the condition for a minimum value of mo- mentum in Eq. (26), vkmin=ω−2σ∆. The energy con- servation determines the angle ϕ0between the vectors k andqas cosϕ0=q 2k+ω−2σ∆ vq/parenleftbigg 1−ω−2σ∆ 2vk/parenrightbigg .(27) The usual condition of \|cosϕ0\|<1 leads to the following restrictions in the integration over qin Eq. (24): a) ifω/2−σ∆<vk<ω −2σ∆ thenvk−\|vk+2σ∆− ω\|<vq<vk +\|vk+2σ∆−ω\|, b) ifω/2−σ∆>vkthen−vk+\|vk+2σ∆−ω\|<vq< vk+\|vk+2σ∆−ω\|. Accounting for all these conditions, the integral over kandqin Eq. (24) can be calculated numerically. We usedthe followingparameters: µ=−3meV, correspond- ing to the Fermi momentum kF≈5×104cm−1and6 5 10 15 20 photon energy [meV]00.010.020.030.041010 K2 [s-3eV-2] R = 10 nm R = 30 nm R = 20 nm FIG. 5: (Color online.) The parameter K2(ω)≡K2↓(ω)− K2↑(ω) for the spin injection rate for diﬀerent R. Other pa- rameters are the same as Fig.4. hole concentration ≈4×108cm−2, ∆ = 1 meV, and 1/τ= 1 meV. These parameters correspond to a rather clean graphene and strong Zeeman splitting of the bands (magnetic ﬁeld of 17 T). For numerical accuracy, the cal- culations were performed with exact Eqs.(A4) and (A6). The results of numerical calculations for the interband spin-ﬂip transitions are presented in Fig. 4. Figure 5 presents the quantity describing the total spin injection rate for the interband transitions in this frequency do- main,K2(ω)≡K2↓(ω)−K2↑(ω). The positive sign of K2(ω) corresponds to the fact that the absolute value of spin density decreases due to the pumping. The peaks correspondtotheabsorptionedgeofdirectopticaltransi- tions, where the transition probability rapidly increases, and the increase at large frequencies is due to the linear energy dependence of the density of states. B. Intraband transitions in the hole subband Using the general formulas (A4) and (A6) we can write down the expression for the spin-ﬂip intraband transitions within the valence band in a low-frequency regionω≪ \|µ\|. Such transitions are associated with a relatively large change of the momentum in the ab- sorption process, and one can expect that they give a smaller transition rate. Upon taking into account that εk1−εk2=−2µ≫ω,and/vextendsingle/vextendsingle/vextendsingleεk2−εk′ 2/vextendsingle/vextendsingle/vextendsingle≪ \|µ\|, we ﬁnd I2σ→2σ′(ω) =−4πσe2A2 c2v4τ2 µ2(1+ω2τ2) ×/summationdisplay kk′C(q)/bracketleftBig f(εk2)−f(εk′ 2)/bracketrightBig ×εk2εk′ 2(k2−kqcosϕ)δ/parenleftBig εk2+ω−εk′ 2/parenrightBig (εk2+ω−εk′ 1)(εk2−εk′ 1).(28) Theδ-function can be presented in the form of Eq.(25), whereϕ0is a solution of the equation vk+2σ∆−ω−v/radicalbig k2−2kqcosϕ+q2= 0.(29)00.0050.010.0151010 K [s-3eV-2] 0 2 4 68 photon energy [meV]00.0040.0081010 K [s-3eV-2] R=40 nm R=20 nmR=30 nmR=40 nm(a) (b)30 nm 20 nm2σ 2σ FIG. 6: (Color online.) The parameter K2↓(ω) for the rate of the spin-ﬂip intraband transitions in the low-frequency do- main (a) and K2↑(ω) (b) for diﬀerent values of the range pa- rameter R. Other parameters are the same as Fig.4. 0 2 4 68 photon energy [meV]00.0040.0081010 K2 [s-3eV-2] R=20 nm R=40 nm FIG. 7: (Color online.) The parameter K2(ω)≡K2↓(ω)− K2↑(ω) for the spin injection rate for diﬀerent Rin the low- frequency domain. Other parameters are the same as Fig.4. Unmarked line corresponds to R= 30 nm. Equations (29) and (26) are similar, with the important diﬀerence in the sign in front of/radicalbig k2−2kqcosϕ+q2.In the case of interband tran- sitions this corresponds to the transitions occurring atω≈2vk,while the intraband transitions occur at lower frequencies determined by the possible momentum transfer due to the randomness of the spin-orbit cou- pling. The solution exists only for vk > ω−2σ∆ and gives Eq.(27). However, the condition \|cosϕ0\|<1 leads here to a diﬀerent restriction. Since only the condition ω−2σ∆<2vkis consistent with vk > ω−2σ∆, the momentum qshould be then in the single range of vk− \|vk+ 2σ∆−ω\|< vq < vk +\|vk+ 2σ∆−ω\|, in contrast to the case of interband transitions. Theresultsofcalculationsfortheintrabandtransitions are presented in Fig. 6. The intensity of such processes is relatively small compared to the interband transitions,7 and they can be seen only at low photon energies. Fig- ure 7 corresponds to spin injection rate by the intraband transitions, K2(ω). V. CONCLUSIONS We haveconsidered certainspin eﬀects associatedwith randomspin-orbitinteractioningraphene. First,wehave calculated the corresponding spin relaxation time, and believe that this mechanism can be dominating when the amplitude ofﬂuctuationsin spin-orbitinteractionis large enough. This may happen in the presence of surface rip- ples with short wavelengths. The other possibility can be related to the absorbed impurities at both surfaces of a free-standinggraphene. One can expect especially strong random spin-orbit coupling for heavy impurity atoms. The second eﬀect concerns the possibility of spin pumping by an external electromagnetic ﬁeld. The re- sults of our calculations show that graphene can be used as a material, in which the electron spin density can be generated by the optical pumping. The mechanism of pumping here is related to the spin-ﬂip transitions asso- ciated with the random Rashba spin-orbit interaction. Acknowledgements This work is partly supported by the FCT Grant PTDC/FIS/70843/2006 in Portugal and by the PolishMinistry of Science and Higher Education as a research project in years 2007 – 2010. This work of EYS was sup- ported by the University of Basque Country UPV/EHU grant GIU07/40, MCI of Spain grant FIS2009-12773- C02-01, and ”Grupos Consolidados UPV/EHU del Gob- ierno Vasco” grant IT-472-10. Appendix A: Formula for the absorption rate Using (22) and (23), after calculating the trace and integrating over energy ε, we ﬁnd out that spin-ﬂip processes can be characterized by the initial state as I1↓(ω),I2↓(ω),I1↑(ω),andI2↑(ω) with the corresponding transition rate Ijσ(ω) deﬁned as Ijσ(ω)≡16e2A2 c2Kjσ(ω), (A1) Kjσ(ω)≡ −Im/integraldisplay v4C(q)Jjσ(q,ω) ω+iδd2kd2k′ (2π)4,(A2) describing transitions from the band and spin states cor- responding to the subscript jσ. HereJjσ(q,ω) have the form: J1↓(q,ω) = [f(ε1k↓)−f(ε1k↓+ω)](ε2 1k↓−∆2+ε1k↓ω)(k2−kqcosϕ) (ε1k↓−ε1k′↑)(ε1k↓−ε2k↓)(ε1k↓−ε2k′↑) ×1 (ε1k↓+ω−ε1k′↑+iδ)(ε1k↓+ω−ε2k↓+iδ)(ε1k↓+ω−ε2k′↑+iδ), (A3) J2↓(q,ω) = [f(ε2k↓)−f(ε2k↓+ω)](ε2 2k↓−∆2+ε2k↓ω)(k2−kqcosϕ) (ε2k↓−ε2k′↑)(ε2k↓−ε1k↓)(ε2k↓−ε1k′↑) ×1 (ε2k↓+ω−ε2k′↑+iδ)(ε2k↓+ω−ε1k↓+iδ)(ε2k↓+ω−ε1k′↑+iδ), (A4) J1↑(q,ω) = [f(ε1k↑)−f(ε1k↑+ω)](ε2 1k↑−∆2+ε1k↑ω)(k2−kqcosϕ) (ε1k↑−ε1k′↓)(ε1k↑−ε2k↑)(ε1k↑−ε2k′↓) ×1 (ε1k↑+ω−ε1k′↓+iδ)(ε1k↑+ω−ε2k↑+iδ)(ε1k↑+ω−ε2k′↓+iδ), (A5) J2↑(q,ω) = [f(ε2k↑)−f(ε2k↑+ω)](ε2 2k↑−∆2+ε2k↑ω)(k2−kqcosϕ) (ε2k↑−ε2k′↓)(ε2k↑−ε1k↑)(ε2k↑−ε1k′↓) ×1 (ε2k↑+ω−ε2k′↓+iδ)(ε2k↑+ω−ε1k↑+iδ)(ε2k↑+ω−ε1k′↓+iδ). (A6) Although the expressions seem to be long, all Jjσ(q,ω) terms have the same simple structure. They con-tain energy-diﬀerencedenominatorscorrespondingto the8 transitions from initial jσstates to all allowed ﬁnal states, and the corresponding resonant terms. Here the single allowed spin-conserving transition arising due to theHAterm in Eq.(19) is the momentum-conserving as well, while the two transitions caused by random spin-orbit term Hk′k soare oﬀ-diagonal both in the spin and momentum subspaces, as illustrated in Fig.2. Taking the imaginary part in each of these terms we get δ−functions corresponding to the energy conservation for the transi- tions to diﬀerent bands. 1K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science306, 666 (2004). 2K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Nature (London) 438, 197 (2005). 3A. K. Geim, K. S. Novoselov, Nature Mater. 6, 183 (2007). 4A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009). 5N. M. R. Peres, J. Phys. Condens. Matter 21, 323201 (2009). 6Analysis of the bandstructure in the presence of spin-orbit coupling can be found in P. Rakyta, A. Korm´ anyos, and J. Cserti, Phys. Rev. B 82, 113405 (2010). 7The Dirac-like electron spectrum in single carbon layers was established by P. R. Wallace Phys. Rev. 71, 622 (1947) and observed experimentally in graphite (I. A. Luk’yanchuk and Y. Kopelevich, Phys. Rev. Lett. 93, 166402 (2004)). Also, the quantum Hall eﬀects in graphene and clean graphite are similar: I. A. Luk’yanchuk and Y. Kopelevich, Phys. Rev. Lett. 97, 256801 (2006). 8M. Nishioka and A. M. Goldman, Appl. Phys. Lett. 90, 252505 (2007). 9S. Cho, Y. F. Chen, and M. S. Fuhrer, Appl. Phys. Lett. 91, 123105 (2007). 10F. S. M. Guimar˜ aes, A. T. Costa, R. B. Muniz, and M. S. Ferreira, Phys. Rev. B 81, 233402 (2010). 11O. V. Yazyev, Rep. Prog. Phys. 73, 056501 (2010). 12D. Huertas-Hernando, F. Guinea, and A. Brataas, Phys. Rev. B74, 155426 (2006). 13H. Min, J. E. Hill, N. A.Sinitsyn, B. R.Sahu, L. Kleinman, and A. H. MacDonald, Phys. Rev. B 74, 165310 (2006). 14Y. Yao, F. Ye, X. L. Qi, S. C. Zhang, and Z. Fang, Phys. Rev. B75, 041401(R) (2007). 15M. Gmitra, S. Konschuh, C. Ertler, C. Ambrosch-Draxl, and J. Fabian, Phys. Rev. B 80, 235431 (2009). 16Spin relaxation times in carbon nanotubes, which can be considered as the rolled graphene sheets are indeed long enough to allow information transfer (L. E. Hueso, J. M. Pruneda, V. Ferrari, G. Burnell, J. P. Vald´ es-Herrera, B. D. Simons, P. B. Littlewood, E. Artacho, A. Fert, and N. D. Mathur, Nature 445, 410 (2007)) due to the very weak spin-orbit coupling. However, theoretical analysis shows that the spin resonance caused by external electric ﬁeld can be observable in these systems: A. De Martino, R. Eg- ger, K. Hallberg, and C. A. Balseiro, Phys. Rev. Lett. 88, 206402 (2002). 17N. Tombros, C. Jozsa, M. Popinciuc, H. T. Jonkman, and B. J. van Wees, Nature (London) 448, 571 (2007). 18N. Tombros, S. Tanabe, A. Veligura, C. Jozsa, M. Popin- ciuc, H. T. Jonkman, and B. J. van Wees, Phys. Rev. Lett. 101, 046601 (2008). 19T. Maassen, F. K. Dejene, M. H. D. Guimar˜ aes, C. J´ ozsa,and B. J. van Wees, preprint arXiv:1012.0526. 20T.-Y. Yang, J. Balakrishnan, F. Volmer, A. Avsar, M. Jaiswal, J. Samm, S.R. Ali, A.Pachoud, M. Zeng, M. Popinciuc, G. G¨ untherodt, B. Beschoten, B. ¨Ozyilmaz, preprint arXiv:1012.1156. 21Wei Han and Roland K. Kawakami, preprint arXiv:1012.3435. 22D. Huertas-Hernando, F. Guinea, and A. Brataas, Eur. Phys. J. Special Topics 148, 177 (2007); D. Huertas- Hernando, F. Guinea, and A. Brataas, Phys. Rev. Lett. 103, 146801 (2009). 23C. Ertler, S. Konschuh, M. Gmitra, and J. Fabian, Phys. Rev. B80, 041405(R) (2009). 24Y. Zhou and M. W. Wu, Phys. Rev. B 82, 085304 (2010). 25E. Ya. Sherman, Phys. Rev. B 67, 161303(R) (2003). 26M. M. Glazov and E. Ya. Sherman, Phys. Rev. B 71, 241312(R) (2005). 27J. C. Meyer, A. K. Geim, M. I. Katsnelson, K. S. Novoselov, T. J. Booth, and S. Roth, Nature (London) 446, 60 (2007). 28M. Ishigami, J. H. Chen, W. G. Cullen, M. S. Fuhrer, and E. D. Williams, Nano Lett. 7, 6 (2007). 29T. O. Wehling, A. V. Balatsky, A. M. Tsvelik, M. I. Kat- snelson, and A. I. Lichtenstein, EPL 84, 17003 (2008). 30E. I. Rashba and V. I. Sheka, Fizika Tverdogo Tela 3, 2369 (1961 [Sov. Phys. Solid State 3, 1718 (1962)]. 31E. I. Rashba and Al. L. Efros, Phys. Rev. Lett. 91, 126405 (2003); Appl. Phys. Lett. 83, 5295 (2003). 32Optical orientation , ed. by F. Mayer and B. Zakharchenya (North-Holland, Amsterdam, 1984). 33S. A. Tarasenko, Phys. Rev. B 73, 115317 (2006). 34O. E. Raichev, Phys. Rev. B 75, 205340 (2007). 35A. A. Perov, L. V. Solnyshkova, and D. V. Khomitsky, Phys. Rev. B 82, 165328 (2010); D. V. Khomitsky, Phys. Rev. B79, 205401 (2009). 36V.K.Dugaev, E.Ya.Sherman, V.I.Ivanov,andJ. Barna´ s, Phys. Rev. B 80, 081301(R) (2009). 37C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005). 38The inﬂuence of the substrate on the spin-orbit coupling is weak for graphene on the top of SiO 2and SiC. An exception is given by graphene on the Ni/Au substrate, where, duetothelarge atomic numberofAu, inducedspin- orbit coupling is very strong: A. Varykhalov, J. Sanchez- Barriga, A. M. Shikin, C. Biswas, E. Vescovo, A. Rybkin, D.Marchenko, andO.Rader, Phys.Rev.Lett. 101, 157601 (2008) However, this extreme case is not of our interest here. 39The detailed analysis done by A. H. Castro Neto and F. Guinea, Phys. Rev. Lett. 103, 026804 (2009) shows that taking intoaccount the bond hybridization by the adatoms can lead to the spin relaxation rate of the order of the observed experimentally. 40S. A. Tarasenko, JETP Letters 84, (2006).9 41A. B. Kuzmenko, I. Crassee, D. van der Marel, P. Blake, and K. S. Novoselov, Phys. Rev. B 80, 165406 (2009). 42Eﬀects of spin-orbit coupling for the optical properties in disorder-free graphene were considered by P. Ingenhoven, J. Z. Bern´ ad, U. Z¨ ulicke, and R. Egger, Phys. Rev. B 81,035421 (2010) 43Magnetic-ﬁeld inducedelectron spin resonance in graphene at frequency 2∆ was analyzed in B. D´ ora, F. Mur´ anyi and F. Simon, EPL 9217002 (2010).
1110.6661v2.Quasiparticle_velocities_in_2D_electron_hole_liquids_with_spin_orbit_coupling.pdf	arXiv:1110.6661v2 [cond-mat.mes-hall] 2 Mar 2012Quasiparticle velocities in 2D electron/hole liquids with spin-orbit coupling D. Aasen, Stefano Chesi, and W. A. Coish Department of Physics, McGill University, Montr´ eal, Qu´ e bec H3A 2T8, Canada (Dated: October 30, 2018) We study the inﬂuence of spin-orbit interactions on quasipa rticle dispersions in two-dimensional electron and heavy-hole liquids in III-V semiconductors. T o obtain closed-form analytical results, we restrict ourselves to spin-orbit interactions with isot ropic spectrum and work within the screened Hartree-Fock approximation, valid in the high-density lim it. For electrons having a linear-in- momentum Rashba (or, equivalently, Dresselhaus) spin-orb it interaction, we show that the screened Hartree-Fock approximation recovers known results based o n the random-phase approximation and we extend those results to higher order in the spin-orbit cou pling. While the well-studied case of electrons leads only to a weak modiﬁcation of quasiparticle properties in the presence of the linear- in-momentum spin-orbit interaction, we ﬁnd two important d istinctions for hole systems (with a leading nonlinear-in-momentum spin-orbit interaction). First, the group velocities associated with the two hole-spin branches acquire a signiﬁcant diﬀerence i n the presence of spin-orbit interactions, allowing for the creation of spin-polarized wavepackets in zero magnetic ﬁeld. Second, we ﬁnd that the interplay of Coulomb and spin-orbit interactions is sig niﬁcantly more important for holes than for electrons and can be probed through the quasiparticle gr oup velocities. These eﬀects should be directly observable in magnetotransport, Raman scatter ing, and femtosecond-resolved Faraday rotation measurements. Our results are in agreement with a g eneral argument on the velocities, which we formulate for an arbitrary choice of the spin-orbit coupling. PACS numbers: 71.10.-w, 71.70.Ej, 71.45.Gm, 73.61.Ey I. INTRODUCTION Semiconductor heterostructures oﬀer the possibility of forming two-dimensional liquids with tunable sheet den- sityns. In an idealized model where the carriers have parabolic dispersion with band mass mand interact with Coulomb forces,1the only relevant quantity is the di- mensionless Wigner-Seitz radius rs= 1//radicalbig πa2 Bns, where aB=/planckover2pi12ǫr/me2is the eﬀective Bohr radius ( ǫris the di- electric constant). Since rsserves as the interaction pa- rameter, changing nsallows for a systematic study of the eﬀects of the Coulomb interaction. In particular, proper- ties of quasiparticle excitations such as their dispersion and lifetime are signiﬁcantly modiﬁed due to electron- electron interactions.1 Great attention has been paid in recent years to band- structure eﬀects involving the spin degree of freedom.2 The strength and form of spin-orbit interaction (SOI) can be controlled in two-dimensional liquids through the choice of materials, the type of carriers (electrons/holes), and details of the conﬁnement potential. For example, it is possible to change the coupling constant with external gates.3–5In addition to detailed studies of single-particle properties, the problem of understanding the eﬀects of SOI in the presence of Coulomb interactions is a topic of ongoing investigations. Quasiparticle properties in the presence of SOI6–10 havebeenexaminedprimarilyaccountingforRashba11,12 and/orDresselhaus SOI,13,14which are dominant in elec- tronic systems. The SOI results in two distinct spin sub- bands, with two associated Fermi surfaces. The eﬀects on quasiparticles are usually very small; at each of the two Fermi surfaces the quasiparticle dispersion6,7,10and lifetime7–9are almost unaﬀected by SOI, except in thecase of very large SOI coupling.9,15In fact, it was found with Rashba SOI that the corrections to these quanti- ties linear in the SOI coupling are absent.7Although ex- plicit calculations are performed within perturbative ap- proximation schemes, notably the random-phase approx- imation (RPA),7–9the SOI leading-order cancellation is valid non-perturbatively (to all orders in rs).16Similar arguments hold for other physical quantities.16,17For ex- ample, values of the ground-state energy obtained with Monte Carlo simulations18for up to rs= 20 could be reproduced with excellent accuracy by simply neglecting SOIcorrectionstotheexchange-correlationenergy.19No- ticeable exceptions exhibiting larger SOI eﬀects are spin- textured broken symmetry phases,20,21non-analytic cor- rections to the spin susceptibility,22,23and the plasmon dispersion.10All these examples involve the presence of spin polarization (either directly20–22or indirectly10), in which case the arguments of Ref. 16 do not apply. Another interesting situation occurs when the SOI has a nonlinear dependence on momentum, thus cannot be written as a spin-dependent gauge potential.17,24,25 Then, the approximate cancellations mentioned above are not expected. Winkler has shown that the domi- nant SOI induced by heterostructure asymmetry is cubic in momentum for heavy holes in III-V semiconductors.26 This theoretical analysis was later shown to be in good agreement with magnetoresistance experiments.27Re- cently, the relevance of this cubic-in-momentum model was supported by the anomalous sign and magnetic- ﬁeld dependence of spin polarization in quantum point contacts.28,29SOIquadraticinmomentumcanalsobein- duced for heavy holes by an in-plane magnetic ﬁeld.30,31 That these band-structure eﬀects can substantially modify standard many-body results is conﬁrmed by re-2 cent Shubnikov-de Haas oscillation measurements in low- density hole systems.33,34For example, a surprisingly small Coulomb enhancement of the g-factor has been reported33and puzzling results have also been obtained for the eﬀective masses m±of the two hole-spin sub- bands (σ=±).34A mechanism for the small g-factor enhancement was suggested in Ref. 31: if SOI strongly distorts the groundstate spin structure, the exchangeen- ergy becomes ineﬀective in promoting full polarization of the hole system. In this paper, we focus on the eﬀective masses m±. We note that m±are directly related to the quasiparti- cle group velocities v±at the Fermi surfaces. With this in mind, we ﬁnd it more transparentto discussthe eﬀects of SOI and electron-electron interactions on m±in terms ofwavepacketmotion, asillustratedin Fig.1. Ifanunpo- larized wavepacket is injected at the Fermi energy (with average momentum along a given direction), the sub- sequent motion is very diﬀerent depending on whether the SOI is linear ( n= 1) or non-linear ( n= 2,3). In the former case ( n= 1), corresponding to electrons, we havev+≃v−and the motion is essentially equivalent to the case without SOI. In contrast, for holes with strong SOI,v+∝negationslash=v−and the two spin components become spa- tially separated. The separation between the two spin components can become quite sizable, and it should be possible to observe such an eﬀect with, e.g., Faraday- rotation imaging techniques.35,36Electron-electron inter- actions, in addition to modifying the average velocity v= (v++v−)/2 (an eﬀect which is well-known without SOI1,37,38), are also reﬂected on the velocity diﬀerence (v+−v−) between the two spin branches. With these motivations in mind, we pursue a study of the quasiparticle group velocity in the presence of linear Rashbaand non-linearSOI.An accurateanalyticaltreat- ment of the electron-electron interactions can be carried out at high density, and we restrict ourselves to this in- teresting limit. The regime of strong electron-electron interactions is much more diﬃcult to treat (see Ref. 38 for a Monte Carlo study without SOI) but it represents a relevant topic for future investigations ( rs∼6−12 in Refs. 33 and 34). This paper is organized as follows: In Sec. II we in- troduce a model Hamiltonian including a generalized SOI31and we brieﬂy review its non-interacting proper- ties, demonstrating that injected holes will separate into spin-polarized wavepackets. The Coulomb interaction is treated in Sec. III by extending the classic treatment of Ref. 39. We describe several results of this screened Hartree-Fock approach in detail, focusing on the quasi- particle group velocities and the interplay of spin-orbit andCoulombinteractioneﬀects. Adiscussionofthesere- sults is givenin Sec. IV. In Appendix A, wegivea general argument showing that corrections to the velocity from any linear-in-momentum SOI can always be neglected to lowest order. Finally, a number of technical details are provided in Appendices B and C./LParen1a/RParen1electrons/LParen1n/Equal1/RParen1 vΤ vFΤ /CapDelΤa∆ t/Equal0 t/EqualΤ/Minus/Plus/LParen1b/RParen1holes/LParen1n/Equal2,3/RParen1 FIG. 1. (Color online) Motion of a wavepacket for a ﬁxed timeτ. In (a) the SOI is not seen, because the diﬀerence in the velocities v±of the two spin components is too small. Panel (a) applies without SOI or to electrons with Rashba SOI (n= 1). In (b) we show the eﬀect of the non-linear SOI present in hole systems ( n= 2,3). Since the two spin branches have signiﬁcantly diﬀerent velocities v±, there is an appreciable separation ∆ = \|v+−v−\|τ. We also illustrate the propagation of non-interacting wavepackets (dashed) f or the same time τ. The eﬀect of interactions (at high density) is to enhance both the average velocity ( v > v F) and the separation of spin-components: ∆ > δ=\|v0 +−v0 −\|τ. II. NONINTERACTING PROBLEM In the high-density limit, rsis small and the system is well-describedbyanon-interactingsingle-particleHamil- tonian. We consider here the following model,31includ- ing a generalized SOI with a linear-, quadratic-, and cubic-in-momentum dependence for n= 1,2,3: H0=p2 2m+iγpn −σ+−pn +σ− 2, (1) wherepis the momentum operator, mthe band mass, p±=px±ipy,σ±=σx±iσy, andγthe generalized spin-orbit coupling, with σthe vector of Pauli matrices. The physical justiﬁcation of this model has been given in Ref. 31: the n= 1 Hamiltonian contains the Rashba SOI present in electronic systems,11,12whilen= 3 includes the analogous term generated by an asymmetric conﬁne- ment potential for holes.26,27,29Finally, the n= 2 case is alsorelevantforholes, inthepresenceofanin-planemag- netic ﬁeld.30,31While SOI terms of diﬀerent form gener- ally coexist (see Appendix A), we assume here that one nvalue is dominant. This greatly simpliﬁes the problem by preserving the isotropy of the electron liquid in the x- y plane and is a good approximation for several relevant situations. For example, it was found for holes26,27,29 that the n= 3 term can be much larger than corrections to the SOI due to bulk inversion asymmetry.2,32Diago-3 nalizingH0yields the energy spectrum E0 σ(k) =/planckover2pi12k2 2m+σγ/planckover2pi1nkn, (2) withσ=±labeling the two chiral spin branches. The corresponding eigenfunctions are31 ϕkσ(r) =eik·r √ 2L2/parenleftbigg1 iσeinθk/parenrightbigg , (3) wherekis a wavevector in the x-y plane, θkis the angle kmakes with the x-axis, and Lis the linear size of the system. It is useful at this point to introduce a dimensionless quantity gcharacterizing the strength of the spin-orbit coupling:31 g=γ/planckover2pi1nkn F EF, (4) whereEF=/planckover2pi12k2 F/2mis the Fermi energy without SOI, written in terms of the Fermi wavevector kF=√2πns=√ 2/(aBrs). While γhas diﬀerent physical dimensions for each value, n= 1,2,3, the dimensionless coupling galways gives the ratio of the spin-orbit energy to the kineticenergy. Thecoupling gthusplaysaroleanalogous to that of rsfor the Coulomb interaction. Taking γto be independent of the density (this is not always the case2), then from Eq. (4) we have g∝kn−2 F∝r2−n s(since EF∝k2 F). This suggests that in the high-density limit (rs→0), the eﬀects of SOI are suppressed for electrons (n= 1), but remain constant ( n= 2) or are enhanced (n= 3) for holes. This simple estimate already indicates a qualitative diﬀerence for holes relative to electrons. We will see that this diﬀerence is indeed signiﬁcant in the following sections. InthepresenceofSOIthetwospinbands( σ=±)have diﬀerent densities n±, giving the total 2D sheet density ns=n++n−. Keeping ns(thuskF) ﬁxed gives a con- straintonthe Fermiwavevectors k±fortheσ=±bands, k2 ++k2 −= 2k2 F. We can then characterize the solution to this equation with a single parameter χ: k±=kF/radicalbig 1∓χ. (5) The parameter χ= (n−−n+)/nsgives the chirality and is determined by both the SOI and electron-electron interaction.40We will assume for deﬁniteness that γ≥0 such that χ≥0 andk+≤k−. For a ﬁxed generalized spin-orbit coupling γ, the non-interacting value of χcan be determined by setting the Fermi energies of the two bands equal, i.e., E0 +(k+) =E0 −(k−). This equation im- mediately gives a relationship between gandχ, g=2χ (1+χ)n/2+(1−χ)n/2. (6) We denote the solution of Eq. (6) by χ0(g), which gives the non-interacting Fermi wavevectors k0 ±=kF√1∓χ0.Explicit expressions for χ0(g) are given in Ref. 31. We only cite here the small- gbehavior, which is easily found from Eq. (6): χ0(g)≃g. (7) We are mainly interested here in the properties of the quasiparticles and, in particular, their group velocities v±. The fact that the dispersion relation (2) is a function only of the magnitude kis a consequence of the model being isotropic with respect to k, which allows us to dis- cuss the magnitude of the group velocity on the Fermi surfaces, v0 ±=1 /planckover2pi1∂E0 ±(k) ∂k/vextendsingle/vextendsingle/vextendsingle k=k0 ±=/planckover2pi1k0 ± m±nγ(/planckover2pi1k0 ±)n−1.(8) Theaboveexpressioncanbeevaluatedexplicitlyinterms of the non-interacting chirality χ0(g): v0 ± vF=/radicalbig 1∓χ0(g)±n 2g[1∓χ0(g)](n−1)/2 ≃1±g 2(n−1), (9) wherevF=/planckover2pi1kF/mistheFermivelocityintheabsenceof SOIandinthesecondlinewehaveexpandedtheresultto lowest order in g, by making use of Eq. (7). Equation (9) shows that there is no relative diﬀerence in group veloc- ity for the σ=±bands when n= 1. In contrast, when n= 2,3 a sizable correction linear in gis present. This diﬀerence reﬂects itself on the evolution of an initially unpolarized wavepacket injected at the Fermi surface, schematically illustrated in Fig. 1. While for electrons the wavepacket remains unpolarized, for holes the two spin components spatially separate with time. Electron- electron interactions modify the non-interacting veloci- tiesv0 ±, but the qualitative diﬀerence introduced by SOI between electrons ( n= 1) and holes ( n= 2,3) remains essentially unchanged. A similar behavior holds for other properties of the electron liquid as well:16vanishing cor- rections to lowest order in gwere found for the quasi- particle lifetime,7,9the occupation,31and the exchange- correlation energy,19if only Rashba SOI ( n= 1) is in- cluded. Finally, anotherrelevantquasiparticleobservableisthe eﬀective mass. This has recently been measured in hole systems through experiments on quantum oscillations.34 This physical quantity is simply given by m±=/planckover2pi1k±/v± and is thus essentially equivalent to v±. III. SCREENED HARTREE-FOCK APPROXIMATION Realistically, charged particles interact through the Coulomb potential so it is interesting to understand how the presence of SOI modiﬁes the behavior of the quasi- particles. The fully interacting Hamiltonian is given by: H=/summationdisplay iH(i) 0+1 2/summationdisplay i/negationslash=je2 ǫr\|ri−rj\|,(10)4 whereH(i) 0(for electron i) is as in Eq. (1) and the pres- ence in (10) of a uniform neutralizing background is un- derstood. Although many sophisticated techniques exist to approach this problem,1,38the simplest approxima- tion to the quasiparticle self-energy is obtained by only including the exchange contribution: Eσ(k) =E0 σ(k)+Σx σ(k), (11) where Σx σ(k) =−/summationdisplay k′σ′(1+σσ′cosnθ′) 2L2nk′σ′V(\|k−k′\|).(12) Here,nk′±= Θ(k±−k′) is the occupation at T= 0 for theσ=±band, respectively, with Θ( x) the Heavi- side step function. The ﬁrst factor in the summation of Eq. (12), involving the angle θ′between k′andk, arises from the scalar product of the non-interacting spinors [Eq. (3)], and takes into account the speciﬁc nature of the spin-orbit interaction ( n= 1,2,3). To lowest order, V(q) is the Fourier transform of the bare Coulomb potential, 2 πe2/(ǫrq). As is well known,1 this form of the Coulomb interaction leads to an unphys- ical divergence in the quasiparticle velocity. By consider- ing an inﬁnite resummation in perturbation theory, the screening of the Coulomb interaction removes the diver- gence, e.g., in the RPA approximation. Finally, by ap- proximating the dielectric function in the eﬀective inter- action by its zero-frequency long-wavelength limit, the RPA self-energy gives Eq. (12) with V(q) =2πe2/ǫr q+√ 2rskF. (13) This screened Hartree-Fock approximation with SOI,6,10 notwithstanding its simplicity, becomes accurate in the high-density limit. A. Renormalized occupation By including the SOI, we can verify that Eq. (12) gives the correct high-density behavior for the Fermi wavevec- torsk±. These are modiﬁed by electron-electron inter- actions from their non-interacting values k0 ±.31,41From Eq. (12), k±can be obtained by equating the chemical potentials in the two spin branches: E+(k+) =E−(k−). (14) After taking the continuum limit, this equation is rewrit- ten in dimensionless form as follows: (yn ++yn −)g=2χ+rs√ 2/summationdisplay σσ′/integraldisplay2π 0dθ 2π/integraldisplayyσ′ 0dy ×y(σ+σ′cosnθ)/radicalbig y2+y2σ−2yyσcosθ+√ 2rs,(15)where we have rescaled the wavevectors k=kFyand deﬁnedy±=√1∓χ. The integral on the right-hand side is the correction from the exchange term and we have veriﬁed that Eq. (15) gives Eq. (6) for rs= 0. We note that, for a given value of the SOI, χ(rs,g) enters in a rather complicated way in Eq. (15), being involved in the integration limits of the exchange term as well as the integrand. In practice, instead of solving Eq. (15) for χ, it is convenient to evaluate gfor a given value of χand numerically invert the function g(rs,χ). The values of k±=kFy±were obtained in Ref. 31 through a diﬀerent procedure, i.e., by minimizing the to- tal energy (including the exchange contribution) of non- interacting states.42Although both methods are unreli- able atrs>1, they both become accurate in the high- density limit, rs<1. In fact, the only diﬀerence in the two approaches is due to the presence of the Thomas- FermiscreeningwavevectorinEq.(15)and, byneglecting√ 2rsin the denominator of the second line, the equation from the variational treatment is recovered. In particu- lar, expanding Eq. (15) at small rsandggives the same result found in Ref. 31: χ(rs,g)≃g 1−√ 2rs πn/summationdisplay j=01 2j−1 .(16) Details of the derivation of Eq. (16) are given in Ap- pendix B. Two salient features of Eq. (16) are:31(i) For n= 1 there is no correction to the noninteracting result χ(rs,g)≃g. On the other hand, the linear dependence ongisactually modiﬁed byelectron-electroninteractions atn= 2,3. (ii) The eﬀect of the electron-electron inter- actions is a reduction of χ(rs,g) from the non-interacting value. This result could be rather surprising, having in mind the well-known enhancement of spin polarization caused by the exchange energy1(when the spin-splitting is generated by a magnetic ﬁeld). However, χdoes not correspondhere to a real spin-polarization, which is zero. Instead, χis simply related to the population diﬀerence of the two chiral spin subbands. B. Quasiparticle velocity InthescreenedHartree-Fockapproximation,thegroup velocities at the Fermi surfaces are given by: v±=1 /planckover2pi1∂Ek± ∂k/vextendsingle/vextendsingle/vextendsingle k=k± =/planckover2pi1k± m±nγ(/planckover2pi1k±)n−1 −/summationdisplay k′σ′(1±σ′cosnθ′) 2L2nk′σ′/bracketleftbigg∂ ∂kV(\|k−k′\|)/bracketrightbigg k=k±.(17) In general, we can discuss all corrections to v±by intro- ducing the following notation: v± vF= 1+δv(rs)+δv0 ±(g)+δv±(rs,g),(18)5 whereδv(rs) is the (spin-independent) correction due to electron-electron interactions at g= 0, which has been the subject of many theoretical and experimental studies (see, e.g., Ref. 1, 34, 37–39, and references therein). In the approximation (17), it is given by39 δv(rs) =−√ 2rs π+r2 s 2+rs(1−r2 s)√ 2πcosh−1(√ 2/rs)/radicalbig 1−r2s/2.(19) As is known,1this approximation gives the correct lead- ing behavior at small rs:δv≃ −(rslnrs)/(√ 2π). The second nontrivial term in Eq. (18) is the non-interacting correction purely due to SOI δv0 ±(g) =v0 ±(g) vF−1, (20) which only depends on g[see Eq. (9)]. Finally, δv±(rs,g) collects all remaining corrections. A pictorial representation of the physical meaning of the three terms is shown in Fig. 1 for an unpolarized wavepacket injected at the Fermi energy. At high den- sity, as seen in Eq. (19) and illustrated in Fig. 1(a), the group velocity is larger than without electron-electron interactions. In Fig. 1(b) we depict the generic situ- ation with SOI. In the presence of SOI, the two spin branches have diﬀerent group velocities. An initially un- polarized wavepacket then splits into its two spin com- ponents. Both the SOI and electron-electron interactions inﬂuence the relative velocity ( v+−v−). The separation after a time τisδ=\|v0 +−v0 −\|τfor the non-interacting case and is modiﬁed by δv±with electron-electron inter- actions: ∆ = \|v+−v−\|τ[see Fig. 1(b)]. The presence of ‘interference’ terms, δv±(rs,g), be- comes clear from Eq. (17). These terms are due to the interplay of many-body interactions with SOI. A ﬁrst contribution to δv±(rs,g), which we refer to as the ‘self- energy contribution’, comes directly from the exchange integral [third line of Eq. (17)]: due to the presence of two distinct Fermi wavevectors k±, the result obviously containscorrectionstoEq.(19)whichdependon g(inad- dition to rs). A second contribution to δv±(rs,g) comes indirectly from the non-interacting part and we refer to it as the ‘repopulation contribution’. Since the Fermi wavevectors k±are modiﬁed from the non-interacting valuesk0 ±, the second line of Eq. (17) gives a result distinct from v0 ±. The sign of this repopulation con- tribution is easily found by noting that, as discussed in Sec.IIIA, theexchangeenergyreducesthe valueof χ(for n= 2,3). This corresponds to an increase (decrease) of k+(k−), i.e., a positive (negative) correction to v+(v−). The eﬀect would be to enhance the diﬀerence in veloc- ity between the two branches, as illustrated in Fig. 1(b) (∆> δ). However, to establish the ultimate form and sign ofδv±(rs,g) requires a detailed calculation of both self-energy and repopulation contributions, which is pre- sented below for some interesting cases. The total spin- dependent part of the velocity, δv0 ±+δv±, can then be compared to the simple non-interacting eﬀect, δv0 ±.C.n= 1: higher-order corrections in g We begin the analysis of Eq. (17) by rewriting it in a more explicit way. To this end, we use ∂ ∂kV(\|k−k′\|) =−ˆk·∂ ∂k′V(\|k−k′\|),(21) whereˆk=k/k. This allows us to integrate Eq. (17) by parts, which leads to: v± vF=y±±g 2nyn−1 ± +rs/summationdisplay σ/integraldisplay2π 0dθ′ 8π√ 2cosθ′(1±σcosnθ′)yσ/radicalBig y2 ±+y2σ−2y±yσcosθ′+rs√ 2 ±rs/integraldisplay2π 0dθ′ 8π/integraldisplayy− y+√ 2nsinnθ′sinθ′dy′ /radicalBig y2 ±+y′2−2y′y±cosθ′+rs√ 2 =L1+L2+L3. (22) Notice that, in the integration by parts of Eq. (17) in the continuum limit, two types of terms enter: those corre- sponding to the second line of Eq. (22) ( L2) involve the derivative of nk′±. This results in a delta function in the dk′integral, which can then be easily evaluated. Thus, only the integral in dθ′is left. The second type of term involves the derivative of cos nθ′: ˆk·∂ ∂k′cosnθ′=n k′sinnθ′sinθ′, (23) and indeed this angular factor appears in the third line of Eq. (22) ( L3). Eq. (22) can alwaysbe evaluated numerically, after ob- tainingthevaluesof χ(thusy±=√1∓χ)fromEq.(15). By specializing to the small g,rslimit for n= 1 SOI, it is known that the linear-in- gcorrection to v±vanishes.7,16 Thus, an expansion to second order in ghas to be per- formed, which has been done in Ref. 7 in the context of the RPA treatment of the quasiparticle properties. To verify the validity of the simpler screened Hartree-Fock procedure,itisinterestingtoperformthesameexpansion for our Eq. (22). In fact, the two calculations bear some similarities sinceL2is the same as the boundary term B(u→0+) boundaryof the RPA treatment, see Eq. (69) of Ref. 7. Thus, we can borrow the expansion in small g,rs: L2≃ −rs√ 2π/parenleftbigg lnrs 2√ 2+2±2 3g−g2 8lng/parenrightbigg ; (n= 1), (24) where terms of order O(r2 s,rsg2) have been omitted. To the same order of approximation, we have for L1: L1≃1−g2 8; (n= 1), (25) which can also be obtained by expanding Eq. (9) (with n= 1). Since χonly receives O(rsg3lng) corrections6 fromelectron-electroninteractions,19itissuﬃcienttouse the non-interacting value, χ0(g), to this order of approx- imation. Thus, the repopulation induced by electron- electron interactions has a negligible eﬀect on v±in this case. The situation will be diﬀerent for n= 2,3. Ex- panding L3for small rs,gyields L3≃ ±√ 2rs 3πg; (n= 1), (26) whichcancelsthelinear-in- gtermofEq.(24),asexpected forn= 1. We note that in the screened Hartree-Fock approxi- mation we are able to obtain an analytic result for the leadingterm of L3, and somedetails ofthe derivationcan be found in Appendix C. In contrast, in the RPA treat- ment of Ref. 7 it was not possible to expand the more involved corresponding term, Bint, in a fully analytic fashion. The cancellation of the linear-in- gcontribution was indicated by a general argument7,16and conﬁrmed through numerical study. Additionally, the absence of higher-order terms which modify Eq. (24) was inferred numerically in Ref. 7. Although the ﬁnal results of both approaches (RPA and screened Hartree-Fock) agree, the situation is clearly more satisfactory within the frame- work of Eq. (22), since cancellation of the linear term in Eq. (24) can be checked exactly and the expansion can be carried out to higher order systematically. The ﬁnal result, computed to higher order in g, reads (for n= 1): v± vF≃1+δv(rs)+δv0 ±(g) +√ 2rs 16π/bracketleftbigg g2/parenleftbigg lng 8+3 2/parenrightbigg ±g3 6/parenleftbigg lng7 85+319 20/parenrightbigg/bracketrightbigg .(27) The second line of Eq. (27) represents the expansion of δv±(rs,g) including all terms up to O(rsg3). Forn= 1, thenon-interactingresultgivesthesamevelocityforboth spin branches [ δv0 ±(g) is actually independent of ±for this particular model]. The O(rsg3) term in Eq. (27) is therefore quite interesting. It shows that a small diﬀer- enceinvelocityexists. Thisisagenuineeﬀectofelectron- electron interactions. The accuracy of Eq. (27) can be seen in Fig. 2: it becomes accurate at very small values of rs(as shown in the inset) while at larger more realistic values of rs it still gives the correct magnitude of the eﬀect. The numerical example of Fig. 2 also shows that δv±(rs,g) is very small. In fact, it is generally much smaller than the non-interacting correction: since δv0 ±(g)≃ −g2/8 [see Eq. (25)] the rsg2lngterm becomes larger only if rslng≫1. This condition is only satisﬁed at extremely small values of g(ifrsis small as well) at which SOI eﬀectsarehardlyofanyrelevance. Thus, Eq.(27)implies that the eﬀects of SOI and electron-electron interactions are essentially decoupled for n= 1. This picture changes substantially for hole systems ( n= 2,3), as we show in the next section./Minus /Plus /Plus0/Minus5 10/Minus5 0.00 0.02 0.04 0.06 0.08 0.10/Minus0.006/Minus0.005/Minus0.004/Minus0.003/Minus0.002/Minus0.0010.000 g102∆v/PlusMinus 0 0.05 0.1 FIG. 2. (Color online) Thick solid lines: the corrections δv±(rs,g) evaluated by numerical integration of Eq. (22). Dashed lines: approximation to δv±(rs,g), given by the sec- ond line of of Eq. (27). In the main plot, we take rs= 0.1, for which Eq. (27) is not very accurate. Very good agreement is obtained at small values of rs(inset, with rs= 0.001). The thinner solid line in the main plot is the total correction du e to spin-orbit coupling: δv0 ±(g) +δv±(rs,g). It is dominated by the non-interacting eﬀect and the spin-splitting is not v is- ible. The value of δv0 ±(g) atg= 0.1, outside the plot range, is∼1% (all corrections are in units of vF). D. Corrections to the velocity for n= 2,3 We now apply the discussion of the previous section to the SOI more appropriate for holes and point out some important diﬀerences. Expansions for small rsandgare given in Appendix C. From Eqs. (C16) and (C17) and usingχ=g+O(rsg,g3) [Eq. (16)], we ﬁnd the following expressions for the self-energy contribution. For n= 2: L2+L3≃δv(rs)+√ 2rs 4π/bracketleftBig ±g 3+g2/parenleftBig lng 8+2/parenrightBig/bracketrightBig ,(28) and forn= 3: L2+L3≃δv(rs)+√ 2rs 4π/bracketleftbigg ±8 15g+g2 4/parenleftbigg 9lng 8+613 30/parenrightbigg/bracketrightbigg . (29) At variance with the case of n= 1, the linear-in- gterm does not vanish here. Thus, we ﬁnd an appreciable cor- rection to the velocity. This correction has opposite sign in the two branches and is positive for the + (higher- energy) branch. A second contribution to the g-linear correction comes fromL1. Expanding L1in terms of χand using Eq. (16) gives, for n= 2, L1≃1+δv0 ±(g)+√ 2rs 4π/parenleftbigg ±2 3g+g2/parenrightbigg ; (n= 2),(30) and forn= 3: L1≃1+δv0 ±(g)+√ 2rs 4π/parenleftbigg ±16 15g+32 15g2/parenrightbigg ; (n= 3). (31)7 /Plus /Minus/Minus /Plus 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35/Minus0.20.00.20.40.60.81.0 g102∆v/PlusMinus FIG. 3. (Color online) Plot of δv±(rs,g) forn= 3 (solid curves) and n= 1 (dashed curves) as a function of the SOI strength g. We have taken rs= 0.3 in both cases. ±indicate the spin branch of each curve. Thus, the repopulation contribution is present in this case and has the sign discussed at the end of Sec. IIIB (it is positive for the + branch). As it turns out, the self-energy, repopulation, and non- interacting contributions to the velocity have the same sign. The three contributions therefore have a cooper- ative eﬀect in enhancing the diﬀerence in velocity be- tween the two spin branches. Of course, based on the high-density theory presented here, we cannot tell if this conclusion holds at all densities. We also note that the g- linear term of the self-energy correction [Eq. (28) or (29)] is always half of the corresponding repopulation correc- tion [Eq. (30) or (31)]. Again, we have not investigated if this curious relation only occurs within this approxi- mation scheme or if it is more general. Finally, we give the complete result for n= 2 v± vF≃1+δv(rs)+δv0 ±(g) +√ 2rs 4π/bracketleftBig ±g+g2/parenleftBig lng 8+3/parenrightBig/bracketrightBig ; (n= 2),(32) and forn= 3 v± vF≃1+δv(rs)+δv0 ±(g) +√ 2rs 4π/bracketleftbigg ±8 5g+g2/parenleftbigg 9lng 8+869 120/parenrightbigg/bracketrightbigg ; (n= 3),(33) and show two numerical examples in Figs. 3 and 4. Fig. 3 is a comparison of δv±(rs,g) forn= 1 and n= 3: it is clear that the dependence on gis very weak for the electron case ( n= 1) and the magnitude of δv±(rs,g) is much larger for holes ( n= 3) with SOI of comparable strength. Fig. 4 shows the separation ∆ between the two spin components of an initially unpolarized wave packet for a ﬁxed travel distance of 1 µm: ∆ = 2\|v+−v−\| v++v−×(1µm). (34)0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.350.00.20.40.60.8 g/CapDelΤa/LParen1Μm/RParen1/CapDelΤa /CapDelΤa0 1Μm FIG. 4. (Color online) Thick solid line: separation ∆ be- tween the two spin components of a hole wavepacket ( n= 3) at a ﬁxed drift distance of 1 µm, see Eq. (34). The dashed line is the non-interacting value ∆ 0. The thin solid line is n= 1, indistinguishable from ∆ = 0. We have assumed here rs= 0.3. The inset schematically illustrates the deﬁnitions of ∆ and ∆ 0. In addition to including both n= 1 and n= 3, in Fig. 4 we plot the non-interacting value ∆ 0, obtained by substituting v±→v0 ±in Eq. (34). For n= 1 the non-interacting velocities v0 ±are the same (∆ 0= 0). The eﬀect of electron-electron interactions is not visi- ble. Thus, the wavepacket remains essentially unsplit and unpolarized (∆ ≃0) and the only signiﬁcant inﬂu- ence is on the averagevelocity( v++v−)/2, from Eq. (19) [see Fig. 1(a)]. In contrast, a large splitting is found forn= 3 where ∆ ,∆0can reach a large fraction of the traveling distance. Since spin-polarized ballistic trans- port is observed in two-dimensional hole systems on µm- scales (e.g., in spin-focusing experiments28,29) and the typical wavepacket traveling time is 1 −50 ps (depend- ing on the density), Fig. 4 suggests that a direct op- tical imaging of the wavepacket separation should be within reach of femtosecond-resolved Faraday rotation measurements.35,36 As for electron-electron interaction eﬀects, we see in Fig. 4 that a visible diﬀerence between ∆ and ∆ 0exists. The diﬀerence is quite small, due to the fact that we are considering here the weak-coupling limit, and all the in- teraction corrections are proportional to rs<1. ∆0is modiﬁed here by ∼2−3% and we collect some represen- tative values in Table I. It can be seen in Table I that, while ∆ 0does not change with rs, the interaction correc- tion ∆−∆0grows at lower densities (see Table I). Since experiments on hole systems can reach values as large as rs= 6−12,33,34it is reasonable to expect signiﬁcant ef- fects from δv±in this low-density regime. As a reference, in electron systems, δvchanges from ∼5% to−30% for rs∼1 to 6.37,38 It might be surprising to see ∆ <∆0in Fig. 4. This is due to the interplay of two competing interaction eﬀects. It is true that, as discussed already, the diﬀerence be- tweenv±is enhanced by δv±. The two spin components8 SOIgrs∆0(nm)∆−∆0(nm) n=10.050.50 0.001 n=30.10.1203 -4.7 n=30.10.3203 -6.2 TABLE I. Separation of wavepackets after 1 µm drift dis- tance for electrons ( n= 1) and holes ( n= 3). The non- interacting value and the electron-electron interaction c orrec- tion are listed, see Eq. (34) and Fig. 4. Typical values for g(∼χ) are/lessorsimilar0.05 for electrons2,4and/lessorsimilar0.2 for holes.2,27 We have assumed in this table that typical values of gare independent of rsand have used high-density values rs<1. therefore split faster in the interacting case. However, the situation shown in Fig. 4 is distinct from that shown in Fig. 1 since a constant traveling distance (and not timeτ) is assumed. At high density the mean velocity (v++v−)/2 is greater for the interacting gas ( δv >0), which allows the wavepackets less time to separate. As it turns out, the latter eﬀect is dominant in Fig. 4. Interestingly, the sign of δvchanges at low density.1 This would imply a cooperative eﬀect of δv±andδvon ∆ ifδv±does not change sign. Unfortunately, to infer the behavior of δv±at low density requires a much more sophisticated approach. IV. CONCLUSION We have presented a theory of the quasiparticle group velocity at high density, in the presence of SOI of dif- ferent types. Contrasting the behavior of electron and hole systems, we ﬁnd several intriguing diﬀerences. We have shown explicitly that the lowest-order cancellations of SOI eﬀects occur only for the electronic case, when the SOI is approximately linear in momentum (e.g., a strong Rashba or Dresslhaus SOI is present). On the other hand, SOI terms non-linear in pare often domi- nant in hole systems.26,27,29Thus, larger eﬀects of the SOI and a non-trivial interplay with electron-electron in- teractions are expected for holes on general grounds. Asanimportantmotivationforfuturetheoreticalstud- ies, hole liquids can be realized in the laboratory with strong SOI and large values of rs. For example, the spin- subband population diﬀerence at zero ﬁeld is of order 15−20% in Ref. 27, with ns≃2−4×1010cm−2. With a hole eﬀective mass m≃0.2m0in GaAs these densities correspond to rs≃9−12. For electrons, materials with strong SOI typically have small eﬀective masses, which results in much lower values of rs. A diluted electron liq- uid with ns≃2×1010cm−2givesrs≃1.2, using InAs parameters ( m= 0.023m0). Discussing the large- rsregime of holes would require extending many-body perturbation theory7–9or Monte Carlo18approaches, so far only applied to linear SOI. The high-density regime studied here would represent a well-controlled limit of these theories for the quasiparti-cle dispersion. In addition to being relevant for trans- port measurements of the eﬀective mass, the signiﬁcant diﬀerence in group velocity at the Fermi surface of the two spin branches could also be addressed by Raman scattering experiments, demonstrated for electron sys- tems in Ref. 43, or via time-resolved Faraday-rotation detection of the spin-polarization.35,36A similar discus- sion should hold with n= 2,3 for other physical observ- ables and many-body eﬀects. For example, studying the compressibility1,44in the presence of SOI and extend- ing then= 1 discussion of the quasiparticle lifetime7–9 would also be topics of interest. Finally, the problem of including in our framework a more general form of SOI than Eq. (1) is clearly of practical relevance. However, as discussed in Appendix A, we expect that our qualitative picture on the diﬀerent role of linear and non-linear SOI remains valid also in this more general situation. ACKNOWLEDGMENTS We thank J. P. Eisenstein and D. L. Maslov for use- ful discussions and acknowledge ﬁnancial support from CIFAR, NSERC, and FQRNT. Appendix A: General spin-orbit coupling A general SOI contains terms with linear and non- linear dependence on momentum, and does not neces- sarily have the isotropic form assumed in Eq. (1). For deﬁniteness, we suppose that there is no magnetic ﬁeld, so that quadratic terms are not present: H=/summationdisplay ip2 i 2m+Hso 1+Hso 3+Hel−el,(A1) whereHso 1,Hso 3, andHel−elgive, respectively, the linear- in-momentum spin-orbit, the cubic-in-momentum spin- orbit, and electron-electron interactions. To show that Hso 1always has a small eﬀect, we consider the uni- tary transformation H′=e−SHeSwithSdeﬁned by [S,/summationtext ip2 i/2m] =Hso 1, giving H′≃p2 2m+Hso 3+Hel−el−[S,Hso 1+Hso 3].(A2) In the same transformed frame, the velocity operator of electron iis given by v′ i=e−S(∂H/∂pi)eS, which yields v′ i≃pi m+∂Hso 3 ∂pi−[S,∂Hso 1 ∂pi+∂Hso 3 ∂pi].(A3) In deriving Eq. (A2) we have used the fact that [S,Hel−el] = 0. For example, for an isotropic SOI as in Eq. (1), S=imγ /planckover2pi1/summationdisplay i(xiσy,i−yiσx,i). (A4)9 The property [ S,Hel−el] = 0 is valid for a general linear-in-momentum SOI, including a combination of Rashba and Dresselhaus SOI.24However, the same iden- tity [S,Hel−el] = 0 does not hold for a transformation with [S,/summationtext ip2 i/2m] =Hso 3, i.e., a transformation that is designed to remove the non-linear component from the non-interacting Hamiltonian. In writing Eq. (A3), we have used [ S,pi/m] =∂Hso 1/∂pi, which implies a cancellation of the Hso 1contribution to v′ ito lowest or- der. Again, this cancellation is only valid for linear-in- momentum SOI. By introducing dimensionless couplings g1,3associated withHso 1,3, in direct analogy with Eq. (4), we see that the commutators in Eqs. (A2) and (A3) are of quadratic or bilinear order in the couplings ( ∼g2 1and∼g1g3). This indicates on general grounds that Hso 3has the largest ef- fect on the quasiparticle velocity if g1/lessorsimilarg3≪1. In this case, if we are interested in lowest-order eﬀects, we can neglect both anticommutators in Eqs. (A2) and (A3), which is equivalent to neglecting Hso 1in the origi- nal Hamiltonian (A1). Thus, to leading order all results we report for the quasiparticle velocities due to a pure cubic-in-momentum spin-orbit interaction also apply in the case of a mixed linear-plus-cubic spin-orbit interac- tion (with the caveat that we consider only the isotropic form of cubic SOI). Appendix B: Derivation of Eq. (16) As discussed in the text, at high density we can neglect the√ 2rsin the integrand of Eq. (15) (second line). For smallg, thevalueof χisalsosmallandwecanperforman expansion of the exchange integral. First notice that the constanttermat χ= 0ismissing,becausetheintegration limits simply become y±= 1 and the integrand vanishes upon the summation on σ,σ′. Therefore we only need to compute the linear term in χ: /bracketleftBigg ∂ ∂χ/summationdisplay σσ′/integraldisplay2π 0dθ 2π/integraldisplayyσ′ 0(σ+σ′cosnθ)ydy/radicalbig y2+y2σ−2yyσcosθ/bracketrightBigg χ=0 =/integraldisplay2π 0dθ 2π/bracketleftBigg/integraldisplay1 02(1−ycosθ)ydy (1+y2−2ycosθ)3 2−cosnθ sinθ 2/bracketrightBigg ,(B1) and after evaluating the dyintegral in the square paren- thesis, Eq. (B1) gives /integraldisplay2π 0dθ 2π/bracketleftBigg 2+2cosθ−cosnθ−1 sinθ 2 −2ln/parenleftBigg 1+1 sinθ 2/parenrightBigg cosθ/bracketrightBigg =4 πn/summationdisplay j=01 2j−1.(B2) Equation (16) is then easily obtained from Eq. (15) by neglecting all the cubic terms in the small parameters g,χ,rs[e.g., the left side of Eq. (15) is ( y2 ++y2 −)g= 2g+O(gχ2)].Appendix C: Small rs,gexpansions We give in this appendix some details on the expan- sions of Eq. (22): vF± vF=y±±g 2nyn−1 ±+√ 2rs 16π(I1+I2+I3),(C1) where we have split L2in its two contributions ( I1,2refer toσ=±) andI3corresponds to L3: I1=/integraldisplay2π 0dθ√ 2y±cosθ(1+cosnθ)√ 2y±sinθ/2+rs, (C2) I2=/integraldisplay2π 0dθ√ 2y∓cosθ(1−cosnθ)/radicalbig 1−y+y−cosθ+rs, (C3) I3=±/integraldisplay2π 0dθ/integraldisplayy− y+dy2nsinnθsinθ/radicalBig y2 ±+y2−2yy±cosθ+rs√ 2,(C4) Notice that these integrals have an explicit dependence onrsandy±=√1∓χ. So, it is easier to perform ﬁrst the expansion in the two small parameters rs,χ. The ﬁnal results in the main text are given in terms of the physical couplings of the hamiltonian: rsandg. Those ﬁnal expression are easily obtained by substituting the value of χin terms of rsandg(χ≃gin ﬁrst approxi- mation). The ﬁrst integral, Eq. (C2), can be evaluated exactly. In particular for n= 1 we obtain I1=−40 3+8πδ(1−δ2)+16δ2 +8(1−3δ2+2δ4)tanh−1√ 1−δ2 √ 1−δ2(C5) whereδ=rs/√ 2y±. This expression can then be easily expanded in rs,χand an analogousprocedure is followed forn= 2,3. To lowest-order in rs, we can set rs= 0 in I2andI3. Similarly to the I1angular integral above, thedθintegrals of I2andI3atrs= 0 can be computed analytically for n= 1,2,3. ForI2this yields directly the desired function of χ. ForI3we still need to perform a last integration in dy. Since the integration region is of size∼χaroundy= 1, we can expand the integrand in thesmallparameter( y−1)andperformtheintegrationin dyorder-by-order, which allows us to extract the leading terms of the expansion in χ. Forn= 1 all this gives δI1≃ ∓4χ−2χ2∓4 3χ3, (C6) δI2≃ ∓4 3χ+χ2/parenleftbigg lnχ 8+13 6/parenrightbigg ±χ3 2/parenleftbigg lnχ 8+3 2/parenrightbigg ,(C7) δI3≃ ±16 3χ+4 3χ2±χ3/parenleftbigg2 3lnχ 16+389 120/parenrightbigg ,(C8) where only the corrections δIα=Iα(χ)−Iα(χ= 0) are listed, sincetermsindependenton χsimplygivethesmall rsexpansion of the well known Eq. (19). Here, terms of10 orderO(rsχ,χ4) are omitted, while it is interesting to keep the O(χ3) terms, since they give the leading spin splitting. Indeed, itiseasilycheckedthatthelinearterms cancel 3/summationdisplay α=1δIα≃χ2/parenleftbigg3 2+lnχ 8/parenrightbigg ±χ3 6/parenleftbigg lnχ7 85+319 20/parenrightbigg ,(C9) which immediately leads to Eq. (27). Forn= 2,3 we can proceed in a similar way. The spin splitting appears now already to linear order in χ. By keeping the ﬁrst subleading correction in χwe have for n= 2: δI1≃ ∓4χ−2χ2, (C10) δI2≃ ±16 15χ+χ2/parenleftbigg 4lnχ 8+134 15/parenrightbigg ,(C11) δI3≃ ±64 15χ+16 15χ2, (C12) and forn= 3: δI1≃ ∓4χ−2χ2, (C13) δI2≃ ±212 105χ+χ2/parenleftbigg 9lnχ 8+899 42/parenrightbigg ,(C14) δI3≃ ±144 35χ+36 35χ2. (C15)The ﬁnal results are for n= 2: 3/summationdisplay α=1δIα≃ ±4 3χ+4χ2/parenleftBig lnχ 8+2/parenrightBig ,(C16) and forn= 3: 3/summationdisplay α=1δIα≃ ±32 15χ+χ2/parenleftbigg 9lnχ 8+613 30/parenrightbigg .(C17) From Eqs. (C16) and (C17) we immediately obtain Eqs. (28) and (29). 1F. Giuliani and G. Vignale, Quantum Theory of the Electron Liquid (Cambridge University Press, Cambridge, 2005). 2R. Winkler, Spin-Orbit Coupling Eﬀects in Two- Dimensional Electron and Hole Systems (Springer, Berlin, 2003). 3S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1989). 4J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, Phys. Rev. Lett. 78, 1335 (1997). 5H. C. Koo, J. H. Kwon, J. Eom, J. Chang, S. H. Han, and M. Johnson, Science 325, 1515 (2009). 6G.-H. Chen and M. E. Raikh, Phys. Rev. B 60, 4826 (1999). 7D. S. Saraga and D. Loss,, Phys. Rev.B 72, 195319 (2005). 8I. A. Nechaev, M. F. Jensen, E. D. L. Rienks, V. M. Silkin, P. M. Echenique, E. V. Chulkov, and P. Hofmann, Phys. Rev. B80, 113402 (2009). 9I. A. Nechaev, P. M. Echenique, and E. V. Chulkov, Phys. Rev. B81, 195112 (2010). 10A. Agarwal, S. Chesi, T. Jungwirth, J. Sinova, G. Vignale, and M. Polini, Phys. Rev. B 83, 115135 (2011). 11Y. A. Bychkov and E. Rashba, JETP Lett. 39, 78 (1984). 12Y. A. Bychkov and E. Rashba, J. Phys. C 17, 6039 (1984). 13G. Dresselhaus, Phys. Rev. 100, 580 (1955). 14M. I. D’yakonov and V.Yu. Kachorovskii, Sov. Phys. Semi- cond.20, 110 (1986). 15D. H. Berman and M. E. Flatt´ e, Phys. Rev. Lett. 105, 157202 (2010). 16S. Chesi and G. F. Giuliani, Phys. Rev. B 83, 235308 (2011).17S. H. Abedinpour, G. Vignale, and I. V. Tokatly, Phys. Rev. B81, 125123 (2010). 18A. Ambrosetti, F. Pederiva, E. Lipparini, and S. Gandolﬁ, Phys. Rev. B 80, 125306 (2009). 19S. Chesi and G. F. Giuliani, Phys. Rev. B 83, 235309 (2011). 20G. F. Giuliani and S. Chesi, Proceedings of Highlights in the Quantum Theory of Condensed Matter (Edizioni della Normale, Pisa, 2005); S. Chesi, G. Simion, and G. F. Giuliani, e-print arXiv:cond-mat/0702060; S. Chesi, Ph.D . thesis, Purdue University, 2007. 21L. O. Juri and P. I. Tamborenea, Phys. Rev. B 77, 233310 (2008). 22R. A.˙Zak, D. L. Maslov, and D. Loss, Phys. Rev. B 82, 115415 (2010). 23R. A.˙Zak, D. L. Maslov, and D. Loss, arXiv:1112.4786. 24I. L. Aleiner and V. I. Falko, Phys. Rev. Lett. 87, 256801 (2001). 25W. A. Coish, V. N. Golovach, J. C. Egues, and D. Loss, phys. stat. sol (b) 243, 3658 (2006). 26R. Winkler, Phys. Rev. B 62, 4245 (2000). 27R. Winkler, H. Noh, E. Tutuc, and M. Shayegan, Phys. Rev. B65, 155303 (2002). 28L. P. Rokhinson, V. Larkina, Y. B. Lyanda-Geller, L. N. Pfeiﬀer, and K. W. West, Phys. Rev. Lett. 93, 146601 (2004). 29S. Chesi, G. F. Giuliani, L. P. Rokhinson, L. N. Pfeiﬀer, and K. W. West, Phys. Rev. Lett. 106, 236601 (2011). 30D. V. Bulaev and D. Loss, Phys. Rev. Lett. 98, 097202 (2007).11 31S. Chesi and G. F. Giuliani, Phys. Rev. B 75, 155305 (2007). 32E. I. Rashba and E. Ya. Sherman, Phys. Lett. A 129, 175 (1988). 33R. Winkler, E. Tutuc, S. J. Papadakis, S. Melinte, M. Shayegan, D. Wasserman, and S. A. Lyon, Phys. Rev. B 72, 195321 (2005). 34Y. T. Chiu, M. Padmanabhan, T. Gokmen, J. Sha- bani, E. Tutuc, M. Shayegan, and R. Winkler, e-print arXiv:1106.4608. 35S. A. Crooker, J. J. Baumberg, F. Flack, N. Samarth, and D. D. Awschalom, Phys. Rev. Lett. 77, 2814 (1996). 36J. M. Kikkawa and D. D. Awschalom, Phys. Rev. Lett. 80, 4313 (1998). 37Y.-W. Tan, J. Zhu, H. L. Stormer, L. N. Pfeiﬀer, K. W. Baldwin, and K. W. West, Phys. Rev. Lett. 94, 016405(2005). 38N. D. Drummond and R. J. Needs, Phys. Rev. B 80, 245104 (2009). 39J. F. Janak, Phys. Rev. 178, 1416 (1969). 40Todeal with extremelylow densities atwhich onlyone spin bandisoccupiedandtheoccupationisaringinmomentum space, it is also useful to generalize χto values larger than one.19However, we do not consider this low-density regime in this paper. 41J. Schliemann, Phys. Rev. B 74, 045214 (2006). 42The method of Ref. 31 is slightly preferable, in that it leads to variational wavefunctions with a lower total energy. 43B. Jusserand, D. Richards, H. Peric, and B. Etienne, Phys. Rev. Lett. 69, 848 (1992). 44J. P. Eisenstein, L. N. Pfeiﬀer, and K. W. West, Phys. Rev. Lett.68, 674 (1992).
2108.06202v2.Coupling_the_Higgs_mode_and_ferromagnetic_resonance_in_spin_split_superconductors_with_Rashba_spin_orbit_coupling.pdf	Coupling the Higgs mode and ferromagnetic resonance in spin-split superconductors with Rashba spin-orbit coupling Yao Lu,1Risto Ojaj arvi,1P. Virtanen,1M.A. Silaev,1, 2, 3and Tero T. Heikkil a1 1Department of Physics and Nanoscience Center, University of Jyv askyl a, P.O. Box 35 (YFL), FI-40014 University of Jyv askyl a, Finland 2Moscow Institute of Physics and Technology, Dolgoprudny, 141700 Russia 3Institute for Physics of Microstructures, Russian Academy of Sciences, 603950 Nizhny Novgorod, GSP-105, Russia (Dated: February 22, 2022) We show that the Higgs mode of superconductors can couple with spin dynamics in the presence of a static spin-splitting eld and Rashba spin-orbit coupling. The Higgs-spin coupling dramatically modies the spin susceptibility near the superconducting critical temperature and consequently enhances the spin pumping eect in a ferromagnetic insulator/superconductor bilayer system. We show that this eect can be detected by measuring the magnon transmission rate and the magnon- induced voltage generated by the inverse spin Hall eect. Superconductors (SC) with broken U(1) symmetry host two kinds of collective modes associated with the order parameter uctuations: the phase mode and the amplitude mode. Coupled to a dynamical gauge eld, the phase mode is lifted up to the plasma frequency [1] due to the Anderson{Higgs mechanism [2, 3]. The other collective mode in SC is the amplitude mode [4, 5] with an energy gap of 2, called the Higgs mode by anal- ogy with the Higgs boson [3] in particle physics. It was commonly believed that unlike the phase mode the Higgs mode usually does not couple linearly to any experimen- tal probe. That is why in earlier experiments, the Higgs mode was only observed in charge-density-wave (CDW) coexisting systems [6{11]. With the advance of terahertz spectroscopy technique [12] it became possible to inves- tigate the Higgs mode through the nonlinear light{Higgs coupling [13{17]. In these experiments, the perturba- tion of the order parameter is proportional to the square of the external electromagnetic eld /E2, so very strong laser pulses are required. Recently, it has been shown that in the presence of a supercurrent the Higgs resonance can actually contribute to the total admittance Y due to the linear coupling of the Higgs mode and the external electromagnetic eld [18{22]. This can be understood from a symmetry ar- gument. Suppose the external electric eld is linearly polarized in the xdirectionE= ^xExei t. The linear coupling of the Higgs mode and the external eld is rep- resented by the susceptibility E=@2S @@Eobtained from the action Sdescribing the electron system con- taining the pair potential eld . Without a supercur- rent, the system preserves the inversion symmetry ( ^I) and the mirror symmetry in the xdirection ( ^Mx). On the other hand Eis odd under both these operations becauseEchanges sign under ^Iand ^Mxwhereas re- mains the same. Therefore Ehas to vanish. In the presence of a supercurrent, the inversion symmetry and the mirror symmetry are both broken and there is no re- striction for Exfrom these symmetries, so Ecan be FIG. 1. System under consideration. A superconductor thin lm is placed on the top of a FI with in-plane magnetization. The SC and FI are coupled via spin exchange interaction. The magnon in FI can be injected into SC in a process known as the spin pumping eect. For magnon frequency = 2 0the SC Higgs mode greatly increases the spin pumping. nonzero. This symmetry argument also explains why the Higgs mode does not couple with an external eld in the direction perpendicular to the supercurrent. Now a natural question arises: without a supercurrent does the Higgs mode couple linearly with other exter- nal probes, such as spin exchange elds? As we show in this Letter it does. The above discussion indicates that the decoupling of the Higgs mode is protected by cer- tain symmetries. In order to couple the Higgs mode to an external eld one needs to break these symmetries. Here we show how it happens in a ferromagnetic insu- lator (FI)/superconductor (SC) bilayer system (Fig. 1). Magnons with momentum qand frequency in the FI can be injected into the SC in a process known as spin pumping [23{29]. We predict that the Higgs mode in the SC couples linearly with the magnon mode in the FI in the presence of Rashba spin-orbit coupling and the magnetic proximity eect into the SC. In this system the symmetries protecting Higgs-spin decoupling are bro- ken: in particular, the (spin) rotation symmetry and thearXiv:2108.06202v2 [cond-mat.supr-con] 21 Feb 20222 time-reversal symmetry. Near the critical temperature, superconductivity is suppressed and 0becomes compa- rable with the magnon frequency . When the magnon frequency matches the Higgs frequency M= 2 0, the Higgs mode is activated and the magnon absorption is hugely enhanced which can be detected through the in- verse spin Hall eect (iSHE) [30{32]. This eect can pos- sibly explain the voltage peak observed in the experiment [33]. We consider a SC/FI bilayer in which the FI and the SC are coupled via the exchange interaction as shown in Fig. 1. For simplicity, we assume that the thickness d of the SC lm is much smaller than the spin relaxation length and the coherence length so that we consider it as a 2D system. The magnetization of the FI can be written asm=m0+m , wherem0is the static manetization polarized in the zdirection and m is the dynamical component perpendicular to m0. When magnons (spin waves) are excited in the FI, they can be injected into the SC in a process known as the spin pumping eect. The DC interface spin current owing from the FI into the SC is polarized in the zdirection and given by [34] Iz=X ;q2JsdIm[~ss( ;q)]m2 ;q; (1) whereJsdis the exchange coupling strength and m is the Fourier amplitude of m . ~ss( ;q) is the total dynamical spin susceptibility ~ ss( ;q) = S+( ;q)=h+( ;q), whereSis the dynamical spin of the SC,his the proximity induced exchange eld h= Jsdm=d[35] and for a vector A= (Ax;Ay;Az) the component is dened as A=AxiAy. One can see that for a xed Jsd, the eciency of the magnon injection is soley determined by ~ ss( ;q). The spin susceptibility of superconductors has been extensively studied [29, 36]. However the previous theories, based on the static mean- eld description, failed to explain the peak of the iSHE signal observed in the spin Seebeck experiment [33]. In this work, we start with the general partition function of the SC,Z=R D[ ; ;;]eSobtained by performing the Hubbard-Stratonovich transformation. The action S is given by S=X K;Q K(i!+kh) K+ Q K+Q K +Q K KQ+QQ U;(2) HereK= (!;k) andQ= ( ;q) are the four-momenta of the electrons and magnons, respectively. != (2n+ 1)Tand = 2nT are the Matsubara frequencies with n2Zand= 1=T.kis the energy dispersion of the electron in the normal state, his the proximity induced exchange eld, and Uis the BCS interaction. In the mean-eld theory, one can ignore the path integral over and replace it by its saddle point value 0which is determined by the minimization of the action@S @j= 0= 0 after integrating out the fermion elds. To include the Higgs mode, we go beyond the mean- eld theory and write the order parameter as = 0+, whereis the deviation of from its saddle point value 0. Here we only consider the amplitude uctuation of , sois real. Expanding the action to the second order inand the strength of the external Zeeman eld h givesS=S0S2with [37] S2=X Q(Q)h(Q) 1 s sss(Q) h+(Q) : (3) Here, all the susceptibilities are functions of Q.S0is the mean-eld action without the external eld. In usual su- perconductors the o-diagonal susceptibilities sand svanish as required by the time-reversal symmetry and the (spin) rotation symmetry because these oper- ations change the sign of h+but have no eect on [38, 39]. In the system under consideration, the proxim- ity induced static exchange eld breaks the time-reversal symmetry and RSOC breaks the (spin) rotation symme- try. Thus the pair-spin susceptibility does not have to vanish, allowing for a nonzero Higgs{spin coupling. Then it is straightforward to calculate the total spin susceptibility ~ ssby integrating out the eld ~ss=ssss: (4) The imaginary part of is sharply peak at the Higgs frequency = 2 dramatically modifying the total spin susceptibility. Phenomenological theory . Before we go to the detailed calculations, we use a simple phenomenological theory to illustrate the eect of RSOC. It has been shown that RSOC can induce a Dzyaloshinskii-Moriya (DM) interac- tion in superconductors described by the DM free energy [40] FDM=X iZ drjj2d;i(hrih); (5) where both = ( r) andh=h(r) are position depen- dent.d;iis the DM vector proportional to the strength of spin-orbit coupling . For RSOC d/[x;z], whereis the spin-orbit coupling strength and is the Pauli matrix acting on the spin space. To nd the pair spin susceptibility we write = 0+(t), h=h0^z+h+(t)(^x+i^y), where ^nis the unit vector in thendirection with n=x;y;z , and generalize the DM free energy to the time dependent DM action. Here we consider the case where the spin wave is propagating in thezdirectionh+(t;r) =P ;qzh+(^x+i^y)ei( tqzz). Focusing on the rst order terms in (t) andh+(t) and3 Fourier transforming them to momentum and frequency space, the DM action can be written as SDM1=X ;qziqz0h0h+( ;qz)( ;qz)~d;z( ;qz) (i^x^y);(6) where ~d;iis the dynamical DM vector, which has the same niteness and spin structure as d;ifrom symmetry analysis. From the above expression, one can see that the Higgs mode couples linearly with the spin degree of freedom in the presence of RSOC. Spin susceptibility . We adopt the quasiclassical ap- proximation to systematically evaluate the susceptibili- ties. In the diusive limit, this system can be described by the Usadel equation [18, 36, 41{45] FIG. 2. Imaginary part of the pair susceptibility. This can be interpreted as the spectral weight of the Higgs mode. A signicant peak emerges when the driving frequency matches the Higgs frequency = 2 0. The inset shows the height of the Higgs peak PHas a function of the inverse of the mo- mentum q. Parameters: 0= 0:8T0,h0= 0:5T0with T00(T= 0). if3@t;^gg=D~r ^g~r^g i[H0;^g] +h Xei( tqzz);^gi : (7) Here ^gis the quasiclassical Green function, D=vF2=3 is the diusion constant and is the disorder scat- tering time. H0=ih03+ 01, whereh0is the proximity induced eective static exchange eld and i is the Pauli matrix acting on the particle-hole space. ~r= (~rz;~rx) is the covariant derivative dened by ~rz=rz+i[x;],~rx=rxi[z;]. The Usadel equation is supplemented by the normalization condition ^g2= 1. In the quasiclassical approximation the approxi- mate PH symmetry of the full Hamiltonian becomes ex-act. In the linear response theory, the external oscillat- ing eldXis small and can be treated as a perturbation. Thus we can write the quasiclassical Green function as ^g= ^g0ei!(t1t2)+ ^gXei(!+ )t1i!t2iqzz, where ^g0is the static Green function and ^ gXis the perturbation of the Green function describing the response to the external eld. Solving the Usadel equation we obtain the quasi- classical Green function, the anomalous Green function F=NeTr [1^g]=4iand the+component of spin in the SChsi=NeTr [3^g]=4i, whereNeis the electron den- sity of states at the Fermi surface and Tr is the trace. The susceptibilities can be evaluated as ^= 1 s sss ="@F @+1 U@F @h+ @hsi @@hsi @h+# : (8) Let us rst set X= 01and consider the pair suscep- tibility. We assume the RSOC is weak and treat as a perturbation. At q= 0 and 0th order in , we have (i ) =" NeT 2X !;42+ 2 s(!)(4!2 2)#1 ;(9) wheres(!) =p (!+ih)2+ 2, with=1. To get the pair susceptibility as a function of real frequency, we need to perform an analytical continuation [38]. Thus i is replaced by + i0+. One can see that the is peaked at the Higgs frequency = 2. We numerically calculate with nite momentum and show the results in Fig. 2 [38, 46]. One can see that the imaginary part of the inverse of the pair suscepti- bility exhibits a sharp peak when the driving frequency equals 2 0. With a nite momentum, the Higgs mode is damped in the sense that the peak in the Higgs spectrum has a nite height and width. FIG. 3. Real part (a) and imaginary part (b) of pair-spin susceptibility. The solid line is the approximate result calcu- lated from Eq. (12) and the circles show the numerical solu- tion from Eq. (7). Parameters used here are: = 0 :8T0 for the blue lines, = T0for the red lines, h0= 0:5T0, Dq2 z=D2= 0:01 T0. To study the response of this system to the external exchange eld we set X=h++3. Again we treat as a perturbation and write the Green function as ^g= ^g0ei!(t1t2)+ (^gh0+ ^gh)ei(!+ )t1i!t2iqzz;(10)4 where ^gh0is 0th order in and ^ghis rst order in . The 0th order solution in is given by [38] ^gh0= ^gh00 +=i[3^g"(1)3^g#(2)]h + s"(1) +s#(2);(11) where ^g"=# =(!ih0)3+1 s"=#ands"=# =p (!ih0)2+ 2. ^gh00is a 22 matrix in the particle-hole space. Without doing detailed calculations, one can immediately see that shas to vanish without RSOC because ^ ghhas no0component. In this case the external exchange eld cannot activate the Higgs mode. To get a nite pair-spin susceptibility we need to consider the rst order terms in which break the spin rotation symmetry. The rst order solution in yields ^gh= diag(^gh";^gh#) with ^gh"=#= 2iD^g0"=# ^gh00;^g0"=# s"=#(!1) +s"=#(!2): (12) FIG. 4. (a) Total spin susceptibility as a function of tem- perature with a xed frequency. (b) Total spin susceptibility as a function of frequency with a xed temperature. The two temperatures have been chosen so that ( T1) = 0:2T0 and (T2) = 0:1T0. The Higgs peak thus shows up when = 2(T). The parameters used here are: h0= 0:5T0, Dq2 z=D2= 0:01 T0. Since the 0th order term does not contribute to the pair-spin susceptibility, we have s= Tr[1^gh]=4ih+. We compare this analytical result with the non- perturbative numerical solution of the Usadel equation in Fig. 3. It shows that the perturbative approach is ac- curate at high temperatures when D20;T, and captures the qualitative behavior of salso at the low temperatures. Another feature of this pair spin suscepti- bility is that at a lower frequency ( = 0 :8T0),sissuppressed at low temperatures because the spin excita- tion is frozen by the pair gap at low temperatures. On the other hand, at higher frequency ( = T0),sis slightly enhanced at low temperatures. We can also get the bare spin susceptibility from ^ gh0, ss= Tr[3^gh0]=4ih+. Then it is straightforward to calculate the total spin susceptibility according to Eq. (4). The results are shown in Fig. 4. The total spin susceptibility exhibits a signicant peak near criti- cal temperature. This is a signature of the Higgs mode with the frequency = 2 0. The dependence of the to- tal susceptibility on the strength of RSOC is studied in the supplementary information [38]. The details depend sensitively on the amount of disorder, as in the disordered case increasing RSOC leads to a stronger spin relaxation. We note that even though the pair-spin susceptibility is linear in momentum qz, the magnon momentum need not be large for the detection of the Higgs mode. This is because the spectral weight of the Higgs mode is propor- tional to 1=q2 zat the Higgs frequency, so that the height of the peak in the total spin susceptibility is independent of the magnon momentum. Experimental detection . We propose that the Higgs mode in Rashba superconductors can be detected in the spin pumping experiment as shown in Fig. 1. Magnons in the FI with momentum qand frequency are injected from one side of FI and propagate in the zdirection to- wards the other end. Due to the spin pumping eect, part of the magnons can be absorbed by the SC on top of it and converted to quasiparticles. This spin injection causes a spin current Is owing in the out-of-plane di- rection. In the presence of RSOC, Isis converted into a charge current Ievia the iSHE Ie=z xzIs, whereis the spin Hall angle [47]. When the width of the SC is smaller than the charge imbalance length the non-equilibrium charge accumulation cannot be totally relaxed resulting into a nite resistance of the SC. Therefore a voltage can be measured across the SC, given by V=z xzX ;q2JsdIm[~ss( ;q)]m2 : (13) Thus by tuning the temperature or the frequency of magnon, one can observe a peak in the voltage [33]. Meanwhile we can also obtain the magnon absorption rate dened as the energy of the absorbed magnons di- vided by time W= 2 X ;q2JsdIm[~ss( ;q)]m2 : (14) This magnon absorption rate results in a dip in the magnon transmission rate which is experimentally mea- surable. Conclusion . In this Letter, we consider a FI/SC bi- layer with RSOC in the bulk of the SC. Using symme-5 try arguments and microscopic theory, we show that the Higgs mode in the SC couples linearly with an exter- nal exchange eld. This Higgs{spin coupling hugely en- hances the total spin susceptibility near a critical phase transition point, which can be detected using iSHE or via strong frequency dependent changes in the magnon transmission. Note that in this work, we consider the dif- fusive limit where the disorder strength is stronger than the RSOC and exchange eld. However, our conclusion on Higgs{spin coupling should still be valid in the case of strong RSOC. In fact, we expect that the coupling is much stronger with strong SOC in the clean limit. In the diusive limit, the RSOC together with disorder ef- fectively generate spin relaxation which reduces the prox- imity induced exchange eld suppressing the Higgs{spin coupling. On the other hand, in the clean case without disorder this eect is absent and hence the Higgs{spin coupling can be stronger. We also compare the Higgs mode in the diusive limit and ballistic limit in the sup- plemental material [38]. This work was supported by Jenny and Antti Wi- huri Foundation, and the Academy of Finland Project 317118 and the European Union's Horizon 2020 Research and Innovation Framework Programme under Grant No. 800923 (SUPERTED). [1] N. Bittner, D. Einzel, L. Klam, and D. Manske, Phys. Rev. Lett. 115, 227002 (2015). [2] P. W. Anderson, Phys. Rev. 130, 439 (1963). [3] P. W. Higgs, Phys. Rev. Lett. 13, 508 (1964). [4] I. Kulik, O. Entin-Wohlman, and R. Orbach, J. Low Temp. Phys. 43, 591 (1981). [5] A. Pashkin and A. Leitenstorfer, Science 345, 1121 (2014). [6] P. B. Littlewood and C. M. Varma, Phys. Rev. B 26, 4883 (1982). [7] P. B. Littlewood and C. M. Varma, Phys. Rev. Lett. 47, 811 (1981). [8] M.-A. M easson, Y. Gallais, M. Cazayous, B. Clair, P. Rodiere, L. Cario, and A. Sacuto, Phys. Rev. B 89, 060503(R) (2014). [9] R. Grasset, Y. Gallais, A. Sacuto, M. Cazayous, S. Ma~ nas-Valero, E. Coronado, and M.-A. M easson, Phys. Rev. Lett. 122, 127001 (2019). [10] R. Grasset, T. Cea, Y. Gallais, M. Cazayous, A. Sacuto, L. Cario, L. Benfatto, and M.-A. M easson, Phys. Rev. B 97, 094502 (2018). [11] T. Cea and L. Benfatto, Phys. Rev. B 90, 224515 (2014). [12] D. N. Basov, R. D. Averitt, D. vanderMarel, M. Dressel, and K. Haule, Rev. Mod. Phys. 83, 471 (2011). [13] R. Matsunaga, Y. I. Hamada, K. Makise, Y. Uzawa, H. Terai, Z. Wang, and R. Shimano, Phys. Rev. Lett. 111, 057002 (2013). [14] R. Matsunaga, N. Tsuji, H. Fujita, A. Sugioka, K. Makise, Y. Uzawa, H. Terai, Z. Wang, H. Aoki, and R. Shimano, Science 345, 1145 (2014). [15] M. Beck, I. Rousseau, M. Klammer, P. Leiderer, M. Mit-tendor, S. Winnerl, M. Helm, G. N. Gol'tsman, and J. Demsar, Phys. Rev. Lett. 110, 267003 (2013). [16] M. Silaev, Phys. Rev. B 99, 224511 (2019). [17] M. A. Silaev, R. Ojaj arvi, and T. T. Heikkil a, Phys. Rev. Research 2, 033416 (2020). [18] A. Moor, A. F. Volkov, and K. B. Efetov, Phys. Rev. Lett. 118, 047001 (2017). [19] S. Nakamura, Y. Iida, Y. Murotani, R. Matsunaga, H. Terai, and R. Shimano, Phys. Rev. Lett. 122, 257001 (2019). [20] S. Nakamura, K. Katsumi, H. Terai, and R. Shimano, Phys. Rev. Lett. 125, 097004 (2020). [21] M. Puviani, L. Schwarz, X.-X. Zhang, S. Kaiser, and D. Manske, Phys. Rev. B 101, 220507(R) (2020). [22] Z. M. Raines, A. A. Allocca, M. Hafezi, and V. M. Gal- itski, Phys. Rev. Research 2, 013143 (2020). [23] I. Zuti c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). [24] J. Linder and J. W. Robinson, Nat. Phys. 11, 307 (2015). [25] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002). [26] E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Lett. 88, 182509 (2006). [27] K. Ando, S. Takahashi, J. Ieda, Y. Kajiwara, H. Nakayama, T. Yoshino, K. Harii, Y. Fujikawa, M. Matsuo, S. Maekawa, et al. , J. Appl. Phys. 109, 103913 (2011). [28] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. B 66, 224403 (2002). [29] T. Kato, Y. Ohnuma, M. Matsuo, J. Rech, T. Jonck- heere, and T. Martin, Phys. Rev. B 99, 144411 (2019). [30] S. Takahashi and S. Maekawa, Phys. Rev. Lett. 88, 116601 (2002). [31] S. Takahashi and S. Maekawa, Sci. Technol. Adv. Mater. 9, 014105 (2008). [32] T. Wakamura, H. Akaike, Y. Omori, Y. Niimi, S. Taka- hashi, A. Fujimaki, S. Maekawa, and Y. Otani, Nat. Mater. 14, 675 (2015). [33] K.-R. Jeon, J.-C. Jeon, X. Zhou, A. Migliorini, J. Yoon, and S. S. Parkin, ACS Nano 14, 15874 (2020). [34] R. Ojaj arvi, T. T. Heikkil a, P. Virtanen, and M. A. Silaev, Phys. Rev. B 103, 224524 (2021). [35] T. T. Heikkil a, M. A. Silaev, P. Virtanen, and F. S. Berg- eret, Progress in Surface Science 94, 100540 (2019). [36] M. A. Silaev, Phys. Rev. B 102, 144521 (2020). [37] T. Cea, C. Castellani, G. Seibold, and L. Benfatto, Phys- ical review letters 115, 157002 (2015). [38] Supplementary material includes details of the symmetry operators and derivation of the susceptibilities. [39] In general h+is inhomogeneous and the gradient of h+ breaks the mirror symmetry locally. Thus we do not need to consider the mirror symmetry of the SC. [40] M. A. Silaev, D. Rabinovich, and I. Bobkova, arXiv preprint arXiv:2108.08862 (2021). [41] K. D. Usadel, Phys. Rev. Lett. 25, 507 (1970). [42] W. Belzig, F. K. Wilhelm, C. Bruder, G. Sch on, and A. D. Zaikin, Superlatt. and Microstruc. 25, 1251 (1999). [43] F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Rev. Mod. Phys. 77, 1321 (2005). [44] F. S. Bergeret and I. V. Tokatly, Phys. Rev. B 89, 134517 (2014). [45] F. S. Bergeret and I. V. Tokatly, Phys. Rev. Lett. 110, 117003 (2013). [46] The code used to obtain the results in this manuscript6 can be found at https://gitlab.jyu./jyucmt/sssc-higgs- rashba . [47] I. V. Tokatly, Phys. Rev. B 96, 060502(R) (2017).
0806.0420v1.Spin_Orbit_Coupling_in_an_f_electron_Tight_Binding_Model.pdf	arXiv:0806.0420v1 [cond-mat.mtrl-sci] 3 Jun 2008Spin-Orbit Coupling in an f-electron Tight-Binding Model M. D. Jones Department of Physics and Center for Computational Researc h, University at Buﬀalo, The State University of New York, Buﬀalo , NY 14260∗ R. C. Albers Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87501† (Dated: October 31, 2018) We extend a tight-binding method to include the eﬀects of spi n-orbit coupling, and apply it to the study of the electronic properties of the actinide eleme nts Th, U, and Pu. These tight-binding parameters are determined for the fcc crystal structure usi ng the equivalent equilibrium volumes. In terms of the single particle energies and the electronic d ensity of states, the overall quality of the tight-binding representation is excellent and of the sa me quality as without spin-orbit coupling. The values of the optimized tight-binding spin-orbit coupl ing parameters are comparable to those determined from purely atomic calculations. PACS numbers: 71.15.Ap, 71.15.Nc, 71.15.Rf, 71.20.Gj,71. 70.Ej I. INTRODUCTION The accurate determination of inter-atomic forces is crucial for almost all aspects of modeling the fundamen- tal behavior of materials. Whether one is interested in static equilibrium properties using Monte Carlo meth- ods, or time dependent phenomena using molecular dy- namics, the essential feature remains the origin, appli- cability, and transferability of the forces acting on the fundamental unit being modeled (atoms or molecules in most cases). First principles methods based on density functional theory have gained wide acceptance for their ease of use, relatively accurate determination of funda- mental properties, and high transferability. These tech- niques, however, are limited in their application by cur- rent computing technology to systems of a few hundred atoms or less (most commonly a few dozen atoms). Po- tentials that are classically derived (i.e., pair potentials) lack directional bonding (or at best add some bond angle information) and other quantum mechanical eﬀects but are computationally far more tractable for larger simu- lations. Recent advances in tight-binding (TB) theory, which include directional bonding, but treat only the most important valence electrons shells, therefore show a great deal of promise. TB models have become a useful method for the computational modeling of materials properties thanks to their ability to incorporate quantum mechanics in a greatly simpliﬁed theoretical treatment, making large accurate simulations possible on modern digital computers1,2. Another advantage of these TB models is their ability to treat a general class of problems that include directional bonding between valence electrons, of particularimportance fortransition metal and f-electron materials. Finally, TB models are widely used in many- bodyformalismsfortheone-electronpartofthe Hamilto- nian. It is therefore a useful representation of the band- structure for a more sophisticated treatment ofelectroniccorrelation, and has so been used3, for example, in dy- namical mean-ﬁeld theory applications for Pu. In this report we present recent developments towards a transferable tight-binding total energy technique appli- cable to heavy metals. With the addition of spin-orbit coupling eﬀects for angular momentum up to (and in- cluding)f-character, we demonstrate the applicability of this technique for the element Pu, of particular interest for its position near the half-ﬁlling point of the 5 fsub- shell in the actinide sequence and the boundary between localized and delocalized f-electrons4. II. TB METHOD The TB model used in this report is similar to that used in the handbook by Papaconstantopoulos5. We have extended the calculations to include f-electrons6 and spin-orbit coupling7. As such, in this report we will elaborate only on those aspects of the technique that are unique to this work. A very brief recapitulation of the underlying TB method and its approximations is in- cluded to create the proper context for the addition of f-electrons and spin-orbit coupling. The Slater-Koster method8consists of solving the sec- ular equation, Hψi,v=ǫi,vSψi,v, (1) for the single-particle eigenvalues and orbitals, under the following restrictions: terms involving more than two centers are ignored, terms where the orbitals are on the same atomic site are taken as constants, and the result- ing reduced set of matrix elements are treated as variable parameters. The Hamiltonian, H, includes the labels for orbitals having generic quantum numbers α,βlocalized on atomsi,j, where the eﬀective potential is assumed to be spherical, and can be represented as a sum over2 atomic centers, Hαi,βj=/angbracketleftigg α,i/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle−∇2+/summationdisplay kVeﬀ k/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleβ,j/angbracketrightigg ,(2) which we further decompose into “on-site” and “inter-site” terms, Hαi,βj=eαδαβδij+Eαi,βj/negationslash=i, (3) where the on-siteterms, eα, representterms in which two orbitals share the same atomic site, and Eαi,βj/negationslash=i=/summationdisplay neik·(Rn+bj−bi)/integraldisplay drψα(r−Rn−bi)Hψβ(r−bj), (4) are the remaining energy integrals involving orbitals lo- cated on diﬀerent atomic sites, and we have used transla- tional invariance to reduce the number of sums over bra- vais lattice points {Rn}, and the bidenote atomic basis vectors within the repeated lattice cells. Note that terms which have both orbitals located on the same site, but the eﬀective potential ( Veﬀ) on other sites have been ig- nored. These contributions are typically taken to be “en- vironmental”correctionstotheon-siteterms, andarenot accounted for in the usual Slater-Koster formalism. For the inter-site terms, the two center approximation also consists of ignoring these additional terms in which the eﬀective potential, Veﬀ, does not lie on one of the atomic sites. Once this approximation has been made, the inter- atomic (i/ne}ationslash=j) matrix elements reduce to a simple sum over angular functions, Gll′m(Ωi,j), and functions which depend only upon the magnitude of the distances be- tween atoms, Hαi,βj=/summationdisplay hll′m(rij)Gll′m(Ωi,j), (5) wherewe havenowadoptedthe usualconventionofusing the familiar l,mangular momentum quantum numbers, and the axis connecting the atoms is the quantization axis. Anequivalentexpressionfor sll′mtermsexistswhen non-orthogonal orbitals are used. The basis set used for theαandβquantum states are the cubic harmonics9 whose functional forms are given in Table I (with ap- propriate normalization factors) where \|±/an}b∇acket∇i}htdenotes the spin-state, which we will need for spin-orbit coupling. The Slater-Kostertables for the sp3d5matrix elements can be found in standard references10, and we have used the tabulated results of Takegahara et al.11for the addi- tional matrix elements involving f-electrons. Typical TB applications are then reduced to using TB as an interpo- lation scheme; the matrix elements ( hll′m,sll′mandeα) are determined by ﬁtting to ab-initio calculated quanti- ties such as the total energy and band energies. In this study we restrict ourselves to the determina- tion of optimal TB parameters at the neighbor distances in the face-centered cubic crystal structure (often used as a surrogate for the more complex ground state crystal structure of the actinides) near the equilibrium volume.Suchtabulationshavebeenextensivelyused5inthestudy of materials with lower atomic number. To the best of our knowledge this is the ﬁrst time that such parame- ters have been presented for light actinide elements that include the f-electron orbitals (although similar param- eters have been determined for the elements Ac and Th in ansp3d5basis5). The TB parameter values so derived are available (on request) from the authors. A. Spin-orbit coupling The primary impact of spin-orbit coupling is to non- trivially couple electrons of diﬀerent spin states, thus doubling the size of the TB Hamiltonian. The spin-orbit contribution to the Hamiltonian is given by Hso=ξ(r)L·S, (6) whereξ(r) = (α2/(2r))(∂V/∂r),Vis the total (crystal) potential. We neglect contributions from more than one center. A new Hamiltonian matrix can then be deﬁned in terms of the spinless one, H=H+Hso=/parenleftbigg H+1 2ξLz1 2ξL− 1 2ξL+H−1 2ξLz/parenrightbigg (7) where ξnl=/planckover2pi1/integraldisplay∞ 0ξ(r)/bracketleftbig R0 nl(r)/bracketrightbig2r2dr, (8) is the spin-orbit coupling parameter between orbitals of orbital angular momentum land primary quantum num- bernlocated on the same atom, L±are the usual raising and lowering operators, and Lzthe azimuthal angular momentum operator, L±Ylm(θ,φ) =/planckover2pi1/radicalbig l(l+1)−m(m±1)Ylm±1 LzYlm(θ,φ) =/planckover2pi1mYlm. The functions R0 nl(r) are the non-relativistic radial wave functions. The spin-orbit contributions to the Hamilto- nian matrix can then be expressed in term of the TB3 TABLE I: TB basis functions used for an sp3d5f7calculation. Note that fl(r) = 1/rl. l=0 l=1 l=2 l=3 \|s±/angbracketright=p 1/4π\|±/angbracketright \| p1±/angbracketright=p 3/4πf1(r)x\|±/angbracketright \| d1±/angbracketright=p 5/16πf2(r)xy\|±/angbracketright \| f1±/angbracketright= 2p 105/16πf3(r)xyz\|±/angbracketright \|p2±/angbracketright=p 3/4πf1(r)y\|±/angbracketright \| d2±/angbracketright= 2p 15/16πf2(r)yz\|±/angbracketright \| f2±/angbracketright=p 7/16πf3(r)x(5x2−3r2)\|±/angbracketright \|p3±/angbracketright=p 3/4πf1(r)z\|±/angbracketright \| d3±/angbracketright= 2p 15/16πf2(r)zx\|±/angbracketright \| f3±/angbracketright=p 7/16πf3(r)y(5y2−3r2)\|±/angbracketright \|d4±/angbracketright=p 15/16πf2(r)(x2−y2)\|±/angbracketright \| f4±/angbracketright=p 7/16πf3(r)z(5z2−3r2)\|±/angbracketright \|d5±/angbracketright=p 5/16πf2(r)(3z2−r2)\|±/angbracketright \| f5±/angbracketright=p 105/16πf3(r)x(y2−z2)\|±/angbracketright \|f6±/angbracketright=p 105/16πf3(r)y(z2−x2)\|±/angbracketright \|f7±/angbracketright=p 105/16πf3(r)z(x2−y2)\|±/angbracketright basis functions listed in Table I. Rather than list contri- butions for the 32x32 matrix, here we list the matrices in the sub-blocks corresponding to each orbital angular mo- mentum. The panddcontributionshavebeen previously discussed in relation to the tight-binding formalism12,13;to the best of our knowledge no fcontribution has yet appeared in the literature. For completeness we detail the spin-orbit contribution for all values of the angular momentum up to l= 3. Hso p=ξnp 2 0−i0 0 0 1 i0 0 0 0 −i 0 0 0 −1i0 0 0−1 0i0 0 0−i−i0 0 1i0 0 0 0 , (9) Hso d=ξnd 2 0 0 0 2 i0 0 1 −i0 0 0 0i0 0 −1 0 0 −i−i√ 3 0−i0 0 0 i0 0 −1√ 3 −2i0 0 0 0 0 i1 0 0 0 0 0 0 0 0 i√ 3−√ 3 0 0 0−1−i0 0 0 0 0 −2i0 1 0 0 −i−i√ 3 0 0 −i0 0 i0 0 1 −√ 3 0i0 0 0 0i−1 0 0 2 i0 0 0 0 0i√ 3√ 3 0 0 0 0 0 0 0 , (10) Hso f=ξnf 4 0 0 0 0 0 0 2 i0 0 0 0 2 i2 0 0 03i 20 0it0 0 0 0 −3 20 0t 0−3i 20 0it0 0 0 0 03i 20 0it 0 0 0 0 0 0 0 03 2−3i 20t it 0 0 0−it0 0−i 20−2i0 0−t0 01 2 0−it0 0i 20 0−2 0 0 −it0 0−i 2 −2i0 0 0 0 0 0 0 −t−it0−1 2i 20 0 0 0 0 2 i−2 0 0 0 0 0 0 0 −2i 0 0 03 20 0−t0 0−3i 20 0−it0 0 0 03i 20 0it03i 20 0−it0 0 0−3 2−3i 20−t it 0 0 0 0 0 0 0 0 −2i0 0t0 0−1 20 0it0 0i 20 2 0 0 −it0 0−i 20it0 0−i 20 0 0t−it01 2i 20 2i0 0 0 0 0 0 , (11) wheret=√ 15/2. B. Fitting the Parameters The values of the TB parameters were determined us- ing standard non-linear least squares optimization rou-4 tinesbymatchingenergybandvaluesderivedfromhighly accurate ﬁrst principles density functional theory (DFT) calculations14. The technique is described in detail in a previous work6, where the DFT calculations in this case used a generalized gradient approximation DFT functional15, and the improved tetrahedron scheme16for Brillouin zone integrations. In this study we use as a starting point high quality ﬁts to the scalar-relativistic energy bands and approximate atomic values of the spin- orbit parameters. The ﬁrst step is to then use this ﬁt for ﬁtting the relativistic energy bands including spin-orbit coupling. Successive optimization steps then relax only the spin-orbit coupling paramaters (step 1), the remain- ing on-site parameters (step 2), and ﬁnally the inter-site terms (step 3). The ﬁt quality through these steps is shown in Figure 1. Note that the quality of the ﬁnal ﬁt is comparable to the original ﬁt quality (open symbols at step 3) when only scalar-relativistic eﬀects were taken into account. 1 2 3 Optimization Step00.1Average rms Fitting Errors [eV]Th U Pu FIG. 1: TB ﬁt quality in terms of the cumulative root mean square (rms) errors at various steps of the optimization pro ce- dure. Step 1 relaxes the spin-orbit parametes ( ξnl), 2 relaxes the remaining on-site parameters, and 3 is a full relaxation of all parameters. Open symbols at Step 3 indicate the original scalar-relativistic ﬁt quality. Note that the cumulative r ms error is over all of the ﬁtted bands (20 bands for Th, U, and Pu). Although the spin-orbit coupling is an atomic quantity , the improvement of our results in step 3 (which relaxes inter - site parameters) indicates some environmental eﬀects shou ld also be taken into account. III. APPLICATION TO THE LIGHT ACTINIDES, TH, U, AND PU A. Energy bands including spin-orbit coupling The ﬁrst comparison between the TB ﬁt and FLAPW calculations are the energy bands shown in Figure 2. Note the excellent agreement between the two sets ofcalculations (the cumulative root mean square error in the TB ﬁts to the ﬁrst 20 energy bands in the irreducible Brillouinzoneis0.013,0.013,and0.072Ry, respectively). Also note that we have included the “semi-core” 6 p (a)Th (b)U (c)Pu FIG. 2: TB energybands for Th ( a= 9.61), U(a= 8.22), and Pu (a= 8.14), shown in comparison with FLAPW valence energy bands (dotted lines). Note the excellent agreement. The abscissa for each calculation has been shifted such that the Fermi energy is at zero. Higher valence states (above the ﬁrst 20) are not ﬁt, hence the poorer ﬁt quality well above the Fermi level. states in the ﬁt to better ﬁx the available pstates in the TB basis. To expand the energy scale comparing the va- lence bands, the ﬁt quality for the semi-core 6p states is5 FIG. 3: TB energy bands (dashed lines) for Pu semi-core 6p states, compared with FLAPW values (solid lines). shown separately in Figure 3 for Pu (all three elements have similarexcellent ﬁt quality for the more localized 6p states). Note that higher energy bands (well above the Fermi level) are not ﬁt, hence the larger discrepancies for those levels. B. Density of states including spin-orbit coupling We also compare the total density of states (DOS) be- tween TB and FLAPW methods in Figure 4. The TB method shown in the ﬁgure used a simple Fermi-Dirac temperature smearing method (with kBT= 500) for integrating over the irreducible wedge of the Brillouin zone, while the FLAPW calculations used the improved tetrahedron16method with Gaussian smear- ing. From the comparison between the TB and FLAPW methods shown in the above ﬁgure, we note that the agreementisexcellent,withallmajorfeaturesintheDOS reproduced by the TB calculations. There is a slight re- duction in the height of some of the larger peaks in the DOS for the TB technique, most likely due to the inabil- ity of the temperature smearing technique to represent the ﬁner grained features as well as the improved tetra- hedron method does. C. Spin-orbit coupling terms It is interesting to compare the spin-orbit coupling pa- rameters,ξnl, predicted by TB theory for the various valence shells relative to the values predicted by accu- rate Hartree-Fock-Slatercalculations ofisolated atoms19. This comparison is shown in Table II. Note the overall agreement between the TB ﬁtted pa- rameters and the atomic values. The overall shift of a few tenths of an eV for the TB values is interesting, and this trend could be representative of crystal ﬁeld eﬀects (this speculation could be checked by performing equiv- alent ﬁts at diﬀerent densities). Equivalently, one can-24 -20 -16 -12 -8 -4 0 4 8 ε−EF [eV]0481216Total DOS [states/eV]FLAPW TB (a)Th -28 -24 -20 -16 -12 -8 -4 0 4 8 ε−EF [eV]04812Total DOS [states/eV]FLAPW TB (b)U -28 -24 -20 -16 -12 -8 -4 0 4 8 ε−EF [eV]04812Total DOS [states/eV]FLAPW TB (c)Pu FIG. 4: TB (dotted lines) and FLAPW (solid lines) total DOS, including spin-orbit coupling. Note that the TB calcu- lation is in quite good agreement with the FLAPW results, despite using a diﬀerent BZ integration method.The absciss a for each calculation has been shifted such that the Fermi en- ergy is at zero. compare the spin-orbit splitting of the electronic energy levels with the purely atomic case. This comparison is also shown in Table II.6 TABLE II: Values of spin-orbit coupling strength, ξnl, and spin-orbit splittings, ∆ nl= (2l+ 1)ξnl/2, for the various valence electron shells predicted by the TB ﬁt compared with purely a tomic values using relativistic density functional theory (DFT)17, a Dirac-Slater atomic code (DIRAC)18, and relativistic Hartree-Fock-Slater (HFS)19atomic calculations. Dashed entries are used for orbitals not populated in the atomic calculations. Values are in eV. Method ξ6p ∆6p ξ5d ∆5d ξ5f ∆5f Th DIRAC 5.29 7.94 0.20 0.51 0.19 0.66 DFT 5.24 7.86 0.21 0.52 – – HFS 4.09 6.14 0.30 0.75 – – TB 4.19 6.29 0.20 0.51 0.18 0.62 U DIRAC 5.96 8.94 0.19 0.47 0.24 0.83 DFT 5.90 8.85 0.20 0.50 0.24 0.84 HFS 4.38 6.57 0.30 0.75 0.35 1.24 TB 4.64 6.96 0.23 0.58 0.42 1.48 Pu DIRAC 6.92 10.38 0.20 0.51 0.31 1.10 DFT – – – – – – HFS 4.60 6.90 – – 0.41 1.43 TB 5.23 7.84 0.59 1.46 0.54 1.90 IV. CONCLUSIONS We have included f-electron and spin-orbit eﬀects in a standard tight-binding method for solids in order to advance simpler simulation methods that are capable of the accuracy of more expensive, full-potential density- functional techniques. We have applied this TB tech- nique to elemental fcc Th, U, and Pu, and have achieved excellent agreement with the electronic properties pre- dicted using a highly accurate FLAPW method. The ﬁtted spin-orbit coupling parameters match very well the values independently predicted by atomic electronic structure calculations. This methodology bodes well for further TB investigations, especially for the study of de- fects, phonons, and dynamical properties. In future work we intend to develop a more transferable model based ona TB total energy formalism6, which should allow the straightforward calculation of detailed materials proper- ties. Acknowledgments This work was carried out under the auspices of the NationalNuclearSecurityAdministrationofthe U.S.De- partment of Energy at Los Alamos National Laboratory under Contract No. DE-AC52-06NA25396. Calculations were performed at the Los Alamos National Laboratory and the Center for Computational Research at SUNY– Buﬀalo. FLAPW calculations were performed using the Wien2k package14. We thank Jian-Xin Zhu for providing helpful remarks. ∗Electronic address: jonesm@ccr.buﬀalo.edu †Electronic address: rca@lanl.gov 1C. M. Goringe, D. R. Bowler, and E. Hernandez, Rep. Prog. Phys. 60, 1447 (1997). 2D. A. Papaconstantopoulos and M. J. Mehl, J. Phys.: Con- dens. Matter 15, R413 (2003). 3J.-X. Zhu, A. K. McMahan, M. D. Jones, T. Durakiewicz, J. J. Joyce, J. M. Wills, and R. C. Albers, Phys. Rev. B 76, 245118 (2007). 4R. C. Albers, Nature 410, 759 (2001). 5D. A. Papaconstantopoulos, Handbook of the Band Struc- ture of Elemental Solids (Plenum Press, New York, 1986). 6M. D. Jones and R. C. Albers, Phys. Rev. B 66, 134105 (2002). 7M. Lach-hab, M. J. Mehl, and D. A. Papaconstantopoulos, J. Phys. Chem. Solids 63, 833 (2002). 8J. C. Slater and G. F. Koster, Phys. Rev. 94, 1498 (1954). 9F. von Der Lage and H. A. Bethe, Phys. Rev. 71, 612 (1947). 10W. A. Harrison, Electronic Structure and the Properties of Solids(Freeman, San Francisco, CA, USA, 1980). 11K. Takegahara, Y. Aoki, and A. Yanase, J. Phys. C 13,583 (1980). 12J. Friedel, P. Lenglart, and G. Leman, J. Phys. Chem. Solids25, 781 (1964). 13D. J. Chadi, Phys. Rev. B 16, 790 (1977). 14P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka, and J. Luitz, WIEN2K, An Augmented Plane Wave + Local Orbitals Program for Calculating Crystal Properties (Karl- heinz Schwartz, Techn. Universitt Wien, Austria, 2001. ISBN 3-9501031-1-2). 15J. P. Perdew, S. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996). 16P. E. Bl¨ ochl, O. Jepsen, and O. K. Andersen, Phys. Rev. B49, 16223 (1994). 17S. Kotochigova, Z. H. Levine, E. L. Shirley, M. D. Stiles, and C. W. Clark, http://math.nist.gov/DFTdata (1996). 18ADF2004.01, SCM, Theoretical Chemistry, Vrije Universiteit, Amsterdam, The Netherlands, http://www.scm.com . 19F. Herman and S. Skillman, Atomic Structure Calculations (Prentice-Hall, Englewood Cliﬀs, NJ, USA, 1963).
1811.09088v1.Enhanced_Rashba_spin_orbit_coupling_in_core_shell_nanowires_by_the_interfacial_effect.pdf	Enhanced Rashba spin-orbit coupling in core-shell nanowires by the interfacial eect Pawe l W ojcik,1,a)Andrea Bertoni,2,b)and Guido Goldoni3, 2,c) 1)AGH University of Science and Technology, Faculty of Physics and Applied Computer Science, 30-059 Krakow, Al. Mickiewicza 30, Poland 2)S3, Istituto Nanoscienze-CNR, Via Campi 213/a, 41125 Modena, Italy 3)Department of Physics, Informatics and Mathematics, University od Modena and Reggio Emilia, Italy (Dated: November 26, 2018) We report on ~k~ pcalculations of Rashba spin-orbit coupling controlled by external gates in InAs/InAsP core-shell nanowires. We show that charge spilling in the barrier material allows for a stronger symmetry breaking than in homoegenous nano-materials, inducing a specic interface-related contribution to spin-orbit coupling. Our results qualitatively agree with recent experiments [S. Futhemeier et al. , Nat. Commun. 7, 12413 (2016)] and suggest additional wavefunction engineering strategies to enhance and control spin-orbit coupling. Understanding and controlling spin-orbit coupling (SOC) is critical in semiconductor physics. In particular, in semiconductor nanowires (NWs)1{7SOC is essential for to the development of a suitable hardware for topo- logical quantum computation8,9, with qubits encoded in zero-mode Majorana states which are supported by hy- brid semiconductor-superconductor NWs10{15. Among other parameters, qubit protection at suciently high temperatures relies on a large SOC which determines the topological gap. Additionally, electrical control of SOC is necessary in the realization of spintronic devices16{26. SOC arises from the absence of inversion symmetry of the electrostatic potential. In semiconductor NWs, typically having a prismatic shape, nite SOC may be induced by distorting the quantum connement (Rashba SOC27) by means of external gates, with the advantage of electrical control. A lattice contribution (Dresselhaus SOC28) is typically small and may vanish in specic crys- tallographic directions - for zincblende NWs, the Dres- selhaus term vanishes along [111] due to the inversion symmetry. The Rashba SOC constant Rhas been investigated experimentally in homogeneous NWs based on the strong SOC materials InSb29,30and InAs31{35. Recently5, we reported on a ~k~ papproach applied to homogeneous NWs which predicts Rfrom compositional and struc- tural parameters only. Our calculations performed for InSb NWs5and InAs NWs36generally conrm recent experiments in homogeneous NWs29,31, exposing values ofRexceeding by one order of magnitude those re- ported for 2D analogous planar systems37{39. Moreover, Rproved to be strongly tunable with external gates in samples and congurations which can be routinely real- ized with current technology. a)Electronic mail: pawel.wojcik@s.agh.edu.pl b)Electronic mail: andrea.bertoni@nano.cnr.it c)Electronic mail: guido.goldoni@unimore.itFor a quantitative prediction of SOC, it is necessary to take into account valence-to-conduction band coupling, the explicit geometry and crystal structure of the NW, and the electron gas distribution which, in turn, must be self-consistently determined by quantum connement eects, interaction with dopants and electron-electron in- teraction. Indeed, in NWs the electron gas localization, and ensuing SOC, is a non-trivial result of competing en- ergy contributions. As a function of doping concentration and ensuing free charge density, the electron gas evolves from a broad cylindrical distribution in the NW core (low density regime) to coupled quasi-1D and quasi-2D chan- nels at the NW edges (large density regime)40{43. Until polygonal symmetry holds, R= 0 regardless. However, external gates easily remove the symmetry; again, how Rmoves from zero under the in uence of the external gates strongly depends on the charge density regime5. In this Letter we extend and apply the ~k~ papproach to core-shell NWs (CSNWs) and expose a novel mechanism through which SOC can be further tailored, and possibly enhanced. Epitaxially overgrown shells are often used in NW technology, either as a passivating layer improv- ing optical performance44, or as a technique to engineer radial heterostructures45. Here we show that CSNWs Figure 1. Schematics of a InAs/InAsP CSNW grown along [111] with a bottom gate.arXiv:1811.09088v1 [cond-mat.mes-hall] 22 Nov 20182 allow for an increased exibility in distorting the elec- tron gas of the NWs, giving rise to a specic, interfacial SOC contribution46,47which substantially increases the total SOC. We make the case for InAs/InAsP CSNWs, a systems of specic interest in photonics48and electri- cal engineering49. Our results qualitatively agree with the recent experiments by Furthmeier et al. in Ref. 50, where the enhancement of SO coupling was measured in GaAs/AlGaAs CSNW, and establish a strategy to in- crease the SOC in Majorana InAs NWs. We consider CSNWs with hexagonal cross-section51 grown along [111] (see Fig. 1), assuming in-wire trans- lational invariance along z. The used ~k~ papproach is described in full in Ref. 5; here we focus on generaliza- tions required to account for the contribution of the in- ternal heterointerface. The 8 8 Kane Hamiltonian is given by16 H88=HcHcv Hy cvHv ; (1) whereHcandHvare the diagonal matrices correspond- ing to the conduction ( 6c) and valence ( 8v, 7v) bands whose expressions are given in Ref. 5. Using the pertur- bative transformation H(E) =Hc+Hcv(HvE)1Hy cv, the Hamiltonian (1) reduces to a 2 2 eective Hamil- tonian for the conduction band electrons. Emphasizing the dependence of material parameters on the position, ~ = (x;y), H= ~2 2r2D1 m(~ )r2D+~2k2 z 2m(~ )+Ec(~ ) +V(~ ) 122+ [^x(~ )x+ ^y(~ )y]kz;(2) wherex(y)are the Pauli matrices and mis the eective mass given by 1 m(~ )=1 m0+2P2 3~22 E0(~ )+1 E0(~ ) + 0(~ ) ;(3) wherePis the conduction-to-valance band coupling pa- rameter. In Eq. (2), ^ x, ^yare the SOC operators ^x=i 3P2^ky(~ )i 3P2(~ )^ky; (4) ^y=i 3P2^kx(~ )i 3P2(~ )^kx; (5) and(~ ) is a material-dependent coecient obtained as follows. In the i-th layer i(~ ) =1 Ec;i+V(~ )E0;iE(6) 1 Ec;i+V(~ )E0;i0;iE; whereEc,E0and 0are the conduction band edge, the energy gap and the split-o band gap, respectively. As- suming that the above parameters change as a step-likefunction at the interfaces (~ ) =X i[i(~ )i+1(~ )] i(~ ); (7) where the sum is carried out over all the layers, and i(~ ) is the shape function, which for the hexagonal section is given by i(~ ) = [(x+xi)(xxi)][(y+yi)(yyi)] [(xy+xi)(xyxi)]; (8) whereis the Heaviside'a function and ( xi;yi) denotes the position of the ( i)-th interface. Further Taylor ex- pansion gives i(~ )1 E0;i+ 0i1 E0;i (9) + 1 E2 0;i1 (E0;i+ 0;i)2! (Ec;i+V(x;y)E): Substituting (9) into Eqs. (4) and (5), the Rashba cou- pling constants can be written as x(y)(~ ) =V x(y)(~ ) +int x(y)(~ ); (10) i.e., the sum of the SOC induced by the electrostatic potential asymmetry, V x(y)(~ )X i1 3P2 1 E2 0;i1 (E0;i+ 0;i)2! @V(~ ) @y(x); (11) and the interface SOC, related to the electric eld at the interfaces between shells, int x(y)(~ )X i1 3P2 ~i~i+1@ i(~ ) @y(x); (12) with ~i=1 E0;i+0;i1 E0;i: Projecting the 3D Hamiltonian (2) on the basis of in- wire states n(~ ) exp(ikzz), where the envelope functions n(~ ) are determined by the strong connement in the lateral direction, leads to SOC matrix elements ;nm x(y)=Z Z n(~ ) x(y)(~ ) m(~ )d~ ; (13) where identies the electrostatic ( =V) or the inter- facial ( =int) contribution. For the NW in Fig. 1 with a single bottom gate, ;nn y= 0 due to inversion symmetry about y. Moreover, here we focus on the lowest intra-subband coecient, n= 1. Below we discuss the SOC constant R=11 x and corresponding interfacial and electrostatic compo- nents, R=V;11 xand R=int;11 x, respectively. The electronic states in the CSNW section, n(~ ), are calculated by a mean-eld self-consistent Sch odinger- Poisson approach40. We neglect the exchange-correlation potential which is substantially smaller than the Hartree3 InAs InAs 0:9P0:1 m[m0]0.0265 0.0308 Ec[eV]0.252 0.3 E0[eV]0.42 0.5 0[eV]0.38 0.35 Table I. Bulk parameters used in calculations56. potential40,52,53. The gradient of the self-consistent po- tentialV(~ ) and the corresponding envelope functions n(~ ) are nally used to determine Rfrom Eq. (13). Material parameters mismatch at the interfaces is taken into account solving the eigenproblem H n=E n with boundary conditions46,54 (i) n(~ k) = (j) n(~ k) (14) ~2 2m(i)r2D (i) n(~ k)~2 2m(j)r2D (j) n(~ k) (15) +[(j)(~ k)(i)(~ k)](x+y)kz (i) n(~ k) = 0; where~ kis the position of the interface between i-th and j-th shells. Equations (14), (15), depend on both the po- tentialV(~ ) at the interface and the energy E. We elim- inate this dependence neglecting the term proportional to (Ec;i+V(~ )E) in the Taylor series, Eq. (9). Then, the interface contributions (12) are determined fully by material parameters. This assumption, justied when (j)(i)is small, neglects the SOC related to the motion of electron in the ~ plane, which in general contributes to the SOC by the boundary conditions. Below we investigate a InAs 50 nm-wide core (mea- sured facet-to-facet) surrounded by a 30 nm InAs 1xPx shell, withx= 0:1 which allows to neglect strain-induced SOC.55Furthermore, as shown below, interfacial SOC is enhanced by the easy penetration of envelope func- tions in low band oset barriers, here only 48 meV high. Simulations have been carried out for a temper- atureT= 4:2 K, in the constant electron concentration regime. The parameters adopted are given in Tab. I. P is assumed to be constant throughout the materials and EP(InAs) = 2m0P2=~2= 21:5 eV. The calculated SOC coecients for the InAs/InAsP CSNW of Fig. 1 as a function of the back gate voltage is reported in Fig. 2(a). The SOC constant is trivially zero ifVg= 0, due to the overall inversion symmetry. At any nite voltage the inversion symmetry is removed, hence R6= 0. As shown in Fig. 2(a), the total Rensues from two dierent contributons, namely interfacial and elec- trostatic, whose magnitude is of the same order. It is thus crucial to include both of them in the assessment of SOC in CSNWs. Note that the electrostatic component almost coincides with the value for an InAs NW with the same geometry, but no overgrown shell57. However, for this specic nanostructure, the largest part of Ris due to the interfacial contribution, which is 50% larger than the electrostatic one. While the ratio between the two contributions is nearly independent of Vg[see the in- Figure 2. (a) Lines: total, electrostatic and interfacial SOC constants vsgate voltage Vg, according to labels. Dots: to- tal SOC constant for an equivalent homegeneous InAs NW. Inset: ratio between interfacial and electrostatic components, int R=V R. Results for ne= 107cm1. See text for structure and material parameters. (b) Electrostatic and interfacial SOC constants vsVgaroundVg= 0 showing shooting up of SO couplings for higher electron density. set in Fig. 2(a)], they are both strongly anisotropic with respect to the eld direction. This is due to the dierent eects on the charge density, as discussed in Ref. 5. This eect can also be grasped from the probability distribu- tion reported in Fig. 3 (top and bottom rows). Indeed, the positive Vgpushes electron states towards the in- terface opposite to the gate, where the gradient of the self-consistent eld is low. On the other hand, at Vg<0 electrons are pulled to the region of the nearest interface with the stronger electric eld, additionally strengthened by the electron-electron interaction5. Figure 3. Top row: Square of the ground state envelope func- tionj 1(x;y)j2. Middle row: linear density of the interfacial SOC constants at interfaces. Bottom row: self-consistent po- tential prole (black line) and j 1(x;y)j2(red line) along the facet-to-facet dashed line marked in the top-middle panel. Re- sults at selected gate voltages Vg=0:1;0;0:1 V for the same structure as in Fig. 2. The value of int Rdepends on the penetration of the4 wave function into the interfaces. As shown in Fig. 3 (middle row) the linear density of interfacial SOC at the interfaces int R= 1(~ )int R(~ ) 1(~ ) is nite almost ev- erywhere, but it has opposite sign at opposite facets.58 For a centro-symmetric system ( Vg= 0) the overall value is zero, since opposite contributions cancel out exactly. We stress a remarkable dierence between CSNWs and analogous planar structures. In a planar asymmetric quantum well, for example, R6= 0. In a CSNW with an embedded quantum well, however, the overall symmetry is recovered even if each facet of the quantum well is indi- vidually asymmetric. Therefore, opposite segments have opposite Rashba contributions and compensate. How- ever, any asymmetric gate potential unbalances opposite contributions, the total eect being related to the amount of envelope function at the interface. Figure 4. Electrostatic V Rand interfacial int Rcontributions of SOC constant vselectron density ne. Results for Vg= 0:1 V. Inset: ratio int R=V Rvsne. Note the almost linear increase of RwithVg. This behaviour is observed in a relatively small charge density regime: the average Coulomb energy is small, most of the charge is located in the core, and it is relatively rigid to an applied transverse electric eld. At larger densi- ties, however, charge moves at the interfaces to minimize Coulomb interaction40, with negligible tunneling energy between opposite facets. In this regime, the symmetric charge density distribution is unstable and it is easily dis- torted by an electric eld5. Accordingly, SOC constant shoots around Vg= 0 as soon as the gate is switched on - see Fig. 2(b). As we show in Fig. 4, both SOC components substan- tially increase in intensity with charge density, while their ratio is weakly aected, it being rather a property of the nano-material (band parameters and band oset). This is explicitly shown in Fig. 5, where the two contributions are plot vsthe stechiometric fraction x. At lowx, pene- tration is very large, and the interfacial eect is dominant (of course for x= 0 the heterostructure is an homoge- neous NW with a larger diameter), while as x= 0:15 the two contributions are comparable, as also shown in theinset. Figure 5. The interfacial int R(blue circles) and electrostatic V R(red circles) SOC constants vsInAs 1xPxalloy composi- tion,x. Inset:int R=V Rvsx. Results for Vg=0:1 V and ne= 107cm1. To summarize, we have shown that Rashba SOC in CSNWs is increased by the eect of the radial heteroint- erface, and its control via external metallic gates may be highly improved by this interfacial eect. Although we did not attempt to optimize Rin the many parameter space allowed by CSNWs, our results suggest that a gen- eral strategy to enhance SOC in CSNWs relies on a mod- ication of the compositional structure exploiting asym- metric penetration of the wave function into the shell layer. This work was partially supported by the AGH UST statutory tasks No. 11.11.220.01/2 within subsidy of the Ministry of Science and Higher Education and in part by PL-Grid Infrastructure. P.W. was supported by Na- tional Science Centre, Poland (NCN) according to deci- sion 2017/26/D/ST3/00109. REFERENCES 1I. A. Kokurin, Physica E 74, 264 (2015). 2I. A. Kokurin, Solid State. Commun. 195, 49 (2014). 3M. Kammermeier, P. Wenk, J. Schliemann, S. Heedt, and T. Sch apers, Phys. Rev. B 93, 205306 (2016). 4G. W. Winkler, D. Varjas, R. Skolasinski, A. A. Soluyanov, M. Troyer, and M. Wimmer, Phys. Rev. Lett. 119, 037701 (2017). 5P. W ojcik, A. Bertoni, and G. Goldoni, Phys. Rev. B 97, 165401 (2018). 6T. Campos, P. E. Faria Junior, M. Gmitra, G. M. Sipahi, and J. Fabian, Phys. Rev. B 97, 245402 (2018). 7J.-W. Luo, S.-S. Li, and A. Zunger, Phys. Rev. Lett. 119, 126401 (2017). 8J. Alicea, Phys. Rev. B 81, 125318 (2010). 9J. Sau, Physics 10, 68 (2017). 10A. Kitaev, Ann. Phys. 303, 2 (2003). 11J. D. Sau, S. Tewari, and S. Das Sarma, Phys. Rev. B 85, 1 (2012).5 12V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A. M. Bakkers, and L. P. Kouwenhoven, Science 336, 1003 (2012). 13J. Mastom aki, S. Roddaro, M. Rocci, V. Zannier, D. Ercolani, L. Sorba, I. J. Maasilta, N. Ligato, A. Fornieri, E. Strambini, and F. Giazotto, Nano Research 10, 3468 (2017). 14S. M. Albrecht, A. P. Higginbotham, M. Madsen, F. Kuemmeth, T. S. Jespersen, J. Nyg aard, P. Krogstrup, and C. M. Marcus, Nature 531, 206 (2016). 15A. Manolescu, A. Sitek, J. Osca, L. m. c. Serra, V. Gudmundsson, and T. D. Stanescu, Phys. Rev. B 96, 125435 (2017). 16J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and I. Zuti c, Acta Physica Slovaca 57, 565 (2007). 17S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1990). 18J. Schliemann, J. C. Egues, and D. Loss, Phys. Rev. Lett. 90, 146801 (2003). 19P. W ojcik and J. Adamowski, Semicond. Sci. Technol. 31, 115012 (2016). 20P. W ojcik, J. Adamowski, B. J. Spisak, and M. Wo loszyn, J. Appl. Phys. 115, 104310 (2014). 21P. W ojcik and J. Adamowski, Sci Rep 7, 45346 (2017). 22P. P. Das, J. Bhandari, N abd Wan, M. Cahay, K. B. Chetry, R. S. Newrock, and S. T. Herbert, Nanotechnology 23, 215201 (2012). 23P. Debray, S. Rahman, and M. Muhammad, Nature Nano Tech- nol.4, 759 (2009). 24M. Kohda, S. Nakamura, Y. Nishihara, K. Kobayashi, T. Ono, J. O. Ohe, Y. Tokura, T. Mineno, and J. Nitta, Nature Com- munications 3, 1082 (2012). 25F. Rossella, A. Bertoni, D. Ercolani, M. Rontani, L. Sorba, F. Beltram, and S. Roddaro, Nat. Nanotechnol. 9, 997 (2014). 26A. Iorio, M. Rocci, L. Bours, M. Carrega, V. Zannier, L. Sorba, S. Roddaro, F. Giazotto, and E. Strambini, ArXiv e-prints (2018), arXiv:1807.04344 [cond-mat.mes-hall]. 27E. I. Rashba, Sov. Phys. Solid State 2, 1109 (1960). 28G. Dresselhaus, Phys. Rev. 100, 580 (1955). 29I. van Weperen, B. Tarasinski, D. Eeltink, V. S. Pribiag, S. R. Plissard, E. P. A. M. Bakkers, L. P. Kouwenhoven, and M. Wim- mer, Phys. Rev. B 91, 201413(R) (2015). 30J. Kammhuber, M. C. Cassidy, F. Pei, M. P. Nowak, A. Vuik, D. Car, S. R. Plissard, E. P. A. M. Bakkers, M. Wimmer, and L. P. Kouwenhoven, Nat Commun. 8, 478 (2017). 31Z. Scher ubl, G. m. H. F ul op, M. H. Madsen, J. Nyg ard, and S. Csonka, Phys. Rev. B 94, 035444 (2016). 32A. E. Hansen, M. T. Bj ork, C. Fasth, C. Thelander, and L. Samuelson, Phys. Rev. B 71, 205328 (2005). 33S. Dhara, H. S. Solanki, V. Singh, A. Narayanan, P. Chaudhari, M. Gokhale, A. Bhattacharya, and M. M. Deshmukh, Phys. Rev. B79, 121311 (2009). 34P. Roulleau, T. Choi, S. Riedi, T. Heinzel, I. Shorubalko, T. Ihn, and K. Ensslin, Phys. Rev. B 81, 155449 (2010). 35S. Est evez Hern andez, M. Akabori, K. Sladek, C. Volk, S. Alagha, H. Hardtdegen, M. G. Pala, N. Demarina, D. Gr utzmacher, and T. Sch apers, Phys. Rev. B 82, 235303 (2010). 36P. W ojcik, A. Bertoni, and G. Goldoni, (2018), unpublished. 37R. L. Kallaher, J. J. Heremans, N. Goel, S. J. Chung, and M. B. Santos, Phys. Rev. B 81, 0035335 (2010). 38R. L. Kallaher, J. J. Heremans, N. Goel, S. J. Chung, and M. B. Santos, Phys. Rev. B 81, 075303 (2010). 39F. Herling, C. Morrison, C. S. Knox, S. Zhang, O. Newell, M. My- ronov, E. H. Lineld, and C. H. Marrows, Phys. Rev. B 95, 155307 (2017).40A. Bertoni, M. Royo, F. Mahawish, and G. Goldoni, Phys. Rev. B84, 205323 (2011). 41S. Funk, M. Royo, I. Zardo, D. Rudolph, S. Morktter, B. Mayer, J. Becker, A. Bechtold, S. Matich, M. Dblinger, M. Bichler, G. Koblmller, J. J. Finley, A. Bertoni, G. Goldoni, and G. Ab- streiter, Nano Letters 13, 6189 (2013). 42J. Jadczak, P. Plochocka, A. Mitioglu, I. Breslavetz, M. Royo, A. Bertoni, G. Goldoni, T. Smolenski, P. Kossacki, A. Kretinin, H. Shtrikman, and D. K. Maude, Nano Letters 14, 2807 (2014). 43M. Royo, A. Bertoni, and G. Goldoni, Phys. Rev. B 89, 155416 (2014). 44F. Jabeen, S. Rubini, V. Grillo, L. Felisari, and F. Martelli, Appl. Phys. Lett. 93, 083117 (2008). 45D. Spirkoska, J. Arbiol, A. Gustafsson, S. Conesa-Boj, F. Glas, I. Zardo, M. Heigoldt, M. H. Gass, A. L. Bleloch, S. Estrade, M. Kaniber, J. Rossler, F. Peiro, J. R. Morante, G. Abstreiter, L. Samuelson, and A. Fontcuberta i Morral, Phys. Rev. B 80, 245325 (2009). 46Z. A. Devizorova and V. A. Volkov, JETP Lett. 98, 101 (2013). 47E. L. Ivchenko, A. Y. Kaminski, and U. R ossler, Phys. Rev. B 54, 5852 (1996). 48J. Treu, M. Bormann, H. Schmeiduch, M. Dblinger, S. Morktter, S. Matich, P. Wiecha, K. Saller, B. Mayer, M. Bichler, M.-C. Amann, J. J. Finley, G. Abstreiter, and G. Koblmller, Nano Letters 13, 6070 (2013). 49X. Liu, P. Liu, H. Huang, C. Chen, T. Jin, Y. Zhang, X. Huang, Z. Jin, X. Li, and Z. Tang, Nanotechnology 24, 245306 (2013). 50S. Furthmeier, F. Dirnberger, M. Gmitra, A. Bayer, M. Forsch, J. Hubmann, C. Sch uller, E. Reiger, J. Fabian, T. Korn, and D. Bougeard, Nat Commun. 7, 12413 (2016). 51A. Sitek, M. Urbaneja Torres, K. Torfason, V. Gudmundsson, A. Bertoni, and A. Manolescu, Nano Letters 18, 2581 (2018). 52B. M. Wong, F. Lonard, Q. Li, and G. T. Wang, Nano Letters 11, 3074 (2011). 53M. Royo, C. Segarra, A. Bertoni, G. Goldoni, and J. Planelles, Phys. Rev. B 91, 115440 (2015). 54E. A. de Andrada e Silva, G. C. La Rocca, and F. Bassani, Phys. Rev. B 55, 16293 (1997). 55Lattice mismatch and strain eld are additional sources of SOC, which are neglected for the present lattice matched heterostruc- tures. We also neglect a SOC contribution resulting from the interface inversion asymmetry related to dierent atomic termi- nation at opposite interfaces59,60which is small. Moreover, ad- ditional calculations with slight smearing of the potential at the interface instead of the step like function did not lead to signi- cantly dierent results. 56I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, J. Appl. Phys. 89, 5815 (2001). 57For the model CSNW under consideration, the gate is attached directly to the NW, so that Ris an upper bound value, while any dielectric interlayer would decrease R. For the NW with no shell, the gate has been placed at the same distance from the core as in the case with the shell, in other to compare the two calculations. 58The sign of the SOC linear density on each facet is decided by the choice of the reference frame. Since ^ xappears combined with the spin matrices in the Hamiltonian (2), the overall sign of the SOC term is unaected by a change in the reference frame. 59E. L. Ivchenko, A. Y. Kaminski, and U. R ossler, Phys. Rev. B 54, 5852 (1996). 60U. R ossler and J. Kainz, Solid State Commun. 121, 313 (2002).
0902.3244v1.Impurity_induced_spin_orbit_coupling_in_graphene.pdf	arXiv:0902.3244v1 [cond-mat.mtrl-sci] 19 Feb 2009Impurity induced spin-orbit coupling in graphene A. H. Castro Neto1and F. Guinea2 1Department of Physics, Boston University, 590 Commonwealt h Ave., Boston MA 02215, USA 2Instituto de Ciencia de Materiales de Madrid, CSIC, Cantobl anco E28049 Madrid, Spain We study the eﬀect of impurities in inducing spin-orbit coup ling in graphene. We show that the sp3distortion induced by an impurity can lead to a large increas e in the spin-orbit coupling with a value comparable to the one found in diamond and other zinc-b lende semiconductors. The spin-ﬂip scattering produced by the impurity leads to spin scatterin g lengths of the order found in recent experiments. Our results indicate that the spin-orbit coup ling can be controlled via the impurity coverage. PACS numbers: 81.05.Uw,71.70.Ej,71.55Ak,72.10.Fk Since the discovery of graphene in 2004 [1] much has been written about its extraordinary charge transport properties [2, 3], such as sub-micron electron mean-free paths, that derive from the speciﬁcity of the carbon σ- bonds against atomic substitution by extrinsic atoms. However,beinganopensurface,itisrelativelyeasytohy- bridizethegraphene’s pzorbitalswith impuritieswithdi- rect consequences in its transport properties [4, 5]. This capability for hybridization with external atoms, such as hydrogen (the so-called graphane), has been shown to be controllable and reversible [6] leading to new doors to control graphene’s properties. Much less has been said about the spin-related trans- port properties such as spin relaxation, although recent experiments show that the spin diﬀusion length scales [7, 8] are much shorter than what one would expect from standard spin-orbit (SO) scattering mechanisms in a sp2 bonded system [9]. In fact, atomic SO coupling in ﬂat graphene is a very weak second order process since it af- fects the πorbitals only through virtual transitions into the deep σbands [10]. Nevertheless, it would be very in- teresting if one could enhance SO interactions because of the prediction of the quantum spin Hall eﬀect in the hon- eycomb lattice [11] and its relation to the ﬁeld of topo- logical insulators [12]. In this paper we argue that impurities (adatoms), such as hydrogen, can lead to a strong enhancement of the SO coupling due to the lattice distortions that they in- duce. In fact, it is well known that atoms that hybridize directly with a carbon atom induce a distortion of the graphene lattice from sp2to sp3[13]. By doing that, the electronicenergyis loweredand the path wayto chemical reactionis enhanced. Nevertheless, it hasbeen knownfor quite sometime [14] that in diamond, a purely sp3carbon bonded system, spin orbit coupling plays an important role in the band structure since it is a ﬁrst order eﬀect, of the order of the atomic SO interaction, ∆at so≈10 meV, in carbon [15]. Here we show that the impurity induced sp3distortion of the ﬂat graphene lattice lead to a signif- icant enhancement of the SO coupling, explaining recent experiments [7, 8] in terms of the Elliot-Yafet mechanism forspin relaxation[16, 17] due to presenceofunavoidableenvironmental impurities in the experiment. Moreover, our predictions can be checked in a controllable way in graphane [6] by the control of the hydrogen coverage. We assume that the carbon atom attached to an impu- rity is raised above the plane deﬁned by its three carbon neighbors (see Fig. 1). The local orbital basis at the po- sition of the impurity (which is assumed to be located at the origin, Ri=0= 0) can be written as: \|πi=0/angbracketright=A\|s/angbracketright+/radicalbig 1−A2\|pz/angbracketright, \|σ1,i=0/angbracketright=/radicalbigg 1−A2 3\|s/angbracketright−A√ 3\|pz/angbracketright+/radicalbigg 2 3\|px/angbracketright, \|σ2,i=0/angbracketright=/radicalbigg 1−A2 3\|s/angbracketright−A√ 3\|pz/angbracketright−1√ 6\|px/angbracketright+1√ 2\|py/angbracketright, \|σ3,i=0/angbracketright=/radicalbigg 1−A2 3\|s/angbracketright−A√ 3\|pz/angbracketright−1√ 6\|px/angbracketright−1√ 2\|py/angbracketright, (1) where\|s/angbracketright, and\|px,y,z/angbracketright, are the local atomic orbitals. Notice that this choice of orbitals interpolates between the sp2conﬁguration, A= 0, to the sp3conﬁguration, A= 1/2. The angle θbetween the new σorbitalsand the direction normal to the plane is cos( θ) =−A/√ A2+2. The energy of the state \|πi/angbracketright,ǫπ, and the energy of the three degenerate states \|σa,i/angbracketright,ǫσ(a= 1,2,3), are given by (see Fig. 2): ǫπ(A) =A2ǫs+(1−A2)ǫp, (2) ǫσ(A) = (1−A2)ǫs/3+(2+ A2)ǫp/3,(3) whereǫs≈ −19.38 eV (ǫp≈ −11.07 eV) is the energy of thes(p) orbital [18]. At the impurity site one has A≈1/2 while away from the impurity A= 0. The Hamiltonian of the problem can be written as, H=Hπ+Hσ+δH, whereHπ(Hσ) describes the π- band (σ-band) of ﬂat graphene, and δHdescribes the localchange in the hopping energies due to the presence2 (b)(a) (a) (b) FIG. 1: (Color online). Top: Top view of the graphene lattice with its orbitals. The orbitals associated with the impurit y and lattice distortion are shown in solid black. (a) sp3or- bital at impurity position; (b) sp2orbital of the ﬂat graphene lattice. of the impurity and sp3distortion: δH=/summationdisplay α=↑,↓/braceleftBig ǫIc† IαcIα+tC−Ic† Iαcπα0 +δǫπc† πα0cπα0+δǫσ/summationdisplay a=1,2,3c† σaα0cσiα0 +Vπσc† πα0(cσ1α0+cσ2α0+cσ3α0)+h.c./bracerightBig (4) where Vπσ(A) =A/radicalbigg 1−A2 3(ǫs−ǫp), (5) cI,α(c† I,α) annihilates (creates) an electron at the impu- rity, and cπαi(cσaαi) annihilates an electron at a carbon site in an orbital π(σa) at position Riwith spin α,ǫIis the electron energy in the impurity, and tC−Ithe tunnel- ing energy between the carbon and impurity, δǫπ(A) = ǫπ(A)−ǫπ(A= 0), and δǫσ(A) =ǫσ(A)−ǫσ(A= 0). In (4) we have not included the change in the hopping between σa,0orbitals(thechangeinenergyduetothedis- tortion is −A2(ǫs−ǫp)/3) and the inter-atomic hopping terms. In this way, we have simpliﬁed the calculations and the interpretation of the results. The inclusion of the other terms do not modify our conclusions. The atomic spin orbit coupling, Hat so= ∆at soL·S, in- duces transitions between porbitals of diﬀerent spin pro- jection [10]. In ﬂat graphene ( A= 0), it leads to tran- sitions between the πandσbands. The change in theground state energy in this case is rather small and given by: (∆at so)2/(ǫπ(A= 0)−ǫσ(A= 0))≈10−2meV [10]. However, the perturbation described by (4) leads to a di- rect local hybridization Vπσbetween the πandσbands that modiﬁes the eﬀective SO coupling acting on the π electrons. The propagator of πelectrons from position Riwith spin αtoRjwith spin βcan be written as: /angbracketleftπi,α\|(ǫ−H)−1\|πj,β/angbracketright ≈ /angbracketleftπi,α\|(ǫ−Hπ)−1\|π0,α/angbracketright ×/angbracketleftπ0,α\|δH\|¯σ0,α/angbracketright×/angbracketleft¯σ0,α\|(ǫ−Hσ)−1\|¯¯σk,α/angbracketright ×/angbracketleft¯¯σk,α\|Hat so\|πk,β/angbracketright/angbracketleftπk,β\|(ǫ−Hπ)−1\|πj,β/angbracketright(6) where\|¯σ0,α/angbracketright= [\|σ10,α/angbracketright+\|σ20,α/angbracketright+\|σ30,α/angbracketright]/√ 3and¯¯σj,α/angbracketright= [\|σ1j,α/angbracketright+eiφ\|σ2j,α/angbracketright+e2iφ\|σ3j,α/angbracketright]/√ 3 where φ= 2π/3. The propagator in (6) can be understood as arising from an eﬀective non-local SO coupling within the πband which goes as: ∆I so(0,i)≈Vπσ/angbracketleft¯σ0,α\|(ǫ−Hσ)−1\|¯¯σi,α/angbracketright∆at so,(7) which allows us to estimate the local value of the SO coupling as: ∆I so(A) ∆atso≈A/radicalbig 3(1−A2). (8) As shown in Fig. 2 the value of the SO coupling depends on the angle (i.e., the value of A) associated with the distortion of the carbon atom away from the graphene plane. Notice that for the sp2case (A= 0) this term vanishes indicating that SO only contributes in second order in ∆at so, while for the sp3case (A= 1/2), the SO coupling is approximately 75% of the atomic value ( ≈7 meV). Also observe that the dependence on the distance from the location of the hydrogen atom is determined by the Green’s function Gσ(0,Rj)) =/angbracketleft¯σ0\|(ǫ− Hσ)−1\|¯¯σj/angbracketright. Thisfunction, calculatedforthesimpliﬁed modelofthe σ bands discussed in ref. [10], showsa signiﬁcant dispersion in Fourier space, ranging from a maximum at the Γ point to zero at the KandK′points. Hence, the range of Gσ(0,R)shouldbe ofthe orderofafew latticeconstants. Based on the previous results we can now calculate the eﬀect of the impurity induced SO coupling in the transport properties. Firstly, we linearize the πband around the K and K’ points in the Brillouin zone and ﬁnd the 2D Dirac spectrum [3]: ǫ±,k=±vFkwherevF (≈106m/s) is the Fermi-Dirac velocity. In this long wavelength limit the impurity potential induced by (7) hascylindricalsymmetryandwecanuseadecomposition of the wavefunction in terms of radial harmonics [19, 20, 21, 22, 23]. A similar analysis, for a system with SO interaction in the bulk has been studied in ref. [9]. We describe the potential scattering by boundary conditions such as one of the components of the spinor vanishes at a distance r=R1(of the order of the Bohr radius) of the impurity [24]. A Rashba-like SO interaction exists in3 the region R1≤r≤R2(region I), and there is neither potential nor spin orbit interaction for r > R2, region II (R2if of the order of the carbon-carbon distance). The wavefunctions in region I can be written as a su- perposition of angular harmonics: Ψn(r,θ)≡A+/bracketleftbigg/parenleftbiggc+Jn(k+r)einθ ic−Jn+1(k+r)ei(n+1)θ/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/arrowtp/angbracketrightBig + +/parenleftbigg ic−Jn+1(k+r)ei(n+1)θ −c+Jn+2(k+r)ei(n+2)θ/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/arrowbt/angbracketrightBig/bracketrightbigg + +B+/bracketleftbigg/parenleftbiggc+Yn(k+r)einθ ic−Yn+1(k+r)ei(n+1)θ/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/arrowtp/angbracketrightBig + +/parenleftbiggic−Yn+1(k+r)ei(n+1)θ −c+Yn+2(k+r)ei(n+2)θ/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/arrowbt/angbracketrightBig/bracketrightbigg + +A−/bracketleftbigg/parenleftbiggc′ −Jn(k−r)eiθ ic′ +Jn+1(k−r)ei(n+1)θ/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/arrowtp/angbracketrightBig − −/parenleftbiggic′ +Jn+1(k−r)ei(n+1)θ −c′ −Jn+2(k−r)ei(n+2)θ/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/arrowbt/angbracketrightBig/bracketrightbigg + +B−/bracketleftbigg/parenleftbiggc′ −Yn(k−r)einθ ic′ +Yn+1(k−r)ei(n+1)θ/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/arrowtp/angbracketrightBig − −/parenleftbigg ic′ +Yn+1(k−r)ei(n+1)θ −c′ −Yn+2(k−r)ei(n+2)θ/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/arrowbt/angbracketrightBig/bracketrightbigg (9) where\| ↑/angbracketrightand\| ↓/angbracketrightare the spin states. The functionsJn(x),Yn(x) are Bessel functions of order n, and: ǫ=±∆I so/2+/radicalBig v2 Fk2 ±+(∆Iso/2)2 (10) c±=/radicalbigg 1/2±∆Iso/(4/radicalBig v2 Fk2 ++(∆Iso/2)2) (11) c′ ±=/radicalbigg 1/2±∆Iso/(4/radicalBig v2 Fk2 −+(∆Iso/2)2) (12) ǫis the energy of the scattered electron ( k±is deﬁned through (10)). The wavefunctions outside the region aﬀected by the impurity, r > R2, can be written as: Ψn(r,θ)≡/parenleftbiggJn(kr)einθ iJn+1(kr)ei(n+1)θ/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/arrowtp/angbracketrightBig + +C↑/parenleftbiggYn(kr)einθ iYn+1(kr)ei(n+1)θ/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/arrowtp/angbracketrightBig + +C↓/parenleftbiggYn+1(kr)ei(n+1)θ iYn+2(kr)ei(n+2)θ/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/arrowbt/angbracketrightBig (13) and:ǫ=vFk. The boundary conditions at r=R1and r=R2lead to the equations: c+A+Jn(k+R1)+c+B+Yn(k+R1)+c′ −A−Jn(k−R1)+c′ −B−Yn(k−R1) = 0 c−A+Jn+1(k+R1)+c−B+Yn+1(k+R1)+c′ +A−Jn+1(k−R1)+c′ +B−Yn+1(k−R1) = 0 c+A+Jn(k+R2)+c+B+Yn(k+R2)+c′ −A−Jn(k−R2)+c′ −B−Yn(k−R2) =Jn(kR2)+C↑Yn(kR2) c−A+Jn+1(k+R2)+c−B+Yn+1(k+R2)+c′ +A−Jn+1(k−R2)+c′ +B−Yn+1(k−R2) =Jn+1(kR2)+C↑Yn+1(kR2) c−A+Jn+1(k+R2)+c−B+Yn+1(k+R2)−c′ +A−Jn+1(k−R2)−c′ +B−Yn+1(k−R2) =C↓Yn+1(kR2) c+A+Jn+2(k+R2)+c+B+Yn+2(k+R2)−c′ −A−Jn+2(k−R2)−c′ −B−Yn+2(k−R2) =C↓Yn+2(kR2) (14) These six equations allow us to obtain the coeﬃcients A±,B±,C↑andC↓. In the absence of the SO inter- action, we have A+=A−,B+=B−,C↓= 0 and C↑=−Jn(kR1)/Yn(kR1). We show in Fig. 3 the results for the cross section for spin ﬂip processes, determined by \|C↓\|2/kF. The main contribution arises from the n= 0 channel. For compari- son, the elastic cross section, calculated in the same way, isσel≈k−1 F. This is about three of magnitude larger than the spin-ﬂip cross section due to the spin orbit cou- pling. Hence, the spin relaxation length is 103times the elastic mean free path [9]. We obtain a mean free path of about 1 µm, in reasonable agreement with the exper- imental results in ref. [7]. This value depends quadrat- ically on ∆I so(A). For a ﬁnite, but small, concentrationof impurities, our results scale with the impurity concen- tration and hence the spin ﬂip processes should increase roughly linearly with impurity coverage in transport ex- periments in systems like graphane [6]. Insummary, wehaveshownthat the impurityinduced, lattice driven, SO coupling in graphene can be of the order of the atomic spin orbit coupling and compara- ble to what is found in diamond and zinc-blend semi- conductors. The value of the SO coupling depends on how much the carbon atom which is hybridized with the impurity displaces from the plane inducing a sp3hy- bridization. Wehavecalculatedthespin-ﬂipcrosssection due to SO coupling for the impurity and shown that it agrees with recent experiments. This results indicates that there are substantial amounts of hybridized impu-4 0.0 0.1 0.2 0.3 0.4 0.5/MinuΣ13.5/MinuΣ13.0/MinuΣ12.5/MinuΣ12.0/MinuΣ11.5/MinuΣ11.0 AΕΠ,ΕΣ/LParen1eV/RParen1 0.0 0.1 0.2 0.3 0.4 0.50.00.10.20.30.40.50.60.7 A/CaΠDΕltaSOI/Slash1/CaΠDΕltaSOat FIG. 2: (Color online).Top: Energy (in eV) of the π(blue) andσ(red)bandsas afunctionof Aaccordingto(3); Bottom: Relative value of the SO coupling at the impurity site relati ve to the atomic value in carbon as a function of Aaccording to (8). 4/MultiΠly10128/MultiΠly1012Ρ/LParen1cm/MinuΣ2/RParen12/MultiΠly10/MinuΣ34/MultiΠly10/MinuΣ3Σso/LParen1nm/RParen1 FIG. 3: (Color online). Cross section for a spin ﬂip process for a defect as described in the text. The parameters used are R1= 1˚AR2= 2˚A and ∆I so= 1meV (blue) and ∆I so= 2meV (red) . rities in graphene, even under ultra-clean high vacuumconditions. Experiments where the impurity coverage is well controlled can provide a “smoking-gun” test of our predictions. We thank illuminating discussions with D. Huertas- Hernando and A. Brataas. AHCN acknowledges the par- tial support of the U.S. Department of Energy under grant DE-FG02-08ER46512. FG acknowledges support from MEC (Spain) through grant FIS2005-05478-C02- 01 and CONSOLIDER CSD2007-00010, by the Comu- nidad de Madrid, through CITECNOMIK, CM2006-S- 0505-ESP-0337. [1] K. S. Novoselov et al., Science 306, 666 (2004). [2] A. K. Geim and K. S. Novoselov, Nature Materials 6, 183 (2007). [3] A. H. Castro Neto et al., Rev. Mod. Phys. 81, 109 (2009). [4] J. H. Chen et al., Nat. Phys. 4, 377 (2008). [5] P. Blake et al.(2008), arXiv:0810.4706. [6] D. C. Elias et al., Science 323, 610 (2009). [7] N. Tombros et al., Nature 448, 571 (2007). [8] N. Tombros et al., Phys. Rev. Lett. 101, 046601 (2008). [9] D. Huertas-Hernando, F. Guinea, and A. Brataas (2008), arXiv:0812.1921. [10] D. Huertas-Hernando, F. Guinea, and A. Brataas, Phys. Rev. B74, 155426 (2006). [11] C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005). [12] C. Kane and E. Mele, Science 314, 1692 (2006). [13] E. J. Duplock, M. Scheﬄer, and P. J. Lindan, Phys. Rev. Lett.92, 225502 (2004). [14] P. Y. Yu and M. Cardona, Fundamentals of Semiconduc- tors: Physics and Materials Properties (Springer, New York, 2005). [15] J. Serrano, M. Cardona, and T. Ruf, Solid St. Commun. 113, 411 (2000). [16] P. G. Elliot, Phys. Rev. 96, 266 (1954). [17] Y. Yafet, in Solid State Physics, vol 13 , edited by ed. by F. Seitz and D. Turnbull (Academic, New York, 1963). [18] W. A. Harrison, Solid State Theory (Dover, New York, 1980). [19] P. M. Ostrovsky, I. V. Gornyi, and A. D. Mirlin, Phys. Rev. B74, 235443 (2006). [20] M. Hentschel and F. Guinea, Phys. Rev. B 76, 115407 (2007). [21] D. S. Novikov, Phys. Rev. B 76, 245435 (2007). [22] M. I. Katsnelson and K. S. Novoselov, Solid State Com- mun.143, 3 (2007). [23] F. Guinea, Journ. Low Temp. Phys. 153, 359 (2008). [24] V. M. Pereira et al., Phys. Rev. Lett. 96, 036801 (2006).
2303.11687v1.Intrinsic_Magnon_Orbital_Hall_Effect_in_Honeycomb_Antiferromagnets.pdf	Intrinsic Magnon Orbital Hall Eﬀect in Honeycomb Antiferromagnets Gyungchoon Go,1Daehyeon An,1Hyun-Woo Lee,2and Se Kwon Kim1 1Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 34141, Korea 2Department of Physics, Pohang University of Science and Technology, Pohang 37673, Korea Wetheoreticallyinvestigatethetransportofmagnonorbitalsinahoneycombantiferromagnet. Weﬁndthatthe magnon orbital Berry curvature is ﬁnite even without spin-orbit coupling and thus the resultant magnon orbital Hall eﬀect is an intrinsic property of the honeycomb antiferromagnet rooted only in the exchange interaction and the lattice structure. Due to the intrinsic nature of the magnon orbital Hall eﬀect, the magnon orbital Nernst conductivity is estimated to be orders of magnitude larger than the predicted values of the magnon spin Nernst conductivity that requires ﬁnite spin-orbit coupling. For the experimental detection of the predicted magnonorbitalHalleﬀect,weinvokethemagnetoelectriceﬀectthatcouplesthemagnonorbitalandtheelectric polarization,whichallowsustodetectthemagnonorbitalaccumulationthroughthelocalvoltagemeasurement. Our results pave a way for a deeper understanding of the topological transport of the magnon orbitals and also its utilization for low-power magnon-based orbitronics, namely magnon orbitronics. Introduction. —Thecollectivelow-energyexcitationsofthe ordered materials are of great interest in condensed matter physics. One of the representative examples is a quantum of spin waves, called a magnon which is a charge-neutral bo- son in magnetic materials. Magnons have been intensively studied for technological applications since they can real- ize Joule-heating-free information transport and processing as well as wave-based computing [1]. In addition, for fun- damental interest, various topological properties of magnon bands have been investigated in the context of the magnon Hall eﬀect [2–7] and the spin Nernst eﬀect [8, 9]. According to the existing theories, the ﬁnite Hall response of magnons canoccurasmaniﬁestationsofspin-orbitcouplingthroughthe Dzyaloshinskii-Moriyainteraction(DMI)[5–9]orthroughthe magnon-phononcoupling[10–13]. TheHallresponsecanalso occurinspintexturesystems[2,3]withthescalarspinchiral- ity, which acts as an eﬀective spin-orbit coupling. In electronics systems, on the other hand, there have been studies showing that electrons can exhibit a Hall eﬀect with- outspin-orbit coupling, owing to their orbital degree of free- dom [14–17]. This discovery evoked a surge of interest in electron-orbital transport phenomena such as the orbital Hall eﬀect[18–22]andtheorbitaltorque[23–25]. Moreover,there have been theoretical suggestions that orbital-dependent elec- tron transport critically aﬀects electron spin dynamics when spin-orbit coupling is present. For instance, it has been sug- gested [16–18] that the orbital Hall eﬀect may play a crucial role in the spin Hall eﬀect. Motivated by the aforementioned advancement of our un- derstandingofelectronorbitals,theorbitalmotionofmagnons hasstartedgatheringattentionrecentlyinmagnetismandspin- tronics. For example, the circulating magnonic modes have been investigated in conﬁned geometries such as whisper- ing gallery mode cavities [26–28], magnetic nanocylinders and nanotubes [29–32]. Also, the orbital magnetization of magnonshasbeenpointedoutastheoriginofweakferromag- netisminanoncollinearkagomeantiferromagnet(AFM)with the DMI [33]. Recently, the orbital-angular-momentum tex- turesofthemagnonbandshavebeenrealizedincollinearmag- netswithnontrivialnetworksofexchangeinteraction[34,35].Furthermore, inspired by achievements in topological meta- materials [36–38], topological magnonic modes carrying the magnon current circulation have been demonstrated in hon- eycomb magnets with exchange-interaction modulation [39]. Despitethestronginterestinmagnonorbitals,studiesontheir transportpropertiesareverylimited[33]. Inparticular,itisan open question whether the magnon orbital degree of freedom can induce a Hall phenomenon withoutspin-orbit coupling, just as its electron counterpart can. In this Letter, we answer this question by investigating the transport of magnon orbitals. For the model system of 2D honeycomb AFMs, we demonstrate that the magnon orbitals can exhibit a Hall eﬀect, namely the magnon orbital Hall ef- fect,withoutspin-orbitcoupling. Themagnonorbitalmoment representsamagnoncurrentcirculationasshowninFig.1(a). We ﬁnd that application of a longitudinal temperature gradi- ent drives thermal magnons to opposite transverse directions depending on their orbital characters [see Fig. 1(b)], giving risetoamagnonHallphenomenon,themagnonorbitalNernst eﬀect. TheestimatedmagnitudeofthemagnonorbitalNernst conductivityisorders-of-magnitudelargerthantheknownval- ues of the magnon spin Nernst conductivity induced by the Dzyaloshinskii-Moriya interaction [8], revealing the hitherto unrecognizedroleofthemagnonorbitalinmagnontransport. Toproposeanexperimentalmethodfordetectingtheaccumu- lation of the magnon orbital at the sample edges induced by themagnonorbitalHalleﬀect,weinvokethemagnetoelectric eﬀect by which a magnonic spin current induces an electric dipole moment. Since the magnon orbital moment can be regarded as a magnonic spin-current circulation, the afore- mentioned magnetoelectric eﬀect dictates that the magnon orbital moment should engender a polarization charge in a two-dimensional space [4, 40, 41] [see Fig. 1(c)] and thus the magnon orbital accumulation should be accompanied by the accumulation of the polarization charge. We estimate the electricvoltageproﬁleinducedbythemagnonorbitalmoment accumulation,whichisshowntobewithincurrentexperimen- tal reach. Owing to strenuous eﬀorts to realize magnetism in various 2D magnetic crystals, our proposal can be tested in a number of transition metal compounds that are known toarXiv:2303.11687v1 [cond-mat.mes-hall] 21 Mar 20232 𝐿𝐿<0 𝐿𝐿>0 right -circular left-circular−𝛻𝛻𝑇𝑇right -circular magnon left-circular magnon𝐿𝐿>0 𝑆𝑆=+ℏelectric polarization (P )(a) (b) (c) magnon+ + + ++ +− − −−− − 𝑞𝑞pol=−∇�𝐏𝐏 polarization charge (𝑞𝑞pol) 𝑆𝑆=−ℏ 𝑆𝑆𝐴𝐴 𝑆𝑆𝐵𝐵 right -handed (𝛽𝛽) 𝑆𝑆=+ℏ left-handed (𝛼𝛼 )𝑆𝑆𝐴𝐴 𝑆𝑆𝐵𝐵 FIG. 1. (a) Schematics of the magnon spin ( S) and magnon orbital ( L) in the honeycomb AFM. The magnon spin and the magnon orbital are determinedby,respectively,theprecessionsofconstituentspinswithinthesitesandtheintersitehopping. (b)Schematicsofthemagnonorbital Nernsteﬀect,whereatemperaturegradient rTinducesanetmagnon-orbitalcurrentinatransversedirectionconsistingofoppositely-moving right-circularmagnonsandleft-circularmagnons. (c)Schematicsofthepolarization Pinducedbythecirculatingmagnonicspincurrentinthe case of orbital L> 0and spinS= +~(left) and the corresponding polarization charge qpol=rP(right). The sign of the polarization charge is determined by the product of the sign of the magnon spin Sand the direction of the spin current circulation, i.e., the sign of L. host honeycomb AFMs [ e:g:;MPX3(M= Fe, Ni, Mn; X=S, Se)] [42–45]. Model construction. —Here we consider a 2D AFM on a honeycomb lattice H=JX hi;jiSiSjKX i(Si;z)2+gBBX iSi;z;(1) whereJ(>0)istheantiferromagneticexchangecouplingand K(>0)is the easy-axis anisotropy, gis the g-factor, Bis theBohrmagneton,and Bistheappliedmagneticﬁeld. Note that our model does not include the DMI which comes from spin-orbitcouplingandthusdoesnotexhibitthemagnonspin Nernsteﬀect[6–9]. InthisLetter,wefocusonthecasewhere a ground state is the collinear Néel state along the z-axis. Performing the Holstein-Primakoﬀ transformation and taking the Fourier transformation, we have H=1 2X k y kHk k; k= (ak;bk;ay k;by k)T;(2) with the momentum-space Hamiltonian Hk=JS0 BB@3 ++ 0 0 fk 0 3 +f k 0 0fk 3 ++ 0 f k 0 0 3 + 1 CCA;(3) where= (2KgBB)=Jandfk=P jeikajwith a1=a 2(p 3;1),a2=a(0;1), and a3=a 2(p 3;1). The magnon excitations can be described by the general- ized Bogoliubov-de Gennes equation in the particle-hole space representation [11, 46]. In this representation, the pseudo-energy-eigenvalue satisﬁes n;k= (3k)nn, where 3= diag(1;1;1;1)is the Pauli matrix acting on the particle-hole space and (k)are the magnon bands given by = k=0 kgBBS; (4)where0 k=JSp (3 +)2jfkj2and= 2K=J(see Sup- plemental Material for calculation details). Here, the indices andstand for two magnonic bands with opposite spin an- gularmomenta[seeFig1(a)]. Thetopologicalpropertyofthe magnonic statejniis characterized by the Berry curvature: n(k) =@An y(k) @kx@An x(k) @ky; (5) where An(k) =hnj3i@kjni=hnj3jniis the Berry connec- tion. Figures 2(a-d) show the magnon band structures n kand the corresponding Berry curvatures n(k)withn=;. In the honeycomb AFM, the broken inversion symmetry al- lows a non-zero Berry curvature without the DMI. Also, be- causeofthecombinedsymmetryofthetime-reversal( T)and a180spin rotation around the x-axis (Cx) of the Hamil- tonian, the energy spectra are even in momentum space (k=k), whereas the magnon Berry curvatures are odd [ n(k) = n(k)][8]. Therefore, the momentum-space integration of the magnon Berry curvature weighed by the Bose-Einsteindistributioniszeroforeachband,indicatingthe absence of the Hall transport of magnons and their spins. Magnon orbital Hall eﬀect. —Although the momentum- spaceintegrationofthemagnonBerrycurvaturesvanishes,the nonvanishingand k-oddstructureoftheBerrycurvatureopens up a possibility for topological transport of certain quantities. If there is a momentum-dependent quantity whose proﬁle is also odd in k, its Hall eﬀect can be present. We show below that this is indeed the case for the magnon orbital moment since it holds the same symmetry property in the momentum space as the Berry curvature [34], i.e., k-odd in the presence of theTCxsymmetry with broken inversion symmetry. From the orbital moment operator ^L=1 4(rvvr), wereadthematrixelementofthemagnonorbitalmoment[21,3 0 𝜋𝜋 −𝜋𝜋−𝜋𝜋0𝜋𝜋 0 𝜋𝜋 −𝜋𝜋−𝜋𝜋0𝜋𝜋 −0.10.1 Ω𝛼𝛼(𝐤𝐤) Ω𝛽𝛽(𝐤𝐤)(c) (d) 0 𝜋𝜋 −𝜋𝜋−𝜋𝜋0𝜋𝜋 0 𝜋𝜋 −𝜋𝜋−𝜋𝜋0𝜋𝜋0 −0.03 Ω𝛼𝛼𝐿𝐿(𝐤𝐤) Ω𝛽𝛽𝐿𝐿(𝐤𝐤)(g) (h) 𝐸𝐸(meV) 012 0 𝜋𝜋 −𝜋𝜋−𝜋𝜋0𝜋𝜋 𝜖𝜖𝐤𝐤𝛽𝛽 0 𝜋𝜋 −𝜋𝜋−𝜋𝜋0𝜋𝜋 𝜖𝜖𝐤𝐤𝛼𝛼 0 𝜋𝜋 −𝜋𝜋−𝜋𝜋0𝜋𝜋 0 𝜋𝜋 −𝜋𝜋−𝜋𝜋0𝜋𝜋 −0.40.4𝐿𝐿(ℏ/𝑚𝑚eff) 𝐿𝐿𝛼𝛼(𝐤𝐤) 𝐿𝐿𝛽𝛽(𝐤𝐤)(a) (b) (e) (f) FIG.2. (a,b)Magnonbandstructures[Eq.(4)],(c,d)theBerrycurvatures[Eq.(5)],(e,f)theequilibriumorbitalmomentstructures[Eq.(6)], and (g, h) the orbital Berry curvatures [Eq. (7)] of two magnonic states denoted by and. For material parameters, we take J= 1:54meV andKS= 0:0086meV, andgBB=J = 0:25. For (e) and (f), meis the magnon eﬀective mass at the Dirac points. 22] hnj^Ljmi=hnjrvvr 4 jmi =1 2~Imh@knjH kj@kmi 1 4~(n;k+ m;k)Imh@knj3j@kmi:(6) By taking the diagonal element of (6), one recovers the well- known formula of the intrinsic orbital moment [47–49]. The magnonorbitalmomentproﬁles Ln(k) =hnj^Lzjniofthetwo magnonsareshowninFig.2(e)and(f)[50]. Inagreementwith the momentum-space texture of the magnon orbital angular momentum in Ref. [34, 35], the evaluated magnon orbital moment has the C3rotation symmetry with k-odd structure. BecauseLn z(k)=Ln z(k)andk=k, the total magnon orbital moment is zero in equilibrium. However, our system canexhibittheintrinsicmagnonorbitalHalltransportbecause bothLn z(k)and n z(k)are odd in k, and thus their product Ln z(k) n z(k)isk-even. Analogous to the generalized Berry curvature[46,51],wewritetheorbitalBerrycurvaturewhich characterizes the magnon orbital Hall transport as follows: L n(k) =X m6=n(3)mm(3)nn2~2Im hnjjL z;yjmihmjvxjni (n;km;k)2; (7) wherejL z;y= (vy3^Lz+^Lz3vy)=4is the magnon orbital current operator, and vi=1 ~@Hk @kiis the velocity operator. Note the summation is performed in the particle-hole space. TheproﬁlesoftheorbitalBerrycurvaturesofthetwomagnon modesareshowninFig.2(g)and(h). Asexpected,theproﬁles areevenin kandthustheirmomentum-spaceintegrationsare ﬁnite,indicatingtheexistenceoftheHalleﬀectofthemagnonorbitals. We emphasize that our model has no spin-orbit cou- pling term such as the DMI. Also, the orbital Berry curvature remainsﬁnitewhenboth BandKapproachzeroaslongasthe antiferromagnetic ground state is maintained. Therefore, the magnon orbital Berry curvature and the resultant Hall eﬀect areintrinsicpropertiesofthehoneycombAFMthatoriginated solely from the exchange interaction and the lattice geometry. The magnon orbital Berry curvature leads to the transverse magnonorbitalcurrentinresponsetoanexternalperturbation, namely the magnon orbital Hall eﬀect. The linear response equation of the transverse magnon orbital current driven by a temperature gradient is given by (JL z)y=L z@xT[46], whereL z=L z;+L z;is the magnon orbital Nernst con- ductivitywith L z;n=2kB ~VP kc1(n) L n(k),wherekBisthe Boltzmann constant, c1() = (1 +)ln(1 +)ln, and n= (en=kBT1)1is the Bose-Einstein distribution. In Fig. 3(a), we show the orbital Nernst conductivity for diﬀer- ent temperatures by using the material parameters of MnPS 3: J= 1:54meV andKS= 0:0086meV [52]. To compare the orbital Nernst conductivity with the spin Nernst conduc- tivity,wetaketheconstanteﬀectivemassapproximationatthe Dirac points where the Berry curvatures are maximized. The magnon orbital Nernst conductivity is estimated to be about 102kB. This value is 103times larger than the estimated values of the magnon spin Nernst conductivity of honeycomb AFMsinthepresenceoftheDMI[8]. Thisisourmainresult: The honeycomb AFM exhibits the magnon orbital Hall ef- fect without spin-orbit coupling and, therefore, its magnitude is orders-of-magnitude larger than the magnon spin Nernst conductivity that requires spin-orbit coupling. Note that the orbital Nernst conductivity is almost independent of the mag- netic ﬁeld in Fig. 3(a), because the magnon eigenstates are unaﬀected by the magnetic ﬁeld. Magnon-orbital-induced polarization. —The magnon or-4 bital Hall eﬀect induces the accumulation of the magnon or- bital at the edges of a system. To propose an experimental method to detect the magnon orbital accumulation, here we develop a phenomenological model for the transverse voltage proﬁleinducedbythelongitudinaltemperaturegradientviathe magnonorbitalNernsteﬀect. Weemphasizethatthefollowing theoryisqualitativeinnatureandthusintendedtoprovidethe order-of-magnitude estimation, not quantitative predictions. Tobeginwith, letusreviewtherelation betweenthespincur- rentandthepolarization. Thespincurrentfromanoncollinear spin conﬁguration is known to induce an electric polarization in magnetic materials by the combined action of the atomic spin-orbit interaction and the orbital hybridization [40, 41]: P=ea ESOe12Is; (8) whereeis the magnitude of the electron charge, ais the dis- tance between the two sites, e12is the unit vector connecting two sites, Is=J(S1S2)is the spin current from site 1 to site 2, and the energy scale ESOis inversely proportional to the spin-orbit coupling strength. In the ground state of a collinearmagnet,thereisnospincurrent (Is= 0)andthusno electric polarization ( P= 0). However, the magnon consists of spatially-varying noncollinear deviations from the ground state. Therefore,amagnoncurrentinacollinearmagnetgives rise to a ﬁnite spin current Is[53, 54]. For the magnonic spin current, we can invoke Eq. (8) to compute the induced polarization since the characteristic time scale of the magnon isgenerallymuchlongerthanthatoftheelectronhoppingpro- cess. The spin current carried by a single magnon is given by Is=S(v=a)^z, which leads to the electric polarization P=eS ESO(v^z); (9) whereS=~is the magnon spin and v=ve12is the magnon velocity. By considering the typical energy scale of ESO, it has been predicted in Refs. [40, 41, 55] that a mea- surable electric polarization can be induced by a magnonic spincurrentinmagneticmaterials[56]. Thismagnetoelectric eﬀect allows us to relate the magnon orbital motion (i.e., the circulatingmagnonicspincurrent)andthepolarizationcharge density. For the qualitative understanding, we schematically depicttheelectricpolarizationproducedbythemagnonicspin- currentcirculationaroundahexagoninahoneycomblatticein Fig. 1(c). The magnonic spin-current circulation induces the electricpolarizationpointingoutwardorinward(andtherefore the positive or negative polarization charge density), depend- ing on the product of the magnon-spin sign and the magnon- orbital-moment sign. Nowletusconsiderthesituationwherethenonequilibrium accumulationofthemagnonorbitalmomentisgeneratedatthe edgesofthesamplebyatemperaturegradientviathemagnon orbital Nernst eﬀect. In the absence of an external ﬁeld along thez-direction, there would be a ﬁnite accumulation of the magnon orbital moment at the edges, but there would be no 0 0.1 0.20123 (a) (b) 0 0.1 0.20123 10 K30 K50 K70 K 10 K30 K50 K70 KFIG. 3. (a) Magnetic-ﬁeld dependence of the magnon orbital Nernst conductivity L zand(b)thespin-polarizedcomponentofthemagnon orbital Nernst conductivity L z;s=L z;L z;. See the main text for the details. induced electric polarization, for spin-up magnons and spin- down magnons are equally populated and thus their contribu- tions to the electric polarization cancel each other. However, when we apply an external magnetic ﬁeld, spin-up magnons andspin-downmagnonsarepopulatedunequallyandthusthe netspindensityofmagnonsbecomesﬁnite. Consequently,the magnonorbitalaccumulationandalsothemagnonorbitalHall currentarespin-polarizedinthepresenceoftheexternalﬁeld. Inparticular,thespin-polarizedcomponentofthemagnonor- bital Nernst conductivity L z;s(=L z;L z;) is zero when the magnetic ﬁeld is zero and becomes ﬁnite as the magnetic ﬁeld is applied as shown in Fig. 3(b). Instead of the magnon orbital accumulation, what is di- rectly related to the observable electric polarization is the spin-polarized magnon orbital accumulation L s=L L . To estimate the spin-polarized magnon orbital accumulation thatisinducedbythemagnonorbitalNernsteﬀect,weusethe phenomenological drift-diﬀusion formalism by following the previousstudiesonelectronorbitaltransport[20,57,58]. For parameters, we use gBB=J = 0:25,ESO= 3eV,T= 70 K and@xT= 1K/m with the constant magnon relaxation time= 30ns and the magnon diﬀusion length = 20 nm. The considered system size is 1 m1m. Figure 4(a) shows the resultant spin-polarized magnon orbital accumula- tion along the y-direction. We also numerically compute the electric voltage proﬁle induced by the spin-polarized magnon orbitalaccumulationbasedonasimpliﬁedmodelfortheinho- mogeneous magnon orbital accumulation (see Supplemental Materialfordetailedcalculation),whichisshowninFig.4(b). Note that the estimated electric voltage at the edges is about 0:1Vwhichiswithinthecurrentexperimentalcapacity. The accumulation of the magnon orbital moment gives rise to the electricvoltageproﬁleviathemagnetoelectriceﬀect,bywhich we can probe the proposed magnon orbital Nernst eﬀect. Discussion. —In this Letter, we have investigated the trans- port of magnon orbital moments in a honeycomb AFM. The k-oddstructuresofboththeBerrycurvatureandthemagnon- orbital texture lead to the k-even structure of the magnon orbitalBerrycurvature,whichgivesrisetothemagnonorbital Nernst eﬀect after momentum-space integration. We empha- size that the magnon orbital Berry curvature does not require spin-orbit coupling and thus is an intrinsic property of the5 (a) (b) 0 0.05 0.10.9 0.95 1−0.10.1 0V (𝜇𝜇V) y(𝜇𝜇m)0 0.05 0.1 y(𝜇𝜇m)0.9 0.95 1𝜌𝜌𝑠𝑠𝐿𝐿(10−4ℏ/nm2) -6-3036 FIG.4. (a)Theproﬁleofthespin-polarizednonequilibriummagnon orbital accumulation L sin theydirection and (b) the corresponding voltageVinduced by the nonequilibrium polarization. The consid- eredsystemsizeis1 m1mandatemperaturegradientisapplied in thexdirection. See the main text for the details. honeycomb AFM originating solely from the exchange inter- actionandthelatticegeometry. Asaresult,themagnonorbital Nernst eﬀect is predicted to be orders-of-magnitudes stronger than the magnon spin Nernst eﬀect that requires spin-orbit coupling. AlthoughherewefocusonthemagnonorbitalHall eﬀectinthehoneycombAFM,wenotethatthemagnonorbital Hall eﬀect is generally expected to be present in systems with broken inversion symmetry such as honeycomb and Kagome ferromagnets with DMI. We also remark that, although we have considered a temperature gradient as a means to drive a magnontransport,onecanalsouseelectronicmeanstopump magnons by using, e.g., the spin Hall eﬀects [59, 60]. For an experimental scheme, we have shown that the magnonorbitalaccumulationcanbedetectedthroughtheelec- tric voltage proﬁle by invoking the magnetoelectric eﬀect. In particular, our theory for the magnon orbital Hall eﬀect pre- dicts that, upon the application of a longitudinal temperature gradient, the electric voltage proﬁle is developed in the trans- verse direction. This phenomenon has the same symmetry as the Nernst eﬀect in metallic layers. A remarkable feature of our results is that the electric voltage is not induced by the conduction electrons but by the circulating spin current asso- ciatedwiththeorbitalmotionofmagnons. Weherenotethat, in addition to the electric polarization, there are several other degrees of freedom that are expected to couple with magnon orbital motions such as photons, phonons, and spin angular momenta as mentioned in Ref. [34], which may lead us to a new detection scheme for magnon orbitals. For the future outlook,consideringaplethoraofmagnon-spin-relatedeﬀects studied in magnonics and spintronics, it is expected that there aremanymagnonorbital-relatedeﬀectsthatawaitthediscov- eryinvolving,e.g.,theinterplayofthemagnonorbitalandthe electron orbital. We thank Kyung-Jin Lee and Giovanni Vignale for use- ful discussions. G.G. acknowledges support by the National Research Foundation of Korea (NRF-2022R1C1C2006578). H.W.L was supported by Samsung Science and Technol- ogy Foundation (SSTF-BA-1501-51). S.K.K. was supported by Samsung Science and Technology Foundation (SSTF- BA2202-04), Brain Pool Plus Program through the National Research Foundation of Korea funded by the Ministry of Sci- ence and ICT (NRF-2020H1D3A2A03099291), and NationalResearch Foundation of Korea funded by the Korea Govern- ment via the SRC Center for Quantum Coherence in Con- densed Matter (NRF-RS-2023-00207732). [1] A.V.Chumak,V.I.Vasyuchka,A.A.Serga,andB.Hillebrands, Magnon spintronics, Nat. Phys. 11, 453 (2015). [2] H. Katsura, N. Nagaosa, and P. A. Lee, Theory of the Thermal Hall Eﬀect in Quantum Magnets, Phys. Rev. Lett. 104, 066403 (2010). [3] J. H. Han and H. Lee, Spin Chirality and Hall-Like Transport Phenomena of Spin Excitations, J. Phys. Soc. Jpn. 86, 011007 (2017). [4] R. Matsumoto and S. Murakami, Theoretical Prediction of a RotatingMagnonWavePacketinFerromagnets,Phys.Rev.Lett. 106, 197202 (2011). [5] Y. Onose, T. Ideue, H. Katsura, Y. Shiomi, N. Nagaosa, and Y.Tokura,ObservationoftheMagnonHallEﬀect,Science 329, 297 (2010). [6] S.K.Kim,H.Ochoa,R.Zarzuela,andY.Tserkovnyak,Realiza- tionoftheHaldane-Kane-MeleModelinaSystemofLocalized Spins, Phys. Rev. Lett. 117, 227201 (2016). [7] S. A. Owerre, Topological honeycomb magnon Hall eﬀect: A calculation of thermal Hall conductivity of magnetic spin exci- tations, J. Appl. Phys. 120, 043903 (2016). [8] R. Cheng, S. Okamoto, and D. Xiao, Spin Nernst Eﬀect of Magnons in Collinear Antiferromagnets, Phys. Rev. Lett. 117, 217202 (2016). [9] V. A. Zyuzin and A. A. Kovalev, Magnon Spin Nernst Eﬀect in Antiferromagnets, Phys. Rev. Lett. 117, 217203 (2016). [10] R. Takahashi and N. Nagaosa, Berry Curvature in Magnon- Phonon Hybrid Systems, Phys. Rev. Lett. 117, 217205 (2016). [11] X. Zhang, Y. Zhang, S. Okamoto, and D. Xiao, Thermal Hall EﬀectInducedbyMagnon-PhononInteractions,Phys.Rev.Lett. 123, 167202 (2019). [12] G. Go, S. K. Kim, and K.-J. Lee, Topological Magnon-Phonon Hybrid Excitations in Two-Dimensional Ferromagnets with Tunable Chern Numbers, Phys. Rev. Lett. 123, 237207 (2019). [13] S. Park, N. Nagaosa, and B.-J. Yang, Thermal Hall Eﬀect, Spin NernstEﬀect,andSpinDensityInducedbyaThermalGradient in Collinear Ferrimagnets from Magnon–Phonon Interaction, Nano Lett. 20, 2741 (2020). [14] T. Tanaka, H. Kontani, M. Naito, T. Naito, D. S. Hirashima, K. Yamada, and J. Inoue, Intrinsic spin Hall eﬀect and orbital Hall eﬀect in 4dand5dtransition metals, Phys. Rev. B 77, 165117 (2008). [15] H. Kontani, T. Tanaka, D. S. Hirashima, K. Yamada, and J. In- oue, Giant Intrinsic Spin and Orbital Hall Eﬀects in Sr2MO4 (M= Ru, Rh, Mo), Phys. Rev. Lett. 100, 096601 (2008). [16] H. Kontani, T. Tanaka, D. S. Hirashima, K. Yamada, and J. In- oue, Giant Orbital Hall Eﬀect in Transition Metals: Origin of Large Spin and Anomalous Hall Eﬀects, Phys. Rev. Lett. 102, 016601 (2009). [17] T.TanakaandH.Kontani,IntrinsicspinandorbitalHalleﬀects in heavy-fermion systems, Phys. Rev. B 81, 224401 (2010). [18] D.Go,D.Jo,C.Kim,andH.-W.Lee,IntrinsicSpinandOrbital HallEﬀectsfromOrbitalTexture,Phys.Rev.Lett. 121,086602 (2018). [19] D. Jo, D. Go, and H.-W. Lee, Gigantic intrinsic orbital Hall eﬀects in weakly spin-orbit coupled metals, Phys. Rev. B 98, 214405 (2018).6 [20] G.SalaandP.Gambardella,GiantorbitalHalleﬀectandorbital- to-spin conversion in 3d,5d, and 4fmetallic heterostructures, Phys. Rev. Res. 4, 033037 (2022). [21] S.BhowalandG.Vignale,OrbitalHalleﬀectasanalternativeto valleyHalleﬀectingappedgraphene,Phys.Rev.B 103,195309 (2021). [22] A.Pezo,D.GarcíaOvalle,andA.Manchon,OrbitalHalleﬀect incrystals: Interatomicversusintra-atomiccontributions,Phys. Rev. B106, 104414 (2022). [23] D. Go and H.-W. Lee, Orbital torque: Torque generation by orbital current injection, Phys. Rev. Res. 2, 013177 (2020). [24] D. Lee, D. Go, H.-J. Park, W. Jeong, H.-W. Ko, D. Yun, D. Jo, S. Lee, G. Go, J. H. Oh, K.-J. Kim, B.-G. Park, B.-C. Min, H. C. Koo, H.-W. Lee, O. Lee, and K.-J. Lee, Orbital torque in magnetic bilayers, Nat. Commun. 12, 6710 (2021). [25] H. Hayashi, D. Jo, D. Go, Y. Mokrousov, H.-W. Lee, and K. Ando, Observation of long-range orbital transport and gi- ant orbital torque, arXiv:2202.13896. [26] J.A.Haigh,A.Nunnenkamp,A.J.Ramsay,andA.J.Ferguson, Triple-Resonant Brillouin Light Scattering in Magneto-Optical Cavities, Phys. Rev. Lett. 117, 133602 (2016). [27] S. Sharma, Y. M. Blanter, and G. E. W. Bauer, Light scattering by magnons in whispering gallery mode cavities, Phys. Rev. B 96, 094412 (2017). [28] A. Osada, A. Gloppe, Y. Nakamura, and K. Usami, Orbital angular momentum conservation in Brillouin light scattering withinaferromagneticsphere,NewJ.Phys. 20,103018(2018). [29] Y. Jiang, H. Y. Yuan, Z.-X. Li, Z. Wang, H. W. Zhang, Y. Cao, andP.Yan,TwistedMagnonasaMagneticTweezer,Phys.Rev. Lett.124, 217204 (2020). [30] A.González,P.Landeros,andÁ.S.Núñez,Spinwavespectrum ofmagneticnanotubes,J.Magn.Magn.Mater. 322,530 (2010). [31] S. Lee and S. K. Kim, Orbital angular momentum and current- induced motion of a topologically textured domain wall in a ferromagnetic nanotube, Phys. Rev. B 104, L140401 (2021). [32] S. Lee and S. K. Kim, Generation of Magnon Orbital Angular Momentum by a Skyrmion-Textured Domain Wall in a Ferro- magnetic Nanotube, Front. Phys. 10, 858614 (2022). [33] R.R.Neumann,A.Mook,J.Henk,andI.Mertig,OrbitalMag- neticMomentofMagnons,Phys.Rev.Lett. 125,117209(2020). [34] R. S. Fishman, J. S. Gardner, and S. Okamoto, Orbital Angular Momentum of Magnons in Collinear Magnets, Phys. Rev. Lett. 129, 167202 (2022). [35] R.S.Fishman,L.Lindsay,andS.Okamoto,Exactresultsforthe orbital angular momentum of magnons on honeycomb lattices, J. Phys. Condens. Matter 35, 015801 (2022). [36] L.-H.WuandX.Hu,SchemeforAchievingaTopologicalPho- tonicCrystalbyUsingDielectricMaterial,Phys.Rev.Lett. 114, 223901 (2015). [37] C. He, X. Ni, H. Ge, X.-C. Sun, Y.-B. Chen, M.-H. Lu, X.-P. Liu, and Y.-F. Chen, Acoustic topological insulator and robust one-way sound transport, Nat. Phys. 12, 1124 (2016). [38] Y. Li, Y. Sun, W. Zhu, Z. Guo, J. Jiang, T. Kariyado, H. Chen, and X. Hu, Topological LC-circuits based on microstrips and observationofelectromagneticmodeswithorbitalangularmo- mentum, Nat. Commun. 9, 4598 (2018). [39] H.Huang,T.Kariyado,andX.Hu,Topologicalmagnonmodes onhoneycomblatticewithcouplingtextures,Sci.Rep. 12,6257 (2022). [40] H. Katsura, N. Nagaosa, and A. V. Balatsky, Spin Current and Magnetoelectric Eﬀect in Noncollinear Magnets, Phys. Rev. Lett.95, 057205 (2005). [41] T. Liu and G. Vignale, Electric Control of Spin Currents and Spin-Wave Logic, Phys. Rev. Lett. 106, 247203 (2011).[42] X.Wang,K.Du,Y.Y.F.Liu,P.Hu,J.Zhang,Q.Zhang,M.H.S. Owen, X. Lu, C. K. Gan, P. Sengupta, C. Kloc, and Q. Xiong, Raman spectroscopy of atomically thin two-dimensional mag- netic iron phosphorus trisulﬁde (FePS3) crystals, 2d Mater. 3, 031009 (2016). [43] K. Kim, S. Y. Lim, J. Kim, J.-U. Lee, S. Lee, P. Kim, K. Park, S. Son, C.-H. Park, J.-G. Park, and H. Cheong, Antiferromag- netic ordering in van der Waals 2D magnetic material MnPS3 probed by Raman spectroscopy, 2d Mater. 6, 041001 (2019). [44] G. Le Flem, R. Brec, G. Ouvard, A. Louisy, and P. Segransan, MagneticinteractionsinthelayercompoundsMPX3(M=Mn, Fe, Ni; X = S, Se), J. Phys. Chem. Solids 43, 455 (1982). [45] X. Jiang, Q. Liu, J. Xing, N. Liu, Y. Guo, Z. Liu, and J. Zhao, Recent progress on 2D magnets: Fundamental mechanism, structural design and modiﬁcation, Appl. Phys. Rev. 8, 031305 (2021). [46] B. Li, S. Sandhoefner, and A. A. Kovalev, Intrinsic spin Nernst eﬀectofmagnonsinanoncollinearantiferromagnet,Phys.Rev. Res.2, 013079 (2020). [47] D. Xiao, J. Shi, and Q. Niu, Berry phase correction to electron density of states in solids, Phys. Rev. Lett. 95, 137204 (2005). [48] T.Thonhauser,D.Ceresoli,D.Vanderbilt,andR.Resta,Orbital magnetizationinperiodicinsulators,Phys.Rev.Lett. 95,137205 (2005). [49] J. Shi, G. Vignale, D. Xiao, and Q. Niu, Quantum theory of or- bitalmagnetizationanditsgeneralizationtointeractingsystems, Phys. Rev. Lett. 99, 197202 (2007). [50] Notethattheorbitalmoment rvdiﬀersfromtheorbitalan- gularmomentum rpbydimensionofmassintheelectronic case [21]. [51] S.ParkandB.-J.Yang,PhononAngularMomentumHallEﬀect, Nano Lett. 20, 7694 (2020). [52] A. R. Wildes, B. Roessli, B. Lebech, and K. W. Godfrey, Spin wavesandthecriticalbehaviourofthemagnetizationin,J.Phys. Condens. Matter 10, 6417 (1998). [53] F. Schütz, M. Kollar, and P. Kopietz, Persistent Spin Currents in Mesoscopic Heisenberg Rings, Phys. Rev. Lett. 91, 017205 (2003). [54] F. Schütz, M. Kollar, and P. Kopietz, Persistent spin currents inmesoscopichaldane-gapspinrings,Phys.Rev.B 69,035313 (2004). [55] J. Xia, X. Zhang, X. Liu, Y. Zhou, and M. Ezawa, Univer- sal quantum computation based on nanoscale skyrmion helic- ity qubits in frustrated magnets, Phys. Rev. Lett. 130, 106701 (2023). [56] The electric polarization can be also caused by the Lorentz transformation of the magnetic dipole from a moving frame to the laboratory P=v c2=g 2e~ mec2(v^z), where vand (=gBz)are the velocity and the magnetic moment of the magnetic dipole, respectively, and meis the electron mass [53, 61, 62]. However, this eﬀect is extremely small because the denominator mec2is order of MeV. [57] Y.-G. Choi, D. Jo, K.-H. Ko, D. Go, K.-H. Kim, H. G. Park, C.Kim,B.-C.Min,G.-M.Choi,andH.-W.Lee,Observationof the orbital Hall eﬀect in a light metal Ti, arXiv:2109.14847. [58] S. Han, H.-W. Lee, and K.-W. Kim, Orbital Dynamics in Cen- trosymmetric Systems, Phys. Rev. Lett. 128, 176601 (2022). [59] L. J. Cornelissen, J. Liu, R. A. Duine, J. B. Youssef, and B. J. vanWees,Long-distancetransportofmagnonspininformation inamagneticinsulatoratroomtemperature,Nat.Phys. 11,1022 (2015). [60] S.T.B.Goennenwein,R.Schlitz,M.Pernpeintner,K.Ganzhorn, M. Althammer, R. Gross, and H. Huebl, Non-local magnetore-7 sistanceinYIG/Ptnanostructures,Appl.Phys.Lett. 107,172405 (2015). [61] J.E.Hirsch,OverlookedcontributiontotheHalleﬀectinferro- magnetic metals, Phys. Rev. B 60, 14787 (1999).[62] F. Meier and D. Loss, Magnetization Transport and Quantized Spin Conductance, Phys. Rev. Lett. 90, 167204 (2003).
1411.3950v1.Spin_orbit_coupling_and_chaotic_rotation_for_eccentric_coorbital_bodies.pdf	Complex Planetary Systems Proceedings IAU Symposium No. 310, 2014 Z. Knezevic & A. Lema^ trec 2014 International Astronomical Union DOI: 00.0000/X000000000000000X Spin-orbit coupling and chaotic rotation for eccentric coorbital bodies Adrien Leleu1, Philippe Robutel1and Alexandre C.M. Correia2;1 1IMCCE, Observatoire de Paris, CNRS, UPMC Univ. Paris 06, Univ. Lille 1, 77Av.Denfert-Rochereau, 75014 Paris, France email: aleleu@imcce.fr, robutel@imcce.fr 2Departamento de F sica, I3N, Universidade de Aveiro, Campus de Santiago, 3810-193 Aveiro, Portugal email: correia@ua.pt Abstract. The presence of a co-orbital companion induces the splitting of the well known Keplerian spin-orbit resonances. It leads to chaotic rotation when those resonances overlap. Keywords. Coorbitals, Rotation, Resonance, Spin-orbit resonance. 1. Introduction and Notations Given an asymmetric body on a circular orbit, denoting its rotation angle in the plane with respect to the inertial frame, the only possible spin-orbit resonance is the synchronous one _=n,nbeing the mean motion of the orbit. On an Keplerian eccentric orbit, Wisdom et al. (1984) showed that there is a whole family of spin-orbit eccen- tric resonances, the main ones being _=pn=2 wherepis an integer. In 2013, Correia and Robutel showed that in the circular case, the presence of a coorbital companion in- duced a splitting of the synchronous resonance, forming a family of co-orbital spin-orbit resonances of the form _=nk=2,being the libration frequency in the coorbital res- onance. Inside this resonance, the dierence of the mean anomaly of the two coorbitals, denoted by , librates around a value close to =3 (around the L4 or L5 Lagrangian equilibrium - tadpole conguration), around (encompassing L3, L4 and L5 - horseshoe conguration) or 0 (quasi-satellite) conguration. We generalize the results of Correia and Robutel (2013) from the case of circular co-orbital orbits to eccentric ones. 2. Rotation The rotation angle satises the dierential equation: +2 2a r3 sin 2(f) = 0;with=nr 3(BA) C; (2.1) whereA<B <C are the internal momenta of the body, ( r;f) the polar coordinates of the center of the studied body and aits instantaneous semi-major axis. Let us consider that the orbit is quasi-periodic. As a consequence, the elliptic elements of the body can be expended in Fourier series whose frequencies are the fundamental frequencies of the planetary system. In other words the time-dependent quantitya r3ei2f that appears in equation (2.1) reads: a r3 ei2f=X j>0je(ijt+j): (2.2) 1arXiv:1411.3950v1 [astro-ph.EP] 14 Nov 20142 Adrien Leleu, Philippe Robutel & Alexandre C.M. Correia Wherejare linear combinations with integer coecients of the fundamental frequen- cies of the orbital motion (here nand) andjtheir phases. Thus (2.1) becomes: =2 2X j>0jsin (2+jt+j): (2.3) For a Keplerian circular orbit, the only spin orbit resonance possible is the synchronous one, since0= 1,0= 2n, andj=j= 0 forj >0. In the general Keplerian case we have the spin-orbit eccentric resonances, j=pnand thejare the Hansen coecients X3;2 p(e) (see Wisdom et al. ). For the circular coorbital case, Correia and Robutel (2013) showed that a whole family results from the splitting of the synchronous resonance of the formj= 2nk. For small amplitudes of libration around L4 or L5 (tadpole), the width of the resonant island decreases as kincreases. In the eccentric coorbital case, each eccentric spin-orbit resonance of the Keplerian case splits in resonant multiplets which are centred in _=pn=2k=2. For relatively low amplitude of libration of , the width of the resonant island decreases as kincreases, see Figure 1 (left). But for higher amplitude, especially for horseshoe orbit, the main resonant island may not be located at k= 0. In Figure 1 (right), the main islands are located at _= 3n=25=2 and _= 3n=26=2. These islands overlap, giving rise to chaotic motion for the spin, while the island located at _= 3n=2 is much thinner. Figure 1. Poincar e surface of section in the plane ( t3n 2;_=n) near the 3 =2 spin-orbit eccen- tric resonance. (left): maxmin= 35- tadpole conguration. (right): maxmin= 336 horseshoe conguration. 3. Conclusion The coorbital spin-orbit resonances populate the phase space between the eccentric resonances. Generalised chaotic rotation can be achieved when harmonics of co-orbital spin-orbit resonances overlap each other, which is a dierent mechanism than the one described by Wisdom et al. (1984), where the eccentricity harmonics overlap. References Correia, A.C.M. C.M., & Robutel, P. 2013, Spin-orbit coupling and chaotic rotation for coorbital bodies in quasi-circular orbits AJ, 779, 20 Wisdom, J., Peale, S. J. & Mignard, F. 1984, The chaotic rotation of Hyperion, Icarus , 58, 137
1505.07667v1.Intrinsic_spin_Hall_effect_in_systems_with_striped_spin_orbit_coupling.pdf	Intrinsic spin Hall eect in systems with striped spin-orbit coupling G otz Seibold,1Sergio Caprara,2Marco Grilli,2and Roberto Raimondi3 1Institut f ur Physik, BTU Cottbus-Senftenberg, PBox 101344, 03013 Cottbus, Germany 2Dipartimento di Fisica - Universit a di Roma Sapienza, piazzale Aldo Moro 5, I-00185 Roma, Italy 3Dipartimento di Matematica e Fisica, Universit a Roma Tre, Via della Vasca Navale 84, 00146 Rome, Italy The Rashba spin-orbit coupling arising from structure inversion asymmetry couples spin and momentum degrees of freedom providing a suitable (and very intensively investigated) environment for spintronic eects and devices. Here we show that in the presence of strong disorder, non- homogeneity in the spin-orbit coupling gives rise to a nite spin Hall conductivity in contrast with the corresponding case of a homogeneous linear spin-orbit coupling. In particular, we examine the inhomogeneity arising from a striped structure for a two-dimensional electron gas, aecting both density and Rashba spin-orbit coupling. We suggest that this situation can be realized at oxide interfaces with periodic top gating. PACS numbers: 72.25.-b, 75.76.+j, 72.25.Rb, 72.15.Gd The spin Hall eect (SHE) [1] is the generation of a transverse spin current by an applied electric eld with the current spin polarization being perpendicular to both the eld and the current ow. Since the SHE allows the control of the spin degrees of freedom even without exter- nal magnetic elds (see e.g. Refs. [2, 3]), it has become a central topic in present spintronics research. [4, 5] The microscopic origin of the SHE lies in the spin-orbit cou- pling (SOC), which in solid-state systems may be due to the potential of the ionic cores of the host lattice, the po- tential of the impurities and the connement potential of the device structure. In a two-dimensional electron gas (2DEG), Bychkov and Rashba [6] have proposed that the lack of inversion symmetry along the direction perpen- dicular to the gas plane leads to a momentum-dependent spin splitting usually described by the so-called Rashba Hamiltonian H=p2 2m+zp; (1) where pis the momentum operator for the motion along the plane hosting the 2DEG, say the xy plane, zis a unit vector perpendicular to it, = (x;y;z) is the vector of the Pauli matrices, and is a coupling con- stant whose strength depends on both the SOC of the material and the eld responsible for the parity break- ing. The Hamiltonian (1), which has been extensively used in the study of the 2DEG in semiconducting sys- tems, has been recently applied also to interface states between dierent metals [7] and between two insulating oxides [9{12]. In the latter systems, higher mobilities, carrier concentration and SOC strengths have led to the expectation of observing stronger SOC-induced eects. The Hamiltonian (1) is deceptively simple, as one realizes when considering transport phenomena. In particular, the intrinsic universal SHE proposed in Ref. [13] turned out to be a non stationary eect, while under station- ary conditions cancellations occur, leading to a vanish- ing spin Hall conductivity (SHC) sH, i.e., the coecientrelating the z-spin current in the ydirection to the ap- plied electric eld, Jz y=sHEx. Here, we show that this is only true for a spatially homogeneous , while considering a space-dependent (x;y) opens the way to a substantial SHE under stationary conditions, even in the presence of strong disorder. We shall rst discuss from a general perspective how this comes about, and shall afterwards demonstrate numerically the eect in the presence of a spatially modulated SOC as it could be re- alized in the 2DEG at the interface of a LaAlO 3/SrTiO 3 (LAO/STO) heterostructure, schematically depicted in Fig. 1. It is experimentally established that the Rashba SOC increases when the electron density in the 2DEG of these heterostructures is increased [8{10, 12]. Since the local electric eld determining is tightly related to the electron density [14], one can naturally infer that the structure in Fig. 1 produces a modulation of the Rashba SOC. dw Jz y 2DEG STOLAO Ex FIG. 1. Schematic view of a possible device in which the SHE is enforced in the 2DEG at the interface of a LAO/STO heterostructure. The yellow stripes represent top-gating elec- trodes of width wand interspacing d. \| General arguments \| The interplay of the intrinsic Rashba SOC with the scattering from impurities makes the dynamics of charge and spin degrees of freedom in- trinsically coupled in a subtle way. This is especially evi-arXiv:1505.07667v1 [cond-mat.mes-hall] 28 May 20152 dent in the vanishing of the SHC with homogeneity in the SOC. Notice that the spin current is a tensor quantity de- pending on the ow direction (lower index) and spin po- larization axis (upper index) and hence the spin current and the electric eld are related by a tensor of third rank a ij. For xed polarization a=z, Onsager's relations require the antisymmetry property z yx=z xy=sH. The vanishing of the SHC manifests via an exact compensation of the contribution originally proposed by Sinova et al. [13] by a further contribution, which arises by the coupling between the spin current and the spin polarization, which is induced in the plane perpendic- ularly to the applied electric eld. This latter eect was almost simultaneously proposed by Edelstein [15] and by Aronov and Lyanda-Geller [16]. It consists of a non-equilibrium spin polarization due to the electric eld Sy 0=eN 0Exfor a eldExalong thexaxis. Heree is the unit charge ( efor electrons), N0=m=2~2is the density of states per spin of the 2DEG described by the rst term of Eq. (1) and is the elastic relaxation time introduced by impurity scattering. In such a 2DEG the standard Drude formula can be written via the Einstein relation as = 2e2N0D, with the diusion coecient D=v2 F=2,vFbeing the Fermi velocity related to the Fermi energy EF=mv2 F=2. To understand the origin of the compensation mentioned above, it is useful to start from a property of the Hamiltonian (1) rst pointed out by Dimitrova [17], which relates the time derivative of theSyspin polarization to the spin current @tSy=2m~1Jz y: (2) Notice that such a relation is not changed by disorder scattering as long as the latter is spin independent. Dis- order is necessary to guarantee a stationary state which implies the left-hand side of Eq. (2) to be zero. Obvi- ously, the corresponding vanishing of the right-hand side entails a vanishing SHC when is a constant. The spe- cic way in which the vanishing of the SHC occurs in a disordered 2DEG via the so-called vertex corrections [18{21] can be heuristically understood by describing the coupling between spin current and spin polarization as a generalized diusion in spin space. By dimensional ar- guments spin current and spin density must be related by the factor LSO=DP, whereLSO=~=(2m) is the spin-orbit precession length originating from the dier- ence of the Fermi momenta of the two branches of the spectrum of Eq. (1), while DPis the Dyakonov-Perel spin relaxation time due to the interplay of SOC and dis- order scattering. In a disordered 2DEG, DPis related toLSOby the diusion coecient L2 SO=DDP, so that one obtains for the spin current Jz y= 2m~1DSy+sH 0Ex: (3) In the above equation, which can be rigorously derived [22, 23], the quantity sH 0is the intrinsic contribution ofRef. [13] in the diusive regime pF=~1, whereas the term proportional to Sycorresponds to the vertex- correction contribution mentioned above. Given the ex- pression for sH 0= (e=8)(2=DP) derived in Ref. 13 it is now apparent that, if we replace Sywith the Edelstein result, the spin current in Eq. (3) vanishes, consistently with the stationarity requirement derived from Eq. (2). The key observation is that this compensation does not necessarily occur for an inhomogeneous SOC, where it is no longer possible to express the time derivative of the spin polarization in terms of the spin current. In such a situation a spin current becomes possible under station- ary conditions. In order to illustrate the physical mechanism by which an inhomogeneous Rashba SOC leads to a nite SHE it is useful to consider a single-interface problem, which is described by Eq. (1) with the replacement !(x) = (x)++(x)(with+>). Clearly as x!1 , one recovers the uniform case with complete cancellation of the spin current for the Rashba model with couplings . On both sides of the interface, the y-spin polariza- tion obeys a diusion-like equation with L, the corre- sponding spin-orbit lengths in the two regions. One can then seek a solution of the form Sy(x) =(x) S0;++s+ex=L+ +(x) S0;+sex=L+ ; whereS0;are the asymptotic values of the y-spin po- larization at1. The constants smust be deter- mined by imposing the appropriate boundary conditions atx= 0. As a result, the spin current is exponentially localized near the interface, where the spin polarization Sy(x) must interpolate between the two asymptotic val- ues and there is no longer complete cancellation between the two terms of Eq. (3). One can imagine to generalize this analysis to a series of interfaces apt to describe a periodic modulation of the SOC. Expectedly, if the spin-orbit length is larger than the distance between two successive interfaces, the spin Hall current should become practically uniform. \| The model and its numerical solution \| The previ- ous arguments within the diusive limit are now substan- tiated by numerical results for a microscopic 2D lattice model (size NxNy) with inhomogeneous Rashba SOC described by the Hamiltonian H=X ijtijcy icj+X i(Vi)cy ici+HRSO;(4) wherecy i(ci) creates (annihilates) an electron with spin projectionon the site identied by the lattice vector Ri, the rst term describes the kinetic energy of electrons on a square lattice (lattice constant a, only nearest-neighbor hopping:tijtforjRiRjj=a) and in the second3 termViis a local disorder potential with a at distribu- tionV0ViV0, andis the chemical potential. The last term is the lattice Rashba SOC, HRSO=iX i0i;i+xh cy iy 0ci+x;0cy i+x;y 0ci;i +iX i0i;i+yh cy ix 0ci+y;0cy i+y;x 0ci;i and the coupling constants i;i+x=y>0 are now dened on the bonds. Note that for constant i;i+x=yEq. (4) takes the usual form [2nd term in Eq. (1)] in momen- tum space [24]. One can show [30] that a \continuity equation" for the local spin density Sy i, (a dot stands for time derivative) _Sy i+ [div Jy]i+i;i+yJz i;i+y+iy;iJz iy;i= 0;(5) holds. For a homogeneous Rashba SOC, where [divJy]i= 0, Eq. (5) corresponds to Eq. (2) and im- plies that the total z-spin current has to vanish under stationary conditions. On the contrary, when varies in space, a cancellation occurs [30] between div Jyand the last two terms of Eq. (5), so that the stationarity condi- tion _S= 0 can be fullled without the vanishing of Jz. This is also clear for a system with periodic boundary conditions, where the total divergence of any current has to vanish, i.e., X i_Sy i=X i i;i+yJz i;i+y+iy;iJz iy;i :(6) Clearly, the left-hand side can vanish without implying Jz= 0, because, if is inhomogeneous, Eq. (6) can be fullled with alternating signs of Jzin regions with dierent[see the top panel of Fig. 4 in Ref. 30]. We exemplify the situation for an inhomogeneous Rashba SOC which varies along the xdirection form- ing a superlattice with d= 20aandw= 10a[see Fig. 1]. In the regions of width d,=a0is smaller than in the regions of width w, where=a1>a 0. The inhomogene- ity ini;i+x;yleads to a concomitant charge modulation which is shown in Fig. 2(a). We have diagonalized the Hamiltonian (4) and calculated the SHC from the Kubo formula (see, e.g., Ref. 13) sH=P ijsH ijwith sH ij2 NX En<EF Em>EFImhnjjz i;i+yjmihmjjch j;j+xjni (EnEm)2+2:(7) Here we have taken the limit of zero temperature and !0 is a small regularization term which can be inter- preted as an inverse electric-eld turn-on time [27] . Note that for the striped system one has already spin currents Jz;0 i;i+y owing in the ground state [25] (see, e.g., Fig. 3 in Ref. 30) whereas the electrically induced spin current is Jz;ind i;i+y=P jsH ijEx. Thus Eq. (6) can be split into a -5 -4 -3 -2 -1 0 chem. pot.00.511.522.5σsH, γ [1/8π]10 20 30x00.0020.0040.006 n(x) 10 20 3000.20.40.60.81α(x)d w σsH=γ (hom.)σsH (stripe) γ (stripe)FIG. 2. Main panel: Spin Hall coecient sHand \station- arity" parameter [both in units of 1 =(8)] as a function of for a homogeneous system with = 0:5t(black solid line) and a striped system ( sH: red solid and : blue dashed) with modulated Rashba SOC as shown in the inset. Here the (red) dashed line displays the variation of (x)i;i+x=i;i+y along thexdirection for stripes along the ydirection and widthw= 10aseparated by a distance d= 20a. The Rashba SOC on the stripes is (x)1= 0:8twhile between the stripes(x)0= 0:2t. The black solid line (dots) in the in- set reports the charge prole at chemical potential =4:3t. System size: 3060 3060 sites. 0 0.02 0.04 0.06n00.51σsH, γ [1/8π]α0=0.2, α1=0.8 α0=0.3, α1=0.7 α0=0.4, α0=0.6 0 0.02 0.04 0.06 0.08 0.1n00.20.40.60.8σsH, γ [1/8π]∆µ=0 ∆µ=0.5a) b) FIG. 3. a): sH(thick lines) and (thin lines) as a function of densityn(average number of electrons per lattice site) for a striped system and parameters are indicated in the panel. b):sH(thick lines) and (thin lines) as a function of density but now with an additional modulation of the local chemical potentialloc ixwhich is set to 0:5ton the stripes ( a0= 0:2t, a1= 0:8t). ground-state contribution (for which of course _Sy i= 0) and a linear-response part. For the latter we dene the quantity = 2P iji;i+ysH ijwhich therefore also de- scribes the linear response ofP i_Sy i= Exto the ap- plied electric eld and which in the following will be used to quantify the \stationarity" of the solution. In fact, Fig. 2 demonstrates that for a constant = 0:5t(black solid line)sHcoincides with and therefore the nite sHis a non-stationary result. On the other hand, the same panel also reports the results for the case a0= 0:2t anda1= 0:8t. In this case one can see that for a non- negligible range of chemical potential near the bottom of4 the band a substantial sH(red solid curve) is present while = 0 (blue dashed curve), marking the occurrence of a SHE in stationary conditions. This indicates the rel- evant role of those states that are still extended along the ydirection, while they are nearly localized inside the po- tential wells arising from the modulation of . As shown in Fig. 3(a), this situation occurs for increasingly large density ranges by increasing the inhomogeneity of . We have also checked that the stationarity is not only global but that in the low density regime _Si0 is fullled at each lattice site. We also investigated the eect of an inhomogeneous chemical potential as it is induced by the striped gating of Fig. 1. In particular, we shift the chemical potential downwards by = 0:5ton the sites below the gate (the regions of width wwith=a1= 0:8t. From Fig. 3(b) one sees that, although sHis reduced, it still remains substantial and the density range with a stationary SHE is even extended (black dashed curves). On the contrary one can see [30] that, in the absence of an inhomogeneous Rashba SOC, a simple charge modulation does not pro- duce any stationary SHE. -4.3 -4.2 -4.1 -4 -3.9 -3.8 chemical potential00.51σsH [1/8π]V0/t=0 V0/t=0.5 V0/t=0.2 V0/t=1.0 -4.2 -4 -3.8 -3.600.05 γ [1/8π]-4.2 -4 -3.8 -3.600.51 σsH[1/8π]η/t=0 η/t=0.02a)b) c)V0/t=0.5 FIG. 4. Main panel (a): Spin Hall conductivity as a function of chemical potential and various values of disorder. Panel (b):sHas a function of including error bars for V0=t= 0:5 and two values of =t= 0 (circles) and =t= 0:02 (triangles). Panel (c) reports the behavior of the \stationarity" parameter . Results in panels (a,c) are obtained for = 0. Finally, we address the quite important issue of the robustness of SHE in the presence of disorder. Previ- ous analyses [26, 27] showed that the SHC for a linear Rashba SOC is rapidly destroyed by disorder. This is easily understood because in the homogeneous Rashba SOC the SHE is a non-stationary eect which cannot survive the relaxation eect of disorder scattering. Here, instead, when = 0 the SHE is present in a stationary state and disorder is much less eective in spoiling it. In the presence of a random potential of nite width V0 the calculations can only be carried out on smaller lat- tices (4040 sites) where we consider stripes of widthw= 4aand distance d= 4a(see Fig. 1). We follow the procedure described in Ref. 26 and diagonalize the Hamil- tonian for dierent disorder congurations and dierent twisted boundary conditions. For each concentration we consider 250 random boundary phases and 50 disorder congurations. As a striking result (main panel of Fig. 4) we nd that the average SHC at low densities ( <4t) is not aected by disorder and only gets suppressed when the chemical potential is within the range of band states ex- tended both along xandydirections. In contrast, and as mentioned above, sHvanishes for a homogeneous, lin- ear Rashba SOC [26, 27] in the presence of disorder and for!0. It has been pointed out in Ref. [27] that the evalua- tion ofsHon nite lattices and taking the limit !0 is complicated by strong uctuations. These strong vari- ances in the SHC are exemplied in panel (b) of Fig. 4 forV0=t= 0:5 where we also show the corresponding re- sult for=t= 0:02. The SHC in the low density regime is not dependent on the small value but one observes a large reduction in the variance which becomes of the or- der of the symbol size. We therefore can safely conclude that our nite size results support a nite SHC at low densities even for strongly disordered systems. Naturally, the system gets more stationary with disor- der, as it is shown in panel (c) of Fig. 4, where a small residual value of for the clean striped system (black curve) is suppressed for all 's. Again, for <4tthis does not imply the vanishing of sH, as would be the case for homogeneous Rashba SOC. \| Discussion and conclusions \| The above analy- sis clearly demonstrates that a system with modulated Rashba SOC can sustain a nite SHE in stationary con- ditions. This occurs for a limited density range, when the chemical potential falls in a region where the states are strongly aected by the modulated (x) and are almost localized in the bottom of the modulating potential (in the direction of the modulation; the states are extended in the perpendicular direction). Therefore the response of the charge current Jch xto the electric eld along the modulation direction is strongly suppressed which can lead to large spin Hall angles eJz y=Jch xfor the striped system. It is important to note that this eect is due to the modulation of the Rashba SOC and cannot arise in a \conventional" charge density wave. In fact, for con- stantand independent of the electronic structure Eq. (6) predicts the vanishing of the SHE under stationary conditions. The implementation of this analysis in a real system is for sure a challenging task for several reasons. First of all the top gating structuring has to be sharp enough to pro- duce a suciently sharp spatial modulation of the 2DEG below the LAO layer (which is at least 20 nm thick): if the modulation of the SOC is not sharp enough on the LSOscale, the 2DEG would feel a nearly uniform and5 the SHE is expected to vanish. One should also con- sider that our analysis is based on a simple one-band model, while the SOC in the 2DEG in the LAO/STO involves several bands [14, 31, 32]. Of course, the basic ideas of this work could be tested and hopefully imple- mented in other, perhaps simpler, systems involving het- erostructures of semiconductors with modulated Rashba SOC which have been already discussed in the literature in dierent contexts [33{35]. We acknowledge interesting discussions with N. Bergeal, V. Brosco, and J. Lesueur. G.S. acknowl- edges support from the Deutsche Forschungsgemein- schaft. M.G. and S.C. acknowledge nancial support from Sapienza University of Rome, project Awards No. C26H13KZS9. APPENDIX Rashba systems in the diusive limit To gain insight on the numerical results discussed in the main text, we discuss here a continuum Rashba model in the diusive limit. The corresponding diusion equa- tions can be derived from a microscopic model by using e.g. the Keldysh technique. Such a derivation is, for in- stance given in Ref. 36. For the following discussion, however, one does not need to know such a microscopic derivation in detail. One main advantage of the diusion equation description is that it contains all the important aspects of the Rashba model and allows an almost an- alytic treatment, which helps in elucidating the physics. We rst provide the diusion equation description for the uniform case. This is very standard and a recent discus- sion can be found in Ref. 23. Then we describe the single-interface problem as the simplest realization of a non-uniform Rashba system with two regions with dif- ferent Rashba SOC. A subsequent subsection reports the two-interface problem. The uniform case In the presence of the Rashba spin-orbit coupling, the spin polarization along the y direction obeys the follow- ing equation in the diusive regime (see Ref. 36 for a derivation) @tsy+D@2 xsy=1 DP(sys0); (8) whereD=v2 F=2 is the standard diusion coecient. 1 DP= (2m)2Dis the inverse Dyakonov-Perel spin re-laxation time. s0=eN0Exis the non-equilibrium spin polarization induced by the electric eld Exapplied along the x direction. Such a non-equilibrium polariza- tion is sometimes called the Edelstein eect or the spin- galvanic eect. Here N0=m=2~2is the 2D density of states and, in the following, we take units such that ~= 1. We consider only the dependence on the x di- rection in the diusion equation to make contact with the numerical calculation. In stationary and uniform cir- cumstances, we must have sy=s0. This leads to the vanishing of the spin current Jz y, as it is well known in the Rashba model (see for instance Ref. 23). To see this, consider that the spin current is given by two terms: a "drift-like" Hall term and a "diusion-like" one. The drift term corresponds to the calculation of Ref. 13, i.e. it is just the Drude formula for the spin Hall conductiv- ity. As for the ordinary Hall conductivity, it is non-zero and nite even in the absence of disorder. The diusion term is usually expected to vanish in uniform situations. However as derived in Ref. 22 and used in Ref. 23, the Rashba interaction can be described in terms of a SU(2) vector potential, which then introduces covariant deriva- tives. The latter are dened by h ~risia =risaabcAb isc(9) whereAa iis the SU(2) vector potential. In the Rashba case,Ax y=Ay x= 2m. The key observation is that the diusion term related to the covariant derivative of the spin density is present even in the uniform case. Speci- cally the spin Hall current we are interested in reads Jz y=sH 0Ex+D LSOsy; (10) whereLSO= (2m)1is the spin-orbit length. Notice the relation L2 SO=Ds, which amounts to say that the Dyakonov-Perel relaxation time is the time over which electrons diuse over a spin-orbit length. sH 0corre- sponds to the expression given in Ref. 13, i.e. the Drude formula evaluated without vertex corrections, of the spin Hall conductivity sH 0=e 82 DP: (11) In the diusive regime pF1, one has DP, whereas in the ballistic one pF1DP. Notice that, by using Eq.(11) and the expression for s0, the spin current (10) can also be written as Jz y=1 2m1 s(sys0): (12) By identifying the y-polarized spin current owing along the x direction Jy x=D@xsy(13)6 the diusion equation becomes @tsy+@xJy x=2mJz y=1 LSOJz y (14) which is the continuity-like equation for syshowing how the torque term associated to syis expressed in terms of Jz y. Single-interface problem In this Section we consider a static time-independent situation. The idea is to analyze a single-interface prob- lem in order to understand the supercell numerical calcu- lation. We then assume the following expression for the Rashba coecient (x) =(x)++(x): (15) Clearly at1, one recovers the uniform case with com- plete cancellation of the spin current for the Rashba model with couplings . The strategy is to solve the diusion equation in the two regions and connect them via the appropriate boundary conditions at x= 0. We seek a solution of the form sy(x) =(x) s0;++s+ex=L+ +(x) s0;+sex=L+ : (16) In the above s0;are the asymptotic values of the y- spin polarization at 1. AlsoLare the corresponding spin-orbit lengths in the two regions. The z-polarized spin current in the two regions reads Jz y;(x) =sH 0;Ex+D L s0;+sex=L =D Lsex=L; (17) where in the last step we used the fact that the constant terms cancel in each region. Dmay dier in the two region because via the Fermi velocity they depend on the electron density. Eq.(17) shows that in the interface region there can be a spin current dierent form zero. To evaluate it we need to know the two values s. To this end we use the continuity of the spin density syand of the spin current Jy xat the interface. Continuity of the spin density gives ss+= s0; (18) where s0=s0;+s0;=eN0(+)Ex. Conti- nuity of the spin current instead gives D Ls+D+ L+s+= 0: (19)After solving the system for sands+,we get, s= s0D L1 D+L1 ++DL1 (20) s+=s0D+ L+1 D+L1 ++DL1 : (21) From this it is clear that the spin Hall current averaged over the spin-orbit length 1 LZ0 1dx Jz y;+1 L+Z1 0dx Jz y;+= 0; (22) which is the analog of Eq.(14). Furthermore the left-hand side correspond to the quantity in the main text. On the other hand we have that the total z-polarized spin current is dierent from zero Z0 1dx Jz y;+Z1 0dx Jz y;+= (23) 2eN0D+Dm(+)2 D+L1 ++DL1 ExsHLeffEx; whereLeffis an eective length determined in terms of LandD. The spin Hall current is localized at the interface within a distance of order L. The above calculation of a single interface suggests that the calculation for the periodic modulation de- scribed in the main text can be analyzed in terms of a series of interfaces. The spin current ows in the interface regions. However, by making the interface separation of the order of the spin relaxation length, one may have a spin current nite everywhere. Two-interface problem Here we consider the problem with two interfaces with the model given by (x) =+(ljxj) +(jxjl): (24) The solution for the spin density sy(x) is of the form sy(x) =(xl)h s0;+sRe(xl)=Li +(xl)h s0;+sLe(x+l)=Li +(ljxj) s0;+s0cosh(x=L +) cosh(l=L +) +sLsinh((lx)=L+) sinh(2l=L +)+sRsinh((l+x)=L+) sinh(2l=L +) where the continuity of syhas already been implemented. The two constants sRandsLmust be determined by imposing the conservation of the longitudinal y-polarized spin current Jy xas in the single-interface problem.7 After going through steps as for the single-interface we get Z1 1dx Jz y(x) = (25) 4eN0D+Dm(+)2tanh(l=L +) D+L1 +tanh(l=L +) +DL1 Ex: The above equation has a simple interpretation in the limitl!1 , when the two interfaces are far apart. In this limit the two interfaces are independent. The total spin current is the sum of the spin currents owing at the two interfaces. Eq.(25) reduces indeed to twice the contribution (23) of a single interface. On the other hand whenlL+the two interfaces interact and the spin Hall current in non zero everywhere. Continuity equations for the Rashba lattice model We provide here a discussion to clarify the role of an in- homogeneous Rashba SOC to allow for a SHE in station- ary conditions. The electron spin Sobeys the equation of motion dS dt=i[S;H] (26) and for the following it is convenient to separate the com- mutator into a term related to the divergence of spin currents and a 'rest' G [S;H] =idivJ+iG: (27) As a result Eq. (26) can be interpreted in terms of a continuity equation G=divJ+dS dt(28) where Gacts as 'source' term which in general is nite due to the non-conservation of spin. From evaluation of the commutators one nds the re- lation Gy i=i;i+yJz i;i+yiy;iJz iy;i:; (29) i.e. a torque for the y-component of the spin is associated with a z-spin current when the Rashba SOC 6= 0. Upon combining Eq. (29) with the y-component of Eq. (28) one nds dSy i dt+ [divJy]i+i;i+yJz i;i+y+iy;iJz iy;i= 0 (30) and in particular for a system with periodic boundaries X idSy i dt+i;i+yJz i;i+y+iy;iJz iy;i = 0 (31)since the total divergence of any current has to vanish. For a homogeneous SOC coupling Eq. (31) implies that the total z-spin current has to vanish under stationary conditions. We exemplify the situation for a inhomogeneous Rashba SOC which varies along the x-direction as i;i+x=1 2 a0+a1+ (a0a1) sgn(sin2ix 2L) (32) i;i+y=i;i+x; (33) i.e. one has stripes of width Lwith Rashba SOC a0alter- nating with L-wide stripes having coupling a1(cf. Fig. 5). The inhomogeneity in i;i+xleads to a concomitant charge modulation which is shown in Fig. 6 for the case L= 4. Clearly the charge accumulates in regions with a0a0a0 a1a1a1 2 L FIG. 5. Stripe-like modulation of the Rashba SOC along the x-direction. Stripes of width Land coupling a0alternate with stripes of the same width and coupling a1as indicated by thin and thick bonds, respectively. large Rashba SOC leading to a CDW prole. We can therefore view this model as an 'eective' model for a density dependent Rashba SOC. 0 5 10 15 20 25 site ix00.040.080.120.160.2charge density 0 5 10 15 20 2500.20.40.60.81 Rashba coupling [t] FIG. 6. Modulation of the coupling constant i;i+x(red) and charge density along the x-direction for the L= 4 stripe-like Rashba SOC. Particle concentration: n= 0:07. Fig. 7 shows the currents owing in the (stationary) ground state. Thus from Eq. (28) the torques Gy iare completely determined by the divergence of the Jyspin currents which are shown by squares in the top panel of Fig. 7. As a consequence of Eq. (29) a large z-spin8 current is owing on sites where also the y-torque is large in contrast to a homogeneous system where Jz= 0. In the ground state the total y-torqueP iGy ivanishes so that from Eq. (29) one obtains 0 =X i i;i+yJz i;i+y+iy;iJz iy;i : (34) 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 FIG. 7. Spin currents (arrows) and torques (squares) in the ground state for a system with alternating L= 4 stripes with Rashba couplings a0= 0:2 anda1= 0:8, respectively. Top panel: y-components; Bottom panel: z-components. Particle concentration: n= 0:07. The total number of a0;1-stripes for a NxNylattice isnstr=Nx=(2L). Denote with Jz 0;1the total z-spin current owing along the bonds of the a0;1-stripes. Then we can rewrite Eq. (34) as 0 =a0Jz 0+a1Jz 1!Jz 1=a0 a1Jz 0 (35) and the total z-spin current of the system is thus givenby Jz tot=nstr(Jz 0+Jz 1) =nstrJz 0 1a0 a1 : (36) Now we can draw the following conclusions: The mod- ulated Rashba SOC causes local torques Gy iwhich are related to local ows of z-spin currents. Provided that the total y-torque vanishes the system thus exhibits a net ow ofJz totfora06=a1in the ground state. The same analysis can now be applied in the presence of an electric eld Ex=@tAxwhich couples to the sys- tem via the charge current, i.e. H0=eP ijch i;i+xAx(i;t) andjch i;i+x. Obviously Eq. (30) holds also in the result- ing non-equilibrium situation which we evaluate in linear response. Each operator ^Oiin Eq. (30) reacts to the electric eld according to ^Oi(!) =ieij(!)Ej(!)=!and the correlation function is given by ij(!) =i NZ1 1dt(tt0)ei!(tt0)hh ^Oi(t);jch j;j+x(t0)i i: 0 5 10 15 20 25 site ix-5e-0505e-050.0001Jz(ix) / E ; dSy(ix) / dt / E 0 5 10 15 20 250.20.40.60.8 Rashba coupling [t] 0 5 10 15 20 25 site ix-0.0002-0.000100.00010.0002Jy(ix) / E ; div Jy / E; γ(ix) / EJy div Jy γ 0 5 10 15 20 250.20.40.60.8 Rashba coupling [t] FIG. 8. Top panel: Modulation of the coupling constant i;i+x(red) and induced Jz-current (black, circles) for a cut along the x-direction and L= 4 stripe-like SOC. The blue line (squares) shows the temporal change of the y-spin com- ponentdSy(ix)=dt. Bottom panel: Induced Jyspin current (black, open circles) and corresponding divergence (black, full circles). The blue line (squares) shows the response of ix= 2i;i+yJz i;i+yalong the same cut. Particle concentra- tion:n= 0:07. The top panel of Fig. 8 shows the induced Jzspin current together with the induced temporal change of9 dSy(ix)=dtalong a cut in x-direction of a L= 4 striped system. The dominant contribution to Jzcomes from the boundary regions between small and large stripes which gives rise to a nite spin Hall conductivity. Moreover, in contrast to the homogeneous case, where the induced Jz current and dSy(ix)=dtare equal @Sy @t=2m ~Jz y: (37) (although with opposite sign), we now observe a much more stationary behavior. The reason can be deduced from the bottom panel which reports the x-dependence of the induced Jyspin current along with its diver- gence, i.e. [ divJy]iJy ix;ix +1(ix)Jy ix1;ix. It turns out that the spatial behavior of the contribution ix= 2i;i+yJz i;i+y(ix) is similar to [ divJy]ibut opposite in sign which from Eq. (30) is responsible for the small value of dSy(ix)=dt. From the above considerations we see that in a ho- mogeneous Rashba system a nite spin Hall conductivity necessarily implies a non-stationary situation with the lo- cal accumulation of Syspin density. On the other hand an inhomogeneous Rashba coupling partially shifts this time dependence to a nite divergence of the Jyspin currents which is stationary due to non-conservation of spin. [1] M. I. Dyakonov and V. I. Perel, Phys. Lett. A 35, 459 (1971). [2] G. Vignale, J. Supercond. Nov. Magn. 23, 3 (2010). [3] T. Jungwirth, J. Wunderlich, and K. Olejn k, Nat. Ma- terials 11, 382 (2012). [4] I. Zuti c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). [5] S. Maekawa, H. Adachi, K. Uchida, J. Ieda, and E. Saitoh, J. Phys. Soc. Japan 82, 102002 (2013). [6] Yu. A. Bychkov and E. I. Rashba, J Phys C: Solid State Phys. 17, 6039 (1984). [7] J. C. Rojas S anchez, L. Vila, G. Desfonds, S. Gambarelli, J. P. Attan e, J. M. De Teresa, C. Mag en, and A. Fert, Nat. Commun. 4, 2944 (2013). [8] J. Nitta, T. Akazaki, and H. Takayanagi, Phys. Rev. Lett. 78, 1335 (1997). [9] A. D. Caviglia, M. Gabay, S. Gariglio, N. Reyren, C. Cancellieri, and J.-M. Triscone, Phys. Rev. Lett. 104, 126803 (2010). [10] A. Fete, S. Gariglio, A. D. Caviglia, J.-M. Triscone, and M. Gabay, Phys. Rev. B 86, 201105 (2012).[11] Lorien H. Hayden, R. Raimondi, M. E. Flatt, and G. Vignale Phys. Rev. B 88, 075405 (2013). [12] S. Hurand, A. Jouan, C. Feuillet-Palma, G. Singh, J. Bis- caras, E. Lesne, N. Reyen, Thales, C. Ulysse, X. Lafosse, M. Pannetier-Lecoeur, S. Caprara, M. Grilli, J. Lesueur, N. Bergeal, arXiv:1503.00967 . [13] Jairo Sinova, Dimitrie Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth, and A. H. MacDonald Phys. Rev. Lett. 92, 126603 (2004). [14] S. Caprara, F. Peronaci, and M. Grilli, Phys. Rev. Lett. 109, 196401 (2012); D. Bucheli, M. Grilli, F. Peronaci, G. Seibold, and S. Caprara, Phys. Rev. B 89, 195448 (2014). [15] V. M. Edelstein, Solid State Communications 73, 233 (1990). [16] A. G. Aronov and Yu. B. Lyanda-Geller, JETP Lett. 50, 431 (1989). [17] Ol'ga V. Dimitrova, Phys. Rev. B 71, 245327 (2005). [18] J. I. Inoue, G. E. W. Bauer, and L. W. Molenkamp, Phys. Rev. B 70, 041303(R) (2004). [19] E. G. Mishchenko, A. V. Shytov, and B. I. Halperin, Phys. Rev. Lett. 93, 226602 (2004). [20] R. Raimondi and P. Schwab, Phys. Rev. B 71, 033311 (2005). [21] A. Khaetskii, Phys. Rev. Lett. 96, 056602 (2006). [22] C. Gorini, P. Schwab, R. Raimondi and A.L. Shelankov, Phys. Rev. B 82, 195316 (2010). [23] R. Raimondi, P. Schwab, C. Gorini, G. Vignale, Ann. Phys. (Berlin) 524, 153 (2012). [24] G. Seibold, D. Bucheli, S. Caprara, and M. Grilli, Euro- phys. Lett. 109, 17006 (2015). [25] E. B. Sonin, Phys. Rev. B 76, 033306 (2007). [26] D. N. Sheng, L. Sheng, Z. Y. Weng, and F. D. M. Hal- dane, Phys. Rev. B 72, 153307 (2005). [27] K. Nomura, Jairo Sinova, N. A. Sinitsyn, and A. H. Mac- Donald, Phys. Rev. B 72, 165316 (2005). [28] A. Agarwal, Stefano Chesi, T. Jungwirth, Jairo Sinova, G. Vignale, and Marco Polini, Phys. Rev. B 83, 115135 (2011). [29] K. C. Hass, B. Velicky, and H. Ehrenreich, Phys. Rev. B 29, 3697 (1984). [30] See Supplemental Material at http://link.aps.org/ sup- plemental/..... for a detailed description of the single and double interface case in the diusive limit, for the deriva- tion of a continuity equation on a lattice in the pres- ence of inhomogeneous Rashba SOC. Numerical results for a spin and spin-current distribution on a lattice with striped Rashba SOC are also reported. [31] G. Khalsa, B. Lee, and A. H. MacDonald, Phys. Rev. B 88, 041302(R) (2013). [32] Z. Zhong, A. T oth, and K. Held, Phys. Rev. B 87, 161102(R) (2013). [33] X. F. Wang, Phys. Rev. B 69, 035302 (2004). [34] G. I. Japaridze, Henrik Johannesson, and Alvaro Ferraz, Phys. Rev. B 80, 041308(R) (2009). [35] Mariana Malard, Inna Grusha, G. I. Japaridze, and Hen- rik Johannesson, Phys. Rev. B 84, 075466 (2011). [36] R. Raimondi et al. Phys. Rev. B 74, 035340 (2006).
1206.1062v1.Von_Neumann_Entropy_Spectra_and_Entangled_Excitations_in_Spin_Orbital_Models.pdf	arXiv:1206.1062v1 [cond-mat.str-el] 4 Jun 2012VonNeumann Entropy Spectra and Entangled Excitations inSp in-Orbital Models Wen-Long You,1,2Andrzej M. Ole´ s,1,3and Peter Horsch1 1Max-Planck-Institut f¨ ur Festk¨ orperforschung, Heisenb ergstrasse 1, D-70569 Stuttgart, Germany 2School of Physical Science and Technology, Soochow Univers ity, Suzhou, Jiangsu 215006, People’s Republic of China 3Marian Smoluchowski Institute of Physics, Jagellonian Uni versity, Reymonta 4, PL-30059 Krak´ ow, Poland (Dated: October 21, 2018) We consider the low-energy excitations of one-dimensional spin-orbital models which consist of spinwaves, orbital waves, and joint spin-orbital excitations. Among t he latter we identify strongly entangled spin-orbital bound states which appear as peaks in the von Neumann entropy (vNE) spectral function introduced in this work. Thestrongentanglement ofboundstatesismanifested byauniversallogarithmicscalingofthevNEwith systemsize,whilethevNEofotherspin-orbitalexcitation ssaturates. Wesuggestthatspin-orbitalentanglement can be experimentally explored by the measurement of the dyn amical spin-orbital correlations using resonant inelasticx-ray scattering, where strongspin-orbit coupl ing associatedwiththe core hole plays a role. PACS numbers: 75.10.Jm,03.65.Ud, 03.67.Mn,75.25.Dk Introduction.— The spin-orbital interplay is one of the im- portant topics in the theory of strongly correlated electro ns [1]. In many cases, the intertwined spin-orbital interacti on is decoupledby mean-ﬁeld approximation,and the spin and or- bital dynamicsare independentfromeach other. Thusa spin- only Heisenberg model can be derived by averaging over the orbitalstate,whichsuccessfullyexplainsmagnetismando pti- calexcitationsinsomematerials,forinstanceinLaMnO 3[2]. But in others,especiallyin t2gsystems[3], the orbitaldegen- eracy plays an indispensable role in understanding the low- energy properties in the Mott insulators of transition meta l oxides (TMOs), such as LaTiO 3[4], LaVO 3and YVO 3[5], and also in recently discussed RbO 2[6]. The well known casesarealsostrongspin-orbitcouplingwhichleadstoloc ally entangled states [7], and entanglement on the superexchang e bondsin K 3Cu2F7[8]. For such models, the mean-ﬁeld-type approximation and the decoupling of composite spin-orbita l correlations fail and generate uncontrolled errors, even w hen the orbitals are polarized [9]. The strong spin-orbital ﬂuc tu- ations on the exchange bondswill induce the violation of the Goodenough-Kanamori rules [10]. Furthermore, the ﬂavors mayformexoticcompositespin-orbitalexcitations. Model and system.— A paradigmatic model derived for a TMO in Mott-insulating limit is the one-dimensional (1D) spin-orbitalHamiltonian,whichreads H=−J/summationdisplay i/parenleftBig /vectorSi·/vectorSi+1+x/parenrightBig/parenleftBig /vectorTi·/vectorTi+1+y/parenrightBig ,(1) where/vectorSiand/vectorTiare spin-1/2 and pseudospin-1/2 operators representing the spin and orbital degrees of freedom locate d at sitei, respectively,and we set below J= 1. It is proposed that ultracold fermions in zig-zag optical lattices can rep ro- duce an effectivespin-orbitalmodel[11]. For general {x,y}, the model (1) has an SU(2) ⊗SU(2) symmetry. An additional Z2bisymmetry occurs by interchanging spin and orbital op- erators when x=y. In the case of x=y=1 4, Hamiltonian (1)reducestoaSU(4)symmetricmodel,whichisexactlysol- uble by the Bethe ansatz [12, 13]. There are three Goldstone modes corresponding to separate spin and orbital excitatio n,aswellascompositespin-orbitalexcitationsincaseof J <0, in contrast to a quadratic dependence of the energy upon the momentum in the long-wave limit for J >0. The spectra of elementary excitations are commonly not analytically so l- uble away from the SU(4) point. We will, however, show that the low-energy excitations can be analytically obtain ed in some speciﬁc phases in the case when J >0, and this offers a platform to study the spin-orbital entanglement. I n this Letter, we go beyond the ideas developed for spin sys- tems[14]. We demonstratethatspin-orbitalentanglemente n- tropy clearly distinguishes weakly correlated spin-orbit al ex- citations from bound states and resonances by its magnitude anddistinct scaling behavior. We proposehow to connectthe entanglemententropywith experimentallyobservablequan ti- tiesofrecentlydevelopedspectroscopies. vonNeumannentropy.— Currently,conceptsfromquantum information theory are being studied with the aim to explore many-body theory from another perspective and vice versa. A particularly fruitful direction is using quantum entangl e- ment to shed light on exotic quantum phases [15, 16]. En- tanglement entropy even distinguishes phases in the absenc e of conventional order parameters [17]. In general a many- body quantum system is subdivided into AandBparts and theentanglemententropyisthevonNeumannentropy(vNE), SvN=−Tr{ρAlog2ρA},whereρA=TrB{ρ}isthereduced density matrix of the subspace Aandρis the full density matrix. The vNE is bounded, SvN≤log2dimρA, and easy to calculate. Experimental determination appears harder, yet thereareproposalsinvolvingtransportmeasurementsinqu an- tumpointcontacts[18]. InterestinglythevNEscalesproportionallytotheboundar y ofthesubregionobtainedbythespatialpartitioning[19]. The dependenceoftheboundaryorarealawcanbetracedbackto studyofblackholephysics[20]andwasextensivelyexploit ed for 1D spin chains [21]. If the block Ais of length lin a sys- temoflength Lwithperiodicboundarycondition,thevNEof gappedgroundstates is boundedas Sl=O(1), while a loga- rithmic scaling Sl=clog2l+O(1) (L≫l≫1)has been provento be universal propertyof the gaplessphases in crit i-2 FIG.1: (color online). Spin-orbital entanglement SvNinthe ground- stateofthespin-orbitalmodel(1)asafunctionof xandyandsystem sizeL= 8. The(red)dashedlinesmarkthecriticallinesdetermined by the ﬁdelity susceptibility (see text). The two-site conﬁ gurations in phases I-IV are shown on the left. The two orbitals per site are degenerate (their splittingis onlyfor clarityof presenta tion). cal systems by the underlyingconformalﬁeld theory [22]. A violationofthe arealaw is expectedforthe low-lyingexcit ed states of critical chains [23]. To date, measurements of the vNE for subdivisionof degrees of freedomother than in spa- tialsegmentationhavenotbeenfullyexplored. Inacomposi te system containing spin and orbital operators, the decompos i- tionofdifferentﬂavorsretainsthereal-spacesymmetries . Phase diagram.— A quantum phase transition (QPT) is identiﬁed as a point of nonanalyticity of the groundstate an d associated expectationvaluesin the thermodynamiclimit. To shedlightonthephaseboundaries,weﬁrst considertwosite s [24],H12=−1 4(/vectorS2 12−/vectorS2 1−/vectorS2 2+2x)(T2 12−/vectorT2 1−/vectorT2 2+2y), where/vectorS12=/vectorS1+/vectorS2and/vectorT12=/vectorT1+/vectorT2. A pair of spins (orbitals)canformeither a singlet with S12= 0(T12= 0)or a triplet with S12= 1(T12= 1), and various combinations of quantumnumberscorrespondto differentphases shown in Fig. 1. In phase I, the state with S12= 1 =T12has the lowest energy, and thus the energy per bond is eI B≥exy= −(x+1/4)(y+1/4). For a largersystem with Lbonds, we haveEI 0(H)≥Lexy. On the other hand, takinga ferro-ferro state\|0∝an}bracketri}htas a variational state, EI 0(H)≤Lexy. Therefore, theenergyofphaseIisexactly EI 0(H) =Lexyandtheferro- ferrostate isthecorrespondinggroundstate. Without prior knowledge of order parameter, variouschar- acterizations from the perspective of quantum information theory can be used to identify phase boundaries. One often usedtoolisthevNE[25]. Tracingorbitaldegreesoffreedom , we obtained the spin-orbital vNE SvNfor the ground state of L= 8chain in the Hilbert subspace of Sz=Tz= 0[25]. However, here we ﬁnd that the vNE of the ground state does notdistinguishphaseIfromphaseIIorIV—allthreephases havingSvN= 0(seeFig. 1). Thereforeweusethequantumﬁ- delitytoquantifythephasediagram[26]. Theﬁdelitydeﬁne d/s48 /s49 /s50 /s51/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48 /s48/s46/s50/s53 /s48/s46/s53/s48 /s48/s46/s55/s53 /s49/s46/s48/s48/s48/s49/s50/s51/s52/s32/s32/s40/s81 /s41 /s81/s40/s97/s41 /s32/s32 /s120/s32/s81/s61/s48/s46/s50/s32 /s32/s81/s61/s48/s46/s53 /s32 /s32/s81/s61/s48/s46/s56/s32/s40/s98/s41 FIG.2: (color online). (a) Energyspectra of 40-site spin-o rbital sys- tem atx=y= 1/4. Dashed lines in the spin-orbital continuum denote the spin, orbital and OBS excitation, all degenerate ; the (red) solid line below corresponds to the BS. (b) The decay rate Γof the OBSfor different momenta Qwithy=xatL→ ∞. asfollows, F(λ,δλ) =\|∝an}bracketle{tΨ0(λ)\|Ψ0(λ+δλ)∝an}bracketri}ht\|, istakenalong a certain path {x(λ),y(λ)}and reveals all phase boundaries. The ﬁdelity susceptibility, χF≡ −(2lnF)/(δλ)2\|δλ→0, ex- hibits a peak at the critical point, and can be treated as a ver - satile order parameter in distinguishing ground states [27 ]. It signals the phase boundaries shown in Fig. 1. Remarkably, the phase diagram found from the ﬁdelity susceptibility for largersystemsisthe sameastheonefor L= 2. Excitations.— In phase I of Fig. 1, with boundaries given by:x+y=1 2,x=−1 4andy=−1 4, the spins and orbitals are fully polarized,and the ferro-ferrogroundstate \|0∝an}bracketri}htis dis- entangled, i.e., can be factorized into spin and orbital sec tor. It is now interesting to ask whether: (i) the vanishing spin- orbitalentanglementinthegroundstateimpliesasuppress ion of joint spin-orbital quantum ﬂuctuations, and (ii) collec tive spin-orbital excitations can form. Using equation of motio n method one ﬁnds spin (magnon) excitations with dispersion ωs(q) = (1 4+y)(1−cosq),andorbital(orbiton)excitations, ωt(q) = (1 4+x)(1−cosq)[28]. Thestabilityoftheorbitons (magnons) implies that x >−1 4(y >−1 4), and determines the QPT between phasesI and II (IV), respectively,while the spin-orbitalcouplingonlyrenormalizesthespectra. For our purpose, it is straightforward to consider the prop- agation of a pair of magnon and orbiton along the ferro-ferro chain, by simultaneously exciting a single spin and a single orbital. The translation symmetry imposes that total momen - tumQ= 2mπ/L(m= 0,···,L−1)is conserved during scattering. The scattering of magnon and orbiton with initi al (ﬁnal)momenta {Q 2−q,Q 2+q}({Q 2−q′,Q 2+q′})andtotal momentum Qisrepresentedbythe Green’sfunction[29], G(Q,ω) =1 L/summationdisplay q,q′/angbracketleftBig/angbracketleftBig S+ Q 2−q′T+ Q 2+q′\|S− Q 2−qT− Q 2+q/angbracketrightBig/angbracketrightBig ,(2) for a combined spin ( S− Q 2−q) and orbital ( T− Q 2+q) excitation.3 /s48 /s49 /s50 /s51/s81/s61/s48/s46/s48 /s81/s61/s48/s46/s50 /s81/s61/s48/s46/s52 /s81/s61/s48/s46/s54 /s81/s61/s48/s46/s56 /s48/s50/s52 /s83 /s118/s78 FIG. 3: (color online). The vNE distribution of 400-site spi n-orbital systeminsubspace PST= 1forx=y= 1/2anddifferentmomenta Q. Isolated vertical lines indicate the BS, with dispersion g iven by the (red) dashed line. The OBSinthe center of spectra is damp ed. The spin-orbital continuum is given by Ω(Q,q) =ωs(Q 2− q) +ωt(Q 2+q). In the noninteracting case, the Green’s function exhibits square-root singularities at the edges o f the continuum [30]. Due to residual, attractive interactions s pin- orbital bound states (BSs) are shifted outside the continuu m [24,31,32], see Fig. 2(a). Thecollectivemodeis determine d by1 +1 2π/integraltextπ −πdq(cosQ 2−cosq)2/[ω−Ω(Q,q)] = 0. The analytic solution of this equation is tedious but straightf or- ward. The collective BS with dispersion ωBS(Q)is well sep- arated from the spinon-orbitoncontinuum[Fig. 2(a)] at lar ge Q. In the long-wave limit the BS energy coincides with the Arovas-Auerbach line [33], i.e., the boundary of the contin - uum,yetthe BS remainsundampedfor x+y >1 2. Inaddition,acollectivemodeofspin-orbitalresonances, \|Ψ(Q)∝an}bracketri}ht=1√ L/summationdisplay m,lal(Q)eiQmS− mT− m+l\|0∝an}bracketri}ht,(3) occurs inside the continuum. Here 0≤l≤L−1de- notes the distance between spin and orbital ﬂips. Remark- ably, the spin and orbital ﬂips are glued together at the same site with al(Q) =δl,0at the SU(4) point [28]. This cou- pled on-site BS (OBS) is a coherent superposition of local modes, all of them with equal weight. It has dispersion ωOBS(Q) =x+y−1 2cosQ, which is degenerate with both ωs(Q)andωt(Q)atx=y=1 4, see Fig. 2(a). This is remi- niscentofthedegeneracyofthethreeGoldstonemodesat the SU(4) point for J=−1[12, 13]. Moving away from the SU(4) point,the OBS decaysdueto residualinteractionsint o magnon-orbitonpairs, and the mean separation ξof spin and orbital excitations increases, i.e., al(Q)∼exp(−l/ξ), lead- ing in the thermodynamic limit to a ﬁnite linewidth deﬁned byΓ =ImG−1(Q,ω)[34]. Thedecayrateofthespin-orbital OBS increases with growing x >1 4and also for decreasing momenta Q,asseeninFig. 2(b). Entropy spectral function.— To investigate the degree of entanglementofexcitedstates,weintroducethevNEspectr al/s55 /s56 /s57 /s49/s48/s53/s54/s55/s56/s57/s49/s48 /s48/s46/s48/s48/s49/s48 /s48/s46/s48/s48/s49/s53 /s48/s46/s48/s48/s50/s48/s51/s46/s54/s52/s46/s48/s52/s46/s52/s40/s97/s41 /s32/s32/s83 /s118/s78 /s108/s111/s103 /s50/s76/s32/s120/s61/s48/s46/s50/s53/s32/s121/s61/s48/s46/s50/s53/s32 /s32/s120/s61/s48/s46/s53/s48/s32/s121/s61/s48/s46/s51/s48/s32 /s32/s120/s61/s48/s46/s53/s48/s32/s121/s61/s48/s46/s53/s48/s32/s81/s61/s48/s46/s56 /s81/s61/s48/s46/s54/s81/s61/s48/s46/s50 /s32/s32/s83 /s118/s78 /s49/s47/s76/s40/s98/s41 /s52 /s54 /s56/s52/s54/s56/s83 /s118/s78 /s108/s111/s103 /s50/s76 FIG.4: (color online). (a) Scalingbehavior of entanglemen t entropy SvNof the spin-orbital BSs for Q= 0.8π. Lines represent loga- rithmicﬁts SvN= log2L+c0,withc0=−0.659,−1.059,−1.251, respectively. (b)Thescalingbehaviorofentanglementent ropyofthe OBS forx=y= 1/2. Lines are ﬁtted by SvN=c1/L+c0, with c0(c1) = 3.69 (380.5), 3.37 (138.4) and 3.31 (47.6) for Q= 0.8π, 0.6πand0.2π. The inset shows the logarithmic behavior of SvNfor the OBSwith Q= 0.8πandx=y= 1/4. functionin theLehmannrepresentation, SvN(Q,ω) =−/summationdisplay nTr{ρ(µ) slog2ρ(µ) s}δ{ω−ωn(Q)},(4) where(µ) = (Q,ωn)denote momentum and excitation en- ergy, and ρ(µ) s=Tro\|Ψn(Q)∝an}bracketri}ht∝an}bracketle{tΨn(Q)\|is obtained by tracing the orbital degreesof freedom. Let us ﬁrst consider the sym- metric case, i.e., x=y. The Hilbert space can be divided into two subspaces characterized by the parity PSTof the in- terchangeof S↔T, which is odd or even. Translation sym- metry allows us to express the reduceddensity matrix ρsin a block-diagonalform,whereeachblockcorrespondstoanirr e- duciblerepresentationlabeledbytotalmomentum Qandpar- ity of exchangesymmetry PST. The vNE can be obtained by diagonalizing separately these blocks. In particular, the non- degenerate eigenstates with odd parity can be explicitly ca st in the form1√ 2(S− Q/2−qT− Q/2+q−S− Q/2+qT− Q/2−q)\|0∝an}bracketri}ht. Con- sequently, the singlet-like pair results in SvN= 1. For other spin-orbital eigenstates with PST= 1,SL≥1, except the pure spin and orbital waves. Interestingly, we ﬁnd that the parity is still conserved in subspace Q= 0forx∝ne}ationslash=y. The strongly entangled spin-orbital BSs are reﬂected by peaks i n the von Neumann spectra SvN(ω), shown in Fig. 3. As mo- mentumQdecreases, the OBS-peak in the center of spectra getsbroader,implyingashorterlifetime. Inspection of vNE spectra shows that the entanglement reaches a local maximum at the BSs. Finite size scaling of vNE of spin-orbital BSs reveals the asymptotic logarith- mic scaling SvN= log2L+c0shown in Fig. 4(a). The same logarithmic scaling is found for the OBS at the SU(4) pointx=y=1 4, as seen in the inset of Fig. 4(b). How- ever, far away from the SU(4) point the scaling is entirely4 /s48 /s49 /s50/s48/s49/s48/s50/s48 /s49 /s50 /s51 /s49 /s50 /s51/s48/s51/s54/s32/s32/s65 /s48/s40/s81/s44 /s41/s32 /s40/s99/s41 /s40/s98/s41 /s40/s97/s41/s32/s65 /s49/s43/s40/s81/s44 /s41/s32 FIG.5: (coloronline). Thespectralfunctionofthe on-site excitation A0(Q,ω)for: (a)x=y= 1/4, (b)x=y= 1/2; (c) the nearest- neighbor A1+(Q,ω)forx=y= 1/2. The momenta range from π/10(bottom) to 9π/10(top); the peak broadening is η= 0.01. Dashed(red)anddotted(green)linescorrespondtotheBSan dOBS, while graydash-dot lines indicate the boundaries of the con tinuum. different and the entropy of the OBS scales as a power law, SvN=c1/L+c0,asseeninFig. 4(b). Thischangeofscaling fromlogarithmictopowerlawin 1/Liscontrolledbythecor- relation length ξmeasuring the average distance of spin and orbital excitations in the OBS wave function (3). From Eq. (3) andal(Q)∼exp(−l/ξ)we obtain, SvN≃log2{L/(1+ξ)}, (5) which yields log2Latx=y= 1/4whereξ= 0. Asξ increases the correction to the vNE is ∝ −log2(1 +ξ). Far awayfromtheSU(4)point,theOBSisdampedand ξbecomes extensive, i.e., ξ/L≈˜c0−˜c1/L, and the vNE approachesa ﬁnite value with a correction ∝1/Las shown in Fig. 4(b). ThisclosecorrespondenceofthevNEofboundstatesandthe correlation length ξsuggests to use the dynamic spin-orbital correlation function as a probe of spin-orbital entangleme nt andasaqualitativemeasureofthe vNEspectra. Spectralfunctions.— Returningto TMOs, onerealizes that joint spin-orbital excitations are not created in the ferro -ferro ground state in photoemission spectroscopy because of spin - conservation. Onthecontrary,resonantinelasticx-raysc atter- ing(RIXS)[35–39]isinprincipleabletomeasurethespectr al functionofthe coupledspin-orbitalexcitationsat distan cel, Al(Q,ω) =1 πlim η→0Im/angbracketleftbigg 0/vextendsingle/vextendsingle/vextendsingle/vextendsingleΓ(l)† Q1 ω+E0−H−iηΓ(l) Q/vextendsingle/vextendsingle/vextendsingle/vextendsingle0/angbracketrightbigg . (6) HereΓ(0) Q=1√ L/summationtext jeiQjS− jT− jis the local excitation op- erator for an on-site spin-orbital excitation. We use as wel l Γ(1±) Q=1√ 2L/summationtext jeiQj(S− j+1±S− j−1)T− jfor the nearest- neighborexcitation. IntheRIXSprocessanelectronwithsp in up is excited by the incoming x-rays from a deep-lying core level into the valence shell. For the time of its existence th ecore hole generates a Coulomb potential and a strong spin- orbit coupling that allows for the non-conservation of spin . Next the hole is ﬁlled by an electron from the occupied va- lence band under the emission of an x-ray. This RIXS pro- cess creates a joint spin-valence excitation with momentum Qin−Qoutand energy ωin−ωout, which can unveil the spec- tralfunctionofthespin-orbitalexcitation. The on-site spectral function A0(Q,ω)shown in Figs. 5(a,b)highlightstheOBS.AttheSU(4)point[Fig. 5(a)]ita p- pearsasa δ-function, A0(Q,ω) =δ{ω−ωOBS(Q)},whereas in Fig. 5(b) the OBS is damped and its intensity decreases strongly with Q. In the latter ﬁgure the BS at the low en- ergysideofthecontinuumappearsasweakadditionalfeatur e, while it is absent in (a), i.e., at the SU(4) point. The neares t neighborspectralfunction A1+(Q,ω)inFig. 5(c)showsboth the spin-orbital continuum and the BS outside of the contin- uum. Notably, comparing with the vNE spectral function in Fig. 3, we ﬁnd the same characteristic energies and similar intensity features as in the RIXS spectra. The spectral func - tion provides information of various correlations, which a re ingredientsto derivethereduceddensitymatrices[40]. Summary.— In this Letter, we study a spin-orbital system and extend the analysis of entanglement to excited states by introducing the vNE spectral function. Our study demon- strates that even in cases where the ground state of a spin- orbitalchainisfullydisentangled,e.g.,intheferro-fer rostate, (i) the spin-orbital excitations are in general entangled, (ii) maximal spin-orbital entanglement occurs for BSs which ap- pearassharppeaksinthevNEspectra,and(iii)thevNEofun- damped BSs exhibits a logarithmic dependence on the chain lengthL. We propose to study the dynamic spin-orbital cor- relation functionas a qualitativemeasure of the vNE spectr a, andsuggestto usehereRIXS asa promisingtechnique. W-L.Y. acknowledges support by the National Natural Science Foundation of China (NSFC) under Grant No. 11004144. A.M.O. acknowledges support by the Polish Na- tionalScienceCenter(NCN)underProjectNo. N202069639. [1] Y. Tokura andN. Nagaosa, Science 288, 462 (2000). [2] L.F. Feiner and A.M. Ole´ s, Phys. Rev. B 59, 3295 (1999); N.N. Kovaleva, A.M. Ole´ s, A.M. Balbashov, A. Maljuk, D.N. Argyriou, G. Khaliullin, and B. Keimer, ibid.81, 235130 (2010). [3] SergeyKrivenko, Phys.Rev. B 85, 064406 (2012). [4] G. Khaliullin and S. Maekawa, Phys. Rev. Lett. 85, 3950 (2000). [5] PeterHorsch,AndrzejM.Ole´ s,LouisFelixFeiner,andG iniyat Khaliullin, Phys.Rev. Lett. 100, 167205 (2008). [6] K. Wohlfeld, M. Daghofer, and A.M. Ole´ s, Europhys. Lett .96, 27001 (2011). [7] G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 102, 017205 (2009). [8] W.Brzezicki and A.M.Ole´ s, Phys.Rev. B 83, 214408 (2011). [9] K. Wohlfeld, M. Daghofer, S. Nishimoto, G. Khaliullin, a nd J. van denBrink, Phys.Rev. Lett. 107, 147201 (2011).5 [10] AndrzejM.Ole´ s,PeterHorsch,LouisFelixFeiner,and Giniyat Khaliullin,Phys.Rev. Lett. 96, 147205 (2006). [11] G. Sun, G. Jackeli, L. Santos, and T. Vekua, arXiv: 1112. 5082 (unpublished). [12] You-Quan Li, Michael Ma, Da-Ning Shi, and Fu-Chun Zhang , Phys.Rev. B 60, 12781 (1999). [13] B.Sutherland, Phys.Rev. B 12, 3795 (1975). [14] Vincenzo Alba, Maurizio Fagotti, and Pasquale Calabre se, J.Stat.Mech. 10, P10020 (2009). [15] Hui Li and F.D.M. Haldane, Phys. Rev. Lett. 101, 010504 (2008). [16] Hong Yao and Xiao-Liang Qi, Phys. Rev. Lett. 105, 080501 (2010). [17] A.Kitaevand J.Preskill,Phys.Rev. Lett. 96, 110404 (2006). [18] I.KlichandL.Levitov, Phys. Rev. Lett. 102, 100502 (2009). [19] J.Eisert,M. CramerandM.B.Plenio, Rev.Mod. Phys. 82, 277 (2010). [20] Luca Bombelli, Rabinder K. Koul, Joohan Lee, and Rafael D. Sorkin, Phys. Rev. D 34, 373 (1986); Mark Srednicki, Phys. Rev. Lett. 71, 666 (1993); S.W. Hawking, J. Maldacena, A. Strominger,J. HighEnergyPhys. 05, 001 (2001). [21] LuigiAmico,RosarioFazio,Andreas Osterloh,andVlat koVe- dral,Rev. Mod. Phys. 80, 517 (2008). [22] Christoph Holzhey, Finn Larsen, and Frank Wilczek, Nuc l. Phys.B424, 443 (1994). [23] Llu´ ısMasanes, Phys.Rev. A 80, 052104 (2009). [24] J. van den Brink, W. Stekelenburg, D.I. Khomskii, G.A. Sawatzky, andK. I.Kugel, Phys.Rev. B 58, 10276 (1998). [25] Yan Chen, Z.D. Wang, Y.Q. Li, and F.C. Zhang, Phys. Rev. B 75, 195113 (2007).[26] Shi-JianGu, Int.J. Mod. Phys.B 24, 4371 (2010). [27] Wen-LongYou,Ying-WaiLi,andShi-JianGu,Phys.Rev.E 76, 022101 (2007). [28] Alexander Herzog, Peter Horsch, Andrzej M. Ole´ s, and J esko Sirker, Phys.Rev. B 83, 245130 (2011). [29] Michael Wortis,Phys.Rev. 132, 85(1963). [30] T. Schneider, Phys.Rev. B 24, 5327 (1981). [31] JanBała,AndrzejM.Ole´ s,andGeorgeA.Sawatzky,Phys .Rev. B63, 134410 (2001). [32] S. Cojocaru and A. Ceulemans, Phys. Rev. B 67, 224413 (2003). [33] D.P.Arovas and A.Auerbach, Phys.Rev. B 52, 10114 (1995). [34] Dysonequationreads G−1(Q,ω) =ω−ωOBS(Q)−Σ(Q,ω), withthetotalself-energy Σ(Q,ω).TherealpartRe G−1(Q,ω) identiﬁes the effective energy pole, and the imaginary part of self-energy isthe inverse of lifetime τ,i.e.,Γ =ImΣ(Q,ω). [35] S.Ishihara andS.Maekawa, Phys.Rev. B 62, 2338 (2000). [36] M.W. Haverkort, Phys.Rev. Lett. 105, 167404 (2010). [37] L.J.P.Ament,M.vanVeenendaal,T.P.Devereaux, J.P.H ill,and J. van denBrink, Rev. Mod. Phys. 83, 705(2011). [38] Filomena Forte, Luuk J.P. Ament, and Jeroen van den Brin k, Phys. Rev. Lett. 101, 106406 (2008). [39] J. Schlappa, K. Wohlfeld, K.J. Zhou, M. Mourigal, M.W. Haverkort, V. N. Strocov, L. Hozoi, C. Monney, S. Nishi- moto, S. Singh, A. Revcolevschi, J.-S. Caux, L. Patthey, H.M . Rønnow, J. van den Brink, and T. Schmitt, Nature 485, 82 (2012). [40] I. Peschel andV. Eisler,J.Phys.A 42, 504003 (2009).
2402.01080v1.Photonic_Spin_Orbit_Coupling_Induced_by_Deep_Subwavelength_Structured_Light.pdf	Photonic Spin-Orbit Coupling Induced by Deep-Subwavelength Structured Light Xin Zhang1,2, Guohua Liu1,2, Yanwen Hu1,2,3∗, Haolin Lin1,2, Zepei Zeng1,2, Xiliang Zhang1,2, Zhen Li1,2,3, Zhenqiang Chen1,2,3, and Shenhe Fu1,2,3∗ 1Department of Optoelectronic Engineering, Jinan University, Guangzhou 510632, China 2Guangdong Provincial Key Laboratory of Optical Fiber Sensing and Communications, Guangzhou 510632, China 3Guangdong Provincial Engineering Research Center of Crystal and Laser Technology, Guangzhou 510632, China We demonstrate both theoretically and experimentally beam-dependent photonic spin-orbit coupling in a two-wave mixing process described by an equivalent of the Pauli equation in quantum mechan- ics. The considered structured light in the system is comprising a superposition of two orthogonal spin-orbit-coupled states defined as spin up and spin down equivalents. The spin-orbit coupling is manifested by prominent pseudo spin precession as well as spin-transport-induced orbital angular momentum generation in a photonic crystal film of wavelength thickness. The coupling effect is sig- nificantly enhanced by using a deep-subwavelength carrier envelope, different from previous studies which depend on materials. The beam-dependent coupling effect can find intriguing applications; for instance, it is used in precisely measuring variation of light with spatial resolution up to 15 nm. I. INTRODUCTION Spin-orbit coupling (SOC), which refers to interaction of a quantum particle’s spin with its momentum, is a fun- damentally important concept. It has been extensively investigated in condensed matter physics [1, 2], atomic and molecular physics [3, 4] and contributes to exciting phenomena such as the spin Hall effect [5] and topolog- ical insulators [6, 7]. Analogous photonic SOC is also demonstrated in a variety of settings [8]. The photonic SOC refers to an interaction between the momentum of light, which also includes spin angular momentum and orbital angular momentum (SAM and OAM). Whereas the SAM is associated with photon circular polarization [9], the OAM is relevant to a helical wavefront of light characterized by a topological number ℓ[10]. The pho- tonic SOC is crucial for the optical Hall effects [11–14], spin-to-orbital angular momentum conversions [15, 16], spin-orbit photonic devices [17–19], etc. The SOC can be engineered in appropriately designed materials. For examples, engineering a tensional strain in graphene shifts the electronic dispersions and induces a controllable vector potential for the electronic SOC [20– 24]. Analogous strategy can be applied to engineer the photonic SOC, by using strained evanescently coupled waveguide arrays [25, 26]. Other approaches for manip- ulating the photonic SOCs are demonstrated by appro- priately designing microcavities [27–32], metamaterials [33–36], photonic crystals [37–39], twisted optical fibers [40, 41], dual-core waveguides [42, 43], etc. The resultant SOCs are material-dependent, determined by geometric configurations of the materials which are often difficult to be tuned once fixed by designs. As a consequence, a tunable photonic SOC process remains elusive. Re- cently, several engineered photonic SOC schemes have been reported, by either embedding a strained honey- ∗Electronic address: huyanwen@jnu.edu.cn;fushenhe@jnu.edu.cncomb metasurface inside a cavity waveguide [44] or us- ing an optical cavity filled with controllable liquid crys- tals [45]. However, the resultant photonic SOCs remain material-dependent. In this work, we report theoretically and experimen- tally a new mechanism for engineering the photonic SOC. We demonstrate this by exploiting analogy be- tween quantum description of a spin-1/2 system and a spin-orbit Hamiltonian derived for structured light in a photonic crystal. The obtained Hamiltonian is closely relevant to structured light, which means that the SOC can be engineered by controlling carrier envelope rather than the structures of materials. Strong SOC is achieved by using deep-subwavelength structured light, as man- ifested by clear pseudo spin precessions. Although the structured light has been extensively investigated in re- cent year [46–51], the dependence of the photonic SOC on its spatial structure remains unnoticed. II. THEORETICAL MODEL We consider a two-wave mixing process involving two interacting photonic states. The SOC takes place in a crystal, represented by its principal refractive index: nx, ny, and nz. With an approximation of the slowly varying envelope along optical axis z, a coupled-wave equation for the process is given by [52] 2iβx∂Ex ∂z+n2 x n2z∂2Ex ∂x2+∂2Ex ∂y2=γy∂2Ey ∂y∂xexp (+ i∆β·z) 2iβy∂Ey ∂z+n2 y n2z∂2Ey ∂y2+∂2Ey ∂x2=γx∂2Ex ∂x∂yexp (−i∆β·z)(1) where Ex,yare linearly polarized fields, and βx,y=k0nx,y denote their propagation constants. k0= 2π/λis free- space wavenumber with λbeing the wavelength. ∆ β= βy−βxis a phase mismatch. We define γx,y= 1−n2 x,y/n2 z as coupling parameters, related to crystal’s polarity. The derivatives ∇2 xyand∇2 yxin Eq. (1) stem from the non- zero term ∇ ·E̸= 0, featuring origin of the SOC [41].arXiv:2402.01080v1 [physics.optics] 2 Feb 20242 FIG. 1: (a) Geometrical representation of spin precession in the presence of SOC. ˆRandˆLdefine the spin-up and spin-down equivalents in the B2direction, respectively; whereas Φ +and Φ −denotes two spin eigenstates in the direction of synthetic fieldB. Spin precession is initiated by a mixing spin Φ = 1 /√ 2(Φ ++iΦ−) located at z0. (b) Bloch-sphere representation of spin-1/2 system, in the presence of external field B. Φ↑and Φ ↓are spin up and spin down in the zdirection, while Φ1/2 +and Φ1/2 −denotes eigenstates of the system, corresponding to direction of B. (c) Polarization states mapped on a longitude line of the first-order ( ℓ= 1) sphere in (a). (d) Corresponding spin vectors to (c). (e) SOC strength as a function of beam width r0. (f)-(i) Theoretical results for the spin vectors under actions of LG beam with different widths. To address the rapid oscillation terms exp( ±i∆β·z), we transform the wave equation to a rotating form, by defining Ex=˜Axexp(+ i∆β·z/2) Ey=˜Ayexp(−i∆β·z/2)(2) respectively. Thus a Hamiltonian of the system is written as H=1 2¯β−∇2 ⊥+ ¯γ∇2 xx,0 0,−∇2 ⊥+ ¯γ∇2 yy +∆β/2,¯γ∇2 yx/(2¯β) ¯γ∇2 xy/(2¯β),−∆β/2 (3) where ∇2 ⊥=∇2 xx+∇2 yydenotes the Laplace operator. We have assumed shallow crystal birefringence, namely ¯β≈(βx+βy)/2 and ¯ γ≈(γx+γy)/2. The second term in Eq. (3), which includes the derivative operators, couples the two polarization components. It means that the SOC is related to spatial structure of light. We study the beam-dependent SOC in a synthetic two- level spin-orbit system. We define right and left circularly polarized vortex states as spin up and spin down equiv- alents in the zdirection. They are written as [53–55]: ˆR= exp(+ iℓϕ)(ˆx−iˆy)/√ 2 ˆL= exp( −iℓϕ)(ˆx+iˆy)/√ 2(4) respectively, where ˆ xand ˆyare unit vectors and ϕ= arctan( y/x). Since the pseudo spins are defined in thecircular basis, the Hamiltonian is modified by a trans- formation from the cartesian coordinate to the circular basis, yielding H′=¯γ−2 4¯β ∇2 ⊥,0 0,∇2 ⊥ +0,∆β/2−i¯γ∇2 yx/(2¯β) ∆β/2 +i¯γ∇2 yx/(2¯β),0 (5) Given an overall field ˜A=˜A(x, y, z )(ΦRˆR+ΦLˆL), where ΦRand Φ Lare weights on ˆRand ˆL, respectively, we reduce Eq. (1) to the Schr¨ odinger-like (Pauli) form i∂Φ(z) ∂z=1 2MP2 ⊥˜A−1 2σ·B Φ(z) (6) where Φ = (Φ R,ΦL)T,P2 ⊥= [−∇2 ⊥,0; 0,−∇2 ⊥], and M= 2¯β˜A/(2−¯γ). Here σis the Pauli matrix vec- tor. The SOC is described by a term −σ·B, where B1=−¯γ∇2 xy˜A/(¯β˜A),B2= 0, and B3=−∆β. It is analogous to a coupling form which describes interaction between a particle’s spin and its angular momentum in a moving frame [1, 2]. More details refer to Appendix A. Since B2is zero, the vector Blies on the purely trans- verse B1B3plane. The SOC Hamiltonian admits eigen- states that point to the vector Band comprise an equal superposition of ˆRandˆL, written as Φ+= 1/√ 2h ˆR+ exp( iφ)ˆLi Φ−= 1/√ 2h ˆR−exp(iφ)ˆLi (7)3 Figure 1(a) visualize the eigenstates and the pure states (ˆRandˆL) in the Poincar´ e sphere, showing close analo- gies to Bloch-sphere representation of the spin-1/2 sys- tem [56, 57] [Fig. 1(b)]. The spin states exhibit cylin- drically symmetric polarization distributions. As illus- tration, Fig. 1(c) displays typical polarizations of states mapped onto a longitude line in the first-order ( ℓ= 1) sphere; while Fig. 1(d) depicts their corresponding spin vectors, represented by an angle arccos( S2), where S2is value of polarization ellipticity. Since the state exhibits identical polarization ellipticity in the transverse plane, the resultant spin vectors are homogeneous. The SOC term shows a dynamical effect, caused by the propagation-variant envelope ˜A. This shows sharp contrast to conventional ones which are usually being in- dependent terms. However, if optical diffraction is ne- glected, the dynamical behavior disappears and the SOC strongly relies on the envelope. In this scenario, a rel- evant beam parameter becomes an important degree of freedom for engineering the SOC. This is demonstrated in Fig. 1(e), showing close relationship between SOC strength and beam width in the phase-matching condi- tion (∆ β=0). Here the Laguerre-Gaussian (LG) envelope is considered as: ˜A(r) =r r0exp(−r2 r2 0) (8) where r= (x2+y2)1/2, and r0features the beam width. At the deep-subwavelenth region ( r0< λ/ 2), the SOC strength is rapidly increasing with a slight decrease of r0. It becomes relatively negligible when r0> λ. This relation suggests that shrinking light to deep-subwavelength scale significantly enhances the SOC. Although the derivations are based on the slowly varying envelope approximation, the model can be ap- plied to deep-subwavelength regime at the early stage of spin evolution. To demonstrate the deep-subwavelength-induced SOC, we set the coupling length to be only one cycle ( z=λ), such that a moderate SOC cannot cause obvious spin transport phenomenon. On the other hand, the SOC strength can be maintained during beam propagation, due to the short coupling length. This results in an adi- abatic spin evolution, represented as a spin precession around B, i.e., dS dz=B×S (9) where S= (S1, S2, S3) is the state vector defined as Sh= Φ†σhΦ (h= 1,2,3). The spin vector is therefore described by S2. We initiate the spin precession from a mixing state: Φ = 1 /√ 2[Φ++iΦ−]. Figure 1(f)-1(i) display theoretical distributions of the spin vectors for different beam parameters. Evidently, for r0=0.05 µm, the spin rotates to an angle about -78o; By comparison, increasing the parameter to r0= 0.13µm causes less significant spin precession, manifested by a spin rotation FIG. 2: (a) Experimental setup. BS: beam splitter; Q: q- plate; FL: flat lens; M: mirror; OB: objective lens; TL: tube lens; QWP: quarter wave plate; P: polarizer; CCD: charge coupled device. The laser is operating at wavelength of λ= 632 .8 nm. The insert in (a) shows that an equatorial mixing spin with equal weight on Φ Rand Φ Lis adiabatically converted to a pure spin down in the presence of the SOC. (b) Layout of the 60-nm-thick flat lens with NA=0.87. (c) Inten- sity distribution of the LG beam at the focal plane ( zf) of the flat lens. (d) Plane-wave interference and (e) y-polarization component of beam at zf, indicating a generation of the ex- pected spin state Φ = 1 /√ 2(Φ ++iΦ−). The scale bar in (c-e) is 250 nm. In color bar, L: low; H: high. angle about -12o. This indicates that the SOC strongly depends on the carrier envelope. Figure 1(h, i) show that a moderate SOC induced by the relatively larger envelope cannot cause spin precession. III. EXPERIMENTAL RESULTS AND DISCUSSION Experiments are carried out to confirm the predictions. A crucial ingredient is to generate the required spin-orbit state at the deep-subwavelength scale. This is challeng- ing since the incident state cannot maintain its prop- erty after tightly focused by the high-numerical-aperture (NA) objective lens [58–60]. To overcome this problem, we fabricate a topology-preserving high-NA flat lens (the thickness is 60 nm) according to a technique reported in [61]. The flat lens [the layout is shown in Fig. 2(b)] has a NA up to 0.87 and a focal length of zf= 8µm. A sys- tem comprising an objective lens (150 ×, NA=0.9) and a tube lens is utilized to characterize the flat lens, see Fig. 2(a). Figure 2(c) presents recorded intensity distribution of light at the focal plane. The focused LG beam exhibits a parameter of r0≃0.32µm. The recorded regular inter- ference [(Fig. 2(d)] and y-polarization component [(Fig.4 FIG. 3: Experimental observation of spin rotation induced by the deep-subwavelength LG beam ( r0= 0.32µm), as mani- fested by spin angular momentum conversion (flipping) from right-handed one (b) to the left-handed one (a). In compar- ison, a larger LG beam parameter r0=2.2 µm is considered, resulting in balanced left-handed (c) and right-handed (d) components. (e, f) The measured spin vectors before and af- ter the crystal film, for (e) r0= 0.32µm, and (f) r0= 2.2 µm. In the color bar (d), L: low; H: high. 2(e)] suggest that the expected initial spin is generated. Theoretical derivation about topology-preserving prop- erty of the flat lens (Appendix B) further confirms the generation. An experimental setup is built for measuring the spin procession. A linearly polarized He-Ne laser ( λ= 632 .8 nm) is divided by a beam splitter. A q-plate with a charge of q= 1/2 is applied to transform the beam into expected spin state carrier by the LG envelope. The pu- rity of the spin state from the q-plate is measured as 95.2% (Appendix C). The LG beam is focused into deep- subwavelength region by the flat lens. A c-cut lithium niobate crystal film (¯ γ=−0.08) with a thickness about one wavelength is placed at the focal plane. The emerg- ing beam, in the presence of the SOC, is expected to accumulate a non-trivial spin phenomenon [see the in- sert in Fig. 2(a)]. Figure 3 presents measurements confirming the spin precession. Since the spin is relevant to the circular po- larization, we measure the right- (spin ↑) and left-handed (spin ↓) circular polarization components. These are achieved by rotating a quarter wave plate to an angle of−π/4 and + π/4 with respect to xaxis, respectively, while inserting a linear polarizer in front of the camera. Figure 3(a) and 3(b) depicts intensity distributions of ΦLand Φ R, respectively. The measured Φ Lcomponent is stronger than the Φ Rone, indicating a spin precession toward south pole of the sphere. Figure 3(e) shows the measured spin rotation by an angle of -5.2o, compared to the initial one [62]. This approximately matches to the simulated result. However, for a larger parameter (r0= 2.2µm), the Φ Land Φ Rcomponents are approxi- mately identical [Fig. 3(c, d)], meaning that the induced SOC is insufficient to flip the spin [Fig. 3(f)]. Slight dif- ference between the experiment and theory can be mainly attributed to the imperfect LG envelope that is closely relevant to the derivative operator ∇2 xy[62]. FIG. 4: Observation of the orbital-angular-momentum state induced by spin precession. (a, c) The experimentally mea- sured plane-wave interference patterns, for two different LG beam parameters: (a) r0= 0.32µm, and (c) r0= 2.2µm. (b, d) The simulated [based on Eq. (1)] interference patterns corresponding to the measurements in (a, c). Experimental conditions are kept the same as those in Fig. 3. We observe non-trivial spin-precession phenomenon, manifested by a generation of the photonic OAM. Ini- tially, both the SAM and OAM of state at the equator are zero. Under the action of the SOC, its intrinsic OAM and SAM are separated simultaneously. This non-trivial phenomenon is observed in Fig. 4(a), showing a clear dislocation in the plane-wave interference fringes for the deep-subwavelength LG beam. This is a manifestation of wavefront helicity with a topological charge being ℓ= 1. The spin precession accompanied by the OAM genera- tion confirms the phenomenon of spin-orbit separation. This effect becomes negligible for larger envelope, since the spin remains at its original position, as indicated by the regular interference fringes [Fig. 4(c)]. Theoretical results correspondingly shown in Fig. 4(b) and 4(d) are in accordance with the measurements. We observe more prominent spin precession by con- sidering the Bessel structured light with deeper subwave- length feature size. The carrier envelope is replaced by ˜A(r) =Jℓ(r/r0) (10) where Jℓdenotes the Bessel function of order ℓ. In practice, we should properly truncate the ideal Bessel beam by using a Gaussian factor. The resultant Bessel- Gaussian (BG) profile exhibits nondiffracting property FIG. 5: Observation of the spin rotation by using the deep- subwavelength BG beam ( r0= 0.12µm). (a, b) Experimen- tally measured intensity distributions of the left- and right- handed circular polarizations. (c, d) Plane-wave interference patterns obtained both in experiment (c) and in simulation (d). (e, f) The measured output spin rotation in comparison with the initial one: (e) experiment; (f) simulation.5 FIG. 6: (a) The simulated [based on Eq. (6)] beam-dependent spin oscillatory modes. (b)(c) The simulated [based on Eq. (1)] phase distributions of the output light states from the barium metaborate crystal film (¯ γ=−0.16), for (b) r0= 95 nm, and (c) r0= 110 nm. over a certain distance. We generate this BG beam us- ing a metasurface whose geometry exhibits cylindrical symmetry. The highly localized BG beam is a result of in-phase interference of many high-spatial-frequency waves [51]. We demonstrate result for a beam param- eter of r0= 0.12µm, while maintaining other param- eters unchanged. Similarly, an initial balance between the left and right-handed components is broken by the SOC [Fig. 5(a, b)]. The output spin rotates to a larger angle of -17.1o, nearly in accordance with the theoreti- cal calculation [Fig. 5(f)]. The measured and simulated interference patterns verify the spin-precession-induced OAM generation, see Fig. 5(c) and 5(d), respectively. Finally, we propose to using the beam-dependent SOC in precision measurement of slight variation of structured light, with measurement accuracy up to 15 nm. This nanometric resolution is usually impossible to be reached by current optical detectors. It requires to realize rapid oscillation between the spin up and spin down. Specifi- cally, we exploit the deep-subwavelength BG beam as car- rier envelope of the spin. In this scenario, the Pauli equa- tion [Eq. (6)] emulates a SOC process for the spin oscil- lation. Figure 6(a) depicts the SOC-supported spin har- monic oscillations along with the coupling distance, for different cases of beam widths. Obviously, the spin oscil- lation is very sensitive to the change of spatial structure of light, giving rising to ultrasensitive beam-dependent oscillatory modes. As a result, a slight change of the beam width leads to significant spin flipping. This al- lows to detect the spatial variation of light as small as 15 nm. To verify the result, we present simulated out- comes [see Fig. 6(b) and 6(c)], clearly showing opposite helical wavefronts of the output states (corresponding to the spin down and spin up), for r0=95 nm and r0=110 nm. Note that one can further increase the measuring sensitivity by properly reducing the beam width. IV. CONCLUSION In summary, we have demonstrated both theoretically and experimentally novel SOC phenomena, caused by the deep-subwavelength spin-orbit structured light. Thisbeam-dependent SOC contrasts to those being material- dependent [44, 45]. The reported SOC is closely rele- vant to the spatial gradient of light field, hence it can be significantly enhanced by using the deep-subwavelength carrier envelopes. We have qualitatively characterized this effect, by measuring the spin precessions under dif- ferent beam parameters. Particularly, based on the deep-subwavelength Bessel beam, a significant spin ro- tation about -17.1o, accompanied by OAM generation, was achieved within a coupling length of only one wave- length. The influence of the phase mismatch on the beam-dependent SOC was also discussed, see Appendix D. These fundamental SOC phenomena may find inter- esting applications in different areas [63–66]. As an ex- ample, we have proposed to use such a strong SOC effect in the precise measurement of slight spatial change of light with nanometric resolution. V. ACKNOWLEDGMENTS We thank Boris Malomed from Tel Aviv University for kind discussions about the SOC. This work was sup- ported by the National Natural Science Fundation of China (62175091, 12304358), and the Guangzhou science and technology project (202201020061). VI. APPENDIX A: ANALOGY OF SPIN-ORBIT COUPLING IN SPIN-1/2 SYSTEM AND SYNTHETIC TWO-LEVEL SYSTEM The spin-1/2 dynamics in the external vector field B can be described by a Hamiltonian term H1/2=σ∗B, where σis the Pauli matrix vector. In a normalized form, it can be expressed as H1/2=1 2 cosθ sinθ·exp (−iφ) sinθ·exp (iφ) −cosθ (11) where θandφare two angles that define a normalized (unit) sphere. The vector Bthen possesses around the sphere, with direction determined by θandφ. This Hamiltonian H1/2admits two spin eigenstates that point along to B, written as Φ1/2 += cosθ 2 Φ↑+ exp( iφ) sinθ 2 Φ↓ Φ1/2 −= sinθ 2 Φ↑−exp(iφ) cosθ 2 Φ↓(12) where Φ ↑= [1 0]Tand Φ ↓= [0 −1]Tare spin-up and spin-down states defined in the zdirection. Figure 1(b) geometrically depicts this picture onto a Bloch sphere. All possible spins of the system can now be mapped onto the sphere, with the spin up Φ ↑and spin down Φ ↓located at the north and south poles of the sphere, respectively. In the presence of the external field B, the initial spin6 TABLE I: Analogies between the presented synthetic spin-1/2 system in the higher-order optical regime and the spin-1/2 system in the quantum mechanics. The direct analogies between these two different settings enable us to emulate intriguing spin transport phenomena in the presence of spin-orbit coupling. Physical parameters Spin-1/2 system Synthetic spin-1/2 system Spins Φ ↑and Φ ↓ˆRandˆL Eigenstates Φ1/2 +and Φ1/2 − Φ+and Φ − Field vector B(real) B=(−¯γ/(¯β˜A)∇2 xy˜A,0,−∆β) Spin-orbit coupling term H1/2=σ·B H SOC=σ·B Space/time coordinates ( x, y, t ) ( x, y, z ) Mass m M = 2¯β˜A/(2−¯γ) Momentum operator P2 ⊥= [−∇2 ⊥,0; 0,−∇2 ⊥] P2 ⊥= [−∇2 ⊥,0; 0,−∇2 ⊥] precesses around the vector B, giving rise to many in- triguing spin transport phenomena such as the geometric phase. In our case, we study spin-orbit coupling of structured light in a photonic crystal. The structured light in the system is comprising a superposition of two orthogonal spin-orbit states with non-trivial topological structures. They can be written as ˆR= exp ( ilϕ)(ˆx−iˆy)/√ 2 and ˆL= exp ( −ilϕ)(ˆx+iˆy)/√ 2, respectively. These topolog- ical states define the spin up and spin down equivalents along the zaxis, respectively, but they are not eigenstates of the analogous spin-orbit Hamiltonian Hsoc=−σ∗B. In the circular basis, a similar Hamiltonian matrix can be written as Hsoc= B2 B3−iB1 B3+iB1−B2 (13) In our case, since B2is zero (see the main text), the effective vector Bobtained here lies on the purely trans- verse B1B3plane, as shown in Fig. 1(a). As a result, the pseudospin eigenstates of Hsocthat point along this transverse vector Bcomprise an equal superposition of ˆRandˆL, express as Φ+= cosπ 4 ˆR+ exp( iφ) sinπ 4 ˆL Φ−= sinπ 4 ˆR−exp(iφ) cosπ 4 ˆL(14) We can now interpret these eigenstates as a mixing of ˆR andˆL. Poincar´ e-sphere representation allows us to visu- alize these spin eigenstates as well as the pure states ˆR andˆL. Clearly, this is analogous to the Bloch-sphere rep- resentation for the spin-1/2 system. The spin-orbit cou- pling makes this state evolves along the Poincar´ e sphere, which can be described by the synthetic Pauli equation, i∂Φ ∂z=1 2MP2 ⊥˜A−1 2σ·B Φ (15) Table I summaries analogous formulas between these two systems.VII. APPENDIX B: THEORETICAL DERIVATION FOR THE TOPOLOGY-PRESERVING FLAT LENS In this section, we theoretically prove that the flat lens used in the experiment does not change the spin-orbit property of the LG beam after tightly focusing. The flat lens is designed by an amplitude-only hologram gener- ated from an interference between an angular cosine wave and a spherical wave (see ref. [61] in the text). When the LG beam ˜A(x, y) carrying a general spin state Φ passes through the flat lens, it is modulated in binary. As a re- sult, the light field behind the flat lens can be expressed as E(x, y, z = 0) = ˜A(x, y)∗t(x, y) Φx(ϕ)ˆx+Φy(ϕ)ˆy (16) where t(x, y) denotes transmission function of the flat lens and ϕ= arctan ( y/x). Within this initial condition, we solve the diffractive problem according to the vectorial Helmholtz wave equation. The diffractive field at the focal plane of the flat lens can be written as E(x, y, z f) =k i2πzfZZ E(x′, y′, z= 0) expik 2zf[(x−x′)2+ (y−y′)2] dx′dy′ (17) Note that owing to the cylindrical symmetry of the flat lens (see the layout in the text, Fig. 2(b)), the transmis- sion function can be also given in a cylindrical form of t(r), where r= (x2+y2)1/2. In this case, the complex amplitude of the initial field is separable in the polar co- ordinates ( r, ϕ). We therefore rewrite the solution in the cylindrical coordinate system and deal with the integrals. We finally obtain the analytical solution for the vectorial light field at the focal plane, given by E(x, y, z f) =f(r) Φx(ϕ)ˆx+ Φy(ϕ)ˆy (18)7 FIG. 7: Modal decomposition results. (a) An experimental setup used to measure the purity of the first-order LG beam emerging from the q-plate. The LG beam is decomposed into LG basis modes. The linearly polarized He-Ne laser operating at the wavelength of 632.8 nm is considered. QWP, quarter wave plate; QP, q-plate with a topological number of q= 1/2; SLM, spatial light modulator; BS, beam splitter; CCD, charge-coupled device. (b) The modal decomposition results at the basis of LG modes with topological charge ranging from l= -5 to 5. where f(r) =−k zfZ∞ 0˜A(r′)t(r′)r′J1krr′ zf expik 2zf(r2+r′2) dr′(19) andJ1indicates the first-order Bessel function. It is ev- ident that the diffractive field at the focal plane shares a similar analytic form to the initial one, except for that the envelope becomes a z-dependent function. It indicates that the flat lens can completely retain the initial spin state when it is focused into the input end of the crystal. The topology-preserving flat lens enables us to detect the pseudo spin precession caused by the deep-subwavelength structured light, which cannot be achieved by using the conventional high NA objective lens. VIII. APPENDIX C: PURITY MEASUREMENT OF THE FIRST-ORDER LG BEAM FROM THE Q-PLATE we perform additional experiment to show that the generated first-order LG beam from the q-plate is of high purity, which is sufficiently enough to detect the photonic spin-orbit coupling effect. We utilize a modal decomposi- tion method [67, 68] to measure the purities of the output LG mode from the q-plate with a topological charge of q= 1/2, see an experimental setup in Fig. 7(a). Two quarter wave plates (QWPs) are used to select a proper polarization of the generated first-order ( l=1) LG beam that matches to the spatial light modulator (SLM). A group of pure LG modes generated from digital holo- grams by using the SLM are considered to decomposed the LG beam. Fig. 7(b) shows the decomposing result depicted in a histogram. It is seen that the measured pu- FIG. 8: Controllable spin-orbit coupling by engineering the phase mismatch in a c-cut electro-optic lithium niobate crys- tal. (a) Experimental scheme for observing the electrically en- gineered spin-orbit coupling. (b-f) Experimentally measured photonic spin states at different applied voltages: (b) U=60 V, (c) U=80 V, (d) U=100 V, (e) U=120 V, and (f) U=140 V. (g) The measured topological charge as a function of ap- plied voltage. Sim: simulation; Expt: experiment. In this experiment, the coupling length of the crystal is set to z=30 mm. rity of the first-order LG beam from the q-plate is 95.2%. IX. APPENDIX D: ENGINEERING PHOTONIC SPIN-ORBIT COUPLING BY TUNING THE PHASE MISMATCH In addition to the beam-dependent photonic spin- orbit coupling which we have shown in the main text, we perform additional experiments confirming that the spin-orbit coupling can be also controlled by engineering the phase mismatch. To this end, we consider electri- cally tuning the phase mismatch in a c-cut electro-optic lithium niobate (LN) crystal, whose optical axis is in ac- cordance with propagation direction of the beam [see Fig. 8(a)]. In the presence of transverse modulation, the phase mismatch can be written as ∆β=−k0n3 oγ22U/d (20) where k0= 2π/λdenotes wavenumber in vacuum with λbeing wavelength, nois the ambient refractive index8 of the crystal, Uis the applied voltage, dis the thick- ness, and γ22= 6.8 pm /V is an electro-optic coefficient of the crystal. In this case, the external knob Uis uti- lized to finely tune the phase mismatch and the resulting spin-orbit coupling. We use the same experimental setup and obtain a voltage-dependent transition between differ- ent spin states in the phase mismatching regime. Panels(b-f) in Fig. 8 show controllable spin states of light by varying the applied voltage. Moreover, we perform de- tailed experiments to measure the topological charge of the output light state as a function of voltage, see Fig. 8(g). These results suggest another important degree of freedom for engineering the spin-orbit coupling. [1] R. Winkler, “Spin-Orbit Coupling Effects in Two- Dimensional Electron and Hole Systems,” Springer Tracts Mod. Phys. 191, Springer, New York (2003). [2] M. Aidelsburger, S. Nascimbene, and N. Goldman, “Ar- tifical gauge fields in materials and engineered systems,” C. R. Phys. 19, 394 (2018). [3] Y. -J. Lin, K. Jim´ enez-Garc ´ia, and I. B. Spielman, “Spin- orbit-coupled Bose-Einstein condensates,” Nature 471, 83 (2011). [4] V. Galitski, and I. B. Spielman, “Spin-orbit coupling in quantum gases,” Nature 494, 49 (2013). [5] Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, “Observation of the spin Hall effect in semi- conductors,” Science 306, 1910 (2004). [6] M. Z. Hasan, and C. L. Kane, “Colloquium: Topological insulators,” Rev. Mod. Phys. 82, 3045 (2010). [7] M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topologial protection,” Nat. Phys. 7, 907 (2011). [8] K. Y. Bliokh, F. J. Rodr ´iguez-Fortu˜ no, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796 (2015). [9] A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Pad- gett, “Intrinsic and Extrinsic Nature of the Orbital An- gular Momentum of a Light Beam,” Phys. Rev. Lett. 88, 053601 (2002). [10] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185 (1992). [11] M. Onoda, S. Murakami, and N. Nagaosa, “Hall Effect of Light,” Phys. Rev. Lett. 93, 083901 (2004). [12] A. Kavokin, G. Malpuech, and M. Glazov, “Optical Spin Hall Effect,” Phys. Rev. Lett. 95, 136601 (2005). [13] O. Hosten, and P. Kwiat, “Observation of the Spin Hall Effect of Light via Weak Measurements,” Science 319, 787 (2008). [14] S. Fu, C. Guo, G. Liu, Y. Li, H. Yin, Z. Li, and Z. Chen, “Spin-Orbit Optical Hall Effect,” Phys. Rev. Lett. 123, 243904 (2019). [15] L. Marrucci, C. Manzo, and D. Paparo, “Optical Spin- to-Orbital Angular Momentum Conversion in Inhomoge- neous Anisotropic Media,” Phys. Rev. Lett. 96, 163905 (2006). [16] Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-Orbital Angular Momentum Conversion in a Strongly Focused Optical Beam,” Phys. Rev. Lett. 99, 073901 (2007). [17] Y. Gorodetski, A. Niv, V. Kleiner, and E. Hasman, “Observation of the Spin-Based Plasmonic Effect in Nanoscale Structures,” Phys. Rev. Lett. 101, 043903 (2008).[18] N. Shitrit, I. Yulevich, E. Maguid, D. Ozeri, D. Veksler, V. Kleiner, and E. Hasman, “Spin-Optical Metamaterial Route to Spin-Controlled Photonics,” Science 340, 724 (2013). [19] L. Fang, H. Wang, Y. Liang, H. Cao, and J. Wang, “Spin- Orbit Mapping of Light,” Phys. Rev. Lett. 127, 233901 (2021). [20] N. Levy, S. A. Burke, K. L. Meaker, M. Panlasigul, A. Zettl, F. Guinea, A. H. C. Neto, and M. F. Crommie, “Strain-Induced Pseudo-Magnetic Feidls Greater Than 300 Tesla in Graphene Nanobubbles,” Science 329, 544 (2010). [21] T. Low, and F. Guinea, “Strain-Induced Pseudomagnetic Field for Novel Graphene Electronics,” Nano Lett. 10, 3551 (2010). [22] F. de Juan, A. Cortijo, M. A. H. Vozmediano, and A. Cano, “Aharonov-Bohm interferences from local defor- mations in graphene,” Nat. Phys. 7, 810 (2011). [23] F. Guinea, M. I. Katsnelson, and A. K. Geim, “Energy gaps and a zero-field quantum Hall effect in graphene by strain engineering,” Nat. Phys. 6, 30 (2010). [24] K. I. Bolotin, F. Ghahari, M. D. Shulman, H. L. Stormer, and P. Kim, “Observation of the fractional quantum Hall effect in graphene,” Nature 462, 196 (2009). [25] M. C. Rechtsman,J. M.Zeuner, A. T¨ unnermann ,S. Nolte, M. Segev, and A. Szameit, “Strain-induced pseudomag- netic field and photonic Landau levels in dielectric struc- tures,” Nat. Photonics 7, 153 (2013). [26] Y. Lumer, M. A. Bandres, M. Heinrich, L. J. Maczewsky, H. Herzig-Sheinfux, A. Szameit, and M. Segev, “Light guiding by artificial gauge fields,” Nat. Photonics 13, 339 (2019). [27] R. O. Umucal1lar, and I. Carusotto, “Artificial gauge field for photons in coupled cavity arrays,” Phys. Rev. A 84, 043804 (2011). [28] S. Longhi, “Effective magnetic field for photons in waveg- uide and coupled resonator lattices,” Opt. Lett. 38, 3570 (2013). [29] K. Fang, Z. Yu, and S. Fan, “Realizing effective mag- netic field for photons by controlling the phase of dy- namic modulation,” Nat. Photonics 6, 782 (2012). [30] K. Fang, Z. Yu, and S. Fan, “Photonic Aharonov-Bohm Effect Based on Dynamic Modulation,” Phys. Rev. Lett. 108, 153901 (2012). [31] K. Fang, and S. Fan, “Controlling the Flow of Light Using the Inhomogeneous Effective Gauge Field that Emerges from Dynamic Modulation,” Phys. Rev. Lett. 111, 203901 (2013). [32] M. Kr´ ol, K. Rechci´ nska, H. Sigurdsson, P. Oliwa, R. Mazur, P. Morawiak, W. Piecek, P. Kula, P. G. Lagoudakis, M. Matuszewski, W. Bardyszewski, B. Pietka, and J. Szczytko, “Realizing Optical Persistent9 Spin Helix and Stern-Gerlach Defection in an Anisotropic Liquid Crystal Microcavity,” Phys. Rev. Lett. 127, 190401 (2021). [33] F. Liu, and J. Li, “Gauge Field Optics with Anisotropic Media,” Phys. Rev. Lett. 114, 103902 (2015). [34] V. G. Sala, D. D. Solnyshkov, I. Carusotto, T. Jacqmin, A. Lemaitre, H. Tercas, A. Nalitov, M. Abbarchi, E. Ga- lopin, I. Sagnes, J. Bloch, G. Malpuech, and A. Amo, “Spin-Orbit Coupling for Photons and Polaritons in Mi- crostructures,” Phys. Rev. X 5, 011034 (2015). [35] S. Ma, H. Jia, Y. Bi, S. Ning, F. Guan, H. Liu, C. Wang, and S. Zhang, “Gauge Field Induced Chiral Zero Mode in Five-Dimensional Yang Monopole Metamaterials,” Phys. Rev. Lett. 130, 243801 (2023). [36] Q. Yan, Z. Wang, D. Wang, R. Ma, C. Lu, G. Ma, X. Hu, and Q. Gong, “Non-Abelian gauge field in optics,” Adv. Opt. Photonics 15, 907 (2023). [37] O. Yesharim, A. Karnieli, S. Jackel, G. Di Domenico, S. Trajtenberg-Mills, and A. Arie, “Observation of the all- optical Stern–Gerlach effect in nonlinear optics,” Nat. Photonics 16, 582 (2022). [38] A. Karnieli, and A. Arie, “All-optical Stern-Gerlach ef- fect,” Phys. Rev. Lett. 120, 053901 (2018). [39] G. Liu, S. Fu, S. Zhu, H. Yin, Z. Li, and Z. Chen, “Higher-order optical rabi oscillations,” Fundamental Research 3, 898 (2023). [40] A. Tomita, and R. Y. Chiao, “Observation of Berry’s topological phase by use of an optical fiber,” Phys. Rev. Lett. 57, 937 (1986). [41] V. S. Liberman, and B. Y. Zel’dovich, “Spin-orbit inter- action of a photon in an inhomogeneous medium,” Phys. Rev. A 46, 5199 (1992). [42] Y. V. Kartashov, B. A. Malomed, V. V. Konotop, V. E. Lobanov, and L. Torner, “Stabilization of spatiotemporal solitons in Kerr media by dispersive coupling,” Opt. Lett. 40, 1045 (2015). [43] H. Sakaguchi, and B. A. Malomed, “One- and two- dimensional solitons in PT-symmetric systems emulating spin-orbit coupling,” New. J. Phys. 18, 105005 (2016). [44] C. Mann, S. A. R. Horsley, and E. Mariani, “Tunable pseudo-magnetic fields for polaritons in strained meta- surfaces,” Nat. Photonics 14, 669 (2020). [45] K. Rechci´ nska, M. Kr´ ol, R. Mazur, P. Morawiak, R. Mirek, K. Lempicka, W. Bardyszewski, M. Matuszewski, P. Kula, W. Piecek, P. G. Lagoudakis, B. Pietka, and J. Szczytko, “Engineering spin-orbit synthetic Hamilto- nians in liquid-crystal optical cavities,” Science 366, 727 (2019). [46] A. Forbes, M. de Oliveira, and M. R. Dennis, “Structured light,” Nat. Photonics 15, 253 (2021). [47] Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singu- larities,” Light: Sci. Appl. 8, 90 (2019). [48] J. Jia, H. Lin, Y. Liao, Z. Li, Z. Chen, and S. Fu, “Pendulum-type light beams,” Optica 10, 90 (2023). [49] A. F. Koenderink, Andrea Al` u, and A. Polman, “Nanophotonics: Shrinking light-based technology,” Sci- ence348, 516 (2015). [50] Y. Hu, S. Wang, J. Jia, S. Fu, H. Yin, Z. Li, and Z. Chen, “Optical superoscillatory waves without side lobes along a symmetric cut,” Adv. Photonics 3, 045002 (2021). [51] Y. Hu, S. Fu, H. Yin, Z. Li, Z. Li, and Z. Chen, “Sub- wavelength generation of nondiffracting structured lightbeams,” Optica 7, 1261 (2020). [52] C. Guo, S. Fu, H. Lin, Z. Li, H. Yin, and Z. Chen, “Dy- namic control of cylindrical vector beams via anisotropy,” Opt. Express 26, 18721 (2018). [53] Y. Hu, G. Mo, Z. Ma, S. Fu, S. Zhu, H. Yin, Z. Li, and Z. Chen, “Vector vortex state preservation in Fresnel cylindrical diffraction,” Opt. Lett. 46, 1313 (2021). [54] D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincar´ e sphere beams from a laser,” Nat. Photonics 10, 327 (2016). [55] G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-Order Poincar´ e Sphere, Stokes Parameters, and the Angular Momentum of Light,” Phys. Rev. Lett. 107, 053601 (2011). [56] M. O. Scully, W. E. Lamb, and A. Barut, “Theory of the Stern-Gerlach apparatus,” Found. Phys. 17, 575 (1987). [57] M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401 (1987). [58] R. Dorn, S. Quabis, and G. Leuchs, “Sharper Focus for a Radially Polarized Light Beam,” Phys. Rev. Lett. 91, 233901 (2003). [59] X. Xie, Y. Chen, K. Yang, and J. Zhou, “Harnessing the Point-Spread Function for High-Resolution Far-Field Op- tical Microscopy,” Phys. Rev. Lett. 113, 263901 (2014). [60] H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501 (2008). [61] X. Zhang, Y. Hu, X. Zhang, S. Zhu, H. Yin, Z. Li, Z. Chen, and S. Fu, “A Global Phase-Modulation Mech- anism for Flat-Lens Design,” Adv. Opt. Mater. 10, 2202270 (2022). [62] In experiment, the spin rotation was obtained by mea- suring S2=\|ΦR\|2− \|ΦL\|2. Ideally, the spatial distri- bution of S2is homogeneous; whereas in experiment it exhibits fluctuations due to the imperfect beam enve- lope. We therefore average it by using an expression: ¯S2=PNx nPNy mS2,n,m/(NxNy). [63] M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Effi- cient separation of the orbital angular momentum eigen- states of light,” Nat. Commun. 4, 2781 (2013). [64] J. Chen, C. Wan, and Q. Zhan, “Engineering photonic angular momentum with structured light: a reivew,” Adv. Photonics 3, 064001 (2021). [65] R. Vijay, C. Macklin, D. H. Slichter, S. J. Weber, K. W. Murch, R. Naik, A. N. Korotkov, and I. Siddiqi, “Stabi- lizing Rabi oscillations in a superconducting qubit using quantum feedback,” Nature 490, 77 (2012). [66] G. Cronenberg, P. Brax, H. Filter, P. Geltenbort, T. Jenke, G. Pignol, M. Pitschmann, M. Thalhammer, and H. Abele, “Acoustic Rabi oscillations between gravita- tional quantum states and impact on symmetron dark energy,” Nat. Phys. 14, 1022 (2018). [67] C. Schulze, A. Dudley, D. Flamm, M. Duparr´ e, and A. Forbes, Measurement of the orbital angular momentum density of light by modal decomposition, New J. Phys. 15, 073025 (2013). [68] D. Wei, Y. Cheng, R. Ni, Y. Zhang, X. Hu, S. Zhu, and M. Xiao, Generating controllable Laguerre-Gaussian laser modes through intracavity spin-orbital angular mo- mentum conversion of light, Phys. Rev. Appl. 11, 014038 (2019).
1408.3753v1.Two_Dimensional_TaSe2_Metallic_Crystals__Spin_Orbit_Scattering_Length_and_Breakdown_Current_Density.pdf	1 Two-Dimensional TaSe 2 Metallic Crystals: Spin-Orbit S cattering Length and Breakdown Current D ensity Adam T. Neal, Yuchen Du , Han Liu, Peide D. Ye* School of Electrical and Computer Engineering and Birck Nanotechnology Center, Purdue University, West Lafayette, IN 47907, USA *correspondence to: yep@purdue.edu 2 Abstract We have determined the spin -orbit scattering length of two-dimensional layered 2H-TaSe 2 metallic crystals by detailed characterization of the weak an ti-localization phenomena in this strong spin -orbit interaction material. By fitting the observed magneto -conductivity , the spin-orbit scattering length for 2H -TaSe 2 is determined to be 17 nm in the few -layer films . This small spin -orbit scattering lengt h is comparable to that of Pt, which is widely used to study the spin Hall effect, and indicates the potential of TaSe 2 for use in spin Hall effect devices . In addition to strong spin -orbit coupling, a material must also support large charge currents to achieve spin -transfer -torque via the spin Hall effect. Therefore, we have characterized the room temperature breakdown current density of TaSe 2 in air, where the best breakdown current density reaches 3.7×107 A/cm2. This large breakdown current further indicates the potential of TaSe 2 for use in spin -torque devices and two -dimensional device interconnect applications. Keywords: tantalum diselenide, transition metal dichalcogenide , spin -orbit scattering, weak anti-localization, breakdown current 3 Although studied for some time ,1,2 the transition metal dichalcogenide (TMD) family of materials has attracted increased attention in the nanoelectronics community due to their two-dimensional layered structure , following the prolific research into graphene .3-6 With graphene’s zero bandgap limit ation for transistor technology, much of the nanoelectronics community’s interest in TMDs has bee n focused on the semiconductors , particularly MoS 2, with the demonstration of single -layer and few-layer field-effect transistors .7-9 Metallic TMDs , on the other hand, have received much less attention in the nanoelectronics community thus far , but recent works on exfoliated TaSe 2 indicate that interest is on the rise .10-12 Notably, single -layer TaSe 2 has been recently characterized by Raman spectroscopy .13 Historically, m etallic TMDs have been intensely studied by material physicists and condensed matter physicists due to their superconducting1,2 and charge density wave14 properties , which remain acti ve areas of research . Obviously , metallic TMDs are not suitable for field-effect transistor channel material s, similar to graphene . One possible nanoelectronics application, previously proposed for graphene ,15 is the use of metallic TMDs as device interconnects for an all two-dimensional ( 2D) material logic technology. Another application of metallic TMDs , particularly of 2H-TaSe 2 on which we will focus in this work, is in spintronics devices. Angle -resolved photoemission spectroscopy (ARPES ) measurements of 2H -TaSe 2 reveal a “d og-bone” like structure of the Fermi surface in the “normal” (not charge density wave) state,16,17 and this structure is attributed to strong spin -orbit coupling in TaSe 2.18 This strong spin -orbit coupling may make TaSe 2 an ideal 2D material for generation of spin currents via the spin Hall effect. Motivated by these potential applications of 2H -TaSe 2, we determine , for the first time, the spin -orbit scattering length of TaSe 2 by characterizing the weak anti -localizat ion phenomena in the material. In addition to strong spin -orbit coupling, a material must also support large charge currents to 4 achieve spin -transfer -torque via the spin Hall effect. The ability to conduct large charge currents is also important for the previously mentioned two -dimensio nal interconnect application. Therefore , we have also characterized the breakdown current density of 2D TaSe 2 crystals for the first time. Results and Discussion First of all, it is important to establish the polytype of the TaSe 2 samples used in this work. The 1T polytype, with octahedral coordination, and the 2H polytype, with trigonal prismatic coordination, are the two most studied in the literature. For the 1T polytype of TaSe 2, the material is in the incommensurate charge d ensity wave state below 600 K and in the commensurate charge density wave state below 473 K .14,19,20 The transition at 473K is accompanied by a stark discontinuity in the resistivity as a function of temperature. The 2H polytype does not transition into the incommensurate charge density wave state until ~120K, and the commensurate charge density wave set s in below 90K. In contrast to the 1T polytype, there are no discontinuities in the resistivity as a function of temper ature, but there is a characteristic change in the slope of the resistivity versus temperature curve at onset of the incommensurate charge density wave state at ~120 K .14,19,20 These properties of the resistivity versus temperature allow one to distinguish between the two polytypes of TaSe 2 electrically. Figure 1(d) shows the resistivity of the TaSe 2 used in this work as a function of temperature from 4 K to 300 K. The characteristic change in the slope of the resistivity versus temperature curve, indicated by the arrow in the figure, confirms that the TaSe 2 used in this work is of the 2H polytype . The crystal structure of 2H -TaSe 2 is shown in Figure 1 (a) -(c) with layered 2D structures as expected . 5 With the polytype of the TaSe 2 established, we determine the electrically active thickness of the flake used to study spin -orbit coupling of TaSe 2. Figure 2 (a) shows atomic force microscopy ( AFM ) image of the TaSe 2 device with the AFM height measurement of the flake overlaid. The 2D crystal has a physical thickness of ~12 nm. A calculation of the Hall coefficient from Hall effect measurements using the thickness measured by AFM yields Hall coefficients which are much too large compared to those published in the literature .21-23 Because of this discrepancy, we conclude that the physical thickness of the flake as measure d by AFM is not electrically active, perhaps due to oxidation of the top and bottom layers of the TaSe 2 flake while exposed to air for long periods of time . We also note that the devices with much thinner flakes cannot be m easured reliably. Considering these observation s, we can estimate the electrically active thickness of our TaSe 2 flake as the Hall coefficients from the literature divided by the Hall slope measured for our TaSe 2 device , ⁄ . The Hall slope is the slope of the Hall resistance , ⁄, as a function of magnetic field . The measured Hall slopes for this device as a function of temperature are shown in Figure 2(b). We perform this thickness estimation at two temperatures, T~5K and T~ 120K, where the Hall slope for our flake at 120K was estimated by linear extrapolation using the measured Hall slopes in Figure 2(b). The electrically active thickness, , is determined to be 0.81 nm and 0.88 nm for 5K and 120K , respectively. Therefore, from this estimation via the Hall effect measurement , we conclude that only one or two atomic layers of the TaSe 2 flake are electrically active. The dependence of the magneto -conductivity on the angle of the magnetic field, shown in Figure 2(c), also confirms this claim that the system studied is a two -dimensional electron system . The resistivity and Hall coefficient plotted in Figure 1 (d) and Figure 2(b) were calculated using th e electric ally active thickness, , that we have determined. Note that the change sign change of the Hall coefficient 6 is expected and is related to the reconstruction of the Fermi surface as the charge density wave state develops with decreasing temperature.21 We now study the spin-orbit coupling strength of 2H-TaSe 2. Indeed, the strong spin-orbit coupling indicated by the “dog’s bone” Fermi surface shape16,17 is confirmed by magneto - transport measurements. Figure 3(a) shows the differential sheet conductance of TaSe 2 as a function of magnetic field for various temperatures . TaSe 2 exhibits a negative magneto - conductivity , characteristic of weak anti -localization, which indicates the strong spin-orbit coupling of TaSe 2. A classical background has been subtracted from the data, determined by fitting the data at T = 8 K and B more th an one Tesla where the localization phenomena is suppressed. To quantitatively determine the spin-orbit scattering length, we must first determine the dimensionality of the weak anti -localization phenomena in the system. Figure 2(c) shows the differential magneto -conductivity for different angles between the magnetic field and the sample. In this case no classical background is subtracted. The angular dependence shows that the weak anti-localization phenomenon behaves two dimensionally. The differential magneto - conductivity , , can be descri bed for 2D weak localization by the following equation:24-26 ( ( ) ( ( ) ( ))) (1) ( ) ( ) ( ) where is the valley degeneracy, for the spin degeneracy, the charge of an electron, is Plan ck’s constant divided by , the phase coherence length, and the spin-orbit scattering length. and are the free parameters which allow fitting of the data. The number of valleys, , can be determined from ARPES performed on TaSe 2 in its commensurate 7 charge den sity wave state.17,21,27 In the commensurate charge density wave state, the TaSe 2 lattice is deformed, effectively increasing the period of the material system in real space. This leads to a smaller Brillouin zone in k -space compared to the undeformed material and also causes reconstruction of the Fermi surface. ARPES indicate s that there are three independent valley’s in the charge density wave Brillouin zone, therefore we choose when fitting the weak anti -localization data. The solid orange line in Figure 3(a) shows an example fit of the weak -anti-localization peak using Equation 1. We can now determine the spin-orbit scattering length of TaSe 2 by fitting the weak anti - localization data in Figure 3(a) , along with others not shown. Figure 3(b) shows the phase coherence length and the spin-orbit scattering length determined from the fittings as a function of temperature. We find that is independent of temperature, while decreases as , which is consistent with dephasing due to electron -electron scattering without too much disorder .28 Because in Figure 3(b) is independent of temperature, we take their average and determine that the spin-orbit scattering length 17 nm for 2H -TaSe 2. This length is comparable to the spin-orbit scattering length of Pt ( 12 nm)29, widely used to study the spin Hall effect, and indicates the potential of TaSe 2 for use in 2D spintronics devices . The weak anti -localization is als o be suppressed by increasing bias current as shown in Figure 4(a) . The higher current bias adds energy to the sample, increasing the electron temperature above that of the helium bath, leading to suppression of the weak anti -localization. Figure 4(b) shows the phase coherence length and spin-orbit scattering length determined from the data in Figure 4(a). The phase coherence length decreases as , which indicates that the electron temperature increases as considering as previous ly determined . The 8 same relationship between bias current and electron temperature has also been observed for a two-dimensional electron gas in the quantum Hall regiem.30 This further confirms that the studied TaSe 2 device is , electrically , an atomic ally-thin material system. Finally, we evaluate the breakdown current density of 2H-TaSe 2 in order to determine its potential to achieve spin Hall effect based spin -transfer -torque by DC characterization of the total 18 fabricated devices. The average room temperature resistivity of the TaSe 2 flakes used for the breakdown current measurements was 1.9×10-4 Ω·cm , determined using four terminal measurements of 8 of the devices. The contact resistance of th e Ni/Au contact to the TaSe 2 flakes was also estimated by subtracting the four terminal resistance from the two terminal resistance and dividing by two. The average contact resistance determined from the 8 four terminal devices was 0.74 Ω·mm, which is one order of magnitude small er compared to metal/MoS 2 contacts.31 This is because TaSe 2 is metallic and forms Ohmic contacts with metals while MoS 2 is a semiconductor and forms Schottky contacts with metals. To ach ieve this low contact resistance, the Ni/Au contact was deposited as soon as possible after TaSe 2 exfoliation to minimize surface oxidation and avoid oxide barriers between the Ni/Au contact and the TaSe 2. Flakes were also stored in a nitrogen box before metal contact deposition to help minimize the surface oxidation. Measurement of t he breakdown current density is performed by continuously increasing the bias voltage across the device until a decrease in current more than one order of magnitude is observed. Figure 5(a) shows the current density versus bias voltage data for the device showing the highest breakdown current density observed among the 18 devices. The breakdown current de nsity i s taken as the current density immediately before the sharp decrease in current was observed. Figure 5(b) shows the histogram of the breakdown currents determined from the 18 devices. The maximum breakdown current density observed is 3.7×107 A/cm2, the 9 average 1.9×107 A/cm2, the minimum 0.5×107 A/cm2 and the standard deviation 0.8×107 A/cm2. The flake thickness as characterized by AFM was used when computing the breakdown current densities , so these reported current densities could be slightly underestimated if we consider the surface oxidation . These breakdown currents are comparable to the charge currents used to induce spin -transfer -torque via the spin Hall effect in Tantalum thin films,32 indicating th e possibility of TaSe 2 based 2D spin-torque devices. The breakdown currents are about one order of magnitude less than those of graphene ;15,33 however , they are comparable to those of MoS 2.34 These large breakdown currents also indicate the potential of TaSe 2 as a 2D interconnect material, particularly if used in conjunction semiconducting TMDs where the similar crystal structure may provide some integration advantage s.35 Conclusions In conclusi on, we have determined the spin -orbit scattering length of 2H-TaSe 2 by detailed characterization of the weak anti -localization phenomena in the material. By fitting the observed magneto -conductivity , the spin orbit scattering length for 2H -TaSe 2 is determined to be 17 nm . This small spin orbit scattering length is compa rable to that of Pt, which is widely used to study the spin Hall effect, and indicates the potential of 2D TaSe 2 for use in spin Hall effect devices. Additionally , we have characterized the room temperature breakdown current density of TaSe 2 in air, where the best breakdown current density observed is 3.7×107 A/cm2. This large breakdown current density further indicates the potential of TaSe 2 for use in 2D spin-torque devices and 2D device interconnect applicat ions. Methods 10 The 2D TaSe 2 devices were fabricated as follows. A bulk Nanosurf TaSe 2 sample was purchased from nanoScience Instruments and confirmed by Raman characterization shown in Figure 1(e) . TaSe 2 flakes were prepared by the method of mechanical exfoliation using adhesive tape, depositing TaSe 2 flakes on an insulating substrate. For the weak anti -localization measurements, the flakes were deposited on insulating SrTiO 3 substrate, while, for the breakdown current measurements, flakes were deposited on 90 nm SiO 2 on Si substrate. Electrical contacts were defined using an electron beam lithography and liftoff process. The metal contacts were deposited by electron beam evaporation. For the weak anti -localization measurements, a 30nm/50nm Ni 0.8Fe0.2/Cu contact was used, and for the breakdown current measurements, a 30nm/50nm Ni/Au contact was used. The weak anti -localization measurements were carried out in a 3He cryostat with a superconducting magne t using a Stanford Research 830 lock-in amplifier. Breakdown currents were measured at room temperature in air using a Keithley 4200 semiconductor characterization system. The authors declare no competing financial interest. Acknowledgements This materia l is based upon work partly supported by NSF under Grant CMMI -1120577 and SRC under Task 2396. A portion of the low temperature measurements was performed at the National High Magnetic Field Laboratory, which is supported by National Science Foundation Cooperative Agreement No. DMR -1157490, the State of Florida, and the U.S. Department of Energy. The authors thank Z. Luo, X. Xu, E. Palm, T. Murphy, J. -H. Park, and G. Jones for experimental assistance. 11 Figure 1: (a) 3D view of the 2H -TaSe 2 crystal structure. (b) Side view of the TaSe 2 crystal cleaved at the ( ̅ ) face. (c) Top view of the TaSe 2 crystal with lattice vectors shown. Purple balls represent Ta atoms, while yellow balls represent Se atoms. (d) Resistivity ρxx and sheet resistance Rsheet as a function of temperate for the TaSe 2 flake used for weak anti -localization measurements. The change in slope at ~120K indicates that the TaSe 2 is of the 2H polytype . (e) Raman characterization of the bulk TaSe 2 from which the flakes were exfol iated. 100 150 200 250 Intensity [A.U.] Raman Shift [cm-1]E2gA1g 0 100 200 300024xx [10-5.cm] Temperature [K]CDW Transition T~120K 100300500 Rsheet [/] a1 a2(a) (b) (c) a3 ̅ face(d) (e)12 Figure 2: (a) AFM image of the TaSe 2 device used to study the spin -orbit scattering via weak anti-localization measurements. The AFM height measurement along the white line is overlaid . (b) Hall coefficient and Hall slope of TaSe 2 as a function of temperature. The sign change results from the reconstruction of the Fermi surface as the charge density wave state develops with decreasing temperature. (c) Differential magneto -conductivity of TaSe 2 at T = 0.4 K with magnetic field appl ied at different angles relative to the sample surface. Zero degrees indicate that the magnetic field is parallel to the TaSe 2 planes, while 90 degrees indicate that the magnetic field is perpendicular to the TaSe 2 planes. 0204060-10-505Rh [10-11 m3/C] Temperature [K]-10-505 Hall Slope [10-2 m2/C] -4-2024-10-8-6-4-202 90 30 xx(B) - xx(0) [10-5 S] Magnetic Field [T]0(b) (c) (a) 12 nm13 Figure 3: (a) Differential magneto -conductivity of 2H -TaSe 2 at temperatures from 1K to 8K. The negative magneto -conductivity shown in the figure is the characteristic of weak anti - localization and indicates the strong spin -orbit coupling of TaSe 2. (b) Phase coherence length (black squares) and spin -orbit scattering length (red circles) extracted from the weak anti - localization data from Figure 3(a). The black solid line indicates the power law decrease of . 1 1010-810-7 L, Lso [meters] Temperature [K] L Lso -2 -1 0 1 2-6-4-202 1 K 2 K 3 K 5 K 8 K 1 K fitxx(B) - xx(0) [10-5 S] Magnetic Field [T](a) (b)14 Figure 4: (a) Differential magneto -conductivity of TaSe 2 for different RMS bias currents as indicated in the figure. The helium bath temperature was 0.4K for these measurements. (b) Phase coherence length (black squares) and spin -orbit scattering length (red circles) extracted from the weak anti -localization data from Figure 4(a). The black solid line indicates the power law decrease of with increasing bias current. -2 -1 0 1 2-6-4-202 10µA 20 µA 50 µA 75 µA 100 µAxx(B) - xx(0) [10-5 S] Magnetic Field [T] 1E-6 1E-5 1E-410-810-7 L LsoL, Lso [meters] Current [A](a) (b)15 Figure 5: (a) DC current density versus voltage characteristic of the TaSe 2 device which show s the highest breakdown current. (b) Histogram of breakdown current densities for 18 TaSe 2 devices measured in air at room temperature. 0.00 0.75 1.50 2.25 3.00 3.7502468 Count Jbreakdown (107 A/cm2) 0 1 2 3 4 501234 Current Density (107 A/cm2) Voltage (V)(a) (b)16 References (1) Hulliger, F. Crystal Chemistry of the Chalcogenides and Pnictides of the Transition Elements. Structure and Bonding 1968 , 4, 83–229. (2) Wilson, J.; Yoffe, A. The Transition Metal Dichalcogenides Discussion and Interpretation of the Observed Optical, Electrical and Structural Properties. Adv. Phys. 1969 , 18, 193–335. (3) Novoselov, K. S.; Geim, A. K.; Morozov, S. V.; Jiang, D.; Zhang, Y.; Dubonos, S. V.; Grigorieva, I. V.; Firsov, A. A. Electric Field Effect in Atomically Thin Carbon Films. Science 2004 , 306, 666 –669. (4) Novoselov, K. S.; Jiang, D.; Schedin , F.; Booth, T. J.; Khotkevich, V. V.; Morozov, S. V.; Geim, A. K. Two -Dimensional Atomic Crystals. Proc. Natl. Acad. Sci. U.S.A. 2005 , 102, 10451 –10453. (5) Novoselov, K.; Geim, A. K.; Morozov, S.; Jiang, D.; Grigorieva, M. K. I.; Dubonos, S.; Firsov, A. Two-Dimensional Gas of Massless Dirac Fermions in Graphene. Nature 2005 , 438, 197–200. (6) Zhang, Y.; Tan, Y.-W.; Stormer, H. L.; Kim, P. Experimental Observation of the Quantum Hall Effect and Berry’s Phase in Graphene. Nature 2005 , 438, 201 –204. (7) Radisavljevic, B.; Radenovic, A.; Brivio, J.; Giacometti, V.; Kis, A. Single -Layer MoS 2 Transistors. Nat. Nanotechnol. 2011 , 6, 147 –150. (8) Ayari, A.; Cobas, E.; Ogundadegbe, O.; Fuhrer, M. S. Realization and Electrical Characterization of Ultrathin Cryst als of Layered Transition -Metal Dichalcogenides. J. Appl. Phys. 2007 , 101, 014507. (9) Liu, H.; Ye, P. D. MoS 2 Dual -Gate MOSFET with Atomic -Layer -Deposited Al 2O3 as Top-Gate Dielectric. IEEE Electron Device Lett. 2012 , 33, 546 –548. (10) Castellanos -Gomez, A.; Navarro -Moratalla, E.; Mokry, G.; Quereda, J.; Pinilla - Cienfuegos, E.; Agraït, N.; Zant, H.; Coronado, E.; Steele, G.; Rubio -Bollinger, G. Fast and Reliable Identification of Atomically Thin Layers of TaSe 2 Crystals. Nano Research 2013 , 6, 191–199. (11) Yan, Z.; Jiang, C.; Pope, T. R.; Tsang, C. F.; Stickney, J. L.; Goli, P.; Renteria, J.; Salguero, T. T.; Balandin, A. A. Phonon and Ther mal Properties of Exfoliated TaS e2 Thin Films. J. Appl. Phys . 2013 , 114, 204301. (12) Renteria, J.; Samnakay , R.; Jiang, C.; Pope, T. R.; Goli, P.; Yan, Z.; Wickramaratne, D.; Salguero, T. T.; Khitun, A. G.; Lake, R. K.; Balandin, A. A. All -metallic electrically gated 2H - TaSe 2 thin-film switches and logic circuits. J. Appl. Phys . 2014 , 115, 034305. (13) Hajiyev , P.; Cong, C.; Qiu, C.; Yu, T. Contrast and Raman Spectroscopy Study of Single - And Few -Layered Charge Density Wave Material: 2H -TaSe 2. Sci. Rep. 2013 , 3, 2593. (14) Wilson, J.; Di Salvo, F.; Mahajan, S. Charge -Density Waves and Superlattices in the Metal lic Layered Transition Metal Dichalcogenides. Adv. Phys. 1975 , 24, 117 –201. (15) Murali, R.; Yang, Y.; Brenner, K.; Beck, T.; Meindl, J. D. Breakdown Current Density of Graphene Nanoribbons. Appl. Phys. Lett. 2009 , 94, 243114 –243114. (16) Rossnagel , K.; Rotenberg, E.; Koh, H.; Smith, N. V.; Kipp, L. Fermi Surface, Charge - Density -Wave Gap, and Kinks in 2H -TaS 2. Phys. Rev. B 2005 , 72, 121103. (17) Borisenko, S. V.; Kordyuk, A. A.; Yaresko, A. N.; Zabolotnyy, V. B.; Inosov, D. S.; Schuster, R.; Büchne r, B.; Weber, R.; Follath, R.; Patthey, L.; Berger, H. Pseudogap and Charge Density Waves in Two Dimensions. Phys. Rev. Lett. 2008 , 100, 196402. 17 (18) Rossnagel, K.; Smith, N. V. Spin -Orbit Splitting, Fermi Surface Topology, and Charge - Density -Wave Gapping in 2H -TaSe 2. Phys. Rev. B 2007 , 76, 073102. (19) Wilson, J. A.; Di Salvo, F. J.; Mahajan, S. Charge -Density Waves in Metallic, Layered, Transition -Metal Dichalcogenides. Phys. Rev. Lett. 1974 , 32, 882 –885. (20) Craven, R. A.; Meyer, S. F. Specific Heat and Resistivity Near The Charge -Density - Wave Phase Transitions in 2H -TaSe 2 And 2H -TaS 2. Phys. Rev. B 1977 , 16, 4583 –4593. (21) Evtushinsky, D. V.; Kordyuk, A. A.; Zabolotnyy, V. B.; Inosov, D. S.; Büchner, B.; Berger, H.; Patthey, L.; Follath, R.; Borisen ko, S. V. Pseudogap -Driven Sign Reversal of the Hall Effect. Phys. Rev. Lett. 2008 , 100, 236402. (22) Lee, H.; Garcia, M.; McKinzie, H.; Wold, A. The Low -Temperature Electrical And Magnetic Properties Of TaSe 2 And NbSe 2. J. Solid State Chem . 1970 , 1, 190 –194. (23) Naito, M.; Tanaka, S. Electrical Transport Properties in 2H -NbS 2, -NbSe 2, -TaS 2 and - TaSe 2. J. Phys. Soc. Jpn. 1982 , 51, 219 –227. (24) Fukuyama, H. Non -Metallic Behaviors of Two -Dimensional Metals and Effect of Intervalley Impurity Scattering. Suppl. Prog. Theor. Phys. 1980 , 69, 220 –231. (25) Fukuyama, H. Effects of Intervalley Impurity Scattering on the Non -Metallic Behavior in Two-Dimensional Systems. J. Phys. Soc. Jpn. 1980 , 49, 649 –651. (26) Knap, W.; Skierbiszewski, C.; Zduniak, A.; Litwi n-Staszewska, E.; Bertho, D.; Kobbi, F.; Robert, J. L.; Pikus, G. E.; Pikus, F. G.; Iordanskii, S. V.; Mosser, V.; Zekentes, K.; Lyanda -Geller, Y. B. Weak Antilocalization and Spin Precession in Quantum Wells. Phys. Rev. B 1996 , 53, 3912 –3924. (27) Kuchin skii, E.; Nekrasov, I.; Sadovskii, M. Electronic Structure of Two -Dimensional Hexagonal Diselenides: Charge Density Waves and Pseudogap Behavior. J. Exp. Theor. Phys. 2012 , 114, 671 –680. (28) Fukuyama, H.; Abrahams, E. Inelastic Scattering Time in Two -Dimensional Disordered Metals. Phys. Rev. B 1983 , 27, 5976 –5980. (29) Niimi, Y.; Wei, D.; Idzuchi, H.; Wakamura, T.; Kato, T.; Otani, Y. Experimental Verification of Comparability between Spin -Orbit and Spin -Diffusion Lengths. Phys. Rev. Lett. 2013 , 110, 016 805. (30) Wei, H. P.; Engel, L. W.; Tsui, D. C. Current Scaling in the Integer Quantum Hall Effect. Phys. Rev. B 1994 , 50, 14609 –14612. (31) Liu, H.; Neal, A. T.; Ye, P. D. Channel Length Scaling of MoS 2 MOSFETs. ACS Nano 2012 , 6, 8563 –8569. (32) Liu, L.; Pai, C.-F.; Li, Y.; Tseng, H. W.; Ralph, D. C.; Buhrman, R. A. Spin -Torque Switching with the Giant Spin Hall Effect of Tantalum. Science 2012 , 336, 555 –558. (33) Liao, A. D.; Wu, J. Z.; Wang, X.; Tahy, K.; Jena, D.; Dai, H.; Pop, E. Thermally Li mited Current Carrying Ability of Graphene Nanoribbons. Phys. Rev. Lett. 2011 , 106, 256801. (34) Lembke, D.; Kis, A. Breakdown of High -Performance Monolayer MoS 2 Transistors. ACS Nano 2012 , 6, 10070 –10075. (35) Chhowalla, M.; Shin, H. S.; Eda, G.; Li, L.-J.; Loh, K. P.; Zhang, H. The Chemistry of Two-Dimensional Layered Transition Metal Dichalcogenide Nanosheets. Nature C hemistry 2013 , 5, 263 –275. 18 TOC Graphics -2 -1 0 1 2-6-4-202 1 K 2 K 3 K 5 K 8 K 1 K fitxx(B) - xx(0) [10-5 S] Magnetic Field [T] 2H-TaSe2
1307.2363v1.Multi_Orbital_Superconductivity_in_SrTiO3__LaAlO3_Interface_and_SrTiO3_Surface.pdf	arXiv:1307.2363v1 [cond-mat.supr-con] 9 Jul 2013Typeset withjpsj3.cls <ver.1.0> FullPaper Multi-OrbitalSuperconductivity inSrTiO 3/LaAlO 3Interface andSrTiO 3Surface YasuharuNAKAMURA1andYouichiYANASE1,2∗ 1Graduate School ofScience and Technology, NiigataUnivers ity, Niigata950-2181, Japan 2Department of Physics, NiigataUniversity, Niigata950-21 81, Japan (Received April23, 2013; accepted May 28, 2013) Weinvestigate thesuperconductivity intwo-dimensional e lectronsystems formedinSrTiO 3nanostruc- tures. Our theoretical analysis is based on the three-orbit al model, which takes into account t2gorbitals of Ti ions. Because of the interfacial breaking of mirror symme try, a Rashba-type antisymmetric spin-orbit coupling arises from the cooperation of intersite and inter orbital hybridyzation and atomic LS coupling. This model shows a characteristic spin texture and carrier d ensity dependence of Rashba spin-orbit cou- pling through the orbital degree of freedom. Superconducti vity is mainly caused by heavy quasiparticles consisting of d yzand dzxorbitals at high carrier densities. We ﬁnd that the Rashba sp in-orbit coupling stabilizes a quasi-one-dimensional superconducting phas e caused by one of the d yzor dzxorbitals at high magnetic ﬁelds along interfaces. This quasi-one-dimensio nal superconducting phase is protected against paramagnetic depairing e ﬀects by the Rashba spin-orbit coupling and realizes a large u pper critical ﬁeld Hc2beyond the Pauli-Clogston-Chandrasekhar limit. This ﬁndi ng is consistent with an extraordinarily large upper critical ﬁeld observed in SrTiO 3/LaAlO 3interfaces and its carrier density dependence. The possible coexistence of superconductivity and ferromagne tism in SrTiO 3/LaAlO 3interfaces may also be attributedtothis quasi-one-dimensional superconductin g phase. KEYWORDS: Non-centrosymmetric superconductivity, two-d imensional electron gas, multi-orbital model Two-dimensional conducting electron systems formed on SrTiO3heterostructures are attracting much attention. For instance, electron gases with a high carrier density on the order of 1013cm−2have been realized in SrTiO 3/LaAlO 3 (STO/LAO) interfaces,1)SrTiO3/LaTiO 3interfaces,2)SrTiO3 (STO)surfaces,3)andδ-dopedSTO.4)Thediscoveryofsuper- conductivity,5)ferromagnetism,6–10)andtheircoexistence7–10) shed light on innovating phenomena in these systems. These quantum condensed phases are controlled by a gate volt- age through the change of carrier density.3,11–14)One of the key issues is the role of Rashba-type antisymmetric spin- orbit coupling15)arising from the interfacial breakingof mir- ror symmetry, which may realize an exotic quantum con- densed phase, such as non-centrosymmetric superconductiv - ity,16)chiral magnetism,17)and their coexistent phase. In this research,we theoreticallystudy the non-centrosymmetric su- perconductivityrealized in STO nanostructuresfrom the mi - croscopicpointofview. It has been shown that a two-dimensional electron gas is conﬁned in a few TiO 2layers of the STO/LAO interface and STO surface in the high-carrier-density region.3,18–23)The conduction bands mainly consist of three t 2gorbitals of Ti ions.18–23)Although the degeneracy of t 2gorbitals signiﬁ- cantly aﬀects the band structure of two-dimensional electron gases,atheoryofsuperconductivitybasedonthemulti-orb ital model has not been conducted. Multiband models have been studied,24,25)but the symmetry of t 2gorbitals is taken into account in this study for the ﬁrst time. We show that the synergy of broken inversion symmetry and orbital degener- acystabilizesanintriguingsuperconductingphaseinthet wo- dimensionalelectrongases. Our study is based on a two-dimensional tight-binding model that reproduces the electronic structure of the STO/LAO interface indicated by ﬁrst principles band struc- ∗E-mail address: yanase@phys.sc.niigata-u.ac.jpturecalculations20–23,26–28)andexperiments.18,19)Weherefo- cus on the STO/LAO interface, which has been intensively investigated,butourmainresultsarealsovalidforotherS TO heterostructures.Themodelisdescribedas H=H0+HI+HZ, (1) wherethesingle-particleHamiltonian H0is H0=Hkin+Hhyb+HCEF+Hodd+HLS, (2) Hkin=/summationdisplay k/summationdisplay m=1,2,3/summationdisplay s=↑,↓(εm(k)−µ)c† k,msck,ms, (3) Hhyb=/summationdisplay k/summationdisplay s=↑,↓[V(k)c† k,1sck,2s+h.c.], (4) HCEF=∆/summationdisplay in3i, (5) Hodd=/summationdisplay k/summationdisplay s=↑,↓[Vx(k)c† k,1sck,3s+Vy(k)c† k,2sck,3s+h.c.],(6) HLS=λ/summationdisplay iLi·Si. (7) Wedenote(d yz,dzx,dxy)orbitalsusingtheindex m=(1,2,3), respectively. The ﬁrst term Hkindescribes the kinetic en- ergy of each orbital and includes the chemical potential µ. Hhybis the intersite hybridization term of d yzand dzxor- bitals.HCEFrepresents the crystal electric ﬁeld of tetrago- nal systems. Because the mirror symmetry is broken near the interface/surface, hybridization is allowed between d xy and dyz/dzxorbitals, and is represented by the “odd par- ity hybridization term” Hodd. The atomic spin-orbit coupling term (LS coupling term) of Ti ions is taken into account inHLS. We here adopt the tight-binding model reproduc- ing ﬁrst principles band structure calculations for STO het - erostructures,26–28)ε1(k)=−2t3coskx−2t2cosky,ε2(k)= −2t2coskx−2t3cosky,ε3(k)=−2t1(coskx+cosky)− 12 J.Phys.Soc. Jpn. FullPaper Author Name 4t4coskxcosky,V(k)=4t5sinkxsinky,Vx(k)=2itoddsinkx, andVy(k)=2itoddsinky. The same tight-binding model has been adopted for the study of surface spin-triplet supercon - ductivity in Sr 2RuO4.29)Recent studies have examined the Rashba-type antisymmetric spin-orbit coupling27,28,30)and magnetotransport30)in STO/LAO interfaces on the basis of thismodel. Inthispaper,wefocusontheroleofRashba-typeantisym- metric spin-orbit coupling in the interface superconducti vity. Intheabovemodel,theRashbaspin-orbitcouplingisinduce d bythecombinationoftheoddparityhybridizationterm, Hodd, and the LS coupling term, HLS. The former arises from the parity mixing of local orbitals, which is a general source of antisymmetric spin-orbit coupling.31,32)For instance, the Vx(k) (Vy(k)) term describes the mixing of d yz(dzx) and d xy orbitals of Ti ions, which mainly occurs through the parity mixingwiththe p yorbital(p xorbital)onoxygenions. We consider the s-wave superconductivity as expected in the bulk STO.33)Unconventional pairing due to the electron correlationhasbeenstudied,34)however,wedonottouchthis possibility. Our reasonable assumption has been justiﬁed b y therecentexperimentonsuperﬂuiddensity.35)Forsimplicity, wetakeintoaccounttheintraorbitalattractiveinteracti onU< 0andtheinterorbitalattractiveinteraction U′<0inthespin- singletchannel; HI=U/summationdisplay i/summationdisplay mni,m↑ni,m↓+U′/summationdisplay i/summationdisplay m/nequalm′ni,m↑ni,m′↓.(8) For the discussion of the superconducting state in the mag- neticﬁeld,we considertheZeemancouplingterm HZ=−/summationdisplay k/summationdisplay m/summationdisplay s,s′µBH·σss′c† k,msck,ms′,(9) in whichσis the Pauli matrix and µBis the Bohr magne- ton. The orbital depairing e ﬀect arising from the coupling of electron motion and vector potential is suppressed by the ge - ometry when we consider the magnetic ﬁeld parallel to the two-dimensional conducting plane, H/bardblˆx. The orbital po- larization due to the magnetic ﬁeld is also ignored since the orbital moment along the plane vanishes for the degenerate dyz/dzxorbitals. Now, we formulate the linearized gap equation, by which we determine the instability to the superconducting phase. First, we diagonalize the noninteracting Hamiltonian ( H0+ HZ) using the unitary matrix ˆU(k)=/parenleftBig ums,j(k)/parenrightBig . Thereby, the basis changes as C† k= Γ† kU†(k), where C† k= (c† k,1↑,c† k,2↑,···,c† k,3↓) andΓ† k=(γ† k,1,γ† k,2,···,γ† k,6). With the use of the operators of quasiparticles, γ† k,jandγk,j, the noninteractingHamiltonianisdescribedas, H0+Hz=/summationdisplay k6/summationdisplay j=1Ej(k)γ† k,jγk,j, (10) whereEj(k)is aquasiparticle’senergyand Ei(k)≥Ej(k)for i>j. Next, we introduce Matsubara Green functions in the or- bitalbasis, Gm′s′,ms(k,iωl)=/integraldisplayβ 0dτeiωlτ/angbracketleftck,m′s′(τ)c† k,ms(0)/angbracketright,(11)=6/summationdisplay j=11 iωl−Ej(k)um′s′,j(k)u∗ ms,j(k),(12) whereωlis the Matsubara frequency. The linearized gap equation is obtained by looking at the divergence of the T- matrix,ˆT(q),whichisgivenby ˆT(q)=ˆT0(q)−ˆT(q)ˆHIˆT0(q). (13) The wave vector qrepresents the total momentumof Cooper pairs. In our model, the matrix element of the irreducible T- matrixˆT0(q)isobtainedas T(mn,m′n′) 0(q) =T/summationdisplay ωl/summationdisplay k[Gm↑,m′↑(q/2+k,iωl)Gn↓,n′↓(q/2−k,−iωl) −Gm↑,n′↓(q/2+k,iωl)Gn↓,m′↑(q/2−k,−iωl)],(14) whereTis the temperature. When we represent the T-matrix using the basis ( mn)=(11,12,13,21,22,23,31,32,33), the interaction term is represented by the 9 ×9 diagonal matrix, ˆHI=(Umδmn)withUm=Uform=1,5,9 andUm=U′ for others. The superconducting transition occurs when the maximum eigenvalue of the matrix, −ˆHIˆT0, is unity. Then, an element of the eigenvector( ψmn) is proportionalto the or- der parameter∆mn=−g/summationtext k/angbracketleftck,m↑c−k,n↓/angbracketright, whereg=Ufor m=nandg=U′form/nequaln. In what follows, we assume a zero total momentum of Cooper pairs, namely, q=0. Al- thoughahelicalsuperconductingstatewith q/nequal0isstabilized in non-centrosymmetricsuperconductorsunder the magneti c ﬁeld,16,36)a ﬁnite momentum qdoes not play any important roleinthefollowingresults.Thisisbecausetheparamagne tic depairing eﬀect is suppressed by the orbital degree of free- dom,aswe showbelow. We choosetheparameters (t1,t2,t3,t4,t5,∆)=(1.0,1.0,0.05,0.4,0.1,2.45),(15) soastoreproducetheelectronicstructureoftwo-dimensio nal electron gases.18–23,26–28)We choose the unit of energy as t1=1. Band structure calculations resulted in t1=300 meV,26)giving rise to an anisotropic Fermi velocity, vF= 7×104−4×105m/s, forn=0.15. For the parameters in eq. (15), the d xyorbital has a lower energy than the d yz/dzx orbitals, as expected in STO heterostructures;18–23,26–28)the level splitting at the Γpoint is−2t2−2t3+4t1+4t4−∆= 1.05∼300meV. The chemical potential µis determined so that the mobile carrier density per Ti ion is n. Although an enormouscarrierdensityof3 .5×1014cm−2correspondingto n=0.5 at the STO/LAO interface was predicted by the “po- larcatastrophe”mechanism,1)recentexperimentshaveshown a rather low density of mobile carriers.11–14,18,19)One of our purposes is to clarify the carrier density dependence of the superconductingstate. ThesourcesofRashba spin-orbitco u- pling are assumed to be todd=0.25 andλ=0.2 unless men- tioned otherwise explicitly. We here assume rather large va l- ues oftoddandλso that the amplitude of Rashba spin-orbit couplingα∼toddλ/∆is larger than the transition tempera- ture of superconductivity. We assume attractive interacti ons U=U′so that the transition temperature at zero magnetic ﬁeld isTc=0.005=17 K. A large transition temperature compared with the experimental Tc=0.3 K is assumed for the accuracy of numerical calculation. Since we discuss theJ.Phys.Soc.Jpn. FullPaper Author Name 3 -1.5-1-0.5 0 0.5 1 1.5 2 2.5 3 -1.5 0 1.5Ej(k) kx 0 0.5 1 1.5 2 0 0.5 1 1.5 2 ky kx 0 0.5 1 1.5 2 0 0.5 1 1.5 2 ky kx(a)(b) (c) (d) n=0.05 n=0.2dxydyz /dzxdyz 0 0.05 0.1 0.15 0.2 0 0.2 0.4 0.6 0.8 1 α1, α2 n α1 α2 Fig. 1. (Color online) (a) Band structure of our model. We sho w the dis- persion relation Ej(k) fork=(kx,0). The origin is the chemical po- tentialµfor a carrier density n=0.15. (b) Carrier density dependence of spin-orbit coupling on the Fermi surface. We show α1=E2(kF2)− E1(kF2) (solid line) and α2=E4(kF4)−E3(kF4) (dashed line) with kFj being the Fermi wave number of the j-th band along the [100] axis. (c) and (d) show Fermi surfaces for n=0.05 and for n=0.2, re- spectively. Other parameters are assumed as ( t1,t2,t3,t4,t5,∆,todd,λ)= (1.0,1.0,0.05,0.4,0.1,2.45,0.25,0.2). normalized µBHc2/Tc,thefollowingresultsarehardlyaltered bythemagnitudeof Tc.Asweshowelsewhere,thesupercon- ductingphaseisalmostindependentofthe ratio U′/U. Figure 1(a) shows the band structure of our model. We see the spin splitting caused by the Rashba spin-orbit cou- pling. Because the Rashba spin-orbit coupling is enhanced around the band crossing points,31)the magnitude of spin splitting shows a nonmonotonic carrier density dependence . Figure 1(b) shows the spin splitting in the lowest pair of bands [α1=E2(kF2)−E1(kF2)] and that in the second low- est pair of bands [ α2=E4(kF4)−E3(kF4)] as a function of carrier density, where kFjis the Fermi wave number of the j-th band along the [100] axis. The nonmonotonic behavior of a spin splitting, α1, is consistent with experimental obser- vationsforSTO/LAOinterfaces.Theseeminglycontradictory carrierdensitydependence13,14)ofRashbaspin-orbitcoupling isprobablycausedbythepeakof α1,aspointedoutbyZhong et al.27)In our model, the Fermi level crosses the bottom of the second lowest pair of bands [ E4(0)=E3(0)=0] at ap- proximately n=0.16. The Fermi surfaces for n=0.05 and n=0.2 are shown in Figs. 1(c) and 1(d), respectively. The isotropic Fermi surfaces mainly consist of the d xyorbital for a low carrier density, n=0.05,while large anisotropicFermi surfaces mainly consist of the d yz/dzxorbitals for a large car- rierdensity, n=0.2. First, we discuss the superconducting state at zero mag- netic ﬁeld. While the superconductivity is mainly caused by the dxyorbital at low carrier densities, n<0.078,the intraor- bitalCooperpairingofd yzanddzxorbitalsisthemainsource of superconductivity at high carrier densities, n>0.078. Thiscrossoverofthesuperconductingstatecoincideswith thechange of quasiparticles on the Fermi surfaces discussed fo r Figs. 1(c) and 1(d). When we assume the attractive interac- tionsU=U′independent of carrier density, the transition temperature monotonically increases with increasing carr ier density. The nonmonotonic carrier density dependence ob- servedinexperiments11)isreproducedbyassumingadecreas- ing function of U=U′against carrier density. In this study, We avoid such a phenomenological assumption and discuss the normalized values such as µBHc2/Tc. Note that the odd- parity hybridization toddand LS coupling λhardly aﬀect the superconductingstate atzeromagneticﬁeld. 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1µBHc2/TC(a) n=0.06 T/TCn=0.1 n=0.12 n=0.15 n=0.2 0 2 4 6 8 10 12 14 0.05 0.1 0.15 0.2 0.25µBHc2/Tc n(todd,λ)=(0.25,0.2)(b) (todd,λ)=(0.0,0.2) (todd,λ)=(0.25,0.0) (todd,λ)=(0.0,0.0) Fig. 2. (Coloronline)(a)Normalizeduppercriticalﬁeld, µBHc2/Tc,forthe ﬁeld parallel to the [100] axis. Solid, dashed, and dash-dot ted lines show the results for high carrier densities, n=0.12, 0.15, and 0.2, respectively. The dotted line is obtained in the crossover region, n=0.1, while dash- two-dotted line assumes a low carrier density, n=0.06. Fermi surfaces mainly consist of the d xyorbital (d yz/dzxorbitals) in the low (high) carrier density region. The other parameters are the same as those in Fig. 1. (b) Carrierdensity dependence of µBHc2/Tcatthelowesttemperature T/Tc= 0.05 [circles]. We also show the results for ( todd,λ)=(0.25,0) [squares], (0,0.2)[diamonds], and (0 ,0) [pluses] for comparison. On the other hand, the Rashba spin-orbit coupling aris- ing from the combination of toddandλleads to an intrigu- ing superconducting phase in the magnetic ﬁeld. Figure 2(a) shows the phase diagram against temperatures and magnetic ﬁelds for various carrier densities. We see an extraordinar ily large normalized upper critical ﬁeld, µBHc2/Tc>9, beyond the Pauli-Clogston-Chandrasekar limit, µBHc2/Tc=1.25,37) aroundn=0.12−0.15.Ithasbeenshownthattheuppercriti- calﬁeldisenhancedbytheRashbaspin-orbitcoupling,38)but that the enhancement is minor in the canonical Rashba-type non-centrosymmetric superconductors as µBHc2/Tc≈2.39) We here ﬁnd that the rather large enhancement of the upper4 J.Phys.Soc. Jpn. FullPaper Author Name criticalﬁeldiscausedbythesynergyoftheRashbaspin-orb it couplingand the orbitaldegreeof freedom.Indeed,whenwe decrease the carrier density to n<0.08,the orbital degree of freedom is quenched and the upper critical ﬁeld is suddenly decreased. As shown in Fig. 2(b), the normalized upper critical ﬁeld µBHc2/Tcshowsa broadpeak at approximately n=0.12and decreases with increasing carrier density for n>0.12 ex- cept for a sharp enhancement at around n=0.16. The de- crease inµBHc2/Tcis attributed to the decrease in Rashba spin-orbit coupling [see Fig. 1(b)]. A sharp peak at around n=0.16 is induced by the appearance of small Fermi sur- faces around theΓpoint, that is, the Lifshitz transition. Be- causetheg-factorofthisbandvanishesat k=(0,0)(Γpoint) inthepresenceofatomicLScoupling λ,Cooperpairinginthe small Fermi surfaces is not disturbed by the magnetic ﬁeld. Thus, a sharp enhancement of the normalized upper critical ﬁeld,µBHc2/Tc, is a signature of the Lifshitz transition. It willbeinterestingtolookforthisLifshitztransitionsin cethe Class D topological superconducting phase is realized near the Lifshitz transition by applying a magnetic ﬁeld.40)Since the renormalization of the g-factor is not due to the broken inversion symmetry, a sharp peak of µBHc2/Tcalso appears for (todd,λ)=(0,0.2) [diamonds in Fig. 2(b)], for which the Lifshitz transition occurs at approximately n=0.1. Aside from this peak, a small upper critical ﬁeld below the Pauli- Clogston-Chandrasekarlimitisobtainedwheneither toddorλ is zero, because the Rashba spin-orbit coupling vanishes. A s expected, the normalized upper critical ﬁeld increases as w e increasetoddorλ. For instance, we obtain µBHc2/Tc∼4.9 for (todd,λ)=(0.1,0.2) andn=0.12, in agreement with the experimentalresultofSTO /LAOinterfaces.13) 0 0.2 0.4 0.6 0.8 1 0 0.4 0.8 1.2 µΒH/TC(1,1) (2,2)(a)\|ψmn\| 0 0.2 0.4 0.6 0.8 1 0 0.4 0.8 1.2 1.6 µΒH/TC(b)\|ψmn\| 0 0.2 0.4 0.6 0.8 1 0 0.4 0.8 1.2 µΒH/TC(c)\|ψmn\| 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 8 9 µΒH/TC(d)\|ψmn\|(1,2) (1,3) (2,3) (3,3) Fig. 3. (Color online) Magnetic ﬁeld dependence of order par ameters for n=0.15.Weshowtheamplitude \|ψmn\|atthetransition temperature, which is proportional to the order parameters \|∆mn\|belowTc. The main compo- nents are\|ψ11\|(thick solid line) and \|ψ22\|(thick dashed line). The other smallcomponentsareshownbythethinlines,asdescribedin Fig.3(d).We assume (todd,λ)=(a) (0,0), (b) (0,0.2), (c) (0.25,0), and (d) (0 .25,0.2). Theother parameters are the sameas those in Fig. 2. Inordertoclarifytherolesoftheorbitaldegreeoffreedom , we show the magnetic ﬁeld dependence of order parametersfor a high carrier density, n=0.15. When both odd parity hybridization, todd, and LS coupling, λ, are ﬁnite [Fig. 3(d)], the magneticﬁeld alongthe x-axissubstantiallyenhancesthe Cooperpairsofthed yzorbitalrepresentedby \|ψ11\|whilethose of the d zxorbital (\|ψ22\|) are suppressed. This means that a quasi-one-dimensional superconducting state dominated b y the dyzorbital is stabilized in the magnetic ﬁeld. Since this high-ﬁeldsuperconductingphaseisrobustagainstthepara m- agneticdepairinge ﬀect,alargeuppercriticalﬁeldisobtained, asshowninFig.2.ItshouldbestressedthattheRashbaspin- orbit couplingplays an essential role in stabilizing the qu asi- one-dimensionalsuperconductingphase. Indeed,we obtain a nearlyisotropictwo-dimensionalsuperconductingphasew ith \|ψ11\|∼\|ψ22\|when either the odd parity hybridization toddor theLS coupling λiszero[Figs.3(a)-3(c)]. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4ky kx Fig. 4. (Color online) g-vector of the lowest band ( l=1),g1(k), which is deﬁned in eq. (18). Arrows show the direction of the g-vector ; the length ofarrowsisproportional to theamplitude oftheg-vector. S olid lines show theFermisurfacesfor n=0.15.Theotherparametersarethesameasthose in Fig. 1. We here illustrate why the quasi-one-dimensional super- conducting phase is protected against the paramagnetic de- pairing eﬀect. For this purpose, we derive the Rashba spin- orbit coupling in the band basis as we have performed in ref. 29. We reduce the single-particle Hamiltonian H0to the three-bandmodelas H0=Hband+HASOC, where Hband=3/summationdisplay l=1/summationdisplay ksξl(k)a† k,lsak,ls, (16) HASOC=3/summationdisplay l=1/summationdisplay kgl(k)·Sl(k), (17) andξl(k)=(E2l(k)+E2l−1(k))/2. The Rashba spin-orbit couplingofthe l-thbandisrepresentedbytheg-vector gl(k)=(E2l(k)−E2l−1(k))Sav 2l(k)/\|Sav 2l(k)\|,(18) whose direction is obtained by calculating the average Sav j(k)=/angbracketleft/summationtext m/summationtext ss′σss′c† k,msck,ms′/angbracketrightjfor thej-th eigenstate. Figure 4 shows the g-vector g1(k) and the Fermi surfaces forn=0.15. It is shown that the momentum dependence of the g-vector is quite di ﬀerent from an often-assumed form, g(k)=α/parenleftBig sinky,−sinkx,0/parenrightBig . This is the characteristic prop- erty of orbitally degenerate non-centrosymmetricsystems .29)J.Phys.Soc.Jpn. FullPaper Author Name 5 In the case of STO heterostructures, the g-vector is nearly parallel to the y-axis for kx>ky, while it is almost along thex-axis for kx<ky. The quasiparticles mainly consist of the dyzorbital (d zxorbital) for the former (latter). Since the Cooperpairingisdisturbedbytheparamagneticdepairinge f- fect when the g-vector is parallel to the magnetic ﬁeld, the ﬁeld along the x-axis suppresses the Cooper pairs of the d zx orbital.Ontheotherhand,theCooperpairsformedbythed yz orbital are protected by the g-vector nearly perpendicular to themagneticﬁeld.Inthisway,thequasi-one-dimensionals u- perconductingphaseisstabilizedbytheorbitaldegreeoff ree- dom so as to avoid the paramagneticdepairinge ﬀect. This is anintuitiveexplanationforthelargeuppercriticalﬁelds hown inFig. 2. Finally, we discuss experimental results of the supercon- ducting phase in STO /LAO interfaces. The superconducting transitiontemperatureshowsanon-monotoniccarrierdens ity dependence11)and its peak at around n=2×1013cm−2co- incides with the crossover from d xy-orbital-dominated Fermi surfaces to d yz/dzx-orbital-dominated Fermi surfaces.19)The Rashba spin-orbit coupling seems to have the maximum am- plitudeinthecrossoverregion,13,14)consistentwiththethree- orbital tight-binding model adopted in this study. Interes t- ingly, a large upper critical ﬁeld, µBHc2/Tc≈4.2, beyond thePauli-Clogston-Chandrasekarlimithasbeenreportedf ora highcarrierdensity n=3×1013cm−2closetothecrossover.13) The decrease in the normalized upper critical ﬁeld µBHc2/Tc with increasing carrier density was also observed for n> 3×1013cm−2.13)Thesebehaviorsareconsistentwithourﬁnd- ing in Fig. 2, although the signature of Lifshitz transition has not been found. This agreement with experimental re- sults indicates that the quasi-one-dimensional supercond uct- ing phase is realized in the STO /LAO interfaces with high carrier densities. In contrast to the theoretical proposal for a helical superconducting phase with a ﬁnite total momen- tumofCooperpairs,15)alargeuppercriticalﬁeldisattributed to the entanglement of orbitals and spins in our three-orbit al model. Indeed, we conﬁrmed that the ﬁnite total momentum of Cooper pairs, namely, the ﬁnite qin the T-matrix, hardly changesourresults.Thecoexistenceofsuperconductivity and ferromagnetism7–10)may also be attributed to the quasi-one- dimensionalsuperconductingphaseprotectedagainstspin po- larization. We would like to stress that such a spin-polariz ed superconducting state is hardly stabilized in the multiban d models,41,42)which phenomenologically assume the Rashba spin-orbitcouplingandneglecttheorbitaldegreeoffreed om. Our proposal for the quasi-one-dimensionalsuperconducti ng phase can be veriﬁed by experiments using a tilted magnetic ﬁeld. For instance, a vortex lattice structure elongated al ong the[010]axiswillbeobservedintheﬁeldslightlytiltedfr om the [100] axis to the [001] axis. As for a quantitative discus - sion, the crossover between low and high carrier density re- gions occurs in our model at around n=0.08, which cor- responds to a carrier density of n=5×1013cm−2. This is in reasonable agreement with experimental carrier density of n=2×1013cm−2,19)anda discrepancyprobablyarisesfrom our inexact choice of tight-binding parameters. Note that a large upper critical ﬁeld has been observed in δ-doped STO thin ﬁlms.43)Although the global inversion symmetry is not broken in this system, surface Rashba spin-orbit couplings play a similar role to the spin-orbit coupling in this study, asdemonstrated for locally non-centrosymmetric supercondu c- tors.44) In summary, we studied the superconductivity in the two- dimensional electron systems formed at the STO /LAO inter- face and STO surface. We analyzed the three-orbital model taking into account t2gorbitals of Ti ions, and found that an unconventional structure of Rashba spin-orbit coupling arisesfromtheorbitaldegeneracyandprotectsthequasi-o ne- dimensionalsuperconductingphaseagainstthe paramagnet ic depairing eﬀect. The orbital degree of freedom plays an es- sential role in the response to the magnetic ﬁeld and leads to alargeuppercriticalﬁeld.Thepeakoftheuppercriticalﬁe ld as a function of carrier density coincides with the crossove r from d xy-orbital-dominatedFermi surfaces to d yz/dzx-orbital- dominatedFermi surfaces. These observationsprovidea sys - tematic understanding of superconducting properties at th e STO/LAOinterface. The authors are grateful to S. Fujimoto and T. Shishido for fruitful discussions. This work was supported by KAK- ENHI(GrantNos.25103711,24740230,and23102709),and by a Grant for the Promotion of Niigata University Research Projects. 1) A.Ohtomo and H.Y. Hwang: Nature 427(2004) 423. 2) J. Biscaras, N. Bergeal, A. Kushwaha, T. Wolf, A. Rastogi, R.C. Bud- hani, and J.Lesueur: Nat. Commun. 1(2010) 89. 3) K. Ueno, S. Nakamura, H. Shimotani, A. Ohtomo, N. Kimura, T . No- jima, H.Aoki, Y.Iwasa, and M. Kawasaki: Nat. Mater. 7(2008) 855. 4) Y. Kozuka, M. Kim, C. Bell, B. G. Kim, Y. Hikita, and H. Y. Hwa ng: Nature462(2009) 487. 5) N. Reyren, S. Thiel, A. D. Caviglia, L. Fitting Kourkoutis , G. Ham- merl, C.Richter, C.W.Schneider, T.Kopp,A.-S.R¨ uetschi, D.Jaccard, M. Gabay, D.A. Muller, J.-M.Triscone, and J. Mannhart: Scie nce317 (2007) 1196. 6) A. Brinkman, M. Huijben, M. van Zalk, J. Huijben, U. Zeitle r, J. C. Maan, W. G. van der Wiel, G. Rijnders, D. H. A. Blank, and H. Hilgenkamp: Nat. Mater. 6(2007) 493. 7) D.A.Dikin, M.Mehta, C.W.Bark,C.M.Folkman, C.B.Eom,an d V. Chandrasekhar: Phys.Rev. Lett. 107(2011) 056802. 8) L. Li, C. Richter, J. Mannhart, and R. C. Ashoori: Nat. Phys .7(2011) 762. 9) J.A.Bert, B.Kalisky, C.Bell, M.Kim,Y.Hikita, H.Y.Hwan g, and K. A.Moler: Nat. Phys. 7(2011) 767. 10) Ariando, X. Wang, G. Baskaran, Z. Q. Liu, J. Huijben, J. B. Yi, A. Annadi, A. Roy Barman, A. Rusydi, S. Dhar, Y. P. Feng, J. Ding, H. Hilgenkamp, and T.Venkatesan: Nat. Commun. 2(2011) 188. 11) A. D. Caviglia, S. Gariglio, N. Reyren, D. Jaccard, T. Sch neider, M. Gabay, S. Thiel, G.Hammerl, J.Mannhart, and J.-M.Triscone : Nature 456(2008) 624. 12) C. Bell, S. Harashima, Y. Kozuka, M. Kim, B. G. Kim, Y. Hiki ta, and H.Y. Hwang: Phys.Rev. Lett. 103(2009) 226802. 13) M. Ben Shalom, M. Sachs, D. Rakhmilevitch, A. Palevski, a nd Y. Da- gan: Phys.Rev. Lett. 104(2010) 126802. 14) A. D. Caviglia, M. Gabay, S. Gariglio, N. Reyren, C. Cance llieri, and J.-M.Triscone: Phys.Rev. Lett. 104(2010) 126803. 15) K. Michaeli, A. C. Potter, and P. A. Lee: Phys. Rev. Lett. 108(2012) 117003. 16)Non-Centrosymmetric Superconductors: Introduction and O verview, ed. by E.Bauer and M.Sigrist (Springer-Verlag, Berlin, 201 2). 17) S.Banerjee, O.Erten, and M. Randeria: arXiv:1303.3275 . 18) A. F. Santander-Syro, O. Copie, T. Kondo, F. Fortuna, S. P ailh` es, R. Weht,X.G.Qiu,F.Bertran,A.Nicolaou,A.Taleb-Ibrahimi, P.LeF` evre, G.Herranz,M.Bibes,N.Reyren,Y.Apertet,P.Lecoeur,A.Ba rth´ el´ emy, and M.J.Rozenberg Nature 469(2011) 189. 19) A.Joshua,S.Pecker,J.Ruhman,E.Altman,andS.Ilani:N at.Commun. 3(2012) 1129. 20) Z.S.Popovi´ c,S.Satpathy,andR.M.Martin:Phys.Rev.L ett.101(2008)6 J.Phys.Soc. Jpn. FullPaper Author Name 256801. 21) R. Pentcheva and W.Pickett: Phys.Rev. B 78(2008) 205106. 22) P. Delugas, A. Filippetti, V. Fiorentini, D. I. Bilc, D. F ontaine, and P. Ghosez: Phys.Rev. Lett. 106(2011) 166807. 23) G.Khalsa and A.H.MacDonald: Phys.Rev. B 86(2012) 125121. 24) Y. Mizohata, M. Ichioka, and K. Machida: Phys. Rev. B 87(2013) 014505. 25) R. M. Fernandes, J. T. Haraldsen, P. W¨ olﬂe, and A. V. Bala tsky: Phys. Rev. B87(2013) 014510. 26) M. Hirayama, T. Miyake, and M. Imada: J. Phys. Soc. Jpn. 81(2012) 084708. 27) Z.Zhong,A. T´ oth, and K.Held: Phys.Rev. B 87(2013) 161102. 28) G.Khalsa, B.Lee, and A.H.MacDonald: arXiv:1301.2784. 29) Y. Yanase: J. Phys. Soc. Jpn. 82(2013) 044711; Y. Yanase and H. Harima: Kotai-Butsuri 47(2012) No.3,1 [in Japanese]. 30) Y. Kim,R.M. Lutchyn, and C. Nayak: arXiv:1304.0464. 31) Y. Yanase and M.Sigrist: J.Phys. Soc.Jpn. 77(2008) 124711. 32) M.Nagano,A.Kodama,T.Shishidou,andT.Oguchi:J.Phys :Condens. Matter21(2009) 064239. 33) G.Binnig, A.Barato ﬀ, H.E.Hoenig,J.G.Bednorz: Phys.Rev.Lett. 45 (1980) 1352.34) K. Yada, S. Onari, Y. Tanaka, and J. Inoue: Phys. Rev. B 80(2009) 140509. 35) J.A.Bert, K.C.Nowack, B.Kalisky,H.Noad,J.R.Kirtley , C.Bell, H. K. Sato, M. Hosoda, Y. Hikita1, H. Y. Hwang, and K. A. Moler: Ph ys. Rev. B86(2012) 060503(R). 36) K.Aoyama and M.Sigrist: Phys.Rev. Lett. 109(2012) 237007. 37) B.S.Chandrasekhar:Appl.Phys.Lett. 1(1962)7;A.M.Clogston:Phys. Rev. Lett. 9(1962) 266. 38) P.A.Frigeri, D.F.Agterberg, A.Koga,and M.Sigrist: Ph ys.Rev.Lett. 92(2004) 097001. 39) Y. Yanase and M.Sigrist: J.Phys.Soc. Jpn. 76(2007) 124709. 40) M. Sato, Y. Takahashi, and S. Fujimoto: Phys. Rev. Lett. 103(2009) 020401. 41) M.H.Fischer,S.Raghu,andE.-A.Kim:NewJ.Phys. 15(2013)023022. 42) S. Caprara, F. Peronaci, and M. Grilli: Phys. Rev. Lett. 109(2012) 196401. 43) M. Kim,Y. Kozuka, C. Bell, Y. Hikita, and H.Y. Hwang: Phys . Rev. B 86(2012) 085121. 44) D. Maruyama, M. Sigrist, and Y. Yanase: J. Phys. Soc. Jpn. 81(2012) 034702.
2001.08794v1.Bright_solitons_in_a_spin_tensor_momentum_coupled_Bose_Einstein_condensate.pdf	arXiv:2001.08794v1 [cond-mat.quant-gas] 23 Jan 2020Bright solitons in a spin-tensor-momentum-coupled Bose-E instein condensate Jie Sun,1Yuanyuan Chen,1,∗Xi Chen,1,2,†and Yongping Zhang1,‡ 1International Center of Quantum Artiﬁcial Intelligence fo r Science and Technology (QuArtist) and Department of Physics, Shanghai University , Shanghai 200444, China 2Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain Synthetic spin-tensor-momentum coupling has recently bee n proposed to realize in atomic Bose- Einstein condensates. Here we study bright solitons in Bose -Einstein condensates with spin- tensor-momentum coupling and spin-orbit coupling. The pro perties and dynamics of spin-tensor- momentum-coupled and spin-orbit-coupled bright solitons are identiﬁed to be diﬀerent. We con- tribute the diﬀerence to the diﬀerent symmetries. I. INTRODUCTION In ultracold neutral atoms, hyperﬁne spin states, cou- pling to linear momentum [ 1–7] or orbital angular mo- mentum [ 8,9], are interesting and signiﬁcant not only in fundamental phenomena of ultracold atoms and con- densed matter physics, but also in the applications in quantum information processing, atom metrology and atomitronics, with the current experimental progress. Particularly, the spin-orbit coupling (SOC) provides the unique dispersion relationship, exhibiting particular fea- tures without analogues in the case of without the SOC. Thecompetitionbetweenatomicmany-bodyinteractions and the dispersion relation generates many fundamental ground state phases [ 10–17] and exotic collective excita- tions [18,19] in spin-orbit-coupled Bose-Einsteinconden- sates (BECs). The interplay between the nonlinearity stemming from atomic interactions and dispersions also gives rise to the existence of bright solitons which are spatially localized states. The interested spin-orbit-coupled dispersions in- evitably change the existence and properties of bright solitons [ 20,21]. In general, solitons follow the symme- tries of spin-orbit-coupled Hamiltonian, which provides a deep insight into the searching of solitons. Moreover, the dynamics of solitons is always accompanied by rich spin dynamics [ 22,23]. The lack of Galilean invariancein spin-orbit-coupled systems [ 24] makes that it is nontriv- ial to ﬁnd movable solitons, one can not directly obtain a movable soliton from its stationary correspondence. Dif- ferent aspects of bright solitons with the SOC have been investigated a lot [ 25–30], ranging from with long-ranged dipole interactions [ 31–33] to in optical lattices [ 34–40]. Very recently, the generation of artiﬁcial spin-tensor- momentum coupling (STMC) into an atomic BEC has been proposed [ 41]. Diﬀerent from the usual spin-orbit coupling where linear momentum is coupled with spin vectors, STMC is the interaction between linear momen- tum and spin tensors. Such emergent interaction can ∗cyyuan@staﬀ.shu.edu.cn †xchen@shu.edu.cn ‡yongping11@t.shu.edu.cnbe applicable to the discovery of exotic topological mat- ters [42,43]. In this paper, we investigate bright solitons in STMC BECs in which the three components of the ground hy- peﬁne states of87Rb are utilized for experimental im- plementation. We ﬁrst apply imaginary-time evolution method to study the stationary properties of STMC soli- ton, and further explore the dynamics by using varia- tional method. By comparing with SOC bright soliton in Refs. [ 44–46], we conclude that the diﬀerence between STMC and SOC bright solitons originates from the dif- ferent symmetry relevant to spin rotation. The paper will be organized as follows. In Sec. IIthe systems and Hamiltonian are introduced for SOC and STMC. Here we present both for completeness and fur- ther comparison. Later, the bright solitons are discussed for both STMC and SOC BECs in Sec. III, to clarify the diﬀerence in the spin rotation and symmetry. Finally, conclusion are made in Sec. IV. II. MODEL AND HAMILTONIAN We ﬁrst consider the experiment of synthetic SOC in three-component BECs [ 47,48], where the three hyper- ﬁne states of87Rb atoms are utilized, with the energy splitting by a bias magnetic ﬁeld, as shown in Fig. 1(a,b). To realize SOC, the atoms are dressed by two counter- propagating Raman laser beams, and the polarizations of lasers are arranged so that two-photon optical transi- tions can be induced, see Fig. 1(b). The transitions in the basis of ( \| ↑∝angbracketright=\|1,−1∝angbracketright,\|0∝angbracketright=\|1,0∝angbracketright,\| ↓∝angbracketright=\|1,1∝angbracketright) are engineered as, HSOC Ram= Ω 0e−i2kRx0 ei2kRx0e−i2kRx 0ei2kRx0 , where Ω is the strength of two-photon Rabi coupling [ 49] andkRis the wavenumber of the Raman beams. Dur- ing the transitions, there is a momentum exchange be- tween the atoms and lasers. Including kinetic energy, the Hamiltonian becomes, HSOC=p2 x/2m+HSOC Ram,, with mbeing atomic mass and pxbeing momentum along the direction of Raman lasers. To explicitly show the existence of SOC, a unitary transformation is needed,2 FIG. 1. Experimental schemes to realize the spin-orbit cou- pling (a,b) and spin-tensor-momentum coupling (c,d). Thre e hyperﬁne states ( \| ↑/angbracketright,\|0/angbracketright,\| ↓/angbracketright) are split by a bias magnetic ﬁeldB0. In (a,b) two laser beams propagate oppositely to couple\|px−2/planckover2pi1kR,↑/angbracketright,\|px,0/angbracketright,\|px+2/planckover2pi1kR,↓/angbracketrightwithpxbeing mo- mentum along laser direction and quasimomentum 2 /planckover2pi1kRrel- evant to the wavenumber of lasers, the quasimoentum diﬀer- ence between hyperﬁne states constitutes the spin-orbit co u- pling. In (c,d) two beams whoes polarizations are parallel t o the bias magnetic ﬁeld propagate along same direction and the third beams in the opposite direction. They can couple \|px−2/planckover2pi1kR,↑/angbracketright,\|px,0/angbracketright,\|px−2/planckover2pi1kR,↓/angbracketright. USOC=ei2kRxFz, such that the Hamiltonian ˜HSOC= USOCHSOCU−1 SOCbecomes ˜HSOC=p2 x 2m−4/planckover2pi1kRpxFz 2m+4(/planckover2pi1kR)2F2 z 2m+√ 2ΩFx.(1) Here (Fx,Fy,Fz) are spin-1 Pauli matrices, and the SOC 2/planckover2pi1kRpxFz/mis involved. Physically, the SOC means thatthereisaquasimomentumdiﬀerence −2/planckover2pi1kRbetween states\| ↑∝angbracketrightand\|0∝angbracketright, and between \|0∝angbracketrightand\| ↓∝angbracketright. Next, the STMC can be introduced artiﬁcially by dressing the atoms with three Raman beams [ 41], see Fig.1(c,d). Two of them with same linear polarization propagate along same direction, and the other propa- gates oppositely. The two-photon transitions accompa- nying momentum transfers become, HSTMC Ram= Ω 0e−i2kRx0 ei2kRx0ei2kRx 0e−i2kRx0 . Note that the diﬀerence between HSOC RamandHSTMC Ram is very slight. To eliminate the spatial dependence in HSTMC Ram, a unitary transformation USTMC=ei2kRxF2 zis performed, and the new total Hamiltonian ˜HSTMC= USTMCHSTMCU−1 STMC, withHSTMC=p2 x/2m+HSTMC Ram is expressed as, ˜HSTMC=p2 x 2m−4/planckover2pi1kRpxF2 z 2m+4(/planckover2pi1kR)2F2 z 2m+√ 2ΩFx.(2)The STMC takes a speciﬁc form as 2 /planckover2pi1kRpxF2 z/m. From the above equation, it is clear that such speciﬁc STMC is just a rearrangement of quasimomentum diﬀerence com- paring with the case of the SOC. The quasimomentum diﬀerence between \| ↑∝angbracketrightand\|0∝angbracketrightis−2/planckover2pi1kR, while it is 2 /planckover2pi1kR between \|0∝angbracketrightand\| ↓∝angbracketright. III. BRIGHT SOLITONS WITH STMC AND SOC Now, we arereadytostudy brightsolitonsin the BECs with both the STMC and SOC whose experimental re- alizations are analyzed in the previous section II. We start from the standard Gross-Pitaevskii (GP) equations and take into consideration the spin-tensor-momentum- coupled and spin-orbit-coupled Hamiltonian in Eq. ( 1) and Eq. ( 2). The dimensionless GP equations for spin- tensor-momentum-coupled BEC are, i∂Ψ ∂t= [−∂2 x+(4i∂x+4+∆)F2 z+√ 2ΩFx+Hint]Ψ,(3) while, the spin-orbit-coupled GP equations are i∂Ψ ∂t= [−∂2 x+4i∂xFz+(4+∆) F2 z+√ 2ΩFx+Hint]Ψ.(4) In the both equations, the units of energy, position co- ordinate and time that we adopt are /planckover2pi12k2 R/2m,1/kRand 2m//planckover2pi1k2 Rrespectively. The additional term ∆ F2 zorigi- nates from quadratic Zeeman eﬀect. Three-component wave functions are Ψ = (Ψ ↑,Ψ0,Ψ↓)T, for convenience, in the following, we relabel the wave functions as Ψ = (Ψ1,Ψ2,Ψ3)T. In above equations, Hint=g0(\|Ψ1\|2+ \|Ψ2\|2+\|Ψ3\|2), forsimplicity, weonlyconsiderinteractions having SU(3) symmetry. Since our aim is to investigate the bright solitons, we focus on attractive interactions of g0<0. The diﬀerence between the spin-tensor-momentum- coupled and spin-orbit-coupled GP equations is the ap- pearance of 4 i∂xF2 zand 4i∂xFz. Such diﬀerence leads to diﬀerent symmetries of GP equations, which aﬀects the properties of bright solitons. We ﬁnd stationary bright solitons by the numerical calculation of GP equa- tions using the imaginary-time evolution method, be- cause of which, the soliton solutions belong to ground states. Typical soliton proﬁles are demonstrated in Fig.2. The upper panel is the proﬁles of spin-tensor- momentum-coupled solitons, and the lower panel is that of spin-orbit-coupled solitons. For further comparison, we adopt same parameters for the GP equations with STMC and SOC. Our general observation is that the imaginarypartsof soliton wavefunctions for both STMC and SOC do not vanish. In contrast, the ground states of ordinary BECs (without STMC or SOC) are real- valued with no node in wave functions [ 12]. This is the unique feature of spin-orbit-coupled [ 21] and spin- tensor-momentum-coupled BECs. At ﬁrst sight, the3 /s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50 /s45/s48/s46/s48/s53/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48 /s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50 /s45/s49/s48 /s48 /s49/s48/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52 /s45/s49/s48 /s48 /s49/s48/s45/s48/s46/s49/s48/s45/s48/s46/s48/s53/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48 /s45/s49/s48 /s48 /s49/s48/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s97/s49 /s97/s50 /s97/s51 /s120/s98/s49 /s120/s82/s101/s97/s108/s73/s109/s97/s103/s98/s50 /s120/s89/s49/s89/s50/s89/s51 /s98/s51 /s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50 /s45/s48/s46/s48/s53/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48 /s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50 /s45/s49/s48 /s48 /s49/s48/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52 /s45/s49/s48 /s48 /s49/s48/s45/s48/s46/s49/s48/s45/s48/s46/s48/s53/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48 /s45/s49/s48 /s48 /s49/s48/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s99/s49 /s99/s50 /s99/s51 /s120/s100/s49 /s120/s100/s50/s51 /s120/s82/s101/s97/s108 /s73/s109/s97/s103/s89/s49/s89/s50/s89/s51 /s100/s51 FIG. 2. Proﬁles of the spin-tensor-momentum-coupled (uppe r panel) and spin-orbit-coupled (lower panel) bright solit ons. In each panel, the ﬁrst (second) row is the real (imaginary) par ts of soliton wave functions Ψ = (Ψ 1,Ψ2,Ψ3)T. Solid-lines are solutions from the imaginary-time evolution method and dot -lines are analytical solutions from the variational metho d. The dimensionless parameters are ∆ = −1,Ω = 0.5 andg0=−2. spin-tensor-momentum-coupled solitons share same pro- ﬁles with spin-orbit-coupled solitons, especially, the real parts of soliton wave functions are almost same. How- ever, there exists an apparent diﬀerence in the imaginary parts. Our solitons as ground states follow symmetries of the systems. The stationary spin-tensor-momentum-coupled GP equations in Eq. ( 3) have a spin rotating symmetry, RSTMC=eiπFx= 0 0−1 0−1 0 −1 0 0 ,(5) which rotates spins along the Fxaxis by the angle of π, and a joint parity symmetry, OSTMC=PK, (6) withPandKbeing the parity and complex conjugate operators. The symmetry RSTMCis relevant to the spin tensorF2 x, sinceF2 x=1 2(I−RSTMC). The eigen-equation isRSTMCΨ =±Ψ. For the +1 eigenstate, Ψ 2(x) = 0, which leads to ∝angbracketleftFx∝angbracketright= 0. Whereas, to minimize energy ofRabicouplingterm√ 2ΩFx, itispreferablethat ∝angbracketleftFx∝angbracketright<0. Therefore, bright solitons select the eigenstate with −1 eigenvalue, RSTMCΨ =−Ψ, the consequence of which is Ψ1(x) = Ψ3(x). Fig.2demonstratesΨ 1(x) = Ψ3(x)from the real and imaginary parts. The symmetry OSTMC determines that the parity of real parts of soliton wave functions Ψ 1,Ψ2and Ψ 3should be opposite to that of imaginary parts. The real parts are even and imaginary parts are odd, see Fig. 2. The symmetry of the stationaryspin-orbit-coupledGP equations in Eq. ( 4) is slightly diﬀerent from the case of the STMC. The spin-orbit-coupled equations possess a particular spin rotating symmetry, RSOC=PeiπFx=P 0 0−1 0−1 0 −1 0 0 ,(7) which must be the joint of spin rotation and parity. The equations also have the symmetry PKwhich is same as the spin-tensor-momentum-coupled case, so the parity of real and imaginary parts of spin-orbit-coupled solitons are even and odd respectively, which can be conﬁrmed4 /s45/s52 /s48 /s52/s48/s46/s48/s49/s46/s48/s50/s46/s48 /s45/s52 /s48 /s52/s48/s46/s48/s49/s46/s48/s50/s46/s48 /s45/s52 /s48 /s52/s48/s46/s53/s49/s46/s48/s49/s46/s53 /s45/s52 /s48 /s52/s45/s52/s46/s48/s45/s50/s46/s48/s48/s46/s48 /s48 /s49 /s50/s49/s46/s56/s50/s46/s48 /s48 /s49 /s50/s49/s46/s54/s49/s46/s56/s50/s46/s48 /s48 /s49 /s50/s48/s46/s53/s48/s46/s54 /s48 /s49 /s50/s45/s52/s46/s48/s45/s51/s46/s48/s68/s97/s49 /s68/s107 /s49 /s107 /s49/s107 /s50 /s107 /s50/s115 /s115/s69 /s69/s97/s50 /s68/s97/s51 /s68/s97/s52 /s87/s98/s49 /s87/s98/s50 /s87/s98/s51 /s87/s98/s52 /s45/s52 /s48 /s52/s48/s46/s48/s49/s46/s48/s50/s46/s48 /s45/s52 /s48 /s52/s48/s46/s48/s49/s46/s48/s50/s46/s48 /s45/s52 /s48 /s52/s48/s46/s53/s49/s46/s48/s49/s46/s53 /s45/s52 /s48 /s52/s45/s52/s46/s48/s45/s50/s46/s48/s48/s46/s48 /s48 /s49 /s50/s49/s46/s56/s50/s46/s48 /s48 /s49 /s50/s49/s46/s54/s49/s46/s56/s50/s46/s48 /s48 /s49 /s50/s48/s46/s53/s48/s46/s54/s48/s46/s55 /s48 /s49 /s50/s45/s52/s46/s48/s45/s51/s46/s53/s45/s51/s46/s48/s68/s99/s49 /s68/s107 /s49 /s107 /s49/s107 /s50 /s107 /s50/s115 /s115/s69 /s69/s99/s50 /s68/s99/s51 /s68/s99/s52 /s87/s100/s49 /s87/s100/s50 /s87/s100/s51 /s87/s100/s52 FIG. 3. Features of the spin-tensor-momentum-coupled (upp er panel) and spin-orbit-coupled (lower panel) bright soli tons characterized from variational wave functions. The variat ional parameters k1,k2,σand total energy ESTMC,ESOCare a function of ∆ and Ω. Solid-lines are from the variational met hod and dots are from the imaginary-time evolution method. I n the ﬁrst (second) row of each panel, Ω = 1 (∆ = −3). The nonlinear coeﬃcient g0=−2. from Fig. 2. The eigen-equation of RSOCisRSOCΨ(x) = ±Ψ(x), taking into account the parity of real and imag- inary parts of wave functions, solitons choose the eigen state with −1 eigenvalue, if they choose the state with +1 eigenvalue, then ∝angbracketleftFx∝angbracketright= 0, the Rabi coupling en- ergy can not be minimized. With −1 eigenvalue, the symmetry RSOCrequires that Ψ 1(x) = Ψ 3(−x) and Ψ2(x) = Ψ 2(−x). Finally, because of the parity from PK, the real parts of Ψ 1(x) and Ψ 3(x) become equal and the imaginary parts of Ψ 1(x) and Ψ 3(x) have opposite signs, while the imaginary part of Ψ 2(x) must disappear. The above symmetry analysis provides a deep insight into the understanding of solitons. So, we are motivated to apply a variational function to stimulate correspond- ing solitons as follows. For the spin-tensor-momentum-coupled soliton, the variational wave function is, ΨSTMC= A[cos(k1x)+iρ0sin(k1x)] B[cos(k2x)+iρ1sin(k2x)] A[cos(k1x)+iρ0sin(k1x)] sech(σx).(8) This trial wave function completely satisﬁes the sym- metries of RSTMCandOSTMC. Variational parameters A,B,k 1,k2,ρ0,ρ1andσwould be determined by the minimization of the total energy ESTMC=/integraltext dx(E0+ ¯ESTMC), with the energy density, E0=\|∂xΨ1\|2+\|∂xΨ2\|2+\|∂xΨ3\|2+(∆+4)( \|Ψ1\|2 +\|Ψ3\|2)+Ω(Ψ 1Ψ∗ 2+Ψ∗ 1Ψ2+Ψ2Ψ∗ 3+Ψ∗ 2Ψ3) +g0 2/parenleftbig \|Ψ1\|2+\|Ψ2\|2+\|Ψ3\|2/parenrightbig2, (9) and ¯ESTMC= 4i(Ψ∗ 1∂xΨ1+Ψ∗ 3∂xΨ3).(10)5 /s48/s53/s48/s49/s48/s48/s48/s50/s48/s52/s48/s54/s48 /s48/s46/s48/s48/s46/s50/s48/s46/s52 /s116 /s124/s89 /s124/s50 /s120/s97 /s45/s50/s48/s48 /s45/s49/s48/s48 /s48 /s49/s48/s48 /s50/s48/s48/s48/s50/s48/s52/s48/s54/s48 /s48/s46/s48/s48/s46/s50/s48/s46/s52 /s116 /s124/s89 /s124/s50 /s120/s98 FIG. 4. The time evolution of an initial spin-tensor-moment um-coupled (a) and spin-orbit-coupled (b) solitons after s witching oﬀ the spin-tensor-momentum coupling and spin-orbit coupl ing respectively, \|Ψ\|2=\|Ψ1\|2+\|Ψ2\|2+\|Ψ3\|2. The parameters are ∆ =−1,Ω = 0.5 andg0=−2. Considering the symmetries of RSOCandPK, the vari- ational wave function for a spin-orbit-coupled soliton might be, ΨSOC= A[cos(k1x)+iρ0sin(k1x)] Bcos(k2x) A[cos(k1x)−iρ0sin(k1x)] sech(σx).(11) All unknown quantities appearing in above function should be determined by the minimization of the energy ESOC=/integraltext dx(E0+¯ESOC), here the spin-orbit-coupled energy density is, ¯ESOC= 4i(Ψ∗ 1∂xΨ1−Ψ∗ 3∂xΨ3).(12) The results from variational approximation approch for both spin-tensor-momentum-coupled and spin-orbit- coupled solitons are shown by dot-lines in Fig. 2. Obvi- ously, the variational wave functions are consistent with the results from the imaginary-timeevolution method, as discussed before. We characterizethe propertiesof brightsolitons by the variational wavefunctions. The features are identiﬁed by the dependence of k1,k2,σand the total energy ESTMC andESOCon the variables of ∆ and Ω. The results are describedin Fig. 3. The magnitudesof k1andk2arerele- vanttothenumberofnodesinsolitonproﬁles. Thelarger k1andk2induce more oscillations in real and imaginary parts of soliton wave functions (see Fig. 2). This type of oscillation is the exotic properties of STMC (4 i∂xF2 z) and SOC (4 i∂xFz). Because of the competition between 4i∂xF2 z(4i∂xFz) and (∆+4) F2 zor√ 2ΩFx, large ∆ and Ω suppress the eﬀect of the STMC and SOC, thus reduc- ing the oscillation nodes. As a result, k1ork2decreases with increasing ∆ or Ω. This somehow explains the ten- dency of lines in Fig. 3(a1-d2). Besides, the modiﬁcation ofk1andk2, the Rabi coupling√ 2ΩFxalso makes soli-ton wave packets more spatially localized to reduce os- cillations. Finally, as shown in Fig. 3(b3,d3), σincreases when increasing Ω. However, the dependence of σon ∆ is not monotonous at all (see Fig. 3(a3,c3)), resulting in two obvious slopes in the total energy as a function of ∆ in Fig. 3(a4,c4). Fig. 3(b4,d4) demonstrates that Ω always reduces the total energy, due to the fact that the Rabi coupling energy is proportional to ∝angbracketleftFx∝angbracketright, satisfying ∝angbracketleftFx∝angbracketright<0. Next, we turn to the dynamics of spin-tensor- momentum-coupled and spin-orbit-coupled solitons. Two diﬀerent kinds of dynamics are presented as follows. Firstofall,thequenchdynamicsisshowninFig. 4, where the initial soliton states are evolved after switching oﬀ the STMC or SOC, by solving the real time evolution of Eqs. ( 3) or Eq. ( 4) but without the STMC (4 i∂xF2 z term) or the SOC (4 i∂xFzterm). Fig. 4(a) and (b) cor- respond to the spin-tensor-momentum-coupled and the spin-orbit-coupled solitons, respectively. After switch- ing oﬀ the STMC or SOC, solitons are not stationary. This provides clear evidence that the solitons are intrin- sically supported by the STMC or SOC. Interestingly, the time evolution of the spin-tensor-momentum-coupled and spin-orbit-coupled solitons are much diﬀerent. The spin-tensor-momentum-coupled soliton moves along one direction, while the spin-orbit-coupled soliton splits into two parts with opposite velocities. This is because that the initial soliton satisﬁes k1=k2,ρ0=ρ1= 1. There- fore, the spin-tensor-momentum-coupled soliton is the spatial conﬁnement of a plane wave, after tuning oﬀ the STMC, it moves in the direction of the plane wave. While, the spin-orbit-coupled solion includes two plane- wave modes due to the component Ψ 2∝cos(k2x) = (eik2x+e−ik2x)/2. The Rabi coupling transfers these two plane-wave modes into other components, leading to the splitting of two branches during the evolution.6 /s48 /s49/s48/s48 /s51/s48/s48 /s51/s53/s48 /s55/s53/s48 /s49/s51/s53/s48 /s50/s49/s53/s48 /s51/s49/s50/s48/s48/s53/s48/s48/s49/s48/s48/s48/s49/s53/s48/s48/s50/s48/s48/s48 /s120/s116/s97 /s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s53/s48/s48 /s55/s48/s48 /s49/s49/s48/s48 /s49/s50/s48/s48/s48/s53/s48/s48/s49/s48/s48/s48/s49/s53/s48/s48/s50/s48/s48/s48 /s116 /s120/s98 FIG. 5. The time evolution of an initial spin-tensor-moment um-coupled (a) and spin-orbit-coupled (b) initiated by a co nstant weak acceleration force which is implemented by adding a lin ear potential −0.001xinto GP equations in Eq. ( 3) and Eq. ( 4).The parameters are ∆ = −1,Ω = 0.5 andg0=−2. Secondly, we shall explore the acceleration of bright solitons. We add a constant weak force to accelerate the initially prepared soliton. The slow adiabatic ac- celeration connects moving bright solitons to station- ary bright solitons [ 21]. Due to the lack of Galilean invariance in spin-tensor-momentum-coupled and spin- orbit-coupled systems, the proﬁles of moving solitons be- comes diﬀerent from these of stationary solitons, there- fore, they are changed during the acceleration, as illus- trated in Fig. 5. The change of the spin-orbit-coupled soliton is more pronounced than that of the spin-tensor- momentum-coupled soliton (see Fig. 5). We provide a simple insight into the understanding of such diﬀerence. The moving bright soliton solutions should be Ψ(x,t) = Φv(x−2vt,t)eivx−iv2t,(13) withvbeing moving velocity. Substituting this ansatz into GP equations in Eq. ( 3) and Eq. ( 4), the resulted equations for Φ v(x−2vt,t) are diﬀerent from the orig- inal ones by additional appearing of −4vF2 zand−4vFz respectively for the spin-tensor-momentum-coupled and spin-orbit-coupled equations. The additional −4vF2 z does not have an eﬀect on the symmetry RSTMC, so the moving spin-tensor-momentum-coupled bright soli- tons still possess RSTMC. In contrast, −4vFzfor the spin-orbit-coupled solitons breaks the symmetry RSOC. The constant acceleration force linearly increases the ve- locities of solitons. However, the symmetry RSTMCman- ages to protect the proﬁles of bright soliton, by avoiding to dramatic change. The initial stationary spin-orbit-coupled bright soliton changes distinctly during the ac- celeration, since the symmetry of the stationary one is so diﬀerent from that of the moving one. IV. CONCLUSION We systematically study bright solitons in three- component BECs with the spin-tensor-momentum cou- pling and spin-orbit coupling, motivated by the rapid development of the research ﬁeld of spin-orbit-coupled ultracoldatomic gasesand by the recent proposalto real- ize the spin-tensor-momentum-coupled BEC. The slight diﬀerence between the STMC and SOC leads to various symmetries,whichgivesrisetodiﬀerentproﬁlesofsoliton wave functions. Moreover, the dynamics of spin-tensor- momentum-coupled and spin-orbit-coupled solitons are diﬀerent during the time-evolution, when they are ini- tiated by switching oﬀ the couplings or by a constant weak acceleration force. We conclude that all diﬀerent properties comes from diﬀerent symmetries. ACKNOWLEDGEMENT We sincerely acknowlege Yong Xu, Biao Wu and Ray- Kuang Lee for helpful discussions. The work supported by the NSF of China (Grant Nos. 11974235, 1174219 and 11474193), the Thousand Young Talents Program of China, SMSTC (18010500400 and 18ZR1415500),7 and the Eastern Scholar and Shuguang (Program No. 17SG39) Program. XC also thanks the support fromRam´ on y Cajal program of the Spanish MINECO (RYC- 2017-22482). [1]Y. -J. Lin, K. Jim´ enez-Garc´ ıa, and I. B. Spielman, Spin- orbit-coupled Bose-Einstein condensates, Nature, 471, 83 (2011). [2]J.-Y. Zhang, S.-C. Ji, Z. Chen, L. Zhang, Z.-D. Du, B. Yan, G.-S. Pan, B. Zhao, Y.-J. Deng, H. Zhai, S. Chen, and J.-W. Pan, Collective dipole oscilla- tions of a spin-orbit coupled Bose-Einstein condensate, Phys. Rev. Lett. 109, 115301 (2012). [3]P. Wang, Z.-Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H. Zhai, and J. Zhang, Spin-orbit coupled degenerate Fermi gases, Phys. Rev. Lett. 109, 095301 (2012). [4]L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yef- sah, W. S. Bakr, and M. W. Zwierlein, Spin- injection spectroscopy of a spin-orbit coupled Fermi gas, Phys. Rev. Lett. 109, 095302 (2012). [5]C. Qu, C. Hamner, M. Gong, C. Zhang, and P. Engels, Observation of Zitterbewegung in a spin-orbit-coupled Bose-Einstein condensate, Phys. Rev. A 88, 021604(R) (2013). [6]A. J. Olson, S.-J. Wang, R. J. Niﬀenegger, C.-H. Li, C. H. Greene, and Y. P. Chen, Tunable Landau-Zener transitions in a spin-orbit-coupled Bose-Einstein conden - sate, Phys. Rev. A 90, 013616 (2014). [7]C. Hamner, C. Qu, Y. Zhang, J. J. Chang, M. Gong, C. Zhang, and P. Engels, Dicke-typephase transition in a spin-orbit-coupled Bose-Einstein condensate, Nat. Com- mun.5, 4023 (2014). [8]H.-R. Chen, K.-Y. Lin, P.-K. Chen, N.-C. Chiu, J.-B. Wang, C.-A. Chen, P.-P. Huang, S.-K. Yip, Y. Kawaguchi, and Y.-J. Lin, Phys. Rev. Lett. 121, 113204 (2018). Spin-orbital-angular-momentum coupled Bose-Einstein condensates. [9]D. Zhang, T. Gao, P. Zou, L. Kong, R. Li, X. Shen, X.-L. Chen, S.-G. Peng, M. Zhan, H. Pu, and K. Jiang, Phys. Rev. Lett. 122, 110402 (2019). Ground-state phase diagram of a spin-orbital-angular-momentum coupled Bose-Einstein condensate. [10]C. Wang, C. Gao, C.-M. Jian, and H. Zhai, Spin-orbit coupled spinor Bose-Einstein condensates, Phys. Rev. Lett. 105, 160403 (2010). [11]T.-L. Ho and S. Zhang, Bose-Einstein condensates with spin-orbit interaction, Phys. Rev. Lett. 107, 150403 (2011). [12]C.-J. Wu, I. Mondragon-Shem, and X.-F. Zhou, Uncon- ventional Bose-Einstein condensations from spin-orbit coupling, Chin. Phys. Lett. 28, 097102 (2011). [13]S. Sinha, R. Nath, and L. Santos, Trapped two- dimensional condensates with synthetic spin-orbit cou- pling, Phys. Rev. Lett. 107, 270401 (2011). [14]Y. Zhang, L. Mao, and C. Zhang, Mean-ﬁeld dy- namics of spin-orbit coupled Bose-Einstein condensates, Phys. Rev. Lett. 108, 035302 (2012). [15]H. Hu, B. Ramachandhran, H. Pu, and X.-J. Liu, Spin- orbit coupled weakly interacting Bose-Einstein conden- sates in harmonic traps, Phys. Rev. Lett. 108, 010402 (2012).[16]Y. Li, L. P. Pitaevskii, and S. Stringari, Quantum tricriticality and phase transitions in spin-orbit cou- pled Bose-Einstein condensates, Phys. Rev. Lett. 108, 225301(2012). [17]J. -R. Li, J. Lee, W. Huang, S. Burchesky, B. Shteynas, F. C ¸. Top, A. O. Jamison, and W. Ketterle, A stripe phase with supersolid properties in spin-orbit-coupled Bose-Einstein condensates, Nature 543, 91 (2017). [18]M. A. Khamehchi, Y. Zhang, C. Hamner, Th. Busch, and P. Engels, Measurement of collective excitations in a spin-orbit-coupled Bose-Einstein condensate, Phys. Rev. A90, 063624 (2014). [19]S. -C. Ji, L. Zhang, X. -T. Xu, Z. Wu, Y. Deng, S. Chen, and J. -W. Pan, Softening of roton and phonon modes in a Bose-Einstein condensate with spin-orbit coupling, Phys. Rev. Lett. 114, 105301 (2015). [20]V. Achilleos, D. J. Frantzeskakis, P. G. Kevrekidis, and D. E. Pelinovsky, Matter-wave bright soli- tons in spin-orbit coupled Bose-Einstein condensates, Phys. Rev. Lett. 110, 264101 (2013). [21]Y. Xu, Y. Zhang, and B. Wu, Bright solitons in spin-orbit-coupled Bose-Einstein condensates, Phys. Rev. A 87, 013614 (2013). [22]O. Fialko, J. Brand, and U. Z¨ ulicke, Soliton magnetiza- tion dynamics in spin-orbit-coupled Bose-Einstein con- densates, Phys. Rev. A 85, 051605(R) (2012). [23]L. Wen, Q. Sun, Y. Chen, D.-S. Wang, J. Hu, H. Chen, W.-M. Liu, G. Juzeli¯ unas, B. A. Malomed, and A.-C. Ji, Motion of solitons in one-dimensional spin-orbit-coupled Bose-Einstein condensates, Phys. Rev. A 94, 061602(R) (2016). [24]Q. Zhu, C. Zhang, and B. Wu, Exotic superﬂuidity in spin-orbit coupled Bose-Einstein condensates, Euro- phys. Lett. 100, 50003 (2012). [25]L. Salasnich, W. B. Cardoso, and B. A. Mal- omed, Localized modes in quasi-two-dimensional Bose- Einstein condensates with spin-orbit and Rabi couplings, Phys. Rev. A 90, 033629 (2014). [26]Y.-C. Zhang, Z.-W. Zhou, B. A. Malomed, and H. Pu, Stable solitons in three dimensional free space without the ground state: self-trapped Bose-Einstein condensates with spin-orbit coupling, Phys. Rev. Lett. 115, 253902 (2015). [27]S. Gautam and S. K. Adhikari, Vector solitons in a spin-orbit-coupled spin-2 Bose-Einstein condensate, Phys. Rev. A 91, 063617 (2015). [28]H. Sakaguchi, E. Ya. Sherman, and B. A. Malomed, Vor- tex solitons in two-dimensional spin-orbit-coupled Bose- Einstein condensates: Eﬀects of the Rashba-Dresselhaus coupling and Zeeman splitting, Phys. Rev. E 94, 032202 (2016). . [29]H. Sakaguchi and B. A. Malomed, Flipping-shuttle os- cillations of bright one- and two-dimensional solitons in spin-orbit-coupled Bose-Einstein condensates with Rabi mixing, Phys. Rev. A 96, 043620 (2017). [30]Y. V. Kartashov and V. V. Konotop, Solitons in Bose- Einstein condensates with helicoidal spin-orbit coupling ,8 Phys. Rev. Lett. 118, 190401 (2017). [31]Y. Xu, Y. Zhang, and C. Zhang, Bright solitons in a two-dimensional spin-orbit-coupled dipolar Bose- Einstein condensate, Phys. Rev. A 92, 013633 (2015). [32]X.Jiang, Z.Fan, Z.Chen, W.Pang, Y.Li, andB.A.Mal- omed, Two-dimensional solitons in dipolar Bose-Einstein condensates with spin-orbit coupling, Phys. Rev. A 93, 023633 (2016); Y. Li, Y. Liu, Z. Fan, W. Pang, S. Fu, and B. A. Malomed, Two-dimensional dipolar gap solitons in free space with spin-orbit coupling, Phys. Rev. A 95, 063613 (2017). [33]E. Chiquillo, Quasi-one-dimensional spin-orbit- and Rabi-coupled bright dipolar Bose-Einstein-condensate solitons, Phys. Rev. A 97013614 (2018). [34]Y. V. Kartashov, V. V. Konotop, and F. Kh. Abdullaev, Gap solitons in a spin-orbit-coupled Bose-Einstein con- densate, Phys. Rev. Lett. 111, 060402 (2013). [35]V. E. Lobanov, Y. V. Kartashov, and V. V. Kono- top, Fundamental, multipole, and half-vortex gap soli- tons in spin-orbit coupled Bose-Einstein condensates, Phys. Rev. Lett. 112, 180403 (2014). [36]Y. Zhang, Y. Xu, and Th. Busch, Gap solitons in spin- orbit-coupled Bose-Einstein condensates in optical lat- tices, Phys. Rev. A 91, 043629 (2015). [37]H. Sakaguchi and B. A. Malomed, One- and two- dimensional gap solitons in spin-orbit-coupled systems with Zeeman splitting, Phys. Rev. A 97013607 (2018). [38]H. Li, S.-L. Xu, M. R. Beli´ c, and J.-X. Cheng, Three-dimensional solitons in Bose-Einstein condensates with spin-orbit coupling and Bessel optical lattices, Phys. Rev. A 98033827 (2018). [39]Sh. Mardonov, V. V. Konotop, B. A. Malomed, M. Mod- ugno, and E. Ya. Sherman, Spin-orbit-coupled soliton in a random potential, Phys. Rev. A 98, 023604 (2018). [40]Y.V.KartashovandD.A.Zezyulin, Stablemultiringand rotating solitons in two-dimensional spin-orbit-coupled Bose-Einstein condensates with a radially periodic po- tential, Phys. Rev. Lett. 122, 123201 (2019). [41]X.-W. Luo, K. Sun, and C. Zhang, Spin-tensor- momentum-coupled Bose-Einstein condensates, Phys. Rev. Lett. 119, 193001 (2017). [42]H. Hu, J. Hou, F. Zhang, and C. Zhang, Topolog- ical triply degenerate points induced by spin-tensor- momentum couplings, Phys. Rev. Lett. 120, 240401 (2018). [43]X. Zhou, X.-W. Luo, G. Chen, S. Jia, C. Zhang, Quan- tum spiral spin-tensor magnetism, arXiv: 1901.00575. [44]Y.-K. Liu and S.-J. Yang, Exact solitons and manifold mixing dynamics in the spin-orbit-coupled spinor con- densates, Europhys. Lett. 108, 30004 (2014). [45]S. Gautam and S. K. Adhikari, Vortex-bright solitons in a spin-orbit-coupled spin-1 condensate, Phys. Rev. A 95, 013608 (2017); S. Gautam and S. K. Adhikari, Three- dimensional vortex-brightsolitons inaspin-orbit-coupl ed spin-1 condensate , Phys. Rev. A 97, 013629 (2018). [46]Y.-E. Li and J.-K. Xue, Stationary and moving solitons in spin-orbit-coupled spin-1 Bose-Einstein condensates, Front. Phys. 13, 130307 (2018). [47]Y.-J. Lin, R. L. Compton, A. R. Perry, W. D. Phillips, J. V. Porto, and I. B. Spielman, Bose-Einstein Con- densate in a Uniform Light-Induced Vector Potential, Phys. Rev. Lett. 102, 130401 (2009). [48]D. Campbell, R. Price, A. Putra, A. Vald´ es-Curiel, D. Trypogeorgos, and I. B. Spielman, Magnetic phases ofspin-1 spin-orbit-coupled Bose gases, Nat. Commun. 7, 10897 (2016). [49]Y. Zhang, M. E. Mossman, Th. Busch, P. Engels, and C. Zhang, Properties of spin-orbit-coupled Bose-Einstein condensates, Front. Phys. 11, 118103, (2016).
1211.0867v2.Tailoring_spin_orbit_torque_in_diluted_magnetic_semiconductors.pdf	arXiv:1211.0867v2 [cond-mat.mes-hall] 3 May 2013Tailoring spin-orbit torque in diluted magnetic semicondu ctors Hang Li, Xuhui Wang, Fatih Doˇ gan, and Aurelien Manchon∗ King Abdullah University of Science and Technology (KAUST) , Physical Science and Engineering Division, Thuwal 23955-6 900, Saudi Arabia (Dated: July 20, 2018) We study the spin orbit torque arising from an intrinsic linearDresselhaus spin-orbit coupling in a single layer III-V diluted magnetic semiconductor. We inv estigate the transport properties and spin torque using the linear response theory and we report he re : (1) a strong correlation exists between the angular dependence of the torque and the anisotr opy of the Fermi surface; (2) the spin orbit torque depends nonlinearly on the exchange coupling. Our ﬁndings suggest the possibility to tailor the spin orbit torque magnitude and angular dependen ce by structural design. PACS numbers: 72.25.Dc,72.20.My,75.50.Pp The electrical manipulation of magnetization is cen- tral to spintronic devices such as high density mag- netic random access memory,1for which the spin trans- fer torque provides an eﬃcient magnetization switch- ing mechanism.2,3Beside the conventional spin-transfer torque, the concept of spin-orbit torque in both metal- lic systems and diluted magnetic semiconductors (DMS) has been studied theoretically and experimentally.4–9In the presence of a charge current, the spin-orbit coupling produces an eﬀective magnetic ﬁeld which generates a non-equilibrium spin density that in turn exerts a torque on the magnetization.4–6Several experiments on magne- tization switching in strained (Ga,Mn)As have provided strong indications that such a torque can be induced by a Dresselhaus-type spin-orbit coupling, achieving criti- cal switching currents as low as 106A/cm2.7–9However, up to date very few eﬀorts are devoted to the nature of the spin-orbit torque in such a complex system and its magnitude and angular dependence remain unaddressed. In this Letter, we study the spin-orbit torque in a di- lutedmagneticsemiconductorsubmittedtoalinearDres- selhaus spin-orbitcoupling. We highlight two eﬀects that have not been discussed before. First, a strong correla- tion exists between the angular dependence of the torque and the anisotropyof the Fermi surface. Second, the spin torquedepends nonlinearly onthe exchangecoupling. To illustrate the ﬂexibility oﬀered by DMS in tailoring the spin-orbit torque, we compare the torques obtained in two stereotypical materials, (Ga,Mn)As and (In,Mn)As. The system under investigation is a uniformly mag- netized single domain DMS ﬁlm made of, for example, (Ga,Mn)As or (In,Mn)As. We assume the system is well below its critical temperature. An electric ﬁeld is applied along the ˆxdirection. It is worth pointing out that we considerhere a large-enoughsystem to allowus disregard any eﬀects arising due to boundaries and conﬁnement. We use the six-band Kohn-Luttinger Hamiltonian todescribe the band structure of the DMS,9 HKL=/planckover2pi12 2m/bracketleftbigg (γ1+5 2γ2)k2−2γ3(k·ˆJ)2 +2(γ3−γ2)/summationdisplay ik2 iˆJ2 i/bracketrightBigg . (1) where the phenomenological Luttinger parameters γ1,2,3 determine the band structure and the eﬀective mass of valence-band holes. γ3is the anisotropy parameter, ˆJis the total angular momentum and kis the wave vector. The bulk inversion asymmetry allows us to augment the Kohn-Luttinger Hamiltonian by a strain-induced spin- orbit coupling of the Dresselhaus type.5,7We assume the growth direction of (Ga,Mn)As is directed along the z- axis, two easy axes are pointed at xandy, respectively.10 In this case, the components of the strain tensor ǫxxand ǫyyare identical. Consequently, we may have a linear Dresselhaus spin-orbit coupling7 HDSOC=β(ˆσxkx−ˆσyky), (2) givenβthe coupling constant that is a function of the axial strain.7,11ˆσx(y)is the 6×6 spin matrix of holes and kx(y)is the wave vector. In the DMS systems discussed here, we incorporate a mean-ﬁeld like exchange coupling to enable the spin angular momentum transfer between the hole spin ( ˆs= /planckover2pi1ˆσ/2) and the localized ( d-electron) magnetic moment ˆΩof ionized Mn2+acceptors,12,13 Hex= 2JpdNMnSaˆΩ·ˆs//planckover2pi1 (3) whereJpdis the antiferromagnetic coupling constant.13,14HereSa= 5/2 is the spin of the ac- ceptors. The hole spin operator, in the present six-band model, is a 6 ×6 matrix.13The concentration of the ordered local Mn2+momentsNMn= 4x/a3is given as a function of xthat deﬁnes the doping concentration of Mn ion.ais the lattice constant. Therefore, the entire system is described by the total Hamiltonian Hsys=HKL+Hex+HDSOC. (4)2 In order to calculate the spin torque, we determine the nonequilibrium spin densities S(of holes) as a linearre- sponse to an external electric ﬁeld,5 S=eEx1 V/summationdisplay n,k1 /planckover2pi1Γn,k∝angbracketleftˆv∝angbracketright∝angbracketleftˆs∝angbracketrightδ(En,k−EF).(5) whereˆvis thevelocityoperator. InEq.(5), thescattering rate of hole carriers by Mn ions is obtained by Fermi’s golden rule,12 ΓMn2+ n,k=2π /planckover2pi1NMn/summationdisplay n′/integraldisplaydk′ (2π)3/vextendsingle/vextendsingle/vextendsingleMk,k′ n,n′/vextendsingle/vextendsingle/vextendsingle2 ×δ(En,k−En′,k′)(1−cosφk,k′),(6) whereφk,k′is the angle between two wave vectors kand k′. The matrix element Mk,k′ n,n′between two eigenstates (k,n) and (k′,n′) is Mkk′ n,n′=JpdSa∝angbracketleftψnk\|ˆΩ·ˆs\|ψn′k′∝angbracketright −e2 ǫ(\|k−k′\|2+p2)∝angbracketleftψnk\|ψn′k′∝angbracketright.(7) Hereǫis the dielectric constant of the host semiconduc- torsandp=/radicalbig e2g/ǫistheThomas-Fermiscreeningwave vector, where gis the density of states at Fermi level. Fi- nally, we calculate the ﬁeld like spin-orbit torque using4 T=JexS×ˆΩ, (8) whereJex≡JpdNMnSa. Throughout this Letter, the results are given in terms of the torque eﬃciency T/eE. The interband transitions, arising from distortions in the distributionfunctioninducedbytheappliedelectricﬁeld, are neglected in our calculation. This implies that the torque extracted from the present model is expected to accommodate only a ﬁeld-like component. The above protocols based on linear response formalism allow us to investigate the spin-orbit torque for a wide range of DMS material parameters. We plot in Fig.1(a) the spin torque as a function of the magnetization angle for diﬀerent values of the band structure anisotropy parameter γ3. The topology of the Fermi surface can be modiﬁed by a linear combination of γ2andγ3: ifγ2=γ3∝negationslash= 0, the Fermi surface around the Γ point is spherical, as shown in Fig.1(c). In this spe- cial case, the angular dependence of the torque is simply proportional to cos θ[red curve in Fig.1(a)], as expected from the symmetry of the k-linear Dresselhaus Hamilto- nian, Eq. (2)4. Whenγ3∝negationslash=γ2, the Fermi surfacedeviates fromasphere[Fig.1(b)and(d)]and,correspondingly,the angular dependence of the torque deviates from a simple cosθfunction [i.e., curves corresponding to γ3= 1.0 and γ3= 2.93 in Fig.1(a)]. In a comparison to the spherical case, the maximal value of the torque at θ= 0 is lower forγ3∝negationslash=γ2. As Eq.(5) indicates, in the linear response treatment formulated here, the magnitude of the spin torque is determined by the transport scattering timeand the expectation values of spin and velocity opera- tors of holes. Qualitatively, as the Fermi surface deviates from a sphere, the expectation value ∝angbracketleftˆsx∝angbracketrightof the heavy hole band, contributing the most to the spin torque, is lowered atθ= 0. More speciﬁcally, as the Fermi surface warps, the an- gular dependence of the spin torque develops, in addition to the cosθenvelop function, an oscillation with a period that is shorter than π. The period of these additional oscillations increases as the Fermi surface becomes more anisotropic in k-space, see Fig. 1(b) and (d). To fur- ther reveal the eﬀect of band warping on spin torque, we plotTy/cosθas a function of the magnetization an- gle in inset of Fig.1(a). When γ3= 2.0 (spherical Fermi sphere),Ty/cosθis a constant, for T∝cosθ. When γ3= 2.93 or 1.0, the transport scattering time of the hole carriers starts to develop an oscillating behavior in θ,15which eventually contributes to additional angular dependencies in the spin torque. The angular dependen- cies in spin-orbittorque shall be detectable by techniques such as spin-FMR9. /s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48 /s49/s53/s48 /s49/s56/s48/s45/s48/s46/s54/s45/s48/s46/s51/s48/s46/s48/s48/s46/s51/s48/s46/s54 /s45/s50 /s45/s49 /s48 /s49 /s50/s45/s50/s45/s49/s48/s49/s50 /s45/s50 /s45/s49 /s48 /s49 /s50/s45/s50/s45/s49/s48/s49/s50 /s45/s50 /s45/s49 /s48 /s49 /s50/s45/s50/s45/s49/s48/s49/s50 /s32/s32 /s32 /s100/s101/s103/s32/s84 /s121/s40/s49/s48/s52 /s47 /s109/s50 /s41 /s88/s90 /s77 /s121 /s32/s32/s32 /s32 /s107 /s120/s40/s110/s109/s45/s49 /s41 /s107 /s120/s40/s110/s109/s45/s49 /s41/s107 /s121/s40/s110/s109/s45/s49 /s41 /s107 /s120/s40/s110/s109/s45/s49 /s41/s40/s98/s41/s40/s97/s41 /s32/s32 /s32/s40/s100/s41/s107 /s121/s40/s110/s109/s45/s49 /s41/s32 /s32/s32 /s32 /s32/s32 /s32/s32/s40/s99/s41/s107 /s121/s40/s110/s109/s45/s49 /s41 FIG. 1. (Color online) (a)The y-component of the spin torque as a function of magnetization direction. Fermi surface int er- section in the kz= 0 plane for (b) γ3= 1.0, (c)γ3= 2.0 and (c)γ3= 2.93. The red, black, orange and blue contours stands for majority heavy hole, minority heavy hole, major- ity light hole and minority light hole band, respectively. I n- set (a) depicts Ty/cosθas a function of magnetization di- rection. The others parameters are ( γ1,γ2) = (6.98,2.0), Jpd= 55 meV nm3andp= 0.2 nm−3. In Fig.2, we compare the angular dependence of spin torque ( Ty) for both (Ga,Mn)As and (In,Mn)As which are popular materials in experiments and device fabrication.16–18Although (In,Mn)As is, in terms of ex- change coupling and general magnetic properties, rather similar to (Ga,Mn)As, the diﬀerence in band structures, lattice constants, and Fermi energies between these two materials gives rise to diﬀerent density of states, strains, and transport scattering rates. For both materials, the3 spin torque decrease monotonically as the angle θin- creasesfrom 0 to π/2. Throughoutthe entire angle range [0,π], the amplitude of the torque in (In,Mn)As is twice larger than that in (Ga,Mn)As. We mainly attribute this to two eﬀects. First of all, the spin-orbit coupling con- stantβin (In,Mn)As is about twice as larger than that in (Ga,Mn)As. Second, for the same hole concentration, the Fermi energy of (In,Mn)As is higher than that of (Ga,Mn)As. /s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48 /s49/s53/s48 /s49/s56/s48/s45/s48/s46/s56/s45/s48/s46/s52/s48/s46/s48/s48/s46/s52/s48/s46/s56 /s32/s32/s32 /s32/s71/s97/s77/s110/s65/s115 /s32/s73/s110/s77/s110/s65/s115/s84 /s121/s40/s49/s48/s52 /s47 /s109/s50 /s41 /s100/s101/s103 FIG. 2. (Color online) Torque Tyas afunctionof themagneti- zation direction for (Ga,Mn)As (black square) and (In,Mn)A s (red dots). For (Ga,Mn)As, ( γ1,γ2,γ3) = (6.98,2.0,2.93); for (In,Mn)As, ( γ1,γ2,γ3) = (20.0,8.5,9.2). The strength of the spin-orbit coupling constant is: for (Ga,Mn)As, β= 1.6meVnm; for (In,Mn)As, β= 3.3meVnm.19Theexchange coupling constant Jpd= 55 meV nm3for (Ga,Mn)As20and 39 meV nm3for (In,Mn)As.21 In the following, we further demonstrate a counter- intuitive feature that, in the DMS system considered in this Letter, the spin orbit torque depends nonlinearly on the exchange splitting. In Fig. 3(a), Tycomponent of the spin torque is plotted as a function of the exchange couplingJpd, for diﬀerent values of β. In the weak ex- change coupling regime, the electric generation of non equilibrium spin density dominates, then the leading role of exchange coupling is deﬁned by its contribution to the transport scattering rate. We provide a simple qualita- tive explanation on such a peculiar Jpddependence. Us- ing a Born approximation, the scattering rate due to the p−dinteraction is proportional to 1 /τJ=bJ2 pd, where parameter bisJpd- independent. When the nonmag- netic scattering rate 1 /τ0is taken into account, i.e., the Coulomb interaction part in Eq.(7), the total scattering time in Eq.(5) can be estimated as 1 /planckover2pi1Γ∝1 bJ2 pd+1 τ0, (9) which contributes to the torque by T∝Jpd/(/planckover2pi1Γ). This explains the transition behavior, i.e., increases linearly then decreases, in the moderate Jpdregime in Fig.3. As the exchange coupling further increases, Eq.(9) is dom- inated by the spin-dependent scattering, therefore thescattering time 1 //planckover2pi1Γ∝1/J2 pd. Meanwhile, the energy splitting due to the exchange coupling becomes signiﬁ- cant, thus ∝angbracketleftˆs∝angbracketright ∝Jpd. In total, the spin torque is insensi- tivetoJpd, explainingtheﬂatcurveinthelargeexchange coupling regime. In Fig. 3(b), we plot the inﬂuence of the exchange coupling on the spin torque for two materi- als. In (In,Mn)As, mainly due to a largerFermi energyin a comparison to (Ga,Mn)As, the peak of the spin torque shiftstowardsalarger Jpd. Thedependence ofthetorque as a function of the exchange in (In,Mn)As is more pro- nounced than in (Ga,Mn)As, due to a stronger spin-orbit coupling. /s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52 /s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56 /s32/s32 /s32/s74 /s112/s100/s40/s109/s101/s86/s32/s110/s109/s51 /s41/s32 /s32/s61/s32/s48/s46/s56/s32/s109/s101/s86/s32/s110/s109 /s32 /s32/s61/s32/s49/s46/s54/s32/s109/s101/s86/s32/s110/s109 /s32 /s32/s61/s32/s50/s46/s52/s32/s109/s101/s86/s32/s110/s109/s84 /s121/s40/s49/s48/s52 /s109/s50 /s41 /s40/s97/s41 /s32/s32 /s32/s40/s98/s41 /s74 /s112/s100/s40/s109/s101/s86/s32/s110/s109/s51 /s41/s32/s71/s97/s77/s110/s65/s115 /s32/s73/s110/s77/s110/s65/s115/s84 /s121/s40/s49/s48/s52 /s109/s50 /s41 FIG. 3. (Color online) The Tycomponent of the spin torque as a function of exchange coupling Jpd. (a)TyversusJpd at various values of β, for (Ga,Mn)As. (b) TyversusJpd, for both (Ga,Mn)As and (In,Mn)As. The magnetization is directed along the z-axis (θ= 0). The other parameters are the same as those in Fig.2. The possibility to engineer electronic properties by doping is one of the deﬁning features that make DMS promising for applications. Here, we focus on the doping eﬀect which allows the spin torque to vary as a function of hole carrier concentration. In Fig. 4(a), the torque is plotted as a function of the hole concentration for diﬀer- entβparameters. With the increase of the hole concen- tration, the torque increases due to an enhanced Fermi energy. In the weak spin-orbit coupling regime (small β), the torque as a function of the hole concentration ( p) fol- lows roughly the p1/3curve as shown in the inset in Fig. 4(a). The spherical Fermi sphere approximation and a simple parabolic dispersion relation allow for an analyti- cal expression ofthe spin torque, i.e., in the leading order4 inβandJex, T=m∗ /planckover2pi1βJex EFσD (10) wherem∗is the eﬀective mass. The Fermi energy EF and the Drude conductivity are given by EF=/planckover2pi12 2m∗(3π2p)2/3, σD=e2τ m∗p, (11) whereτis the transport time. The last two relations im- mediately give rise to T∝p1/3. In the six-band model, theFermisurfacedeviatesfromasphereand, asthevalue ofβincreases, the spin-orbit coupling starts to modify the density of states. Both eﬀects render the torque- versus-hole concentration curve away from the p1/3de- pendence. This eﬀect is illustrated in Fig. 4(b). The former (strong spin-orbit coupling) clearly deviates from p1/3, whereas the latter (weak spin-orbit coupling) fol- lows the expected p1/3trend. /s48/s46/s48 /s48/s46/s52 /s48/s46/s56 /s49/s46/s50 /s49/s46/s54 /s50/s46/s48/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50 /s48/s46/s52 /s48/s46/s56 /s49/s46/s50 /s49/s46/s54 /s50/s46/s48/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48 /s48/s46/s48 /s48/s46/s52 /s48/s46/s56 /s49/s46/s50 /s49/s46/s54 /s50/s46/s48/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50/s49/s46/s54 /s32/s32 /s32/s40/s97/s41/s84 /s121/s40/s49/s48/s52 /s47 /s109/s50 /s41/s84 /s121/s47/s112/s49/s47/s51 /s32/s32 /s32/s61/s32/s48/s46/s56/s109/s101/s86/s32/s110/s109 /s32/s32 /s32/s61/s32/s49/s46/s54/s109/s101/s86/s32/s110/s109 /s32/s32 /s32/s61/s32/s50/s46/s52/s109/s101/s86/s32/s110/s109 /s80/s40/s110/s109/s45/s51 /s41/s84 /s121/s40/s49/s48/s52 /s47 /s109/s50 /s41/s32 /s32 /s32/s32 /s32/s32 /s32/s40/s98/s41 /s80/s40/s110/s109/s45/s51 /s41/s32/s71/s97/s77/s110/s65/s115 /s32/s73/s110/s77/s110/s65/s115FIG. 4. (Color online) The y-component of the spin torque as a function of hole concentration. (a) The y-component of the spin torque versus hole concentration at diﬀerent β. (b) spin torque versus hole concentration in (Ga,Mn)As and (In,Mn)As. For (Ga,Mn)As, Jpd= 55 meV nm3; for (In,Mn)As, Jpd= 39 meV nm3. The other parameters are the same as in Fig.3. In conclusion, in a DMS system subscribing to a lin- ear Dresselhaus spin-orbit coupling, we have found that the angular dependence of the spin-orbit torque has a strong yet intriguing correlation with the anisotropy of the Fermi surface. Our study also reveals a nonlinear dependence of the spin torque on the exchange coupling. From the perspective of material selection, for an equiv- alent set of parameters, the critical switching current needed in (In,Mn)As is expected to be lower than that in (Ga,Mn)As. The results reported here shed light on the design and applications of spintronic devices based on DMS. WhereasthematerialsstudiedinthisworkhaveaZinc- Blende structure, DMS adopting a wurtzite structure, such as (Ga,Mn)N, might also be interesting candidates for spin-orbittorque observationdue to their sizable bulk Rashba spin-orbit coupling. However, these materials usuallypresent asigniﬁcantJahn-Tellerdistortionthat is large enough to suppress the spin-orbit coupling.22Fur- thermore, the formalism developed here applies to sys- tems possessing delocalized holes and long range Mn-Mn interactionsand isnot adaptedto the localizedholescon- trolling the magnetism in (Ga,Mn)N. We are indebted to K. V´ yborn´ y and T. Jungwirth for numerous stimulating discussions. F.D. acknowl- edges support from KAUST Academic Excellence Al- liance Grant N012509-00. ∗aurelien.manchon@kaust.edu.sa 1J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers, and D. C. Ralph, Phys. Rev. Lett. 84, 3149 (2000). 2J. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). 3L. Berger, Phys. Rev. B 54, 9353 (1996). 4A. Manchon and S. Zhang, Phys. Rev. B 78, 212405 (2008); Phys. Rev. B 79, 212405 (2009). 5I. Garate and A. H. MacDonald, Phys. Rev. B 80, 134403 (2009). 6K. M. D. Hals, A. Brataas and Y. Tserkovnyak Eur. Phys. Lett.90, 47002 (2010).7A. Chernyshov, M. Overby, X. Liu, J. K. Furdyna, Y. Lyanda-Geller, and L. P. Rokhinson, Nature Phys. 5, 656 (2009). 8M. Endo, F. Matsukura, and H. Ohno, Appl. Phys. Lett. 97, 222501 (2010). 9D. Fang, H. Kurebayashi, J. Wunderlich, K. V´ yborn´ y, L. P. Zˆ arbo, R. P. Campion, A. Casiraghi, B. L. Gallagher, T. Jungwirth, and A. J. Ferguson, Nature Nanotech. 6, 413 (2011). 10U. Welp, V. K. Vlasko-Vlasov, X. Liu, J. K. Furdyna, and T. Wojtowicz, Phys. Rev. Lett. 90, 167206 (2003).5 11B. A. Bernevig and S.-C. Zhang, Phys. Rev. B 72, 115204 (2005). 12T. Jungwirth, M. Abolfath, J. Sinova, J. Kuˇ cera, and A. H. MacDonald, Appl. Phys. Lett. 81, 4029 (2002). 13M. Abolfath, T. Jungwirth, J. Brum, and A. H. MacDon- ald, Phys. Rev. B 63, 054418 (2001). 14J. van Bree, P. M. Koenraad, and J. Fern´ andez-Rossier, Phys. Rev. B 78, 165414 (2008). 15A. W. Rushforth, K. V´ yborn´ y, C. S. King, K. W. Ed- monds, R. P. Campion, C. T. Foxon, J. Wunderlich, A. C. Irvine, P. Vaˇ sek, V. Nov´ ak, K. Olejn´ ık, Jairo Sinova, T. Jungwirth, and B. L. Gallagher, Phys. Rev. Lett. 99, 147207 (2007).16H. Ohno, H. Munekata, T. Penney, S. von Moln´ ar, and L. L. Chang, Phys. Rev. Lett. 68, 2664 (1992). 17S. Koshihara, A. Oiwa, M. Hirasawa, S. Katsumoto, Y. Iye, C. Urano, H. Takagi, and H. Munekata, Phys. Rev. Lett.78, 4617 (1997). 18T. Jungwirth, Qian Niu and A. H. MacDonald, Phys. Rev. Lett.88, 207208 (2002). 19J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and I. Zutic, Acta Phys. Slov. 57, 565 (2007). 20H. Ohno, J. Magn. Magn. Mater. 200, 110 (1999). 21J. Wang, Master’s thesis (Rice University, Houston, Texas, 2002). 22A. StroppaandG. Kresse, Phys. Rev.B 79, 201201 (2009).
0708.2595v2.Manipulating_the_spin_texture_in_spin_orbit_superlattice_by_terahertz_radiation.pdf	arXiv:0708.2595v2 [cond-mat.mes-hall] 2 Apr 2008Manipulating the spin texture in spin-orbit superlattice b y terahertz radiation D.V. Khomitsky∗ Department of Physics, University of Nizhny Novgorod, 23 Gagarin Avenue, 603950 Nizhny Novgorod, Russian Federat ion (Dated: October 23, 2018) The spin texture in a gate-controlled one-dimensional supe rlattice with Rashba spin-orbit cou- pling is studied in the presence of external terahertz radia tion causing the superlattice miniband transitions. It is shown that the local distribution of the e xcited spin density can be modiﬁed by varying the Fermi level of the electron gas and by changing th e radiation intensity and polarization, allowing the controlled coupling of spins and photons. PACS numbers: 72.25.Fe, 73.21.Cd, 78.67.Pt I. INTRODUCTION The control of the spin degrees of freedom is one of the primary goals of rapidly developing ﬁeld of condensed matter physics known as spintronics.1,2In addition to the methods involving the magnetic ﬁeld which eﬀec- tively governs the spins, the application of non-magnetic spin systems can be considered by taking into account the spin-orbit (SO) interaction. In most widely con- sidered two-dimensional semiconductor heterostructures the SO interaction is usually dominated by the Rashba coupling3coming from the structure inversion asymme- try of conﬁning potential and eﬀective mass diﬀerence. The value of Rashba coupling strength can be tuned by the external gate voltage4and it reaches the value of 2·10−11eVm in InAs-based structures.5One of the im- portant issues of spintronics is the interaction between spins and photons which is promising for further ap- plications in novel electronic and optical devices. The studies of optical properties of SO semiconductor struc- tures have formed a fast growing ﬁeld of studies during the last decade. Some of the research topics included the photogalvanic,6,7,9,10,11,12spin-galvanic8,9and spin- photovoltaic13eﬀects as well as optical spin orientation14 and pure spin current generation.15,16The eﬀects of ter- ahertz radiation onto spin-split states in semiconduc- tors were also the subject of investigation.17,18Another important property of non-uniform spin distributions such as spin coherence standing waves was discussed by Pershin19who found an increase of the spin relaxation time in such structures, making them promising for spin- tronics applications. One of possible ways to create a non-uniform spin distribution in a heterostructure is to apply a metal-gated superlattice with tunable amplitude of electric potential to the two-dimensional electron gas (2DEG) with spin-orbit coupling. The quantum states and spin polarization in this system with standing spin waveswerestudied previously,20andthe problemofscat- tering on such structure has been considered.21 In the present Brief Report we study the problem of the excited spin density creation in a one-dimensional InAs-based superlattice with Rashba spin-orbit coupling by applying an external terahertz radiation. The direc- tion of photon propagation is chosen perpendicular to2DEG plane where both linear and circular polarizations are considered. It is shown that the excited spin texture is sensitive to the position of the Fermi level of the 2DEG and to the radiation intensity. The former can be tuned by the gate voltage, thus providing a new possible way to couple local excited distribution of spins and photons in 2DEG with SO interaction. The obtained results have a qualitative character and are not restricted to one spe- ciﬁc type of semiconductor heterostructure, superlattice period, amplitude of periodic potential, Fermi level posi- tion, etc. Another important issue is the spin relaxation which tends to transform the excited spin distribution back to the equilibrium. The terahertz scale of excita- tionfrequenciesisatleastofanorderofmagnitudelarger than the spin relaxationratesin InAs semiconductorhet- erostructures which can be estimated as 1 /τswhere the spin lifetime τsreaches there the values from 60 ps22to 600 ps23and the spin relaxation time is further increased in non-homogeneous spin textures.19Thus, one can ex- pect that the eﬀects discussed in the currentBriefReport can be experimentally observable. This Brief Report is organized as follows. In Sec.II we brieﬂy describe quantum states in SO superlattice and in Sec.III we calculate the spatial distribution of the ex- cited spin density in a superlattice cell under terahertz radiationwith diﬀerentpositionsofFermilevel, radiation intensity and polarization. The conclusions are given in Sec.IV. II. QUANTUM STATES IN SO SUPERLATTICE We start with the brief description of the quantum states of 2DEG with Rashba SO coupling and one- dimensional (1D) periodic superlattice potential in the absence of external electromagnetic ﬁeld.20The Hamil- tonian is the sum of the 2DEG kinetic energy operator in a single size quantization band with eﬀective mass m, the Rashba SO term with strength αand the periodic electrostatic potential of the 1D superlattice: ˆH=ˆp2 2m+α(ˆσxˆpy−ˆσyˆpx)+V(x),(1)2 where ¯h= 1 and the periodic potential is chosen in the simplest form V(x) =V0cos2πx/awhereais the superlattice period and the amplitude V0is controlled by the gate voltage. The eigenstates of Hamiltonian (1) are two-componentBlochspinorswith eigenvalueslabeledby the quasimomentum kxin a one-dimensional Brillouin zone−π/a≤kx≤π/a, the momentum component ky, and the miniband index m: ψmk=/summationdisplay λnam λn(k)eiknr √ 2/parenleftbigg1 λeiθn/parenrightbigg , λ=±1.(2) Herekn=k+nb=/parenleftbig kx+2π an, ky/parenrightbig andθn= arg[ky−iknx]. The energy spectrum of Hamiltonian (1) consists of pairs of spin-split minibands determined by the SO strength αseparated by the gaps of the order ofV0. An example of the energy spectrum is shown in Fig.1 for the four lowest minibands in the InAs 1D su- perlattice with Rashba constant α= 2·10−11eVm, the electron eﬀective mass m= 0.036m0, the superlattice perioda= 60 nm and the amplitude of the periodic po- tentialV0= 10 meV. It should be mentioned that the spectrum in Fig.1 is limited to the ﬁrst Brillouin zone of the superlattice in the kxdirection while the cutoﬀ in thekydirection is shown only to keep the limits along kxandkycomparable. The Fermi level position EF(Vg) can be varied by tuning the gate voltage which controls the concentration of the 2D electrons. This feature is shown in Fig.1 schematically by the arrows near EF(Vg) as well as the photon energy ¯ hω= 10 meV corresponding toω/2π= 2.43 THz. III. SPIN TEXTURE MANIPULATION The Rashba SO coupling as well as the periodic su- perlattice potential cannot produce the net polarization of 2DEG. Moreover, the two-component eigenvectors of the Rashba Hamiltonian describe the homogeneous lo- cal spin density σi=ψ†ˆσiψ, wherei=x,y,z. In the presence of an additional superlattice potential, how- ever, the local spin density for a given state ( kx,ky) can be inhomogeneous,20as in the spin coherence standing wave,19which gives an idea to obtain a non-uniform spin densitydistributionunderanexternalradiationwhichin- volvesintransitionsthestateswithdiﬀerent( kx,ky)with a varying impact depending on the matrix elements. In this Section we subject the 2DEG with Hamiltonian (1) to the electromagneticradiationpropagatingalong zaxis perpendicular to the 2DEG plane with the electric ﬁeld of the radiation E(t) =eEωexp−iωt+ c.c.with ampli- tudeEω, frequency ωpolarization e= (ex,ey). When the electromagnetic radiation is applied, the excited spin density rate Siat a given point in a real space can be found in the following way:15 FIG. 1: (color online) Energy spectrum of four lowest mini- bands in the InAs 1D superlattice with Rashba constant α= 2·10−11eVm, the electron eﬀective mass m= 0.036 m0, the superlattice period and amplitude a= 60 nm and V0= 10 meV. The Fermi level position EF(Vg) and the pho- ton energy ¯ hω= 10 meV corresponding to ω/2π= 2.43 THz are shown schematically. Si=πe2E2 ω ω2/integraldisplay d2k/summationdisplay jlξjl i(k)¯ejel (3) where ξjl i(k) =/summationdisplay c,m,m′/parenleftBig ψ† m′ˆσiψm/parenrightBig ¯vj m′cvl mc (4) ×[δ(ωmc(k)−ω)+δ(ωm′c(k)−ω)]. (5) Since the structure is completely homogeneous in the ydirection, the Sycomponent vanishes. The other com- ponentsSxandSzof the excited non-equilibrium spin density can be nonzero at a given point in a superlattice evenforthe linear x-polarizedradiation. The coexistence of the axial vector components ( Sx,Sz) in the left side of Eq.(3) together with the polar vector component exin the right side is in agreement with the principles of mag- netic crystal class analysis.24There is a mirror plane of reﬂectionσyin our system which changes yto−yand thus changes the sign of the magnetic moment, leaving theexcomponent of the polarization unchanged. The element of the magnetic crystal class, however, is applied only as a combination σyRwhereRis the time rever- sal operator24which again changes the direction of the3 magnetic moment but does not change the polarization component ex. As a result, the combination σyRleaves both spin projections Sx,zand the polarization compo- nentexinvariant. Another restriction is the absence of total magnetic moment of the sample which means that bothSxandSzmust satisfy to the requirement of zero net polarization: /integraldisplaya 0Sx(x)dx=/integraldisplaya 0Sz(x)dx= 0. (6) The spin density distribution in the superlattice cell is found for three diﬀerent polarizations: linear along x axis, linear along yaxis, and circular in the ( xy) plane of 2DEG. The Fermi level in Fig.1 varies from 4 to 20 meV counted from the bottom of the topmost partially ﬁlled electron size quantization band. According to the spectrum in Fig.1, this variation ﬁlls gradually all four minibands shown there. The excitation energy is chosen to be ¯hω= 10 meV which corresponds to ω/2π= 2.43 THz and provides eﬀective transitions between the occu- pied and vacant minibands, as it can be seen in Fig.1. First, let us considerthe casesofthe excitationlinearly polarized along xaxis and the σ+- circular polarized one. The excited non-equilibrium spin density (3) is shown in Fig.2 as a 2D vector ﬁeld ( Sx(x),Sz(x)). Its magni- tude is proportional to the radiation intensity which is 0.3W/cm2for the present case and the texture shape is varied with respect to the Fermi energy. The mag- nitude of the arrow length in Fig.2, i.e. the maximum excited spin density can be obtained from the value of the excited charge density nex. Taking the data from the experiments with optical excitation17where the volume concentration of the excited carriers reached 1016cm−3 one can estimate the excited surface concentration nexto be oftheorderof1010...1011cm−2. Onecanseein Fig.2 that the spin textures aresimilar for x- andσ+radiation, as well as for σ−(not shown). The explanation is that all these polarizationscontain the xcomponent oftransition matrix elements which causes the most eﬀective transi- tions in the x-oriented superlattice. The transformation of spin density distribution with the Fermi level position in Fig.2 is produced by gradual ﬁlling of the minibands. The small amplitudes of spin density (at EF= 4 meV in Fig.2) correspond to the small ﬁlling factor (see Fig.1). By increasing the Fermi energy the complete ﬁlling of two lowest subbands (at EF= 8 meV and EF= 12 meV in Fig.2) is reached and, ﬁnally, the excited spin den- sity magnitude decreases again with complete ﬁlling of all four nearest miniband in Fig.1 (at EF= 16 meV and EF= 20 meV in Fig.2). It can be seen in Fig.2 that the excited spin density texture have the primary spatial wavelength being close to the superlattice period a. This feature can be explained by the structure of the matrix element of the transitions which reaches maximum am- plitude atkx=±π/a, leading to the eﬀective creation of spin texture (3) with the speciﬁc primary wavelength equal toa. Since the Fermi level position can be varied FIG. 2: Excited spin density distribution ( Sx(x),Sz(x)) along the 1D superlattice elementary cell created under x- andσ+- polarized terahertz excitation with thefrequency ω/2π= 2.43 THz and with the intensity I= 0.3W/cm2. in experiments by tuning the gate voltage which controls the concentration of 2DEG, our model predicts a possi- bleandrealisticmechanismforopticalcreationofvarious spin textures. Another type of the considered polarization is the ra- diation polarized along yaxis. The structure is homoge- neous in this direction and the only reason for the transi- tionprobabilitiestobenonzeroisthe non-parabolicchar- acter of energy spectrum as a function of kydue to the interplay between SO and superlattice potential. How- ever, thisinterplayis reducedwith increasing kyandthus one can expect the probabilities to be smaller for the y- polarized radiation than for the x-polarized one, making the creation of the excited spin texture comparable to the one in Fig.2 possible at higher intensities. This sug- gestion is conﬁrmed by the spin textures in Fig.3 where4 FIG. 3: Excited spin density distribution ( Sx(x),Sz(x)) along the 1D superlattice elementary cell created under y-polarized terahertz excitation with the same parameters as in Fig.2 bu t at higher intensity I= 0.9W/cm2. the similar spin textures as in Fig.2 are created at the intensity 0.9W/cm2which is three times greater than for thex- andσ- polarized light. Nevertheless, all of the intensities considered in the paper are within the range of 0.5−1W/cm2which is accessible in modern experi- mental setups.9,10,12,16,17 Further investigations of gated 2DEG with SO inter-action would require the studies of the spin current con- ventionally described by the operator ˆjs ij= ¯h{ˆvi,σj}/4 where ˆvi=∂ˆH/∂ki. The excited spin current distribu- tion can be analyzed under the same approach as the spin density, and the spin separation distance can be calculated.15The investigations of spin current in our system deserves a separate detailed discussion and will be performed in the forthcoming paper. IV. CONCLUSIONS We have studied the excited spin texture distribution in 2DEG with Rashba spin-orbit interaction subject to 1D tunable superlattice potential and illuminated by the terahertz radiation with diﬀerent polarizations and in- tensities. It was found that in the absence of the net polarization the local excited spin texture can be eﬀec- tively manipulated by varying the Fermi level position in 2DEG as well as the intensity and polarization of the ra- diation at ﬁxed terahertz frequency. The eﬀect of excited spin texture creation discussed in the paper has a quali- tative character and should be observable in a wide class of two-dimensional heterostructures where the spin-orbit coupling energy is more pronounced than the tempera- ture broadening or the smearing caused by edges, defects or impurities. Acknowledgments The author thanks V.Ya. Demikhovskii and A.A. Perov for many helpful discussions. The work was sup- ported by the RNP Program of the Ministry of Educa- tion and Science RF, by the RFBR, CRDF, and by the Foundation ”Dynasty” - ICFPM. ∗Electronic address: khomitsky@phys.unn.ru 1Semiconductor Spintronics and Quantum Computation , edited by D.D. Awschalom, D. Loss, and N. Samarth, Nanoscience and Technology (Springer, Berlin, 2002) 2I. Zˇ uti´ c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). 3E.I. Rashba, Fiz. Tverd. Tela (Leningrad) 2, 1224 (1960) [Sov. Phys. Solid State 2, 1109 (1960)]; Y.A. Bychkov and E.I. Rashba, J. Phys. C 17, 6039 (1984). 4J.B. Miller, D.M. Zumb¨ uhl, C.M. Marcus, Y.B. Lyanda- Geller, D. Goldhaber-Gordon, K. Campman, and A.C. Gossard, Phys. Rev. Lett. 90, 076807 (2003). 5D. Grundler, Phys. Rev. Lett. 84, 6074 (2000). 6L.E. Golub, Phys. Rev. B 67, 235320 (2003). 7S.D. Ganichev, V.V. Bel’kov, Petra Schneider, E.L. Ivchenko, S.A. Tarasenko, W. Wegscheider, D. Weiss, D. Schuh, E.V. Beregulin, and W. Prettl, Phys. Rev. B 68, 035319 (2003).8S.D. Ganichev, Petra Schneider, V.V. Bel’kov, E.L. Ivchenko, S.A. Tarasenko, W. Wegscheider, D. Weiss, D. Schuh, B.N. Murdin, P.J. Phillips, C.R. Pidgeon, D.G. Clarke, M. Merrick, P. Murzyn, E.V. Beregulin, and W. Prettl, Phys. Rev. B 68, 081302(R) (2003). 9S. Giglberger, L.E. Golub, V.V. Bel’kov, S.N. Danilov, D. Schuh, C. Gerl, F. Rohlﬁng, J. Stahl, W. Wegscheider, D. Weiss, W. Prettl, and S.D. Ganichev, Phys. Rev. B 75, 035327 (2007). 10C.L. Yang, H.T. He, Lu Ding, L.J. Cui, Y.P. Zeng, J.N. Wang, and W.K. Ge, Phys. Rev. Lett. 96, 186605 (2006). 11Bin Zhou and Shun-Quing Shen, Phys. Rev. B 75, 045339 (2007). 12K.S. Cho, C.-T. Liang, Y.F. Chen, Y.Q. Tang, and B. Shen, Phys. Rev. B 75, 085327 (2007). 13Arkady Fedorov, Yuriy V. Pershin, and Carlo Piermaroc- chi, Phys. Rev. B 72, 245327 (2005). 14S.A. Tarasenko, Phys. Rev. B 73, 115317 (2006).5 15R.D.R. Bhat, F. Nastos, Ali Najmaie, and J.E. Sipe, Phys. Rev. Lett. 94, 096603 (2005). 16Hui Zhao, Xinyu Pan, Arthur L. Smirl, R.D.R. Bhat, Ali Najmaie, J.E. Sipe, and H.M. van Driel, Phys. Rev. B 72, 201302(R) (2005). 17J.T. Olesberg, Wayne H. Lau, Michael E. Flatt´ e, C. Yu, E. Altunkaya, E.M. Shaw, T.C. Hasenberg, and Thomas F. Boggess, Phys. Rev. B 64, 201301(R) (2001). 18Jacob B. Khurgin, Phys. Rev. B 73, 033317 (2006). 19Yuriy V. Pershin, Phys. Rev. B 71, 155317 (2005). 20V.Ya. Demikhovskii and D.V. Khomitsky, JETP Letters 83, iss.8, p.340 (2006) [Pis’ma v ZhETF 83, iss.8, p.399(2006)]. 21D.V. Khomitsky, Phys. Rev. B 76, 033404 (2007). 22B.N. Murdin, K. Litvinenko, J. Allam, C.R. Pidgeon, M. Bird, K. Morrison, T. Zhang, S.K. Clowes, W.R. Branford, J. Harris, andL.F. Cohen, Phys.Rev.B 72, 085346 (2005). 23K.C.Hall, K.G¨ undoˇ gdu, E.Altunkaya,W.H.Lau, Michael E. Flatt´ e, Thomas F. Boggess, J.J. Zinck, W.B. Barvosa- Carter, and S.L. Skeith, Phys. Rev. B 68, 115311 (2003). 24L.D. Landau and E.M. Lifshitz, Electrodynamics of Con- tinuous Media , Pergamon, New York, 1984.
1012.5572v1.The_multi_state_CASPT2_spin_orbit_method.pdf	arXiv:1012.5572v1 [physics.chem-ph] 27 Dec 2010The multi-state CASPT2 spin-orbit method Zoila Barandiar´ an1,2and Luis Seijo1,2 1Departamento de Qu´ ımica, Universidad Aut´ onoma de Madrid , 28049 Madrid, Spain 2Instituto Universitario de Ciencia de Materiales Nicol´ as Cabrera, Universidad Aut´ onoma de Madrid, 28049 Madrid, Spain (Dated: June 2, 2021) Abstract We propose the multi-state complete-active-space second- order perturbation theory spin-orbit method (MS-CASPT2-SO) for electronic structurecalculati ons. It is a two-step spin-orbit coupling method that does not make use of energy shifts and that intrin sically guarantees the correct characters of the small space wave functions that are used to calculate the spin-orbit couplings, in contrast with previous two-step methods. PACS numbers: 31.15.A-, 31.15.aj, 31.15.am 1I. INTRODUCTION In electronic structure two-step spin-orbit coupling methods, dy namic correlation is han- dled in the ﬁrst step, using the spin-free part of the Hamiltonian and a large conﬁgurational space in variational or perturbational schemes. Then, spin-orbit coupling is handled in the second step, using an eﬀective Hamiltonian and a small conﬁguration al space in spin-orbit conﬁguration interaction (CI) calculations. In these methods, th e eﬀective Hamiltonian con- tains explicit energy shifts, which are a mean to transfer dynamic co rrelation eﬀects from the ﬁrst step to the second step in a simple and eﬀective manner.1–3 It has been found that the energy shifts of the spin-orbit free lev els, which are driven by their energy order within each irreducible representation, can le ad to anomalous results when avoided crossings exist with signiﬁcant change of character o f the wave functions at each side, which take place at diﬀerent nuclear positions in the large a nd in the small electronic conﬁgurational spaces. In these cases, the shifts mu st be assigned according to the characters of the wave functions.4This usually implies analyses of wave functions in both conﬁgurational spaces. The ultimate reason behind these problems, which are present in the available two-step methods,1–3is the diﬀerent nature of the wave functions of the spin-free stat es in the large and in the small conﬁgurational spaces, so that, even when the av oided crossings do not exist or when they take place at the same nuclear positions in the larg e and in the small spaces, such a diﬀerent nature makes the spin-orbit couplings calc ulated in the small space not as accurate (meaning as close to the spin-orbit couplings calcula ted in the large space) as desired. Here, we propose the multi-state complete-active-space second -order perturbation theory spin-orbit method (MS-CASPT2-SO). It is a two-step method that does not make use of energy shifts and that guarantees by construction the correct characters of the small space wave functions that are used to calculate the spin-orbit couplings. 2II. THE MS-CASPT2-SO METHOD Let us assume we have a many electron system with a Hamiltonian ˆHwhich is made of the addition of a spin-free contribution, ˆHSF, and a spin-orbit coupling contribution, ˆHSO: ˆH=ˆHSF+ˆHSO. (1) In the spin-orbit free MS-CASPT2-SO method, the procedure is init ially the same as the MS-CASPT2 procedure:5,6Several state-average complete-active-space self-consisten t- ﬁeld SA-CASSCF (or CASCI) states are calculated, which deﬁne a re ference con- ﬁgurational space called the Pspace. Let us collect them in the row vector ΨCAS= (\|ΨCAS 1/angbracketright,\|ΨCAS 2/angbracketright,...,\|ΨCAS p/angbracketright), wherepis the total number of SA-CASSCF states. These wave functions can be classiﬁed according to their values of s pin quantum numbers and symmetry group irreducible representations and subspecies, SMSΓγ, but we will omit these labels here for simplicity. In spin-orbit free MS-CASPT2 calculations, the SA-CASSCF wave fu nctions are used as a basis to calculate the matrix of a spin-free second order eﬀective Hamiltonian, ˆHSF,eff 2nd, whichisdeﬁnedinEq.30ofRef.5anddependsonlyonthespin-freep artoftheHamiltonian, ˆHSF. This matrix, which is HSF,eff,CAS 2nd = ΨCAS†ˆHSF,eff 2ndΨCAS, is diagonalized in order to compute the MS-CASPT2 energies, as the eigenvalues, and the mod iﬁed SA-CASSCF (or CASCI) states, as the eigenfunctions: HSF,eff,CAS 2nd U=UEMS2, (2) whereEMS2is a diagonal matrix with the MS-CASPT2 energies EMS2 1,EMS2 2,...,EMS2 p, as the diagonal elements and Uis a unitary transformation of the original SA-CASSCF wave functions that preserves the SMSΓγvalues, ΨCAS′= ΨCASU. (3) Obviously, the modiﬁed SA-CASSCF wave functions ΨCAS′also span the Pspace. What is important isthat they aretheappropriatezeroth-order basis fo r asecond-order perturbation theory treatment of the dynamic correlation that leads to the MS- CASPT2 energies6and they have the appropriate characters in correspondence with th ese energies. In the MS-CASPT2 spin-orbit calculations proposed here, we can fo llow two alternative procedures that lead to the same result. Both of them are based o n the use of the spin- 3dependent eﬀective Hamiltonian that results from the addition of th e spin-orbit coupling operator to the spin-free eﬀective Hamiltonian of the MS-CASPT2 m ethod, ˆHeff=ˆHSF,eff 2nd+ˆHSO. (4) In the ﬁrst procedure, which is a formal two-step procedure, th e regular spin-orbit free MS-CASPT2 calculation is completed and the modiﬁed SA-CASSCF wave functions ΨCAS′ are used as a basis for the matrix representation of ˆHeff. The resulting matrix, Heff,CAS′= ΨCAS′†ˆHeffΨCAS′=EMS2+HSO,CAS′, (5) withHSO,CAS′= ΨCAS′†ˆHSOΨCAS′, couples the modiﬁed SA-CASSCF states via spin-orbit coupling. (It couples diﬀerent SMSΓγblocks and it can be factorized according to double group irreducible representations.) Its diagonalization gives the ﬁn al energies and wave functions: Heff,CAS′USO′=USO′EMS2−SO, (6) whereEMS2−SOis a diagonal matrix with the MS-CASPT2-SO target energies EMS2−SO 1,EMS2−SO 2,...,EMS2−SO pas the diagonal elements and USO′is a unitary trans- formation of the modiﬁed SA-CASSCF wave functions that couples t heSMSΓγvalues and gives the target spin-orbit wave functions, ΨMS2−SO= ΨCAS′USO′. (7) Alternatively, in the second procedure, which is a formal one-step procedure, the original SA-CASSCF wave functions ΨCASare used as the basis for the matrix representation of ˆHeff. In order to do this, the regular spin-orbit free MS-CASPT2 calcula tion does not need to be completed, but only the computation of the HSF,eff,CAS 2nd matrix used in Eq. 2, plus the addition of the matrix of ˆHSOin this basis ( HSO,CAS= ΨCAS†ˆHSOΨCAS): Heff,CAS= ΨCAS†ˆHeffΨCAS=HSF,eff,CAS 2nd +HSO,CAS. (8) Its diagonalization gives the same target energies and wave functio ns as the ﬁrst procedure, Heff,CASUSO=USOEMS2−SO, (9) with ΨMS2−SO= ΨCASUSO(10) 4andUSO=UUSO′. The present method is closely related with the restricted active spa ce state interaction approach with spin-orbit coupling of Ref. 3, SO-RASSI. The latter, when it is used in a SA-CASSCF/MS-CASPT2 context, corresponds to diagonalizing th e mixed eﬀective Hamil- tonian matrix EMS2+HSO,CAS, with the spin-orbit free part of Eq. 5 and the spin-orbit coupling part of Eq. 8. The results of both approaches are expect ed to be similar when the CAS and the CAS′wave functions (Eq. 3) are also similar, which is a common case. Diﬀer- ences should show up when the two sets of wave functions are not s o similar in a one-to-one basis, for instance when the dynamic correlation switches the orde r of states. This is the basic advantage of the present method. However, we must note t hat the SO-RASSI method can be used together with general single-state andstate-avera ge RASSCF and CASSCF plus RASPT2 and single-state and multi-state CASPT2 spin-orbit free fr ameworks, whereas the present one can only be used in the spin-orbit free framework of SA -CASSCF plus MS- CASPT2 calculations. Let us now justify why ˆHeff(Eq. 4) is the eﬀective Hamiltonian of choice in the MS- CASPT2-SO method. For this purpose, we recall that the basic idea of two-step methods is to use a spin-orbit eﬀective Hamiltonian made of a spin-free eﬀective Hamiltonian (usually ˆHSF+ˆHshift)plusthespin-orbitcoupling operator ˆHSO, andtochoosethespin-freeeﬀective Hamiltonianbyimposing therequirement that, when used inthesmall s pacePofthesecond step, it has the same eigenvalues that ˆHSFhas in the large space Gof the ﬁrst step.1In the particular case in which the ﬁrst step is a MS-CASPT2 calculation a nd the small space of the second step is deﬁned by the SA-CASSCF wave functions, ˆHSF,eff 2ndis a spin-free eﬀective Hamiltonian that fulﬁlls such a condition. In consequence, ˆHeff(Eq. 4) is the proper spin-orbit eﬀective Hamiltonian. III. CONCLUSION Atwo-step spin-orbit coupling methodformulti-state complete-ac tive-space second-order perturbationtheorycalculationsMS-CASPT2isproposedwhichdoe snotmakeuseofenergy shifts. It intrinsically guarantees the correct characters of the small space wave functions used to calculate the spin-orbit couplings, in contrast with previous two-step spin-orbit coupling methods, where it has to be checked externally. 5Acknowledgments Thisworkwas partlysupported byagrantfromMinisterio deCiencia e Innovaci´ on, Spain (Direcci´ on General de Programas y Transferencia de Conocimient o MAT2008-05379/MAT). 1R. Llusar, M. Casarrubios, Z. Barandiar´ an, and L. Seijo, J. Chem. Phys. 105, 5321 (1996). 2V. Vallet, L. Maron, C. Teichteil, and J.-P. Flament , J. Chem . Phys.113, 1391 (2000). 3P. A. Malmqvist, B. O. Roos, and B. Schimmelpfennig, Chem. Ph ys. Lett. 357, 230 (2002). 4G. S´ anchez-Sanz, Z. Barandiar´ an, and L. Seijo, Chem. Phys . Lett.498, 226 (2010). 5J. Finley, P.- ˚A. Malmqvist, B. O. Roos and L. Serrano-Andr´ es, Chem. Phys. Lett.288, 299 (1998). 6A. Zaitsevskii and J. P. Malrieu, Chem. Phys. Lett. 233, 597 (1995). 6
1411.2990v1.Striped_Ferronematic_ground_states_in_a_spin_orbit_coupled__S_1__Bose_gas.pdf	arXiv:1411.2990v1 [cond-mat.quant-gas] 11 Nov 2014Striped Ferronematic ground states in a spin-orbit coupled S= 1Bose gas Stefan S. Natu,1,∗Xiaopeng Li,1and William S. Cole1 1Condensed Matter Theory Center and Joint Quantum Institute , Department of Physics, University of Maryland, College Park, Maryland 20742-4111 USA We theoretically establish the mean-ﬁeld phase diagram of a homogeneous spin-1, spin-orbit coupled Bose gas as a function of the spin-dependent interac tion parameter, the Raman coupling strength and the quadratic Zeeman shift. We ﬁndthat the inte rplay between spin-orbit coupling and spin-dependent interactions leads to the occurrence of fer romagnetic or ferronematic phases which also break translational symmetry. For weak Raman coupling , increasing attractive spin-dependent interactions (as in87Rb or7Li) induces a transition from a uniform to a stripe XY ferroma gnet (with no nematic order). For repulsive spin-dependent inte ractions however (as in23Na), we ﬁnd a transition from an XYspin spiral phase ( /an}bracketle{tSz/an}bracketri}ht= 0 and uniform total density) with uniaxial nematic order, to a biaxial ferronematic, where the total density, s pin vector and nematic director oscillate in real space. We investigate the stability of these phases aga inst the quadratic Zeeman eﬀect, which generally tends to favor uniform phases with either ferroma gnetic or nematic order but not both. We discuss the relevance of our results to ongoing experimen ts on spin-orbit coupled, spinor Bose gases. INTRODUCTION The interplay between competing orders such as su- perﬂuidity/superconductivity, magnetism, liquid crys- tallinity and density wave order is fundamental to the rich phenomenology of strongly correlated systems. A candidate system for exploring this physics is a spin- orbit coupled Bose condensate [1–4], where the coupling between spin and motional degrees of freedom can lead to a spin textured ground state which breaks rotational symmetry in spin space [5–7], as well as translational symmetry in real space [8, 9]. Indeed, for a pseudospin- 1/2Bosesystem, varyingthespin-orbitcouplingstrength drivesatransitionfromanunmagnetizedphasewithden- sitywaveordertoauniformmagnetizedphase, whichhas been studied both theoreticallyand experimentally[4, 9]. More recently, attention has turned towards exploring the physics of large spin systems, which have no ana- log in condensed matter, such as highly magnetic atoms like Dysprosium, Erbium, Chromium [10–12], and alka- line earthatomswith SU(N) symmetry(See Ref. [13] and referenes therein). The large spin nature of these atoms produces a rich phase diagram with novel topological de- fects, where uniaxial and biaxial nematic and more ex- otic platonic solid orders compete and complement con- ventional magnetically ordered phases [14–16]. In the presence of spin-orbit coupling, the possibility of transla- tional symmetry breaking can lead to textured ground states phases with intertwining magnetic and nematic order. Here we study the simplest, experimentally re- alizable system where such physics is manifest: a spin-1, spin-orbit coupled Bose gas [5, 17], ﬁnding a rich phase diagram. Our main result is summarized in Fig. 1, which shows the schematic, zero-temperaturephase diagramofa spin- orbit coupled spin-1 Bose gas as a function of the spin- dependent interaction and the quadratic Zeeman energy,at ﬁxed Raman coupling and spin-independent interac- tion strength. A new feature of the spin-orbit coupled gas is the appearance of translational symmetry break- ing phases with simultaneous spin and nematic order, which are generically competing orders in this system [15, 16]. Weak repulsive spin-dependent interactions fa- vorauniaxialnematicferromagnet(ferronematic), which supportsaspiralspintextureinthe x−yplane(UN+XY spiral). Large attractive spin-dependent interactions fa- vor a ferromagnetic stripe ( FMstripe), whereas large repulsive spin-dependent interactions favor a biaxialfer- ronematic stripe phase ( BNstripe) where the total den- sity, spin vector and nematic director oscillates in real space. SPIN-1PHENOMENOLOGY In the absence of spin-orbit coupling, the phase dia- gram of a spin-1 Bose gas has been well established the- oretically [18–21] and experimentally [22–27]. Assum- ing short-range (s-wave) interactions, spin rotation in- variance and bosonic statistics forces two-body collisions to occur in the total spin-0 or spin-2 channels, producing the interaction Hamiltonian [18, 19]: Hint=1 2/integraldisplay d3rψ† αψ† βψγψδ(c0δαδδβγ+c2Sαδ·Sβγ),(1) where the greek indices denote the hyperﬁne spin projec- tion, andψσ(r) is the the boson ﬁeld operator. Unlike thepseudospin-1 /2case, thisHamiltonianhasSU(2)spin rotation invariance. The two coupling constants, c0andc2represent spin-independent and spin-dependent interactions re- spectively, and Sis the vector {Sx,Sy,Sz}, whereSi are 3×3, spin-1 matrices. The interactions are expressed in terms of the microscopicscatteringlengths in the spin- 0 (a0) and spin-2 ( a2) channels and atomic mass mas:2 FM/UpTeeFM/VertBar1/VertBar1UN/UpTeeUN/UpTee UN/VertBar1/VertBar1/PlusXY spiralBN stripe FM/VertBar1/VertBar1stripeUN/UpTeeUN/VertBar1/VertBar1 00 q/Slash1c0n0c2/Slash1c0/Minus2/Minus1012 k/Slash1k0/Minus2/Minus1012Ek/Slash1ER /Minus2/Minus1012 FIG. 1: (Color Online) Schematic phase diagram for a spin-orbit coupled spin-1 Bose gas, as a function of spin- dependent interaction strength and quadratic Zeeman shift , showing translation symmetry breaking phases. The underly - ing single-particle dispersion is shown above. For suﬃcien tly large attractive spin-dependent interactions, an XY spin d en- sity wave phase occurs, with oscillations in the total densi ty (FMstripe). For repulsive spin-dependent interactions, an XY spiral phase occurs, which simultaneously has uniaxial nematic order (UN), but the total density remains uniform. For suﬃciently large spin-dependent interactions, a biaxi al nematic phase ( BN) is present where the total density, spin vector and nematic director oscillate in real space. Suﬃ- ciently large positive or negative quadratic Zeeman eﬀect f a- vors homogeneous phases with either uniaxial nematic ( UN⊥ (ψ={0,1,0}) orUN/bardbl(ψ=1√ 2{1,0,1})) or ferromagnetic order (FM⊥(ψ={1,0,0}) orFM/bardbl(ψ=1 2{1,√ 2,1})), but not both. c0= 4π(a0+ 2a2)/3mandc2= 4π(a2−a0)/3m[25]. For87Rb,c2/c0=−0.005, for23Na,c2/c0∼0.05 and for7Li,c2/c0∼ −0.5 [25]. These interactions (and their sign) can however be tuned using optical Feshbach reso- nances [28]. The wave-function of a spin-1 Bose condensate is rep- resented as a spinor ψ=eiθ{ψ1,ψ0,ψ−1}, whereθrepre- sents the broken global U(1) gauge symmetry ofthe Bose condensate, and {1,0,−1}label the three spin states. Owing to the structure of the spin-1 Pauli matrices, this system can exhibit both magnetic order, given by the vector order parameter /an}bracketle{tS/an}bracketri}ht=/an}bracketle{tψ\|S\|ψ/an}bracketri}ht/n, wherenis the density, or nematic order, described by the tensor Nµν= δµν−1 2n/an}bracketle{tψ\|(SµSν+SνSµ)\|ψ/an}bracketri}ht, where{µ,ν} ∈ {x,y,z}, andδµνdenotesthe identity matrix. However,aspointed out by Mueller [16], these orders are not independent of oneanother, butrathercompeting. Diagonalizingthene- matic tensor yields three distinct eigenvalues λ1,λ2,λ3, constrained by λ1+λ2+λ3= 1. Auniaxial nematic has onenon-zeroeigenvalue,whilea biaxialnematichasthree distinct eigenvalues. Attractive spin-dependent interac- tions (c2<0) favor a maximally ferromagnetic phase(/an}bracketle{tψ\|S\|ψ/an}bracketri}ht=ˆz), with no nematic order, which can be represented by unitary rotations of ψ=eiθU{1,0,0}T, whereas repulsive spin-dependent interactions ( c2>0) favor a uniaxial nematic with no spin order, represented by unitary rotations of ψ=eiθU{0,1,0}T[18]. In the presence of spin-orbit coupling, spin-rotation symmetry is broken and the three spin states are no longer degenerate at the single-particle level. We choose thespin-orbitcouplingtobeofequalRashba-Dresselhaus type, whichwasrecentlyrealizedinexperiments[2–4,29– 31], but generalized to the spin-1 case: Hsoc=/planckover2pi12(kx−k0Sz)2 2m+/planckover2pi12k2 ⊥ 2m+Ω 2Sx+δ 2Sz+q 2S2 z(2) wherek0is the wave-vector of the Raman beams, Ω is the strength of the Raman coupling, δandqare the linear and quadratic Zeeman eﬀects respectively. It is convenient to normalize energy by the recoil energy of the Raman lasers ER=/planckover2pi12k2 0 2m. For simplicity, we neglect the linear Zeeman eﬀect term ( δ= 0) in this work, but generallyassumethat q/ne}ationslash= 0, andcantakeonpositiveand negative values, which can be achieved using microwaves [32, 33]. A detailed description of the single-particle physics of a spin-1 spin-orbit coupled gas was recently given by Lan and¨Ohberg [17], and is not repeated here. For weak q and Ω, the low energy spectrum has three local minima atk= 0,±k1, where 0 ≤k1/k0≤1 is determined by diagonalizing Eq. 2, at ﬁxed Ω and q. Increasing positive qresults in a single minimum at k= 0, whereas negative qproduces a two minimum structure, with the minima at±k1. Thedispersionofthelowestbranch(toquadraticorder inkx) is obtained as ǫ(kx) =/planckover2pi12k2 x m/bracketleftbigg 1/2−/planckover2pi12k2 0 m(˜q2+4Ω2)−1/21−z 1+z/bracketrightbigg +O(k4 x), (3) with ˜q=q+/planckover2pi12k2 0 mandz=˜q√ ˜q2+4Ω2. The analytic form of the transition line from three to two minima easily follows as/planckover2pi12k2 0 m(˜q2+4Ω2)−1/21−z 1+z= 1/2. From the triple minimum structure of single-particle dispersion, we make a reasonable variational ansatz for the condensate wave-function: ψ=/radicalbigg N V(χ+eik1xφ++χ0φ0+χ−e−ik1xφ−),(4) whereχ0,χ±are complex numbers, which are deter- mined variationallybelow, Nis the particle number, Vis the volume, and φ±,φ0are the normalized single-particle spinor eigenstates at the minima ±k1,0 respectively. We ﬁx the gaugechoice bychoosingthe eigenvectors( φ±,φ0) to be real, where the respective spin components obey φ±1 ±=φ∓1 ∓, andφ0 +=φ0 −. Particle number conservation3 TABLE I: Orders in spin-orbit coupled spin-1 gas. Order Symbol Order Parameter ferromagnetic FM/bardbl/⊥ /an}bracketle{tSi/an}bracketri}ht /ne}ationslash= 0 Uniaxial nematic UN/bardbl/⊥λ1/ne}ationslash= 0,λ2=λ3= 0 Biaxial nematic BN λ 1<λ2<λ3 Translation stripe, XYspiral /an}bracketle{tSi(r)/an}bracketri}ht ∼cos(k1r) n(r)∼cos(k2r) N=/integraltext d3rn(r) =/summationtext σ∈{−1,0,1}/integraltext d3r\|ψσ(r)\|2imposes the constraint \|χ+\|2+\|χ0\|2+\|χ−\|2= 1. The variational interaction energy takes a suggestive form: E=r(\|χ+\|2+\|χ−\|2)+gµν\|χµ\|2\|χν\|2+g3(χ∗ +χ∗ −χ0χ0+c.c.), (5) whereris the kinetic term and gµνandg3are related to the original interaction parameters multiplied by form factors proportional to the single-particle wave-functions atthethreeminima. Theenergyisinvariantundertrans- formations UC(1) :χµ→eiθχµ,UA(1) :χµ→eiµθχµ andZ2:χµ→χ−µ. Heretheaxial UA(1)originatesfrom translational symmetry, and the Z2symmetry is related to reﬂection where spin transforms as a pseudovector. It is important to emphasize that the Josephson term, proportional to g3is zero throughout the phase diagram of the spin-1 Bose gas without spin-orbit coupling. How- ever, asweshowhere, it playsacrucialroleinthe physics of the spin-orbit coupled gas. A new feature of the spin-orbit coupled Bose gas is the possibility oftranslationalsymmetrybreakingphases [5, 8], which arise because bosons in diﬀerent spin states condense into ﬁnite momentum states. For the spin-1 /2 case, where bosons condense at two minima, the total density develops stripes at a wave-vector 2 k1[5, 8, 9], but the spin density /an}bracketle{tS/an}bracketri}htremains uniform throughout the phase diagram [9, 34]. In the spin-1 case however, in addition to stripe structure in the density [5, 17], the system can display oscillations in the spin and nematic order parameters. This leads to a rich phase diagram, reproduced in Fig. 1, which we now discuss in detail. In Table I, we characterize the condensed phases we ﬁnd, in terms of their order parameters. We minimize the totalenergyperparticle(Eq. 5), with respect to the complex variational parameters χ0,χ±. The spin and nematic order parameters are then com- puted using the resulting mean-ﬁeld ground state wave- function. Normalizing the energy to the laser recoil energyER, and setting δ= 0, yields four dimension- less parameters Ω /ER,q/ER,c0n0/ER,c2n0/ER, where n0=N/Vis the total density. Throughout, we ﬁx the Raman coupling Ω /ERsuch that, in the absence of a quadratic Zeeman eﬀect, the low energy, single-particle dispersion has three local minima. The two minima atk=±k1are always degenerate in the absence of δ. For q>qc1>0 the three minimum structure disappears and only a single minimum at k= 0 is present, whereas for q < qc2<0, the system only has two minima at ﬁnite k. We ﬁx Ω but vary qto access both these regimes in parameter space [17]. ATTRACTIVE SPIN-DEPENDENT INTERACTIONS c2<0 We ﬁrst consider the regime of attractive spin- dependent interactions, which corresponds to87Rb (as in the NIST experiments [2, 29]) and7Li. Absent spin- orbitcoupling, thegroundstateisauniformferromagnet, which is of Ising type ( /an}bracketle{tS/an}bracketri}ht=ˆz) forq <0 and XY type (/an}bracketle{tS/an}bracketri}ht=ˆx) forq >0 (spin rotation symmetry is restored atq= 0). For suﬃciently large q >0, there is a sec- ond order quantum phase transition to a polar ( UN⊥) phase, which has been investigated in detail recently (see Ref. [25] and References therein). In the presence of spin-orbit coupling, at q= 0, spin symmetry is explicitly broken by the Rabi coupling (Ω) term, whichpreferstoalignthetotalmagnetizationalong x. Absent interactions, the single particle wave function is centered around k= 0, and has a small but ﬁnite value of/an}bracketle{tSx/an}bracketri}ht, correspondingtothe explicitly brokensymmetry. Upon turning on c2, the minimum energy state is a ferro- magnet with /an}bracketle{tS/an}bracketri}ht=ˆx. The wave-function for such a state requires all three minima to be occupied, but the relative phaseθ++θ−−2θ0≈0, whereχ±0=\|χ±,0\|eiθ±,0. This state is degenerate with the plane wave Ising ferromag- net formed by occupying a single minimum at k=±k1 with respect to the spin-dependent interaction term, but has stripes in the total density, and is thus penalized by the repulsive density-density interactions (proportional toc0). For weak \|c2\|therefore, an Ising plane wave phase occurs, which has no stripes in the total density. Upon increasing \|c2\|however, the total energy can be lowered by aligning the spin along the xdirection, satisfying the Rabi coupling term, at the expense of producing den- sity (and spin) stripes. This interaction driven transition from a uniform Ising ferromagnet to striped XY ferro- magnet is a new feature of the spin-1, spin-orbit coupled gas [17]. In Fig. 2 we plot the critical value of \|c2\|/c0for the ferromagnetic stripe phase as a function of the Rabi strength Ω/ER. For our parameters, when Ω /ER>1, the system enters the single minimum regime, and the stripe phase is destroyed. Although the stripe phase can occur for arbitrarily small spin dependent interactions, it should be emphasized that the amplitude of the stripes decreases with decreasing \|c2\|/c0, and may be hard to resolve experimentally, particularly for87Rb. The hori- zontal dashed line shows the spin-dependent interaction for7Li [25]; all values of Rabi coupling below the line4 00.20.40.60.8100.10.20.30.40.5 /CapOmega/Slash1ER/VertBar1c2/VertBar1/Slash1c0 /Minus10/Minus505100.00.51.01.5 x/LParen11/Slash1k1/RParen1n/LParen1x/RParen1/Slash1n0 FIG. 2: Critical attractive spin-dependent interaction \|c2\|/c0 for onset of stripe ferromagnetic phase as a function of Rabi coupling Ω/ERatq= 0. We set c0n0/ER= 0.4. Dashed line shows \|c2\|/c0for7Li; all values of Ω /lessorsimilar0.9ERsupport the ferromagnetic phase with stripes in the total density. Inse t shows the density in real space for Ω /ER= 0.8 at\|c2\|/c0 corresponding to7Li. should exhibit a stripe phase. The inset shows the pro- nounced amplitude of the stripes for the7Li interaction parameters at Ω /ER= 0.8, which strongly supports the experimental observability of this phase. As shown in Fig. 1, the striped ferromagnetic phase is destroyed for positive and negative values of the quadratic Zeeman eﬀect. For q >0, we ﬁnd a transition from the stripe ferromagnetic phase to a polar conden- sate, which occurs roughly where the single-particle dis- persion enters the single minimum regime ( qc1), largely independent of c2. Forq <0, the single-particle disper- sion has two minima, and a uniform Ising ferromagnetic phaseoccurswhereonlyoneofthese twodegeneratemin- ima are occupied. The transition from the stripe ferro- magnet to the uniform Ising ferromagnet where trans- lation symmetry is restored, depends on the interaction strength and the magnitude of qas shown in Fig. 1. POLAR REGIME: c2>0 We now turn our attention to the case of repulsive spin-dependent interactions. Absent spin-orbit coupling, repulsivespin-dependentinteractionsyieldapolarphase, where/an}bracketle{tS/an}bracketri}ht= 0, but the system has uniaxial nematic or- der. In the presence of the Rabi term Ω, residual ferro- magneticorderispresentevenfor c2>0, andgenerically, the ground state is ferronematic. For weak spin-dependent interactions and q >0, the system condenses at k= 0, and a uniform ferronematic phase is found. For q <0 however, the system con- denses atk=±k1, and the phases of the condensate at these two points are such that the total density re- mains uniform, but the transverse spin density sponta- neously breaks translational symmetry and develops XY spin density wave order. Similar spiral phases have been predicted to occur in the spin-1 /2 system in the presence of Rashba spin-orbit coupling [5, 7]. This origin of the spin density wave can also be un-xz /Minus10/Minus50510/Minus0.6/Minus0.4/Minus0.20.00.2 x/LParen11/Slash1k0/RParen1S/Slash1n0 FIG. 3: Top: Spatial spin texture in the polar regime of a spin-orbit coupled spin-1 Bose gas for strong spin-depende nt interactions c2/c0= 1.1, and Ω/ER= 0.8, which corresponds to the three minimum regime. We set q= 0 here, although this phase is stable for moderate values of q(see Fig. 1). The arrows indicate the projection of the spin in the x−zplane in spin space at each point in real space. Bottom: x(green dotted),y(red dashed) and z(black solid) components of spin in real space. Oscillations in the zdirection (at wave- vectork=k1) andx−ydirection (at two wave-vectors k= k1,2k1) are diﬀerent with one another. The total density (not shown) also shows anharmonic oscillations at two dominant wave-vectors k=k1andk= 2k1. Note that due to the Raman coupling term, time reversal is explicitly broken, an d the spatially averaged spin along the xdirection is ﬁnite. derstood as follows: For weak Raman coupling, the con- densates at k=±k1,0 are closely related to the original ±1,0 spin states. Thus by applying a gauge transforma- tion, whereby ψ±1→ψ±1e∓ik1x,ψ0→ψ0, we obtain a non spin-orbit coupled Hamiltonian with a spatially os- cillating magnetic ﬁeld along xof the form Ω Sxcos(k1x), leaving all other terms in the Hamiltonian unchanged. This produces an XY spin density wave texture in real space. Asqbecomes more and more negative, the ampli- tude of oscillation of the spiral goes to zero as Ω /\|q\|. The nematic tensor in this phase has only one non-zero eigen- value, which corresponds to a uniaxial nematic. Thus spin-orbitcouplingnaturallyleadstoferronematicphases which break translation symmetry. For even larger c2, the situation becomes more exotic, and the system condenses into all three minima, even at q= 0. The relative phase of the condensates at the three minimaθ++θ−−2θ0≈π. This state has stripes in the total density [17], and concurrently, magnetic order in all x,y,zdirections, as shown in Fig. 3. Interestingly how- ever, the wavelengths of the oscillations in zand those in thex−ydirections are generally diﬀerent from one another. Furthermore, diagonalizing the nematic tensor for this situationyieldsthreedistincteigenvalues,whichisa biax- ialnematicphase. Owingtotheconstraint λ1+λ2+λ3= 1, the degree of biaxiality can be found by taking the diﬀerence between the largest two eigenvalues. As the5 density, spin and nematic order parameters are not inde- pendent of one another, all these orders simultaneously oscillate in real space. In Fig. 4, we plot the spatial oscil- lations in the biaxiality and the total spin, the minimum in the biaxiality coincides with the minimum in the total spin, where a maximally uniaxial nematic phase occurs. EXPERIMENTAL RELEVANCE As we show in Fig. 2, observing the ferromagnetic stripe phase for attractive spin-dependent interactions may be challenging for87Rb, but feasible in7Li. Al- though the contrast in the oscillations in the spin den- sity is large, experiments may not have enough spatial resolution to observe the individual oscillations, which will further be smeared out by spatial averaging eﬀects during expansion and imaging. Martone et al.[35] have recently proposed using Bragg spectroscopy to increase the wavelength of the stripes, making them visible. The XY spiral nematic state occurs even for weak, re- pulsive spin-dependent interactions (as in23Na spinor condensates) by simply tuning the quadratic Zeeman shift to take on negative values, which produces two global minima in the single-particle dispersion. The transverse components of spin can then be probed in situby applying spin echo radio-frequency pulses be- tween individual snapshots [36] to reveal the spin den- sity texture. Nematicity can be probed experimentally by directly measuring spin ﬂuctuations /an}bracketle{tSµSν/an}bracketri}ht, averaged overmanyshots[37]. Alternatively,opticalbirefringence, whereby, the coupling between the nematic order param- eter and the polarization of a probe beam leads to a ro- tation in the polarization of the light ﬁeld can be used to measure the local nematic/biaxial order parameter [38]. Observing the biaxial spin-density wave phase requires strong spin-dependent interactions, which could be in- duced using optical or magnetic Feshbach resonances. This however breaks the SU(2) spin rotation invariance which underpins Eq. 1, and may lead to a qualitatively diﬀerent phase diagram than that discussed here. CONCLUSIONS AND FUTURE DIRECTIONS In conclusion, we ﬁnd that the spin-1 spin-orbit cou- pled Bose gas possesses a rich phase diagram with phases which break translational symmetry, spin symmetry and possess liquid crystalline order. Generally, we ﬁnd that spin-orbit coupling intertwines magnetic and nematic or- der, giving rise to ferronematic phases that break trans- lationalsymmetry. In additionto the usualhomogeneous polarand ferromagneticphasesin the non spin-orbitcou- pled spin-1 gas, we ﬁnd three new phases: a ferromag- netic stripe phase for attractive spin-dependent interac- tions with stripes in the total density and spin, an XY/Minus10/Minus505100.00.10.20.30.40.50.60.7 x/LParen11/Slash1k0/RParen1Order FIG. 4: Total spin (red dashed) and biaxial order parameter (black, solid) shown as a function of space in the biaxial, sp in- density wave phase of a polar spin-orbit coupled gas. The minimum in the biaxiality corresponds to a minimum in the total spin, and here a nearly uniaxial nematic appears, whic h illustrates the competing nature of thenematic and spin ord er parameters. spiral ferronematic phase with uniform total density for weak repulsive spin-dependent interactions and negative quadratic Zeeman shift, and a biaxial ferronematic stripe phase, with spatial oscillations in the total density, spin vector and nematic director. A key diﬀerence between the spin-1 case from the pseudo-spin 1 /2 counterpart is the appearance of spin density wavephases, even when the total density remains uniform for repulsive spin-dependent interactions. We alsoemphasize that although ferronematicphasesarebe- lieved to occur in dipolar fermions [11, 39] and high spin systems such as spin-3 Chromium atoms [15], a crucial diﬀerence here is that the ferronematic ground states we ﬁnd also break translationalsymmetry, due to the under- lying spin-orbit coupling. High spin spin-orbit coupled systems thus oﬀer unique insight into the interplay be- tweencompeting orders, which areubiquitous in strongly correlatedsystems. It will be extremelyinteresting to ex- plore generalizations of this work to even larger spin sys- tems, such asspin-orbitcoupled Dysprosiumand Erbium atoms [11, 12]. Other exciting directions for future work are to explore the nature of the Goldstone modes asso- ciated with broken translation symmetry [40], the topo- logical defects that occur in these striped ferronematic phases [16, 41] and the restoration of diﬀerent symme- tries at ﬁnite temperature [42–44]. Acknowledgements.— We aregrateful to the JQI-NSF- PFC,AFOSR-MURI,andARO-MURI(Atomtronics)for support. ∗Electronic address: snatu@umd.edu [1] V. Galitski and I. B. Spielman, Nature 49549 (2013). [2] Y. K Lin, K. Jimenez-Garcia and I. B. Spielman, Nature, 47183 (2011). [3] J.-Y. Zhang et al., Phys. Rev. Lett. 109115301 (2012). [4] S.-C. Ji, et al.Nature Physics 10314 (2014). [5] C. Wang, C. Gao, C.-M. Jian and H. Zhai, Phys. Rev.6 Lett.105160403 (2010). [6] T. D. Stanescu, B. Anderson and V. Galitski, Phys. Rev. A78023616 (2008). [7] C. Wu, I. Mondragon-Shem and X.-F. Zhou, Chin. Phys. Lett.28097102 (2011). [8] T-L. Ho, S. Zhang, Phys. Rev. Lett. 107150403 (2011). [9] Y. Li, L. P. Pitaevskii and S. Stringari, Phys. Rev. Lett. 108225301 (2012). [10] A. Griesmaier, J. Werner, S. Hensler, J. Stuhler and T. Pfau, Phys. Rev. Lett. 94160401 (2005). [11] K. Aikawa, S. Baier, A. Frisch, M. Mark, C. Ravensber- gen and F. Ferlaino, Science 3451484 (2014). [12] X. Cui, B. Lian, T.-L. Ho, B. L. Lev and H. Zhai, Phys. Rev. A88011601 (R) (2013). [13] M. A. Cazalilla and A. M. Rey, eprint.arXiv: 1403.2792. [14] T.-L. Ho and B. Huang, eprint.arXiv:1401.4513. [15] R. B. Diener and T.-L. Ho, Phys. Rev. Lett. 96190405 (2006). [16] E. J. Mueller, Phys. Rev. A 69033606 (2004). [17] Z. Lan and P. Ohberg, Phys. Rev. A 89023630 (2014). [18] T.-L. Ho, Phys. Rev. Lett. 81742 (1998). [19] T. Ohmi and K. Machida, J. Phys. Soc. Jpn. 671822 (1998). [20] C. K. Law, H. Pu and N. P. Bigelow, Phys. Rev. Lett. 815257 (1998). [21] S. Mukerjee, C. Xu and J.E. Moore, Phys. Rev. Lett. 97, 120406 (2006). [22] J. Stenger et al.Nature396345-348 (1998). [23] D. M. Stamper-Kurn et al.Phys. Rev. Lett. 802027 (1998). [24] L. E. Sadler et al.Nature443312-315 (2006). [25] D. M. Stamper-Kurn and M. Ueda, Rev. Mod. Phys. 85 1191 (2013). [26] D. Jacob et al., Phys. Rev. A 86061601 (R) (2012). [27] M.-S. Chang et al.Phys. Rev. Lett. 92140403 (2004). [28] F. K. Fatemi, K. M. Jones and P. D. Lett, Phys. Rev. Lett.854462 (2000).[29] Y.-J. Lin, R. L. Compton, A. R. Perry, W. D. Phillips, J. V. Porto, and I. B. Spielman, Phys. Rev. Lett. 102 130401 (2009). [30] P. Wang, Z-Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H. Zhai and J. Zhang Phys. Rev. Lett. 109095301 (2012). [31] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah, W. S. Bakr and M. W. Zwierlein Phys. Rev. Lett. 109 095302 (2012). [32] F. Gerbier, A. Widera, S. F¨ olling, O. Mandel and I. Bloch, Phys. Rev. A 73041602 (2006). [33] E. M.Boojkans, A.VinitandC.Raman, Phys.Rev.Lett. 107195306 (2011). [34] In the presence of more exotic Rashba type spin-orbit coupling, the spin density can display spiral order, whichspontaneouslybreakstime-reversalsymmetry.(See Refs. [5, 7]) Here time-reveral symmetry is explicitly bro- ken by the Raman coupling term (see Eq. 2) [35] G. I. Martone, Y. Li and S. Stringari, Phys. Rev. A 90 041604 (R) (2014). [36] J. Guzman, G.-B. Jo, A. N. Wenz, K. W. Murch, C. K. Thomas and D. M. Stamper-Kurn, Phys. Rev. A 84 063625 (2011). [37] C. D. Hamley, C. S. Gerving, T. M. Hoang, E. M. Book- jans and M. S. Chapman, Nature Physics 8305 (2012). [38] I.CarusottoandE.J.Mueller, J.Phys.B 37S115(2004). [39] B. M. Fregoso and E. Fradkin, Phys. Rev. Lett. 103 205301 (2009). [40] Y. Li, G. I. Martone, L. P. Pitaevskii and S. Stringari, Phys. Rev. Lett. 110235302 (2013). [41] C.-M Jian and H. Zhai, Phys. Rev. B 84060508 (R) (2011). [42] J. Radi´ c, S. S. Natu and V. Galitski, Phys. Rev. Lett. 113185302 (2014). [43] S. S. Natu and E. J. Mueller, Phys. Rev. A 84053625 (2011). [44] C. Hickey and A. Paramekanti, eprint arXiv:1409.1216.
1611.08424v2.Spin_orbit_Hamiltonian_for_organic_crystals_from_first_principles_electronic_structure_and_Wannier_functions.pdf	Spin-orbit Hamiltonian for organic crystals from rst principles electronic structure and Wannier functions Subhayan Roychoudhury and Stefano Sanvito School of Physics, AMBER and CRANN Institute, Trinity College Dublin, Dublin 2, Ireland Spin-orbit coupling in organic crystals is responsible for many spin-relaxation phenomena, going from spin diusion to intersystem crossing. With the goal of constructing eective spin-orbit Hamil- tonians to be used in multiscale approaches to the thermodynamical properties of organic crystals, we present a method that combines density functional theory with the construction of Wannier func- tions. In particular we show that the spin-orbit Hamiltonian constructed over maximally localised Wannier functions can be computed by direct evaluation of the spin-orbit matrix elements over the Wannier functions constructed in absence of spin-orbit interaction. This eliminates the prob- lem of computing the Wannier functions for almost degenerate bands, a problem always present with the spin-orbit-split bands of organic crystals. Examples of the method are presented for iso- lated organic molecules, for mono-dimensional chains of Pb and C atoms and for triarylamine-based one-dimansional single crystals. I. INTRODUCTION Spintronics devices operate by detecting the spin of a carrier in the same way as a regular electronic device measures its electrical charge1. These devices are already the state of the art in the design of magnetic sensors such as the magnetic read-head of hard-disk drives2, but also have excellent prospect as logic gate elements3{6. Logic circuits using the spin degree of freedom may oer low energy consumption and high speed owing to the fact that the dynamics of spins takes place at a much smaller energy scale than that of the charge1,3. Recent years have also witnessed a marked increase in interest into investigations of organic molecules and molecular crystals as materials platform, initially for elec- tronics7,8and lately also for spintronics9{11. The main reason behind such interest is that organic crystals, com- ing in a wide chemical variety, are typically much more exible than their inorganic counterparts and they can exhibit an ample range of electronic properties, which are highly tuneable in practice. For example, it is possible to change the conductivity of organic polymers over fteen orders of magnitude12. In addition to such extreme spec- trum of physical/chemical properties organic materials are usually processed at low temperature. This is an ad- vantage over inorganic compounds, which translates into a drastic reduction of the typical manufacturing and in- frastructure costs13. Finally, specic to spintronics is the fact that both the spin-orbit (SO) and hyperne interac- tion are very weak14in organic compounds, resulting in a weak spin scattering during the electron transport15{17. Regardless of the type of media used, either organic or inorganic, spintronics always concerns phenomena re- lated to the injection, manipulation and detection of spins into a solid state environment11. In the prototyp- ical spintronic device, the spin-valve18, a non-magnetic spacer is sandwiched between two ferromagnents. Spins, which are initially aligned along the magnetization vector of the rst ferromagnet, travel to the other ferromagnent through the spacer, and the resistance of the entire devicedepends on the relative orientation of the magnetization vectors of the two magnets. However, if the spin direc- tion is lost across the spacer, the resistance will become independent of the magnetic conguration of the device. As such, in order to measure any spin-dependent eect one has to ensure that the charge carriers maintain their spin direction through the spacer. Notably, this require- ment is not only demanded by spin-valves, but also by any devices based on spins. There are several mechanism for spin-relaxation in the solid state19. In an organic semiconductor (OSC) the unwanted spin- relaxation can be caused by the presence of paramagnetic impurities, by SO coupling and by hyperne interaction. In general paramagnetic impurities can be controlled to a very high degree of precision and they can be almost completely eliminated from an OSC during the chemi- cal synthesis20. The hyperne interaction instead can be usually considered small. This is because there are only a few elements typically present in organic molecules with abundant isotopes baring nuclear spins. The most obvi- ous exception is hydrogen. However, most of the OSC crystals are -conjugated and the -states, responsible for the extremal energy levels, and hence for the elec- tron transport, are usually delocalized. This means that the overlap of the wave function over the H nuclei has to be considered small. Finally, also the SO coupling is weak owing to the fact that most of the atoms composing organic compounds are light. As such, since all the non-spin-conserving interactions are weak in OSCs, it is not surprising that there is con- tradictory evidence concerning the interaction mostly re- sponsible for spin-diusion in organic crystals. Con ict- ing experimental evidence exists supporting either the SO coupling21,22or the hyperne interaction23,24, indicating that the dominant mechanism may depend on the specic material under investigation. For this reason it is impor- tant to develop methods for determining the strength of both the SO and the hyperne coupling in real materi- als. These can eventually be the basis for constructing eective Hamiltonians to be used for the evaluation ofarXiv:1611.08424v2 [cond-mat.mtrl-sci] 9 Feb 20172 the relevant thermodynamics quantities (e.g. the spin diusion length). Here we present one of such methods for the case of the SO interaction. The SO interaction is a relativistic eect arising from the electron motion in the nuclear potential. In the elec- tron reference frame the nucleus moves and creates a magnetic eld, which in turn interacts with the electron spin. This is the spin-orbit coupling25. Since the SO interaction allows the spin of an electron to change di- rection during the electron motion, it is an interaction re- sponsible for spin relaxation. In fact, there exist several SO-based microscopic theories of spin relaxation in solid state systems19. In the case of inorganic semiconductors these usually require knowledge of the band-structure of the material, some information about its mobility and an estimate of the spin-orbit strength. In the case of OSCs the situation, however, is more complex, mostly because the transport mechanism is more dicult to de- scribe. Firstly, the band picture holds true only for a few cases, while for many others one has to consider the material as an ensemble of weakly coupled molecules with a broad distribution of hopping integrals26. Sec- ondly, the typical phonon energies are of the same order of magnitude of the electronic band width, indicating that electron-phonon scattering cannot be treated as a perturbation of the band structure. For all these rea- sons the description of the thermodynamical properties of OSCs requires the construction of a multi-scale the- ory, where the elementary electronic structure is mapped onto an eective Hamiltonian retaining only a handful of the original degrees of freedom27. A rigorous and now standard method for constructing such eective Hamil- tonian consists in calculating the band structure over a set of Wannier functions28,29. These can be constructed in a very general way as the Fourier transform of a linear combination of Bloch states, where the linear combina- tion is taken so to minimize the spatial extension of the Wannier functions. These are the so-called maximally localized Wannier fuctions (MLWFs)30,31. The MLWF method performs best for well-isolated bands. This is indeed the case of OSCs, where often the valence and conduction bands originate respectively from the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) of the gas-phase molecule. In fact, when the MLWF proce- dure is applied to such band structure one obtains Wan- nier orbitals almost identical to the molecule HOMO and LUMO27. Spin-orbit interaction, however, splits such well-dened bands, and in OSCs the split is typically a few tenths of eV. Thus, in this case, one has to apply the MLWF procedure to bands, which are indistinguishable at an energy scale larger then a few eV. In such con- ditions the minimization becomes almost impossible to converge, the MLWFs cannot be calculated for SO-split bands and an alternative scheme must be implemented. Here we describe a method for obtaining the SO ma- trix elements with respect to the Wannier functions cal- culated in the absence of the SO interaction. Since theSO coupling in OSCs is weak, such spin-independent Wannier functions represent a close approximation of those that one could, at least in principle, obtain in the presence of the SO interaction. Furthermore, when the MLWF basis spans the same Hilbert space dened by all the atomic orbitals relevant for describing a given bands manifold, our method provides an accurate description of the system even in the case of heavy elements, i.e. for strong spin-orbit interaction. In particular we implement our scheme together with the atomic-orbital, pseudopo- tential, density functional theory (DFT) code Siesta32. Siesta is used to generate the band structure in absence of the spin-orbit interaction and for calculating the SO potential, while the MLWF procedure is performed with theWannier90 code33. The paper is organized as follows. In the next section we describe our method in detail, by starting from the general idea and then going into the specic numerical implementation. A how-to work ow will also be pre- sented. Next we discuss results obtained for rather di- verse physical systems. Firstly, we evaluate the SO-split energy eigenvalues of a plumbane molecule and show how accurately these match those obtained directly from DFT including SO interaction. Then, we apply our procedure to the calculation of the band structure of a chain of Pb atoms, before moving to materials composed of light ele- ments with low SO coupling. Here we will show that our method performs well for chains made of carbon atoms and of methane molecules. Finally we obtain the SO ma- trix elements for the Wannier functions derived from the HOMO band of a triarylamine-based nanowire, a rela- tively well-known semiconducting material with potential applications in photo-voltaic34and spintronics. II. METHOD A. General idea Here we describe the idea behind our method, which is general and does not depend on the specic implementa- tion used for calculating the band structure. Consider a set ofN0isolated Bloch states, j mki, describing an in- nite lattice. These for instance can be the DFT Kohn- Sham eigenstates of a crystal. One can then obtain the associatedN0Wannier functions from the denition, jwnRi=V (2)3Z BZ2 4N0X m=1Uk mnj mki3 5eikRdk;(1) wherejwnRiis then-th Wannier vector centred at the lattice site R,Vis the volume of the primitive cell and the integration is performed over the rst Brillouin zone (BZ). In Eq. (1) Ukis a unitary operator that mixes the Bloch states and hence denes the specic set of Wan- nier functions. A particularly convenient gauge choice for Ukconsists in minimizing the Wannier functions spread,3 which writes =X n hwn0jr2jwn0ijhwn0jrjwn0ij2 :(2) Such choice denes the so-called maximally localized Wannier functions (MLWFs). In the absence of SO coupling a Wannier function of spins1is composed exclusively of Bloch states with the same spin, s1. By moving from a continuos to a dis- cretek-point representation the spin-polarized version of Eq. (1) becomes31 jws1 nRi=1 NX kX mUs1 mn(k)j s1 mkieikR: (3) Note that this represents either a nite periodic lat- tice comprising Nunit cells or a sampling of Nuni- formly distributed k-points in the Brillouin zone of an innite lattice. Here the Bloch states, which are nor- malized within each unit cell according to the relation h s1 mkj s2 nk0i=Nm;nk;k0s1;s2, obey to the condition pk(r1) = pk(rN+1), where pk(rm) denotes the Bloch function for the p-th band at the wavevector kand posi- tionrm. The projection of a generic Bloch state onto a MLWF in the absence of SO coupling can be written as h s1 qk0jws2 nR2i= =1 NX kX mUs2 mn(k)h s1 qk0j s2 mkieikR= =1 NX kX mUs2 mn(k)eik:RNq;mk;k0s1;s2= =Us2 qn(k0)eik0Rs1;s2:(4) Hence a generic SO matrix element can be expanded over the MLWF basis set as hws1 mR1jVSOjws2 nR2i= =1 N2X p;qX k1;k2hws1 mR1j s1 pk1i(VSO)s1;s2 pk1;qk2h s2 qk2jws2 nR2i= =1 N2X p;qX k1;k2U(s1) pm(k1)eik1R1(VSO)s1;s2 pk1;qk2 Us2 qn(k2)eik2R2; (5) where (VSO)s1;s2 pk1;qk2=h s1 pk1jVSOj s2 qk2i: (6) It must be noted that in the absence of SO coupling, the Bloch states are spin-degenerate, i.e. there are two states corresponding to each spatial wave-function, one with spin up,j "(r)i=j (r)i j"i , and one with spin down,j #(r)i=j (r)i j#i . The same is true for the Wannier functions, i.e. one has always the pair jw"(r)i=jw(r)i j"i ,jw#(r)i=jw(r)i j#i . In the presence of SO coupling, spin mixing occurs and each Bloch and Wannier state is, in general, a linear combination of both spin vectors. Since the Bloch states (or the Wannier ones) obtained in the absence of SO coupling form a complete basis set in the Hilbert space, the SO coupling operator can be written over such basis provided that one takes both spins into account. Therefore we use such spin- degenerate states as our basis for all calculations. B. Numerical Implementation The derivation leading to Eq. (5) is general and the nal result is simply a matrix transformation of the SO operator from the basis of the Bloch states to that of Wannier ones. Note that both basis sets are those calcu- lated in the absence of SO coupling, i.e. we have assumed that the spatial part of the basis function is not modied by the introduction of the SO interaction. For practical purposes we now we wish to re-write Eq. (5) in terms of a localized atomic-orbital basis set, i.e. we wish to make our method applicable to rst-principles DFT cal- culations implemented over local orbitals. In particular all the calculations that will follow use the Siesta pack- age, which expands the wave-function and all the opera- tors over a numerical atomic-orbital basis sets, fjs ;Rjig, wherejs ;Rjidenotes the -th atomic orbital ( is a collective label for the principal and angular momentum quantum numbers) with spin sbelonging to the cell at the position Rj.Siesta uses relativistic pseudopotentials to generate the spin-orbit matrix elements with respect to the basis vectors and truncates the range of the SO inter- action to the on-site terms35. For a nite periodic lattice comprising Nunit cells, a Bloch state is written with respect to atomic orbitals as j pki=NX j=1eikRj X Cp(k)j;Rji! ; (7) where the coecients Cp(k) are in general C-numbers. This state is normalized over unit cell with the allowed k- values beingm NK, where Kis the reciprocal lattice vector andmis an integer. Hence the SO matrix elements written with respect to the spin-degenerate Bloch states calculated in absence of SO interaction are h s1 pk1jVSOj s2 qk2i=X j;lei(k2Rlk1Rj) X ;Cs1 p(k1)Cs2 q(k2)hs1 ;RjjVSOjs2 ;Rli:(8) As mentioned above Siesta neglects all the SO ma- trix elements between atomic orbitals located at dierent atoms. This leads to the approximation hs1 ;RjjVSOjs2 ;Rli=hs1 jVSOjs2 iRj;Rl;(9)4 so that Eq. (8) becomes h s1 pk1jVSOj s2 qk2i=X jei(k2k1)Rj X ;C(s1) p(k1)C(s2) q(k2)hs1 jVSOjs2 i:(10) This can be further simplied by taking into account the relation NX j=1ei(k1k2)Rj=Nk1;k2; (11) which leads to the nal expression for the SO matrix elements h s1 pk1jVSOj s2 qk2i= =NX ;C(s1) p(k1)C(s2) q(k1)hs1 jVSOjs2 ik1;k2: (12) With the result of Eq. (12) at hand we can now come back to the expression for the SO matrix elements written over the MLWFs computed in absence of spin-orbit [see Eq. (5)]. In the case of the Siesta basis set this now reads hws1 mR1jVSOjws2 nR2i= =1 NX p;q;;X kCs1 p(k)Cs2 q(k)U(s1) pm(k)Us2 qn(k) eik(R1R2)hs1 jVSOjs2 i:(13) Finally, we go back to the continuous representation (N!1 ), where the sum over kis replaced by an inte- gral over the rst Brillouin zone hws1 mR1jVSOjws2 nR2i= =V (2)3X p;q;;Z BZCs1 p(k)Cs2 q(k)Us1 pm(k)Us2 qn(k) eik(R1R2)hs1 jVSOjs2 idk:(14) To summarize, our strategy consists in simply evaluat- ing the SO matrix elements over the basis set of the ML- WFs constructed in the absence of SO interaction. These are by denition spin-degenerate and they are in gen- eral easy to compute since associated to well-separated bands. Our procedure thus avoids to run the minimiza- tion algorithm necessary to x the Wannier's gauge over the SO-split bands, which in the case of OSCs have tiny splits. Our method is exact in the case the MLWFs form a complete set describing a particular bands manifold. In other circumstances they constitute a good approxima- tion, as long as the SO interaction is weak, namely when it does not change signicantly the spatial shape of the Wannier functions. However, for a material with strongSO coupling (eg. Pb), if the MLWFs under considera- tion do not span the entire Bloch states manifold, then the SO-split eigenvalues calculated with our method will not match those obtained directly with the rst principles calculation. C. Work ow The following procedure is adopted when calculating the SO-split band structures from the MLWFs Hamilto- nian. The results are then compared to the band struc- ture obtained directly from Siesta including SO interac- tion. 1. We rst run a self-consistent non-collinear spin- DFTSiesta calculation and obtain the band struc- ture. 2. From the density matrix obtained at step (1), we run a non self-consistent single-step Siesta calcu- lation including SO coupling. This gives us the ma- trix elementshs1jVSOjs2i. The band structure obtained in this calculation (from now on this is called the SO-DFT band structure) will be then compared with that obtained over the MLWFs. Note that we do not perform the Siesta DFT cal- culation including spin-orbit interaction in a self- consistent way. This is because the SO interaction changes little the density matrix so that such cal- culation is often not necessary. Furthermore, as we cannot run the MLWF calculation in a self- consistent way over the SO interaction, consider- ing non-self-consistent SO band structure at the Siesta level allows us to compare electronic struc- tures arising from identical charge densities. 3. Since the current version of Wannier90 imple- mented for Siesta works only with collinear spins, we run a regular self-consistent spin-polarized Siesta calculation. This gives us the coe- cientsCs n(k), which are spin-degenerate for a non- magnetic material, C" n(k) =C# n(k). 4. We run a Wannier90 calculation to construct the MLWFs associated to the band structure com- puted at point (3). This returns us the uni- tary matrix, Us pm(k), the Hamiltonian matrix el- ementshws1 mR1jH0jws2 nR2i(H0is the Kohn-Sham Hamiltonian in absence of SO interaction) and the phase factors36eikR. For a non-magnetic mate- rial the matrix elements of H0satisfy the relation hws1 mR1jH0jws2 nR2i=hwmR1jH0jwnR2is1;s2. 5. Fromhs1jVSOjs2iand theCs n(k)'s we calcu- late the matrix elements h s1 pkjVSOj s2 qkiby using Eq. (12). 6. Next we transform the SO matrix ele- ments constructed over the Bloch functions,5 FIG. 1. (Color on line) Atomic structure of (a) a plumbane molecule, (b) a chain of lead atoms and (c) a chain of methane molecules. We have also calculated the electronic structure of a chain of C atoms, which is essentially identical to that presented in (b). Color code: Pb = grey, H = light blue, C = yellow. h s1 pkjVSOj s2 qki, into their Wannier counterparts, hws1 mR1jVSOjws2 nR2i, by using Eq. (14). 7. The nal complete Wannier Hamiltonian now reads hws1 mR1jHjws2 nR2i=hws1 mR1jH0+VSOjws2 nR2i; (15) and the associated band structure can be directly compared with that computed at point (2) directly fromSiesta . III. RESULTS AND DISCUSSION We now present our results, which are discussed in the light of the theory just described. A. Plumbane Molecule We start our analysis by calculating the SO matrix elements and then the energy eigenvalues of a plumbane, PbH 4, molecule [see gure 1(a)]. Due to the presence of lead, the molecular eigenstates change signicantly when the SO interaction is switched on. For this non-periodic system the key relations in Eq. (12) and Eq. (5) reduce to h s1 pjVSOj s2 qi=X ;Cs1 pCs2 qhs1 jVSOjs2 i(16) and hws1 mjVSOjws2 ni=X p;qUs1 pmUs2 qnh s1 pjVSOj s2 qi; (17) respectively, where now the vectors s nare simply the eigenvectors with quantum number nand spins. In Table I we report the rst 10 energy eigenvalues of plumbane, calculated either with or without SO coupling.NonSO SO Siesta MLWF Siesta MLWF -33.93534 -33.93521 -33.93532 -33.93521 -33.93530 -33.93521 -33.93528 -33.93521 -13.02511 -13.02507 -14.69573 -14.69568 -13.02511 -13.02507 -14.69573 -14.69568 -13.02510 -13.02506 -12.64301 -12.64298 -13.02509 -13.02506 -12.64301 -12.64298 -13.02320 -13.02315 -12.64166 -12.64162 -13.02318 -13.02315 -12.64165 -12.64162 -5.75256 -5.75251 -5.75255 -5.75251 -5.75245 -5.75251 -5.75245 -5.75251 MRAD =4:320106MRAD =3:998106 TABLE I. The 10 lowest energy eigenvalues of a plumbane molecule calculated with (SO) and without (NonSO) spin- orbit interaction. The rst and third columns correspond to the SO-DFT Siesta calculation, while the second and the fourth to the MLWFs one. The MRAD for both cases is reported in the last row. These have been computed within the LDA (local den- sity approximation) and a double-zeta polarized basis set. The table compares results obtained with our MLWFs procedure to those computed with SO-DFT by Siesta . Clearly in this case of a heavy ion the SO coupling changes the eigenvalues appreciably, in particular in the spectral region around -13 eV. Such change is well cap- tured by our Wannier calculation, which returns energy levels in close proximity to those computed with SO-DFT bySiesta . In order to estimate the error introduced by our method, we calculate the Mean Relative Absolute Dif- ference (MRAD) , which we dene as1 NPjs iw ij js ijfor a set ofNeigenvalues ( i= 1;:::;N ), wheres iandw iare thei-th eigenvalues calculated from Siesta and the ML- WFs, respectively. Notably the MRAD is rather small both in the SO-free case and when the SO interaction is included. Most importantly, we can report that our procedure to evaluate the SO matrix elements over the MLWFs basis clearly does not introduce any additional error. Before discussing some of the properties of the SO ma- trix elements associated to this particular case of a nite molecule, we wish to make a quick remark on the Wan- nier procedure adopted here. The eigenvalues reported in Table I are the ten with the lowest energies. However, in order to construct the MLWFs we have considered all the states of the calculated Kohn-Sham spectrum. This means that, if our Siesta basis set describes PbH 4with Ndistinct atomic orbitals, then the MLWFs constructed are 2N(the factor 2 accounts for the spin degeneracy). In this case the original local orbital basis set and the constructed MLWFs span the same Hilbert space and the mapping is exact, whether or not the SO interaction is considered. In most cases, however, one wants to construct the MLWFs by using only a subset of the spectrum, for in- stance the rst N0eigenstates. Since in general the SO6 interaction mixes all states, there will be SO matrix ele- ments between the selected N0states and the remaining NN0. This means that a MLWF basis constructed only from the rst N0eigenstates will not be able to pro- vide an accurate description of the SO-split spectrum. Importantly, one in general may expect that the SO in- teraction matrix elements between dierent Kohn-Sham orbitals,h s1pjVSOj s2qi, are smaller than those calcu- lated at the same orbital, h s1njVSOj s2ni. This is be- cause of the short-range of the SO interaction and the fact that the Kohn-Sham eigenstates are orthonormal. In the case of light elements, i.e. for a weak SO potential, one may completely neglect the o-diagonal SO matrix elements. This means that the SO spectrum constructed with the MLWFs associated to the rst N0eigenstates will be approximately equal to the rst N0eigenvalues of the MLWFs Hamiltonian constructed over the entire N-dimensional spectrum. Such property is particularly relevant for OSCs, for which the SO interaction is weak. We now move to discuss a general property of the MLWF SO matrix elements, namely the relations hws mjVSOjws mi= 0 and<[hws mjVSOjws ni] = 0. This means that the SO matrix elements for the same spin and the same Wannier function vanish, while those for the same spin and dierent Wannier functions are purely imaginary. This property can be understood from the following argument. The SO coupling operator is VSO=P RjVRjLRjS, whereVRjis a scalar potential indepen- dent of spin, and LRjis the angular momentum operator corresponding to the central potential of the atom at po- sition Rj. Here Sis the spin operator and the sum runs over all the atoms. By now expanding Sin terms of the Pauli spin matrices one can see that for any vector j s ii=j ii jsi, which can be written as a tensor prod- uct of a spin-independent part, j ii, and a spinorjsi, the following equality holds h s1 mjLSj s2 ni=1 2h h mj^Lzj nis1"s2"+ +h mj^Lj nis1"s2#+h mj^L+j nis1#s2"+ +h mj^Lzj nis1#s2#i :(18) Eq. (18) can then be applied to both the Kohn-Sham eigenstates and the MLWFs, since they are both written asj s ii=j ii jsi. Now, the atomic orbitals used by Siesta have the fol- lowing form jii=jRni;lii jli;Mii;(19) wherejRn;liis a radial numerical function, while the an- gular dependence is described by the real spherial har- monic,37jl;Mi. It can be proved that the real spherical harmonics follow the relation hl;Mij^Lzjl;Mji=iMiMi;Mj:(20) Since any Kohn-Sham eigenstate, j s1pi, can be written asjii js1i, Eq. (18) implies that only the terms in Γπ/ a-25-20-15-10-505E-EF (eV)NonSO SO Γ π/ a(a)(b) σ σ∗πFIG. 2. (Color on line) Bandstructure of a 1D Pb chain cal- culated (a) with Siesta and (b) by diagonalizing the Hamil- tonian matrix constructed over the MLWFs. Black and red lines are for the bands obtained without and with SO cou- pling, respectively. The ,andbands are identied in the picture. ^Lz(or^Lz) contribute to the matrix element between same spins,h s1pjLSj s1pi. Eq. (20) together with the fact that the Kohn-Sham eigenstates are real for a nite molecule further establishes that <[h pj^Lzj qi] = 0. As a consequenceh mj^Lzj mi= 0. Finally, by keeping in mind that the unitary matrix elements transforming the Kohn-Sham eigenstates into MLWFs are real for a molecule, we have also hws1 mjLSjws1 ni=hwmj^Lzjwni= =X p6=qUpmUqnh pj^Lzj qi;(21) which has to be imaginary. Thus we have <hws1mjVSOjws1ni= 0 andhws1mjVSOjws1mi= 0 since VSOmust have real expectation values. B. Lead Chain Next we move to calculating the SO matrix elements for a periodic structure. In particular we look at a 1D chain of Pb atoms with a unit cell length of 2.55 A, which is the DFT equilibrium lattice constant obtained with the LDA. Note that free-standing mono-dimensional Pb chains have been never reported in literature, although7 there are studies of low-dimensional Pb structures encap- sulated into zeolites38. Here, however, we do not seek at describing a real compound, but we rather take the 1D Pb mono-atomic chain as a test-bench structure to apply our method to a periodic structure with a large SO cou- pling. Also in this case we have constructed the MLWFs by taking the entire bands manifold and not a subset of it. For the DFT calculations we have considered a simplesandpsingle-zeta basis set, which, in absence of SO interaction yields three bands with one of them being doubly degenerate [see Fig. 2(a)]. The doubly-degenerate relatively- at band just cuts across the Fermi energy, EF, and it is composed of the pyandpzorbitals orthogonal to the chain axis ( band). The other two bands are sp hybrid (bands). The lowest one at about 25 eV be- lowEFhas mainly scharacter (band), while the other mainlypx(band). Spin-orbit coupling lifts the degeneracy of the p-type band manifold, which is now composed of three distinct bands. In particular the degeneracy is lifted only in the band at the edge of the 1D Brillouin zone, while it also involves the one close to the point (after the band crossing). When the same band structure is calcu- lated from the MLWFs we obtain the plot of Fig. 2(b). This is almost identical to that calculated with SO-DFT demonstrating the accuracy of our method also for peri- odic system. It must be noted that for a periodic structure the Bloch state expansion coecients, Cp(k), and the el- ements of the unitary matrix Uare complex and con- sequently the diagonal elements of VSOwith respect to Wannier functions are not zero in general. However, as expectedhws1 mRjVSOjws2 nR0itends to vanish as the sepa- rationjRR0jincreases. Furthermore, it is clear from Eq. (18) that the SO matrix elements for Wannier func- tions should obey the spin-box anti-hermitian relation hws1 mRjVSOjws2 nR0i=hws2 mRjVSOjws1 nR0i:(22) These two properties can be appreciated in Fig. 3, where we plot the real [panel (a)] and imaginary [panel (b)] part ofhws1 m0jVSOjws2 nRifor some representative band combinations, mandn, as a function of R. C. Carbon Chain Next we look at the case of a 1D mono-atomic car- bon chain with a LDA-relaxed interatomic distance of s1:3A. This has the same structure and electron count of the Pb chain, and the only dierence concerns the fact that the SO coupling in C is much smaller then that in Pb. In this situation we expect that an accurate SO-split band structure can be obtained even when the MLWFs are constructed only for a limited number of bands and not for the entire band manifold as in the case of Pb. This time the DFT band structure is calculated at the LDA level over a double-zeta polarized (DZP) Siesta FIG. 3. (Color on line) The SO matrix elements of a chain of lead atoms calculated with respect to some representative Wannier functions and plotted as a function of the site index, i.e. of the distance between the Wannier function. Panels (a) and (b) show the real and imaginary components respectively. basis set, comprising 13 atomic orbitals per unit cell. In contrast, the MLWFs are constructed only from the rst four bands, which are well isolated in energy from the rest and again describe the spbands with andsym- metry. Since the SO interaction in carbon is small (the band split is of the order of a few meV) it is impossible to visualize the eects of the SO interaction in a stan- dard band plot as that in Fig. 2. Hence, in Fig. 4 we plot the dierence between the band structure calculated in the presence and in the absence of SO coupling. In par- ticular we compare the bands calculated with SO-DFT bySiesta (left-hand side panels in Fig. 4), with those obtained with the MLWFs scheme described here (right- hand side panels in Fig. 4). In the gure we have labelled the bands in order of increasing energy and neglecting the spin degeneracy. Thus, for instance, the 1and 2bands correspond to the two lowest spin sub-bands (note that the band structure of the linear carbon chain is qualita- tively identical to that of the Pb one and we can use Fig. 2 to identify the various bands). We note that the lowest bands, dened as 1and 2, do not split at all due to the SO interaction, exactly as in the case of Pb. This contrasts the behaviour of both the( 3through 6) and( 7and 8) bands, which instead are modied by the SO interaction. Notably the changes in energy of the eigenvalues is never larger then8 0246ESO-ENSO(meV)ψ1 ψ2 ψ3 ψ4 7.5 -5-2.502.55ESO-ENSO(meV)ψ5 ψ6 Γπ /a-7.5-5-2.50ESO-ENSO(meV) ψ7ψ8 Γπ /a FIG. 4. (Color on line) Dierence, ESOENSO, between the band structure of chain of carbon atoms calculated with, ESO, and without, ESO, considering SO interaction. The bands are labelled in increasing energy order without taking into account spin degeneracy. For instance the bands 1and 2 are the two spin sub-bands corresponding to the band (see Fig. 2 for notation). The left-hand side panels show results for the SO-DFT calculations performed with Siesta , while the right-hand side one, those obtained from the MLWFs. 8 meV and it is perfectly reproduced by our MLWFs representation. This demonstrates that truncating the bands selected for constructing the MLWFs is a possi- ble procedure for materials where SO coupling is weak. However, we should note that the truncation still needs to be carefully chosen. Here for instance we have con- sidered all the 2 sand 2pbands and neglected those with either higher principal quantum number (e.g. 3 sand 3p) or higher angular momentum (e.g. bands with dsymme- try originating from the p-polarized Siesta basis), which appear at much higher energies. Truncations, where one considers only a particular orbital of a given shell (say thepzorbital in an npshell), need to be carefully as- sessed since it is unlikely that a clear energy separation between the bands takes place. D. Methane Chain As a rst basic prototype of 1D organic molecular crystal we perform calculations for a periodic chain of methane molecules. We use a double-zeta polarized ba- sis set and a LDA-relaxed unit cell length of 3.45 A (the cell contains only one molecule). Similarly to the previ- ous case, the MLWFs are constructed over only the low- est 4 bands (8 when considering the spin degeneracy). When compared to the bands of the carbon chain, those of methane are much narrower. This is expected, since the bonding between the dierent molecules is small. In Fig. 5 we plot the dierence between the eigenvalues (1D band structure) calculated with, ESO, and without, ENSO, including SO interaction. 00.51ESO-ENSO(meV)ψ1ψ2ψ3ψ4 Γπ /a-101ESO-ENSO(meV)ψ5ψ6ψ7ψ8 Γπ /a FIG. 5. (Color on line) Dierence, ESOENSO, between the band structure of chain of methane molecules calculated with,ESO, and without, ESO, considering SO interaction. The bands are labelled in increasing energy order without taking into account spin degeneracy. The left-hand side pan- els show results for the SO-DFT calculations performed with Siesta , while the right-hand side one, those obtained from the MLWFs. The inset shows an isovalue plot of one of the four MLWFs with the red and blue surfaces denoting positive and negative isovalues, respectively. All the MLWFs have simi- lar structure and they resemble those of the isolated methane molecule because of the small intermolecular chemical bond- ing owing to the large separation. When SO interaction is included the spin-degeneracy is broken and one has now eight bands. These are labeled as min Fig. 5 in increasing energy order. Again we nd no SO split for the lowermost band and then a split, which is signicantly smaller than that found in the case of the C chain. This is likely to originate from the crystal eld of the C atoms in CH 4, which is dierent from that in the C chain (the C-C distance is dierent and there are additional C-H bonds). Again, as in the previous case, we nd that our MLWFs procedure perfectly reproduces the SO-DFT band structure, indicating that in this case of weak SO interaction band truncation does not introduce any signicant error. E. Triarylamine Chain Finally we perform calculations for a real system, namely for triarylamine-based molecular nanowires. These can be experimentally grown through a photo- self-assembly process from the liquid phase39, and have been subject of numerous experimental and theoretical studies34,40. In general, triarylamines can be used as materials for organic light emitting diodes, while their nanowire form appears to possess good transport and spin properties, making it a good platform for organic spintronics41. Triarylamine-based molecular nanowires self-assemble only when particular radicals are attached9 FIG. 6. (Colour on line) Structure of the triarylamine molecule (upper picture) and of the triarylamine-based nanowire investigated here. The radicals associated to the triarylamine derivative are C 8H17, H and Cl, respectively. Colour code: C=yellow, H=light blue, O=red, N=grey, Cl=green. to the main triarylamine backbone and here we consider the case of C 8H17, H and Cl radicals, corresponding to the precursor 1of Ref. [39] (see upper panel in Fig. 6). The nanowire then arranges in such a way to have the central N atoms aligned along the wire axis (see Fig. 6). In general self-assembled triarylamine-based molecular nanowires appear slightly p-doped so that charge trans- port takes place in the HOMO-derived band. This is well isolated from the rest of the valence manifold and has a bandwidth of about 100 meV (see gure Fig. 7 for the band structure). Such band is almost entirely localized on thepzorbital of the central N atoms ( pzis along the wire axis), a feature that has allowed us to construct a pz-sp2model with the spin-orbit strength extracted from that of an equivalent mono-atomic N chain. The model was then used to calculate the temperature-dependent spin-diusion length of such nanowires42. Here we wish to use our MLWFs method to extract the SO matrix ele- ments of triarylamine-based molecular nanowires in their own chemical environment, i.e. without approximating the backbone with a N atomic chain. For this system we use a 1D lattice with LDA- optimized lattice spacing of 4.8 A and run the DFT cal- culations with double-zeta polarized basis and the LDA functional. The MLWFs are constructed by using only the HOMO-derived valence band, i.e. we have a single spin-degenerate Wannier orbital. We can then drop the FIG. 7. (Color on line) Band structure of the 1D triarylamine- based nanowire constructed with the precursor 1of Ref. [39]. This is plotted over the 1D Brillouin zone (Z= =awithathe lattice parameter). The Fermi level is marked with a dashed black line and it is placed just above the HOMO-derived va- lence band (in red). The lower panel is a magnication of the valence band. Note the bandwidth of about 100 meV and the fact that the band has a cosine shape, ngerprint of a single-orbital nearest-neighbour tight-binding-like interac- tion. Only the HOMO band is considered when constructing the MLWFs. band index and write the SO matrix elements as hws1 0jVSOjws2 Ri =V (2)3Z dkU(k)U(k)eik:Rh s1 kjVSOj s2 ki =V (2)3Z dkeik:Rh s1 kjVSOj s2 ki;(23) or in a discrete representation of the reciprocal space hws1 0jVSOjws2 Ri=1 NX keik:Rh s1 kjVSOj s2 ki(24) where the second equality comes from the unitarity of the gauge transformation, U(k). In Fig. 8 we plot the dierence between the band struc- ture computed by including SO interaction and those cal- culated without. Notably our MLWFs band structure is almost identical to that computed directly with SO-DFT, again demonstrating both the accuracy of our method and the appropriateness of the drastic band truncation used here. In this particular case the SO band split is maximized half-way between the point and the edge of the 1D Brillouin zone, where it takes a value of approxi- mately 80eV. Clearly such split is orders of magnitude smaller than the value that one can possibly calculate by a direct construction of the MLWFs from the SO- splitted band structure. Note also that the SO split of10 Γπ /a-0.04-0.0200.020.04ESO -ENonSO (meV) DFT WANNIER FIG. 8. (Color on line) Plot of (E SOENSO) as a function of k in arbitrary unit over a brillouin zone for the highest occupied band of a 1-d chain of triarylamine derivatives. The blue and the red points correspond to calculations with Siesta and Wannier90 respectively. the valence band is calculated here approximately a fac- tor ten smaller than that estimated previously for a N atomic chain42, indicating the importance of the details of the chemical environment in these calculations. Finally we take a closer look at the calculated SO ma- trix elements. As mentioned earlier, in the Siesta on- site approximation35only the matrix elements calculated over orbitals centred on the same atom do not vanish. As a consequence the components hws1 RjVSOjws2 R0idrop to zero asjRR0jgets large. This can be clearly ap- preciated in Fig. 9(a) and Fig. 9(b), where we plot the SO matrix elements for same and dierent spins, respec- tively. From Fig. 9(a) we can observe that <hws1 0jVSOjws1 Ri vanishes for all R. This can be understood in the following way. In general any expectation value of VSO,h s kjVSOj s ki, has to be real. This is in fact anti-symmetric with respect to k, i.e we have h s 0+kjVSOj s 0+ki=h s 0kjVSOj s 0ki, where k=0 denotes the point of the Brillouin zone. Addition- ally,eikRsatises the relation ei(0+k)R= ei(0k)R. Hence, by performing the k-sum over rst Brillouin zone we can write <hws1 0jVSOjws1 Ri=<X keik:Rh s1 kjVSOj s1 ki= 0; (25) wherehws1 0jVSOjws1 0iis the expectation value of VSO and must be real. This implies hws1 0jVSOjws1 0i= 0:(26) We can also see from Fig. 9(b) that for triarylamine the matrix elements hws1 RjVSOjws2 Riare almost zero for s16=s2. This follows directly from Eq. (18). In fact in the particular case of triarylamine nanowires the Wannier FIG. 9. (Color on line) SO matrix elements of a triarylamine- based nanowire calculated with respect to the Wannier func- tions obtained from the HOMO band. Panels (a) and (b) correspond to matrix elements calculated between for same and dierent spins, respectively. functions are constructed from one band only. As such, in order to have a non-zero matrix element, hws1 RjVSOjws2 Ri, we must have non-zero values for hwRj^LjwRi. There- fore, the band under consideration must contain an appreciable mix of components of both the jl;piand jl;p+ 1icomplex spherical harmonics for some landp. As mentioned earlier, the triarylamine HOMO band is composed mostly of pzN orbitals. Hence, it has to be expected that the hws1 RjVSOjws2 Rimatrix elements are small. IV. CONCLUSION We have presented an accurate method for obtaining the SO matrix elements between the MLWFs constructed in absence of SO coupling. Our procedure, implemented within the atomic-orbital-based DFT code Siesta , allows one to avoid the construction of the Wannier functions over the SO-split band structure. In some cases, in par- ticular for organic crystals, such splits are tiny and a di- rect construction is numerically impossible. The method is then put to the test for a number of materials systems, going from isolated molecules, to atomic nanowires, to11 1D molecular crystals. When the entire band manifold is used for constructing the MLWFs the mapping between Bloch and Wannier orbitals is exact and the method can be used for both light and heavy elements. In contrast for weak spin-orbit interaction one can construct the ML- WFs on a subset of the states in the band structures without any loss of accuracy. As such our scheme ap- pears as an important tool for constructing eective spin Hamiltonians for organic materials to be used as input in a multiscale approach to the their thermodynamical properties.ACKOWLEDGEMENT This work is supported by the European Research Council, Quest project. Computational resources have been provided by the supercomputer facilities at the Trin- ity Center for High Performance Computing (TCHPC) and at the Irish Center for High End Computing (ICHEC). Additionally, the authors would like to thank Ivan Rungger and Carlo Motta for helpful discussions and Akinlolu Akande for providing the structure of the triarylamine-based nanowire. 1S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Moln ar, M. L. Roukes, A. Y. Chtchelka- nova, and D. M. Treger, Science 294, 1488 (2001). 2S. Ornes, Proc. Natl. Acad. Sci. USA 110, 3710 (2013). 3B. Behin-Aein, D. Datta, S. Salahuddin, and S. Datta, Nature Nanotech. 5, 266 (2010). 4G. Prinz and K. Hathaway, Phys Today 48, 24 (1995). 5D. D. Awschalom and M. E. Flatt e, Nature Physics 3, 153 (2007). 6I.Zuti c and M. Fuhrer, Nature Physics 1, 85 (2005). 7G. Horowitz, Organic Transistors , edited by H. Klauk (Wiley-VCH Verlag GmbH & Co. KGaA, 2006). 8C. Joachim, J. K. Gimzewski, and A. Aviram, Nature 408, 541 (2000). 9V. Dediu, M. Murgia, F. Matacotta, C. Taliani, and S. Barbanera, Solid State Commun. 122, 181 (2002). 10Z. H. Xiong, D. Wu, Z. Valy Vardeny, and J. Shi, Nature 427, 821 (2004). 11S. Sanvito, Chem. Soc. Rev. 40, 3336 (2011). 12C. K. Chiang, C. R. Fincher, Y. W. Park, A. J. Heeger, H. Shirakawa, E. J. Louis, S. C. Gau, and A. G. MacDi- armid, Phys. Rev. Lett. 39, 1098 (1977). 13S. R. Forrest, Nature 428, 911 (2004). 14S. Sanvito and A. R. Rocha, J. Comput. Theor. Nanosci. 3, 624 (2006). 15S. Pramanik, C. G. Stefanita, S. Patibandla, S. Bandy- opadhyay, K. Garre, H. N., and M. Cahay, Nature Nan- otech. 2, 216 (2007). 16K. Tsukagoshi, B. W. Alphenaar, and H. Ago, Nature 401, 572 (1999). 17G. Szulczewski, S. Sanvito, and J. M. D. Coey, Nature Materials 8, 693 (2009). 18B. Dieny, V. Speriosu, B. Gurney, S. Parkin, D. Wilhoit, K. Roche, S. Metin, D. Peterson, and S. Nadimi, Journal of Magnetism and Magnetic Materials 93, 101 (1991). 19I.Zuti c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). 20V. I. Krinichnyi, Synth. Met. 108, 173 (2000). 21S. Bandyopadhyay, Phys. Rev. B 81, 153202 (2010). 22A. J. Drew et al. , Nature Materials 8, 109 (2009). 23F. J. Wang, C. G. Yang, Z. V. Vardeny, and X. G. Li, Phys. Rev. B 75, 245324 (2007). 24V. Dediu, L. E. Hueso, I. Bergenti, A. Riminucci, F. Bor- gatti, P. Graziosi, C. Newby, F. Casoli, M. P. De Jong, C. Taliani, and Y. Zhan, Phys. Rev. B 78, 115203 (2008). 25C. Cohen-Tannoudji, B. Diu, and F. Lalo_ e, Quantum me- chanics. 2 , Textbook physics (John Wiley & Sons, 1977).26A. Troisi, J. Chem. Phys. 134, 034702 (2011). 27C. Motta and S. Sanvito, J. Chem. Theo. Comp. 10, 4624 (2014). 28G. H. Wannier, Phys. Rev. 52, 191 (1937). 29G. H. Wannier, Rev. Mod. Phys. 34, 645 (1962). 30N. Marzari and D. Vanderbilt, Phys. Rev. B 56, 12847 (1997). 31N. Marzari, A. A. Mosto, J. R. Yates, I. Souza, and D. Vanderbilt, Rev. Mod. Phys. 84, 1419 (2012). 32J. M. Soler, E. Artacho, J. D. Gale, A. Garc a, J. Junquera, P. Ordej on, and D. S anchez-Portal, J. Phys. Condens. Matter 14, 2745 (2002). 33A. A. Mosto, J. R. Yates, Y.-S. Lee, I. Souza, D. Vander- bilt, and N. Marzari, Comput. Phys. Commun. 178, 685 (2008). 34Z. Ning and H. Tian, Chem. Commun. 37, 5483 (2009). 35L. Fern andez-Seivane, M. A. Oliveira, S. Sanvito, and J. Ferrer, J. Phys. Condens. Matter 18, 7999 (2006). 36The correctness of the elements Us pm(k) andeikRis easily veried by ensuring that the following relation is satised hwmR1jwnR2i=1 NX pZ FBZdkhwmR1j pkih pkjwnR2i= =1 NX pZ FBZdkU pm(k)Upn(k)eik(R1R2)= =m;nR1;R2: (27) . 37The real spherical harmonics are constructed from the complex ones,jl;mi, asjl;Mi=1p 2[jl;mi+(1)mjl;mi] andjl;Mi=1 ip 2[jl;mi(1)mjl;mi]. ForM= 0 the real and complex spherical harmonics coincide. 38S. Romanov, Journal of Physics: Condensed Matter 5, 1081 (1993). 39E. Moulin, F. Niess, M. Maaloum, E. Buhler, I. Nyrkova, and N. Giuseppone, Angew. Chem. Int. Ed. 49, 6974 (2010). 40V. Faramarzi, F. Niess, E. Moulin, M. Maaloum, J.-F. Dayen, J.-B. Beaufrand, S. Zanettini, B. Doudin, and N. Giuseppone, Nature Chemistry 4, 485 (2012). 41A. Akande, S. Bhattacharya, T. Cathcart, and S. Sanvito, J. Chem. Phys 140, 074301 (2014). 42S. Bhattacharya, A. Akande, and S. Sanvito, Chem. Com- mun. 50, 6626 (2014).
1502.05683v1.Enhanced_photogalvanic_effect_in_graphene_due_to_Rashba_spin_orbit_coupling.pdf	arXiv:1502.05683v1 [cond-mat.mes-hall] 19 Feb 2015Enhanced photogalvanic eﬀect in graphene due to Rashba spin -orbit coupling M. Inglot,1V. K. Dugaev,1,2E. Ya. Sherman,3,4and J. Barna´ s5,6 1Department of Physics, Rzesz´ ow University of Technology, al. Powsta´ nc´ ow Warszawy 6, 35-959 Rzesz´ ow, Poland 2Departamento de F´ ısica and CeFEMA, Instituto Superior T´ e cnico, Universidade de Lisboa, av. Rovisco Pais, 1049-001 Lisbon, Portugal 3Department of Physical Chemistry, Universidad del Pa´ ıs Va sco UPV-EHU, 48080 Bilbao, Spain 4IKERBASQUE Basque Foundation for Science, Bilbao, Spain 5Faculty of Physics, Adam Mickiewicz University, ul. Umulto wska 85, 61-614 Pozna´ n, Poland 6The Nano-Bio-Medical Centre, Umultowska 85, 61-614 Pozna´ n, Poland (Dated: August 12, 2021) We analyze theoretically optical generation of a spin-pola rized charge current (photogalvanic eﬀect)andspinpolarization ingraphene withRashbaspin-o rbitcoupling. Anexternalmagnetic ﬁeld is applied in the graphene plane, which plays a crucial role i n the mechanism of current generation. We predict a highly eﬃcient resonant-like photogalvanic eﬀ ect in a narrow frequency range which is determined by the magnetic ﬁeld. A relatively less eﬃcient p hotogalvanic eﬀect appears in a broader frequency range, determined by the electron concentration and spin-orbit coupling strength. PACS numbers: 72.25.Fe, 78.67.Wj, 81.05.ue Introduction: Two-dimensional electron systems with spin-orbit (SO) coupling are currently of a broad interest due to the coupled charge and spin dynamics, as revealed in a variety of spin related transport phenomena [1–4]. Owing to the SO coupling, the spin dynamics canbe gen- erated, among others, by a low-frequency electric ﬁeld [5] aswell as opticallyby interband electronictransitions[6– 8]. Moreover,an externalstatic magneticﬁeld canenable the current generation by light absorption, leading to a photogalvanic eﬀect [9]. Many of the spin related phenomena, including also the ones mentioned above, can be observed in two di- mensional graphene monolayers and other graphene-like materials like silicene for instance. The huge inter- est in graphene is related mainly to its natural two- dimensionality, very unusual electronic structure, and high electron mobility which ensures its excellent trans- port properties [10, 11]. Even though the intrinsic SO interaction in free-standing graphene is negligibly small, the Rashba spin-orbit coupling can be rather strong for graphene deposited on certain heavy-element sub- strates [12–14]. Since the electronic band structure of graphene is signiﬁcantly diﬀerent from that of a sim- ple two-dimensional electron gas (2DEG), and the spin- orbit coupling creates a gap in the electronic spectrum, graphene can reveal qualitatively new eﬀects which can not be observed in 2DEG in conventional semiconductor heterostructures. Additionally, the SO-related phenom- enaingraphenearealsoimportantfromthe pointofview of potential applications in all-graphene based spintron- ics devices [15–17]. In this letter we predict an enhanced photogalvanic ef- fect in graphene. To do this we consider the charge and spin currents generated in graphene by optical pump- ing in the infrared photon energy region, and show thatthe optical pumping can be used to generate in graphene not only the spin density [18], but also a spin-polarized net current. An external magnetic ﬁeld applied in the graphene plane plays an important role in the mecha- nism of current generation. We show that the eﬃciency of current generation per absorbed photon can be very high at certain conditions. Apart from this, we also show that one can create spin density without creating electric current, but not vice versa . Model:We consider low-energyelectronic spectrum of graphene in the vicinity of the Dirac points [11]. Addi- tionally, we include the Rashba SO coupling [19] and the Zeemanenergyin aweakexternal in-planemagnetic ﬁeld B. The corresponding Hamiltonian can be then written in the form ˆH=/planckover2pi1v0(±τxkx+τyky)+λ(±τxσy−τyσx)+∆ 2(b·σ),(1) wherev0≃108cm/s is the electron velocity in graphene, τxandτyare the Pauli matrices deﬁned in the sublattice space, ∆ ≡gBis the maximum Zeeman splitting, b≡ B/B,while the + and −signs refer to the KandK′ Dirac points, respectively. Furthermore, g=gLµBwith gL= 2 being the Land´ e factor, and λ=α/2 withα standing for the Rashba coupling constant [20]. The electronicspectrum correspondingto Hamiltonian (1) consists of four energy bands in each valley, Enk, wheren= 1 ton= 4 is the band index. The correspond- ingspectrumfor B= 0isshowninFig.1a, andisgivenby the formula En,k(B= 0) =∓λ±/radicalbig λ2+/planckover2pi12v2 0k2. In turn, the expectation value of the spin in the absence of mag- netic ﬁeld is oriented perpendicularly to the wavevector k[20], similarly as in a semiconductor-based 2DEG with Rashba SO coupling. Contrary to the 2DEG, there are, however, no eigenstates of Hamiltonian (1) with a deﬁ- nite eigenvalue of any spin component. This appears due2 Figure 1: (Color online). (a) Schematic picture of the band structure of graphene with Rashba spin-orbit interaction w ith all possible band transitions for a chosen chemical potenti al, indicated by the vertical arrows. Circles with crosses and dots inside correspond to the opposite spin orientations in the subbands. (b) Dispersion of the low energy states for indicated wavevector orientations. The band index is also marked at the plots. The assumed magnetic ﬁeld B/bardblxis equal to 5 T. to the speciﬁc form of the SO coupling in graphene. The corresponding spin components in the absence of mag- netic ﬁeld and for the wavevector k≡k(cosθ,sinθ) are /angbracketleftσx(k)/angbracketrightB=0=∓vk v0sinθ, (2a) /angbracketleftσy(k)/angbracketrightB=0=±vk v0cosθ, (2b) wherevk=v0×/planckover2pi1v0k//radicalbig λ2+/planckover2pi12v2 0k2istheabsolutevalue of the electron velocity in the absence of magnetic ﬁeld. Here the upper and lower signs correspond to the bands (1,4) and (2,3), respectively. Note, both spin compo- nents given by Eqs. (2a) and (2b) vanish in the limit of k= 0 [20] corresponding to the mixed rather than pure character of the band states in the spin subspace. The electronic spectrum presented in Fig.1a is signif- icantly modiﬁed by an external in-plane magnetic ﬁeld. Assume this ﬁeld is oriented along the axis x. The ex- act electronic spectrum can be then obtained by directdiagonalization of the Hamiltonian (1), and is shown in Fig.1b for wavevectors along the axis xandy. Only the states corresponding to the bands labeled in Fig.1a with the index 2 and 3 are shown there. As one can note, for one propagation orientation the electron bands are shifted vertically, i.e. to higher (lower) energy, while for the second propagation orientation the bands are shifted horizontally, i.e. to left (right) from the point k= 0. The latter separation in the k-space of the bands 2 and 3 is crucial for the enhanced photogalvanic eﬀect. For a weak Zeeman energy, ∆ ≪α, and for electron momenta of our interest, k≫√ α∆//planckover2pi1v0, the ﬁeld-dependent cor- rection to the electron energy, calculatedby perturbation theory, has the form En,k(B)−En,k(B= 0) =∓∆ 2vk v0sinθ,(3) where again the upper (lower) sign corresponds to the bands (1,4) and (2,3), respectively. Assume now that the system is subject to electromag- netic irradiation. Hamiltonian describing interaction of electrons in graphene with the external periodic electro- magnetic ﬁeld, A(t) =A0e−iωt, takes the form ˆHA=∓e cv0(τ·A). (4) As in Eq. (1), diﬀerent signs correspondhere to electrons within the KandK′valleys. The injection rate of a quantity O, related to the inter- subband optical transitions, can be calculated by using the Fermi’s golden rule, O(ω) =/summationdisplay n,n′On→n′(ω), On→n′(ω) =2π /planckover2pi1/integraldisplayd2k (2π)2/vextendsingle/vextendsingle/vextendsingle/angbracketleftΨnk\|ˆHA\|Ψn′k/angbracketright/vextendsingle/vextendsingle/vextendsingle2/hatwideOn→n′ ×δ(Enk+/planckover2pi1ω−En′k)f(Enk)[1−f(En′k)],(5) wheref(Enk) is the Fermi-Dirac distribution function. Since there are two valleys, KandK′, one needs to cal- culate contributions to O(ω) from both of them. The quantities of our interest here are: /hatwideOn→n′≡/hatwide11, (6) forthelightabsorption(where /hatwide11istheidentityoperator), /hatwideOn→n′≡ /angbracketleftΨn′k\|σν\|Ψn′k/angbracketright−/angbracketleftΨnk\|σν\|Ψnk/angbracketright,(7) for the corresponding spin component injection, and /hatwideOn→n′≡ /angbracketleftΨn′k\|ˆIi\|Ψn′k/angbracketright−/angbracketleftΨnk\|ˆIi\|Ψnk/angbracketright(8) for the current injection. Here, ˆIi≡eˆvi, whereeis the electron charge, while ˆ vx≡ ±v0τxand ˆvy≡v0τy. The injected spin current, in turn, can be calculated as [21] /hatwideOn→n′≡ /angbracketleftΨn′k\|ˆJν i\|Ψn′k/angbracketright−/angbracketleftΨnk\|ˆJν i\|Ψnk/angbracketright,(9)3 Figure 2: (Color online) (a) Normalized charge current ˜Iy, in the case of low temperature, T= 1 K (T= 10 K in the in- set). Rashba spin-orbit coupling strength is α= 4 meV (solid red line) and α= 13 meV (dashed blue line). The chemical potential is µ= 5 meV, Bis in the plane of graphene and along the x-axis, while A\|\|B. (b) Contributions of indicated intersubband transitions to the total current presented in (a) forα= 4 meV. whereˆJν i= [σν,ˆvi]+/2. Below we concentrate on the results for injection of charge current and spin density. Results: Using equations for the injection rate one can calculate the charge current and spin polarization induced by the optical pumping. Let us begin with the photogalvanic eﬀect, i.e. charge current generation. Re- sultsfortwodiﬀerentpolarizationsoftheelectromagnetic ﬁeldA(t) are presented in Figs. 2 and 3. Here the in- jection eﬃciency ˜Iiis deﬁned as ˜Ii≡Ii/ev0I0, where I0=πe2Q//planckover2pi1candQis the incident photon ﬂux [22–24]. The transitions start at /planckover2pi1ω≈2µ−αifµ≥αand at /planckover2pi1ω≈2µotherwise. In both cases a strong narrow peak in the injection eﬃciency appears at a resonant energy /planckover2pi1ω≈2µ. This peak is remarkably higher for the electro- magneticﬁeldpolarizedalongthestaticmagneticﬁeld B, compareFigs. 2 and 3. To understand originof this peak lat us consider transitions between the subbands marked withn= 2 and n= 3. First, we determine the shape of the isoenergetical line corresponding to a given Fermi energy,µ≫∆. For the band corresponding to n= 3, one obtains from Eq. (3) the ﬁrst-order correction to theFigure 3: (Color online) Normalized charge current ˜Iy, for a low temperature, T= 1 K ( T= 10 K in the inset). Solid red line is for α= 4 meV and dashed blue line is for α= 13 meV. Chemical potential µ= 5 meV, Bis along the x-axis andA⊥B. Fermi wavevector, /planckover2pi1kF=/radicalbig µ2+2λµ v0∓∆ 2v0sinθ, (10) whichsets the followingboundariesforthe Fermi surface: −/radicalbig µ2+2λµ v0−∆ 2v0</planckover2pi1kF,y</radicalbig µ2+2λµ v0−∆ 2v0 (11a) −/radicalbig µ2+2λµ v0</planckover2pi1kF,x</radicalbig µ2+2λµ v0. (11b) Due to the k-dependent Zeeman term, the Fermi sur- face becomes considerably deformed and anisotropic, as shown in the left panel of Fig.4. The maximum deforma- tion is independent of the chemical potential and spin- orbit coupling. In turn, the resonance line determined by E3,k−E2,k=/planckover2pi1ωis still a circle given by the condition /planckover2pi1kω=/radicalbig /planckover2pi12ω2/4+λ/planckover2pi1ω v0. (12) A part of the resonance line is inside the occupied region. Therefore, we have an interval of the photon energies, (/planckover2pi1ω1,/planckover2pi1ω2), as shown in the right panel of Fig.4, where the transitions occur for positive values of ky, while the transitions with negative ky(which would compensate partly current) are forbidden. As a result, a very eﬃcient currentinjection occurs in this photon energy window, as visible in Fig. 2. In the considered regime of µ≫∆, this photon energy interval is determined by the conditions /planckover2pi1ω1= 2µ−∆vkF v0, (13a) /planckover2pi1ω2= 2µ+∆vkF v0, (13b)4 Figure 4: (Color online). Left: Schematic pictureof the Fer mi line (solid) and resonance line (dashed) for the chosen sub- bands. Optical transitions are possible only at the part of t he dashed line outside the ﬁlled area. Each transition generat es a current of the order of 2 evkF, making the generation highly eﬃcient. Right: Side view on the intersubband transitions, which can occur in the frequency interval ω1≤ω≤ω2. which results in the peak width given by the formula, /planckover2pi1(ω2−ω1) = 2/radicalbig µ2+2λµ µ+λ∆. (14) With the increase in temperature to T >∆, this eﬀect becomessmearedoutbythermalbroadeningoftheFermi distribution, and the injection rate decreases as shown in the insets to Figs. 2 and 3. Similar arguments can be also applied to the transi- tions between n= 1 and n= 4 subbands. As a result, one gets a relatively small negative peak in the injection rate at/planckover2pi1ω≈2µ, see Fig. 2 (b). The weakness of this in- jection channel is due to a relatively small Fermi velocity in the subband 4 at µ−α≪α, while its reversed sign is due to the opposite spin orientation in these subbands, which results (similar to Eq. (11a) and Fig. 4) in a dif- ferent shape of the Fermi surface, where the transitions begin to occur at ky<0. In the limit α≪µ, the positive and negative contributions compensate each other, and the current injection eﬃciency tends to zero, as expected in the absence of spin-orbit coupling. Having discussed the strong peaks in the current injec- tion rate, let us consider now brieﬂy the broad structure. It is formed by momentum dependence of the matrix el- ements and velocity, and has the eﬃciency of the order of ∆/α. The current injection stops at /planckover2pi1ω≈2µ+α, wherethecontributionsduetotransitionsbetween diﬀer- ent bands compensate each other. We also mention that forB/bardblx, the charge current has only the y-component for both polarizations of the incident light. Now let us address brieﬂy the problem of spin and spin current injection. For both incident light polarizations oneobtainsanetspinpolarizationalongthe xandyaxes. The numerical results are presented in Fig. 5 (a) for the total spin polarization Sx, while Fig. 5(b) shows injected spin polarization associated with speciﬁc optical transi- tions. Physical mechanism of the optically injected spin polarization is rather clear, since the spin-ﬂip transitionsFigure 5: (Color online). (a) Total injected normalized spi n polarization ˜Sx=/summationtext n,n′˜Sn→n′ x. Here the Rashba SO cou- plingα= 4 meV (solid red line) and α= 13 meV (dashed red line), µ= 5 meV and B= 5 T. The orientations of A andBare parallel to the x−axis. (b) Transition-related spin injection ˜Sn→n′ x. are related to the above-mentioned fact that the eigen- states of Hamiltonian (1) are not the spin eigenstates, and the broken in magnetic ﬁeld time-reversal symme- try allows one to inject spin density. Since the charge current is along the y-axis, we obtain eﬀectively a spin- polarized current transferring in-plane spin components in they−direction. As concerns the spin current deﬁned in Eq. (9), it is symmetric with respect to the time re- versal and, therefore, magnetic ﬁeld produces there only changes proportional to B2. Summary: We have calculated optical injection of charge current in graphene as the photogalvanic eﬀect due to spin-orbit coupling [25]. The current is injected only in a ﬁnite range of infrared light frequencies, de- termined by the chemical potential µand the spin-orbit coupling strength. The striking feature of the injection is a narrow peak at the resonant frequency /planckover2pi1ω≈2µ, where the current injection can be very eﬃcient. Com- paring the ω-dependence of the current and spin injec- tion, we conclude that, depending on the light frequency, one caninject either spin-polarizednet electric currentor net spin polarization without the current injection. This result can be applied to a controllable current generation in spin-orbit coupled graphene.5 Acknowledgements. This work is supported by the Na- tional Science Center in Poland under Grant No. DEC- 2012/06/M/ST3/00042. The work of MI is supported by the project No. POIG.01.04.00-18-101/12. The work of EYS was supported by the University of Basque Country UPV/EHU under program UFI 11/55, Spanish MEC (FIS2012-36673-C03-01), and ”Grupos Consolida- dos UPV/EHU del Gobierno Vasco” (IT-472-10). [1]Spin Physics in Semiconductors , (M. I. Dyakonov, Ed.) Springer Series in Solid-State Sciences (Springer, Berlin , 2008). [2] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, arXiv:1411.3249. [3] A. G. Aronov and Yu. B. Lyanda-Geller, JETP Lett. 50, 431 (1989). [4] V. M. Edelstein, Solid State Communications 73233 (1990). [5] E. I. Rashba and Al. L. Efros, Phys. Rev. Lett. 91, 126405 (2003). [6] M. J. Stevens, A. L. Smirl, R. D. R. Bhat, A. Najmaie, J. E. Sipe, and H. M. van Driel, Phys. Rev. Lett. 90, 136603 (2003). [7] Y. K. Kato, R. C. Myers, A. C. Gossard and D. D. Awschalom, Phys. Rev. Lett. 93, 176601 (2004). [8] A. Y. Silov, P. A. Blajnov, J. H. Wolter, R. Hey, K. H. Ploog, and N. S. Averkiev, Appl. Phys. Lett. 85, 5929 (2004). [9] S. D. Ganichev, and W. Prettl, J. Phys. Cond. Matter. 15, R935 (2003).[10] A. K. Geim and K. S. Novoselov, Nature Mater. 6, 183 (2007). [11] M. I. Katsnelson, Graphene: Carbon in Two Dimensions (Cambridge Univ. Press, 2012). [12] Y. S. Dedkov, M. Fonin, U. R¨ udiger, and C. Laubschat, Phys. Rev. Lett. 100, 107602 (2008). [13] A. Varykhalov, J. S´ anchez-Barriga, A. M. Shikin, C. Biswas, E. Vescovo, A. Rybkin, D. Marchenko, and O. Rader, Phys. Rev. Lett. 101, 157601 (2008). [14] M. Zarea andN. SandlerPhys. Rev.B 79, 165442 (2009). [15] B. Trauzettel, D. V. Bulaev, D. Loss, and G. Burkard, Nature Phys. 3, 192 (2007). [16] W. Han, R. K. Kawakami, M. Gmitra, and J. Fabian, Nature Nanotechnology 9794 (2014). [17] P. Seneor, B. Dlubak, M.-B. Martin, A. Anane, H. Jaf- fres, and A. Fert, MRS Bulletin 37, 1245 (2012). [18] M. Inglot, V. K. Dugaev, E. Y. Sherman, and J. Barna´ s, Phys. Rev. B 89, 155411 (2014). [19] C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005). [20] E. I. Rashba, Phys. Rev. B 79, 161409 (2009). [21] J. RiouxandG. Burkard, Phys.Rev.B 90, 035210 (2014) [22] V. P. Gusynin and S. G. Sharapov, Phys. Rev. B 73, 245411 (2006). [23] A. B. Kuzmenko, E. van Heumen, F. Carbone, and D. van der Marel, Phys. Rev. Lett. 100, 117401 (2008). [24] P. Ingenhoven, J. Z. Bern´ ad, U. Z¨ ulicke, and R. Egger, Phys. Rev. B 81, 035421 (2010). [25] The proposed mechanism is qualitatively diﬀerent from the coherent control approach of D. Sun, C. Divin, J. Rioux, J. E. Sipe, C. Berger, W. A. de Heer, P. N. First, and T. B. Norris, Nano Lett. 101293 (2010).
1108.1806v3.Spin_Orbital_Locking__Emergent_Pseudo_Spin__and_Magnetic_order_in_Honeycomb_Lattice_Iridates.pdf	arXiv:1108.1806v3 [cond-mat.str-el] 10 May 2012Spin-Orbital Locking, Emergent Pseudo-Spin, and Magnetic order in Honeycomb Lattice Iridates. Subhro Bhattacharjee1,2, Sung-Sik Lee2,3, and Yong Baek Kim1,4 1Department of Physics, University of Toronto, Toronto, Ont ario, Canada M5S 1A7. 2Department of Physics & Astronomy, McMaster University, Ha milton, Ontario, Canada L8S 4M1. 3Perimeter Institute for Theoretical Physics, Waterloo, On tario, Canada N2L 2Y5. 4School of Physics, Korea Institute for Advanced Study, Seou l 130-722, Korea. (Dated: October 25, 2018) The nature of the eﬀective spin Hamiltonian and magnetic ord er in the honeycomb iridates is explored by considering a trigonal crystal ﬁeld eﬀect and sp in-orbit coupling. Starting from a Hubbard model, an eﬀective spin Hamiltonian is derived in te rms of an emergent pseudo-spin-1/2 moment in the limit of large trigonal distortions and spin-o rbit coupling. The present pseudo-spins arise from a spin-orbital locking and are diﬀerent from the jeﬀ= 1/2 moments that are obtained when the spin-orbit coupling dominates and trigonal distor tions are neglected. The resulting spin Hamiltonian is anisotropic and frustrated by further neigh bour interactions. Mean ﬁeld theory suggests a ground state with 4-sublattice zig-zagmagnetic order in a parameter regime that can be relevant to the honeycomb iridate compound Na 2IrO3, where similar magnetic ground state has recently been observed. Various properties of the phase, th e spin-wave spectrum and experimental consequences are discussed. The present approach contrast s with the recent proposals to understand iridate compounds starting from the strong spin-orbit coup ling limit and neglecting non-cubiclattice distortions. I. INTRODUCTION Interplay between strong spin-orbit (SO) coupling and electron-e lectron interaction in correlated electron systems has been a recent subject of intensive study1–24. In particular, 5 dtransition metal (e.g. Iridium (Ir) or Osmium (Os)) oxides are regarded as ideal playgrounds for observing such coop erative eﬀects1–24. Compared to 3 dtransition metal oxides, the repulsive Coulomb energy scale in these systems is reduc ed by the much larger extent of 5 dorbitals, while the SO coupling is enhanced due to high atomic number ( Z= 77 for Ir and z= 76 for Os). Moreover, owing to the extended 5 dorbitals, these systems are very sensitive to the crystal ﬁelds. A s a result, the energy scales mentioned above often become comparable to each other, leading to a variety of competing phases. Precisely for this reason, one expects to see newer emergent quantum phases in such syste ms. Indeed, there have been several theoretical proposals, in context of concrete experimental examples, for sp in liquids1,11,12,14,16,18,19, topological insulators10,19,20, Weyl semimetals21,22, novel magnetically ordered Mott insulators2–6,23,24and other related phases8,9,23in Iridium and Osmium oxides. A typical situation in the iridatesconsist of Ir+4atoms sitting in the octahedralcrystal ﬁeld of a chalcogen, typica lly oxygen or sulphur1–4,7. This octahedral crystal ﬁeld splits the ﬁve 5 dorbitals of Ir into the doubly degenerate eg orbitals and the triply degenerate t2gorbitals(each orbital has a further two-fold spin degeneracy). T heegorbitalsare higher in energy with the energy diﬀerence being approximately 3 eV. There are 5 electrons in the outermost 5 dshell of Ir+4which occupy the low lying t2gorbitals and the low energy physics is eﬀectively described by projec ting out the empty egorbitals25. A characteristic feature of most of the approaches used to und erstand these compounds is to treat the SO coupling as the strongest interactionat the atomic lev el;i.e., by consideringthe eﬀect ofextremely strong SO coupling for electrons occupying the t2gorbitals. This decides the nature of the participating atomic orbitals in the low energy eﬀective theory. In this limit, the orbital angular mom entum, projected to the t2gmanifold, carries an eﬀective orbital angular momentum leﬀ= 1 with a negative SO coupling constant2,3,15. The projected SO coupling splits the t2gmanifold into the lower jeﬀ= 3/2 quadruplet and the upper jeﬀ= 1/2 doublet. Out of the ﬁve valence electrons, four ﬁll up the quadruplet sector leaving the doublet se ctor half ﬁlled. Thus, in the limit of very strong SO coupling, the half ﬁlled doublet sector emerge as the correct low ene rgy degrees of freedom. Considering the eﬀect of coulomb repulsion within a Hubbard model description and perform ing strong coupling expansion, various spin Hamiltonians for jeﬀ= 1/2 are then derived within a strong-coupling perturbation theory11,20,23. In this paper, we, however, consider a diﬀerent limit where the oxyg en octahedra surrounding the Ir+4ions are highly distorted. While the above scenario of half ﬁlled jeﬀ= 1/2 orbitals is applicable to undistorted case, as we shall see, it breaks down in presence of strong distortions of the o ctahedra. In particular, we consider the eﬀect of trigonal distortions, which may be relevant for some of the iridate s ystems including the much debated honeycomb lattice iridate, Na 2IrO3. We show that, in this limit, a diﬀerent “doublet” of orbitals emerge as the low energy degree of freedom. This doublet forms a pseudo-spin-1 /2that results from a kind of (physical)spin-orbital locking, so that the spin and orbital ﬂuctuations are not separable (as discussed below ). We emphasize that this pseudo-spin is diﬀerent2 b a FIG. 1: The zig-zagmagnetic structure as found in Ref.5. The magnetic unit cell has 4 sites. from the jeﬀ= 1/2 doublet discussed above. The spin Hamiltonian for these pseudo-s pins (Eq. 6), on a honeycomb lattice, admits a 4-sub-lattice zig-zag(ﬁg.I) pattern in a relevant parameter regime. Such magnetic order has been recently observed in the experiments5,28on Na 2IrO3and hence our theory may be applicable to this material. The distortion of the octahedron surrounding the Ir+4generates a new energy scale associated with the change in the crystal ﬁeld, which, as we shall see, competes with the SO coup ling. Several kinds of distortion may occur, of which we consider the trigonal distortions of the octahedron wher e it is stretched/compressedalong the body diagonal of the enclosing cube25. In the absence SO coupling, such trigonal crystal ﬁeld splits the t2gmanifold into e′ g(with two degenerate orbitals e′ 1gande′ 2g) and non-degenerate a1g(again there is an added two-fold spin degeneracy for each of these orbitals). The e′ ganda1glevels are respectively occupied by three and two electrons in Ir4+. For large trigonal distortions, the splitting between them is big and the a1gorbitals can be projected out. Now, if one adds SO coupling, the low energy degrees of freedom is described by a sub space of the e′ gorbitals which form an emergent pseudo-spin-1/2 doublet out of \|e′ 1g,↓/angbracketrightand\|e′ 2g↑/angbracketrightstates, where ↑and↓represent the physical spin sz= 1/2,−1/2 (the spins are quantized along the axis of trigonal distortion). The se pseudo-spin-1 /2 is diﬀerent from the jeﬀ= 1/2 andjeﬀ= 3/2 multiplets in the strong SO coupling limit as discussed above. Notice th e (physical)spin-orbital locking for the pseudo-spins, as alluded above. The two approaches, described in the last two paragraphs, of arr iving at the low energy manifold are mutually incompatible. This can be seen as follows: In presence of sizeable trig onal distortions the jeﬀ= 1/2 andjeﬀ= 3/2 multiplets mix with each other and can no longer serve as good low ener gy atomic orbitals. This dichotomy becomes quite evident in the recent studies of Na 2IrO3, where, the Ir+4form a honeycomb lattice. Taking into account the strong SO coupling of Ir4+in Na2IrO3, proposed are a model for a topological insulator in the weak intera ction limit10 and a Heisenberg-Kitaev (HK) model for a possible spin liquid phase in t he strong coupling limit11. These proposals prompted several experimental4–6and theoretical eﬀorts12,26to understand the nature of the ground state in this material. Subsequently, it was found that Na 2IrO3orders magnetically at low temperatures4. However, the magnetic moments form a “zig-zag” pattern (ﬁg. I) which is not consistent with the ones that would be obtained by addin g a weak interaction in a topological insulator (canted antiferromagne t)10or from a nearest neighbour HK model (spin liquid or the so-called stripe antiferromagnet)11. While recent studies27show that a ‘zig-zag’ order may be stabilized within the HK model by including substantial second and third neighbo ur antiferromagnetic interactions, it is hard to justify such large further neighbour exchanges without lattice distortions. (An alternate explanation that we do not pursue here is signiﬁcant charge ﬂuctuations which would mean t hat the compound is close to metal-insulator transition. Theresistivitydataseemstosupportthefactthat th iscompoundisagoodinsulator.4) Iflatticedistortions are responsible for the signiﬁcant further neighbour exchanges, then, there would be sizeable distortion of the oxygen octahedra, which in turn may invalidate the above jeﬀ= 1/2 picture and thus the basic paradigm of the HK model, by mixing the jeﬀ=1/2and jeﬀ= 3/2 subspaces. Recent ﬁnite temperature numerical calculations on the HK model26 also suggest possible inconsistencies with experiments on Na 2IrO3. This necessitates the need for a diﬀerent starting point, to explain the magnetic properties of Na 2IrO3. The rest of this paper is organized as follows. In Section II, we derive the eﬀective spin Hamiltonian in limit of the large trigonal distortion and large SO coupling. This is done by t aking the energy scale associated with trigonal distortion to inﬁnity ﬁrst, followed by that of the SO energ y scale. This order of taking the limit gives a spin Hamiltonian in terms of emergent pseudo-spin −1/2, which is diﬀerent from the HK model. This Hamiltonian3 is both anisotropic and frustrated. It also has further neighbour interactions, the eﬀect of which are enhanced due to anisotropy that makes some of the nearest neighbour bonds we aker. The origin of this anisotropy is trigonal distortion. We argue that this limit may be more applicable for the comp ound Na 2IrO3. Having derived the spin Hamiltonian, we calculate the phase diagram and the spin wave spectr um within mean ﬁeld theory in Sec. III. We see that the ‘zig-zag’ phase occurs in a relevant parameter regime. We also point out the experimental implications of our calculations in context of Na 2IrO3. Finally we summarize the results in Sec. IV. The details of various calculations are given in various appendices. II. THE EFFECTIVE HAMILTONIAN In the cubic environment the t2gorbitals are degenerate when there is no SO coupling. Trigonal disto rtion due to compression or expansion along one of the four C3axes of IrO 6octahedra lifts this degeneracy. Although it is possible that the axes of trigonal distortions are diﬀerent in diﬀerent octa hedra20, we ﬁnd that uniform distortions are more consistent with the experiments (see below) on Na 2IrO3. Hence we consider uniform trigonal distortion. A. The Trigonal Hamiltonian Let us denote the axis of this uniform trigonal distortion of the oct ahedron by the unit vector ˆ n=1√ 3[n1,n2,n3], wherenα=±1. Since there are 2 directions to each of the 4 trigonal axes we may choose a “gauge” to specify ˆ n. This is done by taking n1n2n3= +1. The Hamiltonian for trigonal distortion, when projected in the t2gsector, gives20(in our chosen gauge) Ht2g tri=−/summationdisplay i∆tri 3Ψ† i 0n3n2 n30n1 n2n10 Ψi, (1) where Ψ† i= [d† yz,d† zx,d† xy] and ∆ triis the energy scale for trigonal distortion. The eigenstates are ( ω=eı2π/3) \|a1g/angbracketright=1√ 3[n1\|dyz/angbracketright+n2\|dzx/angbracketright+n3\|dxy/angbracketright], \|e′ 1g/angbracketright=1√ 3/bracketleftbig ωn1\|dyz/angbracketright+ω2n2\|dzx/angbracketright+n3\|dxy/angbracketright/bracketrightbig , \|e′ 2g/angbracketright=1√ 3/bracketleftbig ω2n1\|dyz/angbracketright+ωn2\|dzx/angbracketright+n3\|dxy/angbracketright/bracketrightbig . (2) The trigonal distortion splits the t2gsector into the doubly degenerate e′ gand the non-degenerate a1gwith energies ∆tri/3 and−2∆tri/3 respectively. A description based on Hubbard model for the e′ gorbitals may be systematically derived starting from the t2g orbitals. This is done in A. This has the following general form H′=He′ g SO−/summationdisplay ij/summationdisplay M,M′/summationdisplay σ˜tiM;jM′e† iMσeiM′σ+U 2/summationdisplay i/summationdisplay M,M′/summationdisplay σσ′e† iMσe† iM′σ′eiM′σ′eiMσ, (3) wheree† iMσis the electron creation operator in the e′ gorbital (M= 1,2) with spin σ(=↑,↓);˜tiM;jM′are the eﬀective hopping amplitudes within the subspace and Uis the eﬀective onsite coulomb’s repulsion. We note that the Hund’s coupling (which arises from the orbital dependence of the Coulomb r epulsion) for the t2gorbitals only renormalizes Uin this restricted subspace. (See Afor details). B. The Projected SO coupling The SO coupling, when projected in the e′ gsubspace, yields a block diagonal form (see Bfor details): He′ g SO=−λˆn·/vector siτz i, (4)4 FIG. 2: The e′ gstates split by the SO coupling. where/vector siis the spin operator at the site i,λ≈500meVis the SO coupling parameter and τz= +1(−1) refers to the e′ 1g(e′ 2g) orbital. Thus the projected SO interaction acts as a “Zeeman coupling” whe re the direction of the “magnetic ﬁeld” is along the trigonal axis or opposite to it13. Thus it is natural to choose the direction of spin quantization along the axis of trigonal distortion. This then gives the active atomic orbitals aft er incorporating the SO coupling. These active orbitals are the Krammer’s doublet \|e′ 1g,↓/angbracketrightand\|e′ 2g,↑/angbracketrightas shown in Fig. 2. C. The Spin Hamiltonian Hence the low energy physics may be described by considering only th e above atomic orbitals. The starting point for the calculations is projection of the Hubbard model (Eq. 3) in the space spanned by the Krammers’s doublet \|e′ 1g,↓/angbracketrightand\|e′ 2g,↑/angbracketright. The bandwidth of this projected model is narrow and the eﬀect of the Hubbard repulsion is important. Indeed it can easily render the system insulating. To cap ture the magnetic order in this Mott insulator, we do a strong-coupling expansion in ˜t/Uto get an eﬀective “pseudo-spin” model in terms of the pseudo-sp in-1/2 operators, Sα=1 2e† aρα abeb, (5) where,ρα(α=x,y,z) are the Pauli matrices and a,b= (e′ g1;↓),(e′ g2;↑). The “pseudo-spin” Hamiltonian has the following form up to the quadratic order: H=/summationtext /angbracketleftij/angbracketrightJ(1) ij/vectorSi·/vectorSj+/summationtext /angbracketleft/angbracketleftij/angbracketright/angbracketrightJ(2) ij/vectorSi·/vectorSj+/summationtext /angbracketleft/angbracketleft/angbracketleftij/angbracketright/angbracketright/angbracketrightJ(3) ij/vectorSi·/vectorSj +/summationtext /angbracketleftij/angbracketrightJ(z1) ijSz iSz j+/summationtext /angbracketleft/angbracketleftij/angbracketright/angbracketrightJ(z2) ijSz iSz j+/summationtext /angbracketleft/angbracketleft/angbracketleftij/angbracketright/angbracketright/angbracketrightJ(z3) ijSz iSz j. (6) Here/angbracketleftij/angbracketright,/angbracketleft/angbracketleftij/angbracketright/angbracketrightand/angbracketleft/angbracketleft/angbracketleftij/angbracketright/angbracketright/angbracketrightrefer to summation over ﬁrst, second and third nearest neighbou rs (NNs) respectively. The diﬀerent exchange couplings are given in terms of the underlying parameters of the Hubbard model as J(zα) ij=8 U(T(zα) ij)2, J(α) ij=4 U/bracketleftBig (T(0α) ij)2−(T(zα) ij)2/bracketrightBig =J(0α) ij−1 2J(zα) ij (7) whereα= 1,2,3 denotes that ijare ﬁrst, second or third NNs, respectively and the last expressio n deﬁnes J(0α) ij. T(0α) ijandT(zα) ijare given in terms of the hopping amplitudes (e.g. txy;yz ijfrom the overlap of xyandyzorbitals) of thet2gorbitals as (details are given in C). T(0α) ij=1 3/bracketleftbig/parenleftbig tyz;yz ij+txz;xz ij+txy;xy ij/parenrightbig/bracketrightbig −1 6/bracketleftbig/parenleftbig n1/parenleftbig txz;xy ij+txy;xz ij/parenrightbig +n2/parenleftbig txy;yz ij+tyz;xy ij/parenrightbig +n3/parenleftbig tyz;xz ij+txz;yz ij/parenrightbig/parenrightbig/bracketrightbig (8) T(zα) ij=1 2√ 3/bracketleftbig n1/parenleftbig txz;xy ij−txy;xz ij/parenrightbig +n2/parenleftbig txy;yz ij−tyz;xy ij/parenrightbig +n3/parenleftbig tyz;xz ij−txz;yz ij/parenrightbig/bracketrightbig (9)5 FIG. 3: Section of a honeycomb lattice (shaded in yellow). Ir sites (black) are connected by bonds (orange). The green arr ow is the the [ −1,−1,1] direction of trigonal distortion that makes an angle of ab out 19◦with the plane of the lattice pointing inside the plane. Before moving on to the details of the spin Hamiltonian, we note that, on projecting to the subspace of \|e′ 1g,↓/angbracketrightand \|e′ 2g,↑/angbracketright, the spin and orbitals are no longer independent. Instead at every site there is a pseudo-spin-1 /2 degree of freedom where the spin is lockedto the orbital wave function. This, we refer to as spin-orbital locking . III. APPLICATION TO Na 2IrO3 We now apply the above results to the case of Na 2IrO3. The early X-Ray diﬀraction experiments4suggested a a monoclinic C2/Cstructure for the compound and distorted IrO 6octahedra. However, more recent experiments see a better match for the X-Ray diﬀraction data with the space group C2/m.28,30They also unambiguously conﬁrm the presence of uniform trigonal distortion of the IrO 6octahedra. However, the magnitude of such distortion is not clear at present. Further, experimental measurements suggest: (1 ) The magnetic transition occurs at TN= 15Kwhile the Curie-Weiss temperature is about Θ CW≈ −116K. This indicates presence of frustration. (2) The high temperatur e magnetic susceptibility is anisotropic; the in-plane and out-of-plane susceptibilities are diﬀerent. This may be due to a trigonal distortion of the IrO 6octahedra4. (3) The magnetic speciﬁc heat is suppressed at low temperatures .4(4) Recent resonant X-ray scattering experiment5suggests that the magnetic order is collinear and have a 4-site unit c ell. (5) The magnetic moments have a large projection on the a-axis of the monoclinic crystal5. (6) A combination of these experimental ﬁndings and density functional theory (DFT) calculations strongly suggest that a ‘zig-zag’ pattern for the magnetic moments, as shown in Fig. Iin the ground state5, which has since been veriﬁed independently by two groups using Neutron scattering28,30. Taking these phenomenological suggestions, we try to apply the ab ove calculations to the case of Na 2IrO3. At the outset, we must note that, in the above derivation of the spin H amiltonian we have assumed that the trigonal distortion to be the largest energy scale followed by the SO coupling. While this extreme limit of projecting out thea1gorbitals most likely is not true for Na 2IrO3. However, we expect the real ground state to be adiabatically connected to this limit. With this in mind, we now consider the case of Na 2IrO3. Clearly, the exchanges (Eq. 7) depend both on the direction of the bond and the direction of the t rigonal distortion. So it is important to ask about the direction of the latter. Comparing the crystallographic axes of Na 2IrO3, we ﬁnd that the direction [1 ,1,1] is perpendicular to the honeycomb plane while the other three dire ctions make an acute angle to it. In the monoclinic C2/mstructure, uniform trigonal distortion in these four directions ma y not cost the same energy. In experiments5, the moments are seen to point along the a-axis of the monoclinic crystal which is parallel to the honeycomb plane. This, along with the fact that the m agnetic moment in our model is in the direction of ˆn(explained below) seems to suggest that ˆ n=1√ 3[−1,−1,1] is chosen in the compound (see Fig. 3). In the absence of a better theoretical understanding of the direction of the trig onal distortion, we take this as an input from the experiments. To identity diﬀerent hopping paths (both direct and indirect), we co nsider various overlaps (see C) and ﬁnd, while J(3z)= 0,J(1z)/negationslash=J(2z)/negationslash= 0 are approximately (spatially-)isotropic and antiferromagnetic. For the exchanges of the Heisenberg terms, both J(2)andJ(3)are antiferromagnetic and isotropic (both of them result from indir ect hopping6 FIG. 4: Mean ﬁeld phase diagram for Eq. 6. The two axes are: x0=˜J(1) J(1);y0=J(2) J(1), where J(1)(˜J(1)) are related to the strong(weak) NN exchange and J(2)is the 2ndand 3rdneighbour exchange (see Eq. 7). We take the Ising anisotropy to be 5% ofJ(1). Note that, due to Ising anisotropies, one has zig-zag order aty0= 0. mediated by the Na s-orbitals and are expected to be comparable). For the NN Heisenberg exchanges, the couplings are antiferromagnetic, but, much more spatially anisotropic. We ﬁn d that for the chosen direction of the trigonal distortion, the coupling along one of the NN exchanges ( J(1)) (vizb1in Fig. 3) is diﬀerent from the other two neighbours ( ˜J(1))(b2andb3in Fig.3). A. Mean-Field Theory and Magnetic Order We now consider the mean ﬁeld phase diagram for the above anisotro pic spin Hamiltonian. For J(1)being the largest energy scale, the classical ground state for the model ca n be calculated within mean-ﬁeld theory as a function ofx0=˜J(1)/J(1)andy0=J(2)/J(1)(we have taken J(2)=J(3)). A representativemean-ﬁeld phase diagram is shown in Fig.4. It shows a region of the parameter-space where the zig-zag ord er is stabilized29. The eﬀect of the Ising anisotropies J(1z)andJ(2z)is to pin the magnetic orderingalongthe z-directionofthe pseudo-spinquantizationwhich is also the direction of the trigonal distortion ˆ n. They also gap out any Goldstone mode that arises from the orderin g of the pseudo-spins. The latter results in the exponential suppre ssion of the speciﬁc heat at low temperatures. The other competing phase with a collinear order is the regular two-subla ttice Neel phase. The nature of the ground states may be understood from the follo wing arguments. In the presence of the ˆ nin [−1,−1,1] direction, the NN exchange coupling becomes anisotropic. When it is strong in one direction ( J(1)) and weak in two other directions ( ˜J(1)), for the bonds where the NN coupling becomes weak, the eﬀects o f the small second and third neighbour interactions become signiﬁcant. Since t he latter interactions are antiferromagnetic, they prefer anti-parallel alignment of the spins. As there are more seco nd and third neighbours, their cumulative eﬀect can be much stronger. This naturally leads to the zig-zag state. The NN antiferromagnetic interactions on the weaker bonds compete with the antiferromagnetic second and third neighb our interactions and frustrates the magnet. This suppresses the magnetic ordering temperature far below the Cur ie-Weiss temperature. B. The spectrum for Spin-orbital waves The low energy excitations about this magnetically ordered zig-zag s tate are gapped spin-orbital waves. Signatures of such excitations may be seen in future resonant X-Ray scatter ing experiments. It is important to note that this “pseudo-spin” waves actually contain both orbital and the spin com ponents due to the spin-orbital locking. We calculate the dispersion of such spin-orbital waves to quadratic order using the well-known Holstein-Primakoﬀ methods. Thedetailsarediscussedin D. Arepresentativespin wavespectrum in thezig-zagphaseisshown inFig.5(a) and5(b). The spectrum is gapped and the bottom of the spin-wave dispersio n has some characteristic momentum dependence. C. Experimental Implications Apart from the already discussed exponential suppression of low t emperature magnetic speciﬁc heat, the above calculation predicts an interesting feature in the magnetic suscept ibility. The relation between the magnetic moment7 (a) (b) FIG. 5: The “pseudo-spin” wave spectrum (contours of both th e bands are shown in (a) and a section is shown in (b)). The values used for the parameters are same as that used for the ca lculation of the mean ﬁeld phase diagram (Fig. 4). We note that, as expected, the spectrum is gapped. and the pseudo-spins is /vectorMi=−4µBˆnSz i, (10) whereµBis the Bohr magneton. This follows from the twin facts that, in e′ gsubspace, the angular momentum transverse to ˆ nis quenched and the spins are locked to the orbitals with the axis of qu antization being ˆ nin our pseudo-spin sector (see E). Thus, the magnetization is sensitive to the z-component of the pseudo-spin (the direction of which is shown in Fig. I). Indeed the magnetization has the largest projection along the a-axis of the monoclinic crystal. This was seen in experiments5and was the motivation for choosing the [ −1,−1,1] direction for the trigonal distortion. Along two other axes [ −1,1,−1]and [1,−1,−1], a large component of in-plane magnetization exists, but in diﬀerent directions. Finally the direction [1 ,1,1] is perpendicular to the honeycomb plane and leads to magnetizatio n in the same [1,1,1] direction. While this does not appear to be the case f or Na2IrO3, this may be more relevant for the less-distorted compound Li 2IrO3(see below). Eq. 10suggests that the magnetic susceptibility is highly anisotropic and depends on the cosine of the angle between the direction of mag netic ﬁeld and ˆ n. Indeed signatures of such anisotropy have been already seen in experiments4. We emphasize that within this picture, the in-plane susceptibility also varies with the direction of the magnetic ﬁeld. So the ratio χ⊥/χ/bardblcan be lesser or greater than 1. The current experiments4does nottell the in-planedirectionofmagneticﬁeld andhence wecan notcommentonthe ratiopresently. However, the above picture is strictly based on atomic orbitals. One generally expects that there is also hybridization of the Ir d-orbitals with the oxygen p-orbitals. Such hybridization will contribute to a non-zero isotropic component to the susceptibility24. Also, as remarked earlier, in the actual compound, the SO coupling scaling may not be very small compared to the trigonal distortion limit scale. Additional pert urbation coming from the mixing with the a1g orbitals will also contribute to decrease the anisotropy of the susc eptibility.8 IV. SUMMARY AND CONCLUSION In this paper, we have studied the eﬀect of trigonal distortion and SO coupling and applied it to the case of the honeycomb lattice compound Na 2IrO3. We ﬁnd that, in the limit of large trigonal distortion and SO coupling, a pseudo-spin-1 /2 degree of freedom emerges. Low energy Hamiltonian, in terms of t his pseudo-spin gives a ‘zig-zag’ magnetic order as seen in the recent experiments on Na 2IrO3. We have also calculated the low energy spin-wave spectrum and elucidated various properties of the compound that has been observed in experiments. The pseudo-spin couples the physical spin and the orbitals in a non-trivial manner, sig natures of this may be seen in future inelastic X-ray resonance experiments probing the low energy excitations. While very recent experiments28,30clearly indicate presence of trigonal distortions, their magnitude is yet not conﬁrmed. On the other hand, the only available numerical estimate of the energy scale for trigonal distortion comes from the DFT calculations by Jin et al.13(based on C2/Cstructure). It suggests ∆ tri≈600meV. While, it is not clear if such a large value is in confomity with the experiments, at pres ent, the detection of trigonal distortion in experiments is highly encouraging from the perspective of the pres ent calculations. In these lights of the above calculations, it is tempting to predict the case of Li 2IrO3where recent experiments suggest a more isotropic honeycomb lattice6,31. A possibility is that sizeable trigonal distortion is also present in Li2IrO3(so that the above discussion holds), but, the axis is perpendicular to the plane. What may be the fallouts in such a case ? Our present analysis would then suggest that the ant iferromagnetic exchanges are isotropic and equally strong for the three NNs. This would develop 2-sublattice Neel ord er in the pseudo-spins with the magnetic moments being perpendicular to the plane. Also the further neighbour excha nges are rather weak (compared to Na 2IrO3) and hence frustration is quite small. Indeed recent experiments see or dering very close to the Curie-Weiss temperature, the later being calculated from the high temperature magnetic susc eptibility data6,31. However, present experiments do not rule out the possibility of small or no trigonal distortions in Li 2IrO3, in which case the limit of HK model11,26 may be appropriate. Acknowledgments We acknowledge useful discussion with H. Gretarsson, R. Comin, S. Furukawa, H. Jin, C. H. Kim, Y.-J. Kim, W. Witczak-Krempa, H. Takagi. YBK thanks the Aspen Center for Phy sics, where parts of the research were done. This work was supported by the NSERC, Canadian Institute for Advanc ed Research, and CanadaResearch Chair program. Appendix A: The microscopic model for Na 2IrO3 The generic Hubbard model (for the t2gorbitals) including the trigonal distortions, Hund’s coupling and the S O coupling is H=−λ/summationtext i/vectorli·/vector si+Ht2g tri+/summationtext ij/summationtext mm′/summationtext σσ′/parenleftBig tm;m′ ijd† imσdjm′σ′/parenrightBig +1 2/summationtext i/summationtext mm′/summationtext σσ′Umm′d† imσd† im′σ′dim′σ′dimσ. (A1) Herem,m′=yz,xz,xy andσ=↑,↓andHt2g triis given by Eq. 1. We note that the hopping is diagonal in spin space and in the cubic harmonic basis all hopping are real. Also, the hopping c ontain both the direct and indirect (through Oxygen and Sodium) paths. We have taken Hund’s coupling into accou nt through Umm′, though this is expected to be small in 5 dtransition metals. To a very good approximation the form of Umm′is given by Umm′≡ U0U0−JHU0−JH U0−JHU0U0−JH U0−JHU0−JHU0 , (A2) where the basis is given, as before, by Ψ† i= [d† yz,d† zx,d† xy].U0andJHare the intra orbital Coulomb repulsion and Hund’s coupling term respectively. The transformation between the operators in the trigonal basis, Φ†=/bracketleftBig a† 1g,e′† 1g,e′† 2g/bracketrightBig , andt2gbasis, Ψ†=/bracketleftbig d† yz,d† zx,d† xy/bracketrightbig , is given by Ψ m=Tm,MΦM. The transformation matrix is given by Tm,M=1√ 3 n1n1ω n1ω2 n2n2ω2n2ω n3n3n3 . (A3)9 The transformations for the hopping amplitudes and repulsion term are then given by ˜tiM;jM′ =/summationtext m,m′T∗ m,Mtm;m′ ijTm′,M′; ˜UM1M2=/summationtext m,m′Umm′/parenleftbig T∗ mM1TmM1/parenrightbig/parenleftbig T∗ m′M2Tm′M2/parenrightbig . (A4) Notice that there are contributions to tm;m′ ijfrom both direct and indirect exchanges for the ﬁrst, second and third neighbours, as conﬁrmed from the DFT calculations by H. Jin et al.13. These show that there are contributions from both direct and indirect hoppings for the ﬁrst, second and third ne arest neighbours. Projecting them into the e′ g orbitals we get the eﬀective hopping amplitudes which are then used in Eq.3. As for the Coulomb repulsion term, we ﬁnd that it has the following form ˜UM1M2=U 1 1 1 1 1 1 1 1 1 , (A5) whereU=U0−2JH/3. This form is then used in Eq. 3. The reason for this special form of ˜UM1M2lies in the fact that the e′ gorbitals have equal weight of the three t2gorbitals (see the wave functions in Eq. 2). Appendix B: Projection of Spin-Orbit coupling to the e′ gsubspace The SO coupling, when projected to the t2gorbitals give Ht2g SO=−λ/vectorl·/vector s, (B1) where/vectorlis al= 1 angular momentum operator. We can re-write the t2gcubic harmonics in terms of the spherical harmonics of the eﬀective l= 1 angular momentum operator. These are given by: \|dyz/angbracketright=1√ 2[\|1,−1/angbracketright−\|1,+1/angbracketright]; \|dzx/angbracketright=ı√ 2[\|1,−1/angbracketright+\|1,−1/angbracketright]; \|dxy/angbracketright =\|1,0/angbracketright (B2) The projector for the e′ gspace is: Pe′ g=\|e′ 1/angbracketright/angbracketlefte′ 1\|+\|e′ 2/angbracketright/angbracketlefte′ 2\|. It turns out that /vectorl·/vector sis block diagonal in this subspace. Hence /vectorl·/vector s=\|e′ 1/angbracketright/angbracketlefte′ 1\|/vectorl·/vector s\|e′ 1/angbracketright/angbracketlefte′ 1\|+\|e′ 2/angbracketright/angbracketlefte′ 2\|/vectorl·/vector s\|e′ 2/angbracketright/angbracketlefte′ 2\| (B3) Making the “gauge” choice we get /angbracketlefte′ 1\|/vectorl·/vector s\|e′ 1/angbracketright= ˆn·/vector s;/angbracketlefte′ 2\|/vectorl·/vector s\|e′ 2/angbracketright=−ˆn·/vector s (B4) Appendix C: The hopping parameters 1. Nearest neighbours The nearest neighbours are shown in Fig. 6(a). There are two diﬀerent processes contributing to the hopping.: 1 ) the direct hopping between the Ir atoms and 2) the indirect hopping between the Ir atoms mediated by the oxygen atoms. In presence of the trigonal distortion which has a compone nt along the honeycomb plane (like in this case [−1,−1,1]) the magnitudes of the diﬀerent hopping parameters are diﬀeren t in diﬀerent directions (for both direct and indirect hopping). The results are shown in Table I. We shall make an approximation here. We shall leave out the direction al dependence of the magnitudes on the direction. The argument is that the essential directional depende nce due to the trigonal distortion has been taken care of by the parameter ∆ 1. When the DFT13results are used to ﬁnd the tight-binding parameters29, it is found that (they use ∆ 1= 0) (here tdd1andtdd2are direct hopping and t0is the indirect hopping respectively.) tdd1= −0.5eV;tdd2= 0.15eV;t0= 0.25eV.10 (a) (b) (c) FIG. 6: The 3 nearest neighbours (a), six 2ndnearest neighbours (b) and three 3rdnearest neighbours (c) of the central site. The nomenclature has been used to label the hoppings. (a)NN:tam;b1m′ m′\mdxy dyz dzx dxytdd1(b1) - - dyz- tdd2(b1) −tdd2(b1)+t0(b1)+∆ 1(b1) dzx-−tdd2(b1)+t0(b1)−∆1(b1) tdd2(b1) (b)NN:tam;b2m′ m′\mdxy dyz dzx dxy tdd2(b2) -−tdd2(b2)+t0(b2)+∆ 1(b2) dyz - tdd1(b2) - dzx−tdd2(b2)+t0(b2)−∆1(b2)- tdd2(b2) (c)NN:tam;b3m′ m′\mdxy dyz dzx dxy tdd2(b3) −tdd2(b3)+t0(b3)+∆ 1(b3)- dyz−tdd2(b3)+t0(b3)−∆1(b3) tdd2(b3) - dzx - - tdd1(b3) TABLE I: The hopping paths (both direct and indirect) in the t2gbasis. Performing the transformation to the e′ gbasis, we have T(01) ab1=1 3[tdd1+2tdd2+(tdd2−t0)n3], T(02) ab2=1 3[tdd1+2tdd2+(tdd2−t0)n1], T(03) ab3=1 3[tdd1+2tdd2+(tdd2−t0)n2]. (C1) and T(z1) ab1=−∆1√ 3n3 T(z1) ab2=−∆1√ 3n1 T(z1) ab3=−∆1√ 3n2 (C2) Hence, J(0) ab1=4 3U/bracketleftBig 1 3[tdd1+2tdd2+(tdd2−t0)n3]2−(∆1)2/bracketrightBig , J(0) ab2=4 3U/bracketleftBig 1 3[tdd1+2tdd2+(tdd2−t0)n1]2−(∆1)2/bracketrightBig , J(0) ab3=4 3U/bracketleftBig 1 3[tdd1+2tdd2+(tdd2−t0)n2]2−(∆1)2/bracketrightBig . (C3)11 J(1z) ab1=8(∆1)2 3U; J(1z) ab2=8(∆1)2 3U; J(1z) ab1=8(∆1)2 3U; (C4) where we have taken the direction of the trigonal distortion is take n to be uniform. 2. Second nearest neighbour These are shown in Fig. 6(b). These indirect hoppings are mediated by the Na atoms. In general, in presence of the trigonal distortion in the [ −1,−1,1] direction, the magnitude of the hopping amplitudes are also direct ion dependent. However, since the magnitudes themselves are expected to be sma ll we shall neglect such directional dependence in the magnitudes. The result is summarized in Table II. (a)NNN: tam;a1m′/tam;a4m′ m′\mdxydyzdzx dxy-t2+∆2- dyzt2−∆2-- dzx- --(b)NNN: tam;a2m′/tam;a5m′ m′\mdxydyzdzx dxy--t2+∆2 dyz--- dzxt2−∆2--(c)NNN: tam;a3m′/tam;a6m′ m′\mdxydyzdzx dxy-- - dyz--t2+∆2 dzx-t2−∆2- TABLE II: Hopping paths for the second nearest neighbours So for the e′ gbasis, we have T(02) a,a1=T(02) a,a4=−t2 3n2, T(z2) a,a1=T(z2) a,a4=−∆2√ 3n2 T(02) a,a2=T(02) a,a5=−t2 3n1, T(z2) a,a2=T(z2) a,a5=−∆2√ 3n1 T(02) a,a3=T(02) a,a6=−t2 3n3, T(z2) a,a3=T(z2) a,a6=−∆2√ 3n3 (C5) For example, tight binding ﬁt of the DFT data uses only t2and ﬁnds t2≈ −0.075eV13,29. Therefore we have: J(2) a,aα=4 3U/bracketleftbigg(t2)2 3−(∆2)2/bracketrightbigg , J(2z) a,aα=8(∆2)2 3U. (C6) 3. Third nearest neighbour The third nearest neighbours are listed in Fig. 6(c). The hopping to the third nearest neighbour is mediated by the Na atoms. Again we shall neglect the directional dependence and ta ke these to be in the magnitudes of the hoping amplitudes. The result is summarized in table III. (a)NNNN: tam;b′ 1m′ m′\mdxydyzdzx dxyt3(b′ 1)-- dyz--- dzx---(b)NNNN: tam;b′ 2m′ m′\mdxydyzdzx dxy--- dyz-t3(b′ 2)- dzx---(c)NNNN: tam;b′ 3m′ m′\mdxydyzdzx dxy--- dyz--- dzx--t3(b′ 3) TABLE III: The hoppings for the third nearest neighbours Tight-binding ﬁt to the DFT results13,29indeed show that this hopping energy scale is of the order of t3(b′ α) =tn≈ −0.075eV (C7) Therefore we have: T(03) ab′α=tn 3, T(z3) ab′ 1= 0; (C8)12 or, J(3) ij=4[tn]2 9U;J(3z) ij= 0; (C9) Appendix D: Spin Wave Spectrum To calculate the spin wave spectrum for the zig-zag state we use th e usual Holstein-Primakoﬀ method suited to collinear ordering which may alternate in direction. More precisely we in troduce: Sz=S−a†a;S+=√ 2Sa;S−=√ 2Sa†(D1) for one direction and Sz=−S+a†a;S+=√ 2Sa†;S−=√ 2Sa (D2) for the other direction. Since there are4 sites per unit cell (refer Fig. 1(a) ofthe main text) the quadraticHamiltonian is a 8×8 matrix given by: HQ=Hcl+Hsp, (D3) whereHclis the classical part dealt in the previous section. The spin wave Hamilt onian has the following form Hsp=S 2/summationdisplay kΨ† kHkΨk+Hs (D4) Here Ψ† k=/bracketleftBig a† k,1,a† k,2,a† k,3,a† k,4,a−k,1,a−k,2,a−k,3,a−k,4/bracketrightBig (the subscript 1 ,2,3,4 refers to the four sites in the unit cell as shown in Fig. 1(a) of the main text) and Hs=−S 2[(1−2x+5y)−(2δ2−δ1)]NCell; (D5) Hk =/bracketleftBigg AkBk B† kAk/bracketrightBigg (D6) whereNcellis the number of unit cells and Ak= χk0 0ηk 0χkφk0 0φ∗ kχk0 η∗ k0 0χk ;Bk= 0ξkρk0 ξ∗ k0 0ρk ρ∗ k0 0ξk 0ρ∗ kξk0 (D7) where, χk= (2δ2−δ1)+(1−2x+5y+ycoskx); (D8) ηk =xeıky/parenleftbig 1+eıkx/parenrightbig ; (D9) φk =x/parenleftbig 1+eıkx/parenrightbig ; (D10) ξk = (1+2ycoskx+ye−ıky); (D11) ρk =y(1+eıkx)(1+e−ıky). (D12) Now following usual methods we diagonalize /bracketleftBigg AkBk −B† k−Ak/bracketrightBigg (D13) to get the spin wave spectrum as plotted in Fig. 3(a) and 3(b) of the main text.13 Appendix E: Projection of Zeeman term in the t2gand/braceleftbig \|e′ 1g↓/a\gbracketright,\|e′ 2g↑/a\gbracketright/bracerightbig subspaces. The Zeeman coupling term, when projected to the t2gspace, gives Ht2g Z=µB/parenleftBig −/vectorl+2/vector s/parenrightBig ·/vectorB. (E1) Thus the magnetization after projection is given by: /vectorMt2g=µB/parenleftBig −/vectorl+2/vector s/parenrightBig (E2) This when projected to the subspace \|e′ 1,↑/angbracketrightand\|e′ 2,↓/angbracketrightgives (using the Block diagonal property of the orbital angular momentum as above): ˜HZ= 4µBSzˆn·/vectorB (E3) where/vectorS(note that this is in upper case compared to the physical spin writte n lower case) is the emergent pseudo- spin-1/2 per site. This is the emergent degree of freedom at low ene rgies. Clearly, the magnetic moment is then given by Eq. 7 of the main text. 1Y. Okamoto, M. Nohara, H. Aruga-Katori, and H. Takagi, Phys. Rev. Lett. 99, 137207 (2007). 2B. J. Kim, et al., Phys. Rev. Lett. 101, 076402 (2008). 3B. J. Kim, H. Ohsumi, T. Komesu, S. Sakai, T. Morita, H. Takagi , and T. Arima, Science 323, 1329 (2009). 4Y. Singh and P. Gegenwart, Phys. Rev. B 82, 064412 (2010). 5X. Liu, T. Berlijn, W.-G. Yin, W. Ku, A. Tsvelik, Y.-J. Kim, H. Gretarsson, Y. Singh, P. Gegenwart, and J. P. Hill, Phys. Rev. B 83, 220403(R) (2011). 6Y. Singh, S. Manni, J. Reuther, T. Berlijn, R. Thomale, W. Ku, S. Trebst and P. Gegenwart, Phys. Rev. Lett. 108, 127203 (2012). 7S. Nakatsuji Y. Machida, Y. Maeno, T. Tayama, T. Sakakibara, J. van Duijn, L. Balicas, J. N. Millican, R. T. Macaluso, and Julia Y. Chan, Phys. Rev. Lett. 96, 087204 (2006). 8S. Zhao, J. M. Mackie, D. E. MacLaughlin, O. O. Bernal, J. J. Is hikawa, Y. Ohta, and S. Nakatsuji, Phys. Rev. B 83, 180402(R) (2011). 9F. F. Tafti, J. J. Ishikawa, A. McCollam, S. Nakatsuji, and S. R. Julian, Phys. Rev. B 85, 205104 (2012). 10A. Shitade et al., Phys. Rev. Lett. 102, 256403 (2009). 11J. Chaloupka, G. Jackeli, and G. Khaliullin, Phys. Rev. Lett . 105, 027204 (2010); G. Jackeli and G. Khaliullin, Phys. Rev . Lett. 102, 017205 (2009). 12H.-C. Jiang, Z.-C. Gu, X.-L. Qi, and S. Trebst, Phys. Rev. B 83 , 245104 (2011). 13H. Jin, H. Kim, H. Jeong, C. H. Kim, and J. Yu, arXiv:0907.0743 (unpublished) (2009); C. H. Kim, H. S. Kim, H. Jeong, H. Jin, and J. Yu, Phys. Rev. Lett. 108, 106401 (2012). 14M. J. Lawler, A. Paramekanti, Y. B. Kim, and L. Balents, Phys. Rev. Lett. 101, 197202 (2008). 15G. Chen and L. Balents, Phys. Rev. B 78, 094403 (2008). 16M. R. Norman and T. Micklitz, Phys. Rev. Lett. 102, 067204 (20 09). 17D. Podolsky, A. Paramekanti, Y. B. Kim, T. Senthil, Phys. Rev . Lett. 102, 186401 (2009). 18D. Podolsky, Y. B. Kim, Phys. Rev. B 83, 054401 (2011). 19D. Pesin and L. Balents, Nat Phys 6, 376 (2010). 20B.-J. Yang and Y. B. Kim, Phys. Rev. B 82, 085111 (2010). 21X. Wan, A. Turner, A. Vishwanath, S. Y. Savrasov, Phys. Rev. B 83, 205101 (2011); X. Wan, A. Vishwanath, S. Y. Savrasov, Phys. Rev. Lett. 108, 146601 (2012). 22W. Witczak-Krempa, and Y. B. Kim, Phys. Rev. B 85, 045124 (201 2). 23F. Wang, and T. Senthil, Phys. Rev. Lett. 106, 136402 (2011). 24G. Chen, R. Pereira, and L. Balents, Phys. Rev. B 82, 174440 (2 010). 25P. Fazekas, Lecture notes on electron correlation and magnetism . (Singapore, World Scientiﬁc, 1999). 26J. Reuther, R. Thomale, S. Trebst, Phys. Rev. B 84, 100406(R) (2011). 27I Kimchi, Y.-Z. You, Phys. Rev. B 84, 180407(R) (2011). 28S. K. Choi, R. Coldea, A. N. Kolmogorov, T. Lancaster, I. I. Ma zin, S. J. Blundell, P. G. Radaelli, Yogesh Singh, P. Gegenwart, K. R. Choi, S.-W. Cheong, P. J. Baker, C. Stock, an d J. Taylor, Phys. Rev. Lett. 108, 127204 (2012). 29For example, the parameters calculated from DFT13fall within this regime (Private Communication with H. Jin a nd C. H. Kim). 30Feng Ye, S. Chi, H. Cao, B. C. Chakoumakos, J. A. F-Baca, R. Cus telcean, T. Qi, O. B. Korneta, G. Cao, arXiv:1202.3995 (unpublished) (2012). 31H. Takagi (Private Communication).
1702.03199v1.Orbital_and_spin_order_in_spin_orbit_coupled__d_1__and__d_2__double_perovskites.pdf	Orbital and spin order in spin-orbit coupled d1and d2double perovskites Christopher Svoboda,1Mohit Randeria,1and Nandini Trivedi1 1Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA (Dated: February 13, 2017) We consider strongly spin-orbit coupled double perovskites A 2BB'O 6with B' magnetic ions in eitherd1ord2electronic conguration and non-magnetic B ions. We provide insights into several experimental puzzles, such as the predominance of ferromagnetism in d1versus antiferromagnetism ind2systems, the appearance of negative Curie-Weiss temperatures for ferromagnetic materials and the size of eective magnetic moments. We develop and solve a microscopic model with both spin and orbital degrees of freedom within the Mott insulating regime at nite temperature using mean eld theory. The interplay between anisotropic orbital degrees of freedom and spin-orbit coupling results in complex ground states in both d1andd2systems. We show that the ordering of orbital degrees of freedom in d1systems results in coplanar canted ferromagnetic and 4-sublattice antiferromagnetic structures. In d2systems we nd additional colinear antiferromagnetic and ferromagnetic phases not appearing in d1systems. At nite temperatures, we nd that orbital ordering driven by both superexchange and Coulomb interactions may occur at much higher temperatures compared to magnetic order and leads to distinct deviations from Curie-Weiss law. I. INTRODUCTION Strong SOC in correlated materials has provided a platform for quantum spin liquids, Weyl semimetals, and an ongoing search for high Tcsuperconductivity in the iridates.1,2Among the strongly spin-orbit coupled mate- rials include 4 dand 5ddouble perovskites A 2BB'O 6with electron counts d1-d5on the magnetic B' ion. Here we re- strict our discussion to Bsite ions that are non-magnetic. Due to large distances between the magnetic ions, these materials are often Mott insulators and present a promis- ing class of materials to explore the interplay of strong correlations and spin-orbit coupling. Additionally, the magnetic sites form an FCC lattice leading to frustrated magnetism. Each electron count carries a dierent total angular momentum quantum number providing a new platform for studying novel magnetism. The half lled t2gshells ofd3ions result in an eective spin-3/2 model which are nominally expected to be described as a classically frustrated spin systems.3,4In the opposite limit, d5sys- tems withj= 1=2 are both intrinsically more quantum and are protected from local distortions by time rever- sal symmetry. These systems may oer a route to real- izing Kitaev physics and more generally spin liquids in three dimensions.5,6Thed4case is especially unique since spin-orbit coupling dictates that local moments should be absent and magnetism is forbidden. However several theoretical7{10and experimental11{13studies have exam- ined the possibility of inducing local moments through superexchange interactions. Thed1andd2electron counts stand out in that they combine aspects of the former three electron counts and will be the focus of this paper. First, they possess local angular momenta large enough to support quadrupolar order. Second, they possess unquenched orbital degrees of freedom that result in highly anisotropic interactions between magnetic ions.14,15Both of these aspects will al- low for the orbital degrees of freedom to play a signicantrole in determining the spin, orbital, and spin-orbital or- dering. While both electron counts have similar potential, ex- perimental observations of magnetic properties of the d1 double perovskites have drawn signicant interest. The 4d1compound Ba 2YMoO 6shows no long range magnetic order down to 2K despite having a large Curie-Weiss temperature =160Kand retaining cubic symmetry which leads to the conclusion that the ground state con- sists of valence bonds16{20. Among the 5 d1compounds are ferromagnetic Ba 2NaOsO 621{24, Ba 2MgReO 625,26, and Ba 2ZnReO 626which is unusual since ferromagnetism in Mott insulators is uncommon. There are two addi- tional twists to the story: rst, negative Curie-Weiss temperatures have been observed in these ferromag- nets, and, second, Ba 2LiOsO 6is antiferromagnetic de- spite sharing the same cubic structure as Ba 2NaOsO 6.21 Thed2compounds oer a similar platform to search for unusual magnetism, however experimental stud- ies seem to suggest that antiferromagnetic interactions are more prevalent in d2systems. Phase transitions to antiferromagnetic order are reported in Ca 3OsO 627, Ba2CaOsO 628, and Sr 2MgOsO 629,30while glass-like tran- sitions are reported in Ba 2YReO 631, Ca 2MgOsO 629, and Sr 2YReO 632. There are also several alleged sin- glet ground states: La 2LiReO 631, SrLaMReO 633, and Sr2InReO 632. Many theoretical investigations have been undertaken to understand the magnetism in both d1andd2double perovskites. In the limit of large spin-orbit coupling, the spinS= 1=2 and orbital Le=1 angular momenta add to a total angular momenta of j= 3=2. Within thejj-coupling scheme, magnetic moments are identi- cally zero due to cancellation of the spin and orbital moments, M= 2SL= 0. On the other hand, d2 systems allow for a nonzero moment of M=p 6 2BJ for totalJ= 2 within the LS-coupling scheme. How- ever both systems are experimentally observed to be magnetic. Density functional theory studies have re-arXiv:1702.03199v1 [cond-mat.str-el] 10 Feb 20172 cently revealed the importance of oxygen hybridization in suppressing the orbital moment so that a large non- zero moment results.34,35Other density functional theory studies have pointed out that spin-orbit coupling and hybridized orbitals play a major role in opening a gap within DFT+U.36{38 Model approaches have shed some light on the nature of the magnetically ordered states by using spin-orbital Hamiltonians39, projecting spin-orbital Hamiltonians to the lowest energy total angular momentum multiplet40,41 or lowest energy doublet42, and other approaches43. In both electron counts, Chen et. al.40,41nd canted ferro- magnetism accompanied by quadrupolar order occupies a majority of parameter space. Additionally they nd a novel non-colinear antiferromagnetic phase in d2, but not d1, which was recently found in d1as the most energet- ically favorable antiferromagnetic state.39Proposals for both valence bond ground states39,40and quantum spin liquids40,44also exist. Yet many puzzles remain unsolved. Despite predic- tions for canted ferromagnetic phases40,41in bothd1and d2, many ferromagnetic d1systems exist but few d2fer- romagnets exist. Furthermore, the physical origin of neg- ative Curie-Weiss temperatures in these ferromagnets is still not understood, and there are multiple studies try- ing to reproduce the magnitudes of the eective Curie moments experimentally measured. Here we study magnetic models for the d1andd2cubic double perovskites with strong spin-orbit coupling with both spin and orbital degrees of freedom at nite temper- ature. While we are focusing on applications to ordering in 5dcubic double perovskites, our results may also ap- ply to 4dcompounds and non-cubic double perovskites as well. Despite the greater complexity than the J= 3=2 andJ= 2 multipolar descriptions, the spin-orbital pic- ture actually leads to an intuitive and qualitative under- standing of several aspects of the phenomenology in these double perovskites. In our study of magnetically ordered phases, we arrive at several conclusion which we now list. First, our results emphasize the importance of the orbital degrees of freedom and anisotropic interactions that accompany them. In particular, we show that the anisotropic interactions result in orbital order that sta- bilizes exotic magnetic order. The orbital (quadrupolar) ordering temperature scale is set both by superexchange interactions and by inter-site Coulomb repulsion, and, in several cases, the orbital ordering temperature can be much larger than the magnetic ordering temperature. Second, although we start with the same electronic model for both d1andd2systems, the energetics of the ground states strongly depend on the electron count. This is re ected in how the spin and orbital degrees of freedom order and provides a qualitative understanding for why ferromagnetism has been repeatedly observed in d1systems while antiferromagnetic interactions remain prevalent in d2systems. Third, the onset of orbital order causes changes in mag- netic susceptibility resulting in non-Curie-Weiss behav-ior. Our model gives the appearance of a negative Curie- Weiss temperature for the ferromagnetic phase while still retaining a properly diverging susceptibility at the ferro- magnetic transition. Fourth, if orbital order occurs, hybridization with oxy- gen alone does not reproduce the experimentally deter- mined values of the magnetic moments in d1systems. Corrections are necessary which may arise from dynami- cal Jahn-Teller eects34and more generally with mixing of thej= 3=2 andj= 1=2 states as we propose. Charge transfer from oxygen might also be considered for sys- tems where the Curie moment has measured be in excess of 1B.45 Lastly, we outline where our calculations stand with respect to other work. First, our zero temperature phase diagram for d1contains a 4-sublattice antiferromagnetic phase and a canted ferromagnetic phase which share un- derlying orbital ordering patterns. Our ndings are com- patible with those of Romh anyi et. al.39, and we further provide a clear interpretation of why these orbital order- ing patterns occur, how they dictate the magnetic order- ing, and then extend our calculations to nite tempera- ture. Like Chen et. al., we nd that orbital ordering can occur at temperatures much higher than the magnetic ordering temperature, however, it leads to a dierent in- terpretation of the negative Curie-Weiss temperature in d1ferromagnets. Furthermore, our spin-orbital approach includes mixing between the j= 3=2 andj= 1=2 states induced by orbital order and intermediate spin-orbit cou- pling energy scales. Second, our zero temperature phase diagram for d2diers remarkably from that of Chen et. al.41which we discuss in detail in later sections. How- ever, the most signicant dierence is in the energetics of antiferromagnetism versus ferromagnetism which gives a qualitative explanation for the broadly observed dif- ferences in ordering between 5 d1and 5d2compounds. Finally, we do not consider valence bond or spin liquid phases in this work although both may be applicable to d1andd2systems. Experimentally, many of our ndings can be tested us- ing multiple probes. At the orbital ordering temperature, there will be second order phase transition with a signa- ture in heat capacity as well as changes in the magnetic susceptibility which are relevant for both powder sam- ples and single crystals. NMR/NQR has recently found evidence of time-reversal invariant order above the mag- netic ordering temperature in Ba 2NaOsO 6.24Resonant X-ray scattering may also provide crucial insights into this hidden order as it is sensitive to orbital occupancy. We show that time-reversal invariant orbital order oc- curs in both ferromagnetic and antiferromagnetic phases we nd, and we suggest that experimental probes which are sensitive to such order should also be pointed at the antiferromagnetic compounds as well.3 II.d1DOUBLE PEROVSKITES Here we develop a spin-orbital model for the d1dou- ble perovskites with magnetic B' ions with spin-orbit coupling featuring both spin-orbital superexchange and inter-site Coulomb repulsion between B' ions. We then solve the model within mean eld theory at both zero temperature and nite temperature. At zero tempera- ture, we nd phases with orbital order and show how this ordering restricts the magnetic order. At nite temper- ature, we examine how orbital order modies magnetic susceptibility and the Curie-Weiss parameters. A. Model In the presence of cubic symmetry, the magnetic B0 ions form an FCC lattice and contain one electron in the outermostdshell. The ve degenerate levels are split by the octahedral crystal eld into the higher energy egor- bitals and lower t2gorbitals so that the t2gshell contains one electron. The electronic structure for the t2gorbitals may be approximated by a nearest neighbor tight-binding model where only one of the three orbitals interacts along each direction. HTB=tX X hiji2cy i;cj;+ h:c: (1) Here the sum over is over allyz,zx, andxyplanes in the FCC lattice. For B' sites in plane , theorbital on siteioverlaps with the orbital on site j. Eachorbital has four neighboring orbitals in its plane plane giving a total of twelve relevant B' neighbors per B' site. In addition to the tight-binding term, the unquenched t2g orbital angular momentum L= 1 results in a spin-orbit coupling on each B0ion2HSO=P iLiSi. Here the orbitalL= 1 and spin S= 1=2 operators both satisfy the usual commutation relations for angular momentum (ie.LL=i~L). The on-site multi-orbital Coulomb interaction is given byHU=P iH(i) U H(i) U=U3JH 2Ni(Ni1) +JH5 2Ni2S2 i1 2L2 i (2) whereUis the Coulomb repulsion and JHis Hund's coupling.46Being in the Mott limit, we calculate the eective spin-orbital superexchange Hamiltonian within second order perturbation theory. The superexchange Hamiltonian is given by the following HSE=JSE 4X X hiji2 r1(3 4+SiSj)(n in j)2 +(1 4SiSj) r2(n i+n j)2+4 3(r3r2)n in j (3) whereJSE= 4t2=Uis the superexchange strength and r1= (13)1,r2= (1)1, andr3= (1+2)1with =JH=U.15Here thet2gorbital electron occupationnumbers are written as n i=cy i;ci;The top line of equation (3) contributes a ferromagnetic spin interaction which requires that one of the two orbitals along a bond is occupied while the other is unoccupied. The bottom line of equation (3) contributes an antiferromagnetic spin interaction which is maximized when both orbitals along a bond are occupied. The strength of Hund's coupling, JH=U, determines the strength of the two interactions relative to each other. Due to the large spatial extent of 5 dorbitals from strong oxygen hybridization, we include a term account- ing for the Coulomb repulsion between orbitals on dier- ent sites.40Let (;; ) be a cyclic permutation of the t2gorbitals (yz;zx;xy ). The repulsion is described by the following: HV=VX X hiji2h 9 4n in j4 3(n in i)(n jn j)i (4) While the coecients in (4) are only quantitatively cor- rect in the limit of quadrupolar interactions, they qualita- tively capture the correct repulsion. For example, within thexyplane, a pair of xyorbitals repel each other more than anxyandyzorbital. The total eective magnetic interaction then reads H=HSO+HSE+HV. Of the three parameters, spin- orbit coupling has the largest energy scale 0:4 eV for the 5doxides while superexchange and and intersite Coulomb repulsion are taken to have energy scales on the order of tens of meV. For 4 doxides, the spin-orbit energy scale is reduced to 0 :10:2 eV so that mixing between thej= 3=2 andj= 1=2 states is likely to occur. While our spin-orbit superexchange interaction is calculated in theLS-coupling scheme, recent evidence suggests that the true picture for the 5 doxides lies between the LS andjjlimits.47 We decouple HSEandHVinto all possible on-site mean elds, i.e. Sin iSjn j!Sin ihSjn ji+hSin iiSjn j hSin iihSjn ji. Since the FCC lattice is not bipartite, we decouple into four inequivalent sites shown in Fig. 1(a) where each set of four inequivalent neighbors forms a tetrahedron. Since the mean elds need not factor into the product of spins and orbitals, hSin ii 6=hSiihn ii, there are a total of 15 mean elds per site comprised of three spin operators, three orbital operators, and prod- ucts of the spin and orbital operators. Applying the con- straint that one electron resides on each site, there are 11 independent mean elds per site giving a total of 44 mean elds in the tetrahedron. We then numerically solve for the lowest energy solutions of the mean eld equations. B. Zero Temperature Mean Field Theory In the limit where spin-orbit coupling is the domi- nant energy scale, the magnetically ordered phases can be characterized by an arrangement of ordered J= 3=2 multipoles.40However, when JSEandare comparable,4 0.00 0.05 0.10 0.15 0.200.00.20.40.60.8 kBT/λnyz nzxnxy MTo Tc 0.0 0.1 0.2 0.3 0.4 0.50.000.050.100.150.20 μeff(μB)kBTo/λ ←0.02=V/λ←0.040.06→AFM 4-sublatticeCanted FM V/λ=0 0.01 0.02 0.03 0.0 0.2 0.4 0.6 0.80.100.120.140.160.180.20 JSE/λJH/U LSLS(a) (b) (c) (d) (e) FIG. 1. (a) FCC lattice decoupled into four inequivalent sites shown by four dierent colors. (b) The orbital ordering pattern driven by both JSEandVconstrains the direction of orbital angular momentum. Deviations from the j= 3=2 limit produce a net magnetic moment Mas the spin and orbital components separate. (c) The zero temperature phase diagram shows phases where the moments in each plane of the page (eg. plane containing yellow and black sites) are collinear and the moments between planes are at approximately 90 degrees due to the orbital ordering pattern. Increasing inter-orbital repulsion Vbetween sites reduces minimum strength of Hund's coupling required to induce FM. (d) Mean eld values for the bottom sites (black, yellow) are shown as a function of temperature. The nyzorbital (red) has the largest occupancy followed by the xyorbital (blue). (e) With JSE= 0, we calculate the orbital ordering temperature Toand eective Curie moment enhancement efor dierent values of V. a multipolar description within the j= 3=2 states breaks down and consequently both spin and orbital parts must be considered independently. Furthermore, the orbital contributions come in the forms of both orbital occu- pancynand orbital angular momentum L. Sincen, L, andSare coupled, there is competition between order parameters which results in non-trivial ordering. The zero temperature phase diagram is shown in Fig. 1(c) as a function of the strength of Hund's cou- pling=JH=Uand superexchange JSE=. Large val- ues ofsupport a canted ferromagnetic (FM) structure while smaller values support an antiferromagnetic (AFM) structure. The spin-1/2 and orbital-1 angular momenta order parameters hSiandhLiare shown for each of the four inequivalent sites from Fig. 1(a). In both phases, one of the three directions has no ordered angular momenta, e.g.hLzi=hSzi= 0, so that both magnetic structures are co-planar. Both phases feature some separation of the ordered spin and orbital moments which increases as a function of JSE=. To understand why these magnetic structures emerge, we examine the orbital occupancy or- der parameters, n, separately from the magnetic order parameters. In both the FM and AFM phases, there is an orbital ordering pattern pictured in Fig. 1(b). The two sites in the lower plane of Fig. 1(b) have the yzorbital(red) with the highest electron occupancy while the xy orbital (blue) receives the second highest and the zxor- bital receives the lowest (green, not pictured). The two sites in the upper plane have identical ordering except the roles of the yzandzxorbitals are reversed. Qualita- tively this orbital ordering pattern is favored by both the HVandHSEterms which pushes electrons onto orbitals that have small overlaps. This allows the electron on a green orbital to hop onto an unoccupied green orbital in the plane directly above or below (and similarly for red orbitals). Since these mechanisms work to suppress the overlap of half lled orbitals, ferromagnetic interactions may become energetically favorable. A derivation of the mean eld solution for HVis provided in Appendix A. Once orbital order sets in, the allowed magnetic phases are restricted by the direction of orbital angular momen- tum. Full orbital polarization is time-reversal invariant and would not allow orbital magnetic order. However Fig. 1(d) shows that each site has at least two orbitals with non-negligible occupancy which allows for the devel- opment of an orbital moment. Thus the direction of the orbital moment is determined by the direction common to the two planes of occupied orbitals with the overall sign of the direction (e.g. + xorx) left undetermined. Figure 1(c) shows that the orbital angular momenta be-5 tween planes are close to 90 degrees apart for both FM and AFM phases. As spin and orbital angular momen- tum are coupled together, the spin moments will select which direction the orbital moments choose (i.e. + xor x). The decision to enter an FM or AFM state is then determined by the spin interactions characterized both by the strength of =JH=Uand the magnitude of the orbital order parameter. If is large, then ferromag- netic spin interactions follow and result in both the spin and orbital degrees of freedom aligning within each xy plane producing a net canted FM structure. If is small, then antiferromagnetic spin interactions follow which re- sult in the 4-sublattice AFM structure. We note that the Goodenough-Kanamori-Anderson rules48{50are not enough to determine whether FM or AFM is favored, and the interplay between spin-orbit coupling and the anisotropic orbital degrees of freedom play a crucial role in tipping the energy balance one way or the other. There are two additional factors that determine if the FM or AFM state is selected. The dominant eect is the degree of orbital polarization. When the strength of inter-orbital repulsion Vis increased, the tendency for orbitals to order becomes stronger. This disfavors the overlap of half lled orbitals causing AFM superex- change, and hence promotes FM superexchange. Figure 1(c) shows a dramatic shift toward FM when a small V interaction is included. The second eect comes from the separation of spin and orbital degrees of freedom. When JSEbecomes comparable to , the spin moments can partially break away from the orbital moments tending more toward a regular spin FM state instead of a canted spin FM state. Since a spin AFM state does not bene- t from this separation to the same extent, FM becomes increasingly energetically favorable. Dimer phases have been proposed39,40and oer a way to explain the absence of magnetic order in d1materials. However when =JSEis large, these dimer phases only occur at very small values of =JH=U.39Furthermore, orbital repulsion Vacts to further suppress dimerization. Since our focus is on the magnetically ordered phases of these double perovskites, we will not pursue these possi- bilities in this work. C. Finite Temperature Mean Field Theory We now examine the model at nite temperature. Fig- ure 1(d) shows a characteristic order parameter vs tem- perature curve. At high temperatures all order parame- ters are trivial and each orbital occupancy takes a value ofnyz=nzx=nxy= 1=3. As temperature is low- ered, the rst transition is to a time reversal invariant orbitally ordered state [see Fig. 1(b)] at temperature To whose scale is set both by VandJSE. AtTo, the entropy released is from orbital degeneracy, even when V= 0. Below the second transition at a temperature Tcwhose energy scale is set only by JSE, time reversal symmetry is broken on each site with the development of magneticorder, and the remaining entropy is released. The fundamental question arises of how large the ex- change interaction JSEand repulsion Vare in materi- als systems. Fits to experimental susceptibility21show Ba2LiOsO 6and Ba 2NaOsO 6have relatively small Curie- Weiss temperatures of =40:5 K and=32:5 K respectively indicating that JSEin cubic 5d1double per- ovskites is weak. However integrated heat capacity22of Ba2NaOsO 6shows an entropy release just short of Rln 2 atTcconsistent with the splitting of a local Kramer's doublet with no further anomalies in heat capacity up to 300 K. This suggests ToTcso thatVis the most rele- vant parameter for determining the properties well above Tc. Since the onset of orbital order necessarily alters the angular momenta available to order and respond to an applied magnetic eld, we calculate how the eective Curie-Weiss constant depends on orbital ordering. Using JSE= 0, we calculate the temperature dependent suscep- tibility within mean eld theory as a function of tempera- ture for dierent values of V=. For each value of V=we calculate both the orbital ordering temperature Toand the eective Curie moment e=gBp J(J+ 1) from a t to low temperature inverse susceptibility. Fig. 1(e) gives numerical results from our mean eld theory that shows a linear relationship between Toande. In the absence of orbital order, the projection of the magneti- zation operator to the J= 3=2 space is identically zero. However once orbital order sets in, the j= 1=2 com- ponents of the wavefunction get mixed with the j= 3=2 components. The matrix elements that connect these two Jspaces then acquire expectation values and allow the eective Curie moment to become non-zero. An approx- imate derivation of this relation is provided in Appendix A. In addition to the perturbative separation of LandS due to mixing of the Jmultiplets, hybridization with oxy- gen has been shown to greatly reduce the orbital contri- bution to the moment.34,35Here the magnetization oper- ator assumes the form M= 2S Lwhere = 0:536 and results in an eective Curie moment of 0 :60Bcompared to an experimental value of 0 :67B.21However the onset of quadrupolar order within the j= 3=2 states results in a reduction of the nominal 0 :60Bvalue. In general, the projection of a linear combination of the nyz,nzx, and nxyoperators to the j= 3=2 states is (up to a constant shift) a linear combination of the operators J2 xJ2 yand J2 z. By projecting to the lowest energy doublet induced by these operators, we may calculate the gfactors for this pseudo-spin 1/2 space. While the gfactors are dierent in the three cubic directions due to the anisotropic na- ture of quadrupolar order, the sum of the squares is a con- stant, and the powder average is g2=1 3(g2 x+g2 y+g2 z) = 3. Then splitting of the j= 3=2 states reduces the Curie moment by a factor of ( gp 3=4)=(p 15=4) =p 3=5 which makes the calculated moment 0 :47B. We nd that mix- ing between the j= 3=2 andj= 1=2 states brings the calculated moment closer to experimental values.6 without hybridizationwith hybridization 0 0.04 0.08 0.12012345 kBT/λχ To0 0.04 0.08 0.120246810 kBT/λχ To 0 0.04 0.08 0.1200.20.40.60.81 kBT/λχ-1 To0 0.04 0.08 0.120.0.10.20.30.40.5 kBT/λχ-1 To FIG. 2. Typical susceptibility, =1 3(xx+yy+zz), and inverse susceptibility are plotted against temperature. The susceptibility curves are shown both without the correction due to hybridization, = 1, and with the correction, = 0:536. We have chosen JSE= 0 and left Vnite to illustrate the consequence of orbital order on the susceptibility. By choosingJSE= 0, we show that although Tc= 0 whileTo 6= 0, the tted Curie-Weiss temperature appears to be negative. Note that a single Curie-Weiss t cannot span the entire range belowTo. There are more consequences of orbital ordering that are particularly important for the magnetic susceptibil- ity of this spin-orbital system. The orbital order re- duces symmetry of the system and causes susceptibility to become anisotropic. Since the orbital ordering pat- tern tends to push angular momentum into the order- ing planes, susceptibility is enhanced in these two di- rections while reduced in the third direction. Although anisotropic susceptibility is expected once cubic symme- try is broken, it is an easy test to determine at what tem- perature orbital order occurs. However this is yet a more important eect. When orbital order sets in at To, the ef- fective moment changes as the orbital degrees of freedom tend toward a (partially) quenched state which results in an eective moment which changes with temperature. The non-Curie-Weiss behavior will be critical when inter- preting the observed negative Curie-Weiss temperatures in 5d1ferromagnetic compounds. Within our mean eld theory, we now calculate the susceptibility without the hybridization correction and with the hybridization correction to show this eect. Forclarity, we set JSE= 0 to isolate the contributions from orbital order from those of magnetic interactions. Fig. 2 shows that below the orbital ordering temperature, the susceptibility deviates from the Curie-Weiss law. How- ever the data below Tocan be t over a large range to give a negative Curie-Weiss intercept despite the absence of magnetic interactions. Although the region where the t works the best is just below Towhere the orbital oc- cupation is rapidly changing, there is a quantitative ex- planation for this. We consider the case without hybridization where the eective moment for the j= 3=2 states is identically zero. When orbital order occurs, there is mixing between the j= 3=2 andj= 1=2 states proportional to Vhni=. Then below To, the eective magnetization operator for the lowest energy Kramer's doublet increases in a way proportional tohnidue to the matrix elements between j= 3=2 andj= 1=2. The eective Curie moment goes as the square of magnetization and thus the enhancement is of orderhni2. Since orbital order below Toscales as hni/jToTj1=2within mean eld theory, the eective Curie moment gains a contribution scaling as jToTjjust belowTo. At temperatures far away from Tc, the leading correction to susceptibility and and inverse susceptibil- ity is linear leading to the appearance of a Curie-Weiss law. We note, however, that this is articial and is not indicative of the physical magnetic interactions. Despite using mean eld critical exponents, qualita- tively we have understood how deviations from the Curie- Weiss law occur from changing orbital occupancy. Be- cause we have used a simple model consisting of only andVwith a-priori knowledge of the ideal Curie-Weiss temperature of zero, we have been able to clearly in- terpret the non-Curie-Weiss susceptibility. However the tting procedure must be performed with some caution since both the t region and the chosen value of 0(tem- perature independent term) determine the reported CW ande. In fact, experimental behavior may deviate even more strongly due to the quantitative details of how orbital occupancies change with temperature. In partic- ular, coupling between orbitals and phonons may be a crucial aspect here.34 Reference 40 claimed negative Curie-Weiss tempera- tures were achievable in their model for ferromagnetic ground states, although this crucial result was not ex- plicitly shown. Reference 26 has reproduced that model under the circumstances necessary to generate ferromag- nets with negative Curie Weiss temperatures, and they nd jump discontinuities (nite-to-innite) in the mag- netic susceptibility at Tc. Such jump discontinuities are not seen in Ba 2NaOsO 6, Ba 2MgReO 6, or Ba 2ZnReO 6. We note that our mechanism for shifting the Curie-Weiss temperature is free from these discontinuities and fea- tures a properly diverging susceptibility at Tcfor the fer- romagnetic phase, thereby providing a more accurate and natural description of the transition.7 III.d2DOUBLE PEROVSKITES Here we will modify the d1spin-orbital model to ac- commodate two electrons. Again, we then solve the model within mean eld theory at both zero tempera- ture and nite temperature. At zero temperature, we nd new orbital phases not found in our d1phase di- agram. For completeness, we show susceptibilities and orbital occupancies at nite temperature. A. Model Our model for d2is constructed from the same consid- erations used in d1only changing the electron count. The tight-binding model HTB, the inter-site orbital repulsion HV, and the on-site Coulomb interaction HUare valid for thed2model without modication. However spin-orbit coupling and superexchange will change since the total spin and orbital angular momentum on each site are now composed of two electrons. In the Mott limit, Hund's rules are enforced by HUresulting in a total spin S= 1 and total orbital angular momentum L= 1 on each lat- tice site. Within this space, the spin-orbit interaction takes the form H0 SO= 2P iLiSi. The superexchange Hamiltonian is given by the following H0 SE=JSE 12X X hiji2 r1(2 +SiSj)(n in j)2 (1SiSj) (n i+n j)2+ (3 2r35 2)n in j (5) where the denitions of JSE,r1, andr3correspond to those used previously. As before, the top line in (5) gives a ferromagnetic spin interaction when only one of the two interacting orbitals is occupied while the second line gives an antiferromagnetic spin interaction which is maximized when two half lled orbitals overlap. The total eective magnetic interaction then reads H0=H0 SO+H0 SE+HV. We decouple H0 SEandHVinto all possible on-site mean elds using four inequivalent sites as before and then solve the mean eld equations numerically. B. Zero Temperature Mean Field Theory The zero temperature phase diagram is shown in Fig. 3(b) as a function of the strength of Hund's cou- pling=JH=Uand superexchange JSE=. In the limit of large spin-orbit coupling and the absence of inter-site orbital repulsion, the ground state is predominantly AFM with the moment aligning parallel to the [110] direction within a plane and antiparallel to the [110] direction in the next plane. To see why this phase occupies such a large region of phase space, we analyze the orbital struc- ture that accompanies it, as shown in Fig. 3(a). On each site, one electron moves onto the yzorbital and the other onto thezxorbital. In this conguration both occupiedorbitals overlap with occupied orbitals on neighboring sites and unoccupied orbitals overlap with other unoccu- pied orbitals so that AFM superexchange is maximized. These orbitally controlled AFM interactions then take place between planes and not within planes resulting in AFM between planes while FM interactions prevail in each plane. Since this this orbital pattern is compatible with tetragonal distortion, as observed in Sr 2MgOsO 630, we expect nominally cubic crystal structures to distort. The next phase we nd is the AFM 4-sublattice copla- nar structure previously found in the d1phase diagram. As before, the orbital degrees of freedom are closely aligned with the directions perpendicular to the occupied orbitals, and the spin and orbital moments perturbatively separate from each other with increasing superexchange. It is worth noting that in this region of the phase dia- gram, the next lowest energy phase is AFM [100] that can become a competitive ground state upon inclusion of anisotropy. For large superexchange and Hund's coupling, we nd a ferromagnetic phase with ordering along the [100] di- rection that is best characterized as a \3-up, 1-down" collinear structure where three of the four moments order parallel to each other along the chosen direction and the fourth moment orders anti-parallel to the other three. It is worth noting that the second most energetically favor- able phase in this region of the phase diagram is another \3-up, 1-down" structure where each moment is either approximately parallel or antiparallel to the [110] direc- tion. The energy dierence between the FM [100] and FM [110] phases is negligible and either phase is a suit- able ground state. In addition to these two FM phases, we nd a canted FM solution to the mean eld equa- tions with the same orbital ordering pattern as the d1 canted FM phase. However it is higher enough in energy to rule out as a viable ground state and consequently is not shown in the phase diagram. Unlike the AFM [110] and AFM 4-sublattice struc- tures, the FM/AFM [100] structures features an approx- imately higher degree of degeneracy due to the orbital degrees of freedom. Like the AFM 4-sublattice orbital structure, the FM/AFM [100] orbital structure tends to minimize repulsion between orbitals. Of the four tetra- hedral sites, three of them are able to minimize the re- pulsion and allow occupied orbitals to hop to unoccupied orbitals. While the repulsion is minimized between those three sites, this forces occupied orbitals on each site to point toward the fourth site. Figure 3(a) shows that this fourth site in the FM/AFM [100] orbital pattern chooses one of the orbitals to have a majority occupancy (solid color) and the other two orbitals to have minority occu- pancies (semi-transparent colors). In the FM [110] phase, a similar situation occurs with the main dierence being that now two orbitals have majority occupancy and one orbital has minority occupancy. Before magnetic order sets in, the degeneracy is approximately extensive as the fourth site on every tetrahedron in the lattice has local orbital frustration.8 AFM 110AFM 4-sublattice (AFM 100)FM 100 (FM 110) V/λ=0 0.01 0.02 0.03SL 0.0 0.2 0.4 0.6 0.80.000.050.100.150.20 JSE/λJH/U(a) (b) FIG. 3. (a) Orbital ordering patterns are shown for each type of magnetic order. Orbitals shown in solid colors represent the most occupied orbitals while orbitals not shown or shown transparently have lower occupancy. (b) The zero temperature phase diagram shows three ground state phases: AFM with moments (anti)parallel to [110], AFM 4-sublattice structure, and FM with moments parallel to [100]. Phases shown in parenthesis (AFM [100], FM [110]) show the next lowest energy phase in each region. When inter-site orbital repulsion HVis included, the phase boundaries shift. The most dramatic eect is the recession of the boundary between AFM [110] and the AFM 4-sublattice structure. This becomes apparent by comparing the orbital congurations of the two phases as the AFM [110] structure maximizes the number of AFM singlets which are penalized by the orbital repulsion. Un- like in the d1situation, we nd that the inclusion of V does not enhance FM. While the FM/AFM [100] and FM [110] orbital structures are much more compatible with HVthan the AFM [110] structure, the AFM 4-sublattice structure still dominates. We note that unlike the d1 case, canted FM is not favorable here due to the electron count. The d1case relies on pushing the large majority of the electron weight onto one orbital while retaining a smaller occupancy on a second orbital to generate an orbital moment. However in d2, this second orbital must also be occupied which consequently induces AFM inter- actions within each horizontal plane. Although we have focused on spin-orbital magnetic or- der, it is necessary to remark that exotic singlet ground states are also possible. The Kramer's theorem guaran- tees that trivial ionic singlets will not occur in d1systems, and therefore the experimental observation of singlet be- havior is an indication of a non-trivial ground state. Such considerations do not apply to d2, and experimental ob- servations of singlet behavior may arise from trivial local magnetic singlets. Consequently this local non-magnetic singlet possibility must rst be ruled out when searchingfor exotic singlet behavior. C. Finite Temperature Mean Field Theory Here we consider the model at nite temperature. Fig- ure 4 shows orbital occupations and inverse magnetic susceptibility as a function of temperature for the three ground state phases from the previous section. At high temperature, the orbitals have a uniform occupancy of nyz=nzx=nxy= 2=3. There is a temperature To where time-reversal invariant order sets in through the orbitals and second temperature Tcwhere magnetic or- der sets in. In the case of the AFM [110] phase, Fig. 4(a) shows the two ordering temperatures coincide and that the electrons are pushed onto the nyzandnzxorbitals to maximize antiferromagnetic superexchange. This is dif- ferent from the orbital ordering previously reported be- cause this ordering maximizes orbital repulsion instead of minimizing it, so orbital order itself is not favorable and is entirely driven by antiferromagnetic superexchange. In this situation, the Curie-Weiss law with a negative Curie- Weiss temperature occurs as expected. The transition to an AFM 4-sublattice structure is shown in Fig. 4(b). Above Tosusceptibility follows the Curie-Weiss law with a negative Curie-Weiss constant. BelowTothe orbital occupancies change along with the inverse susceptibility to deviate from the high tempera- ture behavior. Just below To, susceptibility may be t9 (a)AFM 110 (b)AFM 4-sublattice (c)FM 100 0.00 0.02 0.04 0.06 0.08 0.100.000.050.100.150.200.250.30 kBT/λ(χ-χ 0)-1(a.u.)0.00.20.40.60.81.0 nyz,nzx,nxynyz,nzx nxy Tc0.00 0.02 0.04 0.06 0.08 0.100.000.050.100.15 kBT/λ(χ-χ 0)-1(a.u.)0.00.20.40.60.81.0 nyz,nzx,nxynyz nxy nzx Tc To0.00 0.05 0.10 0.15 0.200.000.050.100.150.20 kBT/λ(χ-χ 0)-1(a.u.)0.00.20.40.60.81.0 nyz,nzx,nxynyznxy nzx Tc To FIG. 4. Characteristic inverse susceptibility (blue) and orbital occupation (purple) curves are plotted against temperature for the three phases in Fig. 3: (a) AFM [110], (b) AFM 4-sublattice, and (c) FM [100]. Susceptibility is averaged over all three directions, 1= 3(xx+yy+zz)1, and all sites in the tetrahedra. Orbital occupancies are shown for the site pictured above each plot. to another Curie-Weiss law with another negative Curie- Weiss constant. Similarly to the d1case, there is still deviation from the Curie-Weiss law in this regime, how- ever, the deviations are smaller and so is the enhance- ment of the eective magnetic moment due to mixing of theJ= 2 states with higher energy multiplets. But we note that when JSE= 0, we still nd the appearance of a negative Curie-Weiss constant due to non-Curie-Weiss susceptibility as we did in the d1model. Finally, the transition to an FM [100] structure is shown in Fig. 4(c). Deviations from the Curie-Weiss law are seen below To, and the sign of the Curie-Weiss constant can switch from negative to positive depending which region tted. Unlike the other phases, magnetic or- der appears at Tcwith a rst-order transition marked by the jumps in orbital occupancy and susceptibility. This arises from competition between having the most ener- getically favorable orbital structure at high temperature and the most energetically favorable magnetic structure at low temperature. As in the d1case, we compare values of the the- oretical moments to those from experiment. Oxygen hybridization will result in a Curie moment of e=p 6(1 =2)B. Assuming almost half of the moment resides on oxygen, the calculated moment is then e 1:8B. This is close to the experimentally observed mo- ments in Sr 2MgOsO 6and Ca 2MgOsO 6(both 1:87B)29 but further o from those of Ba 2YReO 6(1:93B)31and La2LiReO 6(1:97B)31. IV. CONCLUSIONS We have studied spin-orbital models for both d1and d2double perovskites where the B' ions are magnetic and have strong spin-orbit coupling. We found several non- trivial magnetically ordered phases characterized both byordering of the spin/orbital angular momentum and or- dering of the orbitals. This orbital ordering shows why ferromagnetism is energetically favorable in these sys- tems when electron count is d1but not when it is d2, particularly at large spin-orbit coupling. Additionally, ordering of the orbital degrees of freedom can produce non-Curie-Weiss behavior which can lead to the appear- ance of a negative Curie-Weiss in the canted ferromag- netic phase. We emphasize that examination of the spin and orbital degrees of freedom separately gives an en- hanced qualitative understanding of the magnetism for this class of spin-orbit coupled double perovskites. V. ACKNOWLEDGEMENTS We thank Patrick Woodward and Jie Xiong for their useful discussions. We acknowledge the support of the Center for Emergent Materials, an NSF MRSEC, under Award Number DMR-1420451. Appendix A: eenhancement and Toford1model To obtain the orbital ordering temperature Toand the eective moment eas a function of V=, we will solve the mean eld equations for HV+HSOanalytically. The relevant mean eld parameters for the four sites from Fig. 1(b) are given below hnxy 1i=hnxy 2i=hnxy 3i=hnxy 4i=1 3+nz (A1) hnyz 1i=hnyz 2i=hnzx 3i=hnzx 4i=1 3+nx (A2) with the conditionP n i= 1 determining the other four parameters. We obtain the single site mean eld10 Hamiltonian for V. H0 V=V (86 3nx+43 3nz)nyz+ (43 3nx+53 3nz)nxy (A3) Since above Tc, the high mean eld Hamiltonian H0 MF= H0 V+HSOis time reversal invariant, we rotate into the basis of total angular momentum Jwhich factors into two 33 blocks of doublets. The upper block may be chosen to be of the form below 0 BB@3 243V(2nx+nz) 3p 67Vnzp 2 43V(2nx+nz) 3p 67Vnz 243V(2nx+nz) 6p 3 7Vnzp 243V(2nx+nz) 6p 37Vnz 21 CCA (A4) where the basis jJ;mJiis given byj1=2;+1=2i, j3=2;3=2i,j3=2;+1=2iin this order. Using = arctan 43p 3 (2nx+nz)=63nz, we diagonalize the Hamiltonian in the j= 3=2 block 0 @3 2x y x 0 y0 1 A (A5) where =Vp 1849nx(nx+nz) + 793n2z 3p 3(A6) x=V43p 3(2nx+nz) cos 2+ 63nzsin 2 9p 2(A7) andyis given byxwith sin!cosand cos! sin applied. The lowest J= 3=2 doublet with energy is mixed with the J= 1=2 doublet with amplitude 2x=3. We project the magnetization operator M= 2SL onto this lowest doublet. Since nominally g= 0 for the j= 3=2 states, the rst non-zero correction to the wave- function comes from mixing of the j= 3=2 andj= 1=2 states. From the projection, we obtain the gfactors for this doublet in all three directions (ie. Mx=gxB 2x, etc) and compute the average gfactor obtained in a pow- der susceptibility measurement g2=1 3 g2 x+g2 y+g2 z to obtain the powder average eective moment for the dou- blet. For the parameter regime we are interested in, nz has a negligible contribution to g, and thegfactor is given approximately by g= 344Vjnxj=9p 3so that the moment ise= 172VjnxjB=9. Now we obtain the mean eld orbital ordering temper- atureTowhich occurs when the j= 3=2 states split. In the limit that nzis negligible, we self consistently solve for the expectation value of the operator the projections of the operator nx!nyz1 3within the 22 sub- space of energies and (ie.jJ= 3=2;Jz=3=2i andjJ= 3=2;Jz= +1=2i). The projection of the nx operator to this subspace is nx! 1 2p 31 6 1 61 2p 3! (A8)so that the mean eld equations for nxread nx=1 2p 3tanh (A9) where 43V 3p 3nx. Then we nd kBTo= 43V=18 which is consistent with Ref. 40. However, in contrast to Ref. 40, our analysis shows that this orbital order is com- patible with both the FM and AFM phases and does not disappear below Tcfor the AFM phase. We can relate the ratios of these results as seen in Fig. 1(e) by kBTo= e=B=1 8nx: (A10) Using the zeroth order approximation for nxas 1=2p 3, this ratio becomes 0.43 which is close to that shown in Fig. 1(e).11 1W. Witczak-Krempa, G. Chen, Y. B. Kim, and L. Balents, Annual Review of Condensed Matter Physics 5, 57 (2014). 2J. G. Rau, E. K.-H. Lee, and H.-Y. Kee, Annual Review of Condensed Matter Physics 7, 195 (2016). 3E. Kermarrec, C. A. Marjerrison, C. M. Thompson, D. D. Maharaj, K. Levin, S. Kroeker, G. E. Granroth, R. Flacau, Z. Yamani, J. E. Greedan, and B. D. Gaulin, Phys. Rev. B91, 075133 (2015). 4A. E. Taylor, R. Morrow, R. S. Fishman, S. Calder, A. I. Kolesnikov, M. D. Lumsden, P. M. Woodward, and A. D. Christianson, Phys. Rev. B 93, 220408 (2016). 5A. M. Cook, S. Matern, C. Hickey, A. A. Aczel, and A. Paramekanti, Phys. Rev. B 92, 020417 (2015). 6A. A. Aczel, A. M. Cook, T. J. Williams, S. Calder, A. D. Christianson, G.-X. Cao, D. Mandrus, Y.-B. Kim, and A. Paramekanti, Phys. Rev. B 93, 214426 (2016). 7G. Khaliullin, Phys. Rev. Lett. 111, 197201 (2013). 8A. Akbari and G. Khaliullin, Phys. Rev. B 90, 035137 (2014). 9K. Pajskr, P. Nov ak, V. Pokorn y, J. Koloren c, R. Arita, and J. Kune s, Phys. Rev. B 93, 035129 (2016). 10O. N. Meetei, W. S. Cole, M. Randeria, and N. Trivedi, Phys. Rev. B 91, 054412 (2015). 11T. Dey, A. Maljuk, D. V. Efremov, O. Kataeva, S. Gass, C. G. F. Blum, F. Steckel, D. Gruner, T. Ritschel, A. U. B. Wolter, J. Geck, C. Hess, K. Koepernik, J. van den Brink, S. Wurmehl, and B. B uchner, Phys. Rev. B 93, 014434 (2016). 12B. Ranjbar, E. Reynolds, P. Kayser, B. J. Kennedy, J. R. Hester, and J. A. Kimpton, Inorganic Chemistry 54, 10468 (2015). 13B. F. Phelan, E. M. Seibel, D. B. Jr., W. Xie, and R. Cava, Solid State Communications 236, 37 (2016). 14K. I. Kugel' and D. I. Khomski, Soviet Physics Uspekhi 25, 231 (1982). 15G. Khaliullin, Progress of Theoretical Physics Supplement 160, 155 (2005). 16M. A. de Vries, A. C. Mclaughlin, and J.-W. G. Bos, Phys. Rev. Lett. 104, 177202 (2010). 17T. Aharen, J. E. Greedan, C. A. Bridges, A. A. Aczel, J. Rodriguez, G. MacDougall, G. M. Luke, T. Imai, V. K. Michaelis, S. Kroeker, H. Zhou, C. R. Wiebe, and L. M. D. Cranswick, Phys. Rev. B 81, 224409 (2010). 18J. P. Carlo, J. P. Clancy, T. Aharen, Z. Yamani, J. P. C. Ru, J. J. Wagman, G. J. Van Gastel, H. M. L. Noad, G. E. Granroth, J. E. Greedan, H. A. Dabkowska, and B. D. Gaulin, Phys. Rev. B 84, 100404 (2011). 19M. A. d. Vries, J. O. Piatek, M. Misek, J. S. Lord, H. M. Rnnow, and J.-W. G. Bos, New Journal of Physics 15, 043024 (2013). 20Z. Qu, Y. Zou, S. Zhang, L. Ling, L. Zhang, and Y. Zhang, Journal of Applied Physics 113, 17E137 (2013). 21K. E. Stitzer, M. D. Smith, and H.-C. zur Loye, Solid State Sciences 4, 311 (2002). 22A. S. Erickson, S. Misra, G. J. Miller, R. R. Gupta, Z. Schlesinger, W. A. Harrison, J. M. Kim, and I. R. Fisher, Phys. Rev. Lett. 99, 016404 (2007). 23A. J. Steele, P. J. Baker, T. Lancaster, F. L. Pratt, I. Franke, S. Ghannadzadeh, P. A. Goddard, W. Hayes, D. Prabhakaran, and S. J. Blundell, Phys. Rev. B 84, 144416 (2011).24L. Lu, M. Song, W. Liu, A. P. Reyes, P. Kuhns, H. O. Lee, I. R. Fisher, and V. F. Mitrovi c, ArXiv e-prints (2017), arXiv:1701.06117 [cond-mat.str-el]. 25K. G. Bramnik, H. Ehrenberg, J. K. Dehn, and H. Fuess, Solid State Sciences 5, 235 (2003), dedicated to Sten An- dersson for his scientic contribution to Solid State and Structural Chemistry. 26C. A. Marjerrison, C. M. Thompson, G. Sala, D. D. Ma- haraj, E. Kermarrec, Y. Cai, A. M. Hallas, M. N. Wilson, T. J. S. Munsie, G. E. Granroth, R. Flacau, J. E. Greedan, B. D. Gaulin, and G. M. Luke, Inorganic Chemistry 55, 10701 (2016). 27H. L. Feng, Y. Shi, Y. Guo, J. Li, A. Sato, Y. Sun, X. Wang, S. Yu, C. I. Sathish, and K. Yamaura, Jour- nal of Solid State Chemistry 201, 186 (2013). 28C. M. Thompson, J. P. Carlo, R. Flacau, T. Aharen, I. A. Leahy, J. R. Pollichemi, T. J. S. Munsie, T. Medina, G. M. Luke, J. Munevar, S. Cheung, T. Goko, Y. J. Uemura, and J. E. Greedan, Journal of Physics: Condensed Matter 26, 306003 (2014). 29Y. Yuan, H. L. Feng, M. P. Ghimire, Y. Matsushita, Y. Tsujimoto, J. He, M. Tanaka, Y. Katsuya, and K. Yamaura, Inorganic Chemistry 54, 3422 (2015), pMID: 25751088, http://dx.doi.org/10.1021/ic503086a. 30R. Morrow, A. E. Taylor, D. J. Singh, J. Xiong, S. Rodan, A. U. B. Wolter, S. Wurmehl, B. B uchner, M. B. Stone, A. I. Kolesnikov, A. A. Aczel, A. D. Christianson, and P. M. Woodward, Scientic Reports 6, 32462 (2016). 31T. Aharen, J. E. Greedan, C. A. Bridges, A. A. Aczel, J. Rodriguez, G. MacDougall, G. M. Luke, V. K. Michaelis, S. Kroeker, C. R. Wiebe, H. Zhou, and L. M. D. Cran- swick, Phys. Rev. B 81, 064436 (2010). 32A. A. Aczel, Z. Zhao, S. Calder, D. T. Adroja, P. J. Baker, and J.-Q. Yan, Phys. Rev. B 93, 214407 (2016). 33C. M. Thompson, L. Chi, J. R. Hayes, A. M. Hallas, M. N. Wilson, T. J. S. Munsie, I. P. Swainson, A. P. Grosvenor, G. M. Luke, and J. E. Greedan, Dalton Trans. 44, 10806 (2015). 34L. Xu, N. A. Bogdanov, A. Princep, P. Fulde, J. v. d. Brink, and L. Hozoi, npj Quantum Materials 1, 16029 (2016). 35K.-H. Ahn, K. Pajskr, K.-W. Lee, and J. Kunes, ArXiv e-prints (2016), arXiv:1611.07749 [cond-mat.str-el]. 36H. J. Xiang and M.-H. Whangbo, Phys. Rev. B 75, 052407 (2007). 37K.-W. Lee and W. E. Pickett, EPL (Europhysics Letters) 80, 37008 (2007). 38S. Gangopadhyay and W. E. Pickett, Phys. Rev. B 91, 045133 (2015). 39J. Romh anyi, L. Balents, and G. Jackeli, ArXiv e-prints (2016), arXiv:1611.00646 [cond-mat.str-el]. 40G. Chen, R. Pereira, and L. Balents, Phys. Rev. B 82, 174440 (2010). 41G. Chen and L. Balents, Phys. Rev. B 84, 094420 (2011). 42T. Dodds, T.-P. Choy, and Y. B. Kim, Phys. Rev. B 84, 104439 (2011). 43H. Ishizuka and L. Balents, Phys. Rev. B 90, 184422 (2014). 44W. M. H. Natori, E. C. Andrade, E. Miranda, and R. G. Pereira, Phys. Rev. Lett. 117, 017204 (2016). 45Private communications. 46A. Georges, L. de' Medici, and J. Mravlje, Annual Review12 of Condensed Matter Physics 4, 137 (2013). 47B. Yuan, J. P. Clancy, A. M. Cook, C. M. Thompson, J. Greedan, G. Cao, B.-C. Jeon, T.-W. Noh, M. H. Upton, D. Casa, T. Gog, A. Paramekanti, and Y.-J. Kim, ArXive-prints (2017), arXiv:1701.06584 [cond-mat.str-el]. 48J. Goodenough, Magnetism and the Chemical Bond (In- terscience Publishers, 1963). 49J. Kanamori, J. Phys. Chem. Solids 10, 87 (1959). 50P. W. Anderson, Phys. Rev. 115, 2 (1959).
1301.3596v1.Mechanical_generation_of_spin_current_by_spin_rotation_coupling.pdf	Mechanical generation of spin current by spin-rotation coupling Mamoru Matsuo1;2, Jun'ichi Ieda1;2, Kazuya Harii1;2, Eiji Saitoh1;2;3;4, and Sadamichi Maekawa1;2 1Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan 2CREST, Japan Science and Technology Agency, Sanbancho, Tokyo 102-0075, Japan 3Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 4WPI, Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan (Dated: October 13, 2018) Spin-rotation coupling, which is responsible for angular momentum conversion between the elec- tron spin and rotational deformations of elastic media, is exploited for generating spin current. This method requires neither magnetic moments nor spin-orbit interaction. The spin current generated in nonmagnets is calculated in presence of surface acoustic waves. We solve the spin diusion equa- tion, extended to include spin-rotation coupling, and nd that larger spin currents can be obtained in materials with longer spin lifetimes. Spin accumulation induced on the surface is predicted to be detectable by time-resolved Kerr spectroscopy. PACS numbers: 72.25.-b, 85.75.-d, 71.70.Ej, 62.25.-g Introduction.\| Spin current, a ow of spins, is a key concept in the eld of spintronics[1, 2]. It can be gen- erated from non-equilibrium spin states, i.e., spin accu- mulation and spin dynamics. The former is routinely produced in nonlocal spin valves[3]. In ferromagnets, the latter is excited by ferromagnetic resonance[4], tem- perature gradient[5], and sound waves in the magnetic insulator[6]. Alternatively, spin currents in nonmagnets have been generated by the spin Hall eect[7], in which a strong spin-orbit interaction (SOI) is utilized. All these existing methods rely on exchange coupling of spins with local magnetization or on SOI. In this Letter, we pursue a new route for generating spin currents by considering spin-rotation coupling[8]: HS=h 2 ; (1) where h=2 is the electron spin angular momentum and is the mechanical rotation frequency. The method re- quires neither magnetic moments nor SOI. In this sense, the mechanism proposed here is particularly relevant in nonmagnets with longer spin lifetime. Nonuniform rotational motion\|. Here, we consider rotational motion of the lattice: =1 2r _u; (2) where uis the displacement vector of the lattice[9]. When the lattice vibration has transverse modes, Eq. (2) does not vanish. In such a case, the mechanical angular mo- mentum of the lattice can be converted into spin angular momentum via HS[10]. However, as shown later, that is nite is insucient to generate spin currents elas- tically. Both the time derivative and the gradient of ro- tational modes are necessary for the generation of spin current. For this purpose, we focus on surface acous- tic waves (SAWs), which induce rotational deformations that vary in space and time (Fig. 1). In presence of SAW, a gradient of mechanical rotation is induced in the attenuation direction. We extend the FIG. 1. Snapshot of mechanical generation of spin current induced by SAW. (a) In presence of a SAW propagating in thex-direction, a gradient of mechanical rotation around the z-axis is induced. The rotation couples to electron spins, and then thez-polarized spin current ows in the y-direction. (b) Spin accumulation induced on the surface. Because spins are polarized parallel to the rotation axis ( z), the striped pat- tern of spin accumulation arises at the surface. spin diusion equation to include the coupling between elastic rotation and spin. By solving the equation in pres- ence of SAW, we can evaluate the induced spin current for metals and semiconductors. Spin diusion equation with spin-rotation coupling.\| First of all, we examine eects of spin-rotation couplingarXiv:1301.3596v1 [cond-mat.mes-hall] 16 Jan 20132 on spin density. When the mechanical rotation, , whose axis is in the z-direction, is applied, the electron spins align parallel to the axis of rotation. This is known as the Barnett eect[11]. In this case, the bottom of the energy band of the electron is shifted by h =2. The number density of up(down) spin electrons is then given by n"(#)=Z"(#) h =2d"N 0("); (3) whereN0is the density of states for electrons, and "and #are chemical potentials for up and down spin electrons, respectively. The z-direction is selected as the quantiza- tion axis. Then, spin density can be estimated as n"n#N0(h ); (4) where="#is spin accumulation. Here, a con- stant density of state is assumed for simplicity. Spin re- laxation occurs in two processes: one is on-site spin ip with the spin lifetime, sf, and the other is spin diusion with the diusion constant, D. Equating these processes leads to@t(n"n#) =1 sfN0+Dr2(N0). Next, we obtain the extended spin diusion equation in presence of spin-rotation coupling: @tDr2+1 sf = h@t ; (5) The R.H.S. of Eq. (5) is a source term originating from spin-rotation coupling. Z-polarized spin current can be calculated from the solution of Eq. (5) as Jz s=0 er; (6) with conductivity 0. If the mechanical rotation is con- stant with time, the source term vanishes. Moreover, even if mechanical rotation depends on time, the uniform rotation in space cannot generate spin currents because spin accumulation is independent of space. Spin accumulation induced by SAW.\| Let us con- sider generation of spin current due to spin-rotation cou- pling of SAWs in nonmagnetic metals or semiconductors. Our setup is shown in Fig. 1 (a). SAWs are generated in thexz-plane and penetrates a nonmagnetic material along they-direction. They then induce mechanical ro- tation around the z-axis, whose frequency = (0;0; ) is given by[9] (x;y;t ) =!2u0 2ctexpfkty+i(kx!t)g; (7) where!andu0are the frequency and amplitude of the mechanical resonator, kis wave number, ctis the transverse sound velocity, and ktis the transverse wave number. The frequency !is related to the wave num- ber as!=ctkand the transverse wave number as kt=kp 12, wheresatises the equation 684+ 83(32c2 t=c2 l)16(1c2 t=c2 l) = 0 andclis the longitu- dinal sound velocity. The Poisson ratio, , is related tothe ratio of velocities as ( ct=cl)2= (12)=2(1), and andare related as (0:875 + 1:12)=(1 +). Spin accumulation generated by the SAW can be eval- uated by solving Eq. (5). By inserting Eq. (7) and =y(y;t)eikxinto Eq. (5), the spin diusion equa- tion can be rewritten as @tD@2 y+ ~1 sf y(y;t) =i!h 0ektyi!t;(8) where ~sf=sf(1+2 sk2)1with the spin diusion length s=pDsfand 0=!2u0=2ct. With the boundary condition@y= 0 on the surface y= 0, the solution is given by y(y;t) =i!h 0Z1 0dt0Z1 0dy0(tt0)e(tt0)=~sf p 4D(tt0) (e(yy0)2 4D(tt0)+e(y+y0)2 4D(tt0))ekty0i!t0:(9) Here, let us consider the time evolution of spin accu- mulation at the surface, y= 0. Because each spin aligns parallel to the rotation axis, i.e., the z-axis, a striped pattern of spin accumulation [shown in Fig. 1 (b)] arises at the surface. The period of spatial pattern is the same as the wavelength of SAW, 2 =k. Recently, spin precession controlled by SAW was ob- served by using the time-resolved polar megneto-optic Kerr eect (MOKE)[12, 13]. In our case, in-plane spin polarization is induced. Therefore, transversal or longi- tudinal MOKE can be used to observe patterns shown in Fig. 1 (b). Spin current from SAW.\| From Eqs. (6) and (9) we obtainz-polarized spin current in the y-direction. In Fig. 2, the SAW-induced spin current is shown. The spin current,Jz s, is plotted as a function of ktyand!tin Fig. 2 (a). The spin current oscillates with the same frequency as that of the mechanical resonator, !. The maximum amplitude, JMax s, is found near the surface, kty1. In Fig. 2 (b), maximum amplitude scaled by !3is plotted as a function of !sf. The amplitude increases linearly when!sf1, whereas it saturates when sf!1. In other words, JMax s/!4for!sf1 whereasJMax s/!3 for!sf1. To clarify material dependence of spin current, we use an asymptotic solution of Eq. (5) for kty1: i!~sf i!~sf+2sk2 t1h 0ekty+i(kx!t);(10) which leads to Jz si!~sf i!~sf+2sk2 t1h0 2e!3u0 c2 tp 12 ekty+i(kx!t) (11) where 0=!2u0=2ct. When spin relaxation is absent, !~sf1, one obtains h 0ekty+i(kx!t)(12)3 FIG. 2. (a) Spin current induced by SAW Jz splotted as a function of ktyand!tfor xedxandz. The spin current oscillates with time. Maximum amplitude is located near the surface,kty1. (b) Maximum amplitude of the spin current scaled by!3plotted as a function of !sf. When!sf1, the scaled amplitude, JMax s!3, increases linearly. On the other hand, it saturates when !sf1. Accordingly, the maximum amplitude, JMax s, is proportional to !4in the former case, whereasJMax s/!3in the latter case. and Jz s(0=e)kth 0ekty+i(kx!t): (13) When!~sf1 andskt1, the spin current be- comes Jz s!sfh0 2e!3u0 c2 tp 12 ekty+i(kx!t+=2):(14) As seen in Eq. (14), the larger spin current can be obtained from materials with the longer spin lifetime, namely, weaker SOI.Let us examine the SAW-induced spin current in typ- ical nonmagnetic materials. Using Eqs. (6) and (9), the maximum value of the spin current for Al, Cu, Ag, Au, and n-doped GaAs normalized by that of Pt, JMax;Pt s , is computed as listed in Table 1. The ratio of the maxi- mum amplitude of the spin current to that of Pt, Js, is dened as Js=JMax s=JMax;Pt s . The ratio depends mainly on the conductivity, 0, and spin lifetime, sf. The or- der, JCu s>JAl s>JAg s>JAu s>JPt s= 1, is unchanged, since these materials well satisfy !sf1. For GaAs, the ratio, JGaAs s, is greater than 1 for !<1GHz, whereas it becomes smaller than 1 for ! >1GHz. This happens because the spin lifetime of GaAs is much longer than that of Pt; i,e., the dimensionless parameter, !sf, be- comes much greater than 1 when ! > 1GHz. In such a case,JMax s=!3for GaAs saturates, whereas that for Pt linearly increases, as shown in Fig. 2 (b). It is worth noting that the spin current generated in a metal with weak spin-orbit interaction such as those of Al and Cu is much larger than that in Pt. In addition, the spin current in n-doped GaAs is comparable to that in Pt. Although conductivities of semiconductors are much smaller than those of metals, the spin lifetime is much longer. Hence, the amplitude of the induced spin current in GaAs is comparable to that in Pt for !GaAs sf<1. Recently, SAWs in the GHz frequency range have been used for spin manipulation[13, 14]. Here, we evaluate spin current at such high frequencies. In case of u0= 109m,!=2= 10GHz, Pt has the maximum amplitude, Jz;Pt s4106A/m2. Conventionally, generation of spin current in nonmag- netic materials has required strong SOI because the spin Hall eect has been utilized. In other words, nonmag- netic materials with short spin lifetimes have been used. On the contrary, the mechanism proposed here requires longer spin lifetimes to generate larger spin currents. Therefore, more options are available for spin-current generation in nonmagnets than ever before. Enhancement of the SAW-induced spin current.\| Very recently, it has been predicted that spin-rotation coupling can be enhanced by an interband mixing of solids[20]: H0 S=(1 +g)h 2 : (15) Here,gis given by g=gg0whereg0= 2 andg are electron gfactors in vacuum and solids, respectively. Considering enhancement, the mechanical rotation, , inserted into the extended spin diusion equation, Eq. (5), is replaced by (1 + g) . Consequently, the spin accumulation, , is modied as !(1 +g), and accordingly, the induced spin current as Jz s!(1+g)Jz s. For lightly doped n-InSb at low temperature, g49 has been employed in a recent experiment[21]. In this case, one obtains g51. Therefore, the amplitude of the spin current can be 50 times larger.4 TABLE I. SAW-induced spin current for Pt, Al, Cu, Ag, Au, and GaAs. The ratio is given by Js=JMax s=JMax;Pt s , where JMax;Pt s is the maximum amplitude of the spin current for Pt. The ratio Jsdepends on the Poisson's ratio, , the transverse velocity,ct, conductivity, 0, and spin lifetime, sf. ct[m/s]0[107( m)1]sf[ps] Js(0.1GHz) Js(1GHz) Js(2.5GHz) Js(10GHz) Ref. Pt 0.377 1730 0.96 0.3 1 1 1 1 [15] Al 0.345 3040 1.7 100 390 290 210 62 [16] Cu 0.343 2270 7.0 42 950 700 650 330 [3] Ag 0.367 1660 2.9 3.5 44 38 34 32 [17] Au 0.44 1220 2.5 2.8 42 35 33 30 [18] GaAs 0.31 2486 3.31041051.6 0.13 0.050 0.013 [19] Discussion and conclusion.\| The method of spin- current generation using spin-rotation coupling is purely of mechanical origin; i.e., it is independent of exchange coupling and SOI. Lattice dynamics directly excites the nonequilibrium state of electron spins, and consequently, spin current can be generated in nonmagnets. As an example, we have theoretically demonstrated that SAW, a situation in which rotational motion of lat- tice couples with electron spins, can be exploited for spin current generation. The spin diusion equation is ex- tended to include eects of spin-rotation coupling. The solution of the equation reveals that the spin current is generated parallel to the gradient of the rotation. More- over, it has been determined that larger spin current can be generated in nonmagnetic materials with longer spin lifetimes. This means that Al and Cu, which have been considered as good materials for a spin conducting chan- nel, are favorable for generating spin current. Spin ac- cumulation induced by the SAW on the surface will be observed by Kerr spectroscopy. These results can be generalized for other lattice dy- namics. SAW discussed above is the Rayleigh wave, which induces rotation with the axis parallel to the sur- face. For instance, the Love wave[22], horizontally polar- ized shear wave, can be utilized to generate spin currents whose spin polarization is perpendicular to the surface. The use of spin rotation coupling, argued here, opens up a new pathway for creating spin currents by elastic waves. The authors thank S. Takahashi for valuable discus- sions. This study was supported by a Grant-in-Aid for Scientic Research from MEXT. [1] S. Maekawa, ed., Concepts in Spin Electronics (Oxford University Press, Oxford, 2006). [2] S. Maekawa, S. Valenzuela, E. Saitoh, and T. Kimura, ed., Spin Current (Oxford University Press, Oxford, 2012).[3] F. J. Jedema, A. T. Filip, and B. J. van Wees, Nature (London), 410, 345 (2001). [4] E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara. Appl. Phys. Lett. 88, 182509 (2006). [5] K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh, Nature (London) 455, 778 (2008). [6] K. Uchida, H. Adachi, T. An, T. Ota, M. Toda, B. Hille- brands, S. Maekawa, and E. Saitoh, Nature Mater. 10, 737 (2011); K. Uchida, T. An, Y. Kajiwara , M. Toda , and E. Saitoh, Appl. Phys. Lett. 99, 212501 (2011); K. Uchida, H. Adachi, T. An, H. Nakayama, M. Toda, B. Hillebrands, S. Maekawa, and E. Saitoh, J. Appl. Phys. 111, 053903 (2012). [7] Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Science 306, 1910 (2004); J. Wunderlich, B. Kaestner, J. Sinova, and T. Jungwirth, Phys. Rev. Lett. 94, 047204 (2005); T. Kimura, Y. Otani, T. Sato, S. Takahashi, and S. Maekawa, Phys. Rev. Lett. 98, 156601 (2007). [8] C. G. de Oliveira and J. Tiomno, Nuovo Cimento 24, 672 (1962); B. Mashhoon, Phys. Rev. Lett. 61, 2639 (1988); J. Anandan, Phys. Rev. Lett. 68, 3809 (1992); B. Mash- hoon, Phys. Rev. Lett. 68, 3812 (1992); F. W. Hehl and W.-T. Ni, Phys. Rev. D42, 2045 (1990). [9] L. D. Landau and E. M. Lifshitz, Theory of Elasticity (Pergamon, New York, 1959). [10] E. M. Chudnovsky, D. A. Garanin, and R. Schilling, Phys. Rev. B 72 , 094426 (2005); C. Calero, E. M. Chud- novsky, and D. A. Garanin, Phys. Rev. Lett. 95, 166603 (2005); C. Calero and E. M. Chudnovsky, Phys. Rev. Lett. 99, 047201 (2007). [11] S. J. Barnett, Phys. Rev. 6, (1915) 239. [12] A. Hern andez-M nguez, K. Biermann, S. Lazi c, R. Hey, and P. V. Santos, Appl. Phys. Lett. 97, 242110 (2010). [13] H. Sanada, T. Sogawa, H. Gotoh, K. Onomitsu, M. Ko- hda, J. Nitta, and P. V. Santos, Phys. Rev. Lett. 106, 216602 (2011). [14] M. Weiler, L. Dreher, C. Heeg, H. Huebl, R. Gross, M. S. Brandt, and S. T. B. Goennenwein, Phys. Rev. Lett. 106, 117601 (2011). [15] L. Vila, T. Kimura, and Y. C. Otani, Phys. Rev. Lett. 99, 226604 (2007). [16] F. J. Jedema, H. B. Heersche, A. T. Filip, J. J. A. Basel- mans, and B. J. van Wees, Nature (London), 416, 7135 (2002). [17] R. Godfrey and M. Johnson, Phys. Rev. Lett. 96, 136601 (2006). [18] J.-H. Ku, J. Chang, H. Kim, and J. Eom, Appl. Phys. Lett. 88, 172510 (2006). [19] J. M. Kikkawa and D. D. Awschalom, Phys. Rev. Lett.80, 4313 (1998). [20] M. Matsuo, J. Ieda, and S. Maekawa, arXiv:1211.0127. [21] C. M. Jaworski, R. C. Myers, E. Johnston-Halperin, and J. P. Heremans, Nature (London) 484, 210 (2012). [22] A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity (Dover, New York, USA, 1967).
1911.12180v2.How_spin_orbital_entanglement_depends_on_the_spin_orbit_coupling_in_a_Mott_insulator.pdf	How spin-orbital entanglement depends on the spin-orbit coupling in a Mott insulator Dorota Gotfryd,1, 2Ekaterina M. P arschke,3, 4Ji r Chaloupka,5, 6Andrzej M. Ole s,2, 7and Krzysztof Wohlfeld1 1Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, ul. Pasteura 5, PL-02093 Warsaw, Poland 2Institute of Theoretical Physics, Jagiellonian University, Prof. S. Lojasiewicza 11, PL-30348 Krak ow, Poland 3Department of Physics, University of Alabama at Birmingham, Birmingham, Alabama 35294, USA 4Institute of Science and Technology Austria, Am Campus 1, A-3400 Klosterneuburg, Austria 5Department of Condensed Matter Physics, Faculty of Science, Masaryk University, Kotl a rsk a 2, CZ-61137 Brno, Czech Republic 6Central European Institute of Technology, Masaryk University, Kamenice 753/5, CZ-62500 Brno, Czech Republic 7Max Planck Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany (Dated: March 3, 2020) The concept of the entanglement between spin and orbital degrees of freedom plays a crucial role in understanding of various phases and exotic ground states in a broad class of materials, including orbitally ordered materials and spin liquids. We investigate how the spin-orbital entanglement in a Mott insulator depends on the value of the spin-orbit coupling of the relativistic origin. To this end, we numerically diagonalize a one-dimensional spin-orbital model with the `Kugel-Khomskii' exchange interactions between spins and orbitals on dierent sites supplemented by the on-site spin-orbit coupling. In the regime of small spin-orbit coupling w.r.t. the spin-orbital exchange, the ground state to a large extent resembles the one obtained in the limit of vanishing spin-orbit coupling. On the other hand, for large spin-orbit coupling the ground state can, depending on the model parameters, either still show negligible spin-orbital entanglement, or can evolve to a highly spin-orbitally entangled phase with completely distinct properties that are described by an eective XXZ model. The presented results suggest that: (i) the spin-orbital entanglement may be induced by large on-site spin-orbit coupling, as found in the 5 dtransition metal oxides, such as the iridates; (ii) for Mott insulators with weak spin-orbit coupling of Ising-type, such as e.g. the alkali hyperoxides, the eects of the spin-orbit coupling on the ground state can, in the rst order of perturbation theory, be neglected. I. INTRODUCTION A. Interacting Quantum Many-body Systems: Crucial Role of Entanglement One of the main questions for a quantum interact- ing many-body system concerns the nature of its ground state. Perhaps the most fundamental question that can be formulated here is as follows: Can the eigenstates of such a system be written in terms of a product of the `lo- cal' (e.g. single-site) basis states? If this is the case, then the ground state can be understood using the classical physics intuition. Moreover, the low lying excited states can then be described as weakly interacting quasiparti- cles, carrying the quantum numbers of the constituents forming the system. Such physics is realized, for instance, by the ions in conventional crystals, spins in ordered mag- nets, or electrons in Fermi liquids [1]. An interesting situation occurs, however, when the an- swer to the above question is negative and we are left with a `fully quantum' interacting many-body problem [2, 3]. In this case, the classical intuition fails, numerical de- scription of the ground state may become exponentially dicult, and the low lying eigenstates cannot be de- scribed as weakly interacting quasiparticles. This can, for example, be found in the spin liquids stabilized in the one-dimensional (1D) or highly frustrated magnets [4] or incommensurate electronic systems with strong electron interactions (the so-called non-Fermi liquids) [5]. In fact,a large number of condensed matter studies are nowadays devoted to the understanding of such `exponentially dif- cult' problems. A way to characterize the fully quantum interacting many-body problem is by introducing the concept of en- tanglement [6{8] and then by dening interesting aspects of its quantum structure through the entanglement en- tropy [9]. It is evident that studying entanglement always requires rst a denition of what is entangled with what, i.e., specifying the division of the system into the sub- systems that may become entangled. Perhaps the most widely performed division has so far concerned splitting a lattice spin system into two subsystems in real space [10]. Such studies enabled to identify the relation between the entanglement of the spin system, its scaling with the sys- tem size, and the product nature of the ground state, cf. Refs. [10{16]. B. Spin-Orbital Entanglement in Transition Metal Compounds Transition metal oxides involve often numerous com- peting degrees of freedom. Examples are the three- dimensional (3D) ground states of LaTiO 3, LaVO 3, YVO 3, Ba 3CuSb 2O9[17{22], where spin and orbital de- grees of freedom are intertwined and entangled. A similar situation occurs in MnP [23], the rst Mn-based uncon- ventional superconductor under pressure, and in some two-dimensional (2D) model systems [24{26]. In all thesearXiv:1911.12180v2 [cond-mat.str-el] 28 Feb 20202 cases the ground state can only be explained by invok- ing the joint spin-orbital uctuations. Consequently, the mean eld decoupling separating interactions into spin and orbital degrees of freedom fails and cannot be used. In this paper we study the spin-orbital entanglement which manifests itself when a quantum many-body sys- tem with interacting spin and orbital degrees of freedom is split into the subsystems with separated degrees of freedom, i.e., one attempts to write interacting spin and orbital wave functions separately [27{29]. The concept of entanglement has been rst introduced in these systems to understand the violation of the so-called Goodenough- Kanamori rules [30, 31] in the ground states of several transition metal oxides with partially lled 3 dorbitals, strong intersite spin-orbital (super)exchange interactions but typically negligible value of the on-site spin-orbit cou- pling. It was also realized that entanglement is impor- tant to understand the excited states where spin and or- bital variables are intertwined. Good examples are the temperature evolution of the low energy excitations in LaVO 3[32, 33] and the renormalization of spin waves by orbital tuning in spin-orbital systems due to the weak interactions with the lattice [34]. Furthermore, the spin- orbital entanglement is also crucial to understand the rst unambiguous observations of the collective orbital excitation (orbiton) in Sr 2CuO 3and CaCu 2O3[35, 36] and their interpretation in terms of the spin and orbital separation in a 1D chain [37{39]. Crucially, the spin-orbital entanglement is expected as well in the oxides with strong on-site spin-orbit coupling\|probably best exemplied by the partially lled 4dand 5dorbitals as found in the ruthenium and iridium oxides [40, 41], or in the recently discovered 5 dTa chlorides [42]. For instance, the concept of spin-orbital entanglement was recently invoked to understand the inelastic x-ray spectrum of Sr 3NiIrO 6[43], the ground state of H 3LiIr2O6[44] or, in a dierent physical set- ting, the photoemission spectra of Sr 2RuO 4[45, 46]. Interestingly, the peculiarities of the interplay between the strong on-site spin-orbit coupling and the spin- orbital (super)exchange interactions allowed for the on- set of several relatively exotic phenomena in this class of compounds\|such as a condensed matter analogue of a Higgs boson in Ca 2RuO 4[47, 48] or the strongly direc- tional, Kitaev-like, interactions between the low energy degrees of freedom (pseudospins) in some of the iridates or ruthenates on a quasi-2D honeycomb lattice (Na 2IrO3, Li2IrO3,-RuCl 3, H3LiIr2O6) [44, 49, 50]. The latter might be described to some extent by the exactly solv- able Kitaev model on the honeycomb lattice which, in- ter alia , supports the onset of a novel spin-liquid ground state with fractionalized `Majorana' excitations [51]. C. Main question(s) and organization of the paper It may come as a surprise that the concept of the spin- orbital entanglement has so far been rigorously investi-gated only for the systems where the spins and orbitals at neighboring sites interact, as a result of the spin, orbital and spin-orbital (super)exchange processes in Mott insu- lators [27{29, 52{57]. This case is physically relevant to all Mott insulators with negligible spin-orbit coupling of relativistic origin and with active orbital degrees of free- dom [58]\|e.g. to the above mentioned case of transition metal oxides with partially lled 3 dorbitals. On the other hand, to the best of our knowledge, such analysis has not been done for the systems with strong on- site coupling between spins and orbitals [59]\|as in the above-discussed case of the transition metal oxides with partially lled 4 dand 5dorbitals and strong spin-orbit coupling. We stress that in this case the spin and orbital degrees of freedom can get entangled as a result of both the nearest neighbor exchange interactions as well as on- site spin-orbit coupling. In fact, one typically implic- itlyassumes that the spin-orbital entanglement should be nonzero, since the spin Sand orbital Loperators cou- ple at each site into a total angular momentum J=S+L [60]. The latter, `spin-orbital entangled' operators (also called pseudospins), then interact as a result of the ex- change processes in the `relativistic' Mott insulators and are best described in terms of various eective pseudospin models, such as for example the Kitaev-like model dis- cussed above [61]. Finally, very few studies discuss the problem of the evolution of a spin-orbital system between the limit of weak and strong spin-orbit coupling [62{65]. Here we intend to bridge the gap between the under- standing of the spin-orbital physics in the above two lim- its. We ask the following questions: (i) what kind of evo- lution does the spin-orbital entanglement develop with increasing spin-orbit coupling? (ii) can one always as- sume that in the limit of strong spin-orbit coupling the spin-orbital entanglement is indeed nonzero? (iii) how does the spin-orbital entanglement arise in the limit of the strong spin-orbit coupling? To answer the above questions we formulate a mini- mal 1D model with S=1=2spin andT=1=2orbital (pseudospin) degrees of freedom. The model has the SU(2) SU(2) intersite interactions between spins and or- bitals which are supplemented by the on-site spin-orbit coupling of the Ising character\|its detailed formulation as well as its relevance is discussed in Sec. II. Using exact diagonalization (ED), the method of choice described in Sec. III, we solve the 1D model and evaluate the vari- ous correlation functions used to study the entanglement. Next, we present the evolution of the ground state prop- erties as a function of the model parameters: in Sec. IV A for dierent values of the three model parameters, and in Sec. IV B for a specic choice of the relation between the two out of the three model parameters. We then show two distinct paths of ground state evolution in Secs. IV B 1 and IV B 2. The evolution of the exact spectra of the periodic L= 4 chain is analyzed in Sec. IV B 3 for increasing. We discuss obtained numerical results uti- lizing mapping of the model onto an eective XXZ model in Sec. V, which is valid in the limit of the strong on-site3 spin-orbit coupling. We use the eective model to un- derstand: (i) how the spin-orbital entanglement sets in the model system and (ii) how it depends on the value of the on-site spin-orbit coupling constant . The pa- per ends with the conclusions presented in Sec. VI and is supplemented by an Appendix which discusses in detail the mapping onto the eective XXZ model in the limit of large spin-orbit coupling !1 . D. Practical note on the organization of the paper We note that, whereas the main results of the paper are given in the extensive Sec. IV, some of the main results can be understood by using a mapping onto an eec- tive XXZ model in Sec. V. Thus, we refer the interested audience looking for the more physical and intuitive un- derstanding of (some of) the obtained numerical results to the latter section. Finally, we stress that allthe im- portant results of the paper are not only listed but also discussed in detail in Sec. VI A. II. MODEL In this paper we study a spin-orbital model Hdened in the Hilbert space spanned by the eigenstates of the spinS=1=2and orbital (pseudospin) T=1=2operators at each lattice site of a 1D chain with periodic bound- ary conditions. The model Hamiltonian consists of two qualitatively distinct terms, H=HSE+HSOC: (1) The rst term HSEdescribes the intersite (su- per)exchange interactions between spins and orbitals. The spin-orbital (`Kugel-Khomskii') exchange reads, HSE=JX i[(SiSi+1+) (TiTi+1+)];(2) whereJ > 0 is the exchange parameter and the con- stantsandare responsible for the relative strengths of the individual spin and orbital exchange interactions. This 1D SU(2) SU(2) symmetric spin-orbital Hamilto- nian has been heavily studied in the literature\|it is ex- actly solvable by Bethe Ansatz at the SU(4) point, i.e., when==1=4[66{68], has a doubly degenerate ground state at the so-called Kolezhuk-Mikeska point ==3=4[69, 70], and was studied using various analytical and numerical methods for several other rele- vant values off;gparameters [39, 71{75]. In partic- ular, the entanglement between spin and orbital degrees of freedom in such a class of Hamiltonians is extremely well-understood [28, 29, 52]. Last but not least, it was suggested that this model may describe the low-energy physics found in NaV 2O5and Na 2Ti2Sb2O [72], CsCuCl 3 and BaCoO 3[76], as well as in the articial Mott insula- tors created in optical lattices [77, 78] and the so-called Coulomb impurity lattices [79].Altogether, this means that the spin-orbital exchange interaction has the simplest possible form [58] that can, nevertheless, describe a realistic situation found in the transition metal oxides. This, as already mentioned in In- troduction, constitutes the main reason behind the choice of this form of spin-orbital intersite interaction. We note that the spin-orbital exchange would often has a more complex form. For instance, this would be the case, if e.g. three instead of two active orbitals were taken into account and the corrections from nite Hund's exchange were included (as relevant for the 5 diridates, whose spin- orbital exchange interactions are given by e.g. Eq. (3.11) of Ref. [19]). The second term in the studied Hamiltonian (1) de- scribes the on-site interaction between the spin and or- bital degrees of freedom and reads HSOC= 2X iSz iTz i: (3) Here the parameter measures the strength of the on- site spin-orbit coupling term (of relativistic origin). The above Ising form of the spin-orbital coupling was chosen as the simplest possible and yet nontrivial one. Moreover, exactly such a form of the spin-orbit coupling is typically realized in systems with twoactive orbitals. This is the case of e.g. the active t2gdoublets in YVO 3[80] and Sr2VO4[81], the molecular orbitals of KO 2[82], or on optical lattices. In fact, such a highly anisotropic form of spin-orbit coupling is valid for any system with an active orbital doublet, either two p(pxandpy) or twot2g(xz andyz) orbitals. III. METHODS AND CORRELATION FUNCTIONS As we are interested in quantum entanglement, and moreover, the exchange Hamiltonian (2) itself bears a rather complex quantum many-body term /(SiSj)(TiTj), we opt for the exact diagonalization (ED) method which preserves the quantum uctuations in the numerically found ground state. More specically, we choose the ED calculations, since: (i) It allows us to investigate the system ground state in a numerically ex- act manner and in a completely unbiased way which for the rst study of its kind is usually selected as the method of choice; (ii) The analytically exact Bethe Ansatz ap- proach can only be applied to a few selected values of the model parameters; (iii) The ED calculations can be relatively easily repeated for a number of model parame- ters and can typically address the qualitative properties of the ground state rather well; (iv) We are interested here in rather local correlations which follow from the local spin-orbit and nearest neighbor exchange interac- tions. We calculate the properties of the ground state of model (1) on nite chains with periodic boundary condi- tions. We utilize chains of length L= 4nsites, where nis4 an integer number, in order to avoid a degenerate ground state appearing in the case of a (4 n+ 2){site chain (see Table 1 of Ref. [67]). For chains L= 4 a standard full ED procedure is performed, while for L= 8, 12, 16, and 20 sites we restrict the ED calculations to the Lanczos method [83]. To capture the changes in the ground state of the spin- orbital model at increasing spin-orbit coupling , we de- ne and investigate the following correlation functions which will be used, besides the von Neumann spin-orbital entanglement entropy [84], to monitor the evolution of the ground state with changing model parameters: (i) The intersite spin-orbital correlation function CSO: CSO=1 LLX i=1h(SiSi+1)(TiTi+1)i 1 LLX i=1hSiSi+1ihTiTi+1i; (4) which measures the intersite (nearest neighbor) correla- tion between the spin and orbital degrees of freedom and has already been used in the literature as a good qualita- tiveestimate for the von Neumann spin-orbital entangle- ment entropy [55, 56]. This correlator can also be used to monitor the failure of the mean eld decoupling between the spins and orbital pseudospins once CSO6= 0 [27]. (ii) The intersite spin Sor orbitalTcorrelation func- tion, S=1 LLX i=1hSiSi+1i; (5) T=1 LLX i=1hTiTi+1i; (6) which measures the intersite (nearest neighbor) correla- tion between the spin (orbital) degrees of freedom and is therefore sensitive to the changes in the ground state properties taking place solely in the spin (orbital) sub- space. We emphasize that these two functions are dened on equal footing in the model with SU(2) SU(2) spin- orbital superexchange. (iii) The {component S of spin scalar product: S =1 LLX i=1 S iS i+1 ; (7) where =x;y;z . This function measures the component of the scalar product and thus allows one to investigate possible anisotropy of the intersite (nearest neighbor) correlations between the spin degrees of freedom. The orbital scalar product component T is dened analo- gously to Eq. (7). (iv) Crucial for the systems with nite spin-orbit cou- pling is the on-site spin-orbit correlation function OSO: OSO=1 LLX i=1hSz iTz ii; (8)which measures the correlations between the zcompo- nents of the spin and orbital operators on the same site. The precise form of this correlator is dictated by the Ising form (3) of the spin-orbit coupling present in Hamiltonian (1). Conveniently, the function (8) is one of the gener- ators of the SU(4) group [68], which proved to be quite useful for examining the range of the SU(4){symmetric ground state. IV. NUMERICAL RESULTS A. von Neumann entropy in a general case The main goal of this paper is to determine how the spin-orbital entanglement changes in the spin-orbital model (1) upon increasing the value of the spin-orbit cou- pling. To this end, we rst dene the entanglement entropy calculated for a system that is bipartitioned into two subsystems: AandB. Typically such a subdivision refers to two distinct parts of the real [10{13] or momen- tum [16] space. Here, however, it concerns spin ( A) and orbital (B) degrees of freedom [27{29, 53]. A standard measure of the entanglement entropy be- tween subsystems AandBin the ground state jGSiof a system of size Lis due to von Neumann [84]. It is dened asSvN=TrAfAlnAg=L, and is obtained by integrating the density matrix, A= TrBjGSihGSjover subsystem B. Consequently, in this paper we use the following denition of the von Neumann spin-orbital en- tanglement entropy: SvN=1 LTrSfSlnSg; (9) where S= TrTjGSihGSj (10) is the reduced spin-only ( S) density matrix with the or- bital (T) degrees of freedom integrated out. The spin-orbital von Neumann entropy is calculated using ED on L{site chain for model (1) and is shown as function of the parameters f;gfor three representative values of the spin-orbit coupling in Fig. 1. In perfect agreement with Refs. [29, 52], the von Neumann entropy SvNis nite in a rather limited region of the f;gpa- rameters for = 0, i.e., in the entangled spin-orbital phase near the origin == 0. The nonzero entan- glement in that case is well-understood and attributed to the onset of the dominant antiferromagnetic (AF) and alternating orbital (AO) uctuations in the ground state without broken symmetry [29, 52], see discussion below in Sec. IV B 4. Interestingly, a nite but `small' spin-orbit coupling < CRIT (CRIT is discussed in more detail in Sec. IV B) does not substantially increase the region in thef;g{parameter space for which the spin-orbital entropy is nonzero cf. Figs. 1(a) and 1(b). A drastic5 -4.0 -2.0 0.0 2.0 4.0 α-4.0-2.00.02.04.0β(a) λ/J=0 -4.0 -2.0 0.0 2.0 4.0 α(b) λ/J=0.1 -4.0 -2.0 0.0 2.0 4.0 α(c) λ/J→∞SvN 0.00.10.20.30.40.5 FIG. 1. The von Neumann spin-orbital entanglement entropy, SvN(9), calculated using ED on L= 12{site periodic chain for the spin-orbital model Eq. (1) and for the increasing value of the spin-orbit coupling : (a)=J= 0, (b)=J= 0:1, and (c)=J!1 . change in the behavior of the spin-orbital von Neumann entropy only happens for the dominant spin-orbit cou- pling> CRIT. In this case the region of nonzero spin- orbital entanglement is not only much larger but also takes place for dierent values of the f;g{parameter space. For instance, it is remarkable that the von Neu- mann entropy in the case of > CRIT almost does not depend onalong the lines of constant +. Moreover, nite entanglement is activated when + >1=2\| however, the value of the von Neumann entropy strongly decreases for andlocated `above' the stripe given by the inequalities 1=2+2 and showing the highest value of entropy. In fact, it will be shown later (see Sec. V) that the von Neumann entropy is expected to vanish in the limit of +!1 . Altogether, we observe that: (i) in the limit of small < CRIT the spin-orbital entanglement entropy does not change substantially w.r.t. the case with vanishing spin-orbit coupling; (ii) in the limit of large > CRIT the spin-orbital entanglement can become nite even if it vanishes for = 0; though it can also happen that (iii) in the limit of large > CRIT the spin-orbital en- tanglement vanishes when + <1=2. B. von Neumann entropy for =: Two distinct evolutions for increasing In order to better understand the physics behind the observations (i) and (ii) discussed in the end of the pre- vious subsection, here we study in great detail the onset of the spin-orbital entanglement once =. As shown in Fig. 1, for these values of the model parameters the region of the nonzero spin-orbital entanglement increases dramatically with the increasing value of the spin-orbit coupling. We present in Fig. 2 the von Neumann spin-orbital en-tanglement entropy SvN(9) and the three spin-orbital correlation functions in the ground state of Hamilto- nian (1) with =, calculated using ED on an L= 12{ site chain. We begin the analysis by comparing the val- ues of the three spin-orbital correlation functions (4), (8), and (5) against the von Neumann spin-orbital entangle- ment entropy, see Figs. 2(b), 2(c), and 2(f). We observe that only the intersite spin-orbital correlation function CSOcan be used as a qualitative measure for the von Neu- mann entropy, consistent with previous studies [55, 56]. In particular, the on-site spin-orbit correlation function OSOcannot be used to `monitor' the entanglement en- tropy, for it measures the correlations between spins and orbitals locally and on the Ising level only. Nevertheless, bothOSOas well asSvNcan be used to identify various quantum phases obtained in the f;gparameter space of the Hamiltonian, as suggested before for system with negligible spin-orbit coupling [56]. Next, we study the evolution of the von Neumann spin-orbital entanglement entropy with increasing spin- orbit coupling for various values of the parameter , see Fig. 2(a). We observe that in the representative 2[1;1] interval there exist three distinct regimes of the value of the von Neumann entropy: (i) two com- pact areas in the f;gparameter space for which the von Neumann entropy is vanishingly small, which exist in the large parameter range of jj&0:1 and=J. 101100[the bottom left and bottom right parts of panel Fig. 2(a)]; (ii) one compact area in the f;gpa- rameter space for which the von Neumann entropy takes maximal possible values, which exists in the large pa- rameter range =J&101101for all values of [the top part of panel Fig. 2(a)]; (iii) the compact area in thef;gparameter space for which the von Neumann entropy is neither negligible nor takes maximal value, which exists in the relatively small parameter range be- tween cases (i) and (ii). In order to understand the on-6 1.0 0.5 0.0 0.5 1.0 α10-310-210-1100101λ/JABA(a)SvN 1.0 0.5 0.0 0.5 1.0 α(b)CSO 1.0 0.5 0.0 0.5 1.0 α(c)OSO 10-310-210-1100101102103 λ/J0.00.10.20.30.40.5SvNA(d) 10-310-210-1100101 λ/JB(e) 1.0 0.5 0.0 0.5 1.0 α10-310-210-1100101λ/J(f)S 0.00.10.20.30.40.5 0.25 0.20 0.15 0.10 0.05 0.00 0.25 0.20 0.15 0.10 0.05 0.00 0.50 0.25 0.000.250.50 FIG. 2. Evolution of the von Neumann spin-orbital entanglement entropy and the three spin-orbital correlation functions in the ground state of Hamiltonian (1) with =, calculated using ED with the periodic L= 12{site chain for logarithmically increasing spin-orbit coupling : (a) the von Neumann spin-orbital entanglement entropy SvN(9) for2[1:0;1:0]; (b) the intersite spin-orbital correlation function CSO(4) for2[1:0;1:0]; (c) the on-site spin-orbit correlation function OSO(8) for 2[1:0;1:0]; (d) the von Neumann spin-orbital entanglement entropy SvN(9) obtained with jj= 0:5 [cut A in panel (a)] and tted with a logistic function (black thin line); (e) the von Neumann spin-orbital entanglement entropy SvN(9) obtained with= 0 [cut B in panel (a)]; (f) the spin correlation function S(5) for2[1:0;1:0]. set of these three distinct regimes, we study below two qualitatively dierent cases of the von Neumann entropy evolution with the increasing spin-orbit coupling: case `A' withjj= 0:5 and case `B' with = 0 [shown with dashed lines in Fig. 2(a)]. 1. From a product state to highly entangled state Whenjj= 0:5 (case A) the evolution of the von Neu- mann spin-orbital entanglement entropy SvN(9) with in- creasing spin-orbit coupling can be well-approximated by a logistic function, see Fig. 2(d). The von Neumann en- tropy has innitesimally small values for =J.101, experiences a rapid growth for =J2(101;101), and saturates at ca. 0 :5 for=J&101. Comparable behav- ior is observed for the intersite spin-orbital correlation (CSO), which, as already discussed, is a good and compu- tationally not expensive qualitative measure for the von Neumann entropy, see Figs. 3(a), 3(c), and 3(e). Cru-cially, the latter calculations are obtained for the spin- orbital chains of dierent length and [as well-visible in Figs. 3(a), 3(c), 3(e)] show relatively small nite-size ef- fects. This means that indeed the von Neumann entropy SvNdepends here mainly on short-range processes and can remain negligibly small for a nite value of the spin- orbit coupling even in the thermodynamic limit. Finally, Figs. 2 and 3 allow us to dene the critical value CRIT for case A as being located in an interval of rapid growth of the spin-orbital entanglement: CRIT=J2(101;101). While the nature of the quantum phase for large spin-orbit coupling > CRIT is discussed in detail in Sec. V B, here we merely mention that in this case the value of the spin-orbital entanglement entropy saturates at about 0:5 (0:504 forL= 12 site chain) per site. Hence, we call this quantum phase a highly entangled state . Be- sides, in this case also the absolute value of the on-site spin-orbit correlation function OSOtakes its maximal value, while the spin (and orbital) correlation function S(T) is weakly AF (AO).7 10-310-210-11001010.25 0.20 0.15 0.10 0.05 0.00A L=8(a) 10-210-1100B(b) 10-310-210-11001010.25 0.20 0.15 0.10 0.05 0.00 L=12(c) 10-210-1100(d) CSO OSO CSU(4) SO OSU(4) SO˜CSO˜OSO10-310-210-1100101 λ/J0.25 0.20 0.15 0.10 0.05 0.00 L=16(e) 10-210-1100 λ/J(f) FIG. 3. The intersite spin-orbital correlation function CSO (green lines) and the on-site spin-orbit correlation function OSO(red lines) as functions of calculated for: case A (left), i.e.,= 0:5 [(a), (c), (e)] and case B (right), i.e., = 0 [(b), (d), (f)]. The ED results are shown for periodic chains of lengthL= 8 (top),L= 12 (middle), and L= 16 (bottom). The dashed lines represent the asymptotic values of the above correlation functions: (i) the exact SU(4){point limit = 0, == 0:25 is denoted byCSU(4) SO andOSU(4) SO (blue and light{ red dashed lines); (ii) the =1,=XY limit\|by ~CSO and ~OSO(dark{green and dark{red dashed lines). For further details see discussion in Secs. IV B 1-IV B 2 and Sec. V B). Next, we focus on the properties of the ground state obtained for small < CRIT. To this end we investi- gate the evolution of the two other correlation functions, the on-site spin-orbit correlation function OSOand the spin correlation function S, for< CRIT andjj= 0:5, see Figs. 2(c), 2(f). We observe that whereas the on-site spin-orbit correlation function shows vanishingly small values in this limit, the spin correlation function S'0:25 (S'0:45) for= 0:5 (=0:5), thus behaving sim- ilarly to the 1D FM (AF) chain, respectively. We note that the (unshown) analogous nearest neighbor orbital 10-210-11001011020.2 0.1 0.00.10.2A (a) Sδδ TδδSzz Tzz/arrownorthwest /arrownorthwest /arrownortheast /arrownorthwest 10-210-1100B (b) L=8 10-210-11001011020.2 0.1 0.00.10.2(c) Sδδ, Tδδ Szz, Tzz ˜Sδδ, ˜Tδδ ˜Szz, ˜Tzz10-210-1100(d) L=12 10-210-1100101102 λ/J0.2 0.1 0.00.10.2(e) 10-210-1100 λ/J(f) L=16FIG. 4. The anisotropic spin correlation function S (7) and orbital correlation function T for increasing . TheSand T(=x;y) components are marked by green color while theSzzandTzzcomponents are marked by red color. The correlation functions are calculated for: case A (left), i.e., = 0:5 [(a), (c), (e)] and case B (right), i.e., = 0 [(b), (d), (f)]. The ED results are shown for periodic chains of lengthL= 8 (top),L= 12 (middle), and L= 16 (bottom). The asymptotic values of the correlation functions in the limit =1are shown for both = 0 and= 0:5 case and denoted as~S,~Szzand~T,~Tzz, see discussion in Sec. V B for further details. correlation function Tcalculated for =0:5 takes complementary values to the spin correlation function for =0:5, i.e.,T=S. Such behavior is again observed for chains of various lengths, with S(T) better approx- imating the expected AF Bethe Ansatz value for larger chains, see Fig. 4. Altogether, this shows that the quan- tum phase that is observed for < CRIT qualitatively resembles the phases obtained in the limit of = 0: the FM AO (AF FO) for= 0:5 (=0:5), respectively. The above discussion contains just one caveat. Let us8 look at the evolution of the anisotropic spin (and orbital) correlation function S (andT ) with the increasing spin-orbit coupling , see Figs. 4(a, c, e). We notice that whenever=J > 0 for= 0:5 there exist an anisotropy between the zz(solid red lines) and the planar ( xx,yy, solid green lines) correlation functions\|which is absent for= 0. However, for =J.3101the anisotropy is only partial, being absent in the strongly AF T cor- relations, in contrast to the S correlations. In fact, S(where=x;y), stay positive as in = 0 case whileSzzbecomes negative. In this way the energy com- ing from the nite spin-orbit coupling is `minimized' in the ground state without qualitatively changing the na- ture of the FM AO and AF FO ground states, allowing however for a very small value of the spin{orbital entan- glement. This is the reason why, in what follows, this quantum phase is called a perturbed FM AO product state. 2. From SU(4) singlet to a highly entangled state We now investigate how the von Neumann spin-orbital entanglement entropy SvNevolves with the spin-orbit coupling once = 0 (case B): i.e., from its nite value for the SU(4){singlet ground state at = 0 [29, 52] to an even higher value obtained in the limit of large =Jin the highly entangled state (i.e., the state already encoun- tered in case A). To this end, we rst note that the von Neumann entropy SvNat= 0 changes with the spin- orbit coupling in a qualitatively dierent manner than in the case ofjj= 0:5, see Fig. 2(e). While we again encounter a monotonically growing function in , which saturates at about 0 :5 for=J&0:2, this function seems to be discontinuous at three particular values of and three `kinks' (for L= 12 sites) that can be easily identi- ed in Fig. 2(e). A similar behavior is encountered in the qualitative measure for the von Neumann entropy\|the spin-orbital correlation function CSO, see Fig. 2(b) and Fig. 3(b, d, f). As a side note let us mention that once = 0 and == 0 the model (1) has an SU(4)-symmetric ground state, as conrmed by the remarkable convergence of the functionsCSOandOSOto their asymptotic values CSU(4) SO andOSU(4) SO calculated at the exact SU(4) point == 1=4, cf. Fig. 3(b, d, f). As the operator in the OSOfunction is one of the generators of the SU(4) group, its zero expectation value in the ground state is not only related to the absence of the spin-orbit coupling but also is a signature of the SU(4)-symmetric singlet [85]. It is clearly visible in Fig. 3(b, d, f) that the CSOand OSOcorrelations split from their SU(4)-singlet asymp- totes in the subsequent kinks, which occur with the in- creasing value of the spin-orbit coupling. Interestingly, the number of kinks grows and their position changes with the system size, see Fig. 3(b, d, f). In fact, L=4 kinks are observed for a chain of length L= 4;8;12;16;20, see panel (a) of Fig. 5. This naturally suggests that in the 0.00 0.05 0.10 0.15 0.20 0.25 λ/J0.25 0.20 0.15 0.10 0.05 0.00OSO(a) 4 8 1216 20 0.0 0.1 0.2 0.3 1/L0.000.020.040.060.080.100.120.14λ/J(b) 4 8 12 1620 0.00 0.01 0.02 1/L20.160.170.180.190.200.21(c) 812 1620FIG. 5. Finite-size scaling of the boundaries of the interme- diate entangled state for the case B in Fig. 3: (a) L=4 kinks forL= 4;8;12;16;20 shown on an example of OSO, (b) the decreasing position of the `rst' kink as a function of 1 =L, (c) the increasing position of the `nal' kink as a function of 1=L2. These ts use the ED numerical results obtained for L= 4;8;12;16;20 periodic chains presented as colorful dots in (b)&(c); the lines are the ts (11) to the numerical data. innite system the number of kinks will be innite. But what about the position of the `rst' and the `last' kink in the thermodynamic limit? To answer this in- triguing question with the available ED data we deduced qualitative values of =J which dene the regime where correlations take intermediate values and the entangled state is not yet dominated by the large spin-orbit cou- pling> CRIT. The nite size scaling performed here, shown in panels (b) and (c) of Fig. 5, uses a polynomial t similar as for instance for the gap in the 1D half-lled Hubbard model [86]. Here the positions of the `rst' and the `last' kink scale dierently with the increasing length of L= 4n chain. Namely, the position of the `rst' kink k1is almost linear in 1=Lwhile the `nal' kink's position kfscales almost linearly with 1 =L2. By performing the ts, we have found that k1= 0:00004 + 0:69712x0:50147x2; kf= 0:201321:13007x2+ 3:14415x4; (11) wherex1=L. As a result, the position of the `rst' kinkk1converges to =J = 0 whenL!1 , and it is indeed reasonable to expect that innitesimal modies weakly spin-orbital correlations in the thermodynamic limit. In contrast, the `nal' kink kfwould then shift to =J'0:201. Therefore, for the case B we dene CRIT as a single number: CRIT=J'0:2. Altogether, this9 10-1100 λ/J-3.5-3.0-2.5-2.0E(a) A 10-310-210-1100101 λ/J-1.2-1.0-0.8-0.6(b) B -0.2 -0.1 0.0 0.1 0.2 α-1.0-0.50.00.51.01.5˜E=E+2λ(c) λ/J=0.1 -1.0 -0.5 0.0 0.5 1.0 α0.010.020.030.040.0(d) λ/J=10 FIG. 6. Top panels\|the energy Eof the ground (blue) and low lying excited states (gray) obtained for model (1) using ED for periodic L= 4-site chain as a function of increasing =J[(a) case A for = 0:5, (b) case B for = 0]. Bottom panels\|complete energy spectra for small and large [(c)=J = 0:1 and (d)=J = 10]. Note that we display here ~E(12) to compare the spectra in similar energy range, independently of the actual value of . means that the quantum phase encountered for 0 << CRIT does not disappear in the thermodynamic limit and that its spin-orbital entanglement grows with the increasing spin-orbit coupling in a continuous way. To contrast this intermediate phase with the one showing the maximal value of entanglement at > CRIT, we call it an intermediate entangled state . To better understand the properties of this phase, we also consider the spin correlation function S, the anisotropic spin S , and the orbital T correlation functions, see Figs. 2(c, f), 3(b, d, f), and 4(b, d, f). Simi- larly to the von Neumann entropy, also OSOorCSOcorre- lation functions show kinks due to nite-size eects which are expected to disappear in the thermodynamic limit. Noticeably, the behavior of S,S , andT is quite dis- tinct w.r.t. the one observed both for the highly entan-gled phase and seemingly for the perturbed FM AO or AF FO phases. This shows that the intermediate entan- gled phase observed at = 0 and for 0 < < CRIT is indeed qualitatively dierent and constitutes a `genuine' quantum phase. 3. Exact spectra for L= 4at increasing We also note that the phase transition to the highly en- tangled phase with increasing is detected by level cross- ing in Fig. 6(b) and by the discontinuity in the derivative@E @ , which appears as the only kink for L= 4 { site chain, cf. Fig. 5(a). Other phase transitions occur by varying\|here at=J= 0:1 a phase transition is found from the FM (FO) phase with Stot= 2 andTtot= 0 (Stot= 0 andTtot= 2) to an entangled SU(4) phase (with all 15 generators being equal to 0) at jj'0:08, see Fig. 6(c). At this latter phase transition one nds also a discontinuous change of the von Neumann entropy [56]. For convenience, we introduce here the energy ~Ewhich does not decrease with increasing as in Fig. 6(b). For a chain of length L= 4 it is dened as follows, ~EE+ 2: (12) To get more insight into the evolution of the spectra with increasing spin-orbit coupling /, we consider in more detail the exact spectra of the L= 4 periodic chain, see Table I. At = 0 the ground state is the SU(4) sin- glet with energy E=0:75. The degeneracies of the excited states follow from the SandTquantum num- bers. Indeed, several states with higher values of Sand Texhibit huge degeneracies. Weak spin-orbit coupling =J= 0:1 perturbation of the superexchange introduces the splittings of degenerate excited states and in fact the TABLE I. The energies of the ground state and eight rst excited states ~E(12), with their degeneracies d, obtained for the periodic L= 4 chain at == 0 describing the spin- orbital model Eq. (1) at = 0 and for two representative values of(=J= 0:1 and 10), standing for weak and strong spin-orbit coupling. = 0 =J= 0:1 =J= 10 ~E d ~E d ~E d 0:75 1 0:55 1 0:45736 1 0:50 28 0:46570 1 0:25 2 0:45711 9 0:37965 4 0 :24367 1 0:43301 12 0:36944 8 0 :24684 2 0:40139 1 0:30 15 0 :24688 4 0:25 48 0:26229 2 0 :24691 1 0:0 76 0:25345 2 0 :25 2 0:25 34 0:25 2 0 :74369 2 0:43301 12 0:24561 2 0 :94779 110 spectrum is quite dense, see Fig. 6(c) and the data in Table I. However, the SU(4) singlet ground state is still robust as shown by the correlation functions CSOand OSOwhich do not change from their = 0 values, see the dotted line in Fig. 5 (a). It indicates that the spin- orbit term does not align here spin Szand orbital Tz components. The energy of the ground state (12) is just moved by 2 from0:75 to ~E=0:55 (Table I), and the spin-orbit coupling does not modify the ground state. AtCRIT = 0:14219Jthe energies of the ground state and of the lowest energy excited state cross and a com- pletely dierent situation arises\|then the von Neumann entropy changes in a discontinuous way to the value cor- responding to the strongly entangled state (for L= 4), and the spin-orbit correlation OSOdrops to0:25, see Fig. 5 (a). Since CRIT is dened based on the changes in the ground state , for > CRIT the full energy spec- trum does change further and consists of several bands of states, separated by gaps of the order of , see Fig. 6(d). The energies and their degeneracies within the low- est band of states are shown in the last two columns of Table I at large =J= 10 for the chain of length L= 4. We have veried that the 16 low energy states dis- played in Table I for =J= 10 collapse to the spectrum of the XY model in the limit of !1 , with the de- generacies 1, 2, 10, 2, 1, as expected and discussed in more detail in Sec. V A. Thus, in general, the spectrum consists of energy bands with the energies increasing in steps of', depending on the number of sites at which the spin-orbit coupling aligns the expectation values of spin and orbital operators, hSz iTz ii, at each site i. In this regime the spectra are dominated by the spin-orbit coupling. Note that the highly entangled phase can in- deed be regarded as a qualitatively unique phase, irre- spectively of the value of \|provided that =and that> CRIT. 4. Summary We have discussed in detail the evolution of the spin- orbital entanglement, and its impact on the quantum phases, with the increasing value of the spin-orbit cou- plingfor two representative values of the parameter . We can now extend the above reasoning to the other values of, keeping=. However, in order to ob- tain a quantum phase diagram of the model we still need to investigate whether the transitions between the ob- tained ground states could be regarded as phase transi- tions or are rather just of the crossover type. Dependence of the ground states energy on the model parameters (see Fig. 6) as well as the analytic characteristics of the von Neumann entropy [see Fig. 2(d-e); cf. Refs [87, 88]] sug- gest that the transitions along cuts A and B [Fig. 2(a)] are of distinct character. Whereas in case A the energy (as well as the von Neumann entropy) shows an analytic behavior across the transition, [Fig. 6(a)], in case B such behavior (both in energy as well as in von Neumann en- α βSU(4)FM dimers⊗AO AF⊗FOFM ⊗AO FM ⊗AO AF⊗F0 log scale/Jλ highly entangled state perturbed perturbed IESFIG. 7. Schematic quantum phase diagram of Hamiltonian (1) in the here{discussed regime of the parameters. The limit of=is depicted by the colorful vertical plane and is based on the results from Sec. IV B: whereas the four distinct phases are depicted with their names and separated by solid lines (IES stands for the intermediate entangled state), the two crossover regimes are denoted by yellow color and sepa- rated by the dashed lines. The limit = 0 is depicted by the horizontal plane and is adopted from Fig. 1 of Ref. [52]\|see text for further details. We note that the shape of the phase boundaries depends on the logarithmic scale of , chosen here for convenience. The schematic phase diagram is based on the ED results on small clusters, see text for the validity of these results in the thermodynamic limit. tropy) is clearly non-analytic [Fig. 6(b)]. This points to a crossover (phase) transition in case A (B), respectively. Altogether, this allows us to draw, on a qualitative level, a quantum phase diagram in the f;gparame- ter space (with =), see Fig. 7 (colorful vertical plane). As already discussed in Sec. IV, there are four distinct ground states (rst two shown in Fig. 7 in blue, and the other two in green and red, respectively): (i) the perturbed FM AO state for &0:08 and < CRIT, (ii) the perturbed AF FO state for .0:08 and < CRIT, (iii) the intermediate entangled state for jj.0:08 and 0< < CRIT, and (iv) the highly entangled state for > CRIT and for all values of . The latter state is discussed in more detailed in Sec. V. The four clearly distinct states are supplemented by two crossover regimes (shown in yellow in Fig. 7), which sep- arate phases (i-ii) from phase (iv)\|see also discussion above. It is instructive to place the above phase diagram in the context of the one already known from the literature11 and obtained for Hamiltonian (1) in the limit of the van- ishing spin-orbit coupling but varying values of both and[29, 39, 52, 66{75]. As can be seen on the hori- zontal plane of Fig. 7, the = 0 phase diagram consists of three simple product phases (AF FO, FM AO and FM FO) as well as two spin-orbital entangled phases (cf. Fig. 1): a phase with previously mentioned `global' SU(4)-symmetric singlet ground state and gapless exci- tations [68] and a phase with the ground state breaking theZ2symmetry and opening a nite gap by forming the two nonequivalent patterns of the spin and orbital dimers [29, 52]. We would like to emphasize at this point that the nite size eects for the spin-orbital model (at = 0) calculated on chains of length L= 16 (the maxi- mal size studied in Ref. [52]) and L= 20 (the maximal size studied here) are already relatively small [52]. This may suggest that the schematic phase diagram of Fig. 7 isqualitatively correct also in the thermodynamic limit. V. DISCUSSION: THE LIMIT OF LARGE A. Eective XXZ model To better understand the numerical results obtained in Sec. IV for the Hamiltonian (1) in the limit of the large spin-orbit coupling, > CRIT, we derive an ef- fective low-energy description of the system. In fact, as already discussed in Introduction, such an approach has become extremely popular in describing the physics of the iridium oxides [61], for it has lead to the description of the latter in terms of eective Heisenberg or Kitaev- like models. To obtain such an eective description for the case of large spin-orbit coupling, > CRIT, we rst obtain the eigenstates of the spin-orbit coupling Hamil- tonian (3): these are two doublets, separated by the gap E=. Next, we restrict the Hilbert space to the lowest doubletfj"i;j#+ig, wherej#i(ji) denotes the state withSz=1=2(Tz=1=2) quantum number. Lastly, we project the intersite Hamiltonian (2) onto the lowest doublet (see Appendix for details) and obtain the follow- ing eective model: He=J 2X i ~Jx i~Jx i+1+~Jy i~Jy i+1+ 2(+)~Jz i~Jz i+1 ;(13) where ~Jz i=1 2 ni;j"i+ni;j#+i is an eective ~Jz=1=2 pseudospin operator. Interestingly, it turns out that this eective Hamilto- nian describes exactly a spin 1=2XXZ chain. Moreover, in the limit of =the Ising interaction in Eq. (13) disappears and we obtain an AF XY model. Thus, re- sembling the iridate case [61], the eective model in the limit of large spin-orbit coupling has a surprisingly simple form. B. Validity of the eective XXZ model:benchmarking =case First, let us show that the eective XXZ model indeed gives the correct description of the ground state of the full spin-orbital model (1) in the limit of > CRIT. To this end, we compare the spin-orbital correlation functions calculated using the eective and the full models. We rst express the spin-orbital correlation function CSO, the on-site spin-orbit correlation function OSO, and the anisotropic spin (orbital) correlation functions S (T ) in the basis spanned by the two lowest doublets per sitefj"i;j#+ig\|see the Appendix for the explicit formula. Next, we compare the values of the correlation functions in the two special =cases, already dis- cussed above: (i) case A with jj= 0:5, and (ii) case B with= 0. As can be seen in Fig. 3 and Fig. 4, the cor- relation functions calculated using the two distinct mod- els agree extremely well once =J&1{100 (=J&0:2) in case A (B), respectively. We note that calculations performed for other values of the f;gparameters (un- shown) also show that the eective model describes the ground state properties in the limit of > CRIT well. Moreover, once =J'106, the ground and lowest lying excited states are quantitatively the same in the full and the eective models. C. Why the spin-orbital entanglement can vanish Having derived the eective model\|and having shown its validity\|we now discuss how it can help us with un- derstanding one of the crucial results of the paper: How can the spin-orbital entanglement vanish in the limit of large spin-orbit coupling > CRIT? We start by expressing the measure for the spin-orbital entanglement for nearest neighbors, the spin-orbital cor- relationCSO, in the basis of the eective model (see Ap- pendix for details): ~CSO=1 2LLX i=1 h~Jx i~Jx i+1+~Jy i~Jy i+1i2h~Jz i~Jz i+1i2+1 8 ; (14) where the averages are calculated in the ground state. To evaluate Eq. (14), we calculate expectation values of the eective pseudospin operators using ED, which we show in Fig. 8. (We note in passing that the presented ED results for an XXZ L= 10 site chain agree well with those which were published earlier, cf. Ref. [89].) The ob- tained ground state of the eective Hamiltonian (13) for + <1=2 is described by a ferromagnetic Ising state, whereh~Jz i~Jz i+1i=1=4and all other correlations vanish. Substituting these into Eq. (14) explains why CSO= 0 in the ground state of model (1) in the limit of large > CRIT and when restricted to + <1=2. In con- clusion, the spin-orbital entanglement for + <1=2 vanishes because not only the on-site interaction between spins and orbitals but also the intersite interactions in the12 -2.0 -1.0 0.0 1.0 2.0 α-2.0-1.00.01.02.0β/angbracketleftbig˜Ji˜Jj/angbracketrightbig XYAF Ising FM IsingAF H(a) -2.0 -1.0 0.0 1.0 2.0 α/angbracketleftbig˜Jx i˜Jx j/angbracketrightbig =/angbracketleftbig˜Jy i˜Jy j/angbracketrightbig XYAF Ising FM IsingAF H(b) -2.0 -1.0 0.0 1.0 2.0 α/angbracketleftbig˜Jz i˜Jz j/angbracketrightbig XYAF Ising FM IsingAF H(c) 0.4 0.2 0.00.2 FIG. 8. The zero-temperature phase diagram of the eective XXZ model (13) as a function of the model parameters and obtained using ED on a L= 10 site chain. The panels present the correlations: (a) h~Ji~Jji; (b)h~J i~J jiwith=x,y; (c)h~Jz i~Jz ji. The labels depict various ground states of the 1D XXZ model: AF H\|the Heisenberg antiferromagnet, AF Ising\|the Ising antiferromagnet, XY\|the XY antiferromagnet, FM Ising\|the Ising ferromagnet. ground state are of purely Ising type, and eectively the ground state is just a product state with no spin-orbital entanglement. D. Why the spin-orbital entanglement can be nite The eective model (13) can also be used to explain the presence of nite spin-orbital entanglement in the limit of large spin-orbit coupling > CRIT while it vanishes in the= 0 limit. Let us rst look at the already discussed in detail=case: In this case and in the = 0 limit, the term in (2) which is explicitly responsible for the spin-orbital entan- glement,/(SiSj)(TiTj), can become relatively small for largeordue to the presence of the TiTjand SiSjterms. Consequently, the region of signicant spin-orbital entanglement is quite small without spin- orbit coupling along the =line, see Fig. 1(a). This situation, however, drastically changes in the limit of large> CRIT, as discussed below. Specically, downfolding the exchange Hamiltonian (2) term by term onto the eective Hamiltonian (13) should reveal the origin of the spin-orbital entanglement in the large spin-orbit coupling limit. First, the TiTjand SiSjterms of Eq. (2) upon projecting onto spin-orbit coupled basis produce ~Jz i~Jz jand~Jz i~Jz j, resulting in the Ising terms in the eective model (13). Note that in the case that=, these Ising terms disappear. Second, the term responsible for the spin-orbital entanglement, i.e., (SiSj)(TiTj) (cf. above), reduces exactly to the XY terms in the eective model. These terms do not vanish once=. In fact, in this special limit the whole eective Hamiltonian is obtained from the term that is fully responsible for the spin-orbital entanglement in theoriginal Hamiltonian. Finally, as the ground state of the XY Hamiltonian carries `spatial entanglement' in pseu- dospins ~J, we expect the spin-orbital entanglement to be nite in the limit of large spin-orbit coupling > CRIT and once=. The above reasoning is conrmed by calculating the two contributions to the intersite spin-orbital correla- tion function ~CSOin the eective model once =. This can be done analytically for the XY model: h~Jz i~Jz i+1i=1=2andh~Jx i~Jx i+1i=h~Jy i~Jy i+1i=1=(2). (These results agree with the correlations calculated us- ing ED and presented in Fig. 8.) The above discussion can now be extended to the case that6=and+ >1=2, for which nite, though increasingly small for large and positive +, spin- orbital entanglement can be observed, see Fig. 1(c). Such result can be understood by using the eective model and by noting that the intersite spin-orbital correlation ~CSOis always nite provided that +is nite and + >1=2. This is because in this limit: (i) the cor- relationsh~Jx i~Jx i+1i=h~Jy i~Jy i+1iare nonzero, see Fig. 8(b); (ii)h~Jz i~Jz i+1i6= 1=4, see Fig. 8(c). It is then only in the limit+!1 that the spin-orbital entanglement can vanish, for the ground state of the XXZ model is `pure' Ising antiferromagnet. (A completely dierent situation occurs once + <1=2, i.e., for the FM ground state of the eective XXZ model, as already discussed in the previous subsection\|that explains why the spin-orbital entanglement can `sometimes' vanish even in the limit of large spin-orbit coupling, > CRIT.)13 VI. CONCLUSIONS A. Entanglement induced by spin-orbit coupling In conclusion, in this paper we studied the spin-orbital entanglement in a Mott insulator with spin and orbital degrees of freedom. We investigated how the spin-orbital entanglement gradually changes with the increasing value of the on-site spin-orbit coupling. The results, obtained by exactly diagonalizing a 1D model with the intersite SU(2) SU(2) spin-orbital superexchange /Jand the on-site Ising-type spin-orbit coupling /, reveal that: 1. For small < CRIT [90]: (a) In general, the spin-orbital entanglement in the ground state is not much more robust than in the= 0 case; (b) If the ground state had nite spin-orbital en- tanglement for = 0, it is driven into a novel spin-orbital strongly entangled phase upon in- creasing; (c) If the ground state did notshow spin-orbital entanglement for = 0, it still shows none or negligible spin-orbital entanglement upon increasing. 2. In the limit of large > CRIT: (a) In general, the spin-orbital entanglement in the ground state is far more robust than in the= 0 case; (b) The ground state may be driven into a novel spin-orbitally entangled phase even if it does not show spin-orbital entanglement for = 0; (c) The ground state may still show vanishing spin-orbital entanglement, but only if the quantum uctuations vanish in the ground state of an eective model (as is the case of an Ising ferromagnet). The statements mentioned under point 2. above, con- cerning> CRIT, constitute, from the purely theoreti- cal perspective, the main results of this paper. In partic- ular, they mean that: (i) the spin-orbital entanglement between spins and orbitals on dierent sites can be trig- gered by a joint action of the on-site spin-orbit coupling (of relativistic origin) and the spin-orbital exchange (of the `Kugel-Khomskii'{type); (ii) and yet, the onset of the spin-orbital entanglement in such a model does not have to be taken `for granted', for it can vanish even in the large spin-orbit coupling limit. Crucially, we have veried that the spin-orbital entan- glement can be induced by the spin-orbit coupling, for the latter interaction may enhance the role played by the spin-orbitally entangled ( SiSj)(TiTj) term by `quench- ing' the bare spin ( SiSj) and orbital ( TiTj) exchange terms in an eective low-energy Hamiltonian valid in thislimit. Interestingly, such mechanism can be valid even if the spin-orbit coupling has a purely `classical' Ising form (as for example in the case discussed in this paper). For a more intuitive explanation of these results, in Sec. V we presented a detailed analysis of the eective low-energy pseudospin XXZ model. B. Consequences for correlated materials The results presented here may play an important role in the understanding of the correlated systems with non- negligible spin-orbit coupling\|such as e.g. the 5 diri- dates, 4druthenates, 3 dvanadates, the 2 palkali hyper- oxides, and other to-be-synthesized materials. To this end, we argue that, even though obtained for a specic 1D model, some of the results presented here are to a large extent valid also for these 2D or 3D systems: First, this is partially the case for the results obtained in the limit of large > CRIT. In particular, the map- ping to the eective XXZ model is also valid in 2D and 3D cases. Moreover, one can easily verify that the spin- orbital correlation function [ ~CSO, Eq. (14)], which mea- sures spin-orbital entanglement never vanishes also in the 2D and 3D cases, unless the quantum uctuations com- pletely disappear (as is the case of the 2D or 3D Ising ferromagnet or antiferromagnet). Therefore, the main conclusions from Secs. IVC and IVD are also valid in 2D and 3D cases and consequently also point 2 of the concluding Section VI A holds. This means that, for ex- ample, the results obtained here would apply to any Mott insulator with two active t2gorbitals with small Hund's coupling and with > CRIT (such as e.g. Sr 2VO4[81]). Naturally, the question remains to what extent one could use the reasoning discussed here to the understand- ing of the spin-orbital ground state of the probably most famous Mott insulators with active orbital degrees of free- dom and large spin-orbit coupling\|the 5 diridates (such as e.g. Sr 2IrO4[41], Na 2IrO3, Li2IrO3, etc. [50]). Here we suggest that, while the situation in the iridates might be quite dierent in detail and requires solving a dis- tinct spin-orbital model with three active t2gorbitals and an SU(2)-symmetric spin-orbit coupling (which is beyond the scope of this work), we expect point 2(b) of the con- cluding Section VI A to hold also in this case: in fact, the quantum nature of the Heisenberg spin-orbit coupling of the iridates (in contrast to the classical Ising spin-orbit coupling studied in this paper), should only facilitate the onset of the spin-orbital entanglement. Thus, we suggest that in principle also for the iridates the ground state may be driven into a novel spin-orbitally entangled phase even if it does not show spin-orbital entanglement for = 0. Second, we suggest that also the fact that the spin- orbit coupling does not induce additional spin-orbital en- tanglement in the limit of small < CRIT will carry on to higher dimensions and to spin-orbital models of lower symmetry\|for a priori there is no reason why the ten- dency observed in a 1D (and highly symmetric) model,14 towards a`more classical' behavior should fail in dimen- sions higher than one (and for more anisotropic mod- els). Thus, in general the spin-orbital entanglement of the systems with weak spin-orbit coupling < CRIT and Ising-like spin-orbit coupling [80], such as e.g. the alkali hyperoxides with two active `molecular' 2 porbitals (e.g. KO 2[82]), should not qualitatively depend on the value of spin-orbit coupling. This means that, to simplify the studies one may, in the rst order of approximation, neglect the spin-orbit coupling in the eective models for these materials. ACKNOWLEDGMENTS We thank Clio Agrapidis, Wojciech Brzezicki, Cheng- Chien Chen, George Jackeli, Juraj Rusna cko, and Takami Tohyama for insightful discussions. The cal- culations were performed partly at the Interdisci- plinary Centre for Mathematical and Computational Modeling (ICM), University of Warsaw, under grant No. G72-9. This research was supported in part by PLGrid Infrastructure (Academic Computer Cen- ter Cyfronet AGH Krak ow). We kindly acknowl- edge support by the Narodowe Centrum Nauki (NCN, Poland) under Projects Nos. 2016/22/E/ST3/00560 and 2016/23/B/ST3/00839. E. M. P. acknowledges funding from the European Union's Horizon 2020 research and in- novation programme under the Maria Sk lodowska-Curie grant agreement No. 754411. J. Ch. acknowledges sup- port by M SMT CR under NPU II project CEITEC 2020 (LQ1601). Computational resources were supplied by the project \e-Infrastruktura CZ" (e-INFRA LM2018140) provided within the program Projects of Large Re- search, Development and Innovations Infrastructures. A. M. Ole s is grateful for an Alexander von Humboldt Foundation Fellowship (Humboldt-Forschungspreis). APPENDIX: EFFECTIVE XXZ MODEL Let us consider the Hamiltonian (1) of the main text: H=HSE+HSOC; (15) where the intersite interaction HSEand on-site spin-orbit coupling are described by HSE=JX i[(SiSi+1+)(TiTi+1+)];(16) HSOC= 2X iSz iTz i: (17) The characteristic scales for HSEandHSOCare intersite exchange parameter Jand on-site SOC , respectively. In the strong spin-orbit coupling limit, > CRIT,HSE can be considered as a perturbation to HSOC. The eigen- states of the full Hamiltonian (15) in zeroth-order arethen obtained by the diagonalization of the on-site spin- orbit partHSOC. In our simple case HSOCis already di- agonal with two doubly{degenerate energies =2. The corresponding eigenstates dened by total momentum ~J form two doublets. The lower energy doublet consists of ~J#=j+#i; ~J"=j"i; while the higher doublet is given by: ~J0 "=j+"i; ~J0 #=j#i: Here,j"i(j+i) denotes the state with Sz=1=2 (Tz=1=2) quantum number. The on-site basis transformation between the spin and orbital fjTz;Szig=fj+"i;j+#i;j"i;j#ig basis and spin- orbit coupledf~J#;~J";~J0 ";~J0 #gbasis consisting of two doublets is described by a unitary matrix U=0 BBB@0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 11 CCCA: (18) We then project the Hamiltonian (16) onto spin-orbit coupled basisf~J,~J0g:HSOC SE=UyHSEU. As we are in- terested in the low-energy physics, we truncate Hilbert space to the lowest doublet ~Jand obtain eective Hamil- tonian (13) from the main text: He=J 2X i ~Jx i~Jx i+1+~Jy i~Jy i+1+ 2(+)~Jz i~Jz i+1 :(19) To analyze the eective model (19) and obtain impor- tant correlation functions, we rst need to establish a link between operators describing correlation functions in originalfjTz;Szigbasis and spin-orbit coupled f~J,~J0g basis. To this end, we project each of the spin/orbital operators,Or=fS r;T rg, =fx;y;zg,r=fi;i+ 1g entering the original correlation functions (4) { (8) onto spin-orbit coupled basis: OSOC r =UyOrU. As most of the correlation functions include intersite terms, the re- sult shall be written as a 16 16 matrix, spanned by f~J,~J0gif~J,~J0gjbasis. We then once again drop out the high-energy doublet on each site and obtain correlation functions as 4 4 matrices dened in Hilbert space of f~Jgif~Jgj: ~S=~T=0 BBB@1 40 0 0 01 40 0 0 01 40 0 0 01 41 CCCA+ =h~Jz i~Jz ji; (20) ~S=~T=0 BBB@0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 CCCA+ = 0; (21)15 where=fx;yg, ~Szz=~Tzz=0 BBB@1 40 0 0 01 40 0 0 01 40 0 0 01 41 CCCA+ =D ~Jz i~Jz jE ; ~CSO=0 BBB@1 160 0 0 01 161 40 01 41 160 0 0 01 161 CCCA+ 2 66640 BBB@1 40 0 0 01 40 0 0 01 40 0 0 01 41 CCCA+3 77752 =1 2h~Jx i~Jx j+~Jy i~Jy ji+1 16h~Jz i~Jz ji2:To express the on-site spin-orbit correlation function OSO, which does not include intersite terms, in the same basis, we multiply it by a 2 2 identity matrix represent- ing the neighboring site: ~OSO;i idj=0 BBB@1 40 0 0 01 40 0 0 01 40 0 0 01 41 CCCA+ =1 4: [1] D. I. Khomskii, Basic Aspects of the Quantum Theory of Solids: Order and Elementary Excitations (Cambridge University Press, Cambridge, 2010). [2] T. Andrade, A. Krikun, K. Schalm, and J. Zaanen, Na- ture Physics 14, 1049 (2018). [3] J. Zaanen, SciPost Phys. 6, 61 (2019). [4] L. Balents, Nature 464, 199 (2010). [5] C. Varma, Z. Nussinov, and W. van Saarloos, Physics Reports 361, 267 (2002). [6] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935). [7] H. Ollivier and W. H. Zurek, Phys. Rev. Lett. 88, 017901 (2001). [8] I. Bentsson and K. Zyczkowski, Geometry of Quan- tum States: An Introduction to Quantum Entanglement (Cambridge University Press, Cambridge, 2006). [9] C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schu- macher, Phys. Rev. A 53, 2046 (1996). [10] G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys. Rev. Lett. 90, 227902 (2003). [11] V. E. Korepin, Phys. Rev. Lett. 92, 096402 (2004). [12] M. M. Wolf, Phys. Rev. Lett. 96, 010404 (2006). [13] D. Gioev and I. Klich, Phys. Rev. Lett. 96, 100503 (2006). [14] J. I. Lattore, E. Rico, and G. Vidal, Quantum Informa- tion & Computation 4, 48 (2004). [15] L. Cincio, J. Dziarmaga, and M. M. Rams, Phys. Rev. Lett. 100, 240603 (2008). [16] R. Thomale, D. P. Arovas, and B. A. Bernevig, Phys. Rev. Lett. 105, 116805 (2010). [17] G. Khaliullin and S. Maekawa, Phys. Rev. Lett. 85, 3950 (2000). [18] G. Khaliullin, P. Horsch, and A. M. Ole s, Phys. Rev. Lett. 86, 3879 (2001). [19] G. Khaliullin, Prog. Theor. Phys. Suppl. 160, 155 (2005). [20] S. Nakatsuji, K. Kuga, K. Kimura, R. Satake, N. Katayama, E. Nishibori, H. Sawa, R. Ishii, M. Hagi- wara, F. Bridges, T. U. Ito, W. Higemoto, Y. Karaki, M. Halim, A. A. Nugroho, J. A. Rodriguez-Rivera, M. A. Green, and C. Broholm, Science 336, 559 (2012). [21] Y. Ishiguro, K. Kimura, S. Nakatsuji, S. Tsutsui, A. Q. R. Baron, T. Kimura, and Y. Wakabayashi, Nature Com- munications 4, 2022 (2013). [22] H. Man, M. Halim, H. Sawa, M. Hagiwara, Y. Wak- abayashi, and S. Nakatsuji, J. Phys.: Condens. Matter30, 443002 (2018). [23] B. Y. Pan, H. Jang, J.-S. Lee, R. Sutarto, F. He, J. F. Zeng, Y. Liu, X. W. Zhang, Y. Feng, Y. Q. Hao, J. Zhao, H. C. Xu, Z. H. Chen, J. P. Hu, and D. L. Feng, Phys. Rev. X 9, 021055 (2019). [24] B. Normand and A. M. Ole s, Phys. Rev. B 78, 094427 (2008); B. Normand, ibid.83, 064413 (2011); J. Chaloupka and A. M. Ole s, ibid.83, 094406 (2011). [25] P. Corboz, M. Lajk o, A. M. L auchli, K. Penc, and F. Mila, Phys. Rev. X 2, 041013 (2012). [26] W. Brzezicki, J. Dziarmaga, and A. M. Ole s, Phys. Rev. Lett. 109, 237201 (2012). [27] A. M. Ole s, P. Horsch, L. F. Feiner, and G. Khaliullin, Phys. Rev. Lett. 96, 147205 (2006). [28] A. M. Ole s, J. Phys.: Condens. Matter 24, 313201 (2012). [29] Y. Chen, Z. D. Wang, Y. Q. Li, and F. C. Zhang, Phys. Rev. B 75, 195113 (2007). [30] J. B. Goodenough, Magnetism and the Chemical Bond (Interscience, New York, 1963). [31] J. Kanamori, Journal of Physics and Chemistry of Solids 10, 87 (1959). [32] S. Miyasaka, Y. Okimoto, and Y. Tokura, J. Phys. Soc. Jpn.71, 2082 (2002). [33] G. Khaliullin, P. Horsch, and A. M. Ole s, Phys. Rev. B 70, 195103 (2004). [34] M. Snamina and A. M. Ole s, New J. Phys. 21, 023018 (2019). [35] J. Schlappa, K. Wohlfeld, K. J. Zhou, M. Mourigal, M. W. Haverkort, V. N. Strocov, L. Hozoi, C. Mon- ney, S. Nishimoto, S. Singh, A. Revcolevschi, J. Caux, L. Patthey, H. M. Rnnow, J. van den Brink, and T. Schmitt, Nature 485, 82 (2012). [36] V. Bisogni, K. Wohlfeld, S. Nishimoto, C. Monney, J. Trinckauf, K. Zhou, R. Kraus, K. Koepernik, C. Sekar, V. Strocov, B. B uchner, T. Schmitt, J. van den Brink, and J. Geck, Phys. Rev. Lett. 114, 096402 (2015). [37] K. Wohlfeld, M. Daghofer, S. Nishimoto, G. Khaliullin, and J. van den Brink, Phys. Rev. Lett. 107, 147201 (2011). [38] K. Wohlfeld, S. Nishimoto, M. W. Haverkort, and J. van den Brink, Phys. Rev. B 88, 195138 (2013). [39] C.-C. Chen, M. van Veenendaal, T. P. Devereaux, and K. Wohlfeld, Phys. Rev. B 91, 165102 (2015). [40] W. Witczak-Krempa, G. Chen, Y. B. Kim, and L. Ba- lents, Annual Review of Condensed Matter Physics 5, 5716 (2014). [41] J. Bertinshaw, Y. K. Kim, G. Khaliullin, and B. J. Kim, Annu. Rev. Condens. Matter Physics 10, 315 (2019). [42] H. Ishikawa, T. Takayama, R. K. Kremer, J. Nuss, R. Dinnebier, K. Kitagawa, K. Ishii, and H. Takagi, Phys. Rev. B 100, 045142 (2019). [43] E. Lefran cois, A.-M. Pradipto, M. Moretti Sala, L. C. Chapon, V. Simonet, S. Picozzi, P. Lejay, S. Petit, and R. Ballou, Phys. Rev. B 93, 224401 (2016). [44] K. Kitagawa, T. Takayama, Y. Matsumoto, A. Kato, R. Takano, Y. Kishimoto, S. Bette, R. Dinnebier, G. Jackeli, and H. Takagi, Nature 554, 341 (2018). [45] C. N. Veenstra, Z.-H. Zhu, M. Raichle, B. M. Lud- brook, A. Nicolaou, B. Slomski, G. Landolt, S. Kittaka, Y. Maeno, J. H. Dil, I. S. Elmov, M. W. Haverkort, and A. Damascelli, Phys. Rev. Lett. 112, 127002 (2014). [46] G. Zhang, E. Gorelov, E. Sarvestani, and E. Pavarini, Phys. Rev. Lett. 116, 106402 (2016). [47] G. Khaliullin, Phys. Rev. Lett. 111, 197201 (2013). [48] A. Jain, M. Krautloher, J. Porras, G. H. Ryu, D. P. Chen, D. L. Abernathy, J. T. Park, A. Ivanov, J. Chaloupka, G. Khaliullin, B. Keimer, and B. J. Kim, Nature Physics 13, 633 (2017). [49] J. Chaloupka, G. Jackeli, and G. Khaliullin, Phys. Rev. Lett. 105, 027204 (2010). [50] S. M. Winter, A. A. Tsirlin, M. Daghofer, J. van den Brink, Y. Singh, P. Gegenwart, and R. Valent , Journal of Physics: Condensed Matter 29, 493002 (2017). [51] A. Kitaev, Annals of Physics 321, 2 (2006). [52] R. Lundgren, V. Chua, and G. A. Fiete, Phys. Rev. B 86, 224422 (2012). [53] W.-L. You, A. M. Ole s, and P. Horsch, Phys. Rev. B 86, 094412 (2012). [54] A. M. L auchli and J. Schliemann, Phys. Rev. B 85, 054403 (2012). [55] W.-L. You, P. Horsch, and A. M. Ole s, Phys. Rev. B 92, 054423 (2015). [56] W.-L. You, A. M. Ole s, and P. Horsch, New J. Phys. 17, 083009 (2015). [57] V. E. Valiulin, A. V. Mikheyenkov, K. I. Kugel, and A. F. Barabanov, JETP Letters 109, 546 (2019). [58] K. I. Kugel and D. I. Khomskii, Sov. Phys. Usp. 25, 231 (1982). [59] Note that in Ref. [52] the impact of relatively small spin-orbit coupling on the entanglement spectra was dis- cussed. [60] E. M. P arschke and R. Ray, Phys. Rev. B 98, 064422 (2018). [61] G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 102, 017205 (2009). [62] M. V. Eremin, J. Deisenhofer, R. M. Eremina, J. Teyssier, D. van der Marel, and A. Loidl, Phys. Rev. B84, 212407 (2011). [63] J. Choukroun, Phys. Rev. B 84, 014415 (2011).[64] C. Svoboda, M. Randeria, and N. Trivedi, Phys. Rev. B 95, 014409 (2017). [65] A. Koga, S. Nakauchi, and J. Nasu, Phys. Rev. B 97, 094427 (2018). [66] B. Sutherland, Phys. Rev. B 12, 3795 (1975). [67] Y. Yamashita, N. Shibata, and K. Ueda, Phys. Rev. B 58, 9114 (1998). [68] Y.-Q. Li, M. Ma, D.-N. Shi, and F.-C. Zhang, Phys. Rev. B60, 12781 (1999). [69] A. K. Kolezhuk and H.-J. Mikeska, Phys. Rev. Lett. 80, 2709 (1998). [70] A. K. Kolezhuk, H.-J. Mikeska, and U. Schollw ock, Phys. Rev. B 63, 064418 (2001). [71] D. P. Arovas and A. Auerbach, Phys. Rev. B 52, 10114 (1995). [72] S. K. Pati, R. R. P. Singh, and D. I. Khomskii, Phys. Rev. Lett. 81, 5406 (1998). [73] C. Itoi, S. Qin, and I. Aeck, Phys. Rev. B 61, 6747 (2000). [74] W. Zheng and J. Oitmaa, Phys. Rev. B 64, 014410 (2001). [75] P. Li and S.-Q. Shen, Phys. Rev. B 72, 214439 (2005). [76] K. I. Kugel, D. I. Khomskii, A. O. Sboychakov, and S. V. Streltsov, Phys. Rev. B 91, 155125 (2015). [77] A. V. Gorshkov, M. Hermele, V. Gurarie, C. Xu, P. S. Julienne, J. Ye, P. Zoller, E. Demler, M. D. Lukin, and A. M. Rey, Nature Physics 6, 289 (2010). [78] X. Zhang, M. Bishof, S. L. Bromley, C. V. Kraus, M. S. Safronova, P. Zoller, A. M. Rey, and J. Ye, Science 345, 1467 (2014). [79] X. Dou, V. N. Kotov, and B. Uchoa, Scientic Reports 6, 31737 (2016). [80] P. Horsch, G. Khaliullin, and A. M. Ole s, Phys. Rev. Lett. 91, 257203 (2003). [81] G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 103, 067205 (2009). [82] I. V. Solovyev, New J. Phys. 10, 013035 (2008). [83] E. Koch, The Lanczos Method, in: The LDA+DMFT approach to strongly correlated materials, edited by E. Pavarini, E. Koch, D. Vollhardt, and A. Lichtenstein (Forschungszentrum J ulich, J ulich, 2011). [84] A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, 110404 (2006). [85] Y. Q. Li, M. Ma, D. N. Shi, and F. C. Zhang, Phys. Rev. Lett. 81, 3527 (1998). [86] A. M. Ole s, G. Tr eglia, D. Spanjaard, and R. Jullien, Phys. Rev. B 32, 2167 (1985). [87] A. Osterloh, L. Amico, G. Falci, and R. Fazio, Nature 416, 608 (2002). [88] X. Jia, A. R. Subramaniam, I. A. Gruzberg, and S. Chakravarty, Phys. Rev. B 77, 014208 (2008). [89] K. Hijii, A. Kitazawa, and K. Nomura, Phys. Rev. B 72, 014449 (2005). [90]CRIT depends on the particular values of the model pa- rameters, see Sec. IV B.
1805.10328v2.Generation_of_Spin_Current_from_Lattice_Distortion_Dynamics__Spin_Orbit_Routes.pdf	arXiv:1805.10328v2 [cond-mat.mes-hall] 20 Jul 2018Journal of the Physical Society of Japan LETTERS Generation of Spin Current from Lattice Distortion Dynamic s: Spin-Orbit Routes Takumi Funato and Hiroshi Kohno Department of Physics, Nagoya University, Nagoya 464-8602 , Japan Generation of spin current from lattice distortion dynamic s in metals is studied with special attention on the e ﬀect of spin-orbit coupling. Treating the lattice distortion by local coordinate transformation, we calculate spin curren t and spin accumulation with the linear response theory. It is fou nd that there are two routes to the spin-current generation: one via the spin Hall e ﬀect and the other via the spin accumulation. The present e ﬀect due to spin-orbit coupling can be comparable to, or even larger than, the one based on the spi n-vorticity coupling in systems with strong spin-orbit coupling. In the ﬁeld of spintronics, spin current occupies a central position for the development of new devices or the discov- ery of novel physical phenomena. To date we know several methods available to generate spin currents, which include spin pumping,1–3)spin Hall eﬀect,4)spin accumulation at the ferromagnet/nonmagnet interface,5)and spin Seebeck eﬀect.6) These are classiﬁed as magnetic, electrical, magnetoelect ric, and thermal means, respectively. Recently, there has also been interest in generating spin currents by mechanical means, namely, by converting angu- lar momentum associated with mechanical motion, such as the rigid rotation of a solid or vorticity of a ﬂuid, into spin an- gular momentum of electrons. In the experiments reported so far, two mechanisms have been considered. One is the acous- tic spin pumping by Uchida et al. ,7, 8)which is based on the magnon-phonon coupling. They succeeded in generating spin current by injecting acoustic waves into yttrium iron garne t (YIG) from the attached piezoelectric element. Theoretica l analyses were given by Adachi and Maekawa,9, 10)Keshtgar et al. ,11)and Deymier et al.12)Another mechanism, proposed by Matsuo et al. ,13, 14)is based on the spin-rotation coupling or the spin-vorticity coupling (SVC). This is the coupling o f the spin to the eﬀective magnetic ﬁeld that emerges in a ro- tating (non-inertial) frame of reference locally ﬁxed on th e material that is in motion. The ﬁrst experiment for the SVC mechanism was conducted on liquid metals.15, 16)To realize the SVC mechanism in solids, it was proposed to use sur- face acoustic waves.17, 18)Nozaki et al. used Py/Cu bilayer and injected surface acoustic waves into Cu from the attache d LiNbO 3(surface acoustic wave ﬁlter).19)The generated AC spin current was detected via the spin-torque ferromagneti c resonance. One of the reasons that the mechanical generation of spin current has attracted attention is that it does not rely on sp in- orbit interaction (SOI). Therefore, previous works did not pay attention to the eﬀects of SOI. However, it is well expected that SOI plays certain roles in the mechanical processes of spin-current generation. For example, the previous experi - ments7, 8)were conducted on systems with an interface, which potentially possesses Rashba SOI. Furthermore, the mechan i- cal generation method may be used in combination with other “conventional” mechanisms that utilize SOI, and thereby en - hance spin current. In this paper, we study a mechanical generation of spin cur-rent by focusing on the e ﬀects of SOI. As a mechanical pro- cess, we consider dynamical lattice deformations of a solid with metallic electrons and with SOI. To treat lattice defor - mations analytically, we use the method of Tsuneto develope d in the context of ultrasonic attenuation in superconductor s,20) which employs a local coordinate transformation. By calcu- lating spin current and spin accumulation induced by dynam- ical lattice deformations, we found two routes to spin-curr ent generation: one via the spin accumulation and the other via the spin Hall eﬀect. As a related work, Wang et al.21)derived the Hamiltonian that includes SOI in a general coordinate sy s- tem starting from the general relativistic Dirac equation, but they did not give an explicit analysis of spin-current gener a- tion. Model: We consider a free-electron system in the presence of random impurities and the associated SOI. The Hamilto- nian is given by H=−∇′2 2m+Vimp(r′)+iλso{[∇′Vimp(r′)]×σ}·∇′. (1) The second term represents the impurity potential, Vimp(r′)= ui/summationtext jδ(r′−Rj), with strength uiand at position Rj(for jth impurity), and the third term is the SOI associated with Vimp, with strength λsoand the Pauli matrices σ=(σx,σy,σz). When the lattice is deformed, e.g., by sound waves, the Hamiltonian becomes Hlab=−∇′2 2m+Vimp(r′−δR(r′,t)) +iλso/braceleftBig/bracketleftBig ∇′Vimp/parenleftbigr′−δR(r′,t)/parenrightbig/bracketrightBig ×σ/bracerightBig ·∇′,(2) whereδR(r′,t) is the displacement vector of the lattice from their equilibrium position r′. Following Tsuneto,20)we make a local coordinate transfor- mation, r=r′−δR(r′,t), from the laboratory (Lab) frame (with coordinate r′) to a “material frame” (with coordinate r) which is ﬁxed to the ‘atoms’ in a deformable lattice. At the same time, the wave function needs to be redeﬁned to keep the normalization condition, ψ(r,t)=[1+∇·δR]1/2ψ′(r′,t)+O(δR2), (3) whereψ′(r′,t) is the wave function in the Lab frame, and ψ(r,t) is the one in the material frame. Up to the ﬁrst order 1J. Phys. Soc. Jpn. LETTERS inδR, the Hamiltonian for ψ(r,t) is given by Hmat=H+H′ K+H′ so, (4) where H=HK+Himp+Hsois the unperturbed Hamiltonian de- ﬁned by HlabwithδR=0. Here, HK=/summationtext k(k2/2m)ψ† kψkis the kinetic energy, with ψk(ψ† k) being the electron annihilation (creation) operator. HimpandHsodescribe the impurity poten- tial and impurity SOI, respectively, Himp=/summationtext k,k′Vk′−kψ† k′ψk, Hso=iλso/summationtext k,k′Vk′−k(k′×k)·ψ† k′σψk, where Vk′−kis the Fourier component of Vimp(r). Assuming a uniformly ran- dom distribution, we average over the impurity positions as /an}b∇acketle{tVkVk′/an}b∇acket∇i}htav=niu2 iδk+k′,0, and/an}b∇acketle{tVkVk′Vk′′/an}b∇acket∇i}htav=niu3 iδk+k′+k′′,0, where niis the impurity concentration. The impurity-averaged retarded/advanced Green function is given by GR/A k(ε)=(ε+ µ−k2/2m±iγ)−1, whereγ=πniu2 iN(µ)(1+2 3λ2 sok4 F) is the damping rate. Here, N(µ) is the Fermi-level density of states (per spin), and kFis the Fermi wave number. In this work, we consider the eﬀects of SOI up to the second order. The eﬀects of lattice distortion are contained in H′ KandH′ so, which come from HKandHso, respectively. In the ﬁrst order inδR, they are given by H′ K=/summationdisplay kWK n(k)un q,ωψ† k+q 2ψk−q 2, (5) H′ so=/summationdisplay k,k′Vk′−kWso ln(k,k′)un q,ωψ† k′+q 2σlψk−q 2. (6) Here, uq,ωis the Fourier component of the lattice velocity ﬁeld, u(r,t)=∂tδR(r,t), and we deﬁned (see Fig. 1), WK n(k)=/bracketleftBigq·k mω−1/bracketrightBig kn, (7) Wso ln(k,k′)=λso iω/bracketleftBig (k×q)lk′ n−(k′×q)lkn/bracketrightBig . (8) The ﬁrst term in WK ndescribes the coupling of the strain ∂iδRn to the stress tensor ∼/summationtext kkiknc† kckof electrons, and modiﬁes the eﬀective mass tensor. Throughout this report, qrepresents the wave vector of the lattice deformation and ωis its fre- quency. We assume that the spatial and temporal variations o f δRare slow and satisfy the conditions q≪ℓ−1andω≪γ, whereℓis the mean free path. Spin and spin-current density operators are given by ˆjα s,0(q)=ˆσα(q)=/summationdisplay kψ† k−q 2σαv0ψk+q 2, (9) ˆjα s,i(q)=/summationdisplay kψ† k−q 2σαviψk+q 2+ˆja,α s,i(q), (10) whereα=x,y,zspeciﬁes the spin direction, i=x,y,zthe current direction, and v0=1. Here, ˆja,α s,i(q)=−iλsoǫαi j/summationdisplay k,k′Vk′−k(k′ j−kj)ψ† k′−q 2ψk+q 2, (11) is the ‘anomalous’ part of the spin-current density, with ǫαi j being the Levi-Civita symbol. We calculate ˆjα s,µin (linear) re- sponse22)tou, /an}b∇acketle{tˆjα s,µ(q)/an}b∇acket∇i}htω=−/bracketleftBig Kss,α µn+Ksj,α µn+Kso,α µn/bracketrightBig q,ωun, (12) where Kss,α µn(Ksj,α µn) is the skew-scattering (side-jump) type contribution in response to H′ K, and Kso,α µndescribes the re- sponse to H′ so. Fig. 1. Two types of vertices associated with the coupling to the lat tice displacement δR, or the velocity ﬁeld u=dδR/dt. Fig. 2. Skew-scattering type contributions to the spin current ( µ=x,y,z) and/or spin accumulation ( µ=0). The black (white) circles represent spin- ﬂip (spin non-ﬂip) vertices. The cross and the dashed line re present an impu- rity and the impurity potential, respectively. The shaded p art represents the impurity ladder vertex corrections. The upside-down diagr ams are also con- sidered in the calculation. Skew-scattering process: The skew-scattering contribution without ladder vertex corrections, shown in Fig. 2, is given by Kss,α µν(ω)=iλsoniu3 i/summationdisplay k1,k2(k1×k2)αv1µWK ν(k2) ×ω iπ/summationdisplay p/bracketleftbigg GR p/parenleftbiggω 2/parenrightbigg −GA p/parenleftbigg −ω 2/parenrightbigg/bracketrightbigg ×GR k1+GA k1−GR k2+GA k2−, (13) with GR/A k±=GR/A k±q 2(±ω 2). By including ladder vertex correc- tions, spin accumulation and spin-current density are calc u- lated as23) /an}b∇acketle{tσα/an}b∇acket∇i}htss=αss SHneτ/parenleftBigg3 5Dq2−iω/parenrightBigg(iq×u)α Dq2−iω+τ−1 sf, (14) /an}b∇acketle{tjα s,i/an}b∇acket∇i}htss=−Diq i/an}b∇acketle{tσα/an}b∇acket∇i}htss+αss SHǫαim/an}b∇acketle{tjm/an}b∇acket∇i}ht, (15) whereαss SH=2π 3k2 FλsoN(µ)uiis the spin Hall angle due to skew scattering,24)ne=2 3mk2 FN(µ) is the electron number density, τ=(2γ)−1is the scattering time, D=1 3v2 Fτis the diﬀusion constant, and τ−1 sf=(4λ2 sok4 F/3)τ−1is the spin relaxation rate due to SOI. In Eq. (15), /an}b∇acketle{tjm/an}b∇acket∇i}htis the charge current, /an}b∇acketle{tjm/an}b∇acket∇i}ht=neτ/braceleftbigg −/parenleftBigg3 5Dq2−iω/parenrightBigg um +/parenleftBigg6 5+1 τ(Dq2−iω)/parenrightBigg Diq m(iq·u)/bracerightbigg , (16) generated by u.20)Here, in the ﬁrst line, the term ∼Dq2um (the term∼iωum) is induced by the ﬁrst (second) term in WK n, Eq. (7), via the spatio-temporal variation of the strain ten sor ∂iδRm(temporal variation of the velocity ﬁeld um). The last term is the diﬀusion current. We see that a spin accumulation (14) is induced by the vorticity of the lattice velocity ﬁeld u. The ﬁrst term and the second terms in Eq. (15) are written with the spin accumulation (Eq. (14)) and the charge current (Eq. (16)), respectively. Side-jump process: The side-jump contributions are ob- 2J. Phys. Soc. Jpn. LETTERS Fig. 3. Side-jump type contribution to the spin-current density ( i,µ= x,y,z) and/or spin accumulation ( µ=0). The diagrams in (a) come from the anomalous velocity, Eq. (11); hence they contribute only to the spin current. The diagrams in (b) can be nonvanishing only when the lattice deformation is nonuniform. The upside-down diagrams are also included in t he calculation. Fig. 4. Response to H′ so, which turned out to vanish. tained from the two types of diagrams in Fig. 3. They give Ksj (a),α in(ω)=iλsoniu2 iǫαi j/summationdisplay k1,k′ 1,k2(k′ 1,j−k1,j)WK n(k2) ×ω iπ/bracketleftBig δk′ 1k2GR k1++δk1k2GA k1′−/bracketrightBig GR k2+GA k2−, (17) Ksj (b),α µν (ω)=λsoniu2 i/summationdisplay k1,k2[(k1−k2)×iq]αv1µWK ν(k2) ×ω iπGR k1+GA k1−GR k2+GA k2−, (18) corresponding to the diagrams in Fig. 3 (a) and (b), respec- tively. With the ladder vertex corrections included, spin a ccu- mulation and spin-current density are calculated as23) /an}b∇acketle{tσα/an}b∇acket∇i}htsj=αsj SHneτ/parenleftBigg3 5Dq2−iω/parenrightBigg(iq×u)α Dq2−iω+τ−1 sf, (19) /an}b∇acketle{tjα s,i/an}b∇acket∇i}htsj (a)=αsj SHǫαim/an}b∇acketle{tjm/an}b∇acket∇i}ht, (20) /an}b∇acketle{tjα s,i/an}b∇acket∇i}htsj (b)=αsj SHneτǫαimDiq m Dq2−iωiq·u−Diq i/an}b∇acketle{tσα/an}b∇acket∇i}htsj. (21) whereαsj SH=−λsom/τis the spin Hall angle due to side-jump processes.24)The diagrams in Fig. 3 (a) give only the spin cur- rent (Eq. (20)) since the left vertices come from the anoma- lous velocity. This contribution is also written with the ch arge current/an}b∇acketle{tjm/an}b∇acket∇i}htgiven by Eq. (16). On the other hand, spin ac- cumulation coming from Fig. 3 (b) is again proportional to the vorticity of the velocity ﬁeld u. In Eq. (21), the ﬁrst term is proportional to the di ﬀusion part of the charge current [the last term in Eq (16)], and the second term is the di ﬀusion spin current. We note that these contributions, coming from the d i- agrams of Fig. 3 (b), vanish when the external perturbation i s uniform, i.e., q=0. Finally, the response to H′ so, shown in Fig. 4, turned out to vanish, Kso,α µn=0. This is also the case when the ladder vertex corrections are included. Result: Taken together, the total spin accumulation and Fig. 5. (Color online) Two routes to the generation of spin current f rom lattice distortion dynamics. The thick arrows indicate the processes governed by SOI. ‘AE’ means acousto-electric e ﬀect. spin-current density arising from the dynamical lattice di stor- tion via SOI have been obtained as /an}b∇acketle{tσα/an}b∇acket∇i}htSOI=αSHneτ/parenleftBigg3 5Dq2−iω/parenrightBigg(iq×u)α Dq2−iω+τ−1 sf, (22) /an}b∇acketle{tjα s,i/an}b∇acket∇i}htSOI=−Diq i/an}b∇acketle{tσα/an}b∇acket∇i}htSOI+αSHǫαim/an}b∇acketle{tjm/an}b∇acket∇i}ht +αsj SHneτǫαimDiq m Dq2−iωiq·u, (23) whereαSH=αss SH+αsj SHis the ‘total’ spin Hall angle. As seen from Eq. (22), spin accumulation is induced by the vorticity of the lattice velocity ﬁeld via SOI. The resulting di ﬀusion spin current contributes to Eq. (23) as the ﬁrst term. In addition , dynamical lattice distortion generates a charge current as well (known as the acousto-electric e ﬀect25)), which is then con- verted to a spin Hall current (in the transverse direction) v ia SOI, as expressed by the second and third terms in Eq. (23). Therefore, there are two routes to the spin-current generat ion in the present mechanism; one is the “di ﬀusion route” caused by the spin accumulation and the other is the “spin Hall route ” that follows the acousto-electric e ﬀect.26)This is illustrated in Fig. 5. In the latter (spin Hall) route, the longitudinal com - ponent of ualso induces spin current via the generation of charge current. Finally, we note that the induced spin accu- mulation (22) and the spin-current density (23) satisfy the spin continuity equation, ∂t/an}b∇acketle{tσα/an}b∇acket∇i}htSOI+∇·/an}b∇acketle{tjα s/an}b∇acket∇i}htSOI=−/an}b∇acketle{tσα/an}b∇acket∇i}htSOI τsf. (24) The term on the right-hand side represents spin relaxation d ue to SOI. The above result does not include the e ﬀects of lattice dis- tortion on the spinorial character of the electron wave func - tion. Such eﬀects are derived from the spin connection in the general relativistic Dirac equation.13)The total spin current and spin accumulation are given by the sum of the contri- butions from the SVC (previous work13)) and SOI (present work). Next, we study the contribution from SVC, an e ﬀect originating from the spin connection. Spin-rotation coupling: For comparing the present result with the previous one that is based on the spin-vorticity cou - pling (SVC),16)we also calculate the spin accumulation and spin-current density in response to the vorticity of the lat tice velocity ﬁeld,ω=∇× u.27)By treating the SVC Hamil- tonian HSV=−1 4σ·ω(q,ω) as a perturbation, one has /an}b∇acketle{tjα s,µ/an}b∇acket∇i}htω=χα µβ(q,ω)ωβ(q,ω), whereχα µβ(q,ω) is the response function. The response function (without vertex correctio ns) is given as χα µβ(q,ω)=1 2N(µ)δαβδµ0+iω 4πδαβ/summationtext kvµGR k+GA k−. With ladder vertex corrections, spin accumulation and spin - 3J. Phys. Soc. Jpn. LETTERS current density are obtained as /an}b∇acketle{tσα/an}b∇acket∇i}htSV=N(µ) 2Dq2+τ−1 sf Dq2−iω+τ−1 sfωα, (25) /an}b∇acketle{tjα s,i/an}b∇acket∇i}htSV=−iωN(µ) 2Diq i Dq2−iω+τ−1 sfωα. (26) They satisfy the spin continuity equation, (∂t+τ−1 sf)/an}b∇acketle{tσα/an}b∇acket∇i}htSV+∇·/an}b∇acketle{tjα s/an}b∇acket∇i}htSV=N(µ) 2τsfωα, (27) with a source term ( ∼ω) on the right-hand side. Alterna- tively, one may deﬁne the “spin accumulation” δµα=µ↑−µ↓ by/an}b∇acketle{tσα/an}b∇acket∇i}htSV=n↑−n↓=N(µ)(δµα+/planckover2pi1ωα/2),17)where the spin quantization axis has been taken along the ˆ αaxis. Then, Eq. (25) leads to (∂t−D∇2+τ−1 sf)δµα=−/planckover2pi1 2˙ωα. (28) This is the basic equation used in Ref. 16 to study spin-curre nt generation. Therefore, in the SVC mechanism, only the trans - verse acoustic waves generate spin current, and the generat ed spin current is purely of di ﬀusion origin. These are in stark contrast with the SOI-induced mechanism. Comparison: To see the magnitude of the present e ﬀect, we estimate the di ﬀusion spin current generated via SOI, Eq. (22), relative to the one due to SVC, Eq. (25), Rdiﬀ(f)≡/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/an}b∇acketle{tjα s,i/an}b∇acket∇i}htdiﬀ SOI /an}b∇acketle{tjα s,i/an}b∇acket∇i}htSV/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=8 3αSHεFτ /planckover2pi1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1+6πi 5D f v2a/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (29) where f=ω/2πis the frequency and va=ω/qis the (phase) velocity of acoustic waves, εF=/planckover2pi12k2 F/2mis the Fermi en- ergy, and/planckover2pi1has been recovered. This ratio is larger for higher frequency f, and for materials with stronger SOI. For CuIr, the spin Hall angle is 2 αSH=2.1±0.6%, indepen- dent of impurity concentration, which is dominated by the ex - trinsic, skew-scattering process.28)In the nearly free electron approximation with the Fermi wave number kF=1.36×1010 m−1, Fermi velocity vF=1.57×106m/s, eﬀective mass m∗=8.66×10−31kg,29)and resistivity ρimp=7.5µΩcm (for 3% Ir), we estimate the scattering time as τimp=5.30×10−15 s, and the diﬀusion constant as Dimp=4.35×10−3m2/s, due to impurities. With the speed of the Rayleigh type surface acou s- tic wave, va=3.80×103m/s, on a single crystal of LiNbO 3,30) we obtain RCuIr diﬀ(f)=1.51/radicalBig 1+(1.14×f)2, (30) where fis expressed in GHz. Therefore the di ﬀusion spin current/an}b∇acketle{tjα s,i/an}b∇acket∇i}htdiﬀ SOIvia SOI is comparable to, or even larger than, that from SVC in metals with strong SOI. It is thus expected that the total contribution /an}b∇acketle{tjα s,i/an}b∇acket∇i}htSOI, which includes both the diﬀusion spin current and the spin Hall current, can be larger than/an}b∇acketle{tjα s,i/an}b∇acket∇i}htSV. The magnitude itself is, however, small; /an}b∇acketle{tjx s,z/an}b∇acket∇i}ht= 1020∼1024m−2s−1=10∼105A/m2forτ−1 sf=0∼5× 1013s−1,δR=1Å, and f=3.8 GHz, as in the case of the SVC mechanism.14) To summarize, we studied the generation of spin current and spin accumulation by dynamical lattice distortion in me t- als with SOI at the impurities. We identiﬁed two routes to the spin-current generation, namely, the “spin Hall route” andthe “spin diﬀusion route.” In the former route, a charge cur- rent is ﬁrst induced by dynamical lattice distortion, which is then converted into a spin Hall current. In the latter route, a spin accumulation is ﬁrst induced from the vorticity of the lattice velocity ﬁeld, which then induces a di ﬀusion spin cur- rent. The result suggests that the spin accumulation (hence the associated diﬀusion spin current) generated via SOI is larger than that due to SVC for systems with strong SOI. Sim- ilar eﬀects are expected in systems with other types of SOI, such as Rashba, Weyl, etc., and such studies will be reported elsewhere. In this connection, we note that Xu et al. recently reported an experiment on the mechanical spin-current gen- eration (due to magnon-phonon coupling) in a system with Rashba SOI.31) Acknowledgment We would like to thank K. Kondou, J. Puebla, and M. Xu for the valuable and informative discussion, and J. Ieda, S. Maekawa, M. Matsuo, M. Mori, and K. Yamamoto for their valuable advice and com- ments. We also thank A. Yamakage, K. Nakazawa, T. Yamaguchi, Y . Imai, and J. Nakane for the daily discussions. This work is support ed by JSPS KAKENHI Grant Numbers 25400339, 15H05702 and 17H02929. TF i s sup- ported by a Program for Leading Graduate Schools: “Integrat ive Graduate Education and Research in Green Natural Sciences.” 1) S. Mizukami, Y . Ando, and T. Miyazaki, J. Magn. Magn. Mater .226- 230, 1640 (2001); Phys. Rev. B 66, 104413 (2002). 2) Y . Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. L ett.88, 117601 (2002); Phys. Rev. B 66, 224403 (2002). 3) E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Lett. 88, 182509 (2006). 4) J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back and T . Jungwirth, Rev. Mod. Phys. 87, 1213 (2015). 5) M. Johnson and R. H. Silsbee, Phys. Rev. Lett. 55, 1709 (1985). 6) K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh Nature 455, 778 (2008). 7) K. Uchida, H. Adachi, T. An, T. Ota, M. Toda, B. Hillebrands , S. Maekawa, and E. Saitoh, Nature Materials 10, 737 (2011). 8) K. Uchida, T. An, Y . Kajiwara, M. Toda, and E. Saitoh, Appl. Phys. Lett. 99, 212501 (2011). 9) H. Adachi, K. Uchida, E. Saitoh, and S. Maekawa, Rep. Prog. Phys. 76, 036501 (2013). 10) H. Adachi, S. Maekawa, Solid State Comm. 198, 22 (2014). 11) H. Keshtgar, M. Zareyan, and G. E. W. Bauer, Solid State Co mm. 198, 30 (2014). 12) P. A. Deymier, J. O. Vasseur, K. Runge, A. Manchon, and O. B ou-Matar, Phys. Rev. B 90, 224421 (2014). 13) M. Matsuo, J. Ieda, E. Saitoh, and S. Maekawa, Phys. Rev. L ett.106, 076601 (2011). 14) M. Matsuo, J. Ieda, E. Saitoh, and S. Maekawa, Appl. Phys. Lett. 98, 242501 (2011). 15) R. Takahashi, M. Matsuo, M. Ono, K. Harii, H. Chudo, S. Oka yasu, J. Ieda, S. Takahashi, S. Maekawa and E. Saitoh, Nature Physics 12, 52 (2016). 16) M. Matsuo, Y . Ohnuma, and S. Maekawa, Phys. Rev. B 96, 020401(R) (2017). 17) M. Matsuo, J. Ieda, K. Harii, E. Saitoh, and S. Maekawa, Ph ys. Rev. B 87, 180402(R) (2013). 18) J. Ieda, M. Matsuo, and S. Maekawa, Solid State Comm. 198, 52-56 (2014). 19) D. Kobayashi, T. Yoshikawa, M. Matsuo, R. Iguchi, S. Maek awa, E. Saitoh, and Y . Nozaki, Phys. Rev. Lett. 119, 077202 (2017). 20) T. Tsuneto, Phys. Rev. 121, 402 (1961). 21) J. Wang, K. Ma, K. Li, and H. Fan, Ann. Phys. 362, 327 (2015). 22) R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957). 23) (Supplemental Material) Some details of the calculatio n of Eqs. (14), (15), (19), (20) and (21), together with all diagrams, are pr ovided on- line. 24) We deﬁne the spin Hall angle as half of that of Ref.4)in order to be consistent with the deﬁnition of spin-current operators. 4J. Phys. Soc. Jpn. LETTERS 25) R. H. Parmenter, Phys. Rev. 89, 990 (1953). 26) These two routes were also identiﬁed in the electrical ge neration of spin current in, K. Hosono, A. Yamaguchi, Y . Nozaki, and G. Tatara , Phys. Rev. B 83, 144428 (2011). 27) While the same Greek letter is used for the external frequ ency (ω) and the vorticity (ωorωα), we hope no confusion will arise since the latter is denoted as a vector. 28) Y . Niimi, M. Morota, D.H. Wei, C. Deranlot, M. Basletic, A . Hamzic,A. Fert, and Y . Otani, Phys. Rev. Lett. 106, 126601 (2011). 29) N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart and Winston, New York, 1981). 30) S. Ashida, Bulletin of the Japan Institute of Metals 17, 1004 (1978) (in Japanese). 31) M. Xu, J. Puebla, F. Auvray, B. Rana, K. Kondou, and Y . Otan i, Phys. Rev. B 97, 180301(R) (2018). 5
1707.02379v2.Interaction_induced_exotic_vortex_states_in_an_optical_lattice_clock_with_spin_orbit_coupling.pdf	Interaction-induced exotic vortex states in an optical lattice clock with spin-orbit coupling Xiaofan Zhou,1, 2Jian-Song Pan,3, 4, 5Wei Yi,3, 4,Gang Chen,1, 2, †and Suotang Jia1, 2 1State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Laser spectroscopy, Shanxi University, Taiyuan 030006, China 2Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, Shanxi 030006, China 3Key Laboratory of Quantum Information, University of Science and Technology of China, CAS, Hefei, Anhui, 230026, China 4Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China 5Wilczek Quantum Center, School of Physics and Astronomy and T. D. Lee Institute, Shanghai Jiao Tong University, Shanghai 200240, China Motivated by a recent experiment [L. F. Livi, et al. , Phys. Rev. Lett. 117, 220401(2016)], we study the ground-state properties of interacting fermions in a one-dimensional optical lattice clock with spin-orbit coupling. As the electronic and the hyperne-spin states in the clock-state manifolds can be treated as eective sites along distinct synthetic dimensions, the system can be considered as multiple two-leg ladders with uniform magnetic ux penetrating the plaquettes of each ladder. As the inter-orbital spin-exchange interactions in the clock-state manifolds couple individual ladders together, we show that exotic interaction-induced vortex states emerge in the coupled-ladder system, which compete with existing phases of decoupled ladders and lead to a rich phase diagram. Adopting the density matrix renormalization group approach, we map out the phase diagram, and investigate in detail the currents and the density-density correlations of the various phases. Our results reveal the impact of interactions on spin-orbit coupled systems, and are particularly relevant to the on-going exploration of spin-orbit coupled optical lattice clocks. I. INTRODUCTION Spin-orbit coupling (SOC) plays a key role in solid- state topological materials such as topological insulators and quantum spin Hall systems [1{3]. The experimental realization of synthetic SOC in cold atomic gases opens up the avenue of simulating synthetic topological mat- ter on the versatile platform of cold atoms [4{18]. In most of the previous studies, synthetic SOC is typically implemented in alkali atoms using a two-photon Raman process, in which dierent spin states in the ground-state hyperne manifold of the atoms are coupled. Thus, as the atoms undergo Raman-assisted spin ips, their center- of-mass momenta also change due to the photon recoil. Alternatively, by considering the atomic spin states as discrete lattice sites, the SOC can also be mapped to ef- fective tunneling in the so-called synthetic dimension [19{ 22]. Such an interpretation has led to the realization of two-leg ladder models with synthetic magnetic ux, and to the subsequent experimental demonstration of chiral edge states using cold atoms [10, 23, 24]. For systems under the synthetic SOC generated by the Raman scheme, a key experimental diculty in reach- ing the desired many-body ground states is the heat- ing caused by high-lying excited states in the Raman process, whose single-photon detuning is limited by the ne-structure splitting [4{6]. This problem can be over- Electronic address: wyiz@ustc.edu.cn †Electronic address: chengang971@163.comcome either by choosing atomic species with large ne- structure splitting [25, 26], or by using alkaline-earth-like atoms [27], which feature long-lived excited states. In- deed, in two recent experiments, synthetic SOCs with signicantly reduced heating have been experimentally demonstrated by directly coupling the ground1S0(re- ferred to asjgi) and the metastable3P0(referred to asjei) clock-state manifolds of87Sr or173Yb lattice clocks [10, 11]. In Ref. [10], the electronic states are fur- ther mapped onto the eective lattice sites along a syn- thetic dimension, such that a two-leg ladder model with uniform magnetic ux is realized and the resulting chiral edge currents are probed. Similar models for bosonic sys- tems have been extensively investigated in the past [28{ 35]. When the ux is small, chiral edge currents emerge at the system boundary, where the currents along the two legs are opposite in direction. This is reminiscent of the Meissner eects of superconductivity [36]. When the ux becomes suciently large, the system undergoes a phase transition as the chiral edge currents are replaced by vortex lattices in the bulk, where currents exist on both the rungs and the edges of the ladder. Furthermore, when taking the hyperne spin states in the clock-state manifolds into account, one can map both the electronic and the spin degrees of freedom into distinct synthetic dimensions, such that the system in Ref. [10] can be extended to model multiple two-leg lad- ders with synthetic magnetic ux penetrating the pla- quettes of each ladder. Here, dierent electronic states label the two legs of each ladder, and dierent spin states label dierent ladders. It would then be interesting to study the ground state of the system as the parame-arXiv:1707.02379v2 [cond-mat.quant-gas] 5 Sep 20172 ters such as the ux or the interactions are tuned. In the clock-state manifolds of alkaline-earth-like atoms, the nuclear and the electronic degrees of freedom are sep- arated, and the short-range two-body interactions oc- cur either in the electronic spin-singlet channel ji= 1 2(jgeijegi) (j#"i+j"#i), or in the electronic spin- triplet channelj+i=1 2(jgei+jegi) (j#"ij"#i ) [37{ 39]. Here,j"iandj#ilabel dierent spin states in the jgiorjeihyperne manifolds. As reported in previ- ous studies, such an inter-orbital spin-exchange interac- tion would induce density-ordered states in either the spin or the charge channel, leading to spin-density wave (SDW), orbital-density order (ODW) or charge-density wave (CDW) phases [40{48]. More interestingly, these interactions would couple the otherwise independent lad- ders, which may induce new patterns of current ow in the system. In this work, adopting the concept of synthetic dimen- sions, we explicitly consider the hyperne spin states in the clock-state manifolds and map the system in Ref. [10] to multiple two-leg ladders (see Fig. 1). Using the density matrix renormalization group (DMRG) ap- proach [49, 50], we then numerically investigate the eect of interactions on the many-body ground-state properties such as the current ow and the density-density correla- tions. Our numerical results reveal a rich phase diagram, where the interactions drastically modify the Meissner and the vortex states in the non-interacting case. In par- ticular, we show the existence of an exotic interaction- induced vortex state, where spin currents emerge be- tween dierent ladders together with SDW in the sys- tem. As the interactions in the clock-state manifolds can be readily tuned by external magnetic eld through the orbital Feshbach resonance [51{53], or by transverse trap- ping frequencies through the connement-induced reso- nance [54, 55], our results have interesting implications for future experiments. The work is organized as follows. In Sec. II, we present the system setup and the mode Hamiltonian. We discuss the phase diagram of the non-interacting case in Sec. III. We then study in detail the impact of interactions on the Meissner state and the vortex state, respectively in Secs. IV and V. The detection and conclusion are given respectively in Secs. VI and VII. II. MODEL AND HAMILTONIAN We consider a similar setup as in the recent experiment on synthetic SOC in optical lattice clocks [10]. As shown in Fig. 1(a), a pair of counter-propagating laser with the \magic" wavelength L= 2=kLis used to generate a one-dimensional (1D) optical lattice potential VL(x) = Vxcos2(kLx), whereVxis the lattice depth. The synthetic SOC is implemented by an ultra-narrow -polarized clock laser with a wavelength C, which drives a single-photon transition between the clock states with the same nuclear (a) (b) VexΩe↓ g↓ g↑e↑λLλC θπ λLFIG. 1: (a) Schematics of the experimental setup. The ul- tracold alkaline-earth-like atoms are trapped in a 1D optical lattice, which is generated by a pair of counter-propagating lasers with the \magic" wavelength L, such that states in the clock-state manifolds1S0and3P0are subject to the same lat- tice potential. An ultranarrow -polarized clock laser with a wavelength Cdrives a single-photon transition between the clock-state manifolds. By introducing an angle between the clock laser and that generating the optical lattice, the pho- ton recoil momentum becomes kC= 2cos=Cand can be tuned experimentally. (b) Energy levels considering two nu- clear spin states j"iandj#iin each manifold. While states in the1S0and3P0manifolds are coherently coupled by the spin-conserving clock laser (the red curves), the interaction in the clock-state manifolds can couple distinct spin states in dierent orbitals (the green curves). spins. Due to the existence of the angle between the wave vector of the clock laser and the alignment of the 1D optical lattice [see Fig. 1(a)], the momentum transfer becomeskC= 2cos=C[10]. A key ingredient in this system is the inter-orbital spin-exchange interaction [37{ 39], as shown by the green curves in Fig. 1(b). The full Hamiltonian is written as ( ~= 1 hereafter) ^HT=^HL+^HC+^HI (1) with ^HL=X Z dx^ † (x) [r2 2m+VL(x)]^ (x);(2) ^HC= R 2X Z dx[^ † g(x)eikCx^ e(x) + H:c:];(3) ^HI=g 2Z dx[ † g" † e# † g# † e"] [ e# g" e" g#] +gZ dxh † g" † e" e" g"+ † g# † e# e# g#i ;(4) where=fg;egis the orbit index, =f";#gis the spin index, and † are the corresponding eld op- erators, Ris the Rabi frequency of the clock laser, g3 are the 1D interaction strengths [51, 55], and H :c:is the Hermitian conjugate. Note that in writing down Hamil- tonian (1), we only consider four nuclear spin states from thejgiandjeimanifolds. In principle, the other nu- clear spin states can be shifted away by imposing spin- dependent laser shifts [24]. When the 1D optical lattice is deep enough and R is not too large [56{58], we may take the single-band approximation and write down the corresponding tight- binding model ^HTB=tX <i;j>;^c† i^cj+ 2X j;(eij^c† jg^cje+H:c:) +UX j(^njg"^nje#+ ^njg#^nje") +U0X j^njg^nje +VexX j(^c† jg"^c† je#^cje"^cjg#+ H:c:); (5) where ^cj(^c† j) is the annihilation (creation) operator for atoms on the ith site of the orbital and the spin , ^nj= ^c† j^cj. The spin-conserving hopping rate t=R dxw(j)h r2 2m+VL(x)i w(j+1), withw(j)being the lowest-band Wannier function on the jth site of the lat- tice potential VL(x),. The spin- ipping hopping rate = RR dxw(j)eikCxw(j).=1 2kCL=Lcos=C is the synthetic magnetic ux per plaquette induced by the SOC, U=1 2(g++g)R dxw(j)w(j)w(j)w(j) andU0=gR dxw(j)w(j)w(j)w(j)are the inter-orbital density-density interaction strengths with the same and dierent nuclear spins, respectively. Vex= 1 2(gg+)R dxw(j)w(j)w(j)w(j)is the inter-orbital spin-exchange interaction strength. Hamiltonian (5) has the advantage that all parameters can be tuned indepen- dently. For example, tcan be controlled by the depth of the optical lattice potential, and can be controlled by the Rabi frequency and the angle of the clock laser, respectively, and fVex,U,U0gcan be tuned through the orbital Feshbach resonance [51{53] or the connement induced resonance [54, 55]. In the following, we take U0=Vex+U, which is dictated by the scattering param- eters of173Yb atoms [54, 55]. From the tight-binding Hamiltonian (5), it is clear that if we drop the interaction terms and map the elec- tronic () and the spin ( ) states onto eective lattice sites along two dierent synthetic dimensions, the non- interacting tight-binding model describes a pair of two- leg ladders. We label the two synthetic dimensions as the orbit- and the spin -directions, respectively, while the optical lattice lies along the x-direction. We may then denote the synthetic dimensions as SD and SD, respectively. As illustrated in Fig. 2, the pair of lad- ders each lie within the ( x;) plane with the legs of both ladders along the x-direction. The rung tunnel- ing in each ladder is facilitated by the SOC, which also induces uniform magnetic ux in each plaquette of the ladder. In the absence of interactions, the ladders are φ(b) φtSD αe↑ g↑e↓ g↓ …… j-1 j j+1 …… 2ieφj SD σ Real dimension x φ(a) SD αt 2ie φj e g …… j-1 j j+1 …… Real dimension xFIG. 2: (a) A two-leg synthetic ladder with a synthetic mag- netic ux=L=Ccosin each plaquette. The orbital states can be treated as an eective synthetic dimension de- noted as SD . The two ladders along the synthetic dimen- sion SDare identical and decoupled. (b) A pair of two-leg synthetic ladders (i.e., jg";e"iandjg#;e#i), with the same ux, are coupled by the inter-orbital spin-exchange interac- tion (green curves). In this lattice, there are two-direction synthetic dimensions, SD and SD. not coupled, as dierent spin states are independent on the single-body level. However, the inter-orbital spin- exchange interaction eectively couples the ladders to- gether [see Fig. 2(b)], which, as we will show later, induce inter-ladder currents along the -direction. An important property here is the current along the legs and the rungs of the ladder. Local and average cur- rents along the x-direction can be dene as [29, 30, 59], Jk j;=i ^c† j+1^cj^c† j^cj+1 ; (6) Jk =1 LX jJk j;: (7) Similarly, currents along the -direction can be dened as [30], J? j;=i(eij^c† je^cjgeij^c† jg^cje); (8) J? =1 LX jJ? j;; (9) Finally, we also dene currents along the -direction as J? j;=i(^c† j#^cj"^c† j"^cj#); (10) J? =1 LX jJ? j;: (11) In the following discussions, we adopt the DMRG formalism to calculate the ground state of the system, from which we characterize currents and density corre- lation functions. For the numerical calculation, we have considered length of chain Lup to 32 sites. We keep4 0.00 .51 .0-0.10.00.10 .00 .51 .00.000.010.020 .00 .51 .0012345 J φ/π J \|\|e /s61555 J\|\|g /s61555 J ⊥(b)( c) φ /π J⊥σ J⊥α Ω/tφ /πVortex IMeissner(a) FIG. 3: (a) Phase diagram in the ( ;) plane. The currents (b)Jkand (c)J?as functions of =. In all subgures, U=U0=Vex= 0 andn= 1, and (b) and (c) have the other parameter =t= 2. the maxstates m= 200 and achieve truncation errors of 1010. We mainly consider the case of half lling, i.e., n=N=(2L) = 1, where Lis the length of chain and N is the total number of atoms. III. PHASES AND CURRENTS IN THE NON-INTERACTING CASE We rst discuss the ground-state phases of the system in the absence of interactions. In this case, dierent spin states are decoupled, and we may identify a pair of two- leg ladders, as illustrated in Fig. 2. From our numerical calculations, we nd that only the Meissner and the vortex states appear in the ground-state phase diagram shown in Fig. 3(a). Typically, when the synthetic ux is small, the ground state is the so-called Meissner state, where the edge currents Jk gandJk e ow in opposite directions along the two legs of each ladder [10, 23, 29], as shown in Fig. 3(b). Upon increasing the ux above a critical value, the ground state features a vortex state with rung currents and vortex lattice structures in the bulk of each ladder, i.e., in the ( x;) plane. The existence of this so-called Vortex I state is conrmed in Fig. 3(c), where nonzero J? in the Vortex I state regime indicates a nite current along the rungs in the -direction for each ladder. The current J? remains zero in the non-interacting case, which is consistent with the picture of two independent ladders of dierent spins. We also note that in the non-interacting case, there are -8-4048-8-4048- 8-40480.00.51.0- 8-4048-0.20.00.2- 8-40480.00.10.2(a)M eissnerCDW U/tV /tVortex IIM eissnere x( d)( c)(b) U/t Order S C U/t J J\|\|e /s61555 J\|\|g /s61555 J ⊥U /t J⊥σ J⊥α FIG. 4: (a) Phase diagram in the ( U;V ex) plane. (b) The SDW orderSand the CDW order Cas well as the currents (c) Jk and (d)J?as functions of U=t. In all subgures, =t= 4, == 0:75, andn= 1, and (b)-(d) have the other parameter Vex=t=6. no density-ordered phases. IV. IMPACT OF INTERACTIONS ON THE MEISSNER STATE We now study the impact of interactions on the Meiss- ner state for =t= 4 and== 0:75. As the interactions are turned on, the system can undergo phase transitions into exotic vortex states or phases with density orders. We map out the phase diagram in the ( U;V ex) plane, while xing other parameters. As shown in Fig. 4(a), the phase diagram consists of three dierent phases: a simple Meissner state, a Meissner state with CDW, and an ex- otic vortex state with SDW, which we label as Vortex II state. The simple Meissner state resembles the Meissner state in the non-interacting case with chiral edge currents and no density-orders in the bulk. The CDW Meissner state features chiral edge currents as well as nite CDW density correlations in the bulk. The most interesting state here is the Vortex II state, which features nite SDW correlations as well as currents and vortex lattice structures in the ( x;) plane. The phase boundaries between these phases can be de- termined from the CDW and the SDW correlations, as well as from the currents' calculations. As illustrated in Fig. 4(b), for a xed Vex=t=6, the CDW order C=1 2LX j(1)jnj (12)5 08162432-0.10.00.10 8162432-0.20.00.2(b) J⊥j ,σs ite( j)(a) nj,↑ − nj,↓s ite( j) e↑ g↑e↓ g↓(c)SD αSD σ Real dimension x FIG. 5: (a) The currents J?and (b) density proles nj;"nj;# for dierent sites. (c) Sketch of currents of the Vortex II state. In (a) and (b), =t= 4,== 0:75,Vex=t=6,U=t= 2, andn= 1. has a nite value in the range 8< U=t < 0, while the SDW order S=1 LX j(1)j(nj;"nj;#) (13) has a nite value for 0 < U=t < 4:7. On the other hand, while the edge currents Jk are always nite and opposite in directions for dierent sites in the direction, the currents J? andJ? only exist within a range, as shown in Figs. 4(c) and 4(d). In particular, the non-vanishing J? indicates inter-ladder currents and vortices in the ( x;) plane. To further characterize the Vortex II state, in Figs. 5(a) and 5(b), we show the spatial distribution of the inter-ladder currents as well as the spin density. Apparently, the SDW and the vortex lattice structure in the ( x;) plane is due to the interplay of the inter-orbital spin-exchange interaction and the synthetic magnetic ux in the ( x;) plane, as shown in Fig. 5(c). V. IMPACT OF INTERACTIONS ON THE VORTEX STATE In this section, we study the impact of interactions on the vortex states. In Fig. 6, we map out the phase dia- gram in the ( U;V ex) plane for =t= 2 and== 0:25. In the absence of interactions, the system is in the vor- tex state. With interactions, the system can undergo phase transitions into various dierent phases. As shown in Fig. 6(a), besides the Vortex II state and the CDW Meissner state, several other exotic phases emerge in the -8-4048-8-4048- 8-40480.00.40.8- 8-4048-0.20.00.2- 8-40480.00.10.2exMeissnerODWVortex IODWV ortex IM eissnerCDW U/tV /tVortex IIMeissnere x V /t Order S Oe x(d)( c)(b)V /t J J\|\|e /s61555 J\|\|g /s61555(a)e x J⊥V /t J⊥σ J⊥α FIG. 6: (a) Phase diagram in the ( U;V ex) plane. (b) The SDW order Sand the ODW order Oas well as the currents (c)Jkand (d)J?as functions of Vex=t. In all subgures, =t= 2,== 0:25, andn= 1, and (b)-(d) have the other parameterU=t= 1. phase diagram. While the Vortex I resembles the vortex state in the absence of interactions, interesting phases with density correlations in the orbital channel appear, which can be further dierentiated by their currents ows as the ODW Meissner state and the ODW Vortex I state, where the vortex occurs in the ( x;) plane, together with density-wave orders in the orbital channel. Here the ODW order is dened as O=1 LX j(1)j(nj;gnj;e): (14) In Figs. 6(b)-6(d), we plot the currents and the den- sity wave orders as functions of Vex=tforU=t = 1. ForVex=t2[8;2:06], the SDW order Shas a nite value, the ODW order vanishes with O= 0, the cur- rentsJk >0 andJ? >0, the corresponding phase is the Vortex II state. When Vex=t2[2:06;1], the SDW order S= 0, the ODW order O= 0, the cur- rentsJk >0,J? >0, andJ? = 0, the corresponding phase is the Vortex I state. When Vex=t2[1;0:77], the SDW order S= 0, the ODW order O > 0, the currents Jk >0,J? >0, andJ? = 0, the corresponding phase is the ODW Vortex I state. When Vex=t2[0:77;8], the SDW order S= 0, the ODW order O= 0, the currents Jk >0,J? = 0, andJ? = 0, the corresponding phase is the Meissner state. According to Hamiltonian (5), there are three dif- ferent interaction parameters U,VexandU0. It is straightforward to see that the CDW order is favored for the attractive interactions ( U < 0,U0<0,Vex<0);6 and that the SDW order is favored for the attractive inter-orbital spin-preserving interaction ( U0<0). This is because the CDW order can decrease the interaction energy of all the attractive on-site interactions. While the SDW order mainly decreases the energy of the at- tractive inter-orbital spin-preserving interactions. These are consistent with previous studies on the SU(2) ladder systems [41, 43]. Note that for the phase diagrams in Figs. 4 and 6, we have xed U0=Vex+U. In the phase diagrams, the CDW state becomes unstable against the Vortex II state when Uis not suciently negative. Apparently, the competition between the CDW and the SDW orders is driven by the interactions associated withUandU0, which is eventually determined by the relative values of UandVex. On the other hand, the emergence of the ODW order in Fig. 6 can be understood as a conguration in which the repulsive energies are minimized in the case of U > 0 andU0>0. A subtlety here is the impact of the magnetic ux in the x-plane on the density ordered phases. From numerical analysis, we see that an increase of magnetic ux can lead to a stronger competition between dierent density-ordered phases, which gives rise to a richer phase diagram. In any case, we emphasize that the exotic Vortex II state with the SDW order is always robust in the negative U0 limit. VI. DETECTION Given the rich phase diagram discussed above, a natu- ral question is how to detect them experimentally. In gen- eral, the dierent phases of the system are characterized by their chiral edge currents as well as the density cor- relations in dierent channels such as CDW, ODW and SDW. Here, the density orders can be probed by state- selective measurements of density distributions. While the CDW order can be identied by oscillations of the total density distribution from site to site, the ODW and the SDW orders can be identied, respectively, by oscil- lations of the density distribution of a given orbital ( jgi orjei) or of a given spin ( j"iorj#i). For the detection of the Meissner and the vortex states, one can in princi- ple follow the approach in Ref. [30], where, by projecting the wave function into isolated double wells along each leg, the chiral currents can be calculated from the oscil- latory density dynamics in the double wells. The vortex state can be identied either from the variation of the chiral currents, as the maximum of the chiral currentsappear at the phase boundary between the Meissner and the vortex states [see Figs. 4(a) and 6(a)]. Alternatively, one should also identify the Vortex I and the Vortex II states, respectively, from the relative phase between the oscillations of dierent orbital and spin states. The rela- tive phase should be in the case of the Meissner state, and smaller than in the case of the vortex states [30]. VII. CONCLUSION We show that by implementing synthetic SOC in alkaline-earth-like atoms, one naturally realizes multiple two-leg ladders with uniform synthetic ux. As inter- actions couple dierent ladders together, the system features a rich phase diagram. In particular, we demon- strate the existence of an interaction-induced vortex state, which possesses SDW in the spin channel. The dierent phases can be experimentally detected based on their respective properties. As many phases in the phase diagram simultaneously feature density order and edge or bulk currents, a potential experimental challenge lies in the ecient detection of the various phases. This is particularly so for the exotic interaction-induced Vortex II state, whose SDW order as well as the bulk currents along SD require spin-selective detections. Nevertheless, with the state of the art quantum control over the clock states in alkaline-earth-like atoms, we expect that these challenges can be overcome with existing experimental techniques. Our results reveal the impact of interactions on spin-orbit coupled systems, and are particularly relevant to the on-going exploration of spin-orbit coupled optical lattice clocks. Acknowledgments This work is supported partly by the Na- tional Key R&D Program of China under Grants No. 2017YFA0304203 and No. 2016YFA0301700; the NKBRP under Grant No. 2013CB922000; the NSFC un- der Grants No. 60921091, No. 11374283, No. 11434007, No. 11422433, No. 11522545, and No. 11674200; \Strate- gic Priority Research Program(B)" of the Chinese Academy of Sciences under Grant No. XDB01030200; the PCSIRT under Grant No. IRT13076; the FANEDD under Grant No. 201316; SFSSSP; OYTPSP; and SSCC. J.-S. P. acknowledges support from National Postdoctoral Program for Innovative Talents of China under Grant No. BX201700156. [1] M. Z. Hasan and C. L. Kane, Topological insulators, Rev. Mod. Phys. 82, 3045 (2010). [2] X.-L. Qi and S.-C. Zhang, Topological insulators and su- perconductors, Rev. Mod. Phys. 83, 1057 (2011).[3] J. Alicea, New directions in the pursuit of Majorana fermions in solid state systems, Rep. Prog. Phys. 75, 076501 (2012). [4] Y.-J. Lin, K. Jim enez-Garc a, and I. B. Spielman, Spin-7 orbit-coupled Bose-Einstein condensates, Nature (Lon- don)471, 83 (2011). [5] P. Wang, Z.-Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H. Zhai, and J. Zhang, Spin-Orbit Coupled Degenerate Fermi Gases, Phys. Rev. Lett. 109, 095301 (2012). [6] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah, W. S. Bakr, and M.W. Zwierlein, Spin-Injection Spec- troscopy of a Spin-Orbit Coupled Fermi Gas, Phys. Rev. Lett. 109, 095302 (2012). [7] Z. Meng, L. Huang, P. Peng, D. Li, L. Chen, Y. Xu, C. Zhang, P. Wang, and J. Zhang, Experimental Ob- servation of a Topological Band Gap Opening in Ultra- cold Fermi Gases with Two-Dimensional Spin-Orbit Cou- pling, Phys. Rev. Lett. 117, 235304 (2016). [8] L. Huang, Z. Meng, P. Wang, P. Peng, S.-L. Zhang, L. Chen, D. Li, Q. Zhou, and J. Zhang, Experimental real- ization of two-dimensional synthetic spin{orbit coupling in ultracold Fermi gases, Nat. Phys. 12, 540 (2016). [9] Z. Wu, L. Zhang, W. Sun, X.-T. Xu, B.-Z. Wang, S.-C. Ji, Y. Deng, S. Chen, X.-J. Liu, and J.-W. Pan, Realization of two-dimensional spin-orbit coupling for Bose-Einstein condensates, Science 354, 83 (2016). [10] L. F. Livi, G. Cappellini, M. Diem, L. Franchi, C. Clivati, M. Frittelli, F. Levi, D. Calonico, J. Catani, M. Ingus- cio, and L. Fallani, Synthetic Dimensions and Spin-Orbit Coupling with an Optical Clock Transition, Phys. Rev. Lett. 117, 220401 (2016). [11] S. Kolkowitz, S. L. Bromley, T. Bothwell, M. L. Wall, G. E. Marti, A. P. Koller, X. Zhang, A. M. Rey, J. Ye, Spin{orbit-coupled fermions in an optical lattice clock, Nature (London) 542, 66 (2017). [12] V. Galitski and I. B. Spielman, Spin{orbit coupling in quantum gases, Nature (London) 494, 49 (2013). [13] N. Goldman, G. Juzeli unas, P. Ohberg, and I. B. Spiel- man, Light-induced gauge elds for ultracold atoms, Rep. Prog. Phys. 77, 126401 (2014). [14] X. Zhou, Y. Li, Z. Cai, and C. Wu, Unconventional states of bosons with the synthetic spin{orbit coupling, J. Phys. B: At. Mol. Opt. Phys. 46, 134001 (2014). [15] H. Zhai, Degenerate quantum gases with spin-orbit cou- pling: a review, Rep. Prog. Phys. 78, 026001 (2015). [16] W. Yi, W. Zhang, and X. Cui, Pairing super uidity in spin-orbit coupled ultracold Fermi gases, Sci. China: Phys. Mech. Astron. 58, 014201 (2015). [17] J. Zhang, H. Hu, X. J. Liu, and H. Pu, Fermi gases with synthetic spin-orbit coupling, Ann. Rev. Cold At. Mol. 2, 81 (2015). [18] Y. Xu and C. Zhang, Topological Fulde Ferrell super u- ids of a spin orbit coupled Fermi gas, Int. J. Mod. Phys. B29, 1530001 (2015). [19] O. Boada, A. Celi, J. I. Latorre, and M. Lewenstein, Quantum Simulation of an Extra Dimension, Phys. Rev. Lett. 108, 133001 (2012). [20] A. Celi, P. Massignan, J. Ruseckas, N. Goldman, I. B. Spielman, G. Juzeli unas, and M. Lewenstein, Synthetic Gauge Fields in Synthetic Dimensions, Phys. Rev. Lett. 112, 043001 (2014). [21] O. Boada, A. Celi, J. Rodr guez-Laguna, J. I. Latorre, and M. Lewenstein, Quantum simulation of non-trivial topology, New J. Phys. 17, 045007 (2015). [22] E. Anisimovas, M. Ra ci unas, C. Str ater, A. Eckardt, I. B. Spielman, and G. Juzeli unas, Semisynthetic zigzag opti- cal lattice for ultracold bosons, Phys. Rev. A 94, 063632(2016). [23] B. K. Stuhl, H.-I. Lu, L. M. Aycock, D. Genkina, and I. B. Spielman, Visualizing edge states with an atomic Bose gas in the quantum Hall regime, Science 349, 1514 (2015). [24] M. Mancini, G. Pagano, G. Cappellini, L. Livi, M. Rider, J. Catani, C. Sias, P. Zoller, M. Inguscio, M. Dalmonte, and L. Fallani, Observation of chiral edge states with neutral fermions in synthetic Hall ribbons, Science 349, 1510 (2015). [25] X. Cui, B. Lian, T.-L. Ho, B. L. Lev, and H. Zhai, Syn- thetic gauge eld with highly magnetic lanthanide atoms, Phys. Rev. A 88, 011601 (2013). [26] N. Q. Burdick, Y. Tang, and B. L. Lev, Long-lived spin- orbit-coupled degenerate dipolar Fermi gas, Phys. Rev. X6, 031022 (2016). [27] M. L. Wall, A. P. Koller, S. Li, X. Zhang, N. R. Cooper, J. Ye, and A. M. Rey, Synthetic Spin-Orbit Coupling in an Optical Lattice Clock, Phys. Rev. Lett. 116, 035301 (2016). [28] E. Orignac and T. Giamarchi, Meissner eect in a bosonic ladder, Phys. Rev. B 64, 144515 (2001). [29] A. Petrescu and K. L. Hur, Bosonic Mott Insulator with Meissner Currents, Phys. Rev. Lett. 111, 150601 (2013). [30] M. Atala, M. Aidelsburger, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch, Observation of chiral currents with ultracold atoms in bosonic ladders, Nat. Phys. 10, 588 (2014). [31] M. Piraud, F. Heidrich-Meisner, I. P. McCulloch, S. Greschner, T. Vekua, and U. Schollw ock, Vortex and Meissner phases of strongly interacting bosons on a two- leg ladder, Phys. Rev. B 91, 140406(R) (2015). [32] S. S. Natu, Bosons with long-range interactions on two- leg ladders in articial magnetic elds, Phys. Rev. A 92, 053623 (2015). [33] F. Kolley, M. Piraud, I. P. McCulloch, U. Schollw ock, and F Heidrich-Meisner, Strongly interacting bosons on a three-leg ladder in the presence of a homogeneous ux, New J. Phys. 17, 092001 (2015). [34] S. Greschner, M. Piraud, F. Heidrich-Meisner, I. P. Mc- Culloch, U. Schollw ock, and T. Vekua, Spontaneous In- crease of Magnetic Flux and Chiral-Current Reversal in Bosonic Ladders: Swimming against the Tide, Phys. Rev. Lett. 115, 190402 (2015). [35] S. Greschner, M. Piraud, F. Heidrich-Meisner, I. P. Mc- Culloch, U. Schollw ock, and T. Vekua, Symmetry-broken states in a system of interacting bosons on a two-leg lad- der with a uniform Abelian gauge eld, Phys. Rev. A 94, 063628 (2016). [36] A. K. Geim, S. V. Dubonos, J. G. S. Lok, M. Henini, and J. C. Maan, Paramagnetic Meissner eect in small superconductors, Nature (London) 396, 144 (1998). [37] X. Zhang, M. Bishof, S. L. Bromley, C. V. Kraus, M. S. Safronova, P. Zoller, A. M. Rey, and J. Ye, Spectroscopic observation of SU(N)-symmetric interactions in Sr orbital magnetism, Science 345, 1467 (2014). [38] G. Cappellini, M. Mancini, G. Pagano, P. Lombardi, L. Livi, M. Siciliani de Cumis, P. Cancio, M. Pizzocaro, D. Calonico, F. Levi, C. Sias, J. Catani, M. Inguscio, and L. Fallani, Direct Observation of Coherent Interorbital Spin-Exchange Dynamics, Phys. Rev. Lett. 113, 120402 (2014); ibid114, 239903 (2015). [39] F. Scazza, C. Hofrichter, M. H ofer, P. C. De Groot, I. Bloch, and S. F olling, Observation of two-orbital spin-8 exchange interactions with ultracold SU(N)-symmetric fermions, Nat. Phys. 10, 779 (2014); ibid11, 514 (2015). [40] K. Kobayashi, M. Okumura, Y. Ota, S. Yamada, and M. Machida, Nontrivial Haldane phase of an atomic two- component Fermi gas trapped in a 1D optical lattice, Phys. Rev. Lett. 109, 235302 (2012). [41] H. Nonne, M. Moliner, S. Capponi, P. Lecheminant, and K. Totsuka, Symmetry-protected topological phases of alkaline-earth cold fermionic atoms in one dimension, Eu- rophys. Lett. 102, 37008 (2013). [42] K. Duivenvoorden and T. Quella, Topological phases of spin chains, Phys. Rev. B 87, 125145 (2013). [43] V. Bois, S. Capponi, P. Lecheminant, M. Moliner, and K. Totsuka, Phase diagrams of one-dimensional half-lled two-orbital SU(N) cold fermion systems, Phys. Rev. B 91, 075121 (2015). [44] A. Roy and T. Quella, Chiral Haldane phases of SU(N) quantum spin chains in the adjoint representation, arXiv: 1512.05229. [45] V. Bois, P. Fromholz, and P. Lecheminant, One- dimensional two-orbital SU(N) ultracold fermionic quan- tum gases at incommensurate lling: A low-energy ap- proach, Phys. Rev. B 93, 134415 (2016). [46] S. Capponi, P. Lecheminant, and K. Totsuka, Phases of one-dimensional SU(N) cold atomic Fermi gases\|From molecular Luttinger liquids to topological phases, Ann. Phys. 367, 50 (2016). [47] X. Zhou, J.-S. Pan, Z.-X. Liu, W. Zhang, W. Yi, G. Chen, and S. Jia, Symmetry-protected topologi- cal states for interacting fermions in alkaline-earth-like atoms, arXiv: 1612.08880. [48] F. Iemini, L. Mazza, L. Fallani, P. Zoller, R. Fazio, and M. Dalmonte, Majorana Quasiparticles Protected by Z2 Angular Momentum Conservation, Phys. Rev. Lett. 118, 200404 (2017). [49] S. R. White, Density matrix formulation for quan- tum renormalization groups, Phys. Rev. Lett. 69, 2863(1992). [50] U. Schollw ok, The density-matrix renormalization group, Rev. Mod. Phys. 77, 259 (2005). [51] R. Zhang, Y. Cheng, H. Zhai, and P. Zhang, Orbital Feshbach Resonance in Alkali-Earth Atoms, Phys. Rev. Lett. 115, 135301 (2015). [52] M. H ofer, L. Riegger, F. Scazza, C. Hofrichter, D. R. Fernandes, M. M. Parish, J. Levinsen, I. Bloch, and S. F olling, Observation of an Orbital Interaction-Induced Feshbach Resonance in173Yb, Phys. Rev. Lett. 115, 265302 (2015). [53] G. Pagano, M. Mancini, G. Cappellini, L. Livi, C. Sias, J. Catani, M. Inguscio, and L. Fallani, Strongly Interacting Gas of Two-Electron Fermions at an Orbital Feshbach Resonance, Phys. Rev. Lett. 115, 265301 (2015). [54] M. Olshanii, Atomic Scattering in the Presence of an External Connement and a Gas of Impenetrable Bosons, Phys. Rev. Lett. 81, 938 (1998). [55] R. Zhang, D. Zhang, Y. Cheng, W. Chen, P. Zhang, and H. Zhai, Kondo eect in alkaline-earth-metal atomic gases with connement-induced resonances, Phys. Rev. A93, 043601 (2016). [56] J.-S. Pan, X.-J. Liu, W. Zhang, W. Yi, and G.-C. Guo, Topological Superradiant States in a Degenerate Fermi Gas, Phys. Rev. Lett. 115, 045303 (2015). [57] L. Zhou and X. Cui, Spin-orbit coupled ultracold gases in optical lattices: High-band physics and insuciency of tight-binding models, Phys. Rev. B 92, 140502(R) (2015). [58] J.-S. Pan, W. Zhang, W. Yi, and G.-C. Guo, Bose- Einstein condensate in an optical lattice with Raman- assisted two-dimensional spin-orbit coupling, Phys. Rev. A94, 043619 (2016). [59] S. Barbarino, L. Taddia, D. Rossini, L. Mazza, and R. Fazio, Magnetic crystals and helical liquids in alkaline- earth fermionic gases, Nat. Commun. 6, 8134 (2015).
1503.06872v2.Spin_Orbit_Torques_in_Two_Dimensional_Rashba_Ferromagnets.pdf	arXiv:1503.06872v2 [cond-mat.mtrl-sci] 13 Jul 2015Spin-Orbit Torques in Two-Dimensional Rashba Ferromagnet s A. Qaiumzadeh,1R.A. Duine,2and M. Titov1 1Radboud University, Institute for Molecules and Materials , 6525 AJ Nijmegen, The Netherlands 2Institute for Theoretical Physics and Centre for Extreme Ma tter and Emergent Phenomena, Utrecht University, 3584 CE Utrecht, The Netherlands (Dated: October 15, 2018) Magnetization dynamics in single-domain ferromagnets can be triggered by a charge current if the spin-orbit coupling is suﬃciently strong. We apply functio nal Keldysh theory to investigate spin- orbit torques in metallic two-dimensional Rashba ferromag nets in the presence of spin-dependent disorders. A reactive, anti-damping-like spin-orbit torq ue as well as a dissipative, ﬁeld-like torque is calculated microscopically, to leading order in the spin -orbit interaction strength. By calculating the ﬁrst vertex correction we show that the intrinsic anti-d amping-like torque vanishes unless the scattering rates are spin-dependent. PACS numbers: 72.15.Gd, 75.60.Jk, 75.70.Tj I. INTRODUCTION Spin-orbitronics1,2has attracted a lot of attention re- cently as a new subﬁeld of spintronics3,4in which the relativistic spin-orbit interaction (SOI) plays a central role. Spin-orbitronics includes generation and detec- tion of spin-polarized currents through the spin Hall eﬀect,5,6the induction of non-equilibrium spin accu- mulations in non-magnetic materials through the Edel- stein eﬀect,7,8the triggering of magnetization dynam- ics in single magnetic systems through spin-orbit torques (SOTs),9–11and magnonic charge pumping by means of inverseSOTs.12Spin-orbitronicsis believed to ultimately enable the faster and more eﬃcient ways of magnetiza- tion switching needed for high density data storage and informationprocessing,thereby providingnovelsolutions to address the essential challenges of spintronics. In this paper we investigate the microscopic origin of SOTs in a two-dimensional (2D) metallic ferromagnet with spin- orbit coupling. The magnetization dynamics in ferromagnets is gov- erned by the seminal Landau-Lifshitz-Gilbert (LLG) equation,13–15 ∂m ∂t=−γm×Heﬀ+αGm×∂m ∂t+T,(1) wheremis a unit vector along the magnetization di- rection\|m\|= 1,γis the gyromagnetic ratio, αGis the Gilbert damping constant and Heﬀis an eﬀective ﬁeld which includes the eﬀects of the external magnetic ﬁeld, exchange interactions, and dipole and anisotropy ﬁelds. The ﬁrst term on the right-hand side of Eq. (1) describes the precession of the magnetization vector maround the eﬀective ﬁeld, while the second term describes the relaxation of magnetization to its equilibrium orienta- tion. Furthermore, Tis a sum of diﬀerent magnetization torques not contained in the eﬀectieve ﬁeld or damping. The spin-polarized current-induced magnetization dy- namics in magnetic materials arises as a result of spin transfer torque (STTs).13–15It is well known that STTmay induce magnetization dynamics in spin-valve struc- tures, and that the exchange interaction between the spin-polarized current and local spins leads, e.g., to domain-wall motion. In uniformly magnetized single- domain systems the transfer of spin angular momen- tum from the spin-current density jsto a local magne- tization is modelled by two diﬀerent STT terms: i) an anti-damping-like (ADL) or Slonczewski in-plane torque T∝m×m×js, and ii) an out-of-plane ﬁeld-like (FL) torqueT∝m×js, which is typically negligible in con- ventional metallic spin valves. On the other hand, in ferromagnets with magnetic domains, in which spin tex- tures such as domain walls are necessarily present, the STT also includes reactive, T∝(js·∇)m, and dissipa- tive,T∝m×(js·∇)m, torques.13–15 Recently, it was demonstrated both theoretically and experimentally that the current-induced nonequilibrium spin polarization7,8in (anti-)ferromagnets with inver- sion asymmetry may exert a so-called SOT on local- ized spins and, consequently, may lead to a non-trivial magnetization dynamics.16–32Unlike STT, the SOT phe- nomenon does not require an injection of spin current or the presence of spatial inhomogeneities in the magneti- zation. The magnetization switching due to SOTs may be achieved with current pulses as short as ∼180ps, while the critical charge current density can be as low as ∼107Acm−2.29 Quite generally Rashba SOTs can be classiﬁed as ei- ther ADL or FL torques33. The ﬁrst theoretical and ex- perimental studies of SOT have demonstrated that the ADL SOT is proportional to the disorder strength and can always be regarded as a small correction to the FL SOT.16–23Ontheotherhand, insomerecentexperiments the opposite statement is made: the torques with ADL symmetry are more likely to be the main source of the observedmagnetization behavior.26–28,34–37These exper- iments are performed with ferromagnetic metals grown on top of a heavy metal with strong SOI and may, in principle, be explained by the spin Hall eﬀect which in- duces a spin-polarized current. This spin current, in turn, exerts a torque on the magnetic layer via the STT2 mechanism35,36,38so that the ADL symmetry term plays the major role in the eﬀect as discussed above. It is, however, a serious experimental challenge to dis- tinguish between SOT and spin-Hall STT in bilayers, since both torques have the same symmetry.35,36Very recently Kurebayashi et al.39conducted an experiment on the bulk of strained GaMnAs, which has an intrinsic crystalline asymmetry. In these experiments, the contri- bution of a possible spin-Hall-eﬀect STT was completely eliminated, while sizable ADL torques were nevertheless detected. This provides a strong argument in favour of the ADL-SOT nature of the observed torque. The au- thorsofRef. 39attribute this torque toan intrinsic Berry curvature, andestimate a scattering-independent, i.e. in- trinsic, ADL-SOT.39–41This intrinsic ADL-SOT has also been reported by van der Bijl and Duine.33 In this paper we calculate both FL- and ADL-SOTs in a 2D Rashba ferromagnetic metal microscopically by us- ing a functional Keldysh theory approach.42By calculat- ing the ﬁrst vertex correction we show that the intrinsic ADL-SOT vanishes unless the impurity scattering is spin dependent. The rest of this paper is organized as follows. Section II introduces the model and method. In Sec. III we calculate SOTs with and without vertex corrections. We conclude our work in Sec. IV. II. MODEL HAMILTONIAN AND METHOD We start with the 2D mean-ﬁeld Hamiltonian ( /planckover2pi1=c= 1), H[ψ†,ψ] =/integraldisplay d2rψ† r,t/bracketleftBig H0+Vimp+ˆj·At/bracketrightBig ψr,t.(2) whereψ†= (ψ∗ ↑,ψ∗ ↓) is the Grassman coherent state spinor. Here, H0is the 2D conducting ferromagnet Hamiltonian density in the presence of Rashba SOI,43 H0=p2 2me+αR(σ×ˆz)·p−1 2∆σ·nr,t−1 2∆Bσz,(3) wherepisthe2Dmomentumoperator, αRisthestrength of the SOI, ∆ and ∆ Bare the exchange energy and the Zeeman splitting due to an external ﬁeld in the z- direction, respectively, nr,tis an arbitrary unit vector that determines the quantizationaxis, and σis the three- dimensional vector of Pauli matrices. The vector potential At=Ee−iΩt/iΩ is included in Eq. (2) to model a dcelectric ﬁeld in the limit Ω →0. It is coupled to the current density operator, which is given by ˆj= (ie/2me)(← −∇−− →∇)−eαRσ×ˆz, whereeis the electron charge and meis the electron eﬀective mass. Finally, the impurity potential Vimpis of the form Vimp(r) =/parenleftbigg V↑0 0V↓/parenrightbigg/summationdisplay iδ(r−Ri), (4)whereV↑(↓)is the strength of spin-up (down) disorder, and the index ilabels the impurity centers Ri. More speciﬁcally, we restrict ourselves to the gaussian limit of the disorder potential. The impurity-averaged retarded Green’s function in the Born approximation is given by44–46 G+ k,ε=/parenleftBig g−1 ↓σ↑+g−1 ↑σ↓+αR(σykx−σxky)/parenrightBig−1 .(5) whereg−1 s=ε−εk+sM+iγs, fors=↑(+) or↓(−), σs= (σ0+sσz)/2,kandεarethewavevectorandenergy, respectively, εk=k2/2me, andM= (∆ + ∆ B)/2. We have also introduced the spin-dependent scattering rate γs=πνnimpV2 s, whereν0=me/2πis the density of states per spin for 2D electron gas, and nimpdenotes the impurity concentration. Here wehaveassumedthat both spin-orbit split bands are occupied, i.e. the Fermi energy is larger than magnetization splitting, εF>M. Following Ref. 42 we minimize the eﬀective action on the Keldysh contour with respect to quantum ﬂuctua- tions ofn. This procedure gives us directly the LLG equation which contains torque terms in linear response with respect to the external ﬁeld E. The eﬀective action is given by S=/integraltext CKdtLF(t), whereCKstands for the Keldysh contour and LF(t) =/integraltext d2r(ˆψ† r,ti∂ ∂tˆψr,t−H) is the mean-ﬁeld Lagrangian. We further assume that we are dealing with a ferro- magnetic metal which is uniformly magnetized in the z- direction. Thus, we can approximate the vector nas nr,t≃ δnx r,t δny r,t 1−1 2(δnx r,t)2−1 2(δny r,t)2) .(6) In order to derive the LLG equation with torque terms it is suﬃcient to expand the eﬀective action up to second order inδnand up to ﬁrst order in the vector poten- tial:Seﬀ=SSOT[O(δn),A] +Srest[O(δn2),A= 0]. A straightforward calculation gives SSOT=/integraldisplay CKdt/integraldisplay CKdt′/integraldisplay d2r/integraldisplay d2r′χa;r−r′;t,t′δna r′,t′,(7) whereχa(a={x,y}) is the response function, χa;r−r′;t,t′=i∆ 4/angbracketleftBig jr,tψ† r′,t′σaψr′,t′/angbracketrightBig ·At,(8) andj=ψ†ˆjψis the charge current density. Note that in the absence of SOI the term SSOTis 0 and only second order terms, Srest42, remain. The ﬁeld δncan be split into the physical magnetization ﬁeld δmand a quantum ﬂuctuation ﬁeld ξasδna r,t±=δma r,t±ξa r,t/2, where + corresponds to the upper and −to the lower branch of the Keldysh contour. At ﬁrst order with respect to the quantum component we obtain SSOT=/integraldisplay dt/integraldisplay dt′/integraldisplay d2r′/integraldisplay d2rχ− a;r−r′;t,t′ξa r′,t′,(9)3 (b) aV(a) bb tcctctccc ttc tAj Aj aVkk 1k1k 2k2k FIG. 1: Feynman diagrams related to the spin-torque re- sponse function Eq. (8): (a) undressed response function, and (b) the ﬁrst vertex correction. The solid line correspon ds toan electron propagator in theBorn approximation, the wig - gly line to the coupling to vector potential and current, and the dashed represents a spin ﬂuctuation. The vertical dotte d line describes the averaging over impurity positions. whereχ−is the advanced component of the correla- tor, and the sum over repeated indices ais assumed. The LLG equation is, then, derived by minimizing the eﬀective action with respect to quantum ﬂuctuations, δSeﬀ/δξ= 0. Thus, the transverse components of the LLG equation in the Fourier space are given by, F/bracketleftbiggδSrest δξa/bracketrightbigg q=0,ε+χ− a;q=0,ε=0= 0,(10) whereF[...] represents the Fourier transformation op- erator. The functional derivative in Eq. (10) gives the precession and Gilbert damping terms of the LLG equation,42while the second term describes the SOT. The dependence of Gilbert damping on SOI is second order inαR,33and we focus below on SOT which is of ﬁrst order in αR. The appearance of the zero-momentum response function χa;q=0,ε=0in the LLG equation shows that the SOT is ﬁnite even for spatially uniform magne- tization, in contrast to the (non-)adiabatic STT which is of the ﬁrst order in the gradient of magnetization. III. CALCULATION OF SOTS In what follows we evaluate the spin-torque response function of Eq. (8), shown diagrammatically in Fig. 1, to derive the SOT in the ballistic limit γs≪kBT, where kBTis the thermal energy. We calculate ﬁrst the bare (undressed) part of the response function, χ(0), depicted in Fig. 1a. The ﬁnal result for spin torque is, then, obtained by adding the ﬁrst vertex correction, χ(1), de- picted in Fig. 1b. Throughout the calculation we assume thatγs≪kBT≪αRkF≪M, wherekFis the Fermi wavevector. The condition αRkF≪Mis normally ful- ﬁlled in the metallic ferromagnets of interest. Whether or not the condition γs≪kBT≪αRkFis fulﬁlled de- pends strongly on the sample quality. The analysis ofspin torques in diﬀusive regime kBT≪γswill require calculation of the full vertex correction and will be done elsewhere. A. Undressed response function The spin-torque response function of Eq. (8) without vertex corrections is given by χ(0) a;t,t′=e∆ 4i/integraldisplayd2k (2π)2Tr[vkˇGk;t,t′σaˇGk;t′,t]·At.(11) wherevk=k/me−αRσ×ˆzis the velocity vector, and ˇGis the Green’s function on the Keldysh contour. From Eq. (11) we ﬁnd retarded and advanced components of the response function in the limit of zero frequency and momentum as χ(0)± a=e∆ 4ilim Ω→0/integraldisplayd2k (2π)2/integraldisplay/integraldisplay dωdω′fω′−fω Ω+ω′−ω±i0 ×1 ΩTr[(vk·E)Ak,ωσaAk,ω′], (12) whereAk,ω=i(G+ k,ω−G− k,ω)/2πis the spectral func- tion andfω= [e(ω−ǫF)/kBT+ 1]−1stands for the Fermi distribution function. In the limit of weak disorder, we can decompose the response function into two parts: the intrinsic part χin, which turns out not to depend on the scattering rate and describesinterbandtransitionsandtheextrinsicpart χex, which essentially depends on disorderand correspondsto intraband contributions. The intrinsic part corresponds to the principal value integration in Eq. (12), while the extrinsic part is given by the corresponding delta- function contribution. To leading order in αRwe ﬁnd χ(0)− in,a=eαR∆ 8Mν0Ea, (13a) χ(0)− ex,a=eαR∆ 8Mν0/bracketleftbiggεF−M γ↓−εF+M γ↑/bracketrightbigg (ˆz×E)a.(13b) The correspondingexpressionsforthe SOTsarethe ADL TADLand FLTFLcontributions, which do not take into account vertex corrections, T(0) ADL=−2eαRν0m×m×(ˆz×E), (14a) T(0) FL=−eαR∆ν0 M/bracketleftbiggεF+M γ↑−εF−M γ↓/bracketrightbigg m×(ˆz×E).(14b) Hence, we ﬁnd that the ADL SOT in the absence of ver- tex corrections has an intrinsic origin, i.e., is disorder- independent. B. Vertex correction Let us now turn to the calculation of the ﬁrst vertex correction to the spin-torque response function depicted4 in Fig. 1b. For the corresponding response function on the Keldysh contour we ﬁnd χ(1) a;t,t′=e∆ 4i/integraldisplaydk1 (2π)2/integraldisplaydk2 (2π)2/integraldisplay cKdt1/integraldisplay cKdt2Tr[At·vk1 ×ˇGk1;t,t1∝an}bracketle{tVimpˇGk2;t1,t′σaˇGk2;t′,t2Vimp∝an}bracketri}htˇGk1;t2,t].(15) The advanced component of χ(1)at zero energy and mo- mentum is, then, given by χ(1)− a=e∆ 4iηb/integraldisplayd2k1 (2π)2/integraldisplayd2k2 (2π)2/integraldisplay/integraldisplay dωdω′fω′−fω Ω+ω−ω′−i0 ×1 ΩTr/bracketleftbig E·vk1/parenleftbig G+ k1,ωσbAk2,ωσaG+ k2,ω′σbAk1,ω′ +G+ k1,ωσbAk2,ωσaAk2,ω′σbG− k1,ω′ +Ak1,ωσbG− k2,ωσaG+ k2,ω′σbAk1,ω′ +Ak1,ωσbG− k2,ωσaAk2,ω′σbG− k1,ω′/parenrightbig/bracketrightbig ,(16) where the summation over the index b={0,z}and the limit Ω→0 are assumed. We have also used the nota- tionsη0=nimp(V↑+V↓)2/4 andηz=nimp(V↑−V↓)2/4. Using the same approximationsas for the undressed part of the response function we obtain the intrinsic contribu- tion as χ(1)− in,a=−eαR∆ 8Mν0γ↑+γ↓ 2(γ↑γ↓)1 2Ea, (17) while the corresponding extrinsic contribution is of the second order in scattering rates and can be neglected. Thus, we obtain the FL and ADL torques in the limit γs≪αRkF≪Mto the leading order in the SOI as TFL=eαRν0∆ M/bracketleftbiggεF−M γ↓−εF+M γ↑/bracketrightbigg m×(ˆz×E),(18) TADL=/bracketleftbiggγ↑+γ↓ 2√γ↑γ↓−1/bracketrightbigg 2eαRν0m×(m×(ˆz×E)).(19) These expressions provide the main result of this paper. IV. CONCLUSIONS The SOTmechanismis basedonthe exchangeofangu- lar momentum between the crystal lattice and the local magnetization via spin-orbit coupling. Here, we foundthe FL- and ADL-SOTs microscopically, Eqs. (18) and (19). The FL-SOT originates from the Fermi surface contribution of the response function Eq. (8), while the ADL-SOT is acquires contributions from the entire bands. Our main result in Eq. (19) immediately shows thattheintrinsiccontributiontoADL-SOTiscompletely canceled in the presence of spin-independent scattering γ↑=γ↓. That is, the intrinsic component of the ADL SOT, which originates from virtual interbranch transi- tions, is canceled by the vertex correction when weak spin-independent impurity scattering is taken into ac- count. Our result, therefore, explicitly elucidates the interplay between intrinsic and extrinsic contributions to ADL SOT. This result resembles the suppression of both spin Hall conductivity in nonmagnetic metals and anomalous Hall conductivity in magnetic metals, in the presence of spin-independent disorder.44–48In these ef- fects the cancelation is model dependent, and occurs for parabolic band dispersion and linear-in-momentum SOI. We expect a similar scenario for intrinsic SOT. The existence of a Rashba eﬀect on the interface be- tween an ultrathin ferromagnet and a heavy metal is the subject of intense discussion. Our results show that the amplitudes of the FL and ADL SOTs can be of the same order of magnitude depending on the rel- ative strengths of the SOI, spin-dependent scattering rates, andexchangeinteraction. Ourresultsmayqualita- tively describe Co/Pt interfaces which are characterised by particularly large Rashba SOI of the magnitude of 1eV˚A.26–28,49Relating the strength of the Rashba cou- pling to the magnitude of the SOTs, however, would re- quire ab-initio modeling and additional experimental in- formation. Which of our results apply to more general models and band structures will be the subject of future investigation. ACKNOWLEDGEMENT We acknowledge Hiroshi Kohno and Dmitry Yudin for useful discussions. The work was supported by Dutch Science Foundation NWO/FOM 13PR3118 and by EU Network FP7-PEOPLE-2013-IRSES Grant No 612624 ”InterNoM”. R.D. isamemberofthe D-ITPconsortium, a program of the Netherlands Organisation for Scientiﬁc Research(NWO), which is funded by the Dutch Ministry of Education, Culture and Science (OCW). 1A. Manchon, Nature Phys. 10, 340 (2014). 2T. Kuschel and G. Reiss, Nat. Nanotechnol. 10, 22 (2015). 3J. Sinova and I. ˇZuti´ c, Nat. Mater. 11, 368 (2012). 4I.ˇZuti´ c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). 5J. Sinova, S.O. Valenzuela, J. Wunderlich, C.H. Back, T. Jungwirth, arXiv:1411.3249 (2014).6N. Nagaosa, J. Sinova, S. Onoda, A.H. MacDonald, and N.P. Ong, Rev. Mod. Phys. 82, 1539 (2010). 7V.M. Edelstein, Solid State Commun. 73, 233 (1990). 8J.-i. Inoue, G.E.W. Bauer, and L.W. Molenkamp, Phys. Rev. B67, 033104 (2003). 9P. Gambardella and I.M. Miron, Phil. Trans. R. Soc. A 369, 3175 (2011).5 10A. Brataas and K.M.D. Hals, Nat. Nanotechnol. 9, 86 (2014). 11K.M.D. Hals and A. Brataas, Phys. Rev. B 88, 085423 (2013). 12C. Ciccarelli, K.M.D. Hals, A. Irvine, V. Novak, Y. Tserkovnyak, H. Kurebayashi, A. Brataas, and A. Fer- guson, Nat. Nanotechnol. 10, 50 (2015). 13N. Locatelli, V. Cros and J. Grollier, Nat. Mater. 13, 11 (2013). 14A. Brataas, A.D. Kent, and H. Ohno, Nat. Mater. 11, 372 (2012). 15D.C. Ralphand M.D. Stiles, J. Magn, Magn, Matter. 320, 1190 (2008). 16B.A. Bernevig and O. Vafek, Phys. Rev. B 72, 033203 (2005). 17A. Manchon and S. Zhang, Phys. Rev. B 78, 212405 (2008);79, 094422 (2009). 18A. Matos-Abiague and R.L. Rodr´ ıguez-Su´ arez, Phys. Rev. B80, 094424 (2009). 19I. Garate, A.H. MacDonald, Phys. Rev. B 80, 134403 (2009). 20X. Wang and A. Manchon, Phys. Rev. Lett. 108, 117201 (2012). 21K.-W. Kim, S.-M. Seo, J. Ryu, K.-J. Lee, and H.-W. Lee, Phys. Rev. B 85, 180404(R) (2012). 22D.A.Pesin andA.H.MacDonald, Phys.Rev.B 86, 014416 (2012). 23A. Chernyshov, M. Overby, X. Liu, J.K. Furdyna, Y. Lyanda-Geller, and L.P. Rokhinson, Nature Phys. 5, 656 (2009). 24I.M. Miron, T. Moore, H. Szambolics, L.D. Buda- Prejbeanu, S. Auﬀret, B. Rodmacq, S. Pizzini, J. Vogel, M. Bonﬁm, A. Schuhl, and G. Gaudin, Nat. Mater. 10, 419 (2011). 25A.V. Khvalkovskiy, V. Cros, D. Apalkov, V. Nikitin, M. Krounbi, K. A. Zvezdin, A. Anane, J. Grollier, A. Fert, Phys. Rev. B 87, 020402(R) (2013). 26I.M. Miron, G. Gaudin, S. Auﬀret, B. Rodmacq, A. Schuhl,S. Pizzini, J. Vogel, and P. Gambardella, Nat. Mater.9, 230 (2010). 27I.M. Miron, K. Garello, G. Gaudin, P.-J. Zermatten, M.V. Costache, S. Auﬀret, S. Bandiera, B. Rodmacq, A. Schuhl, and P. Gambardella, Nature 476, 189 (2011). 28K. Garello, I.M. Miron, C.O. Avci, F.Freimuth, Y. Mokrousov, S. Bl¨ ugel, S. Auﬀret, O. Boulle, G. Gaudin, and P. Gambardella, Nat. Nanotechnol. 8, 587 (2013). 29K. Garello, C.O. Avci, I.M. Miron, M. Baumgartner, A. Ghosh, S. Auﬀret, O. Boulle, G. Gaudin, and P. Gam- bardella, Appl. Phys. Lett. 105, 212402 (2014). 30J.ˇZelezn´ y, H. Gao, K. V´ yborn´ y, J. Zemen, J. Maˇ sek, A. Manchon, J. Wunderlich, J. Sinova, and T. Jungwirth,Phys. Rev. Lett. 113, 157201 (2014). 31P. Wadley, B. Howells, J. ˇZelezn´ y, C. Andrews, V. Hills, R.P. Campion, V. Nov´ ak, F. Freimuth, Y. Mokrousov, A.W. Rushforth, K.W. Edmonds, B.L. Gallagher, T. Jungwirth, arXiv: 1503.03765 (2015). 32J. Linder, Phys. Rev. B 87, 054434 (2013). 33E. van der Bijl and R. A. Duine, Phys. Rev. B 86, 094406 (2012). 34L. Liu, C.-F. Pai, Y. Li, H.W. Tseng, D.C. Ralph, and R.A. Buhrman, Science 336, 555 (2012). 35J. Kim, J. Sinha, M. Hayashi, M. Yamanouchi, S. Fukami, T. Suzuki, S. Mitani, and H. Ohno, Nat. Mater. 12, 240 (2013). 36X. Fan, H. Celik, J. Wu, C. Ni, K.-J. Lee, V.O. Lorenz, and J.Q. Xiao, Nat. Commun. 5, 3042 (2014). 37G. Yu, P. Upadhyaya, Y. Fan, J.G. Alzate, W. Jiang, K.L. Wong, S. Takei, S.A. Bender, L.-T. Chang, Y. Jiang, M. Lang, J. Tang, Y. Wang, Y. Tserkovnyak, P.K. Amiri, and K.L. Wang, Nat. Nanotechnol. 9, 548 (2014); I. M. Miron, ibid.9, 502 (2014). 38P.M. Haney, H.-W. Lee, K.-J. Lee, A. Manchon, and M.D. Stiles, Phys. Rev. B 87, 174411 (2013). 39H. Kurebayashi, J. Sinova, D. Fang, A.C. Irvine, T.D. Skinner, J. Wunderlich, V. Nov´ ak, R.P. Campion, B.L. Gallagher, E.K. Vehstedt, L.P. Zˆ arbo, K. V´ yborn´ y, A.J. Ferguson, and T. Jungwirthn, Nat. Nanotechnol. 9, 211 (2014). 40H. Li, H. Gao, L.P. Zˆ arbo, K. V´ yborn´ y, X. Wang, I. Garate, F. Doˇ gan, A. ˇCejchan, J. Sinova, T. Jungwirth, A. Manchon, Phys. Rev. B 91, 134402 (2015). 41F. Freimuth, S. Bl¨ ugel, and Y. Mokrousov, Phys. Rev. B, 90, 174423 (2014). 42R.A. Duine, A.S. N´ u˜ nez, J. Sinova, and A.H. MacDonald, Phys. Rev. B 75, 214420 (2007). 43D. Bercioux and P. Lucignano, arXiv: 1502.00570 (2015). 44T. Kato, Y. Ishikawa, H. Itoh, and J.-i. Inoue, New J. Phys.9, 350 (2007). 45C.P. Moca and D.C. Marinescu, New J. Phys. 9, 343 (2007). 46T.S. Nunner, N.A. Sinitsyn, M.F. Borunda, V.K. Dugaev, A.A. Kovalev, Ar. Abanov, C. Timm, T. Jungwirth, J.-i. Inoue, A.H. MacDonald, and J. Sinova, Phys. Rev. B 76, 235312 (2007). 47J.-i. Inoue, T. Kato, Y. Ishikawa, H. Itoh, G.E.W. Bauer, and L.W. Molenkamp, Phys. Rev.Lett. 97, 046604 (2006). 48A. Sakai, PhD Thesis (Osake University, 2014); A. Sakai and H. Kohno (unpublished). 49J.-H. Park, C.H. Kim, H.-W. Lee, and J.H. Han, Phys. Rev. B87, 041301(R) (2013).
2103.04960v1.Spin_orbital_model_for_fullerides.pdf	Spin-orbital model for fullerides Ryuta Iwazaki and Shintaro Hoshino Department of Physics, Saitama University, Shimo-Okubo, Saitama 338-8570, Japan (Dated: March 9, 2021) The multiorbital Hubbard model in the strong coupling limit is analyzed for the eectively antifer- romagnetic Hund's coupling relevant to fulleride superconductors with three orbitals per molecule. The localized spin-orbital model describes the thermodynamics of the half-lled (three-electron) state with total spin-1/2, composed of singlon and doublon placed on the two of three orbitals. The model is solved using the mean-eld approximation and magnetic and electric ordered states are claried through the temperature dependences of the order parameters. Combining the model with the band structure from ab initio calculation, we also semi-quantitavely analyze the realistic model and the corresponding physical quantities. In the A15-structure fulleride model, there is an antiferromagnetic ordered state, and subsequently the two orbital ordered state appears at lower temperatures. It is argued that the origin of these orbital orders is related to the Thpoint group symmetry. As for the fcc-fulleride model, the time-reversal broken orbital ordered state is identied. Whereas the spin degeneracy remains in our treatment for the geometrically frustrated lattice, it is expected to be lifted by some magnetic ordering or quantum uctuations, but not by the spin-orbital coupling which is eectively zero for fullerides in the strong-coupling regime. I. INTRODUCTION Strongly correlated electron systems with multiple or- bital degrees of freedom show a variety of intriguing phenomena, and are realized in a wide range of materi- als such as iron-pnictides, heavy-electron materials, and molecular-based organic materials. The alkali-doped ful- lerides are also the typical cases where the strong corre- lation eects with multiorbitals are relevant. This mate- rial has been attracting attention recent years for a lot of experimental ndings. The superconductivity with the high transition temperature 40K is one of the charac- teristic feature [1{8]. While the mechanism is identied as the electron-phonon interaction [9{11], the supercon- ducting dome in the temperature-pressure phase diagram is found to be located near the Mott insulator and an- tiferromagnetic phase, featuring the typical behaviors of the strongly correlated superconductors [7, 12{14]. In the Mott insulating phase, the localized electrons form a low- spin state and the imbalance of the occupancy in orbitals lead to the deformation of the fullerene molecule because of the coupling between electrons and anisotropic molec- ular distortions (Jahn-Teller phonon). Interestingly, such behavior can also be seen in the metallic phase near the Mott insulator but is absent far away from it [15, 16]. This anomalous behavior is called the Jahn-Teller metal where the multiorbital degrees of freedom play an im- portant role. The fullerides are also crystallized on the substrate and the characteristic asymmetry between elec- tron and hole doping is identied [17, 18]. Furthermore, a possible superconducting state has been discussed un- der the excitation by light above the transition temper- ature [19, 20]. Thus, the fulleride materials have been providing the intriguing phenomena up until recently. The alkali-doped fullerides are the systems with triply degenerate t1umolecular orbitals which resembles atomic p-electrons in nature. There, the Hund's coupling, which is usually acting ferromagnetically on the elec-trons located at the dierent orbitals, is eectively an- tiferromagnetic due to the coupling to the anisotropic molecular vibrations [10, 21, 22] and is crucial for the low-temperature physics. The multiorbital Hubbard model with the antiferromagnetic Hund's coupling has been studied theoretically, and the various phase dia- grams are claried using the dynamical mean-eld the- ory suitable for the description of the electronically or- dered states [21{29]. The Jahn-Teller metal has been interpreted as the spontaneous orbital selective Mott state [26, 30] which is an unconventional type of orbital order. The orbital asymmetric feature has also been re- ported in two-dimensional fullerides by using the many- variable variational Monte Carlo method [31]. With the antiferromagnetic Hund's coupling, one of the intra-molecular interaction, pair hopping, plays an important role: it activates the dynamics of the double occupancy in an orbital (doublon). In order to clarify the characters of the existing fulleride materials in detail, we focus our attention on the Mott insulating phase, where the doublon physics can be tackled with reasonable com- putational cost even in the realistic situation. As is well known, for a single-orbital case, the electronic behav- iors in the strong coupling regime are determined by the Heisenberg model of localized electrons. The extension of the Heisenberg model to the multiorbital system is known as the Kugel-Khomskii model which has been de- rived for the ferromagnetic Hund's coupling [32, 33] and describes the degrees of freedom of the spin and orbital. The spin-orbital models have been applied to the egor t2gorbital system [34{37]. On the other hand, the ful- lerides have antiferromagnetic Hund's coupling, so that their strongly correlated eective model diers from the usual Kugel-Khomskii model. While the localized model with antiferromagnetic Hund's coupling have been con- structed for a density-density type interaction [28], here we deal with more complicated but realistic situations. In this paper, we develop the localized spin-orbitalarXiv:2103.04960v1 [cond-mat.str-el] 8 Mar 20212 model for the system with antiferromagnetic Hund's cou- pling. We analyze both the symmetric model and the re- alistic model for fullerides, the former of which is easier to interpret the results and is useful as a reference. By using the mean eld theory, for the spherical model on a bipartite lattice, we obtain the staggered magnetic or- dered state, and also the uniform orbital ordered state at lower temperature regime. This orbital ordered state is not characterised by the ordinary orbital moment but by the doublon's orbital moment. In the A15 fulleride eec- tive model, which is bipartite lattice, we reveal that there are two kinds of orbital ordered states below the antifer- romagnetic transition temperature. The obtained orbital ordered states are interpreted as related to an eective recovery of the four-fold symmetry at low temperatures in theThpoint group. We also analyze the geometri- cally frustrated fcc fulleride model seeking for a spatially uniform ordered state. We reveal that the fcc model has the time-reversal symmetry broken orbital ordered state, where the spin ordered state is absent since the spin-orbit coupling on the fullerene molecule is eectively zero. This paper is organized as follows. We discuss the con- struction of strongly correlated eective models and the theoretical method in Sec. II. In Sec. III, we show numeri- cal results for the model with isotropic hopping (spherical model introduced in Sec. III A). Section IV provides nu- merical results for the spin-orbital model combined with A15 and fcc fulleride band structure. We summarize the results in Sec. V. II. CONSTRUCTION OF MODELS A. Three orbital Hubbard model in strong-coupling limit Let us begin with the three-orbital Hubbard model H=Ht+HU; (1) Ht=X i6=j; ; 0;t 0 ijcy i; ;cj; 0;; (2) HU=U 2X i; ;;0cy i; ;cy i; ;0ci; ;0ci; ; +U0 2X i; 6= 0;;0cy i; ;cy i; 0;0ci; 0;0ci; ; +J 2X i; 6= 0;;0 cy i; ;cy i; 0;0ci; ;0ci; 0; +cy i; ;cy i; ;0ci; 0;0ci; ; ;(3) whereci; ; (cy i; ;) is an annihilation (creation) opera- tor at siteiof fullerenes with the t1umolecular orbital index =x;y;z and spin=";#. We deal with the Hilbert space with a xed number of electrons. We as- sume the condition U0=U2Jfor the local interaction part in the following discussion, which is valid for the TJOHMPOEPVCMPO<latexit sha1_base64="anJOgOHXlpTDDY0YhXC+bSAHO3s=">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</latexit>1p2 <latexit sha1_base64="anJOgOHXlpTDDY0YhXC+bSAHO3s=">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</latexit>1p2 <latexit sha1_base64="anJOgOHXlpTDDY0YhXC+bSAHO3s=">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</latexit>1p2<latexit sha1_base64="z7ibayWUbkOPH61LaKgFQQM9KO8=">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</latexit>\|,ii <latexit sha1_base64="UH9Wfb7wXhWadywsL8z8ujLLONA=">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</latexit>\|x,"ii <latexit sha1_base64="3XX2eVu+Rh9TaIY01XXkIv79q1c=">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</latexit>\|y,"ii <latexit sha1_base64="W0kxyk2VOCVIbX0jCAEyqs20SEk=">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</latexit>\|z,"ii<latexit sha1_base64="Z2pgJTBhiNumLVAFTWJsBALRuPM=">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</latexit>n=3<latexit sha1_base64="izRmVshPXfaqvICjoqW8/FO1zhM=">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</latexit>n=1FIG. 1. Schematic pictures for the ground state wave func- tionsj ;="iiof the local Hamiltonian for n= 3 andn= 1. spherical limit. In this paper, we consider a strong cou- pling regime (HUHt). When we develop the eective model in this limit, the presence of the Hund's coupling Jmakes theoretical treatment complicated since it real- izes quantum-mechanically superposed local wave func- tions. Especially for the negative (antiferromagnetic) J relevant to fullerides, the pair hopping plays an impor- tant role which creates the dynamics of doubly occupied electrons at an orbital (doublon). As shown in the follow- ing, in order to diminish the diculty, we use a symbolic expression without elaborating each intermediate process explicitly. In order to apply the perturbation theory from the strong coupling limit, we rst consider the ground state of the unperturbed Hamiltonian HU. Alkali-doped ful- lerides with half-lled situation (three electrons per t1u orbital) have six-fold degenerate ground states written as j ;ii=1p 2cy i; ;X 06= by i; 0j0i; (4) where we have dened an orbital-dependent doublon- creation operator as by i; =cy i; ;#cy i; ;": (5) The vacuum has been expressed as j0i. These states are uniquely characterized by the spin and orbital of the elec- tron at the singly occupied orbital, which is called `sin- glon' to make contrast against doublons. The schematic picture of the three-electron state j ;="iiis illustrated in Fig. 1. Using the above Hamiltonian, the second-order eec- tive Hamiltonian is written as He=PHt1 HUQHtP; (6)3 wherePis a projection operator to a model space de- scribed by Eq. (4) as P=X i; ;j ;iiih ;j; (7) andQ= 1P. We have used [P;HU] = 0. The energy is measured from the ground state of HU. The size of our model space is 6NwhereN=P i1 is the number of lattice sites. The strategy for obtaining the concrete form of the ef- fective Hamiltonian is to consider the two-site problem. We rst prepare the 212212matrix expressions for the annihilation and creation operators for two-site problem (12 =P i; ;1), and then dene all of the matrix ex- pressions given in Eq. (6). Performing multiplications of such matrices, we obtain the two-site eective Hamilto- nian in the form of the 6262matrix. We expand the above eective hamiltonian by following local operators O idened as O i=X ; 0X ;0j ;ii 0 0ih 0;0j; (8) in the model Hilbert space. =0;x;y;zis Pauli matrix 0= 1 0 0 1 ; x= 0 1 1 0 ; y= 0i i 0 ; z= 1 0 01 ; (9) which represents the degrees of freedom of the spin. An- other matrix =0;;8is given by 0=r 2 30 @1 0 0 0 1 0 0 0 11 A; 1=0 @01 0 1 0 0 0 0 01 A; 2=0 @0i 0 i 0 0 0 0 01 A; 3=0 @1 0 0 0 1 0 0 0 01 A; 4=0 @0 01 0 0 0 1 0 01 A; 5=0 @0 0 i 0 0 0 i 0 01 A; 6=0 @0 0 0 0 01 01 01 A; 7=0 @0 0 0 0 0i 0 i 01 A; 8=r 1 30 @1 0 0 0 1 0 0 021 A; (10) where these matrices are slightly dierent from ordinary denition of the Gell-Mann matrices to make them suit- able forp-electron systems. We note that the above local operators satisfy the orthonormal relation Trh O iO00 ji = 4ij00: (11)Thus, the set of operators O iis regarded as a basis set of the extended Hilbert space (Liouville space). In contrast, the statesj ;iiare the basis in the six-component model Hilbert space. Extending the two-site problem to the full lattice, we obtain the eective Hamiltonian in the strong coupling limit He=X i;jX ;0X ;0I;00 ijO iO00 j: (12) This model is to be analyzed in the rest of this paper. We also comment on the orbital moments in the re- stricted Hilbert space. In terms of the original Hubbard model, the local orbital moment is dened by LiX ; 0;cy i; ;` 0ci; 0;; (13) where the 33 matrices are given by `x=7,`y=5, and`z=2. This angular momentum operator is, how- ever, zero for the restricted Hilbert space: PLiP=0: (14) This anomalous disappearance of the angular momentum is due to the composite nature of the ground state [26] and is very dierent from a singly occupied state. Then the active orbital degrees of freedom are not of the origi- nal electrons but of the three-electron composite involv- ing doublons. This feature also aects the spin-orbit cou- pling which takes the form HSO=1 2SOX iX ; 0X ;0cy i; ;` 00ci; 0;0;(15) in the language of the original multiorbital Hubbard model. The spin-orbit coupling for 2 p-electron in car- bon atom is nearly 2meV, and because of the extended nature of the fullerene molecular the spin-orbit coupling SOfort1uorbitals is one-hundred times smaller than the atomic value ( SO20eV) [38]. Furthermore, for the restricted Hilbert space of n= 3 states, the eect of the spin-orbit coupling enters only through the second-order perturbation contribution as H(2) SO=PHSO1 HUQHSOP (16) =1 2SOX iX ; 0X ;0j ;ii` 00ih 0;0j;(17) where SO=112 SO 20JforJ < 0. Using the values for the antiferromagnetic coupling J 0:03eV for fullerdies [39], we obtain SO1neV which is so tiny. Hence we can safely neglect the spin-orbit coupling in fullerides. It is convenient to recognize that the above three elec- tron state is similar to the singly occupied state jn= 1; ;ii=cy i; ;j0i; (18)4 which is the eigenstate with ni=P ;cy i; ;ci; ; = 1 regardless of the sign of J(see the right column of Fig. 1). In our paper, the number of electrons is xed at each site andniis sometimes simply written as n. In Eq. (18), we explicitly write ` n= 1', and if it is dropped, the state representsn= 3 state dened in Eq. (4). The ground state forn= 3 is obtained by lling the empty orbital in n= 1 state by the doublons as in Eq. (4). We will consider the n= 1 case for reference to il- luminate the characteristics of n= 3 relevant to ful- lerides. When we deal with the second-order eective Hamiltonian for the n= 1 states, we just replace j ;ii byjn= 1; ;iidened in Eq. (18). We note that, in this case, the angular momentum does not vanish as distinct from then= 3 multiplet. For the usual ferromagnetic Hund's coupling ( J >0), the system corresponds to the spin-orbital model considered for the t2gorbitals [37]. B. Mean eld approximations In this paper, we utilize the mean eld approximation (MFA) for the obtained eective Hamiltonian. We apply the external eld for convenience and the full Hamilto- nian is written as He=1 2X i;j~OT i^Iij~OjX i~HT i~Oi (19) X i;j ~HT iij^11 2~MT i ^Iij+^IT ji ~Oj 1 2X i;j~MT i^Iij~MjHMF; (20) where the hat and arrow symbols represent the matrix and vector, respectively, with respect to the intra-site degrees of freedom ( ;). The vector ~Oiis the operator for the order parameter at site i, whose matrix represen- tation is given in Eq. (8). Namely, it is a column vector having 35 components, each of which is a 6 6 matrix where the identity is eliminated. The statistical aver- age~Mi=h~Oiiis the order parameter. In this paper, the coupling constant ^Iijconnects only nearest-neighbor (NN) sites for the spherical model (Sec. III), and NN and next-nearest-neighbor (NNN) site for A15 and fcc fulleride model (Sec. IV). In the following of this section, we concentrate on the bipartite lattice such as A15 struc- ture. Then we introduce two kinds of AB-sublattice to describe staggered orders. For non-bipartite lattice (i.e. fcc), on the other hand, we consider only the uniform so- lution and the similar formula can easily be obtained by regarding the two sublattices as identical.The mean-eld Hamiltonian is then rewritten as HMF=X " ~HT 1 2X 2NN~MT ^I;0+^IT 0; 1 2X 2NNN~MT ^I;0+^IT 0;#N=2X i2~Oi 1 2N 2X "X 2NN~MT ^I;0~M+X 2NNN~MT ^I;0~M# ; (21) where= A;B is the sub-lattice index and is a com- plementary component of , i.e., A = B and B = A.N is the number of site. The number of 2NN isz, 8 or 12 respectively for the spherical, A15 or fcc model. As for2NNN, both the A15 case (and fcc) has six sites. We have used the fact that NN-connected sites belong to the dierent sub-lattices and the NNN-connected sites belongs to the same sub-lattice. Since the coupling con- stants are dependent only on the direction of the vector connecting two sites, we write the interaction parameter as^I;0, where the index 0 represents the site which we focus on. For the bipartite lattice, we introduce the uniform and staggered moments as ~Mu ~Ms =1p 2^1^1 ^1^1~MA ~MB : (22) This expression is useful in analyzing the mean-eld so- lutions shown later. Now we explain the method of numerical calculation. The solutions are obtained by renewing the order param- eters iteratively using the self-consistent equation. The free energy and the self-consistent equation are given by F=TlnZ; (23) ~M=@F @~H; (24) whereZ= Tr eHMFis the partition function made of the mean-eld Hamltonian. For the derivation of the self- consistent equation, the parameters ~Hand~Mmust be regarded as independent variables. The system with the present eective Hamiltonian has 35 kinds of order parameters per site, and there may exist several solutions which take the same free energy as they are connected by symmetries. In the next sections, we show the simplest form of the order parameters among those energetically degenerate solutions. C. Response functions In this subsection, we consider the response function to the weak static eld. We expand the mean-eld Hamil-5 tonian up to rst order of the eld HMF=H(0)+H(1)+O H2 ; (25) H(0)=X i;j ~M(0) iT^Iij~Oj; (26) H(1)=X i;j ~HT iij ~M(1) iT^Iij ~Oj;(27) where the superscript represents the perturbative or- der of the eld and we have neglected the constant term. When we dene the eective eld as~~Hi=~HiP j^IT ji~M(1) jand treatH(1)as perturbation, we obtain the following linear response relation ~M(1) i=X j^(0) ij~~Hj=X j^ij~Hj; (28) where ^is the full susceptibility for the bare external eld ~Hi. According to linear response theory, the zeroth-order susceptibility is obtained by ^(0) ij=Z1=T 0dD T~Oi~OT j()E 0~M(0) i ~M(0) jT ; (29) whereis an imaginary time and Tis imaginary time ordering operator. The Heisenberg picture in an imagi- nary time is expressed as ~Oi() = eH(0)~OieH(0): (30) hi0represents the statistical average with H(0). The susceptibility matrix ^ (0) ijhas only intra-site component since each site is independent under MFA. Substituting the concrete expression to the eective eld in Eq. (28), we obtain X j" ij^1 +X k^(0) ik^IT kj# ~M(1) j=X j^(0) ij~Hj: (31) Then, taking matrix inverse of the left hand side and combining it with Eq. (28), we obtain the susceptibil- ity matrix ^ij. For a bipartite lattice, we introduce the uniform and staggered susceptibilities by ^u=1 NX i;j^ij; (32) ^s=1 NX i;jsisj^ij; (33) wheresi= +1 fori2A andsi=1 fori2B. This quantity will be shown in the next section. Although we focus on the static response functions in this paper, the above argument can easily be generalized for the dy- namical susceptibility which captures the magnetic and electric dynamics of the localized model.From the view point of Landau theory, we can also discuss the stability of the solution based on the suscep- tibilities. We write down the Landau free energy with an order parameter up to second order as FL=1 2X i;j~MT i^aij~MjX i~HT i~Mi; (34) where ^aijis a coecient of the quadratic term. Note that, here,~Mis dened as the deviation from its equilibrium point. Then we obtain the following equation of states: X j^aij~Mj=~Hi: (35) Comparing the linear response function, we nd that the Hessian matrix is identical to the inverse susceptibility: @2FL @~Mi@~Mj= ^aij= (^1)ij: (36) We can consider the necessary and sucient condition for the stable solution. Let "nbe then-th eigenvalue of the matrix ^aij. Each energy corresponds to the eigenenergy of the excitation modes. We must have the condition "n0; (37) for alln, if the system is thermodynamically stable. If "n= 0 is obtained, it indicates the presence of the Nambu-Goldstone mode. With use of Eq. (36), in the actual calculations, we obtain "nby diagonalizing the inverse susceptibility matrix. III. NUMERICAL RESULTS FOR SPHERICAL MODELS In the following of this paper, we will encounter the successive phase transitions with decreasing temperature. There, we denote each transition temperature as Tc1> Tc2>. If there is only one transition temperature is identied, we use Tcto denote it. Note that we use the same symbol for the transition temperatures in dierent models. A. Spherical spin-orbital model First we consider the model in the spherical limit. Namely, we assume the hopping matrix given in Eq. (2) as ^tij=0 @t0 0 0t0 0 0t1 A; (38)6 for a bipartite lattice with the coordination number z. Using the spin-orbital operator O idened in the previ- ous section, we obtain the spherical model as He=X hijih ISSiSj+ILLiLj+IQX Q iQ j +IRX X R; iR; j+ITX X T; iT; j+I0i ; (39) where the sum with hijiis taken over the pairs of the NN sites. The superscript ; (=x;y;z ) and (=x2y2;z2;xy;yz;zx ) are the indices for the polyno- mials, which represents the component of the spin, rank 1 orbital and rank 2 orbital, respectively. We have rewrit- ten the operators in accordance with their symmetries as S i=1 2O0 i; (40) Lx i=1 2O70 i; Ly i=1 2O50 i; Lz i=1 2O20 i; (41) Qx2y2 i =1 2O30 i; Qz2 i=1 2O80 i; Qxy i=1 2O10 i; Qyz i=1 2O60 i; Qzx i=1 2O40 i; (42) Rx; i=1 2O7 i; Ry; i=1 2O5 i; Rz; i=1 2O2 i; (43) Tx2y2; i =1 2O3 i; Tz2; i=1 2O8 i; Txy; i=1 2O1 i; Tyz; i=1 2O6 i; Tzx; i=1 2O4 i:(44) The physical meaning of each order parameter now be- comes clearer with this notation. We call S ia magnetic spin (MS or S),L ia magnetic orbital (MO or L),Q i a electric orbital (EO or Q),R; ia electric spin-orbital (ESO orR) andT; ia magnetic spin-orbital (MSO or T) moments. I0represents energy gain by the second order perturbation process. Obviously, Eq. (39) satises SU(2)SO(3) symmetry in spin-orbital space. We will show the numerical results of the n= 1 and n= 3 spherical models under MFA, both of which have the six states per site in the model space as discussed in Sec. II A. We beforehand introduce the following notation with regard to the coupling constants dened in Eq. (39) as I=X nAnt2 En; (45) for=S;L;Q;R;T; 0, where Enrepresents all pos- sible excitation energies. Its energy corresponds to the denominator of Eq. (6). The coecient Ais summarized in the tables in the following subsections (see Sec. III B or Sec. III C). Before we show the mean-eld results, we discuss the ground state wave function for the two-site problem. Us- ing the single site state dened in Eq. (4) or (18), weTABLE I. Coecients Adened in Eq. (45) for n= 1 spher- ical model. The ground state energy is zero. We add the details for the intermediate state in the main text. EnU3J UJ U + 2J IS2 10=3 2=3 IL 35=3 2=3 IQ 31=32=3 IR 1 5=32=3 IT 1 1=3 2=3 I0610=32=3 obtain the two-site (i.e., sites at iandj) ground state as jgsi=X i;iX j;jC ii; jjj i;iiij j;jij: (46) The explicit form of the matrix Cis written as ^C=0 (iy): (47) This shows that the ground-state wave function is spin- singlet and symmetric on the orbital. This is valid for all the spherical cases considered in this section. For an innite lattice, as in the single-orbital Hubbard model, the inter-site spin-singlet state may favor the antiferro- magnetic state in the ground state for a bipartite lattice. B.n= 1model First of all, we consider the results for the n= 1 model. Although the results are not relevant to the alkali- doped fullerides, the knowledge is useful in interpreting the more complicated model for the spherical n= 3 model (Sec. III C), the realistic A15- (Sec. IV A) and fcc- structure fullerides (Sec. IV B). 1. Coupling constant We begin with the analysis of the intermediate states relevant to the second-order perturbation theory. We show the coecients Adened in Eq. (45) in Table I. We have the three kinds of excited states, whose energy is de- termined by the local Coulomb interaction. For En= U3J, the intermediate states are nine-fold degener- ate spin-triplet states, as expressed, e.g., by cy i;y;"cy i;x;"j0i and1p 2 cy i;y;#cy i;x;"+cy i;y;"cy i;x;# j0i. For En=UJ, the intermediate states are the inter-orbital spin-singlet states such as1p 2 cy i;y;#cy i;x;"cy i;y;"cy i;x;# j0i, and the intra-orbital spin-singlet states with anti-bonding or- bitals written asp 2 3 2by i;zby i;xby i;y j0i. These two kinds of states take the same energy since there is the spherically symmetric condition U0=U2J. For En=U+ 2J, there is only one intermediate state,7 0.40.20.00.2 J/U6420246I⇠/E0ni=1 IS IL IQ IR IT FIG. 2. Hund's coupling ratio J=U dependence of the cou- pling constants for n= 1 spherical model. The vertical axis is normalized by E0=t2=U. which is intra-orbital spin singlet and bonding state writ- ten as1p 3 by i;x+by i;y+by i;z j0i. We show the Hund's coupling dependence of the cou- pling constants in Fig. 2. The perturbation theory is jus- tied for1=2<J=U < 1=3 where the ground states are written in the form of Eq. (18). Taking J= 0, the cou- pling constants become identical. This re ects that the system has SU(6) symmetry and the degrees of freedom of the spin and orbital are equivalent in the absence of Hund's coupling. The largest coupling constant is ISfor the antiferromagnetic case ( J <0) andIQfor the ferro- magnetic Hund's coupling ( J >0). This shows that the system tends to be antiferromagnetic (AFM) or antiferro- orbital (AFO) order depending on the sign of the Hund's coupling. This is understood from the intermediate state. In the case of J > 0, which is relevant to the usual t2g-orbitald-electron systems with n= 1 per atom, the energetically favorable intermediate two-electron state is inter-orbital spin triplet. To realize this intermediate state, the initial state needs to occupy parallel spin con- guration with dierent orbitals such as cy i;x;"cy j;y;"j0i. Therefore, the orbital order should be dominant for J >0 as a leading-order ordering instability. If we take J=U&0:2,IStakes a ferromagnetic coupling constant, which favors parallel spins at two sites. As forJ <0, on the other hand, the intermediate state tends to be intra-orbital spin singlet and bonding state. The corresponding initial state must be antiparallel spin with the same orbital such as cy i;x;"cy j;x;#j0i. Thus, the magnetic order should be dominant for J <0. 0.00.51.01.52.02.53.0T/E0420Internal/Free energy densityni=1, J/U=0.1 UFUSULUQURUT(b) 0.00.51.01.52.02.53.0T/E01.00.50.00.51.0Mu,sni=1, J/U=0.1SzsQz2uTz2,zs <latexit sha1_base64="X52I/2YrE97X48H0XCX0tZyVuPU=">AAAJDHiclVa7btRAFL0hPEx4JIEGiSZiFQRNNIsQIKoopIAG5Z1I2dXK9k6MWY9t2d6F4OwPUCA6BFQgUSBaOkSFhPgBinwCogwSFBQc3/UmIfEjsSX7zp17ztw5M3Ntw3fsMBJic+DI4NFjx09oJ4dOnT5zdnhk9NxS6LUDUy6anuMFK4YeSsd25WJkR45c8QOpK8ORy0brTtK/3JFBaHvuQrTuy7rSLddes009gmtpoRGb1W5jpCImBF9j+41qalQovWa80cE/VKMmeWRSmxRJcimC7ZBOIe5VqpIgH746xfAFsGzul9SlIWDbiJKI0OFt4WmhtZp6XbQTzpDRjxHhAemBpUtjNC6+i/diS3wTH8QP8TeXLWaWJJt1vI0eVvqN4acX5n+XohTeET3YQRUgDETnzyrpD2C3YEeFcYr1WUNUALUsjF4Un6gVIfoWq2RDNZ89iX5mL+/Okxdb87fnxuPL4q34Cd3eiE3xBcq5nV/mu1k595rZXWAescqK5+1iZWP4wzTzLq9nnX39DK+AvwLvGHw72V5N881jVAWMNbSyOGuILmZ1ClhXMznrpXlmzV1lzl4dav5ZCqgcDdShVFCZOqhMJdQeLfJZdXCGOYwbOYwbJZkG8Otg6aZ8vZMWwIppjitEMd7mlpWJv8dnsRifnCQ3E10+9n1WxGGVk7bBuBrXrL6/mGF6z8ghYgJ4pkuRLa5iMd6NjNynGF/M0IR2a7iz5t48kO7tTKx9IM17yA5/D5J2CFxS190SrMVr2rP0tK8YsbO7EstHnmWIcHs/hbyPy8dob++BNlfmALeHyLIV6KOaHO0eAmltI5Pdn3wDgtLK2B/NYa+16wucfzr7mGA3Bv8I1b1/BPuNpWsT1RsTYvZ6ZXIq/VvQ6CJdQsWs0k2apLs0Q4sY9SE9p5f0SnumfdQ+aZ97oUcGUsx5+u/Svv4Dq0vd9g==</latexit>Tc1 <latexit sha1_base64="XlJmCfpEjjRXMdHUfDX7W/sBXjY=">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</latexit>Tc2 0.00.51.01.52.02.53.0T/E001234C/Nni=1, J/U=0.1(d) 0.00.51.01.52.02.53.0T/E00.00.51.01.5S/Nni=1, J/U=0.1 <latexit sha1_base64="iJITrNFThqM9eAjvFM/a/M71vKQ=">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</latexit>ln 6<latexit sha1_base64="8y6Wk8tRec2BIDiGctZQySs6+4Q=">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</latexit>ln 3(c)(a) 0.8350.8401.01.1FIG. 3. Temperature dependence of (a) the order parameter, (b) the decomposed internal energy and total free energy den- sity, (c) entropy and (d) specic heat for n= 1;J=U =0:1 bipartite spherical model. The inset in (c) is enlarged plot aroundTc2. The energy unit of these plots are E0=t2=U. 2. Mean-eld solutions for antiferromagnetic Hund's coupling (J <0) Let us turn our attention to the numerical results using MFA in the spherical model. We take the NN coordina- tion number z= 6 in the numerical calculation by assum- ing a simple cubic lattice in three dimensions. Figure 3 shows the temperature dependence of the physical quan- tities in the bipartite lattice model at J=U =0:1 (an- tiferromagnetic Hund's coupling). We take E0t2=U as the unit of energy. The uniform and staggered order parameters are shown in Fig. 3(a), where the antiferro- magnetic spin (AF- S) order appears rst with decreasing temperature from the high-temperature limit. This cor- responds to the largest coupling constant ISin Fig. 2. At lower temperatures, the ferro (F)-orbital Qmoment ofz2 type appears together with the AF- T(MSO) moments. In order to clarify which is the primary order parameter of the second phase transition at Tc2, we show in Fig. 3(b) the internal energy and free energy per site, where the internal energy is decomposed into each contribution as US=IShSAihSBi; (48) UL=ILhLAihLBi; (49) UQ=IQX hQ AihQ Bi; (50) UR=IRX X hR; AihR; Bi; (51) UT=ITX X hT; AihT; Bi: (52) The total internal energy is given by U=P Ufor =S;L;Q;R;T , where the energy is measured from I0. We see from Fig. 3(b) that the energy UTis gained belowTc2butUQis not. Hence, the AF- Tshould be the8 0123T/E001020301/⌘µ;⌘µu[arb.units]ni=1, J/U=0.1 0123T/E001020301/⌘µ;⌘µs[arb.units]ni=1, J/U=0.1(a)(b) 0.040.020.000.020.040.050.000.05 SxSySzLxLyLzQx2y2Qz2QxyQyzQzxRx,xRy,xRz,xRx,yRy,yRz,yRx,zRy,zRz,zTx2y2,xTz2,xTxy,xTyz,xTzx,xTx2y2,yTz2,yTxy,yTyz,yTzx,yTx2y2,zTz2,zTxy,zTyz,zTzx,z FIG. 4. Temperature dependence of the inverse of (a) uni- form and (b) staggered component of the diagonal suscepti- bilities. The energy unit is E0=t2=U. primary order parameter and F- Qis just induced by the combination of AF- Splus AF-Tmoments. The results are consistent with the magnitude relation IT>IQseen in Fig. 2, where the larger energy gain is obtained from T-moment than the energy loss from Q. Figure 3(c) shows the temperature dependence of the entropy, where all the entropy is released in the ground state. With increasing temperature, the entropy shows a kink atT=E 0'0:84, at which the value of the entropy is close to ln 3 meaning that the orbital degeneracy is lifted below this transition temperature. The inset of (c) shows the magnied picture of the entropy near Tc2, indicating the rst-order transition. The specic heat C=@U=@T is also shown in Fig. 3(d). There are two discontinuity corresponding to the spin and orbital orders. Next we show in Fig. 4 the inverse of the diagonal susceptibilities ; u (uniform) and ; s (staggered) which are dened in Eqs. (32) and (33). First, we observe that the susceptibilities shown here are all positive, in- dicating a stable solution. The AF- Ssusceptibility of x;y;z type diverges at T=E 0'2:3 signaling the on- set of the antiferromagnetic order. Below this transition temperature, the longitudinal zcomponent is decreased while the perpendicular x;ycomponents remain diver- gent. This behavior indicates the presence of the Gold- stone mode, where the excitations are induced by rotat- ing thezcomponent into xy-plane, as in the standard Heisenberg model. Inside this magnetic phase, the or- bital (F-Q) and spin-orbital (AF- T) susceptibility, which arez2type in orbital part, continue to grow and tend to diverge at lower transition point ( Tc2). As shown in Fig. 4(a), the `perpendicular' components, i.e. F- Qyz, F- Qzx, remain divergent below Tc2, indicating the presence of the Goldstone mode even for the orbital order in the spherical model. Namely, because of the symmetry of the spin-orbital space, the energetically equivalent solutions exist and are obtained by rotating the order parameters. Next we discuss the ground state wave function, which includes the information of order parameter at zero tem- perature limit. As is evident from the zero entropy at T= 0, we have the non-degenerate ground state. In the present case, the ground state wave function is very 01234T/E06420Internal/Free energy densityni=1, J/U=0.1 UFUSULUQURUT01234T/E01.00.50.00.51.0Mu,sni=1, J/U=0.1SzuQz2uQz2sTz2,zuTz2,zs <latexit sha1_base64="X52I/2YrE97X48H0XCX0tZyVuPU=">AAAJDHiclVa7btRAFL0hPEx4JIEGiSZiFQRNNIsQIKoopIAG5Z1I2dXK9k6MWY9t2d6F4OwPUCA6BFQgUSBaOkSFhPgBinwCogwSFBQc3/UmIfEjsSX7zp17ztw5M3Ntw3fsMBJic+DI4NFjx09oJ4dOnT5zdnhk9NxS6LUDUy6anuMFK4YeSsd25WJkR45c8QOpK8ORy0brTtK/3JFBaHvuQrTuy7rSLddes009gmtpoRGb1W5jpCImBF9j+41qalQovWa80cE/VKMmeWRSmxRJcimC7ZBOIe5VqpIgH746xfAFsGzul9SlIWDbiJKI0OFt4WmhtZp6XbQTzpDRjxHhAemBpUtjNC6+i/diS3wTH8QP8TeXLWaWJJt1vI0eVvqN4acX5n+XohTeET3YQRUgDETnzyrpD2C3YEeFcYr1WUNUALUsjF4Un6gVIfoWq2RDNZ89iX5mL+/Okxdb87fnxuPL4q34Cd3eiE3xBcq5nV/mu1k595rZXWAescqK5+1iZWP4wzTzLq9nnX39DK+AvwLvGHw72V5N881jVAWMNbSyOGuILmZ1ClhXMznrpXlmzV1lzl4dav5ZCqgcDdShVFCZOqhMJdQeLfJZdXCGOYwbOYwbJZkG8Otg6aZ8vZMWwIppjitEMd7mlpWJv8dnsRifnCQ3E10+9n1WxGGVk7bBuBrXrL6/mGF6z8ghYgJ4pkuRLa5iMd6NjNynGF/M0IR2a7iz5t48kO7tTKx9IM17yA5/D5J2CFxS190SrMVr2rP0tK8YsbO7EstHnmWIcHs/hbyPy8dob++BNlfmALeHyLIV6KOaHO0eAmltI5Pdn3wDgtLK2B/NYa+16wucfzr7mGA3Bv8I1b1/BPuNpWsT1RsTYvZ6ZXIq/VvQ6CJdQsWs0k2apLs0Q4sY9SE9p5f0SnumfdQ+aZ97oUcGUsx5+u/Svv4Dq0vd9g==</latexit>Tc1 <latexit sha1_base64="XlJmCfpEjjRXMdHUfDX7W/sBXjY=">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</latexit>Tc2(a)(b) 0.000.250.500.751.001.25C/N01234T/E00.00.51.01.5S/Nni=1, J/U=0.1SSASBC <latexit sha1_base64="iJITrNFThqM9eAjvFM/a/M71vKQ=">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</latexit>ln 6 <latexit sha1_base64="46zSoqOY4v2X8E4dXP5823PRQJc=">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</latexit>ln 2 <latexit sha1_base64="KOUbV8JEvMfIBzQnLzcUE4PwzHM=">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</latexit>lnp2(c) 01234T/E001020304050eigenvalues [arb.units](d)FIG. 5. Temperature dependence of (a) the order parame- ter, (b) the decomposed internal energy and total free energy density, (c, left axis) the single site entropy, (c, right axis) the specic heat and (d) the eigenvalues of the Hessian matrix ^ a for bipartite spherical model with n= 1;J=U = 0:1. simple and is given using Eq. (4) by j Ai=jn= 1;z;#iA; (53) j Bi=jn= 1;z;"iB; (54) for each sublattice. This corresponds to the staggered spin ordered and uniform orbital ordered state, as is con- sistent with Fig. 3(a). More specically, we can construct the order parameters from the direct product of the wave functions. In the present case, we obtain at sublattice as j ih j=1p 6Sz 1p 3Qz2 1p 3Tz2;z +1 6;(55) where the operators are dened in Eqs. (40){(44). The upper (lower) sign is chosen for = A (= B). The quantities that appear in the right-hand side are identical to the order parameters shown in Fig. 3(a). 3. Mean-eld solutions for ferromagnetic Hund's coupling (J >0) We show the results for the J=U = 0:1 case, where the model is now relevant to materials with d-electrons, to make contrast with behaviors of the systems with anti- ferromagnetic Hund's coupling. Figure 5(a) shows the temperature evolution of the order parameters. As seen in Fig. 2, the largest coupling constant is IQwhich is an- tiferro (IQ<0), and therefore the AF- Qorder ofz2-type appears at the highest transition temperature ( Tc1). The F-Qorder of the same z2-type is simultaneously induced. The rise of the order parameters near the transition tem- perature behaves as pTc1Tfor AF-QandTc1T for F-Q. Hence the AF- Qis the primary order. From the symmetry argument, it can be shown that the F- Qorder9 arises from AF- Qorder since the coupling term in the Landau free energy has the form Qz2 u(Qz2 s)2. The exis- tence of such third-order term can be understood if one considers the symmetry in the plane of Qz2-Qx2y2[26]. At lower temperatures, the magnetic F- Sorder appears, whereT-moments of Tz2;z-type are also nite. From the internal-energy analysis shown in Fig. 5(b), the relevant ordering at Tc2is induced from the interaction ITwhile ISis energetically unfavorable. Thus, comparing with theJ=U =0:1 case, the roles of magnetic order and electric (orbital) order are switched. This switching of the magnetic and orbital ordered states depending on the sign of Jhas also been reported in the two orbital model [25]. We next show the temperature dependence of the en- tropy and specic heat in Fig. 5(c), where we have dened the sublattice-dependent entropy (Shannon entropy) by S=X np nlnp n; (56) wherep nis the probability for the n-th state as calculated from the local partition function Z=P nexp(E n) =P np nZ. Since the entropy at zero temperature is zero at A sublattice and is nite at B sublattice, the two sub- lattices are inequivalent and are not simply connected by symmetry operations. This is due to the presence of the both uniform and staggered orbital order parameters in Fig. 5(a). Indeed, the wave function in the ground state is written for each sublattice as j Ai=jn= 1;z;#iA; (57) ~ BE = jn= 1;x;#iB jn= 1;y;#iB! : (58) The remaining degeneracy at B sublattice is because the x- andy-orbital components are equivalent. Namely, the triply degenerate state at each sublattice splits depending on the sublattice: zorbital becomes energetically higher at A sublattice and lower at B sublattice. Thus the an- tiferro order of this type cannot lift the degeneracy com- pletely. Usually, the degeneracy is lifted by the interaction ef- fects and the unique ground state is expected. Then, one may suspect that the remaining degeneracy might indi- cate the instability of the solutions. In order to show that our degenerate ground states are really stable, we show the energy spectra of the Hessian matrix discussed in Sec. II C. As shown in Fig. 5(d), the excitation energies in terms of Landau theory are all positive or zero, and the system is thus stable. The degeneracy at T= 0 is due to the absence of the relevant interactions, and will be resolved once the other types of the interaction are included in the more realistic situations. We comment on the case where we allow only for the uniform solutions, by having the geometrical frustration eect in mind which does not favor a simple staggered or- ders. Actually, the n= 1 uniform spherical model aroundTABLE II. Coecients Ain Eq. (45) for n= 3 spherical model. The ground state is written as j i;iiij j;jijand its energy is 2(3 U4J). We add the details for the intermediate state in the main text. EnU8J U6J U4J U3J UJ U + 2J IS 1=25=3 25=184=3 20=9 8=9 IL 9=85=4 25=72 210=9 8=9 IQ9=8 1=41=72 22=98=9 IR1=85=1225=72 2=3 10=98=9 IT 1=8 1=12 1=72 2=3 2=9 8=9 I09=2525=18420=98=9 J= 0 has no solution at any temperature because all of the coupling constants are negative (antiferromagnetic) in the spherical model (see Fig. 2). On the other hand, for relatively large jJjregion the uniform solutions can exist. However, since the typical value of Hund's cou- pling isjJj=U0:1 or less, we do not enter the regime with largerjJjin this paper. C.n= 3model Here we consider the model with three electrons per molecule and with the antiferromagnetic Hund's coupling (J < 0). This model is more relevant to the existing fullerides with half-lled t1umolecular orbitals. 1. Coupling constants We show the coecients A, which is dened by Eq. (45), in Table II. Since we consider the half-lled model, the initial and intermediate states for the two- site problem at the sites iandjrelevant to Iijare (ni;nj) = (3;3) and (ni;nj) = (2;4), respectively. Here, ni= 2 andni= 4 states are connected with each other by the particle-hole (PH) transformation. The explicit form forni= 2 state is same as those given in Sec. III B, and thereby the n= 4 can also be constructed from n= 2 accordingly. Below, we list the types of the intermedi- ate states and their energies, specically focusing on the nj= 4 state. The intermediate states with the excited energy En= U8Jare nine kinds of inter-orbital spin triplet state forni= 2 and the PH transformed states fornj= 4 such as by j;zcy j;y;"cy j;x;"j0i. For En= U6J, the intermediate states are the inter-orbital spin triplet states for ni= 2 and the PH trans- formed states which have inter-orbital spin singlet states such as1p 2by j;z cy j;y;#cy j;x;"cy j;y;"cy j;x;# j0ior intra-orbital spin singlet with anti-bonding such asp 2 3 2by j;zby j;yby j;zby j;xby j;yby j;x j0i. For En=U4J, the intermediate states are the inter-orbital spin singlet or intra-orbital spin singlet with anti-bonding states for10 0.50.40.30.20.10.0 J/U6420246I⇠/E0ni=3 IS IL IQ IR IT FIG. 6. Hund's coupling ratio J=U dependence of the cou- pling constants for n= 3 spherical model. ni= 2, and their PH transformed versions for the jsite. For En=U3J, the intermediate states are the intra- orbital spin singlet and bonding states for ni= 2, and the states which have inter-orbital spin triplet for nj= 4. For En=UJ, the intermediate states are the inter- orbital spin singlet or intra-orbital spin singlet with anti- bonding states ( ni= 2), and intra-orbital spin singlet and bonding state such as1p 3 by j;zby j;y+by j;zby j;x+by j;yby j;x j0i fornj= 4. Finally, for En=U+ 2J, which is the low- est among the excited states for J <0, the intermediate state is non-degenerate and is written as the intra-orbital spin singlet with bonding state for ni= 2 and its PH transformed states for nj= 4. Figure 6 shows the Hund's coupling dependence of the coupling constants. The perturbation theory is justied for1=2< J=U < 0 where any level cross for the un- perturbed Hamiltonian does not occur. If we consider J >0, the ground state is a total spin S= 3=2 state (e.g., cy i;z;"cy i;y;"cy i;x;"j0ii) and is dierent from J < 0. This point is in contrast with n= 1 case where the ground state of the local Hamiltonian is not dependent on the sign ofJas shown in Fig. 2. It is notable that the cou- pling constants for n= 3 case are similar to those of the n= 1 spherical model in the region near J=U =0:5, where the same physical behavior is expected. 2. Mean-eld solutions for bipartite lattice We show in Fig. 7(a) the order parameters for the bi- partite lattice model with n= 3 andJ=U =0:1. At Tc1'2:3E0, the system shows the antiferromagnetic or- der, which is consistent with the largest coupling con- stant shown in Fig. 6. With decreasing temperature, the second order at Tc2appears, where the F- Qz2and AF-Tz2;zorder parameters are additionally induced. We emphasize that this orbital order is not of the ordinaryorbital moment of electrons, but of the doublons relevant to the antiferromagnetic Hund's coupling as discussed in Sec. II A. Figure 7(b) shows the temperature dependences of the internal energies and free energy. We show the order- parameter-resolved energies and all the components de- crease upon entering the ordered phase. While this is in contrast to n= 1 cases shown in the previous sub- sections, the largest energy gain arises from the AF- T order. We show in Fig. 7(c) the entropy and specic heat. The clear jump in the specic heat at Tc1indicates the second- order phase transition, and the jump in the entropy at Tc2is the ngerprint of the rst-order phase transition. The wave function in the ground state is j Ai=jz;#iA; (59) j Bi=jz;"iB: (60) The ground state is thus non-degenerate as is consistent with the zero entropy at T= 0. 3. Single-sublattice solution Having the geometrically frustrated lattice in mind, we assume that the spatially modulated solutions are not realized. Then we seek for the spatially uniform solutions (single-sublattice) only. Figure 8(a) shows the order parameter for the single- sublattice model with n= 3,J=U =0:1. The system shows theQz2order atTc=E0'0:28, which is consis- tent with the magnitude of the coupling constant shown in Fig. 6. The entropy and specic heat are shown in Fig. 8(b) with left and right axis, respectively. The resid- ual entropyS= ln 2 remains, which is in accordance with the degeneracy of spin in the absence of the sublattice degrees of freedom. Namely, the wave function of the ground state is degenerated and is written as ~ E = jz;"i jz;#i! : (61) We have conrmed that the eigenvalues of ^ ain Eq. (36) are all non-negative (not shown) and thus the ordered state is stable. We also point out the other interesting possibilities. The above orbital order is induced by the coupling con- stantIQ>0 in Fig. 6. In this gure, it is notable that the values ofIQandIRare very close with each other. Then we try to search for another solutions by introducing the modied coupling constants dened as ~IQ= (1 +r)IQ; (62) ~IR= (1r)IR; (63) where the original spherical model corresponds to r= 0.11 0.00.51.01.52.02.53.0T/E0543210Internal/Free energy densityni=3, J/U=0.1UFUSULUQURUT(b) 0.00.51.01.52.02.53.0T/E01.00.50.00.51.0Mu,sni=3, J/U=0.1SzsQz2uTz2,zs <latexit sha1_base64="X52I/2YrE97X48H0XCX0tZyVuPU=">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</latexit>Tc1 <latexit sha1_base64="XlJmCfpEjjRXMdHUfDX7W/sBXjY=">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</latexit>Tc2 024681012C/N0123T/E00.00.51.01.5S/Nni=3, J/U=0.1SC <latexit sha1_base64="iJITrNFThqM9eAjvFM/a/M71vKQ=">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</latexit>ln 6(c)(a) FIG. 7. Temperature dependence of (a) the order parameter, (b) the decomposed internal energy and total free energy density, (c, left axis) entropy and (c, right axis) the specic heat for n= 3 bipartite spherical model with J=U =0:1. The horizontal axis are normalized by E0=t2=U. 0.00.10.20.3T/E01.00.50.00.51.0Muni=3, J/U=0.1Qz2u 01234C/N0.00.10.20.3T/E00.00.51.01.5S/Nni=3, J/U=0.1SC <latexit sha1_base64="iJITrNFThqM9eAjvFM/a/M71vKQ=">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</latexit>ln 6<latexit sha1_base64="46zSoqOY4v2X8E4dXP5823PRQJc=">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</latexit>ln 2(a)(b) <latexit sha1_base64="f5RI9QClJOPHM2dinkR4cFHy+hU=">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</latexit>Tc<latexit sha1_base64="f5RI9QClJOPHM2dinkR4cFHy+hU=">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</latexit>Tc0.00.10.20.3T/E01.00.50.00.51.0Muni=3, J/U=0.1,r=0.4Rx,xuRy,yuRz,zu <latexit sha1_base64="2hLODGMPDy74G5ldR05CV6wYc7g=">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</latexit>r=0.4 0.00.10.20.3T/E01.00.50.00.51.0Muni=3, J/U=0.1,r=0.4Rx,xuRy,yuRz,zu <latexit sha1_base64="f5RI9QClJOPHM2dinkR4cFHy+hU=">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</latexit>Tc(c) FIG. 8. Temperature dependence of (a) the order parameter, (b, left axis) the entropy and (b, right axis) the specic heat for n= 3 uniform spherical model with J=U =0:1. (c) Similar order-parameter plots for the n= 3 single-sublattice model with coupling constant ratio r=0:4. The energy unit is E0=t2=U. We show the order parameters for n= 3,J=U =0:1 uniform model with the coupling constant ratio r=0:4 in Fig. 8(c). Since the magnitude of the modied cou- pling constants satises ~IR>~IQin the present condition, we obtain the solution for R;moments. Recalling the denition of the Rmoment, we may rewrite the order parameter as R;LSsymbolically. Therefore, it is interpreted that the system has the eective spin-orbit coupling spontaneously. The wave function is written as ~ E =1p 3 jx;"i ijy;"ijz;#i jx;#i ijy;#ijz;"i! ; (64) which indicates that the ground state is entangled with respect to spin and orbital. These doubly degenerate ground states are connected with each other by the time- reversal symmetry. This \spontaneous spin-orbit coupling" splits the six- fold degeneracy into two-fold and four-fold multiplets, and which is realized in the ground state is dependent on the sign of the order parameters. Our solutions show that the ground state is always doubly degenerate, and this should be related to the minimization of the entropy at low temperatures. Thus, although the system at the original parameter shows the doublon-orbital ordering ( Q), the system is located near the parameter range where the intriguing R order occurs. As discussed in Sec. II A the original spin- orbit coupling SOis tiny, but it might enter through the R-type ordering. Such situation is realized only for n= 3 model with the antiferromagnetic Hund's coupling. 020406080100T[K]1.00.50.00.51.0Mµu,sJ/U=0.1SzsQx2y2sQz2uQxyuTx2y2,zuTz2,zsTxy,zs <latexit sha1_base64="X52I/2YrE97X48H0XCX0tZyVuPU=">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</latexit>Tc1 <latexit sha1_base64="XlJmCfpEjjRXMdHUfDX7W/sBXjY=">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</latexit>Tc2 <latexit sha1_base64="U7nRNsthHSA2oaYSY1diN0tj1bk=">AAAJDHiclVa7btRAFL1JeJjwSAINEk3EKgiaaBYQIKoopIAG5Z1Iu6uV7fUasx7bsr0LwdkfoEB0CKhAokC0dIgKCfEDFPkERBkkKCg4vuvdhMSPxJbsO3fuOXPnzMy1Nc+2glCIraHhkSNHjx1XToyePHX6zNj4xNnVwG37urGiu7brr2tqYNiWY6yEVmgb655vqFKzjTWtdSfuX+sYfmC5znK44Rk1qZqO1bR0NYRrdbke6de69fGSmBZ8Te43yolRouSadydG/lCVGuSSTm2SZJBDIWybVApwV6hMgjz4ahTB58OyuN+gLo0C20aUgQgV3haeJlqVxOugHXMGjH6MCBdIFyxdmqQp8V28F9vim/ggfoi/mWwRs8TZbOCt9bCGVx97en7pdyFK4h3Sgx1UDkJDdPas4n4fdgt2mBsnWZ8monyoZWL0vPhYrRDRt1glC6p57In103t5d5682F66vTgVXRJvxU/o9kZsiS9Qzun80t8tGIuvmd0B5hGrLHneDlY2gj9IMu/yetbY18/wMvhL8E7Ct5PtlSTfLEaZw1hFK42ziuh8VjuHtZLKWSvMM23uMnX28lDzT1NAZmggD6WCTNVBpioh92iRzaqCM8hg3Mxg3CzI1IdfBUs34eudNB9WRItcIfLxFrfMVPw9Pov5+PgkOano4rHvsyI2qxy3NcZVuWb1/fkMc3tGDhDjwzNXiGxxFYvwrqfkPsv4fIYGtGviTpt740C6t1Ox1oE07yE7/D2I2wFwcV13CrAmr2nPUpO+fMTO7ootD3kWIYLBfgp4HxeP0R7sgTZXZh+3i8iiFeijGhztHAJpDpDx7o+/AX5hZeyPZrPX3PUFzj6dfYy/G4N/hPLeP4L9xurV6fKNabFwvTQzm/wtKHSBLqJilukmzdBdmqcVjPqQntNLeqU8Uz4qn5TPvdDhoQRzjv67lK//ALyT3fg=</latexit>Tc3 0.040.020.000.020.040.050.000.05 SzsQx2y2sQz2uQxyuTx2y2,zuTz2,zsTxy,zs020406080100T[K]020040060080010001200eigenvalues [K](b)(a)FIG. 9. Temperature dependence of (a) the order parame- ter and (b) the eigenvalues of the matrix ^ afor A15 fulleride model. The blue lled symbol in (b) corresponds to the so- lution given in (a). The red circle represents the solution without the phase transition at Tc3. IV. NUMERICAL RESULTS FOR FULLERIDES We show the numerical results for the fulleride in the strong coupling regime by using the hopping parameters obtained by the rst principles calculation [40]. We take the intra-orbital Coulomb interaction U= 1eV and the Hund's coupling J=U =0:1 in the following. A. A15 structure First of all we show in Fig. 9(a) the temperature depen- dence of order parameters for the strong-coupling limit model of the realistic fulleride material with the A15 structure. The hopping parameters for Cs 3C60is cho-12 sen (A15-Cs( VoptP SC ) in Ref. [40]). The lattice struc- ture is a bipartite lattice, and A and B sublattices are connected with each other by screw transformation (i.e., translation plus four-fold rotation). As shown in the g- ure, atTc1'80K, the antiferromagnetic moment (AF- S) appears by the second-order phase transition. At lower temperatures, we identify the two successive phase tran- sitions (Tc2;3) with orbital moment Qand spin-orbital momentT. These two Q;T moments share the same symmetry under the presence of AF- Szorder. We cannot simply conclude which one is the primary order parame- ter, because the interaction has complicated form for the realistic model and cannot be decomposed to each con- tribution as in the spherical model. We also note that our choice of parameter is not ne-tuned to reproduce correctly the transition temperature in the actual ma- terials, although our results can be compared with the experiments semi-quantitatively. We show in Fig. 9(b) the eigenvalues (lled blue sym- bols) of the Hessian matrix dened in Eq. (36). All the values are non-negative, and therefore the system is stable. On the other hand, we can also calculate the low-temperature solutions by suppressing the ordering at Tc3. The results are plotted as the open red symbols in Fig. 9(b). In this case, the eigenvalues become partially negative and hence the system is not stable although the entropy goes to zero even in this case. Thus, the emer- gence of the order at Tc3is essential in order to reach the stable ground state. We discuss the origin of the second orbital order at Tc3 in more detail. Below, we concentrate on the properties ofQmoments to make the discussion simple, since the symmetry of Qis same as that of Tbelow the transition temperature Tc1. Figure 10(a) shows the orbital order pa- rameters for sublattice A (left panel) and B (right panel) slightly below the transition temperature Tc2(but above Tc3). The three patterns are obtained depending on the initial condition and hence are degenerate solutions. It is seen from Fig. 10(a) that the plane of X=Qz2 and Y=Qx2y2 has a three-fold rotational symmetry and the equilateral triangle points, where the free energy min- ima are located, are tilted from the Xaxis. This tilt angle remains nite at low temperatures below Tc3. This result can be understood from the Landau theory: we can show that, without four-fold rotational symmetry as inThpoint group symmetry in fulleride materials, the Landau free energy is written in the restricted order- parameter space as FL=X =A;Bh c1X(X2 3Y2 ) +c2sY(3X2 Y2 )i ; (65) wheres=A= +1 and s=B=1. We have consid- ered only the third-order term for our purpose. This is consistent with the numerical results and the tilt of the angle is due to the presence of c2term. The tilt angle is estimated with the polar coordinates X=rcosand 0.250.000.25Qz20.40.20.00.20.4Qx2y2Qx2 Qy2↵=A,T.Tc2 0.250.000.25Qz20.40.20.00.20.4Qx2y2↵=B,T.Tc2 0.50.00.5Qz20.500.250.000.250.50Qx2y2↵=A,T!0 0.50.00.5Qz20.500.250.000.250.50Qx2y2↵=B,T!0(a) (b)FIG. 10. (a) Sublattice-dependent order parameters in the plane ofQz2-Qx2y2for A (left) an B (right) sublattices at T= 40:4K (< T c2). The similar plots at low-temperature limit without the transition at Tc3is shown in (b). The dashed circles in (a,b) correspond to the solutions in the system with four-fold symmetries. Each color shows dierent kind of solu- tions, which share the same free energy. The gray arrows with Qx2orQy2are the guide for taking the other quantization axis. Specically, the solution given in Fig. 9(a) corresponds to the blue circle in the present gure (a). The angle in the left panel of (a) is the deviation from the horizontal axis. Y=rsin, leading to another expression of the free en- ergyFL/cos(3+) with= tan1c2=c1being the tilt angle. For example, one can estimate this angle from Fig. 10(a) as = 6:76. The A15 structure has the screw symmetry, i.e., the combination of the translation along [111] and four-fold rotation around x;y;z axes, which relates the order parameters at A and B sublattices. In- deed, the above Landau free energy is invariant under the three-fold rotation and screw transformations. If the four-fold symmetry is present, the condition c2= 0 or= 0 is required. In Fig. 10(b), we show the order parameters at T!0without the second orbital ordering belowTc3, where the four-fold symmetry seems to be eectively recovered since the tilt angle goes to zero when T!0. Hence, the origin of the second orbital order in Fig. 9(a) below Tc3is interpreted as induced from this emergent symmetry at low temperatures which provides an additional free energy gain.13 <latexit sha1_base64="f5RI9QClJOPHM2dinkR4cFHy+hU=">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</latexit>Tc 0510152025T[K]01002003004005006001/⌘µ;⌘µu[K] 0510152025T[K]1.00.50.00.51.0MµuJ/U=0.1LzuQz2u(b)(a) 0.040.020.000.020.040.050.000.05 SxSySzLxLyLzQx2y2Qz2QxyQyzQzxRx,xRy,xRz,xRx,yRy,yRz,yRx,zRy,zRz,zTx2y2,xTz2,xTxy,xTyz,xTzx,xTx2y2,yTz2,yTxy,yTyz,yTzx,yTx2y2,zTz2,zTxy,zTyz,zTzx,z FIG. 11. Temperature dependence of (a) the order parameter and (b) the inverse of the diagonal susceptibility for uniform fcc fulleride model with J=U =0:1. B. fcc structure Finally, we consider the fulleride material with the fcc structure. The spin-orbital model in the strong-coupling limit is obtained by using the hopping parameters for Rb3C60in Ref. [40]. Because of the geometrically frus- trated nature of the fcc lattice, we here seek for only the spatially uniform ordered states. Figure 11(a) shows the temperature evolution of the order parameters. Here the primary order parameter is the uniform Lzmoment which breaks the time-reversal symmetry. The Lz-order arises as ( TcT)1=2, and the Qz2is also induced simultaneously with the linear tem- perature dependence /(TcT). The latter Qmoment is induced from the coupling term with the form ( Lz)2Qz2 in the Landau free energy. We note that Lzis not in- duced when Qz2is a primary order parameter from that coupling, since LzandQz2have dierent time-reversal symmetry [( Lz)2andQz2are same]. The ground-state wave function is written in a simple form as ~ E =1p 2 jx;"i ijy;"i jx;#i ijy;#i! ; (66) where the complex wave function clearly shows the time- reversal symmetry breaking. We note that this or- bital moment is not a simple orbital motion around the fullerene molecule, but a complex motion of the three electron state given in Eq. (4). In our calculations the spinSorder does not occur and the ground state is dou- bly degenerate at each cite. The stability of the solution is checked by the non-negative eigenvalues of the Hessian matrix. The results found here is dierent from the spherical case discussed in Sec. III C 3. The dierence is due to the specic form of the tight-binding hopping parameters. We show one of the coupling constant ILfor the nearestneighbour sites as 0 B@I1I20 I2I30 0 0I41 CA;R=a 2;a 2;0 ; (67) whereRis the direction of the NN molecules and ais the lattice constant for the fcc fulleride. The information for the other NN pairs is constructed from the symme- try operations. The values of the matrix element are I1= 12:5;I2= 9:77;I3= 0:511 andI4= 20:1 in units of K in the present models. The coupling constant has the same symmetry as the hopping parameters in Ref. [40] as required by the space group symmetry. The near- est neighbor coupling constant is largest and is positive, which favors the uniform magnetic orbital moment L. As for the next nearest neighbour site, the coupling con- stant matrices are diagonal and every component of them is smaller than nearest neighbour ones. Since the spin Smoment has the same symmetry as L, it can in general be simultaneously induced under the small but nite spin-orbit coupling. However, as dis- cussed in Sec. II A, the magnitude of the eective spin- orbit coupling for the doublon orbital is SO109eV, which can be regarded as zero in practice. Hence, the spin order can occur independently at low temperatures. The absence of the spin Sorder is interpreted from the point of view of the coupling constant. Figure 11(b) shows that the temperature dependence of the inverse of the diagonal susceptibilities. The blue lines represents the magnetic susceptibility ( S), which indicates that the coupling constants of Sare antiferromagnetic owing to the negative Curie-Weiss temperature. In this case, the transition temperature should be very low due to the ge- ometrical frustration of fcc lattice, but nally the system should show some magnetic ordering [41]. C. Discussion The models in this section are based on the band- structure calculation results. Furthermore, the fulleride materials can be located in the Mott insulator regime depending on the pressure. Hence, our results are poten- tially applied to the real materials. In fulleride materials, the antiferromagnetic orders is experimentally identied at low temperatures, while the orbital orders are not yet reported. Based on our results, we propose that at low temperatures the orbital ordered moments Qare induced with two successive transitions for A15 structures, and Lmoments may appear for fcc structures. Such nger- prints of the orbital orders may be found in thermody- namic quantities in principle. Here the orbital moment is not for a usual electron but for the doublons specic to the systems with antiferromagnetic Hund's coupling as emphasized in the present paper. On the other hand, since the real compounds are polycrystals and the disor- der eects are also present, the orbital orders might be14 smeared out in realistic situations. In this context, the eect of disorders on our spin-orbital model is interesting future issues which make it more direct to compare the theoretical results with experimental observations. More- over, the antiferromagnetic Hund's coupling originates from the electron-phonon coupling. The resultant retar- dation eects are also the parts not included in this paper and an important issue for the more realistic arguments. V. SUMMARY AND OUTLOOK In order to clarify the properties of strongly corre- lated electrons in fulleride superconductors, we have con- structed the spin-orbital model in the strong coupling limit. We begin with the three-orbital Hubbard model with the antiferromagnetic Hund's coupling which is re- alized by the coupling between the electronic degrees of freedom and anisotropic Jahn-Teller molecular vibra- tions. In this case, the pair hopping eect among the dierent orbitals becomes relevant in strong contrast to the multiorbital d-electron systems with the ferromag- netic Hund's coupling. We have mainly considered the half-lledn= 3 case relevant to real materials, where it is composed of the singly-occupied (singlon) plus doubly occupied orbitals (doublon) as illustrated in Fig. 1. The correlated ground state for an isolated fullerene molecule is six-fold degenerate and is characterized by the spin and orbital indices. This is the situation similar to the n= 1 ground states and the analogy between n= 3 andn= 1 helps us for interpreting the results. The usual orbital moment, which is present for the n= 1 case, is absent for n= 3 because of the correlated na- ture of the wave function, and instead the active orbital moment characteristic for doublons exists. As the result, the spin-orbit coupling, which is the order of 1meV for p- electrons, becomes 1neV because of the extended nature of the molecular orbitals and the correlation eects. We have applied the second-order perturbation the- ory with respect to the inter-molecule hopping, and have obtained the localized spin-orbital model specic to the fullerides. The obtained spin-orbital model is analyzed by employing the mean-eld approximation. For refer- ence, we have rst solved the spherical n= 1 model for both ferromagnetic and antiferromagnetic Hund's cou- plings with a spherical limit for the bipartite lattice. We then apply our method to the n= 3 model where the magnetic order is found at relatively high temperatures and the orbital order also occurs at lower temperatures.The temperature dependences of the physical quantities such as order parameters, internal and free energies, spe- cic heat, entropy, and susceptibilities are investigated in detail. The thermodynamic stability is also studied based on the Hessian matrix derived from the inverse susceptibilities, and are checked by conrming that all the eigenvalues are non-negative. We have also considered the realistic situation in alkali- doped fullerides, by using the tight-binding parameters derived from the rst principles calculations. For the choice of the lattice structure, we have taken both the bipartite A15 and fcc structures, whose hopping parame- ters have been derived in Ref. [40]. For the A15 structure, the antiferromagnetic order occurs at high temperatures, and the electric orbital orders arise at lower tempera- tures with two successive transitions. The rst orbital order is already captured in the spherical model, but the second orbital order is characteristic for the Thsymmetry in fulleride materials where only the three-fold rotation symmetry exists. This point has been discussed in de- tail based on the Landau theory. For the fcc model, we have concentrated on the spatially uniform solutions due to the geometrically frustrated nature of the lattice. We have found that the magnetic orbital order occurs. Al- though this orbital moment has the same symmetry as the electronic spin, the spin moment is not induced simul- taneously in fulleride since the spin-orbit coupling is tiny as mentioned above. Thus the spin-moment can order in- dependently, and is expected to be antiferromagnetically ordered in the ground state where the transition tem- perature is expected to be low owing to the geometrical frustration of the fcc lattice. Our formalism itself is constructed in a very general way, and can be applied to any systems in the strong coupling limit with integer llings per atom or molecule. In this context, it would be desirable to develop the gen- eral framework for the strong-coupling-limit spin-orbital model with the combination of the hopping parameters in the Wannier functions obtained from the band-structure calculations. This application is of interest specically in studying the ordered state of the multiorbital elec- tronic systems including transition metals and organic materials. This point remains to be explored and is an intriguing issue in the future. ACKNOWLEDGEMENT This work was supported by JSPS KAKENHI Grants No. JP18K13490 and No. JP19H01842. [1] A. F. Hebard, M. J. Rosseinsky, R. C. Haddon, D. W. Murphy, S. H. Glarum, T. T. M. Palstra, A. P. Ramirez, and A. R. Kortan, Nature (London) 350, 600 (1991).[2] M. J. Rosseinsky, A. P. Ramirez, S. H. Glarum, D. W. Murphy, R. C. Haddon, A. F. Hebard, T. T. M. Palstra, A. R. Kortan, S. M. Zahurak, and A. V. Makhija, Phys. Rev. Lett. 66, 2830 (1991).15 [3] K. Holczer, O. Klein, S. mei Huang, R. B. Kaner, K. jian Fu, R. L. Whetten, and F. Diederich, Science 252, 1154 (1991). [4] K. Tanigaki, T. W. Ebbesen, S. Saito, J. Mizuki, J. S. Tsai, Y. Kubo, and S. Kuroshima, Nature (London) 352, 222 (1991). [5] R. M. Fleming, A. P. Ramirez, M. J. Rosseinsky, D. W. Murphy, R. C. Haddon, S. M. Zahurak, and A. V. Makhija, Nature (London) 352, 787 (1991). [6] A. Y. Ganin, Y. Takabayashi, Y. Z. Khimyak, S. Mar- gadonna, A. Tamai, M. J. Rosseinsky, and K. Prassides, Nat. Mater. 7, 367 (2008). [7] Y. Takabayashi, A. Y. Ganin, P. Jegli c, D. Ar con, T. Takano, Y. Iwasa, Y. Ohishi, M. Takata, N. Takeshita, K. Prassides, and M. J. Rosseinsky, Science 323, 1585 (2009). [8] A. Y. Ganin, Y. Takabayashi, P. Jegli c, D. Ar con, A. Poto cnik, P. J. Baker, Y. Ohishi, M. T. McDonald, M. D. Tzirakis, A. McLennan, G. R. Darling, M. Takata, M. J. Rosseinsky, and K. Prassides, Nature (London) 466, 221 (2010). [9] O. Gunnarsson, Rev. Mod. Phys. 69, 575 (1997). [10] M. Fabrizio and E. Tosatti, Phys. Rev. B 55, 13465 (1997). [11] M. Capone, M. Fabrizio, C. Castellani, and E. Tosatti, Science 296, 2364 (2002). [12] M. Capone, M. Fabrizio, C. Castellani, and E. Tosatti, Rev. Mod. Phys. 81, 943 (2009). [13] Y. Nomura, S. Sakai, M. Capone, and R. Arita, J. Phys. Condens. Matter 28, 153001 (2016). [14] Y. Takabayashi and K. Prassides, Phil. Trans. R. Soc. A 374, 20150320 (2016). [15] R. H. Zadik, Y. Takabayashi, G. Klupp, R. H. Colman, A. Y. Ganin, A. Poto cnik, P. Jegli c, D. Ar con, P. Ma- tus, K. Kamar as, Y. Kasahara, Y. Iwasa, A. N. Fitch, Y. Ohishi, G. Garbarino, K. Kato, M. J. Rosseinsky, and K. Prassides, Sci. Adv. 1, e1500059 (2015). [16] Y. Kasahara, Y. Takeuchi, R. H. Zadik, Y. Takabayashi, R. H. Colman, R. D. McDonald, M. J. Rosseinsky, K. Prassides, and Y. Iwasa, Nat. Commun. 8, 14467 (2017). [17] S. Han, M.-X. Guan, C.-L. Song, Y.-L. Wang, M.-Q. Ren, S. Meng, X.-C. Ma, and Q.-K. Xue, Phys. Rev. B 101, 85413 (2020). [18] M.-Q. Ren, S. Han, S.-Z. Wang, J.-Q. Fan, C.-L. Song, X.-C. Ma, and Q.-K. Xue, Phys. Rev. Lett. 124, 187001 (2020). [19] M. Mitrano, A. Cantaluppi, D. Nicoletti, S. Kaiser, A. Perucchi, S. Lupi, P. D. Pietro, D. Pontiroli, M. Ricc o,S. R. Clark, D. Jaksch, and A. Cavalleri, Nature (Lon- don)530, 461 (2016). [20] A. Cantaluppi, M. Buzzi, G. Jotzu, D. Nicoletti, M. Mi- trano, D. Pontiroli, M. Ricc o, A. Perucchi, P. D. Pietro, and A. Cavalleri, Nat. Phys. 14, 837 (2018). [21] M. Capone, M. Fabrizio, P. Giannozzi, and E. Tosatti, Phys. Rev. B 62, 7619 (2000). [22] Y. Nomura, S. Sakai, M. Capone, and R. Arita, Sci. Adv. 1, e1500568 (2015). [23] A. Koga and P. Werner, Phys. Rev. B 91, 085108 (2015). [24] S. Hoshino and P. Werner, Phys. Rev. B 93, 155161 (2016). [25] K. Steiner, S. Hoshino, Y. Nomura, and P. Werner, Phys. Rev. B 94, 075107 (2016). [26] S. Hoshino and P. Werner, Phys. Rev. Lett. 118, 177002 (2017). [27] K. Ishigaki, J. Nasu, A. Koga, S. Hoshino, and P. Werner, Phys. Rev. B 98, 235120 (2018). [28] K. Ishigaki, J. Nasu, A. Koga, S. Hoshino, and P. Werner, Phys. Rev. B 99, 085131 (2019). [29] C. Yue, S. Hoshino, and P. Werner, Phys. Rev. B 102, 195103 (2020). [30] S. Hoshino, P. Werner, and R. Arita, Phys. Rev. B 99, 235133 (2019). [31] T. Misawa and M. Imada, arXiv:1711.10205 (2017). [32] K. I. Kugel and D. I. Khomskii, ZhETF Pis. Red. 15, 629 (1972), [Sov. Phys. JETP Lett. 15, 446 (1972)]. [33] K. I. Kugel and D. I. Khomskii, Zh. Eksp. Teor. Fiz. 64, 1429 (1973), [Sov. Phys. JETP 37, 725 (1973)]. [34] S. Inagaki, J. Phys. Soc. Jpn. 39, 596 (1975). [35] S. Ishihara, J. Inoue, and S. Maekawa, Phys. Rev. B 55, 8280 (1997). [36] L. F. Feiner and A. M. Ole s, Phys. Rev. B 59, 3295 (1999). [37] B. Normand and A. M. Ole s, Phys. Rev. B 78, 094427 (2008). [38] E. Tosatti, N. Manini, and O. Gunnarsson, Phys. Rev. B 54, 17184 (1996). [39] Y. Nomura and R. Arita, Phys. Rev. B 92, 245108 (2015). [40] Y. Nomura, K. Nakamura, and R. Arita, Phys. Rev. B 85, 155452 (2012). [41] Y. Kasahara, Y. Takeuchi, T. Itou, R. H. Zadik, Y. Tak- abayashi, A. Y. Ganin, D. Ar con, M. J. Rosseinsky, K. Prassides, and Y. Iwasa, Phys. Rev. B 90, 014413 (2014).
2208.09409v1.Spin_triplet_Superconductivity_in_Nonsymmorphic_crystals.pdf	Spin-triplet Superconductivity in Nonsymmorphic crystals Shengshan Qin,1,Chen Fang,2, 1Fu-chun Zhang,1, 3and Jiangping Hu2, 1, 4,y 1Kavli Institute for Theoretical Sciences and CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China 2Beijing National Research Center for Condensed Matter Physics, and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 3Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China 4South Bay Interdisciplinary Science Center, Dongguan, Guangdong Province, China (Dated: August 22, 2022) Spin-triplet superconductivity is known to be a rare quantum phenomenon. Here we show that nonsymmor- phic crystalline symmetries can dramatically assist spin-triplet superconductivity in the presence of spin-orbit coupling. Even with a weak spin-orbit coupling, the spin-triplet pairing can be the leading pairing instability in a lattice with a nonsymmorphic symmetry. The underlining mechanism is the spin-sublattice-momentum lock on electronic bands that are protected by the nonsymmorphic symmetry. We use the nonsymmorphic space group P4=nmm to demonstrate these results and discuss related experimental observables. Our work paves a new way in searching for spin-triplet superconductivity. PACS numbers: Introduction. The spin-triplet superconductors, which are the superconducting analogy of the3He superﬂuid1, have been long-pursued. They have been proposed to be natu- ral candidates for the topological superconductors2–8, host- ing the Majorana modes which are expected to play an es- sential role in the fault-tolerant quantum computations9–11. In the past decades, great efforts have been made in pursu- ing the spin-triplet superconductors12–14. Theoretically, var- ious mechanisms have been proposed in favor of the spin- triplet superconductivity. For instance, the spin-triplet su- perconductivity may arise at ultra low temperature through the Kohn-Luttinger mechanism15; and it can also be induced from the ferromagnetic spin ﬂuctuations or the ferromag- netic exchange coupling13. Experimentally, Sr 2RuO 4has been suggested to be a promising candidate for the spin- triplet superconductors16,17. However, recent experiments raise doubts on this issue18. In many heavy fermion sys- tem such as UPt 319, UTe 220, and the recently synthesized K2Cr3As321,22, signatures for the spin-triplet superconductiv- ity have been observed. During the past decades, the spin-orbit coupled systems have attracted more and more research attentions. The spin- orbit coupling (SOC) has been revealed to play an impor- tant role in various exotic condensed matter systems such as the topological materials23–27. Recent studies suggest that the SOC can help the spin-triplet superconductivity. For instance, it has been predicted the spin-triplet super- conductivity may exist in doped superconducting topological insulators28–32and semimetals33,34. Especially, in the doped topological insulator Bi 2Se3nematic superconductivity has been conﬁrmed experimentally35–40, indicating possible odd- parity spin-triplet superconductivity in the system28,29. Be- sides the topological materials, in the two-dimensional (2D) electron gas formed at the interface between LaAlO 3and SrTiO 341,42, the spin-triplet superconductivity is also pro- posed based on large Rashba SOC43,44. However, all these studies rely on a strong SOC, in which pairing forces may be signiﬁcantly weakened by the SOC as well.In this Letter, we show that the spin-triplet superconduc- tivity can be stabilized by nonsymmorphic symmetries in the spin-orbit coupled systems. Even with a weak SOC, the spin- triplet pairing can be the leading pairing instability in a lattice with a nonsymmorphic symmetry. We specify our study with the nonsymmorphic space group P4=nmm (#:129). Due to the nonsymmorphic symmetries, the sublattice degree always exists in the system. In the presence of the SOC, the sub- lattice degree intertwines with the spin degree. Correspond- ingly, the spin, sublattice and momentum are locked with each other, forming a spin-sublattice-momentum lock texture on the normal-state energy bands. The spin-triplet pairing state is always favored due to spin-sublattice-momentum lock when there is a pairing force between the two sublattices. We ﬁrst brieﬂy review the space group G=P4=nmm , which is nonsymmorphic. There are 16 symmetry operations in its quotient group G=TwithTbeing the translation group. More specially,G=Tcan be written in the following concise direct product form45 G=T=D2d Z2; (1) in a sense that symmetry operations are equivalent if they dif- fer by a lattice translation. We specify the symmetry group with a quasi-2D lattice shown in Fig.1(a), which is similar to the structure of the monolayer FeSe. As shown in the lattice, the ﬁxed point of point group D2din Eq.(1) is at the lattice sites, andZ2is a two-element group including the inversion symmetry which is deﬁned at the bond center between two nearest lattice sites. According to Eq.(1), we can choose the generators of the quotient group G=Tas the inversion sym- metryfIj0g, the mirror symmetry fMyj0gand the rotoin- version symmetry fS4zj0g46, where the symmetry operators have been expressed in the form of the Seitz operators and 0=a1=2 +a2=2witha1anda2being the primitive lattice translations along the xandydirections in Fig.1(a). Low-energy theory near (;).A standard group theory analysis shows that the space group P4=nmm merely has one single 4D irreducible representation at the Brillouin zonearXiv:2208.09409v1 [cond-mat.supr-con] 19 Aug 20222 corner (;), i.e. the M point, in the spinful condition47. The above conclusion straightforwardly leads to three impor- tant implications. (i) For systems respecting the space group P4=nmm , in the presence of SOC all the energy bands are fourfold degenerate at the M point. (ii) All the fourfold de- generate bands respect the same low-energy effective model. (iii) One can use arbitrary orbital to construct the low-energy effective theory near M, and for simplicity we consider one s orbital at each lattice site in Fig.1(a) in the following. With the above preparation, we can construct the low- energy effective theory near M. As all the symmetry opera- tions inG=Tpreserve at (;), we need to derive the matrix form of the symmetry generators of G=T. By a careful analy- sis, we obtain the matrix form of the symmetry operators as, I=s01,My=is23andS4z=eis3=43, whereI,My andS4zstand forfIj0g,fMyj0gandfS4zj0grespectively (details in SM). In the matrix form, siandi(i= 1;2;3) are the Pauli matrices for the spin and the two sublattices respectively, and s0and0the corresponding identity ma- trices. The above matrices are actually a set of irreducible representation matrices for space group P4=nmm at M. Be- sides the crystalline symmetries, the time reversal symmetry isT=is20KwithKthe complex conjugation operation. The low-energy effective theory near M for group P4=nmm is generally depicted by the sixteen =sijma- trices. In deriving the effective model, it is convenient to ﬁrst constrain the system by the time reversal symmetry and the inversion symmetry, and then consider the constraints of other crystalline symmetries. After some algebra, we classify the symmetry allowed matrices along with the k-dependent functions, and obtain the low-energy effective Hamiltonian as follows48(details in SM) Heff(k) =m(k)s00+kxs23+kys13 +t0kxkys01; (2) wherem(k) =t(k2 x+k2 y). Notice that kx=yis deﬁned accord- ing to the M point here. To have a more intuitive impression on the effective theory in Eq.(2), one can understand the pa- rameters in the lattice shown in Fig.1(a). Speciﬁcally, t(t0) describes the hopping between the intrasublattice (intersub- lattice) nearest neighbours, and is the inversion-symmetric Rashba SOC arising from the mismatch between the lattice sites and the inversion center49, i.e. the local inversion- symmetry breaking50. We show the band structures calculated from the effective Hamiltonian in Eq.(2) in Fig.1(b). Due to the presence of both the time reversal and inversion symme- tries, all the energy bands are twofold degenerate. Spin-sublattice-momentum lock. In centrosymmetric sys- tems, the local inversion-symmetry breaking can intertwine the different degrees of freedom51–53. Here, for systems re- specting the space group P4=nmm , symmetries enforce the spin degree locked to the sublattice degree on the energy bands and the sublattice-distinguished spin is nearly fully po- larized for small Fermi surfaces near (;). Before the de- tailed calculations, we ﬁrst consider the symmetry constraints. As shown in Fig.1(a), the inversion symmetry exchanges the two sublattices, while the time reversal symmetry does not FIG. 1: (color online) (a) A sketched quasi-2D lattice structure re- specting the P4=nmm space group: A and B indicate the sublat- tices related by the nonsymmorphic symmetries, the shadow region indicates the unit cell, and the red point is the inversion center lo- cated at the bond center between two nearest neighbouring sites. (b) The band structure near the M point, plotted from the Hamiltonian in Eq.(2) with parameters ft;t0;g=f1:0;0:8;0:12g. (c) and (d) show the spin polarizations on the lower and upper energy bands in (b) respectively: the spin polarization contributed by the A (B) sub- lattice is labeled by the red (blue) arrowed line, with the length of the line indicating the strength of the polarization. The gray lines in (c)(d) show the Fermi surfaces for chemical potential = 0:2. change the spacial position. On the other hand, the inver- sion symmetry preserves the spin but the time reversal sym- metry ﬂips the spin. Therefore, considering the combina- tion of the time reversal symmetry and inversion symmetry, at each kpoint the spin polarizations from the two sublattices are always opposite. Moreover, due to the mirror symmetry fMx=yj0g, the spin is polarized perpendicular to the mirror plane along kx=y= 0, i.e. the Brillouin zone boundary. Based on the low-energy effective theory in Eq.(2), the spin polarizations contributed by the different sublattices, i.e.hsAiandhsBi, can be calculated analytically (details in SM). A direct calculation shows that hsBi=hsAi= (sin;cos;0)=(q 1 +t02k2sin22=42)at point k, with k written as (kx;ky) = (kcos;ksin). We sketch the results in Fig.1(c)(d). As shown, both hsAiandhsBilie in thexy plane and wind around the M point anticlockwise. Moreover, the spin polarization reaches its minimum along kx=ky, and is nearly fully polarized on the bands near the Brillouin zone boundary. It is worth pointing out that, the fully polarized spin along the Brillouin zone boundary satisfying hsAi=hsBi is consistent with the matrix form of the mirror symmetries at M, i.e.Mx=is13andMy=is2354. Superconductivity. For the effective theory in Eq.(2), we3 consider the possible superconductivity induced by the phe- nomenological density-density interactions Hint=Z dq[U2X i=1ni(q)ni(q) + 2Vn1(q)n2(q)];(3) wheren1(q) =P =";#1p NR dkcy k;ck+q;andn2(q) = P =";#1p NR dkdy k;dk+q;are the density operators for the A and B sublattices in Fig.1(a) respectively, and UandVare the intrasublattice and intersublattice interactions respectively. In Eq.(3), we focus on the momentum-independent interac- tions in the weak-coupling condition. Obviously, the negative U(V) correspond to the attractive interaction. Actually, the phenomenological interactions in Eq.(3) arise from the short- range density-density interactions in the real space. Speciﬁ- cally,Uis the onsite interaction, and Vis the leading-order term, i.e. the momentum-independent part, in the intersublat- tice interaction between nearest neighbours (details in SM). From the interactions in Eq.(3), in the mean-ﬁeld level only the momentum-independent superconducting orders are expected55. Due to the fermionic statistics of electrons, the pairing orders are required to satisfy ^(k)is20= (^(k)is20)T, where the pairing term is y(k)^(k) is20 y(k)in the basis y(k) = (cy k;";cy k;#;dy k;";dy k;#). The pairing orders can be further classiﬁed in accordance with the symmetry group of the system, and we classify the momentum-independent pairing orders and present the results in Table.I. As shown, the pairing orders belong to ﬁve differ- ent pairing symmetries in the A1g,B2g,A2u,B2uandEu representations of the D4hgroup, with the AandBrepresen- tations being 1D and the Erepresentation 2D. To show the meaning of the pairing orders clear, we list the explicit form of the superconducting pairing as follows ^A1g:cy k;"cy k;#+dy k;"dy k;#; (4) ^B2g:cy k;"dy k;#+dy k;"cy k;#; ^A2u:icy k;"dy k;#+idy k;"cy k;#; ^B2u:cy k;"cy k;#dy k;"dy k;#; ^Eu: (icy k;"dy k;"icy k;#dy k;#;cy k;"dy k;"+cy k;#dy k;#): As mentioned, the inversion symmetry in space group P4=nmm exchanges the two sublattices. ^A1gand^B2gare spin-singlet pairings with even parity, and ^A1goccurs in the same sublattice while ^B2gis between the different sublat- tices. ^B2uis the intrasublattice spin-singlet pairing with odd parity, whereas ^A2uand^Euare the odd-parity spin-triplet pairings between the different sublattices. To ﬁnd out the superconducting ground state, we solve the following linearized gap equations (details in SM) ^A1g;B2u:UA1g;B2u(Tc) = 1; (5) ^B2g;A2u;Eu:VB2g;A2u;Eu(Tc) = 1; where we have used the fact that ^A1gand^B2ucan only result from the intrasublattice interaction U, and ^B2g,^A2uTABLE I: Classiﬁcation of the possible momentum-independent pairing potentials corresponding to the interactions in Eq.(3), accord- ing to the irreducible representations of the D4hpoint group. Here, the pairing potentials are in the form y(k)^is20 y(k), with the basis being y(k) = (cy k;";cy k;#;dy k;";dy k;#). ES4zI M y^ A1g 1 1 1 1 s00 B2g 1 -1 1 -1 s01 A2u 1 -1 -1 1 s32 B2u 1 1 -1 1 s03 Eu 2 0 -2 0 (s12;s22) and^Euonly arise from the intersublattice interaction V. In Eq.(5),is the ﬁnite-temperature superconducting suscepti- bility for each irreducible representation pairing channel in Table.I, which can be calculated as (Tc) =F(Tc)X sZ dD()X s0=s;sjhus;kj^jus0;kij2:(6) In the above equation, jus;kiis the wavefunction for the state on the Fermi surface contributed by band s, andjus;ki= ITjus;kiis the state degenerate with jus;kidue to the pres- ence of both the inversion symmetry Iand the time re- versal symmetryT.F(Tc) =1 2NR!0 !01 2tanh 2dis a temperature-dependent constant with = 1=kBTcand!0the energy cutoff near the Fermi energy. D() = 2dk0=ds;k0 is the density of states on the Fermi surface. By solving Eq.(5), we can get the superconducting transition temperature for each pairing channel, and the state with the highest Tcis the ground state. FIG. 2: (color online) Superconducting phase diagram versus the chemical potential , the SOC, and the interaction U=V , assuming UandVboth attractive. The colored surface in the ﬁgure is the phase boundary between the different superconducting ground states. In the calculation, the other parameters are ft;t0g=f1:0;0:8g. Here, only the condition for 2jUj>jVjis shown. For the following two conditions, (i) 2jUj<jVjand (ii)U > 0andV < 0, only theB2g andA2ustates can appear in the phase diagram, with their phase boundary always the same with that at 2U=V. According to Eqs.(5)(6), the superconducting instability can merely arise from the attractive interactions. Moreover,4 a direct calculation shows that the superconducting suscep- tibilities always satisﬁes B2u< A1gandA2u= 2Eu (details in SM), meaning that the B2uandEupairing states can never be the ground states. Consequently, in the condi- tion withU < 0andV > 0, i.e. the intrasublattice attractive and intersublattice repulsive interactions, the ground state is always theA1gstate; and in the condition with U > 0and V < 0, theB2gandA2ustates can be the superconducting ground states. If both of the interactions are attractive, the A1g,B2gandA2upairing states can appear in different re- gions in the parameter space, and the corresponding phase di- agram is presented in Fig.2. In the phase diagram, we merely show the condition for U=V > 0:5. Whereas, the phase dia- gram forU=V < 0:5is independent with U=V , and only the B2gandA2ustates can be the ground states with their phase boundary always the same with that at 2U=V. The phase boundary between the B2gandA2ustates at 2U=Valso applies to the condition with U > 0andV < 0. It is worth pointing out that, all the states in the phase diagram are fully gapped. Especially, the B2gstate is actually similar to the nodelessd-wave state in the iron-based superconductors56–58. A remarkable feature in the phase diagram in Fig.2 is that, the spin-triplet A2ustate occupies a large area and it can be the ground state even in the weak SOC limit. The phe- nomenon is closely related to the symmetry-enforced spin- sublattice-momentum lock on the normal-state energy bands shown in Fig.1. As analyzed, near the M point the sublattice- distinguished spin polarization lies in the xyplane and the strength is proportional to 1=q 1 +t02k2sin22=42satisfy- inghsB(k)i=hsA(k)i. In the small chemical potential condition, the spin on the Fermi surface is nearly fully polar- ized. When Cooper pair forms between two electrons with op- posite momenta, the spin-sublattice-momentum lock in Fig.1 enforces the equal-spin pairing state with the spin polarized in thexyplane in the intersublattice channel, which is exactly theA2ustate in Fig.2. Moreover, since the fully polarized spin near M is enforced by symmetries which is regardless of the strength of the SOC, the A2ustate can appear as the ground state in the weak SOC condition as long as the chem- ical potential is small. In the large chemical potential con- dition, the average spin polarization on the large Fermi sur- face is weak. Correspondingly, the spin-singlet A1gandB2g states become more favorable. It is worth mentioning that in the limit!0and!0, the spin-singlet states compete with the spin triplet state, due to the concentric Fermi surface structure arising from the fourfold band degeneracy at M as indicated in Fig.1. Experimental signatures. The different states in the phase diagram in Fig.2 can be distinguished in experiments. In nu- clear magnetic resonance measurements, the temperature de- pendence of the Knight shift Kssand the spin relaxation rate 1=T1can provide essential information on the superconduct- ing orders13,19,59,60. We calculate Kssand1=T1for the dif- ferent superconducting ground states in Fig.2. With a strong SOC, as shown in Fig.3(a) (c) theA2ustate has a distinguish- ing feature in the Knight shift, i.e. the constant Kzzcorre- sponding to magnetic ﬁelds applied along the zdirection, andtheA1gstate is characterized by the Hebel Slichter coherence peak in the spin relaxation rate as presented in Fig.3(d) (f), as the temperature cools down below Tc. For theB2gstate, the Knight shift is always suppressed and the Hebel Slichter coherence peak in the spin relaxation rate is absent. At an ultra low temperature, all the three states show similar expo- nential scaling behavior in the spin relaxation rate as indicated in Fig.3(d)(f), due to their nodeless gap structures. We want to note that the Knight shift results can change if the strength of the SOC is comparable to the pairing order (more details in SM), while the features in the spin relaxation rate always hold for the different states. FIG. 3: (color online) (a) (c) show the Knight shift and (d) (f) show the spin relaxation rate versus the reduced temperature T=Tc, for the three possible superconducting ground states obtained in the phase diagram in Fig.2. The red and blue lines in (a) (c) correspond to the out-of-plane and in-plane Knight shift respectively. In the cal- culations, we set = 0:2and the superconducting order = 0:05 for theA1gandA2ustates, andf;g=f0:1;0:1gfor theB2g state, in accordance with the phase diagram in Fig.2. The other pa- rameters areft;t0;g=f1:0;0:8;0:3g. Another characteristic feature for the A2ustate is its in- plane upper critical ﬁeld exceeding the Pauli limit, which is closely related to the following facts. (i) In the A2ustate, the Cooper pair forms between an electron and its inversion partner, and the magnetic ﬁeld preserves the inversion sym- metry. (ii) The in-plane magnetic ﬁeld only modiﬁes the spin polarization in Fig.1 which is vital for the A2ustate as ana- lyzed in the above. For the A1gandB2gstates, due to the spin-singlet nature, their in-plane upper critical ﬁelds obey the Pauli limit. In the SM, we roughly estimate the in-plane upper critical ﬁelds numerically. Notice that, here we omit the possi- ble superconducting phase transitions, i.e. the phase transition from the even parity state to the odd parity state61,62and the transition to the Fulde-Ferrell-Larkin-Ovchinnikov state63,64, driven by the magnetic ﬁeld; and we also ignore the symme- try breaking effect arising from the in-plane magnetic ﬁeld. In summary, we ﬁnd that the nonsymmorphic lattice sym- metries can greatly assist the spin-triplet superconductivity in the presence of SOC. In a system respecting the space group P4=nmm , the nonsymmorphic symmetry makes the spin- tripletA2ustate be the leading pairing instability because of the spin-sublattice-momentum lock on electronic bands. Topologically, the spin-triplet A2ustate is trivial. The triv-5 iality can be easily understood from the concentric Fermi surface structure arising from the fourfold band degeneracy at M according to the parity criterion for centrosymmetric superconductors28,65. Our work unveils a new way in search- ing for the spin-triplet superconductors. The authors are grateful to Xianxin Wu for fruitful dis- cussions. This work is supported by the Ministry of Sci-ence and Technology of China 973 program (Grant No. 2017YFA0303100), National Science Foundation of China (Grant No. NSFC-12174428, NSFC-11888101 and NSFC- 11920101005), and the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDB28000000 and No. XDB33000000). Electronic address: qinshengshan@ucas.ac.cn yElectronic address: jphu@iphy.ac.cn 1G. E. V olovik, The universe in a helium droplet (Oxford Univer- sity Press, 2003). 2M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators , Rev. Mod. Phys. 82, 3045 (2010). 3X.-L. Qi and S.-C. Zhang, Topological insulators and supercon- ductors , Rev. Mod. Phys. 83, 1057 (2011). 4C.-K. Chiu, J. C. Y . Teo, A. P. Schnyder, and S. Ryu, Classiﬁca- tion of topological quantum matter with symmetries , Rev. Mod. Phys. 88, 035005 (2016). 5J. Alicea, New directions in the pursuit of Majorana fermions in solid state systems , Rep. Prog. Phys. 75, 076501 (2012). 6A. Y . Kitaev, Unpaired Majorana fermions in quantum wires , Phys. Usp. 44, 131 (2001). 7N. Hao and J. P. Hu, Topological quantum states of matter in iron-based superconductors: from concept to material realization , Natl. Sci. Rev. 6, 213 (2019). 8X. X. Wu, R.-X. Zhang, G. Xu, J. P. Hu, and C.-X. Liu, In the Pur- suit of Majorana Modes in Iron-based High-T cSuperconductors , arXiv:2005.03603. 9C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Non-Abelian anyons and topological quantum computation , Rev. Mod. Phys. 80, 1083 (2008). 10S. Vijay, T. H. Hsieh, and L. Fu, Majorana Fermion Surface Code for Universal Quantum Computation , Phys. Rev. X 5, 041038 (2015). 11B. Lian, X.-Q. Sun, A. Vaezi, X.-L. Qi, and S.-C. Zhang, Topo- logical quantum computation based on chiral Majorana fermions , PNAS 115, 10938 (2018). 12M. Sigrist and K. Ueda, Phenomenological theory of unconven- tional superconductivity , Rev. Mod. Phys. 63, 239 (1991). 13A. P. Mackenzie and Y . Maeno, The superconductivity of Sr 2RuO 4 and the physics of spin-triplet pairing , Rev. Mod. Phys. 75, 657 (2003). 14M. Smidman, M. B. Salamon, H. Q. Yuan, and D. F. Agterberg, Superconductivity and spin-orbit coupling in non- centrosymmetric materials: a review , Rep. Prog. Phys. 80, 036501 (2017). 15W. Kohn and J. M. Luttinger, New Mechanism for Superconduc- tivity , Phys. Rev. Lett. 15, 524 (1965). 16Y . Maeno, H. Hashimoto, K. Yoshida, S. Nishizaki, T. Fujita, J. G. Bedonorz, and F. Lichtenberg, Superconductivity in a layered perovskite without copper , Nature 372, 532 (1994). 17T. M. Rice and M. Sigrist, Sr2RuO 4: An electronic analogue of 3He? J. Phys. Condens. Matter 7, L643 (1995). 18A. Pustogow, Y . K. Luo, A. Chronister, Y .-S. Su, D. A. Sokolov, F. Jerzembeck, A. P. Mackenzie, C. W. Hicks, N. Kikugawa, S. Raghu, E. D. Bauer, and S. E. Brown, Constraints on the super- conducting order parameter in Sr 2RuO 4from oxygen-17 nuclear magnetic resonance , Nature 574, 72 (2019). 19R. Joynt and L. Taillefer, The superconducting phases of UPt 3,Rev. Mod. Phys. 74, 235 (2002). 20S. Ran, C. Eckberg, Q.-P. Ding, Y . Furukawa, T. Metz, S. R. Saha, I.-L. Liu, M. Zic, H. Kim, J. Paglione, and N. P. Butch, Nearly ferromagnetic spin-triplet superconductivity , Science 365, 684 (2019). 21J.-K. Bao, J.-Y . Liu, C.-W. Ma, Z.-H. Meng, Z.-T. Tang, Y .-L. Sun, H.-F. Zhai, H. Jiang, H. Bai, C.-M. Feng, Z.-A. Xu, and G.-H. Cao, Superconductivity in quasi-one-dimensional K 2Cr3As3with signiﬁcant electron correlations , Phys. Rev. X 5, 011013 (2015). 22J. Yang, J. Luo, C. J. Yi, Y . G. Shi, Y . Zhou, and G.-Q. Zheng, Spin-Triplet Superconductivity in K 2Cr3As3, Sci. Adv. 7, eabl4432 (2021). 23A. Bansil, H. Lin, and T. Das, Colloquium: Topological band the- ory, Rev. Mod. Phys. 88, 021004 (2016). 24Y . Ando, Topological Insulator Materials , J. Phys. Soc. Jpn. 82, 102001 (2013). 25Y . Ando and L. Fu, Topological Crystalline Insulators and Topo- logical Superconductors: From Concepts to Materials , Annu. Rev. Condens. Matter Phys. 6, 361 (2015). 26N. P. Armitage, E. J. Mele, and A. Vishwanath, Weyl and Dirac semimetals in three-dimensional solids , Rev. Mod. Phys. 90, 015001 (2018). 27B. H. Yan and C. Felser, Topological Materials: Weyl Semimetals , Annu. Rev. Condens. Matter Phys. 8, 337 (2017). 28L. Fu and E. Berg, Odd-Parity Topological Superconductors: The- ory and Application to Cu xBi2Se3, Phys. Rev. Lett. 105, 097001 (2010). 29L. Fu, Odd-parity topological superconductor with nematic order: Application to Cu xBi2Se3, Phys. Rev. B 90, 100509 (R) (2014). 30T. Mizushima, A. Yamakage, M. Sato, and Y . Tanaka, Dirac- fermion-induced parity mixing in superconducting topological in- sulators , Phys. Rev. B 90, 184516 (R) (2014). 31J. W. F. Venderbos, V . Kozii, and L. Fu, Odd-parity superconduc- tors with two-component order parameters: Nematic and chiral, full gap, and Majorana node , Phys. Rev. B 94, 180504 (R) (2016). 32T. Kawakami, T. Okamura, S. Kobayashi, and M. Sato, Topologi- cal Crystalline Materials of J= 3=2Electrons: Antiperovskites, Dirac Points, and High Winding Topological Superconductivity , Phys. Rev. X 8, 041026 (2018). 33S. Kobayashi and M. Sato, Topological Superconductivity in Dirac Semimetals , Phys. Rev. Lett. 115, 187001 (2015). 34T. Hashimoto, S. Kobayashi, Y . Tanaka, and M. Sato, Dirac- fermion-induced parity mixing in superconducting topological in- sulators , Phys. Rev. B 94, 014510 (2016). 35K. Matano, M. Kriener, K. Segawa, Y . Ando, and G.-Q. Zheng, Spin-rotation symmetry breaking in the superconducting state of CuxBi2Se3, Nat. Phys. 12, 852 (2016). 36S. Yonezawa, K. Tajiri, S. Nakata, Y . Nagai, Z. W. Wang, K. Segawa, Y . Ando, and Y . Maeno, Thermodynamic evidence for ne- matic superconductivity in Cu xBi2Se3, Nat. Phys. 13, 123 (2017). 37Y . Pan, A. M. Nikitin, G. K. Araizi, Y . K. Huang, Y . Matsushita, T. Naka, and A. de Visser, Rotational symmetry breaking in the6 topological superconductor Sr xBi2Se3probed by upper-critical ﬁeld experiments , Sci. Rep. 6, 28632 (2016). 38G. Du, Y . F. Li, J. Schneeloch, R. D. Zhong, G. D. Gu, H. Yang, H. Lin, and H.-H. Wen, Superconductivity with two-fold symmetry in topological superconductor Sr xBi2Se3, Sci. China Phys. Mech. Astron. 60, 037411 (2017). 39T. Asaba, B. J. Lawson, C. Tinsman, L. Chen, P. Corbae, G. Li, Y . Qiu, Y . S. Hor, L. Fu, and L. Li, Rotational Symmetry Breaking in a Trigonal Superconductor Nb-doped Bi 2Se3, Phys. Rev. X 7, 011009 (2017). 40R. Tao, Y .-J. Yan, X. Liu, Z.-W. Wang, Y . Ando, Q.-H. Wang, T. Zhang, and D.-L. Feng, Direct Visualization of the Nematic Superconductivity in Cu xBi2Se3, Phys. Rev. X 8, 041024 (2018). 41N. Reyren, S. Thiel, A. D. Caviglia, L. Fitting Kourkoutis, G. Hammerl, C. Richter, C. W. Schneider, T. Kopp, A.-S. R ¨uetschi, D. Jaccard, M. Gabay, D. A. Muller, J.-M. Triscone, and J. Mannhart, Superconducting Interfaces Between Insulating Ox- ides, Science 317, 1196 (2007). 42A. D. Caviglia, M. Gabay, S. Gariglio, N. Reyren, C. Cancellieri, and J.-M. Triscone, Tunable Rashba Spin-Orbit Interaction at Ox- ide Interfaces , Phys. Rev. Lett. 104, 126803 (2010). 43S. Nakosai, Y . Tanaka, and N. Nagaosa, Topological Supercon- ductivity in Bilayer Rashba System , Phys. Rev. Lett. 108, 147003 (2012). 44K. Yada, S. Onari, Y . Tanaka, and J. Inoue, Electrically controlled superconducting states at the heterointerface SrTiO 3/LaAlO 3, Phys. Rev. B 80, 140509 (R) (2009). 45J. P. Hu, Iron-Based Superconductors as Odd-Parity Supercon- ductors , Phys. Rev. X 3, 031004 (2013). 46The point group parts in the three symmetries act on the coor- dinates in Fig.1(a) as, I: (x;y;z )7!(x;y;z),My: (x;y;z )7!(x;y;z), andS4z: (x;y;z )7!(y;x;z). 47S. S. Qin, et al., to appear 48Notice that the effetive theory in Eq.(2) can be transformed to that in Ref.58with a similarity transformation. In constructing the ef- fective theory, though the symmetry group P4=nmm is nonsym- morphic, the matrices and f(k)can be classiﬁed according to the point group D4hdue to the bilinear form of the single-partile Hamiltonian. 49S. S. Qin, C. Fang, F.-C. Zhang, and J. P. Hu, Topological Super- conductivity in an Extended s-Wave Superconductor and Its Impli- cation to Iron-Based Superconductors , Phys. Rev. X 12, 011030 (2022). 50M. H. Fischer, M. Sigrist, D. F. Agterberg, and Y . Yanase, Superconductivity and Local Inversion-Symmetry Breaking , arXiv:2204.02449. 51X. W. Zhang, Q. H. Liu, J.-W. Luo, A. J. Freeman, and A. Zunger, Hidden spin polarization in inversion-symmetric bulk crystals , Nat. Phys. 10, 387 (2014). 52S.-L. Wu, K. Sumida, K. Miyamoto, K. Taguchi, T. Yoshikawa, A. Kimura, Y . Ueda, M. Arita, M. Nagao, S. Watauchi, I. Tanaka,and T. Okuda, Direct evidence of hidden local spin polarization in a centrosymmetric superconductor LaO 0:55F0:45BiS2, Nat. Com- mun. 8, 1919 (2017). 53Y . J. Zhang, P. F. Liu, H. Y . Sun, S. X. Zhao, H. Xu, and Q. H. Liu, Symmetry-Assisted Protection and Compensation of Hidden Spin Polarization in Centrosymmetric Systems , Chinese Phys. Lett. 37, 087105 (2020). 54The matrix form of the mirror symmetry and the nearly fully po- larized spin near the Brillouin zone boundary, is closely related to the nonsymmorphic symmetry. To show this, in the SM we analyze the condition near the Brillouin zone center where the spin polarization turns out to be vanishing samll. This is because the nonsymmorphic symmetry has nontrivial effect merely on the Brillouin boundary, and it is simply equivalent to the point group symmetry at the Brillouin zone center. 55Strictly speaking, the superconducting orders refer to leading or- der approximation of the momentum-dependent superconducting orders. More detailed analysis is shown in the SM. 56I. I. Mazin, Symmetry analysis of possible superconducting states in KxFeySe2superconductors , Phys. Rev. B 84, 024529 (2011). 57P. J. Hirschfeld, M. M. Korshunov, and I. I. Mazin, Gap symmetry and structure of Fe-based superconductors , Rep. Prog. Phys. 74, 124508 (2011). 58D. F. Agterberg, T. Shishidou, J. O’Halloran, P. M. R. Brydon, and M. Weinert, Resilient Nodeless d-Wave Superconductivity in Monolayer FeSe , Phys. Rev. Lett. 119, 267001 (2017). 59L. C. Hebel and C. P. Slichter, Nuclear Relaxation in Supercon- ducting Aluminum , Phys. Rev. 107, 901 (1957). 60L. C. Hebel and C. P. Slichter, Nuclear Spin Relaxation in Normal and Superconducting Aluminum , Phys. Rev. 113, 1504 (1959). 61S. Khim, J. F. Landaeta, J. Banda, N. Bannor, M. Brando, P. M. R. Brydon, D. Hafner, R. K ¨uchler, R. Cardoso-Gil, U. Stockert, A. P. Mackenzie, D. F. Agterberg, C. Geibel, and E. Hassinger, Nonsymmorphic symmetry and ﬁeld-driven odd-parity pairing in CeRh 2As2, Science 373, 1012 (2021). 62D. C. Cavanagh, T. Shishidou, M. Weinert, P. M. R. Brydon, and D. F. Agterberg, Nonsymmorphic symmetry and ﬁeld-driven odd- parity pairing in CeRh 2As2, Phys. Rev. B 105, L020505 (2022). 63P. Fulde and R. A. Ferrell, Superconductivity in a Strong Spin- Exchange Field , Phys. Rev. 135, A550 (1964). 64A. I.Larkin and Y . N. Ovchinnikov, Nonuniform state of super- conductors , Sov. Phys. JETP 20, 762 (1965). 65L. Fu and C. L. Kane, Topological insulators with inversion sym- metry , Phys. Rev. B 76, 045302 (2007). 66L. P. Gorkov, On the energy spectrum of superconductors , J. Ex- ptl. Theoret. Phys. (U. S. S. R.) 34, 735 (1958), Sov. Phys. JETP 7, 505 (1958). 67B. Mhlschlegel, Die thermodynamischen Funktionen des Supraleiters , Z. Phys. 155, 313 (1959). Appendix A: Matrix form for the symmetry operators at G and M The nonsymmorphic symmetry must lead to multiple sublattices in the system, as indicated in the lattice structure respecting the space group P4=nmm in the main text. Due to the sublattice degree of freedom, the matrix form for the symmetry operations isk-dependent. In the following, we take the fS4zj0gsymmetry for instance and construct its matrix form at G and M. To do this, we need to ﬁgure out how the symmetry operations act on the basis (jA(k)i;jB(k)i) = (cy k;dy k)j0i, wherej0iis the vacuum and the spin index has been omitted for convenience. The bases in the reciprocal space and in the real space are related7 TABLE A1: Matrix form for the symmetry operations at G and M. Here, I,My,Mxy,S4zandTstanding forfIj0g,fMyj0g,fMxyj0g, fS4zj0gand the time reversal symmetry respectively. I M yMxy S4zT Gs01is20i(s1+s2)1=p 2eis3=40is20K Ms01is23i(s1+s2)1=p 2eis3=43is20K by the Fourier transform jA(k)i=X jeikRj AjA(Rj A)i;jB(k)i=X jeikRj BjB(Rj B)i; (A1) where Rj A(Rj B) labels the position of the A (B) site in the jthunit cell and Rj ARj B=0=a1=2 +a2=2as shown in in the lattice in the main text. Under fS4zj0g, the G point is left unchanged, i.e. gk=k; however, the M point is transfromed as k!kb1. Accordingly, when fS4zj0gacts on the basis function, we have G:fS4zj0g(jA(k)i;jB(k)i) =eis3=4(jA(k)i;jB(k)i) (A2a) M:fS4zj0g(jA(k)i;jB(k)i) =eis3=4(jA(kb1)i;jB(kb1)i) =X jeis3=4(ei(kb1)Rj AjA(Rj A)i;ei(kb1)Rj Aei(kb1)0jB(Rj B)i) =X jeis3=4(eikRj AjA(Rj A)i;eikRj Aeik0eib10jB(Rj B)i) =X jeis3=4(eikRj AjA(Rj A)i;eikRj BjB(Rj B)i) =eis3=4(jA(k)i;jB(k)i); (A2b) where we have taken use of the fact that Rj Ab1mod 2equals 0 and b10=. Therefore, in the normal state fS4zj0ghas the matrix form eis3=40at G andeis3=43at M. Similar analysis can be applied to other symmetry operations, and we get the results in Table.A1. As pointed out in the main text, for the space group P4=nmm in the spinful condition it merely has one 4D irreducible representation at M, which is contributed by states with angular momenta Jz=1=2andJz=3=2deﬁned according to fS4zj0g(or the fourfold rotation fC4zj0g). However, in constructing the low-energy effective model in the main text we only consider the sorbital. At ﬁrst glance, the sorbital can not contribute states beyond Jz=1=2. According to Eq.(A2b), the additional angular momentum origins from the plane wave part of the Bloch wave function. Appendix B: Effective model at G In this section, we present the detailed construction of the low-energy effective model Heff;G(k)near the Brillouin zone center, i.e. the G point. The effective model near the M point shown in the main text can be constructed in a similar way. We ﬁrst consider the time reversal symmetry and the inversion symmetry, which constrain the system as TH eff;G(k)T1=Heff;G(k);IH eff;G(k)I1=Heff;G(k): (B1) The four-band model Heff;G(k)can be generally expressed in the form of the sixteen =sijmatrices. The constraints in Eq.(B1) merely allow six matrices, i.e. s00,s01,s02,s13,s23ands33, to appear inHeff;G(k). Then, we consider the constraints of the crystalline symmetries. Based on the matrix form of the symmetry operations in Table.A1, one can classify the above six matrices as shown in Table.B1. Therefore, Heff;G(k)must take the following form Heff;G(k) =m(k)s00+t0s01+kzs23+kys13; (B2) withm(k) =t(k2 x+k2 y)andt;t0;all the coefﬁcients. In fact, the effective model at G shown in Eq.(B2) has similar form with the model Hamiltonian considered in Ref.43, but their physical meanings are different. However, if we consider the phenomenological density-density interactions similar to that in the main text, we can expect similar conclusions with that in Ref.43and the spin-triplet superconductivity can appear in the strong SOC condition.8 TABLE B1: Classiﬁcation of the sijmatrices which are allowed by the time reversal symmetry and the inversion symmetry, and the functionsf(k)at G. The classiﬁcation is according to the D4hpoint group. ES4zI M yMxy space sij A1g 1 1 1 1 1 x2+y2s00,s01 B1u 1 1 -1 -1 1 s33 B2u 1 1 -1 1 -1 s02 Eu 2 0 -2 0 0 (x;y) ( s23;s13) Appendix C: Spin polarization on the energy bands In this part, we present the detailed calculations for the spin polarization on the energy bands based on the low-energy effective theory near the M point in the main text. The effective Hamiltonian near M can be solved as j'1(k)i=1p 20 B@1 cos isinei 01 CA;j'2(k)i=1p 20 B@0 isinei cos 11 CA; j'3(k)i=1p 20 B@1 cos isinei 01 CA;j'4(k)i=1p 20 B@0 isinei cos 11 CA; (C1) where sin=kp 2k2+t02k4sin22=4andcos=t0k2sin 2 2p 2k2+t02k4sin22=4withkwritten in the polar coordinates (kx;ky) = (ksin;kcos). In Eq.(C1),j'1(k)iandj'2(k)iare the two degenerate eigenstates corresponding to eigenvalue E= tk2q 2k2+t02k4sin22=4, whilej'3(k)iandj'4(k)iare the two degenerate eigenstates with eigenvalue E+= tk2+q 2k2+t02k4sin22=4. The sublattice-distinguished spin operators are sA=B;i =si(03)=2. Straightforwardly, the spin polarization on theEbands can be calculated as hsA=Bi=h'1jsA=Bj'1i+h'2jsA=Bj'2i, which turns out to be hsBi=hsAi= (sinsin;sincos;0). Obviously,jhsA=Bijatkis proportional to1p 1+t02k2sin22=42. The spin polarization on the E+ bands can be calculated similarly. For the Fermi surfaces near the G point, according to Eq.(B2), the spin polarization can be obtained as jhsA=Bij= 1p 1+t02=42k2. Comparing the results near G and M, one immediately comes to the conclusion, the spin polarization is van- ishing small for small Fermi surfaces near G, while it is nearly fully polarized for small Fermi surfaces near M. Appendix D: Derivation of the superconducting ground state In the main text, we get the superconducting ground states by solving the linearized gap equations. Here, we present the details on the derivation of the linearized gap equation, and present more analysis on the calculations of the superconducting susceptibility. 1. Linearized gap equation In the superconducting state, the Green functions can be deﬁned as Gij(k;) =hTci(k;)cy j(k;0)i; (D1) Fij(k;) =hTci(k;)cj(k;0)i; Fy ij(k;) =hTcy i(k;)cy j(k;0)i: In the mean-ﬁeld level, the superconducting order can be calculated as (k) =1 NX k0U(k;k0)F(k0;= 0) =1 NX k0;nU(k;k0)F(k0;i!n): (D2)9 Notice that the Fourier transformation for Eq.(D1) is as follows: g() =1 P+1 n=1ei!ng(i!n)andg(i!n) =R 0ei!ng() with!n=2n for boson and !n=(2n+1) for fermion. According to Eq.(D2), we must calculate F(k0;= 0) ﬁrstly. To do this, we consider the Gor’kov equations66,i:e:the equation of motion in the superconducting state, which read as G1 0(k;i!)G(k;i!) +(k)Fy(k;i!) = 1; (D3) G1 0(k;i!)F(k;i!)(k)GT(k;i!) = 0; (G1 0)T(k;i!)Fy(k;i!) +y(k)G(k;i!) = 0: According to the Gor’kov equations, we can derive the following equations Fy(k;i!) =GT 0(k;i!)y(k)G(k;i!); (D4) F(k;i!) =G0(k;i!)(k)GT(k;i!); G1(k;i!) =G1 0(k;i!) +(k)GT 0(k;i!)y(k); whereG0(k;i!)is the normal-state Green function. In the weak-coupling condition, is small and we have F(k;i!)'G0(k;i!)(k)GT 0(k;i!) =G0(k;i!)(k)G 0(k;i!); (D5) where we have used the identity G0(k;i!) =Gy 0(k;i!). The normal-state Green function can be written in the band basis G0(k;i!) =1 i!h0(k)=X sjus;kihus;kj i!s;k; (D6) wheresis the band index, and jus;ki(s;k) is the eigenfunction (eigenenergy) for band s. Accordingly, the anomalous super- conducting Green function is F(k;i!) =X s;s0jus;kihus;kj(k)ju s0;kihu s0;kj (i!s;k)(i!s0;k); (D7) where we useju s0;kito lable (jus0;ki). In the weak-coupling condition, the superconductivity is mainly contributed by electrons on the Fermi surfaces. Moreover, since the system in our consideration possesses both the time reversal symmetry and inversion symmetry, the electron on the Fermi surfaces can always form Cooper pair with its time reversal or inversion partner, namelys0= sors0=s, and F(k;i!) =X s;s0jus;kihus;kj(k)ju s0;kihu s0;kj !2+2 s;k: (D8) Correspondingly, the superconducting order parameter in Eq.(D2) is (k) =1 NX k0;nU(k;k0)F(k0;i!n) =1 NX k0;s;s0U(k;k0)jus;k0ihus;k0j(k0)ju s0;k0ihu s0;k0j1 X n1 !2n+2 s;k0 =1 NX k0;sU(k;k0) 2s;k0tanhs;k0 2X s0=s;sjus;k0ihus;k0j(k0)ju s0;k0ihu s0;k0j; (D9) where only the electronic states on the Fermi surfaces are taken into account in the weak-coupling condition. In calculating the frequency summation in Eq.(D9), we have used the relationH jzj!1dz 2i1 2z21 ez+1= 0. For each irreducible representation channel, the superconducting order parameter (k)can be expanded according to the corresponding bases. Considering the orthonormality of the basis functions, we have X ktr[(k)y 0(k)] =X k;tr[(k)y 0(k)] = (D10) =X ktr[1 NX k0;sU(k;k0) 2s;k0tanhs;k0 2X s0=s;sjus;k0ihus;k0j(k0)ju s0;k0ihu s0;k0jy 0(k)] =X 1 NX k;k0;sU(k;k0) 2s;k0tanhs;k0 2X s0=s;shus;k0j(k0)ju s0;k0ihu s0;k0jy 0(k)jus;k0i =X 1 NX k;k0;sU(k;k0) 2s;k0tanhs;k0 2X s0=s;shus;k0j^(k0)jus0;k0ihus0;k0j^y 0(k)jus;k0i;10 where (k) =^(k)is2. Eq.(D10) can be intuitively expressed in the form X=with X;0() =1 NX k;k0;sU(k;k0) 2s;k0tanhs;k0 2X s0=s;shus;k0j^(k0)jus0;k0ihus0;k0j^y 0(k)jus;k0i: (D11) In our consideration the interaction is k-independent, and in the continuum condition we have X;0() =1 NX sZ dk0U 2s;k0tanhs;k0 2X s0=s;shus;k0j^jus0;k0ihus0;k0j^y 0jus;k0i; (D12) =1 NX sZdk0 ds;k0ds;k0dU 2s;k0tanhs;k0 2X s0=s;shus;k0j^jus0;k0ihus0;k0j^y 0jus;k0i; =U NX sZ!0 !0ds;k01 2s;k0tanhs;k0 2Z dD()X s0=s;shus;k0j^jus0;k0ihus0;k0j^y 0jus;k0i; =UF ()X sZ dD()X s0=s;shus;k0j^jus0;k0ihus0;k0j^y 0jus;k0i=U;0(Tc): In Eq.(D12),F() =1 2NR!0 !01 2s;k0tanhs;k0 2ds;k0is a temperature-dependent constant with !0the energy cutoff near the Fermi energy, and D() = 2dk0=ds;k0is the density of states on the Fermi surface. By solving the characteristic equation U(Tc) =I, we can get the superconducting ground state. 2. Superconductivity from density-density interactions In the main text, we consider superconductivity induced from the phenomenological density-density interactions Hint=Z dq[U2X i=1ni(q)ni(q) + 2Vn1(q)n2(q)]; (D13) =1 NZ dkdk0dq(Ucy k0+q;ck0;cy kq;ck;+Udy k0+q;dk0;dy kq;dk;+ 2Vcy k0+q;ck0;dy kq;0dk;0): In the superconducting channel, i.e. k=k0, we have Hint=1 NZ dk0dq(Ucy k0+q;ck0;cy k0q;ck0;+Udy k0+q;dk0;dy k0q;dk0;+ 2Vcy k0+q;ck0;dy k0q;0dk0;0) =1 NZ dkdk0(Ucy k;cy k;ck0;ck0;+Udy k;dy k;dk0;dk0;+ 2Vcy k;dy k;0dk0;0ck0;): (D14) The interactions in Eq.(D14) can be expanded according to the superconducting orders. When we consider Fermi surfaces near the M point, we need to expand the interactions according to the pairing orders classiﬁed at M which is shown in the main text Hint;M =1 NZ dkdk0[U(cy k;cy k;ck0;ck0;+dy k;dy k;dk0;dk0;) + 2Vcy k;dy k;0dk0;0ck0;] (D15) =1 4NZ dkdk0[U(^A1g^y A1g+^B2u^y B2u) +V(^A2u^y A2u+^B2g^y B2g+^(1) Eu^(1)y Eu+^(2) Eu^(2)y Eu)]: Based on Eqs.(D12)(D15), we can ge the linearized gap equations for each irreducible representation channel shown in the main text.11 3. Calculations of the superconducting susceptibility Based on the wave functions in Eq.(C1), we can calculate the superconducting susceptibility for each irreducible representation channel at M shown in the main text straightforwardly A1g=F()X sX s0=s;sZ d[D()jhu s;k0js00ju s0;k0ij2+D+()jhu+ s;k0js00ju+ s0;k0ij2] =F()Z d[D() +D+()]2; B2g=F()Z d[D() +D+()]2 cos2; A2u=F()Z d[D() +D+()]2 sin2; B2u=F()Z d[D() +D+()]2 sin2; Eu=F()Z d[D() +D+()]2 sin2sin2: (D16) In the above equations, Dandju s0iare the density of states and the eigenstates on the Fermi surface contributed by the energy bandErespectively. According to the results in Eq.(D16) it is obvious to notice that, in the intrasublattice pairing channels A1g>B2uand in the intersublattice channels A2u>Eu. Moreover, considering that A2u+B2g=A1g, if we compare theA1gstate with a special case in the intersublattice pairing channels where A2u=B2g, i.e. the phase boundary between the A2uandB2gstate, one can ﬁnd that the A1gstate can never be the ground state for 2jUj<jVjassumingVattractive, which is consistent with the phase diagram in the main text. Appendix E: Lattice Model In the main text, we analyze the k-independent superconducting ground states based on a low-energy effective model. Here, we show that these k-independent states are the leading order approximation, i.e. the superconducting orders preserved to the 0thorder of k, in the minimal lattice model. Taking the lattice structure in the main text into consideration and substituting kx=ywith appropriate trigonometric functions, we can get the corresponding lattice model H(k) = 2t(coskx+ cosky+ 2)s00sinkxs23sinkys13+ 4t0coskx 2cosky 2s01: (E1) The above lattice model respects the space group P4=nmm with the lattice sites located at the D2dinvariant points, and at each lattice site an sorbital is considered as pointed out in the main text. In Eq.(E1), tis the intrasublattice nearest-neighbour hopping, t0is the intersublattice nearest-neighbour hopping, and =2is the inversion-symmetric Rashba SOC. Based on the lattice model, we calculate the sublattice-distinguished spin polarization on the energy bands and plot the results in Fig.E1. As shown, the lattice model captures the essential features analyzed in the main text and in the above sections: for a system respecting the space group P4=nmm , it has fully polarized sublattice-distinguished spin polarization on the energy bands near the Brillouin zone boundary, i.e. kx=y=; while the spin polarization is vanishing small near the Brillouin zone center. FIG. E1: (color online) Sketch for the sublattice-distinguished spin polarizations on the lower energy bands (a) and upper energy bands (b), plotted from the lattice model in Eq.(E1). The parameters are set to be ft;t0;g=f1:0;0:8;0:8g. The symbols in the ﬁgures are the same with that in the main text. For the interacting part, we consider the following electron density-density interaction Hint(k) =UX i;nini+VX hiji;;0ninj0; (E2)12 whereUis the onsite interaction and Vis the intersublattice nearest-neighbour interaction. By doing a Fourier transformation, ci=1p NP kckeikriand1 NP iei(kk0)ri=k;k0, we obtain Hint(k) =U NX kk0q(cy k0+q;ck0;cy kq;ck;+dy k0+q;dk0;dy kq;dk;) +V NX kk0qcy k0+q;ck0;dy kq;0dk;0X =hijieiq(rirj) =U NX kk0q(cy k0+q;ck0;cy kq;ck;+dy k0+q;dk0;dy kq;dk;) +V NX kk0qcy k0+q;ck0;dy kq;0dk;04 cosqx 2cosqy 2: (E3) In the superconducting channel, we set k=k0and get Hint(k) =U NX k0q(cy k0+q;ck0;cy k0q;ck0;+dy k0+q;dk0;dy k0q;dk0;) +V NX k0qcy k0+q;ck0;dy k0q;0dk0;04 cosqx 2cosqy 2 =U NX kk0(cy k;cy k;ck0;ck0;+dy k;dy k;dk0;dk0;) +V NX kk0cy k;dy k;0dk0;0ck0;4 coskxk0 x 2coskyk0 y 2: (E4) Apparently, the onsite interaction can only contribute the constant intrasublattice pairing order. For the intersublattice part, we have Hint;NN (k) =V NX kk0cy k;dy k;0dk0;0ck0;(coskx 2cosky 2cosk0 x 2cosk0 y 2+ sinkx 2sinky 2sink0 x 2sink0 y 2 + coskx 2sinky 2cosk0 x 2sink0 y 2+ sinkx 2cosky 2sink0 x 2cosk0 y 2): (E5) According to the equation, it can be noticed that the term with form factor sinkx 2sinky 2sink0 x 2sink0 y 2dominates the other terms for small Fermi surfaces near the M point ( kx=y), since sinkx=y 2= 1k2 M;x=y 2andcoskx=y 2=kM;x=y 2with kM=kKMandKMbeing the M point ( kMis samll). Therefore, for small Fermi surfaces near M, it is reasonable we only consider the constant pairing orders between different sublattices, i.e. the 0thorder of k-dependent pairing orders contributed by the sinkx 2sinky 2sink0 x 2sink0 y 2term in Eq.(E5), which is exactly the consideration in the main text. Appendix F: Magnetic response In this part, we provide more details on the numerical simulation for the Knight shift and the spin relaxation rate, and present a rough numerical estimation for the in-plane upper critical ﬁeld, for the superconducting ground states in the phase diagram in the main text. 1. Knight shift and spin relaxation rate In the nuclear magnetic resonance, the Knight shift Kssand the spin relaxation rate 1=T1are measured through the static spin susceptibility. In the general condition, the spin susceptibility is deﬁned as st(q;i!) =Z 0dst(q;) =Z 0dhTSs(q)St(q)iei!: (F1) In our consideration, the Knight shift in the nuclear magnetic resonance reads Kss(T)/X ss(0;0)/X k;m;n;jhm(k)jS sjn(k)ij2n(Ekm)n(Ekn) EkmEkn: (F2)13 The spin relaxation rate is 1 T1(T)/lim !!0X q;;sjA(q)j2Im ss(q;!+i0+) !(F3) / X k;k0;m;n;s;jA(kk0)j2jhm(k)jS sjn(k0)ij2@n(E) @E E=Ekm(EkmEk0n): In the above equations, S s(s= 1;2;3and=A;B ) is the spin operator for sublattice in the Nambu space ( y(k);is2 (k)), with y(k)being the basis for the normal-state Hamiltonian shown in the main text. Speciﬁcally, SA s=ss 0+3 2 0andSB s=ss 03 2 0, with0=I22in the Nambu space. Ekmandm(k)are the energy and wavefunction for the mth eigenstate for the superconducting Hamiltonian respectively, and A(q)in Eq.(F3) is the structure factor which is set to be 1. In the numerical calculations, we set the superconducting transition temperature kbTc= 0=3:53 with 0the zero temperature pairing order, and consider the T-dependent superconducting order (T) = 0f(T=Tc)with f(T=Tc)being the BCS-type normalized gap presented in Ref.67. FIG. F1: (color online) The Knight shift for the A1gandA2ustates in different conditions. In the calculations, the other parameters are chosen asft;t0;g=f1:0;0:8;0:3g. The symbols in the ﬁgures are the same with that in the main text. In the main text, we claim that for the system in our consideration, the Knight shift can be affected by the SOC and the interband pairing. Here, we present more numerical results. We mainly consider the A1gandA2ustates, since the B2gstate can only appear in the weak SOC condition (if the SOC is strong, the B2gstate is nodal and cannot be the ground state) where the Knight shift is qualitatively the same with that in the main text. As indicated in Fig.F1 and the results shown in the main text, in the weak pairing condition, i.e. << , the Knight shift for the two states merely changes quantitatively as the SOC varies. TheA2ustate can be distinguished from the other states through the unsuppressed Kzz. However, if the pairing is strong, i.e. , the interband pairing changes the results qualitatively and it suppresses Kzzin the superconducting state. Therefore, in the strong pairing condition, it is not a good choice to use the Knight shift to distinguish the different states. 2. In-plane upper critical ﬁeld For the superconducting ground states in the phase diagram in the main text, the in-plane upper critical ﬁeld of spin-triplet A2ustate is larger than the Pauli limit. Here, we show this by carrying out rough numerical simulations for the pairing orders in presence of the in-plane magnetic ﬁeld. We consider the following Hamiltonian Htotal=Heff+Hint+Hmag; (F4) whereHmag=Bs10is the in-plane Zeeman ﬁeld, and HeffandHintare shown in the main text. In the calculations, we set the parameters as,ft;t0;g=f1:0;0:8;0:3;0:2gfor theA1gandA2ustates, andft;t0;g=f1:0;0:8;0:3;0:1gfor theA1g for theB2gstate. For the Hamiltonian in Eq.(F4), we solve the superconducting gap equation (k) =1 NP k0UF(k0;= 0) itinerantly. In solving the gap equation, for the A1gchannel we choose U=0:8which leads to pairing order in the absence of the Zeeman ﬁeld 0;A1g= 0:06; and for the B2gandA2uchannels, we set V=1:0corresponding to the pairing orders in14 the absence of the Zeeman ﬁeld 0;B2g= 0:11and0;A2u= 0:07. Turning on the in-plane magnetic ﬁeld, we get the pairing orders in Fig.F2. Obviously, for the parameters chosen in the above, the pairing orders for the A1gandB2gstates vanish at B0, while theA2ustate has the upper critical ﬁeld B50. FIG. F2: (color online) By solving the Hamiltonian in Eq.(F4), we get the pairing orders in the presence of the in-plane magnetic ﬁeld.
1403.4265v1.Single_parameter_spin_pumping_in_driven_metallic_rings_with_spin_orbit_coupling.pdf	arXiv:1403.4265v1 [cond-mat.mes-hall] 17 Mar 2014Single-parameter spin-pumping in driven metallic rings wi th spin-orbit coupling J. P. Ramos,1L. E. F. Foa Torres,2P. A. Orellana,3and V. M. Apel1 1Departamento de F´ ısica, Universidad Cat´ olica del Norte, Angamos 0610, Casilla 1280, Antofagasta, Chile 2Instituto de F´ ısica Enrique Gaviola (CONICET) and FaMAF, Universidad Nacional de C´ ordoba, Ciudad Universitaria 50 00, C´ ordoba, Argentina 3Departamento de F´ ısica, Universidad Federico Santa Mar´ ı a, Avenida Vicu˜ na Mackenna 3939, San Joaquin, Santiago, Chil e (Dated: May 8, 2019) Abstract We consider the generation of a pure spin-current at zero bia s voltage with a single time- dependent potential. To such end we study a device made of a me soscopic ring connected to electrodes and clarify the interplay between a magnetic ﬂux , spin-orbit coupling and non-adiabatic driving in the production of a spin and electrical current. B y using Floquet theory, we show that the generated spin to charge current ratio can be controlled by tuning the spin-orbit coupling. 1I. INTRODUCTION Largelyforgottenduringtheearlydecadesofnanoelectronics, t hespindegreeoffreedomis becoming ever closer to the center of the research stage1,2. Indeed, generating and detecting spin-currents is now a fascinating ﬁeld of research3,4with applications in future electronics5, quantum computing6and information storage7. Among the many ways of harnessing the electron spin, spin orbit interaction (SOI) in two-dimensional electr on gases is a promising one since the spin transport properties can be controlled simply by a pplying an electric ﬁeld8,9. Most of the proposals aiming at the control of the spin degree of fr eedom use static elec- tric or magnetic ﬁelds3,4. Here we follow a diﬀerent path and use alternating ﬁelds (ac) as in10,11. The time-dependence introduced by the alternating ﬁelds12provide an avenue for exploring new phenomena including the opening of a laser-induced ban dgap13,14or chiral edge states15and, more generally, the tuning of its topological properties15–18. Another striking phenomena is the coherent generation of a current at zer o bias voltage (termed quantum charge pumping )19–21and, as shown below, the generation of a pure spin-currents throughspinpumping. Quantumpumping isusuallyachieved bydrivinga sampleconnected to electrodes through ac gate voltages. Within the adiabatic appro ximation22, pumping a non-vanishing charge requires the presence of at least two time-d ependent parameters (typ- ically constituted by gate voltages) and has been widely studied in man y systems including pristine23,24and disordered graphene25. But beyond the adiabatic approximation single- parameter pumping is also possible as predicted theoretically26–28(similar to the mesoscopic photovoltaic eﬀect predicted earlier in29) and achieved in careful experiments30–32. Besides reducing the burden of adding more contacts in a nanoscale sample, a single parameter setup could also prove advantageous (as a compared to a two-par ameter one) in reducing capacitive eﬀects and crosstalk between time-dependent gates. Here we address the eﬀect of spin-orbit coupling and its interplay wit h a single time- dependent ﬁeld in the generation of non-adiabatic spin current at z ero bias voltage. To such end we consider a setup as the one represented in Fig.1, where a nan oscale ring is connected to electrodes and has a quantum dot embeded in one of its arms. The time dependence is introduced as an alternating gate applied to the quantum dot and do es not break neither time-reversal nor inversion symmetry (parity). Crucial to the ge neration of pumped current 2is the addition of a magnetic ﬂux threading the ring as shown in Fig. (1) . The spin-orbit coupling is introduced as an additional spin-dependent ﬂux. In this p aper we show how this simple setup is able to provide a minimal model where a pure spin-curre nt can be achieved. FIG. 1. Scheme of setup considered in the text, a quantum ring with a magnetic ﬁeld cross them and a quantum dot embedded in one of its arms, driven by an ac vo ltage source. II. HAMILTONIAN MODEL AND ITS SOLUTION THROUGH FLOQUET THE- ORY Let us start our discussion by presenting our model Hamiltonian for the situation repre- sented in Fig.(1). The total Hamiltonian H(⊔) is written as: H(t) =HC+HQD(t)+HT, (1) whereHCrepresents the left and right contacts and the lower arm of the rin g (represented by sitej= 0 in the notation below), HQD(t) the quantum dot in the upper arm of the ring (which for simplicity is taken to be a single level), and HTthe tunneling Hamiltonian between the quantum-ring and the contacts, which are given by HC=∞/summationdisplay j=−∞,σ(εjc† j,σcj,σ+γc† j,σcj+1,σ)+h.c., (2) HQD(t) =εd(t)/summationdisplay σd† σdσ (3) HT=/summationdisplay σ(Vσ Lc† −1,σdσ+VRc† 1,σdσ)+h.c.. (4) The time dependence is introduced as a modulation of the energy leve ls of the quantum dot. For a single-level quantum-dot, this is is achieved through ε0(t) =ε0+vcos(Ω0t). We 3consider a magnetic and electric ﬁelds in the system, their contribut ions to the Hamiltonian are embedded in the hopping matrix elements Vσ L=V0exp[i2π(φAB+σφSO)/φ0], where φABandφSOare the phases due to the Aharonov-Bohm eﬀect and spin-orbit int eraction respectively, σis the spin index ( σ=↑,↓orσ= 1,−1) andφ0is the ﬂux quantum. Since we are interested in a single-parameter pumping conﬁguration as in28,30,33,34, the calculation of the electrical response requires going beyond the ad iabatic theory. Floquet theory oﬀers a suitable framework12,21. Here we use it in combination with Green’s func- tions, then we have a Floquet-Green function denoted by GFdeﬁned from the Floquet’s Hamiltonian HFas28,35GF= [EI −H F]−1. If the spin-orbit coupling does not couple diﬀerent spin channels, as in our case, the dc component of the current is given by: ¯Iσ=1 τ/integraldisplayτ 0dtIσ(t) (5) ¯Iσ=e h× (6) /summationdisplay n/integraldisplay/bracketleftBig T(n) (R,σ),(L,σ)(ε)fL(ε)−T(n) (L,σ),(R,σ)(ε)fR(ε)/bracketrightBig dε, whereT(n) (R,σ),(L,σ)(ε) is the probability for an electron on the left ( L) with spin σand energy εto be transmitted to the right ( R) reservoir while exchanging nphotons and τ= 2π/Ω0. These probabilities are weighted by the usual Fermi-Dirac distributio n functions fR(L)for each electrode and are given, in terms of Floquet-Green function, by T(n) (R,σ),(L,σ)(ε) = 4Γ R(ε+n/planckover2pi1Ω0)\|G(n) LR,σ(ε)\|2ΓL(ε), (7) where the probability in opposite direction is described exchanging th e indexLwithR, and ΓL(R)is the matrix coupling with left (right) electrode, deﬁned as the imagin ary part of the electrode’s self energies, i. e. Γ L(R)=−Im(ΣL(R)). The associated spin-current is ¯Is=¯I↑−¯I↓while the charge current is ¯I=¯I↑+¯I↓. III. RESULTS AND DISCUSSION Using the model introduced before we now turn to our results for t he pumped electric and spin currents. To start with we consider the system in the absence of spin-orbit coupling. 4We consider the leads in thermodynamic equilibrium ( i. e.fL(ε) =fR(ε) =f(ε)) as a semi- inﬁnite 1d system with nearest neighbor coupling γ, which is used as energy parameter. The ac ﬁeld frequency is set to Ω 0such that /planckover2pi1Ω0=γ/5, and the ﬁeld magnitude is v= 0.07γ/e. The hopping between the contacts and QD is V0=γ/4. FIG. 2. (Color online) (a) Transmission probability from le ft to right as a function of the applied magnetic ﬂux and the Fermi energy (for vanishing spin-orbit interaction). (b) Same as (a) for the pumped current. Note the emergence of local maxima/minima c lose to the parameters where a transmission zero is observed. Figure 2a shows a contour plot of the transmission probability as a fu nction of the Fermi level position and the magnetic ﬂux. There we can observe the pres ence of a region where the transmission is very close to zero (close to the intersection of t he dashed lines). This is due to a destructive quantum interference known asFanoresona nce or antiresonance36–38(for a recent review39). Interestingly, the pumped current shown in Fig. 2b achieves a ma ximum 5intensity whenever the parameters are tuned close to the transm ission zero. Besides, we can see that the sign of the observed maxima is reversed when travers ing the transmission zero. We note that a single-time dependent harmonic potential does not b reak time reversal symmetry (being deﬁned as the existence of a time t0such that the Hamiltonian which is a function of the time tsatisﬁesH(t0+t) =H(t0−t)). It is the magnetic ﬁeld that breaks TRS and allows for pumping to occur. Note, however that th is is true only whenever magnetic ﬂux is diﬀerent fromthe half integer multiples of theﬂux qua ntum. For a magnetic ﬂux ofπfor example, the Hamiltonian does not change upon time-reversal ( the phase in the hopping term VLchanges from exp( iπ) to exp( −iπ) and therefore there is no pumped current as observed in Fig. 2-b. On the other hand one should note that the magnetic ﬁeld alone would not produce pumping. This is the point where the time- dependent ﬁeld enters into the game. Its role in this setup is to provide for additiona l eﬀective channels for transport, thereby circumventing the constraint of phase- rigidity40and allowing for the directional asymmetry in the transmission probabilities. The addition of spin-orbit interaction breaks the spin degeneracy a nd at zero bias both chargeandspincurrentsaregenerated. ByexaminingFig. (2)onecanimaginet hatthespin- orbit phase may be used to tune the working point of our pump for ea ch spinindependently . Indeed, the term φAB+σφSOenters as an eﬀective spin-dependent ﬂux φeﬀ σ. In particular, we could choose this spin-orbit phase so that it cancels out for one s pin direction (leading to a vanishing pumped charge for this spin) and adds up for the othe r, or in such a way as to cancel the charge current while summing up towards the spin cur rent. Figures 3 (a) and (b) show the charge (solid lines) and spin (dashed lin es) currents as a function of the Fermi energy for diﬀerent values of the spin-orb it and Aharanov-Bohm phases, while Figures 3 (c) and (d) show the pumped current for ea ch spin, spin up (black solid lines) and down (red dashed lines). As anticipated, the paramet ers can be chosen so that the currents for each spin direction have opposite signs (Fig. 3 (d)), thereby leading to a pure spin-current as on Fig.3 (b). In this situation, the charge cu rrent cancels out whether the spin-current is maximal. A point that needs to be emphasized in this proposal is that the pump ed current is in- trinsically non-adiabatic (this contrasts for example with Ref.10using a similar setup but with two time-dependent parameters). An adiabatic calculation wou ld actually give a van- ishing response. Going beyond this adiabatic (low-frequency) limit is t herefore mandatory 6 FIG. 3. (Color online) a-b Pumpedcharge ( ¯I, dashed line) and spin currents ( ¯Is, solid line), dashed and solid for diﬀerent values of the applied static ﬂux and spi n-orbit interaction: (a) φAB= 0.2φ0, φSO= 0.1φ0and b)φAB= 0.5φ0,φSO= 0.4φ0. Onecan see that in (b) the charge current vanishes but the spin current is enhanced. The spin-resolved contrib utions to the current for the same cases are shown in (c) and (d). justifying the use of Floquet theory. On the other hand, the pump ed currents in this case emerges as an interplay between photon-assisted processe s and the interference in the Aharanov-Bohm ring28. A similar setup but without contacts to electrodes were considere d in41,42. The spin-orbit coupling allows to obtain spin polarized pumped curren ts and the key role of the time-dependent ﬁeld is to provide for additional paths fo r interference breaking phase-rigidity40, although it does not break time-reversal symmetry. 7The setup discussed here can be realized by using the present tech nologies. A quantum dot inserted in a mesoscopic ring has been fabricated by several lab oratories in the last decades43,44. A particularly interesting case would be an InGAsquantum-dot inserted in a mesoscopic quantum-ring since InGAshas a strong spin-orbit coupling and this coupling can be controlled by an electric ﬁeld45. IV. FINAL REMARKS In summary, we study quantum spin-pumping with a single parameter in a conﬁguration where the eﬀect of the time-dependent ﬁeld is reduced to the esse ntial one: providing for additional channels for transport. The spin-orbit interaction bre aks the spin degeneracy and we exploit it to generate a pure pumping spin-current through the in dependent tuning of the phases dues to Aharonov-Bohm eﬀect and spin-orbit coupling. V. ACKNOWLEDGMENTS We acknowledge support of the SPU-Argentina (Project PPCP-02 5), SeCyT-UNC, ANPCyT-FonCyT, FONDECYT project number 1100560, CONICYT ( Project PSD-65), and the scholarship CONICYT academic year 2012 number 22121816 . 1S. A. Wolf et al., Science 294, 1488 (2001). 2F. Pulizzi, Nature 11, 367 (2012). 3I. Zutic, I., J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). 4J. Sinova and I. Zutic, Nat Mater 11, 368 (2012). 5R. Jansen, Nature Materials 11, 400 (2012). 6D. D. Awschalom et al., Science 339, 1174 (2013). 7D. R. McCamey, J. Van Tol, G. W. Morley, and C. Boehme, Science 330, 1652 (2010). 8S. Datta and B. Das, Applied Physics Letters 56, 665 (1990). 9J. Nitta, F. E. Meijer, and H. Takayanagi, Applied Physics Le tters75, 695 (1999). 10R. Citro and F. Romeo, Phys. Rev. B 73, 233304 (2006). 811B. H. Wu and J. C. Cao, Phys. Rev. B 75, 113303 (2007). 12S. Kohler, J. Lehmann, and P. H¨ anggi, Physics Reports 406, 379 (2005). 13Y. H. Wang, H. Steinberg, P. Jarillo-Herrero, N. Gedik, Scie nce342, 453 (2013). 14T. Oka and H. Aoki, Phys. Rev. B 79, 081406 (2009); H. L. Calvo, H. M. Pastawski, S. Roche, L. E. F. Foa Torres, Appl. Phys. Lett. 98, 232103 (2011). 15P. M. Perez-Piskunow, G. Usaj, C. A. Balseiro, L. E. F. Foa Tor res, Phys. Rev. B 89, 121401(R) (2014). 16N. H. Lindner, G. Refael, V. Galitski, Nature Physics 7, 490 (2011). 17Y. Tenenbaum Katan and D. Podolsky, Phys. Rev. B 88, 224106 (2013). 18M. Thakurathi, K. Sengupta, D. Sen, Majorana edge modes in th e Kitaev model, arxiv: arXiv:1310.4701 [cond-mat.mes-hall] (unpublished). 19D. J. Thouless, Phys. Rev. B 27, 6083 (1983). 20M. B¨ uttiker and M. Moskalets, in Lecture Notes in Physics , edited by J. Asch and A. Joye (Springer Berlin Heidelberg , 2006), Vol. 690, pp. 33–44–. 21M. Moskalets and M. B¨ uttiker, Phys. Rev. B 66, 205320 (2002). 22P. W. Brouwer, Phys. Rev. B 58, R10135 (1998). 23E. Prada, P. San-Jose, and H. Schomerus, Phys. Rev. B 80, 245414 (2009). 24R. Zhu and H. Chen, Applied Physics Letters 95, 122111 (2009). 25L. Ingaramo and L. E. F. Foa Torres, Appl. Phys. Lett. 103, 123508 (2013). 26T. A. Shutenko, I. L. Aleiner, and B. L. Altshuler, Phys. Rev. B61, 10366 (2000). 27L. Arrachea, Phys. Rev. B 72, 121306 (2005). 28L. E. F. Foa Torres, Phys. Rev. B 72, 245339 (2005). 29V. Falko and D. Khmelnitskii, Sov. Phys. JETP 68, 186 (1989). 30B. Kaestner et al., Phys. Rev. B 77, 153301 (2008). 31A. Fujiwara, K. Nishiguchi, and Y. Ono, Appl. Phys. Lett. 92, 042102 (2008). 32B. Kaestner et al., Appl. Phys. Lett. 94, 012106 (2009). 33A. Agarwal and D. Sen, Phys. Rev. B 76, 235316 (2007). 34L. E. F. Foa Torres, H. L. Calvo, C. G. Rocha, and G. Cuniberti, Appl. Phys. Lett. 99, 092102 (2011). 35L. E. F. Foa Torres and G. Cuniberti, Appl. Phys. Lett. 94, 222103 (2009). 36F. Guinea and J. A. Verg´ es, Phys. Rev. B 35, 979 (1987). 937J. L. D’Amato, H. M. Pastawski, and J. F. Weisz, Phys. Rev. B 39, 3554 (1989). 38M. L. Ladron de Guevara and P. A. Orellana, Phys. Rev. B 73, 205303 (2006). 39A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, Rev. Mod. P hys.82, 2257 (2010). 40M. B¨ uttiker, IBM J. Res. Dev. 32, 317 (1988). 41V. E. Kravtsov and V. I. Yudson, Phys. Rev. Lett. 70, 210 (1993). 42V. I. Yudson and V. E. Kravtsov, Phys. Rev. B 67, 155310 (2003). 43E. Buks et al., Phys. Rev. Lett. 77, 4664 (1996). 44K. Kobayashi, H. Aikawa, S. Katsumoto, and Y. Iye, Phys. Rev. B68, 235304 (2003). 45S. Amaha et al., Appl. Phys. Lett. 92, 202109 (2008). 10
1208.5841v1.Radio_frequency_spectroscopy_of_weakly_bound_molecules_in_spin_orbit_coupled_atomic_Fermi_gases.pdf	arXiv:1208.5841v1 [cond-mat.quant-gas] 29 Aug 2012Radio-frequency spectroscopy of weakly bound molecules in spin-orbit coupled atomic Fermi gases Hui Hu1, Han Pu2, Jing Zhang3, Shi-Guo Peng4, and Xia-Ji Liu1∗ 1ARC Centre of Excellence for Quantum-Atom Optics and Centre for Atom Optics and Ultrafast Spectroscopy, Swinburne University of Technology, Melbourne 3122, Austr alia 2Department of Physics and Astronomy and Rice Quantum Instit ute, Rice University, Houston, TX 77251, USA 3State Key Laboratory of Quantum Optics and Quantum Optics De vices, Institute of Opto-Electronics, Shanxi University, Taiyua n 030006, P. R. China 4State Key Laboratory of Magnetic Resonance and Atomic and Mo lecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academ y of Sciences, Wuhan 430071, P. R. China (Dated: October 29, 2018) We investigate theoretically radio-frequency spectrosco py of weakly bound molecules in an ul- tracold spin-orbit-coupled atomic Fermi gas. We consider t wo cases with either equal Rashba and Dresselhaus coupling or pure Rashba coupling. The former sy stem has been realized very recently at Shanxi University [Wang et al., arXiv:1204.1887] and MIT [Cheuk et al., arXiv:1205.3483]. We predict realistic radio-frequency signals for revealing t he unique properties of anisotropic molecules formed by spin-orbit coupling. PACS numbers: 03.75.Ss, 03.75.Hh, 05.30.Fk, 67.85.-d I. INTRODUCTION The coupling between the spin of electrons to their orbital motion, the so-called spin-orbit coupling, lies at the heart of a variety of intriguing phenomena in di- verse ﬁelds of physics. It is responsible for the well- known ﬁne structure of atomic spectra in atomic physics, as well as the recently discovered topological state of matter in solid-state physics, such as topological insu- lators and superconductors [1, 2]. For electrons, spin- orbit coupling is a relativistic eﬀect and in general not strong. Most recently, in a milestone experiment at the National Institute of Standards and Technology (NIST), synthetic spin-orbit coupling was created and detected in an atomic Bose-Einstein condensate (BEC) of87Rb atoms [3]. Using the same experimental technique, non- interacting spin-orbit-coupled Fermi gases of40K atoms and6Li atoms have also been realized, respectively, at Shanxi University [4] and at Massachusetts Institute of Technology (MIT) [5]. These experiments have paved an entirely new way to investigate the celebrated eﬀects of spin-orbit coupling. Owing to the high-controllability of ultracold atoms in atomic species, interactions, conﬁning geometry and pu- rity, the advantage of using synthetic spin-orbit coupling is apparent: (i) The strength of spin-orbit coupling be- tween ultracold atoms can be made very strong, much stronger than that in solids; (ii) New bosonic topological states that have no analogy in solid-state systems may be created; (iii) Ultracold atoms are able to realize topo- logical superﬂuids that are yet to be observed in the solid state; (iv) Strongly correlated topological states can be readily realized, whose understanding remains a grand ∗Electronic address: xiajiliu@swin.edu.auchallenge. At the moment, there has been a ﬂood of the- oretical work on synthetic spin-orbit coupling in BECs [6–18] and atomic Fermi gases [19–31], addressing par- ticularly new exotic superﬂuid phases arising from spin- orbit coupling [8, 9, 13, 23]. In this paper, we investigate theoretically momentum- resolved radio-frequency (rf) spectroscopy of an inter- acting two-component atomic Fermi gas with spin-orbit coupling. The interatomic interactions can be easily ma- nipulated using a Feshbach resonance in40K or6Li atoms [32]. It is known that weakly bound molecules with anisotropic mass and anisotropic wave function (in mo- mentum space) may be formed due to spin-orbit coupling [19, 22, 23]. Here, we aim to predict observable rf signals of these anisotropic molecules. Our calculation is based on the Fermi’s golden rule for the bound-free transition of a stationary molecule [33]. We consider two kinds of spin- orbit coupling: (1) the equal Rashba and Dresselhaus coupling,λkxσy, which has been realized experimentally at Shanxi University [4] and MIT [5], and (2) the pure Rashba coupling, λ(kyσx−kxσy),which is yet to be real- ized. Here σxandσyare the Pauli matrices, kxandky are momenta and λis the strength of spin-orbit coupling. The latter case with pure Rashba spin-orbit coupling is of particular theoretical interest, since molecules induc ed by spin-orbit coupling exist even for a negative s-wave scattering length above Feshbach resonances [19, 22, 23]. The paper is structured as follows. In the next section, we discuss brieﬂy the rf spectroscopy and the Fermi’s golden rule for the calculation of rf transfer strength. In Sec. III, we consider the experimental case of equal Rashba and Dresselhaus spin-orbit coupling. We intro- duce ﬁrst the model Hamiltonian and the single-particle and two-particle wave functions. We then derive an analytic expression for momentum-resolved rf transfer strength following the Fermi’s golden rule. It can be written explicitly in terms of the two-body wave function.2 We discuss in detail the distinct features of momentum- resolved rf spectroscopy in the presence of spin-orbit cou- pling, with the use of realistic experimental parameters. In Sec. IV, we consider an atomic Fermi gas with pure Rashba spin-orbit coupling, a system anticipated to be realized in the near future. Finally, in Sec. V, we con- clude and make some ﬁnal remarks. II. RADIO-FREQUENCY SPECTROSCOPY AND THE FERMI’S GOLDEN RULE Radio-frequency spectroscopy, including momentum- resolved rf-spectroscopy, is a powerful tool to characteri ze interacting many-body systems. It has been widely used to study fermionic pairing in a two-component atomic Fermi gas near Feshbach resonances when it crosses from a Bardeen-Cooper-Schrieﬀer (BCS) superﬂuid of weakly interacting Cooper pairs into a BEC of tightly bound molecules [34–36]. Most recently, it has also been used to detect new quasiparticles known as repulsive polarons [37, 38], which occur when “impurity” fermionic particles interact repulsively with a fermionic environment. The underlying mechanics of rf-spectroscopy is sim- ple. For an atomic Fermi gas with two hyperﬁne states, denoted as \|1/angb∇acket∇ight=\|↑/angb∇acket∇ightand\|2/angb∇acket∇ight=\|↓/angb∇acket∇ight, the rf ﬁeld drives transitions between one of the hyperﬁne states (i.e., \|↓/angb∇acket∇ight) and an empty hyperﬁne state \|3/angb∇acket∇ightwhich lies above it by an energy /planckover2pi1ω3↓due to the magnetic ﬁeld splitting in bare atomic hyperﬁne levels. The Hamiltonian for rf-coupling may be written as, Vrf=V0ˆ dr/bracketleftBig ψ† 3(r)ψ↓(r)+ψ† ↓(r)ψ3(r)/bracketrightBig ,(1) whereψ† 3(r)(ψ† ↓(r)) is the ﬁeld operator which creates an atom in\|3/angb∇acket∇ight(\|↓/angb∇acket∇ight) at the position rand,V0is the strength of the rf drive and is related to the Rabi frequency ωR withV0=/planckover2pi1ωR/2. For the rf-spectroscopy of weakly bound molecules that is of interest in this work, a molecule is initially at rest in the bound state \|Φ2B/angb∇acket∇ightwith energy E0=−ǫB. HereǫBis the binding energy of the molecules. A radio-frequency photon with energy /planckover2pi1ωwill break the molecule and trans- fer one of the atoms to the third state \|3/angb∇acket∇ight. In the case that there is no interaction between the state \|3/angb∇acket∇ightand the spin-up and spin-down states, the ﬁnal state \|Φf/angb∇acket∇ightinvolves a free atom in the third state and a remaining atom in the system. According to the Fermi’s golden rule, the rf strength of breaking molecules and transferring atoms is proportional to the Franck-Condon factor [33], F(ω) =\|/angb∇acketleftΦf\|Vrf\|Φ2B/angb∇acket∇ight\|2δ/bracketleftbigg ω−ω3↓−Ef−E0 /planckover2pi1/bracketrightbigg ,(2) where the Dirac delta function ensures energy conserva- tion andEfis the energy of the ﬁnal state. The inte- grated Franck-Condon factor over frequency should be unity,´+∞ −∞F(ω)dω= 1, if we can ﬁnd a complete setof ﬁnal states for rf transition. Hereafter, without any confusion we shall ignore the energy splitting in the bare atomic hyperﬁne levels and set ω3↓= 0. To calculate the Franck-Condon factor Eq. (2), it is crucial to understand the initial two-particle bound state \|Φ2B/angb∇acket∇ightand the ﬁnal two-particle state \|Φf/angb∇acket∇ight. III. EQUAL RASHBA AND DRESSELHAUS SPIN-ORBIT COUPLING Let us ﬁrst consider a spin-orbit-coupled atomic Fermi gas realized recently at Shanxi University [4] and at MIT [5]. In these two experiments, the spin-orbit coupling is induced by the spatial dependence of two counter propa- gating Raman laser beams that couple the two spin states of the system. Near Feshbach resonances, the system may be described by a model Hamiltonian H=H0+Hint, where H0=/summationdisplay σˆ drψ† σ(r)/planckover2pi12ˆk2 2mψσ(r)+ ˆ dr/bracketleftbigg ψ† ↑(r)/parenleftbiggΩR 2ei2kRx/parenrightbigg ψ↓(r)+H.c./bracketrightbigg (3) is the single-particle Hamiltonian and Hint=U0ˆ drψ† ↑(r)ψ† ↓(r)ψ↓(r)ψ↑(r) (4) is the interaction Hamiltonian describing the contact in- teraction between two spin states. Here, ψ† σ(r)is the creation ﬁeld operator for atoms in the spin-state σ, /planckover2pi1ˆk≡ −i/planckover2pi1∇is the momentum operator, ΩRis the cou- pling strength of Raman beams, kR= 2π/λRis deter- mined by the wave length λRof two Raman lasers and therefore 2/planckover2pi1kRis the momentum transfer during the two- photon Raman process. The interaction strength is de- noted by the bare interaction parameter U0. It should be regularized in terms of the s-wave scattering length as, i.e.,1/U0=m/(4π/planckover2pi12as)−/summationtext km/(/planckover2pi12k2). To solve the many-body Hamiltonian Eq. (3), it is useful to remove the spatial dependence of the Raman coupling term, by introducing the following new ﬁeld op- erators˜ψσvia ψ↑(r) =e+ikRx˜ψ↑(r), (5) ψ↓(r) =e−ikRx˜ψ↓(r). (6) With the new ﬁeld operators ˜ψσ, the single-particle Hamiltonian then becomes, H0=/summationdisplay σˆ dr˜ψ† σ(r)/planckover2pi12/parenleftBig ˆk±kRex/parenrightBig2 2m˜ψσ(r) +ΩR 2ˆ dr/bracketleftBig ˜ψ† ↑(r)˜ψ↓(r)+H.c./bracketrightBig , (7)3 where in the ﬁrst term on the right hand side of the equation we take “ +” for spin-up atoms and “ −” for spin- down atoms. The form of the interaction Hamiltonian is invariant, Hint=U0ˆ dr˜ψ† ↑(r)˜ψ† ↓(r)˜ψ↓(r)˜ψ↑(r).(8) However, the rf Hamiltonian acquires an eﬀective mo- mentum transfer kRex, Vrf=V0ˆ dr/bracketleftBig e−ikRxψ† 3(r)˜ψ↓(r)+H.c./bracketrightBig .(9) For later reference, we shall rewrite the rf Hamiltonian in terms of ﬁeld operators in the momentum space, Vrf=V0/summationdisplay q/parenleftBig c† q−kRex,3cq↓+H.c./parenrightBig , (10)whereψ† 3(r)≡/summationtext qc† q3eiq·rand˜ψ↓(r)≡/summationtext qcq↓eiq·r. Hereafter, we shall denote ck3andckσas the ﬁeld op- erators (in the momentum space) for atoms in the third state and in the spin-state σ, respectively. A. Single-particle solution Using the Pauli matrices, the single-particle Hamilto- nian takes the form, H0=ˆ dr[˜ψ† ↑(r),˜ψ† ↓(r)]/bracketleftBigg /planckover2pi12/parenleftbig k2 R+k2/parenrightbig 2m+hσx+λkxσz/bracketrightBigg/bracketleftbigg˜ψ↑(r) ˜ψ↓(r)/bracketrightbigg , (11) where for convenience we have deﬁned the spin-orbit couplin g constant denoted as λ≡/planckover2pi12kR/mand an “eﬀective” Zeeman ﬁeld h≡ΩR/2. This Hamiltonian is equivalent to the one with equal Rashba and Dresselhaus spin-orbit coupling,λkxσy. To see this, let us take the second transformation and intro duce new ﬁeld operators Ψσ(r)via ˜ψ↑(r) =1√ 2[Ψ↑(r)−iΨ↓(r)], (12) ˜ψ↓(r) =1√ 2[Ψ↑(r)+iΨ↓(r)]. (13) Using these new ﬁeld operators, the single-particle Hamilt onian now takes the form, H0=ˆ dr[Ψ† ↑(r),Ψ† ↓(r)]/bracketleftBigg /planckover2pi12/parenleftbig k2 R+k2/parenrightbig 2m+λkxσy+hσz/bracketrightBigg/bracketleftbigg Ψ↑(r) Ψ↓(r)/bracketrightbigg , (14) which is precisely the Hamiltonian with equal Rashba and Dresselhaus spin-orbit coupling. The single-particle Hamiltonian Eq. (11) can be diag- onalized to yield two eigenvalues Ek±=/planckover2pi12k2 R 2m+/planckover2pi12k2 2m±/radicalbig h2+λ2k2x. (15) Here “±” stands for the two helicity branches. The single- particle eigenstates or the ﬁeld operators in the helicity basis take the form ck+= +ck↑cosθk+ck↓sinθk, (16) ck−=−ck↑sinθk+ck↓cosθk, (17) where θk= arctan[(/radicalbig h2+λ2k2x−λkx)/h]>0(18)is an angle determined by handkx. Note that, cos2θk=1 2/parenleftBigg 1+λkx/radicalbig h2+λ2k2x/parenrightBigg , (19) sin2θk=1 2/parenleftBigg 1−λkx/radicalbig h2+λ2k2x/parenrightBigg . (20) Note also that the minimum energy of the single-particle energy dispersion is given by [24] Emin=/planckover2pi12k2 R 2m−mλ2 2/planckover2pi12−/planckover2pi12h2 2mλ2=−/planckover2pi12h2 2mλ2, (21) ifh<mλ2//planckover2pi12.4 B. The initial two-particle bound state \|Φ2B/angbracketright In the presence of spin-orbit coupling, the wave func- tion of the initial two-body bound state has both spinsinglet and triplet components [19, 22, 23]. The wave function at zero center-of-mass momentum, \|Φ2B/angb∇acket∇ight, may be written as [22], \|Φ2B/angb∇acket∇ight=1√ 2C/summationdisplay k/bracketleftBig ψ↑↓(k)c† k↑c† −k↓+ψ↓↑(k)c† k↓c† −k↑+ψ↑↑(k)c† k↑c† −k↑+ψ↓↓(k)c† k↓c† −k↓/bracketrightBig \|vac/angb∇acket∇ight, (22) wherec† k↑andc† k↓are creation ﬁeld operators of spin-up and spin-down atoms w ith momentum kandC ≡/summationtext k[\|ψ↑↓(k)\|2+\|ψ↓↑(k)\|2+\|ψ↑↑(k)\|2+\|ψ↓↓(k)\|2]is the normalization factor. From the Schrödinger equation (H0+Hint)\|Φ2B/angb∇acket∇ight=E0\|Φ2B/angb∇acket∇ight, we can straightforwardly derive the following equations f or coeﬃcients ψσσ′appearing in the above two-body wave function [22]: /bracketleftbigg E0−/parenleftbigg/planckover2pi12k2 R m+/planckover2pi12k2 m+2λkx/parenrightbigg/bracketrightbigg ψ↑↓(k) = +U0 2/summationdisplay k′[ψ↑↓(k′)−ψ↓↑(k′)]+hψ↑↑(k)+hψ↓↓(k), (23) /bracketleftbigg E0−/parenleftbigg/planckover2pi12k2 R m+/planckover2pi12k2 m−2λkx/parenrightbigg/bracketrightbigg ψ↓↑(k) =−U0 2/summationdisplay k′[ψ↑↓(k′)−ψ↓↑(k′)]+hψ↑↑(k)+hψ↓↓(k), (24) /bracketleftbigg E0−/parenleftbigg/planckover2pi12k2 R m+/planckover2pi12k2 m/parenrightbigg/bracketrightbigg ψ↑↑(k) =hψ↑↓(k)+hψ↓↑(k), (25) /bracketleftbigg E0−/parenleftbigg/planckover2pi12k2 R m+/planckover2pi12k2 m/parenrightbigg/bracketrightbigg ψ↓↓(k) =hψ↑↓(k)+hψ↓↑(k), (26) whereE0=−ǫB<0is the energy of the two-body bound state. Let us introduce Ak≡ −ǫB−(/planckover2pi12k2 R/m+ /planckover2pi12k2/m)<0and diﬀerent spin components of the wave- functions, ψs(k) =1√ 2[ψ↑↓(k)−ψ↓↑(k)], (27) ψa(k) =1√ 2[ψ↑↓(k)+ψ↓↑(k)]. (28) It is easy to see that, ψ↑↑(k) =√ 2h Akψa(k), (29) ψ↓↓(k) =√ 2h Akψa(k), (30) ψa(k) =λkx/bracketleftbigg1 Ak−2h+1 Ak+2h/bracketrightbigg ψs(k),(31) and /bracketleftbigg Ak−4λ2k2 x Ak−4h2/Ak/bracketrightbigg ψs(k) =U0/summationdisplay k′ψs(k′).(32) As required by the symmetry of fermionic system, the spin-singlet wave function ψs(k)is an even function of the momentum k, i.e.,ψs(−k) =ψs(k),and the spin-triplet wave functions are odd functions, satisfy- ingψa(−k) =−ψa(k),ψ↑↑(−k) =−ψ↑↑(k)andψ↓↓(−k) =−ψ↓↓(k). The un-normalized wavefunction ψs(k) = [Ak−4λ2k2 x/(Ak−4h2/Ak)]−1is given by, ψs(k) =1 h2+λ2k2x/bracketleftbiggh2 Ak+λ2k2 xAk A2 k−4(h2+λ2k2x)/bracketrightbigg .(33) Using Eq. (32) and un-normalized wave function ψs(k), the bound-state energy E0or the binding energy ǫBis determined by U0/summationtext kψs(k) = 1, or more explicitly, m 4π/planckover2pi12as−/summationdisplay k/bracketleftBig ψs(k)+m /planckover2pi12k2/bracketrightBig = 0. (34) Here we have replaced the bare interaction strength U0 by the s-wave scattering length asusing the standard regularization scheme mentioned earlier. The normaliza- tion factor of the total two-body wave function is given by, C=/summationdisplay k\|ψs(k)\|2/bracketleftBigg 1+2λ2k2 x (Ak−2h)2+2λ2k2 x (Ak+2h)2/bracketrightBigg . (35) C. The ﬁnal two-particle state \|Φf/angbracketright Let us consider now the ﬁnal state \|Φf/angb∇acket∇ight. For this pur- pose, it is useful to calculate Vrf\|Φ2B/angb∇acket∇ight=V0/summationdisplay qc+ −q−kRex,3c−q↓\|Φ2B/angb∇acket∇ight (36)5 and then determine possible ﬁnal states. It can be readily seen that, Vrf\|Φ2B/angb∇acket∇ight=−V0√ 2C/summationdisplay qc+ −q−kRex,3/braceleftBig [ψ↑↓(q)−ψ↓↑(−q)]c† q↑+[ψ↓↓(q)−ψ↓↓(−q)]c† q↓/bracerightBig \|vac/angb∇acket∇ight. (37) Rewritingψ↑↓andψ↓↑in terms of ψsandψaas shown in Eqs. (27) and (28), and exploiting the parity of th e wave functions, we obtain a general result valid for any type of sp in-orbit coupling, Vrf\|Φ2B/angb∇acket∇ight=−/radicalbigg 1 CV0/summationdisplay qc+ −q−kRex,3/braceleftBig [ψs(q)+ψa(q)]c† q↑+√ 2ψ↓↓(q)c† q↓/bracerightBig \|vac/angb∇acket∇ight. (38) To proceed, we need to rewrite the ﬁeld operators c† q↑andc† q↓in terms of creation operators in the helicity basis. For the case of equal Rashba and Dresselhaus spin-orbit couplin g, using Eqs. (16) and (17), we ﬁnd that c† q↑= cosθqc† q+−sinθqc† q−, (39) c† q↓= sinθqc† q++cosθqc† q−. (40) Thus, we obtain Vrf\|Φ2B/angb∇acket∇ight=−/radicalbigg 1 CV0/summationdisplay qc+ −q−kRex,3/bracketleftBig sq+c† q+−sq−c† q−/bracketrightBig \|vac/angb∇acket∇ight, (41) where sq+= [ψs(q)+ψa(q)]cosθq+√ 2ψ↓↓(q)sinθq, (42) sq−= [ψs(q)+ψa(q)]sinθq−√ 2ψ↓↓(q)cosθq. (43) Eq. (41) can be interpreted as follows. The rf photon breaks a stationary molecule and transfers a spin-down atom to the third state. We have two possible ﬁnal states: (1) we may h ave two atoms in the third state and the upper helicity state, respectively, with a possibility of \|sq+\|2/C; and (2) we may also have a possibility of \|sq−\|2/Cfor having two atoms in the third state and the lower helicity state, respec tively. D. Momentum-resolved rf spectroscopy Taking into account these two ﬁnal states and using the Fermi ’s golden rule, we end up with the following expression for the Franck-Condon factor, F(ω) =1 C/summationdisplay q/bracketleftbigg s2 q+δ/parenleftbigg ω−Eq+ /planckover2pi1/parenrightbigg +s2 q−δ/parenleftbigg ω−Eq− /planckover2pi1/parenrightbigg/bracketrightbigg , (44) where Eq±≡ǫB+/planckover2pi12/parenleftbig k2 R+q2/parenrightbig 2m±/radicalbig h2+λ2q2x+/planckover2pi12(q+kRex)2 2m. (45) The two Dirac delta functions in Eq. (44) are due to energy con servation. For example, the energy of the initial state (of the stationary molecule) is E0=−ǫB, while the energy of the ﬁnal state is /planckover2pi12(q+kRex)2/(2m)for the free atom in the third state and /planckover2pi12(k2 R+q2)/(2m)+/radicalbig h2+λ2q2xfor the remaining atom in the upper branch. Therefore, the rf energy /planckover2pi1ωrequired to have such a transfer is given by Eq+, as shown by the ﬁrst Dirac delta function. It is easy to check that the Franck-Condon factor is integrated to unity,´+∞ −∞F(ω) = 1. Experimentally, in addition to measuring the total number o f atoms transferred to the third state, which is propor- tional toF(ω), we may also resolve the transferred number of atoms for a giv en momentum or wave-vector kx. Such6 a momentum-resolved rf spectroscopy has already been imple mented for a non-interacting spin-orbit coupled atomic Fermi gas at Shanxi University and at MIT. Accordingly, we ma y deﬁne a momentum-resolved Franck-Condon factor, F(kx,ω) =1 C/summationdisplay q⊥/bracketleftbigg s2 q+δ/parenleftbigg ω−Eq+ /planckover2pi1/parenrightbigg +s2 q−δ/parenleftbigg ω−Eq− /planckover2pi1/parenrightbigg/bracketrightbigg , (46) where the summation are now over the wave-vector q⊥≡(qy,qz)and we have deﬁned kx≡qx+kRby shifting the wave-vector qxby an amount kR. This shift is due to the gauge transformation used in Eqs. (5 ) and (6). With the help of the two Dirac delta functions, the summation over q⊥may be done analytically. We ﬁnally arrive at, F(kx,ω) =m 8π2/planckover2pi1C/bracketleftbig s2 q+Θ/parenleftbig q2 ⊥,+/parenrightbig +s2 q−Θ/parenleftbig q2 ⊥,−/parenrightbig/bracketrightbig , (47) whereΘ(x)is the step function and q2 ⊥,±=m /planckover2pi1/parenleftBig ω−ǫB /planckover2pi1/parenrightBig −/parenleftBig k2 R+q2 x+qxkR±m /planckover2pi12/radicalbig h2+λ2q2x/parenrightBig . (48) It is understood that we will use q= (qx,q⊥,+)in the calculation of sq+andq= (qx,q⊥,−)insq−. We may immediately realize from the above expres- sion that the momentum-resolved Franck-Condon factor is an asymmetric function of kx, due to the coexistence of spin-singlet and spin-triplet wave functions in the initia l two-body bound state. Moreover, the contribution from two ﬁnal states or two branches should manifest them- selves in the diﬀerent frequency domain in rf spectra. As we shall see below, these features give us clear signals of anisotropic bound molecules formed by spin-orbit cou- pling. On the other hand, from Eq. (47), it is readily seen that once the momentum-resolved rf spectroscopy is measured with high resolution, it is possible to deter- mine precisely s2 q+ands2 q−and then re-construct the two-body wave function of spin-orbit bound molecules. E. Numerical results and discussions For equal Rashba and Dresselhaus spin-orbit coupling, the bound molecular state exists only on the BEC side of Feshbach resonances with a positive s-wave scattering length,as>0[24]. Thus, it is convenient to take the characteristic binding energy EB=/planckover2pi12/(ma2 s)as the unit for energy and frequency. For wave-vector, we use kR= mλ//planckover2pi12as the unit. The strength of spin-orbit coupling may be measured by the ratio Eλ EB=/bracketleftbigg/planckover2pi12 mλas/bracketrightbigg−2 , (49) where we have deﬁned the characteristic spin-orbit en- ergyEλ≡mλ2//planckover2pi12=/planckover2pi12k2 R/m. Note that, the spin-orbit coupling is also controlled by the eﬀective Zeeman ﬁeld h= ΩR/2. In particular, in the limit of zero Zeeman ﬁeld ΩR= 0, there is no spin-orbit coupling term as shown in the original Hamiltonian Eq. (3), although the char- acteristic spin-orbit energy Eλ/negationslash= 0. UsingkRandEB as the units for wave-vector and energy, we can writea set of dimensionless equations for the binding energy ǫB=−E0, normalization factor C, Franck-Condon factor F(ω)and the momentum-resolved Franck-Condon fac- torF(ω,kx). We then solve them for given parameters Eλ/EBandh/Eλ. In accord with the normalization con- dition´+∞ −∞F(ω) = 1, the units for F(ω)andF(kx,ω) are taken to be 1/EBand1/(EBkR), respectively. /s48 /s49 /s50 /s51 /s52 /s53 /s54/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51 /s48 /s50 /s52 /s54/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51 /s32/s32 /s32/s32 /s69 /s32/s61/s32 /s69 /s66 /s48/s46/s53/s69 /s47/s69 /s66/s61/s50/s70 /s40 /s41 /s47/s69 /s66/s104 /s32/s61/s32/s48/s46/s53 /s69/s49 Figure 1: (color online) Franck-Condon factor of weakly bound molecules formed by equal Rashba and Dresselhaus spin-orbit coupling, in units of E−1 B. Here we take h=Eλ/2 orΩR=/planckover2pi12k2 R/mand set Eλ/EB= 0.5,1, and2. The re- sult without spin-orbit coupling is plotted by the thin dash ed line. The Inset shows the diﬀerent contribution from the two ﬁnal states at Eλ/EB= 1. The one with a remaining atom in the lower (upper) helicity branch is plotted by the dashed (dot-dashed) line. Fig. 1 displays the Franck-Condon factor as a func- tion of the rf frequency at h/Eλ= 0.5and at several ratios ofEλ/EBas indicated. For comparison, we show also the rf line-shape without spin-orbit coupling [33], F(ω) = (2/π)√ω−EB/ω2, by the thin dashed line. In the presence of spin-orbit coupling, the existence of two possible ﬁnal states is clearly revealed by the two peaks in the rf spectra. This is highlighted in the inset for7 Eλ/EB= 1, where the contribution from the two possi- ble ﬁnal states is plotted separately. The main rf response is from the ﬁnal state with the remaining atom staying in the lower helicity branch, i.e., the second term in the Franck-Condon factor Eq. (44). The two peak positions may be roughly estimated from Eq. (48) for the threshold frequencyωc±of two branches, /planckover2pi1ωc±=ǫB+/bracketleftBigg /planckover2pi12/parenleftbig k2 R+q2 x+qxkR/parenrightbig m±/radicalbig h2+λ2q2x/bracketrightBigg min. (50) With increasing spin-orbit coupling, the low-frequency peak becomes more and more pronounced and shifts slightly towards lower energy. In contrast, the high- frequency peak has a rapid blue-shift. /s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50/s48/s49/s50/s51/s52/s53/s54/s32 /s32 /s32/s47/s69 /s66/s40/s97/s41/s32 /s69 /s47/s69 /s66/s32/s61/s32/s48/s46/s49 /s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s32 /s32/s40/s98/s41/s32 /s69 /s47/s69 /s66/s32/s61/s32/s48/s46/s53 /s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s32/s32 /s40/s100/s41/s32 /s69 /s47/s69 /s66/s32/s61/s32/s50/s46/s48 /s107 /s120/s47/s107 /s82/s48 /s48/s46/s49 /s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50/s48/s49/s50/s51/s52/s53/s54/s32 /s32 /s107 /s120/s47/s107 /s82/s47/s69 /s66 /s40/s99/s41/s32 /s69 /s47/s69 /s66/s32/s61/s32/s49/s46/s48 Figure 2: (color online) Linear contour plot of momentum- resolved Franck-Condon factor, in units of (EBkR)−1. Here we takeh=Eλ/2and consider Eλ/EB= 0.1,0.5,1, and2. /s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s104 /s32/s61/s32/s48/s46/s53 /s69 /s44/s32 /s69 /s32/s61/s32 /s69 /s66 /s47/s69 /s32/s61/s32/s49/s46/s48/s70 /s40/s107 /s120/s44/s32 /s41 /s107 /s120/s47/s107 /s82/s51/s46/s48/s50/s46/s53/s50/s46/s48/s49/s46/s53 Figure 3: (color online) Energy distribution curve of the momentum-resolved Franck-Condon factor, in units of (EBkR)−1. We consider several values of the rf frequency ωas indicated, under given parameters h=Eλ/2andEλ=EB. Fig. 2 presents the corresponding momentum-resolved Franck-Condon factor. We ﬁnd a strong asymmetric dis-tribution as a function of the momentum kx. In par- ticular, the contribution from two ﬁnal states are well separated in diﬀerent frequency domains and therefore should be easily observed experimentally. The asymmet- ric distribution of F(kx,ω)is mostly evident in energy distribution curve, as shown in Fig. 3, where we plot F(kx,ω)as a function of kxat several given frequencies ω. In the experiment, each of these energy distribution curves can be obtained by a single-shot measurement. /s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56 /s45/s50 /s48 /s50/s48/s50/s52/s54/s47/s69 /s66 /s32/s32 /s107 /s120/s47/s107 /s82/s104 /s61 /s69 /s48/s46/s52/s48/s46/s54/s48/s46/s56/s104 /s47/s69 /s61/s32/s49/s46/s48/s70 /s40 /s41 /s47/s69 /s66 Figure 4: (color online) Zeeman-ﬁeld dependence of the Franck-Condon factor at Eλ=EB. Here we vary the eﬀective Zeeman ﬁelds, h/Eλ= 0.4,0.6,0.8and1.0. The inset shows the momentum-resolved Franck-Condon factor at h=Eλ. We ﬁnally discuss the eﬀect of the eﬀective Zeeman ﬁeldh= ΩR/2. Fig. 4 shows how the line-shape of Franck-Condon factor evolves as a function of the Zee- man ﬁeld at Eλ/EB= 1. In general, the larger Zeeman ﬁeld the stronger spin-orbit coupling. Therefore, as the same as in Fig. 1, the increase in Zeeman ﬁeld leads to a pronounced peak at about the binding energy. There is a red-shift in the peak position as the binding energy becomes smaller as the Zeeman ﬁeld increases. As antic- ipated, the larger the Zeeman ﬁeld, the more asymmetric F(kx,ω)becomes. In the inset, we show as an example the contour plot of F(kx,ω)ath/Eλ= 1. IV. RASHBA SPIN-ORBIT COUPLING We now turn to the case with pure Rashba spin-orbit coupling,λ(kyσx−kxσy), which may be realized exper- imentally in the near future. The single-particle Hamil- tonian may be written as [23], H0=ˆ dr/bracketleftBig ψ† ↑(r),ψ† ↓(r)/bracketrightBig S/bracketleftbigg ψ↑(r) ψ↓(r)/bracketrightbigg ,(51) where the matrix S=/bracketleftbigg /planckover2pi12/parenleftbig k2 R+k2/parenrightbig /(2m)iλ(kx−iky) −iλ(kx+iky)/planckover2pi12/parenleftbig k2 R+k2/parenrightbig /(2m)/bracketrightbigg .(52) Hereλis the coupling strength of Rashba spin-orbit cou- pling,kR≡mλ//planckover2pi12, and we have added a constant term8 /planckover2pi12k2 R/(2m)to make the minimum single-particle energy zero [24], i.e., Emin= 0. A. Single-particle solution We diagonalize the matrix Sto obtain two helicity eigenvalues [23], Ek±=/planckover2pi12/parenleftbig k2 R+k2/parenrightbig 2m±λk⊥, (53) wherek⊥≡/radicalBig k2x+k2yand “±”stands for the two helicity branches. For later reference, the ﬁeld operators in the original spin basis and in the helicity basis are related by, c† k↑=1√ 2/parenleftBig c† k++ieiϕkc† k−/parenrightBig , (54) c† k↓=1√ 2/parenleftBig ie−iϕkc† k++c† k−/parenrightBig . (55)Hereϕk≡arg(kx,ky)is the azimuthal angle of the wave- vectork⊥in thex−yplane. B. The initial two-particle bound state \|Φ2B/angbracketright In the case of Rashba spin-orbit coupling, the two-body wave function can still be written in the same form as in the previous case, i.e., Eq. (22), and the Schrödinger equation leads to [22], Akψ↑↓(k) = +U0 2/summationdisplay k′[ψ↑↓(k′)−ψ↓↑(k′)]−λ(ky−ikx)ψ↑↑(k)+λ(ky+ikx)ψ↓↓(k), (56) Akψ↓↑(k) =−U0 2/summationdisplay k′[ψ↑↓(k′)−ψ↓↑(k′)]+λ(ky−ikx)ψ↑↑(k)−λ(ky+ikx)ψ↓↓(k), (57) Akψ↑↑(k) =−λ(ky+ikx)ψ↑↓(k)+λ(ky+ikx)ψ↓↑(k), (58) Akψ↓↓(k) = +λ(ky−ikx)ψ↑↓(k)−λ(ky−ikx)ψ↓↑(k), (59) whereAk≡E0−(/planckover2pi12k2 R/m+/planckover2pi12k2/m)<0. It is easy to show thatψa(k) = 0 and /bracketleftbigg Ak−4λ2k2 ⊥ Ak/bracketrightbigg ψs(k) =U0/summationdisplay k′ψs(k′).(60) Thus, we obtain the (un-normalized) wavefunction: ψs(k) =1 2/bracketleftbigg1 E0−2Ek++1 E0−2Ek−/bracketrightbigg , (61) and the equation for the energy E0, m 4π/planckover2pi12as=/summationdisplay k/bracketleftbigg1/2 E0−2Ek++1/2 E0−2Ek−+m /planckover2pi12k2/bracketrightbigg . (62) The spin-triplet wave functions ψ↑↑(k)andψ↓↓(k)are given by, ψ↑↑(k) =/bracketleftBigg −ie−iϕk√ 2λk⊥ E0−2ǫk/bracketrightBigg ψs(k),(63) ψ↓↓(k) =/bracketleftBigg −ie+iϕk√ 2λk⊥ E0−2ǫk/bracketrightBigg ψs(k),(64)whereǫk≡/planckover2pi12k2/(2m). The normalization factor for the two-body wave function is therefore, C=/summationdisplay k\|ψs(k)\|2/bracketleftBigg 1+4λ2k2 ⊥ (E0−2ǫk)2/bracketrightBigg . (65) C. The ﬁnal two-particle state \|Φf/angbracketright To obtain the ﬁnal state, we consider again Vrf\|Φ2B/angb∇acket∇ight. In the present case, we assume that the rf Hamiltonian is given by, Vrf=V0/summationdisplay q/parenleftBig c† q3cq↓+c† q↓cq3/parenrightBig . (66) Following the same procedure as in the case of equal Rashba and Dresselhaus coupling, it is straightforward to show that,9 Vrf\|Φ2B/angb∇acket∇ight=−/radicalbigg 1 CV0/summationdisplay qc+ −q3/bracketleftBig ψs(q)c† q↑+√ 2ψ↓↓(q)c† q↓/bracketrightBig \|vac/angb∇acket∇ight. (67) Using Eqs. (54) and (55) to rewrite c† q↑andc† q↓in terms of c† q+andc† q−, we obtain, Vrf\|Φ2B/angb∇acket∇ight=−/radicalbigg 1 2CV0/summationdisplay q/bracketleftBigg c+ −q3c† q+ E0−2Eq++ieiϕqc+ −q3c† q− E0−2Eq−/bracketrightBigg \|vac/angb∇acket∇ight. (68) Therefore, we have again two ﬁnal states, diﬀering in the hel icity branch that the remaining atom stays. The remaining atom stays in the upper branch with probability (2C)−1(E0−2Eq+)−2, and in the lower branch with probability (2C)−1(E0−2Eq−)−2. D. Momentum-resolved rf spectroscopy Using the Fermi’s golden rule, we have immediately the Franc k-Condon factor, F(ω) =1 C/summationdisplay k/bracketleftBigg δ(ω−Ek+//planckover2pi1) 2(ǫB+2Ek+)2+δ(ω−Ek−//planckover2pi1) 2(ǫB+2Ek−)2/bracketrightBigg , (69) where Ek±≡ǫB+/planckover2pi12k2 R 2m+/planckover2pi12k2 m±λk⊥. (70) For Rashba spin-orbit coupling, it is reasonable to deﬁne th e following momentum-resolved Franck-Condon factor, F(k⊥,ω) =1 C/summationdisplay kz/bracketleftBigg δ(ω−Ek+//planckover2pi1) 2(ǫB+2Ek+)2+δ(ω−Ek−//planckover2pi1) 2(ǫB+2Ek−)2/bracketrightBigg , (71) where we have summed over the momentum kz. Integrating over kzwith the help of the two Dirac delta functions, we ﬁnd that, F(k⊥,ω) =m 16π3/planckover2pi1C/bracketleftBigg Θ/parenleftbig k2 z+/parenrightbig (/planckover2pi1ω+/planckover2pi12k2 R/2m+λk⊥)2kz++Θ/parenleftbig k2 z−/parenrightbig (/planckover2pi1ω+/planckover2pi12k2 R/2m−λk⊥)2kz−/bracketrightBigg , (72) where k2 z±=m /planckover2pi1/parenleftBig ω−ǫB /planckover2pi1/parenrightBig −/parenleftbiggk2 R 2+k2 ⊥±kRk⊥/parenrightbigg . (73) It is easy to see that the threshold frequencies for the two ﬁnal states are given by, /planckover2pi1ωc+=ǫB+/planckover2pi12k2 R 2m, (74) /planckover2pi1ωc−=ǫB+/planckover2pi12k2 R 4m, (75) which diﬀer by an amount of /planckover2pi12k2 R/(4m) =Eλ/4. Near ωc−, we ﬁnd approximately that F(ω)∝Θ(ω−ωc−)/ω2. Thus, the lineshape near the threshold is similar to that of a two-dimensional (2D) Ferm gas [36]. This similarity is related to the fact that at low energy a 3D Fermi gaswith Rashba spin-orbit coupling has exactly the same density of states as a 2D Fermi gas [14]. E. Numerical results and discussions For the pure Rashba spin-orbit coupling, the molec- ular bound state exists for arbitrary s-wave scattering lengthas[19, 22, 23]. We shall take kR=mλ//planckover2pi12and Eλ≡mλ2//planckover2pi12as the units for wave-vector and energy, respectively. With these units, the dimensionless inter- action strength is given by /planckover2pi12/(mλas). The spin-orbit10 eﬀect should be mostly signiﬁcant on the BCS side with /planckover2pi12/(mλas)<0, where the bound state cannot exist with- out spin-orbit coupling. /s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s50 /s47/s40 /s109 /s97 /s115/s41 /s32/s32 /s32/s32 /s32/s32/s70 /s40 /s41 /s47/s69 Figure 5: (color online) Franck-Condon factor of weakly bound molecules formed by Rashba spin-orbit coupling, in units ofE−1 λ. Here we take /planckover2pi12/(mλas) =−1(dashed line), 0 (solid line), and 1(dot-dashed line). In the deep BCS limit, /planckover2pi12/(mλas)→ −∞ , the Franck-Condon factor peaks sharply at/planckover2pi1ω≃Eλ/2and becomes a delta-like distribution. Fig. 5 shows the Franck-Condon factor at three dif- ferent interaction strengths /planckover2pi12/(mλas) =−1,0, and+1. The strong response in the BCS regime ( as<0) or in the unitary limit ( as→ ±∞ ) is an unambiguous signal of the existence of Rashba molecules. In particular, the rf line-shape in the BCS regime shows a sharp peak at about/planckover2pi1ω≃Eλ/2and decays very fast at high frequency. In Fig. 6, we present the corresponding momentum- resolved Franck-Condon factor F(k⊥,ω), in the form of contour plots. We can see clearly the diﬀerent response from the two ﬁnal states. The momentum-resolved rf spectroscopy is particularly useful to identify the contri - bution from the ﬁnal state that has a remaining atom in the upper branch, which, being integrated over k⊥, be- comes too weak to be resolved in the total rf spectroscopy. Finally, we report in Fig. 7 energy distribution curves of F(k⊥,ω)in the unitary limit /planckover2pi12/(mλas) = 0. We ﬁnd two sharp peaks in each energy distribution curve, aris- ing from the two ﬁnal states. When measured experi- mentally, these sharp peaks would become much broader owing to the ﬁnite experimental energy resolution. V. CONCLUSIONS In conclusions, we have investigated theoretically the radio-frequency spectroscopy of weakly bound molecules in a spin-orbit coupled atomic Fermi gas. The wave function of these molecules is greatly aﬀected by spin- orbit coupling and has both spin-singlet and spin-triplet components. As a result, the line-shape of the to- tal radio-frequency spectroscopy is qualitatively diﬀer- ent from that of the conventional molecules at the BEC-/s48 /s49 /s50 /s51/s48/s49/s50/s51/s52/s47/s69/s47/s69/s32 /s32/s47/s69 /s40/s97/s41/s32/s50 /s47/s40 /s109 /s97 /s115/s41/s32/s61/s32 /s48 /s49 /s50 /s51/s48/s49/s50/s51/s52 /s32 /s40/s98/s41/s32/s50 /s47/s40 /s109 /s97 /s115/s41/s32/s61/s32/s48/s46/s48/s48/s49 /s48/s46/s49 /s48 /s49 /s50 /s51/s48/s49/s50/s51/s52 /s32/s32 /s40/s99/s41/s32/s50 /s47/s40 /s109 /s97 /s115/s41/s32/s61/s32 /s107 /s47/s107 /s82 Figure 6: (color online) Contour plot of momentum-resolved Franck-Condon factor of weakly bound molecules formed by Rashba spin-orbit coupling, in units of (EλkR)−1. The in- tensity increases from blue to red in a logarithmic scale. We consider /planckover2pi12/(mλas) =−1(a),0(b), and 1(c). BCS crossover without spin-orbit coupling. In addi- tion, the momentum-resolved radio-frequency becomes highly asymmetric as a function of the momentum. These features are easily observable in current experi- ments with spin-orbit coupled Fermi gases of40K atoms and6Li atoms. On the other hand, from the high- resolution momentum-resolved radio-frequency, we may re-construct the two-body wave function of the bound molecules. We consider so far the molecular response in the radio- frequency spectroscopy. Our results should be quantia- tively reliable in the deep BEC limit with negligible num- ber of atoms, i.e., in the interaction parameter regime with1/(kFas)>2. However, in a real experiment, in or- der to maximize the spin-orbit eﬀect, it is better to work closer to Feshbach resonances, i.e., 1/(kFas)∼0.5. Un- der this situation, the spin-orbit coupled Fermi gas con- sists of both atoms and weakly bound molecules, which may strongly interact with each other. Our prediction for the molecular response is still qualitatively valid, wi th the understanding that there would be an additional pro- nounced atomic response in the rf spectra. A more in- depth investigation of radio-frequency spectroscopy re-11 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56 /s32/s32 /s47/s69 /s32/s61/s32 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s52/s46/s48/s70 /s40/s107 /s44/s32 /s41 /s107 /s47/s107 /s82/s51/s46/s53 /s51/s46/s48 /s50/s46/s53 /s50/s46/s48 /s49/s46/s53 /s49/s46/s48 Figure 7: (color online) Momentum-resolved Franck-Condon factor of Rashba molecules at /planckover2pi12/(mλas) = 0, shown in the form of energy distribution curves at several rf frequencie s as indicated.quires complicated many-body calculations beyond our simple two-body picture pursued in the present work. Acknowledgments We would like to thank Zeng-Qiang Yu and Hui Zhai for useful discussions. HH and XJL are supported by the ARC Discovery Projects (Grant No. DP0984522 and No. DP0984637) and the National Basic Research Program of China (NFRP-China, Grant No. 2011CB921502). HP is supported by the NSF and the Welch Foundation (Grant No. C-1669). JZ is supported by the NSFC Project for Excellent Research Team (Grant No. 61121064) and the NFRP-China (Grant No. 2011CB921601). SGP is supported by the NSFC-China (Grant No. 11004224) and the NFRP-China (Grant No. 2011CB921601). [1] X.-L. Qi and S.-C. Zhang, Physics Today 63, 33 (2010). [2] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010). [3] Y.-J. Lin, K. Jiménez-García, and I. B. Spielman, Nature (London) 471, 83 (2011). [4] P. Wang, Z.-Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H. Zhai, and J. Zhang, Phys. Rev. Lett. 109, 095301 (2012). [5] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah, W. S. Bakr, and M. W. Zwierlein, Phys. Rev. Lett. 109, 095302 (2012). [6] T. D. Stanescu, B. Anderson, and V. Galitski, Phys. Rev. A78, 023616 (2008). [7] J. Larson and E. Sjöqvist, Phys. Rev. A 79, 043627 (2009). [8] C. Wang, C. Gao, G.-M. Jian, and H. Zhai, Phys. Rev. Lett.105, 160403 (2010). [9] C. Wu, I. Mondragon-Shem, and X.-F. Zhou, Chin. Phys. Lett.28, 097102 (2011). [10] T.-L. Ho and S. Zhang, Phys. Rev. Lett. 107, 150403 (2011). [11] X. Q. Xu and J. H. Han, Phys. Rev. Lett. 107, 200401 (2011). [12] S. Sinha, R. Nath, and L. Santos, Phys. Rev. Lett. 107, 270401 (2011). [13] H. Hu, B. Ramachandhran, H. Pu, and X.-J. Liu, Phys. Rev. Lett. 108, 010402 (2012). [14] H. Hu and X.-J. Liu, Phys. Rev. A 85, 013619 (2012). [15] B. Ramachandhran, B. Opanchuk, X.-J. Liu, H. Pu, P. D. Drummond, and H. Hu, Phys. Rev. A 85, 023606 (2012). [16] R. Barnett, S. Powell, T. Gra, M. Lewenstein, and S. Das Sarma, Phys. Rev. A 85, 023615 (2012). [17] Y. Deng, J. Cheng, H. Jing, C.-P. Sun, and S. Yi, Phys. Rev. Lett. 108, 125301 (2012). [18] Q. Zhu, C. Zhang, and B. Wu, eprint arXiv:1109.5811. [19] J. P. Vyasanakere, V. B. Shenoy, Phys. Rev. B 83, 094515 (2011). [20] M. Iskin and A. L. Subas, Phys. Rev. Lett. 107, 050402(2011). [21] S. L. Zhu, L. B. Shao, Z. D. Wang, and L. M. Duan, Phys. Rev. Lett. 106, 100404 (2011). [22] Z. Q. Yu and H. Zhai, Phys. Rev. Lett. 107, 195305 (2011). [23] H. Hu, L. Jiang, X.-J. Liu, and H. Pu, Phys. Rev. Lett. 107, 195304 (2011). [24] L. Jiang, X.-J. Liu, H. Hu, and H. Pu, Phys. Rev. A 84, 063618 (2011). [25] M. Gong, S. Tewari, and C. Zhang, Phys. Rev. Lett. 107, 195303 (2011). [26] L. Han and C. A. R. Sá de Melo, Phys. Rev. A 85, 011606(R) (2012). [27] X.-J. Liu, L. Jiang, H. Pu, and H. Hu, Phys. Rev. A 85, 021603 (2012). [28] X. Cui, Phys. Rev. A 85, 022705 (2012). [29] X.-J. Liu and H. Hu, Phys. Rev. A 85, 033622 (2012). [30] L. He and X.-G. Huang, Phys. Rev. Lett. 108, 145302 (2012). [31] W. Yi and W. Zhang, arXiv:1204.6476. [32] C. Chin, R. Grimm, P. S. Julienne, and E. Tiesinga, Rev. Mod. Phys. 82, 1225 (2010). [33] C. Chin and P. S. Julienne, Phys. Rev. A 71, 012713 (2005). [34] C. Chin, M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, J. Hecker Denschlag, and R. Grimm, Science 305, 1128 (2004). [35] C. H. Schunck, Y. Shin, A. Schirotzek, and W. Ketterle, Nature (London) 454, 739 (2008). [36] Y. Zhang, W. Ong, I. Arakelyan, J. E. Thomas, Phys. Rev. Lett. 108, 235302 (2012). [37] C. Kohstall, M. Zaccanti, M. Jag, A. Trenkwalder, P. Massignan, G. M. Bruun, F. Schreck and R. Grimm, Na- ture (London) 485, 615 (2012). [38] M. Koschorreck, D. Pertot, E. Vogt, B. Fröhlich, M. Feld and M. Köhl, Nature (London) 485, 619 (2012).
1807.07653v1.Three_boson_spectrum_in_the_presence_of_1D_spin_orbit_coupling__Efimov_s_generalized_radial_scaling_law.pdf	arXiv:1807.07653v1 [cond-mat.quant-gas] 19 Jul 2018Three-boson spectrum in the presence of 1D spin-orbit coupl ing: Eﬁmov’s generalized radial scaling law Q. Guan1and D. Blume1 1Homer L. Dodge Department of Physics and Astronomy, The University of Oklahoma, 440 W. Brooks Street, Norman, Ok lahoma 73019, USA (Dated: May 2, 2019) Spin-orbit coupled cold atom systems, governed by Hamilton ians that contain quadratic kinetic energy terms typical for a particle’s motion in the usual Sch r¨ odinger equation and linear kinetic energy terms typical for a particle’s motion in the usual Dir ac equation, have attracted a great deal of attention recently since they provide an alternativ e route for realizing fractional quantum Hall physics, topological insulators, and spintronics phy sics. The present work focuses on the three- boson system in the presence of 1D spin-orbit coupling, whic h is most relevant to ongoing cold atom experiments. In the absence of spin-orbit coupling ter ms, the three-boson system exibits the Eﬁmov eﬀect: the entire energy spectrum is uniquely determi ned by the s-wave scattering length and a single three-body parameter, i.e., using one of the ene rgy levels as input, the other energy levels can be obtained via Eﬁmov’s radial scaling law, which is intimately tied to a discrete scaling symmetry. It is demonstrated that the discrete scaling symm etry persists in the presence of 1D spin- orbit coupling, implying the validity of a generalized radi al scaling law in ﬁve-dimensional space. The dependence of the energy levels on the scattering length , spin-orbit coupling parameters, and center-of-mass momentum is discussed. It is conjectured th at three-body systems with other types of spin-orbit coupling terms are also governed bygeneraliz ed radial scaling laws, provided the system exhibits the Eﬁmov eﬀect in the absence of spin-orbit coupli ng. PACS numbers: I. INTRODUCTION Under which conditions do two, three, or more par- ticles form weakly-bound states, i.e., bound states that are larger than the range of the two-, three-, and higher- body forces that bind the particles together? And under whichconditionsarethecharacteristicsofthesefew-body bound states governed by underlying symmetries? These questions are of utmost importance across physics. For example, the existence of bound tetra-quark systems [1], ﬁrst proposed in 1964 by Gell-Mann [2], has been chal- lenging our understanding of QCD. The existence of the extremely weakly-bound triton has a profound eﬀect on the nuclear chart, including the existence of larger exotic halo nuclei [3, 4]. Historically, the triton has played an important role in the context of the Thomas collapse [5] and the Eﬁmov eﬀect [6, 7], which is intimately tied to a discrete scaling symmetry of the three-body Schr¨ odinger equation. The three-boson system with two-body short-range interactions is considered the holy grail of few-body physics. It has captured physicists’ attention since Eﬁ- mov’s bizarre and counterintuitive predictions in the early 70ies [6, 7] and has spurred a ﬂurry of theoreti- cal and experimental works from nuclear to atomic to condensed matter to particle physics [8–21]. The unique scaling laws exhibited by Eﬁmov trimers can be traced back to the existence of just one large length scale in the problem, namely the two-body s-wave scattering length. The main focus of the present work is on investigating what happens to the three-boson Eﬁmov states in the presence of 1D spin-orbit coupling. Similar to few-bodysystems on the lattice [22], the 1D spin-orbit coupling in- troduces a parametric dependence of the relative Hamil- tonian on the center-of-mass momentum. This center-of- mass momentum dependence leads, as we will show, to a modiﬁcation of the lowest break-up threshold and has a profound eﬀect on the binding energy. Despite this de- pendence on the center-of-mass momentum and despite the fact that the spin-orbit coupling terms depend on three additional parameters (namely, kso, Ω andδ; see below), it is argued that the three-boson system in the presence of 1D spin-orbit coupling possesses, in the zero- range limit, a discrete scaling symmetry and it is shown that the energy spectrum is described by a generalized radial scaling law. The 1D spin-orbit coupling terms, which break the ro- tational symmetry, introduce an unusual single-particle dispersion. The Hamiltonian ˆHjof thej-th particle with massmand momentum operator ˆ/vector pj(with components ˆpj,x, ˆpj,y, and ˆpj,z) is not simply given by ˆ/vector p2 j/(2m) but includes a term that emulates a spin-1/2 particle inter- acting with a momentum-dependent “magnetic ﬁeld” of inﬁnite range [23–27], ˆHj=ˆ/vector p2 j 2mIj+ˆ/vectorB(ˆpj,z)·ˆ/vector σj. (1) Here,Ijdenotes the 2x2 identity matrix that spans the spin degrees of freedom of the j-th particle, the vector ˆ/vector σj contains the three Pauli matrices ˆ σj,x, ˆσj,y, and ˆσj,zof thej-th particle, andˆ/vectorBrepresents the eﬀective magnetic ﬁeld,ˆ/vectorB= (Ω/2,0,/planckover2pi1ksoˆpj,z/m+δ/2), felt by the j-th par- ticle. The Raman coupling Ω, detuning δ, and spin-orbit2 coupling strength kso, which characterize the two-photon Raman transitionthat couples (eﬀectively) two hyperﬁne states of an ultracold atom, describe the deviations from the “normal” quadratic single-particle dispersion curves, Ej,±=/vector p2 j 2m±/radicaligg/parenleftbigg/planckover2pi1ksopj,z m+δ 2/parenrightbigg2 +Ω2 4,(2) where/vector pjandpj,z(both without “hat”) are expecta- tion values of the corresponding operators. For large \|/vector pj\|, the dispersion curves Ej,±approach/vector p2 j/(2m). For small\|/vector pj\|, in contrast, the Ej,±curves deviate appre- ciably from /vector p2 j/(2m). The momenta /vector pjare generalized momenta (sometimes also referred to as quasi-momenta) and not mechanical momenta (sometimes also referred to as kinetic momenta) [28]. Throughout this article, we frequently drop the preﬁx “generalized” and refer to /vector pj as momentum vector of the j-th atom. The Hamiltonian given in Eq. (1) can also be realized by lattice shaking techniques as well as in photonic crystals and mechanical setups [27, 29–31]. If two-body short-range interactions are added, the modiﬁed single-particle dispersion curves can signiﬁ- cantly alter the properties of weakly-bound two- and three-body states. This has been demonstrated exten- sively for two identical fermions for 1D, 2D, and 3D spin-orbit coupling [32–41] and for two identical bosons for 2D and 3D spin-orbit coupling [41–45] but not for the 1D spin-orbit coupling considered in this work. The present work presents the ﬁrst study of how the experi- mentally most frequently realized 1D spin-orbit coupling termsmodify the three-bosonenergyspectrum. Wenote, however, that several three-body studies for bosonic and fermionic systems with other types of spin-orbit coupling exist [46–49]. All of these earlier studies limited them- selvestovanishingcenter-of-massmomentum. Ourwork, in contrast, allows for ﬁnite center-of-mass momenta. The key objective of the present work is to show that the three-boson system in the presence of 1D spin-orbit coupling obeys a generalized radial scaling law, which re- ﬂects the existence of a discrete scaling symmetry in the limit of zero-range interactions. The scaling parameter λ0,λ0≈22.694, is the same as in the absence of the spin-orbit coupling terms. The generalized radial scaling law relates the energy for a given 1 /as,kso, Ω, and ˜δ[˜δ is a generalized detuning that is deﬁned in terms of the detuningδand thez-component of the center-of-mass momentum, see Eq. (21)] to the energy for a scaled set of parameters, namely λ0/as,λ0kso, (λ0)2Ω, and (λ0)2˜δ. Correspondingly, the term “radial” does not refer to the radius in a two-dimensional space as in the usual Eﬁ- mov scenario but to the radius in a ﬁve-dimensional space. The fact that the discrete scaling symmetry “sur- vives” when the spin-orbit coupling terms are added to thethree-bosonHamiltonianwithzero-rangeinteractions can be intuitively understood from the observation that kso, Ωand˜δcanbethoughtofasintroducingﬁnitelength scales into the system. In the standard Eﬁmov scenario,asintroduces a ﬁnite length scale and the radial scaling law holds regardless of whether \|as\|is larger or smaller than the size of the trimer, provided \|as\|is much larger than the intrinsic scalesof the underlying two- and three- body interactions. In the generalized Eﬁmov scenario considered here, the parameters as,kso, Ω, and ˜δeach introduce a ﬁnite length scale. Correspondingly, the gen- eralized radial scaling law holds regardless of whether these length scales are larger or smaller than the size of the trimer, provided the length scales are much larger than the intrinsic scalesof the underlying two- and three- body interactions. Our ﬁndings for the experimentally most frequently realized 1D spin-orbit coupling are consistent with Ref. [48]. References [46, 48] considered an impurity with 3D spin-orbit coupling that interacts with two iden- tical fermions that do not feel any spin-orbit coupling termsandinteractwiththeimpuritythroughshort-range two-body potentials. Restricting themselves to vanish- ing center-of-mass momenta, Ref. [46] stated that the trimersformassratio /greaterorsimilar13.6“nolongerobey the discrete scaling symmetry even at resonance” because the spin- orbitcoupling“introducesanadditionallength scale”. In Ref. [48], the same authors arrive at a seemingly diﬀer- entconclusion, namely“in the presenceofSO[spin-orbit] coupling, the system exhibits a discrete scaling behav- ior” and “the scaling ratio is identical to that without SO [spin-orbit] coupling”. The two statements can be reconciled by noting that the discrete scaling symmetry requires an enlarged parameter space, an aspect that was recognizedin Ref. [48]but notin Ref. [46]. We conjecture that the discrete scaling symmetry holds for any type of spin-obit coupling and all center-of-mass momenta. De- pending on the type of the spin-orbit coupling, the gen- eralized Eﬁmov plot is four- or ﬁve-dimensional and the generalized radial scaling law applies to the entire low- energy spectrum. The dependence of the energy levels on the system parameters has to be calculated explicitly once for each type of spin-obit coupling. The remainder of this article is organized as follows. To set the stage, Sec. II reviews the standard Eﬁmov scenariofor three identical bosons. Section III introduces the system Hamiltonian in the presence of 1D spin-orbit coupling and discussesthe associatedcontinuousand dis- crete scaling symmetries. The generalized radial scaling law for the three-boson system in the presence of 1D spin-orbit coupling is conﬁrmed numerically in Sec. IV. Section V highlights the role of the center-of-mass mo- mentum and discusses possible experimental signatures of this dependence. Finally, Sec. VI presents an outlook. Technical details are relegated to several appendices. II. REVIEW OF STANDARD EFIMOV SCENARIO The relative Hamiltonian for two identical bosons of massminteracting through the zero-range contact inter-3 actionV2b,zr(/vector r), V2b,zr(/vector r) =4π/planckover2pi12as mδ(3)(/vector r)∂ ∂rr, (3) whereasdenotes the two-body s-wave scattering length and/vector rthe internucleardistance vector ( r=\|/vector r\|), possesses a continuous scaling symmetry [8]. Performingthe trans- formation as→λas, /vector r→λ/vector r,andt→λ2t, (4) wheretdenotes the time and λa real number (scal- ing parameter), the relative two-body time-dependent Schr¨ odinger equation remains unchanged. Importantly, the continuous scaling symmetry extends to three identical mass mbosons with position vectors /vector rj that interact through pairwise s-wave zero-rangeinterac- tionsV2b,zr(/vector rjk) [8]. To see this, we consider the time- dependent Schr¨ odinger equation for the relative three- body Hamiltonian ˆHrel, ˆHrel=/summationdisplay j=1,2−/planckover2pi12 2µj∇2 /vector ρj+2/summationdisplay j=13/summationdisplay k=j+1V2b,zr(/vector rjk)+V3b,zr(R), (5) where/vector ρjdenotes the j-th relative Jacobi vector and µj the associated Jacobi mass. We use a “K-tree” (see Ap- pendix A) in which µ1for the two-body system is given bym/2 andµ1andµ2for the three-body system are given bym/2 and 2m/3. The zero-range three-body po- tentialV3b,zr(R), V3b,zr(R) =g3/planckover2pi12 mδ(6)(R), (6) is written in terms of a six-dimensional delta-function in the three-body hyperradius R,R2=r2 12+r2 13+r2 23. Since thecouplingconstant g3hasunits of length4, it can be rewritten as g3=Cκ−4 ∗, whereCis a real constant andκ∗the three-body binding momentum of one of the three-bosonboundstatesatunitarity(inﬁnite as). While V3b,zr(R) has to be regularized in practice, the explicit regularization is irrelevant for our purpose. Performing the transformation as→λas, /vector rjk→λ/vector rjk, t→λ2t,andκ∗→λ−1κ∗,(7) the Schr¨ odinger equation for the relative Hamiltonian given in Eq. (5) remains unchanged, i.e., the three-body system possesses a continuous scaling symmetry. Intriguingly, the three-body systemwith zero-rangein- teractions additionally exhibits an exact discrete scaling symmetry [8]. The discrete transformation is given by as→(λ0)nas, /vector rjk→(λ0)n/vector rjk, t→(λ0)2nt, andκ∗→κ∗,(8) wheren=±1,±2,···,±∞andλ0≈22.694. The dis- crete scaling transformation, which underlies the three- body Eﬁmov eﬀect, is illustrated in Fig. 1(a). Fixing thethree-body parameter κ∗[see Eq. (8)], the Eﬁmov plot depictsKas a function of 1 /as, where K=−/radicalbig m\|E\|//planckover2pi12 (9) andEdenotes the eigen energy of the Hamiltonian ˆHrel givenin Eq. (5). The thick solidline in Fig. 1(a) shows K for one of the three-body eigen energies. The thick solid line merges with the three-atom threshold on the nega- tiveas-side and with the atom-dimer threshold (dashed line) on the positive as-side. The thick solid line is obtained by solving the time-independent Schr¨ odinger equation for the three-body Hamiltonian ˆHrel. Provided the thick solid line is known (a parametrization can be found in Refs. [8, 11]), the thin solid lines—which cor- respond to other three-body eigen energies—can be ob- tained using the discrete scaling symmetry without hav- ing to explicitly solve the Schr¨ odinger equation again. For the construction, it is convenient to switch from the vector/vector y= (1/as,K)Tto a radius y=\|/vector y\|and an angle ξ, K=−ysinξ (10) and (as)−1=ycosξ, (11) whereξgoes fromπ/4 toπ. The limits π/4 andπare setbythe atom-dimerandthree-atomthresholds, respec- tively. To obtain the thin solid lines in Fig. 1(a) from the thick solid line, one ﬁxes the angle ξand reads oﬀ the values of the pair (1 /as,K) corresponding to the solid line. Using y=/radicalbig (as)−2+K2, (12) it can be seen that the discrete scaling transformation as→(λ0)nasandE→(λ0)−2nEimpliesy→(λ0)−ny. Thus, dividing the radius yof the thick solid line by (λ0)±1,(λ0)±2,···and using the scaled value of yin Eqs. (10) and (11), one obtains the values of the vec- tors/vector y= (1/as,K)Tcorresponding to the thin solid lines. This construction, referred to as Eﬁmov’s radial scaling law, is a direct consequence of the discrete scaling sym- metry. If the three-bosonsystemis characterizedby κnew ∗ instead ofκ∗, the entire energy spectrum is scaled, i.e., if/vector y= (1/as,K)Tdescribes a point on the Eﬁmov plot forκ∗, then (κnew ∗/κ∗)/vector ydescribes a point on the Eﬁmov plot forκnew ∗. III. SYMMETRIES IN THE PRESENCE OF 1D SPIN-ORBIT COUPLING This section generalizes the symmetry discussion pre- sentedintheprevioussectiontothetwo-andthree-boson systems in the presence of 1D spin-obit coupling. As a ﬁrst step, we derive the relative two- and three-body Hamiltonian with zero-range interactions in the presence4 -0.2 -0.1 0 0.1 0.2 Sign(as)\|r0/as\|1/2-0.2-0.10-\|E/Esr\|1/41/asKξλ0 λ0(a) (b)(0,0) FIG. 1: (Color online) Radial scaling law for the standard Eﬁmov scenario. (a) The solid lines show the quantity Kas a function of 1 /asfor the zero-range three-boson Hamiltonian. To make this plot, λ0has been artiﬁcially set to 2 instead of 22.694. The dashed line shows the atom-dimer threshold. The thin radially outgoing solid lines and arrows illustrat e the scaling law. Circles and squares mark the critical scatteri ng lengthsa−at which the trimer energy is degenerate with the three-atom threshold and the critical scattering lengths a∗at which the trimer energy is degenerate with the atom-dimer threshold, respectively. (b) Collapse of neighboring ener gy levels for the ﬁnite-range interaction model [ ˆHrelin Eq. (5) withV2b,zrandV3b,zrreplaced by V2b,GandV3b,G, respec- tively;R0=√ 8r0and (κ∗)−1≈66.05r0]. The solid line shows the fourth-root of the energy of the lowest three-boso n state as a function of the square-root of the inverse of the s-wave scattering length. The dashed line shows the asso- ciated atom-dimer threshold. The dots show the energy of the second-lowest three-boson state, with the radial scali ng law applied in reverse so as to collapse the second-lowest le vel (dots) onto the lowest level (solid line). For clarity, the s caled atom-dimer threshold for the second-lowest three-boson st ate is not shown. of 1D spin-orbit coupling. In a second step, it is shown that these systems possess a continuous scaling symme- try. In a third step, it is argued that the three-boson system additionally exhibits a discrete scaling symmetry, suggesting the existence of a generalized radial scaling law. Numerical evidence that supports our claim that the three-boson system with 1D spin-orbit coupling is governed by a generalized radial scaling law is presented in Sec. IV. Westartwiththe ﬁrststep. The N-bosonHamiltonian in the presence of 1D spin-orbit coupling reads ˆH=ˆHni+ˆVint, (13)where the non-interacting and interacting pieces are given by ˆHni=N/summationdisplay j=1ˆ/vector p2 j 2mI1,···,N+ N/summationdisplay j=1/parenleftbigg/planckover2pi1kso mˆpj,z+δ 2/parenrightbigg I1,···,j−1ˆσj,zIj+1,···,N+ N/summationdisplay j=1Ω 2I1,···,j−1ˆσj,xIj+1,···,N (14) and ˆVint= N/summationdisplay j=1,j<kV2b(rjk)+N/summationdisplay j=1,j<k<lV3b(rjkl) I1,···,N. (15) Here,Ij,···,kwithj < kspans the spin degrees of free- dom of particles jthroughk,Ij,···,k=Ij⊗···⊗Ik. For N= 2, onlyV2bcontributes. For N= 3,rjklis equal to the three-body hyperradius R. The interaction model considered throughout this work assumes that the inter- actions are the same in all spin channels. It is, just as in the case without spin-orbit coupling, convenient to use Jacobi coordinates /vector ρjand associated momentum operators ˆ/vector qjinstead of the single-particle quantities/vector rjandˆ/vector pj. Importantly, the N-th Jacobi “quantities” /vector ρNandˆ/vector qNcorrespond to the center-of- mass vector and center-of-mass momentum operator. It can be shown straightforwardly that the Hamiltonian ˆH commutes with the center-of-mass momentum operator ˆ/vector qN[50], i.e., the Schr¨ odinger equation ˆHΨ =EΨ can be solved for each ﬁxed /vector qN. Using this and Jacobi coordi- nates, thenon-interactingﬁxed- /vector qNHamiltonian, denoted byˆ¯Hni, reads ˆ¯Hni=ˆ¯Hni,rel+/vector q2 N 2µNI1,···,N, (16) where ˆ¯Hni,rel=N−1/summationdisplay j=1ˆ/vector q2 j 2µjI1,···,N+ N−1/summationdisplay j=1/planckover2pi1kso mˆqj,zˆΣj,z+ N/summationdisplay j=1Ω 2I1,···,j−1ˆσj,xIj+1,···,N+ /parenleftbigg/planckover2pi1kso µNqN,z+δ 2/parenrightbigg × N/summationdisplay j=1I1,···,j−1⊗ˆσj,z⊗Ij+1,···,N .(17)5 The explicit form of the operators ˆΣj,zwithj= 1,···,N−1 is given in Appendix A. Note that the ﬁrst andsecondlinesofEq.(17)containmomentumoperators while the fourth line of Eq. (17) and the second term on the right hand side of Eq. (16) contain expectation val- ues of the center-of-mass momentum operators (and not operators). As “usual”, the interaction ˆVintdepends on /vector ρ1,···,/vector ρN−1but not on the center-of-mass vector /vector ρN. This implies that the eigen states Ψ can be written as Ψ = Φ cmΦrel, (18) where [51] Φcm= exp/parenleftbiggı/vector qN·/vector ρN /planckover2pi1/parenrightbigg (19) and where the Φ rel, which are eigen states ofˆ¯Hrel, ˆ¯Hrel=ˆ¯Hni,rel+ˆVint, (20) depend on the Jacobi vectors /vector ρ1,···,/vector ρN−1and the spin degrees of freedom. Equation (17) shows that the eigen energies of ˆHde- pend on the generalized detuning ˜δ, ˜δ 2=/planckover2pi1kso µNqN,z+δ 2, (21) i.e.,qN,zandδenter as a combination and not as in- dependent parameters. This observation suggests that the center-of-mass momentum may play a decisive role in determining the characteristics of the weakly-bound two- and three-body states (see also Refs. [35, 37]). The parametric dependence of the Hamiltonianˆ¯Hrelon the z-component qN,zof the center-of-mass momentum is a direct consequence of the fact that the presence of the spin-orbit coupling breaks the Galilean invariance [26]. One immediate consequence of the broken Galilean in- variance is that knowing the energy of an eigen state withqN,z= 0 does not, in general, suﬃce for predicting the energy of an eigen state with qN,z/negationslash= 0. Importantly, the eigen states Ψ depend, in general, explicitly on qN,z andδand not just on ˜δ. We are now ready to address the second step. Parametrizingthe two-body interactions V2bby the zero- range potential V2b,zr, theN= 2 relative Hamiltonian depends on four parameters, namely as,kso, Ω, and ˜δ. It can be readily checked that the corresponding time- dependent Schr¨ odinger equation is invariant under the transformation as→λas, kso→λ−1kso,Ω→λ−2Ω,˜δ→λ−2˜δ, /vector r→λ/vector r,andt→λ2t,(22) i.e., theN= 2 system with zero-range interactions pos- sesses a continuous scaling symmetry. The continuous scalingsymmetryextendstothethree-bosonsystemwithzero-range interactions [ V2b=V2b,zrandV3b=V3b,zrin Eq. (15)] in the presence of spin-orbit coupling since the corresponding time-dependent N= 3 Schr¨ odinger equa- tion is invariant under the transformation as→λas, kso→λ−1kso,Ω→λ−2Ω,˜δ→λ−2˜δ, /vector r→λ/vector r, t→λ2t,andκ∗→λ−1κ∗.(23) Equations (22) and (23) generalize Eqs. (4) and (7) from Sec. II. Paralleling the discussion of Sec. II, step three poses the question whether or not the three-boson system in the presence of spin-orbit coupling additionally possesses a discrete scaling symmetry in the zero-range interaction limit. Our claim is that it does and that the discrete transformation is given by as→(λ0)nas, kso→(λ0)−nkso,Ω→(λ0)−2nΩ, ˜δ→(λ0)−2n˜δ, /vector r→(λ0)n/vector r, t→(λ0)2nt,andκ∗→κ∗, (24) whereλ0is identical to the scaling factor of the stan- dard Eﬁmov scenario, i.e., λ0≈22.694. Since no general analytical solutions exist to the three-boson Schr¨ odinger equation in the presence of spin-orbit coupling, we rely on numerics to support our claim. The claim that the discrete scaling symmetry survives in the presence of the spin-orbit coupling terms can be understood intuitively by realizing that the spin-orbit coupling terms modify the low- but not the high-energy portions of the single- particledispersioncurves. Tosetthestageforthenumer- ical calculations presented in the next section, we discuss a number of consequences of the discrete scaling symme- try. The discrete scaling symmetry suggests a generalized radial scaling law for the three-boson system in the pres- ence of 1D spin-orbit coupling in which the Eﬁmov plot for/vector y= (1/as,K)Tdiscussed in the previous section is replaced by a generalized Eﬁmov plot for /vector y= (1/as,kso,Sign(Ω)/radicalbig m\|Ω\|//planckover2pi12,Sign(˜δ)/radicalig m\|˜δ\|//planckover2pi12,K)T. (25) In the limit that the second, third, and fourth param- eters vanish, each of the usual Eﬁmov energies is four- fold degenerate due to the fact that the spin degrees of freedom enlarge the three-boson Hilbert space by a factor of four (from the 23= 8 independent spin con- ﬁgurations, one can construct four fully symmetric spin functions). For non-vanishing kso, Ω, and ˜δ, we expect that the three-boson system supports four “unique” en- ergy levels. Each of the four energies, collectively re- ferred to as a manifold, is characterized by a vector /vector y. Knowing the dependence of each of these energy curves on 1/as,kso, Sign(Ω)/radicalbig m\|Ω\|//planckover2pi12, and Sign( ˜δ)/radicalig m\|˜δ\|//planckover2pi12, there should exist other energy manifolds for the same κ∗that can be obtained from the manifold that has been6 mapped out without explicitly solving the three-boson Schr¨ odinger equation again. To see how, we switch from the ﬁve parameters given inEq.(25)tothelength y=\|/vector y\|andfourangles ξ1,···,ξ4 for each of the four energy levels in the “reference mani- fold”, K=−ysinξ1sinξ2sinξ3sinξ4,(26) Sign(˜δ)/radicalig m\|˜δ\|//planckover2pi12=ycosξ1sinξ2sinξ3sinξ4,(27) Sign(Ω)/radicalbig m\|Ω\|//planckover2pi12=ycosξ2sinξ3sinξ4, (28) kso=ycosξ3sinξ4, (29) and 1/as=ycosξ4. (30) The full range of possible as,kso, Ω, and ˜δis covered if ξ1,ξ2,ξ3,ξ4∈[0,π]. The range of the angles is further constrained by the energy surfaces of the three-atom and atom-dimer thresholds (see Secs. IV and V). To obtain theKfor other manifolds, one chooses a direction of the vector/vector yby ﬁxing the angles ξ1toξ4and reads oﬀ the values of the components of /vector yfor each of the four known energy levels. Using y=/radicaligg (as)−2+(kso)2+m\|Ω\| /planckover2pi12+m\|˜δ\| /planckover2pi12+K2,(31) it can be seen that the discrete transformation as→ (λ0)nas,kso→(λ0)−nkso, Ω→(λ0)−2nΩ,˜δ→(λ0)−2n˜δ, E→(λ0)−2nEimpliesy→(λ0)−ny. Thus, dividing the “hyperradius” ycorresponding to the j-th energy in the reference manifold by ( λ0)±1, (λ0)±2,···and using the scaled value of yin Eqs. (26)-(30), one obtains the values of the components of /vector yfor thej-th energy level in the other manifolds. The generalized scaling law is tested in the next section by considering two neighboring en- ergy manifolds and conﬁrming that the energy manifolds collapse onto each other if the discrete scaling transfor- mation isapplied to the energylevelsin the moreweakly- bound manifold. IV. NUMERICAL TEST OF THE GENERALIZED RADIAL SCALING LAW To facilitate the numerical calculations, we replace the two-body zero-range potential V2b,zrby an attractive GaussianV2b,Gwith range r0and depthv0, V2b,G(rjk) =v0exp/parenleftigg −r2 jk 2r2 0/parenrightigg , (32) wherev0is negative and adjusted such that V2b,G(rjk) supports at most one two-body s-wave bound state. To reduce ﬁnite-range eﬀects, we aim to work in the regimewhere the absolute value of the free-space s-wavescatter- ing lengthasis notably larger than r0. Parameter com- binations where the absolute value of the free-space p- wave scattering volume is large are excluded. The three- bodyzero-rangepotential V3b,zrisreplacedbyarepulsive GaussianV3b,Gwith range R0and height V0, V3b,G(rjkl) =V0exp/parenleftigg −r2 jkl 2R2 0/parenrightigg . (33) In our numerical calculations, R0is ﬁxed at√ 8r0and V0(V0≥0) is varied to dial in the desired three-body parameterκ∗. Speciﬁcally, we deﬁne κ∗to be the bind- ing momentum of the energetically lowest-lying universal three-body state at unitarity (inﬁnite as) forkso= Ω = ˜δ= 0. Without the repulsive three-body potential, the lowest three-body state is not universal [52]. The repul- sive three-body potential pushes the lowest three-body energyupandweadjust V0, forﬁxedv0(inﬁniteas), such thattheenergyofthelowestthree-bodystateforﬁnite V0 is identical to the energy of the ﬁrst excited three-body state forV0= 0. This corresponds to ( κ∗)−1≈66.05r0, i.e., the trimer is much larger than the intrinsic scales of the two- and three-body interactions. With the repulsive three-bodypotentialturnedon, theradialscalinglawcan be tested using the two lowest-lying energy manifolds. We start the discussion of our numerical results by looking at the three-body spectrum for the standard Eﬁ- mov scenario ( kso= Ω =˜δ= 0). The reason for dis- cussing this “reference system” is two-fold: it illustrates how to check the validity of the radial scaling law for a case where it is known to hold and it gives us a sense for the ﬁnite-range corrections expected in the presence of spin-orbit coupling. The solid line in Fig. 1(b) shows therelativethree-bodyenergyoftheenergeticallylowest- lying state as a function of the inverse of the s-wave scattering length for the Hamiltonian given in Eq. (5) withV2b,zrandV3b,zrreplaced by V2b,GandV3b,G, re- spectively. To “compress” the data, the horizontal and vertical axis employ a square-root and fourth-root repre- sentation. The scattering length is scaled by r0and the energy byEsr, Esr=/planckover2pi12 mr2 0. (34) The trimer energy merges with the three-atom threshold on the negative scattering length side at r0/\|as\| ≈0.01. To get a feeling for the ﬁnite-range eﬀects, we assume that the radial scaling law holds and apply it “in re- verse”. Speciﬁcally, using numerically determined pairs (1/as,K) corresponding to the excited state, the dots in Fig.1(b) showthe points( λ0r0/as,λ0r0K), using—asfor the lowest state—the square-root and fourth-root depic- tion. In the zero-range limit ( r0→0 andR0→0), the dots would lie on top of the solid line. The nearly perfect agreementbetweenthesolidlineandthe dotsin Fig.1(b) indicates that the ﬁnite-range eﬀects are negligibly small for the parameter combinations considered.7 To test the generalized radial scaling law proposed in Sec. III, we calculate the eigen energies of states in the lowest and second-lowest manifolds ofˆ¯Hrel(there are at most four states in each manifold) and scale the energies in the second-lowest manifold assuming that the gener- alized radial scaling law holds. If the energy curves col- lapse, the generalized radial scaling law is validated. In the presence of the 1D spin-orbit coupling, the gen- eralized Eﬁmov plot has ﬁve axes. Clearly, visualizing energy surfaces that depend on four parameters is im- possible and fully mapping out these high-dimensional dependences is computationally demanding. Thus, we consider selected cuts in the ﬁve-dimensional space. Our ﬁrst cut uses ( kso)−1= 50r0, Ω = 2Eso= 0.04Esr, where Eso=(/planckover2pi1kso)2 2m, (35) and˜δ= 0. For these parameters, we calculate the rela- tive energy Eof the states in the lowest energy man- ifold. The solid lines in Fig. 2(a) show the quantity −\|(E−Eaaa th)/Esr\|1/4as a function of Sign( as)\|r0/as\|1/2, whereEaaa thdenotes the energy of the lowest three-atom threshold whose wave function has the same total mo- mentumq3,zalong thez-axis as the three-body system. The determination of Eaaa this discussed in Appendix C. The energy Eaaa this independent of κ∗and referencing E relative to the lowest three-atom threshold does not alter the generalizedradialscalinglaw. Figure2(a) showsthat the lowest energy manifold consists of, depending on the value ofr0/as, zero, one, or two energy levels [the second and third excited states of the lowest manifold exist at largerr0/asthan those shown in Fig. 2(a)]. The lowest three-body energy merges with the three- atom threshold on the negative s-wave scattering length side and with the atom-dimer threshold [dashed line in Fig. 2(a)] on the positive scattering length side. The de- termination of the atom-dimer threshold energy Ead this discussed in Appendix D. Just as the three-atom thresh- old, the atom-dimer threshold is independent of κ∗and determined such that the momentum q3,zof the atom- dimer system is the same as that of the three-body sys- tem. The second lowest state does not merge with the three-atom threshold on the negative asside but with the atom-dimer threshold on the positive asside. Having determined the energies of the states in the lowest energy manifold, the next step is to calculate the energies of the states in the second-lowest energy man- ifold. To map the energies of the states in the second- lowest manifold onto the energies of the states in the lowest energy manifold, we use the same r0,R0, andκ∗ and calculate the energies of the states in the second- lowest energy manifold for a ksothat isλ0times smaller than theksoused to calculate the energiesof the states in the lowest energy manifold [i.e., for ( kso)−1≈1,135r0], for a Ω that is ( λ0)2times smaller than the Ω used to calculate the energies in the lowest energy manifold (i.e., for Ω ≈7.77×10−7Esr), and for ˜δ= 0 (the scal- ing does not change zero) as a function of r0/as. Hav--0.2-0.10-\|(E-Ethaaa)/Esr\|1/4 -0.2-0.10-\|(E-Ethaaa)/Esr\|1/4 -0.2 -0.1 0 0.1 0.2 Sign(as)(r0/as)1/2-0.2-0.10-\|(E-Ethaaa)/Esr\|1/4(a) (b) (c) FIG. 2: (Color online) Testing the generalized radial scal- ing law in the presence of 1D spin-orbit coupling. Pan- els (a)-(c) demonstrate the collapse of two neighboring en- ergy manifolds for the ﬁnite-range interaction model [ˆ¯Hrelin Eq. (20) with V2b=V2b,GandV3b=V3b,G;R0=√ 8r0and (κ∗)−1≈66.05r0]. The energies of states in the lowest man- ifold (solid lines) are obtained for Ω = 2 Eso,˜δ= 0, and (a) (kso)−1= 50r0, (b) (kso)−1= 25r0, and (c) ( kso)−1= 100r0. In all three panels, the dashed lines show the atom-dimer threshold. The dots show the energies of states in the second - lowest manifold, with the generalized radial scaling law ap - plied in reverse so as to collapse the three-body energies of states in the second-lowest manifold (dots) onto the three- body energies of the states in the lowest manifold (solid lin es). For clarity, the scaled atom-dimer thresholds for the secon d- lowest energy manifold are not shown in any of the panels. ing calculated the energies of the states in the second- lowest manifold for the scaled kso, Ω, and ˜δ, the pairs (1/as,E−Eaaa th) are scaled (note that Eaaa thfor the ex- cited state manifold is calculated using the scaled kso, Ω, and˜δvalues). The dots in Fig. 2(a) show the scaled pairs (λ0r0/as,−(λ0)2\|E−Eaaa th\|/Esr), using—as for the lowest energy manifold—the square-root and fourth-root depiction. It can be seen that the solid lines and dots agree very well. Note that the atom-dimer threshold for the second-lowest energy manifold also needs to be re- calculated using the scaled kso, Ω, and ˜δ[the resulting energies lie essentially on top of the dashed line and are not shown in Fig. 2(a)]. The deviation between the solid line and dots is 0 .025% for (r0/as)1/2= 0 and 0.80 % for8 (r0/as)1/2= 0.19. These deviations are comparable to those between the corresponding atom-dimer thresholds [inthiscase, thedeviationsare0 .023%for(r0/as)1/2= 0 and 0.82 % for (r0/as)1/2= 0.19]. We conclude that our numerical results are consistent with the generalized ra- dial scaling law. We emphasize that the scaling law has to be applied to all ﬁve axes of the generalized Eﬁmov plot, i.e., to obtain the dots in Fig. 2(a) it is imperative to not only scale the two axes depicted but also the parameters corresponding to the three axes that are not depicted. The ratio of the lowest energy in neighboring manifolds at unitarity, e.g., is only equal to 22 .6942if the direction of ˆ yis the same for the two energy levels under consideration. The energy scales Eso,\|Ω\|, and\|˜δ\|are much smaller than\|E−Eaaa th\|for a large portion of Fig. 2(a). The region close to the three-atom threshold is an exception. As such it might be argued that the spin-orbit coupling terms are too weak to notably inﬂuence the energy spec- trum, possibly suggesting that the applicability of the generalized radial scaling law is trivial. One fact that speaks against this argumentation is that the shape of the energy levels is notably inﬂuenced by the spin-orbit coupling terms. This is, e.g., reﬂected by the fact that the energy levels in a given energy manifold are not de- generate. To more explicitly demonstrate that the gen- eralized radial scaling law holds when one or more of the energy scales associated with the spin-orbit coupling termsis/arelargerthanthebindingenergy,werepeatthe calculations for larger ksothan those used in Fig. 2(a). Speciﬁcally, to determine the energy of the lowest state in the lowest energy manifold [solid line in Fig. 2(b)], we use (kso)−1= 25r0while keeping r0,R0,κ∗, and˜δun- changed. The Raman coupling strength Ω is set to be equal to 2Eso. To demonstrate the collapse of the ener- gies of the lowest states in the second-lowest and lowest manifolds, we apply the generalized radial scaling law in the same way as in Fig. 2(a). The energy of the lowest state in the second-lowest manifold is shown by dots in Fig. 2(b). The agreement with the solid line is excellent, supporting our claim that the generalized radial scaling law is not limited to the case where the energy scales associated with the spin-orbit coupling are smaller than the binding energy of the trimer, provided these energies are notably smaller than Esr. To show the characteristics of the excited states in the lowest manifold in more detail, we consider a smaller kso, (kso)−1= 100r0, and as before ˜δ= 0 and Ω = 2 Eso. The use of a smaller kso[solid lines in Fig. 2(c)] moves the merging points of the three-body energies corresponding to the excited states with the atom-dimer threshold to the left compared to Fig. 2(a). Again, scaling the pa- rameters appropriately, the dots in Fig. 2(c) show the energies of the states in the second-lowest energy mani- fold. The dots agree nearly perfectly with the solid lines not onlyfor the loweststatein the twomanifolds but also forthe excited statesin the twomanifolds, lending strong numerical support for the validity of the generalized ra-dialscalinglawandhence fortheexistence ofthe discrete scaling symmetry in the presence of 1D spin-orbit cou- pling terms in the zero-range limit. Asalreadymentioned, thethree-bodyparameter κ∗for V0= 0, deﬁned using the energy of the ﬁrst excited state at unitarity in the absence of spin-orbit coupling, is iden- tical to the κ∗for the three-body interaction with ﬁnite V0used throughout this section. Turning on the spin- orbit coupling, we checked that the energies of states in the second-lowestmanifold for V0= 0 agreewell with the energies of states in the lowest manifold for the ﬁnite V0. This providesevidence that the generalizedradialscaling law is, just as the standard radial scaling law, indepen- dent of the details of the underlying microscopic interac- tion model. To conﬁrm the continuous scaling symmetry ofthethree-bodyHamiltonianinthepresenceof1Dspin- orbit coupling, we checked that the energies for diﬀerent κ∗can be mapped onto each other: If /vector ydescribes a point on the Eﬁmov plot for κ∗, then (κnew ∗/κ∗)/vector ydescribes a point on the Eﬁmov plot for the new κnew ∗. V. EXPERIMENTAL IMPLICATIONS: ROLE OF CENTER-OF-MASS MOMENTUM Measuring signatures associated with two consecutive trimer energy levels is challenging, especially for equal- mass bosons, due to the relatively large discrete scaling factor of 22 .694. The reason is that the absolute value of the scattering length should, on the one hand, be no- tably larger than the van der Waals length rvdWand, on the other hand, be smaller than the de Broglie wave lengthλdB[8, 11]. Despite these challenges, the discrete scaling symmetry underlying the standard Eﬁmov sce- nario has been conﬁrmed experimentally by monitoring the atom losses of an ultracold thermal gas of Cs atoms as a function of the s-wave scattering length [20] (for unequal mass mixtures, see Refs. [57, 58]). When the trimer energy is degenerate with the three-atom thresh- old [dots in Fig. 1(a); the corresponding critical scatter- ing lengths are denoted by a(n) −] or with the atom-dimer threshold [squares in Fig. 1(a); the corresponding criti- cal scattering lengths are denoted by a(n) ∗], the losses are enhanced. Since the critical scattering lengths for con- secutive trimer states are related to the scaling factor λ0, these atom-loss measurements provide a direct conﬁrma- tion of the discrete scaling symmetry. In addition, other characteristicsofthe standardEﬁmovscenariohavebeen measured [8, 11–13]. For example, the critical scattering lengthsa(n) −anda(n) ∗for a given trimer level nare re- lated to each other by a universal number. Correspond- ingly, the experimentally determined ratio a(n) −/a(n) ∗can be viewed as a test of the functional form of the en- ergy levels shown in Fig. 1(a). Other experimental tests of the standard Eﬁmov scenario include the determina- tion of the binding energy of an Eﬁmov trimer via radio- frequency spectroscopy [17, 18], the imaging of the quan-9 - -1.2-0.9-0.6-0.3 0 0.300.10.20.30.40.50.6 (a0/as)×104(δ˜/h)/kHz FIG. 3: (Color online) The contours show the negative of the three-boson binding energy, in kHz, of the lowest state i n the second lowest manifold as functions of ( as)−1and˜δfor kso/κ∗≈1.32 and Ω = 2 Eso, whereκ∗denotes the binding momentum of the ﬁrst excited Eﬁmov trimer at unitarity in the absence of spin-orbit coupling. The conversion to a0and kHz is done using the experimentally determined value of a(1) − for Cs (see text). The calculations are performed forˆ¯Hrelwith V2b=V2b,GandV3b= 0 [(κ∗)−1≈66.05r0]. tum mechanical density of the helium Eﬁmov trimer via Coulomb explosion [21], and the observation of four- and ﬁve-body loss features that are universally linked to the critical scattering lengths of the Eﬁmov trimer [53–56]. Directly measuring the discrete scaling symmetry in the presence of spin-orbit coupling requires varying the inverse of the s-wave scattering length by the scaling fac- torλ0, as in the standard Eﬁmov scenario, as well as varying the spin-orbit coupling parameters Ω, Eso, and ˜δby (λ0)2. Covering such a wide range of parameters is expected to be very challenging experimentally. In what follows we instead focus on the situation where the spin- orbit coupling parameters ksoand Ω are held ﬁxed while thes-wave scattering length asand generalized detuning ˜δare varied. An analogous study for the standard Eﬁ- mov scenario would look at the three-boson system for a ﬁxed ﬁnite s-wave scattering length. In this case, the energy spacing would not be ( λ0)2; however, the energies of neighboring states would still be uniquely related to each other. For concreteness, we consider the133Cs system [59], for which the three-atom resonances in the absence of spin-orbitcouplingoccuratthecriticalscatteringlengths a(0) −≈ −936a0anda(1) −≈ −20,190a0[20]. Here, a0 denotes the Bohr radius and the superscripts “(0)” and “(1)”indicatethatthesecriticalscatteringlengthsarefor the ground and ﬁrst excited Eﬁmov trimers, respectively. Applying ournumerical result κ∗a−=−1.505to the ﬁrst excited state, the Cs system is characterized by ( κ∗)−1≈ 13,416a0. Figure 3 shows the negative of the binding energy of the lowest state in the ﬁrst excited manifold for kso/κ∗≈1.32 and Ω = 2 Esoas functions of the inverse of the s- wave scattering length and the generalized detuning ˜δ using, as in the previous sections, that the scattering lengths are the same for all spin channels. Using Cs’s a(1) −, these parameterscorrespondto ( kso)−1≈10,156a0, Eso/h≈0.132kHz, and Ω /h≈0.264kHz. Compari- son with the87Rb experiment at NIST [28], which uses (kso)−1≈3,410a0(corresponding to Eso/h≈1.786kHz) and Ω/hvalues ranging from zero to about 10kHz, sug- gests that the parameter regime covered in Fig. 3 is rea- sonable. Figure 3 shows that the three-boson binding energy for a ﬁxed scattering length is largest for ˜δ= 0 (this is where the three-atom threshold has a degener- acy of six; see Appendix C). In addition, there exists an enhancement of the binding for ˜δ/h≈0.301kHz (this is where the three-atom threshold has a degeneracy of four; see Appendix C). As ˜δgoes to inﬁnity, the trimer in the presenceofthe 1Dspin-orbit couplingbecomesun- bound at the same scattering length as the correspond- ing trimer in the absence of spin-orbit coupling (i.e., at as≈ −20,190a0). The three-boson binding energy shown in Fig. 3 is calculated by enforcing that the three-boson threshold has the same center-of-mass momentum as the trimers (see Appendices C-E). If the detuning δis equal to zero, the generalized detuning ˜δis directly proportional to the z-component q3,zof the center-of-mass momentum [see Eq. (21)]. In this case, the trimer is bound maximally forq3,z= 0. However, for ﬁnite detuning δ, the most strongly bound trimer has a ﬁnite center-of-mass mo- mentum. A similar dependence on the center-of-mass momentum was pointed out in Refs. [35, 37, 60] for the two-fermion system. The dependence ofˆ¯HrelonqN=3,zis a key character- istic of systems with 1D spin-orbit coupling. A similar dependence exists for three-body systems in the pres- ence of 2D or 3D spin-orbit coupling (in these cases, the relative Hamiltonian depends on two or all three compo- nentsof/vector qN=3) andforthree-bodysystemsonalattice (in this case,/vector qN=3is a lattice or quasi-momentum vector). In all works known to us [16, 46–48], the assumption /vector qN=3= 0 is made prior to obtaining concrete results. Table I contrasts studies for systems, which possess a center-of-mass momentum dependence, with the “stan- dard” three-boson Eﬁmov system (ﬁrst row), for which the relative Hamiltonian is independent of /vector qN=3. In the standard Eﬁmov case, the lowest atom-dimer threshold of the relative Hamiltonian is given by the energy Eaof an atom with vanishing atom momentum vector /vector qa(Eais equal to zero) plus the energy Ed(/vector qd= 0) of a dimer with vanishing dimer momentum vector /vector qd. When the relative Hamiltonian depends on /vector qN=3, the atom-dimer threshold needs to be determined carefully, since the trimer with ﬁxed/vector qN=3can break up into an atom with ﬁnite momen- tum and into a dimer with ﬁnite momentum in such a way that the generalized three-body center-of-mass mo- mentum is conserved. Of the many break-up conﬁgu-10 systemˆ¯Hrel=ˆ¯Hrel(/vector qN=3)?restriction? atom-dimer threshold (rel. Ham.) 3-spinless bosons; “standard” Eﬁmov scenario [6] no no Ed(/vector qd= 0)+Ea(/vector qa= 0) FFX; X feels 2D SOC; Borromean binding [47] yes /vector qN=3= 0 min/vector qdEd(/vector qd)+min /vector qaEa(/vector qa) FFX; X feels 3D SOC; universal/Eﬁmov trimers [46, 48] yes /vector qN=3= 0min/vector qd+/vector qa=/vector qN=3[Ed(/vector qd)+Ea(/vector qa)] BBB quasi-particles on lattice; Eﬁmov trimers [16] yes /vector qN=3= 0 Ed(/vector qd= 0)+Ea(/vector qa= 0) BBB; B’s feel 1D SOC (this work) yes no min/vector qd+/vector qa=/vector qN=3[Ed(/vector qd)+Ea(/vector qa)] TABLE I: Summary of three-particle studies. The standard Eﬁ mov scenario (ﬁrst row) is contrasted with three-particle s ystems for which the relative Hamiltonianˆ¯Hreldepends parametrically on the generalized three-body cent er-of-mass momentum vector /vector qN=3. The last column lists the atom-dimer threshold of the relat ive Hamiltonian. The symbols Ed,Ea,/vector qd, and/vector qadenote the energy of the dimer, energy of the atom, generalized mome ntum of the dimer, and generalized momentum of the atom, respectively. “SOC” stands for “spin-orbit coupling”, “F” for “fermion”, “X” for a particle diﬀerent from “F”, and “B” f or “boson”. rations that conserve the three-body center-of-mass mo- mentum, theonewiththelowestenergydeﬁnestheatom- dimer threshold. Table I shows that the deﬁnition of the lowest atom-dimer threshold of the relative Hamil- tonian varies in the literature. The deﬁnitions employedin Refs. [16, 47] disagree with the deﬁnition used in the present work (last row of Table I). While the deﬁnition of Ref. [47] may be meaningful in a many-body context (see also comment [60]), we fail to see how the deﬁnition of Ref. [16] can, in general, be correct. It is proposed that the center-of-mass momentum de- pendence can be observed experimentally by performing atom-loss measurements for ﬁxed kso, Ω, andδon a cold thermalatomic gas. Tuning the s-wavescatteringlength, one expects—just asin the casewhere the spin-orbitcou- pling is absent—enhanced losses when the trimer energy is degenerate with the three-atomthreshold. However, in contrast to the standard Eﬁmov scenario, such a degen- eracy exists for a rangeof scattering lengths provided the trimersembedded inthethermalgashavediﬀerentthree- body center-of-mass momenta (the exact distribution of center-of-mass momenta is set by the temperature of the gas sample). Figure 3 shows that the critical scattering lengtha(1) −changes, for the Cs example, from −20,190a0 for large ˜δto−7,791a0for˜δ= 0. Provided the three- body center-of-mass momenta are spread over the range covered on the vertical axis in Fig. 3, one expects en- hanced losses over the entire scattering length window. The diﬀerence between the losses in the presence and ab- sence of the spin-orbit coupling terms can be interpreted as a few-body probe of the breaking of the Galilean in- variance in the presence of spin-orbit coupling. An important question is whether the changes of the loss features related to the lowest state in the second- lowest manifold will be washed out by ﬁnite tempera- ture eﬀects. A comprehensive answer to this question will require performing three-body recombination calcu- lations, which include thermal averaging, in the pres- ence of spin-orbit coupling. Such calculations are beyond the scope of this work. Given that the energy scales associated with the spin-orbit coupling are, for the ex- ample considered in Fig. 3, comparable to /planckover2pi12κ2 ∗/mand that the binding energy of the trimer near the three-atom threshold is much smaller than /planckover2pi12κ2 ∗/m, we are hopeful that the temperatures realized in previous Cs experiments ( T≥7.7nK) [20] are low enough to ob- serve the impact of the spin-orbit coupling terms on the loss features. For example, without spin-orbit coupling, the loss coeﬃcient L3is maximal around −20,000a0 and reaches about half of its maximum value at around −10,000a0[see Fig. 1(a) of Ref. [20]]. In the pres- ence of spin-orbit coupling, the loss feature is expected to be centered over the range −20,190a0to−7,790a0, leading to an observable modiﬁcation of the shoulder on the less negative scattering length side. The shape of the shoulder is expected to carry a signature of the non-monotonic dependence of the critical scatter- ing length a(1) −on the center-of-mass momentum. For example, three diﬀerent ˜δcorrespond to the same a(1) − fora(1) −∈[−10,330a0,−9,160a0] but each ˜δcorresponds to a unique a(1) −fora(1) −∈[−20,190a0,−10,330a0] and a(1) −∈[−9,160a0,−7,790a0]. For the same spin-orbit coupling parameters, the criti- calscatteringlength a(0) −, associatedwiththe loweststate in the lowest manifold, displays essentially no depen- dence on ˜δ, i.e., the associated three-atom loss feature should only be minimally aﬀected by the spin-orbit cou- pling terms. Intuitively, this can be understood byrealiz- ing that the energy scales associated with the spin-orbit coupling parameters are much smaller than the binding energy of the lowest-lying trimer state. The fact that the loss features for the lowest state in the lowest and second-lowest manifolds are expected to be very diﬀer- ent can also be understood from the generalized radial scaling law. Fixing the spin-orbit coupling parameters11 corresponds to looking at particular cuts in the ﬁve- dimensional parameterspace as opposed to looking along a speciﬁc radial direction. As a consequence, the loss fea- tures for the two manifolds can be very diﬀerent even if the scattering lengths at which the loss features occur are, roughly, spaced by λ0. VI. CONCLUDING REMARKS This work analyzed what happens to the three-boson Eﬁmov spectrum if 1D spin-orbit coupling terms, realiz- able in cold atoms as well as in photonic crystalsand me- chanical setups, are added to the Hamiltonian. The spin- orbit coupling terms introduce a parametric dependence oftherelativeHamiltonianonthecenter-of-massmomen- tum vector. A similar center-of-mass momentum vector dependence exists for few-body systems with short-range interactions on a lattice. The present work mapped out, for the ﬁrst time, the three-boson spectrum as a function of the center-of-mass momentum vector. It was found that the three-boson system in the presence of 1D spin- orbit coupling obeys a generalized radial scaling law in a ﬁve-dimensional parameter space, which is associated with a discrete scaling symmetry. Within the framework of eﬀective ﬁeld theory, the existence of the discrete scal- ing symmetry can be rationalized by scale separation: The discrete scaling symmetry of the standard Eﬁmov scenario “survives” provided the additional length scales are much larger than the ranges of the intrinsic interac- tions. While our work focused on 1D spin-orbit coupling, the discrete scaling symmetry should persist for other types of spin-obit coupling schemes as well. The spin degrees of freedom lead—for the type of spin- orbit coupling considered in this work—to a quadru- pling of each Eﬁmov trimer (manifold of four states). The three-body states in a given manifold are tied to one of the three two-boson states [61]. The point (1/as,kso,Ω,˜δ) = (0,0,0,0) serves as an accumulation point for all four states of the manifold, i.e., in its vicin- ity, there exist inﬁnitely many three-body bound states. The rich structure of two- and three-boson states should be amenable to experimental veriﬁcation. Due to the dependence of the trimers on the center-of-mass momen- tum, the scattering length at which the lowest trimer in the second-lowest manifold merges with the atom-atom- atom threshold is, in fact, a scattering length window. Similar scattering length windows exist for the excited states in the second-lowest manifold. It was argued that these scattering length windows should be observable in cold atom loss experiments, providing a direct few-body signature of the breaking of the Galilean invariance of systems with spin-orbit coupling. If one considers a cut in the generalized ﬁve- dimensional Eﬁmov plot, energy levels are not spaced by the scaling factor ( λ0)2. Let us consider the situa- tion where asis inﬁnitely large and where kso, Ω, and ˜δ are ﬁnite. In this case, the low-energy scales associatedwith the 1D spin-orbit coupling terms lead to a cut-oﬀ of the hyperradial −1/R2Eﬁmov potential curve ( Rde- notes the three-body hyperradius). As a consequence, the number of three-body bound states at unitarity is not inﬁnitely large. Albeit due to a diﬀerent mechanism, this is similar in spirit to the disappearance of Eﬁmov states if an Eﬁmov trimer is placed into a gas of bosons or fermions [62, 63]. This is also similar in spirit to a rather diﬀerent system, namely the H−ion. Taking only Coulomb interactions into account, one obtains a −1/r2 attraction [64], where ris the distance between the ex- tra electron and the atom. Relativistic eﬀects introduce an additional length scale, which renders the number of bound states ﬁnite [64]. The calculationsin this workwereperformedassuming that the interactions between the diﬀerent spin-channels are all equal. If one of the scattering lengths is large and tunable while the others are close to zero, the discrete scaling symmetry should still hold (approximately). To ﬁnd the functional form of the energies for this scenario, the spectrum has to be recalculated. Thestudypresentedshouldbeviewedasaﬁrststepto- ward uncovering the rich three- and higher-body physics that emerges as a consequence of the unique coupling be- tween the relative and center-of-mass degrees of freedom in cold atom systems in the presence of artiﬁcial Gauge ﬁelds. While somewhat diﬀerent in nature, the coupling of these degrees of freedom in the relativistic Klein Gor- don and Dirac equations and quantum ﬁeld theories has captured physicists’ imagination for many decades. VII. ACKNOWLEDGEMENT Support by the National Science Foundation through grant numbers PHY-1509892and PHY-1745142is grate- fully acknowledged. This work used the Extreme Sci- ence and Engineering Discovery Environment (XSEDE), which is supported by NSF Grant No. OCI-1053575. Some of the computing for this project was performed at the OU Supercomputing Center for Education and Re- search (OSCER) at the University of Oklahoma (OU). Appendix A: Jacobi coordinates The single-particleand Jacobicoordinatesemployedin this work are related through the matrix U[65], (/vector ρ1,···,/vector ρN)T=U(/vector r1,···,/vector rN)T, (A1) where U=/parenleftigg 1−1 1/2 1/2/parenrightigg (A2)12 for the equal-mass two-particle system and U= 1−1 0 1/2 1/2−1 1/3 1/3 1/3 (A3) for the equal-mass three-particle system. The transfor- mation matrix Ualso deﬁnes the matrices ˆΣj,z(j= 1,···,N−1). For the two-body system, we have ˆΣ1,z= ˆσ1,z⊗I2−I1⊗ˆσ2,z. (A4) For the three-body system, we have ˆΣ1,z= ˆσ1,z⊗I2⊗I3−I1⊗ˆσ2,z⊗I3(A5) and ˆΣ2,z=1 2(ˆσ1,z⊗I2⊗I3+I1⊗ˆσ2,z⊗I3)− I1⊗I2⊗ˆσ3,z. (A6) Appendix B: Explicitly correlated basis set expansion approach This appendix discusses our approach to solving the few-particle time-independent Schr¨ odinger equation us- ing a basis set expansion in terms of explicitly correlated Gaussian basis functions, which contain non-linear varia- tional parameters that are optimized semi-stochastically. As discussed in the main text, we are interested in boundstates, i.e., eigenstatesthatapproachzeroatlarge interparticle distances. To solve the time-independent Schr¨ odingerequation,weexpandtherelativeportionΦ rel of the eigen state Ψ sought in terms of a set of non- orthogonal eigen functions ψj[65, 66], Φrel=Nb/summationdisplay j=1cjψj. (B1) The basis functions ψjdepend on the relative spatial and the spin degrees of freedom, ψj=ˆS(φj(/vector ρ1,···,/vector ρN−1)χj), (B2) whereχjdenotes anN-particle spin function that is cho- sen from the complete set of 2Npossible spin functions. The spatial parts φjare written in terms of a total of (N−1)(N/2+3) non-linear variational parameters d(j) kl and/vector s(j) k, φj= exp/parenleftigg −N/summationdisplay k<lr2 kl 2d(j) kl+N−1/summationdisplay k=1ı/vector s(j) k·/vector ρk/parenrightigg .(B3) Here, the superscript “( j)” serves to remind us that each basis function is characterized by a set of non-linear vari- ational parameters. The non-linear variational parame- tersd(j) kldetermine the widths of the Gaussian factors ofthe basis functions. These widths are governed, roughly, by the two-body interaction terms in the Hamiltonian. The non-linear parameters /vector s(j) kdetermine the spatial os- cillations due to the kso-dependent one-body terms. We do ﬁnd that the values of /vector s(j) kcan depend quite strongly on the spin basis function considered. This shows that the parameters /vector s(j) kgovern, to leading order, the inter- play between the spatial and spin degrees of freedom. Of course, strictly speaking, the inﬂuence of the single- particle and two-particle interaction terms in the Hamil- tonian on the eigen states cannot be separated; rather, the eigen states are the result of the relative importance of each of these terms and the kinetic energy terms. The N(N−1)/2 non-linear parameters d(j) kland the 3(N−1) non-linear parameters contained in /vector s(j) kare optimized semi-stochastically, i.e., the basis set is constructed so as to minimize the energy of the eigen state under study. The symmetrizer ˆSin Eq. (B2) ensures that the basis functions, and hence the eigen state sought, are fully symmetrized. For the two-boson system, e.g., the sym- metrizer reads ˆS= (1 +ˆP12)/√ 2, where ˆP12exchanges the spatial and spin degrees of freedom of particles 1 and 2. ForNidentical particles, the symmetrizer contains N! terms. The linear parameters cjare determined by solving the generalized eigen value problem that depends on the Hamiltonian matrix and, due to the non-orthogonalityof the basis functions, the overlap matrix [65]. The results presented in this paper utilize basis sets consisting of up toNb= 1,800 basis functions. Key strengths of the nu- mericalapproachemployedare that the Hamiltonian and overlap matrix elements have compact analytical expres- sions and that the non-linear variational parameters can be adjusted so as to capture correlations that occur on scales smaller than r0and larger than ( kso)−1. The energies for the N= 2 system depend on the pa- rametersas,kso, Ω, and ˜δ. ForN= 3, they additionally dependonκ∗. Inpractice,weset /vector qN= 0andscanΩ /Eso, δ/Eso, andaskso(forN= 3, we ﬁx κ∗). To obtain the relative eigen energies Eand eigen states Φ relfor ﬁnite qN,z, we do not need to redo the numerical calculations. Instead, we use the “conversion” implied by Eq. (21), i.e., the relative eigen energy and eigen states are ob- tained by changing from δto˜δ. The full eigen states Ψ are obtained by multiplying the /vector qN= 0 eigen states by the center-of-mass piece Φ cm. As in the case without spin-orbit coupling [67], it tends to be more eﬃcient to describe each relative eigen state Φrelby its own basis set as opposed to constructing one basis set that describes multiple eigen states well. We refer to the eigen state sought—this could be the ground state or one of the excited states—as target state. To construct the basis, we add one basis function at a time. Let us assume that we have a basis set of size Ninitialand that we want to enlarge the basis set by one basis func- tion. To do this, we generate Ntrialtrial basis functions (Ntrialis of the order of a few hundred to a few thou-13 sand) and calculate the energy Ek(k= 1,···,Ntrial) of the target state for each of these enlarged basis sets. As- suming that the generalized eigen value problem for the basis set of size Ninitialhas been solved, the trial ener- gies can be calculated via a root-ﬁnding procedure [65], which is, generally, computationally signiﬁcantly faster than solving the generalized eigen value problem. The basis function to be added is determined by which of the Ntrialtrial energies is lowest, i.e., by looking for the trial basis function that lowers the energy of the target state the most. After the “best” trial function has been added, the generalizedeigen value problemis solvedfor the basis set of sizeNinitial+ 1 and the procedure is repeated to generate a basis set of size Ninitial+2. The convergence of the energy with increasing basis set size can be improved signiﬁcantly by choosing “good” basisfunctions, i.e., bygeneratingbasisfunctions that ef- ﬁciently cover the entire Hilbert space. Conversely, if the basis set is not constructed carefully, the energy may not even converge. In our implementation, the parameters d(j) kland/vector s(j) kthat characterizethespatialpartofthebasis functions are choosen from carefully adjusted parameter windows. For example, the d(j) klare chosen so as to cover length scales ranging from less than r0to a few times the lengthsetbythebindingenergyofthetargetstate. Since the target energy is, in general, not known a priori, the parameter windows are typically reﬁned based on results obtained in preliminary calculations. To choose parame- ter windows for the x-,y- andz-components of /vector s(j) k, we are guided by the non-interacting two- and three-particle dispersion curves. For example, if the non-interacting two-particle dispersion along the z-coordinate exhibits two minima, we choose s(j) 1,zuniformly from the windows [−smax,z,−smin,z] and [smin,z,smax,z], with the windows including the momenta at which the dispersion curve is minimal. The parameters s(j) 1,xands(j) 1,yare selected from windows that include zero. The widths of the parame- ter windows are adjusted through an educated trial and error procedure. Among other things, we check if the pa- rametersselected by the code areclumped in a particular region of the parameter space. Appendix C: Three-atom threshold The three-atom system with center-of-mass momen- tum/vector qN,N= 3,isboundifitsenergyislowerthanthatof three inﬁnitely farseparatedatoms with the samecenter- of-mass momentum and, if a two-body bound state ex- ists, lower than that of an inﬁnitely far separated dimer and atom with the same center-of-mass momentum. To determine the lowest three-body scattering threshold, one thus needs to know the lowest dimer binding energy for all two-body center-of-mass momenta. As a conse- quence, the three-body scattering threshold depends on ˜δ/Esoand Ω/Esoas well as on askso. We deﬁne the scattering threshold Ethusingˆ¯Hreland denote the eigen FIG. 4: (Color online) Lowest non-interacting relative thr ee- atom dispersion curve. The contours show the lowest non- interacting relative three-atom dispersion curve, in unit s of Eso, as functions of q1,zandq2,z. Panels (a)-(c) are for Ω/Eso= 0 and ˜δ/Eso= 0, 8/3, and 3 .5, respectively. Pan- els (d)-(f) are for Ω /Eso= 2 and ˜δ/Eso= 0, 2.278, and 4, respectively. energies ofˆ¯HrelbyE. We start by determining the lowestrelative three-body scattering threshold in the absence of two-body bound states. In this case, the lowest relative scattering thresh- old is determined by the minimum energy of the non- interacting relative dispersion curves for ﬁxed ˜δ/Esoand Ω/Eso. The dispersion curves depend on two relative momenta (namely q1,zandq2,z) and the total number of dispersion curves is eight. Figures 4(a)-4(c) are for the uncoupled case (Ω = 0) and ˜δ/Eso= 0, 8/3, and 3.5, respectively. The number of global minima changes from six for˜δ= 0 [see Fig. 4(a)] to three for 0 <˜δ/Eso<8/3 (not shown) to four for ˜δ/Eso= 8/3 [see Fig. 4(b)] to one for˜δ/Eso>8/3 [see Fig. 4(c)]. A ﬁnite Raman cou- pling strength Ω introduces a coupling between the dif-14 FIG. 5: (Color online) Lowest relative three-atom scatteri ng threshold. Thecontoursshowtheenergy Eaaa th, inunitsof Eso, of the lowest three-atom scattering threshold as functions of Ω/Esoand˜δ/Eso. The thickopencircles andthickdashedline indicate the parameter combinations ( ˜δ/Eso,Ω/Eso) at which the degeneracy of the lowest three-atom scattering thresho ld is six and four, respectively. ferent spin channels. As anexample, Figs. 4(d)-4(f) show the lowest relative non-interacting dispersion curves for Ω/Eso= 2 and ˜δ/Eso= 0, 2.278, and 4, respectively. As for vanishing Raman coupling, the number of global minima changes from six to three (not shown) to four to one with increasing ˜δ. However, the critical general- ized detuning ˜δat which these changes occur diﬀers for Ω/Eso= 2 and Ω = 0. The minimum of the non-interacting relative three- atom dispersion curves deﬁnes, assuming two-body bound states are absent, the lowest three-atom scatter- ing threshold. Figure 5 shows the lowest three-atom scattering threshold energy Eaaa thas functions of ˜δ/Eso and Ω/Eso. The thick open circles and thick dashed line indicate the parameter combinations at which the number of global minima is six and four, respectively. For parameter combinations above the thick open circles and below the thick dashed line the number of global minima of the lowest non-interacting relative dispersion curve is equal to three. Above the thick dashed line the number of global minima is equal to one. If we assume that the three-body binding energy is, approx- imately, largest when the degeneracy of the lowest non- interactingrelativedispersioncurveislargest,then Fig.5 suggeststhatthe three-bodysystemonthenegativescat- tering length side, provided two-body bound states are absent, is enhanced the most compared to the energy of the system without spin-orbit coupling when ˜δ= 0 or q3,z=−3mδ/(2/planckover2pi1kso). The main text shows that this reasoning provides an intuitive understanding for the be- havior of the lowest three-boson state in each manifold. However, the situation for the excited states in a mani- fold is more intricate [61].Appendix D: Atom-dimer threshold As already mentioned, the determination of the low- est atom-dimer scattering threshold requires knowledge of the dimer binding energy and the single-particle dis- persion curve. Since the z-component of the center-of- mass momentum q1,zof the dimer, formed by particles 1 and 2, can be written as a linear combination of q2,z andq3,z,q1,zis not a free parameter. As a consequence, theatom-dimerdispersioncurvesdepend onlyon q2,zbut not onq1,z. Physically, this makes sense since the three- body system breaks up into two units (a dimer and an atom), with the momentum between the two units deter- mining the division of the three-body momentum among the dimer and the atom. To quantify this, we rewrite the Hamiltonianˆ¯Hrelby arbitrarily singling out the third atom and treating the expectation value q2,zof ˆq2,zas a parameter, ˆ¯Hrel/vextendsingle/vextendsingle /angbracketleftˆq2,z/angbracketright=q2,z=ˆH12(q2,z)⊗I3+ I1⊗I2⊗ˆH3(q2,z)+ Vcoupling. (D1) Here, the “dimer Hamiltonian” ˆH12(q2,z) reads ˆH12(q2,z) =/parenleftiggˆ/vector q2 1 2µ1+V2b(r12)/parenrightigg I1⊗I2+ /planckover2pi1ksoq1,z m(ˆσ1,z⊗I2−I1⊗ˆσ2,z)+ /parenleftigg /planckover2pi1ksoq2,z 2m+˜δ 2/parenrightigg (ˆσ1,z⊗I2+I1⊗ˆσ2,z)+ Ω 2(ˆσ1,x⊗I2+I1⊗ˆσ2,x). (D2) Identifying ˜δ12,eﬀ, ˜δ12,eﬀ 2=/planckover2pi1ksoq2,z 2m+˜δ 2, (D3) as a new eﬀective dimer detuning, the eigen energies of ˆH12(q2,z) are the same as those of the two-body Hamil- tonian. The “atom Hamiltonian” ˆH3(q2,z), ˆH3(q2,z) = /vector q2 2 2µ2⊗I3+/parenleftigg −/planckover2pi1ksoq2,z m+˜δ 2/parenrightigg ˆσ3,z+Ω 2ˆσ3,x,(D4) describes the Jacobi particle with mass µ2and eﬀective atom detuning ˜δ3,eﬀ, where ˜δ3,eﬀ 2=−/planckover2pi1ksoq2,z m+˜δ 2. (D5) Note that the eﬀective dimer detuning ˜δ12,eﬀand the eﬀective atom detuning ˜δ3,eﬀdepend on the “true de- tuning”δ, which is ﬁxed by the experimental set-up, on15 thez-component q3,zof the three-body center-of-mass momentum, which is a conserved quantity, and on q2,z, which is treated as a parameter. Assuming that the dis- tance between the center-of-mass of the dimer and the atom is large compared to the size of the dimer and com- pared to the ranges r0andR0of the two- and three-body interactions, the coupling term Vcoupling, Vcoupling= [V2b(r13)+V2b(r23)+V3b(r123)]I1⊗I2⊗I3, (D6) can be set to zero. Thus, the q2,z-dependent relative atom-dimer dispersion curves are obtained by adding the eigen energies of ˆH12andˆH3, which depend parametri- cally onq2,z. Equations(D1)-(D6) assumedthatthe dimerisformed by atoms 1 and 2. Alternatively, the dimer could be formed by atoms 1 and 3 or by atoms 2 and 3. These alternative divisions yield atom-dimer dispersion curves that depend on the z-component of the momentum that is associated with the distance vector between particle 2 and the center-of-mass of the 13-dimer and the z- component of the momentum that is associated with the distance vector between particle 1 and the center-of-mass of the 23-dimer, respectively. Since we are considering three identical bosons, the three divisions are equivalent. In what follows, we use qad,zto reﬂect that we could single out any of the three atoms. The corresponding atom-dimer energy is denoted by Ead th. Since there exist up to three two-boson bound states [61], the three-boson system supports up to six atom-dimer dispersion curves (there could be four or two). As an example, Fig. 6 shows the energy Ead thof the lowest relative atom-dimer dispersion curve as a func- tion ofqad,zfor (askso)−1= 0.01128, Ω/Eso= 2, and various˜δ, i.e.,˜δ= 0,2.287, and 3.5. The system sup- ports, for this Raman coupling strength and scattering length, one weakly-bound two-boson state for all two- body center-of-mass momenta. For ˜δ= 0 (solid line in Fig. 6), the atom-dimer dispersion is symmetric with re- spect toqad,z= 0 and supports two global minima at ﬁniteqad,z. The break-up into a dimer and an atom is energetically most favorable when qad,z/(/planckover2pi1kso) is equal to±0.76. This translates, using Eqs. (D3) and (D5), into˜δ12,eﬀ/Eso=±1.52 and˜δ3,eﬀ/Eso=±3.04. For ˜δ >0 (the dashed and dotted lines in Fig. 6 are for ˜δ/Eso= 2.287 and 3.5, respectively), the atom-dimer dispersions are asymmetric with respect to qad,z= 0 and exhibit a global minimum at negative qad,z, which ap- proachesqad,z= 0 in the ˜δ→ ∞limit. Intuitively, the asymmetry can be understood by realizing that the atom and the dimer already see a detuning. Thus, moving in the positive momentum direction is not equivalent to moving in the negative momentum direction. The mini- mum of the lowest relative atom-dimer dispersion curve decreases with increasing ˜δ.-1.5 -1 -0.5 0 0.5 1 1.5 qad,z/(h_kso)-6-5-4-3Ead th/Eso FIG. 6: (Color online) Relative atom-dimer dispersion curv es for (askso)−1= 0.01128 and Ω /Eso= 2. The solid, dashed, and dotted lines show the energy Ead thas a function of the z-component qad,zof the atom-dimer momentum for ˜δ/Eso= 0,2.287, and 3 .5, respectively. Appendix E: Three-body threshold The three-boson threshold is given by the minimum of the lowest three-atom threshold and the lowest atom- dimer threshold. It depends on the values of Ω, kso, ˜δ, and thes-wave scattering length. Using ksoandEso as units, Fig. 7 shows a contour plot of the lowest rela- tive three-boson threshold as functions of the generalized detuning ˜δ/Esoand the inverse ( askso)−1of thes-wave scattering length for Ω /Eso= 2. As already discussed, the parameter regime in which two-boson bound states exist depends on the value of as. Correspondingly, the thick dotted line, which marks the separation of the re- gion in which the three-atom threshold has the lowest energy (to the left of the thick dotted line) and that in which the atom-dimer threshold has the lowest energy (to the right of the thick dotted line), shows a distinct dependence on the s-wave scattering length. For large ˜δ, the thick dotted line approaches the ( askso)−1= 0 line. For a ﬁxed ˜δ, the energy Ead thof the lowest atom-dimer threshold decreases with increasing ( askso)−1. This can be traced back to the increase of the binding energy of the two-boson ground state with increasing ( askso)−1. The parameter combinations with the largest degener- acy of the scattering threshold are shown by the thick open circles (three-atom threshold; the degeneracyis six) and the thick dash-dotted line (atom-dimer threshold; the degeneracy is two). For all as, the largest degener- acy of the scattering threshold is found for ˜δ= 0. As discussed in the main text, our numerical three-boson calculations show that the binding energy of the most strongly-bound state in each manifold, determined as functions of ( askso)−1and˜δ/Eso, is largest for vanishing ˜δ, i.e., where the degeneracy of the lowest three-boson scattering threshold is maximal.16 FIG. 7: (Color online) Lowest relative three-boson scatter ing threshold for Ω /Eso= 2. The contours show the energy Eth, in units of Eso, of the lowest three-boson scattering threshold as functions of ( askso)−1and˜δ/Eso. To the left of the thick dotted line, bound dimers are not supported and the lowest threshold is given by Eaaa th. In this regime, the lowest scatter- ing threshold is independent of ( askso)−1. To the right of the thick dotted line, a weakly-bound bosonic dimer exists and the lowest threshold is given by Ead th. In this regime, the low- est scatteringthresholddependson( askso)−1. The thickopen circles and thick dashed line mark the (( askso)−1,˜δ/Eso) com- binations for which the lowest three-atom threshold has a de - generacy of six and four, respectively. The thick dash-dott ed line marks the (( askso)−1,˜δ/Eso) combinations for which the lowest atom-dimer threshold has a degeneracy of two. In the region encircled by the thick open circles, the thick dotted line, the thick dashed line, and the left edge of the ﬁgure the degeneracy of the three-atom threshold is equal to three. In the region encircled by the thick dashed line, the thick dott ed line, the upper edge of the ﬁgure, and the left edge of the ﬁgure, the degeneracy of the three-atom threshold is equal t o one. In the region encircled by the thick dash-dotted line, t he right edge of the ﬁgure, the upper edge of the ﬁgure, and the thick dotted line the degeneracy of the atom-dimer threshol d is equal to one. The range of scattering lengths shown on the horizontal axis is, even though diﬀerent units are used, the same as in Fig. 3. [1] H.-X.Chen, W.Chen, X.Liu, andS.-L.Zhu, Thehidden- charm pentaquark and tetraquark states, Phys. Rep. 639, 1 (2016). [2] M. Gell-Mann, A Schematic Model of Baryons and Mesons, Phys. Lett. 8, 214 (1964). [3] E. Epelbaum, H.-W. Hammer, and U.-G. Meißner, Mod- ern theory of nuclear forces, Rev. Mod. Phys. 81, 1773 (2009). [4] I. Tanihata, Neutron halo nuclei, J. Phys. G: Nucl. Part. Phys.22, 157 (1996). [5] L. H. Thomas, The Interaction Between a Neutron and a Proton and the Structure of H3, Phys. Rev. 47, 903 (1935). [6] V. Eﬁmov, Energy levels arising from resonant two-bodyforces in a three-body system, Phys. Lett. B 33, 563 (1970). [7] V. Eﬁmov, Weakly-bound states of three resonantly- interacting particles, Yad. Fiz. 12, 1080 (1970) [Sov. J. Nucl. Phys. 12, 589 (1971)]. [8] E. Braaten and H.-W. Hammer, Universality in few-body systems with large scatteringlength, Phys.Rep. 428, 259 (2006). [9] E. Braaten and H.-W. Hammer, Eﬁmov physics in cold atoms, Annals of Physics 322, 120 (2007). [10] H.-W. Hammer and L. Platter, Eﬁmov States in Nuclear and Particle Physics, Ann. Rev. Nucl. Part. Sci. 60, 207 (2010). [11] P. Naidon and S. Endo, Eﬁmov Physics: a review, Rep.17 Prog. Phys. 80, 056001 (2017). [12] F. Ferlaino and R. Grimm, Trend: Forty years of Eﬁmov physics: Howa bizarre prediction turnedintoa hot topic, Physics3, 9 (2010). [13] C. H. Greene, Universal insights from few-body land, Physics Today 63, 40 (2010). [14] T. Kraemer, M. Mark, P. Waldburger, J. G. Danzl, C. Chin, B. Engeser, A. D. Lange, K. Pilch, A. Jaakkola, H.- C. N¨ agerl, and R. Grimm, Evidence for Eﬁmov quantum states in an ultracold gas of caesium atoms, Nature 440, 315 (2006). [15] M. Zaccanti, B. Deissler, C. D’Errico, M. Fattori, M. Jona-Lasinio, S. M¨ uller, G. Roati, M. Inguscio, and G. Modugno, Observation of an Eﬁmov spectrum in an atomic system, Nat. Phys. 5, 586 (2009). [16] Y. Nishida, Y. Kato, and C. D. Batista, Eﬁmov eﬀect in quantum magnets, Nat. Phys. 9, 93 (2013). [17] T. Lompe, T. B. Ottenstein, F. Serwane, A. N. Wenz, G. Z¨ urn, and S. Jochim, Radio-Frequency Association of Eﬁmov Trimers, Science 330, 940 (2010). [18] S. Nakajima, M. Horikoshi, T. Mukaiyama, P. Naidon, and M. Ueda, Measurement of an Eﬁmov Trimer Binding EnergyinaThree-ComponentMixtureof6Li, Phys.Rev. Lett.106, 143201 (2011). [19] M. Berninger, A. Zenesini, B. Huang, W. Harm, H.-C. N¨ agerl, F. Ferlaino, R. Grimm, P. S. Julienne, and J. M. Hutson, Universality of the Three-Body Parameter for Eﬁmov States in Ultracold Cesium, Phys. Rev. Lett. 107, 120401 (2011). [20] B. Huang, L. A. Sidorenkov, R. Grimm, and J. M. Hut- son, Observation of the Second Triatomic Resonance in Eﬁmov’s Scenario, Phys. Rev. Lett. 112, 190401 (2014). [21] M. Kunitski, S. Zeller, J. Voigtsberger, A. Kalinin, L. P. H. Schmidt, M. Sch¨ oﬄer, A. Czasch, W. Sch¨ ollkopf, R. E. Grisenti, T. Jahnke, D. Blume, and R. D¨ orner, Observation of the Eﬁmov state of the helium trimer, Science348, 551 (2015). [22] D. C. Mattis, The few-body problem on a lattice, Rev. Mod. Phys. 58, 361 (1986). [23] J. Dalibard, F. Gerbier, G. Juzeli¯ unas, and P. ¨Ohberg, Colloquium: Artiﬁcial gauge potentials for neutral atoms, Rev. Mod. Phys. 83, 1523 (2011). [24] V. Galitski and I. B. Spielman, Spin-orbit coupling in quantum gases, Nature 494, 49 (2013). [25] N. Goldman, G. Juzeli¯ unas, P. ¨Ohberg, and I. B. Spiel- man, Light-inducedgauge ﬁeldsfor ultracold atoms, Rep. Prog. Phys. 77, 126401 (2014). [26] H. Zhai, Degenerate quantum gases with spin-orbit cou- pling: a review, Rep. Prog. Phys. 78, 026001 (2015). [27] M. Aidelsburger, S. Nascimbene, and N. Goldman, Ar- tiﬁcial gauge ﬁelds in materials and engineered systems, arXiv:1710.00851. [28] Y.-J. Lin, K. Jim´ enez-Garc´ ıa, and I. B. Spielman, Spi n- orbit-coupled Bose-Einstein condensates, Nature (Lon- don)471, 83 (2011). [29] J. Struck, C. ¨Olschl¨ ager, M. Weinberg, P. Hauke, J. Si- monet, A. Eckardt, M. Lewenstein, K. Sengstock, and P. Windpassinger, Tunable Gauge Potential for Neutral and Spinless Particles in Driven Optical Lattices, Phys. Rev. Lett. 108, 225304 (2012). [30] C. V. Parker, L.-C. Ha, and C. Chin, Direct observa- tion of eﬀective ferromagnetic domains of cold atoms in a shaken optical lattice, Nat. Phys. 9, 769 (2013).[31] T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, Topological Photonics, arXiv:1802.04173 (2018). [32] J. P. Vyasanakere and V. B. Shenoy, Bound states of two spin-1/2 fermions in a synthetic non-Abelian gauge ﬁeld, Phys. Rev. B 83, 094515 (2011). [33] J. P. Vyasanakere, S. Zhang, and V. B. Shenoy, BCS- BEC crossover induced by a synthetic non-Abelian gauge ﬁeld, Phys. Rev. B 84, 014512 (2011). [34] J. P. Vyasanakere and V. B. Shenoy, Rashbons: prop- erties and their signiﬁcance, New. J. Phys. 14, 043041 (2012). [35] J. P. VyasanakereandV. B. Shenoy, Flow-enhanced pair- ing and other unusual eﬀects in Fermi gases in synthetic gauge ﬁelds, Phys. Rev. A 88, 033609 (2013). [36] X. Cui, Mixed-partial-wave scattering with spin-orbi t coupling and validity of pseudopotentials, Phys. Rev. A 85, 022705 (2012). [37] L. Dong, L. Jiang, H. Hui, and H. Pu, Finite-momentum dimer bound state in a spin-orbit-coupled Fermi gas, Phys. Rev. A 87, 043616 (2013). [38] Z. Yu, Short-range correlations in dilute atomic Fermi gases with spin-orbit coupling, Phys. Rev. A. 85, 042711 (2012). [39] Y. WuandZ. Yu, Short-range asymptotic behavior ofthe wave functions of interacting spin-1/2 fermionic atoms with spin-orbit coupling: A model study, Phys. Rev. A 87, 032703 (2013). [40] X. Cui, Multichannel molecular state and rectiﬁed shor t- range boundary condition for spin-orbit-coupled ultra- cold fermions near p-wave resonances, Phys. Rev. A 95, 030701(R) (2017). [41] Q. Guan and D. Blume, Scattering framework for two particles with isotropic spin-orbit coupling applicable t o all energies, Phys. Rev. A 94, 022706 (2016). [42] S.-J. Wang and C. H. Greene, General formalism for ultracold scattering with isotropic spin-orbit coupling, Phys. Rev. A 91, 022706 (2015). [43] D. Luo and L. Yin, BCS-pairing state of a dilute Bose gas with spin-orbit coupling, Phys. Rev. A 96, 013609 (2017). [44] R. Li and L. Yin, Pair condensation in a dilute Bose gas with Rashba spin-orbit coupling, New J. Phys. 16, 053013 (2014). [45] Z.Xu, Z.Yu,andS.Zhang, Evidencefor correlatedstate s in a cluster of bosons with Rashba spin-orbit coupling, New J. Phys. 18, 025002 (2016). [46] Z.-Y. Shi, X. Cui, and H. Zhai, Universal Trimers In- duced by Spin-Orbit Coupling in Ultracold Fermi Gases, Phys. Rev. Lett. 112, 013201 (2014). [47] X. Cui and W. Yi, Universal Borromean Binding in Spin- Orbit-Coupled Ultracold Fermi Gases, Phys. Rev. X 4, 031026 (2014). [48] Z.-Y. Shi, H. Zhai, and X. Cui, Eﬁmov physics and uni- versal trimers inspin-orbit-coupledultracold atomic mix - tures, Phys. Rev. A 91, 023618 (2015). [49] X. Qiu, X. Cui, and W. Yi, Universal trimers emerging from a spin-orbit-coupled Fermi sea, Phys. Rev. A 94, 051604(R) (2016). [50] Q. Guan, One-, two-, and three-body systems with spin- orbit coupling, Ph.D. thesis, Washington State Univer- sity (2017). [51] The normalization factor of the plane wave center-of-18 mass wave function is not written out for notational con- venience. [52] Y. Yan and D. Blume, Energy and structural proper- ties ofN-boson clusters attached to three-body Eﬁmov states: Two-body zero-range interactions and the role of the three-body regulator, Phys. Rev. A 92, 033626 (2015). [53] F. Ferlaino, S. Knoop, M. Mark, M. Berninger, H. Sch¨ obel, H.-C. N¨ agerl, andR. Grimm, Collisions between Tunable Halo Dimers: Exploring and Elementary Four- Body Process with Identical Bosons, Phys. Rev. Lett. 101, 023201 (2008). [54] F. Ferlaino, S. Knoop, M. Berninger, W. Harm, J. P. D’Incao, H.-C. N¨ agerl, and R. Grimm, Evidence for Uni- versal Four-Body States Tied to an Eﬁmov Trimer, Phys. Rev. Lett. 102, 140401 (2009). [55] S. E. Pollack, D. Dries, and R. G. Hulet, Universality in Three- and Four-Body Bound States of Ultracold Atoms, Science326, 1683 (2009). [56] A. Zenesini, B. Huang, M. Berninger, S. Besler, H.-C. N¨ agerl, F. Ferlaino, R. Grimm, C. H. Greene, and J. von Stecher, Resonant ﬁve-body recombination in an ul- tracold gas of bosonic atoms, New J. Phys. 15, 043040 (2013). [57] R. Pires, J. Ulmanis, S. H¨ afner, M. Repp, A. Arias, E. D. Kuhnle, and M. Weidem¨ uller, Observation of Eﬁ- mov Resonances in a Mixture with Extreme Mass Imbal- ance, Phys. Rev. Lett. 112, 250404 (2014). [58] S.-K. Tung, K. Jim´ enez-Garc´ ıa, J. Johansen, C. V. Parker, and C. Chin, Geometric Scaling of Eﬁmov States in a6Li-133Cs Mixture, Phys. Rev. Lett. 113, 240402 (2014). [59] Spin-orbit coupled experiments with Cs, which has tun- able Feshbach resonances, have not yet been conducted. On the other hand, while many experiments utilize Rb to realize spin-orbit coupling, the known Feshbach res- onances in Rb are accompanied by large losses, making the resonances challenging to use. Proposals for imple- menting spin-orbit coupling schemes in Cs exist [68]. [60] When we apply our formalism to the two-identical fermion system, our binding energy agrees with the re- sults obtained by Shenoy [35] but disagrees with the re- sults presented in Ref. [37]. Reference [37] calculated the two-fermion binding energy by allowing the z-component of the center-of-mass momentum of the eigen state of the interacting system and that of the two-atom state associ- ated with thethreshold with respect towhich the binding energy is being measured to be diﬀerent. The deﬁnition employed [see Eq. (14) of Ref. [37]] implies that the z- component of the center-of-mass momentum is not a con-served quantity. A straightforward calculation, however, establishes that the three components of the center-of- mass momentum are individually conserved (see, e.g., Ref. [50]). Correspondingly, the quantity considered in Ref. [37] is inconsistent with the accepted deﬁnition of the term “two-body binding energy”. The quantity de- ﬁned in Ref. [37] does, however, become meaningful in a many-body context. To see this, we consider a two-body bound state with total energy E2(/vector q2) =E+/vector q2 2/(4m) em- bedded into a many-body background (here, Eis the eigen energy of the relative two-particle Hamiltonian; Edepends on q2,z). Assuming that the dimer can re- duce its energy via center-of-mass changing inelastic col- lisions with the background atoms, the lowest energy of the dimer is given by min /vector q2E2(/vector q2). The dimer can then be considered stable if the global energy Eglobal,2, Eglobal,2= min /vector q2E2(/vector q2)−min/vector q2[Eth+/vector q2 2/(4m)] (Ethde- notes the lowest relative atom-atom scattering threshold for ﬁxed q2,z) is negative. For the global energy Eglobal,2, referred to as binding energy in Ref. [37], to be meaning- ful, the dimer needs to be embedded into a background that can, via inelastic collisions, absorb energy on a time scale that is fast compared to the time scales associated with other loss processes. [61] Q. Guan and D. Blume, unpublished. [62] N. T. Zinner, Eﬁmov states of heavyimpurities inaBose- Einstein condensate, EPL 101, 60009 (2013). [63] D. J. MacNeill and F. Zhou, Pauli Blocking Eﬀect on Eﬁmov States near a Feshbach Resonance, Phys. Rev. Lett.106, 145301 (2011). [64] M. Ga ˘ilitis and R. Damburg, Some features of the thresh- old behavior of the cross sections for excitation of hydro- gen by electrons due to the existence of a linear Stark eﬀect in hydrogen, Sov. Phys. JETP 17, 1107 (1963). [65] Y. Suzuki and K. Varga, Stochastic Variational Ap- proach to Quantum-Mechanical Few-Body Problems , 1st ed. (Springer-Verlag Berlin Heidelberg, 1998). [66] J. Mitroy, S. Bubin, W. Horiuchi, Y.Suzuki, L. Adamow- icz, W. Cencek, K. Szalewicz, J. Komasa, D. Blume, and K. Varga, Theory and application of explicitly correlated Gaussians, Rev. Mod. Phys. 85, 693 (2013). [67] D. Rakshit, K. M. Daily, and D. Blume, Natural and un- natural parity states of small trapped equal-mass two- component Fermi gases at unitarity and fourth-order virial coeﬃcient, Phys. Rev. A 85, 033634 (2012). [68] B.-Z. Wang, Y.-H. Lu, W. Sun, S. Chen, Y. Deng, and X.-J. Liu, Dirac-, Rashba, and Weyl-type spin-orbit couplings: Toward experimental realization in ultracold atoms, Phys. Rev. A 97, 011605(R) (2018).
1911.09244v1.Evidence_for_orbital_ordering_in_Ba__2_NaOsO__6___a_Mott_insulator_with_strong_spin_orbit_coupling__from_First_Principles.pdf	Evidence for orbital ordering in Ba 2NaOsO 6, a Mott insulator with strong spin orbit coupling, from First Principles R. Cong1, Ravindra Nanguneri2, Brenda Rubenstein2;y, and V. F. Mitrovi c1;y 1Department of Physics, Brown University, Providence, Rhode Island 02912, USA and 2Department of Chemistry, Brown University, Providence, Rhode Island 02912, USA (Dated: March 7, 2022) We present rst principles calculations of the magnetic and orbital properties of Ba 2NaOsO 6 (BNOO), a 5 d1Mott insulator with strong spin orbit coupling (SOC) in its low temperature emergent quantum phases. Our computational method takes into direct consideration recent NMR results that established that BNOO develops a local octahedral distortion preceding the formation of long range magnetic order. We found that the two-sublattice canted ferromagnetic ground state identied in Lu et al. , Nature Comm. 8, 14407 (2017) is accompanied by a two-sublattice staggered orbital ordering pattern in which the t2gorbitals are selectively occupied as a result of strong spin orbit coupling. The staggered orbital order found here using rst principles calculations asserts the previous proposal of Chen et al. , Phys. Rev. B 82, 174440 (2010) and Lu et al. , Nature Comm. 8, 14407 (2017), that two-sublattice magnetic structure is the very manifestation of staggered quadrupolar order. Therefore, our results arm the essential role of multipolar spin interactions in the microscopic description of magnetism in systems with locally entangled spin and orbital degrees of freedom. I. INTRODUCTION The competition between electron correlation and spin orbit coupling (SOC) present in materials containing 4- and 5dtransition metals is an especially fruitful ten- sion predicted to lead to the emergence of a plethora of exotic quantum phases, including quantum spin liq- uids, Weyl semimetals, Axion insulators, and phases with exotic magnetic orders1{11. There has been an active quest to develop microscopic theoretical models to de- scribe such systems with comparably strong correlations and SOC to enable the prediction of their emergent quan- tum properties1{5. In strong Mott insulators, mean eld theories predict strong SOC to partially lift the degen- eracy of total angular momentum eigenstates by entan- gling orbital and spin degrees of freedom to produce highly nontrivial anisotropic exchange interactions2,3,5,7. These unusual interactions are anticipated to promote quantum uctuations that generate such novel quantum phases as an unconventional antiferromagnet with dom- inant magnetic octuple and quadrupole moments and a noncollinear ferromagnet whose magnetization points along the [110] axis and possesses a two-sublattice struc- ture. Because their SOC and electron correlations are of comparable magnitude6, 5ddouble perovskites (A2BB0O6) are ideal materials for testing these predic- tions. Indeed, recent NMR measurements on a repre- sentative material of this class, Ba 2NaOsO 6(BNOO), revealed that it possesses a form of exotic ferromag- netic order: a two-sublattice canted ferromagnetic (cFM) state, reminiscent of theoretical predictions12. Speci- cally, upon lowering its temperature, BNOO evolves from a paramagnetic (PM) state with perfect fcccubic sym- metry into a broken local symmetry (BLPS) state. This BLPS phase precedes the formation of long-range mag- netic order, which at suciently low temperatures, co- exists with the two-sublattice cFM order, with a netmagnetic moment of 0:2Bper osmium atom along the [110] direction. One key question that remains is whether such cFM order implies the existence of com- plex orbital/quadrupolar order. In this paper, we report a two-sublattice orbital order- ing pattern that coexists with cFM order in BNOO, as revealed by DFT+U calculations. Evidence for this or- der is apparent in BNOO's selective occupancy of the t2g orbitals and spin density distribution. More specically, the staggered orbital pattern is manifest in BNOO's par- tial density of states and band structure, which possesses a distinctt2gorbital contribution along high symmetry lines. This staggered orbital pattern is not found in the FM[110] phase. The results of our rst principles cal- culations paint a coherent picture of the coexistence of cFM order with staggered orbital ordering in the ground state of BNOO. Therefore, the staggered orbital order discovered here validates the previous proposal that the two-sublattice magnetic structure, which denes the cFM order in BNOO, is the very manifestation of staggered quadrupolar order with distinct orbital polarization on the two-sublattices2,12. Furthermore, our results arm that multipolar spin interactions are an essential ingredi- ent of quantum theories of magnetism in SOC materials. This paper is organized as follows. The details of our rst principles (DFT) simulations calculations of NMR observables is rst described in Section II. In Section III, we present our numerical results for the nature of the or- bital order for a given imposed magnetic state. Lastly, in Section IV, we conclude with a summary of our cur- rent ndings and their bearing on the physics of related materials. II. COMPUTATIONAL APPROACH All of the following computations were performed using the Vienna Ab initio Simulation Package (VASP), com-arXiv:1911.09244v1 [cond-mat.str-el] 21 Nov 20192 plex version 5.4.1/.4, plane-wave basis DFT code13{16 with the Generalized-Gradient Approximation (GGA) PW9117functional and two-component spin orbit cou- pling. We used 500 eV as the plane wave basis cuto en- ergy and we sampled the Brillouin zone using a 10 105 k-point grid. The criterion for stopping the DFT self- consistency cycle was a 105eV dierence between suc- cessive total energies. Two tunable parameters, Uand J, were employed. Udescribes the screened-Coulomb density-density interaction acting on the Os 5 dorbitals andJis the Hund's interaction that favors maximizing Sz total18. In this work, we set U= 3:3 eV andJ= 0:5 eV based upon measurements from Ref. 12 and calculations in Ref. 21. We note that these parameters are similar in magnitude to those of the SOC contributions we observe in the simulations presented below, which are between 1-2 eV. This is in line with previous assertions that the SOC and Coulomb interactions in 5 dperovskites are sim- ilar in magnitude. Projector augmented wave (PAW)19,20 pseudopotentials (PPs) that include the psemi-core or- bitals of the Os atom, which are essential for obtaining the observed electric eld gradient (EFG) parameters21, were employed to increase the computational eciency. A monoclinic unit cell with P2 symmetry is required to realize cFM order (see the Supplemental Information). The lattice structure with BLPS characterized by the or- thorhombic Q2 distortion mode that was identied as being in the best agreement with NMR ndings and re- ferred to as Model A.3 in Ref.21was imposed. The general outline of the calculations we performed is described in the following. We rst carried out single self- consistent or `static' calculations with GGA+SOC+U with a xed BLPS structure for Model A, representing the orthorhombic Q2 distortion mode. Then, a mag- netic order with [110] easy axes, as dictated by experi- mental ndings12,22, is imposed on the osmium lattices. Typically, we found that the nal moments converged to nearly the same directions as the initial ones. Speci- cally, two types of such initial order are considered: (a) simple FM order with spins pointing along the [110] di- rection; and, (b) non-collinear, cFM oder in which ini- tial magnetic moments are imposed on the two osmium j~Sj(S) j~Lj(L)M(M) cFM Os1 0.55 -41.56 0.44 90+46.29 0.12 -34 Os2 0.55 90+28.73 0.43 -31.07 0.11 110 FM110 Os 0.83 45 0.52 225 0.31 45 TABLE I. cFM and FM[110] magnetic moments for the im- posed representative BLPS structure using GGA+SOC+U. The angles, , are in degrees and measured anti-clockwise with respect to the + xaxis. The magnitudes of spin, orbital, and total moments are denoted by j~Sj,j~Lj, andj~Mj, in units ofB, respectively. The small net magnetic moment is due to the anti-aligment of ~Sand~Lein theJe=3 2state. As of now, the FM110 state has not been experimentally identied in a 5ddouble perovskite. Sx Sy SzSublattice 1 Sublattice 2FIG. 1. Contour plots of the spin density on two distinct sublattices of the BLPS structure (Model A.3 in Ref.21) from GGA+SOC+U calculations. The S x(top row), Sy (center row), and Sz(bottom row) components of the spin density on a single Os octahedron from sublattice 1 (left column) and sublattice 2 (right column) are plotted. The dierent colors denote the signs of the Sx;y;z projections. The isovalues are blue for positive Sx;y;z, 0:001, and yellow for negative Sx;y;z,0:001. On the top left, the negative Sx density is sandwiched between the lobes of the positive Sx densities on the Os atom, and vice-versa for the Os atom on the top right. On the top left, four of the O atoms have a cloverleaf spin density pattern with alternating positive and negative Sxdensities, while on the top right, only the two axial O atoms have this pattern. The other O atoms in the top two OsO 6octahedra have spin densities that are uniformly polarized. sublattices in the directions determined in Ref. 12. We used the Methfessel-Paxton (MP) smearing technique23 to facilitate charge density convergence. For the density of states and band structure calculations, we employed the tetrahedron smearing with Bl ochl corrections24and Gaussian methods, respectively. III. ORBITAL ORDERING WITH IMPOSED MAGNETIC CFM AND FM110 ORDERS In the following subsections, we report our results for the orbital order, band structure, and density of states of BNOO when we impose magnetic order with [110] easy axes and the local orthorhombic distortion that best matched experiments21. In Table I, we summarize the converged orbital and spin magnetic moments. In3 total on Sublattice 1 & 2 c aSx b c b ab b a c b a c b ac b accFM FM [110] b a c FIG. 2. Two views of the Sx-component of the spin density for imposed FM order and an orthorhombic Q2 distortion (Model A.3 in Ref. 21 on both sublattices as viewed along the -aand -cdirections. This component shows only the Sx-projection of the spin vector eld. The isovalues are blue, positive S x: 0:001, and yellow, negative S x: -0:001. BNOO,M= 2S+Le= 0, since the t 2glevel can be re- garded as a pseudospin with Le=1. The magnitude of the spin moment, j~Sj, is in the vicinity of 0:5B, while the orbital moment, j~Lj, is0:4B. These values are reduced from their purely local moment limit due to hybridization with neighboring atoms, and, in the case of~L, by quenching caused by the distorted crystal eld. For imposed cFM order, we nd that the relative angle within the two sublattices is in agreement with our NMR ndings in Ref. 12. Indeed, rst principles cal- culations, performed outside of our group, taking into account multipolar spin interactions found that the re- ported canted angle of 67corresponds to the global energy minimum25. Next, we will explore the nature of the orbital order- ing. Previous rst principles works hinted at the presence of orbital order in BNOO, but did not fully elucidate its nature26. Since we imposed cFM order and SOC, we were able to obtain a more exotic orbital order than uni- form ferro-order. We report below evidence for a type of layered, anti-ferro-orbital-order (AFOO) that has been shown to arise in the mean eld treatment of multipolar Heisenberg models with SOC5. First, we analyze the nature of the orbital order by computing the spin density. The spin density is a contin- uous vector eld of the electronic spin, and can point in non-collinear directions. Its operators are the product of the electrons' density and their spin-projection operators, such as z(~ r) =P i(~ ri~ r)Sz i: The spin densities are given by the expectation value, hz(~ r)i=Tr[dz(~ r)]; (1) wheredis the 5d-shell single-particle density matrix obtained from DFT+U calculations. In Fig. 1, thehx;y;z(~ r)i, obtained via GGA+SOC+U calculations, are displayed for two distinct Os sublattices. The re- sults illustrate that the spins are indeed localized about the Os atoms, and that there is a noticeable imbalance in the distribution of the n"andn#spin densities, which manifests in their dierence, hz(~ r)in"(~ r)n#(~ r). The dierence in the spatial distribution between the two sublattice spin densities is indicative of the orbital order- ing. The net spin moments are obtained by integrating the spin density over the volume of a sphere enclosing the Os atoms. In Fig. 1, it is visually clear that: I.TheSx(top) andSy(center) spin density components are overwhelm- ingly of a single sign, which gives rise to net moments in the (a;b) plane; and II.The signs of SxandSybe- tween the two sublattices are reversed, indicating that the sublattice spins are canted symmetrically about the [110] direction and the angle between them exceeds 90. In contrast, for Sz(bottom), both signs of Szcontribute equally, so that the net Sz0 after integrating over the sphere. In Fig. 2, we plot the total Sx-component of the spin density over two sublattices for both types of imposed magnetic order. It is evident that the staggered orbital E (eV)-1 1.5 -0.5 0.0 0.50 1.0dxy dyzdz dzx dx - y2 2 2My My Total Total E (eV)-1 1.5 -0.5 0.0 0.50 1.0 E (eV)-1 1.5 -0.5 0.0 0.50 1.0E (eV)-1 1.5 -0.5 0.0 0.50 1.0E (eV)-1 1.5 -0.5 0.0 0.50 1.0 E (eV)-1 1.5 -0.5 0.0 0.50 1.0Mx Mx DOS (states/eV/cell)-2-1012 -3dxy dyz dz dzx dx - y2 2 2 dxy dyz dz dzx dx - y2 2 2dxy dyz dz dzx dx - y2 2 2dxy dyzdz dzx dx - y2 2 2DOS (states/eV/cell)-2-1012 -3 DOS (states/eV/cell) 00.51.01.52.02.53.03.5DOS (states/eV/cell) 00.51.01.52.02.53.03.5 DOS (states/eV/cell)-2-1012 -3DOS (states/eV/cell)-2-1012 -3 dxy dyzdz dzx dx - y2 2 2a) b) c) d) e) f)Os1 Os1 Os1 Os2Os2 Os2 FIG. 3. The partial density of states (PDOS) for spin de- composed parts of the Q2 orthorhombic distortion (Model A.3 in Ref. 21) for the Os atom in each sublattice, Os1 and Os2.4 pattern only arises when cFM order is imposed. There- fore, we demonstrate that the staggered orbital order can solely coexist with cFM order. We note also that in Figs. 1 and 2 there is non- negligible spin density on the O atoms of the OsO 6oc- tahedra. It is usually thought that atoms with closed shells, like O in stoichiometric compounds, possess neg- ligible spin densities. This is an unexpected feature in BNOO that has been previously noted in Ref. 26, and is due to the stronger 5 d-2phybridization, which results in OsO 6cluster orbitals. The spin imbalance is a quan- tity associated with the cluster rather than the individual atoms, which is why we see the spin densities on the O atoms. For non-collinear systems, the orbital character of each osmium's 5 dmanifold can be further decomposed into the Cartesian components of the spin magnetization: hSiiMi,i=x;y;z . Since the spins lie in the ( xy) plane and the Mzcomponent is zero for both sublattices, we only plotted Mx,My, and the total PDOS for the two sublattices. We see in Fig. 3 that, rstly, for both sub- lattices, only the t2gorbitals have an appreciable density of states consistent with the fact that the calculated d occupation at the Os sites is hndi<6. Secondly, below the band gap, the dyzorbital has the same occupation on both sublattices, while the dxyorbital is occupied on one sublattice and the dzxorbital on the other. This pattern in which certain dorbitals are preferentially oc-cupied at dierent sites deviates from the case without orbital ordering, in which each of the dxy,dyz, anddzx orbitals have the same occupancies on both Os sites, as shown in Ref. 27. These orbital occupations are con- sistent with mean eld predictions of the occupancy of the Osdorbitals at zero temperature, which also pre- dict a staggered pattern5. This staggered pattern arises from BNOO's distinctive blend of cFM order with strong SOC. To study this ordering in greater depth, we can com- pute the occupation matrices, which after diagonaliza- tion, yield the occupation number (ON) eigenvalues and corresponding natural orbital (NO) eigenvectors. For a given Os atom, the 5 dspin-orbitals have unequal ampli- tudes in each NO, as expected for the AFOO. The NOs also all have dierent occupation numbers. Regardless of their precise occupations, the unequal spin-orbital super- positions in the NOs endow the Os with a net non-zero spin and orbital moment. We moreover note that, due to 5d-2phybridization, there is signicant charge transfer from O to Os, such that the charge on the 5 dshell of Os- mium ishndi5-6, which is very dierent from the nom- inal heptavalent 5 d1lling from simple valence counting. Furthermore, the ten NOs are fractionally occupied with the largest ON close to hn1i1:0 and the other nine NOs having occupations ranging from 0 :370:56. For the NO,j1i, with the occupation hn1i1:0, the coe- cients of the egorbitals are an order of magnitude smaller than those for the t2gorbitals. 0.000.050.100.150.200.250.300.35 0.00.10.20.30.4 X1RΓ Γ X2M2M1X1RΓ Γ X2M2M1X1RΓ Γ X2M2M1X1RΓ Γ X2M2M1X1RΓ X2M2M1X1RΓ Γ X2M2M1X1RΓ Γ X2M2M1X1RΓ Γ X2M2M1total totalE (eV) -0.4-0.40.00.20.40.6E (eV) -0.4-0.40.00.20.40.6E (eV) -0.4-0.40.00.20.40.6E (eV) -0.4-0.40.00.20.40.6E (eV) -0.4-0.40.00.20.40.6E (eV) -0.4-0.40.00.20.40.6E (eV) -0.4-0.40.00.20.40.6E (eV) -0.4-0.40.00.20.40.6 Os1 Os1 Os1 Os1 Os2 Os2 Os2 Os2dxy dxydyz dyzdzx dzxΓ FIG. 4. The band structures of the two sublattice Os atoms near the Fermi level with 5 dpartial characters for the Q2 orthorhombic distortion (Model A.3 in Ref.21): Os1 sublattice (top), Os2 sublattice (bottom). The projection of each 5 d orbital onto the Kohn-Sham bands is represented by the color shading. The color bar on the left shows the color scaling for the partial characters of the t2gorbitals, while the color bar on the right shows the scaling for all of the orbitals. The chosen high symmetry points are =(0,0,0), X 1=(1 2,0,0), M 1=(0,1 2,0), M 2=(1 2,0,1 2), X 2=(0,0,1 2), and R=(0,1 2,1 2). In Fig. 4, we plot the band structures of the two sublattice Os atoms along the high symmetry direc- tions of the monoclinic cell, with the total partial char-acters of the Os 5 dbands color-coded proportional to their squared-amplitude contributions to the Kohn-Sham eigenvectors, the so-called fat-bands. The total par-5 tial character is the root of the sum of the squares of the partial characters of the Cartesian spin projections,q M2x+M2y+M2z. We plot the partial characters of the spin projections of MxandMyin the Supplemental In- formation Figs. 3 and 4, but not Mzbecause it is two orders of magnitude smaller than the other two. For both Os atoms along these high symmetry directions, only t2g orbitals are occupied, consistent with hndi<6. Thet2g andegare irreducible representations of perfect cubic, octahedral, or tetrahedral symmetry. Because these sym- metries are broken in the structure with Q2 distortion, there are no pure t2goregorbitals, nor a ( egt2g) en- ergy splitting, and there will be a small mixing between the two sets of orbitals. We see that for both sublattices, below the band gap, the d yzorbital is most heavily occupied (as denoted by the brighter green color), especially along the X 1-M2di- rection, while the d xyand d zxorbitals are less occupied. However, around the point, the dxyorbital obviously has the largest occupancy. We point out that here the dorbital character contribution is only for the selected high symmetry directions. Thus, it can not be directly compared with the PDOS result. Nevertheless, the dier- ent orbital character contributions re ected in the color can also be observed for all three t2gorbitals, especially along the X 2- line. We can also see that the dispersions are largest along the X 1-M2and X 1- paths, while the bands are atter from M 2to R. The band gap is indirect and0:06 eV in magnitude. Finally, we have computed the gaps for the imposed cFM phase. We found that the gaps for the cFM phase with the DFT+U parameters of U= 3:3 eV andJ= 0:5 eV are nite, but too small to be considered Mott insu- lating gaps. However, we nd that the gap opens dra- matically as we raise Uto 5.0 eV, as shown in detail in the Supplemental Information. In fact, even a \small" increase to U= 4:0 eV is sucient to open the gap to Egap= 0:244 eV. This indicates that the true value ofUfor the osmium 5 dshell in BNOO could plausi- bly approach 4 :0 eV, but not exceed it. Previously, it was found that LDA+U with U= 4 eV> W was in- sucient to open a gap27. Here, we demonstrated that GGA+SOC+U is sucient to open a gap for U4:0 eV. IV. CONCLUSIONS In this work, we carried out DFT+U calculations on the magnetic Mott insulator Ba 2NaOsO 6, which hasstrong spin orbit coupling. Our numerical work is in- spired by our recent NMR results revealing that this ma- terial exhibited a broken local point symmetry (BLPS) phase followed by a two-sublattice exotic canted ferro- magnetic order (cFM). The local symmetry is broken by the orthorhombic Q2 distortion mode21. The question we addressed here is whether this distortion is accom- panied by the emergence of orbital order. It was pre- viously proposed that the two-sublattice magnetic struc- ture, revealed by NMR, is the very manifestation of stag- gered quadrupolar order with distinct orbital polariza- tion on the two sublattices arising from multipolar ex- change interactions2,12. Moreover, it was indicated via a dierent mean eld formalism that the anisotropic inter- actions result in orbital order that stabilizes exotic mag- netic order5. Therefore, distinct mean eld approaches2,5 with a common ingredient of anisotropic exchange inter- actions imply that exotic magnetic order, such as the cFM reported in Ref. 12, is accompanied/driven by an orbital order. Motivated by the cFM order detected in NMR exper- iments, here we investigated BNOO's orbital ordering pattern from rst principles. We found two-sublattice orbital ordering, illustrated by the spin density plots, within the alternating planes in which the total magnetic moment resides. An auxiliary signature of the orbital or- dering is revealed by the occupancies of the t2gorbitals in the density of states and band structures. Our rst principles work demonstrates that this two-sublattice or- bital ordering mainly arises from cFM order and strong SOC. Moving forward, it would be worthwhile to more thoroughly investigate the cFM order observed in this work using other functionals or methods more adept at handling strong correlation to eliminate any ambiguities that stem from our specic computational treatment. V. ACKNOWLEDGMENTS The authors thank Jeong-Pil Song and Yiou Zhang for enlightening discussions. We are especially grate- ful to Ian Fisher for his long term collaboration on the physics of Ba 2NaOsO 6. This work was supported in part by U.S. National Science Foundation grants DMR- 1608760 (V.F.M.) and DMR-1726213 (B.M.R.). The cal- culations presented here were performed using resources at the Center for Computation and Visualization, Brown University, which is supported by NSF Grant No. ACI- 1548562. yCorresponding authors V. F. M. (vemi@brown.edu) & B. R. (brenda rubenstein@brown.edu) 1B. J. Kim, H. Jin, S. J. Moon, J.-Y. Kim, B.-G. Park, C. S. Leem, J. Yu, T. W. Noh, C. Kim, S.-J. Oh, J.-H. Park, V. Durairaj, G. Cao, and E. Rotenberg, Phys. Rev. Lett. 101, 076402 (2008).2G. Chen, R. Pereira, and L. Balents, Phys. Rev. B 82, 174440 (2010). 3G. Chen and L. Balents, Physical Review B 84, 094420 (2011).6 4H. Ishizuka and L. Balents, Physical Review B 90, 184422 (2014). 5C. Svoboda, M. Randeria, and N. Trivedi, arXiv:1702.03199v1 (unpublished) (2017). 6W. Witczak-Krempa, G. Chen, Y. B. Kim, and L. Balents, Annual Review of Condensed Matter Physics 5, 57 (2014). 7J. Romh anyi, L. Balents, and G. Jackeli, Phys. Rev. Lett. 118, 217202 (2017). 8B. J. Kim, H. Ohsumi, T. Komesu, S. Sakai, T. Morita, H. Takagi, and T. Arima, Science 323, 1329 (2009). 9H. Zhang, K. Haule, and D. Vanderbilt, Phys. Rev. Lett. 111, 246402 (2013). 10S. J. Moon, H. Jin, K. W. Kim, W. S. Choi, Y. S. Lee, J. Yu, G. Cao, A. Sumi, H. Funakubo, C. Bernhard, and T. W. Noh, Phys. Rev. Lett. 101, 226402 (2008). 11X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, Phys. Rev. B 83, 205101 (2011). 12L. Lu, M. Song, W. Liu, A. P. Reyes, P. Kuhns, H. O. Lee, I. R. Fisher, and V. F. Mitrovi c, Nature Communications 8, 14407 EP (2017). 13G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993). 14G. Kresse and J. Hafner, Phys. Rev. B 49, 251 (1994). 15G. Kresse and J. Furthmuller, Comput. Mat. Sci. 6, 15 (1996).16G. Kresse and J. Furthmuller, Phys. Rev. B 54, 11169 (1996). 17J. P. Perdew and Y. Wang, Phys. Rev. B 45, 13244 (1992). 18F. Z. Hund, Zeitschrift f ur Physik 33, 345. 19P. Bl ochl, Phys. Rev. B 50, 17953 (1994). 20G. Kresse and J. Joubert, Phys. Rev. B 59, 1758 (1999). 21R. Cong, R. Nanguneri, B. M. Rubenstein, and V. F. Mitrovic, arXiv e-prints , arXiv:1908.09014 (2019), arXiv:1908.09014. 22A. Erickson, S. Misra, G. J. Miller, R. Gupta, Z. Schlesinger, W. Harrison, J. Kim, and I. Fisher, Phys- ical Review Letters 99, 016404 (2007). 23M. Methfessel and A. T. Paxton, Phys. Rev. B 40, 3616 (1989). 24P. E. Bl ochl, O. Jepsen, and O. K. Andersen, Phys. Rev. B49, 16223 (1994). 25D. F. Mosca, \Master Thesis - Universit a di Bologna: Quantum Magnetism in Relativistic Osmates from First Principles," (2019). 26S. Gangopadhyay and W. E. Pickett, Phys. Rev. B 91, 045133 (2015). 27K.-W. Lee and W. E. Pickett, EPL (Europhysics Letters) 80, 37008 (2007).
1608.02230v1.Spin_orbit_coupling_in_Fe_based_superconductors.pdf	Noname manuscript No. (will be inserted by the editor) Spin-orbit coupling in Fe-based superconductors M.M. Korshunov Yu.N. Togushova I. Eremin P.J. Hirschfeld Received: date / Accepted: date Abstract We study the spin resonance peak in re- cently discovered iron-based superconductors. The res- onance peak observed in inelastic neutron scattering experiments agrees well with predicted results for the extendeds-wave (s) gap symmetry. Recent neutron scattering measurements show that there is a dispar- ity between longitudinal and transverse components of the dynamical spin susceptibility. Such breaking of the spin-rotational invariance in the spin-liquid phase can occur due to spin-orbit coupling. We study the role of the spin-orbit interaction in the multiorbital model for Fe-pnictides and show how it aects the spin resonance feature. Keywords Fe-based superconductors Spin-resonance peakSpin-orbit coupling The nature of the superconductivity and gap sym- metry and structure in the recently discovered Fe-based superconductors (FeBS) are the most debated topics in condensed matter community [1]. These quasi two- dimensional systems shows a maximal Tcof 55 K plac- ing them right after high- Tccuprates. Fe d-orbitals form M.M. Korshunov E-mail: korshunov@phys.u .edu L.V. Kirensky Institute of Physics, Krasnoyarsk 660036, Rus- sia M.M. Korshunov and Yu.N. Togushova Siberian Federal University, Svobodny Prospect 79, Krasno- yarsk 660041, Russia P.J. Hirschfeld Department of Physics, University of Florida, Gainesville, Florida 32611, USA I. Eremin Institut f ur Theoretische Physik III, Ruhr-Universitat Bochum, D-44801 Bochum, Germany Kazan Federal University, Kazan 420008, Russiathe Fermi surface (FS) which in the undoped systems consists of two hole and two electron sheets. Nesting between these two groups of sheets is the driving force for the spin-density wave (SDW) long-range magnetism in the undoped FeBS and the scattering with the wave vector Qconnecting hole and electron pockets is the most probable candidate for superconducting pairing in the doped systems. In the spin- uctuation studies [2,3, 4], the leading instability is the extended s-wave gap which changes sign between hole and electron sheets (sstate) [5]. Neutron scattering is a powerful tool to measure dynamical spin susceptibility (q;!). It carries infor- mation about the order parameter symmetry and gap structure. For the local interactions (Hubbard and Hund's exchange),can be obtained in the RPA from the bare electron-hole bubble 0(q;!) by summing up a series of ladder diagrams to give (q;!) = [IUs0(q;!)]10(q;!), whereUsandIare interaction and unit matrices in or- bital space, and all other quantities are matrices as well. Scattering between nearly nested hole and electron Fermi surfaces in FeBS produce a peak in the normal state magnetic susceptibility at or near q=Q= (;0). For the uniform s-wave gap, sign k= signk+Qand there is no resonance peak. For the sorder parameter as well as for an extended non-uniform s-wave symme- try,Qconnects Fermi sheets with the dierent signs of gaps. This fullls the resonance condition for the in- terband susceptibility, and the spin resonance peak is formed at a frequency below c= min (jkj+jk+qj) (compare normal and ssuperconductor's response in Fig. 1) [6,7,8]. The existence of the spin resonance in FeBS was predicted theoretically [6,7] and subsequently discovered experimentally with many reports of well- dened spin resonances in 1111, 122, and 11 systems [9,10,11].arXiv:1608.02230v1 [cond-mat.supr-con] 7 Aug 20162 M.M. Korshunov et al. 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5Im χ(q=[π,0],ω) ω/∆0non-SC, χ+- non-SC, 2 ×χzz s++, χ+- s++, 2×χzz s±, χ+- s±, 2×χzz Fig. 1 Fig. 1. Calculated Im (Q;!) in the normal state, and for thes++andspairing symmetries. In the latter case, the resonance is clearly seen below != 20. Spin-orbit coupling constant= 100 meV, intraorbital Hubbard U= 0:9 eV, Hund'sJ= 0:1 eV, interorbital U0=U2J, and pair-hopping termJ0=J. One of the recent puzzles in FeBS is the discov- ered anisotropy of the spin resonance peak in Ni-doped Ba-122 [12]. It was found that +and 2zzare dif- ferent. This contradicts the spin-rotational invariance (SRI)hS+Si= 2hSzSziwhich have to be obeyed in the disordered system. One of the solution to the puzzle is the spin-orbit (SO) interaction which can break the SRI like it does in Sr 2RuO 4[13]. Here we incorporate the eect of the SO coupling in the susceptibility cal- culation for FeBS to shed light on the spin resonance anisotropy. The simplest model for pnictides in the 1-Fe per unit cell Brillouin zone comes from the three t2gd- orbitals. The xzandyzcomponents are hybridized and form two electron-like FS pockets around ( ;0) and (0;) points, and one hole-like pocket around = (0;0) point. Thexyorbital is considered to be decoupled from them and form an outer hole pocket around point. The one-electron part of the Hamiltonian is given by H0=P k;;l;m"lm kcy klckm, wherelandmare orbital indices,ckmis the annihilation operator of a particle with momentum kand spin. This model for pnic- tides is similar to the one for Sr 2RuO 4and, in particu- lar, thexyband does not hybridize with the xzandyz bands. Keeping in mind the similarity to the Sr 2RuO 4 case, for simplicity we consider only the Lz-component of the SO interaction [13]. Due to the structure of the Lz-component, the interaction aects xzandyzbands only. Following Ref. [14], we write the SO coupling term, HSO=P fLfSf, in the second-quantized form asHSO= i 2P l;m;nlmnP k;;0cy klckm0^n 0, wherelmnis the completely antisymmetric tensor, indices fl;m;ng take valuesfx;y;zg$fdyz;dzx;dxyg$f 2;3;1g, and ^n 0are the Pauli spin matrices. The matrix of the Hamiltonian H=H0+HSOis then ^"k=0 @"1k 0 0 0"2k"4k+ i 2sign 0"4ki 2sign " 3k1 A (1) As for Sr 2RuO 4, eigenvalues of ^ "kdo not depend on spin, therefore, spin-up and spin-down states are still degenerate in spite of the SO interaction. We calculated both + (longitudinal) and zz(trans- verse) components of the spin susceptibility and found that in the normal state +>2zzat small frequen- cies, see Fig. 1. As expected, for the s++supercon- ductor (conventional isotropic s-wave) there is no res- onance peak and the disparity between +and 2zz is very small. For the ssuperconductor, however, the situation is opposite { we observe a well dened spin resonance and +is larger than 2 zzby about 15% near the peak position (Fig. 1). In summary , we have shown that the spin resonance peak in FeBS gains anisotropy in the spin space due to the spin-orbit coupling. This result is in qualitative agreement with experimental ndings. We do not ob- serve changes in the peak position but this may be due to the simple model that we studied. Acknowledgements Partial support was provided by DOE DE-FG02-05ER46236 (P.J.H. and M.M.K.) and NSF-DMR- 1005625 (P.J.H.). M.M.K. acknowledge support from RFBR (grants 09-02-00127, 12-02-31534 and 13-02-01395), Presid- ium of RAS program \Quantum mesoscopical and disordered structures" N20.7, FCP Scientic and Research-and-Educational Personnel of Innovative Russia for 2009-2013 (GK 16.740.12.0731 and GK P891), and President of Russia (grant MK-1683.2010.2), Siberian Federal University (Theme N F-11), Program of SB RAS #44, and The Dynasty Foundation and ICFPM. I.E. ac- knowledges support of the SFB Transregio 12, Merkur Foun- dation, and German Academic Exchange Service (DAAD PPP USA No. 50750339). References 1. P. J. Hirschfeld, M. M. Korshunov, and I. I. Mazin, Rep. Prog. Phys. 74, 124508 (2011). 2. S. Graser et al. , New. J. Phys. 11, 025016 (2009). 3. K. Kuroki et al. , Phys. Rev. Lett. 101, 087004 (2008). 4. S. Maiti et al. , Phys. Rev. Lett. 107, 147002 (2011). 5. I. I. Mazin et al. , Phys. Rev. Lett. 101, 057003 (2008). 6. M. M. Korshunov and I. Eremin, Phys. Rev. B 78, 140509(R) (2008). 7. T. A. Maier and D. J. Scalapino, Phys. Rev. B 78, 020514(R) (2008). 8. T. A. Maier et al. , Phys. Rev. B 79, 134520 (200).Spin-orbit coupling in Fe-based superconductors 3 9. A. D. Christianson et al. , Nature 456, 930 (2008). 10. D. S. Inosov et al. , Nature Physics 6, 178 (2010). 11. D. N. Argyriou et al., Phys. Rev. B 81, 220503(R) (2010). 12. O. J. Lipscombe et al. , Phys. Rev. B 82, 064515 (2010). 13. I. Eremin, D. Manske, and K. H. Bennemann, Phys. Rev. B65, 220502(R) (2002). 14. K. K. Ng and M. Sigrist, Europhys. Lett. 49, 473 (2000).
2311.16268v2.Gilbert_damping_in_two_dimensional_metallic_anti_ferromagnets.pdf	Gilbert damping in two-dimensional metallic anti-ferromagnets R. J. Sokolewicz,1, 2M. Baglai,3I. A. Ado,1M. I. Katsnelson,1and M. Titov1 1Radboud University, Institute for Molecules and Materials, 6525 AJ Nijmegen, the Netherlands 2Qblox, Delftechpark 22, 2628 XH Delft, the Netherlands 3Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20, Uppsala, Sweden (Dated: March 29, 2024) A finite spin life-time of conduction electrons may dominate Gilbert damping of two-dimensional metallic anti-ferromagnets or anti-ferromagnet/metal heterostructures. We investigate the Gilbert damping tensor for a typical low-energy model of a metallic anti-ferromagnet system with honeycomb magnetic lattice and Rashba spin-orbit coupling for conduction electrons. We distinguish three regimes of spin relaxation: exchange-dominated relaxation for weak spin-orbit coupling strength, Elliot-Yafet relaxation for moderate spin-orbit coupling, and Dyakonov-Perel relaxation for strong spin-orbit coupling. We show, however, that the latter regime takes place only for the in-plane Gilbert damping component. We also show that anisotropy of Gilbert damping persists for any finite spin-orbit interaction strength provided we consider no spatial variation of the N´ eel vector. Isotropic Gilbert damping is restored only if the electron spin-orbit length is larger than the magnon wavelength. Our theory applies to MnPS 3monolayer on Pt or to similar systems. I. INTRODUCTION Magnetization dynamics in anti-ferromagnets con- tinue to attract a lot of attention in the context of possible applications1–4. Various proposals utilize the possibility of THz frequency switching of anti- ferromagnetic domains for ultrafast information storage and computation5,6. The rise of van der Waals magnets has had a further impact on the field due to the pos- sibility of creating tunable heterostructures that involve anti-ferromagnet and semiconducting layers7. Understanding relaxation of both the N´ eel vector and non-equilibrium magnetization in anti-ferromagnets is recognized to be of great importance for the function- ality of spintronic devices8–13. On one hand, low Gilbert damping must generally lead to better electric control of magnetic order via domain wall motion or ultrafast do- main switching14–16. On the other hand, an efficient con- trol of magnetic domains must generally require a strong coupling between charge and spin degrees of freedom due to a strong spin-orbit interaction, that is widely thought to be equivalent to strong Gilbert damping. In this paper, we focus on a microscopic analysis of Gilbert damping due to Dyakonov-Perel and Elliot-Yafet mechanisms. We apply the theory to a model of a two- dimensional N´ eel anti-ferromagnet with a honeycomb magnetic lattice. Two-dimensional magnets typically exhibit either easy-plane or easy-axis anisotropy, and play crucial roles in stabilizing magnetism at finite temperatures17,18. Easy-axis anisotropy selects a specific direction for mag- netization, thereby defining an axis for the magnetic or- der. In contrast, easy-plane anisotropy does not select a particular in-plane direction for the N´ eel vector, allowing it to freely rotate within the plane. This situation is anal- ogous to the XY model, where the system’s continuous symmetry leads to the suppression of out-of-plane fluc- tuations rather than fixing the magnetization in a spe- cific in-plane direction19,20. Without this anisotropy, themagnonic fluctuations in a two-dimensional crystal can grow uncontrollably large to destroy any long-range mag- netic order, according to the Mermin-Wagner theorem21. Recent density-functional-theory calculations for single-layer transition metal trichalgenides22, predict the existence of a large number of metallic anti-ferromagnets with honeycomb lattice and different types of magnetic order as shown in Fig. 1. Many of these crystals may have the N´ eel magnetic order as shown in Fig. 1a and are metallic: FeSiSe 3, FeSiTe 3, VGeTe 3, MnGeS 3, FeGeSe 3, FeGeTe 3, NiGeSe 3, MnSnS 3, MnSnS 3, MnSnSe 3, FeSnSe 3, NiSnS 3. Apart from that it has been predicted that anti-ferromagnetism can be induced in graphene by bringing it in proximity to MnPSe 323or by bringing it in double proximity between a layer of Cr 2Ge2Te6and WS224. Partly inspired by these predictions and recent technological advances in producing single-layer anti- ferromagnet crystals, we propose an effective model to study spin relaxation in 2D honeycomb anti-ferromagnet with N´ eel magnetic order. The same system was studied by us in Ref. 25, where we found that spin-orbit cou- pling introduces a weak anisotropy in spin-orbit torque and electric conductivity. Strong spin-orbit coupling was shown to lead to a giant anisotropy of Gilbert damping. Our analysis below is built upon the results of Ref. 25, and we investigate and identify three separate regimes of spin-orbit strength. Each regime is characterized by qualitatively different dependence of Gilbert damping on spin-orbit interaction and conduction electron transport time. The regime of weak spin-orbit interaction is dom- inated by exchange field relaxation of electron spin, and the regime of moderate spin-orbit strength is dominated by Elliot-Yafet spin relaxation. These two regimes are characterized also by a universal factor of 2 anisotropy of Gilbert damping. The regime of strong spin-orbit strength, which leads to substantial splitting of electron Fermi surfaces, is characterized by Dyakonov-Perel relax- ation of the in-plane spin component and Elliot-Yafet re-arXiv:2311.16268v2 [cond-mat.dis-nn] 28 Mar 20242 FIG. 1. Three anti-ferromagnetic phases commonly found among van-der-Waals magnets. Left-to-right: N´ eel, zig-zag, and stripy. laxation of the perpendicular-to-the-plane Gilbert damp- ing which leads to a giant damping anisotropy. Isotropic Gilbert damping is restored only for finite magnon wave vectors such that the magnon wavelength is smaller than the spin-orbit length. Gilbert damping in a metallic anti-ferromagnet can be qualitatively understood in terms of the Fermi surface breathing26. A change in the magnetization direction gives rise to a change in the Fermi surface to which the conduction electrons have to adjust. This electronic re- configuration is achieved through the scattering of elec- trons off impurities, during which angular momentum is transferred to the lattice. Gilbert damping, then, should be proportional to both (i) the ratio of the spin life-time and momentum life-time of conduction electrons, and (ii) the electric conductivity. Keeping in mind that the con- ductivity itself is proportional to momentum life-time, one may conclude that the Gilbert damping is linearly proportional to the spin life-time of conduction electrons. At the same time, the spin life-time of localized spins is inversely proportional to the spin life-time of conduc- tion electrons. A similar relation between the spin life- times of conduction and localized electrons also holds for relaxation mechanisms that involve electron-magnon scattering27. Our approach formally decomposes the magnetic sys- tem into a classical sub-system of localized magnetic mo- ments and a quasi-classical subsystem of conduction elec- trons. A local magnetic exchange couples these sub- systems. Localized magnetic moments in transition- metal chalcogenides and halides form a hexagonal lat- tice. Here we focus on the N´ eel type anti-ferromagnet that is illustrated in Fig. 1a. In this case, one can de- fine two sub-lattices A and B that host local magnetic moments SAandSB, respectively. For the discussion of Gilbert damping, we ignore the weak dependence of both fields on atomic positions and assume that the modulus S=\|SA(B)\|is time-independent. Under these assumptions, the magnetization dynamics of localized moments may be described in terms of two fields m=1 2S SA+SB ,n=1 2S SA−SB , (1) which are referred to as the magnetization and staggeredmagnetization (or N´ eel vector), respectively. Within the mean-field approach, the vector fields yield the equations of motion ˙n=−Jn×m+n×δs++m×δs−, (2a) ˙m=m×δs++n×δs−, (2b) where dot stands for the time derivative, while δs+and δs−stand for the mean staggered and non-staggered non- equilibrium fields that are proportional to the variation of the corresponding spin-densities of conduction electrons caused by the time dynamics of nandmfields. The en- ergy Jis proportional to the anti-ferromagnet exchange energy for localized momenta. In Eqs. (2) we have omitted terms that are propor- tional to easy axis anisotropy for the sake of compact- ness. These terms are, however, important and will be introduced later in the text. In the framework of Eqs. (2) the Gilbert damping can be computed as the linear response of the electron spin- density variation to a time change in both the magneti- zation and the N´ eel vector (see e. g. Refs.25,28,29). In this definition, Gilbert damping describes the re- laxation of localized spins by transferring both total and staggered angular momenta to the lattice by means of conduction electron scattering off impurities. Such a transfer is facilitated by spin-orbit interaction. The structure of the full Gilbert damping tensor can be rather complicated as discussed in Ref. 25. However, by taking into account easy axis or easy plane anisotropy we may reduce the complexity of relevant spin configurations to parameterize δs+=α∥ m˙m∥+α⊥ m˙m⊥+αmn∥×(n∥×˙m∥),(3a) δs−=α∥ n˙n∥+α⊥ n˙n⊥+αnn∥×(n∥×˙n∥), (3b) where the superscripts ∥and⊥refer to the in-plane and perpendicular-to-the-plane projections of the corre- sponding vectors, respectively. The six coefficients α∥ m, α⊥ m,αm,α∥ n,α⊥ n, and αnparameterize the Gilbert damp- ing. Inserting Eqs. (3) into the equations of motion of Eqs. (2) produces familiar Gilbert damping terms. The damping proportional to time-derivatives of the N´ eel vec- tornis in general many orders of magnitude smaller than that proportional to the time-derivatives of the magneti- zation vector m25,30. Due to the same reason, the higher harmonics term αmn∥×(n∥×∂tm∥) can often be ne- glected. Thus, in the discussion below we may focus mostly on the coefficients α∥ mandα⊥ mthat play the most important role in the magnetization dynamics of our system. The terms proportional to the time-derivative of ncorrespond to the transfer of angular momentum between the sub- lattices and are usually less relevant. We refer to the results of Ref. 25 when discussing these terms. All Gilbert damping coefficients are intimately related to the electron spin relaxation time. The latter is rel- atively well understood in non-magnetic semiconductors3 with spin-orbital coupling. When a conducting electron moves in a steep potential it feels an effective magnetic field caused by relativistic effects. Thus, in a disordered system, the electron spin is subject to a random magnetic field each time it scatters off an impurity. At the same time, an electron also experiences precession around an effective spin-orbit field when it moves in between the collisions. Changes in spin direction between collisions are referred to as Dyakonov-Perel relaxation31,32, while changes in spin-direction during collisions are referred to as Elliot-Yafet relaxation33,34. The spin-orbit field in semiconductors induces a char- acteristic frequency of spin precession Ω s, while scalar disorder leads to a finite transport time τof the con- ducting electrons. One may, then, distinguish two limits: (i) Ω sτ≪1 in which case the electron does not have sufficient time to change its direction between consec- utive scattering events (Elliot-Yafet relaxation), and (ii) Ωsτ≫1 in which case the electron spin has multiple pre- cession cycles in between the collisions (Dyakonov-Perel relaxation). The corresponding processes define the so-called spin relaxation time, τs. In a 2D system the spin life-time τ∥ s, for the in-plane spin components, appears to be dou- ble the size of the life-time of the spin component that is perpendicular to the plane, τ⊥ s32. This geometric ef- fect has largely been overlooked. For non-magnetic 2D semiconductor one can estimate35,36 1 τ∥ s∼( Ω2 sτ,Ωsτ≪1 1/τ, Ωsτ≫1, τ∥ s= 2τ⊥ s. (4) A pedagogical derivation and discussion of Eq. 4 can be found in Refs. 35 and 36. Because electrons are con- fined in two dimensions the random spin-orbit field is always directed in-plane, which leads to a decrease in the in-plane spin-relaxation rate by a factor of two compared to the out-of-plane spin-relaxation rate as demonstrated first in Ref. 32 (see Refs. 36–40 as well). The reason is that the perpendicular-to-the-plane component of spin is influenced by two components of the randomly changing magnetic field, i. e. xandy, whereas the parallel-to-the- plane spin components are only influenced by a single component of the fluctuating fields, i. e. the xspin pro- jection is influenced only by the ycomponent of the field and vice-versa. The argument has been further general- ized in Ref. 25 to the case of strongly separated spin-orbit split Fermi surfaces. In this limit, the perpendicular-to- the-plane spin-flip processes on scalar disorder potential become fully suppressed. As a result, the perpendicular- to-the-plane spin component becomes nearly conserved, which results in a giant anisotropy of Gilbert damping in this regime. In magnetic systems that are, at the same time, con- ducting there appears to be at least one additional energy scale, ∆ sd, that characterizes exchange coupling of con- duction electron spin to the average magnetic moment of localized electrons. (In the case of s-d model descriptionit is the magnetic exchange between the spin of conduc- tionselectron and the localized magnetic moment of d orfelectron on an atom.) This additional energy scale complicates the simple picture of Eq. (4) especially in the case of an anti-ferromagnet. The electron spin precession is now defined not only by spin-orbit field but also by ∆sd. As the result the conditions Ω sτ≪1 and ∆ sdτ≫1 may easily coexist. This dissolves the distinction between Elliot-Yafet and Dyakonov-Perel mechanisms of spin re- laxation. One may, therefore, say that both Elliot-Yafet and Dyakonov-Perel mechanisms may act simultaneously in a typical 2D metallic magnet with spin-orbit coupling. The Gilbert damping computed from the microscopic model that we formulate below will always contain both contributions to spin-relaxation. II. MICROSCOPIC MODEL AND RESULTS The microscopic model that we employ to calculate Gilbert damping is the so-called s–dmodel that couples localized magnetic momenta SAandSBand conducting electron spins via the local magnetic exchange ∆ sd. Our effective low-energy Hamiltonian for conduction electrons reads H=vfp·Σ+λ 2 σ×Σ z−∆sdn·σΣzΛz+V(r),(5) where the vectors Σ,σandΛdenote the vectors of Pauli matrices acting on sub-lattice, spin and valley space, respectively. We also introduce the Fermi velocity vf, Rashba-type spin-orbit interaction λ, and a random im- purity potential V(r). The Hamiltonian of Eq. (5) can be viewed as the graphene electronic model where conduction electrons have 2D Rashba spin-orbit coupling and are also cou- pled to anti-ferromagnetically ordered classical spins on the honeycomb lattice. The coefficients α∥ mandα⊥ mare obtained using linear response theory for the response of spin-density δs+to the time-derivative of magnetization vector ∂tm. Impu- rity potential V(r) is important for describing momen- tum relaxation to the lattice. This is related to the an- gular momentum relaxation due to spin-orbit coupling. The effect of random impurity potential is treated pertur- batively in the (diffusive) ladder approximation that in- volves a summation over diffusion ladder diagrams. The details of the microscopic calculation can be found in the Appendices. Before presenting the disorder-averaged quantities α∥,⊥ m, it is instructive to consider first the contribution to Gilbert damping originating from a small number of electron-impurity collisions. This clarifies how the num- ber of impurity scattering effects will affect the final re- sult. Let us annotate the Gilbert damping coefficients with an additional superscript ( l) that denotes the number of scattering events that are taken into account. This4 01234(i) ?["] (0) ?(1) ?(2) ? (1) ? 102101100101 01234(i) k["] (0) k(1) k(2) k(1) k FIG. 2. Gilbert damping in the limit ∆ sd= 0. Dotted (green) lines correspond to the results of the numerical evaluation of ¯α(l) m,⊥,∥forl= 0,1,2 as a function of the parameter λτ. The dashed (orange) line corresponds to the diffusive (fully vertex corrected) results for ¯ α⊥,∥. m. means, in the diagrammatic language, that the corre- sponding quantity is obtained by summing up the ladder diagrams with ≤ldisorder lines. Each disorder line cor- responds to a quasi-classical scattering event from a sin- gle impurity. The corresponding Gilbert damping coeffi- cient is, therefore, obtained in the approximation where conduction electrons have scattered at most lnumber of times before releasing their non-equilibrium magnetic moment into a lattice. To make final expressions compact we define the di- mensionless Gilbert damping coefficients ¯ α∥,⊥ mby extract- ing the scaling factor α∥,⊥ m=A∆2 sd πℏ2v2 fS¯α∥,⊥ m, (6) where Ais the area of the unit cell, vfis the Fermi ve- locity of the conducting electrons and ℏ=h/2πis the Planck’s constant. We also express the momentum scat- tering time τin inverse energy units, τ→ℏτ. Let us start by computing the coefficients ¯ α∥,⊥(l) m in the formal limit ∆ sd→0. We can start with the “bare bub- ble” contribution which describes spin relaxation without a single scattering event. The corresponding results read ¯α(0) m,⊥=ετ1−λ2/4ε2 1 +λ2τ2, (7a) ¯α(0) m,∥=ετ1 +λ2τ2/2 1 +λ2τ2−λ2 8ε2 , (7b) where εdenotes the Fermi energy which we consider pos- itive (electron-doped system).In all realistic cases, we have to consider λ/ε≪1, while the parameter λτmay in principle be arbitrary. For λτ≪1 the disorder-induced broadening of the electron Fermi surfaces exceeds the spin-orbit induced splitting. In this case one basically finds no anisotropy of “bare” damping: ¯ α(0) m,⊥= ¯α(0) m,∥. In the opposite limit of substan- tial spin-orbit splitting one gets an ultimately anisotropic damping ¯ α(0) m,⊥≪¯α(0) m,∥. This asymptotic behavior can be summarized as ¯α(0) m,⊥=ετ( 1 λτ≪1, (λτ)−2λτ≫1,(8a) ¯α(0) m,∥=ετ( 1 λτ≪1, 1 2 1 + (λτ)−2 λτ≫1,(8b) where we have used that ε≫λ. The results of Eq. (8) modify by electron diffusion. By taking into account up to lscattering events we obtain ¯α(l) m,⊥=ετ( l+O(λ2τ2) λτ≪1, (1 +δl0)/(λτ)2λτ≫1,(9a) ¯α(l) m,∥=ετ( l+O(λ2τ2) λτ≪1, 1−(1/2)l+1+O((λτ)−2)λτ≫1,(9b) where we have used ε≫λagain. From Eqs. (9) we see that the Gilbert damping for λτ≪1 gets an additional contribution of ετfrom each scattering event as illustrated numerically in Fig. 2. This leads to a formal divergence of Gilbert damping in the limit λτ≪1. While, at first glance, the divergence looks like a strong sensitivity of damping to impurity scatter- ing, in reality, it simply reflects a diverging spin life-time. Once a non-equilibrium magnetization mis created it becomes almost impossible to relax it to the lattice in the limit of weak spin-orbit coupling. The formal diver- gence of α⊥ m=α∥ msimply reflects the conservation law for electron spin polarization in the absence of spin-orbit coupling such that the corresponding spin life-time be- comes arbitrarily large as compared to the momentum scattering time τ. By taking the limit l→ ∞ (i. e. by summing up the entire diffusion ladder) we obtain compact expressions ¯α⊥ m≡¯α(∞) m,⊥=ετ1 2λ2τ2, (10a) ¯α∥ m≡¯α(∞) m,∥=ετ1 +λ2τ2 λ2τ2, (10b) which assume ¯ α⊥ m≪¯α∥ mforλτ≫1 and ¯ α⊥ m= ¯α∥ m/2 forλτ≪1. The factor of 2 difference that we observe when λτ≪1, corresponds to a difference in the elec- tron spin life-times τ⊥ s=τ∥ s/2 that was discussed in the introduction32. Strong spin-orbit coupling causes a strong out-of-plane anisotropy of damping, ¯ α⊥ m≪¯α∥ mwhich corresponds to5 a suppression of the perpendicular-to-the-plane damping component. As a result, the spin-orbit interaction makes it much easier to relax the magnitude of the mzcompo- nent of magnetization than that of in-plane components. Let us now turn to the dependence of ¯ αmcoefficients on ∆sdthat is illustrated numerically in Fig. 3. We consider first the case of absent spin-orbit coupling λ= 0. In this case, the combination of spin-rotational and sub- lattice symmetry (the equivalence of A and B sub-lattice) must make Gilbert damping isotropic (see e. g.25,41). The direct calculation for λ= 0 does, indeed, give rise to the isotropic result ¯ α⊥ m= ¯α∥ m=ετ(ε2+∆2 sd)/2∆2 sd, which is, however, in contradiction to the limit λ→0 in Eq. (10). At first glance, this contradiction suggests the exis- tence of a certain energy scale for λover which the anisotropy emerges. The numerical analysis illustrated in Fig. 4 reveals that this scale does not depend on the values of 1 /τ, ∆sd, orε. Instead, it is defined solely by numerical precision. In other words, an isotropic Gilbert damping is obtained only when the spin-orbit strength λis set below the numerical precision in our model. We should, therefore, conclude that the transition from isotropic to anisotropic (factor of 2) damping occurs ex- actly at λ= 0. Interestingly, the factor of 2 anisotropy is absent in Eqs. (8) and emerges only in the diffusive limit. We will see below that this paradox can only be re- solved by analyzing the Gilbert damping beyond the in- finite wave-length limit. One can see from Fig. 3 that the main effect of finite ∆sdis the regularization of the Gilbert damping diver- gency ( λτ)−2in the limit λτ≪1. Indeed, the limit of weak spin-orbit coupling is non-perturbative for ∆ sd/ε≪ λτ≪1, while, in the opposite limit, λτ≪∆sd/ε≪1, the results of Eqs. (10) are no longer valid. Assuming ∆sd/ε≪1 we obtain the asymptotic expressions for the results presented in Fig. 3 as ¯α⊥ m=1 2ετ(2 3ε2+∆2 sd ∆2 sdλτ≪∆sd/ε, 1 λ2τ2 λτ≫∆sd/ε,(11a) ¯α∥ m=ετ(2 3ε2+∆2 sd ∆2 sdλτ≪∆sd/ε, 1 +1 λ2τ2λτ≫∆sd/ε,(11b) which suggest that ¯ α⊥ m/¯α∥ m= 2 for λτ≪1. In the op- posite limit, λτ≫1, the anisotropy of Gilbert damping grows as ¯ α∥ m/¯α⊥ m= 2λ2τ2. The results of Eqs. (11) can also be discussed in terms of the electron spin life-time, τ⊥(∥) s = ¯α⊥(∥) m/ε. For the inverse in-plane spin life-time we find 1 τ∥ s= 3∆2 sd/2ε2τ λτ ≪∆sd/ε, λ2τ ∆sd/ε≪λτ≪1, 1/τ 1≪λτ,(12) that, for ∆ sd= 0, is equivalent to the known result of Eq. (4). Indeed, for ∆ sd= 0, the magnetic exchange 103102101100101 101101103105m;k;?["] sd="= 0:1sd="= 0m;k m;?FIG. 3. Numerical results for the Gilbert damping compo- nents in the diffusive limit (vertex corrected)as the function of the spin-orbit coupling strength λ. The results correspond toετ= 50 and ∆ sdτ= 0.1 and agree with the asymptotic expressions of Eq. (11). Three different regimes can be dis- tinguished for ¯ α∥ m: i) spin-orbit independent damping ¯ α∥ m∝ ε3τ/∆2 sdfor the exchange dominated regime, λτ≪∆sd/ε, ii) the damping ¯ α∥ m∝ε/λ2τfor Elliot-Yafet relaxation regime, ∆sd/ε≪λτ≪1, and iii) the damping ¯ α∥ m∝ετfor the Dyakonov-Perel relaxation regime, λτ≫1. The latter regime is manifestly absent for ¯ α⊥ min accordance with Eqs. (12,13). plays no role and one observes the cross-over from Elliot- Yafet ( λτ≪1) to Dyakonov-Perel ( λτ≫1) spin relax- ation. This cross-over is, however, absent in the relaxation of the perpendicular spin component 1 τ⊥s= 2( 3∆2 sd/2ε2τ λτ ≪∆sd/ε, λ2τ ∆sd/ε≪λτ,(13) where Elliot-Yafet-like relaxation extends to the regime λτ≫1. As mentioned above, the factor of two anisotropy in spin-relaxation of 2 Dsystems, τ∥ s= 2τ⊥ s, is known in the literature32(see Refs.36–38as well). Unlimited growth of spin life-time anisotropy, τ∥ s/τ⊥ s= 2λ2τ2, in the regime λτ≪1 has been described first in Ref. 25. It can be qual- itatively explained by a strong suppression of spin-flip processes for zspin component due to spin-orbit induced splitting of Fermi surfaces. The mechanism is effective only for scalar (non-magnetic) disorder. Even though such a mechanism is general for any magnetic or non- magnetic 2D material with Rashba-type spin-orbit cou- pling, the effect of the spin life-time anisotropy on Gilbert damping is much more relevant for anti-ferromagnets. In- deed, in an anti-ferromagnetic system the modulus of m is, by no means, conserved, hence the variations of per- pendicular and parallel components of the magnetization vector are no longer related. In the regime, λτ≪∆sd/εthe spin life-time is de- fined by exchange interaction and the distinction between Dyakonov-Perel and Elliot-Yafet mechanisms of spin re- laxation is no longer relevant. In this regime, the spin- relaxation time is by a factor ( ε/∆sd)2larger than the momentum relaxation time. Let us now return to the problem of emergency of the6 106410541044103410241014 12k=?n= 32 n= 64n= 96 n= 128 FIG. 4. Numerical evaluation of Gilbert damping anisotropy in the limit λ→0. Isotropic damping tensor is restored only ifλ= 0 with ultimate numerical precision. The factor of 2 anisotropy emerges at any finite λ, no matter how small it is, and only depends on the numerical precision n, i.e. the number of digits contained in each variable during computa- tion. The crossover from isotropic to anisotropic damping can be understood only by considering finite, though vanishingly small, magnon qvectors. factor of 2 anisotropy of Gilbert damping at λ= 0. We have seen above (see Fig. 4) that, surprisingly, there ex- ists no energy scale for the anisotropy to emerge. The transition from the isotropic limit ( λ= 0) to a finite anisotropy appeared to take place exactly at λ= 0. We can, however, generalize the concept of Gilbert damping by considering the spin density response function at a finite wave vector q. To generalize the Gilbert damping, we are seeking a response of spin density at a point r,δs+(r) to a time derivative of magnetization vectors ˙m∥and ˙m⊥at the point r′. The Fourier transform with respect to r−r′ gives the Gilbert damping for a magnon with the wave- vector q. The generalization to a finite q-vector shows that the limits λ→0 and q→0 cannot be interchanged. When the limit λ→0 is taken before the limit q→0 one finds an isotropic Gilbert damping, while for the oppo- site order of limits, it becomes a factor of 2 anisotropic. In a realistic situation, the value of qis limited from below by an inverse size of a typical magnetic domain 1/Lm, while the spin-orbit coupling is effective on the length scale Lλ= 2πℏvf/λ. In this picture, the isotropic Gilbert damping is characteristic for the case of suffi- ciently small domain size Lm≪Lλ, while the anisotropic Gilbert damping corresponds to the case Lλ≪Lm. In the limit qℓ≪1, where ℓ=vfτis the electron mean 2 0 2 k[a.u.]2:50:02:5energy [a.u.]=sd= 4 2 0 2 k[a.u.]=sd= 2 2 0 2 k[a.u.]=sd= 1FIG. 5. Band-structure for the effective model of Eq. (5) in a vicinity of Kvalley assuming nz= 1. Electron bands touch for λ= 2∆ sd. The regime λ≤2∆sdcorresponds to spin-orbit band inversion. The band structure in the valley K′is inverted. Our microscopic analysis is performed in the electron-doped regime for the Fermi energy above the gap as illustrated by the top dashed line. The bottom dashed line denotes zero energy (half-filling). free path, we can summarize our results as ¯α⊥ m=ετ ε2+∆2 sd 2∆2 sdλτ≪qℓ≪∆sd/ε, 1 3ε2+∆2 sd ∆2 sdqℓ≪λτ≪∆sd/ε, 1 2λ2τ2 λτ≫∆sd/ε,, (14a) ¯α∥ m=ετ ε2+∆2 sd 2∆2 sdλτ≪qℓ≪∆sd/ε, 2 3ε2+∆2 sd ∆2 sdqℓ≪λτ≪∆sd/ε, 1 +1 λ2τ2λτ≫∆sd/ε,(14b) which represent a simple generalization of Eqs. (11). The results of Eqs. (14) correspond to a simple behav- ior of Gilbert damping anisotropy, ¯α∥ m/¯α⊥ m=( 1 λτ≪qℓ, 2 1 +λ2τ2 qℓ≪λτ,(15) where we still assume qℓ≪1. III. ANTI-FERROMAGNETIC RESONANCE The broadening of the anti-ferromagnet resonance peak is one obvious quantity that is sensitive to Gilbert damping. The broadening is however not solely defined by a particular Gilbert damping component but depends also on both magnetic anisotropy and anti-ferromagnetic exchange. To be more consistent we can use the model of Eq. (5) to analyze the contribution of conduction electrons to an easy axis anisotropy. The latter is obtained by expanding the free energy for electrons in the value of nz, which has a form E=−Kn2 z/2. With the conditions ε/λ≫1 and ε/∆sd≫1 we obtain the anisotropy constant as K=A 2πℏ2v2( ∆2 sdλ 2∆sd/λ≤1, ∆sdλ2/2 2∆ sd/λ≥1,(16)7 where Ais the area of the unit cell. Here we assume both λand ∆ sdpositive, therefore, the model natu- rally gives rise to an easy axis anisotropy with K > 0. In real materials, there exist other sources of easy axis or easy plane anisotropy. In-plane magneto-crystalline anisotropy also plays an important role. For example, N´ eel-type anti-ferromagnets with easy-axis anisotropy are FePS 3, FePSe 3or MnPS 3, whereas those with easy plane and in-plane magneto-crystalline anisotropy are NiPS 3and MnPSe 3. Many of those materials are, how- ever, Mott insulators. Our qualitative theory may still apply to materials like MnPS 3monolayers at strong elec- tron doping. The transition from 2∆ sd/λ≥1 to 2∆ sd/λ≤1 in Eq. (16) corresponds to the touching of two bands in the model of Eq. (5) as illustrated in Fig. 5. Anti-ferromagnetic magnon frequency and life-time in the limit q→0 are readily obtained by linearizing the equations of motion ˙n=−Jn×m+Km×n⊥+n×(ˆαm˙m), (17a) ˙m=Kn×n⊥+n×(ˆαn˙n), (17b) where we took into account easy axis anisotropy Kand disregarded irrelevant terms m×(ˆαn˙n) and m×(ˆαm˙m). We have also defined Gilbert damping tensors such as ˆαm˙m=α∥ m˙m∥+α⊥ m˙m⊥, ˆαn˙n=α∥ n˙n∥+α⊥ n˙n⊥. In the case of easy axis anisotropy we can use the lin- earized modes n=ˆz+δn∥eiωt,m=δm∥eiωt, hence we get the energy of q= 0 magnon as ω=ω0−iΓ/2, (18) ω0=√ JK, Γ =Jα∥ n+Kα∥ m (19) where we took into account that K≪J. The expression forω0is well known due to Kittel and Keffer42,43. Using Ref. 25 we find out that α∥ n≃α⊥ m(λ/ε)2and α⊥ n≃α∥ m(λ/ε)2, hence Γ≃α∥ m K+J/2 ε2/λ2+ε2τ2 , (20) where we have simply used Eqs. (10). Thus, one may often ignore the contribution Jα∥ nas compared to Kα∥ m despite the fact that K≪J. In the context of anti-ferromagnets, spin-pumping terms are usually associated with the coefficients α∥ nin Eq. (3b) that are not in the focus of the present study. Those coefficients have been analyzed for example in Ref. 25. In this manuscript we simply use the known results forαnin Eqs. (17-19), where we illustrate the effect of both spin-pumping coefficient αnand the direct Gilbert damping αmon the magnon life time. One can see from Eqs. (19,20) that the spin-pumping contributions do also contribute, though indirectly, to the magnon decay. The spin pumping contributions become more important in magnetic materials with small magnetic anisotropy. The processes characterized by the coefficients αnmay also be 103102101100101 0:000:010:021=k m="= 0:04 ="= 0:02 ="= 0:01FIG. 6. Numerical evaluation of the inverse Gilbert damping 1/¯α∥ mas a function of the momentum relaxation time τ. The inverse damping is peaked at τ∝1/λwhich also corresponds to the maximum of the anti-ferromagnetic resonance quality factor in accordance with Eq. (21). interpreted in terms of angular momentum transfer from one AFM sub-lattice to another. In that respect, the spin pumping is specific to AFM, and is qualitatively differ- ent from the direct Gilbert damping processes ( αm) that describe the direct momentum relaxation to the lattice. As illustrated in Fig. 6 the quality factor of the anti- ferromagnetic resonance (for a metallic anti-ferromagnet with easy-axis anisotropy) is given by Q=ω0 Γ≃1 α∥ mr J K. (21) Interestingly, the quality factor defined by Eq. (21) is maximized for λτ≃1, i. e. for the electron spin-orbit length being of the order of the scattering mean free path. The quantities 1 /√ Kand 1 /¯α∥ mare illustrated in Fig. 6 from the numerical analysis. As one would ex- pect, the quality factor vanishes in both limits λ→0 andλ→ ∞ . The former limit corresponds to an over- damped regime hence no resonance can be observed. The latter limit corresponds to a constant α∥ m, but the reso- nance width Γ grows faster with λthan ω0does, hence the vanishing quality factor. It is straightforward to check that the results of Eqs. (20,21) remain consistent when considering systems with either easy-plane or in-plane magneto-crystalline anisotropy. Thus, the coefficient α⊥ mnormally does not enter the magnon damping, unless the system is brought into a vicinity of spin-flop transition by a strong external field. IV. CONCLUSION In conclusion, we have analyzed the Gilbert damping tensor in a model of a two-dimensional anti-ferromagnet on a honeycomb lattice. We consider the damping mech- anism that is dominated by a finite electron spin life-time8 due to a combination of spin-orbit coupling and impu- rity scattering of conduction electrons. In the case of a 2D electron system with Rashba spin-orbit coupling λ, the Gilbert damping tensor is characterized by two com- ponents α∥ mandα⊥ m. We show that the anisotropy of Gilbert damping depends crucially on the parameter λτ, where τis the transport scattering time for conduction electrons. For λτ≪1 the anisotropy is set by a geo- metric factor of 2, α∥ m= 2α⊥ m, while it becomes infinitely large in the opposite limit, α∥ m= (λτ)2α⊥ mforλτ≫1. Gilbert damping becomes isotropic exactly for λ= 0, or, strictly speaking, for the case λ≪ℏvfq, where qis the magnon wave vector. This factor of 2 is essentially universal, and is a geomet- ric effect: the z-component relaxation results from fluctu- ations in two in-plane spin components, whereas in-plane relaxation stems from fluctuations of the z-component alone. This reflects the subtleties of our microscopic model, where the mechanism for damping is activated by the decay of conduction electron momenta, linked to spin-relaxation through spin-orbit interactions. We find that Gilbert damping is insensitive to mag- netic order for λ≫∆sd/ετ, where ∆ sdis an effective exchange coupling between spins of conduction and local- ized electrons. In this case, the electron spin relaxation can be either dominated by scattering (Dyakonov-Perel relaxation) or by spin-orbit precession (Elliot-Yafet re- laxation). We find that the Gilbert damping component α⊥ m≃ε/λ2τis dominated by Elliot-Yafet relaxation irre- spective of the value of the parameter λτ, while the other component crosses over from α∥ m≃ε/λ2τ(Elliot-Yafet relaxation) for λτ≪1, to α∥ m≃ετ(Dyakonov-Perel re- laxation) for λτ≫1. For the case λ≪∆sd/ετthe spin relaxation is dominated by interaction with the exchange field. Crucially, our results are not confined solely to the N´ eel order on the honeycomb lattice: we anticipate a broader applicability across various magnetic orders, including the zigzag order. This universality stems from our focus on the large magnon wavelength limit. The choice of the honeycomb lattice arises from its unique ability to main- tain isotropic electronic spectra within the plane, coupled with the ability to suppress anisotropy concerning in- plane spin rotations. Strong anisotropic electronic spec- tra would naturally induce strong anisotropic in-plane Gilbert damping, which are absent in our results. Finally, we show that the anti-ferromagnetic resonance width is mostly defined by α∥ mand demonstrate that the resonance quality factor is maximized for λτ≈1. Our microscopic theory predictions may be tested for systems such as MnPS 3monolayer on Pt and similar heterostruc- tures.ACKNOWLEDGMENTS We are grateful to O. Gomonay, R. Duine, J. Sinova, and A. Mauri for helpful discussions. This project has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No 873028. Appendix A: Microscopic framework The microscopic model that we employ to calculate Gilbert damping belongs to a class of so-called s–dmod- els that describe the physical system in the form of a Heisenberg model for localized spins and a tight-binding model for conduction electrons that are weakly coupled by a local magnetic exchange interaction of the strength ∆sd. Our effective electron Hamiltonian for a metallic hexagonal anti-ferromagnet is given by25 H0=vfp·Σ+λ 2[σ×Σ]z−∆sdn·σΣzΛz,(A1) where the vectors Σ,σandΛdenote the vectors of Pauli- matrices acting on sub-lattice, spin and valley space re- spectively. We also introduce the Fermi velocity vf, Rashba-type spin-orbit interaction λ. To describe Gilbert damping of the localized field n we have to add the relaxation mechanism. This is pro- vided in our model by adding a weak impurity potential H=H0+V(r). The momentum relaxation due to scat- tering on impurities leads indirectly to the relaxation of Heisenberg spins due to the presence of spin-orbit cou- pling and exchange couplings. For modeling the impurity potential, we adopt a delta- correlated random potential that corresponds to the point scatter approximation, where the range of the im- purity potential is much shorter than that of the mean free path (see e.g. section 3.8 of Ref. 44), i.e. ⟨V(r)V(r′)⟩= 2πα(ℏvf)2δ(r−r′), (A2) where the dimensionless coefficient α≪1 characterizes the disorder strength. The corresponding scattering time for electrons is obtained as τ=ℏ/παϵ , which is again similar to the case of graphene. The response of symmetric spin-polarization δs+to the time-derivative of non-staggered magnetization, ∂tm, is defined by the linear relation δs+ α=X βRαβ\|ω=0˙mβ, (A3) where the response tensor is taken at zero frequency25,45. The linear response is defined generally by the tensor Rαβ=A∆2 sd 2πSZdp (2πℏ)2 Tr GR ε,pσαGA ε+ℏω,pσβ ,(A4)9 where GR(A) ε,pare standing for retarded(advanced) Green functions and the angular brackets denote averaging over disorder fluctuations. The standard recipe for disorder averaging is the diffu- sive approximation46,47that is realized by replacing the bare Green functions in Eq. (A4) with disorder-averaged Green functions and by replacing one of the vertex op- erators σxorσywith the corresponding vertex-corrected operator that is formally obtained by summing up ladder impurity diagrams (diffusons). In models with spin-orbit coupling, the controllable dif- fusive approximation for non-dissipative quantities may become, however, more involved as was noted first in Ref. 48. For Gilbert damping it is, however, sufficient to consider the ladder diagram contributions only. The disorder-averaged Green function is obtained by including an imaginary part of the self-energy ΣR(not to be confused here with the Pauli matrix Σ 0,x,y,z) that is evaluated in the first Born approximation Im ΣR= 2παv2 fZdp (2π)2Im1 ε−H0+i0. (A5) The real part of the self-energy leads to the renormaliza- tion of the energy scales ε,λand ∆ sd. In the first Born approximation, the disorder-averaged Green function is given by GR ε,p=1 ε−H0−iIm ΣR. (A6) The vertex corrections are computed in the diffusive approximation. The latter involves replacing the vertex σαwith the vertex-corrected operator, σvc α=∞X l=0σ(l) α, (A7) where the index lcorresponds to the number of disorder lines in the ladder. The operators σ(l) αcan be defined recursively as σ(l) α=2ℏv2 f ετZdp (2π)2GR ε,pσ(l−1) αGA ε+ℏω,p, (A8) where σ(0) α=σα. The summation in Eq. (A7) can be computed in the full operator basis, Bi={α,β,γ}=σαΣβΛγ, where each index α,βandγtakes on 4 possible values (with zero standing for the unity matrix). We may always normalize TrBiBj= 2δijin an analogy to the Pauli matrices. The operators Biare, then, forming a finite-dimensional space for the recursion of Eq. (A8). The vertex-corrected operators Bvc iare obtained by summing up the matrix geometric series Bvc i=X j1 1− F ijBj, (A9)where the entities of the matrix Fare given by Fij=ℏv2 f ετZdp (2π)2Tr GR ε,pBiGA ε+ℏω,pBj .(A10) Our operators of interest σxandσycan always be de- composed in the operator basis as σα=1 2X iBiTr (σαBi), (A11) hence the vertex-corrected spin operator is given by σvc α=1 2X ijBvc iTr(σαBi). (A12) Moreover, the computation of the entire response tensor of Eq. (A4) in the diffusive approximation can also be expressed via the matrix Fas Rαβ=α0ετ 8ℏX ij[TrσαBi]F 1− F ij[TrσβBj],(A13) where α0=A∆2 sd/πℏ2v2 fSis the coefficient used in Eq. (6) to define the unit of the Gilbert damping. It appears that one can always choose the basis of Bioperators such that the computation of Eq. (A13) is closed in a subspace of just three Bioperators with i= 1,2,3. This enables us to make analytical computa- tions of Eq. (A13). Appendix B: Magnetization dynamics The representation of the results can be made some- what simpler by choosing xaxis in the direction of the in-plane projection n∥of the N´ eel vector, hence ny= 0. In this case, one can represent the result as δs+=c1n∥×(n∥×∂tm∥) +c2∂tm∥+c3∂tm⊥+c4n, where ndependence of the coefficients cimay be param- eterized as c1=r11−r22−r31(1−n2 z)/(nxnz) 1−n2z, (B1a) c2=r11−r31(1−n2 z)/(nxnz), (B1b) c3=r33, (B1c) c4= (r31/nz)∂tmz+ζ(∂tm)·n. (B1d) The analytical results in the paper correspond to the evaluation of δs±up to the second order in ∆ sdusing perturbative analysis. Thus, zero approximation corre- sponds to setting ∆ sd= 0 in Eqs. (A1,A5). The equations of motion on nandmare given by Eqs. (2), ∂tn=−Jn×m+n×δs++m×δs−, (B2a) ∂tm=m×δs++n×δs−, (B2b)10 It is easy to see that the following transformation leaves the above equations invariant, δs+→δs+−ξn, δ s−→δs−−ξm, (B3) for an arbitrary value of ξ. Such a gauge transformation can be used to prove that the coefficient c4is irrelevant in Eqs. (B2). In this paper, we compute δs±to the zeroth order in \|m\|– the approximation which is justified by the sub- lattice symmetry in the anti-ferromagnet. A somewhat more general model has been analyzed also in Ref. 25 to which we refer the interested reader for more technical details. Appendix C: Anisotropy constant The anisotropy constant is obtained from the grand po- tential energy Ω for conducting electrons. For the model of Eq. (A1) the latter can be expressed as Ω =−X ς=±1 βZ dε g(ε)νς(ε), (C1) where β= 1/kBTis the inverse temperature, ς=±is the valley index (for the valleys KandK′),GR ς,pis the bare retarded Green function with momentum pand in the valley ς. We have also defined the function g(ε) = ln (1 + exp[ β(µ−ε)]), (C2) where µis the electron potential, and the electron density of states in each of the valleys is given by, νς(ε) =1 πZdp (2πℏ)2Im Tr GR ς,p, (C3) where the trace is taken only over spin and sub-lattice space, In the metal regime considered, the chemical potential is assumed to be placed in the upper electronic band. In this case, the energy integration can be taken only for positive energies. The two valence bands are always filled and can only add a constant shift to the grand potential Ω that we disregard. The evaluation of Eq. (C1) yields the following density of states ντ(ε) =1 2πℏ2v2 f 0 0 < ε < ε 2 ε/2 +λ/4ε2< ε < ε 1, ε ε > ε 1,(C4)where the energies ε1,2correspond to the extremum points (zero velocity) for the electronic bands. These energies, for each of the valleys, are given by ε1,ς=1 2 +λ+p 4∆2+λ2−4ς∆λnz , (C5a) ε2,ς=1 2 −λ+p 4∆2+λ2+ 4ς∆λnz (C5b) where ς=±is the valley index. In the limit of zero temperature we can approximate Eq. (C1) as Ω =−X ς=±1 βZ∞ 0dε(µ−ε)νς(ε). (C6) Then, with the help of Eq. (C1) we find, Ω =−1 24πℏ2v2 fX ς=± (ε1,ς−µ)2(4ε1,ς−3λ+ 2µ) +(ε2,ς−µ)2(4ε2,ς+ 3λ+ 2µ) . (C7) By substituting the results of Eqs. (C5) into the above equation we obtain Ω =−1 24πℏ2v2 fh (4∆2−4nz∆λ+λ2)2/3 +(4∆2+ 4nz∆λ+λ2)2/3−24∆µ+ 8µ3i .(C8) A careful analysis shows that the minimal energy cor- responds to nz=±1 so that the conducting electrons prefer an easy-axis magnetic anisotropy. By expanding in powers of n2 zaround nz=±1 we obtain Ω = −Kn2 z/2, where K=1 2πℏ2v2( \|∆2λ\| \| λ/2∆\| ≥1, \|∆λ2\|/2\|λ/2∆\| ≤1.(C9) This provides us with the easy axis anisotropy of Eq. (16). 1S. A. Siddiqui, J. Sklenar, K. Kang, M. J. Gilbert, A. Schleife, N. Mason, and A. Hoff- mann, Journal of Applied Physics 128, 040904 (2020), https://pubs.aip.org/aip/jap/article-pdf/doi/10.1063/5.0009445/15249168/040904 1online.pdf. 2V. V. Mazurenko, Y. O. Kvashnin, A. I. Lichtenstein, and M. I. Katsnelson, Journal of Experimental and Theoretical Physics 132, 506 (2021).11 3L.ˇSmejkal, J. Sinova, and T. Jungwirth, Phys. Rev. X 12, 040501 (2022). 4B. A. Bernevig, C. Felser, and H. Beidenkopf, Nature 603, 41 (2022). 5T. G. H. Blank, K. A. Grishunin, B. A. Ivanov, E. A. Mashkovich, D. Afanasiev, and A. V. Kimel, Phys. Rev. Lett. 131, 096701 (2023). 6W. Wu, C. Yaw Ameyaw, M. F. Doty, and M. B. Jungfleisch, Journal of Applied Physics 130, 091101 (2021), https://pubs.aip.org/aip/jap/article- pdf/doi/10.1063/5.0057536/13478272/091101 1online.pdf. 7M. Gibertini, M. Koperski, A. F. Morpurgo, and K. S. Novoselov, Nature Nanotechnology 14, 408 (2019). 8K. M. D. Hals, Y. Tserkovnyak, and A. Brataas, Physical Review Letters 106, 107206 (2011), publisher: American Physical Society. 9R. Cheng, D. Xiao, and A. Brataas, Physical Review Let- ters116, 207603 (2016), publisher: American Physical So- ciety. 10S. Urazhdin and N. Anthony, Physical Review Letters 99, 046602 (2007), publisher: American Physical Society. 11R. Cheng, M. W. Daniels, J.-G. Zhu, and D. Xiao, Phys- ical Review B 91, 064423 (2015), publisher: American Physical Society. 12R. Khymyn, I. Lisenkov, V. Tiberkevich, B. A. Ivanov, and A. Slavin, Scientific Reports 7, 43705 (2017), number: 1 Publisher: Nature Publishing Group. 13R. Cheng, J. Xiao, Q. Niu, and A. Brataas, Physical Review Letters 113, 057601 (2014), publisher: American Physical Society. 14A. Mougin, M. Cormier, J. P. Adam, P. J. Metaxas, and J. Ferr´ e, Europhysics Letters (EPL) 78, 57007 (2007), pub- lisher: IOP Publishing. 15A. A. Thiele, Physical Review Letters 30, 230 (1973), pub- lisher: American Physical Society. 16R. Weber, D.-S. Han, I. Boventer, S. Jaiswal, R. Lebrun, G. Jakob, and M. Kl¨ aui, Journal of Physics D: Applied Physics 52, 325001 (2019), publisher: IOP Publishing. 17V. Y. Irkhin, A. A. Katanin, and M. I. Katsnelson, Phys. Rev. B 60, 1082 (1999). 18D. V. Spirin, Journal of Magnetism and Magnetic Materi- als264, 121 (2003). 19J. Kosterlitz, Journal of Physics C: Solid State Physics 7, 1046 (1974). 20J. M. Kosterlitz and D. J. Thouless, in Basic Notions Of Condensed Matter Physics (CRC Press, 2018) pp. 493–515. 21N. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 (1966). 22B. L. Chittari, D. Lee, N. Banerjee, A. H. MacDonald, E. Hwang, and J. Jung, Phys. Rev. B 101, 085415 (2020). 23P. H¨ ogl, T. Frank, K. Zollner, D. Kochan, M. Gmitra, and J. Fabian, Phys. Rev. Lett. 124, 136403 (2020).24K. Dolui, M. D. Petrovi´ c, K. Zollner, P. Plech´ aˇ c, J. Fabian, and B. K. Nikoli´ c, Nano Lett. 20, 2288 (2020). 25M. Baglai, R. J. Sokolewicz, A. Pervishko, M. I. Katsnel- son, O. Eriksson, D. Yudin, and M. Titov, Physical Re- view B 101, 104403 (2020), publisher: American Physical Society. 26M. F¨ ahnle and D. Steiauf, Physical Review B 73, 184427 (2006). 27H. T. Simensen, A. Kamra, R. E. Troncoso, and A. Brataas, Physical Review B 101, 020403 (2020), pub- lisher: American Physical Society. 28A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Physical Review Letters 101, 037207 (2008), publisher: American Physical Society. 29H. Ebert, S. Mankovsky, D. K¨ odderitzsch, and P. J. Kelly, Physical Review Letters 107, 066603 (2011), publisher: American Physical Society. 30Q. Liu, H. Y. Yuan, K. Xia, and Z. Yuan, Physical Review Materials 1, 061401 (2017), publisher: American Physical Society. 31M. Dyakonov and V. Perel, Sov. Phys. Solid State, Ussr 13, 3023 (1972). 32M. Dyakonov and V. Kachorovskij, Fizika i tehnika poluprovodnikov 20, 178 (1986). 33R. Elliott, Phys. Rev. 96, 266 (1954). 34Y. Yafet, in Solid State Physics , Vol. 14, edited by F. Seitz and D. Turnbull (Elsevier, 1963) pp. 1–98. 35M. Dyakonov, arXiv:cond-mat/0401369 (2004). 36M. I. Dyakonov, ed., Spin Physics in Semiconductors , 2nd ed., Springer Series in Solid-State Sciences (Springer Inter- national Publishing, 2017). 37N. Averkiev, L. Golub, and M. Willander, Semiconductors 36, 91 (2002). 38A. Burkov, A. S. N´ u˜ nez, and A. MacDonald, Phys. Rev. B70, 155308 (2004). 39A. A. Burkov and L. Balents, Physical Review B 69, 245312 (2004), publisher: American Physical Society. 40N. A. Sinitsyn and Y. V. Pershin, Reports on Progress in Physics 79, 106501 (2016), publisher: IOP Publishing. 41A. Kamra, R. E. Troncoso, W. Belzig, and A. Brataas, Phys. Rev. B 98, 184402 (2018). 42C. Kittel, Phys. Rev. 82, 565 (1951). 43F. Keffer and C. Kittel, Phys. Rev. 85, 329 (1952). 44J. Rammer and H. Smith, Rev. Mod. Phys. 58, 323 (1986). 45I. Ado, O. A. Tretiakov, and M. Titov, Phys. Rev. B 95, 094401 (2017). 46J. Rammer, Quantum Transport Theory (CRC Press, New York, 2018). 47G. D. Mahan, Many-particle physics (Springer Science & Business Media, 2013). 48I. Ado, I. Dmitriev, P. Ostrovsky, and M. Titov, EPL 111, 37004 (2015).
1705.09659v1.Role_of_the_spin_orbit_coupling_in_the_Kugel_Khomskii_model_on_the_honeycomb_lattice.pdf	arXiv:1705.09659v1 [cond-mat.str-el] 26 May 2017Role of the spin-orbit coupling in the Kugel-Khomskii model on the honeycomb lattice Akihisa Koga, Shiryu Nakauchi, and Joji Nasu Department of Physics, Tokyo Institute of Technology, Megu ro, Tokyo 152-8551, Japan (Dated: May 30, 2017) We study the eﬀective spin-orbital model for honeycomb-lay ered transition metal compounds, ap- plyingthe second-order perturbation theory tothethree-o rbital Hubbardmodel with the anisotropic hoppings. This model is reduced to the Kitaev model in the str ong spin-orbit coupling limit. Com- bining the cluster mean-ﬁeld approximations with the exact diagonalization, we treat the Kugel- Khomskii type superexchange interaction and spin-orbit co upling on an equal footing to discuss ground-state properties. We ﬁnd that a zigzag ordered state is realized in the model within nearest- neighbor interactions. We clarify how the ordered state com petes with the nonmagnetic state, which is adiabatically connected to the quantum spin liquid state realized in a strong spin-orbit coupling limit. Thermodynamic properties are also addressed. The pr esent work should provide another route to account for the Kitaev-based magnetic properties i n candidate materials. Orbital degrees of freedom have been studied as a cen- tral topic of strongly correlated electron systems as they possess own quantum dynamics and are strongly entan- gled with other degrees of freedom such as charge and spin [1]. Recently, multiorbital systems with strong spin- orbit (SO) couplings have attracted considerable atten- tion [2, 3]. One of the intriguing examples is the series of the Mott insulators with honeycomb-based structures such asA2IrO3(A= Na,Li) [4–6], and β-Li2IrO3[7]. In these compounds, a strong SO coupling for 5 delectrons lifts the triply degenerate t2glevels and the low-energy Kramersdoublet, which is referred to as an isospin, plays an important role at low temperatures. Furthermore, anisotropic electronic clouds intrinsic in the t2gorbitals result in peculiar exchange couplings and the system is well described by the Kitaev model for the isospins [8, 9]. The ground state of this model is a quantum spin liquid (QSL), and hence a lot of experimental and theoretical works have been devoted to the iridium oxides in this context [10–18]. Very recently, the ruthenium compound α-RuCl 3with 4delectrons has been studied actively as another Kitaev candidate material [19–27]. In general, the SO coupling in 4 dorbitals is weaker than that in 5dorbitals and is comparable with the exchange energy. Therefore, it is highly desired to deal with SO and ex- change couplings on an equal footing although the mag- netic properties for honeycomb-layered compounds have been mainly discussed within the isospin model with the Kitaev and other exchange couplings including longer- range interactions [10, 28–32]. In this Letter, we study the role of the SO cou- pling in the Mott insulator with orbital degrees of free- dom. We examine the localized spin-orbital model with the Kugel-Khomskii type superexchange interactions be- tween nearest-neighbor sites and onsite SO couplings on the two-dimensional honeycomb lattice. In the strong SO coupling limit, this model is reduced to the Kitaev model and the QSL state is realized. On the other hand, a conventional spin-orbital ordered state may be stabi- lized in the small SO coupling case. To examine thecompetition between the magnetically disordered and or- dered states in the intermediate SO coupling region, we ﬁrst use the cluster mean-ﬁeld (CMF) theory [33] with theexactdiagonalization(ED).Wedeterminetheground statephasediagraminthemodelandclarifythatazigzag magneticallyorderedstate is realizeddue to the competi- tion between distinct exchanges. Calculating the speciﬁc heat and entropy in terms of the thermal pure quantum (TPQ) state [34], we discuss how thermodynamic prop- erties characteristic of the Kitaev model appear in the intermediate SO coupling region. We start with the three-orbital Hubbard model on the honeycomb lattice. This should be appropriate to de- scribe the electronic state of the t2gorbitals in the com- poundsA2IrO3andα-RuCl 3since there exists a large crystalline electric ﬁeld for the dorbitals. The transfer integraltbetweenthe t2gorbitalsvialigand porbitalsare evaluated from the Slater-Koster parameters, where the neighboringoctahedra consisting of six ligands surround- ing transition metal ions share their edges. Note that the transfer integrals involving one of the three t2gorbitals vanish due tothe anisotropicelectronicclouds[9]. We re- fer to this as an inactive orbital and the other orbitals as active ones. These depend on three inequivalent bonds, which are schematically shown as the distinct colored lines in Fig. 1. Moreover, we consider the onsite intra- and inter-orbital Coulomb interactions, UandU′, Hund coupling K, and pair hopping K′in the conventional manner. In the following, we restrict our discussions to the conditions U=U′+2KandK′=K, which are lead by the symmetry argument of the degenerate orbitals. We use the second-order perturbation theory in the strong coupling limit since the Mott insulating state is realized in the honeycomb-layered compounds. We then obtainthe Kugel-Khomskii-typeexchangemodel, assum- ing that ﬁve electrons occupy the t2gorbitals in each site. By taking the SO coupling into account, the eﬀective Hamiltonian is explicitly given as H=/summationdisplay /angbracketleftij/angbracketrightγHex(γ) ij−λ/summationdisplay iLi·Si, (1)2 whereλis the SO coupling, and SiandLiare spin and orbital angular-momentum operators at the ith site, re- spectively. The exchange Hamiltonian Hex(γ) ij, which de- pends on the bond γ(=x,y,z) of the honeycomb lattice(see Fig. 1), is given as Hex(γ) ij=H(γ) 1;ij+H(γ) 2;ij+H(γ) 2;ij, (2) with H(γ) 1;ij= 2J1/parenleftbigg Si·Sj+3 4/parenrightbigg/bracketleftbigg τ(γ) ixτ(γ) jx−τ(γ) iyτ(γ) jy−τ(γ) izτ(γ) jz+1 4τ(γ) i0τ(γ) j0−1 4/parenleftBig τ(γ) i0+τ(γ) j0/parenrightBig/bracketrightbigg , (3) H(γ) 2;ij= 2J2/parenleftbigg Si·Sj−1 4/parenrightbigg/bracketleftbigg τ(γ) ixτ(γ) jx−τ(γ) iyτ(γ) jy−τ(γ) izτ(γ) jz+1 4τ(γ) i0τ(γ) j0+1 4/parenleftBig τ(γ) i0+τ(γ) j0/parenrightBig/bracketrightbigg , (4) H(γ) 3;ij=−4 3(J2−J3)/parenleftbigg Si·Sj−1 4/parenrightbigg/bracketleftbigg τ(γ) ixτ(γ) jx+τ(γ) iyτ(γ) jy−τ(γ) izτ(γ) jz+1 4τ(γ) i0τ(γ) j0/bracketrightbigg , (5) (a) (b) FIG. 1. Honeycomb lattice. (a) Eﬀective cluster model with ten sites, which are treated in the framework of the CMF method. (b) Twelve-site cluster for the TPQ states. where we follow the notation of Ref. [35], and J1= 2t2/U[1−3K/U]−1,J2= 2t2/U[1−K/U]−1,J3= 2t2/U[1+2K/U]−1are the exchange couplings between nearest neighbor spins. Here, we have newly introduced the orbital pseudospin operators τ(γ) lwithl=x,y,z,0. Note that its deﬁnition depends on the direction of the bond (γ-bond) between the nearest neighbor pair /angbracketleftij/angbracketright. τ(γ) lis represented by the 3 ×3 matrix based on the three orbitals: the 2 ×2 submatrix on the two active orbitals is given by σl/2 forl=x,y,zand the identity matrix for l= 0, and the other components for one in- active orbital are zero, where σlis the Pauli matrix. We here note that Hamiltonian H1enhances ferromagnetic correlations, while H2andH3lead to antiferromagnetic correlations. Therefore, spin frustration should play an important role for the ground state in the small K/U region, where J1∼J2∼J3. What is the most distinct from ordinary spin-orbital modelsisthatthe presentsystemdescribesnotonlyspin- orbital orders but also the QSL state realized in the Ki- taev model. When the SO coupling is absent, the system is reduced to the standard Kugel-Khomskii type Hamil-tonian. In the large Hund coupling case, the Hamilto- nianH(γ) 1;ijis dominant. Then, the ferromagnetically or- dered ground state should be realized despite the pres- ence of orbital frustration. In the smaller case of the Hund coupling, the ground state is not trivial due to the existence of spin frustration, discussed above. On the other hand, in the case λ→ ∞, the SO coupling lifts the degeneracy at each site and the lowest Kramers doublet, \|˜σ/angbracketright= (\|xy,σ/angbracketright ∓ \|yz,¯σ/angbracketright+i\|zx,¯σ/angbracketright)/√ 3, plays a crucial role for low temperature properties. Then, the model Hamiltonian Eq. (1) is reduced to the exactly solvable Kitaev model with the spin-1/2 isospin opera- tor˜S, asHeﬀ=−˜J/summationtext /angbracketleftij/angbracketrightγ˜Siγ˜Sjγ(γ=x,y,z), where ˜J[= 2(J1−J2)/3] is the eﬀective exchange coupling [8]. It is known that, in this eﬀective spin model, the QSL ground state is realized with the spin gap. At ﬁnite temperatures, a fermionic fractionalization appears to- gether with double peaks in the speciﬁc heat [15, 16]. In the following, we set the exchange coupling J1as a unit of energy. We then study ground-state and ﬁnite- temperature properties in the spin-orbital system with parameters K/Uandλ/J1. First, we discuss ground state properties in the spin- orbital model by means of the CMF method [33]. In the method, the original lattice model is mapped to an eﬀective cluster model, where spin and orbital corre- lations in the cluster can be taken into account prop- erly. Intercluster correlations are treated through sev- eral mean-ﬁelds at ith site,/angbracketleftSik/angbracketright,/angbracketleftτ(γ) il/angbracketrightand/angbracketleftSikτ(γ) il/angbracketright, wherek=x,y,zandl=x,y,z,0. These mean-ﬁelds aredetermined via the self-consistent conditions imposed on the eﬀective cluster problem. The method is compa- rable with the numerically exact methods if the cluster size is large, and has successfully been applied to quan- tum spin [33, 36–38] and hard-core bosonic systems [39– 41]. To describe some possible ordered states such as the zigzag and stripy states [10], we introduce two kinds of clustersinthehoneycomblattice, whichareshownasdis- tinct colors in Fig. 1(a). Using the ED method, we self-3 0.4 0.42 0.44 0.46 0.48 0.5 0.1 0.15 0.2 0.25 0.3mS K/U-1.43-1.42-1.41-1.4-1.39 0.1 0.2 0.3Eg/J1Nλ/J1=0.0 λ/J1=0.2 FIG.2. ThespinmomentsasafunctionoftheHundcoupling K/U. Solidandopencircles (squares)representtheresultsfor the ferromagnetically and zigzag ordered states in the syst em withλ/J1= 0.0 (0.2). The ground state energy is shown in the inset. consistently solve two eﬀective cluster problems. To dis- cuss magnetic properties at zero temperature, we calcu- late spin and orbital moments, mα S=\|/summationtext i(−1)δα i/angbracketleftSi/angbracketright\|/N andmα L=\|/summationtext i(−1)δα i/angbracketleftLi/angbracketright\|/N, whereNis the number of sites and δα iis the phase factor for an ordered state α. Whenλ= 0, the spin and orbital degrees of freedom are decoupled. Here, we show in Fig. 2 the spin mo- mentsmf Sandmz Sfor the ferromagnetically and zigzag orderedstates, respectively, whichareobtainedby means of the ten-site CMF method (CMF-10). Namely, we have conﬁrmedthatotherorderedstatessuchasantiferromag- netic and stripy states are never stabilized in the present calculations, and thereby we do not show them in Fig. 2. Meanwhile, the local orbital moment disappears in the caseλ= 0. In the system with the large Hund coupling, the exchange coupling J1is dominant, and the ferromag- netically ordered ground state is realized with the fully- polarized moment mf S= 0.5, as shown in Fig. 2. On the other hand, in the smaller Kregion, the exchange cou- plingsJ2andJ3are comparable with J1. SinceH2and H3should enhance antiferromagnetic correlations, the ferromagnetically ordered state becomes unstable. We ﬁnd that a zigzag magnetically ordered state is realized with ﬁnite mz SaroundK/U∼0.12. To study the compe- tition between these ordered states, we show the ground state energies in the inset of Fig. 2. We clearly ﬁnd the hysteresis in the curves, which indicates the existence of the ﬁrst-order phase transition. By examining the cross- ing point, we clarify that the quantum phase transition between ferromagnetically and zigzag ordered states oc- curs atK/U∼0.15. In the case with K/U <0.1, due to strong frustration, it is hard to obtain the converged solutions. This will be interesting to clarify this point in a future investigation. 0 0.5 1 1.5 0 0.5 1(a) K/U=0.12 mz µ mz Lmz Smf µ mf L mf Sm λ/J1 0 0.5 1 1.5 2(b) K/U=0.3 mf µ mf L mf S λ/J1 FIG. 3. Total magnetic moment mµ, spin moment mS, and orbital moment mLin the spin-orbital systems with (a) K/U= 0.12 and (b) K/U= 0.3. The introduction of λcouples the spin and orbital de- grees of freedom. The spin moments slightly decrease in both states, as shown in Fig. 2. The zigzag and ferro- magnetically ordered states are stable against the small SO coupling and the ﬁrst-order transition point has little eﬀectontheSOcoupling. Todiscussthestabilityofthese states against the strong SO coupling, we calculate the spinand orbitalmomentsin the systemwith K/U= 0.12 and 0.3, as shown in Fig. 3. The introduction of the SO coupling slightly decreases the spin moment, as dis- cussedabove. Bycontrast,theorbitalmomentisinduced parallel to the spin moment. Therefore, the total mag- netic moment mα µ=\|/summationtext i(−1)δα i/angbracketleft2Si+Li/angbracketright\|/Nincreases. WhenK/U= 0.12, the zigzag ordered state becomes unstable and the ﬁrst-order phase transition occurs to the ferromagnetically ordered state at λ/J1∼0.4. Fur- ther increase of the SO coupling decreases the total mo- mentmf µ. Finally, a jump singularity appears around λ/J1∼0.8(1.8) in the system with K/U= 0.12(0.3). It is also found that the magnetic moment is almost zero and each orbital is equally occupied as in the isospin states\|˜σ/angbracketrightin the larger SO coupling region. Therefore, we believe that this state is essentially the same as the QSL state realized in the Kitaev model. By performing similar calculations, we obtain the ground state phase diagram, as shown in Fig. 4. The disordered (QSL) state is realized in the region with largeλ/J1. The ferromagnetically ordered state is re- alized in the region with small λ/J1and large K/U. The decrease of the Hund coupling induces spin frustration, which destabilizes the ferromagnetically ordered state. We wish to note that the zigzag ordered state is stable in the small SO coupling region, which is not directly taken into account in the Kitaev model. Next, we discuss thermodynamic properties in the sys-4 0 0.5 1 1.5 2 0.1 0.15 0.2 0.25 0.3 Ferro ZigzagDisorder λ/J 1 K/U FIG. 4. The ground state phase diagram of the spin-orbital model. Transition points are obtained by the CMF-10. tem. It is known that, in the Kitaev limit ( λ→ ∞), the excitations are characterized by two energy scales, which correspond to localized and itinerant Majoranafermions. This clearly appears in the speciﬁc heat as two peaks at T/˜J= 0.012 and 0 .38 [16]. To clarify how the double peak structure appears in the intermediate SO coupling region, we make use of the TPQ state for the twelve-site cluster with the periodic boundary condition [see Fig. 1(b)]. According to the previous study [30], the dou- ble peak structure appears in the spin-1/2 Kitaev model even with the twelve-site cluster. Therefore, we believe that thermodynamic properties in the system can be dis- cussed, at least, qualitatively in our calculations. Here, we ﬁx the Hund coupling as K/U= 0.3 to dis- cuss ﬁnite temperature properties in the system with the intermediate SO coupling. Figure 5 shows the speciﬁc heat and entropy in the system with λ/J1= 0,1,2,4 and 10. In this calculation, the quantities are deduced by the statistical average of the results obtained from, at least, twenty independent TPQ states. When λ= 0, we ﬁnd a broad peak around T/J1= 0.4 in the curve of the spe- ciﬁc heat. In addition, most of the entropy is released atT/J1∼0.1, as shown in Fig. 5(b). This can be ex- plained by the fact that ferromagnetic correlations are enhanced and spin degrees of freedom are almost frozen. The appearance of the large residual entropy should be an artifact in the small cluster with the orbital frustra- tion. The introduction of the SO coupling leads to in- teresting behavior. It is clearly found that the broad peak shifts to higher temperatures. This indicates the formation of the Kramers doublet and a part of the en- tropyS= log(6) −log(2) is almost released, as shown in Fig. 5(b). In addition, we ﬁnd in the case λ/J1≥2, two peaksinthespeciﬁcheatatlowertemperatures. Thecor- responding temperatures are little changed by the mag- nitude of the SO coupling and the curves are quantita- tively consistent with the results for the isospin Kitaev model on the twelve sites, which are shown as dashed 0 0.5 1 1.5 0.01 0.1 1 10 100log 2log 6(b) S T/J1 0 0.5 1(a) Cλ/J1=0 λ/J1=1 λ/J1=2 λ/J1=4 λ/J1=10 FIG. 5. The speciﬁc heat (a) and entropy (b) as a function of the temperature for the system with λ/J1= 0,1,2,4, and 10. Dashed lines represent the results for the isospin Kitae v model with twelve sites. lines. Therefore, we believe that the Kitaev physics ap- pears in the region. On the other hand, when λ/J1= 1, a single peak structure appears in the speciﬁc heat, indi- cating that the Kitaev physics is hidden by the formation of the Kramers doublet due to the competition between the exchange interaction and SO coupling. We have used the TPQ states to clarify how the double peak structure inherent in the Kitaev physics appears, in addition to the broad peak for the formation of the Kramers doublet at higher temperatures. To conclude, we have studied the eﬀective spin-orbital model obtained by the second-order perturbation the- ory. Combining the CMF theory with the ED method, we have treated the Kugel-Khomskii type superexchange interaction and SO coupling on an equal footing to de- termine the ground-state phase diagram. We have clar- iﬁed how the magnetically ordered state competes with the nonmagnetic state, which is adiabatically connected to the QSL state realized in a strong SO coupling limit. Particularly, we have revealed that a zigzag orderedstate is realized in this eﬀective spin-orbital model with ﬁnite SO couplings. The present study suggests another mech- anism to stabilize the zigzag ordered phase close to the QSL in the plausible situation, and also will stimulate further experimental studies in the viewpoint of the SO coupling eﬀect on magnetic properties in Kitaev candi- date materials.5 The authors would like to thank S. Suga for valuable discussions. Parts of the numerical calculations are per- formed in the supercomputing systems in ISSP, the Uni- versity of Tokyo. This work was partly supported by the Grant-in-Aid for Scientiﬁc Research from JSPS, KAK- ENHI Grant Number JP17K05536, JP16H01066 (A.K.), and JP16K17747, JP16H00987 (J.N.). [1] Y. Tokura and N. Nagaosa, Science 288, 462 (2000). [2] W. Witczak-Krempa, G. Chen, Y. Baek Kim, and L. Balents, Ann. Rev. Condens. Matter Phys. 5, 57 (2014). [3] H. Watanabe, T. Shirakawa, and S. Yunoki, Phys. Rev. Lett.105, 216410 (2010). [4] Y. Singh and P. Gegenwart, Phys. Rev. B 82, 064412 (2010). [5] Y. Singh, S. Manni, J. Reuther, T. Berlijn, R. Thomale, W. Ku, S. Trebst, and P. Gegenwart, Phys. Rev. Lett. 108, 127203 (2012). [6] R. Comin, G. Levy, B. Ludbrook, Z.-H. Zhu, C. N. Veen- stra, J. A. Rosen, Y. Singh, P. Gegenwart, D. Stricker, J. N. Hancock, D. van der Marel, I. S. Elﬁmov, and A. Damascelli, Phys. Rev. Lett. 109, 266406 (2012). [7] T. Takayama, A. Kato, R. Dinnebier, J. Nuss, H. Kono, L. S. I. Veiga, G. Fabbris, D. Haskel, and H. Takagi, Phys. Rev. Lett. 114, 077202 (2015). [8] A. Kitaev, Ann. Phys. 321, 2 (2006). [9] G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 102, 017205 (2009). [10] J. Chaloupka, G. Jackeli, and G. Khaliullin, Phys. Rev. Lett.110, 097204 (2013). [11] V. M. Katukuri, S. Nishimoto, V. Yushankhai, A. Stoy- anova, H.Kandpal, S.Choi, R.Coldea, IRousochatzakis, L. Hozoi, and J. van den Brink, New J. Phys. 16, 013056 (2014). [12] J. Knolle, D. L. Kovrizhin, J. T. Chalker, and R. Moess- ner, Phys. Rev. Lett. 112, 207203 (2014). [13] Y. Yamaji, Y. Nomura, M. Kurita, R. Arita, and M. Imada, Phys. Rev. Lett. 113, 107201 (2014). [14] J. Knolle, G.-W. Chern, D. L. Kovrizhin, R. Moessner, and N. B. Perkins, Phys. Rev. Lett. 113, 187201 (2014). [15] J. Nasu, M. Udagawa, and Y. Motome, Phys. Rev. Lett. 113, 197205 (2014). [16] J. Nasu, M. Udagawa, and Y. Motome, Phys. Rev. B 92, 115122 (2015). [17] T. Suzuki, T. Yamada, Y. Yamaji, and S. I. Suga, Phys. Rev. B92, 184411 (2015). [18] J. Yoshitake, J. Nasu, and Y. Motome, Phys. Rev. Lett. 117, 157203 (2016). [19] K. W. Plumb, J. P. Clancy, L. J. Sandilands, V. V. Shankar, Y. F. Hu, K. S. Burch, H.-Y. Kee, and Y.-J. Kim, Phys. Rev. B 90, 041112 (2014).[20] Y. Kubota, H. Tanaka, T. Ono, Y. Narumi, and K. Kindo, Phys. Rev. B 91, 094422 (2015). [21] J. A. Sears, M. Songvilay, K. W. Plumb, J. P. Clancy, Y. Qiu, Y. Zhao, D. Parshall, and Y.-J. Kim, Phys. Rev. B 91, 144420 (2015). [22] M. Majumder, M. Schmidt, H. Rosner, A. A. Tsirlin, H. Yasuoka, and M. Baenitz, Phys. Rev. B 91, 180401 (2015). [23] L. J. Sandilands, Y. Tian, K. W. Plumb, Y.-J. Kim, and K. S. Burch, Phys. Rev. Lett. 114, 147201 (2015). [24] R. D. Johnson, S. C. Williams, A. A. Haghighirad, J. Singleton, V. Zapf, P. Manuel, I. I. Mazin, Y. Li, H. O. Jeschke, R. Valent´ ı, and R. Coldea, Phys. Rev. B 92, 235119 (2015). [25] J. Nasu, J. Knolle, D. L. Kovrizhin, Y. Motome, and R. Moessner, Nat. Phys. 12, 912 (2016). [26] A. Koitzsch, C. Habenicht, E. M¨ uller, M. Knupfer, B. B¨ uchner, H. C. Kandpal, J. van den Brink, D. Nowak, A. Isaeva, and T. Doert, Phys. Rev. Lett. 117, 126403 (2016). [27] L. J. Sandilands, Y. Tian, A. A. Reijnders, H.-S. Kim, K. W. Plumb, Y.-J. Kim, H.-Y. Kee, and K. S. Burch, Phys. Rev. B 93, 075144 (2016). [28] H.-S. Kim, Vilay Shankar V., A. Catuneanu, and H.-Y. Kee, Phys. Rev. B 91, 241110 (2015). [29] A. Banerjee, C. A. Bridges, J.-Q. Yan, A. A. Aczel, L. Li, M. B. Stone, G. E. Granroth, M. D. Lumsden, Y. Yiu, J. Knolle, S. Bhattacharjee, D. L. Kovrizhin, R. Moessner, D. A. Tennant, D. G. Mandrus, and S. E. Nagler, Nat. Mater.15, 733 (2016). [30] Y. Yamaji, T. Suzuki, T. Yamada, S. I. Suga, N. Kawashima, and M. Imada, Phys. Rev. B 93, 174425 (2016). [31] S.-H. Do, S.-Y. Park, J. Yoshitake, J. Nasu, Y. Motome, Y. S. Kwon, D. T. Adroja, D. J. Voneshen, K. Kim, T.-H. Jang, J.-H. Park, K.-Y. Choi, and S. Ji, preprint arXiv:1703.01081. [32] S. M. Winter, K. Riedl, A. Honecker, and R. Valenti, preprint arXiv:1702.08466. [33] T. Oguchi, Prog. Theor. Phys. 13, 148 (1955). [34] S. Sugiura and A. Shimizu, Phys. Rev. Lett. 108, 240401 (2012); Phys. Rev. Lett. 111, 010401 (2013). [35] G. Khaliullin, Prog. Theor. Phys. 160155 (2005). [36] T. Sakai and M. Takahashi, J. Phys. Soc. Jpn. 58, 3131 (1989). [37] A. Kawaguchi, A. Koga, K. Okunishi, and N. Kawakami, Phys. Rev. B 65, 214405 (2002). [38] D. Yamamoto, Phys. Rev. B 79, 144427 (2009). [39] S. R. Hassan, L. de Medici, and A.-M. S. Tremblay, Phys. Rev. B76, 144420 (2007). [40] D. Yamamoto, I. Danshita, and C. A. R. S´ a de Melo, Phys. Rev. A 85, 021601 (2012). [41] R. Suzuki and A. Koga, JPS Conf. Proc. 3, 016005 (1014); J. Phys. Soc. Jpn. 83, 064003 (2014).
1803.05158v2.Inter_orbital_topological_superconductivity_in_spin_orbit_coupled_superconductors_with_inversion_symmetry_breaking.pdf	arXiv:1803.05158v2 [cond-mat.supr-con] 31 May 2018Inter-orbital topological superconductivity in spin-orb it coupled superconductors with inversion symmetry breaking Yuri Fukaya,1Shun Tamura,1Keiji Yada,1Yukio Tanaka,1Paola Gentile,2and Mario Cuoco2 1Department of Applied Physics, Nagoya University, Nagoya 4 64-8603, Japan 2SPIN-CNR, I-84084 Fisciano (Salerno), Italy and Dipartime nto di Fisica ”E. R. Caianiello”, Universitá di Salerno, I-84084 Fisciano (Salerno), Italy We study the superconducting state of multi-orbital spin-o rbit coupled systems in the presence of an orbitally driven inversion asymmetry assuming that the i nter-orbital attraction is the dominant pairing channel. Although the inversion symmetry is absent , we show that superconducting states that avoid mixing of spin-triplet and spin-singlet conﬁgur ations are allowed, and remarkably, spin- triplet states that are topologically nontrivial can be sta bilized in a large portion of the phase diagram. The orbital-dependent spin-triplet pairing gene rally leads to topological superconductivity with point nodes that are protected by a nonvanishing windin g number. We demonstrate that the disclosed topological phase can exhibit Lifshitz-type tra nsitions upon diﬀerent driving mechanisms and interactions, e.g., by tuning the strength of the atomic spin-orbit and inversion asymmetry couplings or by varying the doping and the amplitude of order parameter. Such distinctive signatures of the nodal phase manifest through an extraordinary recons truction of the low-energy excitation spectra both in the bulk and at the edge of the superconductor . I. INTRODUCTION Spin-triplet pairing is at the core of intense investiga- tion especially because of its foundational aspect in un- conventional superconductivity1–4and owing to its tight connection with the occurrence of topological phases with zero-energy surface Andreev bound states5–10marked by Majorana edge modes11–19. Some of the fundamental essences of topological spin-triplet superconductivity a re basically captured by the Kitaev model20and its gener- alized versions where non-Abelian states of matter and their employment for topological quantum computation can be demonstrated20–24. Another remarkable element of odd-parity superconductivity is given by the poten- tial of having active spin degrees of freedom making such states of matter also appealing for superconduct- ing spintronics applications based on spin control and coherent spin manipulation of Cooper pairs25–31. The interplay of magnetism and spin-triplet superconductiv- ity can manifest within diﬀerent unconventional physical scenarios, such as the case of the emergent spin-orbital interaction between the superconducting order parame- ter and interface magnetization32,33, the breakdown of the bulk-boundary correspondence34, and the anomalous magnetic35,36and spin-charge current37eﬀects occurring in the proximity between chiral or helical p-wave and spin-singlet superconductors. Achieving spin-triplet ma - terials platforms, thus, sets the stage for the development of emergent technologies both in nondissipative spintron- ics and in the expanding area of quantum devices. Although embracing strong promises, spin-triplet su- perconductivity is quite rare in nature and the mecha- nisms for electron pairs gluing are not completely set- tled. The search for spin-triplet superconductivity has been performed along diﬀerent routes. For instance, sci- entiﬁc exploration has been focused on the regions of the materials phase diagram that are in proximity to ferro-magnetic quantum phase transitions38,39, as in the case of heavy fermions superconductivity, i.e., UGe2, URhGe, andUIr2, or in materials on the verge of a magnetic in- stability, e.g., ruthenates2,40. Another remarkable route to achieve spin-triplet pair- ing relies on the presence of a source of inversion symme- try breaking, both at the surface/interface and in the bulk, or alternatively, in connection with noncollinear magnetic ordering41–51. Paradigmatic examples along these directions are provided by quasi-one-dimensional heterostructures whose interplay of inversion and time- reversal symmetry breaking or noncollinear magnetism have been shown to convert spin-singlet pairs into spin- triplet ones and in turn to topological phases48,52–55. Similar mechanisms and physical scenarios are also en- countered at the interface between spin-singlet super- conductors and inhomogeneous ferromagnets with even and odd-in time spin-triplet pairing that are generally generated25. Semimetals have also been indicated as fun- damental building blocks to generate spin-triplet pairing as theoretically proposed and demonstrated in topologi- cal insulators interfaced with conventional superconduc- tors or by doping Dirac/Weyl phases56, e.g., in the case of Cu-doped Bi2Se357–63in anti-perovskites materials64, as well as Cd 3As265,66. Generally, there are two fundamental interactions to take into account in inversion asymmetric microscopic environments: i) the Rashba spin-orbit coupling67due to inversion symmetry breaking at the surface or inter- face in heterostructures, and ii) the Dresselhaus coupling arising from the inversion asymmetry in the bulk of the host material68. For the present analysis, it is worth noting that typically in multi-orbital materials, it is the combination of the atomic spin-orbit interaction with the inversion symmetry-breaking sources that eﬀectively gen- erates both Rashba and Dresselhaus emergent interac- tions within the electronic manifold close to the Fermi level. Another general observation is that the lack of in-2 version symmetry is expected to lead to a parity mixing of spin-singlet and spin-triplet conﬁgurations69–71with an ensuing series of unexpected features ranging from anomalous magneto-electric72eﬀects to unconventional surface states73, topological phases74–76, and non-trivial spatial textures of the spin-triplet pairs77. Such symme- try conditions in intrinsic materials are, however, funda- mentally linked to the momentum dependent structure of the superconducting order parameter. In contrast, when considering multi-orbital systems, more channels are possible with emergent unconventional paths for elec- tron pairing that are expected to be strongly tied to the orbital character of the electron-electron attraction and of the electronic states close to the Fermi level. Orbital degrees of freedom are key players in quantum materials when considering the degeneracy of d-bands of the transition elements not being completely removed by the crystal distortions or due to the intrinsic spin-orbita l entanglement78triggered by the atomic spin-orbit cou- pling. In this context, a competition of diﬀerent and complex types of order is ubiquitous in realistic mate- rials, such as transition metal oxides, mainly owing to the frustrated exchange arising from the active orbital degrees of freedom. Such scenarios are commonly en- countered in materials where the atomic physics plays a signiﬁcant role in setting the character of the electronic structure close to the Fermi level. As the d-orbitals have an anisotropic spatial distribution, the nature of the elec - tronic states is also strongly dependent on the system’s dimensionality. Indeed, two-dimensional (2D) conﬁned electron liquids originating at the interface or surface of materials generally manifest a rich variety of spin-orbita l phenomena79. Along this line, understanding how elec- tron pairing is settled in quantum systems exhibiting a strong interplay between orbital degrees of freedom and inversion symmetry breaking represents a fundamental problem in unconventional superconductivity, and it can be of great relevance for a large class of materials. In this study, we investigate the nature of the super- conducting phase in spin-orbit coupled systems in the absence of inversion symmetry assuming that the inter- orbital attractive channel is dominant and sets the elec- trons pairing. We demonstrate that the underlying in- version symmetry breaking leads to exotic spin-triplet superconductivity. Isotropic spin-triplet pairing conﬁg - urations, without any mixing with spin-singlet, gener- ally occur among the symmetry allowed solutions and are shown to be the ground-state in a large part of the pa- rameters space. We then realize an isotropic spin-triplet superconductor whose orbital character can make it topo- logically non trivial. Remarkably, the topological phase exhibits an unconventional nodal structure with unique tunable features. An exotic ﬁngerprint of the topologi- cal phases is that the number and k-position of nodes can be controlled by doping, orbital polarization, through the competition between spin-orbit coupling and lattice dis- tortions, and temperature (or equivalently, the amplitude of the order parameter).The paper is organized as follows. In Sec. II, we in- troduce the model Hamiltonian and present the classi- ﬁcation of the inter-orbital pairing conﬁgurations with respect to the point-group and time-reversal symmetries. Section III is devoted to an analysis of the stability of the various orbital entangled superconducting states and the energetics of the isotropic superconducting states. Sec- tion IV focuses on the electronic spectra of the energet- ically most favorable phases and the ensuing topological conﬁgurations both in the bulk and at the boundary. Fi- nally, in Sec. V, we provide a discussion of the results and few concluding remarks. II. MODEL AND SYMMETRY CLASSIFICATION OF SUPERCONDUCTING PHASES WITH INTER-ORBITAL PAIRING One of the most common crystal structures of transi- tion metal oxides is the perovskite structure, with tran- sition metal (TM) elements surrounded by oxygen (O) in an octahedral environment. For cubic symmetry, ow- ing to the crystal ﬁeld potential generated by the oxy- gen around the TM, the ﬁvefold orbital degeneracy is removed and dorbitals split into two sectors: t2g, i.e., yz,zx, andxy, andeg, i.e.,x2−y2and3z2−r2. In the present study, the analysis is focused on two-dimensional (2D) electronic systems with broken out-of-plane inver- sion symmetry and having only the t2gorbitals (Fig. 1) close to the Fermi level to set the low energy excitations. For highly symmetric TM-O bonds, the three t2gbands are directional and basically decoupled, e.g., an electron in thedxyorbital can only hop along the yorxdirection through the intermediate pxorpyorbitals. Similarly, the dyzanddzxbands are quasi-one-dimensional when con- sidering a 2D TM-O bonding network. Furthermore, the atomic spin-orbit interaction (SO) mixes the t2gorbitals thus competing with the quenching of the orbital angu- lar momentum due to the crystal potential. Concerning the inversion asymmetry, we consider microscopic cou- plings that arise from the out-of-plane oxygen displace- ments around the TM. Indeed, by breaking the reﬂection symmetry with respect to the plane placed in between the TM-O bond80, a mixing of orbitals that are even and odd under such a transformation is generated. Such crys- tal distortions are much more relevant and pronounced in 2D electron gas forming at the interface of insulating po- lar and nonpolar oxide materials or on their surface and they result in the activation of an eﬀective hybridization, which is odd in space, of dxyanddyzordzxorbitals along theyorxdirections, respectively. Although the polar environment tends to amplify the out-of-plane oxy- gen displacements with respect to the position of the TM ion, such types of distortions can also occur at the inter- face of nonpolar oxides and in superlattices81. Thus, the model Hamiltonian, including the t2ghop- ping connectivity, the atomic spin-orbit coupling, and the3 inversion symmetry breaking term, reads as H=/summationdisplay kˆC(k)†H(k)ˆC(k), (1) H(k) =H0(k)+HSO(k)+His(k), (2) whereˆC†(k) =/bracketleftBig c† yz↑k,c† zx↑k,c† xy↑k,c† yz↓k,c† zx↓k,c† xy↓k/bracketrightBig is a vector whose components are associated with the elec- tron creation operators for a given spin σ[σ= (↑,↓)], orbitalα[α= (xy,yz,zx )], and momentum kin the Brillouin zone. In Fig. 1(a), we report a schematic illus- FIG. 1. (a) dyz,dzx, anddxy-orbitals with L= 2orbital angu- lar momentum. (b) Schematic image of the orbital dependent hopping amplitudes for εyz,εxy, and the orbital connectivity associated with the inversion asymmetry term ∆is. Here, we do not explicitly indicate the intermediate p-orbitals of the oxygen ions surrounding the transition metal element that e n- ter the eﬀective d−dhopping processes. εzxis obtained from εyzby rotating π/2aroundz-axis.∆iscorresponds to the odd-in-space hopping amplitude from dxytodzxalong they- direction. Similarly, the odd-in-space hopping amplitude from dxytodyzalong thex-direction is obtained by π/2rotation around the z-axis. (c) Sketch of the orbital mixing through the spin-orbit coupling term in the Hamiltonian. σdenotes the spin state, and ¯σis the opposite spin of σ.∆tgives the level splitting between dxy-orbital and dyz/dzx-orbitals. (d) Schematic illustration of inter-orbital interaction. tration of the local orbital basis for the t2gstates.H0(k), HSO(k), andHis(k)indicate the kinetic term, the spin- orbit interaction, and the inversion symmetry breaking term, respectively. In the spin-orbital basis, H0(k)isgiven by H0(k) =−µ/bracketleftBig ˆl0⊗ˆσ0/bracketrightBig + ˆεk⊗ˆσ0, (3) ˆεk= εyz0 0 0εzx0 0 0εxy , εyz= 2t1(1−cosky)+2t3(1−coskx), εzx= 2t1(1−coskx)+2t3(1−cosky), εxy= 4t2−2t2coskx−2t2cosky+∆t, whereˆl0andˆσ0are the unit matrices in orbital and spin space, respectively. Here, µis the chemical potential, and t1,t2, andt3are the orbital dependent hopping ampli- tudes as schematically shown in Fig. 1(b). ∆tdenotes the crystal ﬁeld potential owing to the symmetry lower- ing from cubic to tetragonal symmetry. The symmetry reduction yields a level splitting between dxyorbital and dyz/dzxorbitals.HSO(k)denotes the atomic L·Sspin- orbit coupling, HSO(k) =λSO/bracketleftBig ˆlx⊗ˆσx+ˆly⊗ˆσy+ˆlz⊗ˆσz/bracketrightBig ,(4) withˆσi(i=x,y,z)being the Pauli matrix in spin space. In order to write down the L·Sinteraction, it is conve- nient to introduce the matrices ˆlx,ˆlyandˆlz, which are the projections of the L= 2angular momentum operator onto thet2gsubspace, i.e., ˆlx= 0 0 0 0 0i 0−i0 , (5) ˆly= 0 0−i 0 0 0 i0 0 , (6) ˆlz= 0i0 −i0 0 0 0 0 , (7) assuming {(dyz,dzx,dxy)}as orbital basis. Finally, as mentioned above, the breaking of the mirror plane in be- tween the TM-O bond, due to the oxygen displacements, leads to an inversion symmetry breaking term His(k)of the type His(k) = ∆is/bracketleftBig ˆly⊗ˆσ0sinkx−ˆlx⊗ˆσ0sinky/bracketrightBig .(8) This contribution gives an inter-orbital process, due to the broken inversion symmetry, that mixes dxyanddyz ordzxalongxoryspatial directions [Fig. 1(b)]. Hisre- sembles a Rashba-type Hamiltonian that, however, cou- ples the momentum to the orbital angular momentum rather than the spin. Its origin is due to distortions or other sources of inversion symmetry breaking that lead to local asymmetries deforming the orbital lobes and in turn antisymmetric hopping terms within the orbitals in4 thet2gsector. In this respect, it is worth pointing out that it is the combination of the local spin-orbit cou- pling and the antisymmetric inversion symmetry inter- action that leads to a nontrivial momentum dependent spin-orbital splitting. While the original Rashba eﬀect67 for the single-band system describes a linear spin split- ting and is typically very small, the multi-band char- acter of the model Hamiltonian yields a more complex spin-orbit coupled structure with signiﬁcant splitting82. Indeed, near the Γpoint of the Brillouin zone, one can have a linear spin splitting with respect to the momentum for the lowest energy bands, but a cubiclike splitting in momentum83,84for the intermediate ones with enhanced anomalies when the ﬁlling is close to the transition from two to four Fermi surfaces. The Rashba-like eﬀects due to the combined atomic spin-orbit coupling and the or- bitally driven inversion-symmetry term can be inﬂuenced by the application of an external electric ﬁeld (e.g., via gating) in a dual way. On one hand, the gating directly modiﬁes the ﬁlling concentration and, on the other, it can aﬀect the deformation of the orbital lobes by chang- ing the amplitude of the polar distortion85,86. In this paper, we set t1=t2≡tas a unit of energy for convenience and clarity of presentation. The analysis is performed for a representative set of hopping param- eters, i.e.,t3/t= 0.10and∆t/t=−0.50. The primary reason for the choice of the electronic parameters is that we aim to model superconductivity in transition-metal based layered materials with low electron concentration in thet2gsector at the Fermi level both in the presence of atomic spin-orbit and inversion symmetry breaking cou- plings. In this framework, the set of selected parameters is representative of a general physical regime where the hierarchy of the energy scales is such that ∆t>∆is>λSO and∆t∼t. The choice of this regime is also motivated by the fact that this relation can be generally encoun- tered in 3d(or4d) layered oxides or superlattices in the presence of tetragonal distortions with ﬂat octahedra and interface driven inversion-symmetry breaking potential. For instance, in the case of the two-dimensional elec- tron gas (2DEG) forming at the interface of two band insulators [e.g., the n-type 2DEG in LaAlO 3/SrTiO 387 (LAO/STO)] or the 2DEG at the surface of a band in- sulator [e.g., in SrTiO 3(STO)], the energy scales for the electronic parameters, as given by abinitio80,83,88or spectroscopic studies89, are such that the bare ∆t∼50- 100meV,∆is∼20meV, andλSO∼10meV, while the eﬀective main hopping amplitudes (i.e., t) can be in the 200-300meV range. Similar electronic energies can be also encountered in 4dlayered oxides. Slight variations of these parameters are expected; however, they do not alter the qualitative aspects of the achieved results. We also point out that our analysis is not intended for a spe- ciﬁc material case and that variations in the amplitude of the electronic parameters that keep the indicated hi- erarchy do not alter the qualitative outcome and do not lead to signiﬁcant changes in the results. The electronic structure of the examined model sys-−0.5−0.2500.250.5 Γ Μ XE / t Μ0 ππ −π0 ΓΜ XY kxky 0 ππ −π0 ΓΜ XY kxky 0 ππ −π0 ΓΜ XY kxky(a) (b) (c) (d)(b)(c)(d) FIG. 2. (a) Band structure close to the Fermi energy in the normal state at λSO/t= 0.10and∆is/t= 0.20. (b)-(d) Fermi surfaces at (b) µ/t=−0.25, (c)µ/t= 0.0, and (d)µ/t= 0.35. tem can be accessed by direct diagonalization of the ma- trix Hamiltonian. Representative dispersions for λSO/t= 0.10and∆is/t= 0.20are shown in Fig. 2(a). We ob- serve six non degenerate bands due to the presence of bothHSO(k)andHis(k). Once the dispersions are de- termined, one can immediately notice that the number of Fermi surfaces and the structure can be varied by tun- ing the chemical potential µ. Indeed, for µ/t=−0.25, µ/t= 0.0, andµ/t= 0.35one can single out all the main possible cases with two, four, and six Fermi sheets, as given in Figs. 2(b), (c), and (d), respectively. For the explored regimes of low doping, all the Fermi surfaces are made of electron-like pockets centered around origin of the Brillouin zone ( Γ). The dispersion of the lowest occupied band has weak anisotropy as it has a dominant dxycharacter (Fig. 2(a)); moreover, moving to higher electron concentrations, the outer Fermi sheets exhibit a highly anisotropic proﬁle that becomes more pronounced when the chemical potential crosses the bands mainly arising from the dyzanddzx-orbitals. After having considered the normal state properties, we concentrate on the possible superconducting states that can be realized, their energetics and their topolog- ical behavior. The analysis is based on the assumption that the inter-orbital local attractive interaction is the only relevant pairing channel that contributes to the for- mation of Cooper pairs. Then, the intra-orbital pair- ing coupling is negligible. Such a hypothesis can be physically applicable in multi-orbital systems because th e intra-band Coulomb interaction is typically larger than the inter-band one. Indeed, in the t2grestricted sector the Coulomb interaction matrix elements of low-energy5 lattice Hamiltonian can be evaluated by employing the Hubbard-Kanamori parametrization90in terms of U,U′ andJH, after symmetrizing the Slater-integrals91within thet2gshell assuming a cubic splitting of the t2gand egorbitals.Ucorresponds to the intra-orbital Coulomb repulsion, whereas U′(withU′=U−2JHin a cu- bic symmetry) is the inter-orbital interaction which is reduced by Hund exchange, JH. Hence, one has that the inter-orbital Coulomb repulsion is generally always smaller than the intra-orbital one. Estimates for transi- tion metal oxide materials in d1,d2ord3conﬁgurations, being relevant for the t2gshell and thus for our work, indicate that U∼3.5eV andU′∼2.5eV92. Thus, it is plausible to expect that the Coulomb repulsion tends to further suppress the electron pairing that occurs within the same band. In addition, in the case of having the electron-phonon coupling as a source of electrons attrac- tion, it is shown that the eﬀective inter- and intra-orbital attractive interaction can be of the same magnitude (see Appendix for more details). In this framework, we point out that topological super- conductivity is proposed to occur, owing to inter-orbital pairing, in Cu-doped Bi2Se3for an inversion symmetric crystal structure60. Here, although similar inter-orbital pairing conditions are considered, we pursue the super- conductivity in low-dimensional conﬁgurations, e.g., at the interface of oxides, with the important constraint of having a broken inversion symmetry. Concerning the or- bital structure of the pairing interaction, owing to the tetragonal crystalline symmetry, the coupling between thedxy-orbital and dyz/dzx-orbital is equivalent, and thus one can assume that only two independent chan- nels of attraction are allowed, as shown in Fig. 1(d). Indeed,Vxydenotes the interaction between the dxyand dyz/dzx-orbitals, while Vzrefers to the coupling between thedyzanddzx-orbitals. Then, the pairing interaction is given by HI=Vxy/summationdisplay i[nxy,inyz,i+nxy,inzx,i] +Vz/summationdisplay inyz,inzx,i, (9) nα,i=c† α↑icα↑i+c† α↓icα↓i, (10) whereidenotes the lattice site. A. Irreducible representation and symmetry classiﬁcation In this subsection, we classify the inter-orbital super- conducting states according to the point group symme- try. The system upon examination has a tetragonal sym- metry associated with the point group C4v, marked by four-fold rotational symmetry C4and mirror symmetries MyzandMzx. Based on the rotational and reﬂection symmetry transformations, all the possible inter-orbital isotropic pairings can be classiﬁed into ﬁve irreduciblerepresentations of the C4vpoint group as summarized in Table I. For our purposes, only solutions that do not TABLE I. Irreducible representation of the inter-orbital isotropic superconducting states for the tetragonal group C4v. In the columns, we report the sign of the order parameter upon a four-fold rotational symmetry transformation, C4, and the reﬂection mirror symmetry Myz, as well as the explicit spin and orbital structure of the gap function. In the E repre - sentation, +and−of the subscript mean the doubly degen- erate mirror-even ( +) and mirror-odd ( −) solutions, respec- tively. C4vC4Myzorbital basis function (dxy,dyz)d(xy,yz) y A1+ + (dxy,dzx)d(xy,zx) x=−d(xy,yz) y (dyz,dzx)d(yz,zx) z A2+−(dxy,dyz)d(xy,yz) x (dxy,dzx)d(xy,zx) y=d(xy,yz) x B1−+(dxy,dyz)d(xy,yz) y (dxy,dzx)d(xy,zx) x=d(xy,yz) y (dxy,dyz)d(xy,yz) x B2− − (dxy,dzx)d(xy,zx) y=−d(xy,yz) x (dyz,dzx)ψ(yz,zx) E±i±(dxy,dyz)ψ(xy,yz),d(xy,yz) z (dxy,dzx)ψ(xy,zx) +=∓id(xy,yz) z+ d(xy,zx) z−=∓iψ(xy,yz) − (dyz,dzx)d(yz,zx) x ,d(yz,zx) y break the time-reversal symmetry are considered and are reported in Table I. Then, the superconducting order parameter associated to bands αandβcan be classi- ﬁed as an isotropic ( s-wave) spin-triplet/orbital-singlet d(α,β)-vector and s-wave spin-singlet/orbital-triplet with amplitudeψ(α,β)or as a mixing of both conﬁgurations. With these assumptions, one can generally describe the isotropic order parameter with spin-singlet and triplet components as ˆ∆α,β=iˆσy/bracketleftBig ψ(α,β)+ˆσ·d(α,β)/bracketrightBig , (11) withαandβstanding for the orbital index, and having for each channel three possible orbital ﬂavors. Further- more, owing to the selected tetragonal crystal symmetry, one can achieve three diﬀerent types of inter-orbital pair- ings. The spin-singlet conﬁgurations are orbital triplets and can be described by a symmetric superposition of op- posite spin states in diﬀerent orbitals. On the other hand, spin-triplet components can be expressed by means of the following d-vectors: d(xy,yz)=/parenleftBig d(xy,yz) x,d(xy,yz) y,d(xy,yz) z/parenrightBig , d(xy,zx)=/parenleftBig d(xy,zx) x,d(xy,zx) y,d(xy,zx) z/parenrightBig , d(yz,zx)=/parenleftBig d(yz,zx) x,d(yz,zx) y,d(yz,zx) z/parenrightBig ,6 withd(α,β)indicating the spin-triplet conﬁguration built withαandβ-orbitals. In general, independently of the orbital mixing, spin-triplet pairing can be expressed in a matrix form as ∆T=/parenleftBigg ∆↑↑∆↑↓ ∆↓↑∆↓↓/parenrightBigg =/parenleftBigg −dx+idydz dzdx+idy/parenrightBigg ,(12) where the d-vector components are related to the pair- ing order parameter with zero spin projection along the corresponding symmetry axis. The three components dx=1 2(−∆↑↑+∆↓↓),dy=1 2i(∆↑↑+∆↓↓)anddz= ∆↑↓ are expressed in terms of the equal spin ∆↑↑and∆ ↓↓, and the anti-aligned spin ∆↑↓gap functions. As the compo- nents of the d-vector are associated with the zero spin projection of spin-triplet conﬁguration, if the d-vector points along a given direction, the parallel spin conﬁg- urations lie in the plane perpendicular to the d-vector orientation. In the presence of time-reversal symmetry, the superconducting order parameter should satisfy the following relations: ∆↓↓ α,β=/bracketleftBig ∆↑↑ α,β/bracketrightBig∗ , (13) ∆↑↓ α,β=−/bracketleftBig ∆↓↑ α,β/bracketrightBig∗ , (14) with the appropriate choice of the U(1) gauge. In addi- tion, the pairing order parameter has four-fold rotational symmetry and mirror reﬂection symmetry with respect to theyzandzxplanes as dictated by the point group C4v. Thus, it has to be transformed according to the following relations: C4ˆ∆Ct 4=einπ 2ˆ∆, Myzˆ∆Mt yz=±ˆ∆, wherenequals to 0for A representation, 2for B repre- sentation, 1, and3for E representation. Such properties are very important to distinguish the symmetry of the so- lutions obtained by the Bogoliubov-de Gennes equation. The energy gap functions are then explicitly constructed by taking into account the corresponding irreducible rep- resentations. For the one-dimensional representations, theA1state is given by d(xy,zx) x=−d(xy,yz) y, ∆↑↑ xy,yz= ∆↓↓ xy,yz=id(xy,yz) y, ∆↑↑ xy,zx=−∆↓↓ xy,zx=−d(xy,zx) x, ∆↑↓ yz,zx= ∆↓↑ yz,zx=d(yz,zx) z, while for the A2representation, d(xy,zx) y=d(xy,yz) x, ∆↑↑ xy,yz=−∆↓↓ xy,yz=−d(xy,yz) x, ∆↑↑ xy,zx= ∆↓↓ xy,zx=id(xy,zx) y,theB1representation, d(xy,zx) x=d(xy,yz) y, ∆↑↑ xy,yz= ∆↓↓ xy,yz=id(xy,yz) y, ∆↑↑ xy,zx=−∆↓↓ xy,zx=−d(xy,zx) x, and theB2representation, d(xy,zx) y=−d(xy,yz) x, ∆↑↑ xy,yz=−∆↓↓ xy,yz=−d(xy,yz) x, ∆↑↑ xy,zx= ∆↓↓ xy,zx=id(xy,zx) y, ∆↑↓ yz,zx=−∆↓↑ yz,zx=ψ(yz,zx). Finally, for the E representation, there are doubly degen- erate mirror-even (+)and mirror-odd (−)solutions: ψ(xy,zx) +=∓id(xy,yz) z+, d(xy,zx) z−=∓iψ(xy,yz) −, ∆↑↓ xy,yz=α−ψ(xy,yz) −+α+d(xy,yz) z+, ∆↓↑ xy,yz=−α−ψ(xy,yz) −+α+d(xy,yz) z+, ∆↑↓ xy,zx=α+ψ(xy,zx) ++α−d(xy,zx) z−, ∆↓↑ xy,zx=−α+ψ(xy,zx) ++α−d(xy,zx) z−, ∆↑↑ yz,zx=−α−d(yz,zx) x−+iα+d(yz,zx) y+, ∆↓↓ yz,zx=α−d(yz,zx) x−+iα+d(yz,zx) y+, whereα+andα−denote arbitrary constants for the lin- ear superposition. As a consequence of the symmetry constraint and of the inter-orbital structure of the or- der parameter, diﬀerent types of isotropic spin-triplet and singlet-triplet mixed conﬁgurations can be obtained. Equal spin-triplet and opposite spin-triplet pairings are mixed in the A1representation. On the other hand, in theB2representation, equal spin-triplet and spin-singlet pairings are mixed. For the A2andB1representations, only equal spin-triplet pairings are allowed, and all types of pairings can be realized in the E representation. It is worth noting that A1,B2, and E representations have pairings between all the orbitals in the yz-zxandxy- yz/zx channels, while A2andB1can make electron pair- ings only in the xy-yz/zx channel, that is, by mixing thedxyanddyz/dzx-orbitals as shown in Table I. This symmetry constraint is important when searching for the ground-state conﬁguration. III. ENERGY GAP EQUATION AND PHASE DIAGRAM In order to investigate which of the possible symmetry- allowed solutions is more stable energetically, we solve th e Eliashberg equation within the mean ﬁeld approximation by taking into account the multi-orbital eﬀects near the7 transition temperature. The linearized Eliashberg equa- tion within the weak coupling approximation is given by Λ∆στ α,γ=−kBT NVα,γ/summationdisplay k′,iεmFασ,γτ(k′,iεm), (15) Vxy,yz=Vxy,zx=Vyz,xy=Vzx,xy≡Vxy, Vyz,zx=Vzx,yz≡Vz, Fασ,γτ(k′,iεm) (16) =/summationdisplay β,δ/summationdisplay σ′,τ′∆σ′τ′ β,δGσσ′ α,β(k′,iεm)Gττ′ γ,δ(−k′,−iεm), whereΛis the eigenvalue of the linearized Eliashberg equation. Here, σ,τ,σ′, andτ′denote the spin states and α,β,γ, andδstand for the orbital in- dices.Fασ,γτ(k′,iεm)is the anomalous Green’s func- tion. As we assume an isotropic Cooper pairing, which isk-independent, the summation over momentum and Matsubara frequency in Eq. (16) gets simpliﬁed. Fi- nally, the problem is reduced to the diagonalization of the24×24matrix. We then study the relative stabil- ity of the irreducible representations as listed in Table I. An analysis of the energetically most favorable super- conducting states is performed as a function of Vz/Vxy, assuming that Vxy/t=−1.0and for a given temperature T/t= 5.0×10−5. When we keep the ratio Vz/Vxy, the eigenvalue Λis proportional to Vxywithin the mean-ﬁeld approximation. The choice of the representative coupling Vxy/t=−1.0is guided by the fact that one aims to ac- cess a physical regime for the superconducting phase that in principle can be compared to realistic superconduct- ing materials in the weak coupling limit. For instance, if one chooses t∼200−300meV, which is common in ox- ides, and considering that the superconducting transition temperature Tcis obtained when the magnitude of the greatest eigenvalue gets close to 1, then one would ﬁnd Tcto be of the order of 100−300mK, which is reasonable for the 2DEG superconductivity at the oxide interface. Figure 3 shows the superconducting phase diagram for representative amplitudes of the spin-orbit coupling, λSO/t= 0.10, and inversion asymmetry interaction, ∆is/t= 0.20, while varying both the chemical potential and the ratio of the pairing couplings Vz/Vxy. Owing to the inequivalent mixing of the orbitals in the paired con- ﬁgurations, it is plausible to expect a signiﬁcant compe- tition between the various symmetry allowed states and that such an interplay is sensitive not only to the pairing orbital anisotropy, but also to the structure and the num- ber of Fermi surfaces. A direct observation is that for Vz larger than Vxy, theA1phase is stabilized with respect to theB1phase because it contains a d(yz,zx) z channel of a spin-triplet pairing in the yz-zxsector that is absent inB1phase. However, such a simple deduction does not directly explain why the A1phase wins the competition with other superconducting phases, e.g., the B2and E phases, which also can gain condensation energy by pair- ing electrons in the yz-zxsector. As a diﬀerent type0.5 0.75 1 1.25−0.3−0.2−0.100.10.20.30.4 µ / t Vz / Vxy# of FS = 2# of FS = 4# of FS = 6B1 B1A1 A1 A1B1 FIG. 3. Phase diagram as a function of Vz/VxyatλSO/t= 0.10,∆is/t= 0.20,T/t= 5.0×10−5, andVxy/t=−1.0. The brown solid line is the border between A1andB1states. The black solid line indicates the value of the chemical po- tential for which the number of Fermi surfaces changes. The black dotted lines correspond to the values of the chemical potentials used in Fig. 2 for the normal state Fermi surfaces . ofd-vector orientation enters into the A1and E conﬁg- urations, yet in the B2state, theyz-zxchannel has a spin-singlet pairing, one can deduce that the interplay between the spin of the Copper pairs and that of the single-electron states close to the Fermi level is relevant to single out the most favorable superconducting phase. The boundary between the A1andB1phases exhibits a sudden variation when one tunes the chemical potential across the value for which the number of Fermi surfaces changes. Such an abrupt transition is, however, plau- sible when passing through a Lifshitz point in the elec- tronic structure of the normal state because other pairing channels get activated at the Fermi level. The relation between the modiﬁcation of the superconducting state and electronic topological or Lifshitz transition93that the Fermi surface can undergo is a subject of general interest. Indeed, there are many theoretical studies and experi- mental signatures pointing to a subtle interplay of Lif- shitz transitions and superconductivity in cuprates94,95, heavy-fermion superconductors96and more recently in iron-based superconductors97–100. In those cases, major changes of the superconducting state seem to occur when going through a Lifshitz transition because Fermi pockets can appear or disappear at the Fermi level and in turn lead to diﬀerent physical eﬀects. Here, along this line of investigation, the role of the electron ﬁlling is also quite important and sets the com- petition between the energetically most stable phases. Indeed, one can notice that the A1(B1) phase is sta- bilized for higher (lower) Vz/Vxyand lower (higher) µ. Furthermore, we ﬁnd that, in the case of two Fermi sur- faces, the A1state is further stabilized by decreasing the chemical potential and moving to a regime of extremely low concentration. On the other hand, a transition to8 theB1phase is achieved by electron doping. In the dop- ing regime of four bands at the Fermi level, the A1-B1 boundary evolves approximately as a linear function of Vz/Vxy. This implies that the A1conﬁguration tends to be less stable and a higher ratio Vz/Vxyis needed to achieve such a conﬁguration at a given chemical poten- tial. Finally, approaching the doping regime of six Fermi surfaces, the A1-B1boundary becomes independent of the amplitude of µ. It is remarkable that the doping can substantially alter the competition between the A1and B1phases, thus manifesting the intricate consequences of the spin-orbital character of the electronic structure close to the Fermi level. To explicitly and quantitatively demonstrate the en- ergy competition among all the symmetry allowed phases, one can follow the behavior of the eigenvalues of the linearized Eliashberg equations as a function of the ratioVz/Vxy(Fig. 4). Figures. 4(a)-(c) show the 0 0.5 1 1.5 2 2.500.511.522.5 0 0.5 1 1.5 2 2.500.511.52 0 0.5 1 1.5 2 2.500.20.40.60.8 Vz / VxyThe eigenvalueA1 representation B1 representation (a) (b) (c) The eigenvalueB2 representationA2 representation E representation The eigenvalue Vz / Vxy Vz / Vxy FIG. 4. Evolution of the eigenvalue of the Eliashberg matrix equation as a function of Vz/Vxyat (a)µ/t=−0.25, (b) µ/t= 0.0, and (c)µ/t= 0.35atλSO/t= 0.10,∆is/t= 0.20, T/t= 5.0×10−5, andVxy/t=−1.0. (a)A1representation is dominant. (b)and(c) B1is dominant for small amplitude of the ratioVz/Vxy. eigenvalues of the Eliashberg matrix equation for all the irreducible representations as a function of Vz/Vxywhen the number of Fermi surfaces is (a) two ( µ/t=−0.25), (b) four (µ/t= 0.0), and (c) six ( µ/t= 0.35) as indicated by the dotted lines in Fig. 3. With the increase in Vz, the magnitude of the eigenvalues of the irreducible represen- tations including the yz-zxchannel, i.e., A1,B2, and E representations, increases in all the cases with two, four, and six Fermi surfaces. On the other hand, the eigen- values of the A 2and B 1representations are independent ofVz, asVzis irrelevant for this pairing channel. When the number of Fermi surfaces is two, the A1representa- tion is the most dominant pairing for all Vz. Although the magnitude of the eigenvalues for the B2andErep- resentations also increases with Vz, these solutions never become dominant as compared with the A1state. When the number of Fermi surfaces is four or six, the eigenvalueof theB1phase is larger than that of the A1representa- tion for lower Vz. Finally, we have investigated the phase diagram by scanning a larger range of temperatures for few repre- sentative cases of pairing interaction and ﬁlling concen- tration (see Appendix). The results are not signiﬁcantly changed except in a region of extremely high tempera- ture, corresponding to an unphysically large amplitude of the pairing interaction. There, although B 1keeps be- ing the most stable state, the largest eigenvalues indicate a competition between the B 1and B2rather than the B 1 and A 1conﬁgurations. IV. TOPOLOGICAL PROPERTIES AND ENERGY EXCITATION SPECTRUM IN THE BULK AND AT THE EDGE In the previous section, we conﬁrmed that both the A1andB1pairings can be energetically stabilized in a large region of the parameter space. Thus, it is relevant to further consider the nature of the electronic structure of these superconducting phases in order to provide key elements and indications that can be employed for the detection of the most favorable inter-orbital supercon- ductivity. The analysis is based on the solution of the Bogoliubov-de Gennes (BdG) equation for the evalua- tion of the low-energy spectral excitations both in the bulk and at the edge of the superconductor for both the A1andB1phases. The matrix Hamiltonian in momen- tum space is given by HBdG(k) =/parenleftBigg H(k)ˆ∆ ˆ∆†−H∗(−k)/parenrightBigg . (17) withH(k)being the normal state Hamiltonian. A. Bulk energy spectrum and topological superconductivity In order to determine the excitation spectrum, we solve the BdG equations for both the A1andB1conﬁgurations. For convenience, we introduce the gap amplitude \|∆0\|, and we set the components of the d-vectors to be d(xy,yz) y=−d(xy,zx) x=d(yz,zx) z=\|∆0\|, (18) forA1and d(xy,yz) y=d(xy,zx) x=\|∆0\|, (19) forB1state. Here, the parameter \|∆0\|/t= 1.0×10−3is set as a scale of energy. We start focusing on the doping regime of four bands at the Fermi level. In this case, the A1state has a fully gapped electronic structure for all the bands at the Fermi level as demonstrated by the inspection of the in-plane9 10−1100101 10−1100101 10−1100101 10−1100101kxkyΓπ/2 −π/2 −π/2 π/2Fermi surface 00 π0 −π θπ0−π θgap / \| ∆0\| gap / \| ∆0\| π0 −π θgap / \| ∆0\| π0 −π θgap / \| ∆0\|(a) (b) (c) (d)(e)θ(b) (c) (d) (e) FIG. 5. (a) Fermi surfaces at µ/t= 0.0in the normal state. [(b)-(e)] A1quasi-particle energy gap along the Fermi surface as a function of the polar angle θas shown in (a) for λSO/t= 0.10,∆is/t= 0.20, and\|∆0\|/t= 1.0×10−3corresponding to the Fermi surfaces in (a). angular dependence of the gap magnitude [Figs. 5(b)- (e)]. In particular, we notice that the gap amplitude is not isotropic and orbital dependent when moving from the outer to the inner Fermi surface [Figs. 5(b)-(e)]. The nodal state (Fig. 6), on the other hand, exhibits a more regular behavior of the gap amplitude which is basically orbital independent and point nodes occurring only along the diagonal of the Brillouin zone on the various Fermi surfaces. It is interesting to further investigate the nature of the nodalB1phase by determining whether the existence of the nodes is related to a non-vanishing topological invari- ant. As the model Hamiltonian owes particle-hole and time-reversal symmetry, one can deﬁne a chiral operator ˆΓas a product of the particle-hole ˆCand time-reversal ˆΘoperators. As the chiral symmetry operator anticom- mutes with HBdG(k), by employing a unitary transforma- tion rotating the basis in the eigenbasis of ˆΓ, the Hamilto- nian can be put in an oﬀ-diagonal form with antidiagonal blocks. Hence, the determinant of each block can be put in a complex polar form and, as long as the eigenvalues are non-zero, it can be used to obtain a winding number by evaluating its trajectory in the complex plane. On a10−1010−5100 10−1010−5100 10−1010−5100 10−1010−5100kxky Fermi surface1D winding number = +1π/2 −π/2 −π/2 π/21D winding number = −1 00 0−π πgap / \|∆0\| θ0 −π πgap / \|∆0\| θ 0−π πgap / \|∆0\| θ0−π πgap / \|∆0\| θ(a) (b) (c) (d) (e)θΓ(b) (c) (d) (e) FIG. 6. (a) Fermi surfaces and position of the nodes at µ/t= 0.0. We indicate the winding numbers deﬁned at each node. (b)-(e) indicate the quasi-particle energy spectra for the B1 state withλSO/t= 0.10, ∆is/t= 0.20, and\|∆0\|/t= 1.0× 10−3at the corresponding Fermi surfaces shown in (a). general ground, we point out that the number of singu- larities in the phase of the determinant is a topological invariant101because it cannot change without the am- plitude going to zero, thus implying a gap closing and a topological phase transition. For this symmetry class, then, one can associate and determine the winding num- ber around each node by following, for instance, the ap- proach already applied successfully in Refs. [102–104]. The chiral, particle-hole, and time-reversal operators ar e expressed as ˆΓ =−iˆCˆΘ, (20) ˆC=/parenleftBigg 0ˆI6×6 ˆI6×60/parenrightBigg =ˆl0⊗ˆσx⊗ˆτ0, (21) ˆΘ =ˆl0⊗iˆσy⊗ˆτ0. (22) Here,ˆI6×6andˆτ0denote the 6×6unit matrix and the identity matrix in the particle-hole space, respectively. As we consider time-reversal symmetric pairings, the chi- ral operator anticommutes with the Hamiltonian: {HBdG(k),ˆΓ}= 0. (23) One can then introduce a unitary matrix ˆUΓthat diago-10 0.5 0.6 0.70.50.60.7 0π / 2 π / 2 kxky kxky C(a) (b) +1 : −1 : FIG. 7. (a) Fermi surfaces at λSO/t= 0.10,∆is/t= 0.20, µ/t= 0.0, and point nodes position (winding number) at ∆0/t= 1.0×10−3. (b) Zoomed view of the plot in (a) and a contour of the integral C. nalizes the chiral operator ˆΓ: ˆU† ΓˆΓˆUΓ=/parenleftBigg ˆI6×60 0−ˆI6×6/parenrightBigg , (24) ˆU† Γ=ˆUΓ=1√ 2/parenleftBigg ˆI6×6ˆl0⊗ˆσy ˆl0⊗ˆσy−ˆI6×6/parenrightBigg .(25) In this basis the BdG Hamiltonian is block antidiagonal- ized byˆUΓ, ˆU† ΓHBdG(k)ˆUΓ=/parenleftBigg 0 ˆq(k) ˆq†(k) 0/parenrightBigg , (26) ˆq=H(k)/bracketleftBig ˆl0⊗ˆσy/bracketrightBig −ˆ∆. (27) Then, the determinant of the ˆq(k)matrix block can be put in a complex polar form, and as long as the eigen- values are nonzero, it can be used to obtain the winding numberWby evaluating its trajectory in the complex plane as W=1 2π/contintegraldisplay Cdθ(k), (28) θ(k)≡arg[det ˆq(k)]. Cin Eq. (28) is a closed line contour that encloses a given node as schematically shown in Fig. 7(b). From the explicit calculation, we ﬁnd that the amplitude of Wis±1[see Figs. 6(a) and 7(a)]. If the nodes have a nonzero winding number, edge states appear due to the bulk-edge correspondence. It is known103that the follow- ing index theorem is satisﬁed: for any one-dimensional cut in the Brillouin zone that is indicated by a given momentum k/bardblthat is parallel to the edge, one has that w(k/bardbl) =n+−n−, withn+andn−being the number of the eigenstates associated to the eigenvalues +1and−1 of the chiral operator ˆΓ, respectively. The number of edge states is equal to \|w(k/bardbl)\|when considering a boundary conﬁguration with a conserved k/bardbl. We can easily show thatWwhich is given in Eq. (28) and w(k/bardbl)are deeply10−310−210−1−0.3−0.2−0.100.10.20.30.410−310−210−1−0.3−0.2−0.100.10.20.30.4 10−310−210−1−0.3−0.2−0.100.10.20.30.410−310−210−1 10−310−210−1 0.05 0.1 0.150.10.20.3µ / t \| ∆ 0 \| / tFS = 6 FS = 4 FS = 2 FS = 2FS = 4FS = 6 n = 6(a) (b) (c) FS = # of FS n = # of point nodes ( Γ−M)(d) (e) n = 6 n = 4 FS = 2FS = 4FS = 6FS = 6 FS = 4 FS = 2 FS = 6 FS = 4 FS = 2n = 2n = 6 n = 4µ / t µ / tn = 4 n = 2 n = 2n = 0 n = 4 n = 2 n = 2n = 0n = 4n = 4 n = 6n = 4 n = 4n = 2 n = 2 n = 2n = 0 n = 0n = 2n = 2 n = 0 n = 0 n = 0n = 0n = 2 n = 2n = 2n = 2n = 4n = 6 n = 4n = 2 n = 0 n = 0 λSO / t∆is / t (a)(b)(c) (d) (e)(f)\| ∆ 0 \| / t \| ∆ 0 \| / t\| ∆ 0 \| / t\| ∆ 0 \| / t FIG. 8. Phase diagram of the Lifshitz transitions for the nodalB1phase at (a) λSO/t= 0.10and∆is/t= 0.10, (b)λSO/t= 0.10and∆is/t= 0.20, (c)λSO/t= 0.10and ∆is/t= 0.30, (d)λSO/t= 0.05and∆is/t= 0.20, and (c) λSO/t= 0.15and∆is/t= 0.20. (f) Schematic plot of the cor- respondence between the spin-orbit and the inversion asym- metry couplings and the panels (a)-(e). Black and red labels denote the number of Fermi surfaces in the normal state and the point nodes in the superconducting phase that are locate d along the diagonal of the Brillouin zone from Γto M, respec- tively. The black dotted line and the orange solid line indi- cate the two-to-four Fermi surface separation and the four- to- six Fermi surface boundary in the normal state. Circles and squares set the transition lines for the nodal superconduct or between conﬁgurations having diﬀerent number of nodes in the excitation spectrum. linked:w(k1)−w(k2) =−Wsgn(k1−k2)whereWis the total winding number around the nodes between k/bardbl=k1 andk/bardbl=k2. Thus, nonzero Won the nodes means nonzerow(k/bardbl)and the existence of the zero energy edge state with appropriate choice of the crystal plane. Such relation sets the main physical connection between the winding number and the properties of the topological su- perconductor. We generally ﬁnd that two to six point nodes can oc-11 cur along the Γ-M direction, and their number is related to that of the Fermi surfaces. Interestingly, the position of the point nodes is not ﬁxed and pinned to the lines of the Fermi surface in the normal state. In general, their position along the diagonal of the Brillouin zone depends on the amplitude \|∆0\|and indirectly on the val- ues of the spin-orbit and inversion asymmetry couplings. Thus, two adjacent point nodes with opposite winding numbers can, in principle, be moved until they merge and then disappear by opening a gap in the excitation spectrum. This behavior is generally demonstrated in Fig. 8. A phase diagram can be determined in terms of the amplitude \|∆0\|and the chemical potential µ. The nodal superconductor can undergo diﬀerent types of Lif- shitz transitions, and in general, those occurring in the normal state are not linked to the nodal merging in the superconducting phase. Indeed, one of the characteristic features of the nodal superconductor is that, by changing the ﬁlling, through µ, one can drive a transition from two to four and six point nodes independently of the number of bands crossing the Fermi level in the normal state. It is rather the strength of \|∆0\|that plays an important role in tuning the nodal superconductor. An increase in \|∆0\|tends to reduce the number of nodes until a fully gapped phase appears. As the critical lines are sensitive to the spin-orbit λSOand inversion asymmetry ∆iscou- plings, one can get line crossings that allow for multiple merging of nodes such that the superconductor can un- dergo a direct transition from six to two at µ/t∼0.40 [Figs. 8(e) and (d)] or from four to zero point nodes, as for instance nearby the crossing between the blue and orange lines at µ/t∼0.10in the Fig. 8. As the positions of the point nodes are ﬁxed, each Fermi surface in the limit of small \|∆0\|and its distance in the Brillouin zone increases with level splitting by ∆isandλSO, a larger \|∆0\|is required to annihilate the point nodes when both ∆isandλSOgrow in amplitude as demonstrated by the shift of the green and blue critical lines in Figs. 8(a)-8(c) for diﬀerent values of ∆is, and Figs. 8(d), 8(b), and 8(e) in terms of λSO. When considering these results in the context of two-dimensional superconductors that emerge at the surface or interface of band insulators we observe that the achieved topological transitions can be driven by gate voltage and temperature, as µand∆isare tun- able by electric ﬁelds, and the amplitude of \|∆0\|can be controlled by the temperature and the electric ﬁeld as well. B. Local density of states at the edge of the superconductor Having established that the nodes in the B1conﬁg- uration are protected by a nonvanishing winding num- ber, one can expect that ﬂat zero-energy surface Andreev bound states (SABS) occur at the boundary of the su- perconductor. In this section, we investigate the SABS and the localdensity of states (LDOS) for two diﬀerent terminations of the two-dimensional superconductor, i.e., the (100) and (110) oriented edges. We start by discussing the LDOS for the (100) and (110) edges at representative values of λSO/t= 0.10,∆is/t= 0.20, and\|∆0\|/t= 1.0×10−3, and by varying the chemical potential in order to compare the cases with a diﬀerent number of point nodes in the bulk energy spectrum at µ/t=−0.25,µ/t= 0.0, and µ/t= 0.35as shown in Figs. 9(a)-9(c), respectively. As expected, the momentum-resolved LDOS indicates that zero-energy SABS can be observed but only for spe- ciﬁc orientations of the edge. Indeed, as reported in Figs. 9(d)-9(i), one has zero-energy SABS (ZESABS) for the (110) boundary while they are absent for the (100) edge. The reason for having inequivalent SABS edge modes is directly related to the presence of a nontrivial winding number that is protecting the point nodes. For the (110) edge, isolated point nodes exist in the surface Brillouin zone, and they have winding numbers with opposite sign. Thus, the ZESABS, which connects the nodes with a pos- itive and negative winding number, emerge in the gap. On the other hand, when considering the (100) oriented termination, the winding numbers for positive kxand negativekxare completely opposite in sign, and they cancel each other when projected on the (100) surface Brillouin zone. Thus, ﬂat zero-energy states cannot oc- cur for the (100) edge. Nevertheless, helical edge modes are observed inside the energy gap as demonstrated in Fig. 9(d). This is because the Majorana edge modes with positive and negative chirality can couple, get split, and acquire a dispersion. The diﬀerences in the edge ABS also manifest in the momentum integrated LDOS. For the (110) edge, owing to the presence of the ZESABS, the LDOS normalized by its normal state value at E= 0 shows pronounced zero-energy peaks (see dash-dotted line in Figs. 9(j), (k), and (l)). On the other hand, for the (100) boundary, they lead to a broad peak or exhibit many narrow spectral structures reﬂecting the complex dispersion of the edge states. Finally, we discuss the \|∆0\|dependence of LDOS at zero energy, i.e., E= 0. For the (110) edge, the zero- energy peak mainly originates from the zero-energy ﬂat band. The height of the zero-energy peak can then be characterized by (i) the strength of the localization of the edge state and (ii) the total length of the ZESABS within the surface Brillouin zone. The strength of the localiza- tion is deﬁned by the inverse of the localization length 1/ξ and1/ξ∝ \|∆0\|. In other words, the peak height gener- ally increases with \|∆0\|. On the other hand, as shown in Fig. 8, the extension in the momentum space of the zero- energy ﬂat states becomes shorter with increasing \|∆0\|. For simplicity, one can focus on the two Fermi surface conﬁguration. In this case, the total length of the zero- energy ﬂat band is roughly estimated as δk(1−\|∆0\|/\|∆c 0\|) for\|∆0\|<\|∆c 0\|and zero for \|∆0\|>\|∆c 0\|, whereδkis the Fermi surface splitting along the Γ-M direction and \|∆c 0\| is a critical value above which the point-nodes disappear. Then, the height of the zero-energy peak is proportional12 FIG. 9. Momentum-resolved and angular averaged LDOS for B 1representation. The Fermi surfaces and the position of the point nodes are shown for (a) µ/t=−0.25, (b)µ/t= 0.0, and (c)µ/t= 0.35. The momentum ( k/bardbl) resolved LDOS at the (100) oriented surface for (d) µ/t=−0.25, (e)µ/t= 0.0, and (f)µ/t= 0.35. The momentum ( k/bardbl) resolved LDOS at the (110) oriented surface for (g) µ/t=−0.25, (h)µ/t= 0.0, and (i)µ/t= 0.35. LDOS normalized by its normal state value at E= 0 [(DOS N(E= 0)] at the (100) and (110) oriented surfaces, and in the bulk for (j)µ/t=−0.25, (k)µ/t= 0.0, and (l)µ/t= 0.35. The red solid line, blue dash-dotted line, and black dashed l ine denote the LDOS at the (100) oriented surface, (110) orie nted surface, and in the bulk, respectively. Other parameters ar eλSO/t= 0.10,∆is/t= 0.20, and\|∆0\|/t= 1.0×10−3.13 0 50 100 150 200 250 300 350 10-310-210-1(110) surface µ/tDOSSC(E=0)/DOSN(E=0) \|∆0\|/t0.35 0 -0.25 FIG. 10. The LDOS at E= 0for the (110) oriented surface for theB1representation as a function of \|∆0\|/tatλSO/t= 0.10 and∆is/t= 0.20. The red solid line, blue dotted line, and green dashdotted line correspond to µ/t=−0.25,µ/t= 0.0, andµ/t= 0.35. to\|∆0\|(1− \|∆0\|/\|∆c 0\|)for\|∆0\|<\|∆c 0\|and vanishes for \|∆0\|>\|∆c 0\|. This is a nonmonotonic dome-shaped be- havior of the ZELDOS as a function of \|∆0\|. The ex- plicit proﬁle can be seen in Fig. 10 at µ/t=−0.25and µ/t= 0.0. Forµ/t= 0.35, the point nodes still exist in this parameter regime, and the height of the zero en- ergy peak develops with \|∆0\|. Thus, we have that the ZESABS get strongly renormalized and are tunable by a variation in the electron ﬁlling ( µ) and amplitude of the order parameter \|∆0\|as shown in Fig. 8. V. DISCUSSION AND SUMMARY We investigated and determined the possible super- conducting phases arising from inter-orbital pairing in an electronic environment marked by spin-orbit coupling and inversion symmetry breaking while focusing on mo- mentum independent paired conﬁgurations. One remark- able aspect is that, although the inversion symmetry is absent, one can have symmetry-allowed solutions that avoid mixing of spin-triplet and spin-singlet conﬁgura- tions. Importantly, states with only spin-triplet pairing s can be stabilized in a large portion of the phase diagram. Within those spin-triplet superconducting states, we unveiled an unconventional type of topological phase in two-dimensional superconductors that arises from the interplay of spin-orbit coupling and orbitally driven inversion-symmetry breaking. For this kind of a model system, atomic physics plays a relevant role and in- evitably tends to yield orbital entanglement close to the Fermi level. Thus we assumed that local inter-orbital pairing is the dominant attractive interaction. As already mentioned, this type of pairing in the presence of inver- sion symmetry breaking allows for solutions that do not mix spin-singlet and triplet conﬁgurations. The orbital- singlet/spin-triplet superconducting phase can have atopological nature with distinctive spin-orbital ﬁnger- prints in the low-energy excitations spectra that make it fundamentally diﬀerent from the topological conﬁgu- ration that is usually obtained in single band noncen- trosymmetric superconductors. Here, a remarkable ﬁnd- ing is that, contrary to the common view that an isotropic pairing structure leads to a fully gapped spectrum, a nodal superconductivity can be achieved when consid- ering an isotropic spin-triplet pairing. Although in a dif- ferent context, we noticed that akin paths for the genera- tion of an anomalous nodal-line superconductor can also be encountered when local spin-singlet pairing occur in antiferromagnetic semimetals105. In the present study, for a given symmetry, the super- conducting phase can exhibit point nodes that are pro- tected by a nonvanishing winding number. The most striking feature of the disclosed topological superconduc - tivity is expressed by its being prone to both topological and Lifshitz-type transitions upon diﬀerent driving mech- anisms and interactions, e.g., when tuning the strength of intrinsic spin-orbit and orbital-momentum couplings or by varying doping and the amplitude of order pa- rameter by, for example, varying the temperature. The essence of such a topologically and electronically tunable superconductivity phase is encoded in the fundamental observation of having control of the nodes position in the Brillouin zone. Indeed, the location of the point nodes is not determined by the symmetry of the order parame- ter in the momentum space, as occurs in the single band noncentrosymmetric system, but rather it is a nontrivial consequence of the interplay between spin-triplet pair- ing and the spin-orbital character of the electronic struc- ture. In particular, their position and existence in the Brillouin zone can be manipulated through various types of Lifshitz transitions, if one varies the chemical poten- tial, the amplitude of the spin-triplet order parameter, the inversion symmetry breaking term, and the atomic spin-orbit coupling. While electron doping can induce a change in the number of Fermi surfaces, such electronic transition is not always accompanied by a variation in the number of nodes within the superconducting state. This behavior allows one to explore diﬀerent physical sce- narios that single out notable experimental paths for the detection of the targeted topological phase. Owing to the strong sensitivity of the topological and Lifshitz transi- tions with respect to the strength of the superconducting order parameter, one can foresee the possibility of observ- ing an extraordinary reconstruction of the superconduct- ing state both in the bulk and at the edge by employing the temperature to drive the pairing order parameter to a vanishing value, i.e., at the critical temperature, starti ng from a given strength at zero temperature. Then, a sub- stantial thermal reorganization of the superconducting phase can be obtained. While a variation in the number of nodes in the low energy excitations spectra cannot be easily extracted by thermodynamic bulk measurements, we ﬁnd that the electronic structure at the edge of the su- perconductor generally undergoes a dramatic reconstruc-14 tion that manifests into a non-monotonous behavior of the zero bias conductance or in an unconventional ther- mal dependence of the in-gap states. Another impor- tant detection scheme of the examined spin-triplet su- perconductivity emerges when considering its sensitivity to the doping or to the strength of the inversion symme- try breaking coupling, which can be accessed by applying an electrostatic gating or pressure. Such gate/distortive control can ﬁnd interesting applications, especially when considering two-dimensional electron gas systems. Another interesting feature of the multiple-nodes topo- logical superconducting phase is given by the strong sen- sitivity of the edge states to the geometric termination, as demonstrated in Fig. 9. This is indeed a consequence of the presence of nodes with an opposite sign winding num- ber within the Brillouin zone. Hence, when considering the electronic transport along a proﬁle that is averaging diﬀerent terminations, it is natural to expect multiple in-gap features. Owing to the multi-orbital character of the supercon- ducting state, we expect that non-trivial odd-in-time pair amplitudes are also generated106–110. In particu- lar, we predict that both local odd-in-time spin-singlet and triplet states can be obtained in the bulk and at the edge. The local spin-singlet odd-in-time pair correlation s are an exquisite consequence of the multi-orbital super- conducting phase. Accessing the nature of their competi- tion/cooperation and its connection to the nodal super- conducting phase is a general and relevant problem in relation to the generation, manipulation, and control of odd-in-time pair amplitudes. It is also relevant to comment on the impact of an intra-orbital pairing on the achieved results. Here, there are few fundamental observations to make. Firstly, one may ask whether the topological B 1phase is robust to the adding of an extra pairing component which in the intra-orbital channel is most likely to have a spin- singlet symmetry. For this circumstance, one can start by pointing out that for any intra-orbital pairing com- ponent that does not break the chiral symmetry protect- ing the nodal structure of the superconducting state, the B1conﬁguration can only undergo a Lifshitz-type tran- sition associated with the merging of nodes having oppo- site sign in the winding number. Moreover, speciﬁcally for the B 1irreducible representation, the intra-orbital spin-singlet component would have a dx2−y2-wave sym- metry (∼coskx−cosky) and thus its amplitude would be vanishing along the Γ-M direction of the Brillouin zone where the nodes of the B 1phase are placed. Hence, the intra-orbital component cannot aﬀect at all the nodal structure of the B 1phase. From this perspective, the B1phase is remarkably robust to the inclusion of spin- singlet intra-orbital pairing components. In Appendix, the intra-orbital spin-singlet pairings other than B 1rep- resentation ( dx2−y2-wave) are discussed. Concerning the experimental consequences of the topo- logical superconducting phase, one can observe that, apart from the direct spectroscopic access to the tem-perature dependence of the edge states, the use of a superconductor-normal metal-superconductor (S-N-S) junction can also contribute to design of experiments to directly probe the peculiar behavior of the B 1phase. In particular, by scanning its temperature dependent prop- erties, since the B 1state can undergo a series of Lifshitz transitions within the superconducting phase by gapping out part of the nodes, a dramatic modiﬁcation of the Andreev spectrum at the S-N boundary is expected to occur. Hence, upon the application of a phase diﬀerence between the superconductors in the S-N-S junction, the Josephson current is expected to exhibit an anomalous temperature behavior. In particular, the abrupt changes in the Andreev bound states will drive a rapid variation in the Josephson current through the S-N-S junction when the superconductor undergoes transitions in the number of nodes. Finally, we point out that the examined model Hamil- tonian is generally applicable to two-dimensional lay- ered materials, in the low/intermediate doping regime, havingt2gd-bands at the Fermi level and subjected to both atomic spin-orbit coupling and inversion symme- try breaking, for instance owing to lattice distortions and bond bending. Many candidate material cases can be encountered in the family of transition metal ox- ides. There, unconventional low-dimensional quantum liquids with low electron density can be obtained by engineering a 2DEG at polar/nonpolar interfaces be- tween two band insulators, on the surface of band insu- lators (i.e., STO) or by designing single monolayer het- erostructures, ultrathin ﬁlms or superlattices. A paradig - matic case of superconducting 2DEG is provided by the LAO/STO heterostructure111–114. Recent experimental observations by tunneling spectroscopy have pointed out that the superconducting state can be unconventional owing to the occurrence of in-gap states with peaks at zero and ﬁnite energies115. Although these peaks may be associated with a variety of concomitant physical mechanisms, e.g., surface Andreev bound states5–10, the anomalous proximity eﬀect by odd-frequency spin-triplet pairing15,116–128, and bound states owing to the presence of magnetic impurities129, their nature can provide key information about the pairing symmetry of the super- conductor. Furthermore, the observation of Josephson currents130across a constriction in the 2DEG conﬁrms a fundamental unconventional nature of the superconduct- ing state131–133. A common aspect emerging from the two diﬀerent spectroscopic probes is that the supercon- ducting state seems to have a multi-component character. Although it is not easy to disentangle the various contri- butions that may aﬀect the superconducting phase in the 2DEG, we speculate that the proposed topological phase can be also included within the possible candidates for addressing the puzzling properties of the superconduc- tivity of the oxide interface.15 ACKNOWLEDGMENTS This work was supported by a JSPS KAKENHI (Grants No. JP15H05853, No. JPH06136, and No. JP15H03686), and the JSPS Core-to-Core program "Ox- ide Superspin", and the project Quantox of QuantERA ERA-NET Cofund in Quantum Technologies, imple- mented within the EU H2020 Programme. VI. APPENDIX In this section we address three diﬀerent issues related to the presented results. Firstly, we investigate how a modiﬁcation of the pairing interaction aﬀects the phase diagram and the relative competition between the var- ious conﬁgurations by scanning a larger range of tem- peratures at representative cases of ﬁlling concentration . Then, we consider the classiﬁcation of the irreducible rep- resentations of the superconducting phases in the pres- ence of an intra-orbital attractive interaction. Moreover , we demonstrate that the intra-orbital and inter-orbital pairing interactions mediated by phonons have the same amplitude. Starting from the impact of the pairing interaction on the phase diagram, in Fig. 11 we show that at a given temperature the maximal eigenvalue in the various irreducible representations scales with the values of Vz andVxyatλSO/t= 0.10,∆is/t= 0.20andµ/t= 0.0. When we keep the ratio Vz/Vxy, the eigenvalue Λis pro- portional to Vxywithin the mean ﬁeld approximation. Hence, the phase diagram is basically determined by the ratioVz/Vxy. In addition, since the transition tempera- tureTcis achieved when the magnitude of the greatest eigenvalue gets close to 1, then, according to this relation , one can identify the regime of temperatures which is close to the superconducting transition by suitably scaling the pairing interactions. In this way, the corresponding irre- ducible representation with the largest eigenvalue is the most stable according to the solution of the gap equation. In order to understand how a change in the criti- cal temperature can aﬀect the relative stability, in Fig. 12 we report the eigenvalues for the various irreducible representations at λSO/t= 0.10,∆is/t= 0.20and µ/t= 0.0as a function of T/tat two diﬀerent ratio (a) Vz/Vxy= 1.0and (b)Vz/Vxy= 0.70. We notice that the most stable conﬁguration is not aﬀected by a change in temperature or the strength of the pairing coupling. However, the eigenvalues of the B 2and E representation become larger than that of A 1aboveT/t∼1.0×10−2 (see Fig. 12(b)), thus aﬀecting the competition between the A1and B1conﬁgurations. Otherwise, the analysis at diﬀerent temperatures demonstrate that even for larger values of the pairing interaction the phase diagram is not much aﬀected. Concerning the role of the intra-orbital spin-singlet pairing, in Ref. [1], we can classify the possible ir- reducible representations for the tetragonal group C4v0 1 200.050.10.150.2 0 1 200.511.52 0 1 200.250.50.751 Vz / Vxy Vz / Vxy(a) (c)the eigenvalueVxy / t = −0.10 Vxy / t = −1.0 A1 representation B1 representationA2 representation B2 representation E representation Vz / Vxy(b)Vxy / t = −0.5 FIG. 11. The eigenvalues for various irreducible represen- tations as a function of Vz/Vxyat (a)Vxy/t=−0.10, (b) Vxy/t=−0.50and (c)Vxy/t=−1.0, assuming that T/t= 1.0×10−5,λSO/t= 0.10,∆is/t= 0.20, andµ/t= 0.0. 10−510−410−310−210−110000.20.40.60.81 10−510−410−310−210−1100 T / tthe eigenvalueA1 representation B1 representationA2 representation B2 representation E representation (a) (b)Vz / Vxy = 1.00 Vz / Vxy = 0.70 T / t FIG. 12. The eigenvalues for various irreducible represent a- tions as a function of T/tat (a)Vz/Vxy= 1.0and (b)Vz/Vxy= 0.70, assuming that Vxy/t=−1.0,λSO/t= 0.10,∆is/t= 0.20, andµ/t= 0.0. assuming both isotropic inter-orbital pairing and intra- orbital ones with isotropic and anisotropic structures compatible with the symmetry conﬁguration as shown in Table II. Finally, we consider the relative strength of the attrac- tive interaction in the inter- and intra-orbital channel as due to electron-phonon coupling in a t2gmulti-orbital system. Consider the electron phonon coupling in t2gsystem, Hep=1√ N/summationdisplay k,q,m,l,l′σαm ll′(q)c† k+q,l,σck,l′,σ,(29) wheremdenotes the phonon mode, αm ll′(q)is the electron-phonon coupling constant, and landl′stands for orbital indices in the basis of yz,zx, andxy. Here,16 TABLE II. Irreducible representation of isotropic inter-o rbital superconducting states and intra-orbital spin-singlet on es with isotropic and anisotropic structures for the tetragon al groupC4v. In the columns, we report the sign of the order parameter upon a four-fold rotational symmetry transforma - tion,C4, and the reﬂection mirror symmetry Myz, as well as the explicit spin and orbital structure of the gap function. In the E representation, +and−of the subscript mean the dou- bly degenerate mirror-even ( +) and mirror-odd ( −) solutions, respectively. C4vC4Myzorbital basis function A1+ +(dyz,dyz) ψ(yz,yz)=const. (dzx,dzx) ψ(zx,zx)=ψ(yz,yz) (dxy,dxy) ψ(xy,xy)=const. (dxy,dyz) d(xy,yz) y (dxy,dzx) d(xy,zx) x=−d(xy,yz) y (dyz,dzx) d(yz,zx) z A2+−(dyz,dyz)ψ(yz,yz)= sinkxsinky(coskx−cosky) (dzx,dzx)ψ(zx,zx)(kx,ky) =ψ(yz,yz)(ky,−kx) (dxy,dxy)ψ(xy,xy)= sinkxsinky(coskx−cosky) (dxy,dyz) d(xy,yz) x (dxy,dzx) d(xy,zx) y=d(xy,yz) x B1−+(dyz,dyz)ψ(yz,yz)∝coskx−cosky (dzx,dzx)ψ(zx,zx)(kx,ky) =−ψ(yz,yz)(ky,−kx) (dxy,dxy)ψ(xy,xy)∝coskx−cosky (dxy,dyz) d(xy,yz) y (dxy,dzx) d(xy,zx) x=d(xy,yz) y B2− −(dyz,dyz)ψ(yz,yz)∝sinkxsinky (dzx,dzx)ψ(zx,zx)(kx,ky)=−ψ(yz,yz)(ky,−kx) (dxy,dxy)ψ(xy,xy)∝sinkxsinky (dxy,dyz) d(xy,yz) x (dxy,dzx) d(xy,zx) y=−d(xy,yz) x (dyz,dzx) ψ(yz,zx) E±i±(dxy,dyz) ψ(xy,yz),d(xy,yz) z (dxy,dzx)ψ(xy,zx) +=∓id(xy,yz) z+ d(xy,zx) z−=∓iψ(xy,yz) − (dyz,dzx) d(yz,zx) x ,d(yz,zx) ywe consider only the diagonal elements, which are rele- vant to the attractive interaction. Oﬀ-diagonal ones are relevant to the pair hopping, which enhance the transi- tion temperature. Hep=1√ N/summationdisplay k,q,m,l,σαm ll(q)c† k+q,l,σck,l,σ.(30) The eﬀective interaction in the Eliashberg equation due to this electron-phonon coupling is given by Vm ll′(q,ωn) =−αm ll(q)αm l′l′(q)Dm(q,ωn),(31) whereDm(q,ωn)is the Green’s function of phonon Dm(q,ωn) =2ωm(q) ω2m(q)+ω2n, (32) with phonon’s frequency ωm(q)and bosonic Matsubara frequencyωn= 2nπkBT. Here, we suppose the A 1modes are the most relevant to the interaction. In A 1modes, αm yz,yz(q) =αm zx,zx(q) (33) in the tetragonal symmetry. Then, we have the relation Vm yz,yz(q,ωn) =Vm zx,zx(q,ωn) =Vm yz,zx(q,ωn) =Vm zx,yz(q,ωn). This means that the eﬀective inter and intraorbital at- tractive interaction in these two orbitals are the same. We can also include xy-orbitals into this relation when ∆tand∆isis small and the symmetry goes toward the cubic. 1M. Sigrist and K. Ueda, Rev. Mod. Phys. 63, 239 (1991). 2Y. Maeno, H. Hashimoto, K. Yoshida, S. Nishizaki, T. Fu- jita, J. G. Bednorz, and F. Lichtenberg, Nature 372, 532 (1994). 3H. Tou, Y. Kitaoka, K. Ishida, K. Asayama, N. Kimura, Y. Onuki, E. Yamamoto, Y. Haga, and K. Maezawa, Phys. Rev. Lett. 80, 3129 (1998). 4S. Kashiwaya, H. Kashiwaya, H. Kambara, T. Furuta, H. Yaguchi, Y. Tanaka, and Y. Maeno, Phys. Rev. Lett. 107, 077003 (2011). 5L. J. Buchholtz and G. Zwicknagl, Phys. Rev. B 23, 5788 (1981).6J. Hara and K. Nagai, Prog. Theor. Phys. 76, 1237 (1986). 7C. R. Hu, Phys. Rev. Lett. 72, 1526 (1994). 8Y. Tanaka and S. Kashiwaya, Phys. Rev. Lett. 74, 3451 (1995). 9S. Kashiwaya and Y. Tanaka, Rep. Prog. Phys. 63, 1641 (2000). 10T. Löfwander, V. S. Shumeiko, and G. Wendin, Super- cond. Sci. Technol. 14, R53 (2001). 11H. Kwon, K. Sengupta, and V. Yakovenko, Eur. Phys. J. B37, 349 (2004). 12A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, Phys. Rev. B 78, 195125 (2008).17 13S. Ryu, A. P. Schnyder, A. Furusaki, and A. Ludwig, New J. Phys. 12, 065010 (2010). 14X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011). 15Y. Tanaka, M. Sato, and N. Nagaosa, J. Phys. Soc. Jpn. 81, 011013 (2012). 16M. Leijnse and K. Flensberg, Semiconductor Science and Technology 27, 124003 (2012). 17C. W. J. Beenakker, Annu. Rev. Condens. Matter Phys. 4, 113 (2013). 18M. Sato and S. Fujimoto, J. Phys. Soc. Jpn. 85, 072001 (2016). 19M. Sato and Y. Ando, Rep. Prog. Phys. 80, 076501 (2017). 20A. Y. Kitaev, Usp. Fiz. Nauk (Suppl.) 171, 131 (2001). 21C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Rev. Mod. Phys. 80, 1083 (2008). 22F. Wilczek, Nat. Phys. 5, 614 (2009). 23J. Alicea, Rep. Prog. Phys. 75, 076501 (2012). 24R. Aguado, La Rivista del Nuovo Cimento 40, 523 (2017). 25F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Rev. Mod. Phys. 77, 1321 (2005). 26A. I. Buzdin, Rev. Mod. Phys. 77, 935 (2005). 27T. S. Khaire, M. A. Khasawneh, W. P. Pratt, and N. O. Birge, Phys. Rev. Lett. 104, 137002 (2010). 28W. A. Robinson, J. D. S. Witt, and M. G. Blamire, Sci- ence329, 59 (2010). 29M. Eschrig, J. Kopu, J. C. Cuevas, and G. Schön, Phys. Rev. Lett. 90, 137003 (2003). 30M. Eschrig and T. Löfwander, Nature Physics 4, 138 (2008). 31J. Linder and J. W. A. Robinson, Nat. Phys. 11, 307 (2015). 32P. Gentile, M. Cuoco, A. Romano, C. Noce, D. Manske, and P. M. R. Brydon, Phys. Rev. Lett. 111, 097003 (2013). 33D. Terrade, D. Manske, and M. Cuoco, Phys. Rev. B 93, 104523 (2016). 34M. T. Mercaldo, M. Cuoco, and P. Kotetes, Phys. Rev. B94, 140503(R) (2016). 35A. Romano, P. Gentile, C. Noce, I. Vekhter, and M. Cuoco, Phys. Rev. Lett. 110, 267002 (2013). 36A. Romano, P. Gentile, C. Noce, I. Vekhter, and M. Cuoco, Phys. Rev. B 93, 014510 (2016). 37A. Romano, C. Noce, I. Vekhter, and M. Cuoco, Phys. Rev. B 96, 054512 (2017). 38D. Fay and J. Appel, Phys. Rev. B 22, 3173 (1980). 39C. Pﬂeiderer, Rev. Mod. Phys. 81, 1551 (2009). 40A. P. Mackenzie and Y. Maeno, Rev. Mod. Phys. 75, 657 (2003). 41I. Martin and A. F. Morpurgo, Phys. Rev. B 85, 144505 (2012). 42K. Pöyhönen, A. Westström, J. Röntynen, and T. Oja- nen, Phys. Rev. B 89, 115109 (2014). 43F. Pientka, L. I. Glazman, and F. von Oppen, Phys. Rev. B88, 155420 (2013). 44B. Braunecker and P. Simon, Phys. Rev. Lett. 111, 147202 (2013). 45J. Klinovaja, P. Stano, A. Yazdani, and D. Loss, Phys. Rev. Lett. 111, 186805 (2013). 46Y. Kim, M. Cheng, B. Bauer, R. M. Lutchyn, and S. Das Sarma, Phys. Rev. B 90, 060401 (2014). 47F. Pientka, L. I. Glazman, and F. von Oppen, Phys. Rev. B89, 180505 (2014). 48S. Nadj-Perge, I. K. Drozdov, B. A. Bernevig, and A. Yaz-dani, Phys. Rev. B 88, 020407 (2013). 49S. Nakosai, Y. Tanaka, and N. Nagaosa, Phys. Rev. B 88, 180503 (2013). 50A. Heimes, P. Kotetes, and G. Schön, Phys. Rev. B 90, 060507 (2014). 51D. Mendler, P. Kotetes, and G. Schön, Phys. Rev. B 91, 155405 (2015). 52J. D. Sau, R. M. Lutchyn, S. Tewari, and S. DasSarma, Phys. Rev. Lett. 104, 040502 (2010). 53J. Alicea, Phys. Rev. B 81, 125318 (2010). 54R. M. Lutchyn, J. D. Sau, and S. DasSarma, Phys. Rev. Lett.105, 077001 (2010). 55Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev. Lett. 105, 177002 (2010). 56G. Volovik, The Universe in a Helium Droplet (Oxford Science Publications, New York, 2003). 57Y. S. Hor, A. J. Williams, J. G. Checkelsky, P. Roushan, J. Seo, Q. Xu, H. W. Zandbergen, A. Yazdani, N. P. Ong, and R. J. Cava, Phys. Rev. Lett 104, 057001 (2010). 58L. A. Wray, S. Y. Xu, Y. Xia, Y. S. Hor, D. Qian, A. V. Fedorov, H. Lin, A. Bansil, R. J. Cava, and M. Z. Hasan, Nature Phys. 6, 855 (2010). 59M. Kriener, K. Segawa, Z. Ren, S. Sasaki, and Y. Ando, Phys.Rev. Lett. 106, 127004 (2011). 60L. Fu and E. Berg, Phys. Rev. Lett. 105, 097001 (2010). 61S. Sasaki, M. Kriener, K. Segawa, K. Yada, Y. Tanaka, M. Sato, and Y. Ando, Phys. Rev. Lett. 107, 217001 (2011). 62L. Hao and T. K. Lee, Phys. Rev. B 83, 134516 (2011). 63A. Yamakage, K. Yada, M. Sato, and Y. Tanaka, Phys. Rev. B 85, 180509 (2012). 64M. Oudah, A. Ikeda, J. N. Hausmann, S. Yonezawa, T. Fukumoto, S. Kobayashi, M. Sato, and Y. Maeno, Nat. Commun. 7, 13617 (2016). 65L. Aggarwal, A. Gaurav, G. S. Thakur, Z. Haque, A. K. Ganguli, and G. Sheet, Nat. Matter 15, 32 (2015). 66H. Wang, H. Wang, H. Liu, H. Lu, W. Yang, S. Jia, X.-J. Liu, X. C. Xie, J. Wei, and J. Wang, Nat. Matter 15, 38 (2015). 67E. I. Rashba, Sov. Phys. Solid State 2, 1109 (1960). 68G. Dresselhaus, Phys. Rev. 100, 580 (1955). 69L. P. Gorkov and E. I. Rashba, Phys. Rev. Lett. 87, 037004 (2001). 70P. A. Frigeri, D. F. Agterberg, A. Koga, and M. Sigrist, Phys. Rev. Lett. 92, 097001 (2004). 71K. Yada, S. Onari, Y. Tanaka, and J.-i. Inoue, Phys. Rev. B 80, 140509 (2009). 72C. K. Lu and S. Yip, Phys. Rev. B 78, 132502 (2008). 73A. B. Vorontsov, I. Vekhter, and M. Eschrig, Phys. Rev. Lett.101, 127003 (2008). 74Y. Tanaka, T. Yokoyama, A. V. Balatsky, and N. Na- gaosa, Phys. Rev. B 79, 060505(R) (2009). 75M. Sato and S. Fujimoto, Phys. Rev. B 79, 094504 (2009). 76A. P. Schnyder, P. M. R. Brydon, D. Manske, and C. Timm, Phys. Rev. B 82, 184508 (2010). 77Z.-J. Ying, M. Cuoco, C. Ortix, and P. Gentile, Phys. Rev. B 96, 100506 (R) (2017). 78A. M. Oleś, J. Phys.: Condens. Matter 24, 313201 (2012). 79H. Y. Hwang, Y. Iwasa, M. Kawasaki, B. Keimer, N. Na- gaosa, and Y. Tokura, Nat. Mater. 11, 103 (2012). 80G. Khalsa, B. Lee, and A. H. MacDonald, Phys. Rev. B 88, 041302 (2013). 81C. Autieri, M. Cuoco, and C. Noce, Phys. Rev. B 89, 075102 (2014).18 82R. Winkler, Spin-Orbit Coupling Eﬀects in Two- Dimensional Electron and Hole Systems (Springer Berlin Heidelberg, 2003). 83Z. Zhong, A. Tóth, and K. Held, Phys. Rev. B 87, 161102(R) (2013). 84Y. Kim, R. M. Lutchyn, and C. Nayak, Phys. Rev. B 87, 245121 (2013). 85A. Joshua, S. Pecker, J. Ruhman, E. Altman, and S. Ilani, Nat. Comm. 3, 1129 (2012). 86K. Steﬀen, F. Loder, and T. Kopp, Phys. Rev. B 91, 075415 (2015). 87A. Ohtomo and H. Y. Hwang, Nature (London) 427, 423 (2004). 88J. Zabaleta, V. S. Borisov, R. Wanke, H. O. Jeschke, S. C. Parks, B. Baum, A. Teker, T. Harada, K. Syassen, T. Kopp, N. Pavlenko, R. Valentí, and J. Mannhart, Phys. Rev. B 93, 235117 (2016). 89M. Salluzzo, J. C. Cezar, N. B. Brookes, V. Bisogni, G. M. D. Luca, C. Richter, S. Thiel, J. Mannhart, M. Hui- jben, A. Brinkman, G. Rijnders, and G. Ghiringhelli, Phys. Rev. Lett. 102, 166804 (2009). 90Sugano, Y. Tanabe, and H. Kamimura, Multiplets of transition-metal ions in crystal (Academic Press, New York London, 1970). 91J. Slater, Quantum Theory of Atomic Structure (McGraw- Hill, New York, 1960). 92L. Vaugier, H. Jiang, and S. Biermann, Phys. Rev. B 86, 165105 (2012). 93I. M. Lifshitz, Sov. Phys. JETP 11, 1130 (1960). 94S. Benhabib, A. Sacuto, M. Civelli, I. Paul, M. Cazayous, Y. Gallais, M.-A. Méasson, R. D. Zhong, J. Schneeloch, G. D. Gu, D. Colson, and A. Forget, Phys. Rev. Lett. 114, 147001 (2015). 95M. R. Norman, J. Lin, and A. J. Millis, Phys. Rev. B 81, 180513(R) (2010). 96E. A. Yelland, J. M. Barraclough, W. Wang, K. V. Kamenev, and A. D. Huxley, Nat. Phys. 7, 890 (2011). 97X. Shi, Z.-Q. Han, X.-L. Peng, P. Richard, T. Qian, X.- X. Wu, M.-W. Qiu, S. Wang, J. P. Hu, Y.-J. Sun, and H. Ding, Nat. Comm. 8, 14988 (2017). 98M. Ren, Y. Yan, X. Niu, R. Tao, D. Hu, R. Peng, B. Xie, J. Zhao, T. Zhang, and D.-L. Feng, Sci. Adv. 3, 1603238 (2017). 99S. N. Khan and D. D. Johnson, Phys. Rev. Lett. 112, 156401 (2014). 100C. Liu, A. D. Palczewski, R. S. Dhaka, T. Kondo, R. M. Fernandes, E. D. Mun, H. Hodovanets, A. N. Thaler, J. Schmalian, S. L. Bud’ko, P. C. Canﬁeld, and A. Kamin- ski, Phys. Rev. B 84, 020509(R) (2011). 101S. Tewari and J. D. Sau, Phys. Rev. Lett. 109, 150408 (2012). 102K. Yada, M. Sato, Y. Tanaka, and T. Yokoyama, Phys. Rev. B. 83, 064505 (2011). 103M. Sato, Y. Tanaka, K. Yada, and T. Yokoyama, Phys. Rev. B 83, 224511 (2011). 104P. M. R. Brydon, A. P. Schnyder, and C. Timm, Phys. Rev. B 84, 020501 (2011). 105W. Brzezicki and M. Cuoco, Phys. Rev. B 97, 064513 (2018). 106A. M. Black-Schaﬀer and A. V. Balatsky, Phys. Rev. B 88, 104514 (2013).107A. M. Black-Schaﬀer and A. V. Balatsky, Phys. Rev. B 87, 220506 (2013). 108L. Komendová, A. V. Balatsky, and A. M. Black-Schaﬀer, Phys. Rev. B 92, 094517 (2015). 109Y. Asano and A. Sasaki, Phys. Rev. B 92, 224508 (2015). 110L. Komendová and A. M. Black-Schaﬀer, Phys. Rev. Lett. 119, 087001 (2017). 111N. Reyren, S. Thiel, A. D. Caviglia, L. F. Kourkoutis, G. Hammerl, C. Richter, C. W. Schneider, T. Kopp, A.- S. Ruetschi, D. Jaccard, M. Gabay, D. A. Muller, J.-M. Triscone, J. Mannhart, F. L. Kourkoutis, and G. Ham- merl, Science 317, 1196 (2007). 112C. Cen, S. Thiel, G. Hammerl, C. W. Schneider, K. E. Andersen, C. S. Hellberg, J. Mannhart, and J. Levy, Nat. Mater. 7, 298 (2008). 113A. D. Caviglia, S. Gariglio, N. Reyren, D. Jac- card, T. Schneider, M. Gabay, S. Thiel, G. Hammerl, J. Mannhart, and J.-M. Triscone, Nature 456, 624 (2008). 114T. Schneider, A. D. Caviglia, S. Gariglio, N. Reyren, and J.-M. Triscone, Phys. Rev. B 79, 184502 (2009). 115L. Kuerten, C. Richter, N. Mohanta, T. Kopp, A. Kampf, J. Mannhart, and H. Boschker, Phys. Rev. B 96, 014513 (2017). 116X. Liu, J. D. Sau, and S. Das Sarma, Phys. Rev. B 92, 014513 (2015). 117Z. Huang, P. Woelﬂe, and A. V. Balatsky, Phys. Rev. B 92, 121404(R) (2015). 118P. Burset, B. Lu, G. Tkachov, Y. Tanaka, E. Hankiewicz, and B. Trauzettel, Phys. Rev. B 92, 205424 (2015). 119S.-P. Lee, R. M. Lutchyn, and J. Maciejko, Phys. Rev. B 95, 184506 (2017). 120O. Kashuba, B. Sothmann, P. Burset, and B. Trauzettel, Phys. Rev. B 95, 174516 (2017). 121J. Cayao and A. M. Black-Schaﬀer, Phys. Rev. B 96, 155426 (2017). 122Y. Tanaka and S. Kashiwaya, Phys. Rev. B 70, 012507 (2004). 123Y. Tanaka, S. Kashiwaya, and T. Yokoyama, Phys. Rev. B71, 094513 (2005). 124Y. Asano, Y. Tanaka, and S. Kashiwaya, Phys. Rev. Lett. 96, 097007 (2006). 125Y. Tanaka, Y. Asano, A. A. Golubov, and S. Kashiwaya, Phys. Rev. B 72, 140503(R) (2005). 126Y. Tanaka and A. A. Golubov, Phys. Rev. Lett. 98, 037003 (2007). 127Y. Tanaka, A. A. Golubov, S. Kashiwaya, and M. Ueda, Phys. Rev. Lett. 99, 037005 (2007). 128Y. Tanaka, Y. Tanuma, and A. A. Golubov, Phys. Rev. B76, 054522 (2007). 129H. Ebisu, K. Yada, H. Kasai, and Y. Tanaka, Phys. Rev. B91, 054518 (2015). 130D. Stornaiuolo, D. Massarotti, R. Di Capua, P. Lu- cignano, G. P. Pepe, M. Salluzzo, and F. Tafuri, Phys. Rev. B 95, 140502 (2017). 131Y. Tanaka and S. Kashiwaya, Phys. Rev. B 53, R11957 (1996). 132Y. S. Barash, H. Burkhardt, and D. Rainer, Phys. Rev. Lett.77, 4070 (1996). 133Y. Tanaka and S. Kashiwaya, Phys. Rev. B 56, 892 (1997).
1703.08405v2.Gate_control_of_the_spin_mobility_through_the_modification_of_the_spin_orbit_interaction_in_two_dimensional_systems.pdf	Gate control of the spin mobility through the modication of the spin-orbit interaction in two-dimensional systems M. Luengo-Kovac,1F. C. D. Moraes,2G. J. Ferreira,3A. S. L. Ribeiro,2 G. M. Gusev,2A. K. Bakarov,4V. Sih,1and F. G. G. Hernandez2, 1Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, United States 2Instituto de F sica, Universidade de S~ ao Paulo, S~ ao Paulo, SP 05508-090, Brazil 3Instituto de F sica, Universidade Federal de Uberl^ andia, Uberl^ andia, MG 38400-902, Brazil 4Institute of Semiconductor Physics and Novosibirsk State University, Novosibirsk 630090, Russia (Dated: November 14, 2021) Spin drag measurements were performed in a two-dimensional electron system set close to the crossed spin helix regime and coupled by strong intersubband scattering. In a sample with uncom- mon combination of long spin lifetime and high charge mobility, the drift transport allows us to determine the spin-orbit eld and the spin mobility anisotropies. We used a random walk model to describe the system dynamics and found excellent agreement for the Rashba and Dresselhaus couplings. The proposed two-subband system displays a large tuning lever arm for the Rashba constant with gate voltage, which provides a new path towards a spin transistor. Furthermore, the data shows large spin mobility controlled by the spin-orbit constants setting the eld along the direction perpendicular to the drift velocity. This work directly reveals the resistance experienced in the transport of a spin-polarized packet as a function of the strength of anisotropic spin-orbit elds. The pursuit for a new active electronic component based on ow of spin, rather than that of charge, strongly motivates research in semiconductor spintronics [1{5]. Since the Datta-Das proposal for a ballistic spin tran- sistor, full electrical control of the spin state was sug- gested using the gate-tunable Rashba spin-orbit interac- tion (SOI) [6{10]. Further studies, including the Dressel- haus SOI [11], were made to assure a nonballistic tran- sistor robust against spin-independent scattering [12{14]. For example, it has been demonstrated that SU(2) spin rotation symmetry, preserving the spin polarization, can be obtained in the persistent spin helix (PSH) formed when the strengths of the Rashba and Dresselhaus SOI are equal (=) [15{19]. This is possible because the uniaxial alignment of the spin-orbit eld suppresses the relaxation mechanism when the spins precess about this eld while experiencing momentum scattering [20]. Gate control of this symmetry point was experimentally ob- served [21{23] and allowed to produce a transition to the PSH(=) in the same subband [24]. Drift in those systems showed surprising properties [25, 26] such as the current-control of the temporal spin-precession frequency [27]. Although the helical spin-density texture could be even transported without dissipation under certain con- ditions [15], the spin transport suers additional resis- tance from the spin Coulomb drag [28{32]. These fric- tional forces appear as a lower mobility for spins than for charge and studies in new systems are still necessary to understand this important constraint for future devices. A two-dimensional electron gas (2DEG) hosted in a quantum well (QW) with two occupied subbands oers unexplored opportunities for the study of spin transport [33, 34]. Theoretically, the inter- and intra-subband spin- orbit couplings (SOCs) have been extensively studied[35{38]. In terms of a random walk model (RWM) [39], the spin drift and diusion was recently developed for these systems displaying two possible scenarios regarding the intersubband scattering (ISS) rate [40]. The interplay between the two subbands may introduce new features to the PSH dynamics, for example, a crossed persistent spin helix [41] may arise when the subbands are set to orthog- onal PSHs (i.e., 1=1and2=2) in the weak ISS limit. In this report, we experimentally study spin drag in a system with the two-subbands individually set close to the PSH+and PSH, but with strong ISS, where the dynamics is given by the averaged SOCs of both sub- bands. The combination of long spin lifetime and high charge mobility allows us to determine the spin mobility and the spin-orbit eld anisotropies with the application of an accelerating in-plane voltage. We are able to con- trol the SOCs in both subbands and to show a linear dependence for the sum of the Rashba constants with gate voltage. Finally, we determine an inverse relation for the spin mobility dependence on the SOCs directly revealing the resistance experienced in the transport of a spin-polarized packet as a function of the strength of anisotropic spin-orbit elds. The sample consists of a single 45 nm wide GaAs QW grown in the [001] ( z) direction and symmetrically doped. Due to the Coulomb repulsion of the electrons, the charge distribution experiences a soft barrier inside the well. Figure 1(a) shows the calculated QW band prole and charge density for both subbands. The electronic system has a conguration with symmetric and antisymmetric wave functions for the two lowest subbands with sub- band separation of SAS = 2 meV. The subband den- sity (n 1= 3.7, n 2= 3.31011cm2) was obtained from the Shubnikov-de Hass (SdH) oscillations as shown inarXiv:1703.08405v2 [cond-mat.mes-hall] 18 May 20172 0102030405060 00.511.522.53 00.511.522.53Rxx (:)Rxy (k:) Bext (T)6789FFT f (T)n2n1246-15-10-50-0.100.10.2n (x1011 cm-2)Vg (V)Ez (V/Pm)n1n2n1+n2-20-100-0.100.10.2E (meV)Ez (V/Pm)H1H2H3-20020z (nm)(a)(b) (d)(c) FIG. 1. (a) Longitudinal ( Rxx) and Hall ( Rxy) magnetoresis- tance of the two-subband QW. From the SdH periodicity, one can obtain the subbands density nin the lower inset. The top inset shows the potential prole and subbands charge den- sity calculated from the self-consistent solution of Schr odinger and Poisson equations for Ez=0. (b) Subband energy levels and (c) electron concentration dependence on V gandEz. (d) Geometry of the device and contacts conguration. Fig. 1(a) and the low-temperature charge mobility was 2.2106cm2/Vs [42]. A device was fabricated in a cross- shaped conguration with width of w=270m and chan- nels along the [1 10] (x) and [110] (y) directions. Lateral Ohmic contacts deposited l=500m apart were used to apply an in-plane voltage ( Vip) in order to induce drift transport. For the ne tuning of the subband SOCs, a semitransparent contact on top of the mesa structure (Vg) was used to modify structural symmetry and sub- band occupation. The eect of Vgon the subband energy levels () and densities ( n) is shown in Fig. 1(b) and (c) as a function of the out-of-plane electric eld ( Ez). Note that the total density changes linearly with Vgand thatVg=0 corresponds to a built-in electric eld of 0.15 V/m. Figure 1(d) displays the experimental scheme with the connection of VipandVg[43]. To describe the magnetization dynamics and the mea- sured SO elds for our two-subband system, we combine the calculated SOCs with RWM [39, 40, 44]. For a [001] GaAs 2DEG, the x and y components of the SO elds for each subband =f1;2gare BSO;(k) =2 gB0 B@ ++1;+ 23;k2 xk2 y k2 ky +1;23;k2 xk2 y k2 kx1 CA; (1) plus corrections due to the intersubband SOCs [23, 35{ 38, 41]. Above, g=0:44 is the electron g-factor for GaAs andBis the Bohr magneton. The SOCs are the usual Rashba and linear1;and cubic3;Dressel- haus terms. Considering the strong intersubband scat- tering (ISS) regime of the RWM [40], the randomization of the momenta k(within the Fermi circle k=kF) and subbandis much faster than the spin precession. Con- sequently, the dynamics is governed by an averaged SOC -3-2-10123 -0.3-0.2-0.100.10.20.3DX/EX Ez (V/Pm)2ndsubband 1stsubbandPSH+PSH--40-2002040-40-2002040x (Pm)y (Pm)-40-2002040-40-2002040x (Pm)y (Pm) -40-2002040-40-2002040 x (Pm)y (Pm)-40-2002040-40-2002040 x (Pm)y (Pm)-40-2002040-40-2002040 x (Pm)y (Pm)-2-10123 -0.3-0.1500.150.3SOC (meVA)D1D2K6DXEz (V/Pm)-1012 -0.3-0.1500.150.3E1,2E1,1Ez (V/Pm)0.40.60.81 -0.3-0.1500.150.3Ez (V/Pm)E3,1E3,26EX(a)(b) (d)(c) (e)1stPSH+2ndPSH- Ez=0Ez=0.15Ez=0.3FIG. 2. (a)-(c) Calculated SOCs for the Rashba ( ), linear (1;) and cubic ( 3;) Dresselhaus for each subband =f1;2g, as well as intersubband SOCs and as a func- tion ofEz. The purple lines give the sum of and . (d) The ratio==1 when the subband is set to the PSHregime. The insets show the single-subband magne- tization maps on the xyplane for the PSHregimes, and the self-consistent potentials and subband densities for the respectiveEz. (e) Two-subband magnetization maps in the strong ISS regime for dierent Ez. AtEz= 0 the well is sym- metric (= 0) and the magnetization shows an isotropic Bessel pattern. For nite Ezthe broken symmetry leads to the stripped PSH pattern in accordance with the pos- itive ratioP=P[purple line in panel (d)]. The ar- rows in the Fermi circle show the rst harmonic component ofPBSO;(k), illustrating the transition from isotropic to uniaxial eld with increasing Ez. All the xy maps are frames of the spin pattern at t = 13 ns. eldhBSOi= (hBx SOi;hBy SOi) transverse to the drift ve- locity vdr= (vx dr;vy dr) Namely, the eld components read hBx SOi=hm ~gB2X =1(++ )i vy dr; (2) hBy SOi=hm ~gB2X =1(+ )i vx dr; (3) where =1;23;, andm= 0:067m0is the eec- tive electron mass for GaAs and ~is Planck's constant. Since Bx(y) SO/vy(x) dr, it is convenient to analyze the linear coecients bx(y)= By(x) SO=vx(y) dr, which are given by the3 terms between square brackets above. The intra- and intersubband SOCs are calculated within the self-consistent Hartree approximation [35{38] for GaAs quantum wells tilted by Ez. The chemical potential is set to return the density n=n1+n2= 71011cm2forEz= 0, while it varies linearly for niteEzin Fig. 1(c). The SOCs are dened from the matrix elements ;0=hjwV0+HV0 Hj0iand ;0= hjk2 zj0i, wherejiis the eigenket for subband ,w= 3:47A2andH= 5:28A2are bulk coecients [35{38, 45], V0=@zV(z) andV0 H=@zVH(z) are the derivatives of the heterostructure and Hartree potentials alongz, = 11 eV A3is the bulk Dresselhaus constant, andkzis the z-component of the momentum. The usual intrasubband Rashba and linear Dresselhaus SOCs are =;and1;= ;. The non-diagonal terms are the intersubband SOCs =12and = 12. The calcu- lated SOCs, plotted in Fig. 2(a)-(c) as a function of Ez, show agreement with previous studies [46, 47]. The high- densitynmakes the cubic Dresselhaus 3; n=2 comparable with 1;, strongly aecting the PSH tuning [17]=, with=1;3;. NearEz0:04 V/m, the SOCs reach almost simul- taneously the balanced condition for the PSH+in the rst subband ( 1=1= +1) and for the PSHin the second subband ( 2=2=1), as shown by the ratio =in Fig. 2(d). The expected magnetization pat- terns for the single-subband PSH is shown in the inset of Fig. 2(d). The PSHshows more stripes than the PSH+due to the higher value of , which grows quickly within the Ezrange. However, the ratio of the aver- aged SOCs (P)=(P) approaches the PSH regimes only forjEzj>0:3 V/m. As we will see next, the experimental data matches well the strong ISS regime of the RWM, therefore the dynamics is governed by the averaged SOCs. In this case, the expected magnetiza- tion patterns are shown in Fig. 2(e). With increasing Ez the system transitions from isotropic ( Ez= 0) to uniax- ial (Ez>0:3 V/m), as indicated by the formation of stripes and the orientation of the rst harmonic compo- nent of the total eldPBSO;(k) [arrows in Fig. 2(e)]. We are interested in the determination of the anisotropy for the coecients bx(y), estimated in one or- der of magnitude in Fig. 3(a) and (b). We measured the spin polarization using time-resolved Kerr rotation as function of the space and time separation of pump and probe beams. All optical measurements were per- formed at 10 K. A mode-locked Ti:Sapphire laser with a repetition rate of 76 MHz tuned to 816.73 nm was split into pump and probe pulses. The polarization of the pump beam was controlled by a photoelastic modulator and the intensity of the probe beam was modulated by an optical chopper for cascaded lock-in detection. An electromagnet was used to apply an external magnetic eld in the plane of the QW. The spatial positioning of the pump relative to the probe (d) was controlled using 00.511.52 -50-250255075A(d) (a.u.)d (Pm)dc = 27.1 Pm0255075vdr \|\| yVip (mV) 0123 -40-30-20-10010203040Bext (mT)0IK (a.u.)255075vdr \|\| yVip (mV)0246810 0246810 00.511.522.53i (mA)BSO (mT)vdr (Pm/ns) by=(3.0 0.3) mTns/Pmbx=(0.4 0.2) mTns/Pmvdr \|\| yvdr \|\| x00.511.522.53 01020304050607080vdr (Pm/ns)Pys=(1.4 0.1) x105cm2/VsPxs=(2.2 0.1)x105cm2/VsEip (V/m)020406080100120140160 Vip (mV)(a)(b) (f)(e)± ± ±±(d)(c)bxbyFIG. 3. Calculated coecients b with vdrparallel to (a) x and (b) y for each subband (colored) and total eld (black). (c) Amplitude of the drifting spin polarization in space show- ing, for example, the center of the packet d cfor 75 mV. (d) Linear dependence of v drwith the channel V ip. The slope gives the spin mobility along v drin x or y. (e) Field scan ofKfor several V ipmeasured at d c. (f) By(x) SOas function of vx(y) drand the current owing in that channel. The slopes bx(y)give the strength of the SOCs that generate the eld along y(x) for drift in x(y). The solid lines are gaussian (c) and linear (d and f) ttings. Scans taken at t=13 ns. a scanning mirror. We dened the spin injection point to be x=y=0 at t=0. The application of an in-plane electric eld (E ip=Vip/l), in the x or y-oriented channel, adds a drift velocity to the 2DEG electrons and allows us to determine the spin mobility and the spin-orbit eld components [48{50]. The sample was rotated such that each channel un- der study was oriented parallel to the external magnetic eldBextkvdrfor all measurements reported here. From the SOI form in k-space, we expected BSO?vdrimply- ing that the observable BSOdirection will be BSO?Bext. Considering this orientation, we can model the Kerr rota- tion signal as K(Bext;d) = A(d) cos ( !t) with the pre- cession frequency given by != (gB=~)p B2 ext+B2 SO, where A(d) is the amplitude at a given pump-probe spa- tial separation and B SOis the internal SO eld compo- nent perpendicular to Bext(and to vdr). Figure 3 shows the results of the spin drag experiment with the gate contact open. Scanning the pump-probe separation in space at xed long time delay (13 ns), we determined the central position d cof the spin packet4 amplitude for several V ipin a given crystal orientation. From the values of d cin Fig. 3(c), we calculated the drift velocity as v dr=dc/t and plotted it as a function of V ip in Fig. 3(d). The slope of the linear t give us spin mo- bilities (x;y s) in the range of 105cm2/Vs. Values in the same order of magnitude have been measured by Doppler velocimetry for the transport in single subband samples [32]. Nevertheless, in those systems the spin lifetimes were restricted to the picosecond range and the trans- port was limited to the nanometer scale. Following the drifting spin packet in space, Fig. 3(e) displays a B extscan from where changes in the amplitude of zeroth resonance determined B SOstrength at d c. As explained above, the data conrmed the perpendicular orientation between BSOandvdrand did not show a component parallel to Bextwithin the experimental res- olution [51]. From the Lorenztian shape of the B extscan [52, 53], we evaluated a spin lifetime of 7 ns at V ip=0. This experiment was only possible due to the nanosecond spin lifetime in our sample that extends the spin trans- port to several tens of micrometers [54, 55]. Figure 3(f) shows the tted values of B SOfor sev- eral Vipapplied along x and y. We observed highly anisotropic spin-orbit elds in the range of several mT as expected from Fig 3(a) and (b). The B SOorientation was aligned primary with the x axis in agreement with the simulation in Fig 2(e). The slopes bx(y)=By(x) SO/vx(y) dr give the strength of the SOCs that generate the eld ac- cording to Eqs. 2 and 3. For this condition of the sample as-grown, we foundP= 0.57 meV A andP = 0.75 meVA. Note the inverse behaviour on V ipfor the mobility and for BSOstrength in perpendicular directions. In Fig. 3(c) and (d), the axis with the largest mobility is also the axis with smaller spin-orbit eld in the perpendic- ular direction. This result may be related to the spin Coulomb drag observed previously in the transport of spin-polarized electrons [31, 32]. Next, we demonstrate the direct control of the spin mobility through the gate modication of the subband SOCs. Figure 4(a) shows that the magnitude and the orienta- tion with the largest scan be tuned by E z. BSOdisplays anisotropic components with Bx SObeing larger in all the studied range, which conrms the preferential alignment towards the PSH+in Fig. 2(e). The variation of Bx SO has a minimum (indicated by an arrow) close to position when the second subband attains the PSH(with BSO along y). Dividing Fig. 4(a) panels, the values for b are plotted in Fig. 4(b). The lines plotted together with the data are the expected values using Eq. 2 and 3 with the SOCs from Fig. 2(a)-(c). When the QW approaches the symmetric condition (E z=0), bx(y)decreases remov- ing the anisotropy of B SOas simulated in Fig. 2(e). The addition and subtraction of bxandbygive the sum of the Rashba and Dressselhaus SOCs displayed in Fig. 4(c). Dashed lines corresponding to the purple curves in -0.200.20.40.60.81 00.050.10.15SOC (meVA)6DXEz (V/Pm)6EX00.511.522.53 -0.0500.050.10.15b (mT/Pm/ns)vdr \|\| yvdr \|\| xEz (V/Pm)-4.4 eÅ235.0 eÅ20123456-0.0500.050.10.15BSO (mT)Ez (V/Pm)BSO \|\| xBSO \|\| y00.511.522.53-15-10-50Ps (x105cm2/Vs)Vg (V)vdr \|\| yvdr \|\| x(a)(b) (d)(c) 01234 00.20.40.60.811.26DX+EX), 6DX+E*X) (meVA)Ps (x105cm2/Vs)vdr \|\| yvdr \|\| xP0s-1.1x105𝑐𝑚2𝑉𝑠𝑚𝑒𝑉Å-3.4x105 𝑐𝑚2𝑉𝑠𝑚𝑒𝑉ÅFIG. 4. (a) Spin mobility and B SOas a function of the gate-tunable E z. (b) Ratio bx(y)from (a), showing a crossing at Ez=0. (c) SOCs obtained from the addition and subtrac- tion ofbxandbyin (b). (d) Spin mobility as function of the SOCs that dene the B SOstrength along the direction per- pendicular to v dr. The solid lines are linear ttings and the dashed lines (b,c) are the theoretical results from the RWM combined with the self-consistent calculation of the SOCs. Fig.2(a) and (c) are plotted together displaying excellent agreement. The slope for the Rashba SOI indicates a tun- ing lever arm of 35 e A2. This value is considerably larger than those reported in recent studies for single subband samples, typically below 10 e A2[17, 23]. Finally, Fig. 4(d) presents x(y) s[from (a)] against the SOCs den- ing By(x) SO:P(+ ) andP(+ ), respectively. This last plot illustrates the inverse dependence, with negative slope, for the spin mobility and strength of the SOCs perpendicular to the drift direction. The dierent slopes for x and y channels give us a hint that this eect depends not only on how B SOchanges with v dr(given by the SOCs) but also in the magnitude of the elds. A common maximum value 0 s=3105cm2/Vs was found independent of vdrorientation. In conclusion, we have studied a 2DEG system with two subbands set close to the crossed PSH regime under strong intersubband scattering and successfully described it using a random walk model. In the spin transport with nanosecond lifetimes over micrometer distances, we demonstrate the control of the subbands spin-orbit cou- plings with gate voltage and observed spin mobilities in the range of 105cm2/Vs. Specically, the sum of the Rashba SOCs presents a linear behaviour with remark- ably large tunability lever arm with gate voltage. We5 tailored the spin mobility by controlling the strength of the spin-orbit interaction in the direction perpendicular to the drift velocity. Our ndings provided evidence of the rich physical phenomena behind multisubband sys- tems and experimentally demonstrated relevant proper- ties required for the implementation of a nonballistic spin transistor. This work is a result of the collaboration initiative SPRINT No. 2016/50018-1 of the S~ ao Paulo Research Foundation (FAPESP) and the University of Michi- gan. F.G.G.H also acknowledges nancial support from FAPESP Grants No. 2009/15007-5, No. 2013/03450- 7, and No. 2014/25981-7 and 2015/16191-5. G.J.F. acknowledges the nancial support from CNPq and FAPEMIG. The work at the University of Michigan is supported by the National Science Foundation under Grant No. DMR-1607779. Corresponding author. Electronic address: felixggh@if.usp.br [1] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Moln ar, M. L. Roukes, A. Y. Chtchelkanova, and D. M. Treger, Science 294, 1488 (2001). [2] I. ^Zuti c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). [3] D. D. Awschalom and M. E. Flatt e, Nat. Phys. 3, 153 (2007). [4] J. Wunderlich, B.-G. Park, A. C. Irvine, L. P. Z^ arbo, E. Rozkotov a, P. Nemec, V. Nov ak, J. Sinova, and T. Jungwirth, Science 330, 1801 (2010). [5] J. C. Egues, G. Burkard, and D. Loss, Appl. Phys. Lett. 82, 2658 (2003). [6] E. I. Rashba, Sov. Phys. Solid State 2, 1109 (1960); Y. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6039 (1984); JETP Lett. 39, 78 (1984). [7] S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1990). [8] J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, Phys. Rev. Lett. 78, 1335 (1997). [9] V. Lechner, L. E. Golub, P. Olbrich, S. Stachel, D. Schuh, W. Wegscheider, V. V. Bel'kov and S. D. Ganichev, Appl. Phys. Lett. 94, 242109 (2009). [10] P. Chuang, S.-C. Ho, L. W. Smith, F. Sgakis, M. Pep- per, C.-H. Chen, J.-C. Fan, J. P. Griths, I. Farrer, H. E. Beere, G. A. C. Jones, D. A. Ritchie, and T.-M. Chen, Nat. Nanotechnol. 10, 35 (2014). [11] G. Dresselhaus, Phys. Rev. 100, 580 (1955). [12] J. Schliemann, J. C. Egues, and D. Loss, Phys. Rev. Lett . 90, 146801 (2003). [13] M. Ohno and K. Yoh, Phys. Rev. B 77, 045323 (2008). [14] Y. Kunihashi, M. Kohda, H. Sanada, H. Gotoh, T. So- gawa, and J. Nitta, Appl. Phys. Lett. 100, 113502 (2012). [15] B. A. Bernevig, J. Orenstein, and S.-C. Zhang, Phys. Rev. Lett. 97, 236601 (2006). [16] J. D. Koralek, C. Weber, J. Orenstein, B. Bernevig, S.- C. Zhang, S. Mack, and D. Awschalom, Nature 458, 610 (2009). [17] M. P. Walser, C. Reichl, W. Wegscheider, and G. Salis,Nat. Phys. 8, 757 (2012). [18] J. Schliemann, Rev. Mod. Phys. 89, 011001 (2017). [19] M. Kohda and G. Salis, Semiconductor Science and Tech- nology (2017). at press: https://doi.org/10.1088/1361- 6641/aa5dd6 [20] M. I. D'yakonov and V. I. Perel', Sov. Phys. Solid State 13, 3023 (1972). [21] M. Kohda, V. Lechner, Y. Kunihashi, T. Dollinger, P. Olbrich, C. Sch onhuber, I. Caspers, V. V. Bel'kov, L. E. Golub, D. Weiss, K. Richter, J. Nitta, and S. D. Ganichev, Phys. Rev. B 86, 081306(R) (2012). [22] J. Ishihara, Y. Ohno, and H. Ohno, Appl. Phys. Exp. 7, 013001 (2014). [23] F. Dettwiler, J. Fu, S. Mack, P. J. Weigele, J. C. Egues, D. D. Awschalom, and D. M. Zumb uhl, arXiv:1702.05190 [cond-mat.mes-hall]. [24] K. Yoshizumi, A. Sasaki, M. Kohda, and J. Nitta, Appl. Phys. Lett. 108, 132402 (2016). Here, the PSHis la- beled as inverse PSH (iPSH). [25] L. Yang, J. D. Koralek, J. Orenstein, D. R. Tibbetts, J. L. Reno, and M. P. Lilly Phys. Rev. Lett. 109, 246603 (2012). [26] Y. Kunihashi, H. Sanada, H. Gotoh, K. Onomitsu, M. Kohda, J. Nitta, and T. Sogawa, Nat. Commun. 7, 10722 (2016). [27] P. Altmann, F. G. G. Hernandez, G. J. Ferreira, M. Ko- hda, C. Reichl, W. Wegscheider, and G. Salis, Phys. Rev. Lett.116, 196802 (2016). [28] I. D'Amico and G. Vignale, Europhys. Lett. 55, 566 (2001). [29] I. D'Amico and G. Vignale, Phys. Rev. B 68, 045307 (2003). [30] W.-K. Tse and S. Das Sarma, Phys. Rev. B 75, 045333 (2007). [31] C. P. Weber, N. Gedik, J. E. Moore, J. Orenstein, J. Stephens, and D. D. Awschalom, Nature 437, 1330 (2005). [32] L. Yang, J. D. Koralek, J. Orenstein, D. R. Tibbetts, J. L. Reno, and M. P. Lilly Nat. Phys. 8, 153 (2012). [33] F. G. G. Hernandez, L. M. Nunes, G. M. Gusev, and A. K. Bakarov, Phys. Rev. B 88, 161305(R) (2013). [34] F. G. G. Hernandez, G. M. Gusev, and A. K. Bakarov, Phys. Rev. B 90, 041302(R) (2014). [35] E. Bernardes, J. Schliemann, M. Lee, J. C. Egues, and D. Loss, Phys. Rev. Lett. 99, 076603 (2007). [36] E. Bernardes, J. Schliemann, C. Egues, and D. Loss, Phys. Status Solidi C 3, 4330 (2006). [37] R. S. Calsaverini, E. Bernardes, J. C. Egues, and D. Loss, Phys. Rev. B 78, 155313 (2008). [38] J. Fu and J. C. Egues, Phys. Rev. B 91, 075408 (2015). [39] L. Yang, J. Orenstein, and D.-H. Lee, Phys. Rev. B 82, 155324 (2010). [40] G. J. Ferreira, F. G. G. Hernandez, P. Altmann, G. Salis, Phys. Rev. B 95, 125119 (2017). [41] J. Fu, P. H. Penteado, M. O. Hachiya, D. Loss, and J. C. Egues, Phys. Rev. Lett. 117, 226401 (2016). [42] The barriers were made of short-period AlAs/GaAs su- perlattices in order to shield the doping ionized impuri- ties and eciently enhance the mobility, more details in K.-J. Friedland, R. Hey, H. Kostial, R. Klann, and K. Ploog, Phys. Rev. Lett. 77, 4616 (1996). [43] The measured low-temperature resistance was 10 for the sample and 100 for the lateral Ohmic contacts. Thus, the eective V ipin the channel is about 10 times6 smaller than the applied voltage. The values of V ipused for the plots and calculations are the eective ones. [44] The model neglects changes in the intersubband scattering-time that depends on the QW symmetry as studied in: N. C. Mamani, G. M. Gusev, T. E. Lamas, and A. K. Bakarov, O. E. Raichev, Phys. Rev. B 77, 205327 (2008); N. C. Mamani, G. M. Gusev, E. C. F. da Silva, O. E. Raichev, A. A. Quivy, and A. K. Bakarov, Phys. Rev. B 80, 085304 (2009). [45] R. Winkler, Spin-Orbit Coupling Eects in Two- Dimensional Electron and Hole Systems , Springer Tracts in Modern Physics (Springer, Berlin, 2003). [46] M. Studer, M. P. Walser, S. Baer, H. Rusterholz, S. Sch on, D. Schuh, W. Wegscheider, K. Ensslin, and G. Salis Phys. Rev. B 82, 235320 (2010). [47] L. Meier, G. Salis, I. Shorubalko, E. Gini, S. Sch on and K. Ensslin, Nat. Phys. 3, 650 (2007). [48] J. M. Kikkawa and D. D. Awschalom, Nature 397, 139 (1999).[49] Y. Kato, R. C. Myers, A. C. Gossard and D. D. Awschalom, Nature 427, 50 (2004). [50] B. M. Norman, C. J. Trowbridge, J. Stephens, A. C. Gossard, D. D. Awschalom, and V. Sih, Phys. Rev. B 82, 081304(R) (2010). [51] V. K. Kalevich and V. L. Korenev, Zh. Eksp. Teor. Fiz . 52, 859 (1990) [JETP Lett. 52, 230 (1990)]. [52] R. I. Dzhioev, K. V. Kavokin, V. L. Korenev, M. V. Lazarev, B. Ya. Meltser, M. N. Stepanova, B. P. Za- kharchenya, D. Gammon, and D. S. Katzer, Phys. Rev. B66, 245204 (2002). [53] S. A. Crooker, M. Furis, X. Lou, P. A. Crowell, D. L. Smith, C. Adelmann, and C. J. Palmstrm, J. Appl. Phys. 101, 081716 (2007). [54] S. Ullah, G. M. Gusev, A. K. Bakarov, and F. G. G. Hernandez, J. Appl. Phys. 119, 215701 (2016). [55] F. G. G. Hernandez, S. Ullah, G. J. Ferreira, N. M. Kawa- hala, G. M. Gusev, and A. K. Bakarov, Phys. Rev. B 94, 045305 (2016).
1908.07766v2.Spin_orbit_coupled_quantum_memory_of_a_double_quantum_dot.pdf	Spin-orbit-coupled quantum memory of a double quantum dot L. Chotorlishvili,1A. Gudyma,2J. Wätzel,1A. Ernst,2,3and J. Berakdar1 1Institut für Physik, Martin-Luther Universität Halle-Wittenberg, D-06120 Halle/Saale, Germany 2Max Planck Institute of Microstructure Physics, Weinberg 2, 06120 Halle/Saale, Germany 3Institute for Theoretical Physics, Johannes Kepler University, Altenberger Strasse 69, 4040 Linz, Austria (Dated: October 29, 2019) The concept of quantum memory plays an incisive role in the quantum information theory. As conﬁrmed by several recent rigorous mathematical studies, the quantum memory inmate in the bipartite system ρABcan reduce uncertainty about the part B, after measurements done on the partA. In the present work, we extend this concept to the systems with a spin-orbit coupling and introducea notionof spin-orbitquantum memory. We self-consistently exploreUhlmann ﬁdelity, pre and post measurement entanglement entropy and post measurement conditional quantum entropy of the system with spin-orbit coupling and show that measurement performed on the spin subsystem decreases the uncertainty of the orbital part. The uncovered eﬀect enhances with the strength of the spin-orbit coupling. We explored the concept of macroscopic realism introduced by Leggett and Garg and observed that POVM measurements done on the system under the particular protocol are non-noninvasive. For the extended system, we performed the quantum Monte Carlo calculations and explored reshuﬄing of the electron densities due to the external electric ﬁeld. I. INTRODUCTION Let us consider a typical setting of a bipartite quantum system1, described by the density matrix ˆρABshared by two parties Alice ( A) and Bob ( B). Suppose Aperforms two consecutive measurements of Hermitian observables XandY. The uncertainty relation in the Robertson’s form2,3states that the product of the standard devia- tions is larger or equal to the expectation value of com- mutator/triangleX/triangleY≥1 2/vextendsingle/vextendsingle/angbracketleftψ/vextendsingle/vextendsingle/bracketleftbig X,Y/bracketrightbig/vextendsingle/vextendsingleψ/angbracketright/vextendsingle/vextendsinglewith respect to the shared quantum state \|ψ/angbracketright. From Bob’s perspective, the uncertainty in Alice measurement results depends how- ever on the nature of the quantum state/vextendsingle/vextendsingleψ/angbracketright, meaning on whether/vextendsingle/vextendsingleψ/angbracketrightis entangled or separable. Berta et al.4 showed that, for Bob, entanglement may decrease the lower bound for the uncertainty in Alice measurements outcome. That means Bob may become more certain about the results of Alice measurements done on her part A, if Bob subsystem Bis entangled with A. More specif- ically, Bob uncertainty concerning Alice measurements is determined by the quantum conditional entropy de- ﬁned as follows S/parenleftbig A\|B/parenrightbig =S/parenleftbig ˆ/rho1AB/parenrightbig −S/parenleftbig trA/parenleftbig ˆ/rho1AB/parenrightbig/parenrightbig . Here ˆ/rho1ABis the post-measurement density matrix of the bi- partite system and S/parenleftbig ˆ/rho1AB/parenrightbig is the von Neumann entropy S/parenleftbig ˆ/rho1AB/parenrightbig =−tr/parenleftbig ˆ/rho1ABln/parenleftbig ˆ/rho1AB/parenrightbig/parenrightbig . A negative quantum con- ditional entropy is in contrast to conventional wisdom regarding entropy of a classical system, as classically, en- tropy is an extensive quantity and hence the entropy of the whole system should not be lower than the entropy of the subsystem. Physical realizations of the subsystems AandBare diverse1. For instance, AandBcould be two electrons in a double quantum dot, each hosting spin and orbital degrees of freedom. The spin and orbital degrees of free- dom of electrons may be entangled. However, due to the SOinteraction, spindegreesoffreedommaybeentangled with orbital degrees as well. Thus, in a double quantumdot the quantum state of the system may hold spin, or- bitalandspin-orbitentanglement. Suchsolid-state-based systems are very attractive due to their scalability and the various tools at hand to control, read and write in- formation. When the two dots are in close proximity tunneling sets in, as well as orbital correlation mediated by the Coulomb interaction. In the presence of a spin or- bital (SO) interaction, of the Rashba type5for instance, the spin becomes aﬀected by the orbital motion. Our in- terest in this work is devoted to the information obtain- able on the orbital subsystem through a measurements done on the spin subsystem and how the quality of this information is aﬀected by the SO interaction. We note in this context, that the strength of SO interaction in a semiconductor-based quantum structures can be tuned to certain extent by a static electric ﬁeld. The orbital part can be assessed for example by exploiting the diﬀer- ent relaxation times of the electrons pair to a reservoir depending on their spin state6,7. In what follows, we show that measurement done on the spin subsystem reduces the uncertainty about the orbital part, meaning that information about one sub- system can be extracted indirectly through the measure- ment done on another subsystem. We also study the uncertainty of two incompatible measurements done on the spin subsystem and explore factor of quantum mem- ory. Namely, we prove that when the system is in a pure state, quantum memory reduces the uncertainty of two incompatible spin measurements. Our focus here is on the case when Alice does two in- compatible quantum measurements on one of the parts of the bipartite system. Say, Alice measures two non- commuting spin components of the qubit at her hand. The concept of quantum memory states that the entan- glement between qubits of Alice and Bob permits Bob to reduce the upper limit of the uncertainty bound of the measurements done by Alice. In what follows we highlight and illustrate by direct numerical examples thearXiv:1908.07766v2 [quant-ph] 28 Oct 20192 subtle eﬀects of spin-orbital coupling on quantum mem- ory. In particular, spin-orbit-coupled systems may store three diﬀerent types of entanglements related to spin- spin, spin-orbit, and orbit-orbit parts. We prove that only the entire entanglement allows a reduction of the upper bound for the uncertainty. After the elimination of the spin-orbit and the orbit-orbit parts, the residual spin-spin entanglement is not enough to reduce the un- certainty. Our result is generic and is expected to apply to a broad class of materials with spin-orbital coupling. The paper is structured as follows: In section IIwe re- view the experimental studies relevant to our work. Sec- tionIIIpresents the theoretical model, in section IVwe describe measurement procedures and explore the post- measurement states, in section Vwe study the Uhlmann ﬁdelity between pre and post measurement states of the spin subsystem and evaluate post-measurement quantum conditional entropy, in the section VIwe study eﬀect of the SO interaction on the quantum discord, in the sec- tionVIIwestudynon-invasivemeasurements, inthesec- tionVIIIwe explore the impact of Coulomb interaction, in the section IXwe present results of quantum Monte Carlo calculations for the electron density obtained for the extended system, in the section Xwe discuss the problem of quantum memory and conclude the work. II. EXPERIMENTALLY FEASIBLE POVM PROTOCOLS IN QUANTUM DOTS Quantum dots are assumed as an experimental real- ization to the theory below, similar to the ﬁrst quan- tum computing scheme based on spins in isolated quan- tum dots which was proposed by D. Loss and D. Di- Vincenzo8, see also9–11and references therein. An ulti- mate goal of a quantum gate and a quantum information protocol is to read out and record the outcome state. Several types of local spin measurements were realized experimentally12–15. In quantum dots, the spin can be measured selectively through the spin-to-charge conver- sion16–19. Our focus is on the experimentally feasible spin POVM (positive operator-valued measure) measure- ments see20, the only measurement considered through- outthepresentwork. Fundamentallimitsfornondestruc- tive measurement of a single spin in a quantum dot was studied recently21. Here we brieﬂy look back to experimental and concep- tual aspects of the POVM spin measurement in quan- tum dots. The spin-resolved ﬁlter (barrier) permits to pass through the gate only electrons with particular spin orientation, i.e., transmits \|1/angbracketrightand bans the\|0/angbracketright. Thus if particlepasses,forsureweknowtheprojectionofitsspin. However, what is detected in the experiment is not a spin projection but a charge. Through the change in the elec- tric charge recognized by the electrometer, we infer the information that electron has passed through the ﬁlter. The beauty of this scheme is simpleness that allows in- troducing POVM projectors ΠA 0=\|0/angbracketright/angbracketleft0\|A,ΠA 1=\|1/angbracketright/angbracketleft1\|Afor a quantum dot in the formal theoretical discussion. Of interest is also the single-shot measurement scheme that can selectively access the singlet or the triplet two-electron states in a quantum dot7. The scheme exploits the diﬀerent coupling strengths of the triplet and singlet states to the reservoir. Therefore, charge relaxation times are diﬀerent too 1/ΓT<1/ΓS. A nondestructive measurement is achieved by an electric pulse of duration τthat shifts temporally the chemical potential of the dot with respect to the Fermi level of the reservoir, where 1/ΓT< τ < 1/ΓSis chosen. For the dot in the singlet electron state, the time is too short for tunneling, but the triplet state may tunnel. If two consecutive measurements are done within a time interval shorter than relaxation time T1, the measurement procedure is invasive, meaning that the outcome of the second measurement depends on the ﬁrst measurement. The measurement procedure is noninvasive if the time interval between measurements exceeds ∆τ1,2> T 1. In the experiment7values of the parameters for GaAs/Al xGa1−xheterostructure read: 1/ΓT= 5µs,τ= 20µs, and 1/ΓS= 100µs. III. MODEL OF THE SYSTEM The issue of quantum memory has already been ad- dressed for a number of model systems22–29. Here, we focus particularly on the interacting two-electron dou- ble quantum dots5,30–40. We self-consistently explore the Uhlmann ﬁdelity, pre and post measurement entangle- ment entropy, and post measurement conditional quan- tum entropy of the system and show that a measure- ment performed on the spin subsystem decreases the un- certainty of the orbital part. This eﬀect becomes more prominent with increasing the strength of SO coupling. We consider a double quantum dot characterized by a rather strong quantum conﬁnement potential in the yandzdirections, see pictorial Fig.(1). For sin- gle particle we use the orbitals Ψnx,ny,nz(x,y,z ) = Nφnx(x)Yny(y)Znz(z)where/angbracketleftφnx\|φn/primex/angbracketright=δnx,n/primex, /angbracketleftYny\|Yn/primey/angbracketright=δny,n/primeyand/angbracketleftZnz\|Zn/primez/angbracketright=δnz,n/primez. We consider a situation with a strong conﬁnement in yandzdirec- tions such that only the lowest subbands with ny= 0 andnz= 0are occupied. The relevant dynamics takes place in the xdirection only, subject to the eﬀective one- dimensionalpotential V(x). TheHamiltonianofconﬁned electrons reads ˆH0=−¯h2 2m∗N/summationdisplay n=1∂2 ∂x2n+N/summationdisplay n<mVC(rrrn,rrrm) +ˆHSO +N/summationdisplay n=1(V(xn) +eExn). (1) Here,V(x) =m∗ω2min/bracketleftbig (x−∆/2)2,(x+ ∆/2)2/bracketrightbig /2is3 FIG. 1. Schematic representation of the considered double quantum dot system in the presence of an external electric ﬁeld and spin-orbit coupling. In a quasi-one-dimensional con- ductive channel, two quantum dots are created and controlled by two local gates: ”gate 1” and ”gate 2”. The quantum con- ﬁnement in y-direction is strong enough such that the only lowest subband state ny= 0is occupied. The ”gate 0” is used to control tunnel junction between the dots (that mimics the changing of the interdot distance). The applied constant electric ﬁeld, polarized in x-direction, is represented by the sky blue arrow. the double-dot conﬁnement potential, ˆHSO=−iαN/summationdisplay n=1∂ ∂xnˆσy n+BN/summationdisplay n=1ˆσz n,(2) is the Rashba SO term with the magnetic ﬁeld, ∆ =/lscriptd0 is the inter-dot distance with the dimensionless scaling factor/lscript,m∗is the electron eﬀective mass, eis the ab- solute value of the electron charge, and the strength of the constant static electric ﬁeld is E. The trial magnetic ﬁeldBapplied along the z-axis has no particular eﬀect on the phase of the wave function in 1D case but speciﬁes the quantization axis and shifts energy levels. Note that the Coulomb potential VC(rrr1,rrr2) =e2/κ\|rrr1−rrr2\|, where κis the dielectric constant, still depends on the six co- ordinatesrrr1= (x1,y1,z1)andrrr2= (x2,y2,z2)and will be reduced to the (x1,x2)-variables later in the text. In what follows we introduce dimensionless units by setting x1,2→x1,2/d0,β=m∗ωd2 0/¯h,ˆH0→ˆH0/(¯h2/(m∗d2 0)), E0=m∗ed3 0E/¯h2. We adopt parameters of the semicon- ductormaterialGaAs, β= 1inthiscasecorrespondstoa conﬁnement energy ¯hω= 11.4meV,d0= 10nm. Dimen- sionless electric ﬁeld E0= 1is equivalent to the applied external ﬁeld of the strength E= 1.1V/µm. The single particleenergylevelsofasingledot( /lscript= 0)aregiventhen byεn=β(n+ 1/2). For the sake of simplicity to start, we neglect the Coulomb term and treat SO coupling per- turbatively. The antisymmetric total wave-functions are presented as direct products of the orbital and spin parts Ψ(1) n=ψS n⊗χA/parenleftbig 1,2/parenrightbig andΨ(2) n=ψA n⊗χS/parenleftbig 1,2/parenrightbig , where/vextendsingle/vextendsingleχT+ S/parenleftbig 1,2/parenrightbig/angbracketrightbig =\|1↑/angbracketrightbig \|2↑/angbracketrightbig ,/vextendsingle/vextendsingleχT− S/parenleftbig 1,2/parenrightbig/angbracketrightbig =\|1↓/angbracketrightbig \|2↓/angbracketrightbig , \|χT0 S/parenleftbig 1,2/parenrightbig/angbracketrightbig =1√ 2/parenleftbig \|1↑/angbracketright\|2↓/angbracketright+\|1↓/angbracketright\|2↑/angbracketright/parenrightbig and the asym-metric spin function read/vextendsingle/vextendsingleχA/parenleftbig 1,2/parenrightbig/angbracketrightbig =1√ 2/parenleftbig \|1↑/angbracketright\|2↓/angbracketright− \|1↓/angbracketright\|2↑/angbracketright/parenrightbig . We deﬁne the two-electron symmetric and antisymmetric coordinate wave functions of a double quantum well as follows: /vextendsingle/vextendsingleψS,A n,n/prime/angbracketrightbig =1/radicalbig 2(1±S2)/bracketleftbig ψL,n/parenleftbig x1/parenrightbig ψR,n/prime/parenleftbig x2/parenrightbig ±ψL,n/parenleftbig x2/parenrightbig ψR,n/prime/parenleftbig x1/parenrightbig/bracketrightbig ,(3) whereψL,n/parenleftbig x/parenrightbig is the single particle wave function cor- responding to the left dot and quantum state n, while ψR,n/prime/parenleftbig x/parenrightbig is associated with the right dot and quantum staten/primeandS=/angbracketleftψL,n\|ψR,n/prime/angbracketrightistheoverlapintegral. The results of the exact numerical calculations (not shown) have conﬁrmed that for the large values of the parameter β/greatermuch1overlap integral is zero S= 0and tunneling pro- cesses are not activated. As will be shown bellow eﬀect of the Coulomb term in this case is less relevant and can be neglected safely. In the double quantum dot, the equilib- rium positions of electrons shifts along the x-axis by the distance±d0/2. The harmonic oscillator eigenfunctions follow Heitler-London ansatz41,42and readψL(R),n/parenleftbig x/parenrightbig = φL(R),n/parenleftbig x/parenrightbig , whereφL(R),n/parenleftbig x/parenrightbig =1√ 2nn!/parenleftbigg β π/parenrightbigg1/4 × exp/parenleftBig −β(x±1/2−d)2 2/parenrightBig Hn/parenleftbig x√β/parenrightbig ,Hn/parenleftbig x√β/parenrightbig is Hermite polynomial and d=eE/d 0m∗ω2. The energy spectrum of unperturbed system is de- scribed by the sum of energies of non-interacting oscil- latorsEN=β/parenleftbig n+ 1/2/parenrightbig +β/parenleftbig n/prime+ 1/2/parenrightbig , N = (n,n/prime) and we introduced the following notations for brevity./vextendsingle/vextendsingleΦN/angbracketrightbig =/vextendsingle/vextendsingleψS,A n,n/prime/angbracketrightbig ⊗/vextendsingle/vextendsingleχA,S/angbracketrightbig , see Eq. (3). The presence of the SO term mixes diﬀerent spin sec- tors and spin and orbital states. Considering/vextendsingle/vextendsingleΨM/angbracketrightbig =/vextendsingle/vextendsingleψA 0,1/angbracketrightbig ⊗/vextendsingle/vextendsingleχT+ S/angbracketrightbig as an unperturbed wave function we ob- tain: /vextendsingle/vextendsingleΦM/angbracketrightbig =/vextendsingle/vextendsingleψA 0,1/angbracketrightbig ⊗/vextendsingle/vextendsingleχT+ S/angbracketrightbig +α 2√β/parenleftbigg1 2/vextendsingle/vextendsingleψS 1,1/angbracketrightbig −/vextendsingle/vextendsingleψS 0,0/angbracketrightbig/parenrightbigg ⊗/vextendsingle/vextendsingleχA/angbracketrightbig . (4) Using Eq. (4) and tracing out orbital (spin) parts we construct reduced density matrix of the spin (or- bital) subsystem respectively: ˆρS=1 Z/braceleftbig/vextendsingle/vextendsingleχT+ S/angbracketrightbig/angbracketleftbig χT+ S/vextendsingle/vextendsingle+ 5α2 16β/vextendsingle/vextendsingleχA/angbracketrightbig/angbracketleftbig χA/vextendsingle/vextendsingle/bracerightbig ,ˆρor=1 Z/parenleftbigg/vextendsingle/vextendsingleψA 0,1/angbracketrightbig/angbracketleftbig ψA 0,1/vextendsingle/vextendsingle+α2 16β/vextendsingle/vextendsingleψS 1,1/angbracketrightbig/angbracketleftbig ψS 1,1/vextendsingle/vextendsingle+ α2 4β/vextendsingle/vextendsingleψS 0,0/angbracketrightbig/angbracketleftbig ψS 0,0/vextendsingle/vextendsingle/parenrightbigg , whereZ= 1 +5α2 16β. IV. POVM MEASUREMENTS AND POST-MEASUREMENT STATES The generic state of two non-interacting particles is a product state. Therefore a density matrix of a system can be factorized as a direct product of density matrices of individual particles. After tracing out states of one4 particle, product state leaves a system in a pure state with a zero entropy. However, in the case of fermions, the Pauli principle imposes quantum correlation even in the absence of interaction. As for an interacting bipartite system in most of the cases, the state is entangled43–50. Tracing out part of bipartite entangled state results in a mixed state and ﬁnite entropy. Quantum correlation manifests in continuous variables systems as well51–58. Therefore under certain conditions, we expect the or- bital part to be entangled. After setting theoretical ma- chinery of the problem, we proceed with the information measures of uncertainty and quantum correlations in the system. In particular, we specify pre-measurement von Neumann entropy of the orbital and spin subsystems: S/parenleftbig ˆρor/parenrightbig =−tr/parenleftbig ˆρorln/parenleftbig ˆρor/parenrightbig/parenrightbig =−α2 16βln/parenleftbigα2 16β/parenrightbig −α2 4βln/parenleftbigα2 4β/parenrightbig andS/parenleftbig ˆρs/parenrightbig =−tr/parenleftbig ˆρsln/parenleftbig ˆρs/parenrightbig/parenrightbig =−5α2 16βln/parenleftbig5α2 16β/parenrightbig respec- tively, where we assumed that Z≈1. Spin and or- bital von Neumann entropies increase with the Rashba SO coupling constant β. Let us assume that Alice per- forms POVM measurement59on the ﬁrst qubit at her hand (in what follows we use notations A= 1andB= 2 for the ﬁrst and second qubit). After measurement the initial sate collapses either to the post-measurement state \|Ψ(1) AB/angbracketrightbig =/parenleftbig ΠA 0/circlemultiplytextIB/parenrightbig/vextendsingle/vextendsingleΦM/angbracketrightbig /radicalBig/angbracketleftbig ΦM/vextendsingle/vextendsingle/parenleftbig ΠA 0/circlemultiplytextIB/parenrightbig/vextendsingle/vextendsingleΦM/angbracketrightbig,(5) with probability ΓA 0=/angbracketleftΦM/vextendsingle/vextendsingle/parenleftbig ΠA 0/circlemultiplytextIB/parenrightbig/vextendsingle/vextendsingleΦM/angbracketrightbig /angbracketleftΦM/vextendsingle/vextendsingleΦM/angbracketrightbig, (6) or to the post-measurement state \|Ψ(2) AB/angbracketrightbig =/parenleftbig ΠA 1/circlemultiplytextIB/parenrightbig/vextendsingle/vextendsingleΦM/angbracketrightbig /radicalBig/angbracketleftbig ΦM/vextendsingle/vextendsingle/parenleftbig ΠA 1/circlemultiplytextIB/parenrightbig/vextendsingle/vextendsingleΦM/angbracketrightbig,(7) with probability ΓA 1=/angbracketleftΦM/vextendsingle/vextendsingle/parenleftbig ΠA 1/circlemultiplytextIB/parenrightbig/vextendsingle/vextendsingleΦM/angbracketrightbig /angbracketleftΦM/vextendsingle/vextendsingleΦM/angbracketrightbig. (8) POVM operators have form: ΠA 0=\|0/angbracketright/angbracketleft0\|A,ΠA 1= \|1/angbracketright/angbracketleft1\|A,IBis the identity operator acting on the qubitB. Easy to see that ΓA 0= 5α2//parenleftbig 10α2+ 32β/parenrightbig ;ΓA 1=/parenleftbig 5α2+ 32β/parenrightbig //parenleftbig 10α2+ 32β/parenrightbig and ΓA 1> ΓA 0. After involved calculations we derive explicit expressions for the post-measurement reduced or- bital ˆ/rho1(1,2) AB = trs/parenleftbig \|Ψ(1,2) AB/angbracketrightbig/angbracketleftbig Ψ(1,2) AB\|/parenrightbig and spin ˆσ(1,2) AB = tror/parenleftbig \|Ψ(1,2) AB/angbracketrightbig/angbracketleftbig Ψ(1,2) AB\|/parenrightbig density matrices: ˆσ(1) AB=/vextendsingle/vextendsingle1↓/angbracketright\|2↑/angbracketright/angbracketleft2↑\|/angbracketleft1↓\|, ˆσ(2) AB=1 1 + 5α2/32β/parenleftbigg \|1↑/angbracketright\|2↑/angbracketright/angbracketleft2↑\|/angbracketleft1↑\| +5α2 32β\|1↑/angbracketright\|2↓/angbracketright/angbracketleft2↓\|/angbracketleft1↑\|/parenrightbigg , (9)and ˆ/rho1(1) AB=4 5/parenleftbigg1 4\|ψs 1,1/angbracketright/angbracketleftψs 1,1\|+\|ψs 0,0/angbracketright/angbracketleftψs 0,0\| −1 2\|ψs 1,1/angbracketright/angbracketleftψs 0,0\|−1 2\|ψs 0,0/angbracketright/angbracketleftψs 1,1\|/parenrightbigg , ˆ/rho1(2) AB=1 1 + 5α2/32β/parenleftbigg \|ψA 0,1/angbracketright/angbracketleftψA 0,1\|+α2 8β/parenleftbigg1 4\|ψs 1,1/angbracketright/angbracketleftψs 1,1\| +\|ψs 0,0/angbracketright/angbracketleftψs 0,0\|−1 2\|ψs 1,1/angbracketright/angbracketleftψs 0,0\|−1 2\|ψs 0,0/angbracketright/angbracketleftψs 1,1\|/parenrightbigg/parenrightbigg .(10) Sinceα/√βis the small parameter, with high accuracy we set 1 + 5α2/32β≈1. V. THE UHLMANN FIDELITY AND THE POST-MEASUREMENT QUANTUM CONDITIONAL ENTROPY Before study the entropy of the system we explore the ﬁdelity between pre and post measurement states of the spin subsystem. In its most general form, the ﬁdelity problem was formulated by Uhlmann. For de- tails about the Uhlmann ﬁdelity, we refer to59. At ﬁrst, let us perform the standard puriﬁcation proce- dure of the pre ˆρsand post measurement spin den- sity matrices ˆσAB. We adopt spectral decompositions ˆρs=/summationtext xPX/parenleftbig x/parenrightbig \|x/angbracketright/angbracketleftx\|AB,ˆσAB=/summationtext xQY/parenleftbig y/parenrightbig \|y/angbracketright/angbracketlefty\|AB, as- sociated with the ensembles {PX,\|x/angbracketright},{QY,\|Y/angbracketright}where random variables x,ybelong to the diﬀerent alphabets. A puriﬁcation with respect to the reference system R we deﬁne as follows: \|φρ/angbracketrightR,AB=/summationtext x/radicalBig PX/parenleftbig x/parenrightbig \|x/angbracketrightR\|x/angbracketrightAB, \|φσ/angbracketrightR,B= trA/parenleftbigg/summationtext y/radicalBig QY/parenleftbig y/parenrightbig \|y/angbracketrightR\|y/angbracketrightAB/parenrightbigg . The Uhlmann ﬁdelity between two mixed states read: F/parenleftbig trA/parenleftbig ˆσAB/parenrightbig ,ˆρs/parenrightbig = max (Uσ,Uρ)/vextendsingle/vextendsingle/angbracketleftφσ\|/parenleftbig U† ρUσ/parenrightbigR⊗IAB\|φσ/angbracketrightR,AB\|2.(11) The Uhlmann theorem59facilitates calculation of Uhlmann ﬁdelity and ﬁnally, we deduce: F/parenleftbig trA/parenleftbig ˆσ(2) AB/parenrightbig ,ˆρs/parenrightbig =/parenleftbigg 1 +5α2 16β√ 2/parenrightbigg ×1 1 + 5α2/32β×1 1 + 5α2/16β. (12) For the small SO coupling we obtain asymptotic esti- mation:F/parenleftbig trA/parenleftbig ˆσ(2) AB/parenrightbig ,ˆρs/parenrightbig ≈1−5/parenleftbig 3−√ 2/parenrightbig α2/32β. As we see the distance between pre and post-measurement states decays with SO constant α. Taking into account Eq. (9), Eq. (10) and probabili- ties Eq. (6), Eq. (8) we deduce the expression of the post measurement von Neumann entropy of the spin subsys- temS/parenleftbig ˆσ(2) AB/parenrightbig =−ΓA 15α2 32βln/parenleftbig5α2 32β/parenrightbig . The diﬀerence between pre and post measurement entropies of the spin subsys- temS/parenleftbig ˆσS/parenrightbig −S/parenleftbig ˆσAB/parenrightbig =−5α2 16βln/parenleftbig5α2 16β/parenrightbig + ΓA 15α2 32βln/parenleftbig5α2 32β/parenrightbig5 FIG. 2. Dependence of the von Neumann entropy on the system’s and ﬁeld parameters: (a) Planes describe the pre (green) and post (orange) measurement entropies as a function of the spin-orbit coupling strength αand the applied external electric ﬁeldE0. The eﬀective inter-dot distance is ∆ = 0.8d0. (b) The diﬀerence between pre and post measurement entropies of the orbital subsystem S/parenleftbig ˆρor/parenrightbig −S/parenleftbig ˆ/rho1AB/parenrightbig as a function of the spin-orbit coupling α, plotted for diﬀerent inter-dot distances. is positive for any ΓA 1<1and that means POVM measurement decreases the entropy of the spin subsys- tem. The post measurement von Neumann entropy of the orbital subsystem S/parenleftbig ˆ/rho1AB/parenrightbig =−ΓA 15α2 32βln/parenleftbig5α2 32β/parenrightbig . The diﬀerence between pre and post measurement en- tropies of the orbital subsystem S/parenleftbig ˆρor/parenrightbig −S/parenleftbig ˆ/rho1AB/parenrightbig = −α2 16βln/parenleftbigα2 16β/parenrightbig −α2 4βln/parenleftbigα2 4β/parenrightbig + ΓA 15α2 32βln/parenleftbig5α2 32β/parenrightbig . Easy to see that−α2 4βln/parenleftbigg α2 4β/parenrightbigg >5α2 32βln/parenleftbig5α2 32β/parenrightbig and the entropy after measurement decreases S/parenleftbig ˆρor/parenrightbig −S/parenleftbig ˆ/rho1AB/parenrightbig >0. An inter- esting observation is that POVM measurement done on the spin subsystem through the SO interaction decreases the von Neumann entropy of the orbital part. As larger is SO coupling constant α, larger is a decrement of the orbital entropy. Even more surprising is that measure- ment equates post-measurement von Neumann entropies of the spin and orbital subsystems S/parenleftbig ˆσ(2) AB/parenrightbig =S/parenleftbig ˆ/rho1AB/parenrightbig . The pair concurrence of the spin subsystem is deﬁned as follows:C=max/parenleftbig 0,√R1−√R2−√R3−√R4/parenrightbig , with the eigenvalues Rn, n= 1,...4of the following matrix R= ˆρS/parenleftbig ˆσy 1⊗ˆσy 2/parenrightbig/parenleftbig ˆρS/parenrightbig∗/parenleftbig ˆσy 1⊗ˆσy 2/parenrightbig . For the pre and post measurement concurrence we obtain: C/parenleftbig ˆρS/parenrightbig = 5α2/16β, C/parenleftbig ˆσAB/parenrightbig = 0. The measurement disentangles the system. Taking into account Eq. (9), for the von Neumann en- tropy of the subsystem Bwe deduce S/parenleftbig trA/parenleftbig ˆσ(2) AB/parenrightbig/parenrightbig = −ΓA 15α2 32βln/parenleftbig5α2 32β/parenrightbig . Therefore for the post-measurement conditional quantum entropy we obtain: S/parenleftbig A\|B/parenrightbig = S/parenleftbig ˆσAB/parenrightbig −S/parenleftbig trA/parenleftbig ˆσAB/parenrightbig/parenrightbig = 0. Notethattheconditionalquantumentropyofthepost-measurement state quantiﬁes the uncertainty that Bob has about the outcome of Alice’s measurement. The zero value ofS/parenleftbig A\|B/parenrightbig means that Bob has precise information about the measurement result. The same eﬀect we see in the post-measurement entropy of the orbital subsystem. Due to the SO coupling, the measurement done on the qubitAreduces the post-measurement entropy of the or- bital subsystem. The eﬀect of the electric ﬁeld, Coulomb interaction and tunneling processes activated in case of small inter-dot distance may modify this picture. In case of a short inter-dot distance (i.e., parameter βis an order of1<β < 10) eﬀect of the quantum tunneling processes assisted by the Coulomb interaction becomes important. We explore this problem using numeric methods. VI. QUANTUM GENERALIZATION OF CONDITIONAL ENTROPY The quantum mutual information quantiﬁes all cor- relations in the quantum bipartite system, and at least part of these correlations can be classical. Vedral, Zurek, and others asked the question: whether it is possible to have a more subtle notion of quantum correlations rather than the entanglement60–63. For pure states, the quan- tum discord is equivalent to the quantum entanglement but is distinct when the state is mixed. The central issue for the quantum discord is a quantum generalization of conditional entropy (the quantity that is distinct from the conditional quantum entropy). Quantum discord is quantiﬁed as follows: DA/parenleftbig ˆρs/parenrightbig = min/braceleftbig ΠB j/bracerightbig/braceleftbigg S(A)−S(A,B)−S/parenleftbigg/summationdisplay jpjtr/braceleftbigg/parenleftbig ΠB jˆρsΠB j/parenrightbig /pj/bracerightbigg ln/braceleftbigg/parenleftbig ΠB jˆρsΠB j/parenrightbig /pj/bracerightbigg/parenrightbigg/bracerightbigg . (13)6 We omit details of calculations and present result for the diﬀerence between pre and post-measurement quantum discordsDA/parenleftbig ˆρs/parenrightbig −DA/parenleftbig ˆσAB/parenrightbig =5α2 32βln 4. From this re- sult we see that similar to the pre and post-measurement von Neumann entropy, quantum discord decreases after measurement. VII. QUANTUM WITNESS AND NON-INVASIVE MEASUREMENTS The concept of macroscopic realism introduced by Leggett and Garg64postulates criteria of noninvasive measurability. In the sequence of two measurements, the ﬁrst blind measurement has no consequences on the out- come of the second measurement if a system is classi- cal. However, in the case of quantum systems, any mea- surement alters the state of the system independently from the fact was the ﬁrst measurement either blind (i.e., the measurement result is not recorded) or not. Simi- lar to the Bell’s inequalities, quantumness (i.e., entangle- ment) may violate the macroscopic realism and Leggett- Garg inequalities. This eﬀect is widely discussed in the literature65–67. Quantum witness introduced in68is the central characteristic of invasive measurements. In this section we discus particular type of non-invasive mea- surement protocol. The directly measured probability we deﬁne in terms of the following expression PB/parenleftbig 1/parenrightbig = tr/braceleftbig ΠB 1N/parenleftbig ˆρs/parenrightbig/bracerightbig . Here ΠB 1=\|1/angbracketright/angbracketleft1\|Bis the operator of the projective measure- ment done on the second qubit, N/parenleftbig ˆρs/parenrightbig =/summationtext i=1,2ˆLiˆρsˆL† i is the trace preserving quantum channel with Kraus operatorsL1=\|0/angbracketright/angbracketleft1\|A,L2=\|1/angbracketright/angbracketleft0\|A. The blind- measurement probability we deﬁne as follows: GB/parenleftbig 1/parenrightbig = tr/braceleftbig ΠB 1N/parenleftbigˆΞs/parenrightbig/bracerightbig , where density matrix of the system after blind measurement is given by ˆΞs=/summationtext i=0,1ΠA iˆρsΠA i. The quantum witness that quantiﬁes invasiveness of the quantum measurements is given by the formula: W=\|tr/braceleftbig ΠB 1/parenleftbig N/parenleftbig ˆρs/parenrightbig −N/parenleftbigˆΞs/parenrightbig/parenrightbig/bracerightbig \|. (14) Direct calculations for our system shows that GB/parenleftbig 1/parenrightbig =PB/parenleftbig 1/parenrightbig =1 Z/parenleftbig 1 + 5α2/32β/parenrightbig .(15) The quantum witness is zero W= 0indicating that mea- surements done on the system within this particular pro- cedure are noninvasive. VIII. THE EFFECT OF THE COULOMB INTERACTION We study the case of a short inter-dot distance and the eﬀect of the Coulomb interaction. We utilize the conﬁgurational interaction (CI) ansatz and perform ex- tensive numerical calculations. Utilizing the single par- ticle orbitals we solve the stationary one-dimensionalSchrödinger equation in absence of the Coulomb term. By means of numerical diagonalization of the single par- ticle Hamiltonian ˆHSP=−∂2 x/2+V(x)+xE0discretized on a ﬁne space grid we obtain the single-particle orbitals φi(x) =ci,Lφi,L(x)+ci,Rφi,R(x)andenergies εi. Wecon- structed the symmetric and anti-symmetric two-electron wave functions labeled as (+,−)and evaluate matrix el- ements of ˆH0including the Coulomb term: /angbracketleftΥn/prime 0\|ˆH0\|Υn 0/angbracketright=/epsilon10 nδn,n/prime+/angbracketleftΥn/prime 0\|VC\|Υn 0/angbracketrightδb,b/prime,(16) wherebis a part of the index n={i,j,b = (+,−)}. Note that two-electron wave-functions \|Υn 0/angbracketrightaccounts the ef- fect of doubly-occupied states as well. We diagonalize the matrix Eq. (16) and obtain the fully correlated two- electron eigenstates and eigenvalues {\|Ψn/angbracketright,/epsilon1n}. For a goodconvergenceandreliabilityofthespectrum, weused 80 single-particle orbitals \|φi/angbracketright. In the last step we add the Rashba SOC term to Eq. (1). The matrix elements of the total Hamiltonian including the SO term read /angbracketleftΨn/primeχ/prime\|ˆH0+ˆHSO\|Ψnχ/angbracketright=/epsilon1nδn,n/primeδχχ/prime −iα2/summationdisplay i=1/angbracketleftΨ/prime n\|∂xi\|Ψn/angbracketright/angbracketleftχ/prime\|σy i\|χ/angbracketright. (17) Here the last term corresponds to the Rashba SO interac- tion in the matrix form. The spin-resolved two-electron eigenstates\|Φn/angbracketrightand the corresponding energies Enwe obtain by means of numerical diagonalization of Eq. (17). In Fig. 2 (a) pre and post measurement von Neu- mann entropies are plotted for the ﬁxed inter-dote dis- tance ∆ = 0.8d0. The values of the applied electric ﬁeld and SO coupling are in the range of 0< E 0<8 and 0< α < 1, i.e.E0= 1corresponds to a static electric ﬁeld≈1.1V/µm,β= 1is equivalent to the realistic parameters adopted for GaAs ¯hω= 11,4meV, m∗= 0,067me,d0= 10nm. The post-measurement von Neumann entropy S(ˆρAB)is always smaller than the pre-measurement entropy S(ˆρor). Electric ﬁeld en- hances both pre and post-measurement entropies and forE0>2we see the saturation eﬀect. The diﬀerence between pre and post measurement entropies of the or- bital subsystem S(ˆρor)−S(ˆρAB)at diﬀerent inter-dot distances is plotted in Fig. 2 (b). As we see measure- ment done on the spin subsystem reduces the entropy of the orbital part. Reduction of entropy increases with the strength of SO coupling term α. On the other hand at small inter-dot distances the diﬀerences between pre- and post-measurement entropies of the orbital subsystem S(ˆρor)−S(ˆρAB)is smaller due to the Coulomb term. We note that when SO coupling is zero, the reduced density matrix of the orbital subsystem corresponds to the pure state, and therefore von Neumann entropy is zero see Fig. 2 (a) and Fig. 2 (b). The maximum value of the von Neumann entropy depends on the number of the quan- tum states involved in the process and reaches the peak7 for the maximally mixed state. Strong electric ﬁeld in- creases the amount of the involved quantum states, and von Neumann entropy reaches its saturation value. Nu- merical calculations frankly conﬁrm the validity and cor- rectness of analytical results. IX. QUANTUM MONTE CARLO CALCULATIONS ELECTRON DENSITY Here we consider the extended system 1 of four elec- trons in the four-dot conﬁnement potential V(x) =m∗ω2 2min/bracketleftbigg (x−3∆ 2)2,(x−∆ 2)2,(x+∆ 2)2, (x+3∆ 2)2/bracketrightbigg . (18) We perform numerical simulations with the modi- ﬁed continuous spin Variational Monte Carlo (CSVMC) algorithm69,70. We introduce auxiliary spinor vector χ†(s) =N/productdisplay n=1⊗[eisn,e−isn], (19) wheresnare auxiliary variables deﬁned on [0,2π)with the periodic boundary conditions. We construct eﬀec- tive scalar wave-function as a scalar product of the wave- functions and vectors χ†(s)follows ψ(x,s) =χ†(s)·Ψ(x) (20) The inverse transformation is done through the integra- tion over the auxiliary variables Ψ(x) =1 (2π)N/integraldisplayN/productdisplay n=1dsnψ(x,s)χ(s).(21) We write the eﬀective Schrödinger equation for the scalar wave-function i¯h∂ ∂tψ(x,s) =ˆHeffψ(x,s), (22) where ˆHeffis the eﬀective Hamiltonian. We construct the eﬀective Hamiltonian replacing the spinor operators by the following operators: ˆσx= cos(2s)−sin(2s)∂ ∂s, (23a) ˆσy= sin(2s) + cos(2s)∂ ∂s. (23b) ˆσz=−i∂ ∂s, (23c)This transformation expands Hilbert space of the prob- lem from the particular spin sector s=1 2to arbitrary spin. Toselectthedesiredsolutionfromthesetofallpos- sible solutions we introduce equality constraints s2=3 4 ands2 z=1 4. First of these constraints fulﬁls automati- cally while second one in introduced directly into the La- grange function L=/angbracketleftBig ˆH/angbracketrightBig +λ/parenleftbig/angbracketleftbig ˆσ2 z/angbracketrightbig −1/parenrightbig . The Lagrange function is constructed through minimization of the ef- fective Hamiltonian with the additional spin-variable ki- netic energy term. doing an importance sampling with a guiding wave function ψT. We use trial wave-function in the Slater-Jastrow form ψT=DeJ, (24) whereJis the Jastrow factor which takes into ac- count correlations introduced through the many-body interaction71. The none-interacting part is chosen to be a Slater determinant spanned in the lowest lying single- particle orbitals. Single particle orbitals are approxi- mated with product of Heitler-London functions41,42and phase calculated from the homogeneous system. In Fig. 3 the pair distribution function is shown for dif- ferent values of the trapping parameter β= 1,3and10. The Rashba constant is equal to α= 0.4. In the regime, β/greatermuch1the electronic density is localized in the vicin- ity of minimums of the trapping potential and the over- lap between neighboring trapping gaps is small (Fig. 3c). With the decrease of trapping barrier, electrons delocal- ize (Figs. 3a-b). The eﬀect of the electric ﬁeld is pre- sented in Fig. 4. Pair distribution function for β= 1,3 and10andE0= 1is plotted in Fig. 4. Coordinates x1 andx2are centered at the minimums of the V(x) +eEx. At the ﬁnite electric ﬁeld minimums in the direction of the ﬁeld are energetically preferable and total density shifts towards the direction of the applied ﬁeld. X. QUANTUM MEMORY We already showed that measurement done on the spin subsystemreducestheorbitalentanglement. Nowwedis- cuss a diﬀerent scheme when Alice does two incompatible quantum measurements on one of the parts of the bipar- tite system, and we try to answer the question: whether the spin-orbit interaction can reduce Bob’s total uncer- tainty about measurements done by Alice? We consider two cases: in the ﬁrst case Alice and Bob share the total density matrix of a bipartite SO system Eq. (4)8 FIG. 3. The pair distribution function ρ(x1,x2)at zero magnetic and electric ﬁelds for various values of the trapping parameter β. The Rashba constant α= 0.4. Parameter βdeﬁnes the inverse localization length of wave-function. When the localization length exceeds the distance between minimums of trapping potential V(x)the electronic wave-function is delocalized. (3a-b) Delocalized pair distribution function for β= 1andβ= 3. With the increase of βpotential barrier between minimums of the potential increases and electrons become localized in the minimums of the potential. (3c) Localized pair distribution function forβ= 10. FIG. 4. The pair distribution function. The applied electric ﬁeld steers the electronic density to the edge of the sample in the direction of the ﬁeld. The pair distribution function of the ﬁrst two particles ρ(x1,x2)is centered at the minimums of V(x) +eExforE0= 1. Various values of the trapping parameter are considered β= 1,3,10. The Rashba constant is equal to α= 0.4. ˆρAB=1 Z/braceleftbigg/vextendsingle/vextendsingleψA 0,1/angbracketrightbig/angbracketleftbig ψA 0,1/vextendsingle/vextendsingle⊗/vextendsingle/vextendsingleχT+ S/angbracketrightbig/angbracketleftbig χT+ S/vextendsingle/vextendsingle+5α2 16β/parenleftbigg1√ 5/vextendsingle/vextendsingleψS 1,1/angbracketrightbig −2√ 5/vextendsingle/vextendsingleψS 0,0/angbracketrightbig/parenrightbigg/parenleftbigg1√ 5/angbracketleftbig ψS 1,1/vextendsingle/vextendsingle−2√ 5/angbracketleftbig ψS 0,0/vextendsingle/vextendsingle/parenrightbigg ⊗/vextendsingle/vextendsingleχA/angbracketrightbig/angbracketleftbig χA/vextendsingle/vextendsingle +√ 5α 4√β/parenleftbigg/vextendsingle/vextendsingleψA 0,1/angbracketrightbig/parenleftbigg1√ 5/angbracketleftbig ψS 1,1/vextendsingle/vextendsingle−2√ 5/angbracketleftbig ψS 0,0/vextendsingle/vextendsingle/parenrightbigg ⊗/vextendsingle/vextendsingleχT+ S/angbracketrightbig/angbracketleftbig χA/vextendsingle/vextendsingle+/parenleftbigg1√ 5/vextendsingle/vextendsingleψS 1,1/angbracketrightbig −2√ 5/vextendsingle/vextendsingleψS 0,0/angbracketrightbig/parenrightbigg/angbracketleftbig ψA 0,1/vextendsingle/vextendsingle⊗/vextendsingle/vextendsingleχA/angbracketrightbig/angbracketleftbig χT+ S/vextendsingle/vextendsingle/parenrightbigg/bracerightbigg ,(25) or they share the mixed state formed after trac- ing the orbital subsystem ˆρS AB=1 Z/braceleftbig/vextendsingle/vextendsingleχT+ S/angbracketrightbig/angbracketleftbig χT+ S/vextendsingle/vextendsingle+ 5α2 16β/vextendsingle/vextendsingleχA/angbracketrightbig/angbracketleftbig χA/vextendsingle/vextendsingle/bracerightbig , where/vextendsingle/vextendsingleχT+ S/parenleftbig 1,2/parenrightbig/angbracketrightbig =\|1↑/angbracketrightbig \|2↑/angbracketrightbig ,/vextendsingle/vextendsingleχA/parenleftbig 1,2/parenrightbig/angbracketrightbig =1√ 2/parenleftbig \|1↑/angbracketright\|2↓/angbracketright−\| 1↓/angbracketrightbig \|2↑/angbracketrightbig/parenrightbig ,Z= 1 +5α2 16β and the functions/vextendsingle/vextendsingleψA 0,1/angbracketrightbig ,/vextendsingle/vextendsingleψS 1,1/angbracketrightbig ,/vextendsingle/vextendsingleψS 0,0/angbracketrightbig are deﬁned in the section III. Bob sends Alice subsystem Aand Alice does two incompatible measurements (she measures σz Aand σx A). The post-measurement states are given by4: ˆρRB=/summationdisplay n\|ψn/angbracketright/angbracketleftψn\|⊗IBˆρS AB\|ψn/angbracketright/angbracketleftψn\|⊗IB, ˆρQB=/summationdisplay n\|φn/angbracketright/angbracketleftφn\|⊗IBˆρS AB\|φn/angbracketright/angbracketleftφn\|⊗IB.(26)HereIBis the identity operator acting on the subsystem B, and\|ψ1/angbracketright=\|1/angbracketright,\|ψ2/angbracketright=\|0/angbracketright,\|φ1,2/angbracketright=1√ 2(\|0/angbracketright±\|1/angbracketright) are the eigenfunctions of σz A,σx A. Bob has not precise information about the measurements of Alice. The un- certainty about outcomes of measurements is quantiﬁed through the entropy measure: S/parenleftbig R\|B/parenrightbig +S/parenleftbig Q\|B/parenrightbig ≥ln/parenleftbigg1 c/parenrightbigg +S(A\|B/parenrightbig .(27) Herec=maxn,m\|/angbracketleftψm\|φn/angbracketright\|2,S/parenleftbig R\|B/parenrightbig =−ˆρRBln ˆρRB+ trR(ˆρRB) ln trR(ˆρRB)is the conditional quantum infor- mation, and the last term S(A\|B/parenrightbig describes the eﬀect of the quantum memory, meaning that for a negative S(A\|B/parenrightbig <0quantum memory reduces the uncertainty.9 Note that negative conditional quantum entropy points to entanglement in the system. The inverse statement is not always true, i.e., not for all entangled states, con- ditional quantum entropy is negative. Nevertheless, for a pure state Eq. (25) shared by Alice and Bob ˆρAB, the conditionalquantumentropycanbecalculatedexplicitly, and it reads: S(A\|B/parenrightbig ˆρAB=5α2 32βZln/parenleftbigg5α2 32βZ/parenrightbigg +5α2+ 32β 32βZln/parenleftbigg5α2+ 32β 32βZ/parenrightbigg . (28)Easy to see that for any 0< α <√βconditional quan- tum entropy is negative for a pure state S(A\|B/parenrightbig ˆρAB<0. This fact means that correlations stored in the spin-orbit system work as quantum memory and reduce the uncer- tainties of measurements. However in case of the mixed state ˆρS ABsituation is diﬀerent. All entropy measures can be calculated analytically, and we deduce: S(A\|B)ˆρS AB=−1 Zln/parenleftbigg1 Z/parenrightbigg +5α2 32βZln/parenleftbigg5α2 32βZ/parenrightbigg −5α2 16βZln/parenleftbigg5α2 16βZ/parenrightbigg +5α2+ 32β 32βZln/parenleftbigg5α2+ 32β 32βZ/parenrightbigg ,(29) S(R\|B) =−1 Zln/parenleftbigg1 Z/parenrightbigg −5α2 32βZln/parenleftbigg5α2 32βZ/parenrightbigg +5α2+ 32β 32βZln/parenleftbigg5α2+ 32β 32βZ/parenrightbigg , (30) S(Q\|B) =5α2 32βZln/parenleftbigg5α2 32βZ/parenrightbigg +5α2+ 32β 32βZln/parenleftbigg5α2+ 32β 32βZ/parenrightbigg −5α2+ 16β+/radicalbig 25α4+ 256β2 32βZln/parenleftBigg 5α2+ 16β+/radicalbig 25α4+ 256β2 32βZ/parenrightBigg −5α2+ 16β−/radicalbig 25α4+ 256β2 32βZln/parenleftBigg 5α2+ 16β−/radicalbig 25α4+ 256β2 32βZ/parenrightBigg . (31) For strong conﬁnement potential and realistic SO couplingα/√β < 1,Z= 1 +5α2 16β≈1. Apparently S(A\|B)ˆρS AB>0meaning that spin orbit coupling in case of a mixed states enhances uncertainties of measurements. The reason for this nontrivial eﬀect is the following. The total entanglement between subsystems AandBstored in the state ˆρABconsists of spin-spin, spin-orbit, and orbit-orbit contributions.Averaging over the orbital states eliminates part of entanglement. The residual spin-spin entanglement is not enough to reduce the uncertainty of measurements done by Alice. To support this statement, we compare the entanglement stored in the states ˆρABand ˆρS AB. The reduced density matrix ˆρA=trB(ˆρAB)has theform: ˆρA=/parenleftBig1 2Zα2 32β/vextendsingle/vextendsingleψL,1/angbracketrightbig/angbracketleftbig ψL,1/vextendsingle/vextendsingle+1 2Zα2 32β/vextendsingle/vextendsingleψR,1/angbracketrightbig/angbracketleftbig ψR,1/vextendsingle/vextendsingle+1 2Z4α2 32β/vextendsingle/vextendsingleψL,0/angbracketrightbig/angbracketleftbig ψL,0/vextendsingle/vextendsingle+1 2Z4α2 32β/vextendsingle/vextendsingleψR,0/angbracketrightbig/angbracketleftbig ψR,0/vextendsingle/vextendsingle/parenrightBig ⊗/vextendsingle/vextendsingle1↓/angbracketrightbig/angbracketleftbig 1↓/vextendsingle/vextendsingle +/parenleftbigg1 2Zα2 32β/vextendsingle/vextendsingleψL,1/angbracketrightbig/angbracketleftbig ψL,1/vextendsingle/vextendsingle+1 2Z/parenleftbigg 1 +α2 32β/parenrightbigg/vextendsingle/vextendsingleψR,1/angbracketrightbig/angbracketleftbig ψR,1/vextendsingle/vextendsingle+1 2Z/parenleftbigg 1 +4α2 32β/parenrightbigg/vextendsingle/vextendsingleψL,0/angbracketrightbig/angbracketleftbig ψL,0/vextendsingle/vextendsingle+1 2Z4α2 32β/vextendsingle/vextendsingleψR,0/angbracketrightbig/angbracketleftbig ψR,0/vextendsingle/vextendsingle/parenrightbigg ⊗/vextendsingle/vextendsingle1↑/angbracketrightbig/angbracketleftbig 1↑/vextendsingle/vextendsingle (32) The corresponding von Neumann entropy: S(ˆρA) =−3 2Zα2 32βln/parenleftbigg1 2Zα2 32β/parenrightbigg −3 2Z4α2 32βln/parenleftbigg1 2Z4α2 32β/parenrightbigg −1 2Z/parenleftbigg 1 +α2 32β/parenrightbigg ln/parenleftbigg1 2Z/parenleftbigg 1 +α2 32β/parenrightbigg/parenrightbigg −1 2Z/parenleftbigg 1 +4α2 32β/parenrightbigg ln/parenleftbigg1 2Z/parenleftbigg 1 +4α2 32β/parenrightbigg/parenrightbigg . (33)10 The von Neumann entropy for the state ˆρs A=trB(ˆρs AB): S(ˆρS A) =−1 Z/parenleftbigg 1 +5α2 32β/parenrightbigg ln/parenleftbigg1 Z/parenleftbigg 1 +5α2 32β/parenrightbigg/parenrightbigg −5α2 32βZln/parenleftbigg5α2 32βZ/parenrightbigg . (34) Apparently S(ˆρA)> S(ˆρs A)and part of entanglement is lost after averaging over the orbital states. XI. CONCLUSIONS Combining the analytical method with extensive nu- meric calculations, in the present work, we studied the inﬂuence of the spin-orbit interaction on the eﬀect of quantum memory. We observed that measurement done on the spin subsystem through the spin-orbit channel al- lows to extract information about the orbital subsystem and reduce the entropy of the orbital part. On the handresult of two incompatible measurements done on the spin subsystem, depends on the fact whether the den- sity matrix of the system is pure or mixed. In the case of pure states, the spin-orbit coupling works as quantum memory and reduces the uncertainty about the measure- ment results, whereas, in the case of mixed states, the spin-orbit coupling enhances uncertainty. XII. ACKNOWLEDGMENT We acknowledge ﬁnancial support from DFG through SFB 762. 1G. Alber, T. Beth, M. Horodecki, P. Horodecki, R. Horodecki, M. Rötteler, H. Weinfurter, R. Werner, and A. Zeilinger, Quantum information: An introduction to basic theoretical concepts and experiments , Vol. 173 (Springer, 2003). 2W. Heisenberg, Zeitschrift für Physik 43, 172 (1927). 3H. P. Robertson, Phys. Rev. 34, 163 (1929). 4M. Berta, M. Christandl, R. Colbeck, J. M. Renes, and R. Renner, Nature Physics 6, 659 (2010). 5A. Brataas and E. I. Rashba, Phys. Rev. B 84, 045301 (2011). 6J.R.Petta, A.C.Johnson, A.Yacoby, C.M.Marcus, M.P. Hanson, and A. C. Gossard, Phys. Rev. B 72, 161301 (2005). 7T.Meunier,I.T.Vink,L.H.WillemsvanBeveren, F.H.L. Koppens, H. P. Tranitz, W. Wegscheider, L. P. Kouwen- hoven, and L. M. K. Vandersypen, Phys. Rev. B 74, 195303 (2006). 8D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120 (1998). 9B. E. Kane, Nature 393, 133 (1998). 10P. Recher, E. V. Sukhorukov, and D. Loss, Phys. Rev. Lett.85, 1962 (2000). 11R. Hanson and G. Burkard, Phys. Rev. Lett. 98, 050502 (2007). 12J. Elzerman, R. Hanson, L. W. Van Beveren, B. Witkamp, L. Vandersypen, and L. P. Kouwenhoven, nature 430, 431 (2004). 13R. Hanson, L. H. W. van Beveren, I. T. Vink, J. M. Elz- erman, W. J. M. Naber, F. H. L. Koppens, L. P. Kouwen- hoven, and L. M. K. Vandersypen, Phys. Rev. Lett. 94, 196802 (2005). 14R.HansonandD.D.Awschalom,Nature 453,1043(2008). 15R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha, and L. M. K. Vandersypen, Rev. Mod. Phys. 79, 1217(2007). 16T. Fujisawa, D. G. Austing, Y. Tokura, Y. Hirayama, and S. Tarucha, Nature 419, 278 (2002). 17R. Hanson, B. Witkamp, L. M. K. Vandersypen, L. H. W. van Beveren, J. M. Elzerman, and L. P. Kouwenhoven, Phys. Rev. Lett. 91, 196802 (2003). 18J. A. Folk, R. M. Potok, C. M. Marcus, and V. Umansky, Science 299, 679 (2003), https://science.sciencemag.org/content/299/5607/679.full.pdf. 19W. Lu, Z. Ji, L. Pfeiﬀer, K. West, and A. Rimberg, Nature 423, 422 (2003). 20D. D. Awschalom, D. Loss, and N. Samarth, Semicon- ductor Spintronics and Quantum Computation (Springer Science & Business Media, 2002). 21D. Scalbert, Phys. Rev. B 99, 205305 (2019). 22F. Adabi, S. Haseli, and S. Salimi, EPL (Europhysics Let- ters)115, 60004 (2016). 23G. Adesso, T. R. Bromley, and M. Cianciaruso, Journal of Physics A: Mathematical and Theoretical 49, 473001 (2016). 24A. K. Pati, M. M. Wilde, A. R. U. Devi, A. K. Rajagopal, and Sudha, Phys. Rev. A 86, 042105 (2012). 25P. J. Coles and M. Piani, Phys. Rev. A 89, 022112 (2014). 26T.Nemoto, F.Bouchet, R.L.Jack, andV.Lecomte,Phys. Rev. E 93, 062123 (2016). 27P. J. Coles, M. Berta, M. Tomamichel, and S. Wehner, Rev. Mod. Phys. 89, 015002 (2017). 28S. Wehner and A. Winter, New Journal of Physics 12, 025009 (2010). 29R. Kumar and S. Ghosh, Journal of Physics B: Atomic, Molecular and Optical Physics 51, 165301 (2018). 30E. I. Rashba and A. L. Efros, Phys. Rev. Lett. 91, 126405 (2003). 31E. I. Rashba and A. L. Efros, Applied Physics Letters 83, 5295 (2003), https://doi.org/10.1063/1.1635987.11 32D. V. Khomitsky and E. Y. Sherman, Phys. Rev. B 79, 245321 (2009). 33D. V. Khomitsky, L. V. Gulyaev, and E. Y. Sherman, Phys. Rev. B 85, 125312 (2012). 34E. Y. Sherman and D. Sokolovski, New Journal of Physics 16, 015013 (2014). 35S. Faniel, T. Matsuura, S. Mineshige, Y. Sekine, and T. Koga, Phys. Rev. B 83, 115309 (2011). 36P. I. Tamborenea and H. Metiu, Phys. Rev. Lett. 83, 3912 (1999). 37D. Stepanenko, M. Rudner, B. I. Halperin, and D. Loss, Phys. Rev. B 85, 075416 (2012). 38J. Pawłowski, P. Szumniak, A. Skubis, and S. Bednarek, Journal of Physics: Condensed Matter 26, 345302 (2014). 39Q. Li, L. Cywiński, D. Culcer, X. Hu, and S. Das Sarma, Phys. Rev. B 81, 085313 (2010). 40X. Hu and S. Das Sarma, Phys. Rev. A 61, 062301 (2000). 41W. Heitler and F. London, Zeitschrift für Physik 44, 455 (1927). 42P. Fazekas, Series in Modern Condensed Matter Physics: Lecture Notes on Electron Correlation and Magnetism (World Scientiﬁc Publishing Co. Pte. Ltd.: River Edge, NJ, 1999). 43J. Stolze and D. Suter, Quantum Computing: a short course from Theory to Experiment (John Wiley & Sons, 2008). 44K. Życzkowski, P. Horodecki, A. Sanpera, and M. Lewen- stein, Phys. Rev. A 58, 883 (1998). 45W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998). 46H. Ollivier and W. H. Zurek, Phys. Rev. Lett. 88, 017901 (2001). 47R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. 81, 865 (2009). 48L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 84, 2722 (2000). 49L.Amico, R.Fazio, A.Osterloh, andV.Vedral,Rev.Mod. Phys. 80, 517 (2008). 50S. L. Braunstein and P. van Loock, Rev. Mod. Phys. 77, 513 (2005).51R. Simon, Phys. Rev. Lett. 84, 2726 (2000). 52C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Rev. Mod. Phys. 84, 621 (2012). 53C.Silberhorn, P.K.Lam, O.Weiß, F.König, N.Korolkova, and G. Leuchs, Phys. Rev. Lett. 86, 4267 (2001). 54G. Adesso and F. Illuminati, Journal of Physics A: Math- ematical and Theoretical 40, 7821 (2007). 55F. Toscano, A. Saboia, A. T. Avelar, and S. P. Walborn, Phys. Rev. A 92, 052316 (2015). 56G. Adesso, A. Seraﬁni, and F. Illuminati, Phys. Rev. A 70, 022318 (2004). 57E. Shchukin and W. Vogel, Phys. Rev. Lett. 95, 230502 (2005). 58R. F. Werner and M. M. Wolf, Phys. Rev. Lett. 86, 3658 (2001). 59M. M. Wilde, Quantum information theory (Cambridge University Press, 2013). 60V. Vedral, Rev. Mod. Phys. 74, 197 (2002). 61L. Henderson and V. Vedral, Journal of Physics A: Math- ematical and General 34, 6899 (2001). 62H. Ollivier and W. H. Zurek, Phys. Rev. Lett. 88, 017901 (2001). 63W. H. Zurek, Phys. Rev. A 67, 012320 (2003). 64A.J.LeggettandA.Garg,Phys.Rev.Lett. 54,857(1985). 65G. Schild and C. Emary, Phys. Rev. A 92, 032101 (2015). 66K. Wang, G. C. Knee, X. Zhan, Z. Bian, J. Li, and P. Xue, Phys. Rev. A 95, 032122 (2017). 67D. Avis, P. Hayden, and M. M. Wilde, Phys. Rev. A 82, 030102 (2010). 68C.-M. Li, N. Lambert, Y.-N. Chen, G.-Y. Chen, and F. Nori, Scientiﬁc reports 2, 885 (2012). 69C. A. Melton, M. Zhu, S. Guo, A. Ambrosetti, F. Pederiva, and L. Mitas, Phys. Rev. A 93, 042502 (2016). 70C. A. Melton and L. Mitas, Phys. Rev. E 96, 043305 (2017). 71N. D. Drummond, M. D. Towler, and R. J. Needs, Phys. Rev. B 70, 235119 (2004).
1812.08884v1.Non_local_Spin_charge_Conversion_via_Rashba_Spin_Orbit_Interaction.pdf	Non-local Spin-charge Conversion via Rashba Spin-Orbit Interaction Junji Fujimoto Institute for Chemical Research, Kyoto University, Uji, Kyoto 611-0011, Japan and RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan Gen Tatara RIKEN Center for Emergent Matter Science (CEMS) and RIKEN Cluster for Pioneering Research (CPR), Wako, Saitama 351-0198, Japan (Dated: December 24, 2018) Abstract We show theoretically that conversion between spin and charge by spin-orbit interaction in metals occurs even in a non-local setup where magnetization and spin-orbit interaction are spatially separated if electron diusion is taken into account. Calculation is carried out for the Rashba spin-orbit interaction treating the coupling with a ferromagnet perturbatively. The results indicate the validity of the concept of eective spin gauge eld (spin motive force) in the non-local conguration. The inverse Rashba-Edelstein eect observed for a trilayer of a ferromagnet, a normal metal and a heavy metal can be explained in terms of the non-local eective spin gauge eld. 1arXiv:1812.08884v1 [cond-mat.mes-hall] 20 Dec 2018(F)Ferromagnetmagnetization Rasbha spin-orbit interaction at the NM-HM interface Nonmagnetic Metal (NM) Heavy Metal (HM)FIG. 1. Schematic gure of a trilayer of a ferromagnet (F), a normal metal (NM) and a heavy metal (HM). The Rashba spin-orbit interaction is localized at the NM-HM interface. I. INTRODUCTION The objective of spintronics is to manipulate spins by electric means and vice versa. For gen- erating spin accumulation and spin current, several methods have been experimentally established in the last two decades, including the spin pumping eect1{4, where magnetization precession of a ferromagnet (F) is used to generate spin current into a normal metal (NM) in a F-NM junction. For electric detection of spin current, so called the inverse spin Hall eect induced by spin-orbit interaction of heavy metal is widely used5. Another electric detection of spin is by interfacial Rashba spin-orbit interaction, called the inverse Rashba-Edelstein eect. The eect, the reciprocal eect of the current-induced spin po- larization studied theoretically by Edelstein6, has been experimentally demonstrated in a trilayer of a ferromagnet, a normal metal and a heavy metal (HM) (Fig. 1)7. The Rashba interaction is expected to be localized at the interface of NM and HM, and the normal metal works as a spacer to separate the magnetization and the Rashba interaction. The current observed was argued to sup- port the spin current picture, in which a spin current generated by spin pumping eect propagates through the normal metal, forming spin accumulation at the NM-HM interface, nally resulting in a current as a result of inverse Rashba-Edelstein eect. Theoretically, current generation indicates existence of an eective electric eld (motive force). In the present magnetic systems, it is the one driving electron spin, namely, eective spin gauge eld or spin motive force. Eective spin gauge eld has been known to arise for a slowly-varying magnetic texture in ferromagnets, as the texture gives rise to a phase (spin Berry's phase) for electron spin wave function as a result of sdexchange interaction8,9. The spin Berry's phase generates a current when magnetization has dynamics besides spatial texture. The concept of eective spin gauge eld was shown to be generalized to include spin-orbit interactions that is linear in the wave vector, like the Rashba interaction10{12. It was demonstrated that magnetization Mand the Rashba eldgive rise to an eective spin gauge eld proportional to M, leading to an eective 2electric eld and current proportional to _M. It was also pointed out that spin relaxation leads to a perpendicular eective spin electric eld, (M_M)13. The latter component leads to a direct current (DC) for a precessing magnetization, while the former induces only alternating current (AC). The DC is in the direction perpendicular to both and the average of M, which agrees with the geometry of the experimentally-observed inverse Rashba-Edelstein eect. The above theories, however, do not directly apply to the experimental situations with a spacer layer, as the coexistence of magnetization and Rashba interaction is assumed in the theories. The objective of the paper is to demonstrate theoretically that the concept of eective electric eld can be generalized to describe non-local congurations where the magnetization and the Rashba interaction are spatially separated by a nonmagnetic metal. For charge transport in metals, longer distance than the electron mean free path is possible due to electron diusion; Current generation in disordered metals can therefore occur non-locally. As far as the diusion is induced by elastic scattering conserving spin, the spin information is expected to be equally transported long distance. In fact, long-range diusive component of spin current induced by magnetization dynamics was studied in Ref.14. The spin-charge conversion eect was brie y mentioned there, but assuming uniform Rashba interaction. It was also pointed out that long-range spin chirality contributes to the anomalous Hall eect in disordered ferromagnet if the spatial size is less than spin diusion length15, indicating that spin Berry's phase has long-range components. Moreover, spin Hall and inverse spin Hall eects were recently formulated in terms of non-local conversion of spin and electric current including electron diusion16. In this representation, the observed spin density (or current) in the direct (inverse) spin Hall eect is directly related to the driving electric eld (or spin pumping eld) via a non-local response function of spin and electric current. In this paper, we calculate electric current generated in the system of conduction electron with sdexchange interaction with a dynamic magnetization and the Rashba interaction having spatial distributions. Although experiments are carried out in trilayers with rather sharp interfaces, we here describe the slowly-varying case, which can be accessible straightforwardly by an expansion with respect to the wave vectors of the two interactions. The vertex corrections (VCs) to the correlation function representing the eect turns out to contain a singular pole at slowly-varying limit, indicating the diusive nature. The diusion propagator arising from this pole is shown to connect the information of the magnetization and the Rashba interaction even when they are spatially separated, resulting in a non-local current generation. The eect is interpreted in terms of the non-local component of spin electric eld. 3II. MODEL AND GREEN FUNCTIONS We consider the following model Hamiltonian, H=H0+HR+Hsd(t) with H0=Z dr y(r) ~2r2 2me+uVNiX i=1(rRi)! (r); (1a) HR=i~ 2ijkZ dri(r) y(r)k rj (r) rj y(r) k (r) ; (1b) Hsd(t) =Z dr y(r) M(r;t) (r); (1c) where (r) =t( "(r); #(r)) is the spinor form of the annihilation operator of electron with the mass being me, andH0consists of the kinetic term and non-magnetic impurity potential with the strength being u.HRis the Rashba spin-orbit interaction with the spatial-dependent Rashba eld denoted by(r) = (x(r);y(r);z(r)). Thesdexchange interaction is given by Hsd(t), where M(r;t) is the magnetization vector including the sdinteraction strength. We deal with H0as the unperturbed Hamiltonian and treat HRandHsdperturbatively. Here, Vis the volume of the system,= (x;y;z) is the vector form of the Pauli matrices, ~is the Planck constant divided by 2, andijkis the Levi-Civita symbol. We consider slowly-varying case with weak spatial dependencies of M(r) and(r). Our particular interest is the case where M(r) and(r) do not coexist, such as a tri-layer structure composed of F, NM and HM. The charge current density operator of the system is given by j(r) =e~ 2mei y(r) r (r) r y(r) (r) +e(r)s(r); (2) the rst two terms of which we call the current density for the normal velocity and the last term is called that for the anomalous velocity, where e(>0) is the elementary charge, and s(r) = y(r) (r) is the spin density. In the Fourier forms, the Hamiltonians of Eqs. (1) are given as H0=X kkcy kck+uX k;qNiX i=1eiqRicy k+q 2ckq 2; (3a) Hsd(t) =Zd! 2ei!tX qS(q)M(q;!); (3b) HR=X k;q(q~k)cy k+q 2ckq 2; (3c) wherek=~2k2=2me, ands(q) is the Fourier component of the spin operator given by s(q) =1 VX kcy kq 2ck+q 2(4) 4The current density in the Fourier form is given as j(q) =e VX k~k mecy kq 2ck+q 2+eX q0q0s(qq0); (5) where the rst and second terms correspond to the currents of the normal and anomalous velocities, respectively. We denote the thermal Green function for the Hamiltonian H0+HRasGk;k0(in), which is evaluated up to the rst order with respect to HRas Gk;k0(in)'gk(in)k;k0+~ 2gk(in) kk0(k+k0) gk0(in); (6) wheren= (2n+ 1)kBTis the Matsubara frequency of fermion, gk(in) is the thermal Green function for the Hamiltonian H0given by gk(in) =1 ink+isgn(n)~=(2)(7) with the signum function sgn( x). The lifetime of electron evaluated within the Born approximation is given as ~ 2=niu2 (8) with=(F) being the density of states (DOS) at the Fermi energy Fof NM. III. NON-LOCAL EFFECTIVE ELECTRIC FIELDS By evaluating the non-local charge current induced by the magnetization dynamics, and using the Drude conductivity, we show that the charge current is driven by the non-local eective electric elds. We consider the exchange interaction up to the second order and the Rashba interaction in the rst order in this section. A. Linear response to exchange interaction For the linear response of the charge current hj(r;t)i(1)to the external eld n(r0;t0), where the external Hamiltonian is given by Hsd(t0), the current is calculated based on the Kubo formula17 (see Appendix A 2) as hji(r;t)i(1)=i ~Zt 1dt0 [ji(r;t);Hsd(t0)] =Z dr0Z1 1dt0(1) ij(r;r0;tt0)Mj(r0;t0); (9) 5whereji(r;t) is the Heisenberg representation of Eq. (2), [ A;B] =ABBAis the communicator, hi is the thermal average for H0+HR, and the linear response coecient (1) ij(r;r0;tt0) is the retarded correction function between the charge current density and the spin density, (1) ij(r;r0;tt0) =i ~(tt0) [ji(r;t);sj(r0;t0)] (10) with(t) being the Heaviside step function and sj(r0;t0) being the Heisenberg representation of the spin density. Note that, since the Rashba eld in the system has the spatial dependence, the linear response coecient cannot be expressed as (1) ij(rr0;tt0), which also means that the space translational symmetry is not assumed in the system. In the Fourier form, the charge current is given as hji(q;!)i(1)=X q0R;(1) ij(q;q0;!)Mj(q0;!) (11) Here,R;(1) ij(q;q0;!) can be calculated from the following correlation function in the Matsubara formalism, (1) ij(q;q0;i!) =VZ 0dei! Tji(q;)sj(q0;0) ; (12) by taking the analytic continuation, i!!~!+i0 as R;(1) ij(q;q0;!) =(1) ij(q;q0;!+i0); (13) where= 1=kBTis the inverse temperature with the Boltzmann constant kB, and!= 2= (= 0;1;) is the Matsubara frequency of boson. Note that the Matsubara frequencies are dened as in unit of energy instead of frequency. By means of the thermal Green function for (1, n) inin+iωλ k+q/2 k+q/2 k−q/2 k−q/2ki mσj (1, a) inin+iωλ k−q k+q−qσjσmk kilmαq,l FIG. 2. The Feynman diagrams of (1;n) ij and(1;a) ij. The solid lines with two arrows denote the Green functions including the Rashba interaction given by Eq. (6), the lled circle represents the spin vertex, the unlled triangle describes the normal velocity vertex, and the dashed wavy line indicates the anomalous velocity vertex without the Pauli matrix. 6H0+HR, Eq. (12) is expressed as (1) ij=(1;n) ij+(1;a) ij+(1;a)(df) ij +(1;n)(df) ij; (14) where(1;n) ij and(1;a) ijare the contributions from the normal and anomalous velocities without vertex corrections, which are given as (1;n) ij(q;q0;i!) =e VX nX k;k0~ki metrh Gk+q 2;k0+q0 2(in+i!)jGk0q0 2;kq 2(in)i ; (15a) (1;a) ij(q;q0;i!) =e VX nX q00ilmq00;lX k;k0tr mGk;k0(in+i!)jGk0q0;k+q00q(in) (15b) Figure 2 depicts (1;n) ij and(1;a) ij. Contributions (1;n)(df) ij and(1;a)(df) ij contain the diusion ladder VCs, whose diagrams and expressions are given in Appendix B (Fig. 4d-j). In order to evaluate them up to the rst order of the Rashba interaction, we expand the Green functions in Eq. (15a) by using Eq. (6). As Eq. (15b) is already the rst order of the Rashba interaction, the Green functions there can be approximated as Gk;k0(in) =gk(in)k;k0. We take the analytic continuation, i!!~!+i0, and calculate the !-linear contribution, which leads to a contribution proportional to _M. We also expand them up to the second order with respect to qandq0. The details of the calculations are shown in Appendix C. Finally, we obtain R;(1) ij(q;q0;!) =emljqq0;l (1) q;q0;im+i!'(1) q;q0;im+ ; (16) '(1) q;q0;im=2 q02 im(qq0)q0(qiq0 i)q0 m ; (17) where(1) q;q0;imis the static response to the magnetization, and '(1) q;q0;imis the dynamical response of our interest. In the real space, using the Drude conductivity D= 2e2F=(3me) =e2D0with =(F) being DOS at the Fermi energy of NM and D0being the diusion constant, we nd the linear-order current as hj(r;t)i(1)=el2 3Z dr0D(rr0) rr0rr [(r)_M(r0;t)]rr rr0[(r)_M(r0;t)] (18) =el2 3Z dr0D(rr0) rr0 rr[(r)_M(r0;t)] ; (19) where D(r)1 VX qeiqr D0q2=3 4l21 r; (20) is the diusion propagator, lp3D0being the elastic mean free path. 7B. Second order response to exchange interaction For the second order response to the exchange Hamiltonian, the charge current is given by hji(r;t)i(2)=i ~2Zt 1dt0Zt0 1dt00Dh [ji(r;t);Hsd(t0)];Hsd(t00)iE =ZZ dr0dr00ZZ1 1dt0dt00R;(2) ijk(r;r0;r00;tt0;t0t00)Mj(r0;t0)Mk(r00;t00);(21) where the second order response coecient R;(2) ijk(r;r0;r00;tt0;t0t00) is expressed as R;(2) ijk(r;r0;r00;tt0;t0t00) =1 2 Qijk(r;r0;r00;tt0;t0t00) +Qikj(r;r00;r0;tt00;t00t0) ; (22) Qijk(r;r0;r00;tt0;t0t00) =i ~2 (tt0)(t0t00)Dh [ji(r;t);sj(r0;t0)];sk(r00;t00)iE (23) Here,R;(2) ikj(r;r00;r0;tt00;t00t0) =R;(2) ijk(r;r0;r00;tt0;t0t00). In the Fourier form, the current is given as hji(q;!)i(2)=X q0;q00Z1 1d!0 2R;(2) ijk(q;q0;q00;!;!0)Mj(q0;!!0)Mk(q00;!0) (24) From Appendix A 3, the non-linear response coecient R;(2) ijk(q;q0;q00;!;!0) is evaluated from (2) ijk(q;q0;q00;i!;i!0) =V2 2ZZ 0dd0ei!+i!00 Tji(q;+0)sj(q0;0)sk(q00;0) (25) by taking the analytic continuations as i!!~!+ 2i0; i!0!~!0+i0 (26) Note that, in order to obtain the precise response coecient, the order of the analytic continuations fori!andi!0must be specied; taking i!0to the real frequency ~!0and then taking i!to ~!from the upper plane ( !(0)>0). Hence, we have set Eq. (26). We should emphasise that the Matsubara Green function method can apply to the non-linear responses as demonstrated by Jujo18and by Kohno and Shibata19. (See Appendix A for general cases.) Here, we separate Eq. (25) into three components in the similar way of the calculation of (1) ij [Eq. (14)], (2) ijk=(2;n) ijk+(2;a) ijk+(2;n)(df) ijk+(2;a)(df) ijk; (27) 8+(2, n) σj σkki mk+q 2k+q 2 k−q 2 k+q 2 k−q 2k−q 2in+iωλ in+iωλ in σjσk ki m k+q 2 k−q 2k+q 2 k−q 2k+q 2 k−q 2 in−iωλin−iωλin (2, a) +σj σm σkin+iωλ in+iωλ inkk k−q k k−qk−q+qilmαq,lσmk k k−qk−q+q σjσk in−iωλin−iωλink k−q ilmαq,lFIG. 3. The Feynman diagrams of (2;n) ijkand(2;a) ijk. The lines and symbols are dened in the caption of Fig. 2. where the rst two terms are the contributions from the normal and anomalous velocities without VCs, respectively, given by (2;n) ijk(q;q0;q00;i!;i!0) =e 2VX nX k;k0;k00~ki m trh Gk+q 2;k0+q0 2(in+i!)jGk0q0 2;k00+q00 2(in+i!0)kGk00q00 2;kq 2(in) +Gk+q 2;k00+q00 2(in)kGk00q00 2;k0+q0 2(ini!0)jGk0q0 2;kq 2(ini!)i ; (28) (2;a) ijk(q;q0;q00;i!;i!0) =e 2VX nX k;k0;k00ilmX q000q000;l trh mGk;k0(in+i!)jGk0q0;k00(in+i!0)kGk00q00;kq+q000(in) +mGk;k00(in)kGk00q00;k0(ini!0)jGk0q0;kq+q000(ini!)i (29) The last two terms in Eq. (27) include the ladder type VCs of (2;n) ijkand(2;a) ijk, which are given in Appendix B. We evaluate them up to the rst order of the Rashba interaction. For (2;n) ijk, we expand the Green functions in the rst term of Eq. (15) by using Eq. (6). As (2;a) ijkis already the rst order of the Rashba interaction, the Green functions in the second term can be approximated 9asGk;k0(in) =gk(in)k;k0. We take the analytic continuation as shown by Eq. (26) and evaluate the!0-linear contribution, which leads to a contribution proportional to M_M. Then, we expand them up to the second order with respect to q,q0andq00. The details of the calculations are shown in Appendix C. After all, we have the followings: R;(2) ijk(q;q0;q00;!;!0) = 2ieolmmjkqq0q00;l (2) q;q0;q00;io+i!#(2) q;q0;q00;io+i!0'(2) q;q0;q00;io+ ; (30) '(2) q;q0;q00;io=2i2 ~io(qq0q00)(q0+q00)(qiq0 iq00 i)(q0 o+q00 o) (q0+q00)2; (31) where(2) q;q0;q00;iois static response to the magnetization, #(2) q;q0;q00;iois the dynamical response pro- portional to d( MjMk)=dt, which are negligible. '(2) q;q0;q00;iois the dynamical response of our interest, which is proportional to such as Mjd(Mk)=dt. In the real space, dynamically-induced current is hj(r;t)i(2)=4el22 3~Z dr0D(rr0) rr0 rr (r) M(r0;t)_M(r0;t) :(32) IV. RESULTS AND DISCUSSION The generated charge current to the second order responses to the exchange interaction is thereforej(r;t) =hj(r;t)i(1)+hj(r;t)i(2). The current is expressed as a response to a non-local eective electric eld as j(r;t) =DEe(r;t), where Ee(r;t) =mel2 2eFZ dr0D(rr0) rr0 rr (r) _M(r0;t) +4 ~[M(r0;t)_M(r0;t)] : (33) Note that the magnetization M(r;t) is dened as including the sdinteraction strength. The linear response term, E(1) e, is written as E(1) e=_A(1) e, where A(1) e(r;t) =mel2 2eFZ dr0D(rr0) rr0 rr (r)M(r0;t) (34) is a non-local extension of eective gauge eld discussed in Refs.10{12. In contrast, the second-order contribution, proportional to spin damping, M_M, does not have the corresponding gauge eld like in the local case13. For junctions like a trilayer homogeneous in the xy-plane, the spatial derivative is nite only in thez-direction. The in-plane current, which is of experimental interest, in this case reads jk(r;t) =mel2 2eFDZ dr0D(rr0)(rz r(r))rz r0 _M(r0;t) +4 ~[M(r0;t)_M(r0;t)] :(35) 10This result indicates that the spatially-inhomogeneity of precessing spin at the F-NM interface drives an in-plane eective motive force at the NM-HM interface as a result of electron diusion. This motive force is an alternative and direct interpretation of inverse Rashba-Edelstein eect. For describing the case of a spacer thicker than spin diusion length, spin relaxation eect needs to be included in the diusion. As was discussed in Ref.16, the result in this case becomes Eq. (33) with diusion D(r) replaced by the one including spin diusion length, Ds(r)1 VX qeiqr D0q2+ s; (36) where s, proportional to spin relaxation rate, is related to spin diusion length lsasls=l=p3 s. The non-local eective electric eld found in the present study is an electric counterpart of the non-local eective magnetic eld (non-local spin Berry's phase) discussed in the context of the anomalous Hall eect15. Although the spin Berry's phase itself arises from static magnetization textures, calculation of the non-local contribution in the present formalism requires including an external eld with a nite or innitesimal frequency, as the electron diusion applies to non- equilibrium situations only. ACKNOWLEDGMENTS JF would like to thank A. Shitade and S.C. Furuya for giving informative comments. JF is supported by a Grant-in-Aid for Specially Promoted Research (No. 15H05702). GT thanks a Grant-in-Aid for Exploratory Research (No.16K13853) and a Grant-in-Aid for Scientic Research (B) (No. 17H02929) from the Japan Society for the Promotion of Science, a Grant-in-Aid for Scientic Research on Innovative Areas (No.26103006) from The Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan, and the Graduate School Materials Science in Mainz (MAINZ) (DFG GSC 266) for nantial support. Appendix A: Matsubara formalism for non-linear responses Here, we show the formulation of the non-linear response theory based on the Matsubara for- malism. Some textbooks20,21explain that it is not possible. However, by a careful treatment of analytic continuations, this formulation can be done and leads to the exactly same result from the Keldysh formalism. In this Appendix, we show the way to evaluate the responses up to the second order with respect to the external force. We also discuss brie y the evaluations for the higher order responses. 111. Setup In this Appendix, we assume that the system we consider is expressed by the Hamiltonian H, and the mechanical external force is F(t) (is index), which couples to the physical quantity ^A, hence the external Hamiltonian given by ^H0(t) =^AF(t) (A1) (One should presume that the dummy index sums over all the external forces.) We introduce > 0 as an innitesimal quantity to ensure that the system is in thermal equilibrium and the external force is zero at the time t!1 , and the external force is turned on adiabatically from the time: F(t) =etZ1 1d! 2ei!tF(!) =Z1 1d! 2ei(!+i)tF(!) (A2) Following the paper by Kubo17, the response of the physical quantity ^Bto the external force F is given byh^Bi(t) =h^Bi0+1B(t) +2B(t) ++kB(t) +with thek-th order response kB(t) =1 i~kZt 1dt1Zt1 1dt2Ztk1 1dtk Tr ^A1(t1);h ^A2(t2); ;[^Ak(tk);^] i ^B(t) F1(t1)F2(t2)Fk(tk); (A3) whereh^Bi0is the expectation value without any external elds, ^A(t) =ei^Ht=~^Aei^Ht=~is the Heisenberg representation of ^A, [^A;^B] = ^A^B^B^Ais communicator, ^ is the density matrix operator for ^H, and ^=e^H=Tre^H=e( ^H)(A4) with= 1=kBTand with =kBTln Trfe^Hgbeing the thermodynamic potential. Introducing jnias the eigenstates of the Hamiltonian, ^Hjni=Enjni, Trfg is given as Trf^Ag=X nhnj^Ajni (A5) The thermal average hi for the systemHin the temperature Tis dened by D ^AE = Trf^^Ag=X ne( En)hnj^Ajni (A6) We also note that the time translational symmetry is held in thermal equilibrium. 122. Linear response We rst look at the linear response ( k= 1). Using the cyclic relation, Tr f^A^B^Cg= Trf^B^C^Ag= Trf^C^A^Bg, we nd 1B(t) =1 i~Zt 1dt0Trn [^A(t0);^]^B(t)o F(t0) =Z1 1dt0QR (tt0)F(t0); (A7) whereQR (t) is the retarded two-point Green function, QR (t) =i ~(t)D [^B(t);^A(0)]E (A8) By using the Fourier transformation22,1B(!) =QR (!)F(!), where QR (!) =i ~Z1 0dtei(!+i0)tD [^B(t);^A(0)]E (A9) Here,!+i0 stands for lim !0+!+i. The Matsubara Green function corresponding to QR (!) is Q(i!) =1 ~Z~ 0dei!D Tf^B()^A(0)gE ; (A10) where!= 2=~(= 0;1;2;) is the Matsubara frequency of bosons, is the imaginary time, and ^A() =e^H=~^Ae^H=~is the so-called Heisenberg representation of ^Ain the imaginary time (t=i) and dened in the region ~~. Tfg is the time ordering operator of . Note that as one shows the periodicity hTf^B(~)^A(0)gi=hTf^B()^A(0)gifor0 using Eq. (A6), it can be expressed by means of the Fourier series of ei!. The correspondence between QR (!) andQ(i!) is proven easily by representing them in the Lehmann representation and taking the analytic continuation, i!!!+i0, resulting to QR (!) =Q(!+i0) (A11) From these, we can evaluate the linear response coecient QR (!) from the corresponding Matsub- ara Green function Q(i!) by taking the analytic continuation, i!l!!+i0. 3. Second order response Next, we show the way to evaluate the second order response precisely. This procedure is similar to the evaluation of the linear response; (1) nd the correlation function in the Matsubara 13formalism corresponding to the response coecient, (2) calculate the correlation function, and (3) take the precise analytic continuation. The procedures (1) and (3) are of the central theme in this Appendix since the procedure (2) is same as the well-known procedure. Fork= 2 in Eq. (A3), the second order response is given as 2B(t) =1 i~2Zt 1dt1Zt1 1dt2Trnh ^A(t1);^A(t2);^i ^B(t)o F(t1)F(t2) =Z1 1dt1Z1 1dt2QR (t;t1;t2)F(t1)F(t2) =1 2!Z1 1dt1Z1 1dt2 QR (t;t1;t2) +QR (t;t2;t1) F(t1)F(t2); (A12) whereQR (t;t0;t00) is the retarded three-points correlation function given by QR (t;t0;t00) =1 ~2(tt0)(t0t00)D [^B(t);^A(t0)];^A(t00)E (A13) Here, we should point out that we treated the external forces F(t1) andF(t2) symmetrically; we added the term interchanging andas well ast1andt2and divided them by 2! as in the last equal of Eq. (A12). From the following relation by using Eq. (A5), QR (t;t1;t2) =1 ~2(tt1)(t1t2)X n;m;le( En)(1e(EnEl)) n ei(EnEm)(tt1)=~+i(EnEl)(t1t2)=~hnj^Bjmihmj^Ajlihlj^Ajni+ (c:c:)o ; (A14) one ndQR (t;t1;t2) =QR (tt1;t1t2) with QR (t;t0) =1 ~2(t)(t0)D [^B(t+t0);^A(t0)];^A(0)E (A15) By means of QR (t;t0), Eq. (A12) is rewritten as 2B(t) =Z1 1dt1Z1 1dt2R (tt1;t1t2)F(t1)F(t2); (A16) R (t;t0) =1 2! QR (t;t0) +QR (t+t0;t0) (A17) Then, the second order response [Eq. (A12)] is expressed in the Fourier space23as 2B(!) =Z1 1d!0 2R (!;!0)F(!!0)F(!0); (A18) R (!;!0) = (QR (!;!0) +QR (!;!!0))=2!; (A19) QR (!;!0) =1 ~2Z1 0dtZ1 0dt0ei(!+i)t+i(!0+i0)t0D [^B(t+t0);^A(t0)];^A(0)E (A20) 14We introduced the convergence factor and0, but these two must have the relation >0because ofQR (!;!!0). Hence, we use 2 i0 as the convergence factor for !andi0 as that for !0. As we show in Appendix A 4, the corresponding correlation function in the Matsubara formu- lation toR (!;!0) (not toQR (!;!0)) is given as '(i!;i!0) =1 2!~2Z~ 0dZ~ 0d0ei!(0)+i!00D T;0f^B()^A(0)^A(0)gE (A21) Taking the analytic continuation i!!!+ 2i0 andi!0!!0+i0, the following relation is held R (!;!0) ='(!+ 2i0;!0+i0) (A22) Hence, we can evaluate the second order response [Eq. (A18)] from the corresponding correlation function in the Matsubara formalism, '(i!;i!0), by taking the analytic continuations. 4. Correspondence between '(i!;i!0)andR (!;!0) Here, we show Eq. (A22). First, we perform the integrals of tandt0inQR (!;!0). Introducing !+=!+ 2i0 and!0 +=!0+i0, QR (!;!0) =1 ~2X n;m;le( En)(1e(EnEl))Z1 0dtZ1 0dt0ei!+t+i!0 +t0 n ei(EnEm)t=~+i(EnEl)t0=~hnj^Bjmihmj^Ajlihlj^Ajni +ei(EnEm)t=~i(EnEl)t0=~hmj^Bjnihnj^Ajlihlj^Ajmio =X n;m;le( En)(1e(EnEl)) ( hnj^Bjmihmj^Ajlihlj^Ajni (~!++EnEm)(~!0 ++EnEl)+hmj^Bjnihnj^Ajlihlj^Ajmi (~!+En+Em)(~!0 +En+El)) ; (A23) and forQR (!;!!0), by interchanging n$min the above equation, we obtain QR (!;!!0) =X n;m;le( En)e(EnEm)(1e(EmEl)) ( hmj^Bjnihnj^Ajlihlj^Ajmi (~!+En+Em)(~!+~!0 ++EmEl)+hnj^Bjmihmj^Ajlihlj^Ajni (~!++EnEm)(~!+~!0 +Em+El)) 15Here,!+!0 +=!!0+i(0), and>0is needed for the convergence in the limit t!1 , hence putting = 20. From these, R (!;!0) =QR (!;!0) +QR (!;!!0) is given as R (!;!0) =1 2X n;m;le( En) hnj^Bjmihmj^Ajlihlj^Ajni ~!++EnEma(!+;!0 +) +hmj^Bjnihnj^Ajlihlj^Ajmi ~!++EnEma(!+;!0 +)! ; (A24) where a(!+;!0 +)1e(EnEl) ~!0 ++EnEl+e(EnEm)(1e(EmEl)) ~!+~!0 +Em+El(A25) Next, we perform the integrals of the correlation function in the Matsubara formalism [Eq. (A21)]. From the time-ordering operator, for >0>0, D T;0f^B()^A(0)^A(0)gE =X m;n;le( En)e(EmEl)0=~hnj^Bjmihmj^Ajlihlj^Ajnie(EnEm)=~; (A26) hence we nd 1 ~2Z~ 0d0Z~ 0dei!(0)+i!00D T;0f^B()^A(0)^A(0)gE =X m;n;le( En)hnj^Bjmihmj^Ajlihlj^Ajni1 ~2Z~ 0d0ei(!0!)0e(EmEl)0=~Z~ 0de(i~!+EnEm)=~ =X m;n;le( En)hnj^Bjmihmj^Ajlihlj^Ajni i~!+EnEma(i!;i!0) (A27) Also for0> > 0, D T;0f^B()^A(0)^A(0)gE =X m;n;le( El)e(ElEm)0=~hmj^Bjnihnj^Ajlihlj^Ajmie(EmEn)=~; (A28) and then, we obtain 1 ~2Z~ 0d0Z0 0dei!(0)+i!00D T;0f^B()^A(0)^A(0)gE =X m;n;le( El)hmj^Bjnihnj^Ajlihlj^Ajmi1 ~2Z~ 0d0ei(!0!)0e(ElEm)0=~Z0 0de(i~!+EmEn)=~ =X m;n;le( En)hmj^Bjnihnj^Ajlihlj^Ajmi i~!Em+Ena(i!;i!0) (A29) 16Therefore, Eq. (A21) is rewritten as '(i!;i!0) =1 2X n;m;le( En) hnj^Bjmihmj^Ajlihlj^Ajni i~!+EnEma(i!;i!0) +hmj^Bjnihnj^Ajlihlj^Ajmi i~!+EnEma(i!;i!0)! ; (A30) and as compared with Eq. (A24), it is obvious that Eq. (A22) is held. 5. Third and higher order responses The third order response, k= 3 for Eq. (A3), reads 3B(t) =Z1 1dt1Z1 1dt2Z1 1dt3R (t;t1;t2;t3)F(t1)F(t2)F(t3); (A31) whereR (t;t1;t2;t3) is a symmetrized response coecient given as R (t;t1;t2;t3) =1 3! QR (t;t1;t2;t3) +QR (t;t1;t3;t2) +QR (t;t2;t1;t3) +QR (t;t2;t3;t1) +QR (t;t3;t1;t2) +QR (t;t3;t2;t1) ; (A32) QR (t;t1;t2;t3) =1 i~3 (tt1)(t1t2)(t2t3)Dh^B(t);^A(t1)];^A(t2) ;^A(t3)iE : (A33) One can see QR (t;t1;t2;t3) =QR (tt1;t1t2;t2t3) by using Eq. (A5), and the Fourier form is shown as QR (!;!0;!00) =1 i~3ZZZ1 0dtdt0dt00ei(!+i)t+i(!0+i0)t0+i(!00+i00)t00 Dh^B(t+t0+t00);^A(t0+t00)];^A(t00) ;^A(0)iE : (A34) Hence, the third order response in the Fourier space is given as 3B(!) =1 3!Z1 1d!0 2Z1 1d!00 2R (!;!0;!00)F(!!0)F(!0!00)F(!00) (A35) with R (!;!0;!00) =QR (!;!0;!00) +QR (!;!0;!0!00) +QR (!;!+!00!0;!00) +QR (!;!+!00!0;!!00) +QR (!;!!00;!0!00) +QR (!;!!0;!!00): (A36) Equation (A36) leads that the convergence factors need to have the relation >0>00. Hence, we put= 300,0= 200. 17There is the corresponding correlation function in the Matsubara formalism to R (!;!0;!00) given by '(i!;i!0;i!00) =1 3!~3ZZZ~ 0dd0d00ei!(0)+i!0(000)+i!0000D Tf^B()^A(0)^A(00)^A(0)gE : (A37) By taking the analytic continuations, i!!!+ 3i0,i!0!!0+ 2i0,i!00!!00+i0, we have R (!;!0;!00) ='(!+ 3i0;!0+ 2i0;!00+i0): (A38) From thek= 1;2;3-th order responses, it is expected that the k-th order response is evaluated as follows: the response of ^Bto the external forces is expressed as kB(t) =ZZ Z1 1dt1dt2dtkR 12k(tt1;t1t2;;tk1tk)F1(t1)F2(t2)Fk(tk); (A39) whereR 12k(tt1;t1t2;;tk1tk) is the response coecient already symmetrized, whose Fourier component R 12k(!;! 1;!2;;!k1) is evaluated from the corresponding correlation function in the Matsubara formalism '12k(i!;i!1;i!2;;i!k1) =1 k!~kZZ ZZ~ 0dd1dk2dk1ei!(1)+i!1(12)++i!k2(k2k1)+i!k1k1 D Tf^B()^A1(1)^A2(2)^Ak2(k2)^Ak1(k1)^Ak(0)gE (A40) by taking the analytic continuations, i!!!+ki0,i!1!!1+(k1)i0,,i!k2!!k2+2i0, i!k1!!k1+i0; R 12k(!;! 1;!2;;!k1) ='12k(!+ki0;!1+ (k1)i0;!2+ (k2)i0;;!k1+i0): (A41) Appendix B: Expressions of diagrams In this Appendix, we show the expressions of all the diagrams contributing the non-local emer- gent electric elds shown in Fig. 4 for the linear response and in Fig. 6 for the second order response. 18(a) kk+q k+qmloαq−q,lk+q+q 2 m σo(b) k−q k−qk σo mloαq−q,lk−q+q 2 m(c) kk+q σmilmαq−q,l (d) kk+qk+q k+q k(e) k−q k−qk k−qk (f) kk+qk+q k(g) k−qkk k+q k−q k−q (h) (i) (j) k kk+q k+qFIG. 4. The Feynman diagrams of (1) ij(i!) in the rst order of the Rashba spin-orbit interaction; (a)- (c) without the ladder type VCs and (d)-(j) with the VCs. The solid lines with arrows denote the Green functions without the Rashba interaction, given by Eq. (6), the lled circle represents the spin vertex, the unlled triangle describes the normal velocity vertex, the dashed wavy line indicates the anomalous velocity vertex, and the solid wavy line depicts the Rashba-interaction vertex without the spin component. niu u kkk+qk+q inin+iωλ kk+q FIG. 5. The diagrammatic description for the four-point vertex of the diusion ladder. The dotted lines denote the impurity potential, and the cross symbol represents the impurity concentration. The solid lines without arrows are for the external momentums. Equations (15a) and (15b) in the rst order of the Rashba interaction are given respectively by 19Fig. 4 (a)-(c), which reads (1;n) ij(q;q0;i!) =emljqq0;l1 X nX = j;zim; q;q0(i+ n;in) +j;?im; q;q0(i+ n;in) + (i+ n$in) ; (B1a) (1;a) ij(q;q0;i!) =eiljqq0;l1 X nX = j;z q0(i+ n;in) +j;? q0(i+ n;in) ; (B1b) wherei+ n=in+i!andj;?= (1j;z), and im;0 q;q0(im;in) =1 VX k~2 me k+q 2 i k+q+q0 2 mgk+q;(im)gk+q0;0(im)gk;(in);(B2) 0 q0(im;in) =1 VX kgk+q0;(im)gk;0(in) (B3) Here, we used gk;(in) =gk;(in) for calculating the diagram of Fig.4 (b), resulting in the term which is interchanged i+ nandinfor the rst two terms in Eq. (B1a). The Green function gk;(in) is here expressed depending spin =, but we will evaluate it as gk;(in) =gk(in) in the next section. The four-point vertex of the diusion ladder is given by Fig. 5, 0 q(i+ n;in) =niu2+(niu2)2 VX kgk+q;(i+ n)gk;0(in) +(niu2)3 V X kgk+q;(i+ n)gk;0(in)!2 + =niu2 1niu20 q(i+n;in): (B4) The diusion ladder VCs of (1;n) ijare given by Fig. 4 (d)-(i), which read (1;n)(df) ij (q;q0;i!) =(d)+(e) ij +(f)+(g) ij +(h)+(i) ij (B5) with (d)+(e) ij =emljqq0;l1 X nX =h j;zim; q;q0(i+ n;in) q0(i+ n;in) q0(i+ n;in) +j;?im; q;q0(i+ n;in) q0(i+ n;in) q0(i+ n;in) + (i+ n$in)i ; (B6a) (f)+(g) ij =emljqq0;l1 X nX =h j;zi; q(i+ n;in) q(i+ n;in)m; q;q0(i+ n;in) +j;?i; q(i+ n;in) q(i+ n;in)m; q;q0(i+ n;in) + (i+ n$in)i ; (B6b) (h)+(i) ij =emljqq0;l1 X nX =h j;zi; q(i+ n;in) q(i+ n;in)m; q;q0(i+ n;in) q0(i+ n;in) q0(i+ n;in) +j;?i; q(i+ n;in) q(i+ n;in)m; q;q0(i+ n;in) q0(i+ n;in) q0(i+ n;in) + (i+ n$in)i ; (B6c) 20where i;0 q;q0(im;in) =~ VX k k+q+q0 2 igk+q;(im)gk+q0;0(im)gk;(in); (B7) i; q(im;in) =1 VX k~ me k+q 2 igk+q;(im)gk;(in); (B8) and the diusion ladder VCs of (1;a) ijis given by Fig. 4 (j), which reads (1;a)(df) ij (q;q0;i!) =eiljqq0;l1 X nX =h j;z q0(i+ n;in) q0(i+ n;in) q0(i+ n;in) +j;? q0(i+ n;in) q0(i+ n;in) q0(i+ n;in)i :(B9) We calculate all the above quantities in Appendix C. i+ n,k+q in,ki+ n,k+q+q i+ n,k+qin,k+qin,k+q+q in,k+q i− n,ki+ n,k in,k−q−qin,k−qi+ n,k−qin,k i− n,k−qin,k−q i− n,k−q−q in,ki+ n,k+q+q i+ n,k+qin,k+q+q in,k+q i− n,ki+ n,k+q−q i+ n,k+qi+ n,k+q in,kin,k+q i− n,kin,k+qin,k+q−q(a) (b) (c) (d) (h) (g) (f) (e) FIG. 6. The Feynman diagrams of (2) ijk(i!;i!0) in the rst order of the Rashba interaction without the diusion ladder VCs. The lines and symbols are dened in the caption of Fig. 4. (a)-(f): The contributions of the normal velocity term (2;n) ijk(i!;i!0) and (g)-(h): that of the anomalous velocity term (2;a) ijk(i!;i!0). Note that (b), (d), (f) and (h) are same contributions as that which are obtained by replacing j$k, q0$q00,n! n,+ n!n, and0+ n!0 nin (a), (c), (e), and (g), respectively. For the second order response discussed in Sec. III B, expanding Eqs. (28) and (29) in the rst order of the Rashba interaction, we have the diagrams in Fig. 6. The diagrams shown in Fig. 6 (a)-(f) and (g)-(h) are obtained from Eqs. (28) and (29), respectively, which reads (2;n) ijk(i!;i!0) =ieolmmjkqq0q00;l1 X nX = h m;z io; q;q0;q00(i+ n;i0+ n;in) + io; q;q00;q0(in;i0+ n;i+ n)io; q;q0;q00(i+ n;i0+ n;in) +j;z io; q;q0;q00(i+ n;i0+ n;in) + io; q;q00;q0(in;i0+ n;i+ n)io; q;q0;q00(i+ n;i0+ n;in) 21(i) (j) (k) (n) (o) (p)(l) (m) in,k,σim,k+q,σim,k+q,σ in,k,σ(q) (r)FIG. 7. All the Feynman diagrams of (2) ijk(i!;i!0) in the rst order of the Rashba interaction; (i), (j), (k) and (n) include the diagrams shown in Fig. 6 (a)-(b), (c)-(d), (g)-(h) and (e)-(f), respectively. The three diagrams surrounded by a thick line include main contributions to the non-local emergent electric elds. In the diagrams (i)-(p), the momentums and Matsubara frequencies are not displayed for readability; they are same as in the diagrams of Fig. 6 for (i), (j), (k), and (n). The momentums and Matsubara frequencies in (l), (m), (o) and (p) are expected from (i), (j) and (n) by using Fig. 5. The lled triangle and double circle are the full vertexes of the normal velocity and the spin, respectively, given by (q) and (r). The momentums and Matsubara frequencies of the arrowed lines are shared with the corresponding diagrams in Fig. 6. +k;z io; q;q0;q00(i+ n;i0+ n;in) + io; q;q00;q0(in;i0+ n;i+ n)io; q;q0;q00(i+ n;i0+ n;in) + (j$k;q0$q00;n! n;+ n!n;0+ n!0 n)i ; (B10a) (2;a) ijk(i!;i!0) =ieilmmjkqq0q00;l1 X nX =h m;z q0;q00(i+ n;i0+ n;in) +j;z q0;q00(i+ n;i0+ n;in) +k;z q0;q00(i+ n;i0+ n;in) + (j$k;q0$q00;n! n;+ n!n;0+ n!0 n)i ; (B10b) wherei n=ini!,i0 n=ini!0, and ij;000 p;q;r(il;im;in) =1 VX k~2 me k+p 2 i k+p+q+r 2 j gk+p;(il)gk+q+r;0(il)gk+r;00(im)gk;(in); (B11a) ij;000 p;q;r(il;im;in) =1 VX k~2 me k+p 2 i k+pq+r 2 j gk+p;(il)gk+pq;0(im)gk+r;00(im)gk;(in); (B11b) 22000 q;r(il;im;in) =1 VX kgk+q+r;(il)gk+r;0(im)gk;00(in) (B11c) The normal velocity terms containing diusion ladder VCs are shown in Fig. 6 (i)-(j), (l)-(p) and given as (2;n)(df) ijk(i!;i!0) =(i)+(j) ijk+(l)+(m) ijk+(n)+(o)+(p) ijk; (B12) (2;a)(df) ijk(i!;i!0) =(k) ijk(B13) where (l)+(m) ijk=ieolmmjkqq0q00;l1 X nX = h m;z io; q;q0+q00(i+ n;in) q0+q00(i+ n;in) q0;q00(i+ n;i0+ n;in) + (i+ n$in) +j;z io; q;q0+q00(i+ n;in) q0+q00(i+ n;in) q0;q00(i+ n;i0+ n;in) + io; q;q0+q00(in;i+ n) q0+q00(i+ n;in) q0;q00(in;i0+ n;i+ n) +k;z io; q;q0+q00(i+ n;in) q0+q00(i+ n;in) q0;q00(i+ n;i0+ n;in) + io; q;q0+q00(in;i+ n) q0+q00(i+ n;in) q0;q00(in;i0+ n;i+ n) + (j$k;q0$q00;n! n;+ n!n;0+ n!0 n)i ; (B14) (k) ijk=ieilmmjkqq0q00;l1 X nX =h m;z q0+q00(i+ n;in) q0+q00(i+ n;in) q0;q00(i+ n;i0+ n;in) +j;z q0+q00(i+ n;in) q0+q00(i+ n;in) q0;q00(i+ n;i0+ n;in) +k;z q0+q00(i+ n;in) q0+q00(i+ n;in) q0;q00(i+ n;i0+ n;in) + (j$k;q0$q00;n! n;+ n!n;0+ n!0 n)i ; (B15) and(i)+(j) ijkand(n)+(o)+(p) ijkcontains dierent types of diusion from (k)+(l)+(m) ijk. Appendix C: Calculation details In this Appendix, we show the details of the calculations of the response coecients at absolute zero,T= 0. In the present perturbative approach, the free Green functions are spin unpolarized, which means gk;(in) is equivalent to gk(in) dened as in Eq. (7). First, we calculate the liner response coecient. Equations. (B1a) and (B1b) are reduced to the following simple forms, (1;n) ij(q;q0;i!) = 2emljqq0;l1 X n im q;q0(i+ n;in) + im q;q0(in;i+ n) ; (C1a) (1;a) ij(q;q0;i!) = 2eiljqq0;l1 X nq0(i+ n;in); (C1b) 23and their diusion VCs which mainly contribute to the non-local emergent electric elds are given as (1;n)(df) ij (q;q0;i!)'(d)+(e) ij = 2emljqq0;l1 X nh im q;q0(i+ n;in)q0(i+ n;in)q0(i+ n;in) + (i+ n$in)i ; (C1c) (1;a)(df) ij (q;q0;i!) = 2eiljqq0;l1 X nh q0(i+ n;in)q0(i+ n;in)q0(i+ n;in)i ; (C1d) where im q;q0(i+ n;in),q0(i+ n;in), andq0(i+ n;in) are given respectively by that dropped the spin dependences in Eqs. (B2), (B3), and (B4). Using the following relations q0(i+ n;in)q0(i+ n;in) =1 +1 1niu2q0(i+n;in); (C2) andq0(i+ n;in) =q0(in;i+ n) due togk(in) =gk(in), we nd (1) ij(q;q0;i!)'2emljqq0;l1 X nim q;q0(i+ n;in) + im q;q0(in;i+ n) +imq0(i+ n;in) 1niu2q0(i+n;in)(C3) We rewrite the Matsubara summation of into the contour integral, and then change the contour path as the two path of [ 1i0;+1i0] and [1i!i0;+1i!i0]. After taking the analytic continuation i!!!+i0, we obtain R;(1) ij(q;q0;!) = 2emljqq0;l (1) q;q0;im+i!'(1) q;q0;im+ ; (C4) where(1) q;q0;imis the zeroth order term of !. The!-linear term '(1) q;q0;imis obtained as '(1) q;q0;im=~ 22Re im q;q0(+i0;i0) +imq0(+i0;i0) 1niu2q0(+i0;i0); (C5) im q;q0(+i0;i0) =1 VX k~2 me k+q 2 i k+q+q0 2 mgR k+qgR k+q0gA k; (C6) q0(+i0;i0) =1 VX kgR k+q0gA k; (C7) wheregR k= (k++i~=2)1andgA k= (gR k). Here, we remained the terms which contains both the retarded and advanced Green functions in Eq. (C5). Expanding im q;q0(+i0;i0) and 24q0(+i0;i0) up toq2andq02and performing the k-summations, we obtain Re im q;q0(+i0;i0) 'im 1 2I011+~2(q2+q02) 3meRe I131+4 5I241 +4~2qq0 15meRe I241 +~2 3me(q+q0)i(q+q0)mRe I131+4 5I241 +4~2 15me(qiqm+q0 iq0 m)Re I241 +~2 4meqi(qm+q0 m)Re I021+4 3I131 (C8) ' ~ im 1 +D0qq0 D0(qiq0 i)q0 m ; (C9) q0(+i0;i0)'I011+~2q02 2me I021+4 3I131 '2 ~ 1D0q02 ; (C10) whereD0= 2F=3meis the diusion constant, and we used Eqs. (D8) and neglected the higher order contributions of ~=F(1). Hence, using Eq. (8), we nd '(1) q;q0;im= q02 im(qq0)q0(qiq0 i)q0 m ; (C11) which leads to Eq. (16). It should be noted that '(1) q;q0;ijmdoes not contain any terms proportional to 1=D0q02, which means that there is no contribution such as hj(r;t)i(1)/Z(r)_M(r0;t) jrr0jdr0: (C12) Next, we calculate the second order response coecient. The coecient is also simplied in the case of+=. Equations (B10a) and (B10b) are given as (2;n) ijk(i!;i!0) = 2ieolmmjkqq0q00;l1 X nh io q;q0;q00(i+ n;i0+ n;in) + io q;q00;q0(in;i0+ n;i+ n) io q;q00;q0(in;i0 n;i n)io q;q0;q00(i n;i0 n;in) + io q;q0;q00(i+ n;i0+ n;in)io q;q00;q0(in;i0 n;i n)i ; (C13) (2;a) ijk(i!;i!0) = 2ieilmmjkqq0q00;l1 X nh q0;q00(i+ n;i0+ n;in)q00;q0(in;i0 n;i n)i ;(C14) where io q;q0;q00(i+ n;i0+ n;in), io q;q0;q00(i+ n;i0+ n;in), and q0;q00(i+ n;i0+ n;in) are same respectively as that dropped the spin indexes in Eqs. (B11a), (B11b), and (B11c). However, Eqs. (C13) does not contribute the non-local emergent electric elds of our interest because it is canceled by the diusion ladder VCs shown in Fig. 6 (i), (j), and (n). We also nd that Eq. (C14) is canceled by the diusion ladder VCs of the spin vertex jork, which gives rise to 1 =D0q002or 1=D0q02and do not contribute the emergent electric eld we focus in this paper. 25The main contributions of the diusion VCs [Eqs. (B12) and (B13)] to the non-local emergent electric eld are given as (2;n)(df) ijk(i!;i!0) = 2ieolmmjkqq0q00;l1 X nh io q;q0+q00(i+ n;in)q0+q00(i+ n;in)q0;q00(i+ n;i0+ n;in) io q;q0+q00(in;i n)q0+q00(in;i n)q00;q0(in;i0 n;i n) + io q;q0+q00(in;i+ n)q0+q00(in;i+ n)q0;q00(in;i0+ n;i+ n) io q;q0+q00(i n;in)q0+q00(i n;in)q00;q0(i n;i0 n;in)i ; (C15) (2;a)(df) ijk(i!;i!0) = 2ieilmmjkqq0q00;l1 X nh q0+q00(i+ n;in)q0+q00(i+ n;in)q0;q00(i+ n;i0+ n;in) q0+q00(in;i n)q0+q00(in;i n)q00;q0(in;i0 n;i n)i ; (C16) where q(i+ n;in) and q(i+ n;in) are dened respectively by that dropped the spin indexes in Eqs. (B3) and (B4). For Eqs. (C15) and (C16), we perform the similar procedures as in the calculations of the linear response coecient; rewriting the Matsubara summation of into the contour integral, changing the integral path into the three paths [ 1iPii0;+1iPii0] with P1=!,P2=!0andP3= 0 for the terms which depend on the frequencies set ( i+ n;i0+ n;in), and withP1= 0,P2=!0andP3=!for the terms which depend on the frequencies set (in;i0 n;i n). Then, taking the analytic continuations as Eq. (26), we obtain R;(2) ijk(q;q0;q00;!;!0) = 2ieolmmjkqq0q00;l (2) q;q0;q00;io+i!#(2) q;q0;q00;io+i!0'(2) q;q0;q00;io+ ; (C17) where the rst term gives rise to the current depending on the magnetization Mj(t)Mk(t), not on its dynamics, and the second term leads to the current due to the total derivative of the magnetizationsd(Mj(t)Mk(t))=dt. The third term of Eq. (C17) is the component that we are focusing, which is obtained as '(2) q;q0;q00;io='(1) q;q0+q00;ioniu2 2q0;q00; (C18) q0;q00=X s;t=sq0;q00(it0;is0;it0) =2i VX kIm (gR kgA k)gR k+q0gA kq00 (C19) where'(1) q;q0+q00;iois given by Eq. (C5), we neglected the terms which contains only retarded/advanced Green functions because they are just higher order contributions with respect to ~=F, and we 26used q00;q0(i0;i0;i0) = q0;q00(i0;i0;i0). Here, expanding q0;q00with respect to q0andq00 up to the second order, we have q0;q00=2iIm 2I012+~2(3q02+ 4q0q00+ 3q002) 2meI013+4~2(2q02+ 3q0q00+ 2q002) 3meI114 '8i2 ~2 1D0(2q02+ 3q0q00+ 2q002) ; (C20) whereIlmnis given by Eq. (D1) and left in the leading order of F=~in the second equal by means of Eqs. (D8). Hence, using Eqs. (C9), (C10), and (8), we nally nd '(2) q;q0;q00;io=2i2 ~io(qq0q00)(q0+q00)(qiq0 iq00 i)(q0 o+q00 o) (q0+q00)2+O(q2;q02;q002);(C21) which leads to Eq. (32). Appendix D: Integrals In this Appendix, we show k-integralsIlmnthat we use in this paper; Ilmn=1 VX k~2k2 2mel gR km gA kn; (D1) where we set m+nl+ 3=2 to its convergence. We rewrite the summation over kinto the energy integral, Ilmn=Z1 Fd(+F)(+F)l (+i~=2)m(i~=2)n; (D2) where()/pis DOS. For evaluating Ilmnwith respect to the leading order of ~=F, it is valid to approximate +F'F, which means that DOS and the energy approximates to the values at the Fermi level, and to regard the lower limit of the integral as 1. However, we need to evaluate the higher order contributions precisely such as in Eq. (16). Hence, we calculate Ilmnwithout any approximations. Considering (x)/x1=2, the analyticity is as follows: Supposed that z=x+iyandw=pz= X+iY, and in the polar coordinate, z=rei(+2n)with< andn= 0;1;2;, w=e1 2logz=e1 2logr+i 2+ni=8 < :prei 2(n= 0); prei 2(n= 1);(D3) wheren= 0;1 means the n-th Riemann surface: [ ;) forn= 0, and [;3) forn= 1. For the Riemann surface of n= 0, X=prcos 2; Y =prsin 2; (D4) 27and, the condition < , usingx=rcos=r(2 cos2(=2)1),y=rsin= 2rsin(=2) cos(=2), cos 2=r 1 2 1 +x r ;sin 2= sign(y)r 1 2 1x r Similarly, for the Riemann surface of n= 1, cos 2=r 1 2 1 +x r ;sin 2=sign(y)r 1 2 1x r These are collectively expressed as w=8 < :px(c+(y=x) +isign(y)c(y=x)) (n= 0); px(c+(y=x) +isign(y)c(y=x)) (n= 1);(D5) where c() =p 1 +2 p 1 +21 2!1 2 (D6) From Eq. (D5), we can rewrite the path of the integral in Eq. (D2) as (see Fig. 8) 1 2Z1 Fd(+F)(+F)(+F)l (+i~=2)m(i~=2)n =1 4Z1+i0 F+i0+Z CR+ZFi0 1i0+Z C0 d(+F)(+F)l (+i~=2)m(i~=2)n =i 2X =1Res=i~=2(+F)(+F)l (+i~=2)m(i~=2)n ; (D7) where the path CRis given by =Rei, 02, (changing the Riemann surfaces at =), C0is given by =ei, 0 <2, andc=c(~=2F). Noted that the sign of the Residue at =i~=2is minus, because we gather it of the Riemann surface of n= 1. Here, we show the results of the integrals Ilmnwithm= 1. I011=2 ~c+; (D8a) I012=22 ~2 i 21 1i c++ 2ic 1i ; (D8b) I013=23 ~3 1 +1 2i 1i+1 42 (1i)2 c++ 2c 1i 11 2i 1i ; (D8c) I112=2F2 ~2 i3 2 c++i 2c ; (D8d) I113=2F3 ~3 1 +3i 23 42 1i c++ 2 1 +3 2i 1i c ; (D8e) 28FIG. 8. The path of the integral is described. The solid and dashed line of the path CRandC0denote the pathes on the Riemann surfaces n= 0 andn= 1, respectively. The contributions from C0andCRvanish in the limit of R!1 . I114=2F4 ~4 i+3 2+3 4i2 1i+1 43 (1i)2 c+ 2 i3 2 1i+1 2i2 (1i)2 c ; (D8f) I214=22 F4 ~4 i+5 2+9i2 45 43 1i c+ 2 i5 25i2 21 1i c ; (D8g) I215=22 F5 ~5 1 +5i 292 45 4i3 1i5 164 (1i)2 c++ 2 1 +5i 25 22 1i+5 8i3 (1i)2 c ; (D8h) where=~=2F(1). For small (>0), c+() = 1 + (5=8)2(13=128)4+; (D9a) c() ==2 + (3=16)3(17=256)5 (D9b) E-mail address: fujimoto.junji.s8@kyoto-u.ac.jp 1R. H. Silsbee, A. Janossy, and P. Monod, Phys. Rev. B 19, 4382 (1979). 2S. Mizukami, Y. Ando, and T. Miyazaki, J. Magn. Magn. Mater. 226-230 , 1640 (2001). 3Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002). 4M. V. Costache, M. Sladkov, S. M. Watts, C. H. van der Wal, and B. J. van Wees, Phys. Rev. Lett. 97, 216603 (2006). 5E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Lett. 88, 182509 (2006). 6V. Edelstein, Solid State Communications 73, 233 (1990). 297J. C. R. S anchez, L. Vila, G. Desfonds, S. Gambarelli, J. P. Attan e, J. M. De Teresa, C. Mag en, and A. Fert, Nat. Commun. 4, 2944 (2013). 8G. E. Volovik, J. Phys. C Solid State Phys. 20, L83 (3 1987). 9G. Tatara, Physica E: Low-dimensional Systems and Nanostructures 106, 208 (2019). 10K.-W. Kim, J.-H. Moon, K.-J. Lee, and H.-W. Lee, Phys. Rev. Lett. 108, 217202 (2012). 11A. Takeuchi and G. Tatara, J. Phys. Soc. Jpn. 81, 033705 (2 2012). 12N. Nakabayashi and G. Tatara, New J. Phys. 16, 015016 (2014). 13G. Tatara, N. Nakabayashi, and K.-J. Lee, Phys. Rev. B 87, 054403 (2013). 14A. Takeuchi, K. Hosono, and G. Tatara, Phys. Rev. B 81, 144405 (2010). 15K. Nakazawa and H. Kohno, J. Phys. Soc. Jpn. 83, 073707 (2014). 16G. Tatara, Phys. Rev. B 98, 174422 (2018). 17P. D. R. Kubo, P. D. M. Toda, and P. D. N. Hashitsume, in Statistical Physics II , Springer Series in Solid-State Sciences No. 31 (Springer Berlin Heidelberg, 1991) pp. 146{202. 18T. Jujo, J. Phys. Soc. Jpn. 75, 104709 (2006). 19H. Kohno and J. Shibata, J. Phys. Soc. Jpn. 76, 063710 (2007). 20A. M. Zagoskin, Quantum Theory of Many-Body Systems: Techniques and Applications (Springer- Verlag, 1998). 21G. Rickayzen, Green's Functions and Condensed Matter , reprint ed. (Dover Publications, New York, 2013). 22As we have introduced the convergence factor as in Eq. (A2), 1B(t) is also assumed to be expressed as1B(t) =etR 1B(!)ei!td!=2for the time-translational symmetry. 23Considering the convergence factors, we have assumed 2B(t) =e(+0)tR 2B(!)ei!tdt=2. 30
1111.5466v1.Rashba_spin_torque_in_an_ultrathin_ferromagnetic_metal_layer.pdf	arXiv:1111.5466v1 [cond-mat.mtrl-sci] 23 Nov 2011Rashba spin torque in an ultrathin ferromagnetic metal laye r Xuhui Wang∗and Aurelien Manchon† Physical Science & Engineering Division, KAUST, Thuwal 239 55-6900, Kingdom of Saudi Arabia (Dated: August 24, 2018) In a two-dimensional ferromagnetic metal layer lacking inv ersion symmetry, the itinerant electrons mediate the interaction between the Rashba spin-orbit inte raction and the ferromagnetic order parameter, leading to a Rashba spin torque exerted on the mag netization. Using Keldysh technique, in the presence of both magnetism and a spin-orbit coupling, we derive a spin diﬀusion equation that provides a coherent description to the diﬀusive spin dy namics. The characteristics of the spin torque and its implication on magnetization dynamics are di scussed in the limits of large and weak spin-orbit coupling. PACS numbers: 75.60.Jk,75.70.Tj,72.25.-b,72.10.-d I. INTRODUCTION By transferring angular momentum between the elec- tronic spin and the orbital, spin-orbit coupling ﬁlls the needforelectricalmanipulationofspindegreeoffreedom. Outstanding examples are the electrically generated bulk spin polarization1,2and the well-known spin Hall eﬀect (SHE)3–5in a two dimensional electron gas where the spin-orbit interaction, particularly of the Rashba-type,6 plays the leading role. Rashba spin-orbit interaction not only introduces an eﬀective ﬁeld perpendicular to the lin- ear momentum but also provides the backbone to the spin-relaxation through the so-called D’yakonov-Perel mechanism,7which is dominant in a two-dimensional system. Besides its prominent role in semiconductors, Rashba spin-orbit coupling is believed to exist at ferro- magnetic/heavy metal as well as ferromagnetic/metal- oxide interfaces, in which the inversion symmetry break- ing oﬀers a potential gradient empowering the spin-orbit coupling. Meanwhile, magnetism continuously stimulates the in- dustrial and academic appetite. In the pursuit of fast magnetization switching, Slonczewski-Berger spin trans- fer torque8employs a polarized spin current instead of a cumbersome magnetic ﬁeld. This celebrated scheme demands non-collinear magnetic textures in forms of, for example, spin valves or domain wall structures.9 In the presence of inversion symmetry breaking (such asasymmetricinterfaces),aferromagneticmetallayeras- sembles both magnetism and spin-orbit coupling, hence oﬀering an alternative switching mechanism:10,11Spin- orbit coupling transfers the orbital angular momentum carried by an electric current to the electronic spin, thus creating an eﬀective magnetic ﬁeld (Rashba ﬁeld). As long as the eﬀective ﬁeld is mis-aligned with the magne- tization direction, the so-called Rashba torque emerges, thus exciting the magnetization. Current-driven magnetization dynamics by spin-orbit torque has been demonstrated by several experiments on metal-oxide based systems.12–14In fact, the Rashba torque can be categorized into to a broader family of spin-orbit interaction induced torque that has been observed in diluted magnetic semiconductors.16–18Re-cently, Miron et al.,15has demonstrated the current- inducedmagnetizationswitchingusinga singleferromag- net in Pt/Co/AlO xtrilayers, which further consolidates the feasibility of the Rashba torque. The same type of spin-orbit coupling induced torque is predicted to im- prove current-driven domain wall motion,11,19which is supported by experimental observations.20At this stage, we are aware of an alternative explanation, as pointed out by Liu et al.,21in terms of the spin Hall eﬀect (SHE) occurring in the underlying heavy metal layer, such as Pt or Ta. The distinction between the spin Hall induced eﬀect and the Rashba one is discussed in the last section of this article. In searching for a general form of the Rashba torque in ferromagnetic metal layers,10we found an expression that consists of two components:22An in-plane torque (∝m×(ˆy×m)) and an out-of-plane one ( ∝ˆy×m), givenˆyisthein-planedirectiontransversetotheinjected current and mis the magnetization direction. Numerical solution on a two-dimensional nano-wire with one open transport direction has been carried out to appreciate the signiﬁcance of diﬀusive motion on the spin torque. We found that the in-plane component of the torque in- creases when narrowing the magnetic wire22. In the present article, we give a full theoretical deriva- tion of the coupled diﬀusive equation for spin dynam- ics in a ferromagnetic metal layer and describe the form of the Rashba torque in both weak and strong Rashba limits. In Sec. II, we combine the Keldysh formalism and the gradient expansion technique to derive a cou- pled diﬀusion equation for charge and non-equilibrium spin densities. To demonstrate that the diﬀusion equa- tion provides a coherent framework to describe the spin dynamics, we dedicate Sec. III to the spin diﬀusion in a ferromagnetic metal, which shows an excellent agree- ment to early result on the same system. In Sec.IV, we illustrate that the absence of magnetism (in our diﬀusion equation) describes the well-know phenomenon of elec- trically induced spin polarization. The cases of a weak and a strong spin-orbit coupling are discussed in Sec.V and Sec. VI, respectively, where we provide an analytical form of the Rashba torque in an inﬁnite medium. In Sec. VII, we discuss the implication of the Rashba torque on2 magnetization dynamics as well as its distinction from spin Hall eﬀect induced torque. II. DIFFUSION EQUATIONS The system of interest is deﬁned as a quasi-two- dimensional ferromagnetic metal layer rolled out in the xy-plane. Two asymmetric interfaces provide a conﬁne- ment in z-direction, along which the potential gradi- ent generates a Rashba spin-orbit coupling. Therefore a single-particle Hamiltonian for an electron of momen- tumˆkis (/planckover2pi1= 1 is assumed throughout) ˆH=ˆk2 2m+αˆσ·(ˆk×ˆz)+1 2∆xcˆσ·m+Hi(1) whereˆσis the Pauli matrix, mthe eﬀective mass, andmthe magnetization direction. The ferromag- netic exchange splitting is given by ∆ xcandαrepre- sents the Rashba constant (parameter). Hamiltonian ˆHi=/summationtextN j=1V(r−rj) sums the contribution of the non- magnetic impurity scattering potential V(r) localized at rj. To derive a diﬀusion equation for the non- equilibriumchargeandspindensities, weemployKeldysh formalism.23Using Dyson equation, in a 2 ×2 spin space, we obtain a kinetic equation that assembles the retarded (advanced) Green’s function ˆGR(ˆGA), the Keldysh com- ponent of the Green function ˆGK, and the self-energy ˆΣK, i.e., [ˆGR]−1ˆGK−ˆGK[ˆGA]−1=ˆΣKˆGA−ˆGRˆΣK,(2) where all Green’s functions are full functions with inter- actions taken care of by the self-energies ˆΣR,A,K. The retarded (advanced) Green’s function in momentum and energy space is ˆGR(A)(k,ǫ) =1 ǫ−ǫk−ˆσ·b(k)−ˆΣR(A)(k,ǫ),(3) whereǫk=k2/(2m) is the single-particle energy. We have introduced a k-dependent eﬀective ﬁeld b(k) = ∆xcm/2+α(k×z) of the magnitude bk=\|∆xcm/2+ α(k×z)\|and the direction ˆb=b(k)/bk. Neglecting localization eﬀect and electron-electron in- teractions, we assume a short-range δ-function type im- purity scattering potential. At a low concentration and a weak coupling to electrons, the second-order Born ap- proximation is justiﬁed,23i.e., the self-energy is24 ˆΣR,A,K(r,r′) =δ(r,r′) mτˆGR,A,K(r,r) (4) where the momentum relaxation time reads 1 τ≈2π/integraldisplayd2k′ (2π)2\|V(k−k′)\|2δ(ǫk−ǫk′),(5)whereV(k) is the Fourier transform of the scattering potential and the magnitude of momentum kandk′is evaluated at Fermi vector kF. The quasi-classical distribution function ˆ g≡ ˆgk,ǫ(T,R), deﬁned as the Wigner transform of the Keldysh function ˆGK(r,t;r′,t′), is obtained by inte- grating out the relative spatial-temporal coordinates while retaining the center-of-mass ones R= (r+r′)/2 andT= (t+t′)/2. As long as the spatial proﬁle of the quasi-classical distribution function is smooth at the scale of Fermi wave length, we may apply the gradient expansion technique on Eq.(2),25which gives us a transport equation associated with macroscopic quantities. The left-hand side of the kinetic equation in gradient expansion becomes [ˆGR]−1ˆGK−ˆGK[ˆGA]−1 ≈[ˆg,ˆσ·b(k)]+i τˆg+i∂ˆg ∂T +i 2/braceleftbiggk m+α(ˆz×ˆσ),∇Rˆg/bracerightbigg ,(6) where{·,·}denotes the anti-commutator. The relax- ation time approximation indulges the right-hand side of Eq.(2) as ˆΣKˆGA−ˆGRˆΣK ≈1 τ/bracketleftBig ˆρ(ǫ,T,R)ˆGA(k,ǫ)−ˆGR(k,ǫ)ˆρ(ǫ,T,R)/bracketrightBig (7) wherewe haveintroducedthe densitymatrix by integrat- ing out the momentum kin ˆg, i.e., ˆρ(E,T,R) =1 2πN0/integraldisplayd2k′ (2π)2ˆgk′,ǫ(T,R).(8) For the convenience of discussion, time variable is changed from Ttot. At this stage, we have a kinetic equation depending on ˆ ρas well as on ˆ g i[ˆσ·b(k),ˆg]+1 τˆg+∂ˆg ∂t+1 2/braceleftbiggk m+α(ˆz×ˆσ),∇Rˆg/bracerightbigg =i τ/bracketleftBig ˆGR(k,ǫ)ˆρ(ǫ)−ˆρ(ǫ)ˆGA(k,ǫ)/bracketrightBig . (9) A Fouriertransformationon temporal variableto the fre- quency domain ωleads to Ωˆg−bk[ˆUk,ˆg] =iˆK, (10) where Ω = ω+i/τand the operator ˆUk≡ˆσ·ˆbsatisﬁes ˆUkˆUk= 1. The right hand side of Eq.(10) is partitioned according to ˆK=−1 2/braceleftbiggk m+α(ˆz×ˆσ),∇Rˆg/bracerightbigg /bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright ˆK(1) +i τ/bracketleftBig ˆGR(k,ǫ)ˆρ(ǫ)−ˆρ(ǫ)ˆGA(k,ǫ)/bracketrightBig /bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright ˆK(0).(11)3 The equilibrium part is denoted by ˆK(0)while the gradi- ent term ˆK(1)is regarded as perturbation. Functions ˆ g and ˆρare both in frequency domain. We solve Eq. (10) formally to ﬁnd a solution to ˆ g ˆg=i(2b2 k−Ω2)ˆK+2b2 kˆUkˆKˆUk−Ωbk[ˆUk,ˆK] Ω(4b2 k−Ω2)≡L[ˆK]. (12) An iterationprocedureto solveEq.(12) hasbeenoutlined by Mishchenko et al.,in Ref.[24]. We follow this proce- dure here: According to the partition scheme on ˆK, we useˆK(0)to obtain the zero-th order approximation as ˆg(0)≡L[ˆK(0)(ˆρ)], which replaces ˆ ginˆK(1)to generate a correction due to the gradient term, i.e., ˆK(1)(ˆg(0)). We further insert ˆK(1)(ˆg(0)) back to Eq.(12) to obtain a correction given by L[ˆK(1)(ˆg(0))], then we obtain the ﬁrst order approximation to the quasi-classical distribu-tion function, ˆg(1)= ˆg(0)+L[ˆK(1)(ˆg(0))]. (13) The above procedure is repeated to any desired order, i.e., ˆg(n)= ˆg(n−1)+L[ˆK(1)(ˆg(n−1))].(14) In this paper, the second order approximation is suﬃ- cient. The full expression of the second orderapproxima- tion istedious thusnot included in the following. Adiﬀu- sion equation is derived by an angle averagingin momen- tum space, which allowsalltermsthat areoddorderin ki (i=x,y) to vanish while the combinations such as kikj contribute to the averaging by a factor k2 Fδij, givenkF the Fermi wave vector.25Further more, a Fourier trans- form from frequency domain back to the real time brings a diﬀusion type equation for the density matrix, ∂ ∂tˆρ(t) =D∇2ˆρ−1 τxcˆρ+1 2τxc(ˆz×ˆσ)·ˆρ(ˆz×ˆσ)+iC[ˆz×ˆσ,∇ˆρ]−B{ˆz×ˆσ,∇ˆρ} +Γ[(m×∇)zˆρ−ˆσ·m∇ˆρ·(ˆz×ˆσ)−(ˆz×ˆσ)·∇ˆρˆσ·m] +1 2Txc(ˆσ·mˆρˆσ·m−ˆρ)−i˜∆xc[ˆσ·m,ˆρ]−2R{ˆσ·m,(m×∇)zˆρ}, (15) where all quantities are evaluated at Fermi energy ǫF. In a two-dimensional system, the diﬀusion constant D= τv2 F/2 is given in terms of Fermi velocity vFand mo- mentum relaxation time τ. The renormalized exchange splitting reads ˜∆xc= (∆xc/2)/(4ξ2+ 1) where ξ2= (∆2 xc/4+α2k2 F)τ2. The other parameters are C=αkFvFτ (4ξ2+1)2,Γ =α∆xcvFkFτ2 2(4ξ2+1)2, R=α∆2 xcτ2 2(4ξ2+1), 1 τxc=2α2k2 Fτ 4ξ2+1, B=2α3k2 Fτ2 4ξ2+1,1 Txc=∆2 xcτ 4ξ2+1. τxcis the relaxation time due to the so-called D’yakonov- Perel mechanism.1Equation (15) is valid in the dirty limitξ≪1, whichenablestheapproximation1+4 ξ2≈1. Charge density nand the non-equilibrium spin density S are introduced by the vector decomposition on the den- sity matrix ˆ ρ=n/2 +S·ˆσ. In a real experimental setup,12,15,20spin transport in ferromagnetic layers suf- fersfromrandommagneticscatterers, forwhichweintro- duce an isotropic spin-ﬂip relaxation S/τsfphenomeno- logically. Eventually, we obtain a set of diﬀusion equations for the charge and spin densities, i.e., ∂n ∂t=D∇2n+B∇z·S +Γ∇z·mn+R∇z·m(S·m),(16)and ∂S ∂t=D∇2S−1 τ/bardblS/bardbl−1 τ⊥S⊥ −∆xcS×m−1 Txcm×(S×m) +B∇zn+2C∇z×S+2R(m·∇zn)m +Γ[m×(∇z×S)+∇z×(m×S)],(17) where∇z≡ˆz×∇. The spin density S/bardbl≡Sxˆx+Syˆyis relaxed at a rate 1 /τ/bardbl≡1/τxc+ 1/τsfwhileS⊥≡Szˆz has a rate 1 /τ⊥≡2/τxc+1/τsf. Forabroadrangeoftherelativestrengthbetweenspin- orbit coupling and the exchange splitting, i.e., αkF/∆xc, Eq.(16) and Eq.(17) describe the spin dynamics in a fer- romagnetic layer. When the magnetism vanishes (∆ xc= 0), theB-term provides a source that generates spin den- sity electrically.2,24On the other hand, when the spin- orbit coupling is absent ( α= 0), the ﬁrst two lines in Eq.(17) describe a diﬀusive motion of spin density in a ferromagnetic metal, which, to be shown in the next section, agrees excellently with early results in the cor- responding limit.26C-term describes the coherent pre- cession of the spin density around the eﬀective Rashba ﬁeld. The precession of the spin density (induced by the Rashba ﬁeld) around the exchange ﬁeld is described by the Γ-term, thus a higher order (compared to C) in the4 dirty limit for Γ = ∆ xcτC/2. TheR-term contributes to the magnetization renormalization. III. SPIN DIFFUSION IN A FERROMAGNET Spin diﬀusion in a ferromagnet has been discussed ac- tively in the ﬁeld of spintronics.26–29In this section we show explicitly that, by suppressing the spin-orbit cou- pling, Eq.(17) describes the spin diﬀusion equation in the corresponding limits. In the present model, vanishing Rashba spin-orbit cou- pling means α= 0, then Eq.(17) reduces to ∂ ∂tS=D∇2S+1 τ∆m×S −1 τsfS−1 Txcm×(S×m),(18) whereτ∆≡1/∆xcis the time scale of the coherent pre- cession of the spin density around the magnetization. This equation diﬀers from the result of Zhang et al.,27 only by a dephasing of the transverse component of the spin density that is set by the time scale Txc. In a fer- romagnetic metal, we may divide the spin density into alongitudinal component that follows the magnetization direction adiabatically, and a deviation that is perpendic- ularto the magnetization, i.e., S=s0m+δSwheres0 is the local equilibrium spin density. Such a partition, after restoring the electric ﬁeld by ∇→∇+eE∂ǫ, gives rise to ∂ ∂tδS+∂ ∂ts0m =s0D∇2m+D∇2δS+DePFNFE·∇m −δS τsf−s0m τsf−δS Txc+∆xcm×δS,(19) where the magnetic order parameteris allowedto be spa- tial dependent, i.e., m=m(r,t). The energy derivative is treated as ∂ǫS≈PFNFmgivenPFthe spin polariza- tion and NFthe density of state, both at Fermi energy. For a smooth magnetic texture in which the character- istic length scale of the magnetic proﬁle is much larger than the length scale for electron transport, we discard the contribution D∇2δS.26The diﬀusion of the equilib- rium spin density follows s0D∇2m≈s0m/τsf. In this paper, we retain only terms that are ﬁrst order in tem- poral derivative, which simpliﬁes Eq.(19) to −1 τ∆m×δS+/parenleftbigg1 τsf+1 Txc/parenrightbigg δS= −s0∂ ∂tm+DePFNFE·∇m.(20)The last equation can be solved exactly δS=τ∆ 1+ς2/bracketleftbiggPF em×(je·∇)m+ςPF e(je·∇)m −s0m×∂m ∂t−ςs0∂m ∂t/bracketrightbigg (21) whereς=τ∆(1/τsf+ 1/Txc) and the electric current je=e2nτE/mis given in terms of electron density n. Apart from the inclusion of the dephasing of trans- versecomponentasimplementedinparameter ς, thenon- equilibrium spin density Eq.(21) agrees excellently with Eq.(8) in Ref.[26]. Given the knowledge of the spin density, the spin torque, deﬁned as T=−1 τ∆m×δS+1 TxcδS, (22) is given by T=1 1+ς2/bracketleftbigg −ηs0∂m ∂t+βs0m×∂m ∂t +ηPF e(je·∇)m−βPF em×(je·∇)m/bracketrightbigg (23) whereη= 1 +ςτ∆/Txcandβ=τ∆/τsf. Assum- ing a long dephasing time of the transverse component (i.e.,Txc→ ∞), thenη≈1 and Eq. (23) reproduces the Eq.(9) in Ref.[26]. On the other hand, a short dephasing time (of the transverse component) enhances parameter ηtherefore increases the temporal spin torque (i.e., the ﬁrst term in Eq.(23)). IV. ELECTRICALLY GENERATED SPIN DENSITY The eﬀect of an electrically generated non-equilibrium spin density due to spin-orbit coupling2can be ex- tractedfrom Eq.(17) bysetting exchangeinteractionzero (i.e., ∆ xc= 0). Retaining D’yakonov-Perel as the only spin relaxation mechanism and letting τsf=∞, Eq.(17) ends up in D∇2S−1 τxc(S+Szˆz) +2C(ˆz×∇)×S+B(ˆz×∇)n= 0 (24) which reduces to the results in the well-known spin Hall eﬀect.24,30,31In the case of an inﬁnite medium along transport direction, i.e., ˆx-direction, Eq.(24) gives rise to a solution to the spin density S=τxcBeE1 ǫFnˆy=eEζ πvFˆy, where only the linear term in electric ﬁeld has been re- tained. On the right hand side, we have used the charge5 density in a 2D system n=k2 F/(2π) and introduced the parameter ζ=αkFτas used in Ref. [24]. In the following sections, we explore the spin torque in the presence of both exchange and Rashba ﬁeld in an in- ﬁnite medium. The primary focus is on two cases: Weak and a strong spin-orbit coupling, when comparing to the magnitude of exchange splitting. In general, Eq.(17) is applicable through a broad range of relative strength be- tween spin-orbit coupling and exchange splitting. A full scale numerical simulation on the diﬀusion equation is beyond the scope of this paper, we refer the readers to Ref.[22] for further interests. V. WEAK SPIN-ORBIT COUPLING A weak Rashba spin-orbit coupling implies a small D’yakonov-Perel relaxation rate 1 /τxc∝α2, such that τxc≫τsf,τ∆, which allows spin relaxation to be dom- inated by random magnetic impurities. In this regime, when comparing to the magnitudes of Cand Γ, the con- tribution from BandRare at a higher order in α, thus to be disregarded. We consider a stationary state where ∂S/∂t= 0. An electric ﬁeld applied along ˆx-direction, i.e.,E=Eˆx. In an inﬁnite medium,10all the spatial derivatives vanishes ( ∇→0) and the dynamic equation reads −1 τ∆m×S+1 Txcm×(S×m)+1 τsfS =2eECˆy×∂ǫS +eEΓ[ˆy×(m×∂ǫS)+m×(ˆy×∂ǫS)].(25) In addition to the spin density induced by exchange splitting, a weak spin-orbit interaction leads to a devi- ation in spin density that can be considered as a per- turbation. Therefore, we may well apply the partition S=S⊥+S/bardblmto separate the longitudinal and the transverse components. Eq.(25) is thus reduced to 1 τ∆m×S⊥−1 T⊥S⊥−1 τsfS/bardblm =−2eECP FNFˆy×m −eEΓPFNFm×(ˆy×m) (26) where 1/T⊥≡1/Txc+ 1/τsfand we have again em- ployedthe approximationon the energy derivative ∂ǫS≈ PFNFmand replaced the energy derivative of the charge density by the density of states at Fermi energy (i.e.,∂ǫn≈n/ǫF=NF). We solve Eq.(26) to obtain a solution to the non-equilibrium spin density S⊥=τ∆ 1+ς2eEPFNF[(2C+ςΓ)m×(ˆy×m) −(Γ−2ςC)(ˆy×m)]. (27) andS/bardbl= 0. In Eq.(27), the second component, oriented along the direction ˆy×m, is actually perpendicular totheplanespannedbythemagnetizationdirectionandthe eﬀective Rashba ﬁeld (along ˆy), which, as to be shown below, contributes to a Rashba torque that fulﬁls the symmetry described in a recent experiment.15The deﬁ- nition Eq.(22) leads to a general expression for the spin torque T=T⊥ˆy×m+T/bardblm×(ˆy×m),(28) which consists of an out-of-plane and anin-plane com- ponents with magnitudes determined by T⊥=eEPFNF 1+ς2(2ηC+βΓ), (29) T/bardbl=eEPFNF 1+ς2(ηΓ−2βC). (30) Theplaneis deﬁned by the magnetization direction m and the direction of the eﬀective Rashba ﬁeld that in the present setting is aligned along ˆy-direction. Note that the sign of the in-plane torque, Eq. (30), can change depending on the interplay between spin relaxation and precession. To compare directly with the results in Ref.[10], we al- low the spin relaxation time τsf→ ∞, therefore β≈0. We also consider the transverse dephasing time to be inﬁnite.26,27Under these assumptions, η≈1 andς≈0 and we have T⊥≈2eEPFNFCandT/bardbl≈eEPFNFΓ. In the dirty limit, Γ ≪Cdue to ∆ xcτ≪1, therefore mak- ing use of the relation for the polarization PF= ∆xc/ǫF and the Drude relation je=e2nτE/m, we obtain an out-of-plane torque T= 2αm∆xc eǫFjeˆy×m, (31) which agrees excellently with the spin torque in an in- ﬁnite system in the corresponding limit as derived in Ref.[10]. VI. STRONG SPIN-ORBIT COUPLING The opposite limit to Sec.V is a strong spin-orbit coupling. In this case, we consider the scenario that αkF≫∆xcand the D’yakonov-Perel relaxation mech- anism is dominating, i.e., 1 /τxc≫1/τsf, due to the fact 1/τxc∝α2. Therefore, it is not physical to sim- ply assume that the direction of spin density is domi- nantlyalignedalongthemagnetizationdirection, aswhat is treated in the case of a weak spin-orbit coupling. A self-consistent solution from Eq.(17) to the spin density is more justiﬁed. Again, as in Sec.V, we consider an inﬁnite system where an electric ﬁeld Eis applied at ˆx-direction. The magnetization direction is left arbitrary. We approxi- mate the energy derivative by ∂ǫ≈1/ǫF. The above6 assumptions simplify Eq.(17) to 1 τ∆S×m+1 Txcm×(S×m)−2eEC ǫFˆy×S +1 τxc(S+Szˆz) =eE ǫFnBˆy,(32) wherea strongspin-orbitcouplingrendersΓ and Rterms negligible. By considering Txc≫τ∆,τxc, Eq. (32) re- duces to 1 τ∆S×ˆm+1 τxc(S+Szˆz) −2eEC ǫFˆy×S=eE ǫFnBˆy,(33) which is a set of linear equations for the non-equilibrium spin density. We are interested in the linear response regime, which implies that at the distance as deﬁned by the Fermi wave length 1 /kF, we have eE/kF≪αkF. Therefore up to the ﬁrst order in exchange splitting, we extract the spin density from the above equation to be S=eE ǫFnτxcB/parenleftBig ˆy−χˆy×m−χ 2mxˆz/parenrightBig (34) whereχ≡τxc/τ∆we have used the identity ˆy×m= mzˆx−mxˆz. This yields a spin torque T=αm∆xc eǫFje(ˆy×m +χm×(ˆy×m)−χ 2mxˆz×m/parenrightBig .(35) This torque is slightly diﬀerent from the weak Rashba limit and has a strong implication in terms of magneti- zation dynamics. The torque is dominated by a ﬁeld-like torque along ˆy, similarly to the weak Rashba case. First, in contrast to the weak Rashba case [see Eq. (30)], the sign of the in-plane torque remains positive. Secondly, the anisotropic spin relaxation coming from D’yakonov- Perel mechanism yields an additional component of spin accumulation that is oriented along ˆz. The implication of this torque on the current-driven magnetization dy- namics is discussed in the next section. VII. DISCUSSION Current-induced magnetization dynamics in a sin- gle ferromagnetic layer has been observed in vari- ous structures that involve interfaces between transi- tion metal ferromagnets, heavy metals and/or metal- oxide insulators. Existing experimental systems are Pt/Co/AlO x,12,13,15,20Ta/CoFeB/MgO,14Pt/NiFe and Pt/Co bilayers.21Besides the structural complexity in such systems, an unclear form of spin-orbit coupling in the bulk and interfaces places a challenge to understand the nature of the torque.A. Validity of Rashba model The celebrated Rashba-type eﬀective interfacial spin- orbit Hamiltonian was pioneered by E. I. Rashba to model the inﬂuence of asymmetric interfaces in semicon- ducting 2DEG:6A sharp potential drop, emerging at the interface (say, in the xy-plane) between two materials, givesrisetoapotentialgradient ∇Vthatisnormaltothe interface, i.e., ∇V≈ξ(r)ˆz. In case a rotational symme- try exists in the interface plane, a spherical Fermi surface assumptionallowsthe spin-orbitinteractionHamiltonian tohavetheform ˆHR=αˆσ·(p×ˆz), whereα≈ ∝an}b∇acketle{tξ∝an}b∇acket∇i}ht/4m2c2. As a matter of fact, in semiconducting interfaces where the transport is described by a limited number of bands around a high symmetry point, the Rashba form can be recovered through k·ptheory.32 As far as metallic interfaces areconcerned, a spin-orbit splitting of the Rashba-type in the conduction band has been observedat Au surfaces,33Gd/GdO interfaces,34Bi surfacesandcompounds,35andmetallicquantumwells.36 The presence of a Rashba interaction in graphene37and at oxide hetero-interfaces38has also been reported re- cently. It is quite interesting to notice that the sym- metry breaking-induced spin splitting of the conduction band seems rather general and might not be restricted to heavy metal interfaces36. In the case of transition metals, however, the free elec- tron approximation fails to characterize the band struc- ture accuratelydue to both alargenumberofband cross- ingattheFermienergyandastronghybridizationamong s,panddorbitals. Density functional theory (DFT) is a successful tool to investigate the nature of spin-orbit in- teraction at metallic surfaces. For example, in Refs.[39], the authors observe a band splitting that possesses simi- larpropertiesasRashbaspin-orbitinteractionanddecays exponentially away from the surface.39Alternatively, the spin-orbit interaction at metallic surfaces has been ad- dressed using tight-binding models for the porbitals.40,41 At such sharp interfaces, the magnitude of the orbital angular momentum (OAM) is considered to play a dom- inant role at the onset of a Rashba-type spin splitting. This ﬁnding is consistent with the long stand- ing work on interfacial magnetic anisotropy at a ferromagnet/heavy metal,42and more recently, ferromagnetic/metal-oxide interface.43In such systems, a perpendicular magnetic anisotropy arises from the orbital overlap between the 3 dstates of the ferromagnets and the spin-orbit coupled states of the normal metal. The observation of perpendicular magnetic anisotropy at Co/metal-oxide interfaces tends to support the major role of large interfacial OAM in the onset of interfacial spin-orbit eﬀects.41,43The presence of interfacial Rashba spin-orbit coupling has also been shown to produce interfacial perpendicular magnetic anisotropy.44 All these previoustheoreticaland experimental studies strongly suggest that the interfacial spin splitting exists in the presence of a large OAM and potential gradient. However, a microscopic description of realistic interfaces7 is still missing. Although the Rashba spin-orbit interac- tion is a convenient Hamiltonian to extract qualitative behaviors, its applicability to realistic metallic interfaces with complex band structures remains to be tested. B. Spin Hall eﬀect versus Rashba torque Recently, Liu et al.,21proposedtomanipulatethemag- netization of a Pt/Co or Pt/NiFe bilayer using the spin current generatedby spin Hall eﬀect in the underlying Pt layer. When injecting a charge current jeinto a normal metal accommodating a strong spin-orbit coupling, the asymmetricspinscatteringinducesatransversepurespin current that has the form J= (αH/e)je×ˆµ⊗ˆµ, where αHis the spin Hall angle and ˆµis the spin direction.45 When impinging on the ferromagnetic layer deposited on top of the Pt layer, the spin current transverse to the local magnetization is absorbed and generates a torque TSHE= (bH/e)(1−βm×)m×(ˆµ×m) (to be called SHE torque thereafter). Here, bH=αHjeµB/eis the spintorqueamplitudewheretheregularspinpolarization Pis replaced by the spin Hall angle αH.βis the non- adiabaticity parameter proposed by Zhang and Li26and it stems from the presence of spin-ﬂip scattering in the system. In the conﬁguration adopted by Liu et al.,the charge current is injected along ˆxand the torque is given by TSHE=αHµBje e(m×(ˆy×m)+βˆy×m).(36) Note that a more realistic model should account for spin diﬀusion in Co and Pt, as discussed in Ref. [46]. An important conclusion is that, besides the correction in the case of a strong Rashba coupling, both Rashba and SHE produce the same type of torque, see Eq.(28) and Eq.(35) in this article. Nevertheless, distinctions can be made. First, in the absence of the corrections due to spin-ﬂip and spin pre- cession, the Rashba torque reduces to the ﬁeld-like term, ˆy×m, whereas the SHE torque reduces to the (anti- )dampingterm m×(ˆy×m). Thisassertionmustbescru- tinizede carefully since the actual relative magnitude be- tween the ﬁeld-like and the damping torques depends on the width of the magnetic wire as well as on the detailed spin dynamics in presence of spin-ﬂip and precession.22 Furthermore, for such an ultra-small system the spin-ﬂip scattering giving rise to the non-adiabaticity parameter (β) might be signiﬁcantly diﬀerent from the one mea- sured in a more conventional thin ﬁlm. A second important diﬀerence arises from the fact that the Rashba torque arises from spin-orbit ﬁelds generated byinterfacial currents, whereas the SHE torque is due to the current ﬂowing in the bulkof the Pt layer. Therefore, foraconstantexternalelectricﬁeld, varyingthethickness of Pt layer shall enhance the SHE torque, while keeping the Rashba torque unchanged.The torques as a function of the Co layer thickness is more diﬃcult to foresee. Although one could claim that Rashba spin-orbit interaction is expected to be localized at the interface, where the potential gradient is large, numerical simulations show that the Rashba-type inter- actionsurvivesafew monolayers39(which istypically the thickness of the Co layer under consideration). In addi- tion, the presenceofquantumwellstatesmightalsomod- ifythenatureofthespin-orbitinteractionintheultrathin magnetic layer in a system such as Pt/Co/AlO x.36 The same is true for the SHE torque. The injection of spin current into a Co layer is accompanied by spin precession that takes place over a very short decoherence length. This decoherence length has been studied experi- mentally and theoretically in spin valves and found to be of the order of a few monolayers.47In the typical case of 3 or 4 monolayer-thickferromagnets, the SHE torque can not be considered as a purely interfacial phenomenon. C. Magnetization Dynamics In Pt/Co/AlO xtrilayers, Miron et alhave observed a current-driven domain wall nucleation,12an enhanced current-driven domain wall velocity20and a current- driven magnetization switching.15The symmetry of the spin torque required to explain the experimental ﬁndings agree well with Rashba torque proposed in Ref. 10. On a similar structure, Pi et al13and Suzuki et al14also ob- served an eﬀective ﬁeld torque that could be interpreted intermsoftheRashbatorque. Recently,Liu et al21inter- preted their experiments on Pt/NiFe and Pt/Co bilayers using SHE in the underlying Pt layer. 1. Magnetization switching According to our previous discussions, both Rashba torque and SHE torque have a general form T=T⊥ˆy× m+T/bardblm×(ˆy×m). The ﬁrst term acts like a ﬁeld oriented along the direction transverse current direction whereas the second term acts like an (anti-)damping term, mimicking a conventional spin transfer torque that would arise from a polarizer pointing to ˆy. Asaconsequence,bothRashbatorqueandSHEtorque possess the appropriate symmetry to excite the magne- tization of a single ferromagnet and induce switching, as observedby Miron et al15and Liuet al.21In the case of a large Rashba spin-orbit coupling, the torque acquires an additional component that acts like an eﬀective magnetic ﬁeld along ˆz, vanishing as the magnetization component mxis zero (see Section VI), which provides an additional torque that helps destabilize the magnetization.8 2. Current-driven domain wall motion The inﬂuence of Rashba/SHE torque on a domain wall can be illustrated within the rigid Bloch wall approxima- tion. The perpendicularly magnetized Bloch wall is pa- rameterized by m= (cosφsinθ,sinφsinθ,cosθ) where φ=φ(t) andθ(x,t) = 2tan−1[e(x−xc(t))/∆], where xc refers to the center of the domain wall and ∆ is deﬁned as the domain wall width. To describe the dynamics of a Bloch wall, Landau-Lifshitz-Gilbert (LLG) equation ∂tm=−γm×Heff+αG∂tm×m+τ(37) has to be augmented by the current induced torque τ τ=bJ∇m−βbJm×∇m +bJ(τ⊥ˆy×m+τ/bardblm×(ˆy×m) +τzmxˆz×m).(38) The torque τis written in the most general form, where the ﬁrst two terms are the regular adiabatic and the so- called non-adiabatic torques; the next two terms ( τ/bardbland τ⊥) emerge from the presenceof Rashbaand/orspin Hall eﬀect and the last term τzappears only in large Rashba limit (see Sec. VI). The magnitude of the adiabatic torque is bJ=µBPje/e. The eﬀective ﬁeld is given by Heff=2A Ms∇2m+HKmxˆx+H⊥mzˆz.(39) Parameter γin LLG is the gyromagnetic ratio, αGis the Gilbert damping, Ais the exchange constant, Msis the saturation magnetization, HKis the in-plane magnetic anisotropy and H⊥is the combination of an out-of-plane anisotropyand ademagnetizingﬁeld. Themagnetization dynamics can be obtained readily from Eqs. (37)-(39) by integrating over the magnetic volume ∂tφ+αG∂txc ∆=/bracketleftbigg∆π 2(τ/bardbl−τz 2)cosφ−β/bracketrightbiggbJ ∆(40) αG∂tφ−∂txc ∆=−γHK 2sin2φ+/parenleftbigg 1+∆π 2τ⊥cosφ/parenrightbiggbJ ∆. (41) We observe that the in-plane torque τ/bardbldistorts the do- main wall texture, while the perpendicular torque τ⊥ drives the domain wall motion. The additional torque τz, arising in the large Rashba limit, only contributes to the in-plane torque. Therefore, in the following, we will refer to the in-plane torque as τ∗ /bardbl=τ/bardbl−τz/2. Below the Walker breakdown ( ∂tφ= 0), the velocity is given by ∂txc=−/parenleftbigg β−∆π 2τ∗ /bardblcosφ/parenrightbiggbJ αG(42) γHK 2sin2φ=/bracketleftbigg αG−β+∆π 2(αGτ⊥+τ∗ /bardbl)cosφ/bracketrightbiggbJ αG∆, (43)where the tilting angle φis given by the competition be- tween the magnetic anisotropy, the non-adiabatictorque, and the Rashba/SHEtorque. In the case ofweak Rashba (τz= 0), assuming τ/bardbl=βτ⊥and omitting the correction to the spin precession, we recover the results of Ref. [48]. When neglecting the in-plane torque and accounting for the perpendicular Rashba torque ( τ∗ /bardbl= 0), the Rashba torque only acts like an eﬀective transverse ﬁeld and en- hances the Walker breakdown limit20[see Eq. (43)]. Accounting for the in-plane component τ/bardblarising ei- ther from corrections to Rashba torque or from the SHE, this torque appears to modify the domain wall veloc- ity. Therefore, depending on the strength and the sign of Rashba/SHE torque as well as on the resulting tilt- ing angle φ, it is possible to obtain a vanishing or even a reversed domain wall velocity, as has been shown nu- merically in Ref. [48] and illustrated in Eq. (42). A full scale numerical investigation is beyond the scope of this article, but it will help understand the profound eﬀect of Rashba and SHE torque on the domain wall structures. VIII. CONCLUSION Using Keldysh technique, in the presence of both mag- netism and a Rashba spin-orbit coupling, we derive a spin diﬀusion equation that provides a coherent descrip- tion to the diﬀusive spin dynamics. In particular, we have derived a general expression for the Rashba torque in the bulk of a ferromagnetic metal layer, at both weak and strong Rashba limits. We ﬁnd that the torque is in general composed of two components, a ﬁeld-like torque and the other (anti-)damping one. Being aware of the recent alternative interpretation on the current-induced magnetization switching in a single ferromagnet, we have discussedthe diﬀerencebetweentheRashbaandtheSHE torques. While exploring the common features, we found that the magnetization dynamics driven by the Rashba torque presents several interesting similarities to that in- duced by SHE torque. Nevertheless, further investiga- tion involving structural modiﬁcation of the system is expected to provide a deeper knowledge on the nature of the interfacial spin-orbit interaction as well as the current-induced magnetization switching in a single fer- romagnet. Acknowledgments We thank G. E. W. Bauer, J. Sinova, M. D. Stiles, X. Waintal, and S. Zhang for stimulating discussions. We are specially grateful to K. -J. Lee and H. -W. Lee for inspiring discussion about the magnetization dynamics.9 ∗Electronic address: xuhui.wang@kaust.edu.sa †Electronic address: aurelien.manchon@kaust.edu.sa 1M. I. D’yakonov and V. I. Perel, Sov. Phys. JETP Lett. 13, 467 (1971); Phys. Lett. A 35, 459 (1971). 2V. M. Edelstein, Solid. Stat. Comm. 73, 233 (1989). 3S. Murakami et al.,Science301, 1348 (2004). 4J. Sinova et al., Phys. Rev. Lett. 92, 126603 (2004). 5Y. K. Kato et al.,Science306, 1910 (2004). 6Yu. A. Bychkov and E. I. Rashba, J. Phys. C: Solid. Stat. Phys.17, 6039 (1984). 7M. I. D’yakonov and V. I. Perel, Sov. Phys. Solid State 13, 3023 (1971). 8J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996); L. Berger, Phys. Rev. B. 54, 9353 (1996). 9M. D. Stiles and J. Miltat, Top. Appl. Phys. 101, 225 (2006); D. C. Ralph and M. D. Stiles, J. Magn. Magn. Mater.320, 1190 (2008); J. Z. Sun and D. C. Ralph, ibid. 320, 1227 (2008). 10A. Manchon and S. Zhang, Phys. Rev. B. 78, 212405 (2008);ibid.79, 094422 (2009). 11A. Matos-Abiague and R. L. Rodriguez-Suarez, Phys. Rev. B80, 094424 (2009); I. Garate and A. H. MacDonald, ibid. 80, 134403 (2009); P. M. Haney, and M. D. Stiles, Phys. Rev. Lett. 105, 126602 (2010). 12I. M. Miron et al.,Nature Materials 9, 230 (2010). 13U. H. Pi et al.,Appl. Phys. Lett. 97, 162507 (2010). 14T. Suzuki et al.,Appl. Phys. Lett. 98, 142505 (2011). 15I. M. Miron et al.,Nature (London) 476, 189 (2011). 16A. Chernyshov et al.,Nature Physics 5, 656 (2010). 17D. Fang et al.,Nature Nanotechnology 6, 413 (2011). 18M. Endo et al.,Appl. Phys. Lett. 97, 222501 (2010). 19K. Obata, and G. Tatara, Phys Rev. B 77, 214429 (2008). 20I. M. Miron et al.,Nature Materials 10, 419 (2011). 21L. Liu et al.,Phys. Rev. Lett. 106, 036601 (2011); arXiv:1110.6846 (2011). 22X. Wang and A. Manchon, arXiv:1111.1216 (2011). 23J. Rammer and H. Smith, Rev. Mod. Phys. 58, 323 (1986). 24E. G. Mishchenko et al.,Phys. Rev. Lett. 93, 226602 (2004). 25J. Rammer, Quantum Field Theory of Non-equilibrium States(Cambridge University Press, Cambridge, 2007). 26S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004). 27S. Zhang et al.,Phys. Rev. Lett. 88, 236601 (2002).28Y. Tserkovnyak et al.,Rev. Mod. Phys. 77, 1375 (2005). 29Y. Tserkovnyak et al.,Phys. Rev. B 79, 094415 (2009). 30A. A. Burkov et al.,Phys. Rev. B 70, 155308 (2004). 31˙I. Adagideli and G. E. W. Bauer, Phys. Rev. Lett. 95, 256602 (2005). 32R. Winkler, Spin-Orbit Coupling Eﬀects in Two- Dimensional Electron and Hole Systems , Springer, Berlin, (2003). 33S. LaShell et al.,Phys. Rev. Lett. 77, 3419 (1996); F. Reinertet al.,Phys. Rev. B 63, 115415 (2001); G. Nicolay et al., ibid .65, 033407 (2001); M. Hoesch et al., ibid .69, 241401(R) (2004). 34O. Krupin et al.,Phys. Rev. B 71, 201403(R) (2005); O Krupinet al.,New. J. Phys. 11, 013035 (2009). 35Ph. Hofmann, Prog. Surf. Sci. 81, 191 (2006); T. Hirahara et al.,Phys. Rev. B 76, 153305 (2007); H. Mirhosseini et al., ibid .79, 245428 (2009); A. Takayama et al.,Phys. Rev. Lett. 106, 166401 (2011); K. Ishizaka et al.,Nature Materials 10, 521 (2011). 36A. Varykhalov et al.,Phys. Rev. Lett. 101, 256601 (2008); J. Hugo Dil et al., ibid .101, 266802 (2008); A. G. Rybkin et al.,Phys. Rev. B 82, 233403 (2010). 37Yu. S. Dedkov et al.,Phys. Rev. Lett. 100, 107602 (2008); O. Rader et al., ibid .102, 057602 (2009). 38A. D. Caviglia et al.,Phys. Rev. Lett. 104, 126803 (2010). 39G. Bihlmayer et al.,Phys. Rev. B 75, 195414 (2007); G. Bihlmayer et al.,Surf. Sci. 600, 3888 (2006); M. Nagano et al.,J. Phys.: Cond. Mater. 21, 064239 (2009). 40L. Petersen and P. Hedeg˚ ard, Surf. Sci. 459, 49 (2000). 41S. R. Park et al.,Phys. Rev. Lett. 107, 156803 (2011). 42K. Kyuno et al.,J. Phys. Soc. Jpn. 61, 2099 (1992); D. Welleret al.,Phys. Rev. B 49, 12888 (1994); G. H. O. Daalderop et al., ibid .50, 9989 (1994). 43H. X. Yang et al.,Phys. Rev. B 84, 054401 (2011); C. Nistoret al., ibid .84, 054464 (2011). 44A. Manchon, Phys. Rev. B 83, 172403 (2011). 45J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999); S. Zhang, ibid.85, 393 (2000). 46A. Vedyayev et al.,arXiv:1108.2589 (2011). 47S. Wang et al.,Phys. Rev. B 77, 184430 (2008). 48K.-W. Kim et al.,arXiv:1111.3422 (2011).
1908.01236v2.Experimental_evidence_for_Zeeman_spin_orbit_coupling_in_layered_antiferromagnetic_conductors.pdf	Experimental evidence for Zeeman spin-orbit coupling in layered antiferromagnetic conductors R. Ramazashvili,1,∗P. D. Grigoriev,2, 3, 4,†T. Helm,5, 6, 7,‡F. Kollmannsberger,5, 6M. Kunz,5, 6,§W. Biberacher,5E. Kampert,7H. Fujiwara,8A. Erb,5, 6J. Wosnitza,7, 9R. Gross,5, 6, 10and M. V. Kartsovnik5,¶ 1Laboratoire de Physique Th´ eorique, Universit´ e de Toulouse, CNRS, UPS, France 2L. D. Landau Institute for Theoretical Physics, 142432 Chernogolovka, Russia 3National University of Science and Technology MISiS, 119049 Moscow, Russia 4P. N. Lebedev Physical Institute, 119991 Moscow, Russia 5Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, Walther-Meißner-Strasse 8, D-85748 Garching, Germany 6Physik-Department, Technische Universit¨ at M¨ unchen, D- 85748 Garching, Germany 7Hochfeld-Magnetlabor Dresden (HLD-EMFL) and W¨ urzburg-Dresden Cluster of Excellence ct.qmat, Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany 8Department of Chemistry, Graduate School of Science, Osaka Prefecture University, Osaka 599-8531, Japan 9Institut f¨ ur Festk¨ orper- und Materialphysik, TU Dresden, 01062 Dresden, Germany 10Munich Center for Quantum Science and Technology (MCQST), D-80799 Munich, Germany (Dated: December 23, 2020) Most of solid-state spin physics arising from spin-orbit coupling, from fundamental phenomena to industrial applications, relies on symmetry-protected degeneracies. So does the Zeeman spin- orbit coupling, expected to manifest itself in a wide range of antiferromagnetic conductors. Yet, experimental proof of this phenomenon has been lacking. Here, we demonstrate that the N´ eel state of the layered organic superconductor κ-(BETS) 2FeBr 4shows no spin modulation of the Shubnikov- de Haas oscillations, contrary to its paramagnetic state. This is unambiguous evidence for the spin degeneracy of Landau levels, a direct manifestation of the Zeeman spin-orbit coupling. Likewise, we show that spin modulation is absent in electron-doped Nd 1.85Ce0.15CuO 4, which evidences the presence of N´ eel order in this cuprate superconductor even at optimal doping. Obtained on two very diﬀerent materials, our results demonstrate the generic character of the Zeeman spin-orbit coupling. INTRODUCTION Spin-orbit coupling (SOC) in solids intertwines elec- tron orbital motion with its spin, generating a variety of fundamental eﬀects1,2. Commonly, SOC originates from the Pauli termHP=¯h 4m2 eσ·p×∇V(r) in the electron Hamiltonian3,4, where ¯his the Planck constant, methe free electron mass, pthe electron momentum, σits spin andV(r) its potential energy depending on the position. Remarkably, N´ eel order may give rise to SOC of an en- tirely diﬀerent nature, via the Zeeman eﬀect5,6: Hso Z=−µB 2/bracketleftbig g/bardbl(B/bardbl·σ) +g⊥(k)(B⊥·σ)/bracketrightbig ,(1) whereµBis the Bohr magneton, Bthe magnetic ﬁeld, whileg/bardblandg⊥deﬁne the g-tensor components with respect to the N´ eel axis. In a purely transverse ﬁeld B⊥, a hidden symmetry of a N´ eel antiferromagnet pro- tects double degeneracy of Bloch eigenstates at a spe- cial set of momenta in the Brillouin zone(BZ)5,6: at such momenta,g⊥must vanish. The scale of g⊥is set by g/bardbl, which renders g⊥(k) substantially momentum depen- dent, and turnsHso Zinto a veritable SOC5–8. This cou- pling was predicted to produce unusual eﬀects, such as spin degeneracy of Landau levels in a purely transverse ﬁeldB⊥9,10and spin-ﬂip transitions, induced by an AC electric rather than magnetic ﬁeld10. Contrary to the textbook Pauli spin-orbit coupling, this mechanism doesnot require heavy elements. Being proportional to the applied magnetic ﬁeld (and thus tunable!), it is bound only by the N´ eel temperature of the given material. In addition to its fundamental importance as a novel spin- orbit coupling mechanism, this phenomenon opens new possibilities for spin manipulation, much sought after in the current eﬀort11–13to harness electron spin for future spintronic applications. While the novel SOC mechanism may be relevant to a vast variety of antiferromagnetic (AF) conductors such as chromium, cuprates, iron pnic- tides, hexaborides, borocarbides, as well as organic and heavy-fermion compounds6, it has not received an exper- imental conﬁrmation yet. Here, we present experimental evidence for the spin degeneracy of Landau levels in two very diﬀerent lay- ered conductors, using Shubnikov–de Haas (SdH) os- cillations as a sensitive tool for quantifying the Zee- man eﬀect14. First, the organic superconductor κ- (BETS) 2FeBr 4(hereafterκ-BETS)15is employed for testing the theoretical predictions. The key features mak- ing this material a perfect model system for our purposes are (i) a simple quasi-two-dimensional (quasi-2D) Fermi surface and (ii) the possibility of tuning between the AF and paramagnetic (PM) metallic states, both showing SdH oscillations, by a moderate magnetic ﬁeld15,16. We ﬁnd that, contrary to what happens in the PM state, the angular dependence of the SdH oscillations in the AF state of this compound is notmodulated by the ZeemanarXiv:1908.01236v2 [cond-mat.str-el] 21 Dec 20202 splitting. We show that such a behavior is a natural con- sequence of commensurate N´ eel order giving rise to the Zeeman SOC in the form of Eq. (1). Having established the presence of the Zeeman SOC in an AF metal, we utilize this eﬀect for probing the electronic state of Nd 2−xCexCuO 4(NCCO), a pro- totypical example of electron-doped high- Tccuprate superconductors17. In these materials, superconduc- tivity coexists with another symmetry-breaking phe- nomenon manifested in a Fermi-surface reconstruction as detected by angle-resolved photoemission spectroscopy (ARPES)18–21and SdH experiments22–25. The involve- ment of magnetism in this Fermi-surface reconstruction has been broadly debated26–40. Here, we present detailed data on the SdH amplitude in optimally doped NCCO, tracing its variation over more than two orders of mag- nitude with changing the ﬁeld orientation. The oscil- lation behavior is found to be very similar to that in κ-BETS. Given the crystal symmetry and the position of the relevant Fermi-surface pockets, this result is ﬁrm evidence for antiferromagnetism in NCCO. Our ﬁnding not only settles the controversy in electron-doped cuprate superconductors—but also clearly demonstrates the gen- erality of the novel SOC mechanism. Theoretical background — Before presenting the ex- perimental results, we recapitulate the eﬀect of Zee- man splitting on quantum oscillations. The superposi- tion of the oscillations coming from the conduction sub- bands with opposite spins results in the well-known spin- reduction factor in the oscillation amplitude14:Rs= cos/parenleftBig πg 2m me/parenrightBig ; heremeandmare, respectively, the free electron mass and the eﬀective cyclotron mass of the relevant carriers. We restrict our consideration to the ﬁrst-harmonic oscillations, which is fully suﬃcient for the description of our experimental results. In most three- dimensional (3D) metals, the dependence of Rson the ﬁeld orientation is governed by the anisotropy of the cy- clotron mass. At some ﬁeld orientations Rsmay vanish, and the oscillation amplitude becomes zero. This spin- zeroeﬀect carries information about the renormalization of the product gmrelative to its free-electron value 2 me. For 3D systems, this eﬀect is obviously not universal. For example, in the simplest case of a spherical Fermi surface, Rspossesses no angular dependence whatsoever, hence no spin zeros. By contrast, in quasi-2D metals with their vanishingly weak interlayer dispersion such as in layered organic and cuprate conductors, spin zeros are41–46a ro- bust consequence of the monotonic increase of the cy- clotron mass, m∝1/cosθ, with tilting the ﬁeld by an angleθaway from the normal to the conducting layers. In an AF metal, the g-factor may acquire a kdepen- dence through the Zeeman SOC mechanism. It becomes particularly pronounced in the purely transverse geome- try, i.e., for a magnetic ﬁeld normal to the N´ eel axis. In this case,Rscontains the factor ¯ g⊥averaged over the cy- clotron orbit, see Supplementary Information (hereafter SI) for details. As a result, the spin-reduction factor ina layered AF metal takes the form: Rs= cos/bracketleftbiggπ cosθ¯g⊥m0 2me/bracketrightbigg , (2) wherem0≡m(θ= 0◦). Often, the Fermi surface is centered at a point k∗, where the equality g⊥(k∗) = 0 is protected by symmetry6– as it is for κ-BETS (see SI). Such a k∗belongs to a line node g⊥(k) = 0 crossing the Fermi surface. Hence, g⊥(k) changes sign along the Fermi surface, and ¯ g⊥in Eq. (2) vanishes by symmetry of g⊥(k). Consequently, the quantum-oscillation amplitude is predicted to have no spin zeros9. For pockets with Fermi wave vector kFwell below the inverse AF coheren- ce length 1/ξ,g⊥(k) can be described by the leading term of its expansion in k. For such pockets, the present result was obtained in Refs.9,10,47. According to our estimates in the SI, both in κ-BETS and in optimally doped NCCO (x= 0.15), the product kFξconsiderably exceeds unity. Yet, the quasi-classical consideration above shows that forkFξ >1 the conclusion remains the same: ¯ g⊥= 0, see SI for the explicit theory. We emphasize that centering of the Fermi surface at a point k∗withg⊥(k∗) = 0 – such as a high-symmetry point of the magnetic BZ boundary6– is crucial for a vanishing of ¯ g⊥. Otherwise, Zeeman SOC remains inert, as it does in AF CeIn 3, whosedFermi surface is centered at the Γ point (see SI and Refs.48–50), and in quasi-2D EuMnBi 2, with its quartet of Dirac cones centered away from the magnetic BZ boundary51,52. With this, we turn to the experiment. RESULTS AF Organic Superconductor κ-(BETS) 2FeBr 4—This is a quasi-2D metal with conducting layers of BETS donor molecules, sandwiched between insulating FeBr− 4- anion layers15. The material has a centrosymmetric or- thorhombic crystal structure (space group Pnma ), with theacplane along the layers. The Fermi surface consists of a weakly warped cylinder and two open sheets, sep- arated from the cylinder by a small gap ∆ 0at the BZ boundary, as shown in Fig. 115,16,53. The magnetic properties of the compound are mainly governed by ﬁve localized 3 d-electron spins per Fe3+ion in the insulating layers. Below TN≈2.5 K, theseS= 5/2 spins are ordered antiferromagnetically, with the unit cell doubling along the caxis and the staggered magnetiza- tion pointing along the aaxis15,54. Above a critical mag- netic ﬁeldBc∼2−5 T, dependent on the ﬁeld orien- tation, antiferromagnetism gives way to a saturated PM state55. The SdH oscillations in the high-ﬁeld PM state and in the N´ eel state are markedly diﬀerent (see Fig. 1b). In the former, two dominant frequencies corresponding to a classical orbit αon the Fermi cylinder and to a large magnetic-breakdown (MB) orbit βare found, in agreement with the predicted Fermi surface16,53. The3 0 2 4 6 8 10 12 140.130.140.150.16R (W) B (T)Bc 0 2 40.010.1110 b F (kT)a 0.05 0.100.010.1110 FFT ampl. (Arb. units) F (kT)d gꓕ(k) kcka kc0AF d QAF /2c /c /2c /c 0 ba /c /c 0ka kc 0(a) (b) (c)Paramagnetic phaseAntiferromagnetic phase AF PM FIG. 1. 2D Fermi surface of κ-BETS in the para- and antiferromagnetic phases. (a) Fermi surface of κ-BETS in the PM state15,53(blue lines). The blue arrows show the classical cyclotron orbits αand the red arrows the large MB orbit β, which involves tunneling through four MB gaps ∆ 0in a strong magnetic ﬁeld. (b)Interlayer magnetoresistance of the κ-BETS sample, recorded at T= 0.5 K with ﬁeld applied nearly perpendicularly to the layers ( θ= 2◦). The vertical dash indicates the transition between the low-ﬁeld AF and high-ﬁeld PM states. The insets show the fast Fourier transforms (FFT) of the SdH oscillations for ﬁeld windows [2 −5] T and [12 −14] T in the AF and PM state, respectively. (c)The BZ boundaries in the AF state with the wave vector QAF= (π/c,0) and in the PM state are shown by solid-black and dashed-black lines, respectively. The dotted-blue and solid-orange lines show, respectively, the original and reconstructed Fermi surfaces16. The shaded area in the corner of the magnetic BZ, separated from the rest of the Fermi surface by gaps ∆ 0and ∆ AF, is theδpocket responsible for the SdH oscillations in the AF state. The inset shows the function g⊥(k). Antiferromagnetic phase 0 1 2 3 40510 5 5 0°16°34° 60°65.4° Rosc/Rbackg (%) B cosq (T)52° 3 65.4° 0° 16° 34° 52° 03060901200246AFFT (arb. unit) F cos (q) (T)2 10 0 1 2 3 4 5 6051015202530 0°R (W) B (T)16°34°52°q : 65.4° k-BETSBc(b) (a) FIG. 2. SdH oscillations in the antiferromagnetic phase of κ-BETS .(a)Examples of the ﬁeld-dependent interlayer resistance at diﬀerent ﬁeld orientations, at T= 0.42 K. The AF – PM transition ﬁeld Bcis marked by vertical dashes. Inset: the orientation of the current Jand magnetic ﬁeld Brelative to the crystal axes and the N´ eel axis N.(b)Oscillating component, normalized to the non-oscillating B-dependent resistance background, plotted as a function of the out-of-plane ﬁeld component B⊥=Bcosθ. The curves corresponding to diﬀerent tilt angles θare vertically shifted for clarity. For θ≥52◦the ratio Rosc/Rbackg is multiplied by a constant factor, as indicated. The vertical dashed lines are drawn to emphasize the constant oscillation phase in these coordinates; Inset: FFT spectra of the SdH oscillations taken in the ﬁeld window [3 −4.2] T. The FFT amplitudes at θ= 52◦and 65.4◦are multiplied by a factor of 2 and 10, respectively. oscillation amplitude exhibits spin zeros as a function of the ﬁeld strength and orientation, which is fairly well described by a ﬁeld-dependent spin-reduction factor Rs(θ,B), with theg-factorg= 2.0±0.2 in the presence of an exchange ﬁeld BJ≈−13 T, imposed by PM Fe3+ionson the conduction electrons45,56. In the SI, we provide further details of the SdH oscillation studies on κ-BETS. BelowBc, in the AF state, new, slow oscillations at the frequencyFδ≈62 T emerge, indicating a Fermi-surface reconstruction16. The latter is associated with the fold-4 ing of the original Fermi surface into the magnetic BZ, andFδis attributed to the new orbit δ, see Fig. 1c. This orbit emerges due to the gap ∆ AFat the Fermi-surface points, separated by the N´ eel wave vector ( π/c,0)57. Figure 2 shows examples of the ﬁeld-dependent inter- layer resistance of κ-BETS, recorded at T= 0.42 K, at diﬀerent tilt angles θ. The ﬁeld was rotated in the plane normal to the N´ eel axis (crystallographic aaxis). In ex- cellent agreement with previous reports16,58, slow oscil- lations with frequency Fδ= 61.2 T/cosθare observed belowBc, see inset in Fig. 2b. Thanks to the high crys- tal quality, even in this low-ﬁeld region the oscillations can be traced over a wide angular range \|θ\|≤70◦. The angular dependence of the δ-oscillation amplitude Aδis shown in Fig. 3. The amplitude was determined by fast Fourier transform (FFT) of the zero-mean oscil- lating magnetoresistance component normalized to the monotonicB-dependent background, in the ﬁeld window between 3.0 and 4.2 T, so as to stay below Bc(θ) for all ﬁeld orientations. The lines in Fig. 3 are ﬁts using the Lifshitz-Kosevich formula for the SdH amplitude14: Aδ=A0m2 √ BRMBexp(−KmT D/B) sinh(KmT/B )Rs(θ), (3) whereA0is a ﬁeld-independent prefactor, B= 3.5 T (the midpoint of the FFT window in 1 /Bscale),m the eﬀective cyclotron mass ( m= 1.1meatθ= 0◦16, growing as 1 /cosθwith tilting the ﬁeld as in other quasi-2D metals59,60),K= 2π2kB/¯he,T= 0.42 K,TDthe Dingle temperature, and RMBthe MB factor. For κ-BETS,RMBtakes the form RMB =/bracketleftbig 1−exp/parenleftbig −B0 Bcosθ/parenrightbig/bracketrightbig /bracketleftbig 1−exp/parenleftbig −BAF Bcosθ/parenrightbig/bracketrightbig , with two cha- racteristic MB ﬁelds B0andBAFassociated with the gaps ∆ 0and ∆ AF, respectively. The Zeeman splitting eﬀect is encapsulated in the spin factor Rs(θ). In Eq. (1), the geometry of our experiment implies B/bardbl= 0, thus in the N´ eel state Rs(θ) takes the form of Eq. (2). ExcludingRs(θ), the other factors in Eq. (3) decrease monotonically with increasing θ. By contrast, Rs(θ) in Eq. (2), generally, has an oscillating angular dependence. For ¯g⊥=g= 2.0 found in the PM state45, Eq. (2) yields two spin zeros, at θ≈43◦and 64◦. Contrary to this, we observe nospin zeros, but rather a monotonic decrease of Aδby over two orders of magnitude as the ﬁeld is tilted away from θ= 0◦to±70◦, i.e., in the entire angular range where we observe the oscillations. The diﬀerent curves in Fig. 3 are our ﬁts using Eq. (3) with A0and TDas ﬁt parameters, and different values of the g-factor. We used the MB ﬁeld values B0= 20 T and BAF= 5 T. While the exact values of B0andBAFare unknown, they have virtually no eﬀect on the ﬁt quality, as we demon- strate in the SI. The best ﬁt is achieved with g= 0, i.e. with an angle-independent spin factor Rs= 1. The ex- cellent agreement between the ﬁt and the experimental data conﬁrms the quasi-2D character of the electron con- duction, with the 1 /cosθdependence of the cyclotron mass. Comparison of the curves in Fig. 3 with the data rules /s45/s54/s48 /s45/s52/s48 /s45/s50/s48 /s48 /s50/s48 /s52/s48 /s54/s48/s48/s46/s48/s49/s48/s46/s49/s49/s49/s48 /s69/s120/s112/s101/s114/s105/s109/s101/s110/s116/s124/s65 /s100/s124/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41 /s113 /s32/s40/s100/s101/s103/s114/s101/s101/s115/s41/s32/s32/s32/s32/s70/s105/s116/s115/s32/s119/s105/s116/s104/s32/s69/s113/s46/s32/s40/s51/s41/s58 /s32/s32/s32/s32 /s32/s32/s32 /s103 /s32/s61/s32/s50/s46/s48 /s32/s32/s32/s32 /s32/s32/s32 /s103 /s32/s61/s32/s49/s46/s48 /s32/s32/s32/s32 /s32/s32/s32 /s103 /s32/s61/s32/s48/s46/s53 /s32/s32/s32/s32 /s32/s32/s32 /s103 /s32/s61/s32/s48/s46/s50 /s32/s32/s32/s32 /s32/s32/s32 /s103 /s32/s61/s32/s48 /s107 /s45/s66/s69/s84/s83FIG. 3. Angular dependence of the SdH amplitude Aδ in the AF state of κ-BETS. The lines are ﬁts using Eq. (3) with diﬀerent values of the g-factor. out ¯g⊥>0.2. Given the experimental error bars, we cannot exclude a nonzero ¯ g⊥<∼0.2, yet even such a small ﬁnite value would be in stark contrast with the textbook g= 2.0, found from the SdH oscillations in the high-ﬁeld, PM state45. Below we argue that, in fact, ¯ g⊥in the N´ eel state is exactly zero. Optimally doped NCCO — This material has a body-centered tetragonal crystal structure (space group I4/mmm ), where (001) conducting CuO 2layers alter- nate with their insulating (Nd,Ce)O 2counterparts17. Band-structure calculations61,62predict a holelike cylin- drical Fermi surface, centered at the corner of the BZ. However, angle-resolved photoemission spectroscopy (ARPES)18,19,21,63reveals a reconstruction of this Fermi surface by a ( π/a,π/a ) order. Moreover, magnetic quan- tum oscillations23–25show that the Fermi surface remains reconstructed even in the overdoped regime, up to the critical doping xc(≈0.175 for NCCO), where the super- conductivity vanishes64. The origin of this reconstruc- tion remains unclear: while the ( π/a,π/a ) periodicity is compatible with the N´ eel order observed in strongly un- derdoped NCCO, coexistence of antiferromagnetism and superconductivity in electron-doped cuprates remains controversial. A number of neutron-scattering and muon- spin rotation studies31–34have detected short-range N´ eel ﬂuctuations, but no static order within the superconduct- ing doping range. However, other neutron scattering35,36 and magnetotransport37–39experiments have produced evidence of static or quasi-static AF order in supercon- ducting samples at least up to optimal doping xopt. Al- ternative mechanisms of the Fermi-surface reconstruc- tion have been proposed, including a d-density wave28, a charge-density wave29, or coexistent topological and ﬂuctuating short-range AF orders30.5 (a) (b) 20 30 40 50 60 700123 0.0 0.3 0.6 0.90.00.20.40.60.8Rosc/Rbackg (%) B cosq (T)0°37°55°60°64° ´ 2´ 5 67° AFFT (arb. unit) F cos q (kT)´ 10 ´ 15 0 20 40 60 800.00.20.4 67° 55° 37°R (W) B (T)q : 0° NCCOc b FIG. 4. Examples of MR and angle-dependent SdH oscillations in optimally doped NCCO. (a) Examples of the B-dependent interlayer resistance at diﬀerent ﬁeld orientations, at T= 2.5 K;(b)oscillating component, normalized to the non-oscillating B-dependent resistance background, plotted as a function of the out-of-plane ﬁeld component B⊥=Bcosθ. The curves corresponding to diﬀerent tilt angles θare vertically shifted for clarity. For θ= 64◦and 67◦the ratioRosc/Rbackg is multiplied by a factor of 2 and 5, respectively. The vertical dashed lines are drawn to emphasize the constant oscillation phase in these coordinates; Inset: FFT spectra of the SdH oscillations taken in the ﬁeld window 45 to 64 T. To shed light on the relevance of antiferromagnetism to the electronic ground state of superconducting NCCO, we have studied the ﬁeld-orientation dependence of the SdH oscillations of the interlayer resistance in an opti- mally doped, xopt= 0.15, NCCO crystal. The overall magnetoresistance behavior is illustrated in Fig. 4(a). At low ﬁelds the sample is superconducting. Imme- diately above the θ-dependent superconducting critical ﬁeld the magnetoresistance displays a non-monotonic feature, which has already been reported for optimally doped NCCO in a magnetic ﬁeld normal to the lay- ers22,65. This anomaly correlates with an anomaly in the Hall resistance and has been associated with mag- netic breakdown through the energy gap, created by the (π/a,π/a )-superlattice potential64. With increasing θ, the anomaly shifts to higher ﬁelds, consistently with the expected increase of the breakdown gap with tilting the ﬁeld. SdH oscillations develop above about 30 T. Figure 4(b) shows examples of the oscillatory component of the mag- netoresistance, normalized to the ﬁeld-dependent non- oscillatory background resistance Rbackg, determined by a low-order polynomial ﬁt to the as-measured R(B) de- pendence. In our conditions, B<∼65 T,T= 2.5 K, the only discernible contribution to the oscillations comes from the hole-like pocket αof the reconstructed Fermi surface22. This pocket is centered at the reduced BZ boundary, as shown in the inset of Fig. 5. While mag- netic breakdown creates large cyclotron orbits βwith the area equal to that of the unreconstructed Fermi surface, even in ﬁelds of 60-65 T the fast βoscillations are more than two orders of magnitude weaker than the αoscilla- tions24,64.The oscillatory signal is plotted in Fig. 4(b) as a func- tion of the out-of-plane ﬁeld component B⊥=Bcosθ. In these coordinates the oscillation frequency remains con- stant, indicating that F(θ) =F(0◦)/cosθand thus con- ﬁrming the quasi-2D character of the conduction. In the inset we show the respective FFTs plotted against the cosθ-scaled frequency. They exhibit a peak at Fcosθ= 294 T, in line with previous reports. The relatively large width of the FFT peaks is caused by the small number of oscillations in the ﬁeld window [45 −64] T. This re- strictive choice is dictated by the requirement that the SdH oscillations be resolved over the whole ﬁeld window at all tilt angles up to θ≈72◦. In the SI we provide an additional analysis of the amplitude at ﬁxed ﬁeld values in the SI conﬁrming the FFT results. Furthermore, in Fig. 4(b) one can see that the phase of the oscillations is not inverted, and stays constant in the studied angular range. This is fully in line with the absence of spin zeros, see Eq. (2). The main panel of Fig. 5 presents the angular depen- dence of the oscillation amplitude (symbols), in a ﬁeld rotated in the ( ac) plane. The amplitude was determined by FFT of the data taken at T= 2.5 K in the ﬁeld win- dow 45 T≤B≤64 T. The lines in the ﬁgure are ﬁts using Eq. (3), for diﬀerent g-factors. The ﬁts were performed using the MB factor RMB= [1−exp(−B0/B)]24,64, the reported values for the MB ﬁeld B0= 12.5 T, and the ef- fective cyclotron mass m(θ= 0◦) = 1.05m064, while tak- ing into account the 1 /cosθangular dependence of both B0andm. The prefactor A0and Dingle temperature TD were used as ﬁt parameters, yielding TD= (12.6±1) K, close to the value found in the earlier experiment64. Note that, contrary to the hole-doped cuprate YBa 2Cu3O7−x,6 /s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48/s48/s46/s48/s49/s48/s46/s49/s49/s49/s48 /s32/s32 /s32/s32 /s103 /s32/s61/s32/s50/s46/s48 /s32/s32 /s103 /s32/s61/s32/s49/s46/s48 /s32/s32 /s103 /s32/s61/s32/s48/s46/s50 /s32/s32 /s103 /s32/s61/s32/s48/s124/s65 /s97/s124/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41 /s113 /s32/s40/s100/s101/s103/s114/s101/s101/s115/s41/s32/s69/s120/s112/s101/s114/s105/s109/s101/s110/s116 /s70/s105/s116/s115/s32/s119/s105/s116/s104/s32/s69/s113/s46/s32/s40/s51/s41/s58 /s78/s67/s67/s79/s97 FIG. 5. Angular dependence of the SdH amplitude in optimally doped NCCO. The lines are ﬁts using Eq. (3) with diﬀerent g-factor values. Inset: The ﬁrst quadrant of the BZ with the Fermi-surface reconstructed by a superlattice potential with wave vector Q= (π/a,π/a ). If this potential involves N´ eel order, the function g⊥(k) (red line in the inset) vanishes at the reduced BZ boundary (dashed line). The SdH oscillations are associated with the oval hole pocket αcentered at (π/2a,π/2a)22. where the analysis of earlier experiments46,66was compli- cated by the bilayer splitting of the Fermi cylinder46, the single-layer structure of NCCO poses no such diﬃculty. Similar toκ-BETS, the oscillation amplitude in NCCO decreases by a factor of about 300, with no sign of spin zeros as the ﬁeld is tilted from θ= 0◦to 72.5◦. Again, this behavior is incompatible with the textbook value g= 2, which would have produced two spin zeros in the interval 0◦≤θ≤70◦, see the green dash-dotted line in Fig. 5. A reduction of the g-factor to 1.0 would shift the ﬁrst spin zero to about 72◦, near the edge of our range (blue dotted line in Fig. 5). However, this would simultaneously suppress the amplitude at small θby a factor of ten, contrary to our observations. All in all, our data rule out a constant g>0.2. DISCUSSION In both materials, our data impose on the eﬀective g- factor an upper bound of 0 .2. At ﬁrst sight, one could simply view this as a suppression of the eﬀective gto a small nonzero value. However, below we argue that, in fact, our ﬁndings imply ¯ g⊥= 0 and point to the im- portance of the Zeeman SOC in both materials. The quasi-2D character of electron transport is crucial for this conclusion: as mentioned above, in three dimensions, the mere absence of spin zeros imposes no bounds on the g- factor.Inκ-BETS the interplay between the crystal symmetry and the periodicity of the N´ eel state5,6,47guarantees that g⊥(k) vanishes on the entire line kc=π/2cand is an odd function of kc−π/2c, see the inset of Fig. 1c and Fig. S2 in the SI. The δorbit is centered on the line kc=π/2c; hence ¯g⊥in Eq. (2) vanishes, implying the absence of spin zeros, in agreement with our data. At the same time, quantum oscillations in the PM phase clearly reveal the Zeeman splitting of Landau levels with g= 2.0. Therefore, we conclude that ¯ g⊥= 0 is an intrinsic property of the N´ eel state. In optimally doped NCCO, as already mentioned, the presence of a (quasi)static N´ eel order has been a sub- ject of debate. However, if indeed present, such an order leads tog⊥(k) = 0 at the entire magnetic BZ boundary (see SI). For the hole pockets, producing the observed Fα/similarequal300 T oscillations, ¯ g⊥= 0 by symmetry of g⊥(k) (see inset of Fig. 5 and Fig. S3 in SI). Such an inter- pretation requires that the relevant AF ﬂuctuations have frequencies below the cyclotron frequency in our experi- ment,νc∼1012Hz at 50 T. Finally, we address mechanisms – other than Zeeman SOC of Eq. (1) – which may also lead to the absence of spin zeros. While such mechanisms do exist, we will show that none of them is relevant to the materials of our interest. When looking for alternative explanations to our ex- perimental ﬁndings, let us recall that, generally, the eﬀectiveg-factor may depend on the ﬁeld orientation. This dependence may happen to compensate that of the quasi-2D cyclotron mass, m/cosθ, in the expres- sion (2) for the spin-reduction factor Rs, and render the latter nearly isotropic, with no spin zeros. Obviously, such a compensation requires a strong Ising anisotropy [g(θ= 0◦)/greatermuchg(θ= 90◦)] – as found, for instance, in the heavy-fermion compound URu 2Si2, with the values gc= 2.65±0.05 andgab= 0.0±0.1 for the ﬁeld along and normal to the caxis, respectively67,68. However, this scenario is irrelevant to both materials of our interest: In κ-BETS, a nearly isotropic g-factor, close to the free- electron value 2.0, was revealed by a study of spin zeros in the paramagnetic state45. In NCCO, the conduction electrong-factor may acquire anisotropy via an exchange coupling to Nd3+local moments. However, the low- temperature magnetic susceptibility of Nd3+in the basal plane is some 5 times larger than along the caxis69,70. Therefore, the coupling to Nd3+may only increase gab relative togc, and thereby only enhance the angular de- pendence of Rsrather than cancel it out. Thus, we are lead to rule out a g-factor anisotropy of crystal-ﬁeld ori- gin as a possible reason behind the absence of spin zeros in our experiments. As follows from Eq. (2), another possible reason for the absence of spin zeros is a strong reduction of the ratio gm/2me. However, while some renormalization of this ratio in metals is commonplace, its dramatic suppression (let alone nulliﬁcation) is, in fact, exceptional. Firstly, a vanishing mass would contradict m/m e≈1, experimen-7 tally found in both materials at hand. On the other hand, a Landau Fermi-liquid renormalization14g→g/(1+G0) would require a colossal Fermi-liquid parameter G0≥10, for which there is no evidence in NCCO, let alone κ- BETS with its already mentioned g≈2 in the paramag- netic state45. A suﬃcient diﬀerence of the quantum-oscillation am- plitudes and/or cyclotron masses for spin-up and spin- down Fermi surfaces might also lead to the absence of spin zeros. Some heavy-fermion compounds show strong spin polarization in magnetic ﬁeld, concomitant with a substantial ﬁeld-induced diﬀerence of the cyclotron masses of the two spin-split subbands71,72. As a result, for quantum oscillations in such materials, one spin am- plitude considerably exceeds the other, and no spin zeros are expected. Note that this physics requires the presence of a very narrow conduction band, in addition to a broad one. In heavy-fermion compounds, such a band arises from thefelectrons, but is absent in both materials of our interest. Another extreme example is given by the single fully polarized band in a ferromagnetic metal, where only one spin orientation is present, and spin zeros are obviously absent. Yet, no sign of ferro- or metamagnetism has been seen in either NCCO or κ-BETS. Moreover, in κ- BETS,the spin-zero eﬀect has been observed in the para- magnetic state45, indicating that the quantum-oscillation amplitudes of the two spin-split subbands are compara- ble. However, for NCCO one may inquire whether spin polarization could render interlayer tunneling amplitudes for spin-up and spin-down diﬀerent enough to lose spin zeros, especially in view of an extra contribution of Nd3+ spins in the insulating layers to spin polarization. In the SI, we show that this is notthe case. Thus, we are lead to conclude that the absence of spin zeros in the AF κ-BETS and in optimally doped NCCO is indeed a manifestation of the Zeeman SOC. Our ex- planation relies only on the symmetry of the N´ eel state and the location of the carrier pockets, while being in- sensitive to the mechanism of the antiferromagnetism or to the orbital makeup of the relevant bands. METHODS Crystals of κ-(BETS) 2FeBr 4were grown electrochem- ically and prepared for transport measurements as re- ported previously45. The interlayer ( I\|\|b) resistance was measured by the standard four-terminal a.c. technique using a low-frequency lock-in ampliﬁer. Magnetoresis- tance measurements were performed in a superconduct- ing magnet system at ﬁelds of up to 14 T. The samples were mounted on a holder allowing in-situ rotation of the sample around an axis perpendicular to the external ﬁeld direction. The orientation of the crystal was deﬁned by a polar angle θbetween the ﬁeld and the crystallographic baxis (normal to the conducting layers). Optimally doped single crystals of Nd 1.85Ce0.15CuO 4,grown by the traveling solvent ﬂoating zone method, were prepared for transport measurements as reported previ- ously23. Measurements of the interlayer ( I\|\|c) resistance were performed on a rotatable platform using a standard four-terminal a.c. technique at frequencies of 30 −70 kHz in a 70 T pulse-magnet system, with a pulse duration of 150 ms, at the Dresden High Magnetic Field Laboratory. The raw data were collected by a fast digitizing oscillo- scope and processed afterwards by a digital lock-in pro- cedure22. The orientation of the crystal was deﬁned by a polar angle θbetween the ﬁeld and the crystallographic caxis (normal to the conducting layers). ACKNOWLEDGMENTS It is our pleasure to thank G. Knebel, H. Sakai, I. Sheikin, and A. Yaresko for illuminating discus- sions. This work was supported by HLD at HZDR, a member of the European Magnetic Field Labora- tory (EMFL). P. D. G. acknowledges State Assignment #0033-2019-0001 for ﬁnancial support and the Labo- ratoire de Physique Th´ eorique, Toulouse, for the hos- pitality during his visit. R. G. acknowledges ﬁnan- cial support from the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) via Germany’s Excellence Strategy (EXC-2111-390814868). J. W. ac- knowledges ﬁnancial support from the DFG through the W¨ urzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter - ct.qmat (EXC 2147, project-id 390858490). AUTHOR CONTRIBUTIONS T.H., F.K., M.K., E.K., W.B., and M.V.K performed the experiments and analyzed the data. R.R., P.D.G., and M.V.K. initiated the exploration and performed the theoretical analysis and interpretation of the experimen- tal results. H.F. and A.E. provided high-quality single crystals. R.R., P.D.G., T.H., W.B., E.K., J.W., R.G., and M.V.K. contributed to the writing of the manuscript. DATA AVAILABILITY The authors declare that all essential data supporting the ﬁndings of this study are available within the paper and its supplementary information. The complete data set in ASCII format is available from the corresponding authors upon reasonable request. ADDITIONAL INFORMATION Supplementary Information: Supplementary In- formation is available at [URL to be provided by the publisher].8 Competing interests: The authors declare no com- peting interests. REFERENCES ∗revaz@irsamc.ups-tlse.fr †grigorev@itp.ac.ru ‡t.helm@hzdr.de §Present address: TNG Technology Consulting GmbH, 85774 Unterf¨ ohring, Germany ¶mark.kartsovnik@wmi.badw.de 1R. Winkler, Spin-orbit Coupling Eﬀects in Two- dimensional Electron and Hole Systems, Springer Tracts in Modern Physics, Vol. 191, Berlin – Heidelberg, 2003. 2M. I. Dyakonov (Editor), Spin Physics in Semiconductors, Springer Series in Solid-State Sciences, Vol. 157, Berlin – Heidelberg, 2017. 3V. B. Berestetski˘ ı, E. M. Lifshitz and L. P. Pitaevski˘ ı, “Course of Theoretical Physics”, Vol. 4, Quantum Elec- trodynamics , Pergamon Press (1982). 4C. Kittel, Quantum Theory of Solids , John Wiley & Sons, Inc., New York – London (1963). 5R. Ramazashvili, Kramers Degeneracy in a Magnetic Field and Zeeman Spin-Orbit Coupling in Antiferromagnetic Conductors , Phys. Rev. Lett. 101, 137202 (2008). 6R. Ramazashvili, Kramers degeneracy in a magnetic ﬁeld and Zeeman spin-orbit coupling in antiferromagnetic con- ductors , Phys. Rev. B 79, 184432 (2009). 7S. A. Brazovskii and I. A. Luk’yanchuk, Symmetry of elec- tron states in antiferromagnets , Sov. Phys. JETP 69, 1180- 1184 (1989). 8S. A. Brazovskii, I. A. Luk’yanchuk, and R. R. Ramaza- shvili, Electron paramagnetism in antiferromagnets , JETP Lett. 49, 644-646 (1989). 9V.V. Kabanov and A. S. Alexandrov, Magnetic quantum oscillations in doped antiferromagnetic insulators , Phys. Rev. B 77, 132403 (2008); 81, 099907(E) (2010). 10R. Ramazashvili, Electric excitation of spin resonance in antiferromagnetic conductors , Phys. Rev. B 80, 054405 (2009). 11E. I. Rashba and V. I. Sheka, ”Electric-Dipole Spin Res- onances”, in Landau Level Spectroscopy , edited by G. Landwehr and E. I. Rashba (Elsevier, New York, (1991). 12M. Shaﬁei et al. ,Resolving Spin-Orbit- and Hyperﬁne- Mediated Electric Dipole Spin Resonance in a Quantum Dot, Phys. Rev. Let. 110, 107601 (2013). 13J. W. G. van den Berg et al. ,Fast Spin-Orbit Qubit in an Indium Antimonide Nanowire , Phys. Rev. Let. 110, 066806 (2013). 14D. Shoenberg, Magnetic Oscillations in Metals , Cambridge University Press (1984). 15H. Fujiwara et al. ,A Novel Antiferromagnetic Or- ganic Superconductor κ-(BETS) 2FeBr 4[Where BETS = Bis(ethylenedithio)tetraselenafulvalene] , J. Am. Chem. Soc.123, 306-314 (2001). 16T. Konoike et al. ,Fermi surface reconstruction in the magnetic-ﬁeld-induced superconductor κ-(BETS) 2FeBr 4, Phys. Rev. B 72, 094517 (2005).17N. P. Armitage, P. Fournier, and R. L. Greene, Progress and perspectives on electron-doped cuprates , Rev. Mod. Phys. 82, 2421-2487 (2010). 18A. F. Santander-Syro et al. ,Two-Fermi-surface supercon- ducting state and a nodal d-wave energy gap of the electron- doped Sm 1.85Ce0.15CuO 4−δcuprate superconductor , Phys. Rev. Lett. 106, 197002 (2011). 19D. Song et al. ,Electron Number-Based Phase Diagram of Pr1−xLaCexCuO 4−δand Possible Absence of Disparity be- tween Electron- and Hole-Doped Cuprate Phase Diagrams , Phys. Rev. Lett. 118, 137001 (2017). 20J.-F. He et al. ,Fermi surface reconstruction in electron- doped cuprates without antiferromagnetic long-range order , Proc. Natl. Acad. Sci. USA 116, 3449-3453 (2019). 21H. Matsui et al. ,Evolution of the pseudogap across the magnet-superconductor phase boundary of Nd2−xCexCuO 4, Phys. Rev. B 75, 224514 (2007). 22T. Helm et al. ,Evolution of the Fermi surface of the electron-doped high-temperature superconductor Nd2−xCexCuO 4revealed by Shubnikov - de Haas oscilla- tions , Phys. Rev. Lett. 103, 157002 (2009). 23T. Helm et al. ,Magnetic breakdown in the electron- doped cuprate superconductor Nd 2−xCexCuO 4: The recon- structed Fermi surface survives in the strongly overdoped regime , Phys. Rev. Lett. 105, 247002 (2010). 24M. V. Kartsovnik et al. ,Fermi surface of the electron-doped cuprate superconductor Nd 2−xCexCuO 4probed by high- ﬁeld magnetotransport , New J. Phys. 13, 015001 (2011). 25J. S. Higgins et al. ,Quantum oscillations from the recon- structed Fermi surface in electron-doped cuprate supercon- ductors , New J. Phys. 20, 043019 (2018). 26T. Das, R. S. Markiewicz, and A. Bansil, Superconductivity and topological Fermi surface transitions in electron-doped cuprates near optimal doping , J. Phys. Chem. Solids 69, 2963-2966 (2008). 27S. Sachdev, Where is the quantum critical point in the cuprate superconductors? , Phys. Status Solidi B 247, 537- 543 (2010). 28S. Chakravarty, R. B. Laughlin, D. K. Morr, and C. Nayak, Hidden order in cuprates , Phys. Rev. B 63, 094503 (2001). 29E. H. da Silva Neto et al. ,Charge ordering in the electron- doped superconductor Nd 2–xCexCuO 4, Science 347, 282- 285 (2015). 30S. Sachdev, Topological order, emergent gauge ﬁelds, and Fermi surface reconstruction , Rep. Prog. Phys. 82, 014001 (2019); S. Sachdev, H. D. Scammell, M. S. Scheurer, and G. Tarnopolsky, Gauge theory for the cuprates near optimal doping , Phys. Rev. B 99, 054516 (2019). 31G. M. Luke et al. ,Magnetic order and electronic phase diagrams of electron-doped copper oxide materials , Phys. Rev. B 42, 7981-7988 (1990). 32E. M. Motoyama et al. ,Spin correlations in the electron-doped high-transition-temperature superconductor9 Nd2−xCexCuO 4−δ, Nature 445, 186-189 (2007). 33P. K. Mang et al. ,Phase decomposition and chemical inho- mogeneity in Nd 2−xCexCuO 4±δ, Phys. Rev. B 70, 094507 (2004). 34H. Saadaoui et al. ,The phase diagram of electron-doped La2−xCexCuO 4−δ, Nat. Commun. 6, 6041 (2015). 35K. Yamada et al. ,Commensurate Spin Dynamics in the Superconducting State of an Electron-Doped Cuprate Su- perconductor , Phys. Rev. Lett. 90, 137004 (2003). 36H. J. Kang et al. ,Electronically competing phases and their magnetic ﬁeld dependence in electron-doped nonsupercon- ducting and superconducting Pr 0.88LaCe 0.12CuO 4±δ, Phys. Rev. B 71, 214512 (2005). 37Y. Dagan et al. ,Origin of the Anomalous Low Temperature Upturn in the Resistivity of the Electron-Doped Cuprate Superconductors , Phys. Rev. Lett. 94, 057005 (2005). 38W. Yu, J. S. Higgins, P. Bach, and R. L. Greene, Trans- port evidence of a magnetic quantum phase transition in electron-doped high-temperature superconductors , Phys. Rev. B 76, 020503(R) (2007). 39A. Dorantes et al. ,Magnetotransport evidence for irre- versible spin reorientation in the collinear antiferromag- netic state of underdoped Nd2−xCexCuO 4, Phys. Rev. B 97, 054430 (2018). 40R. L. Greene, P. R. Mandal, N. R. Poniatowski, and T. Sarkar, The Strange Metal State of the Electron-Doped Cuprates , Annu. Rev. Condens. Matter Phys. 11, 213-229 (2020). 41J. Wosnitza et al. ,de Haas–van Alphen studies of the organic superconductors α-(ET) 2NH4Hg(SCN) 4 andκ-(ET) 2Cu(NCS) 2with ET = bis(ethelenedithio)- tetrathiafulvalene , Phys. Rev. B 45, 3018-3025 (1992). 42A. E. Kovalev, M. V. Kartsovnik, and N. D. Kushch, Quan- tum and quasi-classical magnetoresistance oscillations in a new organic metal (BEDT-TTF) 2TlHg(SeCN) 4, Solid State Commun. 87, 705-708 (1993). 43F. A. Meyer et al. ,High-Field de Haas-Van Alphen Studies ofκ-(BEDT-TTF) 2Cu(NCS) 2, Europhys. Lett. 32, 681- 686 (1995). 44S. Uji et al. ,Fermi surface and internal magnetic ﬁeld of the organic conductors λ-(BETS) 2FexGa1−xCl4, Phys. Rev. B 65, 113101 (2002). 45M. V. Kartsovnik et al. ,Interplay Between Conducting and Magnetic Systems in the Antiferromagnetic Organic Super- conductorκ-(BETS) 2FeBr 4, J. Supercond. Nov. Magn. 29, 3075-3080 (2016). 46B. J. Ramshaw et al. ,Angle dependence of quantum os- cillations in YBa 2Cu3O6:59 shows free-spin behaviour of quasiparticles , Nat. Phys. 7, 234-238 (2011). 47R. Ramazashvili, Quantum Oscillations in Antiferromag- netic Conductors with Small Carrier Pockets , Phys. Rev. Lett. 105, 216404 (2010). 48R. Settai et al. ,De Haas – van Alphen studies of rare earth compounds , J. Magn. Magn. Mater. 140–144 , 1153-1154 (1995). 49T. Ebihara, N. Harrison, M. Jaime, S. Uji, and J. C. Lash- ley,Emergent Fluctuation Hot Spots on the Fermi Surface of CeIn 3in Strong Magnetic Fields , Phys. Rev. Lett. 93, 246401 (2004). 50L. P. Gor’kov and P. D. Grigoriev, Antiferromagnetism and hot spots in CeIn 3, Phys. Rev. B 73, 060401(R) (2006). 51H. Masuda et al. ,Impact of antiferromagnetic order on Landau-level splitting of quasi-two-dimensional Dirac fermions in EuMnBi 2, Phys. Rev. B 98, 161108(R) (2018).52S. Borisenko et al. ,Time-reversal symmetry breaking type- II Weyl state in YbMnBi 2, Nat. Commun. 10, 3424 (2019). 53S. Uji et al. ,Two-dimensional Fermi surface for the or- ganic conductor κ-(BETS) 2FeBr 4, Physica B 298, 557-561 (2001). 54T. Mori and M. Katsuhara, Estimations of πd-interactions in organic conductors including magnetic anions , J. Phys. Soc. Jpn. 71, 826-844 (2002). 55T. Konoike et al. ,Magnetic-ﬁeld-induced superconductiv- ity in the antiferromagnetic organic superconductor κ- (BETS) 2FeBr 4, Phys. Rev. B 70, 094514 (2004). 56O. C´ epas, R. H. McKenzie, and J. Merino, Magnetic- ﬁeld-induced superconductivity in layered organic molecu- lar crystals with localized magnetic moments , Phys. Rev. B65, 100502(R) (2002). 57R. E. Peierls, Quantum Theory of Solids , Oxford Univer- sity Press (2001). 58T. Konoike et al. ,Anomalous Magnetic-Field-Hysteresis of Quantum Oscillations in κ-(BETS) 2FeBr 4, J. Low Temp. Phys. 142, 531-534 (2006). 59J. Wosnitza, Fermi Surfaces of Low-Dimensional Organic Metals and Superconductors , Springer Verlag, Berlin – Hei- delberg (1996). 60M. V. Kartsovnik, High Magnetic Fields: A Tool for Studying Electronic Properties of Layered Organic Metals , Chem. Rev. 104, 5737-5782 (2004). 61S. Massidda, N. Hamada, J. Yu, and A. J. Freeman, Elec- tronic structure of Nd-Ce-Cu-O, a Fermi liquid supercon- ductor , Physica C 157, 571-574 (1989). 62O. K. Andersen, A. I. Liechtenstein, O. Jepsen, and F. Paulsen, LDA energy bands, low-energy Hamiltonians, t/prime, t/prime/prime,t⊥(k), andJ⊥, J. Phys. Chem. Solids 56, 1573-1591 (1995). 63N. P. Armitage et al. ,Doping dependence of an n-type cuprate superconductor investigated by angle-resolved pho- toemission spectroscopy , Phys. Rev. Lett. 88, 257001 (2002). 64T. Helm et al. ,Correlation between Fermi surface trans- formations and superconductivity in the electron-doped high-Tcsuperconductor Nd 2−xCexCuO 4, Phys. Rev. B 92, 094501 (2015). 65P. Li, F. F. Balakirev, and R. L. Greene, High-Field Hall Resistivity and Magnetoresistance of Electron-Doped Pr2−xCexCuO 4−δ, Phys. Rev. Lett. 99, 047003 (2007). 66S. E. Sebastian et al. ,Spin-Order Driven Fermi Surface Reconstruction Revealed by Quantum Oscillations in an Underdoped High TcSuperconductor , Phys. Rev. Lett. 103, 256405 (2009). 67M. M. Altarawneh et al. ,Superconducting Pairs with Ex- treme Uniaxial Anisotropy in URu 2Si2, Phys. Rev. Lett. 108, 066407 (2012). 68G. Bastien et al. ,Fermi-surface selective determination of theg-factor anisotropy in URu 2Si2, Phys. Rev. B 99, 165138 (2019). 69M. F. Hundley, J. D. Thompson, S-W. Cheong, Z. Fisk, and S. B. Oseroﬀ, Speciﬁc heat and anisotropic magnetic susceptibility of Pr 2CuO 4, Nd 2CuO 4and Sm 2CuO4 crys- tals, Physica C 158, 102-108 (1989). 70Y. Dalichaouch, M. C. de Andrade, and M. B. Maple, Synthesis, transport, and magnetic properties of Ln2−xCexCu0 4−ysingle crystals (Ln = Nd, Pr, Sm) , Physica C 218, 309-315 (1993). 71N. Harrison, D. W. Hall, R. G. Goodrich, J. J. Vuillemin, and Z. Fisk, Quantum Interference in the Spin-Polarized10 Heavy Fermion Compound CeB 6: Evidence for Topologi- cal Deformation of the Fermi Surface in Strong Magnetic Fields , Phys. Rev. Lett. 81, 870 (1998). 72A. McCollam et al. ,Spin-dependent masses and ﬁeld-induced quantum critical points , Physica B 359-361 , 1-8 (2005).1 SUPPLEMENTARY INFORMATION I. SPIN-REDUCTION FACTOR Rs(θ)IN A MAGNETIC FIELD PERPENDICULAR TO THE N´EEL AXIS For a quasi-2D metal, the phase ϕof the ﬁrst quantum- oscillation harmonic (i.e., of the fundamental frequency oscillations) is, up to a constant, ϕ=F B=¯h eF Bcosθ, whereθis the tilt angle between the ﬁeld and the nor- mal to the conducting plane, Fstands for the oscil- lation frequency at θ= 0◦, andFthe corresponding Fermi-surface area. The Zeeman eﬀect splits the spin- degenerate Fermi surface, breaking up FintoF+and F−, withδF=F+−F−∝B. Adding the two harmonic oscillations at close frequencies F+andF−is equivalent to a single oscillation at frequency F, with an amplitude modulated by the spin-reduction factor Rs(θ) = cos/bracketleftbigg¯h 2eδF Bcosθ/bracketrightbigg . (1) For the ﬁeld perpendicular to the N´ eel axis, B=B⊥, Eq. (1) of the main text yields the single-particle Hamil- tonian:H=E(k)−1 2µBg⊥(k)(B⊥·σ), whereE(k) is the zero-ﬁeld dispersion near the Fermi surface, and σ= (σx,σy,σz) is a vector composed of the three Pauli matrices. Upon turning on the ﬁeld, a given point kof the Fermi surface undergoes a small shift δksuch that δk·∇kE(k) =±1 2µBg⊥(k)B⊥, where the±signs cor- respond to the ‘up’ and ‘down’ spin projections on B⊥ and to the subscript of the resulting Fermi-surface areas F±. As shown in Fig. S1, upon the shift by δkan ele- mentdkof the Fermi surface contributes the shaded area dk(δk·ˆlk) =±1 2µBB⊥dkg⊥(k)/\|∇kE(k)\|to the variation of the total area, enclosed by the Fermi surface. Here, ˆlk=∇kE(k)/\|∇kE(k)\|is the local unit vector, normal to the Fermi surface. Therefore, to linear order in B⊥, the areasF±of the two spin-split Fermi surfaces diﬀer kδ dkl FIG. S1. Fermi surface of a two-dimensional conductor in zero ﬁeld (dashed line) and in a transverse ﬁeld B ⊥ (solid line). Upon turning on B⊥, a Fermi-surface element dkshifts by a small momentum δk(see the main text), adding the shaded trapezoid of the area dF=dk(δk·ˆlk) to the area enclosed by the Fermi surface, where ˆlk=∇kE(k)/\|∇kE(k)\| is the local unit vector, normal to the Fermi surface. The total variation of the Fermi-surface area is given by integrating dF over the Fermi surface, as explained in the main text.by δF=F+−F−=µBB⊥/contintegraldisplay FSdkg⊥(k) \|∇kE(k)\|, (2) where the line integral is taken along the zero-ﬁeld Fermi surface. It is convenient to introduce the transverse g- factor, averaged over the Fermi surface: ¯g⊥=/contintegraldisplay FS¯h2dk 2πmg⊥(k) \|∇kE(k)\|, (3) where m=1 2π/contintegraldisplay FS¯h2dk \|∇kE(k)\|(4) is the (θ= 0◦) cyclotron mass. Substituting µB=e¯h 2me into Eq. (2), we combine it with Eq. (1) to arrive at the expression for the spin-reduction factor: Rs(θ) = cos/bracketleftbiggπ cosθ¯g⊥m 2me/bracketrightbigg , (5) that is Eq. (2) of the main text. For a momentum- independent g⊥, Eq. (5) matches the textbook expres- sion1for the spin-reduction factor in two dimensions. II. SYMMETRY ANALYSIS OF THE N ´EEL STATES OF κ-(BETS) AND NCCO A. Symmetry analysis of the N´ eel state of κ-(BETS) 2FeBr 4in a transverse ﬁeld The existence of a special set of momenta in the Bril- louin zone (BZ), where Bloch eigenstates of a N´ eel an- tiferromagnet remain degenerate in transverse magnetic ﬁeld, is a general phenomenon. However, the precise geo- metry of this set depends on the interplay between the periodicity of the N´ eel order and the symmetry of the underlying crystal lattice2,3. Here, we describe this set forκ-(BETS) 2FeBr 4, hereafter referred to as κ-BETS. Upon transition from the paramagnetic to N´ eel state, the lattice period of κ-BETS along the caxis doubles, and the symmetry of the paramagnetic state with respect to both the time reversal ˆθand the elementary transla- tion ˆTcalong thecaxis is broken. Yet, the product ˆθˆTc remains a symmetry operation, along with the spin rota- tion ˆUn(φ) around the N´ eel axis nby an arbitrary angle φ. Applied transversely to n, a magnetic ﬁeld breaks the symmetry with respect to both ˆθˆTcandˆUn(φ); however, ˆUn(π)ˆθˆTcremains a symmetry operation2,3. It maps a Bloch eigenstate \|k/angbracketrightat wave vector konto a degenerate orthogonal eigenstate ˆUn(π)ˆθˆTc\|k/angbracketrightat wave vector−k, as shown in Fig. S2. Upon combination with reﬂection ˆRa: (kc,ka)→(kc,−ka), the resulting symmetry opera- tion ˆRaˆUn(π)ˆθˆTcmaps\|k/angbracketrightat wave vector k= (kc,ka)arXiv:1908.01236v2 [cond-mat.str-el] 21 Dec 20202 ka /aπ kc /cπ π/2c π/2cδ k Qg ( ) ck kc U ( ) T nπθckR U ( ) T a n πθck FIG. S2. Schematic view of the BZ of κ-BETS. The BZ in its paramagnetic state (dashed line) and in the N´ eel state with wave vector Q= (π/c,0) (solid line). The δpocket is centered at the corner k= (±π/2c,±π/a) of the magnetic BZ. The arrows show an exact Bloch eigenstate \|k/angbracketrightat a wave vector kon the vertical segment kc=π/2cof the magnetic BZ boundary – and its symmetry partners Un(π)θTc\|k/angbracketrightand RaUn(π)θTc\|k/angbracketright. The orthogonality /angbracketleftk\|RaUn(π)θTc\|k/angbracketright= 0 implies that g⊥(k) vanishes on the segment kc=π/2c. The inset illustrates the g⊥(k) being an odd function of kc−π/2c. onto a degenerate orthogonal eigenstate ˆRaˆUn(π)ˆθˆTc\|k/angbracketright at wave vector (−kc,ka)2,3. For an arbitrary k= (kc,ka) at the vertical segment kc=π/2cof the magnetic BZ boundary, the wave vectors (−kc,ka) and (kc,ka) diﬀer by the reciprocal wave vector Q= (π/c,0) of the N´ eel state; in the nomenclature of the magnetic BZ, they are one and the same vector. The degeneracy of such a \|k/angbracketrightwith ˆθˆTcˆRaˆUn(π)\|k/angbracketrightmeans thatg⊥(k) vanishes at the entire segment kc=π/2c. Theδpocket is centered at ( ±π/2c,±π/a) and is sym- metric with respect to reﬂection around the line kc= ±π/2c, as shown in Figs. 1 and S2. At the same time, as shown in Supplemental Material IV and illustrated in the insets of Figs. 1 and S2, g⊥(k) is odd under reﬂection around the same line. As a result, for the δpocket ¯g⊥in Eq. (3) vanishes, as stated in the main text. B. Symmetry analysis of the N´ eel state of Nd2−xCexCuO 4in a transverse ﬁeld In the antiferromagnetic state of Nd 2−xCexCuO 4 (hereafter NCCO), the Cu2+spins point along the lay- ers. At zero ﬁeld, they form a so-called non-collinear structure: the staggered magnetization vectors of adja- cent layers are normal to each other, pointing along the crystallographic directions [100] and [010], respectively (see Ref.4for a review). However, an in-plane ﬁeld above 5 T transforms this spin structure into a collinear one, with the staggered magnetization in all the layers aligned transversely to the ﬁeld. Therefore, in our experiment, π/a π/akQ ky kx kg ( )kxkx R U ( ) T ynθa π U ( ) T nπ θ akFIG. S3. Schematic view of the BZ of NCCO. The BZ in its paramagnetic state (dashed line) and in the N´ eel state with wave vector Q= (π/a,π/a ) (solid line). The carrier pocketsαare centered at the midpoints k= (±π/2a,±π/2a) of the magnetic BZ boundary. The blue arrows show an ex- act Bloch eigenstate \|k/angbracketrightat a wave vector kon the magnetic BZ boundary – and its symmetry partners Un(π)θTa\|k/angbracketrightand RyUn(π)θTa\|k/angbracketright. The orthogonality /angbracketleftk\|RyUn(π)θTa\|k/angbracketright= 0 implies that g⊥(k) vanishes on the segment kxa=±π/√ 2. The inset illustrates g⊥(k) being an odd function of kxa− π/√ 2. with the ﬁeld B > 45 T rotated around the [100] axis, the staggered magnetization is normal to the ﬁeld at all tilt angles except for a narrow interval 0◦<\|θ\|<∼5◦. Thus, we can restrict ourselves to the purely transverse-ﬁeld geometry, with the ﬁeld normal to the N´ eel axis, which makes the analysis similar to that forκ-BETS. The only diﬀerence is that, given the tetragonal symmetry of NCCO, the triple product ˆUn(π)ˆθˆTacan now be combined with reﬂections ˆRx: (kx,ky)→(−kx,ky) and ˆRy: (kx,ky)→(kx,−ky). As a result, for any wave vector kat the magnetic Brillouin-zone boundary, one ﬁnds /angbracketleftk\|ˆRyˆUn(π)ˆθˆTa\|k/angbracketright= /angbracketleftk\|ˆRxˆUn(π)ˆθˆTa\|k/angbracketright= 0. This guarantees double degen- eracy of Bloch eigenstates, hence the equality g⊥(k) = 0 at the entire boundary of the magnetic BZ, as shown in Fig. S3. The charge-carrier pockets of our interest are believed to be centered at ( ±π/2a,±π/2a), and are symmetric with respect to reﬂections RxandRyaround the kxand kyaxes, as shown in Figs. 4 and S3. At the same time, as shown in Section IVand illustrated in the inset of Figs. 4 and S3,g⊥(k) is odd under the very same reﬂections. As a result, ¯g⊥in Eq. (3) vanishes for these pockets, as stated in the main text.3 III. ESTIMATING THE PRODUCT kFξ A.δpocket in κ-BETS Looking only for a crude estimate, we assume a parabolic energy dispersion and treat the δpocket as circular of radius kFand areaFδ= 2πeFδ/¯h. Deﬁning the antiferromagnetic coherence length as ξ= ¯hvF/∆AF, we ﬁnd: kFξ= ¯hkFvF/∆AF= 2εF/∆AF. (6) The Fermi energy εFin Eq. (6) can be expressed via the Shubnikov-de Haas (SdH) frequency Fδ= 61 T: εF=¯h2k2 F 2m=¯h2Fδ 2πm=¯heFδ m≈6 meV. (7) Assuming a BSC-like relation between the N´ eel tem- perature,TN≈2.5 K, and the antiferromagnetic gap ∆AFin the electron spectrum, we evaluate the latter as ∆ AF/similarequal1.8kBTN≈0.4 meV. A similar estimate is obtained from the critical ﬁeld, Bc≈5 T, required to suppress the N´ eel state: ∆ AF∼µBBc≈0.3 meV. Thus, we ﬁnd kFξ/similarequal2εF ∆AF∼30−40/greatermuch1, which means thatg⊥(k) is nearly constant over most of the Fermi sur- face, except in a small vicinity of kc=π/2c, where it changes sign, cf. Figs. S2 and S4. B. Small hole pocket of the reconstructed Fermi surface in NCCO In NCCO, the small Fermi-surface pocket α, responsi- ble for the observed oscillations, is far from being circu- lar. Therefore, we can no longer estimate kFξthe same way as we did for the δpocket inκ-BETS. Instead, we evaluate the relevant Fermi wave vector and the antifer- romagnetic coherence length separately. The value of the Fermi wave vector in the direction normal to the magnetic BZ boundary can be found from ARPES maps of the Fermi surface5–7:kF= 0.4± 0.1 nm−1. The coherence length can be estimated using the MB gap value, ∆ AF≈16 meV, and parameters of the (ap- proximately circular) large parent Fermi surface obtained from the analysis of MB quantum oscillations8. Using the corresponding SdH frequency F= 11.25 kT and cy- clotron mass mc= 3.0me, we estimate the Fermi velocity, vF∼¯hkF/mc≈√ 2¯heF/mc≈2.2×105m/s, which leads to the coherence length ξ∼¯hvF/∆AF≈9 nm. This yields the product kFξ∼3−5, which implies g⊥(k) being piecewise nearly constant over most of the Fermi surface, except in a small vicinity of the magnetic BZ boundary, where g⊥(k) changes sign, cf. Figs. S3 and S4.IV. SYMMETRY PROPERTIES OF g⊥(k) In Section II A, we have shown that in κ-BETS the factorg⊥(k) vanishes at the entire kc=±π/2csegment of the magnetic BZ boundary. Here, we establish an im- portant general symmetry property of g⊥(k). In the case ofκ-BETS, this property implies that g⊥(k) is an odd function of kc−π/2c. Theδpocket, responsible for the observed SdH oscillations, is centered on this segment, at the corner of the magnetic BZ (see Fig. 1). As a re- sult, for this pocket the “eﬀective g-factor” ¯g⊥in Eq. (3) vanishes by symmetry. Without loss of generality, we consider the simplest case of double commensurability, relevant to both ma- terials of our interest. In both of them, the underlying non-magnetic state is centrosymmetric, with the relevant electron band having the spectrum ε(k). Spontaneous N´ eel magnetization with wave vector Qinteracts with the conduction-electron spin σvia the exchange term (∆AF·σ), coupling the states at wave vectors kand k+Q. In the N´ eel phase, subjected to magnetic ﬁeld B, the electron Hamiltonian takes the form9 Hk=/bracketleftbigg ε(k)−g(B·σ) ( ∆AF·σ) (∆AF·σ)ε(k+Q)−g(B·σ)/bracketrightbigg ,(8) where the factor µB/2 has been absorbed into the deﬁ- nition of B, and ∆AF=JSis the product of the an- tiferromagnetically ordered moment Sand its exchange couplingJto the conduction electrons. In a purely trans- verse ﬁeld B⊥⊥∆AF, the Hamiltonian (8) can be easily diagonalized2,10to yield the spectrum E(k) =ε+(k)±/radicalBig ∆2 AF+ [ε−(k)−g(B⊥·σ)]2,(9) /s45/s49 /s107/s103 /s40/s107 /s41 FIG. S4. Theg-factor for magnetic ﬁeld normal to the N´ eel axis. Schematic plot of g⊥(k) as a function of the momentum component k, normal to the line g⊥(k) = 0. At smallk < 1/ξ, the function g⊥(k) is linear: g⊥(k)≈gξk. Beyondk≈1/ξ,g⊥(k) is nearly constant: g⊥(k)≈g. Here, ξ= ¯hvF/∆AFis the antiferromagnetic coherence length, and ∆AFis the energy gap in the electron spectrum (9) of the N´ eel state.4 whereε±(k) =1 2[ε(k)±ε(k+Q)]. Equation (9) shows that ∆ AFis the energy gap in the electron spectrum of the N´ eel state. From Eq. (9), one easily ﬁnds the eﬀective transverseg-factorg⊥(k)2,11 g⊥(k) =gε−(k)/radicalBig ∆2 AF+ε2 −(k), (10) plotted in Fig. S4 as a function of momentum component k, normal to the line g⊥(k) = 0. The parent paramagnetic state is invariant under time reversal, thus ε(k) =ε(−k). Also, in a doubly- commensurate antiferromagnet with N´ eel wave vector Q, the wave vector 2 Qis a reciprocal lattice vector of the underlying non-magnetic state; thus, ε(k+ 2Q) =ε(k). From these properties, it follows that E(k) =E(−k+Q) andg⊥(k) =−g⊥(−k+Q)2. We will now show how this symmetry property leads to ¯ g⊥= 0. In NCCO as well as in the N´ eel state of κ-BETS, the relevant Fermi surface consists of two symmetric parts, which map onto each other under transformation k→−k+Q. Contributions of these two parts to the integral in the right-hand side of Eq. (2) cancel each other exactly; hence, Eq. (3) yields ¯g⊥= 0. In other words, Eq. (2) yields δF= 0, and thus, in Eqs. (2) and (3) one ﬁnds Rs(θ) = 1: in a transverse ﬁeld, the amplitude of magnetic quantum oscillations has no spin zeros. The arguments above rely on a quasi-classical descrip- tion. Note that the key conclusion, the absence of spin zeros in a transverse ﬁeld, holds regardless of how the Fermi wave vector kFcompares with the inverse antifer- romagnetic coherence length 1 /ξ∼∆AF/¯hvF, where the behavior of g⊥(k) crosses over from linear to constant as illustrated in Fig. S4. In the limit of kFξ<∼1, the problem can be analyzed by reducing the Hamiltonian to the leading terms of its mo- mentum expansion around the band extremum. The con- clusion remains intact: in a purely transverse ﬁeld, the Zeeman term of Eq. (1) does notlift the spin degeneracy of Landau levels12; hence, the quantum-oscillation am- plitude has no spin zeros. The present work extends the validity range of this result from a small Fermi-surface pocket to an arbitrarily large Fermi surface. V. QUANTUM OSCILLATIONS IN CeIn3 As we pointed out in the main text, for certain Fermi surfaces Zeeman spin-orbit coupling may not manifest it- self in quantum oscillations: this is sensitive to where the Fermi surface is centered. An illustrative exam- ple is provided by CeIn 3, a heavy-fermion compound of simple-cubic Cu 3Au structure, with a moderately en- hanced Sommerfeld coeﬃcient, γ= 130 mJ/K2mol. Be- lowTN≈10 K, it develops a type-II antiferromagnetic structure with wave vector Q= (π a,π a,π a) and an ordered moment of about (0.65 ±0.1)µBper Ce atom13, shown in Fig. S5(a). The material remains a metallic down Ce (a) L /CapSigma WX /LParen1b/RParen1FIG. S5. Geometry of CeIn 3in real and reciprocal space. (a) Cubic unit cell of CeIn 3in its N´ eel state, show- ing Ce atoms and their magnetic moments13. Indium atoms (not shown) are located at the face centers of the unit cell. (b) Cubic BZ of paramagnetic CeIn 3shown by dashed lines and, inside, the magnetic BZ. Hexagonal faces (dark gray) form the reciprocal-space surface of the symmetry-protected degeneracy g⊥(k) = 0 [2]. [H] FIG. S6. Schematic view of the dsheet of CeIn 3in the N´ eel state. We only show a single pair of necks for clarity. The outer sheet illustrates the Fermi-surface reconstruction with broadened necks, the inner sheet shows a reconstruction with the necks truncated. The red lines represent two cy- clotron trajectories: a smaller cyclotron-mass trajectory in a magnetic ﬁeld pointing along ΓL, and a larger cyclotron-mass trajectory, approaching the neck of the Fermi surface. to the lowest temperatures. The BZs in the paramag- netic and in the N´ eel state are shown in Fig. S5(b). In a transverse magnetic ﬁeld, anti-unitary symmetry pro- tects double degeneracy (and thus the equality g⊥(k) = 0) on hexagonal faces of the magnetic BZ in Fig. S5(b)2. We will focus on the dbranch of the Fermi surface of CeIn 3, a nearly spherical sheet centered at the zone center Γ, with a radius close to√ 3 2π a, whereais the lattice constant. In the paramagnetic state, this sheet has necks protruding out near the Lpoints in Fig. S5(b), connecting it to another sheet centered at the corner of5 the paramagnetic BZ14,15. Thedsheet has been studied in detail by quantum oscillations, which revealed a large enhancement of the cyclotron mass upon tilting the ﬁeld, from m≈2m0for cyclotron trajectories passing far from the necks (e.g., for magnetic ﬁeld B/bardbl/angbracketleft100/angbracketright) tom> 12m0for trajectories approaching the necks (such as B/bardbl/angbracketleft110/angbracketright)16,17. This mass enhancement can be explained18by the Fermi-surface ge- ometry, illustrated in Fig. S6: approaching a neck, the cyclotron trajectory inevitably runs into a saddle point with its concomitant logarithmic enhancement of the cy- clotron mass. Upon transition to the N´ eel state, the Fermi surface undergoes a reconstruction by folding into the magnetic BZ. Depending on the neck size and on the value of the N´ eel gap in the electron spectrum, the reconstructed sheets may have their necks broadened or truncated al- together18, as shown in Fig. S6. The cyclotron-mass enhancement for those ﬁeld orien- tations, for which the quasiclassical trajectories approach a neck16,17is consistent with the Fermi sheet having necks in the N´ eel state. So is the observation of spin zeros16: the enhancement of m/m 0alone produces spin zeros as the cyclotron trajectory approaches a neck with tilting the ﬁeld. Crucially for the Zeeman eﬀect that we are interested in, the average of g⊥(k) over a cyclotron trajectory on the dsheet does notvanish – simply be- cause most (if not all) of the cyclotron trajectory in ques- tion lies within the ﬁrst magnetic BZ, and thus g⊥(k) keeps its sign over most (or all) of the trajectory. Indeed, being centered at the Γ point, the dsheet cannot possi- bly be symmetric with respect to the surface g⊥(k) = 0 in Fig. S5(b), and thus our symmetry argument for the vanishing ¯g⊥inevitably breaks down. The observation of spin zeros16in CeIn 3is thus perfectly consistent with the physics of the Zeeman spin-orbit coupling described in the main text. VI. SdH OSCILLATIONS IN THE HIGH-FIELD PARAMAGNETIC STATE OF κ-BETS The SdH oscillations in the high-ﬁeld, paramagnetic (PM) state of κ-BETS have been described in detail in a number of publications19–23, revealing a Fermi sur- face largely consistent with that obtained from band- structure calculations24. Here, we give a brief overview, referring to our own data, which show perfect agreement with the previous reports. The measurements were per- formed on the same crystal as discussed in the main text. Figure 1(b) in the main text shows an example of the low-temperature interlayer resistance measured in a mag- netic ﬁeld up to 14 T, directed nearly perpendicularly to the conducting layers. The kink around Bc≈5.2 T re- ﬂects the transition from the low-ﬁeld AF to the high- ﬁeld PM state. The slow SdH oscillations associated with the small pocket δ(shown in Fig. 1c) of the re- constructed FS in the AF state collapse at Bc. In thePM state, the oscillation spectrum is composed of two fundamental frequencies Fα≈840 T andFβ≈4200 T and their combinations, as shown in the inset in Fig.1(b). The dominant contribution comes from the αoscillations associated with the closed portion of the Fermi surface centered on the BZ boundary shown in Fig. 1(a). The β oscillations, which are considerably weaker than the αos- cillations, originate from the large magnetic-breakdown (MB) orbit enveloping the whole Fermi surface [dashed orange line in Fig. 1(b)]. Additionally, there are sizable contributions of the second harmonic of the αoscillations and combination frequencies, β−pαwithp= 1,2, in the SdH spectrum, which are often observed in clean, highly two-dimensional organic metals in the intermediate MB regime25. The oscillation amplitude shows spin-zeros not only in the angular dependence but also with changing the ﬁeld strength at a ﬁxed orientation. This is due to the presence of an exchange ﬁeld BJimposed on the con- duction electrons by saturated paramagnetic Fe3+ions, although some details of its behavior are still to be under- stood1,23,26. The analysis of the spin-zero positions in the PM state yields the g-factor very close to that of free elec- trons,g= 2.0±0.2 and the exchange ﬁeld BJ≈−13 T pointing against the external magnetic ﬁeld23. Note that both αandβoscillations are rapidly sup- pressed with lowering the ﬁeld and cannot be resolved near the transition ﬁeld Bc. This rapid suppression is due to the relatively high cyclotron masses: compared to theδ-oscillation mass, the αandβmasses are some 5 and 7 times higher, respectively. The exponential dependence of the SdH amplitude on the cyclotron mass, see Eq. (3) of the main text, renders the amplitude factor for the α oscillations approximately 2 orders of magnitude smaller than that for the slow δoscillations at ﬁelds near Bc. In addition to the higher cyclotron mass, the MB gap ∆ 0 contributes to the damping of the βoscillations via the MB damping factor. This is why the βoscillations are only seen at highest ﬁelds, ≥10 T, as weak distortions of theα-oscillation wave form and as a small peak in the FFT spectrum. VII. DETAILS OF THE SdH FIT FOR κ-BETS IN THE AF STATE In the main text, we noted a large uncertainty of B0 andBAF. Here, we show that the quality of our SdH amplitude ﬁts is insensitive to the exact values of B0and BAF. Equation (3) for the amplitude Aδcontains the MB factor R[δ] MB=/bracketleftbigg 1−exp/parenleftbigg −B0 Bcosθ/parenrightbigg/bracketrightbigg/bracketleftbigg 1−exp/parenleftbigg −BAF Bcosθ/parenrightbigg/bracketrightbigg , (11) which must be taken into account when analyzing the angular dependence Aδ(θ). A rough estimate of B0can be obtained from the ratio between the α- andβ-oscillation amplitudes at a certain ﬁeld and temperature, using the LK formula6 [Eq. (3) of the main text] with the MB factors R[α] MB= [1−exp (−B0/B)] andR[β] MB= exp (−2B0/B) for the αandβoscillations, respectively. From the data in Fig. 1(b) we estimate the ratio between the FFT am- plitudes of the αandβoscillations in the ﬁeld interval 10 to 14 T, at T= 0.5 K, asAβ/Aα≈0.03. To com- plete the estimation, we also need to know the cyclotron masses, which have been determined in earlier experi- ments:mα= 5.2m0andmβ= 7.9m020, and the Dingle temperature TD. The latter arises from scattering on crystal imperfections1and, therefore, varies from sample to sample. As shown below, for the present crystal a reasonable estimate of the Dingle temperature in the AF state,TD/similarequal0.7 K, can be obtained from the angular de- pendence of the δ-oscillation amplitude. Assuming that TDis momentum-independent and remains the same in the PM state, we substitute this value in the LK formula, arriving at the MB ﬁeld value B0≈12 T. This is three times higher than the upper end of the ﬁeld interval used for the SdH oscillation analysis in the AF state. There- fore, the ﬁrst factor in the right-hand side of Eq. (11) is close to unity and does not contribute signiﬁcantly to the angular dependence Aδ(θ)27. To conﬁrm this, we have checked how our ﬁts are aﬀected by varying B0in the range between 5 T and 50 T, as will be presented below. The MB ﬁeld BAFis due to magnetic ordering and can be estimated from the gap ∆ AFwith the help of the Blount criterion1,28, BAF∼mc ¯he·∆2 AF/εF/similarequal0.15 T. (12) Here, we estimated the Fermi energy εFfrom SdH os- cillations in the paramagnetic state: εF∼¯h2k2 F/2m∼ ¯heFβ/mc,β, with the SdH frequency Fβ= 4280 T and corresponding cyclotron mass mc,β= 7.9m020. Of course, these are only rough estimates. Moreover, the observation of the δoscillations in ﬁelds up to Bc/similarequal 5 T implies that the relevant MB ﬁeld must be in the range of a few tesla, to provide a non-vanishing second factor in the right-hand side of Eq. (11). On the other hand, the MB ﬁeld cannot be much higher than the ﬁelds we applied ( B<∼Bc). We have tentatively set the upper estimate for BAFto about 5 T and checked how our ﬁts are aﬀected by varying BAFfrom 0.15 T to 5 T. The results are summarized in Fig. S7, which presents several ﬁts to the experimental data for κ-BETS (the same as in Fig. 3 of the main text), with values of B0 andBAFvarying in a broad range: 5 T ≤B0≤50 T and 0.15 T≤BAF≤5 T. All the ﬁts assume g⊥= 0; as shown in the main text, a ﬁnite value for g⊥would simply lead to sharp spin zeros, insensitive to the monotonic θdependence of RMB. One can clearly see that the ﬁts are nearly indistinguishable and virtually insensitive to variation of B0within the given range, whereas the variation of BAFbarely results in a 15% change of the Dingle temperature: TD= (0.69±0.05) K. The parameter A0in Eq. (3) changes roughly in inverse /s45/s56/s48 /s45/s54/s48 /s45/s52/s48 /s45/s50/s48 /s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48/s48/s46/s48/s49/s48/s46/s49/s49/s49/s48 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s70/s105/s116/s115/s32/s119/s105/s116/s104/s32/s103 /s32/s61/s32/s48/s58 /s32/s32 /s66 /s65/s70/s32/s61/s32/s53/s32/s84/s44/s32 /s32/s32/s32/s32/s32/s66 /s48/s32/s61/s32/s32/s53/s32/s84/s59/s32/s32/s32 /s84 /s68/s32/s61/s32/s48/s46/s54/s57/s50/s32 /s32/s48/s46/s48/s49/s56/s32/s75 /s32/s32 /s66 /s65/s70/s32/s61/s32/s53/s32/s84/s44/s32/s32/s32/s32/s32/s32 /s66 /s48/s32/s61/s32/s50/s48/s32/s84/s59/s32/s32 /s84 /s68/s32/s61/s32/s48/s46/s54/s51/s56/s32 /s32/s48/s46/s48/s49/s54/s32/s75 /s32/s32 /s66 /s65/s70/s32/s61/s32/s53/s32/s84/s44/s32/s32/s32/s32/s32/s32 /s66 /s48/s32/s61/s32/s53/s48/s32/s84/s59/s32/s32 /s84 /s68/s32/s61/s32/s48/s46/s54/s51/s55/s32 /s32/s48/s46/s48/s49/s54/s32/s75 /s32/s32 /s66 /s65/s70/s32/s61/s32/s48/s46/s49/s53/s32/s84/s44/s32 /s66 /s48/s32/s61/s32/s50/s48/s32/s84/s59/s32/s32 /s84 /s68/s32/s61/s32/s48/s46/s55/s52/s48/s32 /s32/s48/s46/s48/s49/s55/s32/s75 /s32/s32 /s66 /s65/s70/s32/s61/s32/s48/s46/s49/s53/s32/s84/s44/s32 /s66 /s48/s32/s61/s32/s53/s48/s32/s84/s59/s32/s32 /s84 /s68/s32/s61/s32/s48/s46/s55/s51/s57/s32 /s32/s48/s46/s48/s49/s55/s32/s75/s65 /s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41 /s32/s40/s100/s101/s103/s46/s41/s32/s69/s120/s112/s101/s114/s105/s109/s101/s110/s116/s97/s108/s32/s100/s97/s116/s97FIG. S7. Angle-dependent δoscillation amplitude in κ- BETS compared to theoretical ﬁts. Black dots are the experimental data and lines are ﬁts using Eq. (3) of the main text with diﬀerent ﬁxed values of the MB ﬁelds B0andBAF. The angle-independent Dingle temperature TDand amplitude prefactorA0are the ﬁtting parameters and g⊥is set to zero. The plot demonstrates insensitivity of the angular dependence on the concrete choice of B0andBAF. proportion to BAF. However, A0is largely an empirical parameter, irrelevant to our study. Thus, we conclude that the mentioned uncertainty of the MB ﬁelds has no eﬀect on the quality of our ﬁts, as stated in the main text. VIII. AMPLITUDE ANALYSIS FOR NCCO In our experiment on NCCO we were restricted to a rel- atively narrow ﬁeld interval, between 45 and 64 T where the oscillations were observable over the whole angular range. Due to the low frequency, this interval contains only a few oscillation periods, see Fig. S8 and Fig. 4 in the main text. Under these conditions the FFT spectrum is sensitive to the choice of the ﬁeld window and details of the background subtraction. The choice of the upper bound of the ﬁeld window is most important here: this af- fects the largest-amplitude oscillation and thus gives the dominant contribution to the overall error. In the course of our analysis we always ensured that the order of the polynomial ﬁt (≤4) to the non-oscillating B-dependent background did not aﬀect the height of the FFT peak cor- responding to the SdH oscillations, within our resolution. The main source of error is the variation of the oscillation phase at the (ﬁxed) upper end of the ﬁeld window caused by the changing frequency F(θ): this disturbs the shape and amplitude of the largest oscillation. In order to re- duce this parasitic eﬀect, we kept the FFT window more narrow than that of the background ﬁt; the upper Bbor- der of the FFT window was at least 1 T lower than that of the background ﬁt. In our analysis we found that the7 30 40 50 60 70-0.010.000.010.020.030.04Rosc/Rbackg B (T)0°50° 37° 20°54°54.5°55°(a) (b) (c) 40 50 60 70-20246810Rosc/Rbackg (´10-3) B (T)x10 x3 57.8°58.2°60.1°67.2°70.7° 64.2° 62.8° -10 010 20 30 40 50 60 70 800.010.1110\|Aa\| (arb. unit) q (degree) 52.8 T 55 T 60 T AFFT [45 - 64] T LK Fit FIG. S8. Amplitude analysis of the angle-dependent αoscillations in NCCO. (a) and (b) oscillating resistance component, normalized to the non-oscillating B-dependent background, plotted as a function of magnetic ﬁeld. The curves corresponding to diﬀerent tilt angles θare vertically shifted for clarity. For θ= 67.2◦and 70.7◦the ratioRosc/Rbackg is multiplied by a factor of 3 and 10, respectively. The vertical dashed lines indicate the ﬁeld values at which we determine A(θ) shown in (c). (c) Normalized peak-to-peak oscillation amplitude determined for B= 52.8 T, 55 T, and 60 T in comparison to the amplitude determined by FFT for B= [45−64] T. Red dashed line is the ﬁt with g= 0, see Fig. 5 in the main text. associated error bar for the height of the FFT peak did not exceed±10 %. While this uncertainty may be impor- tant, for example, for an exact evaluation of the eﬀective cyclotron mass, in our case, when the oscillation ampli- tude is expected to change an order of magnitude near a spin-zero, it is not signiﬁcant. Only at highest angles, the error becomes comparable to the signal, which is related to the low signal-to-noise ratio at these conditions. To crosscheck our FFT analysis we present an alterna- tive amplitude analysis in Fig. S8. For each Rosc/Rbackg curve, we determine the peak-to-peak amplitude from the linearly interpolated envelopes (dashed black lines in Fig. S8a,b) at a ﬁxed ﬁeld value. We chose the middle of the FFT window in 1 /Bscale, i.e., B= [(1/45 + 1/64)/2]−1T= 52.8 T, as well as the values 55 T and 60 T, with oscillations discernible up to 67 °. Fig. S8c presents these points plotted versus the tilt angle θand compares them to the FFT values and LK ﬁt from Fig. 3shown in the main text. The extracted data from both approaches match very well. Clearly, our main conclusion holds, that is, the overall angular dependence exhibits no indication of a spin-zero eﬀect for angles of up to at least 70°. IX. ESTIMATING THE DISPARITY OF SPIN-UP AND SPIN-DOWN TUNNELING AMPLITUDES IN NCCO The Nd3+spins in the insulating layers of NCCO are polarized by strong magnetic ﬁeld. Consequently one may wonder whether this spin polarization could ren- der interlayer tunneling amplitudes for spin-up and spin- down diﬀerent enough to lose spin zeros in the c-axis magnetoresistance oscillations. In the following we will show that this possibility can be ruled out. The ratio of interlayer the conductivities σ↑andσ↓can be crudely estimated via the tunneling amplitudes w↑andw↓as σ↑ σ↓≈w2 ↑ w2 ↓= exp/parenleftbigg −2 ¯h/integraldisplay dz/bracketleftBig/radicalbig 2m(U(z)−EF−EZ/2)−/radicalbig 2m(U(z)−EF+EZ/2)/bracketrightBig/parenrightbigg , whereU(z) is the tunneling potential and EZthe Zee- man splitting, responsible for the diﬀerence between σ↑andσ↓. For a rough estimate, it suﬃces to re- placeU(z) by a rectangular potential barrier of spa- tial widthd, the unperturbed tunneling amplitude beingw= exp/parenleftBig −d/radicalbig 2m(U(z)−EF)/¯h/parenrightBig . The expression in the exponent above can then be expanded in small EZ to yield σ↑ σ↓≈exp/parenleftBigg EZ U−EF/radicalbig 2m(U−EF)d ¯h/parenrightBigg = exp/bracketleftbigg −EZ U−EFlnw/bracketrightbigg .8 To put in the numbers, notice that the deviation of the experimental data in Fig. 5 of the main text from the theoretical ﬁt with Rs= 1 does not exceed 20%. Assum- ing this deviation to be entirely due to an interference of unequal spin-up and spin-down oscillation amplitudes would imply σ↑/σ↓∼20. Let us estimate the EZre- quired for such a behavior. Recall that wis essentially the ratio of a typical interlayer hopping amplitude tz∼10−2eV to the in-plane Fermi scale U−EF∼1 eV, and thus lnw≈−4.6. The ratio σ↑/σ↓∼20 would then mean EZ>∼0.6 eV: an unrealistic bound, which allows us to rule out this scenario. REFERENCES 1D. Shoenberg, Magnetic Oscillations in Metals (Cambridge University Press, Cambridge, 1984). 2R. Ramazashvili, Phys. Rev. B 79, 184432 (2009). 3R. Ramazashvili, Phys. Rev. Lett. 101, 137202 (2008). 4N. P. Armitage, P. Fournier, and R. L. Greene, Rev. Mod. Phys. 82, 2421-2487 (2010). 5J.-F. He, C. R. Rotundu, M. S. Scheurer, Y. He, M. Hashimoto, K. Xu, Y. Wang, E. W. Huang, T. Jia, S.-D. Chen, B. Moritz, D.-H. Lu, Y. S. Lee, T. P. Dev- ereaux, and Z.-X. Shen, Proc. Natl. Acad. Sci. USA 116, 3449-3453 (2019). 6N. P. Armitage, F. Ronning, D. H. Lu, C. Kim, A. Dam- ascelli, K. M. Shen, D. L. Feng, H. Eisaki, Z.-X. Shen, P. K. Mang, N. Kaneko, M. Greven, Y. Onose, Y. Taguchi, and Y. Tokura, Phys. Rev. Lett. 88, 257001 (2002). 7H. Matsui, T. Takahashi, T. Sato, K. Terashima, H. Ding, T. Uefuji, and K. Yamada, Phys. Rev. B 75, 224514 (2007). 8T. Helm, M. V. Kartsovnik, C. Proust, B. Vignolle, C. Putzke, E. Kampert, I. Sheikin, E.-S. Choi, J. S. Brooks, N. Bittner, W. Biberacher, A. Erb, J. Wosnitza, and R. Gross, Phys. Rev. B 92, 094501 (2015). 9N. I. Kulikov and V. V. Tugushev, Sov. Phys. Usp. 27, 954-976 (1984). 10S. A. Brazovskii, I. A. Luk’yanchuk, and R. R. Ramaza- shvili, JETP Lett. 49, 644-646 (1989). 11V. V. Kabanov and A. S. Alexandrov, Phys. Rev. B 77, 132403 (2008); 81, 099907(E) (2010) 12R. Ramazashvili, Phys. Rev. B 80, 054405 (2009). 13J. M. Lawrence and S. M. Shapiro, Phys. Rev. B 22, 4379- 4388 (1980). 14M. Biasini, G. Ferro, and A. Czopnik, Phys. Rev. B 68, 094513 (2003). 15J. Rusz and M. Biasini, Phys. Rev. B 71, 233103 (2005). 16R. Settai, T. Ebihara, M. Takashita, H. Sugawara, N. Kimura, K. Motoki, Y. Onuki, S. Uji, and H. Aokii,J. Magn. Magn. Mat. 140-144 , 1153-1154 (1995). 17T. Ebihara, N. Harrison, M. Jaime, S. Uji, and J. C. Lash- ley, Phys. Rev. Lett. 93, 246401 (2004). 18L. P. Gor’kov and P. D. Grigoriev, Phys. Rev. B 73, 060401 (R) (2006). 19L. Balicas, J. S. Brooks, K. Storr, D. Graf, S. Uji, H. Shina- gawa, E. Ojima, H. Fujiwara, H. Kobayashi, A. Kobayashi, and M. Tokumoto, Solid State Commun. 116, 557-562 (2000). 20S. Uji, H. Shinagawa, Y. Terai, T. Yakabe, C. Terakura, T. Terashima, L. Balicas, J. S. Brooks, E. Ojima, H. Fu- jiwara, H. Kobayashi, A. Kobayashi, and M. Tokumoto, Physica B 298, 557-561 (2001). 21T. Konoike, S. Uji, T. Terashima, M. Nishimura, S. Ya- suzuka, K. Enomoto, H. Fujiwara, E. Fujiwara, B. Zhang, and H. Kobayashi, Phys. Rev. B 72, 094517 (2005). 22T. Konoike, S. Uji, T. Terashima, M. Nishimura, T. Ya- maguchi, K. Enomoto, H. Fujiwara, B. Zhang, and H. Kobayashi, J. Low Temp. Phys. 142, 531-534 (2006). 23M. V. Kartsovnik, M. Kunz, L. Schaidhammer, F. Koll- mannsberger, W. Biberacher, N. D. Kushch, A. Miyazaki, and H. Fujiwara, J. Supercond. Nov. Magn. 29, 3075-3080 (2016). 24H. Fujiwara, E. Fujiwara, Y. Nakazawa, B. Zh. Narymbe- tov, K. Kato, H. Kobayashi, A. Kobayashi, M. Tokumoto, and P. Cassoux, J. Am. Chem. Soc. 123, 306-314 (2001). 25M. V. Kartsovnik, Chem. Rev. 104, 5737-5782 (2004). 26O. C´ epas, R. H. McKenzie, and J. Merino, Phys. Rev. B 65, 100502 (2002). 27One may expect that TDin the PM state is somewhat lower than the value TD≈0.7 K obtained for the AF state, due to the absence of scattering on defects of the AF order (e.g., domain walls). If so, the actual MB ﬁeld is even higher than 12 T, hence the corresponding MB factor is even closer to unity. 28E. I. Blount, Phys. Rev. 126, 1636-1653 (1962).
1409.5600v1.Angular_dependence_of_spin_orbit_spin_transfer_torques.pdf	arXiv:1409.5600v1 [cond-mat.mtrl-sci] 19 Sep 2014Angular dependence of spin-orbit spin transfer torques Ki-Seung Lee1,∗∗, Dongwook Go2,∗∗, Aur´ elien Manchon3, Paul M. Haney4, M. D. Stiles4, Hyun-Woo Lee2,∗and Kyung-Jin Lee1,5† 1Department of Materials Science and Engineering, Korea Uni versity, Seoul 136-701, Korea 2PCTP and Department of Physics, Pohang University of Scienc e and Technology, Pohang 790-784, Korea 3Physical Science and Engineering Division, King Abdullah U niversity of Science and Technology (KAUST), Thuwal 23955-6900, Saud i Arabia 4Center for Nanoscale Science and Technology, National Inst itute of Standards and Technology, Gaithersburg, Maryland 20899 , USA 5KU-KIST Graduate School of Converging Science and Technolo gy, Korea University, Seoul 136-713, Korea In ferromagnet/heavy metal bilayers, an in-plane current g ives rise to spin-orbit spin transfer torque which is usually decomposed into ﬁeld-like and dampi ng-like torques. For two-dimensional free-electron and tight-bindingmodels with Rashba spin-o rbit coupling, the ﬁeld-like torque acquires nontrivial dependence on the magnetization direction when the Rashba spin-orbit coupling becomes comparable to the exchange interaction. This nontrivial an gular dependence of the ﬁeld-like torque is related to the Fermi surface distortion, determined by th e ratio of the Rashba spin-orbit coupling to the exchange interaction. On the other hand, the damping- like torque acquires nontrivial angular dependence when the Rashba spin-orbit coupling is comparab le to or stronger than the exchange interaction. It is related to the combined eﬀects of the Ferm i surface distortion and the Fermi sea contribution. The angular dependence is consistent wit h experimental observations and can be important to understand magnetization dynamics induced by spin-orbit spin transfer torques. I. INTRODUCTION In-plane current-induced spin-orbit spin transfer torques in ferromagnet/heavy metal bilayers provide an eﬃcient way of inducing magnetization dynamics and may play a role in future magnetoelectronic devices.1–14 Two mechanisms for spin-orbit torques have been pro- posed to date; the bulk spin Hall eﬀect in the heavy metal layer15–19and interfacial spin-orbit coupling eﬀect at the ferromagnet/heavymetal interface20–30frequently referred to as the Rashba eﬀect. Substantial eﬀorts have been expended in identifying the dominant mechanism forthespin-orbittorque.29–34Forthispurpose, oneneeds to go beyond qualitative analysis since both the mecha- nismsresultinqualitativelyidenticalpredictions,i.e. two vector components of spin-orbit torque (see Eq. (1)). For quantitative analysis, we adopt the commonly used de- composition of the spin-orbit torque T, T=τfˆM×ˆy+τdˆM×(ˆM×ˆy), (1) where the ﬁrst term is commonly called the ﬁeld-like spin-orbit torque, the second term the damping-like spin- orbit torque or the Slonczewski-like spin-orbit torque, ˆM= (cosφsinθ,sinφsinθ,cosθ) is the unit vector along the magnetization direction, ˆyis the unit vector perpen- dicular to both current direction ( ˆx) and the direction in which the inversion symmetry is broken ( ˆz),τfand τddescribe the magnitude of ﬁeld-like and damping-like spin-orbit torque terms, respectively. Since Tshould be orthogonal to ˆM, the two terms in Eq. (1), which are or- thogonal to ˆMand also to each other, provide a perfectly general description of the spin-orbit torque regardless of the detailed mechanism of T. The quantitative analysis ofTthen amounts to the examination of the propertiesofτfandτd. Oneofintriguingfeaturesofspin-orbittorqueobserved in some experiments is the strong dependence of τfand τdon the magnetization direction.35,36Comparing the measured and calculated angular dependence will pro- videcluestothemechanismofthespin-orbittorque. The detailed angular dependence also determines the magne- tization dynamics and hence is important for device ap- plications based on magnetization switching1,3,10–13, do- main wall dynamics2,5–9,14, and magnetic skyrmion mo- tion37. Theories based on the bulk spin Hall eﬀect com- bined with a drift-diﬀusion model or Boltzmann trans- port equation29predict no angular dependence of τfand τd, which is not consistent with the experimental re- sults.35,36For theories based on the interfacial spin-orbit coupling, the angular dependence is subtle. Based on the Rashba model including D’yakonov-Perel spin relax- ation, Pauyac et al.38studied the angular dependence of spin-orbit torque perturbatively in weak Rashba regime (r≡αRkF/J≪1) and strong Rashba regime ( r≫1) whereαRis the strength of the Rashba spin-orbit cou- pling,kFis the Fermi wave vector, and Jis the exchange coupling. They found that both τfandτdare almost independent of the angular direction of ˆMin the weak Rashba regime. In the strong Rashba regime, on the other hand, they found that τdexhibits strong angular dependence. Theoriginoftheangulardependencewithin this model is the anisotropy of the spin relaxation, which arises naturally since the Rashba spin-orbit interaction is responsible for the anisotropic D’yakonov-Perel spin relaxation mechanism. For τf, in contrast, they found it to be almost constant in the strong Rashba regime even when the spin relaxationis anisotropic. Experimen- tally,35,36both the damping-like and the ﬁeld-like contri-2 butions depend strongly on the magnetization direction. Herewe reexaminethe angulardependence ofthe spin- orbit torque based on the Rashba interaction motivated by the following two observations. The ﬁrst motivation comes from a ﬁrst-principles calculation39of Co/Pt bi- layers, according to which both the spin-orbit potential and the exchange splitting are large near the interface between the heavy metal and the ferromagnet. This im- plies that the problem of interest is not in the analyti- cally tractableweak Rashbaor strongRashbaregime but in the intermediate Rashba regime ( r≈1). We exam- ine this intermediate regime numerically and ﬁnd that in contrast to both the strong and weak Rashba regimes, τf has a strong angular dependence. The second motivation comes from a recent calculation28,30showing that the interfacial spin-orbit coupling can generate τdthrough a Berry phase contribution40. In contrast, earlier the- ories25–27of the interfacial spin-orbit coupling found a separate contribution to τdfrom spin relaxation. More- over those calculations30suggest that the Berry phase contribution to τdis much largerthan the spin relaxation contribution. Here, we examine the angular dependence of the Berry phase contribution. To be speciﬁc, we examine the angular dependence of the spin-orbit torques for a free-electron model of two- dimensionalferromagneticsystemswiththeRashbaspin- orbit coupling. When an electric ﬁeld is applied to gener- ate an in-plane current, the spin-orbit torque arises from the two types of changes caused by the electric ﬁeld. One is the electron occupation change. For a small electric ﬁeld, the net occupation change is limited to the Fermi surface so that the spin-orbit torque caused by the oc- cupation change comes from the Fermi surface. For this reason, this contribution is referred to as the Fermi sur- face contribution. The other is the state change. The electric ﬁeld modiﬁes the potential energy of the system, which in turn modiﬁes wavefunctions of all single parti- cle states. Thus the spin-orbit torque caused by the state changecomesnotonlyfromthestatesneartheFermisur- face but also from all states in the entire Fermi sea. This contribution is referred to as the Fermi sea contribution and often closely related to the momentum-space Berry phase30. We ﬁnd that in the absence of spin relaxation, the Fermi surface contribution to τdis vanishingly small, whileτfremains ﬁnite. The τfhas a substantial angu- lar dependence in the intermediate Rashba regime. This nontrivial angular dependence of τfis related to Fermi surface distortion, which becomes signiﬁcant when the Rashba spin-orbit coupling energy ( ∼αRkF) is compa- rabletotheexchangecoupling( ∼J). Ontheotherhand, the Fermi sea contribution generates primarily τdwhich exhibits strong angular dependence in both the interme- diate and strong Rashba regimes. The nontrivial angular dependence of τdis caused by the combined eﬀects of the Fermi surface distortion and the Fermi sea contribution. We also compute the angular dependence of the spin- orbit torques for a tight-binding model and ﬁnd that theresults are qualitatively consistent with those for a free- electron model. II. SEMICLASSICAL MODELS In this section, we use subscripts (1) and (2) to de- note the Fermi surface and the Fermi sea contributions, respectively. The model Hamiltonian for an electron in the absence of an external electric ﬁeld is H0=p2 2m+αRσ·(k×ˆ z)+Jσ·ˆM,(2) wherek= (kx,ky) is the two-dimensionalwavevector, m is the electronmass, J(>0) is the exchangeparameter, p is the momentum, and σis the vector of Pauli matrices, andMx,My, andMzare thex-,y-, andz-components ofˆM, respectively. When ˆMis position-independent, which will be assumed all throughout this paper, kis a good quantum number. For each k, there are two energy eigenvalues since the spin may point in two diﬀerent di- rections. Thus the energy eigenvalues of H0form two energy bands, called majority and minority bands. The one-electron eigenenergy of H0is E± k=¯h2k2 2m∓ǫk, (3) where the upper (lower) sign corresponds to the majority (minority)band, k2=k2 x+k2 y, andǫk=\|JˆM+αR(k×ˆ z)\|. To determine the spin state of the majorityand minor- ity bands, it is useful to combine the last two terms of H0 into an eﬀective Zeeman energy term (= −µBBeﬀ,k·σ), where the eﬀective magnetic ﬁeld is k-dependent and given by Beﬀ,k=−J µBˆM−αR µB(k×ˆ z). (4) HereµBis the Bohr magneton. Beﬀ,kﬁxes the spin direction of the majority and minority bands. For the eigenstate \|ψk,±∝angb∇acket∇ightof an eigenstate in the major- ity/minority band, its spin expectation value s± k(1)≡ (¯h/2)∝angb∇acketleftψk,±\|σ\|ψk,±∝angb∇acket∇ightis given by s± k(1)=±¯h 2ˆBeﬀ,k, (5) whereˆBeﬀ,kis the unit vector along Beﬀ,k. In terms of thek-dependent angle θkandφk, which are deﬁned byˆBeﬀ,k= (sinθkcosφk,sinθksinφk,cosθk), the eigen- state\|ψk,±∝angb∇acket∇ightis given by \|ψk,+∝angb∇acket∇ight= eik·r/parenleftbiggcos(θk/2) sin(θk/2)eiφk/parenrightbigg (6) \|ψk,−∝angb∇acket∇ight= eik·r/parenleftbiggsin(θk/2) −cos(θk/2)eiφk/parenrightbigg (7)3 Together with the energy eigenvalue E± kin Eq. (3), the eigenstate \|ψk,±∝angb∇acket∇ightcompletely speciﬁes properties of the equilibriumHamiltonian. Thegroundstateofthe system isthenachievedbyﬁllingupallsingleparticleeigenstates \|ψk,±∝angb∇acket∇ight, below the Fermi energy EF. When an electric ﬁeld E=Eˆ xis applied, one of the eﬀects is the modiﬁcation of the state occupation. This eﬀect generates the non-equilibrium spin density s± (1)as s± (1)=/integraldisplaydk2 (2π)2/bracketleftbigg f±/parenleftbigg k−eEτ ¯hˆx/parenrightbigg −f±(k)/bracketrightbigg s± k(1),(8) where−eis the electron charge, τis the relaxation time, andf±(k) = Θ(EF−E± k) is the zero-temperature elec- tron occupation function where Θ( x) is the Heaviside step function. Note that the net contribution to s± (1) arises entirely from the states near EFdue to the can- cellation eﬀect between the two occupation functions in Eq. (8). Thus s± (1)is aFermi surface contribution. The total spin density generated by the occupation change becomes s(1)=s+ (1)+s− (1). This is related to the spin- orbit torque T(1)generatedby the occupation change via T(1)= (J/¯h)s(1)×ˆM. In Eq. (8), we use the relaxation time approximation with the assumption that the scat- tering probability is isotropic and spin-independent. The other important eﬀect of the electric ﬁeld other changing than the occupation is that it modiﬁes the po- tential energy that the electrons feel, and hence modiﬁes their wave functions, generating in turn a correction to s± k(1). We call this correction s± k(2). It is calculated in Appendix A and given by s± k(2)=±¯h 2αReE/bracketleftbiggJ 2ǫ3 k(ˆM×ˆy)+αR 2ǫ3 k(ˆx×k)/bracketrightbigg .(9) Summing over all occupied states in the major- ity/minority band, gives the total spin density s± (2)gen- erated by the state change in that band, and is given by s± (2)=/integraldisplaydk2 (2π)2f±(k)s± k(2). (10) Note that the equilibrium occupation function fappears in Eq. (10) rather than the diﬀerence between the two occupation functions. The occupation change eﬀect is ignoredin Eq.(10)sinceweareinterestedin lineareﬀects of the electric ﬁeld Eands± k(2)is already ﬁrst order in E. Note that all the occupied single particle states in the Fermi sea contribute to s± (2). Thuss± (2)amounts is a Fermi sea contribution. The total spin density generated by the state change becomes s(2)=s+ (2)+s− (2). This is related to the spin-orbit torque T(2)generated by the state change via T(2)= (J/¯h)s(2)×ˆM. Afewremarksareinorder. In Eq.(8), the twooccupa- tion functions cancel each other for most kvalues. They do not cancel for kpoints that correspond to electronexcitation slightly above the Fermi surface or the hole excitation slightly below the Fermi surface. Thus the di- rection of s± (1)can be estimated simply by evaluating the diﬀerence of Beﬀ,kbetween two k’s of electron-like and hole-like excitations. This shows that s± (1)points along Eˆx×ˆz=−Eˆy. Thus the spin-orbit torque T(1)should be proportional to Eˆy×ˆM, which is nothing but the ﬁeld-like spin-orbit torque. Thus the Fermi surface con- tribution T(1)contributes mostly to τf. To be precise, however, this statement is not valid for spin-dependent scattering, which we neglect in deriving Eq. (8). If the scattering is spin-dependent, T(1)producesτdas well as τfas demonstrated in Refs. 25–27. In this paper, we neglect the contribution to the angular dependence of τd fromT(1)andspin-dependent scatteringsinceithasbeen already treated in Ref. 38. The contribution to τdin our study comes from the Fermi sea contribution T(2). One can easily verify that the ﬁrst term in Eq. (9) generates the spin-orbit torque proportionalto ( ˆM×ˆy)×ˆM, which has the form of the damping-like spin-orbit torque. The second term in Eq. (9) on the other hand almost vanishes uponkintegration in Eq. (10). This demonstrates that the Fermi sea contribution T(2)contributes mostly to τd. We also compute the spin-orbit torques based on a tight-binding model because free electron models with linear Rashba coupling, like we use here, can exhibit pathological behavior when accounting for vertex correc- tions to the impurity scattering. For example, the intrin- sic spin Hall eﬀect40, that has the same physical origin of the Fermi sea contribution to spin-orbit torque, gives a universalresult that vanishes when vertexcorrectionsare included.41–45However, the intrinsic spin Hall eﬀect does not vanish when the electron dispersion deviates from free electron behavior or the spin-orbit coupling is not linear in momentum.46–49Since we neglect vertex correc- tions in the calculations presented in this paper, it is nec- essary to check whether or not the angular dependence of spin-orbit torque obtained in a free-electron model is qualitatively reproduced in a tight-binding model, where the electron dispersion deviates from free electron be- havior and the spin-orbit coupling is not strictly linear in momentum. To compute spin-orbit torque in the two- band (majority and minority spin bands) tight-binding model on a square lattice with the lattice constant a, we replacekxandkyby sin(kxa)/aand sin(kya)/a, respec- tively. The corresponding spin density is then calculated by integrating the electric ﬁeld-induced spin expectation value up to the point of band ﬁlling. For most cases in a tight-binding model, the result converges for a k-point mesh with mesh spacing dk=0.052 nm−1and 80,000 k- points, wherethe convergencecriteriais 1percentchange ofresults with a ﬁner mesh by factor of 2. All results pre- sented in this paper are converged to this criteria.4 /s45/s49/s46/s56/s45/s49/s46/s50/s45/s48/s46/s54/s48/s46/s48/s48/s46/s54/s49/s46/s50/s49/s46/s56 /s45/s49/s46/s56 /s45/s49/s46/s50 /s45/s48/s46/s54 /s48/s46/s48 /s48/s46/s54 /s49/s46/s50 /s49/s46/s56 /s45/s49/s46/s56 /s45/s49/s46/s50 /s45/s48/s46/s54 /s48/s46/s48 /s48/s46/s54 /s49/s46/s50 /s49/s46/s56/s45/s49/s46/s56/s45/s49/s46/s50/s45/s48/s46/s54/s48/s46/s48/s48/s46/s54/s49/s46/s50/s49/s46/s56 /s45/s49/s46/s56 /s45/s49/s46/s50 /s45/s48/s46/s54 /s48/s46/s48 /s48/s46/s54 /s49/s46/s50 /s49/s46/s56/s69/s108/s101/s99/s116/s114/s105/s99/s32/s102/s105/s101/s108/s100/s32/s100/s105/s114/s101/s99/s116/s105/s111/s110/s122 /s72/s101/s97/s118/s121/s32/s109/s101/s116/s97/s108/s32/s32 /s40/s97/s41 /s72/s101/s97/s118/s121/s32/s109/s101/s116/s97/s108/s70/s101/s114/s114/s111/s109/s97/s103/s110/s101/s116/s120 /s50/s45/s100/s105/s109/s101/s110/s115/s105/s111/s110/s97/s108/s32/s82/s97/s115/s104/s98/s97/s32/s105/s110/s116/s101/s114/s102/s97/s99/s101/s40/s102/s41 /s32/s32/s40/s98/s41 /s32/s32 /s40/s100/s41/s32 /s32/s107 /s121/s32/s40/s120/s49/s48/s49/s48 /s32/s109/s45/s49 /s41 /s32/s107 /s120/s32/s40/s120/s49/s48/s49/s48 /s32/s109/s45/s49 /s41/s40/s101/s41 /s32/s32/s77/s97/s106/s111/s114/s105/s116/s121 /s32/s98/s97/s110/s100 /s32/s114/s32/s61/s32/s48 /s32/s114/s32/s61/s32/s49/s46/s48/s40/s99/s41 /s32 FIG. 1: (color online) Fermi surface and spin direction for a free-electron model. (a) r= 0 (only exchange splitting), (b) r=∞(only Rashba spin-orbit coupling and non-magnetic), (c) co mparison of Fermi surfaces (majority band) for r= 0 and r= 1.0, (d)r= 0.8, (e)r= 1.0, and (f) r= 1.2. We assume ˆM= (0,1,0),EF= 10 eV, m=m0, andJ= 1 eV. Here m0 is the free electron mass. The outer red (inner blue) Fermi su rface corresponds to majority (minority) band. Arrows are t he eigendirections of spins on the Fermi surface. The coordina te system is shown on the right. III. RESULTS AND DISCUSSION We ﬁrst discuss Fermi surface distortion as a function ofr(=αRkF/J). Figure 1 shows the Fermi surface and the spin direction at each k-point for various values of the ratior. Without Rashba spin-orbit coupling ( r= 0), the spin directiondoesnot depend on kforferromagnetic systems (Fig. 1(a)). Without exchange coupling (non- magnetic Rashba system ( r=∞)), on the other hand, the spins point in the azimuthal direction (Fig. 1(b)). For these extreme cases, the Fermi surfaces of two bands are concentric circles. The Fermi surfaces distort signiﬁcantly when r≈1. Figure 1(c) comparestwo Fermi surfaces(majorityband) forr= 0andr= 1.0whenˆM= (0,1,0). Whenthemag- netizationhasanin-planecomponentasinthiscase,each sheet of the Fermi surface shifts in a diﬀerent direction and distorts from perfect circularity(Fig. 1(c): note that the dotted Fermi surfaceis for r= 0 andis a circle). This distortion arises because the k-dependent eﬀective mag- netic ﬁeld (Eq. (4)) contains contributions both from the exchange and Rashba spin-orbit couplings. An eﬀective ﬁeld from the exchange is aligned along ˆMand uniform regardless of k, whereas that from the Rashba spin-orbit coupling lies in the x−yplane and is k-dependent. For example, for ˆM= (0,1,0) and majority band, an eﬀec- tive ﬁeld from the Rashba spin-orbit coupling is parallel (anti-parallel) to that from the exchange at k= (kF,1,0) (k= (kF,2,0)), wherekF,1(>0) andkF,2(<0) are the Fermi wave vectors corresponding to the electric ﬁeld-induced electron-like and hole-like excitations, respec- tively. The k-dependent eﬀective ﬁeld distorts the Fermi surface distortion as demonstrated in Fig. 1(c)-(f). This Fermi surface distortion also aﬀects the spin di- rection at each k-point because the spin eigendirection is k-dependent due to the Rashba spin-orbit coupling. In the weak (strong) Rashba regime, the spin landscape is similar with that in Fig. 1(a) (Fig. 1(b)). In these ex- treme cases, the spin landscape is not signiﬁcantly mod- iﬁed by the change in the magnetization direction as one of the eﬀective ﬁelds (either from the exchange or from the Rashba spin-orbit coupling) is much stronger than the other. As a result, τfhas almost no angular distor- tionintheseregimes. Thespinlandscapefor r≈1onthe other hand becomes highly complicated (Fig. 1(d)-(f)) as the Fermi surface distortion is maximized. One can easily verify that the spin landscape for r≈1 varies sig- niﬁcantly with the magnetization direction because the Fermi surface distortion is closely related to the in-plane component of the magnetization as explained above. As the non-equilibrium spin density corresponding to τf(i.e.s(1)) is obtained from the integration of the spins on the Fermi surface, this magnetization-angle- dependent change in the spin landscape generates a non- trivial angular dependence of τf. A similar argument is valid for τd(i.e.s(2)) which comes from the Fermi sea contribution because the Fermi surface distortion af- fects the interval of the integration. Therefore, the re- sults shown in Fig. 1 suggest that the spin-orbit torque originatingfromtheinterfacialspin-orbitcouplingshould have a strong dependence on the magnetization angles θ5 /s48/s49/s50 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s49/s50/s48/s51/s54/s57 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s51/s54/s57/s45/s50/s46/s48 /s45/s49/s46/s54 /s45/s49/s46/s50 /s45/s48/s46/s56 /s32/s32/s102/s32/s40/s120/s49/s48/s50/s51 /s32/s101/s69 /s41 /s70/s114/s101/s101/s45/s101/s108/s101/s99/s116/s114/s111/s110 /s32/s61/s32/s48/s114/s32/s61/s32 /s82/s107 /s70/s47/s74/s32/s61/s32/s45/s48/s46/s52 /s40/s100/s41 /s45/s50/s46/s48 /s45/s49/s46/s54 /s45/s49/s46/s50 /s45/s48/s46/s56 /s114/s32/s61/s32 /s82/s107 /s70/s47/s74/s32/s61/s32/s45/s48/s46/s52/s102/s32/s40/s120/s49/s48/s50/s51 /s32/s101/s69 /s41 /s80/s111/s108/s97/s114/s32/s97/s110/s103/s108/s101/s32 /s47/s70/s114/s101/s101/s45/s101/s108/s101/s99/s116/s114/s111/s110 /s32/s61/s32 /s47/s50/s40/s97/s41 /s84/s105/s103/s104/s116/s45/s98/s105/s110/s100/s105/s110/s103 /s32/s61/s32/s48/s82/s32/s40/s101/s86/s46/s110/s109/s41 /s32/s45/s48/s46/s48/s50/s32/s32/s32 /s32/s45/s48/s46/s48/s52 /s32/s45/s48/s46/s48/s54/s32/s32/s32 /s32/s45/s48/s46/s48/s56 /s32/s45/s48/s46/s50/s56/s102/s32/s40/s120/s49/s48/s50/s51 /s32/s101/s69 /s41 /s82/s32/s40/s101/s86/s46/s110/s109/s41 /s32/s45/s48/s46/s48/s50/s32/s32/s32 /s32/s45/s48/s46/s48/s52 /s32/s45/s48/s46/s48/s54/s32/s32/s32 /s32/s45/s48/s46/s48/s56 /s32/s45/s48/s46/s50/s56/s40/s99/s41 /s84/s105/s103/s104/s116/s45/s98/s105/s110/s100/s105/s110/s103 /s32/s61/s32 /s47/s50/s40/s98/s41 /s102/s32/s40/s120/s49/s48/s50/s51 /s32/s101/s69 /s41 /s80/s111/s108/s97/s114/s32/s97/s110/s103/s108/s101/s32 /s47 FIG. 2: (color online) Polar angle ( θ) dependence of ﬁeld- like spin-orbit torque coeﬃcient τf. Free-electron model (a and b): (a) azimuthal angle of magnetization φ= 0 and (b) φ=π/2. Tight-binding model (c and d): (c) φ= 0 and (d)φ=π/2. For a free-electron model, we use EF= 10 eV, m=m0, andJ= 1 eV. For a tight-binding model, we use m=m0,J= 1 eV, a= 0.3 nm, and normalized electron densityn=N/Nmax= 0.5 where Nis the electron density of ﬁlled bands and Nmax(= 2.2×1019m−2) is the maximum electron density. For a free-electron model, the results fo r r= 1 are excluded because it is singular. For a tight-binding model, the results for -0.28 eV ·nm< αR<-0.08 eV ·nm are not included because of bad convergence. andφwhenris close to 1. Several additional remarks for the Fermi surface dis- tortionareasfollows. First, the twoFermi surfacestouch exactly for r= 1 (Fig. 1(e)) and they anticross for r>1 (Fig. 1(f)). As a result, the spin landscape rapidly changes when rvaries around 1 so that a similar dras- tic change in the angular dependence of the spin-orbit torqueisexpected. Second, alleﬀectsfromtheFermisur- face distortion, described for a free-electronmodel above, should also aﬀect the results obtained for a tight-binding model. However, as the shape of the Fermi surface is diﬀerent for the two models (i.e. for J= 0 andαR= 0, the Fermi surface for a free-electron model is a circle, whereas that for a tight-binding model with half band- ﬁlling is a rhombus), the results for the two models are quantitatively diﬀerent. We next show the angular dependence of τfandτd for the two models. Here we do not attempt to analyze the detailed angular dependence quantitatively, because it is very parameter sensitive. In contrast, our intention is to identify the general trends that emerge from these numerical calculations. Figure 2 shows the angular de- pendence of τffor a free-electron model ((a) and (b)) and a tight-binding model ((c) and (d)). In both models, we obtain nontrivial angular dependence of τfin certain pa- rameter regimes. In the free-electron model, we ﬁnd τf is almost constant in weak ( r≪1) and strong ( r≫1)/s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s50/s52/s54 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s53/s49/s48/s49/s53/s50/s48/s80/s111/s108/s97/s114/s32/s97/s110/s103/s108/s101/s32 /s47/s114/s32/s61/s32 /s82/s107 /s70/s47/s74 /s32/s45/s48/s46/s52 /s32/s45/s48/s46/s56 /s32/s45/s49/s46/s50 /s32/s45/s49/s46/s54 /s32/s45/s50/s46/s48 /s32/s32/s100/s32/s40/s120/s49/s48/s56 /s32/s101/s69/s41/s70/s114/s101/s101/s45/s101/s108/s101/s99/s116/s114/s111/s110 /s32/s61/s32/s48 /s114/s32/s61/s32 /s82/s107 /s70/s47/s74 /s32/s45/s48/s46/s52/s32/s32 /s32/s45/s48/s46/s56 /s32/s45/s49/s46/s50/s32/s32 /s32/s45/s49/s46/s54 /s32/s45/s50/s46/s48/s40/s100/s41/s100/s32/s40/s120/s49/s48/s56 /s32/s101/s69/s41 /s65/s122/s105/s109/s117/s116/s104/s97/s108/s32/s97/s110/s103/s108/s101/s32 /s47/s70/s114/s101/s101/s45/s101/s108/s101/s99/s116/s114/s111/s110 /s32/s61/s32 /s47/s50/s40/s97/s41 /s80/s111/s108/s97/s114/s32/s97/s110/s103/s108/s101/s32 /s47/s84/s105/s103/s104/s116/s45/s98/s105/s110/s100/s105/s110/s103 /s32/s61/s32/s48/s82/s32/s40/s101/s86/s46/s110/s109/s41 /s32/s45/s48/s46/s48/s50/s32/s32/s32 /s32/s45/s48/s46/s48/s54 /s32/s45/s48/s46/s49/s48/s32/s32/s32 /s32/s45/s48/s46/s49/s52 /s32/s45/s48/s46/s49/s56/s100/s32/s40/s120/s49/s48/s56 /s32/s101/s69/s41 /s82/s32/s40/s101/s86/s46/s110/s109/s41 /s32/s45/s48/s46/s48/s50/s32/s32/s32 /s32/s45/s48/s46/s48/s54 /s32/s45/s48/s46/s49/s48/s32/s32/s32 /s32/s45/s48/s46/s49/s52 /s32/s45/s48/s46/s49/s56/s40/s99/s41 /s84/s105/s103/s104/s116/s45/s98/s105/s110/s100/s105/s110/s103 /s32/s61/s32 /s47/s50/s40/s98/s41 /s100/s32/s40/s120/s49/s48/s56 /s32/s101/s69/s41 /s65/s122/s105/s109/s117/s116/s104/s97/s108/s32/s97/s110/s103/s108/s101/s32 /s47 FIG. 3: (color online) Angular dependence of damping-like spin-orbit torque coeﬃcient τd. Free-electron model (a and b): (a) polar angle dependence at φ= 0 and (b) azimuthal angle dependence at θ=π/2. Tight-binding model (c and d): (c) polar angle dependence at φ= 0 and (d) azimuthal angle dependence at θ=π/2. Same parameters are used as in Fig. 2. Rashba regimes, consistent with earlierworks.22,23In the intermediate Rashba regimes, however, τfis not a con- stant. We ﬁnd that τfdepends not only on the polar angleθbut also the azimuthal angle φ, as expected from the Fermi surface distortion (Fig. 1). In Fig. 2(b), τf forr <1 (r >1) is maximal (minimal) at θ=π/2, which is caused by the anticrossing of the two Fermi sur- faces (Fig. 1(d)-(f)). Despite the strong angular depen- dence, the sign of τfis preserved since the spin direction of nonequilibrium spin density is unambiguously deter- mined once the direction of electric ﬁeld and the sign of αRareﬁxed. Theseoveralltrendsarequalitativelyrepro- duced in a tight-binding model (Fig. 2(c) and (d)). The magnitude and angular dependence of τfdiﬀer quantita- tively from those of the free electron model, due to the diﬀerent shape of the Fermi surfaces for the two models. Figure 3 shows the angular dependence of τdfor a free- electron model ((a) and (b)) and a tight-binding model ((c) and (d)). This τdresults from the Fermi sea contri- bution (Eqs. (9) and (10)). In both models, we obtain nontrivial angular dependence of τdboth in the inter- mediate and strong Rashba regimes (Fig. 3(a) and (c)). This is in contrast to τfwhich exhibits nontrivial angular dependence only in the intermediate Rashba regime. To understand this diﬀerence, we derive an approximate τd by expanding up to third order inαRkF Jand assuming no Fermi surface distortion (i.e. the Fermi wave vector kF does not depend on the direction of k), which is analyt- ically tractable. By integrating Eq. (10) with these as- sumptions, weﬁnd τd∝(16J2−3α2 Rk2 F−9α2 Rk2 Fcos(2θ)). Therefore, the Fermi sea contribution induces an intrin- sicangulardependence in τd, which increaseswith\|αR\|kF J6 irrespective of the Fermi surface distortion. The results in Fig. 3, which are obtained numerically, include the Fermi surface distortion, so that the nontrivial angular dependence of τdresults from the combined eﬀects of the intrinsic Fermi sea contribution and the Fermi surface distortion. For example, Fig. 3(a) shows a sharp diﬀer- ence in the angular dependence of τdforr>1 andr<1. This is qualitatively similar to the results of τfshown in Fig. 2(b), showing that the Fermi surface distortion also has a role in the angular dependence of τd. The sign of τddoes not change with the magnetization angle despite the strong angular dependence, similar to the behavior of τf. Whenθ=π/2 (Fig. 3(d)), a steep increase of τdis obtained at φ=π/4 and 3π/4, origi- nating from the shape of the Fermi surface. We expect that this strong dependence of τdonφcan be observed in epitaxial bilayers but may be absent in sputtered bi-layers as sputtered thin ﬁlms consist of small grains with diﬀerent lattice orientation in the ﬁlm plane. However, the dependence of τdon the polar angle θ(Fig. 3(a) and (c)) is irrelevant to this in-plane crystallographic issue so we expect that it will be observable in experiments when the interfacial spin-orbit coupling is comparable to or stronger than the exchange coupling. We note that a strong dependence of τdonθ(but a very weak depen- dence onφ) was experimentally observed in sputtered bilayers.35 We ﬁnally illustrate the connection between τd(i.e. T(2)) and the Berry phase. This examination is mo- tivated by Ref. 30, which called T(2)the Berry phase contribution. To clarify the connection, it is useful to ex- pressthe Fermi seacontribution s(2)in the Kubo formula form, s(2)=1 2eE¯h2AIm/summationdisplay ab/integraldisplayd2k (2π)2[fa(k)−fb(k)]∝angb∇acketleftk,a\|σ\|k,b∝angb∇acket∇ight∝angb∇acketleftk,b\|vx\|k,a∝angb∇acket∇ight [Ea(k)−Eb(k)+2iδ]2, (11) wherea,bare band indices, and δis an inﬁnitesimally small positive constant. In the present case, \|k,a∝angb∇acket∇ightis ei- ther\|ψk,+∝angb∇acket∇ightor\|ψk,−∝angb∇acket∇ight. One then uses the relations vx=1 ¯h∂H0(k,M) ∂kx, σα=1 J∂H0(k,M) ∂Mα,(12) where the notation H0(k,M) emphasizes that the un-perturbed Hamiltonian H0is a function of the momen- tumkand the magnetization M. Note that here we use Minstead of ˆMsince one needs to relax the constraint \|ˆM\|= 1toestablishtheconnectionwiththeBerryphase. Equation (12) allows one to convert the numerator of Eq. (11) as follows, ∝angb∇acketleftk,a\|σ\|k,b∝angb∇acket∇ight=−1 J[Ea(k)−Eb(k)]∝angb∇acketleftk,a\|∇M\|k,b∝angb∇acket∇ight,∝angb∇acketleftk,b\|vx\|k,a∝angb∇acket∇ight=−1 ¯h[Eb(k)−Ea(k)]∝angb∇acketleftk,b\|∂ ∂kx\|k,a∝angb∇acket∇ight.(13) Thus the numerator of Eq. (11) acquires the factor [Ea(k)−Eb(k)]2, which cancels the denominator in thelimitδ→0. Then one of the two summations for the band indices aandbcan be performed to produce [s(2)]α=1 2eE¯hA J/summationdisplay a/integraldisplayd2k (2π)2fa(k)/bracketleftbigg∂ ∂kxAa Mα(k)−∂ ∂MαAa kx(k)/bracketrightbigg , (14) where the spin-space Berry phase Aa Mα(k) and the momentum-space Berry phase Aa kx(k) are deﬁned by Aa Mα(k) =i∝angb∇acketleftk,a\|∂ ∂Mα\|k,a∝angb∇acket∇ight (15) Aa kx(k) =i∝angb∇acketleftk,a\|∂ ∂kx\|k,a∝angb∇acket∇ight.7 Here these Berry phases are manifestly real. Equa- tion (14) establishes the connection between T(2)and the spin-momentum-space Berry phase. A few remarks are in order. First, through an explicit evaluation of the Berry phases, one can verify that the integrand of Eq. (14) generates s± k(2)in Eq. (9) precisely. Second, Eq. (14) contains the occupation function fa(k) itself rather than diﬀerence between the occupation func-tions or derivatives of the occupation function. Thus s(2) may be classiﬁed as a Fermi sea contribution. We note, however, that the Fermi sea contribution Eq. (14) may be converted to a diﬀerent form50, where the net contri- bution is evaluated only at the Fermi surface. To demon- strate this point, we integrate Eq. (14) by parts, which generates [s(2)]α=1 2eE¯hA J/summationdisplay a/integraldisplayd2k (2π)2/bracketleftbigg −∂fa(k) ∂kxAa Mα(k)+∂fa(k) ∂MαAa kx(k)/bracketrightbigg . (16) Notethatinthezerotemperaturelimit, both ∂fa(k)/∂kx and∂fa(k)/∂Mαare nonzero only at the Fermi surface, and thus the net contribution to s(2)depends only on properties evaluated at the Fermi surface. In this sense, this Fermi surface contribution is analogous to Friedel oscillations. Friedel oscillations form near surfaces when electrons reﬂect and the incoming and outgoing waves interfere. Then, each electron below the Fermi energy makes an oscillatory contribution to the density with a wavelength that depends on the energy. However, inte- grating up from the bottom of the band to the Fermi energy gives a result that only depends on the proper- ties of the electrons at the Fermi energy where there is a sharp cut-oﬀ in the integration. IV. SUMMARY We use simple models to examine the angular depen- denceofspin-orbittorquesasafunctionoftheratioofthe spin-orbit interaction to the exchange interaction. We ﬁnd that both the ﬁeld-like and damping-like torques are angle independent when the spin-orbit coupling is weak but become angle-dependent when the spin-orbit coupling becomes comparable to the exchange coupling. When the spin-orbit coupling becomes much stronger than the exchange coupling, the angular dependence of the ﬁeld-like torque goes away, but that of the damping- like torque remains. The angulardependence of the ﬁeld- like torque becomes signiﬁcant when the spin-orbit cou- pling becomes strong enough to distort the Fermi surface so that it changes when the direction of the magnetiza- tion changes. On the other hand, the angular depen- dence of the damping-like torque is caused by the com- bined eﬀects of the intrinsic Fermi sea contribution and the Fermi surface distortion. We expect that these qual- itative conclusions will hold for more realistic treatments of the interface. The strong angular dependence of the spin-orbit torques will signiﬁcantly impact their role in largeamplitudemagnetizationdynamicslikeswitchingor domain wall motion. This suggests caution when com-paring measurement of the strength of torques with the magnetizations in diﬀerent directions. Acknowledgments K.-J.L. acknowledges support from the NRF (2011- 028163, NRF-2013R1A2A2A01013188) and under the Cooperative Research Agreement between the Univer- sity of Maryland and the National Institute of Stan- dards and Technology Center for Nanoscale Science and Technology, Award 70NANB10H193, through the University of Maryland. H.-W.L. was supported by NRF (2013R1A2A2A05006237) and MOTIE (Grant No. 10044723). A.M. acknowledges support by the King Ab- dullah University of Science and Technology. D.G. ac- knowledges support from the Global Ph.D. Fellowship Program funded by NRF (2014H1A2A101). Appendix A: Derivation of Eq. (9) Here, we derive the Fermi sea contribution of the spin- orbit torque. First, we calculate the change of the eigen- statesinthepresenceofanexternalelectricﬁeld. Second, we calculate the resulting spin accumulation. We use time-dependent perturbation theory, adiabatically turn- ing on the electric ﬁeld, which is treated as the pertur- bation. Weadopttime-dependent perturbationapproach insteadofthe Kuboformulaforpedagogicalreasonssince it directly shows how the states change due to the per- turbation. One can show that both approaches give the same result. Let us consider an in-plane electric ﬁeld E′(t) = Eexp(δt) , where exp( δt) gives the adiabatic turning-on process. The electric ﬁeld starts toincreasefrom t=−∞ untilt= 0, for very small δwhich will be set to be zero at the end of the calculation. This is represented by the vectorpotential A=−texp(δt)EsinceE=−∂A/∂t. In the presence of a vector potential, the momentum opera- torpis replaced by p+eA. Thus, the total Hamiltonian8 becomes H=(p+eA)2 2m+αR ¯hσ·[(p+eA)×ˆz]+Jσ·ˆM =H0+H1(t)+O(E2) (A1) where H1(t) =−e(E·p) mtexp(δt) (A2) +αR ¯h[(eE×σ)·ˆz]texp(δt). Here, the ﬁrst term comes from the kinetic energy and the the second term from the Rashbaspin-orbit coupling. In the interaction picture, the propagator of the order of O(E1) is U(I) 1=−i ¯h/integraldisplay0 −∞dtH(I) 1(t) (A3) where H(I) 1(t) =eiH0t/¯hH1(t)e−iH0t/¯h =−e(E·p) mtexp(δt) +αR ¯h[(eE×σ(I)(t))·ˆz]texp(δt),(A4)and σ(I)(t) =eiH0t/¯hσe−iH0t/¯h =σcos/parenleftbigg2ǫkt ¯h/parenrightbigg +(ˆn×σ)sin/parenleftbigg2ǫkt ¯h/parenrightbigg +ˆn(ˆn·σ)/bracketleftbigg 1−cos/parenleftbigg2ǫkt ¯h/parenrightbigg/bracketrightbigg .(A5) Here we deﬁne ǫk=/vextendsingle/vextendsingle/vextendsingleJˆM+αRk×ˆz/vextendsingle/vextendsingle/vextendsingle, (A6) and ˆn=1 ǫk/parenleftBig JˆM+αRk×ˆz/parenrightBig . (A7) Thus, U(I) 1(k) =U(I) 1(a)(k)+U(I) 1(b)(k), (A8) where U(I) 1(a)(k) =−ie(E·k) m1 δ2, (A9) and U(I) 1(b)(k) =−i ¯hαR ¯h/braceleftbigg1 2[(eE×σ)·ˆz]/parenleftbigg1 (2ǫk/¯h+iδ)2+1 (2ǫk/¯h−iδ)2/parenrightbigg +1 2i[(eE×(ˆn×σ))·ˆz]/parenleftbigg1 (2ǫk/¯h+iδ)2−1 (2ǫk/¯h−iδ)2/parenrightbigg +[(eE×ˆn)·ˆz](ˆn·σ)/bracketleftbigg −1 δ2−1 2/parenleftbigg1 (2ǫk/¯h+iδ)2+1 (2ǫk/¯h−iδ)2/parenrightbigg/bracketrightbigg/bracerightbigg . (A10) Thus, the change in the state due to the adiabatically turned on electric ﬁeld is given by δ\|ψk,±∝angb∇acket∇ight=U(I) 1(k)\|ψk,±∝angb∇acket∇ight. (A11) Now, the spin accumulation arising from the changes in the occupied states is s± k(2)=¯h 2[(δ∝angb∇acketleftψk,±\|)σ\|ψk,±∝angb∇acket∇ight+∝angb∇acketleftψk,±\|σ(δ\|ψk,±∝angb∇acket∇ight)]δ→0 =¯h 2×2Re/bracketleftBig ∝angb∇acketleftψk,±\|σU(1) 1(k)\|ψk,±∝angb∇acket∇ight/bracketrightBig δ→0 =±¯h 2αReE/braceleftbiggJ 2ǫ3 k[ˆM×(ˆz×ˆE)]+αR 2ǫ3 k(ˆE×k)/bracerightbigg , (A12)where±indicates majority/minoritybands, respectively. When the electric ﬁeld is applied along the ˆxdirection, we arrive at Eq. (6). Note that U(I) 1(a)does not contribute to the spin expectation value since it is purely imaginary. (∗∗) These authors equally contributed to this work.9 ∗Electronic address: hwl@postech.ac.kr †Electronic address: kj˙lee@korea.ac.kr 1I.M. Miron, K.Garello, G.Gaudin, P.-J. Zermatten, M.V. Costache, S. Auﬀret, S. Bandiera, B. Rodmacq, A. Schuhl, and P. Gambadella, Nature (London) 476, 189 (2011). 2I. M. Miron, T. Moore, H. Szambolics, L. D. Buda- Prejbeanu, S. Auﬀret, B. Rodmacq, S. Pizzini, J. Vogel, M. Bonﬁm, A. Schuhl, and G. Gaudin, Nat. Mater. 10, 419 (2011). 3L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph and R. A. Buhrman, Science 4, 555 (2012). 4L. Liu, C.-F. Pai, D. C. Ralph and R. A. Buhrman, Phys. Rev. Lett. 109, 186602 (2012). 5S.-M. Seo, K.-W. Kim, J. Ryu, H.-W. Lee, and K.-J. Lee, Appl. Phys. Lett. 101, 022405 (2012). 6A. Thiaville, S. Rohart, ´E. Ju´ e, V. Cros, and A. Fert, Eu- rophys. Lett. 100, 57002 (2012). 7P. P. J. Haazen, E. Mur´ e, J. H. Franken, R. Lavrijsen, H. J. M. Swagten, and B. Koopmans, Nat. Mater. 12, 299 (2013). 8S. Emori, U. Bauer, S.-M. Ahn, E. Martinez, and G. S. D. Beach, Nat. Mater. 12, 611 (2013). 9K.-S. Ryu, L. Thomas, S.-H. Yang, and S. S. P. Parkin, Nat. Nanotech. 8, 527 (2013). 10K.-S. Lee, S.-W. Lee, B.-C. Min, and K.-J. Lee, Appl. Phys. Lett. 102, 112410 (2013). 11A. van den Brink, S. Cosemans, S. Cornelissen, M. Man- frini, A. Vaysset, W. van Roy, T. Min, H. J. M. Swagten, and B. Koopmans, Appl. Phys. Lett. 102, 112410 (2013). 12K. Garello, C. O. Avci, I. M. Miron, O. Boulle, S. Auﬀret, P. Gambadella, and G. Gaudin, arXiv:1310.5586. 13K.-S. Lee, S.-W. Lee, B.-C. Min, and K.-J. Lee, Appl. Phys. Lett. 104, 072413 (2014). 14Y. Yoshimura, T. Koyama, D. Chiba, Y. Nakatani, S. Fukami, M. Yamanouchi, H. Ohno, K.-J. Kim, T. Moriyama, and T. Ono, Appl. Phys. Express 7, 033005 (2014). 15M. I. D’yakonov and V. I. Perel, JETP Lett. 13, 467 (1971). 16J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999). 17S. F. Zhang, Phys. Rev. Lett. 85, 393 (2000). 18K. Ando, S. Takahashi, K. Harii, K. Sasage, J. Ieda, S. Maekawa, and E. Saitoh, Phys. Rev. Lett. 101, 036601 (2008). 19L. Liu, T. Moriyama, D. C. Ralph and R. A. Buhrman, Phys. Rev. Lett. 106, 036601 (2011). 20Yu. A. Bychkov and E. I. Rashba, JETP. Lett. 39, 78 (1984). 21V. M. Edelstein, Solid State Commun. 73, 233 (1990). 22K. Obata, and G. Tatara, Phys Rev. B 77, 214429 (2008). 23A. Manchon and S. Zhang, Phys. Rev. B 78, 212405 (2008). 24A. Matos-Abiague and R. L. Rodriguez-Suarez, Phys. Rev. B80, 094424 (2009).25X. Wang and A. Manchon, Phys. Rev. Lett. 108, 117201 (2012). 26K.-W. Kim, S.-M. Seo, J. Ryu, K.-J. Lee, and H.-W. Lee, Phys. Rev. B 85, 180404(R) (2012). 27D. A. Pesin and A. H. MacDonald, Phys. Rev. B 86, 014416 (2012). 28E. van der Bijl and R. A. Duine, Phys. Rev. B 86, 094406 (2012). 29P. M. Haney, H.-W. Lee, K.-J. Lee, A. Manchon, and M. D. Stiles, Phys. Rev. B 87, 174411 (2013). 30H. Kurebayashi, J. Sinova, D. Fang, A. C. Irvine, T. D. Skinner, J. Wunderlich, V. Nov´ ak, R. P. Campion, B. L. Gallagher, E. K. Vehstedt, L. P. Zarobo, K. Vyborny, A. J. Ferguson, and T. Jungwirth, Nat. Nanotech. 9, 211 (2014). 31J. Kim, J. Sinha, M. Hayashi, M. Yamanouchi, S. Fukami, T. Suzuki, S. Mitani, and H. Ohno, Nat. Mater. 12, 240 (2013). 32X. Qiu, K. Narayanapillai, Y. Wu, P. Deorani, X. Yin, A. Rusydi, K.-J. Lee, H.-W. Lee, and H. Yang, arXiv:1311.3032. 33X. Fan, H. Celik, J. Wu, C. Ni, K.-J. Lee, V. O. Lorenz, and J. Q. Xiao, Nat. Commun. 53042 (2014). 34R. H. Liu, W. L. Lim, and S. Urazhdin, Phys. Rev. B 89, 220409(R) (2014). 35K. Garello, I. M. Miron, C. O. Avci, F. Freimuth, Y. Mokrousov, S. Bl¨ ugel, S. Auﬀret, O. Boulle, G. Gaudin, and P. Gambardella, Nat. Nanotech. 8, 587 (2013). 36X. Qiu, P. Deorani, K. Narayanapillai, K.-S. Lee, K.-J. Lee, H.-W. Lee, and H. Yang, Sci. Rep. 4, 4491 (2014). 37J. Sampaio, V. Cros, S. Rohart, A. Thiaville, and A. Fert, Nat. Nanotech. 8, 839 (2013). 38C. O. Pauyac, X. Wang, M. Chshiev, and A. Manchon, Appl. Phys. Lett. 102, 252403 (2013). 39P. M. Haney, H.-W. Lee, K.-J. Lee, A. Manchon, and M. D. Stiles, Phys. Rev. B 88, 214417 (2013). 40J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth, and A. H. MacDonald, Phys. Rev. Lett. 92, 126603 (2004). 41J.-i. Inoue, G. E. W. Bauer, and L. W. Molenkamp, Phys. Rev. B67, 033104 (2003). 42J. Schliemann and D. Loss, Phys. Rev. B 69, 165315 (2004). 43A. A. Burkov, A. S. N´ u˜ nez and A. H. MacDonald, Phys. Rev. B70, 155308 (2004). 44E. G. Mishchenko, A. V. Shytov, and B. I. Halperin, Phys. Rev. Lett. 93, 226602 (2004). 45O. V. Dimitrova, Phys. Rev. B 71, 245327 (2005). 46S. Murakami, Phys. Rev. B 69, 241202(R) (2004). 47B. A. Bernevig and S.-C. Zhang, Phys. Rev. Lett. 95, 016801 (2005). 48A. V. Shytov, E. G. Mishchenko, H.-A. Engel, and B. I. Halperin, Phys. Rev. B 73, 075316 (2006). 49A. Khaetskii, Phys. Rev. B 73, 115323 (2006). 50F. D. M. Haldane, Phys. Rev. Lett. 93, 206602 (2004)
1501.04022v3.Entanglement_Driven_Phase_Transitions_in_Spin_Orbital_Models.pdf	arXiv:1501.04022v3 [cond-mat.str-el] 23 Jun 2015Entanglement driven phase transitions in spin-orbital models Wen-Long You1,2, Andrzej M. Ole´ s1,3and Peter Horsch1 1Max-Planck-Institut f¨ ur Festk¨ orperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany 2College of Physics, Optoelectronics and Energy, Soochow Universit y, Suzhou, Jiangsu 215006, People’s Republic of China 3Marian Smoluchowski Institute of Physics, Jagiellonian University, prof. S. /suppress Lojasiewicza 11, PL-30348 Krak´ ow, Poland E-mail:a.m.oles@fkf.mpg.de Abstract. To demonstrate the role played by the von Neumann entropy spect ra in quantum phase transitions we investigate the one-dimensional an isotropic SU(2)⊗XXZ spin-orbital model with negative exchange parameter. In the cas e of classical Ising orbital interactions we discover an unexpected n ovel phase with Majumdar-Ghosh-like spin-singlet dimer correlations triggered by spin-orbital entanglement and having k=π/2 orbital correlations, while all the other phases are disentangled. For anisotropic XXZ orbital interactions both spin-orbital entanglement and spin-dimer correlations extend to the antiferro-spin/alterna ting-orbital phase. This quantum phase provides a unique example of two coupled order p arameters which change the character of the phase transition from ﬁrst-order t o continuous. Hereby we have established the von Neumann entropy spectral function as a valuable tool to identify the change of ground state degeneracies and of the spin- orbital entanglement of elementary excitations in quantum phase transitions. PACS numbers: 75.25.Dk, 03.67.Mn, 05.30.Rt, 75.10.Jm Submitted to: New J. Phys.Entanglement driven phase transitions in spin-orbital mod els 2 1. Spin-orbital physics and von Neumann entropy spectra In the Mott-insulating limit of a transition metal oxide the low-energy physics can be described by Kugel-Khomskii-type models [1], where both spin and orb ital degrees of freedom undergo joint quantum ﬂuctuations and novel types of s pin-orbital order [2] or disorder [3] may emerge. Following the microscopic derivation from the multiorbital Hubbard model, the generic structure of spin-orbital superexch ange takes the form of a generalized Heisenberg model [4,5], H=/summationdisplay /angbracketleftij/angbracketright/bardblγ/braceleftBig J(γ) ij(/vectorTi,/vectorTj)/vectorSi·/vectorSj+K(γ) ij(/vectorTi,/vectorTj)/bracerightBig , (1) as indeed found not only for the simplest systems with S= 1/2 spins: KCuF 3[1], the RTiO3perovskites [6], LiNiO 2and NaNiO 2[7], Sr 2CuO3[8], or alkali RO 2hyperoxides [9], but also for larger spins as e.g. for S= 2 in LaMnO 3[10]. In such models the parameters that determine the spin- SHeisenberg interactions stem from orbital operatorsJ(γ) ijandK(γ) ij— they depend on the bond direction and are controlled by the orbital degree of freedom which is described by pseudospin ope rators{/vectorTi}. That is, these parameters are not necessarily ﬁxed by rigid orbital orde r [3,11], but quantum ﬂuctuations of orbital occupation [12,13] may strongly inﬂuence t he form of the orbital operators, particularly in states with spin-orbital entanglement ( SOE) [14,15]. As a consequence, amplitudes and even the signs of the eﬀective excha nge can ﬂuctuate in time. Such entangled spin-orbital degrees of freedom can form ne w states of matter, as for instance the orbital-Peierls state observed at ﬁnite temper ature in YVO 3[16,17]. Another example are the collective spin and orbital excitations in a on e-dimensional (1D) spin-orbital chain under a crystal ﬁeld which can be universally described by fractionalized fermions [18]. It is challenging to ask which measure of S OE would be the most appropriate one to investigate quantum phase transition s in such systems. The subject is rather general and it has become clear that entang lement and other concepts from quantum information provide a useful perspective for the understanding of electronic matter [19–23]. Other examples of entangled system s are: topologically nontrivial states [24], relativistic Mott insulators with 5 dions [25], ultracold alkaline- earth atoms [26], and skyrmion lattices in the chiral metal MnSi [27]. One well-known characterization of a quantum system is the entang lement entropy (EE) determined by bipartitioning a system into AandBsubsystems. This subdivision can refer for example to space [19,28], momentum [28,29 ], or diﬀerent degrees of freedom such as spin and orbital [30]. A standard measu re is the von Neumann entropy (vNE), S0 vN≡ −TrA{ρ0 Alog2ρ0 A}, for the ground state \|Ψ0∝an}bracketri}htwhich is obtained by integrating the density matrix, ρ0 A= TrB\|Ψ0∝an}bracketri}ht∝an}bracketle{tΨ0\|, over subsystem B. Another important measure is the entanglement spectrum (ES) int roduced by Li and Haldane [31], which has been explored for gapped 1D spin systems [32 ], quantum Heisenbergladders[33], topologicalinsulators[34], bilayersandspin- orbitalsystems[28]. The ES is a property of the ground state and basically represents the eigenvalues piof the reduced density matrix ρ0 Aobtained by bipartitioning of the system. InterestinglyEntanglement driven phase transitions in spin-orbital mod els 3 a correspondence of the ES and the tower of excitations relevant for SU(2) symmetry breaking has been pointed out recently [35]. It was also noted that t he ES can exhibit singular changes, although the system remains in the same phase [36 ]. This suggests that the ES has less universal character than initially assumed [37]. In this paper we explore a diﬀerent entanglement measure, namely t he vNE spectrum which monitors the vNE of ground and excited states of the system, for instance of a spin-orbital system as deﬁned in equation (1). In this case we consider the entanglement obtained from the bipartitioning into spin and orbit al degrees of freedom in the entire system [30]. Here the vNE is obtained from the d ensity matrix, ρ(n) s= Tro\|Ψn∝an}bracketri}ht∝an}bracketle{tΨn\|, by taking the trace over the orbital degrees of freedom (Tr o) for each eigenstate \|Ψn∝an}bracketri}ht. We show below that the vNE spectrum, SvN(ω) =−/summationdisplay nTrs{ρ(n) slog2ρ(n) s}δ{ω−ωn}, (2) reﬂects the changes of SOE entropy for the diﬀerent states at p hase transitions. The excitation energies, ωn=En−E0, of eigenstates \|Ψn∝an}bracketri}htare measured with respect to the ground state energy E0. It has already been shown that the vNE spectra uncover a surprisingly large variation of entanglement within elementary excit ations [30]. Also certain spectral functions have been proposed, that can be det ermined by resonant inelastic x-ray scattering [38], and provide a measure of the vNE sp ectral function. Here we generalize this function to arbitrary excitations \|Ψn∝an}bracketri}ht, i.e.,beyondelementary excitations which refer to a particular ground state. We demonstr ate that focusing on general excited states opens up a new perspective that sheds ligh t on quantum phase transitions and the entanglement in spin-orbital systems. The paper is organized as follows: In section 2 we introduce the 1D sp in-orbital model with ferromagnetic exchange, and in section 3 we present its phase diagrams for the Ising limit of orbital interactions and for the anisotropic SU( 2)⊗XXZmodel with enhanced Ising component. SOE is analyzed in section 4 using bot h the spin- orbital correlation function and the entanglement entropy and we show that these two measures are equivalent. In section 5 we present the entanglemen t spectra and discuss their relation to the quantum phase transitions. The main conclusion s and summary are given in section 6. The distance dependence ofspin correlations in th eantiferromagnetic phase is explored in the Appendix. 2. Ferromagnetic SU(2) ⊗XXZspin-orbital model The motivation for our theoretical discussion of spin-orbital phys ics comes from t2g electron systems in which orbital quantum ﬂuctuations are enhanc ed by an intrinsic reduction of the dimensionality of the electronic structure [12]. Exa mples of strongly entangled quasi-1D t2gspin-orbital systems due to dimensional reduction arewell known and we mention here just LaTiO 3[6], LaVO 3and YVO 3[13], where the latter two involve{yz,zx}orbitals along the ccubic axis; as well as pxandpyorbital systems in 1D fermionic optical lattices [39–41]. This motivates us to consider th e 1D spin-orbitalEntanglement driven phase transitions in spin-orbital mod els 4 model forS= 1/2 spins and T= 1/2 orbitals with anisotropic XXZinteraction, i.e., with reduced quantum ﬂuctuation part in orbital interactions. The \|+∝an}bracketri}htand\|−∝an}bracketri}htorbital states are a local basis at each site and play a role of yzandzxstates int2gsystems, H(x,y,∆) =−JL/summationdisplay j=1/parenleftBig /vectorSj·/vectorSj+1+x/parenrightBig/parenleftBig [/vectorTj·/vectorTj+1]∆+y/parenrightBig , (3) [/vectorTj·/vectorTj+1]∆≡∆/parenleftbig Tx jTx j+1+Ty jTy j+1/parenrightbig +Tz jTz j+1, (4) whereJ >0 andweuse periodicboundaryconditions fora ringof Lsites, i.e.,L+1≡1. The parameters of this model are {x,y}and ∆. At x=y= 1/4 and ∆ = 1 the model has SU(4) symmetry. Hund’s exchange coupling does not only modify xandybut also leads to the XXZanisotropy (∆ <1), a typical feature of the orbital sector in real materials [5,12]. The antiferromagnetic model ( J=−1) is Bethe-Ansatz integrable at the SU(4) symmetric point [42] and its phase diagram is well establish ed by numerical studies [43,44]. It includes two phases with dimer correlations [45] wh ich arise near the SU(4) point. Some of its ground states could be even determined ex actly at selected (x,y,∆) points [46–50]. Here we are interested in the complementary and less explored mode l with negative (ferromagnetic) coupling ( J= 1), possibly realized in multi-well optical lattices [51], which has been studied so far only for SU(2) orbital interaction (∆ = 1) [30]. This model is physically distinct from the antiferromagnetic ( J=−1) model, except for the Ising limit (∆ = 0) where the two models can be mapped onto each other , but none was investigated so far. The phase diagrams for J= 1, see ﬁgure 1, determined using the ﬁdelity susceptibility [52] display a simple rule that the vNE (2) vanis hes for exact ground states of rings of length Lwhich can be written as products of spin ( \|ψs∝an}bracketri}ht) and orbital (\|ψo∝an}bracketri}ht) part,\|Ψ0∝an}bracketri}ht=\|ψs∝an}bracketri}ht⊗\|ψo∝an}bracketri}ht. 3. Phase transitions in the spin-orbital model To understand the role played by the SOE in the 1D spin-orbital mode l (3) and (4) we consider the phase diagrams for ∆ = 0 and ∆ = 0 .5, see ﬁgure 1. In the case ∆ = 0 all trivial combinations of ferro (F) and antiferro (A) spin-orbital phases labeled I-IV haveS0 vN= 0, i.e., spins and orbitals disentangle in all these ground states: FS/ FO, AS/FO, AS/AO, FS/AO. If both subsystems exhibit quantum ﬂuctu ations, the ground state\|Ψ0∝an}bracketri}htcan no longer be written in the product form. This occurs for the AS /AO phase III at ∆ >0. Most remarkable is the strongly entangled phase V at ∆ = 0 and y <0, see ﬁgure 1(a). This phase occurs near x≃ −∝an}bracketle{t/vectorSj·/vectorSj+1∝an}bracketri}htAF≡ln2−1/4, i.e., when the uniform antiferromagnetic spin correlations in phase III are compensated by the parameter x, so that the energy associated with Hamiltonian (3) de facto disappears . This triggers state V with strong SOE (see below) as the only option for the system to ga in substantial energy in this parameter range by nonuniform spin-orbital correla tions. The analysis ofEntanglement driven phase transitions in spin-orbital mod els 5 /s45/s49 /s48 /s49/s45/s49/s48/s49/s40/s97/s41 /s120/s121/s73/s73/s86 /s73/s73/s73/s73/s73/s86 /s45/s49 /s48 /s49/s45/s49/s48/s49/s40/s98/s41 /s86 /s120/s121/s73/s86 /s73 /s73/s73/s73/s73/s73/s86/s73 Figure 1. Phase diagrams of the spin-orbital model [equation (3)] obtained b y two methods, ﬁdelity susceptibility or an exact diagonalization of an L= 8 site model, for: (a) ∆ = 0, and (b) ∆ = 0 .5. The spin-orbital correlations in phases I-IV correspond to FS/FO, AS/FO, AS/AO, FS/AO order (see text). At ∆ = 0 only the gr ound state of a novel phase V has ﬁnite EE, S0 vN>0 (shaded), whereas at ∆ >0 the EE in phases III and VI is also ﬁnite. phase V in terms of the longitudinal equal-time spin (orbital) structu re factors Ozz(k) =1 L2L/summationdisplay m,n=1e−ik(m−n)∝an}bracketle{tOz mOz n∝an}bracketri}ht, (5) whereO=SorT, reveals in ﬁgure 2(a) at ∆ = 0 and y=−1/4 for the spin structure factorSzz(k)∝(1−cosk). This is a manifestation of nearest neighbour correlations, while further neighbour spin correlations vanish and moreover we ﬁn d a quadrupling in theorbitalsector, seeﬁgure2(b). Thusthespincorrelationsind icateeitherashort-range spin liquid or a translational invariant dimer state. Thehidden spin-dimer order [53] can be detected by the four-spin correlator (we use periodic boundary conditions), D(r) =1 LL/summationdisplay j=1/bracketleftbigg/angbracketleftBig (/vectorSj·/vectorSj+1)(/vectorSj+r·/vectorSj+1+r)/angbracketrightBig −/angbracketleftBig /vectorSj·/vectorSj+1/angbracketrightBig2/bracketrightbigg . (6) At ∆ = 0 we ﬁnd \|D(r)\|with long-range dimer correlations in phase V, but not in III. Phase III is a state with alternating ( k=π) spin (orbital) correlations in the range x <0.17 shown in ﬁgures 2(a,b). Interestingly for ∆ >0 the dimer spin correlations \|D(r)\|are not only present in phase V but also appear in phase III. Moreov er a phase VI emerges, complementary to phase V, with interchanged role of s pins and orbitals, see ﬁgure 1(b). The order parameters for phase VI follow from the fo rm of structure factors which develop similar but complementary momentum dependence to th at for phase V seen in ﬁgure 2(a), i.e., maxima at π/2 forSzz(k) and atπforTzz(k). We remark that phases V and VI are unexpected and they were overlooked before for the SU(2) ⊗SU(2) model at ∆ = 1 [30]. From the size dependence of \|D(r)\|in ﬁgures 2(c,d) we concludeEntanglement driven phase transitions in spin-orbital mod els 6 /s48 /s49 /s50/s48/s46/s48/s48/s46/s53/s49/s46/s48 /s49 /s50/s48/s49/s50/s40/s97/s41 /s32/s32/s83/s122/s122 /s40/s107/s41 /s107/s47/s32 /s120/s60/s48/s46/s49/s55 /s32 /s32/s120 /s48/s46/s49/s55 /s32/s40/s49/s45/s99/s111/s115/s32/s107/s41/s47/s50/s40/s98/s41/s84/s122/s122 /s40/s107/s41/s32/s32 /s107/s47/s32 /s32/s120/s60/s48/s46/s49/s55 /s32 /s32/s120 /s48/s46/s49/s55 /s48/s46/s49 /s48/s46/s51 /s48/s46/s53/s48/s46/s48/s48/s46/s49 /s48/s46/s49 /s48/s46/s51 /s48/s46/s53/s40/s100/s41/s40/s99/s41 /s120/s62 /s48/s46/s55/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s53 /s48/s46/s51/s32/s32/s32/s32/s32 /s48/s46/s52 /s32/s32/s32/s32/s32 /s32/s32/s32/s32/s32/s32/s124/s68/s40/s114/s41/s124 /s49/s47/s114/s32/s120/s61 /s48/s46/s54 /s32/s32/s32/s32/s32 /s32 /s32/s32 Figure 2. Top— Spin [ Szz(k)] and orbital [ Tzz(k)] structure factors (5) for the 1D spin-orbital model (3) of L= 8 sites at ∆ = 0 and y=−1/4: (a)Szz(k) and (b) Tzz(k). Bottom— Spin dimer correlations D(r) equation (6) found at ∆ = 0 .5 for decreasing 1 /rfor clusters of (c) L= 12 and (d) L= 16 sites. that the dimer correlations are long-ranged at ∆ = 0 .5 in phase V, but also in III, as seen from the data for x∈[0.0,0.4), where they coexist with the AS correlations. These results suggest thatthegroundstateVinﬁgure1(a)isfor medby spin-singlet product states \|ΦD 1∝an}bracketri}ht= [1,2][3,4][5,6]···[L−1,L], \|ΦD 2∝an}bracketri}ht= [2,3][4,5][6,7]···[L,1], (7) where [l,l+1] = (\|↑↓∝an}bracketri}ht−\|↓↑∝an}bracketri}ht )/√ 2 denotes a spin singlet. They arenot coupled to orbital singlets on alternating bonds as it happens for the AFantiferromag netic SU(2) ⊗SU(2) spin-orbital chain in a diﬀerent parameter regime [46], but to Ising co nﬁgurations in theEntanglement driven phase transitions in spin-orbital mod els 7 orbital sector. The four-fold ( k=π/2) periodicity of orbital correlations is consistent with four orbital states: \|Ψz 1∝an}bracketri}ht=\|++−−++···−−∝an}bracketri}ht, \|Ψz 2∝an}bracketri}ht=\|−++−−+···+−∝an}bracketri}ht, \|Ψz 3∝an}bracketri}ht=\|−−++−−···++∝an}bracketri}ht, \|Ψz 4∝an}bracketri}ht=\|+−−++−···−+∝an}bracketri}ht. (8) The decoupling of singlets is complete for y=−1/4 and ∆ = 0, where (++) and ( −−) bonds yield vanishing coupling in equation (3), and the phase boundar ies of region V arexc III,V= 3/4 + 2∝an}bracketle{t/vectorSj·/vectorSj+1∝an}bracketri}htAF≃0.136 andxc V,II= 3/4 in the thermodynamic limit; moreover we ﬁnd perfect long-range order of spin singlets, i.e., D(r) = (3/8)2(−1)r. The dimerized spin-singlet state at ∆ = 0 has the same spin structure as the Majumdar-Ghosh (MG) state [54], however its origin is diﬀerent. While the MG state in aJ1-J2Heisenberg chain results from frustration of antiferromagnetic e xchange (at J2=J1/2), here the spin singlets are induced by the SOE. At ∆ = 0 the only pha se with ﬁnite SOE S0 vN= 1 is phase V, see ﬁgure 3. In contrast, for ∆ >0 one ﬁnds ﬁnite EE also in phase III, when the original product ground state changes into a more complex superposition ofstatesandjointspin-orbital ﬂuctuations[14]a ppear. These correlations control the SOE and give equivalent information to S0 vN, see section 4. Furthermore, EE increases with xtowards phase V where it is further ampliﬁed andexceeds S0 vN= 1. The related softening of orbital order will be discussed below. Interes tingly we ﬁnd a one- to-one correspondence of ﬁnite EE and long-range order in the sp in dimer correlations /s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s49/s50/s32 /s32/s83 /s118/s78 /s120/s32 /s32 /s32 /s32 Figure 3. Spin-orbital entanglement entropy S0 vNin the ground state of the spin- orbital model (3) for the three phases III, V and II as a function ofxfor various ∆. Solid line for ∆ = 0 stands for the k= 0 ground state in the limit of ∆ →0. Parameters: y=−0.5 andL= 8.Entanglement driven phase transitions in spin-orbital mod els 8 \|D(r)\|. The superstructure of phase V emerges from the interplay of spin and orbitals, where orbitals modulate the interaction of spins in equation (3), and vice versa . It is important to distinguish this from the Peierls eﬀect, where the coup ling to the lattice is an essential mechanism. The orbital Peierls eﬀect observed in van adates [16,17] or the orbital-selective Peierls transition studied recently [55] fall into the former category, yet, as they involve orbital singlets — they are distinct from the cas e discussed here. 4. Spin-orbital entanglement Thedescriptionofspin-orbitalentanglement intermsofthevNEen tropy, asdiscussed in section 3, is a very convenient measure of entanglement. But it is als o a highly abstract measure. To capture its meaning, one has to refer to mathematica l intuition, namely to the fact that any product state, \|Ψ∝an}bracketri}ht=\|ψs∝an}bracketri}ht⊗\|ψo∝an}bracketri}ht, has zero vNE. That is, an entangled state is a state that cannot be written as a single product. A more p hysical measure are obviously spin-orbital correlation functions relative to their me an-ﬁeld value [14]. Such correlation functions vanish for product states where mean -ﬁeld factorization of the relevant product is exact, i.e., spins and orbitals are disentangle d. To detect spin-orbital entanglement in the ground state we evalua te here the joint spin-orbital bond correlation function C1for the SU(2) ⊗XXZmodel (3), deﬁned as follows for a nearest neighbour bond ∝an}bracketle{ti,i+1∝an}bracketri}htin the ring of length L[14], C1≡1 LL/summationdisplay i=1/braceleftBig/angbracketleftBig (/vectorSi·/vectorSi+1)(/vectorTi·/vectorTi+1)/angbracketrightBig −/angbracketleftBig /vectorSi·/vectorSi+1/angbracketrightBig/angbracketleftBig /vectorTi·/vectorTi+1/angbracketrightBig/bracerightBig .(9) The conventional intersite spin- and orbital correlation functions are: Sr≡1 LL/summationdisplay i=1/angbracketleftBig /vectorSi·/vectorSi+r/angbracketrightBig , (10) Tr≡1 LL/summationdisplay i=1/angbracketleftBig /vectorTi·/vectorTi+r/angbracketrightBig . (11) The above general expressions imply averaging over the exact (tr anslational invariant) ground state found from Lanczos diagonalization of a ring. While SrandTrcorrelations indicate the tendency towards particular spin and orbital order, C1quantiﬁes the spin- orbital entanglement — if C1∝ne}ationslash= 0 spin and orbital degrees of freedom are entangled and the mean-ﬁeld decoupling in equation (3) cannot be applied as it ge nerates uncontrollable errors. Figures 4(a) and 4(b) show the nearest neighbour correlation fun ctionsS1,T1and C1aty=−0.5, for ∆ = 0 and ∆ = 0 .5, respectively, as functions of x. The nearest neighbour spin correlation function S1is antiferromagnetic (negative) in all phases III, V and II shown in ﬁgure 4, while ( negative)T1indicates AO correlations in phase III andferro-orbital ( positive) inphase II. Finite ∆ = 0 .5 triggers orbital ﬂuctuations whichEntanglement driven phase transitions in spin-orbital mod els 9 /s45/s48/s46/s52/s48/s46/s48/s48/s46/s52 /s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s45/s48/s46/s52/s48/s46/s48/s48/s46/s52/s40/s97/s41 /s84 /s122/s61/s48/s84 /s122/s61/s54/s73/s73/s73 /s86/s73/s73 /s32/s32/s73/s73/s86 /s84 /s122/s61/s54 /s120/s32/s67 /s49 /s32/s84 /s49 /s32/s83 /s49/s84 /s122/s61/s48/s73/s73/s73 /s40/s98/s41 Figure 4. Nearest neighbour spin S1(10), orbital T1(11), and joint spin-orbital C1(9) correlations as obtained for a spin-orbital ring (3) with L= 12 sites and y=−0.5, as functions of xfor: (a) ∆ = 0, and (b) ∆ = 0 .5. The III-V phase boundary (dotted vertical line) in (b) has been determined by the m aximum of the ﬁdelity susceptibility [52]. lowerT1below the classical value of 0.25 found at ∆ = 0. In the intermediate sp in dimer phaseT1is negative for all ∆ >0, while it is zero for ∆ = 0. It is surprising that C1is positive in phase V at ∆ = 0 in spite of the classical Ising orbital interactions, see ﬁgure 4(a). It is also positive in phases II I and V at ∆ = 0 .5 [see ﬁgure 4(b)]. Note that positiveC1is found in the present spin-orbital chain with J >0, whileC1isnegative whenJ <0 [42]. In phase II C1vanishes in the entire parameter range as then the ground state can be written as a product. The s ame is true for phase III at ∆ = 0. We emphasize that the dependence of C1onxis completely analogous to that of the von Neumann entropy in ﬁgure 3, which also displays a bro ad maximum in the vicinity of the III-Vphase transition at ∆ = 0 .5, and a step-like structure in phase VEntanglement driven phase transitions in spin-orbital mod els 10 at ∆ = 0. Thus we conclude here that the vNE yields a faithful measur e of SOE in the ground state that is qualitatively equivalent to the more direct entanglement measure via the spin-orbital correlation function C1[14]. 5. Entanglement spectra and quantum phase transitions Figure 3 stimulates the question about the origin and the understan ding of the sudden or gradual EE changes at phase transitions. This can be resolved b y exploring the vNE spectral function deﬁned in equation (2) and shown in ﬁgures 5 (a) and 5(b) for ∆ = 0 and 0.5, where colors encode the vNE of states. The excita tion energies ωn(x) =En(x)−E0(x) are plotted here as function of the parameter x. Only the lowest excitationsareshownthatarerelevantforthephasetransitions andthelow-temperature physics. They include: ( i) the elementary excitations of the respective ground state, and (ii) the many-body excited states that are relevant for the phase t ransition(s) and may become ground states or elementary excitations in neighbourin g phases when the parameterxis varied. The AS/FO ground state of phase II in ﬁgure 5(a) obtained for a rin g ofL= 8 sites is an AS singlet ( S= 0) with a maximal orbital quantum number, T=L/2 = 4, and a twofold ( k= 0,π) degeneracy at ∆ = 0. The spin excitation spectrum appears as horizontal (red) lines and consists of gapless triplet S= 1 excitations. The low- lying excitations of the Bethe-Ansatz-solvable antiferromagnetic Heisenberg chain form a two-spinon ( s-¯s) continuum, whose lower bound is given by ε(k) =π\|sink\|/2 in the thermodynamic limit [56]. For the L= 8 ring the spectrum is discrete with a ∆ k=π/4 spacing, and it is known that the energy of triplet excitations εS(π) will scale to zero as 1/L[57–59]. Red lines in phase II with ﬁnite slope are orbital excitations. T he x-dependence is due to the spin part of H(3) which determines both the orbital energy scale and the dispersion, JT≡(x+∝an}bracketle{t/vectorSj·/vectorSj+1∝an}bracketri}htAF)(1−∆cosk). This energy changes with xand at ﬁnite ∆ also with momentum k, see ﬁgure 5(b). While the orbitons are gapped, the low-lying excitations are either magnons or x-dependent spin-orbital excitations. It is remarkable that the latter are entangled in general, although the ground state II is disentangled. Withdecreasing xa ﬁrst-orderphase transitionfromIItoVoccursby level crossin g of disentangled (red) and entangled (green) ground states. The spin-singlet ( S= 0) ground state of phase V has degeneracy 4 at ∆ = 0, and its compone nts are labeled by the momenta k= 0,±π/2,π. This is reﬂected by ﬁnite ϕTorder parameter in ﬁgure 6(a). Note that at ∆ >0 this four-fold ground state degeneracy is lifted. In the spin-dimer phase a gap opens in the spectrum of elementary spin exc itations [60,61]. The one-magnon triplet gap ∆ S(δ)∝δ3/4depends on yvia the dimerization parameter δ≡1/\|4y\|. In phase V even the pure magnetic excitations are entangled [see h orizontal green lines in ﬁgure 5(a)]. The lowest excitations in the vicinity of the p hase transitions have orbital character. From ﬁnite EE in ﬁgure 5(a) one recognize s that these states are inseparable spin-orbital excitations.Entanglement driven phase transitions in spin-orbital mod els 11 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52 /s83/s79/s83/s79 /s107/s61/s48/s47/s107/s61/s48/s47 /s32/s32 /s122 /s40 /s41/s32/s115 /s40 /s41 /s115 /s40 /s41 /s115 /s40 /s41/s69 /s110/s45/s69 /s48 /s120/s73/s73/s73/s86/s73/s73/s115 /s40 /s41 /s107/s61/s48/s47/s83/s61/s48/s32/s84/s61/s48/s40/s97/s41 /s48/s46/s48/s48/s48/s46/s50/s53/s48/s48/s46/s53/s48/s48/s48/s46/s55/s53/s48/s49/s46/s48/s48/s49/s46/s50/s53/s49/s46/s53/s48/s49/s46/s55/s53/s50/s46/s48/s48 /s83/s61/s48/s32/s84/s61/s48 /s83/s61/s48/s32/s84/s61/s52 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s50/s52/s122 /s40 /s41 /s83/s79 /s40 /s41/s32/s84 /s40 /s41 /s83/s61/s48/s32/s32/s32 /s32/s32/s84 /s40 /s41 /s83/s61/s48/s32/s84/s61/s52/s32 /s107/s61/s84 /s40 /s41 /s32/s32 /s122 /s40 /s41 /s122 /s40 /s41/s115 /s40 /s41/s69 /s110/s45/s69 /s48 /s120/s73/s73/s73 /s86 /s73/s73/s48/s46/s48/s48/s48/s46/s51/s50/s48/s48/s46/s54/s52/s48/s48/s46/s57/s54/s48/s49/s46/s50/s56/s49/s46/s54/s48/s49/s46/s57/s50/s50/s46/s50/s52/s50/s46/s53/s54/s50/s46/s56/s56 /s83/s61/s48/s32/s107/s61/s48/s40/s98/s41 Figure 5. vNE-spectrum of lowest energies En(x) (relative to the ground state energy E0(x)) versusxwith colors representing the size of the vNE of individual states. Da ta for the three phases III, V and II is shown for y=−0.5,L= 8 and: (a) ∆ = 0, and (b) ∆ = 0.5. HereεS(k) [εT(k)] denotes spin (orbital) excitation, εz(k) corresponds to an elementary excitation having the same SandTas the ground state, and εSO(k) stands for the spin excitation under simultaneous ﬂipping of orbitals . The phase transition from the dimer phase V to the AS phase III ( S= 0) appears singular in the sense that it is ﬁrst order at ∆ = 0 and continuous othe rwise [ﬁgures 5(a,b)]. Tolocatethecenter ofthecontinuous phasetransitionbe tween phases IIIandVEntanglement driven phase transitions in spin-orbital mod els 12 /s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s69 /s48/s45/s69/s86 /s48/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48 /s32/s67/s83 /s49 /s32/s84 /s32/s111/s114/s100/s101/s114/s32/s112/s97/s114/s97/s109/s101/s116/s101/s114/s40/s97/s41 /s32 /s120/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s40/s98/s41 /s32/s67/s83 /s49 /s32/s32/s84 /s32/s32 Figure 6. Ground state energy relative to phase V, E0(x)−EV 0(x) (dots), orbital order parameters (dashed), ψT= [Tzz(π)]1 2,ϕT= [Tzz(π/2)]1 2, and bond spin correlations \|S1\|=\|∝an}bracketle{t/vectorS1·/vectorS2∝an}bracketri}ht\|(solid line), for phases III, V and II (from left to right), for: (a) ∆ = 0.0 and (b) ∆ = 0 .5. Parameters: y=−0.5 andL= 12. at ∆>0, we have selected the peak of the ﬁrst derivative of the entangle ment entropy, see ﬁgure 3. Yet also the peaks in the derivatives of the ﬁdelity susc eptibility, the orbital correlation function T1[see ﬁgure 4(b)] and the orbital order parameters ψTandϕTin ﬁgure 6(b) may be used. Finally, we note that the scaling of entangle ment with system size has quite diﬀerent behaviour in phases III and V, indicating that a phase transition separates them. Furthermore, the peculiar feature of the AS/AO phase III manife sts itself in a twofold degeneracy and zero SOE at ∆ = 0 in contrast to the nondeg enerate ground state and ﬁnite SOE at ﬁnite ∆. The entanglement has two sources, namely: (i) the interplayofquantumﬂuctuationsinthespinandorbitalsectorsan d(ii)thedimerization order which coexists with antiferromagnetic spin correlations in pha se III at ﬁnite ∆. The latter is the origin of the nondegenerate ground state as it yield s a coupling to theεSO(π) excitation (nearly horizontal in x), and the emergence of the spin-dimer correlations D(r) leads to a faster decay of the spin correlations in phase III than in the 1D antiferromagnetic Heisenberg chain, see the Appendix. The orb ital order parameters ψTandϕTcompete in phases III and V, see ﬁgure 6(b), near the phase boun dary inEntanglement driven phase transitions in spin-orbital mod els 13 ﬁgure 1(b). This also explains why the transition from phase V to III is smooth at ﬁnite ∆ in terms of both the vNE (ﬁgure 3) and the nearest neighbour spin correlations \|S1\|. 6. Conclusions and summary Summarizing, we have studied the quantum phases and the spin-orb ital entanglement of the 1D ferromagnetic SU(2) ⊗XXZmodel by means of the Lanczos method. We have discovered a previously unknown translational invariant phase V wit h long-range spin singlet order and four-fold periodicity in the orbital sector. Its me chanism is distinct from the dimer phases found in the 1D antiferromagnetic spin-orbit al model near the SU(4) symmetric point [45]. Both III-V and II-V phase transitions arise from the spin-orbital entanglement in the case of Ising orbital interactions . When the orbital interactions change from Ising to anisotropic XXZ-type, the entanglement develops in phase III, where antiferromagnetic spin correlations and long-r ange spin dimer order coexist, changing the quantum phase transition from ﬁrst- order to continuous. Furthermore in the regime of ﬁnite orbital ﬂuctuations (∆ >0) another phase VI emerges, which is complementary in many aspects to phase V, but wit h the important diﬀerence that phase VI disappears in the limit ∆ = 0. We have shown that the von Neumann entropy spectral function SvN(ω) (2) is a valuable tool that captures the spin-orbital entanglement SOE o f excitations and explains the origin of the entanglement entropy change at a phase t ransition. From the perspective of spin-orbital entanglement we encounter ( i) ﬁrst-order transitions between disentangled (II) and entangled (V) phases, ( ii) a continuous transition involving two competing order parameters between two entangled phases, III and V, and ( iii) trivial ﬁrst-order transitions between two disentangled phases. Case ( ii) goes beyond the commonly accepted paradigm of a single order parameter to charac terize a quantum phase. Moreover, we have presented two simple measures of entanglemen t in the ground stateandshownthattheyarebasicallyequivalent —thedirectmeas ureviathe(quartic) spin-orbital bond correlation function C1(9) and the von Neumann entropy S0 vN. The latterisdeﬁnedbyseparating globallyspinfromorbitaldegreesoff reedomintheground state. Acknowledgments We thank Bruce Normand and Krzysztof Wohlfeld for insightful disc ussions. W-L You acknowledges support by the Natural Science Foundation of Jiang su Province of China under Grant No. BK20141190 and the NSFC under Grant No. 11474 211. A M Ole´ s kindly acknowledges support by Narodowe Centrum Nauki (NCN, Na tional Science Center) under Project No. 2012/04/A/ST3/00331.Entanglement driven phase transitions in spin-orbital mod els 14 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s50/s52/s54/s56 /s48/s46/s53 /s49/s46/s48/s124/s60/s83 /s48/s83 /s114/s62/s124 /s49/s47/s114/s32/s120/s61/s45/s49/s46/s48 /s32/s120/s61/s45/s48/s46/s53 /s32/s120/s61/s48/s46/s48 /s32/s120/s61/s48/s46/s49 /s32/s120/s61/s48/s46/s50 /s32/s120/s61/s48/s46/s51 /s32/s120/s61/s48/s46/s52 /s32/s120/s61/s48/s46/s53 /s32/s120/s61/s48/s46/s54 /s32/s120/s40/s97/s41/s40/s98/s41 /s49/s47/s114 Figure 7. Modulus of spin correlations Srequation (10) versus the inverse distance 1/ras obtained for a spin-orbital ring (3) with L= 16 sites, for: (a) ∆ = 0, and (b) ∆ = 0.5. Parameter: y=−0.5. Appendix: Distance dependence of the antiferromagnetic sp in correlations Here we explore in more detail the competition of the antiferromagn etic (AF) spin correlations of the spin-orbital chain in the AS/AO phase III and th e Majumdar-Ghosh like spin-singlet dimer correlations that coexist at ﬁnite ∆, as we foun d in our work. For ∆ = 0 the spin correlations in phase III are those of an AF Heisenb erg spin chain, /angbracketleftBig /vectorSi·/vectorSi+r/angbracketrightBig ∼(−1)r/radicalbig ln\|r\| \|r\|, (12) whichreveal thetypical1 /r-powerlawdecaycombined withlogarithmiccorrectionsthat were ﬁrst predicted by conformal ﬁeld theory [62,63] as well as by renormalization group methods [64], and subsequently conﬁrmed [65] by numerical density matrix method [66]. In ﬁgure 7(a) we present our numerical data for the spin-correla tion function Sr equation (10) (i.e., for translational invariant ground states) for several values of x, and for ∆ = 0 and y=−0.5. In the ∆ = 0 case there are only two distinct types of behaviour of Sr, namely exponential decay in phase V and the power law decay of the 1D quantum N´ eel spin liquid state, which are the same in phases II an d III. Figure 7(b) displays Srat ∆ = 0 .5 for diﬀerent x-values. Here again the unperturbed AF correlations of the 1D N´ eel spin-liquid state appe ar in phase II (x≥0.7). It is evident that in phase III the AF spin correlations are stron gly reduced, due to the competition with the coexisting long-range ordered spin- singlet correlations. The spin singlet order increases with xin phase III, and as a consequence we observe here that the decay of Srbecomes stronger as xapproaches the III/V transition. In ﬁgure 8 we present a logarithmic plot which highlights the diﬀerent d ecays ofSrEntanglement driven phase transitions in spin-orbital mod els 15 /s48 /s50 /s52 /s54 /s56/s45/s52/s45/s50/s48/s50/s108/s110/s40/s124/s60/s83 /s48/s83 /s114/s62/s124/s41 /s114/s32/s120/s61/s48/s46/s55 /s32/s108/s105/s110/s101/s97/s114/s32/s102/s105/s116 /s32/s120/s61/s48/s46/s51 /s32/s108/s105/s110/s101/s97/s114/s32/s102/s105/s116 /s32/s120/s61/s48/s46/s54 /s32/s108/s105/s110/s101/s97/s114/s32/s102/s105/s116 Figure 8. Logarithm of modulus of spin correlations Srequation (10) for increasing distanceras obtained for the spin-orbital model equation (3) on a ring of L= 16 sites for ∆ = 0.5,y=−0.5, and three values of x. Exponential decay of Srwith increasing ris obtained for phase V ( x= 0.6). for ∆ = 0.5 in the three diﬀerent phases: III, V, and II. We have selected th e values for x= 0.3, 0.6 and 0.7, respectively, for greater transparency. The log-plot shows c learly the exponential decay of Srin phase V. It also shows that the L= 16 system reveals strong ﬁnite size eﬀects in phase II where Srhas power law decay. Nevertheless it is clear already from the L= 16 data that the AF spin correlations in phase III (here shown forx= 0.3) are strongly suppressed and approach the exponential decay ofSrin phase V (x= 0.5 and 0.6) when xapproaches the III-V phase boundary from the left. Summarizing, we ﬁnd that in phase III the AF spin correlations of the 1D N´ eel spin liquid state decay much more rapidly as the competing spin-singlet order emerges. This eﬀect is particularly strong near the boundary of phase III to the spin-singlet dimer phase V. Whether in the thermodynamic limit the correlations Sralso decay exponentially in phase III as in V cannot be decided here, and this que stion is beyond the scope of the present work. References [1] Kugel K I and Khomskii D I 1973 JETP37725 Kugel K I and Khomskii D I 1982 Sov. Phys. Usp. 25231 [2] Brzezicki W, Ole´ s A M and Cuoco M 2015 Phys. Rev. X 5011037 [3] Corboz P, Lajk´ o M, L¨ auchli A M, Penc K and Mila F 2012 Phys. Rev. X 2041013 [4] Tokura Y and Nagaosa N 2000 Science288462 [5] Ole´ s A M, Khaliullin G, Horsch P and Feiner L F 2005 Phys. Rev. B 72214431 [6] Khaliullin G and Maekawa S 2000 Phys. Rev. Lett. 853950Entanglement driven phase transitions in spin-orbital mod els 16 Mochizuki M and Imada M 2004 New J. Phys. 6154 Pavarini E, Yamasaki A, Nuss J and Andersen O K 2005 New J. Phys. 7188 [7] Reitsma A, Feiner L F and Ole´ s A M 2005 New J. Phys. 7121 [8] Wohlfeld K, Nishimoto S, Haverkort M W and van den Brink J 2013 Phys. Rev. B 88195138 [9] Solovyev I V 2008 New J. Phys. 10013035 Wohlfeld K, Daghofer M and Ole´ s A M 2011 Europhys. Lett. 9627001 [10] Feiner L F and Ole´ s A M 1999 Phys. Rev. B 593295 [11] Zhou J-S, Ren Y, Yan J-Q, Mitchell J F and Goodenough J B 2008 Phys. Rev. Lett. 100046401 [12] Khaliullin G 2005 Prog. Theor. Phys. Suppl. 160155 [13] Horsch P, Ole´ s A M, Feiner L F and Khaliullin G 2008 Phys. Rev. Lett. 100167205 [14] Ole´ s A M, Horsch P, Feiner L F and Khaliullin G 2006 Phys. Rev. Lett. 96147205 [15] Ole´ s A M 2012 J. Phys.: Condens. Matter 24313201 [16] Ulrich U, Khaliullin G, Sirker J, Reehuis M, Ohl M, Miyasaka S, Tokura Y and Keimer B 2003 Phys. Rev. Lett. 91257202 Horsch P, Khaliullin and Ole´ s A M 2003 Phys. Rev. Lett. 91257203 [17] Sirker J, Herzog A, Ole´ s A M and Horsch P 2008 Phys. Rev. Lett. 101157204 Herzog A, Horsch P, Ole´ s A M and Sirker J 2011 Phys. Rev. B 83245130 [18] Chen C-C, van Veenendaal M, Devereaux T P and Wohlfeld K 2015 Phys. Rev. B 91165102 [19] Amico L, Fazio R, Osterloh A and Vedral V 2008 Rev. Mod. Phys. 80517 [20] Zhou H-Q, Barthel T, Fjaerestad J O and Schollw¨ ock U 2006 Phys. Rev. A 74050305(R) [21] Byczuk K, Kunes J, Hofstetter W and Vollhardt D 2012 Phys. Rev. Lett. 108087004 [22] Held K and Mauser N J 2013 Eur. Phys. J. B 86328 [23] Veenstra C N, Zhu Z-H, Raichle M, Ludbrook B M, Nicolaou A, Sloms ki B, Landolt G, Kittaka S, Maeno Y, Dil J H, Elﬁmov I S, Haverkort M W and Damascelli A 2014 Phys. Rev. Lett. 112 127002 [24] Hasan M Z and Kane C L 2010 Rev. Mod. Phys. 823045 [25] Jackeli G and Khaliullin G 2009 Phys. Rev. Lett. 102017205 [26] Gorshkov A V, Hermele M, Gurarie V, Xu C, Julienne P S, Ye J, Zoller P, Demler E, Lukin M D and Rey A M 2010 Nature Phys. 6289 [27] M¨ uhlbauer S, Binz B, Jonietz F, Pﬂeiderer C, Rosch A, Neubaue r A, Georgii R and B¨ oni P 2009 Science323915 [28] Lundgren R, Chua V and Fiete G A 2012 Phys. Rev. B 86224422 [29] Thomale R, Arovas D P and Bernevig B A 2010 Phys. Rev. Lett. 105116805 [30] You W-L, Ole´ s A M and Horsch P 2012 Phys. Rev. B 86094412 [31] Li H and Haldane F D M 2008 Phys. Rev. Lett. 101010504 [32] Pollmann F, Turner A M, Berg E and Oshikawa M 2010 Phys. Rev. B 81064439 Alba V, Haque M and L¨ auchli A M 2012 Phys. Rev. Lett. 108227201 [33] Poilblanc D 2010 Phys. Rev. Lett. 105077202 [34] Fidkowski L 2010 Phys. Rev. Lett. 104130502 Regnault N and Bernevig B A 2011 Phys. Rev. X 1021014 [35] Kolley F, Depenbrock S, McCulloch I P, Schollw¨ ock U and Alba V 201 3Phys. Rev. B 88144426 [36] Lundgren R, Blair J, Greiter M, L¨ auchli A, Fiete G A and Thomale R 2014Phys. Rev. Lett. 113 256404 [37] Chandran A, Khemani V and Sondhi S L 2014 Phys. Rev. Lett. 113060501 [38] Ament L J P, van Veenendaal M, Devereaux T P, Hill J P and van de n Brink J 2011 Rev. Mod. Phys.83705 [39] Zhao E and Liu W V 2008 Phys. Rev. Lett. 100160403 Wu C 2008 Phys. Rev. Lett. 100200406 Wu C and Das Sarma 2008 Phys. Rev. B 77235107 [40] Sun G, Jackeli G, Santos L and Vekua T 2012 Phys. Rev. B 86155159 [41] Zhou Z, Zhao E and Liu W V 2015 Phys. Rev. Lett. 114100406Entanglement driven phase transitions in spin-orbital mod els 17 [42] Li Y-Q, Ma M, Shi D-N and Zhang F-C 1998 Phys. Rev. Lett. 813527 Frischmuth B, Mila F and Troyer M 1999 Phys. Rev. Lett. 82835 [43] Itoi C, Qin S and Aﬄeck I 2000 Phys. Rev. B 616747 Orignac E, Citro R and Andrei N 2000 Phys. Rev. B 61, 11533 [44] Yamashita Y, Shibata N and Ueda K 2000 J. Phys. Soc. Jpn. 69242 [45] Li Peng and Shen Shun-Qing 2005 Phys. Rev. B 72214439 [46] Kolezhuk A K and Mikeska H-J 1998 Phys. Rev. Lett. 802709 [47] Kolezhuk A K, Mikeska H-J and Schollw¨ ock U 2001 Phys. Rev. B 63064418 [48] Martins M J and Nienhuis B 2000 Phys. Rev. Lett. 854956 [49] Kumar B 2013 Phys. Rev. B 87195105 [50] Brzezicki W, Dziarmaga J and Ole´ s A M 2014 Phys. Rev. Lett. 112117204 [51] Belemuk A M, Chtchelkatchev N M and Mikheyenkov A V 2014 Phys. Rev. A 90023625 [52] You W-L, Li Y-W, and Gu S-J 2007 Phys. Rev. E 76022101 [53] Yu Y, M¨ uller G and Viswanath V S 1996 Phys. Rev. B 549242 [54] Majumdar C K and Ghosh D 1969 J. Math. Phys. 101388 [55] Streltsov S V and Khomskii D I 2014 Phys. Rev. B 89161112(R) [56] des Cloizeaux J and Pearson J J 1962 Phys. Rev. 1282131 [57] Horsch P and von der Linden W 1988 Z. Phys. B 72181 [58] Aﬄeck I 1993 Rev. Math. Phys. 6887 [59] Koma T and Tasaki H 1994 J. Stat. Phys. 76745 [60] Spronken G, Fourcade B and L´ epine Y 1986 Phys. Rev. B 331886 [61] Uhrig G S and Schulz H J 1996 Phys. Rev. B 54R9624 [62] Aﬄeck I, Gepner D, Schulz H and Ziman T 1989 J. Phys. A: Math. and Theor. 22511 [63] Lukyanov S 1998 Nuclear Physics B 522533 [64] Giamarchi T and Schulz H J 1989 Phys. Rev. B 394620 Singh R R, Fisher M E and Shankar R 1989 Phys. Rev. B 392562 [65] Hallberg K A, Horsch P and Mart´ ınez G 1995 Phys. Rev. B 52R719 [66] Schollw¨ ock U 2005 Rev. Mod. Phys. 77259
1610.02927v2.The_Role_of_Interaction_in_the_Pairing_of_Two_Spin_orbit_Coupled_Fermions.pdf	Role of interaction in the binding of two Spin-orbit Coupled Fermions Chong Ye,1, 2Jie Liu,2, 3Li-Bin Fu1, 1Graduate School, China Academy of Engineering Physics, Beijing 100193, China 2National Laboratory of Science and Technology on Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China 3HEDPS, CAPT, and CICIFSA MoE, Peking University, Beijing 100871, China We investigate role of an attractive s-wave interaction with positive scattering length in the binding of two spin-orbit coupled fermions in the vacuum and on the top of a Fermi sea in the single impurity system, motivated by current interests in exploring exotic binding properties in the appearance of spin-orbit couplings. For weak spin-orbit couplings where the density of states is not signicantly altered, we analytically show that the high-energy states become more important in determining the binding energy when the scattering length decreases. Consequently, tuning the interaction gives rise to a rich behavior, including a zigzag of the momentum of the bound state or inducing transitions among the meta-stable states. By exactly solving the two-body quantum mechanics for a spin-orbit coupled Fermi mixture of40K-40K-6Li, we demonstrate that our analysis can also apply to the case when the density of states is signicantly modied by the spin-orbit coupling. Our ndings pave a way for understanding and controlling the binding of fermions in the presence of spin orbit couplings. PACS numbers: 03.65.Ge, 71.70.Ej, 67.85.Lm I. INTRODUCTION In ultracold physics, many schemes have been proposed to generated various types of synthetic spin-orbit cou- plings (SOC) by controlling atom-light interaction [1]. In 2011, I. B. Spielman's group in NIST had gener- ated an equal weight combination of Rashba-type and Dresselhaus-type SOC in87Rb [2]. Afterwards, SOC has triggered a great amount of experimental interest [3{5]. In the appearance of SOC, the ultracold atomic gases have been altered dramatically [6{8]. One basic issue is the binding of two spin-orbit cou- pled fermions in the vacuum [9{16] where SOC has given rise to the change of binding energy and the appearance of nite-momentum dimer bound states. Another rele- vant issue is the binding of two fermions on the top of a Fermi sea (the molecular state) for the case where a single impurity is immersed in a noninteracting Fermi gas [16{21]. In the appearance of SOC, the center-of- mass (c.m.) momentum of the molecular state becomes nite [16, 21]. All of these can be understood from the perspective of two-body quantum mechanics. It general contains three components: the threshold energy associ- ated with the c.m. momentum, the density of states, and the interacting strength. For extremely weak attractive interaction, changes of two-body properties under SOC came from the dierent threshold behavior of the density of states [9, 10, 14]. However, in the strong interacting regime, the binding of two fermions presents a rich behav- ior [13, 16] such as the variation of the c.m. momentum and the competition between two meta-stable states with the tuning of interacting strength. These phenomena can lbfu@gscaep.ac.cnnot simply owe to the threshold behavior of the density of states. Therefore, the mechanism as to how all these three components cooperate with each other in deter- mining the novel two-body properties is pressing needed. The establishment of such a comprehensive picture will shed light on ongoing explorations of the intriguing be- havior of spin-orbit coupled Fermi gases [9{16]. Below, we report a theoretical contribution to address this issue, which also allows predictions of new phenomena. We investigate the two-body quantum mechanisms of the binding of two spin-orbit coupled fermions in the vac- uum and on the top of a Fermi sea in the single impurity Fermi gas. We consider an attractive s-wave interaction with positive scattering length, the strength of which can be tuned in a wide range via a Feshbach resonance [22]. From Sec. II to Sec. IV, we give analyses which do not dependent on the concrete type of SOC. In Sec. II, by decomposing the two-body energy (molecular energy) into the threshold energy and the binding energy, both of which depend on the c.m. momentum of two fermions, we establish a direct relation between the interaction, the density of states, and the binding energy. In Sec. III, with the rst-order perturbation analysis in the weak SOC limit, we reveals that the low-energy states play a decisive role in determining the binding energy when the scattering length is large, in contrast to the small scattering length case where the high-energy states can dominate. This allows us to elucidate the mechanism underlying interesting phenomena such as a zigzag be- havior of the two-body ground state momentum and the competition between two meta-stable states in Sec. IV. In Sec. V, we illustrate our analysis with an interact- ing Fermi mixture of40K-40K-6Li with40K containing an (kxz+hx)-type SOC, which can be realized by the state of the art experimental techniques using cold atoms [4, 23]. Remarkably, by exactly solving the two-arXiv:1610.02927v2 [cond-mat.quant-gas] 24 Feb 20182 body problem for this system, we show that our analysis aords insights into the main properties of the binding of two spin-orbit coupled fermions, even when the density of states is signicantly altered by SOC. Our ndings re- veal the role of interaction in the binding of two spin-orbit coupled fermions and allow deep physical understandings of the rich two-body properties in the presence of SOC. II. BINDING OF TWO FERMIONS WITH SOC We consider two dierent spin-orbit coupled fermionic species in three dimensions (3D) at zero temperature: atom A (B) has Na(Nb) components, with the corre- sponding non-interacting Hamiltonian Ha(Hb). We con- sider an attractive s-wave contact interaction with posi- tive scattering length between the two fermionic species as described by Hint=U VX Q;k;k0ay k;l0by Qk;m0ak0;l0bQk0;m0;(1) withQthe c.m. momentum of two scattering fermions. Hereay k;l0(by k;m0) denotes the creation operator of a SOC-free atom A (B) in the l0-th (m0-th) spin compo- nent with momentum k,Uis the bare interaction, and Vis the quantization volume. The total Hamiltonian is thusH=Ha+Hb+Hint. With this Hamiltonian, we address to the binding of two fermionic atoms A and B (i) in the vacuum [9{16] and (ii) on the top of a non-interacting Fermi sea of atoms Ain the situation where a single impurity of Bis im- mersed in a non-interacting Fermi gas of A[17{21]. The ansatz wave function of the two-body bound state and the molecular state can be expressed in a general form j Qi=X i;j0X k i;j Q;ky k;iy Qk;jj?i; (2) where Qis the c.m. momentum of two particles. For (i), j?iis the vacuum state and the summationP0 kincludes all the states. For (ii), j?iis the non-interacting spin- orbit coupled Fermi sea of Aand the summationP0 k excludes the states below the Fermi surfaces, re ecting the eect of Pauli blocking. Here, y k;i=P i0i;i0 kay k;i0 (y k;i=P i0i;i0 kby k;i0) is the creation operator of an atom A (B) in the i-th eigen-state of Hamiltonian Ha(Hb) with momentum kand energy "a k;i("b k;i) and i;j Q;kdenotes the variational coecient. The coecients i;i0 kandi;i0 k are xed by SOC. Solving the eigen-equation Hj Qi= EQj Qigives i;j Q;k=(i;l0 kj;m0 Qk) EQEij Q;kU V0X k0;i0;j0 i0;j0 Q;k0i0;l0 k0j0;m0 Qk0;(3) withEij Q;k="a k;i+"b Qk;j. Rearranging Eq. (3), we obtain a self-consistent equation for two-body energy(molecular energy) EQin the momentum-space repre- sentation, i.e., 1 U=1 VX i;jX k0ji;l0 kj2jj;m0 Qkj2 EQEij Q;k: (4) A key step of our treatment next constitutes a decom- position ofEQ: Dening the threshold energy associated with the c.m. momentum QbyEQ thmini;j;kfEij Q;kg, we writeEij Q=EQ th+". The rest of the two-body energy (molecular energy) is therefore Esc QEQEQ th. While EQ this only aected by SOC, Esc Qencodes the eect of interaction. Such decomposition of EQin terms of Esc Q andEQ th, as we shall see, allows a transparent correspon- dence to the SOC-free counterpart. Following a standard procedure, we obtain the self-consistent equation for Esc Q in the energy domain of "as in Ref.[24] Z1 0 " Qd" Esc Q"=1 U: (5) Here, " Qis dened by " Q=X iX jZ0 ji;l0 kj2jj;m0 Qkj2jJjdd; (6) which describes the density of states in 3D[25]. For (i), the integrationR0dd includes all the states. For (ii), the integrationR0dd excludes the states below the Fermi surfaces. In Eq. (6), andlabel the degrees of freedom other than ", andJdenotes the standard Ja- cobian. These formulas can be also easily adapted to de- scribing the binding of two homo-nuclear fermions where AandBare the same fermionic species. Equation (5) establishes a direct relation between the interaction U, the density of states " Q, andEsc Q. Intu- ition behind it can be gained in the limit of vanishing SOC in case (i), where EQ th=Q2=(2m) withmthe reduced mass of two fermions, and " Q= " 0= 2p2m". Then,Esc Qis independent of Qas ensured by Eq. (5), and can be identied as Esc Q"b=1=(2ma2 s) [~1] withas>0 the s-wave scattering length, i.e., the binding energy at rest. In this case, Eq. (5) reduces to, in the momentum space representation, the well known renor- malization equation for two scattering particles, i.e., 1 U=m 2as1 VX k2m k2: (7) Equation (5) thus extends the standard prescription for two interacting fermions to the presence of SOC, where Esc Qis the counterpart of the binding energy "b. III. ROLE OF INTERACTION Based on above treatment, below we elucidate how the interaction cooperates with the eect of SOC in deter-3 mining the behavior of Esc Q, when the interaction strength a1 sis tuned in a wide range via Feshbach resonance [22]. To compare to the SOC-free case, we introduce the quan- tityQEsc Q"b. For weak SOC that does not signi- cantly alter the density of states, the leading term of Q can be derived from Eq. (5) as [26]: Q=hZ1 0 " Q (""b)2d"i1Z1 0 " Q " 0 ""bd": (8) Here we have ignored the modication of the renormal- ization relation by SOC [27{31]. In discussing the eect of interaction on Q, we will be interested in (i)@Q @a1 s and (ii) QQ0Q0Q: The sign of the former re ects howQfor xed Qchanges with interaction, while that of the latter tells whether a large or small Qis energet- ically favored for a given interaction. Using Eq. (8), we nd QQ0'[R1 0 " Q (""b)2d"]1R1 0 " Q0 " Q ""bd". Both of Qand QQ0rely crucially on " Q. Thus, while the form of " Qvaries with specic setups [see Eq. (6)], its qualita- tive analysis aords insights into generic behavior of Q, as we elaborate next. In order to give some analyses, we apply the further approximation Q'hZ1 0 " 0 (""b)2d"i1Z1 0 " Q " 0 ""bd" /p"bZ1 0 " Q " 0 ""bd": (9) Consider rst the simplest case where " Q " 0>0 for all energy levels "[32], i.e., SOC induces an increase in the number of available scattering states at all energies. From Eq. (9), we see Q<0, hence binding with nite Q leads to an energy decrease as compared to the SOC-free case, irrespective of the interacting strength. Such energy drop, following from@Q @a1 s>0, can be further enhanced by increasing a1 s. If, moreover, " Qincreases monotoni- cally with Q, we have QQ0<0, i.e.,Qdecreases with increasing Qfor xed scattering length. The amplitude of this decrease can be controlled by tuning the scattering length, which enhances with increased a1 s. In contrast, if the eect of SOC is such that " Q " 0alters sign depending on the energy "of the state, Qcan exhibit a very rich behavior. To demonstrate it, consider " Q " 0has opposite sign in the low- and high- energy regimes, with a sign ip occurring at the energy "0. Applying the mean value theorem to Eq. (9), we nd Z1 0 " Q " 0 ""bd"=fl=("1"b) +fh=("2"b);(10) with"12(0;"0), and"22("0;1). Herefl=R"0 0( " Q " 0)d"andfh=R1 "0( " Q " 0)d"are the number of scat- tering states in the low- and high-energy regimes, respec- tively. Since flandfhhave opposite signs, the contribu- tion from the high-energy states to Qis suppressed by the smaller pre-factor compared to the low-energy states.Yet, such suppression becomes less signicant when a1 s increases, following similar reasoning as before. We thus expect the sign of Qto be mainly determined by the low-energy states for large as, whereas the high-energy states can become decisive for small as. This has in- teresting physical implications: by tuning the scattering length and hence the sign of Qand QQ0, we can con- trol whether a bound pair favors nonzero Q, and even the specic choice of Q. IV. TYPICAL BEHAVIORS OF TWO-BODY GROUND STATES We now show that, combining EQ th, above insights into the cooperative eects of interaction and SOC on Esc Q allows predictions on generic features of the dispersion EQ. This can be best illustrated in two following cases. (i) IfEQ thhas only one minimum, without interaction, the two-body (molecular) ground state c.m. momentum Qgwill locate at Q1whereEQ this minimized. By con- trast, adding interaction can strongly modify Esc Qand thusEQ, according to previous analysis, which renders Qgto deviate from Q1. Such deviation intimately de- pends on the behavior of Esc Q: IfEsc Qvaries monotonically withQfor a xed scattering length, Qgshifts from Q1 in such a way that a smaller Esc Qcan be reached. Such shift can be further enhanced by increasing a1 s, provided it does not qualitatively alter the behavior of Esc Q, i.e,. Esc Qstays increasing (or decreasing) with Qwhen varying a1 s[c.f. inset of Fig. 1(d)]. If, instead, the behavior of Esc Qundergoes a qualitative change when a1 sincreases, e.g. from increasing to decreasing with Q[see inset of Fig. 2(c)], Qgwill rst exhibit a zigzag away from Q1 before increasing above Q1monotonically [see inset of Fig. 2(d)]. (ii) In general EQ thcan have multiple local minima, each corresponding to a meta-stable state. For individ- ual meta-stable state, the associated c.m. momentum ex- hibits similar behavior as in (i). An interesting question then concerns how two-body (molecular) ground state transits among multiple meta-stable states when the in- teraction is tuned. To address it, suppose for simplicity thatEQ thhas two degenerate local minima at Q1and Q2respectively, and Esc Qvaries monotonically with Q for a xed scattering length. The two-body (molecular) ground state c.m. momentum Qis expected to be close to Q1orQ2, depending on which corresponds to a smaller Esc Q. If the behavior of Esc Qcan be changed qualitatively by tuninga1 s, say from increase to decrease with Q, a transition of the system between the two meta-stable states can be induced. This phenomenon also occurs when the two local minima EQ thbecome non-degenerate, due to the competition between EQ thandEsc Q, which is the origin of the transition discussed in Ref. [16]. In addition, with the increasing of a1 s,Esc Qwill dominate over EQ thin4 � �� (c)EEth,+Q Eth,-Q- - 1(a)- Q (b)0kas( )-1 (d) - - - - - 1 - - - - - - - - - - - - - - - - 1 0kas( )-1 0kas( )-1 Figure 1. Binding of spin-orbit coupled fermions in the vacuum. (a) The distribution of " Q " 0. (b)Qas a function of Q with dierent ( k0as)1according to Eq.(8). (c) The helicity-dependent threshold energy EQ th;+(EQ th;) is the minimum energy of two particles with A in the upper (lower) helicity branch and a c.m. momentum Q. (d) The two-body energy with dierent interacting strengths by exactly solving Eq. (4). The inset shows the variation of the ground state c.m. momentum. Here Q0=1:5k0ex. determining the dispersion of EQ. This may qualitative change the dispersion of two-body (molecular) energy, say from a double-well type with two meta-stable states to a single-well type with one meta-stable state, which may cause the disappear of the transition. V. SPIN-ORBIT COUPLED THREE-COMPONENT FERMI MIXTURE Previous discussions from Sec. II to Sec. IV are not dependent on the concrete type of the SOC. To give an example, below we present concrete calculations by solv- ing Eq. (4) for a system of interacting Fermi mixture of40K-40K-6Li (A-A-B), where the atom40K couples to SOC and the atom6Li is spinless. Here, we choose an (kxz+hx)-type SOC which can be readily realized experimentally in40K [4]. In this three-component mix- ture, the6Li fermions are tuned close to a wide Feshbach resonance with spin up species of40K [23]. The Hamil-tonian for the system reads H=X k;"a kay k;ak;+X k(hay k;"ak;#+hay k;#ak;") +X k"b kby kbk+U VX k;k0;qay q 2+k;"by q 2kbq 2k0aq 2+k0;" +X k(kxay k;"ak;"kxay k;#ak;#): (11) Hereak;(=";#) denotes the annihilation operator of a SOC-free particle Awith spinand momenta k, while the operator bkannihilates a particle Bwith momenta k. In addition, "a(b) k=k2=(2ma(b)) is the kinetic energy of particleA(B). The SOC parameters handare respec- tively proportional to the Raman coupling strength and the momentum transfer in the Raman process generating the SOC [4]. We also note that via a global pseudo-spin rotation such SOC can be transformed to an equal weight combination of Rashba-type and Dresselhaus-type SOC (kxy+hz) which is the rst SOC generated in ultra- cold atomic gases [2]. Therefore, the ( kxz+hx)-type SOC can be interpreted as an equal weight combination5 QEth,+Q Eth,-QEQ Q (c)3 1 2(b)Q1 Q3Q20.8 0.6 -1.72 -1.68 -1.64Q1 0.3 0.4 0.5 0.60kas( )-1 (d)0.3 0.4 0.5 0.60kas( )-1 0kas( )-1Q1Q 0.2 0.4 0.6-1.7-1.5-1.3 Q1Q2 Q3(a)Q1 Q2 Q3- Figure 2. Binding of spin-orbit coupled fermions on top of a Fermi sea in single impurity system. (a) The distribution of " Q " 0. (b)Qas a function of Qwith dierent ( k0as)1according to Eq.(8). The inset shows Qin the region near Q1. (c) The threshold EQ th= minfEQ th;;EQ th;+g. The helicity-dependent threshold energy EQ th;+(EQ th;) is the minimum energy of two particles with A in the upper (lower) helicity branch and a c.m. momentum Q. (d) The two-body energy with dierent interacting strengths by exactly solving Eq. (4). The inset shows the variation of the ground state c.m. momentum. Here Q0=2k0ex. of Rashba-type and Dresselhaus-type SOC [4]. In the presence of SOC, the single-particle eigenstates ofAin the helicity basis are created by operators ay k;= ;" kay k;"+;# kay k;#, with;" k= k,;# k= k, and k= [p h2+2k2xkx]1=2=p 2[h2+2k2 x]1=4, with +() labelling the upper (lower) helicity branch. The single particle dispersions of two helicity branches are "a k;="a kp h2+2k2x. Here we have measured the energy in the unit of E0= 22ma=~2, the momentum in the unit of k0= 2ma=~2, andh= 0:4E0. We rst present our results for the binding of AandB in the vacuum, as summarized in Fig. 1. The density of states [see Fig. 1(a)] exhibits a monotonic decrease with both Qand". As expected, Esc Qwill change monoton- ically with respect to both Qanda1 s[see Fig. 1(b)]. Together with EQ th[see Fig. 1(c)], we see that the actual ground state c.m. momenta will be pulled to the direc- tion with a smaller magnitude than Q1and the increase ofa1 swill enhance this tendency [see Fig. 1(d)]. We now turn to the binding of AandBon the top of the Fermi sea of Ain the situation where a single impu-rity ofBimmerses in a non-interacting Fermi sea of spin- orbit coupled Awith the Fermi energy Eh=1:5E0, as illustrated in Fig. 2. There, both the density of states [see Fig. 2(a)] and Esc Q[see Fig. 2(b)] exhibit a rich be- havior. In addition, from EQ thin Fig. 2(c), we see that there exist two meta-stable states near Q1andQ2, re- spectively. Let us rst analyze the c.m. momenta asso- ciated with the meta-stable states, e.g., the one formed nearQ1. Seen from Fig. 2(a), " Qfor c.m. momentum nearQ1decreases with Qin the low energy region (e.g. 0< " < 2E0), but increases in the high energy region (e.g. 6E0< " < 10E0). In addition, near Q1,EQ sc[see the inset of Fig. 2(b)] shows a qualitative change with in- creasing of a1 s. We thus expect from earlier discussions a zigzag behavior of c.m. momenta of the meta-stable state, as conrmed by our results plotted in the inset of Fig. 2(d). Next, we discuss which of the two meta-stable states is energetically favored. Due to the degeneracy of the two local minima of EQ th, this is determined by the density of states, which is larger near Q1than that near Q2[see Fig. 2(a)]. Hence the meta-stable state near Q16 (b)-Q Q0Q1Q2 Q30.8 1.0 1.2 1.40kas( )-1 (c)EEth,+Q Eth,-Q Q1Q2Q3 (d)Q1Q2Q30.8 1.0 1.2 1.40kas( )-1(a)- Q1 Q2 Q3 Figure 3. (a) The distribution of " Q " 0. (b)Qas a function of Qwith dierent ( k0as)1according to Eq.(8). (c) The threshold energy EQ th= minfEQ th;;EQ th;+g. The helicity-dependent threshold energy EQ th;+(EQ th;) is the minimum energy of two particles with A in the upper (lower) helicity branch and a c.m. momentum Q. (d) The two-body energy with dierent interacting strengths by exactly solving Eq. (4). Here Q0=3k0ex. is energetically favored by EQ sc[see Fig. 2(b)]. We thus expect the molecular ground state c.m. momentum to be nearQ1, well agreeing with Fig. 2(d). Comparing the binding of AandBin the vacuum and on top of the lled Fermi sea, we observe that the pres- ence of Fermi sea not only elevates EQ thin the regime Q1<Q<Q2, giving rise to two minima, but also en- hances the density of states there. Consequently, the minimum of EQ scoccurs at Q3, and the two meta-stable states merge together [see Fig. 2(d)] following from pre- vious analysis. We remark that, while the SOC here has dramatically modied the density of states compared to the SOC-free case, our analyses based on perturbation treatment agree remarkably well with the exact numeri- cal results. VI. CONCLUDING DISCUSSIONS AND SUMMARY When the Fermi sea has only one Fermi surface, the two meta-stable states formed near Q1andQ2are fa- vored by the threshold energy and the density of states, respectively, see Fig. 3. In Ref. [16] with high Fermienergy, tuning the interaction can induce a transition between the two meta-stable states. In contrast, as il- lustrated in Fig. 3 where the Fermi energy Eh= 0, such transition is missing and the increase of a1 swill eventu- ally cause a merge of the two meta-stable states. With an increase of the Fermi energy, our case crossovers to that discussed in Ref. [16]. In addition, we note that for the single impurity Fermi system we only consider the low- est energy state within our ansatz, the ground state of the system should be given by connecting the molecular ground state to the polaron ground state which describes the particle-hole excitations above the Fermi sea. Summarizing, we have investigated how the tuning of interacting strength of an attractive s-wave interaction aects two-body energy under certain distribution of the density of states. Combining with the dispersion of the threshold energy, we can predict typical behavior of the two-body bound state when tuning the scattering length and hence the interaction, including the change of the c.m. momentum of the two-body ground state and the competition between multiple meta-stable states. Our perturbation analyses are not dependent on the concrete type of SOC and corroborated by the exact numerical solution of the two-body problem for a spin-orbit coupled7 Fermi mixture of40K-40K-6Li, even though the density of states is signicantly altered by the eect of SOC.VII. ACKNOWLEDGMENTS We thanks Ying Hu for helpful discussion. The work is supported by the National Basic Research Program of China (973 Program) (Grants No. 2013CBA01502 and No. 2013CB834100), the National Natural Science Foun- dation of China (Grants No. 11374040, No. 11475027, No. 11575027, and No. 11274051). [1] Jean Dalibard, Fabrice Gerbier, Gediminas Juzeli unas, and Patrik Ohberg, Rev. Mod. Phys. 83, 1523 (2011). [2] Y.-J. Lin, K. Jimen ez-Garc a, and I. B. Spielman, Nature (London) 471, 83 (2011). [3] Jin-Yi Zhang, Si-Cong Ji, Zhu Chen, Long Zhang, Zhi- Dong Du, Bo Yan, Ge-Sheng Pan, Bo Zhao, YouJin Deng, Hui Zhai, Shuai Chen, and Jian-Wei Pan, Phys. Rev. Lett. 109, 115301 (2012). [4] Pengjun Wang, Zeng-Qiang Yu, Zhengkun Fu, Jiao Miao, Lianghui Huang, Shijie Chai, Hui Zhai, and Jing Zhang, Phys. Rev. Lett. 109, 095301 (2012). [5] Lawrence W. Cheuk, Ariel T. Sommer, Zoran Hadz- ibabic, Tarik Yefsah, Waseem S. Bakr, and Martin W. Zwierlein, Phys. Rev. Lett. 109, 095302 (2012); Lianghui Huang, Zengming Meng, Pengjun Wang, Peng Peng, Shao-Liang Zhang, Liangchao Chen, Donghao Li, Qi Zhou and Jing Zhang, Nat. Phys. 10, 1038 (2016). [6] H. Zhai, Int. J. Mod. Phys. B 26, 1230001 (2012). [7] V. Galitski and I. B. Spielman, Nature, 494, 49 (2013). [8] H. Zhai, Rep. Prog. Phys., 78, 026001 (2015). [9] Jayantha P. Vyasanakere, and Vijay B. Shenoy, Phys. Rev. B 83, 094515 (2011). [10] Jayantha P. Vyasanakere, Shizhong Zhang, and Vijay B. Shenoy, Phys. Rev. B 84, 014512 (2011). [11] Hui Hu, Lei Jiang, Xia-Ji Liu, and Han Pu, Phys. Rev. Lett.107, 195304 (2011). [12] Zeng-Qiang Yu and Hui Zhai, Phys. Rev. Lett. 107, 195305 (2011). [13] Ren Zhang, Fan Wu, Jun-Rong Tang, Guang-Can Guo, Wei Yi, and Wei Zhang, Phys. Rev. A. 87. 033629 (2013). [14] Vijay B. Shenoy, Phys. Rev. A. 88. 033609 (2013). [15] Fan Wu, Ren Zhang, Tian-Shu Deng, Wei Zhang, Wei Yi, and Guang-Can Guo, Phys. Rev. A. 89.063610 (2014). [16] Lihong Zhou, Xiaoling Cui, and Wei Yi, Phys. Rev. Lett. 112, 195301 (2014). [17] F. Chevy, Phys. Rev. A 74, 063628 (2006). [18] R. Combescot, A. Recati, C. Lobo and F. Chevy, Phys. Rev. Lett, 98, 180402 (2007).[19] Sascha Zollner, G. M. Bruun, and C. J. Pethick, Phys. Rev. A 83, 021603(R) (2011). [20] Marco Koschorreck, Daniel Pertot, Enrico Vogt, Bernd Frohlich, Michael Feld and Michael Kohl, Nature 485, 619 (2012). [21] Wei Yi and Wei Zhang, Phys. Rev. Lett 109, 140402 (2012). [22] Cheng Chin, Rudolf Grimm, Paul Julienne, and Eite Tiesinga, Rev. Mod. Phys. 82, 1225 (2010). [23] Andr e Schirotzek, Cheng-Hsun Wu, Ariel Sommer, and Martin W. Zwierlein, Phys. Rev. Lett. 102, 230402 (2009); M. Koschorreck, D. Pertot, E. Vogt, B. Fr ohlich, M. Feld, and M. K ohl, Nature (London) 485, 619 (2012). [24] W. Ketterle and M. W. Zwierlein, Rivista del Nuovo Ci- mento 31, 247-422 (2008). [25] We note that for homo-nuclear systems, our derivation for " Qreduces to the singlet density of states as discussed in Ref. [14]. [26] With Eq.(5) and renomolization eqaution, we haveR1 0 " 0 "b"d"=R1 0 " Q Q+"b"d". The right hand of the equa- tion can be writen as followR1 0 " Q Q+"b"d"=R1 0[ " Q "b" Q " Q ("b")2+2 Q " Q ("b")3+:::]d". To rst order of Q(j"Q "bj<<1), we arrive Eq.(8). [27] Xiaoling Cui, Phys. Rev. A 85, 022705 (2012). [28] Peng Zhang, Long Zhang, and Wei Zhang, Phys. Rev. A 86, 042707 (2012). [29] Peng Zhang, Long Zhang, and Youjin Deng, Phys. Rev. A86, 053608 (2012). [30] Yuxiao Wu and Zhenhua Yu, Phys. Rev. A, 87, 032703 (2013). [31] Hao Duan, Li You, and Bo Gao, Phys. Rev. A, 87, 052708 (2013). [32] the analysis for the case with " Q " 0>0 is similar.
0908.2961v2.Theory_of_anisotropic_exchange_in_laterally_coupled_quantum_dots.pdf	arXiv:0908.2961v2 [cond-mat.mes-hall] 23 Feb 2010Theory of anisotropic exchange in laterally coupled quantu m dots Fabio Baruﬀa1, Peter Stano2,3and Jaroslav Fabian1 1Institute for Theoretical Physics, University of Regensbu rg, 93040 Regensburg, Germany 2Institute of Physics, Slovak Academy of Sciences, 84511 Bra tislava, Slovak Republic 3Physics Department, University of Arizona, 1118 E 4thStreet, Tucson, AZ 85721, USA The eﬀects of spin-orbit coupling on the two-electron spect ra in lateral coupled quantum dots are investigated analytically and numerically. It is demonstr ated that in the absence of magnetic ﬁeld the exchange interaction is practically unaﬀected by spin- orbit coupling, for any interdot coupling, boosting prospects for spin-based quantum computing. The a nisotropic exchange appears at ﬁnite magnetic ﬁelds. A numerically accurate eﬀective spin Hamil tonian for modeling spin-orbit-induced two-electron spin dynamics in the presence of magnetic ﬁeld is proposed. PACS numbers: 71.70.Gm, 71.70.Ej, 73.21.La, 75.30.Et The electron spins in quantum dots are natural and viable qubits for quantum computing,[1] as evidenced by the impressive recent experimental progress [2, 3] in spin detection and spin relaxation,[4, 5] as well as in coherent spin manipulation.[6, 7] In coupled dots, the two-qubit quantumgatesarerealizedbymanipulatingtheexchange couplingwhichoriginatesin theCoulombinteractionand the Pauliprinciple.[1, 8] How is the exchangemodiﬁed by the presence of the spin-orbit coupling? In general, the usual (isotropic) exchange changes its magnitude while a new, functionally diﬀerent form of exchange, called anisotropic, appears, breaking the spin-rotational sym- metry. Such changes are a nuisance from the perspective of the error correction,[9] although the anisotropic ex- change could also induce quantum gating.[10, 11] Theanisotropicexchangeofcoupledlocalizedelectrons hasaconvolutedhistory[12–18]. Thequestionboilsdown to determining the leading order in which the spin-orbit coupling aﬀects both the isotropic and anisotropic ex- change. At zero magnetic ﬁeld, the second order was suggested,[19] with later revisions showing the eﬀects are absent in the second order.[12, 20] The analytical com- plexities make a numerical analysis particularly useful. Here we perform numerically exact calculations of the isotropic and anisotropic exchange in realistic GaAs cou- pled quantum dots in the presence of both the Dres- selhaus and Bychkov-Rashba spin-orbit interactions.[21] The numerics allowsus to make authoritative statements about the exchange. We establish that in zero magnetic ﬁeld the second-order spin-orbit eﬀects are absent at all interdot couplings. Neither is the isotropic exchange af- fected, nor is the anisotropic exchange present. At ﬁnite magnetic ﬁelds the anisotropic coupling appears. We de- rive a spin-exchange Hamiltonian describing this behav- ior, generalizing the existing descriptions; we do not rely on weak coupling approximations such as the Heitler- London one. The model is proven highly accurate by comparison with our numerics and we propose it as a re- alistic eﬀective model for the two-spin dynamics in cou- pled quantum dots. Our microscopic description is the single band eﬀec-tive mass envelope function approximation; we neglect multiband eﬀects.[22, 23] We consider a two electron double dot whose lateral conﬁnement is deﬁned electro- statically by metallic gates on the top of a semiconduc- tor heterostructure. The heterostructure, grown along [001] direction, provides strong perpendicular conﬁne- ment, such that electrons are strictly two dimensional, with the Hamiltonian (subscript ilabels the electrons) H=/summationdisplay i=1,2(Ti+Vi+HZ,i+Hso,i)+HC.(1) The single electron terms are the kinetic energy, model conﬁnement potential, and the Zeeman term, T=P2/2m= (−i¯h∇+eA)2/2m, (2) V= (1/2)mω2[min{(x−d)2,(x+d)2}+y2],(3) HZ= (g/2)(e¯h/2me)B·σ=µB·σ, (4) and spin-orbit interactions—linear and cubic Dressel- haus, and Bychkov-Rashba[21], Hd= (¯h/mld)(−σxPx+σyPy), (5) Hd3= (γc/2¯h3)(σxPxP2 y−σyPyP2 x)+Herm. conj. ,(6) Hbr= (¯h/mlbr)(σxPy−σyPx), (7) which we lump together as Hso=w·σ. The posi- tionrand momentum Pvectors are two dimensional (in-plane); m/meis the eﬀective/electron mass, eis the proton charge, A=Bz(−y,x)/2 is the in-plane vector potential to magnetic ﬁeld B= (Bx,By,Bz),gis the electrong-factor, σarePaulimatrices,and µistherenor- malized magnetic moment. The double dot conﬁnement is modeled by two equal single dots displaced along [100] by±d, each with a harmonic potential with conﬁnement energy ¯hω. The spin-orbit interactions are parametrized by the bulk material constant γcand the heterostruc- ture dependent spin-orbit lengths lbr,ld. Finally, the Coulomb interaction is HC= (e2/4πǫ)\|r1−r2\|−1, with the dielectric constant ǫ. The numerical results are obtained by exact diagonal- ization(conﬁgurationinteractionmethod). Thetwoelec- tron Hamiltonian is diagonalized in the basis of Slater2 500 250 100 50 25 10 5tunneling energy [ µeV] 0246energy [meV] 0 25 50 75 interdot distance [nm]JΨ-Ψ+ 2T∆ FIG. 1: Calculated double dot spectrum as a function of the interdot distance/tunneling energy. Spin is not considere d and the magnetic ﬁeld is zero. Solid lines show the two elec- tron energies. The two lowest states are explicitly labeled , split by the isotropic exchange Jand displaced from the near- est higher excited state by ∆. For comparison, the two lowest single electron states are shown (dashed), split by twice th e tunneling energy T. State spatial symmetry is denoted by darker (symmetric) and lighter (antisymmetric) lines. determinants constructed from numerical single electron states in the double dot potential. Typically we use 21 single electron states, resulting in the relative error for energies of order 10−5. We use material parameters of GaAs:m= 0.067me,g=−0.44,γc= 27.5 meV˚A3, a typical single dot conﬁnement energy ¯ hω= 1.1 meV, and spin-orbit lengths ld= 1.26µm andlbr= 1.72µm from a ﬁt to a spin relaxation experiment.[24, 25] Let us ﬁrst neglect the spin and look at the spectrum in zero magnetic ﬁeld as a function of the interdot dis- tance (2d)/tunneling energy, Fig. 1. At d= 0 our model describes a single dot. The interdot coupling gets weaker as one moves to the right; both the isotropic exchange Jand the tunneling energy Tdecay exponentially. The symmetry of the conﬁnement potential assures the elec- tronwavefunctionsaresymmetricorantisymmetricupon inversion. The two lowest states, Ψ ±, are separated from the higher excited states by an appreciable gap ∆, what justiﬁes the restriction to the two lowest orbital wave- functions for the spin qubit pair at a weak coupling. Our further derivations are based on the observation PΨ±=±Ψ±, I 1I2Ψ±=±Ψ±,(8) whereIf(x,y) =f(−x,−y) is the inversion operator andPf1g2=f2g1is the particle exchange operator. Functions Ψ ±in the Heitler-Londonapproximationfulﬁll Eq. (8). However, unlike Heitler-London, Eq. (8) is valid generally in symmetric double dots, as we learn from nu- merics (we saw it valid in all cases we studied).Let us reinstate the spin. The restricted two qubit subspace amounts to the following four states ( Sstands for singlet, Tfor triplet), {Φi}i=1,...,4={Ψ+S,Ψ−T+,Ψ−T0,Ψ−T−},(9) Within this basis, the system is described by a 4 by 4 Hamiltonian with matrix elements ( H4)ij=/an}bracketle{tΦi\|H\|Φj/an}bracketri}ht. Without spin-orbit interactions, this Hamiltonian is di- agonal, with the singlet and triplets split by the isotropic exchange J,[1, 8] and the triplets split by the Zeeman en- ergyµB. It is customary to refer only to the spinor part of the basis states, using the sigma matrices, resulting in the isotropic exchange Hamiltonian, Hiso= (J/4)σ1·σ2+µB·(σ1+σ2).(10) A naive approach to include the spin-orbit interaction is to consider it within the basis of Eq. (9). This gives the Hamiltonian H′ ex=Hiso+H′ aniso, where H′ aniso=a′·(σ1−σ2)+b′·(σ1×σ2),(11) with the six real parameters given by spin-orbit vectors a′= Re/an}bracketle{tΨ+\|w1\|Ψ−/an}bracketri}ht,b′= Im/an}bracketle{tΨ+\|w1\|Ψ−/an}bracketri}ht.(12) The form of the Hamiltonian follows solely from the in- version symmetry Iw=−wand Eq. (8). The spin-orbit coupling appears in the ﬁrst order. The Hamiltonian H′ exfares badly with numerics. Fig- ure 2 shows the energy shifts caused by the spin-orbit coupling for selected states, at diﬀerent interdot cou- plings and perpendicular magnetic ﬁelds. The model is completely oﬀ even though we use numerical wavefunc- tions Ψ ±in Eq. (12) without further approximations. To improve the analytical model, we remove the linear spin-orbit terms from the Hamiltonian using transformation[20, 26, 27] U= exp[−(i/2)n1·σ1−(i/2)n2·σ2],(13) wheren= (x/ld−y/lbr,x/lbr−y/ld,0). Up to the second order in small quantities (the spin- orbit and Zeeman interactions), the transformed Hamil- tonianH=UHU†is the same as the original, Eq. (1), except for the linear spin-orbit interactions: Hso=−(µB×n)·σ+(K−/¯h)Lzσz−K+,(14) whereK±= (¯h2/4ml2 d)±(¯h2/4ml2 br). In the unitarily transformed basis, we again restrict the Hilbert space to the lowest four states, getting the eﬀective Hamiltonian Hex= (J/4)σ1·σ2+µ(B+Bso)·(σ1+σ2) +a·(σ1−σ2)+b·(σ1×σ2)−2K+.(15) The operational form is the same as for H′ ex. The qual- itative diﬀerence is in the way the spin-orbit enters the parameters. First, a contribution to the Zeeman term, µBso=ˆ z(K−/¯h)/an}bracketle{tΨ−\|Lz,1\|Ψ−/an}bracketri}ht, (16)3 -0.6-0.4-0.200.2 0 50 100 interdot distance [nm]-0.6-0.4-0.200.2 energy shift [ µeV] 0 0.5 1 magnetic field [T]ab cdHex numericalHex' FIG. 2: The spin-orbit induced energy shift as a function of the interdot distance (left) and perpendicular magnetic ﬁe ld (right). a) Singlet in zero magnetic ﬁeld, c) singlet at 1 Tes la ﬁeld, b) and d) singlet and triplet T+at the interdot distance 55 nm corresponding to the zero ﬁeld isotropic exchange of 1µeV. The exchange models H′ ex(dashed) and Hex(dot- dashed) are compared with the numerics (solid). appears due to the inversion symmetric part of Eq. (14). Second, the spin-orbit vectors are linearly proportional to both the spin-orbit coupling and magnetic ﬁeld, a=−µB×Re/an}bracketle{tΨ+\|n1\|Ψ−/an}bracketri}ht, (17a) b=−µB×Im/an}bracketle{tΨ+\|n1\|Ψ−/an}bracketri}ht. (17b) The eﬀective model and the exact data agree very well for all interdot couplings, as seen in Fig. 2. At zero magnetic ﬁeld, only the ﬁrst and the last term in Eq. (15) survive. This is the result of Ref. [20], where primed operators were used to refer to the fact that the Hamiltonian Hexrefers to the transformed basis, {UΦi}. Note that if a basis separable in orbital and spin part is required,undoing UnecessarilyyieldstheoriginalHamil- tonian Eq. (1), and the restriction to the four lowest states gives H′ ex. Replacing the coordinates ( x,y) by mean values ( ±d,0)[12] visualizes the Hamiltonian Hex as an interaction through rotated sigma matrices, but this is just an approximation, valid if d,lso≫l0. One of our main numerical results is establishing the validity of the Hamiltonian in Eq. (15) for B= 0, con- ﬁrming recent analytic predictions and extending their applicability beyond the weak coupling limit. In the transformed basis, the spin-orbit interactions do not lead to any anisotropic exchange, nor do they modify the isotropic one. In fact, this result could have been antic- ipated from its single-electron analog: at zero magnetic ﬁeld there is no spin-orbit contribution to the tunneling energy,[28] going opposite to the intuitive notion of the spin-orbit coupling induced coherent spin rotation and spin-ﬂip tunneling amplitudes. Figure 3a summarizes this case, with the isotropic exchangeas the only nonzeroFIG. 3: a) The isotropic and anisotropic exchange as func- tions of the interdot distance at zero magnetic ﬁeld. b) The isotropic exchange J, anisotropic exchange c/c′, the Zeeman splitting µB, and its spin-orbit part µBsoat perpendicular magnetic ﬁeld of 1 Tesla. c) Schematics of the exchange-spli t four lowest states for the three models, Hiso,H′ ex, andHex, which include the spin-orbit coupling in no, ﬁrst, and secon d order, respectively, at zero magnetic ﬁeld (top). The latte r two models are compared in perpendicular and in-plane mag- netic ﬁelds as well. The eigenenergies are indicated by the solid lines. The dashed lines show which states are coupled by the spin-orbit coupling. The arrows indicate the redistr i- bution of the couplings as the in-plane ﬁeld direction chang es with respect to the crystallographic axes (see the main text ). parameterof model Hex. In contrast, model H′ expredicts a ﬁnite anisotropic exchange.[36] From the concept of dressed qubits[29] it follows that the main consequence of the spin-orbit interaction, the transformation Uof the basis, is not a nuisance for quan- tum computation. We expect this property to hold also for aqubit array, since the electrons areat ﬁxed positions withoutthepossibilityofalongdistancetunneling. How- ever, a rigorous analysis of this point is beyond the scope of this article. If electrons are allowed to move, Uresults in the spin relaxation.[30] Figure 3b shows model parameters in 1 Tesla perpen- dicular magnetic ﬁeld. The isotropic exchange again decays exponentially. As it becomes smaller than the Zeeman energy, the singlet state anticrosses one of the polarized triplets (seen as cusps on Fig. 2). Here it is T+, due to the negative sign of both the isotropic ex- change and the g-factor. Because the Zeeman energy always dominates the spin-dependent terms and the sin- glet and triplet T0are never coupled (see below), the anisotropic exchange inﬂuences the energy in the sec- ond order.[12] Note the diﬀerence in the strengths. In H′ extheanisotropicexchangefallsoﬀexponentially, while Hexpredicts non-exponential behavior, resulting in spin- orbit eﬀects larger by orders of magnitude. The eﬀective magnetic ﬁeld Bsois always much smaller than the real magnetic ﬁeld and can be neglected in most cases.4 Figure3ccomparesanalyticalmodels. Inzeroﬁeldand no spin-orbit interactions, the isotropic exchange Hamil- tonianHisodescribes the system. Including the spin- orbit coupling in the ﬁrst order, H′ ex, gives a nonzero coupling between the singlet and triplet T0. Going to the second order, the eﬀective model Hexshows there are no spin-orbit eﬀects (other than the basis redeﬁnition). The Zeeman interaction splits the three triplets in a ﬁ- nite magnetic ﬁeld. Both H′ exandHexpredict the same type of coupling in a perpendicular ﬁeld, between the singlet and the two polarized triplets. Interestingly, in in-plane ﬁelds the two models diﬀer qualitatively. In H′ exthe spin-orbit vectors are ﬁxed in the plane. Rota- tion of the magnetic ﬁeld “redistributes” the couplings among the triplets. (This anisotropy with respect to the crystallographic axis is due to the C2vsymmetry of the two-dimensional electron gas in GaAs, imprinted in the Bychkov-Rashba and Dresselhaus interactions.[21]) In contrast, the spin-orbit vectors of Hexare always per- pendicular to the magnetic ﬁeld. Remarkably, aligning the magnetic ﬁeld along a special direction (here we al- low anarbitrarypositioned dot, with δthe anglebetween the main dot axis and the crystallographic xaxis), [lbr−ldtanδ,ld−lbrtanδ,0], (18) all the spin-orbit eﬀects disappear once again , as ifB were zero. (An analogous angle was reported for a sin- gle dot in Ref. [31]). This has strong implications for the spin-orbit induced singlet-triplet relaxation. Indeed, S↔T0transitions are ineﬀective at any magnetic ﬁeld , as these two states are never coupled in our model. Sec- ond,S↔T±transitions will show strong (orders ofmag- nitude) anisotropy with respect to the ﬁeld direction, reaching minimum at the direction given by Eq. (18). This prediction is straightforwardly testable in experi- ments on two electron spin relaxation. Our derivation was based on the inversion symmetry of the potential only. What are the limits of our model? We neglected third order terms in Hsoand, restricting the Hilbert space, corrections from higher excited orbital states. (Among the latter is the non-exponential spin- spin coupling[12]). Compared to the second order terms we keep, these are smaller by (at least) d/lsoandc/∆, respectively.[35] Apart from the analytical estimates, the numerics, which includes all terms, assures us that both of these are negligible. Based on numerics we also con- clude our analytical model stays quantitatively faithful even at the strong coupling limit, where ∆ →0. More involved is the inﬂuence of the cubic Dresselhaus term, which is not removed by the unitary transformation. This term is the main source for the discrepancy of the model and the numerical data in ﬁnite ﬁelds. Most im- portantly, it does not change our results for B= 0. Concluding, we studied the eﬀects of spin-orbit cou- pling on the exchange in lateral coupled GaAs quantumdots. Wederiveandsupportbyprecisenumericsaneﬀec- tive Hamiltonian for two spin qubits, generalizing the ex- isting models. The eﬀective anisotropic exchange model shouldbeusefulinpreciseanalysisofthephysicalrealiza- tions of quantum computing schemes based on quantum dot spin qubits, as well as in the physics of electron spins in quantum dots in general. Our analysis should also improve the current understanding of the singlet-triplet spin relaxation [32–35]. ThisworkwassupportedbyDFGGRK638,SPP1285, NSF grant DMR-0706319, RPEU-0014-06, ERDF OP R&D “QUTE”, CE SAS QUTE and DAAD. [1] D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120 (1998). [2] R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha, and L. M. K. Vandersypen, Rev. Mod. Phys. 79, 1217 (2007). [3] J. M. Taylor, J. R. Petta, A. C. Johnson, A. Yacoby, C. M. Marcus, andM. D.Lukin, Phys. Rev.B 76, 035315 (2007). [4] J. M. Elzerman, R. Hanson, L. H. Willems van Beveren, B. Witkamp, L. M. K. Vandersypen, and L. P. Kouwen- hoven, Nature 430, 431 (2004). [5] F. H. L. Koppens, K. C. Nowack, and L. M. K. Vander- sypen, Phys. Rev. Lett. 100, 236802 (2008). [6] J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, Science 309, 2180 (2005). [7] K. C. Nowack, F. H. L. Koppens, Y. V. Nazarov, and L. M. K. Vandersypen, Science 318, 1430 (2007). [8] X. Hu and S. Das Sarma, Phys. Rev. A 61, 062301 (2000). [9] D. Stepanenko, N. E. Bonesteel, D. P. DiVincenzo, G. Burkard, and D. Loss, Phys. Rev. B 68, 115306 (2003). [10] D. Stepanenko and N. E. Bonesteel, Phys. Rev. Lett. 93, 140501 (2004). [11] N. Zhao, L. Zhong, J.-L. Zhu, and C. P. Sun, Phys. Rev. B74, 075307 (2006). [12] S. Gangadharaiah, J. Sun, and O. A.Starykh, Phys. Rev. Lett.100, 156402 (2008). [13] L. Shekhtman, O. Entin-Wohlman, and A. Aharony, Phys. Rev. Lett. 69, 836 (1992). [14] A. Zheludev, S. Maslov, G. Shirane, I. Tsukada, T. Ma- suda, K. Uchinokura, I. Zaliznyak, R. Erwin, and L. P. Regnault, Phys. Rev. B 59, 11432 (1999). [15] Y. Tserkovnyak and M. Kindermann, Phys. Rev. Lett. 102, 126801 (2009). [16] S. Chutia, M. Friesen, and R. Joynt, Phys. Rev. B 73, 241304(R) (2006). [17] L. P. Gorkov and P. L. Krotkov, Phys. Rev. B 67, 033203 (2003). [18] S. D. Kunikeevand D. A. Lidar, Phys. Rev. B 77, 045320 (2008). [19] K. V. Kavokin, Phys. Rev. B 64, 075305 (2001). [20] K. V. Kavokin, Phys. Rev. B 69, 075302 (2004). [21] J. Fabian, A. Matos-Abiagus, C. Ertler, P. Stano, and I.ˇZuti´ c, Acta Phys. Slov. 57, 565 (2007).5 [22] S. C. Badescu, Y. B. Lyanda-Geller, and T. L. Reinecke, Phys. Rev. B 72, 161304(R) (2005). [23] M. M. Glazov and V. D. Kulakovskii, Phys. Rev. B 79, 195305 (2009). [24] P. Stano and J. Fabian, Phys. Rev. Lett. 96, 186602 (2006). [25] P. Stano and J. Fabian, Phys. Rev. B 74, 045320 (2006). [26] I. L. Aleiner and V. I. Fa ˇlko, Phys. Rev. Lett. 87, 256801 (2001). [27] L. S. Levitov and E. I. Rashba, Phys. Rev. B 67, 115324 (2003). [28] P. Stano and J. Fabian, Phys. Rev. B 72, 155410 (2005). [29] L.-A. Wu and D. A. Lidar, Phys. Rev. Lett. 91, 097904 (2003). [30] K. V. Kavokin, Semicond. Sci. Technol. 23, 114009(2008). [31] V. N. Golovach, A. Khaetskii, and D. Loss, Phys. Rev. B77, 045328 (2008). [32] K. Shen and M. W. Wu, Phys. Rev. B 76, 235313 (2007). [33] E. Y. Sherman and D. J. Lockwood, Phys. Rev. B 72, 125340 (2005). [34] J. I. Climente, A. Bertoni, G. Goldoni, M. Rontani, and E. Molinari, Phys. Rev. B 75, 081303(R) (2007). [35] O. Olendski and T. V. Shahbazyan, Phys. Rev. B 75, 041306(R) (2007). [36] The spin-orbit vectors are determined up to the relativ e phase of states Ψ +and Ψ −. The observable quantity is c=√a2+b2and analogously for c′=√a′2+b′2.
1709.07948v1.Quantification_of_spin_accumulation_causing_spin_orbit_torque_in_Pt_Co_Ta_stack.pdf	1 Quantification of spin accumulation causing spin-orbit torque in Pt/Co/Ta stack Feilong Luo, Sarjoosing Goolaup, Christian Engel, and Wen Siang Lew* School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371 Abstract Spin accumulation induced by spin-orbit coupling is experimentally quantified in stack with in -plane magnetic anisotropy via the contribution of spin accumulation to Hall resistances. Using a biasing direct current the spin accumulation within the structure can be tuned, enabling quantification. Quantification shows the spin accumulation can be more than ten percentage of local magnetization , when the electric current is 1011 Am−2. The spin accumulation is dependent of the thickness of Ta layer , the trend agrees with that of spin Hall angle indicating the capability of Ta and Pt in generating spins . *Corresponding author: wensiang@ntu.edu.sg 2 Introduction Current -induced spin accumulation causes spin-orbit torque (SOT) on the magnetization of a ferromagnetic metal (FM) layer sandwiched by two heavy metal (HM) layers , via exchange interaction [1]. The spin accumulation originates from t wo spin-orbit coupling effects: Rashba effect and spin Hall effect [2-10]. The SOT is reflected in the revised Landau –Lifshitz –Gilbert equation by the term 0 JMs , where 0 is the gyromagnetic coefficient, M is the magnetization of the FM layer, s is the spin accumulation , and J is a coefficient related to spin diffusion length of accumulated spin s in the FM layer. The term 0 JMs can be decomposed into a fieldlike torque FF H τ Mp and a damping like torque DD H τ M m p , where p represents the spin orientation of the electrons diffusing into the FM layer , and m is the unit vector of M [1, 10 -14]. The corresponding effective field s arising from SOT can be written as the fieldlike term FFHHp and damping like term DDHH m p , alternatively, FD J H H s p m p [6, 8, 10, 12, 15 -20]. The effective field , Js, which is a combination of spin accumulation and a spin-diffusion related coefficient, has been widely characterized via current -induced domain wall motion [8, 21, 28 -30], ferromagnetic resonance (FMR) techniques [31 -38], and SOT -assisted magnetization switching [6, 20, 22, 35, 39] . Quantification of the spin accumulation, which plays a crucial role in the origins of the SOT, has remained e lusive . In this letter , we provide a concise solution to quantify the spin accumulation in the sandwich ed structure with in -plane magnetic anisotropy (IMA) . We propose the spin accumulation s contribute s to the second harmonic Hall resistance in the harmonic Hall 3 voltage scheme , in addition to the SOT effective field Js as expected. Applying a biasing direct current (DC) enables the extraction of the contribution of the spin accumulation from the second harmonic Hall resistances . Analogized to first harmonic Hall resistance which is induced by the magnetization, modulation of the second Hall resistance via DC current can be used to compute the spin accumulation . Results of the computation show the spin accumulation is dependent of the thickness of HM layers . This quantification allows us to understand the anatomy of Js and distinguish the role s of J and s. Main body Following the transfer of momentum to the local magnetization, the accumulated spins s adopt similar polarization as the magne tization orientation of the FM layer . The structure comprises of Ta/Co/Pt multilayer, where the FM layer exhibits IMA . The initial polarization of s is induced by R ashba effect due to the asymmetric HM/FM interface and spin Hall effect within the Ta and Pt layers [2-10, 8, 15, 18, 19, 21 -27]. The Rashba effect re-orientates the spin with in the conduction electrons of FM layer to provide a net resultant spin in the FM layer [5]. Additionally, t he spin Hall effect induces a spin -selective separation of electron s in the HM layer; the spin polarized electrons then diffuses into the FM layer [10]. In the Co layer, the transfer of spin torque from the spin polarized electron to the FM layer occurs on the nanosecond time scale [ 5]. A schematic of the spin transfer process is depicted in Fig 1(a) . At the end of the spin transfer, s is in relaxation state, hence it adopt s the same orientation as the local magnetic moment Mm as depicted in Fig. 1(b). In experiment, w ithin the low frequency regime of hundreds of Hertz, corresponding to period of oscillation of alternating current (AC) in millisecond scale, it is reasonable t o 4 consider that the accumulated spins s follow the orientation of Mm. Similarly, e xtending to direct current ( DC) bias regime , an identical approximation can be made and the spins s similarly aligns along m. Therefore , after the electron spins have transferred the momentum to the local magnetization , the resultant polarization direction of the electron is along the magnetization orientation of the FM layer. Thus, the spin accumulation can be written as sm. Consequently , the total magnetization of the stack becomes to Mm+sm from Mm. We propose that t he spin accumulation sm results in additional planar Hall resistance, analogized to the local magnetization Mm. The magnitude of planar Hall resistance, RPHE due to the local magnetic moment Mm is parabolic with respect to magnitude of the local magnetization M via a coefficient k, 2 PHER kM [40-42]. The Hall resistance induced by sm as an ext ra magnetization should present the same behavior of that induced by Mm. The planar Hall resistance due to Mm is expressed as P PHE sin 2 RR , where φ is the azimuthal angle of magnetization Mm [7, 12, 43, 44 ]. Analogically, the planar Hall resistance, 𝑟p due to the extra magnetization, sm, should follow a similar trend as P PHE sin 2 rr and 2 PHEr ks . Obtaining k from the expression of RPHE, we derive the expression of 𝑟PHE as 2 PHE PHE 2=RrsM . As such, 𝑟PHE can be used to calculate the magnitude of spin accumulation. Applying a biasing DC increases the magnitude of 𝑟PHE to measureable levels . When AC and DC are applied in the wire concurrently , the harmonic Hall voltage induced by sm can be written as 2 PHE s,Hall AC DC 2sin 2 sinRv s j t jM , where jAC and ω are 5 the amplitude and frequency of AC density respectively , and jDC is the magnitude of DC density. At steady -state which is the rate of spin decay equaling to that of spin generating , s is proportional to the current density and can be written as AC DCsin s j t j , where ζ is the coefficient constant. Substituting s by AC DCsinj t j in vs,Hall gives 2 3 2 2 2 3 3 s,Hall PHE DC DC AC AC DC AC 2sin 2 3 sin 3 sin sin v R j j j t j j t j tM . (1) In Eq. (1), we substitute 2sin t with 1 cos2 22t , as such, eliminate the constant 1 2 item to obtain a second harmonic Hall voltage as 2 2 s,2ndHall PHE AC DC 23sin 2 cos22v R j j tM . Consequently, sm induces a second harmonic Hall resistance 𝑟𝛼±𝛽 as s,2ndHall ACsin 2vrzj , (2) Where 2 PHE DC AC 23 2z R j jM , α and β correspond to the factors α× 1010 Am−2 for jAC and β× 1010 Am−2 for jDC, ± indicat es the sign of DC. Compared with the expression of 2 PHE PHE 2=RrsM , the expression 2 PHE DC AC 23 2z R j jM is the other expression of rPHE which includes the electric current. Similarly, Eq. (2) is the other expression of rP. For a constant amplitude of jAC, the resistance 𝑧𝛼±𝛽 is proportional to the amplitude of applied DC. As such, jDC provides a way to modulate the resistance from the baseline provided by jAC. 6 This resistance 𝑟𝛼±𝛽 can be obtained , by subtracting the second harmonic Hall resistance ℜ𝛼±𝛽 which is measured in experiment by DC -biased AC from that of ℜ𝛼0 which is solely due to AC only. The measured second harmonic Hall resistance is induced by both sm and Mm concurrently . As such, ℜ𝛼±𝛽 is the sum of 𝑟𝛼±𝛽 due to sm and 𝑅𝛼±𝛽 due to Mm, The second harmonic Hall resistance 𝑅𝛼±𝛽 due to Mm is ,, 42 AHE PHE extcos 2cos cos2D AC F AC xHHR R RHH , (3) where HꞱ is the effective perpendicular anisotropy field, RAHE is the amplitude of anomalous Hall effect (AHE) resistance [7, 12, 43, 44, 45 ]. DC has no effect on 𝑅𝛼±𝛽, alternatively, 0RR [Appendix ]. Both HD,AC and HF,AC are only determined by the AC component of electric current , HD,AC is along z-axis while HF,AC is along y-axis, for the wires with IMA [45]. Hence, according to Eq. (2), the measured second harmonic Hall resistance by AC only, is 000R . While , the measured second harmonic Hall resistance by DC -biased AC is 0Rr , where β is not equal to 0 . Therefore, subtract ion of ℜ𝛼0 from ℜ𝛼±𝛽, 𝛥ℜ𝛼±𝛽(ℜ𝛼±𝛽−ℜ𝛼0), equals to 𝑟𝛼±𝛽. Based on Eq. (2), we conclude 2 PHE DC AC 23sin 22R j jM . As such, theoretically , the coefficient indicating the magnitude of spin accumulation can be quantified by 𝛥ℜ𝛼±𝛽. Measurements of the second harmonic Hall resistances ℜ𝛼±𝛽with respect to the azimuthal angle of magnetization were carried out in magnetic wire with stack s of Ta( 8 nm)/Co(2 nm)/Pt(5 nm) . The wire has IMA as evidenced by hysteresis loop measurements using both Kerr and anomalous Hall effects [45]. For all measurements, a lock -in amplifier 7 was used to obtain the harmonic Hall voltage signals. T he second harmonic Hall resistance ℜ𝛼±𝛽 is calculated by dividing the second harmonic Hall voltage with the magnitude of the AC. Only the Hall resistance mo dulation has been considered removing the offset resistance for each measurement. Each measured ℜ𝛼±𝛽 as well as the following Δℜ𝛼±𝛽 has been moved to be around 0 Ω by eliminating a constant offset for easy comparison . A schematic of the measurement setup is shown in Fig. 1(c) . The azimuthal angle of magnetization of the wire depends on the applied field s as ext extarctany xH H , where transvers Hy-ext sweep s from −1800 Oe to +1 800 Oe along y-axis while ±Hx-ext keeps ± 560 Oe to orientate Mm along ±x-axis. In the following, we do not distinguish Hy-ext from φ as φ is equivalent to Hy-ext, since the φ corresponds to unique Hy-ext for constant Hx-ext. The longitudinal magnetic field Hx-ext was used to ensure a uniform magnetization along the wire axis [45]. The second harmonic Hall resistances , ℜ40,±6, with respect to the azimuthal angle of magnetization, measured at Hx-ext = −560 Oe are presented in Fig. 1( d). For AC-Js only, the derived equation to represent the second Hall resistance is given in Eq. 3 . Through substituting HꞱ, RAHE and RPHE in Eq. 3 with experimental values of 5790 Oe, 26 mΩ and 6 mΩ , respectively, the measured 𝑅40 is in good agreement with Eq. 3. A damping like term HD,AC of 29 Oe and fieldlike term HF,AC of 4 Oe are obtained , as shown in Fig. 1( d). This good agreement suggests that AC-Js ,,F AC D ACHHp m p result s in the symmetric behavior of ℜ40 with respect to Hy-ext. For the second harmonic Hall resistance obtained with using both DC and AC concurrently , an asymmetric behavior around Hy-ext = 0 Oe is observed for ℜ4±6. ℜ4+6 and ℜ4−6 are mirror symmetric to each other at Hy-ext = 0 Oe. These 8 variations indicate that both magnitude and sign of the DC bias in the wire contributes to the corresponding signals , ℜ4±6. The derived equation for the second harmonic Hall resistance with DC bias is still Eq. 3 [45]. As such, w e may expect to include DC induced HD,DC and HF,DC in Eq. 3 as offset s of HꞱ and Hy-ext to explain the behavior of ℜ4±6. However, the revised Eq. 3 fails to fit ℜ4±6 with fitting root-mean -square error ( RMSE ) reaches minimum as shown in Fig. 1(d) , as the measured ℜ4±6 and the fitted ℜ4±6 by Eq. 3 do not overlap . The failure clarifies the negligible role of DC -Js in the behavior of ℜ4±6. The differences, Δℜ4±6, are explored to investigate the behavior of ℜ4±6 with respect to Hy-ext. Δℜ𝛼±𝛽 is computed by subtracting the second harmonic Hall resistance obtained without DC bias ( 𝑅𝛼0) from that with DC bias as Δℜ𝛼±𝛽=ℜ𝛼±𝛽−ℜ𝛼0, consequently, Δℜ4±6=ℜ4±6−ℜ40. As shown in Fig. 1(d), Δℜ4±6 adapts the behavior similar to the first harmonic Hall resistance with respect to Hy-ext as shown in Fig. 1( f). The analytical expression of the first harmonic Hall resistance , which is mainly from PHE, is 1stHall PHE sin 2 RR . We use 66 44 sin 2Z to fit Δℜ4±6, where 2𝑍4±6 equals to the difference between maximum and minimum values of Δℜ4±6, 60 µΩ . The experimental Δℜ4±6 are in good agreement with 6 4sin 2Z as shown in Fig. 1( d). R1stHall is due to the magnetization of the wire. Analogically, Δℜ𝛼±𝛽 is due to an extra magnetization and sin 2Z , (4) where 𝑍𝛼±𝛽 is the amplitude of Δℜ𝛼±𝛽. ℜ𝛼0 is given by Eq . 3. Respective of the expression of Δℜ𝛼±𝛽=ℜ𝛼±𝛽−ℜ𝛼0, ℜ𝛼±𝛽 is determined by both the pre -known AC -Js and the extra magnetization. 9 In the following, w e confirm the extra magnetization is sm, as we further show that Δℜ𝛼±𝛽 follows 𝑟𝛼±𝛽 with respect to the orientation of Mm, 𝑍𝛼±𝛽 is equal to the derived 𝑧𝛼±𝛽 of Eq. (2) , 𝑍𝛼±𝛽 with respect to JDC and JAC follows the predict ed behavior of or 2 PHE DC AC 23 2z R j jM in experiments . The proposal is validated through analyzing 𝛥ℜ𝛼±𝛽 obtained with varying magnetic vector of Hx-ext. Figure 2(a) shows 𝛥ℜ4±6 measured with Hx-ext = +560 Oe. As Hy-ext varies from −1800 Oe to +1800, the magnitude of 𝛥ℜ4+6 changes from −20 µΩ to +20 µΩ . In Fig. 1(d) which represents 𝛥ℜ4+6 with Hx-ext = −560 Oe, 𝛥ℜ4+6 changes fr om +20 µΩ to −20 µΩ. The change of sign for 𝛥ℜ𝛼±𝛽 is present ed in the 𝛥ℜ4+6, as well as in 𝛥ℜ4−6. 𝛥ℜ𝛼±𝛽 follow R1stHall or 𝑟𝛼±𝛽 to change their signs when Hx-ext is orientated along opposite directions. An alternative approach to substantiate the proposal is that 𝛥ℜ𝛼±𝛽 should follow R1stHall or 𝑟4±6 to present extremum at Hy-ext = ±Hx-ext. In experiments, the extremum s of R1stHall , ±6 mΩ, are at Hy-ext = ± 360 Oe and ± 1000 Oe, when the applied constant field Hx-ext equals to +360 Oe and +1000 Oe, respectively, as shown in Fig. 2(b). Similarly, the extremum values of 𝛥ℜ4+6, which are ± 30 µΩ, are at Hy-ext = ± 360 Oe as shown in Fig. 2(c) where Hx-ext is applied as +360 Oe and at ± 1000 Oe as shown in Fig. 2(d) where Hx-ext is applied as +1000 Oe. As such, experimentally on condition that 𝑍𝛼±𝛽 follows a linear function of jDC and jAC as predicted by 2 PHE DC AC 23 2R j jM , it is allowed to conclude th e extra magnetization is from the spin accumulation sm. For different DC biases, ℜ4±[1 to 5] were measured to 10 compute Δℜ4±[1 to 5]. ℜ4±3 and Δℜ4±3 are exhibited in Fig. 3(a) with Hx-ext = −560 Oe. ℜ4±3 and Δℜ4±3 show the behavior similar to ℜ4±6 and Δℜ4±6, respectively. 2𝑍4±3 is calculated to be 30 µΩ less than 2𝑍4±6. Figure 3(b) shows all the computed 2𝑍4±[1 𝑡𝑜 5]. We find that 2𝑍4 is as a linear function of the DC density. For different AC biases, ℜ6,5,4,3,2,1±4 were measured to compute 𝛥ℜ6,5,4,3,2,1±4. ℜ4±4 and Δℜ4±4 as examples are presented, as shown in Fig. 3(c) with Hx-ext = −560 Oe. ℜ4±4 and Δℜ4±4 show the behavior similar to ℜ4±6 and Δℜ4±6, respectively. Figure 3(d) shows the computed resistances 2𝑍6,5,4,3,2,1±4. 2𝑍4±4 is ca lculated to be 4 4 µΩ lower than 2𝑍6±4 of 62 µΩ. 2𝑍±4 is as a linear function of the AC density. Hence , we conclude th e extra magnetization is sm, and r as well as 2 PHE DC AC 23 2Z z R j jM . ( 5) The coefficient, ζ, which indicates the capability of electric current inducing spin accumulation can be extracted from the measurement . ζ2 is proportional to 𝑍𝛼±𝛽 as shown in Eq. ( 5). 𝑍𝛼±𝛽 is extracted from Hall resistances measured at various combinations of AC and DC current densities. For each AC current density, the extracted Z shows a linear behavior with respect to the DC current density. Through carry ing out partial derivative of 𝑍𝛼±𝛽 over jDC for each AC density , DCZ j is obtained as shown in the inset of Fig. 4( a). DCZ j is a linear function to jAC. The slope of DCZ j with respect to or as a function of jAC, AC DCZ jj is calculated to be 1.3 µΩ∙[1010 Am−2]−2. We obtain 11 2 2 AC DC3 2PHEZRj j M from Eq. (5) . By substituting RPHE and M with 5.7 mΩ and 458 emu∙cc [45], respectively, ζ is computed to be 56 emu/cc per 1011 Am−2. For the sample stack with t=4, DCZ j is shown in Fig. 4( b). AC DCZ jj is obtained as 0.8 µΩ∙[1010 Am−2]−2, similarly, ζ is computed to be 45 emu/cc per 1011 Am−2, with RPHE=5.6 mΩ and M=466 emu /cc [45]. For the sample stacks with t=2, 6, 10, DCZ j is obtained at jAC=4× 1010 Am−2 as shown in Fig. 4(c). The expression of DCZ j is 2 PHE AC 2 DC3 2ZRjjM . Therefore, substituting jAC, RPHE and M with the ex perimental values in Fig. 4(c), we obtain ζ for the sample stacks with t=2, 6, 10, as shown in Fig. 4(d ) where ζ of t=4 and 8 are also included. For samples with t≤6, ζ is ~47 emu/cc per 1011 Am−2. For samples with t>6, ζ increases from 56 emu/cc per 1011 Am−2, reaching a maximum of 107 emu/cc per 1011 Am−2 at t = 10. The critical current density for SOT induced magnetization switching and domain wall driving is in the order of 1×1011 Am−2. The total current density in our experiment is in the same order for wires investigated. For films with t=2, 4, 6, 8, and 10, the measured spin accumulation is 54 emu/cc, 45 emu/cc, 44 emu/cc, 56 emu/cc and 107 emu/cc , respectively. The corresponding local magnetization is 581 emu/cc, 446 emu/cc, 436 emu/cc, 458 emu/cc, and 592 emu/cc. Therefore, t he ratio of the spin accumulation s to the local magnetization M, is ~10% for the films with t=2, 4 and 6, 12% for t=8, and 18% for 10. The p ercentage indicates that the spin density induced by a current density of 1×1011 Am−2 is in the comparable order to the local magnetization in all the stacks , as we firstly 12 claim. As the initial orientation of spin accumulation is along y-axis, when the local magnetization orientates along y-axis, the spin accumulation can vary the magnetization by as much as 13% in the wire with t=10. Therefore, to switch magnetization and drive domain wall motion by a current density of 1×1011 Am−2 is within expectation. If current density is increased to 1×1012 Am−2 which is the critical current for spin -transfer torque to switch magnetization and drive domain wall motion, the spin accumulation can be 540 emu/cc, 450 emu/cc, 440 emu/cc, 560 emu/cc and 1070 emu/cc for films with t=2, 4, 6, 8, and 10, respectively. The magnitudes of spin accumulation are close to the magnitudes of the corresponding local magnetization. In this case, the magnetic moments can be reorganized in the wires. Hence, the orientation of magnetization could be determined by the spin accumulation . As shown in Fig. 4(d), ζ follows the spin Hall angles of the samples which have been reported in our previous work with respect to the thickness of Ta layer [45], irrespective of their magnitudes. ζ represents the extra magnetization generated by an electric current, while spin Hall angle of Ta/Pt indicates the perc entage of spin current converted by Ta/Pt in an electric current. As such, the same trend of ζ and spin Hall angle with respect to the thickness of Ta is expected , since the extra magnetization should be as a linear function to magnitude of spin current. Therefore , the same trend confirms the spin accumulation in Co laye r is from the Ta and Pt layers. Conclusion 13 We have experimentally quantified the s pin accumulation induced by electric current in series stack s of Ta/Co/Pt. We find that spin accumulation is around dozens of percentage of local magnetization when the current density is 1011 Am−2. As our results demonstrate for the first time, when the spins from Ta and Pt are in relaxation state, they still contribute to secon d harmonic Hall resistance, instead when they are in initial state only as expected . The coefficient of spin accumulation over electric current is consistent with spin Hall angle . This consistency suggests that the coefficient can be used to evaluate the c apability of a heavy metal converting electric current to spin current. As the measurements are easily carried out, w e offer a concise solution to estimate spin accumulation. Acknowledgements This work was supported by the Singapore National Research Foundation, Prime Minister's Office, under a Competitive Research Programme (Non -volatile Magnetic Logic and Memory Integrated Circuit Devices, NRF -CRP9 -2011 -01), and an Industry -IHL Partnership Program (NRF2015 -IIP001 -001). The work was also supported by a MOE - AcRF Tier 2 Grant (MOE 2013 -T2-2-017). WSL is a member of the Singapore Spintronics Consortium (SG -SPIN). References [1] A. Manchon and S. Zhang, Phys. Rev. B 78, 212405 (2008); 79, 094422 (2009). [2] E.I. Rashba, Sov. Phys. Solid State 2, 1224 (1960). 14 [3] M. I. Dyakonov and V. I. Perel, Phys. Lett. A 35, 459 (1971). [4] Y. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6039 (1984). [5] J. Stohr and H.C. Siegmann, Magnetism from Fundamentals to Nanoscale Dynamics (Springer -Verlag Berlin, Heidelberg, 2006), p. 107. [6] I. M. Miron, G. Gaudin, S. Auffret, B. Rodmacq, A. Schuhl, S. Pizzini, J. Vogel, and P. Gambardella, Nat. Mater. 9, 230 (2010). [7] I. M. Miron, K. Garello, G. Gaudin, P. -J. Zermatten, M. V. Costache, S. Auffret, S. Bandiera, B. Rodmac q, A. Schuhl, and P. Gambardella, Nature 476, 189 (2011). [8] S. Emori, U. Bauer, S -M, Ahn, E. Martinez, and G. S. D. Beach, Nat. Mater. 12, 611 (2013). [9] X. Fan, H. Celik, J. Wu, C. Ni, K. -J. Lee, V. O. Lorenz, and J. Q. Xiao, Nat. Commun. 5, 3042 (2014 ). [10] T. D. Skinner, K. Olejní k, L. K. Cunningham, H. Kurebayashi, R. P. Campion, B. L. Gallagher, T. Jungwirth, and A. J. Ferguson, Nat. Commun. 6, 6730 (2015). [11] L. Berger, Phys. Rev. B 54, 9353 (1996). [12] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996); 247, 324 (2002). [13] S. Zhang, P. M. Levy, and A. Fert, Phys. Rev. Lett. 88, 236601 (2002). [14] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004). [15] K. Garello, I. M. Miron, C. O. Avci, F. Freimuth, Y. Mokrousov, S. Blu ¨ gel, S. A uffret, O. Boulle, G. Gaudin, and P. Gambardella, Nat. Nanotechnol. 8, 587 (2013). 15 [16] X. Wang and A. Manchon, Phys. Rev. Lett. 108, 117201 (2012). [17] S. -M. Seo, K. -W. Kim, J. Ryu, H. -W. Lee, and K. -J. Lee, Appl. Phys. Lett. 101, 022405 (2012). [18] J. Kim, J. Sinha, M. Hayashi, M. Yamanouchi, S. Fukami, T. Suzuki, S. Mitani, and H. Ohno, Nat. Mater. 12, 240 (2013). [19] M. Jamali, K. Narayanapillai, X. Qiu, L. M. Loong, A. Manchon, and H. Yang, Phys. Rev. Lett. 111, 246602 (2013). [20] M. Hayashi, J. Ki m, M. Yamanouchi, and H. Ohno, Phys. Rev. B 89, 144425, (2014). [21] Satoru Emori, David C. Bono, and Geoffrey S. D. Beach, Appl. Phys. Lett. 101, 042405 (2012). [22] L. Q. Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett. 109, 096602 (2012). [23] S. Woo, M. Mann, A. J. Tan, L. Caretta, and G. S. D. Beach, Appl. Phys. Lett. 105, 212404 (2014). [24] H. R. Lee, K. Lee, J. Cho, Y. H. Choi, C. Y. You, M. H. Jung, F. Bonell, Y. Shiota, S. Miwa, and Y. Suzuki, Sci. Rep. 4, 6548 (2 014). [25] X. Qiu, P. Deorani, K. Narayanapillai, K. -S. Lee, K. -J. Lee, H. -W. Lee, and H. Yang, Sci. Rep. 4, 4491 (2014). [26] M. Montazeri, P. Upadhyaya, M. C. Onbasli, G. Yu, K. L. Wong, M. Lang, Y. Fan, X. Li, P. K. Amiri, R. N. Schwartz, C. A. Ross, a nd K. L. Wang, Nat. Commun. 6, 8958 (2015). 16 [27] X. Qiu, K. Narayanapillai, Y. Wu, P. Deorani, D. H. Yang, W. S. Noh, J. H. Park, K. J. Lee, H. W. Lee, and H. Yang, Nat. Nanotechnol. 10, 333 (2015). [28] D. Chiba, M. Kawaguchi, S. Fukami, N. Ishiwata, K. Shimamura, K. Kobayashi, and T. Ono, Nat. Commun. 3, 888 (2012). [29] P. P. J. Haazen, E. Mure, J. H. Franken, R. Lavrijsen, H. J. M. Swagten, and B. Koopmans, Nat. Mater. 12, 299 (2013). [30] K. S. Ryu, L. Thomas, S. H. Yang, and S. Parkin, Nat. Nanotechn ol. 8, 527 (2013). [31] T. Kimura, Y. Otani, T. Sato, S. Takahashi, and S. Maekawa, Phys. Rev. Lett. 98, 156601 (2007). [32] K. Ando, S. Takahashi, K. Harii, K. Sasage, J. Ieda, S. Maekawa, and E. Saitoh, Phys. Rev. Lett. 101, 036601 (2008). [33] O. Mosend z, J. E. Pearson, F. Y. Fradin, G. E. W. Bauer, S. D. Bader, and A. Hoffmann, Phys. Rev. Lett. 104, 046601 (2010). [34] L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett. 106, 036601 (2011). [35] L. Liu, C. F Pai, Y. Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Science 336, 6081 (2012). [36] G. Allen, S. Manipatruni, D. E. Nikonov, M. Doczy, and I. A. Young, Phys. Rev. B 91, 144412 (2015). [37] K. Ueda, C. F. Pai, A. J. Tan, M. Mann, and G. S. D. Beach, Appl. Phys. Lett. 108, 232405 (2016). 17 [38] X. Qiu, W. Legrand, P. He, Y. Wu, J. Yu, R. Ramaswamy, A. Manchon, and H. Yang, Phys. Rev. Lett. 117, 217206 (2016). [39] T. Suzuki , S. Fukami, N. Ishiwata, M. Yamanouchi, S. Ikeda, N. Kasai, and H. Ohno, Appl. Phys. Lett. 98, 142505 (2011). [40] V. D. Ky, and I. A. N SSR, Ser. Fiz. 29(4), 576 (1965). [41] M. L. Yu, and J. T. H. Chang, J. Phys. Chem. Solids 31, 1997 (1970). [42] V. D. Ky, Phys. Stat. Sol. 26, 565 (1968). [43] T. R. McGuire and R. I. Potter, IEEE Trans. Magn. 11, 1018 (1975). [44] H. X. Tang, R. K. Kawakami, D. D. Awschalom, and M. L. Roukes, Phys. Rev. Lett. 90, 107201 (2003). [45] F. Luo, S. Goolaup, W. C. Law, S. Li, F. Tan, C. Engel, T. Zhou, W. S . Lew, Phys. Rev. B. 95,174415 (2017). Figures and captions 18 Fig. 1 (a) the transfer of spin torque from s to M in dozens of nanoseconds; (b) s relaxes to be along M after SOT transferring; (c) schematic of measurement setup; ( d) measured ℜ4±6 and obtained Δℜ4±6 with respect to Hy-ext, fit of Δℜ4±6 indicate that Δℜ4±6 is fitted by Eq. (2) where Hx-ext is 560 Oe; ( e) measured ℜ4±6 with respect to Hy-ext, fit of Δℜ4+6 indicate that Δℜ4+6 is fitted to reaching minimum RMSE by Eq. (1) with changing Hx-ext to 560+69 Oe and Hy-ext to 560+38 Oe, fit of Δℜ4−6 indicate that Δℜ4−6 is fitted to reaching minimum RMSE by Eq. (1) with changing Hx-ext to 560+60 Oe and Hy-ext to 560−39 Oe ; (f) measured first harmonic Hall resistances with respect to Hy-ext when obtaining ℜ4+6 with Hx-ext =± 560 Oe. In (d), the magenta and violet lines show the fit to the experimental data . 19 Fig. 2 ( a) measured ℜ4±6 and obtained Δℜ4±6 with respect to Hy-ext when Hx-ext=−560 Oe; (b) obtained first harmonic Hall resistance with respect to Hy-ext when measuring corresponding ℜ4±6 with Hx-ext equating to +360 Oe and +1000 Oe; (c) measured ℜ4±6 and obtained Δℜ4±6 with respect to Hy-ext when Hx-ext=+360 Oe; (d) measured ℜ4±6 and obtained Δℜ4±6 with respect to Hy-ext when Hx-ext=+1000 Oe. In (a), (c) and (d), the magenta and violet lines show the fit to the experimental data . 20 Fig. 3 (a) measured ℜ4±3 and obtained Δℜ4±3 with respect to Hy-ext when Hx-ext=−560 Oe; (b) calculated 𝑍4±𝛽 with respect to the DC offset β when the AC density is fixed to be 4× 1010 Am−2; (c) measured ℜ4±4 and obtained Δℜ4±4 with respect to Hy-ext when Hx- ext=−560 Oe; (d) calculated 𝑍𝛼±4 with respect to the AC α when the DC densities are fixed to be ± 4× 1010 Am−2. In (a) and (c), the magenta and violet lines show the fit to the experimental data. 21 Fig. 4 calculated 𝑍𝛼±𝛽 with respect to the DC offset β under different AC densities, inset is the slope of 𝑍𝛼±𝛽 to β with respect to α for sample (a) Ta(8 nm)/Co(2 nm)/Pt(5 nm) and sample (b) Ta(4 nm)/Co(2 nm)/Pt(5 nm); (c) calculated 𝑍4±𝛽 with respect to the DC offset β for the samples of Ta( t nm)/Co(2 nm)/Pt(5 nm) with t=2, 6 and 10; (d) calculated ζ (black line) and reported spin Hall angle (red line) with respect to the thickness of Ta.
2002.02415v1.Correlated_motion_of_particle_hole_excitations_across_the_renormalized_spin_orbit_gap_in___rm_Sr_2_Ir_O_4_.pdf	arXiv:2002.02415v1 [cond-mat.str-el] 5 Feb 2020Correlated motion of particle-hole excitations across the renormalized spin-orbit gap in Sr2IrO4 Shubhajyoti Mohapatra1and Avinash Singh1,∗ 1Department of Physics, Indian Institute of Technology, Kanpu r - 208016, India (Dated: February 7, 2020) The high-energy collective modes of particle-hole excitat ions across the spin-orbit gap in Sr 2IrO4are investigated using the transformed Coulomb interactio n terms in the pseudo-spin-orbital basis constituted by the J= 1/2 and 3/2 states arising from spin-orbit coupling. With appropriate interaction st rengths and renormalized spin-orbit gap, these collective modes yield two well-deﬁn ed propagating spin-orbit exciton modes, with energy scale anddispersioninexcellen t agreement with resonant inelastic X-ray scattering (RIXS) measurements. PACS numbers: 75.30.Ds, 71.27.+a, 75.10.Lp, 71.10.Fd2 I. INTRODUCTION The iridium based transition-metal oxides exhibiting novel J=1/2 Mott insulating states have attracted considerable interest in recent years in view of the ir potential for host- ing collective quantum states such as quantum spin liquids, topologica l orders, and high- temperature superconductors.1The eﬀective J=1/2 antiferromagnetic (AFM) insulating stateiniridates arises fromanovel interplay between crystal ﬁeld , spin-orbit coupling (SOC) and intermediate Coulomb correlations. Exploration of the emerging quantum states in the iridate compounds therefore involves investigation of the correlat ed spin-orbital entangled electronic states and related magnetic properties. Amongtheiridiumcompounds, thequasi-two-dimensional (2D)squa re-latticeperovskite- structured iridate Sr 2IrO4is of special interest as the ﬁrst spin-orbit Mott insulator to be identiﬁed and because of its structural and physical similarity with L a2CuO4.2,3It exhibits canted AFM ordering of the pseudospins below N´ eel temperature TN≈240 K. The canting of the in-plane magnetic moments tracks the staggered IrO 6octahedral rotations about the caxis. The eﬀectively single (pseudo) orbital ( J=1/2) nature of this Mott insulator has motivated intensive ﬁnite doping studies aimed at inducing the superc onducting state as in the cuprates.4–10 Technological advancements and improved energy resolution in res onant inelastic X-ray scattering (RIXS) have been instrumental in the elucidation of the pseudospin dynamics in Sr2IrO4. Recent measurements point to a partially resolved ∼30 meV magnon gap at the Γ point,11which has been further resolved via high-resolution RIXS and inelast ic neutron scattering (INS), bothof which indicate another magnongapbetw een 2 to 3 meV at ( π,π).12 These low-energyfeaturescorrespondtodiﬀerent magnonmode sassociatedwithbasal-plane and out-of-plane ﬂuctuations, indicating the presence of anisotr opic spin interactions. In addition to magnon modes, RIXS experiments have also revealed a hig h-energy dispersive feature in the energy range 0.4-0.8 eV. Attributed to electron-ho le pair excitations across the spin-orbit gap between the J=1/2 and 3/2 bands, this distinctive mode is referred to as the spin-orbit exciton.13–17 Among the theoretical approaches, the spin-orbit exciton was ide ntiﬁed as a bound state inthespectral functionofthetwo-particleGreen’sfunctionwithin themulti-orbitalitinerant electron picture.16However, the full dispersion was not obtained, and the original t2gbasis3 was employed instead of the more natural SOC-split Jstates with intrinsic spin-orbit gap. In another approach, the exciton dispersion was obtained in analog y with hole motion in an AFM background.13,15However, the bare exciton dispersion was neglected, and an appro ach which allows for a uniﬁed description of both magnon and spin-orbit ex citon on the same footing will be desirable as both excitations are observed in the same RIXS measurements. In this paper, we therefore plan to investigate the correlated mot ion of inter-orbital particle-hole excitations across the renormalized spin-orbit gap (b etweenJ=1/2 andJ=3/2 sectors), along with detailed comparison with RIXS data for the spin -orbit exciton modes in Sr2IrO4. Similar comparison for the magnon dispersion involving intra-orbital (J=1/2) particle-hole excitations has provided experimental evidence of se veral distinctive features associated with the rich interplay of spin-orbit coupling, Coulomb inte raction, and realistic multi-orbital electronicbandstructure, such as(i)ﬁnite- Uandﬁnite-SOCeﬀects, (ii)mixing and coupling between the J=1/2 and 3/2 sectors, and (iii) Hund’s-coupling-induced true magnetic anisotropy and magnon gap.18–20 Thestructure ofthepaper isasfollows. After a briefaccount oft hetransformedCoulomb interaction terms in the pseudo-spin-orbital basis in Sec. II, the A FM state of the three orbital model is discussed in Sec. III. The spin-orbit gap renormaliz ation due to the relative energy shift between the J=1/2 and 3/2 sectors arising from the density interaction terms is discussed in Sec. IV. The spin-orbit exciton as a resonant state f ormed by the correlated propagation of the inter-orbital, spin-ﬂip, particle-hole excitation across the renormalized spin-orbit gap is investigated in Sec. V. Finally, conclusions are prese nted in Sec. VI. II. COULOMB INTERACTION IN THE PSEUDO-SPIN-ORBITAL BASIS Due to large crystal-ﬁeld splitting ( ∼3 eV) in the IrO 6octahedra, the low-energy physics ind5iridates is eﬀectively described by projecting out the empty e glevels which are well above thet2glevels. Spin-orbit coupling (SOC) further splits the t 2gstates into (upper) J=1/2 doublet and (lower) J=3/2 quartet with an energy gap of 3 λ/2. Four of the ﬁve electrons ﬁll the J=3/2 states, leaving one electron for the J=1/2 sector, rendering it mag- netically active in the ground state. The three Kramers pairs above correspond to pseudo orbitals (l= 1,2,3) withpseudo4 FIG. 1: The pseudo-spin-orbital energy level scheme for the three Kramers pairs along with their orbital shapes. The colors represent the weights of real spi n↑(red) and ↓(blue) in each pair. spins(τ=↑,↓) each, with the \|J,mj/angbracketrightand corresponding \|l,τ/angbracketrightstates having the form: \|l= 1,τ=σ/angbracketright=/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 2,±1 2/angbracketrightbigg = [\|yz,¯σ/angbracketright±i\|xz,¯σ/angbracketright±\|xy,σ/angbracketright]/√ 3 \|l= 2,τ=σ/angbracketright=/vextendsingle/vextendsingle/vextendsingle/vextendsingle3 2,±1 2/angbracketrightbigg = [\|yz,¯σ/angbracketright±i\|xz,¯σ/angbracketright∓2\|xy,σ/angbracketright]/√ 6 \|l= 3,τ= ¯σ/angbracketright=/vextendsingle/vextendsingle/vextendsingle/vextendsingle3 2,±3 2/angbracketrightbigg = [\|yz,σ/angbracketright±i\|xz,σ/angbracketright]/√ 2 (1) where\|yz,σ/angbracketright,\|xz,σ/angbracketright,\|xy,σ/angbracketrightare the t 2gstates and the signs ±correspond to spins σ=↑/↓. The coherent superposition of diﬀerent-symmetry t2gorbitals, with opposite spin polariza- tion between xz/yzandxylevels implies spin-orbital entanglement, and also imparts unique extended 3D shape to the pseudo-orbitals l= 1,2,3, as shown in Fig 1. Inverting the above transformation, the three real-spin-orbita l basis states can be repre- sented in terms of the pseudo-spin-orbital basis states, given be low in terms of the corre- sponding creation operators: a† yzσ a† xzσ a† xyσ = 1√ 31√ 61√ 2 iσ√ 3iσ√ 6−iσ√ 2 −σ√ 3√ 2σ√ 30 a† 1τ a† 2τ a† 3τ (2) where,σ=↑/↓andτ=σ. We consider the on-site Coulomb interaction terms: Hint=U/summationdisplay i,µniµ↑niµ↓+U′/summationdisplay i,µ<ν,σniµσniνσ+(U′−JH)/summationdisplay i,µ<ν,σniµσniνσ +JH/summationdisplay i,µ/negationslash=ν(a† iµ↑a† iν↓aiµ↓aiν↑+a† iµ↑a† iµ↓aiν↓aiν↑) (3)5 in the real-spin-orbital basis ( µ,ν=yz,xz,xy ), including the intra-orbital ( U) and inter- orbital (U′) density interaction terms, the Hund’s coupling term ( JH), and the pair hopping term (JH). Herea† iµσandaiµσare the creation and annihilation operators for site i, orbital µ, spinσ=↑,↓, and the density operator niµσ=a† iµσaiµσ. Using the transformation from the t2gbasis to the pseudo-spin-orbital basis given above, and keeping the Hubbard, density, and Hund’s coupling like interactio n terms which are relevant for the present study, we obtain (for site i): Hint(i) =1 2/summationdisplay m,m′,τ,τ′Uττ′ mm′nmτnm′τ′+/parenleftbiggU−U′ 3/parenrightbigg/summationdisplay τa† 1τa† 2τa1τa2τ +/parenleftbiggU−2JH−U′ 6/parenrightbigg/summationdisplay τ/parenleftig a† 2τa† 3τa2τa3τ+2a† 3τa† 1τa3τa1τ/parenrightig (4) where the transformed interaction matrices Uττ′ mm′in the new basis ( m,m′= 1,2,3): Uττ mm′= 0U′U′−2 3JH U′0U′−1 3JH U′−2 3JHU′−1 3JH0 , Uττ mm′= 1 3(U+2U′)1 3(U+2U′−3JH)1 3(U+2U′−JH) 1 3(U+2U′−3JH)1 2(U+U′)1 6(U+5U′−4JH) 1 3(U+2U′−JH)1 6(U+5U′−4JH)1 2(U+U′) (5) for pseudo-spins τ′=τandτ′=τ, whereτ=↑,↓. Similar transformation to the Jbasis has been discussed recently, focussing only on the density interac tion terms.21 Using the spherical symmetry condition ( U′=U-2JH), the transformed interaction Hamil- tonian (4) simpliﬁes to: Hint(i) =/parenleftbigg U−4 3JH/parenrightbigg n1↑n1↓+(U−JH)[n2↑n2↓+n3↑n3↓] −4 3JHS1.S2+2JH[Sz 1Sz 2−Sz 1Sz 3] +/parenleftbigg U−13 6JH/parenrightbigg [n1n2+n1n3]+/parenleftbigg U−7 3JH/parenrightbigg n2n3. (6) The symmetry features of the interaction terms above are consis tent with a general pseudo- spin rotation symmetry analysis which shows that the Hund’s coupling (JH) and pair- hopping (JH) interaction terms in Eq. (3) explicitly break this symmetry systema tically, while the Hubbard ( U) and density ( U′) interaction terms do not.226 III. ANTIFERROMAGNETIC STATE OF THE THREE-ORBITAL MODEL We consider the various interaction terms in Eq. (6) in the Hartree- Fock (HF) approxi- mation, focussing on the staggered ﬁeld terms corresponding to ( π,π) ordered AF state on the square lattice. The charge terms corresponding to density co ndensates will be discussed in the next section. For general ordering direction with component s∆l= (∆x l,∆y l,∆z l), the staggered ﬁeld term for sector lin the pseudo-orbital basis is given by: Hsf(l) =/summationdisplay ksψ† kls/parenleftig −sτ.∆l/parenrightig ψkls=/summationdisplay ks−sψ† kls ∆z l∆x l−i∆y l ∆x l+i∆y l−∆z l ψkls (7) whereψ† kls= (a† kls↑a† kls↓),s=±1 for the two sublattices A/B, and the staggered ﬁeld components ∆α=x,y,z l=1,2,3are self-consistently determined from: 2∆α 1=U1mα 1+2JH 3mα 2+JH(mα 3−mα 2)δαz 2∆α 2=U2mα 2+2JH 3mα 1−JHmα 1δαz 2∆α 3=U3mα 3+JHmα 1δαz (8) in terms of the staggered pseudo-spin magnetization components mα=x,y,z l=1,2,3. In practice, it is easier to choose set of ∆l=1,2,3and self-consistently determine the Hubbard-like interaction strengths Ul=1,2,3such that U1=U−4 3JHandU2=U3=U−JHusing Eq. (8). Transforming the staggered-ﬁeld term back to the three-orbita l basis (yzσ,xzσ,xy ¯σ), and including the SOC and band terms,23the full HF Hamiltonian considered in our band structure and spin ﬂuctuation analysis is given by HHF=HSO+Hband+Hsf, where, Hband=/summationdisplay kσsψ† kσs ǫyz k′0 0 0ǫxz k′0 0 0ǫxy k′ δss′+ ǫyz kǫyz\|xz k0 −ǫyz\|xz kǫxz k0 0 0 ǫxy k δ¯ss′ ψkσs′ (9) in the composite three-orbital, two-sublattice basis, showing the d iﬀerent hopping terms connecting the same and opposite sublattice(s). Corresponding t o the hopping terms in the7 tight-binding model, the various band dispersion terms in Eq. (9) are given by: ǫxy k=−2t1(coskx+cosky) ǫxy k′=−4t2coskxcosky−2t3(cos2kx+cos2ky)+µxy ǫyz k=−2t5coskx−2t4cosky ǫxz k=−2t4coskx−2t5cosky ǫyz\|xz k=−2tm(coskx+cosky). (10) Heret1,t2,t3are respectively the ﬁrst, second, and third neighbor hopping ter ms for the xyorbital, which has energy oﬀset µxyfrom the degenerate yz/xzorbitals induced by the tetragonal splitting. For the yz(xz) orbital,t4andt5are the NN hopping terms in y(x) andx(y) directions, respectively. Mixing between xzandyzorbitals is represented by the NN hopping term tm. We have taken values of the tight-binding parameters ( t1,t2,t3,t4, t5,tm,µxy,λ) = (1.0, 0.5, 0.25, 1.028, 0.167, 0.0, -0.7, 1.35) in units of t1, where the energy scalet1= 280 meV. Using above parameters, the calculated electronic band structure shows AFM insulating state and mixing between pseudo-orbital sectors.18,23As the pseudo-spin canting is not relevant for the following discussion, we have set tmto zero by going to the locally rotated coordinate frame. To illustrate the AF state calculation, we have taken staggered ﬁeld values ∆x l=1,2,3= (0.92,0.08,−0.06) in units of t1, which ensures self-consistency for all three orbitals, with the given relations U2=U3=U1+JH/3. Using the calculated sublattice magnetization val- uesmx l=1,2,3=(0.65,0.005,-0.038), we obtain Ul=1,2,3=(0.80,0.83,0.83) eV, which ﬁnally yields U=U1+4 3JH=0.93 eV for JH=0.1 eV. For these parameter values, the calculated magnon dispersion and energy gap are in very good agreement with RIXS mea surements.11–13,15The easyx-yplane anisotropy arising from Hund’s coupling results in energy gap ≈40 meV for the out-of-plane ( z) magnon mode.19 The electron ﬁllings are obtained as nl=1,2,3≈(1.064,1.99,1.946) in the three pseudo orbitals. Finitemixing betweenthe J=1/2and3/2sectorsisreﬂectedinthesmalldeviations from ideal ﬁllings and also in the very small magnetic moment values for l= 2,3 as given above, which play a crucial role in the expression of true magnetic an isotropy and magnon gap in view of the Hund’s coupling induced anisotropic interactions in Eq . (6). The values λ=0.38 eV,U=0.93 eV, and JH=0.1 eV taken above lie well within the estimated parameter range for Sr 2IrO4.16,248 IV. RENORMALIZED SPIN-ORBIT GAP In this section, we obtain the relative energy shift between the J=1/2 and 3/2 states arising from the transformed density interaction terms in Eq. (6. T his relative shift eﬀec- tively renormalizes the spin-orbit gap and plays an important role in de termining the energy scale of the spin-orbit exciton, as discussed in the next section. Co rresponding to the total density condensate /angbracketleftnl↑+nl↓/angbracketrightin the HF approximation of the density interaction terms, the spin-independent self-energy contributions for the three pseud o orbitals are obtained as: Σl=1 dens=U/angbracketleftbigg1 2n1+n2+n3/angbracketrightbigg −JH/angbracketleftbigg2 3n1+13 6n2+13 6n3/angbracketrightbigg Σl=2 dens=U/angbracketleftbigg n1+1 2n2+n3/angbracketrightbigg −JH/angbracketleftbigg13 6n1+1 2n2+7 3n3/angbracketrightbigg Σl=3 dens=U/angbracketleftbigg n1+n2+1 2n3/angbracketrightbigg −JH/angbracketleftbigg13 6n1+7 3n2+1 2n3/angbracketrightbigg (11) The formally unequal contributions will result in relative energy shift s between the three orbitals depending on the electron ﬁlling. With /angbracketleftn1/angbracketright=1 and/angbracketleftn2/angbracketright=/angbracketleftn3/angbracketright=2 for thed5system having nominally half-ﬁlled and ﬁlled orbitals, the relative energy shift: ∆dens= Σl=1 dens−Σl=2,3 dens=U−3JH 2(12) betweenl=1 and (degenerate) l=2,3 orbitals. ForU >3JH, the relative energy shift enhances the energy gap between J=1/2 and 3/2 sectors, eﬀectively resulting in a correlation-induced renormaliza tion of the spin-orbit gap and the spin-orbit coupling. The SOC strength is renormalized as ˜λ=λ+ 2∆dens/3 by the relative energy shift. With ∆ dens= (U−3JH)/2≈0.3 eV for the parameter values consideredearlier, weobtain ˜λ≈0.6eV,whichisinagreementwiththecorrelation-enhanced SOC strength obtained in a recent DFT study of Sr 2IrO4.24Ford4systems with nominally /angbracketleftn1/angbracketright=0, the relative energy shift increases to U−3JH. This enhancement of the spin-orbit gap renormalization is seen in recent DFT study of the hexagonal irid ates Sr 3LiIrO6and Sr4IrO6with Ir5+(5d4) and Ir4+(5d5) ions, respectively.25 V. SPIN-ORBIT EXCITON Magnon excitations modes in Sr 2IrO4essentially involve collective modes of intra-orbital, spin-ﬂip, particle-hole excitations within the magnetically active J=1/2 sector.18,19In anal-9 FIG. 2: Propagator of inter-orbital, spin-ﬂip, particle-h ole excitations across the renormalized spin-orbit gap between the nominally ﬁlled J=3/2 sector and the half-ﬁlled J=1/2 sector. ogy, we will investigate here the collective modes of inter-orbital, pa rticle-hole excitations across the renormalized spin-orbit gap between the nominally ﬁlled J=3/2 sector and the half-ﬁlledJ=1/2 sector. We will consider both pseudo-spin-ﬂip and non-pseud o-spin-ﬂip cases for these spin-orbit exciton modes. Starting ﬁrst with the s pin-ﬂip case, we consider the composite pseudo-spin-orbital ﬂuctuation propagator in the z-ordered AFM state: χ−+ so(q,ω) =/integraldisplay dt/summationdisplay ieiω(t−t′)e−iq.(ri−rj)/angbracketleftΨ0\|T[S− i,m,n(t)S+ j,m,n(t′)]\|Ψ0/angbracketright (13) involving the inter-orbital spin-raising and -lowering operators S+ j,m,n=a† jm↑ajn↓and S− i,m,n=a† in↓aim↑at lattice sites jandi, describing the propagation of a spin-ﬂip, particle-hole excitation between diﬀerent pseudo orbitals mandn. Althoughthe most general propagator would involve S− i,m,nandS+ j,m′,n′, the above simpliﬁed propagator is a good approximation in view of the orbital restrictions on the particle-hole states as disc ussed below. Also, we have considered the z-ordered AFM state for simplicity as the JH-induced weak easy-plane anisotropy has negligible eﬀect on the spin-orbit exciton. In the ladder-sum approximation, the spin-orbital propagator is o btained as: [χ−+ so(q,ω)] =[χ0 so(q,ω)] 1−U[χ0 so(q,ω)](14) where the relevant interactions U=Uττ mnfor the spin-ﬂip particle-hole pair are given in Eq. (5). The ladder-sum approximation with repeated (attractive ) interactions (as shown in Fig. (2) for the retarded case) represents resonant scatter ing of the particle-hole pair, resulting in a resonant state split-oﬀ from the particle-hole continu um, which we identify as the spin-orbit exciton modes.10 The bare particle-hole propagator in the above equation: [χ0 so(q,ω)]mn ss′=/summationdisplay k/bracketleftigg /angbracketleftϕn k−q\|τ−\|ϕm k/angbracketrights/angbracketleftϕm k\|τ+\|ϕn k−q/angbracketrights′ E+ k−q−E− k+ω−iη+/angbracketleftϕn k−q\|τ−\|ϕm k/angbracketrights/angbracketleftϕm k\|τ+\|ϕn k−q/angbracketrights′ E+ k−E− k−q−ω−iη/bracketrightigg (15) wasevaluatedinthetwo-sublattice basisbyintegratingoutthefer mionsinthe( π,π)ordered state. Here Ekandϕkare the eigenvalues and eigenvectors of the Hamiltonian matrix in the pseudo-spin-orbital basis, and the Eksuperscript +( −) refers to particle (hole) energies above (below) the Fermi energy. The projected amplitudes ϕm kτabove were obtained by projecting the kstates in the three-orbital basis \|µ,σ/angbracketrighton to the pseudo-orbital basis \|m,τ/angbracketright corresponding to the J= 1/2 and 3/2 sector states, as given below: ϕ1 k↑=1√ 3/parenleftbig φyz k↓−iφxz k↓+φxy k↑/parenrightbig ϕ1 k↓=1√ 3/parenleftbig φyz k↑+iφxz k↑−φxy k↓/parenrightbig ϕ2 k↑=1√ 6/parenleftbig φyz k↓−iφxz k↓−2φxy k↑/parenrightbig ϕ2 k↓=1√ 6/parenleftbig φyz k↑+iφxz k↑+2φxy k↓/parenrightbig ϕ3 k↑=1√ 2/parenleftbig φyz k↓+iφxz k↓/parenrightbig ϕ3 k↓=1√ 2/parenleftbig φyz k↑−iφxz k↑/parenrightbig (16) in terms of the amplitudes φµ kσin the three-orbital basis ( µ=yz,xz,xy ). The [χ0(q,ω)] matrix was evaluated by performing the ksum over the 2D Brillouin zone divided into a 300×300 mesh. The dominant contribution to [ χ0 so(q,ω)] above will correspond to particle (+) states in the nominally half-ﬁlled pseudo-orbital m=1 (J=1/2 sector) and hole ( −) states in the nom- inally ﬁlled pseudo-orbitals n=2,3 (J=3/2 sector). Due to these restrictions, the bare propa- gatoressentially becomes diagonalinthecomposite particle-holeor bitalbasis( m′=m,n′=n), which justiﬁes the simpliﬁed propagator considered above. In orde r to focus exclusively on the high-energy spin-orbit exciton modes, particle-hole excitation s within the J=1/2 sector (which yield the low-energy magnon modes) have been excluded. Fig. 3 shows the spin-orbit exciton spectral function: Aq(ω) =1 πIm Tr/bracketleftbig χ−+ so(q,ω)/bracketrightbig (17) as an intensity plot for qalong the high symmetry directions of the BZ. For clarity, we have considered here the particle-hole propagator in Eq. (15) separat ely for (m,n)=(1,3) and (1,2). The relevant interaction terms for these two cases are: Uττ 13=U-5JH/3 andUττ 12=U- 7JH/3. Here, we have taken U=0.93 eV and JH=0.1 eV as obtained in Sec. III, and the renormalized spin-orbit gap (Sec. IV) has been incorporated.11 (π/2,π/2)(π,0) (π,π) (π/2,π/2) (0,0) ( π,0) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9ω (eV) 0.001 0.01 0.1 1 10 n = 3 (a) (π/2,π/2)(π,0) (π,π) (π/2,π/2) (0,0) ( π,0) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9ω (eV) 0.001 0.01 0.1 1 10 n = 2 (b) FIG. 3: The spin-orbit exciton spectral function Aq(ω) for the two cases: (a) ( m,n)=(1,3) and (b) (m,n)=(1,2), showing well deﬁned dispersive modes near the lowe r edge of the continuum. The exciton represents collective spin-orbital excitations a cross the renormalized spin-orbit gap. Thespin-orbitexcitonspectralfunctioninFig. 3(a)clearlyshowsa welldeﬁnedpropagat- ing mode near the lower edge of the continuum with signiﬁcantly higher intensity compared to the continuum background. With increasing interaction strengt h, this mode progressively shifts to lower energy further away from the continuum, and beco mes less dispersive and more prominent in intensity, indicating enhanced localization of the sp in-orbit exciton. Fig. 3(b) shows a similar exciton mode for the other case ( m,n)=(1,2), with slightly higher energy and reduced dispersion as well as signiﬁcant damping. The relatively reduced interaction strength Uττ 12for this mode accounts for the slightly higher energy. We have similarly obtained the spectral functions for the non-spin-ﬂip case s by considering operators nτ j,m,n=a† jmτajnτandnτ† i,m,n=a† inτaimτinstead ofS+ j,m,nandS− i,m,nin Eq. (13) with appropriate interactions Uττ mn. The spectral functions for these cases are nearly identical, as e xpected from the non-magnetic character of the ﬁlled J=3/2 sector. The calculated dispersion and energy scale of the two spin-orbit exc iton modes are in excellent agreement with RIXS measurements in Sr 2IrO4.11,13Comparison of the calculated Aq(ω) with the observed RIXS intensity and its momentum dependence is b eyond the scope of this work. The basic RIXS mechanism involved in the creation of the spin-orbit exciton, whose propagation is considered in Eq. (14), is explained below. TheL3-edge RIXS essentially involves second-order dipole-allowed transit ions between 2p3/2core level and t2glevels. The incoming photon resonantly excites a 2 p3/2electron to the unﬁlled t2gstates (upper Hubbard band of the nominally J= 1/2 sector). In the12 FIG. 4: The optical (i) excitation ( i→2p3/2) and (ii) de-excitation (2 p3/2→f) processes (in the hole picture) involved in the RIXS mechanism for the particl e-hole excitation across the renormal- ized spin-orbit gap. The real spin is conserved in optical tr ansitions. subsequent radiative de-excitation, an electron from the ﬁlled t2gstates ﬁlls the 2 p3/2core hole, the loss in photon energy thereby corresponding to the over all particle-hole excitation in thet2gmanifold. The magnon and spin-orbit exciton cases correspond to t he ﬁnal-state t2ghole created in the J= 1/2 and 3/2 sectors, respectively. In the magnon case, with both initial and ﬁnal hole states in the J= 1/2 sector (in the hole picture), the dipole matrix elements /angbracketleft2p3/2\|Dǫ\|i/angbracketrightand/angbracketleftf\|D† ǫ′\|2p3/2/angbracketrightinvolving pseudo- spin-ﬂip have been shown to be ﬁnite,26,27implying that RIXS is fully allowed, and the observed low-energy RIXS spectrum corresponds to the magnon excitation. In the spin- orbit exciton case, with ﬁnal hole state in the J=3/2 sector, the optical excitation and de-excitation processes are shown in Fig. 4. These processes invo lve no change in real spin which is conserved in optical transitions.27However, due to the spin-orbital entangled nature of the Jstates, both pseudo-spin-ﬂip and non-pseudo-spin-ﬂip cases ar e allowed with respect to the initial and ﬁnal hole states. For example, the pseud o-spin-ﬂip case is realized ifi→2p3/2involves excitation of ( xy,σ=↑) hole from \|l= 1,τ=↑/angbracketrightstate and 2 p3/2→f involves de-excitation of hole to the ( yz,σ=↑) component of \|l= 3,τ=↓/angbracketrightstate.13 VI. CONCLUSIONS Well-deﬁned propagating spin-orbit exciton modes were obtained re presenting collective modes of inter-orbital, particle-hole excitations across the renor malized spin-orbit gap, with both dispersion and energy scale in excellent agreement with RIXS st udies. The relevant interaction terms for the two exciton modes as well as the renorma lized spin-orbit gap, which play an important role in the spin-orbit exciton energy scale, we re obtained from the transformation of the various Coulomb interaction terms to the ps eudo-spin-orbital basis formed by the J=1/2 and 3/2 states. The approach presented here allows for a un iﬁed description of magnons and spin-orbit excitons in spin-orbit coupled systems. ∗Electronic address: avinas@iitk.ac.in 1W. Witczak-Krempa, G. Chen, Y. B. Kim, and L. Balents, Annu. R ev. Condens. Matter Phys. 5, 57-82 (2014). 2J. G. Rau, E. Kin-Ho Lee, and H.-Y. Kee, Annu. Rev. Condens. Ma tter Phys. 7, 195-221 (2016). 3J. Bertinshaw, Y. K. Kim, G. Khaliullin, and B. J. Kim, Annu. R ev. Condens. Matter Phys. (in press). 4F. Wang and T. Senthil, Phys. Rev. Lett. 106, 136402 (2011). 5Y. K. Kim, O. Krupin, J. D. Denlinger, A. Bostwick, E. Rotenbe rg, Q. Zhao, J. F. Mitchell, J. W. Allen, and B. J. Kim, Science 345, 187190 (2014). 6A. de la Torre, S. McKeown Walker, F. Y. Bruno, S. Ricc´ o, Z. Wa ng, I. Gutierrez Lezama, G. Scheerer, G. Giriat, D. Jaccard, C. Berthod, T. K. Kim, M. Hoe sch, E. C. Hunter, R. S. Perry, A. Tamai, and F. Baumberger, Phys. Rev. Lett. 115, 176402 (2015). 7Y. K. Kim, N. H. Sung, J. D. Denlinger, and B. J. Kim, Nature Phy sics12, 3741 (2016). 8H. Gretarsson, N. Sung, J. Porras, J. Bertinshaw, C. Dietl, J an A. N. Bruin, A. F. Bangura, Y. K. Kim, R. Dinnebier, J. Kim, A. Al-Zein, M. Moretti Sala, M . Krisch, M. Le Tacon, B. Keimer, and B. J. Kim, Phys. Rev. Lett. 117, 107001 (2016). 9X. Chen, J. L. Schmehr, Z. Islam, Z. Porter, E. Zoghlin, K. Fin kelstein, J. P. C. Ruﬀ, and S. D. Wilson, Nat. Commun. 9, 103 (2018). 10S. Bhowal, J. M. Kurdestany and S. Satpathy, J. Phys.: Conden s. Matter 30235601 (2018).14 11D. Pincini, J. G. Vale, C. Donnerer, A. de la Torre, E. C. Hunte r, R. Perry, M. Moretti Sala, F. Baumberger, and D. F. McMorrow, Phys. Rev. B 96, 075162 (2017). 12J. Porras, J. Bertinshaw, H. Liu, G. Khaliullin, N. H. Sung, J .-W. Kim, S. Francoual, P. Steﬀens, G. Deng, M. Moretti Sala, A. Eﬃmenko, A. Said, D. Casa , X. Huang, T. Gog, J. Kim, B. Keimer, and B. J. Kim, Phys. Rev. B 99, 085125 (2019). 13J. Kim, D. Casa, M. H. Upton, T. Gog, Y.-J. Kim, J. F. Mitchell, M. van Veenendaal, M. Daghofer, J. van den Brink, G. Khaliullin, and B. J. Kim, Phys . Rev. Lett. 108, 177003 (2012). 14B. H. Kim, G. Khaliullin, and B. I. Min, Phys. Rev. Lett. 109, 167205 (2012). 15J. Kim, M. Daghofer, A. H. Said, T. Gog, J. van den Brink, G. Kha liullin, and B. J. Kim, Nat. Commun. 5, 4453 (2014). 16J-i Igarashi and T. Nagao, Phys. Rev. B 90, 064402 (2014). 17X. Lu, P. Olalde-Velasco, Y. Huang, V. Bisogni, J. Pelliciar i, S. Fatale, M. Dantz, J. G. Vale, E. C. Hunter, J. Chang, V. N. Strocov, R. S. Perry, M. Grioni, D. F . McMorrow, H. M. Rønnow, and T. Schmitt, Phys. Rev. B 97, 041102(R) (2018). 18S. Mohapatra, J. van den Brink, and A. Singh, Phys. Rev. B 95, 094435 (2017). 19S. Mohapatra and A. Singh, arXiv: 1903.03360 (2019). 20J-i Igarashi and T. Nagao, J. Phys. Soc. Jpn. 83, 053709 (2014). 21C. Martins, M. Aichhorn, and S. Biermann, J. Phys.: Condens. Matter29, 263001 (2017). 22S. Mohapatra and A. Singh, arXiv: 2001.00190 (2020). 23H. Watanabe, T. Shirakawa, and S. Yunoki, Phys. Rev. Lett. 105, 216410 (2010). 24S. Zhou, K. Jiang, H. Chen, and Z. Wang, Phys. Rev. X 7, 041018 (2017). 25X. Ming, X. Wan, C. Autieri, J. Wen, and X. Zheng, Phys. Rev. B 98, 245123 (2018). 26L. J. P. Ament, G. Khaliullin, and J. van den Brink, Phys. Rev. B,84, 020403 (2011). 27S. Boseggia, Ph.D. Thesis, Magnetic order and excitations in perovskite iridates stud ied with resonant X-ray scattering techniques (2014).
1612.03447v4.The_Spinon_Fermi_Surface_U_1__Spin_Liquid_in_a_Spin_Orbit_Coupled_Triangular_Lattice_Mott_Insulator_YbMgGaO4.pdf	The spinon Fermi surface U(1) spin liquid in a spin-orbit-coupled triangular lattice Mott insulator YbMgGaO 4 Yao-Dong Li1, Yuan-Ming Lu2, and Gang Chen1;3 1State Key Laboratory of Surface Physics, Department of Physics, Center for Field Theory & Particle Physics, Fudan University, Shanghai, 200433, China 2Department of Physics, The Ohio State University, Columbus, OH, 43210, United States and 3Collaborative Innovation Center of Advanced Microstructures, Nanjing, 210093, China (Dated: August 24, 2017) Motivated by the recent progress on the spin-orbit-coupled triangular lattice spin liquid candi- date YbMgGaO 4, we carry out a systematic projective symmetry group analysis and mean-eld study of candidate U(1) spin liquid ground states. Due to the spin-orbital entanglement of the Yb moments, the space group symmetry operation transforms both the position and the orientation of the local moments, and hence brings dierent features for the projective realization of the lattice symmetries from the cases with spin-only moments. Among the eight U(1) spin liquids that we nd with the fermionic parton construction, only one spin liquid state, that was proposed and analyzed in Yao Shen, et. al. , Nature 540, 559-562 (2016) and labeled as U1A00 in the present work, stands out and gives a large spinon Fermi surface and provides a consistent explanation for the spectro- scopic results in YbMgGaO 4. Further connection of this spinon Fermi surface U(1) spin liquid with YbMgGaO 4and the future directions are discussed. Finally, our results may apply to other spin- orbit-coupled triangular lattice spin liquid candidates, and more broadly, our general approach can be well extended to spin-orbit-coupled spin liquid candidate materials. I. INTRODUCTION The interplay between strong spin-orbit coupling (SOC) and strong electron correlation has attracted a signicant attention in recent years1. At the mean time, the abundance of strongly correlated materials with 5 d and 4felectrons, such as iridates and rare-earth mate- rials1,2, brings a fertile arena to explore various emer- gent and exotic phases that arise from such an inter- play3{32. The recently discovered quantum spin liquid (QSL) candidate YbMgGaO 433, where the rare-earth Yb atoms form a perfect triangular lattice, is an ideal sys- tem that involves strong spin-orbital entanglement in the strong Mott insulating regime of the Yb electrons34{41. In YbMgGaO 4, the thirteen 4 felectrons of the Yb3+ ions are well localized and form a spin-orbit-entangled total moment JwithJ= 7=234,35. The eight-fold de- generacy of the J= 7=2 moment is further split by the D3dcrystal electric elds. The resulting ground state Kramers doublet of the Yb3+ion, whose two-fold de- generacy is protected by the time-reversal symmetry, is well separated from the excited doublets and is re- sponsible for the low-temperature magnetic properties of YbMgGaO 4. No signature of time-reversal symmetry breaking is observed for YbMgGaO 4down to the lowest measured temperature36{38. Applying the recent theoret- ical result on spin-orbit-coupled Mott insulators42, two of us and collaborators have proposed YbMgGaO 4to be the rst QSL candidate in the spin-orbit-coupled Mott in- sulator with odd electron llings34{36,39. More broadly, YbMgGaO 4represents a new class of rare-earth materi- als where the strong spin-orbit entanglement of the local moments meets with the geometrical frustration of the triangular lattice such that exotic quantum phases may be stabilized.Apart from the absence of magnetic ordering, the heat capacity was found to be Cv/T0:7at low tempera- tures33,34,37,43, and is close to the well-known T2=3heat capacity44{46. The latter was the one obtained within a random phase approximation for the spinon-gauge cou- pling in a spinon Fermi surface U(1) QSL44{46. More substantially, the broad continuum36,37of the magnetic excitation with a clear dispersion for the upper excitation edge agrees reasonably with the particle-hole continuum of the spinon Fermi surface36. However, due to the scat- tering with the phonon degrees of freedom, the thermal transport measurement in YbMgGaO 4was unable to ex- tract the intrinsic magnetic contribution to the thermal conductivity43. Partly motivated by the spin liquid be- haviors in YbMgGaO 4and more broadly by the families of rare-earth magnets with identical structures, in this paper, we carry out a systematic projective symmetry group (PSG) analysis for a triangular lattice Mott insu- lator with spin-orbital-entangled local moments. Unlike the cases for the spin-only moments in the pioneering work by X.-G. Wen47, the space group symmetry opera- FIG. 1. (a) The intralayer symmetries of the R 3m space group for YbMgGaO 435. (b) The same lattice symmetry group with a dierent complete set of elementary transformations. Here S6C1 3I. The bold arrow is the axis for the C2rotation (see Appendix).arXiv:1612.03447v4 [cond-mat.str-el] 23 Aug 20172 tion, in particular, the rotation, transforms both the po- sition and the orientation of the Yb local moments35,39. We nd that, among the eight U(1) QSL states, the spinon mean-eld state that was introduced in Ref. 36 and labeled as the U1A00 state in our PSG classication, contains a large spinon Fermi surface and gives a large spinon scattering density of states that is consistent with the inelastic neutron scattering (INS) results. The following part of the paper is organized as follows. In Sec. II, we describe the space group symmetry and the the multiplication rules for the symmetry transformation. In Sec. III, we introduce the fermionic spinon construc- tion and the fermionic spinon mean-eld Hamiltonian. In Sec. IV, we explain the scheme for the projective sym- metry group classication when the spin-orbit coupling is present. In Sec. V, we explain the relationship between the spinon band structure and the projective symmetry group of the spinon mean-eld states. In Sec. VI, we focus on the U1A00 state and study the spectroscopic properties of this state. Finally in Sec. VII, we discuss the experimental relevance and remark on the thermal transport result and the competing scenarios and pro- posals. The details of the calculation are presented in the Appendices. II. SPACE GROUP SYMMETRY It was pointed out that the intralayer symmetries in- volves two translations, T1andT2, one two-fold rota- tion,C2, one three-fold rotation, C3, and one spatial in- versionI(see Fig. 1(a))35,39. Here we use a dierent complete set of elementary transformations for the space group symmetries that involve two translations, T1and T2, one two-fold rotation, C2, and one more operation, S6(see the denition in Fig. 1(b)). It is ready to con- rmI=S3 6;C3=S2 6with the denition S6C1 3I. The multiplication rules of this symmetry group is given as T1 1T2T1T1 2=T1 1T1 2T1T2= 1; (1) C1 2T1C2T1 2=C1 2T2C2T1 1= 1; (2) S1 6T1S6T2=S1 6T2S6T1 2T1 1= 1; (3) (C2)2= (S6)6= (S6C2)2= 1: (4) Due to the presence of time reversal in YbMgGaO 434,36{38, we further supplement the symmetry group with the time reversal Tsuch thatO1TOT = 1 andT2= 1, whereOis a lattice symmetry operation. III. FERMIONIC PARTON CONSTRUCTION To describe the U(1) QSL that we propose for YbMgGaO 4, we introduce the fermionic spinon opera- torfr(=";#) that carries spin-1/2, and express the Yb local moment as Sr=1 2X ;fy rfr; (5)U(1) QSLWT1rWT2rWC2rWS6r U1A00I22I22I22I22 U1A10I22I22iyI22 U1A01I22I22I22iy U1A11I22I22iyiy TABLE I. List of the gauge transformations for the four U1A PSGs. For the time reversal, all PSGs here have WT r=I22. The last two letters in the labels of the U(1) QSLs are extra quantum numbers in the PSG classication48. where= (x;y;z) is a vector of Pauli matrices. We further impose a constraintP fy rfr= 1 on each site to project back to the physical Hilbert space of the spins. The choice of fermionic spinons allows a local SU(2) gauge freedom47. As a direct consequence of the spin-orbital entangle- ment, the spinon mean-eld Hamiltonian for the U(1) QSL should generically involve both spin-preserving and spin- ipping hoppings, and has the following form HMF=X (rr0)X trr0;fy rfr0+h:c: ; (6) wheretrr0;is the spin-dependent hopping. The choice of the mean-eld ansatz in Eq. (6) breaks the local SU(2) gauge freedom down to U(1). Here, to get a more com- pact form for Eq. (6), we follow Ref. 49 and introduce the extended Nambu spinor representation for the spinons such that r= (fr";fy r#;fr#;fy r")Tand HMF=1 2X (r;r0) y rurr0 r0+h:c: ; (7) whereurr0is a hopping matrix that is related to trr0;. With the extended Nambu spinor, the spin operator Sr and the generator Grfor the SU(2) gauge transformation are given by47,50{53 Sr=1 4 y r( I22) r; (8) Gr=1 4 y r(I22 ) r; (9) whereI22is a 22 identity matrix. Under the symme- try operationO, rtransforms as r!UOGO O(r) O(r)=GO O(r)UO O(r); (10) whereGO O(r)is the local gauge transformation that cor- responds to the symmetry operation O, and we add a spin rotationUObecause the spin components are trans- formed whenOinvolves a rotation. In Eq. (10), the gauge transformation and the spin rotation are commu- tative54simply because [ S r;G r] = 0. Moreover, from Eq. (9), the gauge transformation GO ris block diagonal withGO r=I22 WO r, whereWO ris a 22 matrix (see Appendix).3 IV. PROJECTIVE SYMMETRY GROUP CLASSIFICATION For the spinon mean-eld Hamiltonian in Eq. (6), the lattice symmetries are realized projectively and form the projective symmetry group (PSG). To respect the lat- tice symmetry transformation O, the mean-eld ansatz should satisfy urr0=GOy O(r)Uy OuO(r)O(r0)UOGO O(r0): (11) The ansatz itself is invariant under the so-called invariant gauge group (IGG) with urr0=G1y rurr0G1 r0. The IGG can be regarded as a set of gauge transformations that correspond to the identity transformation. For an U(1) QSL, IGG = U(1). A general group relation O1O2O3O4= 1 for the lattice symmetry turns into the following group relation for the PSG UO1GO1 rUO2GO2 O2O3O4(r)UO3GO3 O3O4(r)UO4GO4 O4(r) =UO1UO2UO3UO4GO1 rGO2 O2O3O4(r)GO3 O3O4(r)GO4 O4(r)(12) 2IGG; (13) where we used the fact that the gauge transformation commutes with the spin rotation. As the series of rota- tionsO1O2O3O4either rotate the spinons by 0 or 2 , UO1UO2UO3UO4=I44; (14) whereI44is a 44 identity matrix. Since fI44gIGG, then GO1 rGO2 O2O3O4(r)GO3 O3O4(r)GO4 O4(r)2IGG: (15) This immediately indicates that, to classify the PSGs for a spin-orbit-coupled Mott insulator, we only need to fo- cus on the gauge part, rst nd the gauge transformation with the same procedures as those for the conventional Mott insulators with spin-only moments47, and then ac- count for the spin rotation. For the mean-eld ansatz in HMF, we choose the \canonical gauge" for the IGG with IGG =fI22 eizj2[0;2)g: (16) Under the canonical gauge, the gauge transformation as- sociated with the symmetry operation Otakes the form of GO r=I22 WO r I22 (ix)nOeiO[r]z ; (17) wherenO= 0;1. For translations, one can always choose a gauge such that WT1 r= (ix)n1; (18) WT2 r= (ix)n2ei2[x;y]z(19) withn1;n2= 0;1 and2[0;y] = 0. The group rela- tion in Eq. (3) further demands n1=n2= 0. Thus the FIG. 2. (a,b,c) The mean-eld spinon bands along the high- symmetry momentum lines (see (d)) of the U1A00, U1A01 and U1A11 states, where t1;t0 1andt2are hoppings in their spinon mean-eld Hamiltonians (see Appendix). The Dirac cones are highlighted in dashed circles. The dashed line refers to the Fermi level. (d) The Brioullin zone of the triangular lattice. group relation in Eq. (1) gives WT1r= 1;WT2r=eix1z, where1is the ux through each unit cell of the triangu- lar lattice and takes the value of 0 or (see Appendix). The PSGs with 1= 0 () are labeled by U1A (U1B). Among the sixteen algebraic PSGs that we nd, eight un- physical solutions have T2= 1 for the spinons and give vanishing spinon hoppings everywhere. In Tab. I and the Appendix, we list the remaining eight PSGs that have T2=1 consistent with the fact that fermionic spinons are Kramers doublets (see Appendix). V. MEAN-FIELD STATES Here we obtain the spinon mean-eld Hamiltonian from Tab. I and explain why the U1A00 state stands out as the candidate ground state for YbMgGaO 4. We start with the U1A states. Among the four U1A states, the U1A10 state gives a vanishing mean-eld Hamilto- nian for the spinon hoppings between the rst and the second neighbors, the remaining ones except the U1A00 state all have symmetry protected band touchings at the spinon Fermi level (see Fig. 2). To illustrate the idea55, we consider the U1A01 state where the spinon Hamilto- nian has the form HU1A01 MF =P kh(k)fy kfkin the momentum space and h(k) is a 22 matrix with h(k) =d0(k)I22+3X =1d(k): (20) For this band structure there are nondegenerate band touchings at , M and K points that are protected by the PSG of the U1A01 state. Under the operation S6,4 the PSG demands that spinons to transform as fk"!ei=3fy S1 6k;#; (21) fk#!ei=3fy S1 6k;": (22) ApplyingS6three times and keeping HMFinvariant, we require h(k) =[yh(k)y]T(23) which forces d0(k) = 0. The time reversal sym- metry (T=iy I22K) further requires that d(k) =d(k). Thus we have symmetry pro- tected band touchings with h(k) = 0 at the time reversal invariant momenta and M. The K points are invariant underC2andS6because the spinon partile-hole trans- formation is involved for S6(see Appendix). Using those two symmetries, we further establish the band touching at the K points. Likewise, for the U1A11 state, the PSG demands the band touchings at and M points. Because there are only two spinon bands for the U1A states, these band touchings generically occur at the spinon Fermi level. Due to the Dirac band touchings at the Fermi level, the low-energy dynamic spin structure factor, that measures the spinon particle-hole continuum, is concentrated at a few discrete momenta that correspond to the intra-Dirac- cone and the inter-Dirac-cone scatterings36. Clearly, this is inconsistent with the recent INS result that observes a broad continuum covering a rather large portion of the Brillouin zone36,37. For the U1B states, the spinons experience a back- ground ux in each unit cell. The direct consequence of thebackground ux is that the U1B states support an enhanced periodicty of the dynamic spin structure in the Brillouin zone47,56,57. Such an enhanced periodicity is absent in the INS result36,37. In particular, unlike what one would expect for an enhanced periodicity, the spec- tral intensity at the point is drastically dierent from the one at the M point in the existing experiments36,37. The above analysis leads to the conclusion that the U1A00 state is the most promising candidate U(1) QSL for YbMgGaO 4, and this conclusion is independent from any microscopic model. The spinon mean eld Hamilto- nian, allowed by the U1A00 PSG, is remarkably simple and is given as58 HU1A00 MF =t1X hrr0i;fy rfrt2X hhrr0ii;fy rfr;(24) where the spinon hopping is isotropic for the rst and the second neighbors. This mean-eld state only has a single band that is 1/2-lled, so it has a large spinon Fermi surface. From HU1A00 MF , we construct the mean- eld ground state by lling the spinon Fermi sea, j U1A00 MFi=Y k<Ffy k"fy k#j0i (25)wherekis the spinon dispersion and Fis the spinon Fermi energy. The mean-eld variational energy is Evar=h U1A00 MFjHspinj U1A00 MFi; (26) where Hspin=X hrr0iJzzSz rSz r0+J(S+ rS r0+S rS+ r0) +J( rr0S+ rS+ r0+ rr0S rS r0) i 2Jz ( rr0S+ r rr0S r)Sz r0 +Sz r( rr0S+ r0 rr0S r0) (27) is the microscopic spin model that was introduced in Refs. 34 and 35, and rr0is a bond-dependent phase fac- tor due to the spin-orbit-entangled nature of the Yb mo- ments35. The anisotropic nature of the spin interaction has been clearly supported by the recent polarized neu- tron scattering measurement59. For the specic choice withJ= 0:915Jzz, we nd the minimum variational en- ergyEvar=0:39Jzzand occurs at t2= 0:2t1(see Ap- pendix). Here, the expectation values of the JandJz interactions simply vanish, and this is an artifact of the free spinon mean-eld theory with the isotropic hoppings in Eq. (24). We here establish that the U1A00 state is a spinon Fermi surface U(1) QSL. VI. SPECTROSCOPIC PROPERTIES For the U1A00 state, the dynamic spin structure essen- tially detects the spinon particle-hole excitation across the Fermi surface. The information about the Fermi surface is encoded in the prole of the dynamic spin structure factor. We evaluate the dynamic spin struc- ture factor within the free spinon mean-eld theory (see Appendix) (see Fig. 3(a)). Qualitatively similar to the mean-eld theory with only rst neighbor spinon hop- pings, the improved free-spinon mean-eld theory of HU1A00 MF captures the crucial features of the INS re- sults36,37. The spinon particle-hole continuum covers a large portion of the Brillouin zone, and vanishes beyond the spinon bandwidth. More importantly, the \V-shape" upper excitation edge near the point in Fig. 3(a) was clearly observed in the experiments36,37, and the slope of the \V-shape" is the Fermi velocity. Due to the isotropic spinon hoppings, HU1A00 MF does not explicitly re ect the absence of spin-rotational symmetry that is brought by the JandJzinteractions. To incorporate the JandJzinteractions, we follow the phenomenological RPA treatment for the \ t-J" model in the context of cuprate superconductors60and consider H=HU1A00 MF +H0 spin; (28) whereH0 spinare theJandJzinteractions (see Ap- pendix). While the free spinon results from HU1A00 MF al- ready capture the main features of the neutron scatter- ing data36,37, the anisotropic spin interaction H0 spin, in- cluded by RPA, merely redistributes the spectral weight5 FIG. 3. (a)S(q;!) along the high-symmetry momentum lines from HU1A00 MF witht2= 0:2t1. The spinon bandwidth B= 9:6t1. (b) The RPA corrected SRPA(q;!) along the high symmetry momentum lines. We have set the parameters in the spin model to be J=Jzz= 0:915,J=Jzz= 0:35, and Jz=Jzz= 0:2. The ratio Jzz=t1is obtained from Refs. 34 and 36 and xed to be 1 :0 for concreteness. in the momentum space. We nd in Fig. 3(b) that, the low-energy spectral weight at M is slightly enhanced, a feature observed in Refs. 36 and 37. From our choice of the parameters, it is plausible that this peak results from the proximity to a phase with a stripe-like magnetic order35,36,39. VII. DISCUSSION We have demonstrated that the spinon Fermi surface U(1) QSL gives a consistent explanation of the INS re- sult in YbMgGaO 4. Moreover, the anisotropic spin in- teraction, slightly enhances the spectral weight at the M points. The U(1) gauge uctuation in the spinon Fermi surface U(1) QSL44,45was suggested to be the cause for the sublinear temperature dependence of the heat capac- ity in YbMgGaO 435,36,39,46. In YbMgGaO 4, the coupling between the Yb moments is relatively weak34. It is feasible to fully polarize the spin with experimentally accessible magnetic elds35,37,39,61 and to study the evolution of the magnetic properties un- der the magnetic eld. Recently, two of us have predicted the spectral weight shift of the INS for YbMgGaO 4under a weak magnetic eld41, and the predicted spectral cross- ing at the point and the dispersion of the spinon con- tinuum have actually been conrmed in the recent INS measurement62. Numerically, it is useful to perform nu- merical calculation with xed JandJzzthat are close to the ones for YbMgGaO 4, and obtain the phase dia- gram of our spin model by varying JandJz35,39,63. More care needs to be paid to the disordered region of the mean-eld phase diagram35where quantum uctuation is found to be strong35. The \2kF" oscillation in the spin correlation would be the strong indication of the spinon Fermi surface. Noteworthily recent DMRG works64,65 have actually provided some useful information about the ground states of the system, in particular, Ref. 65 suggested the scenario of exchange disorders. Certain amount of exchange disorder may be created by the crys- tal electric eld disorder that stems from the Mg/Gamixing in the non-magnetic layers37,61, but recent polar- ized neutron scattering measurement did not nd strong exchange disorder59. Regardless of the possibilities of exchange disorders, the spin quantum number fraction- alization, that is one of the key properties of the QSLs, could survive even with weak disorders. The approach and results in our present work are phenomenologically based and are independent of the microscopic mechanism for the possible QSL ground state in YbMgGaO 4. Ref. 43 claimed the absence of the magnetic thermal conductivity in YbMgGaO 4by extrapolating the low- temperature thermal conductitivity data in the zero mag- netic eld. Here, we provide an alternative understand- ing for this thermal transport result. The hint lies in the eld dependence of the thermal conductivity. It was found that, when strong magnetic elds are applied to YbMgGaO 4, the thermal conductivity xx=Tat 0.2K is increased compared with the one at zero eld43. If one ig- nores the disorder eect and assumes the zero-eld ther- mal conductivity is a simple addition of the magnetic contribution and the phonon contribution with xx=spin;xx+phonon;xx; (29) the strong magnetic eld almost polarizes the spins com- pletely and creates a spin gap for the magnon excitation, hence suppress the magnetic contribution. The high- eld thermal conductivity would be purely given by the phonon contribution, and we would expect a decreasing of the thermal conductivity in the strong eld compared to the zero eld result. This is clearly inconsistent with the experimental result. Therefore, the zero-eld ther- mal conductivity is not a simple addition of the magnetic contribution and the phonon contribution, i.e., xx6=spin;xx+phonon;xx: (30) This also strongly suggests the presence rather than the absence of magnetic excitations in the thermal conductiv- ity result at zero magnetic eld. If there is no magnetic excitation in the system at low temperatures, the low- temperature thermal conductivity at zero eld should just be the phonon contribution, and we would expect the zero-eld thermal conductivity to be the same as the one in the strong eld limit, (although the intermediate eld regime could be dierent). This is again inconsis- tent with the experiments. This means that the magnetic excitation certainly does not have a large gap and could just be gapless as we propose from the spinon Fermi sur- face state. In fact, the gapless nature of the magnetic excitation is consistent with the power-law heat capac- ity results in YbMgGaO 4. What suppresses xxcould arise from the mutual scattering between the magnetic excitations and the the phonons. In fact, similar eld de- pendence of thermal conductivity xxhas been observed in other rare-earth systems such as Tb 2Ti2O766{68and Pr2Zr2O769. It was suggested there67{69that the spin- phonon scattering is the cause. The Yb local moment, that is a spin-orbit-entangled object, involves the orbital6 degree of freedom. The orbital degree of freedom is sen- sitive to the ion position, and thus couples to the phonon strongly. This is probably the microscopic origin for the strong coupling between the magnetic moments and the phonons in the rare-earth magnets. This is quite dif- ferent from the organic spin liquid candidates and the herbertsmithite kagome system where the orbital degree of freedom does not seem to be involved70{73. If the ground state of YbMgGaO 4is a QSL with the spinon Fermi surface, the eld-driven transition from the QSL ground state to the fully polarized state is neces- sarily a unconventional transition beyond the traditional Landau's paradigm and has not been studied in the pre- vious spin liquid candidates70{73. The smooth growth of the magnetization with varying external elds indicates a continuous transition34. Since we propose YbMgGaO 4 to be a spinon Fermi surface U(1) QSL and gapless, the transition would be associated with the openning of the spin gap at the critical eld. The continuous nature of the transition suggests the spin gap to open in a contin- uous manner. Moreover, the spinon connement would be concomitant with the spin gap that suppresses the spinon density of states and allows the instanton events of the U(1) gauge eld to proliferate. Therefore, it might be interest to identify the critical eld and obtain the critical properties of the eld-driven transition. Thermo- dynamic, spectroscopic, and thermal transport measure- ments with ner eld variation would be helpful. Finally, several families of rare-earth triangular lattice magnets have been discovered recently35,39,74{79. Their properties have not been studied carefully. Our general classication results and the prediction of the spectro- scopic properties would apply to the QSL candidates that may emerge in these families of materials. It is certainly exciting if one nds the new QSL candidates in these families behave like YbMgGaO 435. VIII. ACKNOWLEDGEMENTS We thank one anonymous referee for the suggestion for improvement to this paper, and Zhu-Xi Luo for point- ing out some typos. G.C. acknowledges the discussion with Xuefeng Sun from USTC and Yuji Matsuda about thermal transports in rare-earth magnets, and the dis- cussion with Professor Sasha Chernyshev about the re- lated matters. This work is supported by the Min- istry of Science and Technology of China with the Grant No.2016YFA0301001 (G.C.), the Start-Up Funds of OSU (Y.M.L.) and Fudan University (G.C.), the National Sci- ence Foundation under Grant No. NSF PHY-1125915 (Y.M.L and G.C.), the Thousand-Youth-Talent Program (G.C.) of China, and the rst-class university construc- tion program of Fudan University.Appendix A: The coordinate System and space group symmetry Following our convention in Fig. 1 in the main text, we choose the coordinate system of the triangular lattice to be a1= (1;0); (A1) a2= (1 2;p 3 2): (A2) We label the triangular lattice sites by r=xa1+ya2. Restricted to the triangular layer, the space group con- tains two translations T1along thea1direction,T2along thea2direction, a counterclockwise three-fold rotation C3around the lattice site, a two-fold rotation C2around a1+a2, and the inversion Iat the lattice site. Their actions on the lattice indices are T1: (x;y)!(x+ 1;y); (A3) T2: (x;y)!(x;y+ 1); (A4) C3: (x;y)!(y;xy); (A5) C2: (x;y)!(y;x); (A6) I: (x;y)!(x;y): (A7) In the formulation introduced in the main text, we consider an equivalent set of generators, fT1;T2;C2;S6g, where the operation S6isdened asS6C1 3Iand acts on the lattice indices as S6: (x;y)!(xy;x): (A8) It is evident that these two sets of generators are equiva- lent, since we merely redene the symmetry rather than introducing any new symmetry. The multiplication rule of this symmetry group is given in the main text. For the convenience of the presentation below, we also list these rules here, T1 1T2T1T1 2=T1 1T1 2T1T2= 1; (A9) C1 2T1C2T1 2=C1 2T2C2T1 1= 1; (A10) S1 6T1C6T2=S1 6T2C6T1 2T1 1= 1;(A11) (C2)2= (C6)6= (S6C2)2= 1: (A12) Including the time reversal symmetry, we further have T1 1TT1T=T1 2TT2T= 1; (A13) C1 2TC2T=S1 6TS6T= 1; (A14) T2= 1: (A15) Appendix B: Projective symmetry group classication As we describe in the main text, we consider the U(1) QSL. The spinon mean-eld Hamiltonian has the follow- ing form HMF=X (rr0)X trr0;fy rfr0+h:c: ;(B1)7 U(1) QSLWT1rWT2rWC2r WS6r U1A00I22I22I22 I22 U1A10I22I22iyI22 U1A01I22I22I22 iy U1A11I22I22iyiy U1B00I22(1)xI22(1)xyI22(1)xyy(y1) 2I22 U1B10I22(1)xI22iy(1)xy(1)xyy(y1) 2I22 U1B01I22(1)xI22(1)xyI22iy(1)xyy(y1) 2 U1B11I22(1)xI22iy(1)xyiy(1)xyy(y1) 2 TABLE II. List of the gauge transformations for the sym- metry operations of the eight U(1) PSGs, where ( x;y) is the coordinate in the oblique coordinate system. For time rever- sal symmetry, all PSGs have the same gauge transformation WT r=I22. wheretrr0;is the spin-dependent hopping. With the extended Nambu spinor representation49 r= (fr";fy r#;fr#;fy r")T,HMFhas a more compact form HMF=1 2X (r;r0) y rurr0 r0+h:c: ; (B2) whereurr0is a hopping matrix that is related to trr0;, urr0=0 BBBBB@trr0;"" 0trr0;"# 0 0t rr0;## 0t rr0;#" trr0;#" 0trr0;## 0 0t rr0;"# 0t rr0;""1 CCCCCA:(B3) 1. Spatial symmetry First of all, the gauge transformation and spin rotation are commutative. So in the PSG classication, we only need to focus on the gauge part of the PSG transforma- tion. In the canonical gauge IGG = fI22 eizj2 [0;2)g, the gauge transformation associated with a given symmetry operation Otakes the form GO r=I22 WO rI22 (ix)nOeiO[r]z ;(B4) wherenO= 0;1. For the symmetry multiplication rule O1O2O3O4= 1 whereOiis an unitary transformation, the corresponding PSG relation becomes GO1 rGO2 O2O3O4(r)GO3 O3O4(r)GO4 O4(r)2IGG (B5) or equivalently, WO1 rWO2 O2O3O4(r)WO3 O3O4(r)WO4 O4(r) 2feizj2[0;2)g: (B6)We start with T1andT2, where WT1 r= (ix)nT1; (B7) WT2 r= (ix)nT2eiT2[r]z: (B8) Through Eq. (A10) that connects T1andT2, one imme- diately has nT1=nT2. From Eq. (A11) where the to- tal number of T1andT2is odd, one immediately has nT1=nT2= 0. So we have WT1 r= 1; (B9) WT2 r=eiT2[x;y]z: (B10) Using Eq. (A9), we have [WT1T1]1[WT2T2][WT1T1][WT2T2]1 =T1 1(WT1)1WT2T2WT1T1T1 2W1 T2 2feizj2[0;2)g;(B11) which leads to the result T2[x+ 1;y]T2[x;y]1 (B12) with1to be determined. Since it is always possible to choose a gauge such that T2[0;y] = 0, then we have T2[x;y] =1x. Similarly,T1 1T1 2T1T2= 1 leads to T2[x+ 1;y+ 1]T2[x;y+ 1] =2:(B13) It is ready to nd 2=1. We continue to nd WS6randWC2r. For the operation S6withWS6r= (ix)nS6eiS6[x;y]z, Eq. (A11) leads to S6[T1(r)] +S6[r] =1y+3; (B14) S6[T2(r)] +S6[r] =41x+1y;(B15) fornS6= 0, and S6[T1(r)] +S6[r] =1y+3 (B16) S6[T2(r)] +S6[r] =4+1x+1y:(B17) fornS6= 1. So we obtain whennS6= 0; S6[r] =1xy3x4y1y(y1) 2(B18) whennS6= 1; S6[r] =1xy3x4y1y(y1) 2:(B19) FornS6= 1, we further require 1= 0;.S6 6= 1 is automatically satised with the above relations for both nS6= 0 andnS6= 1. ForWC2rwithWC2r= (ix)nC2eiC2[x;y]z, we need to consider two separate cases with nc2= 0;1, respectively. IfnC2= 0, Eq. (A10) leads to T2[C1 2T1(r)]C2[T1(r)] +C2[r] =5;(B20) C2[T2(r)] +T2[T2(r)] +C2[r] =6:(B21)8 So we obtain C2[x;y] =5x6yxy 1and1= 0; fornC2= 0. Similary, for nC2= 1, we obtain C2[x;y] = 5x6yxy 1. UsingC2 2= 1, we further have 6=5fornC2= 0, and6=6fornC2= 1. So we arrive at the result nC2= 0; C2[x;y] =5(xy)xy 1;(B22) nC2= 1; C2[x;y] =5(x+y)xy 1:(B23) Here, to simplify the above expression, we choose a pure gauge tranformation ~Wa r=eixz5. Under the pure gauge transformation, the gauge part of the PSG trans- forms as WO r!~Wa rWO r~Way O1(r): (B24) Clearly ~Wa ronly modies WT1andWT2by an overall phase shift, but WC2rbecomes WC2 r= (ix)nC2eixy 1z(B25) for bothnC2= 0;1, except that we require 1= 0;for nC2= 0. For the relation ( S6C2)2= 1, we need to consider the four cases with nS6= 0;1 andnC2= 0;1. FornS6=nC2= 0, we have 1=, and (S6C2)2= 1 gives3+ 24= 0. We then introduce a pure gauge transformation ~Wb r, ~Wb r=ei(x+y)4z: (B26) After applying ~Wb r, we have C2=xy 1; (B27) S6=xy 11y(y1) 2(B28) with1= 0;. FornS6= 0 andnC2= 1, we obtain 3= 0. We introduce a pure gauge transformation ~Wc r, ~Wc r=ei(xy)4z: (B29) After applying ~Wb r, we have C2=xy 1; (B30) S6=xy 11y(y1) 2: (B31) FornS6= 1 andnC2= 0, we obtain 3= 0. We apply a pure gauge transformation ~Wb rand obtain C2=xy 1; (B32) S6=xy 11y(y1) 2: (B33) FornS6= 1 andnC2= 1, we obtain 3+ 24= 0. We apply a pure gauge transformation ~Wc rand obtain C2=xy 1; (B34) S6=xy 11y(y1) 2: (B35)In summary, we have WT1 r= 1; WT2 r=ei1x: (B36) and WC2 r= (ix)nC2ei1xyz; (B37) WS6 r= (ix)nS6ei1[xyy(y1) 2]z; (B38) where1= 0;fornC2= 0 ornS6= 1. 2. Time reversal symmetry Because time reversal is an antiunitary symmetry, the productO1T1OTbecomes (WO r)y[(WT r)yWO rWT O1(r)](B39) for the PSGs, where WTis the gauge transformation associated with the time reversal. We here redene WT r=WT r(iy); (B40) so that O1T1OT ! (WO r)y(WT r)yWO rWT O1(r):(B41) WT rhas the general form WT r= (ix)nTeiT[r]z. We start with nT= 0. The relation in Eq. (A13) leads to T[x;y]T[x1;y] =7; (B42) T[x;y+ 1]T[x;y] =8; (B43) so we have T[x;y] =7x8y. Applying this result to Eq. (A14), we have C2[y;x]T[y;x] +C2[y;x] +T[x;y] =9; S6[x;y]T[x;y] +S6[x;y] +T[y;x+y] =10;(B44) fornC2=nS6= 0. The above equations give 7=8= 0, so we have WT r= 1. Other cases can be obtained likewise. We nd that for both nT= 0 and nT= 1, there is T[x;y] = 0 and1= 0;. So we have WT r= 1;iy; (B45) where we have used a global and uniform rotation ei 4z to rotatexto the basis of y. Including the time reversal, there are 16 PSG solutions. But for WT r= 1, the mean-eld ansatz is found to vanish everythere. This makes sense as these PSGs have T2= 1 for the fermionic spinons that are expected to Kramers doublets. So only 8 of them with T2=1 for the spinons survive. Replacing ei1zwith1, we present the PSG solutions in the table of the main text.9 Appendix C: Spinon band structures and mean-eld Hamiltonians As we establish in the previous section and the main text, there are four U1A PSGs and four U1B PSGs. In the main text, we have argued that the experimental re- suls in YbMgGaO 4is against the U1B states. So here we focus on the U1A states. From the U1A PSGs, it is straight to obtain the spinon transformations. We list the results in Tab. III. 1. Spinon band structures Using Tab. III, we obtain the spinon mean-eld Hamil- tonian. In particular, the U1A10 state gives vanishing spinon hoppings on the rst and second neighbors, and the U1A01 state gives an isotropic spinon hopping on both rst and second neighbors. The U1A10 state, as we described in the main text, has symmetry protected band touchings at the , M and K points. The U1A11 state has symmetry protected band touchings at the and M points. For the U1A10 state, the spinon mean-eld Hamilto- nian has the form HU1A01 MF =X kh(k)fy kfk; (C1) whereh(k) is given by h(k) =d0(k)I22+3X =1d(k): (C2) In the main text, we have used ( S6)3andTto showd0(k) = 0 and the band touchings at and M. To account for the band touching at the K point, we need to use S6 andC2. UnderS6, S6HS1 6=X k ei2 3h(S1 6(k))"#fy k"fk#+h:c: =H; (C3) whereh(k)"#=dx(k)idy(k). Since K is invariant un- derS6, dx(K)idy(K) =ei2 3[dx(K)idy(K)];(C4) hencedx(K) =dy(K) = 0. TheC2symmetry constraints the dzterm, we have C2HC1 2=X kdz(C1 2(k))fy k#fk#dz(C1 2(k))fy k"fk" =H: (C5) Since K is also invariant under C2, we obtain dz(K) = dz(K). Hence dz(K) = 0. We conclude that h(K) = 0 and there exists a band touching at K. For the U1A11 state, TandS6are implemented in the same way as the U1A01 state, and we arrive at the same conclusion that there are band touchings at the and M points. At the K point, however, the band structure is generally gapped due to a nonzero dz. 2. Spinon mean-eld Hamiltonians The U1A00 state has the isotropic spinon hoppings on rst and second neighboring bonds, and the mean-eld Hamiltonain HU1A00 MF has already been given in the main text. This states gives a large spinon Fermi surface in the Brioullin zone. The spinon mean-eld states of the U1A01 state and the U1A11 state are given by HU1A01 MF =X x;yt1h ify (x+1;y);"f(x;y);#ify (x+1;y);#f(x;y);"ei 6fy (x;y+1);"f(x;y);# +ei 6fy (x;y+1);#f(x;y);"ei 6fy (x+1;y+1);"f(x;y);#+ei 6fy (x+1;y+1);#f(x;y);"+h:c:i +t2h ei2 3fy (x+1;y1);"f(x;y);#+ei 3fy (x+1;y1);#f(x;y);"+fy (x+1;y+2);"f(x;y);# fy (x+1;y+2);#f(x;y);"+ei 3fy (x+2;y+1);"f(x;y);#+ei2 3fy (x+2;y+1);#f(x;y);"+h:c:i ; (C6)10 TABLE III. The transformation for the spinons under four U1A PSGs that are labeled by U1A nC2nS6. U(1) PSGs T1 T2 C2 S6 U1A00f(x;y);"!f(x+1;y);" f(x;y);#!f(x+1;y);#f(x;y);"!f(x;y+1);" f(x;y);#!f(x;y+1);#f(x;y);"!ei 6f(y;x);# f(x;y);#!ei5 6f(y;x);"f(x;y);"!ei 3f(xy;x);" f(x;y);#!e+i 3f(xy;x);# U1A10f(x;y);"!f(x+1;y);" f(x;y);#!f(x+1;y);#f(x;y);"!f(x;y+1);" f(x;y);#!f(x;y+1);#f(x;y);"!ei 6fy (y;x);" f(x;y);#!ei 6fy (y;x);#f(x;y);"!ei 3f(xy;x);" f(x;y);#!e+i 3f(xy;x);# U1A01f(x;y);"!f(x+1;y);" f(x;y);#!f(x+1;y);#f(x;y);"!f(x;y+1);" f(x;y);#!f(x;y+1);#f(x;y);"!ei 6f(y;x);# f(x;y);#!ei5 6f(y;x);"f(x;y);"!ei 3fy (xy;x);# f(x;y);#!e+i 3fy (xy;x);" U1A11f(x;y);"!f(x+1;y);" f(x;y);#!f(x+1;y);#f(x;y);"!f(x;y+1);" f(x;y);#!f(x;y+1);#f(x;y);"!ei 6fy (y;x);" f(x;y);#!ei 6fy (y;x);#f(x;y);"!ei 3fy (xy;x);# f(x;y);#!e+i 3fy (xy;x);" and HU1A11 MF =X x;yt1h ify (x+1;y);"f(x;y);"ify (x+1;y);#f(x;y);#+ify (x;y+1);"f(x;y);" ify (x;y+1);#f(x;y);#ify (x+1;y+1);"f(x;y);"+ify (x+1;y+1);#f(x;y);#+h:c:i +t0 1h fy (x+1;y);"f(x;y);#+fy (x+1;y);#f(x;y);"+ei 3fy (x;y+1);"f(x;y);# +ei2 3fy (x;y+1);#f(x;y);"+ei2 3fy (x+1;y+1);"f(x;y);#+ei 3fy (x+1;y+1);#f(x;y);"+h:c:i +t2h ei 6fy (x+1;y1);"f(x;y);#+ei5 6fy (x+1;y1);#f(x;y);"ify (x+1;y+2);"f(x;y);# ify (x+1;y+2);#f(x;y);"+ei5 6fy (x2;y1);"f(x;y);#+ei 6fy (x2;y1);#f(x;y);"+h:c:i ; (C7) where in both Hamiltonians t1,t0 1denote the rst neigh- bor hoppings and t2denotes the second neighbor hop- ping. The band structures for specic choices of the hopping parameters are plotted in the main text. Clearly, we observe the band touchings at the , M and K points for the U1A01 state, and band touchings at the and M points for the U1A11 state. Appendix D: The U1A00 state and the spectroscopic results 1. Free spinon mean-eld theory The spinon mean-eld Hamiltonian of the U1A00 state is HU1A00 MF =t1X hrr0i;fy rfrt2X hhrr0ii;fy rfr;(D1) from which we compute the dynamic spin structure factor for dierent choices t2=t1. The dynamic spin structurefactor is given by S(q;!) =1 NX r;r0eiq(rr0)Z dtei!t h U1A00 MFjS r(t)S+ r0(0)j U1A00 MFi =X n(!nq)jhnjS+ qj U1A00 MFij2;(D2) whereNis the total number of spins, the summation is over all mean-eld states with the spinon particle-hole excitation,nqis the energy of the n-th excited state with the momentum q. The results are depicted in Fig. 4(a-e) and are consistent with the inelastic neutron scattering results36,37. All the results so far are independent from any microscopic spin interaction.11 FIG. 4. (a-e) Dynamic spin structure factor for the free spinon theory of the U1A00 state with dierent values of t2=t1. (f-h) The evolution of SRPA(q;!) as a function of J. In all subgures, the energy transfer is normalized against the corresponding bandwidth B. The parameter is dened as Jzz=t1. 2. Variational calculation and random phase approximation Here we consider the microscopic spin Hamiltonian that was introduced in Refs. 34 and 35, Hspin=X hrr0iJzzSz rSz r0+J(S+ rS r0+S rS+ r0) +J( rr0S+ rS+ r0+ rr0S rS r0) i 2Jz[( rr0S+ r rr0S r)Sz r0 +Sz r( rr0S+ r0 rr0S r0)]; (D3) where rr0= 1;ei2=3;ei2=3forrr0along thea1;a2 anda3bonds, respectively. Here, a3=a1a2. It was suggested and demonstrated that the anisotropic JandJzinteractions compete with the XXZ part of the Hamiltonian and may lead to disordered state34,35,39. Our calculation does show the enhancement of quantum uctuation in certain regions of the phase diagram35. Here we comment about the choices of the exchange cou- plings in the main text and in the following calculation. TheJzzandJcouplings can be determined by theCurie-Weiss temperature measurement on a single crys- tal sample. The complication comes from the subtraction of the Van Vleck susceptibility. Due to the Ga3+/Mg2+ exchange disorder in the non-magnetic layers, although these ions do not directly enter the Yb exchange path, it may modify the local crystal eld environment of the Yb3+ion and thus lead to some complication and varia- tion of the Van Vleck susceptibility. As a result, the very precise determination of the JzzandJcouplings can be an issue. That may explain some dierences of the Jzz andJcouplings that were obtained34{37,39. Partly for the same reason, the results on JandJzmay also be aected. However, quantum spin liquid, if it exists as the ground state of our generic model, is expected to be a phase that covers a nite region of the phase diagram. Therefore, the very precise value of the couplings may not be quite necessary from this point of view. There- fore, we here rely on our previous results of the quantum uctuation for the mean-eld phase diagram that indi- cates strong uctations in certain parameter regimes. We choose the exchange parameters from these disordered re- gions. For this spin Hamiltonian, the mean-eld variational energy is given as Evar=h U1A00 MFjHspinj U1A00 MFi=1 L2X qh U1A00 MFjJzz(q)Sz qSz q+ 2J(q)S+ qS qj U1A00 MFi =1 L2X q" Jzz(q)X nhnjSz qj U1A00 MFi2+ 2J(q)X nhnjS+ qj U1A00 MFi2# =1 L4X q2 4Jzz(q) 4X n;khnjfy k+q;"fk;"fy k+q;#fk;#j U1A00 MFi2 + 2J(q)X n;khnjfy k+q;"fk;#j U1A00 MFi23 5;(D4)12 where we have omitted JandJzbecause they do not conserve spin, therefore their contribution to Evaris zero. This is an artifact of the free spinon theory of HU1A00 MF that only includes isotropic spinon hoppings for the rst two neighbors. Due to the isotropic spinon hoppings, HU1A00 MF does not explicitly re ect the absence of spin-rotational symmetry that is brought by the JandJzinteractions. To in- corporate the JandJzinteractions, as we describe in the main text, we followed the phenomenological treat- ment for the \ t-J" model in the context of cuprate super- conductors60and consider H=HU1A00 MF +H0 spin, where H0 spinare theJandJzinteractions. In the parton construction, H0 spinis treated as the spinon interactions and thus introduces the spin rotational symmetry break- ing. With a random phase approximation for the inter- actionH0 spin, we obtain the dynamic spin susceptibility60 RPA(q;!) = 10(q;!)J(q)10(q;!);(D5) where0is the free-spinon susceptibility, and J(q) is the spin exchange matrix from H0 spin, J(q) = 0 BB@2(uqvq)J2p 3wqJp 3wqJz 2p 3wqJ 2(uq+vq)J(uqvq)Jz p 3wqJz (uqvq)Jz 01 CCA(D6) withuq= cos(qa1),vq=1 2(cos(qa2) + cos(qa3)), andwq=1 2(cos(qa2)cos(qa3)). The renormalized SRPA(q;!) can be read o from RPAviaSRPA(q;!) = 1 Im RPA(q;!)+and is plotted in Fig. 3(b) in the main text. The very precise values of JandJzcannot be de- termined from the existing data-rich neutron scattering experiment in a strong eld normal to the triangular plane. This is partly due to the experimental resolu- tion, and is also due to the fact that the linear spin wave spectrum for the eld normal to the plane is indepen- dent ofJzand is not quite sensitive to J35,39. In Fig. 3(b) of the main text, instead, we choose ( J;Jz) to fall into the disordered region of the phase diagram in Ref. 35 where the quantum uctuations are expected to be strong35. Appendix E: The U1B states In this section we use PSG to determine the free spinon mean-eld Hamiltonian for the U1B states to the rst and second spinon hoppings. In Fig. 5, we further present their spectroscopic features for comparison. Like the notation for U1As, the U1B states are also labeled by U1BnC2nS6.1. The U1B00 state For the- ux states, the dynamic spin structure fac- tor has an enhanced periodicity due to anticommutative lattice translations. One direct consequence of the pe- riodicity is that and M become equivalent, and the V-shaped upper excitation edge in Ref. 36 cannot be re- produced for the U1B states. We choose the spinon basis in the momentum space fk;I= (fA;k;";fB;k;";fA;k;#;fB;k;#)T, whereAandBde- note the two inequivalent sites in each unit cell due to the ux. The Hamiltonian is written in terms of the Dirac ma- trices aand their anticommutators ab= [a;b]=(2i): (E1) The representation is chosen to be (1;2;3;4;5)= (x 1;z 1;y x;y y;y z). aand abis odd under time reversal except when a= 4 orb= 4. The Hamiltonian is thus h(k) =5X a=1da(k)a+5X a<b=1dab(k)ab: (E2) For the U1B00 state, we have d3(k) =t0 1sin(kx=2p 3ky=2); d4(k) =t0 1cos(kx=2 +p 3ky=2); d5(k) =2t0 1sin(kx); d13(k) =2t1sin(kx=2p 3ky=2); d14(k) =2t1cos(kx=2 +p 3ky=2); d15(k) =2t1sin(kx); d23(k) =p 3t0 1sin(kx=2p 3ky=2); d24(k) =p 3t0 1cos(kx=2 +p 3ky=2); d34(k) = 2t2cos(p 3ky); d35(k) = 2t2sin(3kx=2p 3ky=2); d45(k) = 2t2cos(3kx=2 +p 3ky=2): (E3) 2. The U1B01 state d3(k) =t2sin(3kx=2 +p 3ky=2); d4(k) =t2cos(3kx=2p 3ky=2); d5(k) = 2t2sin(p 3ky); d23(k) =p 3t2sin(3kx=2 +p 3ky=2); d24(k) =p 3t2cos(3kx=2p 3ky=2): (E4)13 FIG. 5. Dynamic spin structure factor for six free spinon mean-eld states other than U1A00. Note the U1A10 Hamiltonian is identically zero for the rst and second neighbor hoppings. None of them is consistent with the spinon Fermi surface picture. In all subgures, the energy transfer is normalized against the corresponding bandwidth B. 3. The U1B10 state d3(k) =p 3t1sin (kxp 3ky)=2 ; d4(k) =p 3t1cos (kx+p 3ky)=2 ; d23(k) =t1sin (kxp 3ky)=2 ; d24(k) =t1cos (kx+p 3ky)=2 ; d25(k) = 2t1sinkx: (E5) 4. The U1B11 state d3(k) =p 3t2sin (3kx+p 3ky)=2 ; d4(k) =p 3t2cos (3kxp 3ky)=2 ; d23(k) =t2sin (3kx+p 3ky)=2 ; d24(k) =t2cos (3kxp 3ky)=2 ; d25(k) =2t2sin(p 3ky); d34(k) = 2t1cos(kx); d35(k) =2t1sin (kx+p 3ky)=2 ; d45(k) =2t1cos (kxp 3ky)=2 : (E6)14 gangchen.physics@gmail.com 1William Witczak-Krempa, Gang Chen, Yong Baek Kim, and Leon Balents, \Correlated Quantum Phenomena in the Strong Spin-Orbit Regime," Annual Review of Con- densed Matter Physics 5, 57{82 (2014). 2Jerey G. Rau, Eric Kin-Ho Lee, and Hae-Young Kee, \Spin-Orbit Physics Giving Rise to Novel Phases in Cor- related Systems: Iridates and Related Materials," Annual Review of Condensed Matter Physics 7, 195{221 (2016). 3G. Jackeli and G. Khaliullin, \Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models," Phys. Rev. Lett. 102, 017205 (2009). 4Gang Chen and Leon Balents, \Spin-orbit eects in Na4Ir3O8: A hyper-kagome lattice antiferromagnet," Phys. Rev. B 78, 094403 (2008). 5Ji r Chaloupka, George Jackeli, and Giniyat Khaliullin, \Kitaev-Heisenberg Model on a Honeycomb Lattice: Pos- sible Exotic Phases in Iridium Oxides A2IrO3," Phys. Rev. Lett. 105, 027204 (2010). 6Dmytro Pesin and Leon Balents, \Mott physics and band topology in materials with strong spin{orbit interaction," Nature Physics 6, 376{381 (2010). 7Shigeki Onoda and Yoichi Tanaka, \Quantum melting of spin ice: Emergent cooperative quadrupole and chirality," Phys. Rev. Lett. 105, 047201 (2010). 8Lucile Savary and Leon Balents, \Coulombic quantum liq- uids in spin-1 =2 pyrochlores," Phys. Rev. Lett. 108, 037202 (2012). 9Kate A. Ross, Lucile Savary, Bruce D. Gaulin, and Leon Balents, \Quantum Excitations in Quantum Spin Ice," Phys. Rev. X 1, 021002 (2011). 10Yi-Ping Huang, Gang Chen, and Michael Hermele, \Quantum Spin Ices and Topological Phases from Dipolar- Octupolar Doublets on the Pyrochlore Lattice," Phys. Rev. Lett. 112, 167203 (2014). 11Gang Chen and Leon Balents, \Spin-orbit coupling in d2 ordered double perovskites," Phys. Rev. B 84, 094420 (2011). 12Hamid R. Molavian, Michel J. P. Gingras, and Benjamin Canals, \Dynamically Induced Frustration as a Route to a Quantum Spin Ice State in Tb 2Ti2O7via Virtual Crys- tal Field Excitations and Quantum Many-Body Eects," Phys. Rev. Lett. 98, 157204 (2007). 13Gang Chen and Yong Baek Kim, \Anomalous enhance- ment of the Wilson ratio in a quantum spin liquid: The case of Na 4Ir3O8," Phys. Rev. B 87, 165120 (2013). 14R. Applegate, N. R. Hayre, R. R. P. Singh, T. Lin, A. G. R. Day, and M. J. P. Gingras, \Vindication of Yb 2Ti2O7as a Model Exchange Quantum Spin Ice," Phys. Rev. Lett. 109, 097205 (2012). 15K. A. Ross, J. P. C. Ru, C. P. Adams, J. S. Gard- ner, H. A. Dabkowska, Y. Qiu, J. R. D. Copley, and B. D. Gaulin, \Two-Dimensional Kagome Correlations and Field Induced Order in the Ferromagnetic XY Pyrochlore Yb2Ti2O7," Phys. Rev. Lett. 103, 227202 (2009). 16Gang Chen and Michael Hermele, \Magnetic orders and topological phases from f-dexchange in pyrochlore iri- dates," Phys. Rev. B 86, 235129 (2012). 17Lucile Savary, Xiaoqun Wang, Hae-Young Kee, Yong Baek Kim, Yue Yu, and Gang Chen, \Quantum spin ice onthe breathing pyrochlore lattice," Phys. Rev. B 94, 075146 (2016). 18SungBin Lee, Eric Kin-Ho Lee, Arun Paramekanti, and Yong Baek Kim, \Order-by-disorder and magnetic eld re- sponse in the Heisenberg-Kitaev model on a hyperhoney- comb lattice," Phys. Rev. B 89, 014424 (2014). 19SungBin Lee, Shigeki Onoda, and Leon Balents, \Generic quantum spin ice," Phys. Rev. B 86, 104412 (2012). 20Eric Kin-Ho Lee, Robert Schaer, Subhro Bhattacharjee, and Yong Baek Kim, \Heisenberg-kitaev model on the hy- perhoneycomb lattice," Phys. Rev. B 89, 045117 (2014). 21Shigeki Onoda and Yoichi Tanaka, \Quantum uctua- tions in the eective pseudospin-1 2model for magnetic py- rochlore oxides," Phys. Rev. B 83, 094411 (2011). 22Lieh-Jeng Chang, Shigeki Onoda, Yixi Su, Ying-Jer Kao, Ku-Ding Tsuei, Yukio Yasui, Kazuhisa Kakurai, and Martin Richard Lees, \Higgs transition from a magnetic Coulomb liquid to a ferromagnet in Yb 2Ti2O7," Nature Communications 3, 992 (2012). 23J. S. Gardner, S. R. Dunsiger, B. D. Gaulin, M. J. P. Gin- gras, J. E. Greedan, R. F. Kie , M. D. Lumsden, W. A. MacFarlane, N. P. Raju, J. E. Sonier, I. Swainson, and Z. Tun, \Cooperative Paramagnetism in the Geometrically Frustrated Pyrochlore Antiferromagnet Tb 2Ti2O7," Phys. Rev. Lett. 82, 1012{1015 (1999). 24Fei-Ye Li, Yao-Dong Li, Yue Yu, Arun Paramekanti, and Gang Chen, \Kitaev materials beyond iridates: order by quantum disorder and weyl magnons in rare-earth double perovskites," Phys. Rev. B 95, 085132 (2017). 25Gang Chen, \\Magnetic monopole" condensation of the pyrochlore ice U(1) quantum spin liquid: Application to Pr2Ir2O7and Yb 2Ti2O7," Phys. Rev. B 94, 205107 (2016). 26M J P Gingras and P A McClarty, \Quantum spin ice: a search for gapless quantum spin liquids in pyrochlore mag- nets," Reports on Progress in Physics 77, 056501 (2014). 27Yao-Dong Li and Gang Chen, \Symmetry enriched U(1) topological orders for dipole-octupole doublets on a py- rochlore lattice," Phys. Rev. B 95, 041106 (2017). 28E. Lhotel, S. Petit, S. Guitteny, O. Florea, M. Ciomaga Hatnean, C. Colin, E. Ressouche, M. R. Lees, and G. Balakrishnan, \Fluctuations and All-In All-Out Ordering in Dipole-Octupole Nd 2Zr2O7," Phys. Rev. Lett. 115, 197202 (2015). 29Owen Benton, Olga Sikora, and Nic Shannon, \Seeing the light: Experimental signatures of emergent electromag- netism in a quantum spin ice," Phys. Rev. B 86, 075154 (2012). 30Anson W. C. Wong, Zhihao Hao, and Michel J. P. Gingras, \Ground state phase diagram of generic xypyrochlore magnets with quantum uctuations," Phys. Rev. B 88, 144402 (2013). 31Gang Chen, Rodrigo Pereira, and Leon Balents, \Exotic phases induced by strong spin-orbit coupling in ordered double perovskites," Phys. Rev. B 82, 174440 (2010). 32S. H. Curnoe, \Structural distortion and the spin liquid state in Tb 2Ti2O7," Phys. Rev. B 78, 094418 (2008). 33Yuesheng Li, Haijun Liao, Zhen Zhang, Shiyan Li, Feng Jin, Langsheng Ling, Lei Zhang, Youming Zou, Li Pi, Zhaorong Yang, Junfeng Wang, Zhonghua Wu, and Qing- ming Zhang, \Gapless quantum spin liquid ground state in the two-dimensional spin-1/2 triangular antiferromag-15 net YbMgGaO 4," Scientic Reports 5, 16419 (2015). 34Yuesheng Li, Gang Chen, Wei Tong, Li Pi, Juanjuan Liu, Zhaorong Yang, Xiaoqun Wang, and Qingming Zhang, \Rare-Earth Triangular Lattice Spin Liquid: A Single- Crystal Study of YbMgGaO4," Phys. Rev. Lett. 115, 167203 (2015). 35Yao-Dong Li, Xiaoqun Wang, and Gang Chen, \Anisotropic spin model of strong spin-orbit-coupled trian- gular antiferromagnets," Phys. Rev. B 94, 035107 (2016). 36Yao Shen, Yao-Dong Li, Hongliang Wo, Yuesheng Li, Shoudong Shen, Bingying Pan, Qisi Wang, H. C. Walker, P. Steens, M Boehm, Yiqing Hao, D. L. Quintero-Castro, L. W. Harriger, Lijie Hao, Siqin Meng, Qingming Zhang, Gang Chen, and Jun Zhao, \Spinon Fermi surface in a tri- angular lattice quantum spin liquid YbMgGaO 4," Nature 540, 559{562 (2016). 37Joseph A. M. Paddison, Zhiling Dun, Georg Ehlers, Yao- hua Liu, Matthew B. Stone, Haidong Zhou, and Martin Mourigal, \Continuous excitations of the triangular-lattice quantum spin liquid YbMgGaO 4," Nature Physics, arXiv preprint 1607.03231 (2016). 38Yuesheng Li, Devashibhai Adroja, Pabitra K. Biswas, Pe- ter J. Baker, Qian Zhang, Juanjuan Liu, Alexander A. Tsirlin, Philipp Gegenwart, and Qingming Zhang, \Muon Spin Relaxation Evidence for the U(1) Quantum Spin- Liquid Ground State in the Triangular Antiferromagnet YbMgGaO4," Phys. Rev. Lett. 117, 097201 (2016). 39Yao-Dong Li, Yao Shen, Yuesheng Li, Jun Zhao, and Gang Chen, \The eect of spin-orbit coupling on the eective-spin correlation in YbMgGaO 4," arXiv preprint 1608.06445 (2016). 40Yao-Dong Li, Xiaoqun Wang, and Gang Chen, \Hidden multipolar orders of dipole-octupole doublets on a trian- gular lattice," Phys. Rev. B 94, 201114 (2016). 41Yao-Dong Li and Gang Chen, \Detecting spin fractional- ization in a spinon fermi surface spin liquid," Phys. Rev. B96, 075105 (2017). 42Haruki Watanabea, Hoi Chun Po, Ashvin Vishwanath, and Michael Zaletel, \Filling constraints for spin-orbit cou- pled insulators in symmorphic and nonsymmorphic crys- tals," PNAS 112, 14551{14556 (2015). 43Y. Xu, J. Zhang, Y. S. Li, Y. J. Yu, X. C. Hong, Q. M. Zhang, and S. Y. Li, \Absence of Magnetic Ther- mal Conductivity in the Quantum Spin-Liquid Candidate YbMgGaO4," Phys. Rev. Lett. 117, 267202 (2016). 44Olexei I. Motrunich, \Variational study of triangular lat- tice spin-1=2 model with ring exchanges and spin liquid state in-(ET)2Cu2(CN)3," Phys. Rev. B 72, 045105 (2005). 45Sung-Sik Lee and Patrick A. Lee, \U(1) Gauge Theory of the Hubbard Model: Spin Liquid States and Possible Ap- plication to -(BEDT-TTF)2Cu2(CN)3," Phys. Rev. Lett. 95, 036403 (2005). 46Patrick A. Lee and Naoto Nagaosa, \Gauge theory of the normal state of high- Tcsuperconductors," Phys. Rev. B 46, 5621{5639 (1992). 47Xiao-Gang Wen, \Quantum orders and symmetric spin liq- uids," Phys. Rev. B 65, 165113 (2002). 48See the Supplementary information. 49Johannes Reuther, Shu-Ping Lee, and Jason Alicea, \Clas- sication of spin liquids on the square lattice with strong spin-orbit coupling," Phys. Rev. B 90, 174417 (2014). 50Gang Chen, Andrew Essin, and Michael Hermele, \Majo- rana spin liquids and projective realization of SU(2) spinsymmetry," Phys. Rev. B 85, 094418 (2012). 51Michael Hermele, \SU(2) gauge theory of the Hubbard model and application to the honeycomb lattice," Phys. Rev. B 76, 035125 (2007). 52Samuel Bieri, Claire Lhuillier, and Laura Messio, \Projec- tive symmetry group classication of chiral spin liquids," Phys. Rev. B 93, 094437 (2016). 53Yi-Zhuang You, Itamar Kimchi, and Ashvin Vishwanath, \Doping a spin-orbit Mott insulator: Topological super- conductivity from the Kitaev-Heisenberg model and pos- sible application to (Na 2/Li2)IrO 3," Phys. Rev. B 86, 085145 (2012). 54Robert Schaer, Subhro Bhattacharjee, and Yong Baek Kim, \Spin-orbital liquids in non-Kramers magnets on the kagome lattice," Phys. Rev. B 88, 174405 (2013). 55Yuan-Ming Lu, \Symmetry protected gapless Z2spin liq- uids," arXiv preprint 1606.05652 (2016). 56Xiao-Gang Wen, \Quantum order: a quantum entangle- ment of many particles," Physics Letters A 300, 175 { 181 (2002). 57Andrew M. Essin and Michael Hermele, \Spectroscopic signatures of crystal momentum fractionalization," Phys. Rev. B 90, 121102 (2014). 58In the previous work36, only the nearest-neighbor spinon hopping is included. 59Sandor Toth, Katharina Rolfs, Rolfs, Andrew R. Wildes, and Christian Ruegg, \Strong exchange anisotropy in YbMgGaO 4from polarized neutron diraction," arXiv preprint arXiv:1705.05699 (2017). 60Jan Brinckmann and Patrick A. Lee, \Slave Boson Ap- proach to Neutron Scattering in YBa 2Cu3O6+ySupercon- ductors," Phys. Rev. Lett. 82, 2915{2918 (1999). 61Yuesheng Li, Devashibhai Adroja, Robert I. Bewley, David Voneshen, Alexander A. Tsirlin, Philipp Gegenwart, and Qingming Zhang, \Crystalline Electric-Field Randomness in the Triangular Lattice Spin-Liquid YbMgGaO4," Phys. Rev. Lett. 118, 107202 (2017). 62Yao Shen, Yao-Dong Li, H. C. Walker, P. Steens, M. Boehm, Xiaowen Zhang, Shoudong Shen, Hongliang Wo, Gang Chen, and Jun Zhao, \Fractionalized exci- tations in the partially magnetized spin liquid candidate YbMgGaO 4," arXiv preprint arXiv:1708.06655 (2017). 63Changle Liu, Rong Yu, and Xiaoqun Wang, \Semiclassical ground-state phase diagram and multi- qphase of a spin- orbit-coupled model on triangular lattice," Phys. Rev. B 94, 174424 (2016). 64Qiang Luo, Shijie Hu, Bin Xi, Jize Zhao, and Xiao- qun Wang, \Ground-state phase diagram of an anisotropic spin-1 2model on the triangular lattice," Phys. Rev. B 95, 165110 (2017). 65Z. Zhu, P.A. Maksimov, S.R. White, and A.L. Cherny- shev, \Disorder-induced Mimicry of a Spin Liquid in YbMgGaO 4," arXiv preprint arXiv:1703.02971 (2017). 66M. Hirschberger, J. W. Krizan, R. J. Cava, and N. P. Ong, \Large thermal hall conductivity of neutral spin ex- citations in a frustrated quantum magnet," Science 348, 106{109 (2015). 67S. J. Li, Z. Y. Zhao, C. Fan, B. Tong, F. B. Zhang, J. Shi, J. C. Wu, X. G. Liu, H. D. Zhou, X. Zhao, and X. F. Sun, \Low-temperature thermal conductivity of Dy2Ti2O7 and Yb 2Ti2O7single crystals," Phys. Rev. B 92, 094408 (2015). 68Q. J. Li, Z. Y. Zhao, C. Fan, F. B. Zhang, H. D. Zhou, X. Zhao, and X. F. Sun, \Phonon-glass-like behavior of16 magnetic origin in single-crystal Tb 2Ti2O7," Phys. Rev. B 87, 214408 (2013). 69Yuji Matsuda, (2017), Talk at workshop on topological materials, Kyoto University. 70Tian-Heng Han, Joel S Helton, Shaoyan Chu, Daniel G Nocera, Jose A Rodriguez-Rivera, Collin Broholm, and Young S Lee, \Fractionalized excitations in the spin-liquid state of a kagome-lattice antiferromagnet," Nature 492, 406{410 (2012). 71Y. Shimizu, K. Miyagawa, K. Kanoda, M. Maesato, and G. Saito, \Spin liquid state in an organic mott insulator with a triangular lattice," Phys. Rev. Lett. 91, 107001 (2003). 72T. Itou, A. Oyamada, S. Maegawa, M. Tamura, and R. Kato, \Quantum spin liquid in the spin-12 triangular antiferromagnet EtMe 3Sb[Pd(dmit)2]2," Phys. Rev. B 77, 104413 (2008). 73Satoshi Yamashita, Yasuhiro Nakazawa, Masaharu Oguni, Yugo Oshima, Hiroyuki Nojiri, Yasuhiro Shimizu, Kazuya Miyagawa, and Kazushi Kanoda, \Thermodynamic prop- erties of a spin-1/2 spin-liquid state in a kappa-type organic salt," Nature Physics 4, 459{462 (2008). 74Y. Q. Liu, S. J. Zhang, J. L. Lv, S. K. Su, T. Dong, Gang Chen, and N. L. Wang, \Revealing a triangular latticeising antiferromagnet in a single-crystal CeCd 3As3," arXiv preprint 1612.03720 (2016). 75Shohei Higuchi, Yuki Noshima, Naoki Shirakawa, Masami Tsubota, and Jiro Kitagawa, \Optical, transport and mag- netic properties of new compound CeCd 3P3," Materials Research Express 3, 056101 (2016). 76Andr T. Nientiedt and Wolfgang Jeitschko, \The Series of Rare Earth Zinc Phosphides RZn 3P3(R=Y, LaNd, Sm, GdEr) and the Corresponding Cadmium Compound PrCd 3P3," Journal of Solid State Chemistry 146, 478 { 483 (1999). 77A. Yamada, N. Hara, K. Matsubayashi, K. Munakata, C. Ganguli, A. Ochiai, T. Matsumoto, and Y. Uwatoko, \Eect of pressure on the electrical resistivity of CeZn 3P3," J. Phys.: Conf. Ser. 215, 012031 (2010). 78Stanislav S. Stoyko and Arthur Mar, \Ternary Rare-Earth Arsenides REZn 3As3(RE = LaNd, Sm) and RECd 3As3 (RE = LaPr)," Inorganic Chemistry 50, 11152{11161 (2011). 79M. B. Sanders, F. A. Cevallos, and R. J. Cava, \Mag- netism in the KBaRE(BO 3)2(RE= Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, Lu) series: materials with a triangular rare earth lattice," arXiv preprint 1611.08548 (2016).
0905.0611v1.Transport_through_a_band_insulator_with_Rashba_spin_orbit_coupling__metal_insulator_transition_and_spin_filtering_effects.pdf	arXiv:0905.0611v1 [cond-mat.mes-hall] 5 May 2009Transport through a band insulator with Rashba spin-orbit coupling: metal-insulator transition and spin-ﬁltering eﬀects T. Jonckheere,1G.I. Japaridze,2T. Martin,1,3and R. Hayn4 1Centre de Physique Th´ eorique, UMR 6207, case 907, Campus de Luminy, 13288 Marseille Cedex 9, France 2Andronikashvili Institute of Physics, Tamarashvili str. 6, 0177 Tbilisi, Georgia 3Universit´ e de la Mediterran´ ee, Campus de Luminy, 13288 Ma rseille Cedex 9, France 4Institut Mat´ eriaux Micro´ electronique Nanosciences de P rovence, Facult´ e St. J´ erˆ ome, Case 142, F-13397 Marseille Cedex 20 , France (Dated: November 20, 2018) Abstract Wecalculatethecurrent-voltage characteristic ofaone-d imensionalbandinsulatorwithmagnetic ﬁeld and Rashba spin-orbit coupling which is connected to no nmagnetic leads. Without spin-orbit coupling we ﬁnd a complete spin-ﬁltering eﬀect, meaning that the electric transport occurs in one spin channel only. For a large magnetic ﬁeld which closes the band gap, we show that spin-orbit coupling leads to a transition from metallic to insulating b ehavior. The oscillations of the diﬀerent spin-components of the current with the length of the transp ort channel are studied as well. PACS numbers: 72.25.-b, 71.70.Ej 1I. INTRODUCTION There is a great interest today to study the phenomena of quantu m transport in low dimensional systems, both from a technological and a fundamenta l point of view. Especially important are questions of spin polarized transport, also known as spintronics.1A famous example is the proposition of the Datta-Das transistor2which uses the rotation of the elec- tron spin due to spin-orbit (SO) coupling. There are two sources of spin-orbit coupling in quasi one-dimensional systems (1D), an intrinsic one due to the lac k of inversion symmetry in certain crystal structures (Dresselhaus term)3and an external one triggered by an applied voltage to surface gates (the Rashba SO coupling).4 Several works studied the SO coupling and electronic transport in q uasi 1D metallic systems.5,6,7,8,9,10,11,12,13,14,15,16In contrast, the inﬂuence of SO coupling and magnetic ﬁeld on the transport in 1D band insulators is unexplored, and it can be ex pected to be funda- mentally diﬀerent. In the letter band insulators, we will report on tw o interesting eﬀects: the complete spin ﬁltering eﬀect and the SO induced metal-insulator t ransition. An in- complete spin ﬁltering eﬀect is possible in 1D metallic systems with a pote ntial step or additional impurities,7,14,16but the complete spin ﬁltering as well as the spin-orbit induced metal-insulator transition which will be reported below are speciﬁc to 1D band insulators and cannot be observed (in principle) in 1D metals. A prototype model for a one-dimensional (1D) band insulator is a ha lf-ﬁlled ionic chain with alternating on-site energies (energy diﬀerence ∆). Such an ion ic chain will be used in our study, however the obtained results are expected to be gene ric to any kind of 1D band insulators, including charge transfer insulators and realized in diato mic polymers,17as well as the 1D Peierls insulators, such as polyacetylene.18In a wider sense, one-dimensional band insulators may also be realized in carbon-nanotubes. These nanotu bes have the advantage that the value of the gap may be tuned in a very wide range from 600 m eV (for (12,0) nanotubes) up to 8 meV (for (13,0) nanotubes) or even smaller valu es.19 Before presenting detailed calculations, let us start with some qualit ative arguments. We ﬁrst discuss transport in 1D band insulators in a magnetic ﬁeld Band in absence of SO interaction. Although the magnetic ﬁeld induced metal-insulator a nd insulator-metal transitions have been the subject of studies for decades,20in the context of transport in mesoscopic systems these eﬀects have not been investigated in de tail. As we show in this 2paper, in the limit of ultra-low temperatures ( T≪∆) and strong magnetic ﬁeld ( B≥∆) the ﬁeld induced insulator-metal transitions lead to the almost comp lete spin ﬁltering eﬀect, since in this case only one spin channel is open for transport at the F ermi level. However, the metallic phase reached at B≥∆ shows unconventional and substantially diﬀerent properties compared to a normal metal. As we will show, co ntrary to the usual 1D metallic phase, the Rashba spin-orbit coupling opens up a gap again , leading to a spin- orbit induced metal-insulator transition. It is important to note tha t both eﬀects, i.e. the complete spin ﬁltering eﬀect and the metal-insulator transition induc ed by the Rashba spin- orbit coupling are very speciﬁc to 1D band insulators, and may not be observed in 1D metals. Rather than analyzing the eﬀect of these transitions by computing the bulk transport properties of the chain, such as the conductivity, we choose to co mpute the current of a ﬁnite chain of such a material, whose extremities are connected to metallic electrodes. A bias is imposed between the electrodes in order to induce current ﬂow. On the one hand it allows to probe the spin ﬁltering eﬀects in a setup which is close to experimen tal situations, on the other hand it also allows to investigate potential ﬂuctuations of the current as a function of the chain length in the presence of SO coupling. In particular, we w ill show that a complex behavior, with several periods and a complicated energy de pendence is obtained in the presence of a band gap ∆ and a magnetic ﬁeld; this is totally diﬀere nt from the simple harmonic oscillations, with a period inversely proportional to the SO c oupling strength, obtained in the metallic case. The paper is organized as follows. In Sec. II, we introduce the mode l and in Sec. III we discuss the spectrum of the inﬁnite chain. In Sec. IV, we discuss th e method which is used to obtain the transport properties as well as physical results. We conclude in Sec. V. II. THE MODEL We note ﬁrst that the spin-orbit coupling can be generated by a volt ageVGapplied to external gates perpendicular to the current. This is known as Ras hba spin-orbit coupling,4 and deﬁnes the device studied in the present paper (Fig. 1). We con sider a ﬁnite chain (oriented in the ˆ xdirection) connected to metallic leads. Lateral metallic gates are pla ced so that to create an electric ﬁeld which is perpendicular to both the c hain and the magnetic 3GATEQuantum wire VDVGH FIG. 1: (Color online) Schematic ﬁgure of the transport proc ess studied in the paper. The SO coupling parameter αRis proportional to VG. ﬁeld(ˆz)direction. Withtheseconventions thefollowingHamiltoniandescribe s themolecular chain: H=−t/summationdisplay n,σ/parenleftbig c† n,σcn+1,σ+h.c./parenrightbig +∆ 2/summationdisplay n,σ(−1)nc† n,σcn,σ −gµBH 2/summationdisplay n,σσc† n,σcn,σ +αR/summationdisplay n/parenleftig c† n,↑cn+1,↓−c† n,↓cn+1,↑+h.c./parenrightig . (1) Here the ﬁrst contribution describes the kinetic energy in the tight binding model, the second one accounts for alternating on-site energies, the third t erm is the Zeeman coupling (magnetic ﬁeld B=gµBH) and the last term is the Rashba SO coupling (strength αR). We consider a ﬁnite chain of length Lwhich is connected to left and right leads by tunneling amplitudes TlandTr, respectively. Note that we investigate here the case of nonmagnetic leads. We assume that the SO coupling vanishes in the leads and that the ma gnetic ﬁeld only aﬀects the central region signiﬁcantly. III. THE SPECTRUM To understand the magneto transport results it is useful to ﬁrst consider the spectrum of (1). For clarity all spectra are plotted in the reduced Brillouin zon ek∈[−π/2a,π/2a] associated with the presence (possibly small) of alternating on site e nergies. Typically this 4/MinusΠ 2/MinusΠ 4Π 4Π 2 /Minus2/Minus112 /MinusΠ 2/MinusΠ 4Π 4Π 2 /Minus2/Minus112 /MinusΠ 2/MinusΠ 4Π 4Π 2 /Minus2/Minus112 /MinusΠ 2/MinusΠ 4Π 4Π 2 /Minus2/Minus112ΑR/EquΑl0 ΑR/EquΑl0.4 /CΑΠDeltΑ/EquΑl0 /CΑΠDeltΑ/EquΑl0.6 FIG. 2: Spectrum of the tight-binding chain (see Eqs. (1)-(2 )) with magnetic ﬁeld ( B=gµBH= 1.3), with and without Rashba coupling αRand ionicity ∆ ( t= 1 has been taken as unit of energy). spectrum consists of 4 branches and it can be obtained exactly: E± 1/2(k) =±/radicalig 4α2 Rsin2k+B2 4+∆2 4+4t2cos2k±W W=/radicalig 16α2 Rt2sin2(2k)+4B2t2cos2k+B2∆2 4(2) in the general case with spin-orbit coupling αRand in the presence of a magnetic ﬁeld. It is shown in Fig. 2 for diﬀerent cases of ∆ and αR, with a non-zero magnetic ﬁeld B. The upper left corner of Fig. 2 depicts the trivial case of a non dimer ized tight binding chain (∆ = 0) in the presence of a magnetic ﬁeld. The latter gives rise t o a splitting between the spin up and spin down bands. The spectrum has been folded in this reduced Brillouin zone to serve as a point of comparison for the other cases, with dim erisation. We now consider the case of a non-zero value for ∆ (bottom left plot of Fig. 2). For αR= 0, the spin up and down bands are still separated, but the dimerisa tion opens a gap for each spin band at the boundaries of the Brillouin zone. This implies t hat for energies close to the Fermi level only one spin channel will be open for the tra nsport (complete spin ﬁltering eﬀect, see next Section). As shown on the Figure, the mag netic ﬁeld can be so strong that the gap closes and the system can become metallic. We n ow switch on the Rashba coupling in the presence of dimerisation (bottom right corne r of Fig. 2). In this case, the coupling between spin up and spin down gives rise to an antic rossing, so that the spin-orbit coupling opens up a gap again. On the other hand, there is no spin ﬁltering eﬀect for a homogeneou s, metallic chain 5(∆ = 0, top row of Fig. 2). Without magnetic ﬁeld (not shown), the sp in-orbit coupling can be taken exactly into account by a shift of k→k+arctan(αR/t). As can be easily inferred from the spin split band structure in a magnetic ﬁeld (left plot) the de nsity of states for spin up and spin down electrons is the same in that case. And the intro duction of spin-orbit coupling (right plot) does not open a gap. This proves that both eﬀe cts, i.e. the complete spin ﬁltering eﬀect and the spin-orbit driven metal-insulator transit ion cannot be observed in a metallic system (∆ = 0). IV. TRANSPORT THROUGH A FINITE CHAIN Intheabsence of electronic interactions, the current througha ﬁnite chain oflength Lcan be cast exactly in a Landauer type formula, written here for zero t emperature. This current depends on the orientation of electrons spin at the input lead and th e output lead: the currentIss′for instance, corresponds to electrons which enter with spin s(withs=↑or↓) from the left lead and leave the current channel with spin s′to the right lead. With this convention, Iss′(VD) = ΓLΓR/integraldisplayµR µLdE/vextendsingle/vextendsingle/vextendsingleGss′ ab(E)/vextendsingle/vextendsingle/vextendsingle2 . (3) The integration is peformed between the chemical potentials of the left and right leads (µL=−VDandµR= 0). The energy dependent transmission is simply proportional to t he square modulus ofthe total retarded Greenfunction of the chain (which include the coupling with the leads) between both endpoints, noted here aandb. The tunneling rates on the left and right side are deﬁned as Γ j≡2πρjT2 j(j=L,R), whereρjis the (constant) density of states of lead j, andTjthe tunneling amplitude to lead j. The total Green function of the chain between the end sites aandb,Gss′ abcan be obtained from the Green function of the bare chain (uncoupled to leads) gss′ abby solving the Dyson equations: G↑↑ ab G↓↑ ab G↑↑ bb G↓↑ bb = g↑↑ ab g↓↑ ab g↑↑ bb g↓↑ bb + g↑↑ aag↑↓ aag↑↑ abg↑↓ ab g↓↑ aag↓↓ aag↓↑ abg↓↓ ab g↑↑ bag↑↓ bag↑↑ bbg↑↓ bb g↓↑ bag↓↓ bag↓↑ bbg↓↓ bb ΣaG↑↑ ab ΣaG↓↑ ab ΣbG↑↑ bb ΣbG↓↑ bb andsimilar equations for theopposite spins, andwhere Σ j=−iΓjis theretarded self-energy coming from the coupling to lead j=L,R. The Green functions of the bare chain gss′ abare 6obtained simply by computing the eigenvalues and eigenstates of the ﬁnite chain, and using a spectral representation: gss′ ab(E) =/summationdisplay nψs n(a)/parenleftbig ψs′ n(b)/parenrightbig∗ E−En+i0+(4) Here all the Green functions, and consequently the current in Eq. (3), are 2x2 matrices in spin space. This is a consequence of the Rashba SO coupling, which co uples the spin-up and spin-down channels. Without SO coupling all quantities become dia gonal in spin space, and the formula for the total Green function reduces to: Gss ab=gss ab (1−ΣLgssaa)(1−ΣRgss bb)−ΣLΣRgss abgss ba(5) Letusstartthediscussionofournumerical resultswiththecurre nt-voltagecharacteristics in a magnetic ﬁeld with ∆ /negationslash= 0, but without SO coupling (see Fig. 3). The magnetic ﬁeld B=Bcis chosen such that it just closes the gap, but the exact value of th is parameter is nevertheless not important for the spin-ﬁltering eﬀect . The transport for drain voltages betweenVD= 0 andVD≃0.6tis only possible for one spin channel. It means that we ﬁnd complete spin polarization in the transport channel (connected to nonmagnetic leads) and a complete spin-ﬁltering. The spin polarization of the current is deﬁn ed in the general case as7,14 P=I↑↑+I↓↑−I↑↓−I↓↓ I↑↑+I↓↑+I↑↓+I↓↓. (6) As shown on Fig. 3, the spin polarization remains ﬁnite (but smaller tha n unity) for larger voltages (between approximatively 0.6 tand2.25t) anddisappears at approximatively 2.25 t where the current reaches saturation (all the electrons of the t ight-binding band contribute). A ﬁnite spin polarization means also that the current creates a tota l magnetization M in the transport channel of length L. The value of the total magnetization is given by M/µB=L(I↑↑+I↓↑−I↑↓−I↓↓)//angbracketleftv/angbracketright, where/angbracketleftv/angbracketrightmeans the average velocity of the electrons which are active in the transport process (ballistic transport). This spin-ﬁltering eﬀect is expected to work for a wide range of gap v alues. The voltage region where only one spin channel is open is determined by the applied magnetic ﬁeld. This works also if the magnetic ﬁeld is not suﬃciently strong to close th e gap. Therefore, even materials with gap values of about 0.5 eV are possible candidates to show the complete spin-ﬁltering eﬀect. The onset of the minority spin channel (at zer o energy in Fig. 3) is given by the relative position of the chemical potential with respect to the upper band edge 70.51.01.52.02.5VD 0.000.020.040.060.080.10I 01200.0250.05 0.51.01.52.02.5VD0.250.500.751.P FIG. 3: Upper plot: total current as a function of the bias vol tageVD, in the spin ﬁltering conﬁguration. ∆ = 0 .6,B= 0.6, ΓL= ΓR= 0.1 andt= 1, for a chain of 500 sites.The inset shows the separate contributions from the spin-up and spin-down c urrent. Lower plot: spin polarization (Eq. (6)) for the same parameters. of the valence band which may vary from one experimental situation to another. We now consider the case of non-zero SO coupling. The transition fr om metallic to insulating behavior driven by SO coupling is shown in Fig. 4. The magnetic ﬁeld is the same as in Fig. 3, i.e. it just closes the gap B=Bc= ∆, and the Rashba SO coupling isαR= 0.2t. It is created by an external gate voltage (see Fig. 1). The SO cou pling leads to an insulating behavior, as seen in the spectrum (Fig. 2) and in the current-voltage characteristics (Fig. 4). In contrast to Fig. 3, the presence of t he SO coupling αRleads to a current on-set at VD≃0.25tcorresponding to half of the gap value for our choice of the chemical potential. The diﬀerent current components Iss′are now all diﬀerent, and the spin polarization (Eq. 6) is diﬀerent from zero but not complete (0 <P <1). Note that the relative values of the diﬀerent spin-components of t he current in Fig. 4 are 80.51.01.52.02.5VD 0.000.020.040.060.080.10I 0120.0.020.04 0.51.01.52.02.5VD /Minus0.4/Minus0.20.2P FIG. 4: (Color online) Upper plot: total current as a functio n of the bias voltage VD, in the the presence of Rashba spin-orbit coupling, with αR= 0.2, ∆ = 0.6,B= 0.6, ΓL= ΓR= 0.1 andt= 1 for a chain of 500 sites. The inset shows the four spin compone nts of the current (in this order from top to bottom near VD= 2.5):I↓↓(red),I↑↑(black), I↓↑(orange), and I↑↓(blue). Lower plot: spin polarization (Eq. (6)) for the same parameters. dependent on the chain length. This is due to the Rashba SO coupling, which is known to induce spin precession. Here, this spin precession is made more comp lex due to the presence of the magnetic ﬁeld Band the ionicity ∆. The oscillations of the current components, as a function of the chain length L, are shown in Fig. 5, for Lvarying between 500 and 600. These oscillations have a rather small contrast, show severa l periods and a complicated dependence on bias voltage VDin the general case (a dominating period seems to be present for the oﬀ diagonal components of the current though). This has to be contrasted with the pure metallic case ( B= 0 and ∆ = 0, shown in the inset of Fig. 5), where only one period Lp=π/αis present independently on VD, and where the contrast is maximum. 95005205405605806000.0050.0100.0150.0200.025 5205400.0.020.04I/DownArrow/DownArrow I/UΠArrow/UΠArrow I/DownArrow/UΠArrow I/UΠArrow/DownArrow L FIG. 5: (Color online) Oscillations of the spin components o f the current as a function of the chain length (lengths between 500 and 600), for VD= 2.0, when SO Rashba coupling is present (αR= 0.2, ∆ = 0 .6,B= 0.6, ΓL= ΓR= 0.05 andt= 1). Inset: the same plot with B= 0 and ∆ = 0, where I↑↑=I↓↓andI↑↓=I↓↑ V. CONCLUSIONS In studying the combined eﬀect of magnetic ﬁeld and SO interaction o n the transport in 1D band insulators we found two interesting eﬀects. First, already without SO coupling, the presence of a magnetic ﬁeld leads to complete spin ﬁltering. We studie d this eﬀect here by connecting the conduction channels to nonmagnetic leads but the e ﬀect of magnetic leads is easy to imagine, at least qualitatively. Then, spin ﬁltering means high conductance for parallel magnetization in the leads and low conductance for antipara llel arrangement. We speculate that the voltage region of the spin ﬁltering eﬀect may b e dramatically enhanced by the presence of magnetic impurities in the band insulato r, due to the giant Zeemann eﬀect. This might be important for the experimental veriﬁ cation of our proposal. The second striking eﬀect of this study appears in band insulators w ith small band gap that may be closed by a magnetic ﬁeld. In that situation, the SO coup ling leads again to an insulating behavior. That is especially interesting for the Rashba s pin orbit coupling which is tuned by a gate voltage. Therefore, we may propose a devic e in which the metal- 10insulator transition is controlled by the gate voltage via the Rashba S O term. This is in sharp contrast with 1D metallic systems, where the SO coupling doe s not lead to any metal-insulator transition. We also showed the oscillations of the diﬀerent current components with the chain length. Whereas the simple oscillations in metallic systems are easy to underst and, the oscillations are much more complex for band insulators. We have let a detailed ana lysis of these oscil- lations for further studies. In our calculations the band insulator w as simulated by an ionic term of alternating on-site energies in the Hamiltonian. But we think t hat our results are generic to any kind of band insulator. On the other hand, the way in w hich Coulomb corre- lations inﬂuence our results may be diﬀerent from one microscopic Ha miltonian to another. We expect that the Coulomb correlation just scales the band gap (e ither to larger or to smaller values) and that the presented results should remain valid wit h eﬀective parameters, however. The authors thank Marc Bescond and Alvaro Ferraz for useful dis cussion. 1A. Fert, Rev. Mod. Phys. 80, 1517 (2009). 2S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1990). 3G. Dresselhaus, Phys. Rev. 100, 580 (1955). 4E.I.Rashba, Fiz. Tverd.Tela(Leningrad) 2, 1224 (1960) [Sov. Phys.SolidState 2, 1109(1960)]. 5A.V. Moroz, K.V. Samokhin, and C.H.W. Barnes, Phys. Rev. B 62, 16900 (2000); ibidPhys. Rev. B62, 16900 (2000). 6W. H¨ ausler, Phys. Rev. B 63, 121310(R) (2001). 7P. St˘ reda and P. S˘ eba, Phys. Rev. Lett. 90, 256601 (2003). 8A. Iucci, Phys. Rev. B 68, 075107 (2003). 9V. Gritsev, G.I. Japaridze, M. Pletyukhov, and D. Baeriswyl , Phys. Rev. Lett. 94, 137207 (2005). 10P. Foldi, B. Molnar, M.G. Benedict, and F.M. Peeters, Phys. R ev. B71, 033309 (2005). 11F. Cheng and G. Zhou, Journal of Physics: Condensed Matter 19, 136215 (2007). 12M. Scheid, M. Kohda, Y. Kunihashi, K. Richter, and J. Nitta, P hys. Rev. Lett. 101, 266401 (2008). 1113S. Bellucci and P. Onorato Phys. Rev. B 78, 235312 (2008). 14J.E. Birkholz and V. Meden, Jour. Phys.: Condensed Matter 20, 085226 (2008); ibid, Phys. Rev. B79, 085420 (2009). 15G.I. Japaridze, H. Johannesson and A. Ferraz, unpublished, arXiv:0904.1846 (2009). 16Z. Ristivoyevic, G.I.Japaridze and T. Nattermann, unpubli shed (2009). 17M.J. Rice and E.J. Mele, Phys. Rev. Lett. 49, 1455 (1982). 18W.P. Su, J.R. Schrieﬀer, and A.J. Heeger, Phys. Rev. Lett. 42, 1698 (1979). 19N. Hamada, S.I. Sawada, and A. Oshiyama, Phys. Rev. Lett. 68, 1579 (1992). 20N.B. Brandt and E.A. Svistunova, Sov. Phys. Uspekhi 101249 (1970). 12
1107.2627v3.BCS_BEC_crossover_in_spin_orbit_coupled_two_dimensional_Fermi_gases.pdf	arXiv:1107.2627v3 [cond-mat.quant-gas] 4 Jan 2012BCS-BEC crossover in spin-orbit coupled two-dimensional F ermi gases Gang Chen,1,2,3Ming Gong,1and Chuanwei Zhang1,∗ 1Department of Physics and Astronomy, Washington State Univ ersity, Pullman, Washington, 99164 USA 2Department of Physics, Shaoxing University, Shaoxing 3120 00, P. R. China 3State Key Laboratory of Quantum Optics and Quantum Optics De vices, College of Physics and Electronic Engineering, Shanxi Univ ersity, Taiyuan 030006, P. R. China The recent experimental realization of spin-orbit couplin g for ultra-cold atoms has generated much interest in the physics of spin-orbit coupled degenera te Fermi gases. Although recently the BCS-BEC crossover in three-dimensional (3D) spin-orbit co upled Fermi gases has been intensively studied, the corresponding two-dimensional (2D) crossove r physics has remained unexplored. In this paper, we investigate, both numerically and analytica lly, the BCS-BEC crossover physics in 2D degenerate Fermi gases in the presence of a Rashba type of spi n-orbit coupling. We derive the mean ﬁeld gap and atom number equations suitable for the 2D spin-o rbit coupled Fermi gases and solve them numerically and self-consistently, from which the dep endence of the ground state properties (chemical potential, superﬂuid pairing gap, ground state e nergy per atom) on the system parameters (e.g., binding energy, spin-orbit coupling strength) is ob tained. Furthermore, we derive analytic expressions for these ground state quantities, which agree well with our numerical results within a broad parameter region. Such analytic expressions also agr ee qualitatively with previous numerical results for the 3D spin-orbit coupled Fermi gases, where ana lytic results are lacked. We show that with an increasing SOC strength, the chemical potential is s hifted by a constant determined by the SOC strength. The superﬂuid pairing gap is enhanced signiﬁc antly in the BCS limit for strong SOC, but only increases slightly in the BEC limit. PACS numbers: 03.75.Ss, 05.30. Fk, 74.20.Fg I. INTRODUCTION Spin-orbitcoupling(SOC),theinteractionbetweenthe spin and orbital degrees of freedom of a particle, has played important roles in condensed matter as well as atomic and nuclear physics. For instance, it is known that the coupling between electron spin and its linear momentum in solids leads to many important condensed matter phenomena such as spin and anomalous Hall ef- fects [1, 2], topological insulators and and superconduc- tors [3], spintronics [4], etc. In atomic physics, the cou- pling between the electron spin and its motion around an atomic nucleus is responsible for many of the details of atomic structure [5]. Superﬂuidity and superconductivity are another im- portant phenomena in physics and have been widely studied in many physical systems, including solids, He- lium liquids, as well as ultra-cold atomic gases [6]. Al- though particles in superﬂuids and superconductors usu- ally possess spins or pseudospins, the eﬀects of SOC on superﬂuidity and superconductivity have remained largely unexplored. In this context, the recent experi- mental realization of SOC for ultra-cold atoms [7] pro- videsacompletelynewplatformforexploringmany-body phenomena in spin-orbit coupled superﬂuids, including both Bose-Einsteincondensates (BECs) [8, 9] and degen- erate Fermi gases [10, 11]. In the presence of SOC, vari- ousnew andexoticsuperﬂuid phenomenamayexist since ∗Electronic address: cwzhang@wsu.eduspins are not conserved during their motion. In particu- lar, in spin-orbit coupled BECs, the ground state phase diagrams as well as the collective excitations have been studied [8]. In spin-orbitcoupled degenerateFermi gases, the crossoverphysicsfromthe Bardeen-Cooper-Schrieﬀer (BCS) superﬂuids of loosely bounded Cooper pairs to the BEC of tightly bounded molecules has been inves- tigated extensively, in both uniform and trapped three- dimensional (3D) gases [10, 11]. However, the study of spin-orbit coupled two-dimensional (2D) Fermi gases is still lacked. Onthe otherhand, the 2DdegenerateFermigas(with- out SOC) in itself is one of the most active topics in ultra-cold atomic physics [12]. Experimentally, the 2D degenerate Fermi cold atomic gases have been realized recently using a highly anisotropic pancake-shaped po- tential [13], and the interaction energy in this system has been measured using the radio-frequency spectroscopy [14]. In this system, tunable interactions between atoms through Feshbach resonance [15] allow the exploration of the crossover physics from the BCS superﬂuids to the BEC of molecules in 2D [16]. In addition to the study of many-body physics such as quantum ﬂuctuations, non- Fermi liquid behavior, etc. [17], the 2D Fermionic cold atomic gases are particularly interesting because of the existence of exotic topological excitations such as Majo- rana fermions with non-Abelian exchange statistics [18]. Motivated by these recent experimental breakthroughs in the realization of SOC and 2D Fermi gases, in this paper, we investigate, both numerically and analytically, the BCS-BEC crossover physics in 2D degenerate Fermi gases in the presence of a Rashba type of SOC. Our main2 results are the following: 1) We derive the mean ﬁeld gap and atom number equations suitable for the 2D spin-orbit coupled Fermi gases. 2) Thesetwoequationsaresolvednumericallyand self- consistently, from which the dependence of the ground state properties (chemical potential, superﬂuid pairing gap, groundstateenergyperatom)onthesystemparam- eters (e.g., binding energy, SOC strength) is obtained. 3) We derive analytic expressions for these ground state physical quantities, which agree well with our nu- merical results within a broad parameter region. Such analytic expressions also agree qualitatively with previ- ous numerical results for the 3Dspin-orbit coupled Fermi gases [10], where analytic results are lacked. 4)We ﬁnd that with anincreasingSOCstrength α, the chemical potential is shifted by a constant −mα2deter- mined by α. The superﬂuid pairing gap and the ground state energy per atom are aﬀected at the order of α4 rather than α2, which means that weak SOC does not aﬀect the pairing gap and the ground state energy per atom. In the strong SOC regime, the pairing gap and the ground state energy are enhanced signiﬁcantly in the BCS limit, but only increase slightly in the BEC limit. Although these analytic results are obtained for the 2D Fermi gases, they also provide qualitative understanding for the numerical results in 3D Fermi gases [10], where similar changes of the chemical potential and superﬂuid pairing order with respect to the SOC strength are ob- served but analytic results are lacked. The paper is organized as follows. Section II describes the physical system: the spin-orbit coupled 2D degen- erate Fermi gases, and the corresponding Hamiltonian. In section III, we derive the mean ﬁeld gap and atom number equations. These equations are self-consistently solved in section IV, both numerically and analytically (through perturbative methods), to obtain the ground state properties (chemical potential, superﬂuid order pa- rameter, and ground state energy per atom) of the spin- orbit coupled Fermi gases in the BCS-BEC crossover. Section V consists of discussion and conclusion. II. THE PHYSICAL SYSTEM AND THE HAMILTONIAN The physical system in consideration is a 2D de- generate Fermi gas. Experimentally, the 2D degener- ate Fermi gas has been realized using a 1D deep op- tical lattice along the third dimension, where the tun- neling between diﬀerent layers is suppressed completely [13, 14]. The 1Doptical lattice potential V0sin2(2πz/λw) can be generated using two counter-propagating laser beams (parallel to the zaxis with a wavelength λw). In this case, the two-body binding energy is given byEb≃0.915/planckover2pi1ωLexp(√ 2πlL/as)/π, where ωL=/radicalbig 8π2V0/(mλ2w) is the eﬀective trapping frequency along thezaxis,lL=/radicalbig /planckover2pi1/(mωL), andasis the 3D s-wavescattering length [19]. Therefore the two-body binding energyEbcan be tuned by varying the s-wave scattering lengthasvia the Feshbash resonance for the study of the BCS-BEC crossover physics [16]. For a small attractive as→0−,Eb→0, correspondingtotheBCSlimit. While for a small repulsive as→0+,Eb→ ∞, corresponding to the BEC limit. When Ebincreases from 0 to ∞, the system evolves continuously from a BCS superﬂuid to a BEC of molecules. The SOC for cold atoms can be generated by the in- teraction between atoms and laser beams, as shown in many previous literatures [20], and demonstrated in a recent benchmark experiment [7]. In this paper, we con- sider only a Rashba type of SOC, and the eﬀects of other types of SOC (e.g. Dresselhaus or the combination of both) can be investigated similarly. For simplicity we consider a uniform Fermi gas and neglect the weak har- monic trap in the 2D plane, whose eﬀects can be incor- porated using the local density approximation. The Hamiltonian for this uniform 2D spin-orbit cou- pled degenerate Fermi gases can be written as H=HF+HI+Hsoc, (1) where HF=/summationdisplay k,σζkC† kσCkσ (2) is the single atom Hamiltonian, C† kσis the creation oper- ator for a Fermi atom with the momentum k= (kx,ky), σ=↑,↓are the pseudospins of atoms. ζk=ǫk−µwith the kinetic energy ǫk=k2/2m, the chemical potential µ, and the atom mass m. Henceforth, we take the Planck constant /planckover2pi1= 1. The s-wave scattering interaction be- tween atom HI=g/summationdisplay kC† −k↑C† k↓Ck↓C−k↑, (3) wheregis the eﬀective scattering interaction parameter. In a 2D Fermi gas, 1 g=−/summationdisplay k1 (2ǫk+Eb). (4) TheHamiltonianfortheRashbatypeofSOCforatoms can be written as Hsoc=α/summationdisplay k[(ky+ikx)C† k↑Ck↓+(ky−ikx)C† k↓Ck↑],(5) whereαis the SOC strength. In solid state materials, α is generally much smaller than KF/(2m) withKFas the Fermi vector. However, in ultra-cold neutral atoms, α can reach the order of KF/(2m) [20]. Such strong SOC, together with tunable interactions through the Feshbach resonance, may yield some exotic many-body phenom- ena that have not been explored in solid state systems. Recently, a generalized SOC with the Rashba and the Dresselhaus terms in the Hamiltonian (1) has been con- sidered [11].3 III. MEAN FIELD GAP AND ATOM NUMBER EQUATIONS As the ﬁrst step for the eventual understanding of the 2D spin-orbit coupled Fermi gas, we consider the zero temperature superﬂuid physics under the mean ﬁeld ap- proximation [16]. Generally, the mean ﬁeld approxima- tion can give qualitatively but not quantitatively correct results [21]. In the mean ﬁeld approximation, the super- ﬂuid order parameter is taken as ∆ =g/summationdisplay k/angbracketleftCk↓C−k↑/angbracketright. (6) With this pairing order parameter, we can rewrite the two-body interaction Hamiltonian (3) as HI=−∆2/g+∆/summationdisplay k(Ck↓C−k↑+C† −k↑C† k↓).(7) Therefore the total Hamiltonian can be rewritten as HB=1 2/summationdisplay kΨ†(k)MkΨ(k)−∆2 g+/summationdisplay kζk(8) under the Nambu spinor basis Ψ( k) = (Ck↑,Ck↓,C† −k↓,−C† −k↑)T, where Bogoliubov-de-Gennes operator Mk= ζkαk+∆ 0 αk−ζk0 ∆ ∆ 0 −ζk−αk+ 0 ∆ −αk−−ζk (9) preserves the particle-hole symmetry, k±=ky±ikx. The quasiparticle excitation spectrum Eλ k,±=λ/radicalbig (ǫk−µ±αk)2+∆2 (10) is the eigenvalue of the matrix Mk,λ=±correspond to the particle and hole branches of the spectrum. For each branch, there are two diﬀerent excitations due to the existence of the SOC. From Eq. (8), we see the total ground-state energy is EG=−∆2 g+/summationdisplay k[ζk−1 2(E+ k,++E+ k,−)].(11) Without the SOC, the term ( E+ k,++E+ k,−)/2 reduces to the well-known form Ek=/radicalbig (ǫk−µ)2+∆2. (12) in the BCS theory. The ground-state properties of the 2D spin-orbit cou- pled Fermi gases can be obtained from the atom number equation n=−∂EG ∂µ=/summationdisplay k[1+1 2(∂E+ k+ ∂µ+∂E+ k− ∂µ)],(13) and the superﬂuid gap equation ∂EG ∂∆=/summationdisplay k[∆ 2ǫk+Eb−1 4(∂E+ k+ ∂∆+∂E+ k− ∂∆)] = 0.(14)IV. GROUND STATE PROPERTIES A. Numerical results We numerically solve the above atom number equa- tion (13) and the gap equation (14) self-consistently to obtain various ground state quantities. In Fig. 1, we plot the dependence of the ground state quantities: the chemical potential µ, the superﬂuid order parameter ∆, and the ground state energy per atom E=EG/n, on the physical parameters: the SOC strength αand the binding energy Eb. We see that the chemical potential µ decreaseswiththeincreasingspin-orbitcouplingstrength α. With increasing binding energy, the chemical poten- tial also decreases, signaling the crossover physics from the BCS superﬂuids to the BEC molecules. One inter- esting feature shown in Fig. 1(a2) is that the shift of the chemical potential induced by the SOC depends only on the SOC strength. In Fig. 1(b1), we see the superﬂuid order parameter ∆ increases with increasing α. Here ∆ 0is the superﬂuid order parameter without SOC. For a small α, the change of ∆/∆0is very small. The growth of ∆ becomes signiﬁ- cant only when αKFis large than the Fermi energy EF. -1.2-0.8-0.400.40.81.2µ (EF) -4.0-3.0-2.0-1.00.01.0 0.951.001.051.101.151.20∆/∆0 1.001.011.02 0.0 0.5 1.0 1.5 2.0 αKF (EF)-0.6-0.4-0.20.00.2E (EF) 0 2 4 6 8 10 Eb (EF)-0.50-0.49-0.48-0.47-0.46-0.45(a1) (a2) (b1) (b2) (c1) (c2)E b = 0.1 αK F = 0.5E b = 1.0αK F = 1.0 Figure 1: (Color online) (a1-c1) Plot of the chemical poten- tialµ, the dimensionless superﬂuid pairing gap ∆ /∆0, and the ground-state energy per atom Ewith respect to the SOC strength αfor the two-body binding energy Eb= 0.1EF (Black line) and 1 .0EF(Red line). (a2-c2) Plot of µ, ∆/∆0, andEwith respect to EbforαKF= 0.5EF(Black line) and 1.0EF(Red line). In all ﬁgures, the solid lines represent the approximate analytical results obtained in the paper, whil e the open symbols correspond to the exact numerical results.4 Furthermore, the eﬀects of the SOC are clearly seen for a smallEb(the BCS side), but not important for a large Eb(the BEC side). The ground state energy per atom Ehas a similar be- havior as ∆ /∆0. It increases with α, but the growth is only important for a large α. Without SOC E=−1 2EF, and the SOC induces a small correction. The change of Eis signiﬁcant in the BCS side (small Eb), but negligible in the BEC side (large Eb). We note that similar changes ofthe chemical potential and the superﬂuid orderparam- eter have been observed numerically in a 3D spin-orbit coupled Fermi gases. All these numerical observed phe- nomena will be explained in the next subsection where the analytic expressions for these ground state quantities are derived. B. Analytic results Although the above numerical results give the depen- dence of the ground state quantities on the SOC strength for certain parameters, analytic results are desired for a better understanding of the underlying physics. In the following, we present a perturbative method (with αas the small parameter) to analytically extract the funda- mental ground-state properties of the spin-orbit coupled Fermi gases. For this purpose, we rearrange Eq. (11) into two parts EG=E0+Esoc (15) with E0=−∆2 g+/summationdisplay k(ζk−Ek) (16) as the ground-state energy without SOC. For a 2D Fermi gas,E0can be obtained exactly, yielding [16] E0=m 4π[∆2ln(/radicalbig µ2+∆2−µ Eb)−µ(/radicalbig µ2+∆2+µ)−∆2 2]. (17) The second term in Eq. (15) is given by Esoc=/summationdisplay k[Ek−1 2(E+ k,++E+ k,−)],(18) which describes the contribution from the SOC. Because it is diﬃcult to derive a simple analytic ex- pression for Esocdirectly, we ﬁrst perform a Taylor ex- pansion with respect to the SOC strength, and then do the summation over k. Formally, Esoccan be written as Esoc=m 4πCi(µ,∆)ηi(19) withη=mα2/2. The coeﬃcients Ci(µ,∆) can be ob- tained, in principle, for any order. Here, we only give theﬁrst six orders C1=−4(/radicalbig µ2+∆2+µ), (20) C2=−8 3(1+µ/radicalbig µ2+∆2), (21) C3=−16∆2 15(µ2+∆2)3/2, (22) C4=32µ∆2 35(µ2+∆2)5/2, (23) C5=64∆2(∆2−4µ2) 315(µ2+∆2)7/2, (24) C6=−128µ∆2(3∆2−4µ2) 693(µ2+∆2)9/2. (25) Although the expression for Esocis very complicate, the expressions for the chemical potential µand the su- perﬂuid pairing order ∆ are very simple, as we will show later in the paper. With the ground-state energy EG, the superﬂuid pair order and the chemical potential can be derived by self-consistently solving the corresponding gap and number equations ln(/radicalbig µ2+∆2−µ Eb) =G∆(η,∆,µ), (26) /radicalbig µ2+∆2+µ= 2EF−Gµ(η,∆,µ) (27) analytically, where G∆(η,∆,µ) =/summationtext i∂Ci(µ,∆) ∂∆ηi, and Gµ(η,∆,µ) =/summationtext i∂Ci(µ,∆) ∂µηi.EFis the Fermi energy for a 2D non-interacting Fermi gas without SOC and with the density n=mEF/π. Without SOC ( η= 0), G∆(η,∆,µ) =Gµ(η,∆,µ) = 0, and Eqs. (26) and (27) become /radicalbig µ2+∆2−µ=Eb,/radicalbig µ2+∆2+µ= 2EF. Inthis case,the superﬂuidpairingorderandthe chemical potential are given exactly by ∆ 0=√2EbEFandµ0= EF−Eb/2 [16], as expected. In the presence of SOC ( η/negationslash= 0), the nonlinear equa- tions (26) and (27) cannot be solved exactly. However, approximate solutions can be derived for the physical parameters within current experimentally achievable re- gion. Since the ground-state energy depends on ∆2, the solutions of Eqs. (26) and (27) can be assumed to be µ=/summationtext i=0µiηiand ∆2=/summationtext i=0Λiηi. Substituting these expressions into the nonlinear equations (26) and (27) and then comparing the coeﬃcients for the same order of ηi, we obtain Λ iandµirespectively. 1. Chemical potential After a straightforwardbut tedious calculation, the co- eﬃcients for the chemical potential are given by µ1=5 −2,µ2= 4Eb/[3(Eb+ 2EF)2] andµ3=−64Eb(Eb− 4EF)/[15(Eb+ 2EF)4]. Note that the second order µ2 is already very small for all diﬀerent values of Ebwhen η < EF, therefore the high-order terms do not aﬀect the chemical potential signiﬁcantly. The chemical potential can then be written as µ≃EF−Eb 2−2η. (28) From Fig. 1a, we see Eq. (28) agrees well with the exact values of the chemical potential obtained from numeri- cally solving Eqs. (13) and (14) self-consistently. Equation (28) shows clearly that with the increasing SOC strength α, the chemical potential µis decreased by 2η=mα2, as shown in Fig. 2. If we deﬁne an eﬀec- tive chemical potential µ=µ+2η=µ+mα2, Eq. (28) can be rewritten as µ=EF−Eb/2. Similar to the previ- ous discussion without SOC, we ﬁnd that in the asymp- totic BCS limit with a weak bound state ( Eb≪EF), the chemical potential µ≃EF. However, in the deep BEC regime with a strong bound state ( Eb≫EF), the chemical potential µ=−Eb/2. The BCS-BEC crossover may occur around the crossover point µ= 0 [16]. Fur- thermore, in the presence of SOC, Eq. (28) can also be written as µ=µf+µ∆, where µf=EF−2η is the chemical potential for the free Fermi gas and µ∆=−Eb/2 +/summationtext i=1µ∆i(Eb)ηireﬂects the revision of the chemical potential induced by the two-body binding energyEb. Hereµ∆1= 0 implies that the binding energy Ebhas no eﬀect on the chemical potential µat the order ofη=mα2/2. It is important to point out that although the chemi- cal potential µonly has a simple shift from that without SOC, the underlying physics is quite diﬀerent. Without SOC, the pairing wave function is simply singlet. How- ever, in the presence of SOC, each energy band contains both spin up and down components. As a result, the pairing wave function has a more complicated structure withbothsingletandtripletcomponents[22]. Thetriplet pairing correlations in s-wave superﬂuids may be used todetecttheanisotropicFulde-Ferrell-Larkin-Ovchinikov states[23]. Inaddition, the2DFermigaseswithSOCcan /s48/s107/s40/s98/s41 /s107/s69/s40/s107/s41 /s48/s69/s40/s107/s41 /s40/s97/s41 /s61/s109 Figure 2: (Color online) (a) The chemical potential without SOC in the BCS limit. (b) The chemical potential with SOC in the BCS limit. For weak SOC, their diﬀerence ∆ µ=mα2.exhibitp-wave character in the helicity bases. If a Zee- man ﬁeld in Hamiltonian (8) is added, a novel topolog- ical phase transition from non-topological superﬂuids to topologicalsuperﬂuidscanbe induced. Inthe topological superﬂuid phase, there exist Majorana fermions and the associated non-Abelian statistics, which are the critical ingredients for implementing topological quantum com- puting [18]. 2. Superﬂuid order gaps The eﬀects of SOC are more interesting for the super- ﬂuid pairing gap ∆. Applying the same procedure as that for the chemical potential, we ﬁnd ∆2≃2EbEF+16EbEF 3(Eb+2EF)2η2.(29) Thereisno ﬁrst-ordercorrectionwith respect to η(∼α2) for ∆2, and the second-order η2(∼α4) is the leading cor- rection. Moreover,thesecond-ordercoeﬃcient ∂2∆2/∂η2 is alwayspositive and hasa maximum 4 η2/3(=m2α4/3) whenEb= 2EF. The high-order coeﬃcients for η3and η4are Λ3=−512EbEF(Eb−EF)/[15(Eb+2EF)4] and Λ4= 64EbEF(1017E2 b−3508EbEF+1076E2 F)/[315(Eb+ 2EF)6] respectively. In order to see the eﬀect of SOC on the superﬂuid pairing gap more clearly, we introduce a dimensionless quantity ∆d=∆ ∆0≃/radicaligg 1+8 3(Eb+2EF)2η2,(30) where ∆ 0is the superﬂuid pairing gap without SOC. In the asymptotic BCS limit with a weak bound state (Eb≪EF), ∆d≃/radicalbig 1+2η2/E2 F. For a weak SOC (η≪EF) in typical solid-state materials, ∆ d≃1, which means that the SOC does not aﬀect the super- ﬂuid pairing gap signiﬁcantly. However, for a strong SOCη∼EFthat has been achieved for ultracold atoms [7], this gap can be enhanced greatly. In the deep BEC regime with a strong binding energy ( Eb≫EFand Eb≫η), ∆d≃/radicalbig 1+8η2/(3E2 b), therefore the super- ﬂuid pairing gap increases only slightly with the increas- ing SOC strength. These analytic results agree well with the numericalresults shownin Fig. 1b. Note that similar behavior for ∆ dis also observed in the numerical results in 3D [10]. 3. Ground-state energy per atom IntermsofEqs. (28)and(29), theground-stateenergy per Fermi atom E=EG/ncan be obtained E≃ −1 2EF+8EF 3(Eb+2EF)2η2. (31)6 The comparison of Eq. (31) with the direct numerical simulation results is shown in Fig. 1c. The ground- state energy, like the superﬂuid pairing gap, depends on η2(∼α4). In the asymptotic BCS limit ( Eb≪EF), E≃ −EF/2+2η2/(3EF), which means that the ground- state energy can be enhanced signiﬁcantly only for a largeη. In the deep BEC regime ( Eb≫EFand Eb≫η),E≃ −EF/2 + 8EFη2/(3E2 b), therefore the ground-state energy only increases slightly even when η∼EF. In addition, the high-order coeﬃcients for η3 andη4are given by E3=−128EbEF/[5(Eb+ 2EF)4] andE4= 128EbEF(491Eb−538EF)/[315(Eb+ 2EF)6] respectively. Note that although such mean ﬁeld the- ory gives qualitatively correct results, it may not agree quantitatively with the experimental results in the deep BEC regime as shown in recent Monte Carlo numerical simulation for the 2D Fermi gases without SOC [21]. Finally, we want to remarkthat if the high-orderterms in the chemical potential µ, the superﬂuid pairing gap ∆, and the ground-state energy per Fermi atom E=EG/n are included, the analytical results in Eqs. (28), (30) and (31) ﬁt better with the numerical results even for EF< αKF<3.0EF. V. DISCUSSION AND CONCLUSION The mean ﬁeld zero temperature BCS-BEC crossover physics discussed above provides the ﬁrst critical step towards the understanding of the 2D spin-orbit coupled degenerate Fermi gases. Clearly, many issues need be further explored in the future, as demonstrated by the development along this direction after the initial submis- sionofourpaper. Inparticular,theﬁnitetemperatureef- fect need be taken into account in a realistic experiment. Without SOC, it is known that there are no superﬂu- ids for 2D degenerate Fermi gas at a ﬁnite temperature[24], where relevant physics is the Berezinskii-Kosterlitz- Thouless(BKT)transition[25], leadingto the generation of the vortex-antivortex pairs. Recent some interesting eﬀects of the SOC on the BKT transition [26] has been investigated. In experiments, the Fermi gases are con- ﬁned in a 2D harmonic trap, whose eﬀects may be taken into account using the local density approximation, as shown in the recent preprint [26]. The trap geometry for the Fermi gases may also be quais-2D instead of the strict 2D. In this case, the conﬁnement along the third direction may aﬀect the critical transition temperature, which need be further explored. In summary, motivated by the recent experimental breakthrough for the realization of the SOC for cold atoms and the 2D degenerate Fermi gas, we investigate the zero temperature BCS-BEC crossover physics in 2D spin-orbitcoupleddegenerateFermigasesusingthemean ﬁeld approximation. By solving the corresponding gap and atom number equations both numerically and an- alytically, we reveal the ground state properties of the spin-orbit coupled 2D Fermi gases. Our analytic results agreequantitativelywith ournumericalresults in 2Dand qualitatively with previous numerical results for the 3D spin-orbit coupled Fermi gases, where analytic results are lacked. The analytic expressions for various physi- cal quantities may provide a powerful tool for engineer- ing and probing many new topological phenomena in 2D Fermi gases, including the intriguing Majorana physics and the associated non-Abelian statistics. Acknowledgements We thank Yongping Zhang, Li Mao, and Wei Yi for helpful discussion. This work is supported by DARPA-YFA (N66001-10-1-4025),ARO (W911NF-09-1- 0248), NSF (PHY-1104546), and AFOSR (FA9550-11-1- 0313). GC is also supported by the 973 program under Grant No. 2012CB921603, the NNSFC under Grant No. 11074154, and the ZJNSF under Grant No. Y6090001. [1] N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Rev. Mod. Phys. 82, 1539 (2010). [2] D. Xiao, M. -C. Chang, and Q. Niu, Rev. Mod. Phys. 82, 1959 (2010). [3] X. -L. Qi, and S. -C. Zhang, Rev. Mod. Phys. 83, 1057 (2011) [4] I.˘Zuti´ c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). [5] V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum electrodynamics , (Butterworth- Heinemann, 1982). [6] A. J. Leggett, Quantum Liquids: Bose condensation and Cooper pairing in condensed-matter systems (Oxford University Press Inc., New York, 2006). [7] Y.-J.Lin, K.Jim´ enez-Garc´ ıa, andI.B.Spielman, Natu re (London) 471, 83 (2011); Y. -J. Lin, R. L. Compton1, K. Jim´ enz-Garc´ ıa, J. V. Porto, and I. B. Spielman, Nature (London) 462, 628 (2009). [8] C. Wang, C. Gao, C. -M. Jian, and H. Zhai, Phys. Rev.Lett.105, 160403 (2010); T.-L. Ho and S. Zhang, Phys. Rev. Lett. 107, 150403 (2011); Y. Zhang, L. Mao, and C. Zhang, arXiv:1102.4045; H. Hu, H. Pu, and X.-J.Liu, arXiv: 1108.4233. [9] S. K. Yip, Phys. Rev. A 83, 043616, (2011); Z. F. Xu, R. L¨ u, and L. You, Phys. Rev. A 83, 053602 (2011); T. Kawakami, T. Mizushima, and K. Machida, Phys. Rev. A84, 011607(R) (2011); C. Jian and H. Zhai, Phys. Rev. B84, 060508 (2011); C. Wu, I. Mondragon-Shem, and X.-F. Zhou, Chin. Phys. Lett. 28, 097102 (2011). [10] J. P. Vyasanakere, S. Zhang, and V. B. Shenoy, Phys. Rev. B84, 014512 (2011); M. Gong, S. Tewari, and C. Zhang, Phys. Rev. Lett. 107, 195303 (2011); Z. -Q. Yu, and H. Zhai, Phys. Rev.Lett. 107, 195305 (2011); H. Hu, L. Jiang, X. -J. Liu, and H. Pu, Phys. Rev. Lett. 107, 195304 (2011); M. Iskin, and A. L. Subasi, Phys. Rev. Lett.107, 050402 (2011); Phys. Rev. A 84, 041610 (R) (2011); Phys. Rev. A 84, 043621 (2011); W. Yi, and G. -C. Guo, Phys. Rev. A 84, 031608 (2011); L. Han, and7 C. A. R. S´ a de Melo, arXiv:1106.3613. [11] L. Dell’Anna, G. Mazzarella, and L. Salasnich, Phys. Rev. A84, 033633 (2011). [12] D. S. Petrov, M. Holzmann, and G. V. Shlyapnikov, Phys. Rev. Lett. 84, 2551 (2000); S. S. Botelho, and C. A. R. S´ a de Melo, Phys. Rev. Lett. 96, 040404 (2006); W. Zhang, G. -D. Lin, and L. -M. Duan, Phys. Rev. A 77, 063613 (2008); Phys. Rev. A 78, 043617 (2008); I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80, 885 (2008). [13] K. Martiyanov, V. Makhalov, and A. Turlapov, Phys. Rev. Lett. 105, 030404 (2010). [14] B. Fr¨ ohlich, M. Feld, E. Vogt, M. Koschorreck, W. Zw- erger, and M. K¨ ohl, Phys. Rev. Lett. 106, 105301 (2011). [15] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Rev. Mod. Phys. 82, 1225 (2010). [16] M. Randeria, J. -M. Duan, and L. -Y. Shieh, Phys. Rev. Lett.62, 981 (1989); Phys. Rev. B 41, 327 (1990). [17] S. Sachdev, Science 288, 475 (2000); S. E. Korshunov, Phys. Usp. 49, 225 (2006). [18] S. Tewari, S. Das Sarma, C. Nayak, C. Zhang, and P. Zoller, Phys. Rev. Lett. 98, 010506 (2007); C. Zhang, S. Tewari, R. M. Lutchyn, and S. Das Sarma, Phys. Rev. Lett.101, 160401 (2008); C. Nayak, S. H. Simon, A. Stern, M, Freedman, and S. D. Sarma, Rev. Mod. Phys.80, 1083 (2008). [19] J. Tempere, M. Wouters, and J. T. Devreese, Phys. Rev. B75, 184526 (2007). [20] J. Ruseckas, G. Juzeli¯ unas, P. ¨Ohberg, and M. Fleis- chhauer, Phys. Rev. Lett. 95, 010404 (2005); T. D. Stanescu, C. Zhang, andV. Galitski, Phys.Rev.Lett. 99, 110403 (2007); C. Zhang, Phys. Rev. A 82, 021607(R) (2010); G. Juzeli¯ unas, J. Ruseckas, and J. Dalibard, Phys. Rev. A 81, 053403 (2010); J. D. Sau, R. Sensarma, S. Powell, I. B. Spielman, and S. Das Sarma, Phys. Rev. B83, 140510 (2011); D. L. Campbell, G. Juzeli¯ unas, and I. B. Spielman, Phys. Rev. A 84, 025602 (2011). J. Dal- ibard, F. Gerbier, G. Juzeliunas, and P. Ohberg, Rev. Mod. Phys. (accepted for publications) [21] G. Bertaina, and S. Giorgini, Phys. Rev. Lett. 106, 110403 (2011). [22] L. P. Gor’kov, and E. I. Rashba, Phys. Rev. Lett. 87, 037004 (2001). [23] I. Zapata, F. Sols, and E. Demler, arXiv:1106.1605. [24] P. C. Hohenberg, Phys. Rev. 158, 383 (1967). [25] V. L. Berezinskii, Sov. Phys. JETP 32, 493 (1971); J. M. Kosterlitz and D. Thouless, J. Phys. C 5, L124 (1972). [26] L. He, and X. -G. Huang, arXiv:1109.5577.
0709.3429v2.Weak_and_strong_coupling_limits_of_the_two_dimensional_Fröhlich_polaron_with_spin_orbit_Rashba_interaction.pdf	arXiv:0709.3429v2 [cond-mat.str-el] 25 Jan 2008Weak and strong coupling limits of the two-dimensional Fr¨ o hlich polaron with spin-orbit Rashba interaction C. Grimaldi Max-Planck-Institut f¨ ur Physik komplexer Systeme, N¨ oth nitzer Srt.38, D-01187 Dresden Germany LPM, Ecole Polytechnique F´ ed´ erale de Lausanne, Station 1 7, CH-1015 Lausanne, Switzerland The continuous progress in fabricating low-dimensional sy stems with large spin-orbit couplings has reached a point in which nowadays materials may display s pin-orbit splitting energies ranging from a few to hundreds of meV. This situation calls for a bette r understanding of the interplay between the spin-orbit coupling and other interactions ubi quitously present in solids, in particular when the spin-orbit splitting is comparable in magnitude wi th characteristic energy scales such as the Fermi energy and the phonon frequency. In this article, the two-dimensional Fr¨ ohlich electron-p honon problem is reformulated by introduc- ing the coupling to a spin-orbit Rashba potential, allowing for a description of the spin-orbit eﬀects on the electron-phonon interaction. The ground state of the resulting Fr¨ ohlich-Rashba polaron is studied in the weak and strong coupling limits of the electro n-phonon interaction for arbitrary val- ues of the spin-orbit splitting. The weak coupling case is st udied within the Rayleigh-Schr¨ odinger perturbation theory, while the strong-coupling electron- phonon regime is investigated by means of variational polaron wave functions in the adiabatic limit. It is found that, for both weak and strong coupling polarons, the ground state energy is systematical ly lowered by the spin-orbit interaction, indicating that the polaronic character is strengthened by the Rashba coupling. It is also shown that, consistently with the lowering of the ground state, th e polaron eﬀective mass is enhanced compared to the zero spin-orbit limit. Finally, it is argued that the crossover between weakly and strongly coupled polarons can be shifted by the spin-orbit i nteraction. PACS numbers: 71.38.-k, 71.38.Fp, 71.70.Ej I. INTRODUCTION The Fr¨ ohlich Hamiltonian describing a single electron coupled to longitudinal optical phonons is a paradig- matic model of the electron-phonon (el-ph) interaction,1 and has represented in the past, in addition to its in- terest for the solid-state physics, an ideal problem for testing many mathematical methods in quantum ﬁeld theory.2Because of the coupling with the phonon ﬁeld, the resulting quasi-particle, the polaron, has an eﬀec- tive mass larger, and a ground state energy lower, than the free electron. These quantities have been investi- gated for the three-dimensional (3D) case by means of perturbation theory for the weak-coupling limit,3and of variational treatments for the intermediate,4and strong- coupling cases.5,6The path-integral variational calcula- tions of Feynman,7and subsequent reﬁnements of this method,8have provided a solid description for all val- ues of the coupling, veriﬁed also by improved variational methods,9and by quantum Monte-Carlo studies.10,11 The interest aroused some time ago on semiconduc- tor heterojunctions, or other low-dimensional systems, prompted to modify the Fr¨ ohlich model to accounting for two-dimensional(2D) and quasi-2Dsystems.12By apply- ingthe samemethodsderivedforthe3Dcase, theground statepropertiesforthestrictly2Dcasewereevaluatedfor weak, strong and intermediate couplings,13,14,15,16and the obtained systematic lowering of the ground state en- ergy and the enhancing of the eﬀective mass compared to the 3D case has pointed out the role of dimensionality in enhancing the polaronic character.12,17Concerning the el-ph problem in low dimensions, recent progresses in developing high-quality low- dimensional systems and in material engineering provide hints that, for a vast class of low-dimensional materi- als, the usual 2D Fr¨ ohlich model, as considered in lit- erature, may be incomplete. This concern comes about by considering 2D systems exhibiting strong spin-orbit (SO) splitting of the electronic states due to the inver- sion asymmetry in the direction orthogonal to the con- ducting plane (Rashba SO mechanism). This situation is encountered in semiconductor quantum wells with asym- metricconﬁningpotentials,18inthesurfacestatesofmet- als and semi-metals,19,20,21and in surface alloys such as Li/W(110),22Pb/Ag(111),23,24and Bi/Ag(111),25with SO splitting energies ranging from a few meV in GaAs quantum wells to about 0.2 eV in Bi/Ag(111).25In these systems, therefore, the SO energy may be of the same order or even much larger than the typical phonon fre- quency, rising the question of how such state of aﬀair aﬀects the el-ph interaction, in general, and the Fr¨ ohlich coupling, in particular. As pointed out in several works,26,27,28,29,30,31the Rashba interaction describing the SO coupling can have profound eﬀects on the low energy properties of the itin- erant electrons. Namely, in the low density regime, the Rashba SO coupling induces a topological change of the Fermi surface of the free electrons, leading to an eﬀective reduction of the dimensionality in the electronic density of states (DOS). In this situation, a 2D low density elec- tron gas would develop, in the presence of SO Rashba coupling, a phenomenology similar to one-dimensional2 (1D) systems, triggered by the square-root divergence of the (eﬀectively 1D) DOS at low energies. Some interesting consequences of this scenario on the el-ph problem have already been discussed in Ref.[30], concerning the superconducting transition, and in Ref.[31] for the eﬀective mass and the spectral proper- ties. The picture arising from these works, although be- ing limited to the momentum-independent Holstein el-ph interaction and to weak-to-moderate couplings, conﬁrms that, for suﬃciently low electron densities, the coupling tothephononsisampliﬁedbytheSOinteractionthrough the 1D-like divergence of the DOS. Notwithstanding the relevance of these results for the Holstein model, the use of a local el-ph interaction may however result inadequate in the extremely low elec- tron density regime, where the SO eﬀects are more evident.30,31Indeed, the lack of eﬀective screening in this case would rather suggest a long-range interaction as be- ing a more appropriate description of the el-ph coupling. It is therefore natural to consider the 2D Fr¨ ohlich po- laron, and its coupling to the SO interaction, as a model better describing the unscreened el-ph interaction in 2D Rashba systems in the low density limit. Inthisarticle,asingleelectronmovingwithaparabolic dispersion in the two-dimensional x-yplane is coupled simultaneously to the Rashba SO potential and to the phonon degrees of freedom through a Fr¨ ohlich interac- tion term. The total system is then described by the 2D Fr¨ ohlich-Rashba Hamiltonian H=Hel+Hph+Hel−ph, where (/planckover2pi1= 1) Hel=p2 2m+Ω(p)·σ (1) is the Hamiltonian for an electron with mass mand mo- mentum operator p=−i∇with components ( px,py,0), σis the spin-vector operator with components given by the Pauli matrices, and Ω(p) is the SO vector ﬁeld which in the case of Rashba coupling reduces to: Ω(p) =γ −py px 0 , (2) whereγis the SO coupling parameter. The phonon part of the Hamiltonian is given by Hph=ω0/summationdisplay qa† qaq, (3) wherea† q(aq) is the creation (annihilation) operator for a phonon with momentum q= (qx,qy) and optical fre- quencyω0. The el-ph interaction Hamiltonian for the 2Delectroncoupledtolongitudinaloptical(LO)phonons is:12,14 Hel−ph=1√ A/summationdisplay q1√q(M0eiq·raq+M∗ 0e−iq·ra† q) (4)with M0=iω0/parenleftbigg2π2α2 mω0/parenrightbigg1/4 , (5) whereα=e2(ǫ−1 ∞−ǫ−1 0)/radicalbig m/2ω0is the dimensionless el- ph coupling constant, with ebeing the electron charge, andǫ∞andǫ0the high frequency and static dielectric constants, respectively. It is worth clarifying here the signiﬁcance of the 2D Fr¨ ohlich interaction of Eq.(4) with respect to the char- acteristics of speciﬁc materials. For quantum wells and 2D heterostructures, where the electron wave function is assumed here to be conﬁned in a sheet of zero thickness, Eq.(4) describes the coupling of the electron to bulk LO phonons, while the coupling to interface phonon modes is neglected. The inclusion of such interface phonon contri- butions may be important in describing speciﬁc materi- als, but it is unnecessary for the present study, where the focus is on the SO eﬀects on the unscreened (long-range) el-ph interaction, for which Eq.(4) is a paradigm for the 2D case. Concerning the el-ph coupling of electronic sur- face states, Eq.(4) coincides (apart from a redeﬁnition ofM0) with the coupling to 2D surface phonons when the coupling to bulk phonons extending below the sur- face is negligible.32Such approximation is coherent with the ideal 2D assumption for the electron wave function, which is physically realized when the electronic surface stateshavenegligiblecouplingtothe bulk. Afurther mo- tivation of using the 2D Fr¨ ohlich model (4) is that, in the absence of SO interaction, the ground state polaron en- ergyEPand eﬀective mass m∗havealready been studied by several authors,12,13,14,15,16and the exact results ob- tained for the weak ( α≪1) and strong ( α≫1) coupling limits provide a useful reference for the eﬀect of nonzero SO coupling. In the present work, the 2D Fr¨ ohlich-Rashba Hamil- tonian is studied by considering the weak and strong coupling limits of the el-ph interaction, with arbitrary strength of the SO coupling γ. Forα≪1 the polaron energyEPand the eﬀective mass m∗are obtained from second order perturbation theory in Sec.II, where nu- merical and exact analytical results are presented. It is shown that the eﬀect of γ∝negationslash= 0 is qualitatively similar to that observed in the Holstein model,30,31namely, the SO coupling enhances the eﬀective coupling to the phonons. Inparticular, EPisloweredby γand, simultaneously, the eﬀective mass m∗is enhanced. In Sec.III the strong cou- pling limit α≫1 is treated by the variational method, providing a rigorous upper bound of the ground state energy for arbitrary values of the SO interaction. As for the weak el-ph coupling case, it is found that EP(m∗) is lowered (enhanced) by the SO interaction, implying that the Rashba coupling always ampliﬁes the polaronic char- acter, regardless of whether the el-ph interaction is weak or strong.3 II. WEAK COUPLING In the presence of SO interaction, the electron wave function is a spinor and its Green’s function is conve- niently represented by a 2 ×2 matrix in the spin sub- space. For α= 0 the free electron Green’s function G0 is readily obtained from Hel: G0(k,ω) =/parenleftbigg ω−k2 2m−Ω(k)·σ/parenrightbigg−1 =1 2/summationdisplay s=±[1+sˆΩ(k)·σ]Gs 0(k,ω),(6) wherekis a 2D electron wavenumber, ˆΩ(k) = Ω(k)/\|Ω(k)\|and Gs 0(k,ω) =1 ω−k2/2m−sγk(7) is the free electronpropagatorfor the two( s=±1)chiral states characterized by two distinct bands with shifted parabolic dispersions k2/2m±γk. The lowest band has its minimum value −E0atk=k0, wherek0andE0are the Rashba momentum and energy deﬁned respectively by: k0=mγ, E 0=m 2γ2. (8) For later convenience, it is useful to express the electron energy relative to E0, so that the poles of Eq.(7) appear at energies: E±(k) =1 2m(k±k0)2. (9) The free electron ground state is then given by the elec- tron occupying the lower band at wavenumber k=k0 with energy ω= 0. In the weak el-ph coupling limit ( α≪1) the ground state properties are obtained by the electron self-energy evaluated in the second order perturbation theory. At zerotemperature, the resultingsingleelectronself-energy is therefore: Σ(k,ω) =\|M0\|2/integraldisplaydk′ (2π)21 \|k−k′\|G0(k′,ω−ω0).(10) Because of the momentum dependence of the Fr¨ ohlich interaction, and contrary to the Holstein el-ph case con- sidered in Ref.[31], the self-energy is not diagonal in the spin subspace. However, since the momentum depen- dence enter only through the modulus of the momentum transfer, equation (10) can be rewritten in a quite simple form. By using ( ˆΩ(k)·σ)2= 1 and (ˆΩ(k)·σ)(ˆΩ(k′)·σ) =ˆk·ˆk′+(ˆk×ˆk′)σxσy,(11) then the quantity ˆΩ(k′)·σappearing in Eq.(10) through G0(k′,ω−ω0) can be replaced simply by ( ˆΩ(k)·σ)ˆk·ˆk′because the second term of Eq.(11) vanishes after the integration over k′. In this way, the resulting self-energy reduces to: Σ(k,ω) = Σd(k,ω)1+Σo(k,ω)ˆΩ(k)·σ,(12) where1is the unit matrix and Σ dand Σ oare, respec- tively, the diagonal and oﬀ-diagonal contributions to the self-energy, both depending solely on the modulus of k.33 Their explicit expressions are: Σd(k,ω)=\|M0\|2 2/integraldisplaydk′ (2π)2/summationdisplay s1 \|k−k′\|1 ω−ω0−Es(k′), (13) Σo(k,ω)=\|M0\|2 2/integraldisplaydk′ (2π)2/summationdisplay s1 \|k−k′\|sk·k′ ω−ω0−Es(k′). (14) In the limit of zero SO coupling, since Es(k)→k2/2m, Σo(k,ω) vanishes because of the summation over s= ±1 in Eq.(14). Notice also that, independently of γ, Σo(k,ω) = 0 when the factor 1 /\|k−k′\|in Eq.(14) is re- placed by a constant, as in the momentum-independent Holstein el-ph coupling model. By using Eq.(12) the Dyson equation for the interact- ing propagator Greduces to G−1(k,ω) =G−1 0(k,ω)−Σ(k,ω) =ω−k2 2m−Σd(k,ω)−E0 −[γk+Σo(k,ω)]ˆΩ(k)·σ,(15) and the poles ω±ofGare then given by: ω±=E±(k)+Σd(k,ω±)±Σo(k,ω±).(16) Now, the Rayleigh-Schr¨ odinger perturbation theory per- mits to evaluate the lower energy pole ω−at the lowest order in the el-ph coupling α. This is accomplished by replacingω−by the unperturbed energy E−(k) in the energy variables of Σ dand Σ o. In this way, the lower pole reduces to ω−=E−(k)+Σ−(k)+O(α2), where Σ−(k) = Σd(k,E−(k))−Σo(k,E−(k)).(17) Finally, by expanding Σ −(k) up to the second order in k−k0, the polaron dispersion in the vicinity of k0can be written as: ω−=EP+1 2m∗(k−k∗ 0)2, (18) where the polaron ground-state energy EP, the eﬀective massm∗, and the eﬀective Rashba momentum k∗ 0are4 given respectively by: EP= Σ−(k0)−m∗ 2Σ′ −(k0)2 = Σ−(k0)+O(α2), (19) m∗ m= [1+mΣ′′ −(k0)]−1 = 1−mΣ′′ −(k0)+O(α2), (20) k∗ 0 k0= 1−m∗ k0Σ′ −(k0) = 1−m k0Σ′ −(k0)+O(α2). (21) Let us ﬁrst consider EPandm∗. In the zero SO limit, Eqs. (19) and (20) at k0= 0 lead respectively to EP= παω0/2 andm∗/m= 1 +πα/8, which correspond to the results already reported in Refs.[13,14,15]. For ﬁnite values of the SO coupling the ground state energy and /s49/s46/s48/s48/s49/s46/s48/s53/s49/s46/s49/s48/s49/s46/s49/s53 /s32/s32 /s32/s102 /s69 /s80 /s40 /s48/s41/s40/s97/s41 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s49/s50/s51/s52/s53/s32/s102 /s109/s42/s40 /s48/s41 /s48/s40/s98/s41/s48 /s50 /s52 /s54 /s56 /s49/s48/s49/s46/s48/s49/s46/s50/s49/s46/s52/s49/s46/s54 /s32 /s32/s102 /s69 /s80/s40 /s48/s41 /s48 /s48 /s50 /s52 /s54 /s56 /s49/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48 /s32 /s32/s102 /s109/s42/s40 /s48/s41 /s48 FIG. 1: (a): ground state energy factor fEP(ε) as a function of the SO parameter ε0=E0/ω0. The solid line is the nu- merical calculation, while the dashed line is the weak SO lim it Eq.(25). Inset: fEP(ε) plotted for a wider range of ε0. (b): the eﬀective mass factor fm∗(ε0) from numerical integration (solid line) and from Eq.(26) (dashed line). Inset: fm∗(ε0) plotted for a wider range of ε0.the eﬀective mass can be expressed as EP=−π 2αω0fEP(ε0), (22) m∗ m= 1+π 8αfm∗(ε0), (23) where the factors fEP(ε0) andfm∗(ε0) contain all the eﬀects of the SO interaction and depend solely on the dimensionless SO parameter ε0≡E0 ω0=mγ2 2ω0. (24) In the weak SO limit, the self-energy terms (13) and (14) canbeexpandedinpowersoftheSOinteraction,allowing for an analytical evaluation of the integrals. In this way, up to the linear order in ε0,fEP(ε0) andfm∗(ε0) are found to be: fEP(ε0) = 1+ε0 4+O(ε2 0), (25) fm∗(ε0) = 1+9 8ε0+O(ε2 0), (26) indicating that the polaronic character is strengthened by the SO interaction since, through Eqs. (22) and (23), the polaron energy EPis lowered and, simultaneously, the eﬀective mass m∗is enhanced when ε0>0. This fea- ture is not limited to the small ε0limit, but holds true for arbitrary strengths of the SO coupling. This is shown in Fig. 1 where fEP(ε0) andfm∗(ε0), obtained from a numerical integration of Eqs.(13) and (14), are plotted as a function of ε0by solid lines and compared with Eqs. (25) and (26) (dashed lines). The same quantities calcu- lated for a wider range of ε0are plotted in the insets of Fig.1 and conﬁrm that the ground state energy EPand the eﬀective mass m∗are continuous functions of ε0and are, respectively, further lowered and enhanced by the SO coupling. In the strong SO limit ε0≫1, it is found thatfEP(ε0)growsasln( ε0)whilefm∗(ε0)growslinearly. It is interesting to note that the Holstein-Rashba model studied in Ref.[31] predicts results qualitatively similar to the Fr¨ ohlich model, indicating that the SO interaction strengthen the polaronic character independently of the speciﬁc form of the el-ph interaction.34 In addition to EPandm∗, the interplay between the el-ph coupling and the SO interaction modiﬁes also the Rashba momentum k0through Eq.(21). In the weak SO limit, the eﬀective quantity k∗ 0is found to be k∗ 0 k0≃1−π 32αε0, (27) indicating a reduction of the bare Rashba momentum k0, conﬁrmed also by the numerical calculation of Eq.(21) reported in Fig. 2 by the solid line. As shown in the inset, for ﬁxed el-ph coupling α,k∗ 0however does not deviate much from its bare limit k0, even for large values of the SO parameter ε0. Let us compare now the present results with those appeared recently in literature. In Ref.[35] the ground5 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s45/s48/s46/s48/s51/s45/s48/s46/s48/s50/s45/s48/s46/s48/s49/s48/s46/s48/s48 /s32/s32/s40/s107 /s48/s42/s47/s107 /s48/s45/s49/s41/s47 /s48/s48 /s50 /s52 /s54 /s56 /s49/s48/s45/s48/s46/s48/s51/s45/s48/s46/s48/s50/s45/s48/s46/s48/s49/s48/s46/s48/s48 /s32 /s32/s40/s107 /s48/s42/s47/s107 /s48/s45/s49/s41/s47 /s48 FIG. 2: Eﬀective Rashba momentum k∗ 0as a function of the SO parameter ε0=E0/ω0. The numerical integration of Eq.(21) (solid line) is compared with the weak SO result (27) (dashed line). Inset: the same quantity plotted for a wider range of ε0. state energy of a polaron near a polar-polar semiconduc- tor interface with Rashba SO coupling has been evalu- ated with the Lee-Low-Pines method.4As a function of the SO splitting, the polaron ground state is found to be lowered, in qualitative agreement therefore with the present results. A more quantitative comparison is how- ever precluded by the diﬀerent model of Ref.[35], where contributions from interface phonon modes and conﬁn- ing potentials are considered as well. In another work,36 the Rayleigh-Schr¨ odinger perturbation theory has been applied to the polaron ground state of the 2D Fr¨ ohlich- Rashba model, permitting therefore a direct comparison with the analysis presented here. Despite that the au- thors of Ref.[36] ﬁnd that the polaron ground state is lowered by ε0, their values of EPdiﬀer from those plot- ted in Fig. 1(a). In Ref.[36], in fact, the ground state energy factor fEPis found to be fEP(ε0) = 1/√1−ε0, which implies a small ε0expansiondiﬀerent from Eq.(25) and, more importantly, a divergence of EPatε0= 1. In Fig. 1(a), instead, nothing of special happens at ε0= 1. This discrepancy is easily traced back in the fact that in Ref.[36] the expansion of Σ −(k), Eq.(17), is made around k= 0, instead of k=k0asdone here, which doesnot cor- respond to a perturbative calculation of the ground-state energy. The results presented in this section have been derived by assuming a weak coupling to the phonons. However, as it is clear from the plots in Fig. 1, the enhancement of the polaronic character driven by ε0for ﬁxedαunavoid- ably renders the perturbative approach invalid for suﬃ- ciently large ε0values. For example, from Eq.(23), the validity of the weak coupling results for m∗/mare sub- jected to the condition αfm∗(ε0)≪1, otherwise higher order el-ph contributions should be considered for a con- sistent description of the SO eﬀects. The question re- mains therefore whether the SO enhancement of the po-laronic character survives also for large αvalues, or it is instead limited to the weak coupling limit. In the next section, this problem is studied for the limiting case of strong el-ph interaction α≫1, providing therefore, to- gether with the weak coupling results, a global under- standing of the SO eﬀects on the Fr¨ ohlich polaron. III. STRONG COUPLING It is well known that a perturbative scheme such that employed in the previous section fails to describe the Fr¨ ohlich polaron ground state when the el-ph coupling is very large. This is due to the fact that for α≫1 the lattice polarization, and resulting “self-trapping” ef- fect experienced by the the electron,37renders the plane waverepresentationofthe unperturbed electroninappro- priate for obtaining the polaron ground state. Instead, as originally proposed in Ref.[5] and rigorously proved in Refs.[38,39], the asymptotic description of the polaron wave function in the strong coupling limit α≫1 is that of a product between purely electronic, ψ(r), and purely phononic, \|ξ∝angbracketright, wave functions. Within such adi- abatic limit, the ground state energy and the eﬀective mass of a 2D Fr¨ ohlich polaron have been calculated in Refs.[14,15] by using the variational method with diﬀer- ent ansatz wave functions. From Ref.[14], one realizes thatexponential,gaussianandPekar-typewavefunctions provide increasingly better estimates of EPwith accura- cies respectively of 14%, 0 .3%, and 0.03% with respect to the exact ground state energy EP/ω0=−0.40474α2, ob- tained by a numerical solution of the integro-diﬀerential equation for the electron wave function.16In the follow- ing, the variational method is used to evaluate the SO eﬀects on the polaron ground state. A. trial wave functions For the nonzero SO case, due to the presence of the Pauli matrices in Eq.(1), suitable ansatz wave functions must take into account the electron spin degrees of free- dom. Hence, in full generality, the strong-coupling po- laron wave function may be represented as: \|Ψ,ξ∝angbracketright= Ψ(r)\|ξ∝angbracketright, whereΨ(r) is a two-components spinor for the electron. The corresponding expectation value of the to- tal Hamiltonian His: ∝angbracketleftΨ,ξ\|H\|Ψ,ξ∝angbracketright=∝angbracketleftΨ\|Hel\|Ψ∝angbracketright+∝angbracketleftξ\|Hph\|ξ∝angbracketright +1√ A/summationdisplay q1√q(M0ρ(q)∝angbracketleftξ\|aq\|ξ∝angbracketright+h.c.), (28) where ρ(q) =∝angbracketleftΨ\|eiq·r\|Ψ∝angbracketright=/integraldisplay dreiq·r\|Ψ(r)\|2.(29)6 The form of Eq.(28) permits to integrate out the phonon wave function in the usual way. Hence, by introducing the phonon coherent state \|ξ∝angbracketright=NeP qξqa† q\|0∝angbracketright, whereN is a normalization factor and ξqa variational parame- ter, minimization of (28) with respect to ξqleads to the functional E[Ψ] =∝angbracketleftΨ\|Hel\|Ψ∝angbracketright−\|M0\|2 ω0/integraldisplaydq (2π)21 q\|ρ(q)\|2,(30) where the continuum limit A−1/summationtext q→/integraltext dq/(2π)2has been performed. By choosing an appropriate functional form for Ψ(r), and by minimizing E[Ψ] with respect to the variational parameters deﬁning Ψ(r), an upper bound for the ground state energy is then E[Ψ0], where Ψ0(r) is such that E[Ψ0] = min(E[Ψ]). As done in the previous section, the polaron energy is then obtained from EP=E[Ψ0]+E0, (31) whereE0is the free-electronSOenergydeﬁned in Eq.(8). Of course, the functional form of Ψ(r) is decisive for obtaining accurate estimates of the ground state energy, and a suitable choice must be guided by looking at the properties of the true ground state spinor ΨG(r). These can be deduced by a formal minimization of the func- tionalE[Ψ] with respect to Ψ. By introducing the La- grange multiplier ǫto ensure that the wave function is normalized to unity, minimization of (30) leads to: HelΨ(r)+V(r)Ψ(r) =ǫΨ(r), (32) where, by using the deﬁnition of ρ(q) given in in Eq.(29): V(r) =−2\|M0\|2 ω0/integraldisplaydq (2π)2ρ(q)∗ qeiq·r =−\|M0\|2 πω0/integraldisplay dr′\|Ψ(r′)\|2 \|r−r′\|. (33)From the above expression of V(r), the functional (30) can be rewritten as E[Ψ] =∝angbracketleftΨ\|Hel\|Ψ∝angbracketright+¯V/2, where ¯V=∝angbracketleftΨ\|V(r)\|Ψ∝angbracketright. Now, if ΨGis the exact ground state wave function, with ground state energy EG=E[ΨG], then, from (32) and EG=∝angbracketleftΨ\|Hel\|Ψ∝angbracketright+¯V/2, it is found thatǫ=EG+¯V/2, so that Eq.(32) reduces to: HelΨG(r)+[V(r)−¯V/2]ΨG(r) =EGΨG(r).(34) As noted in Ref.[29] (see also Refs.[40,41]), the ground- state wave function of a 2D electron subjected to a SO Rashba interaction and to a 2D central potential ( i.e.a potential depending only upon r=\|r\|) is of the form ΨG(r) =/parenleftbigg ψ1(r) ψ2(r)eiϕ/parenrightbigg , (35) whereϕis the azimuthal angle of r. Now, if Eq.(35) is used in Eq.(33), the resulting self-consistent potential depends only upon r,V(r)→V(r), so that Eq.(35) is consistently alsothe correct form for the polaron ground- state wave function. Hence, passing to polar coordi- nates, Eq.(34) can be rewritten as a system of integro- diﬀerential equations for the spinor components ψ1and ψ2: /bracketleftbigg −1 2m/parenleftbiggd2 dr2+1 rd dr/parenrightbigg +U(r)/bracketrightbigg ψ1(r)−γ/parenleftbiggd dr+1 r/parenrightbigg ψ2(r) =EGψ1(r), (36) /bracketleftbigg −1 2m/parenleftbiggd2 dr2+1 rd dr−1 r2/parenrightbigg +U(r)/bracketrightbigg ψ2(r)+γd drψ1(r) =EGψ2(r), (37) whereU(r) =V(r)−¯V/2 and the polaron energy is ob- tained from EP=EG+E0. By introducing the dimen- sionless variable ρ=r/ℓP, whereℓP= 1/α(mω0)1/2is a measure of the polaron spatial extension in the zero SO limit, and by noticing that EGdoes not depend on the sign ofγ, it is straightforward to realize from Eqs.(36)and (37) that the polaron ground state energy scales as EP=F/parenleftBigε0 α2/parenrightBig α2ω0, (38) whereε0=E0/ω0is the dimensionless SO energy intro- duced in Eq.(24) and Fis a generic function. It is found therefore from Eq.(38) that the dependence of EPon the SO interaction is through the eﬀective parameter ε0/α2,7 which is treated in the following as an independent vari- able. Although ε0/α2is then formally allowed to vary from 0 to ∞, it is nevertheless important to estimate the range over which ε0/α2is expected to vary for reason- able values of the microscopic parameters E0,ω0, andα. To this end, it must be reminded that the strong cou- pling limit of a 2D Fr¨ ohlich polaron (in the absence of SO interaction) is appropriate only for α/greaterorapproxeql5,12and that the typical phonon energy scale is of the order of few to tens meV, say ω0≈5−10 meV. The largest value of the Rashba energy E0reported so far is of about 0 .2 eV,25 so thatε0/α2/lessorsimilar1−2 is a rather conservative estimate compatiblewith materialparametersand with the strong coupling polaron hypothesis. Letusnowevaluatethebehaviorof ψ1(r) andψ2(r) for r≪ℓPandr≫ℓP. By requiring a regular solution at the origin, it turns out by inspection of Eqs.(36) and (37) that the spinor components of (35) behave as ψ1(r) = const.andψ1(r)∝rasr→0, while the behavior for r≫ℓPis obtained from the large rlimit of Eqs.(36) and (37): −1 2md2ψ1(r) dr2−γdψ2(r) dr=Wψ1(r),(39) −1 2md2ψ2(r) dr2+γdψ1(r) dr=Wψ2(r),(40) where the quantity W=EG+¯V/2 is negative for bound states. Solutions of Eqs.(39) and (40) which are ﬁnite forr→ ∞are linear combination of exp( −λ+r) and exp(−λ−r) with λ±=/radicalBig −2m(EP+¯V/2)±ik0, (41) implying an exponential decay of the polaron wave func- tion, accompanied by periodic oscillations of wavelength 2π/k0. The informations gathered on the limiting behaviors of the ground state wave function are suﬃcient for guess- ing some appropriate trial wave functions to be used in Eq.(30). By assuming that for zero SO coupling the elec- tronisinaspin-upstate,thenasimpleansatzcompatible with the limits discussed above is Ψ(r) =f(r)/parenleftbigg cos(br) sin(br)eiϕ/parenrightbigg , (42) wherebis a variationalSO parametervanishing for γ= 0 andf(r) isanexponentiallydecayingfunction for r→ ∞ and such that f(0)∝negationslash= 0. The advantage of Eq.(42) is that one can use exponential or Pekar-type functions for f(r), automatically recovering therefore the known results for the zero SO case.14It should be noted, however, that in theU(r)→0 limit Eq.(42) does not reproduce cor- rectly the behavior of the exact ground state wave func- tion, which is instead given by Eq.(35) with ψ1(r) and ψ2(r) proportional to the Bessel functions J0(k0r) and J1(k0r), respectively.40,41Hence, Eq.(42) is not expected toprovideareliablegroundstateenergyin the strongSOregime, for which U(r) can be treated as a perturbation. To remedy to this deﬁciency, the following alternative form of the polaron ansatz is proposed: Ψ(r) =f(r)/parenleftbigg J0(br) J1(br)eiϕ/parenrightbigg , (43) where, as before, bis a variational SO parameter. As it willbeshownbelow, thelowestvalueof EPisgiveneither by Eq.(42) or by Eq.(43), depending on the speciﬁc form considered for f(r) and on the value of the SO coupling. B. ground state energy To evaluate the polaron ground state energy, three dif- ferent trial wave functions for f(r) are considered: ex- ponential, Gaussian and Pekar-type. As shown below, the Gaussian ansatz will provide results comparable to those coming from the exponential and Pekar functions, despite its faster decay for r→ ∞compared to Eq.(41). These three trial wave functions will be used in combi- nation with the sinuisodal and the Bessel-type spinors of Eqs.(42) and (43), respectively, giving a total of six diﬀerent ansatzes for the Fr¨ ohlich-Rashba polaron wave function. Exponential ansatz . Let us start by evaluating the functionalE[Ψ], Eq.(30), byusingtheexponentialansatz f(r) =Aexp(−ar), whereais a variational parameter andAis a normalization factor, in combination with the sinuisodal trial wave function (42). By introducing the dimensionlessquantities ˜ a=aℓP,˜b=bℓP, and ˜γ=k0ℓP, for nonzero SO interaction the functional (30) evaluated with the exponential-sinuisodal ansatz reduces to E[Ψ] α2ω0=1 2/bracketleftBigg ˜a2+˜b2+˜a2ln/parenleftBigg 1+˜b2 ˜a2/parenrightBigg/bracketrightBigg −˜γ˜b/parenleftbigg 1+˜a2 ˜a2+˜b2/parenrightbigg −3π˜a 8√ 2.(44) For weak SO couplings, Eq.(44) has its minimum at ˜b= ˜γ=√2ε0/αand ˜a= 3√ 2π/16, so that the resulting polaron energy EP=E[Ψ0]+E0becomes EP α2ω0=−/parenleftbigg3π 16/parenrightbigg2 −ε0 α2+O/parenleftbiggε2 0 α4/parenrightbigg .(45) In theε0= 0 limit, Eq.(45) reduces to EP/α2ω0= −(3π/16)2≃ −0.3469, recovering therefore the result of Ref.[14], while for ε0>0 the polaron energy is lowered by the SO interaction, in qualitative analogy with the weak electron-phonon behavior discussed in Sec.II. The lowering of EPis conﬁrmed by a numerical minimization of Eq.(44) whose results are plotted in Fig.3(a) (open cir- cles). Forε0/α2= 1, the polaron energy has dropped to EP/α2ω0≃ −0.65, that is about two times lower than the zero SO case. However, upon increasing ε0/α2,EP displays a minimum at ε0/α2≃3.98 [inset of Fig.3(a)]8 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s45/s49/s46/s48/s45/s48/s46/s57/s45/s48/s46/s56/s45/s48/s46/s55/s45/s48/s46/s54/s45/s48/s46/s53/s45/s48/s46/s52/s45/s48/s46/s51 /s40/s97/s41 /s32/s32/s69 /s80/s47/s40 /s48/s50 /s41 /s48/s47/s50/s69/s120/s112/s111/s110/s101/s110/s116/s105/s97/s108/s32/s97/s110/s115/s97/s116/s122 /s32/s115/s105/s110/s117/s105/s115/s111/s100/s97/s108 /s32/s66/s101/s115/s115/s101/s108 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s45/s49/s46/s48/s45/s48/s46/s57/s45/s48/s46/s56/s45/s48/s46/s55/s45/s48/s46/s54/s45/s48/s46/s53/s45/s48/s46/s52/s45/s48/s46/s51 /s40/s98/s41 /s32/s32 /s48/s47/s50/s71/s97/s117/s115/s115/s105/s97/s110/s32/s97/s110/s115/s97/s116/s122 /s32/s115/s105/s110/s117/s105/s115/s111/s100/s97/s108 /s32/s66/s101/s115/s115/s101/s108 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s45/s49/s46/s48/s45/s48/s46/s57/s45/s48/s46/s56/s45/s48/s46/s55/s45/s48/s46/s54/s45/s48/s46/s53/s45/s48/s46/s52/s45/s48/s46/s51 /s40/s99/s41 /s32/s32 /s48/s47/s50/s80/s101/s107/s97/s114/s45/s116/s121 /s112/s101/s32/s97/s110/s115/s97/s116/s122 /s32/s115/s105/s110/s117/s105/s115/s111/s100/s97/s108 /s32/s66/s101/s115/s115/s101/s108/s48/s46/s49 /s49 /s49/s48 /s49/s48/s48/s45/s50/s46/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48 /s32 /s32 /s69 /s80 /s47/s40 /s48/s50 /s41 /s48/s47/s50/s48/s46/s49 /s49 /s49/s48 /s49/s48/s48/s45/s50/s46/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48 /s32 /s32 /s69 /s80 /s47/s40 /s48/s50 /s41 /s48/s47/s50/s48/s46/s49 /s49 /s49/s48 /s49/s48/s48/s45/s50/s46/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48 /s32 /s32 /s69 /s80 /s47/s40 /s48/s50 /s41 /s48/s47/s50 FIG. 3: Polaron ground state energy as a function of ε0/α2for diﬀerent trial wave functions for f(r). (a): exponential; (b): Gaussian; (c): Pekar. The sinuisodal and the Bessel type of a nstazes are given respectively by Eq.(42) and Eq.(43). Inse t: the polaron energy for a wider range of ε0/α2values. and for larger values of the SO interaction the polaron energy increases. Eventually, for ε0/α2/greaterorsimilar14 the calcu- lated ground state energy becomes larger than the zero SO valueEP/α2ω0=−(3π/16)2. Such upturn of EP for largeε0stems from the inadequacy of the sinuiso- dal components of (42) in treating the oscillatory be- havior in the strong SO regime which, as pointed out above, should instead be given by Bessel-type functions. Indeed when the exponential ansatz for f(r) is used in Eq.(43), rather than in Eq.(42), not only the resulting EPis lower than the previous case, but also the upturn ofEPdisappears, leading to a monotonous lowering of the polaron energy as ε0/α2increases [ﬁlled circles in Fig.3(a)]. As ε0/α2→ ∞, however, the polaron energy does not decrease indeﬁnitely but rather approaches a limiting value. Although an accurate numerical evalua- tion ofEPforε0/α2>100 has turned out to be diﬃcult, the asymptotic value of EPcan nevertheless be obtained analytically from the strong SO limit of the exponential- Bessel expression for E[Ψ]: E[Ψ] α2ω0=˜a2+˜b2 2−˜b˜γ−π√ 2˜a, (46) whose minimum is at ˜b= ˜γand ˜a=π/√ 2, leading to lim ε0/α2→∞EP α2ω0=−π2 4≃ −2.467.(47) Gaussian ansatz . The results obtained by using a Gaussian wavefunction of the form f(r) =Aexp(−a2r2) are plotted in Fig. 3(b). Compared to the exponentialwave function, the Gaussian ansatz gives an overall low- eringofthe polaronenergyfor both sinuisodaland Bessel forms of the spinors. In the ε0/α2≪1 limit, and inde- pendently of which particular spinor is used, the ground state polaron energy is found to be: EP α2ω0=−π 8−ε0 α2+O/parenleftbiggε2 0 α4/parenrightbigg , (48) conﬁrminginthisregimethelineardependenceontheSO coupling ofEq.(45). For largervalues ofthe SO coupling, and contrary to the case shown in Fig. 3(a), the sinuiso- daland Bessel-typespinorsgivebasicallythe samevalues ofEPfor all SO couplings up to ε0/α2≃1. Beyond this value, as for the case with the exponential wave function, the polaron energy obtained from the sinuisodal ansatz becomes largerthan that obtained from the Bessel spinor and, as shown in the inset of Fig. 3(b), rapidly increases while the Gaussian-Bessel anstaz gives a monotonous lowering of EP. Forε0/α2≫1, the Gaussian-Bessel energy functional has the same form of Eq.(46) with the latter term substituted by −2.279˜a, which implies lim ε0/α2→∞EP α2ω0≃ −2.579. (49) Pekar-type ansatz . Let us now evaluate EPby using in Eqs.(42) and (43) the Pekar-type ansatz f(r) =A(1+ a1r+a2r2)exp(−ar). For zero SO coupling, this ansatz givesEP/α2ω0≃ −0.4046,14which is a lower energy than those obtained from the exponential and Gaussian trialwavefunctionsandonly0 .03%higherthanthe exact result−0.40474 of Ref.[16]. As shown in Fig. 3(c), the9 Pekar-type ansatz gives slightly better estimates of EP also for nonzero SO couplings, with an overall behavior similar to the previous cases. Namely, in the weak SO regime one ﬁnds EP α2ω0=−0.4046−ε0 α2+O/parenleftbiggε2 0 α4/parenrightbigg ,(50) and, as before, for stronger SO couplings the energy ob- tained from the sinuisodal spinor increases indeﬁnitely withε0/α2. However, contrary to the exponential and Gaussian ansatzes, the Pekar-type wave function may give a lower polaron energy when used in combination with the sinuisodal spinor. This holds true as long as ε0/α2/lessorsimilar2.72, while for stronger SO couplings it is the Bessel-type spinor which gives the lower EP[inset of Fig. 3(c)]. A numerical minimization of the asymptotic limit of the Pekar-Bessel functional for ε0/α2≫1 gives lim ε0/α2→∞EP α2ω0≃ −2.91, (51) whichislowerthanthe asymptoticvaluesofEqs.(47) and (49). The results plotted in Fig. 3 clearly demonstrate that, sincethe variationalmethod providesanupper bound for true ground state polaron energy, the lowering of EPin- ducedbythe SOcouplingisarobustfeatureofthe strong coupling Fr¨ ohlich-Rashba polaron. Among the diﬀerent ansatzes studied, the lower polaron energy is obtained by using a Pekar-type wavefunction for f(r) in combination with the sinuisodal spinor for weak to moderate values of ε0/α2orwith the Bessel-typespinor for strongerSO cou- plings. Given that, as discussed above, reasonable values ofε0/α2for strongly-coupled polarons fall in the range 0≤ε0/α2/lessorsimilar1−2, the Pekar-sinuisodal wave function provides therefore the best description of the Fr¨ ohlich- Rashba polaron in this regime. C. eﬀective mass As demonstrated in Sec.II, the eﬀective mass m∗of a weakly-coupled polaron is enhanced by the SO in- teraction and, given the results above, the same phe- nomenon is reasonably expected to occur also for the strong-coupling case. To quantify the polaron mass en- hancement within the localized wave function formal- ism, it is useful to follow the approach of Refs.[42,43,44], brieﬂy described below, where a moving wave packet is constructed from the localized wave function. The quan- tity to minimize is Jυ[Ψ′,ξ′] =∝angbracketleftΨ′,ξ′\|H−υ·P\|Ψ′,ξ′∝angbracketright,(52) whereυis a Lagrange multiplier, which will turn out to be the mean polaron velocity, and P=p+/summationtext qqa† qaqis thetotalmomentumoperator. Thewavefunction \|Ψ′,ξ′∝angbracketright is given by the product Ψ′(r)\|ξ′∝angbracketrightwhere Ψ′(r) =eip0·rΨ(r) (53)is the electron wave packet with p0being a variational momentum, Ψ(r) is the ansatz localized wave function, and\|ξ′∝angbracketright=NeP qξ′ qa† q\|0∝angbracketright. Minimization of (52) with re- spect toξ′ qgives now the functional Jυ[Ψ′] =∝angbracketleftΨ′\|Hel−υ·p\|Ψ′∝angbracketright −\|M0\|2/integraldisplaydq (2π)2\|ρ(q)′\|2 q1 ω0−q·∝angbracketleftΨ′\|υ\|Ψ′∝angbracketright, (54) wherepis the electron momentum operator and ρ(q)′= ∝angbracketleftΨ′\|eiq·r\|Ψ′∝angbracketright. By using Eq.(53), it is easily shown that Jυ[Ψ′] reduces to Jυ[Ψ′] =∝angbracketleftΨ\|Hel\|Ψ∝angbracketright+p2 0 2m−p0·υ −\|M0\|2/integraldisplaydq (2π)2\|ρ(q)\|2 q1 ω0−q·υ,(55) whereρ(q) =∝angbracketleftΨ\|eiq·r\|Ψ∝angbracketright. Equation (55) is minimized withrespectto p0bysetting p0=mυand, byexpanding the last term of Eq.(55) up to the second order in υ, the corresponding minimum Jυ[Ψ] becomes:42 Jυ[Ψ] =E[Ψ]−m 2υ2/bracketleftbigg 1+2\|M0\|2 mω3 0/integraldisplaydq (2π)2(q·ˆu)2 q\|ρ(q)\|2/bracketrightbigg , (56) whereE[Ψ] is given in Eq.(30). From the above expres- sion, it is clear that Jυ[Ψ] diﬀers from J0[Ψ] at least to orderυ2. Hence, if ΨυandΨ0are the wave functions which minimize Jυ[Ψ] andJ0[Ψ], respectively, then the diﬀerence Ψυ−Ψ0is also of order υ2. As a consequence, the minimum of (56), Jυ[Ψυ], diﬀers from Jυ[Ψ0] only to order(Ψυ−Ψ0)2=O(υ4) sothat, by neglectingtermsof higher order than υ2, minimization of (56) is achieved by thebestwavefunctionwhichminimizes E[Ψ]. Therefore, by usingE[Ψ0] =EP−E0and evaluating ∝angbracketleftΨ0\|P\|Ψ0∝angbracketright, from Eqs.(52) and (56) it turns out that EP(υ) =EP+m 2υ2/bracketleftbigg 1+2\|M0\|2 mω3 0/integraldisplaydq (2π)2(q·ˆu)2 q\|ρ0(q)\|2/bracketrightbigg , (57) permitting us to identify the quantity within square brackets as the mass enhancement factor m∗/m. By in- tegrating over the direction of qand by using (5), m∗/m becomes in the strong-coupling limit m∗ m=√ 2πα (mω0)3/2/integraldisplay∞ 0dq 2πq2\|∝angbracketleftΨ0\|eiq·r\|Ψ0∝angbracketright\|2,(58) which, by replacing the momentum variable by the di- mensionless quantity ˜ q=qℓP, gives a mass enhancement proportional to α4in the zero SO case. By using the ex- ponential, Gaussian, and Pekar-type ansatzes in Eq.(58), the resulting mass enhancement factor becomes m∗/m= (3/16)3π4α4≃0.6421α4,m∗/m= (π/4)2α4≃0.617α4, andm∗/m≃0.73α4, respectively.45 The results for nonzero SO coupling are plotted in Fig.4 for the sinuisodal (open circles) and Bessel (ﬁlled10 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53 /s40/s97/s41 /s32/s32/s109 /s42/s47/s40/s109 /s52 /s41 /s48/s47/s50/s69/s120/s112/s111/s110/s101/s110/s116/s105/s97/s108/s32/s97/s110/s115/s97/s116/s122 /s32/s115/s105/s110/s117/s105/s115/s111/s100/s97/s108 /s32/s66/s101/s115/s115/s101/s108 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53 /s40/s98/s41 /s32/s32 /s48/s47/s50/s71/s97/s117/s115/s115/s105/s97/s110/s32/s97/s110/s115/s97/s116/s122 /s32/s115/s105/s110/s117/s105/s115/s111/s100/s97/s108 /s32/s66/s101/s115/s115/s101/s108 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53 /s40/s99/s41 /s32/s32 /s48/s47/s50/s80/s101/s107/s97/s114/s45/s116/s121 /s112/s101/s32/s97/s110/s115/s97/s116/s122 /s32/s115/s105/s110/s117/s105/s115/s111/s100/s97/s108 /s32/s66/s101/s115/s115/s101/s108 /s48/s46/s48/s49 /s48/s46/s49 /s49 /s49/s48 /s49/s48/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48/s54/s48 /s32 /s32 /s109/s42/s47/s40/s109/s52 /s41 /s48/s47/s50/s48/s46/s48/s49 /s48/s46/s49 /s49 /s49/s48 /s49/s48/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48/s54/s48 /s32 /s32 /s109/s42/s47/s40/s109/s52 /s41 /s48/s47/s50/s48/s46/s48/s49 /s48/s46/s49 /s49 /s49/s48 /s49/s48/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48/s54/s48 /s32 /s32 /s109/s42/s47/s40/s109/s52 /s41 /s48/s47/s50 FIG. 4: Polaron mass enhancement m∗/min units of α4as a function of ε0/α2for diﬀerent ansatz wave functions. (a): exponential; (b): Gaussian; (c): Pekar. Inset: m∗/mα4is plotted for a wider range of SO values. circles) spinors evaluated with exponential (a), Gaus- sian (b), and Pekar-type (c) wave functions. For all cases,m∗/mincreases with ε0/α2without much quan- titative diﬀerences between the various ansatzes as long asε0/α2/lessorsimilar2. As shown in the insets of Fig. 4, for larger values of the SO coupling the use of the sinuisodal spinor largely overestimates the increase of the eﬀective mass compared to the Bessel-type spinor results. How- ever, despite of the weaker enhancement of m∗/m, the Bessel-type spinors give nevertheless an inﬁnite eﬀective mass atε0/α2=∞. Indeed, independently of the par- ticular form of f(r), forε0/α2→ ∞the expectation value∝angbracketleftΨ0\|eiq·r\|Ψ0∝angbracketrightappearing in Eq.(58) goes like a/q forq→ ∞, rendering the integral over qof Eq.(58) di- vergent. IV. DISCUSSION AND CONCLUSIONS The results presented in the previous sections consis- tently show that, for both the weak and strong coupling limits of the el-ph interaction, the ground state energy EPof the Fr¨ ohlich-Rashba polaron is lowered by the SO interaction and the mass is enhanced, leading to the conclusion that the Rashba coupling ampliﬁes the polaronic character. This scenario suggests also that a weak-coupling polaron at ε0= 0 may be turned into a strong-coupling one for ε0>0 or, more generally, that the crossover between weakly and strongly coupled po- larons may be shifted by the SO interaction. This pos- sibility can be tested by looking at the curves plotted in the main panel of Fig. 5, where the weak and strong coupling results for EP/ω0are reported as a function ofthe el-ph coupling αfor diﬀerent ε0values. For ε0= 0, the polaron energy follows EP/ω0≃ −πα/2 for small α andEP/ω0≃ −0.4046α2for largeα. These two limiting behaviors are plotted in Fig. 5 by the uppermost curves and compared with a numerical solutions of the Feyn- man variational path integral for the 2D polaron (ﬁlled circles). The largest deviation of the path integral solu- tions from the weak and strong coupling approximations falls in the range of intermediate values of αand signals a region of crossover between the weakly and strongly coupled polaron. A rough estimate of the crossover po- sition is given by a “critical” coupling, say α∗, obtained by equating the weak and strong coupling results. For ε0= 0 therefore one has πα/2 = 0.4046α2, which gives α∗≃3.9. Now,asshowninFig. 5for ε0= 5andε0= 20, the increase ofthe SO interaction systematically reduces, for ﬁxedα, the polaron ground state energy and, at the same time, shifts the intersection point between the weak and strong coupling curves towards smaller values of the el-ph interaction. The “critical”value α∗of the crossover is therefore reduced by the SO interaction. For ε0= 5 andε0= 20 it is found that α∗≃3.6 andα∗≃2.7, respectively. The systematic reduction of the crossover coupling by the SO interaction is made evident in the inset of Fig. 5, where α∗is plotted as a function of ε0. From Fig. 5 it is also expected that, beside the reduc- tionofα∗, the crossoverregionislikelytobe narrowedby ε0. Indeed, the intersection between the weak and strong coupling solutions for ε0= 20 is apparently smoother than the case for ε0= 0, suggesting that the true ground state energy would deviate less, and in a narrower region aroundα∗, from the weak and strong coupling solutions. The scenario illustrated above, and in particular the11 /s48 /s50 /s52 /s54 /s56 /s49/s48/s45/s51/s53/s45/s51/s48/s45/s50/s53/s45/s50/s48/s45/s49/s53/s45/s49/s48/s45/s53/s48 /s32/s32/s69 /s80/s47 /s48/s32/s32 /s48/s32/s61/s32/s48 /s32/s32 /s48/s32/s61/s32/s53 /s32/s32 /s48/s32/s61/s32/s50/s48 /s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53/s52/s46/s48 /s119/s101/s97/s107/s32 /s99/s111/s117 /s112/s108/s105/s110 /s103 /s32/s32/s42 /s48/s115/s116/s114/s111/s110 /s103/s32 /s99/s111/s117 /s112/s108/s105/s110 /s103 FIG. 5: (Color online). Ground state polaron energy EPas a function of the el-ph coupling αfor diﬀerent values of the dimensionless SO parameter ε0=E0/ω0. The straight lines at small αrefer to the weak coupling results, while the curves at largeαare the solution of the strong-coupling theory. The ﬁlled circles are the solution of the Feynman path integral ansatz (see text). The point of intersection between the wea k and strong coupling curves is a measure of the crossover el-p h coupling α∗. Inset:α∗is plotted as a function of ε0. SO eﬀect on the crossover coupling, may be veriﬁed by quantum Monte-Carlo calculations of the Fr¨ ohlich- Rashba action or, more simply, by generalizing the Feyn- man ansatz for the retarded interaction to ε0>0.7The results presented here on the limiting cases α≪1 and α≫1 may then serve as a reference for such more gen-eral calculations schemes for arbitrary values of the el-ph coupling and of the SO interaction. Let us discuss, before concluding, possible generaliza- tions of the Fr¨ ohlich-Rashba model employed here and the consequences on the polaronic character. Let us re- mind that in Ref.[31] it has been demonstrated that also for a momentum independent el-ph interaction model, the Rashba SO term leads to an eﬀective enhancement ofthe el-phcoupling. TheSOinduced loweringofthe po- laron ground state is therefore robust against the speciﬁc form of the el-ph interaction, so that a similar behavior is expected to occur also when considering the contribu- tions from interface or surface phonon modes. However, a diﬀerent form of the SO interaction term may lead to a much weaker eﬀect. Consider for example the situa- tion in which, in addition to the Rashba SO coupling, the system lacks also of bulk inversion symmetry, as in III-V semiconductor heterostructures, leading to an ex- tra SO term of the Dresselhaus type.18,46When both SO contributions are present, the square root divergence of the DOS at the bottom of the band of the free electron disappears, and it is replaced by a weaker logarithmic divergence at higher energies. In this situation therefore, at least for weak el-ph couplings, the SO interaction is expected to have a weaker eﬀect on the polaron ground state, which tends to vanish as the Dresselhaus term be- comes comparable to the Rashba one. Let us conclude by noticing that, recently, the possi- bility of varying the coupling of 2D Fr¨ ohlich polarons in a controlled way has been experimentally demonstrated by acting on the dielectric polarizability of organic ﬁeld- eﬀect transistors.47The results presented here suggest that tunable 2D Fr¨ ohlich polarons may be achieved also by acting on the SO coupling, which can be tuned by applied gate voltages in quasi-2D structured materials. Acknowledgments The author thanks Emmanuele Cappelluti and Frank Marsiglio for valuable comments. 1H. Fr¨ ohlich, Adv. Phys. 3, 325 (1954). 2T. K. Mitra, A. Chatterjee, and S. Mukhopadhyay, Phys. Reports. 153, 91 (1987). 3H. Fr¨ ohlich, H. Pelzer, and S. Zienau, Phil. Mag. 41, 221 (1950). 4T.D.Lee, F.Low, andD.Pines, Phys.Rev. 90, 297(1953). 5S. I. Pekar, Zh. Eksp. Teor. Fiz. 16, 355 (1946); 16, 341 (1946). 6S. J. Miyake, J. Phys. Soc. Japan 38, 181 (1975); 41, 747 (1976). 7R. P. Feynman, Phys. Rev. 97, 660 (1955). 8G. Ganbold and G. V. Eﬁmov, Phys. Rev. B 50, 3733 (1994); J. Phys.: Condens. Matter 10, 4845 (1998). 9G. De Filippis, V. Cataudella, V. Marigliano Ramaglia, C.A. Perroni, andD. Bercioux, Eur.Phys. J. B 36, 65(2003). 10C. Alexandrou, W. Fleischer, and R. Rosenfelder, Phys. Rev. Lett. 65, 2615 (1990). 11J. T. Titantah, C. Pierleoni, and S. Ciuchi, Phys. Rev. Lett.87, 206406 (2001). 12S. Das Sarma and B. A. Mason, Ann. Phys. (N.Y.), 163, 78 (1985). 13W. J. Huybrechts, Solid Sate Commun. 28, 95 (1978). 14X. Wu, F. M. Peeters, and J. T. Devreese, Phys. Rev. B 31, 3420 (1985). 15F. M. Peeters, X. Wu, and J. T. Devreese, Phys. Rev. B 37, 933 (1988). 16C. Qinghu, F. Minghu, Z. Qirui, W. Kelin, and W. Shao- long, J. Phys.: Condens. Matter 8, 7139 (1996).12 17J. T. Devreese, J. Phys.: Condens. Matter. 19, 255201 (2007). 18I.ˇZuti´ c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). 19S. LaShell, B. A. McDougall, and E. Jensen, Phys. Rev. Lett.77, 3419 (1996). 20Yu. M. Koroteev, G. Bihlmayer, J. E. Gayone, E. V. Chulkov, S. Blugel, P. M. Echenique, and P. Hofmann, Phys. Rev. Lett. 93, 046403 (2004). 21K. Sugawara, T. Sato, S. Souma, T. Takahashi, M. Arai, and T. Sasaki, Phys. Rev. Lett. 96, 046411 (2006). 22E. Rotenberg, J. W. Chung, and S. D. Kevan, Phys. Rev. Lett.82, 4066 (1999). 23D. Pacil´ e,C. R. Ast, M. Papagno, C. Da Silva, L. Mores- chini, M. Falub, Ari P. Seitsonen, and M. Grioni, Phys. Rev. B73245429 (2006). 24C. R. Ast, G. Wittich, P. Wahl, R. Vogelgesang, D. Pacil´ e, M. C. Falub, L. Moreschini, M. Papagno, M. Grioni, and K. Kern, Phys. Rev. B 75, 201401(R) (2007). 25C. R. Ast, J. Henk, A. Ernst, L. Moreschini, M. C. Falub, D. Pacil´ e, P. Bruno, K. Kern, and M. Grioni, Phys. Rev. Lett.98, 186807 (2007). 26I. I. Boiko and E. I. Rashba, Sov. Phys. Solid State 2, 1692 (1960). 27A. G. Galstyan and M. E. Raikh, Phys. Rev. B 58, 6736 (1998). 28C. Grimaldi, Phys. Rev. B 72, 075307 (2005). 29A. V. Chaplik and L. I. Magarill, Phys. Rev. Lett. 96, 126402 (2006); Physica E 34, 344 (2006). 30E. Cappelluti, C. Grimaldi, and F. Marsiglio, Phys. Rev. Lett.98, 167002 (2007). 31E. Cappelluti, C. Grimaldi, and F. Marsiglio, Phys. Rev. B76, 085334 (2007). 32E. Evans and D. L. Mills, Phys. ev. B 8, 4004 (1973); H. Sun and S. -W. Gu, Phys. Rev. B 40, 11576 (1989). 33The form of the self-energy reported in Eq.(12) is not lim- ited to the weak-coupling limit. For a more general deriva-tion see Ref.[31]. 34There are however quantitative diﬀerences. Indeed, in the strong SO limit, the Holstein-Rashba model gives an eﬀec- tive mass enhancement proportional to√ε0, i.e., a weaker increase with respect to the linear dependence found here for the Fr¨ ohlich-Rashba model (see Ref.[31]). 35J. Liu and J. -L. Xiao, Commun. Theor. Phys. 46, 761 (2006). 36Z. Li, Z. Ma, A. R. Wright, and C. Zhang, Appl. Phys. Lett.90, 112103 (2007). 37The term “trapping” is used here in a loose way, since no real trapping eﬀect exist for the continuum Fr¨ ohlich polaron. 38M. Donsker and S. R. S. Varadhan, Commun. Pure Appl. Math.36, 505 (1983). 39E. H. Lieb and L. E. Thomas, Commun. Math. Phys. 183, 511 (1997). 40E. N. Bulgakov and A. F. Sadreev, JETP Lett. 73, 505 (2001). 41E. Tsitsishvili, G. S. Lozano, and A. O. Gogolin, Phys. Rev. B70, 115316 (2004). 42G. R. Allcock, Adv. Phys. 5, 412 (1956). 43R. Parker, G. Whitﬁeld, and M. Rona, Phys. Rev. B 10, 698 (1974). 44J. da Providencia, M. da Concei¸ c˜ ao Ruivo, and C. A. de Sousa, Ann. Phys. 91, 366 (1975). 45Note that the result for the exponential ansatz diﬀers from that of Ref.[15]. The diﬀerence stems from an incorrect expression of the exponential decay factor a: in Ref.[15] a/ℓP= 3π/8 while instead it should be given by a/ℓP= 3π/16, as in Ref.[14] and in the present paper. 46G. Dresselhaus, Phys. Rev. 100, 580 (1955). 47I. N. Hulea, S. Fratini, H. Xie, C. L. Mulder, N. N. Iossad, G. Rastelli, S. Ciuchi, and A. F. Morpurgo, Nat. Mater. 5, 982 (2006).
2401.08966v1.Spin_Orbit_Torque_on_a_Curved_Surface.pdf	1 1 Spin Orbit Torque on a Curve d Surface Seng Ghee Tan1†, Che Chun Huang2, Mansoor B.A.Jalil3, Zhuobin Siu3 (1) Department of Optoelectric Physics, Chinese Culture University, 55 Hwa -Kang Road, Yang - Ming -Shan, Taipei 11114, Taiwan (2) Department of Physics, National Taiwan University, Taipei 10617, Taiwan (3) Department of Electrical and Computer Engineering, National University of Singapore, 4 Engineering Drive 3, Singapore 117576 ABSTRACTS We provide a general formulation of the spin -orbit coupling on a 2D curved surface. Considering the wide applicability of spin -orbit effect in spinor -based condensed matter physics, a general spin -orbit formulation could aid the study of spintronics , Dirac graphene, topological systems, and quantum information on curved surfaces. Particular attention is then devoted to the development of an impo rtant spin -orbit quantity known as the spin-orbit torque. As devices trend smaller in dimension, the physics of local geometries on spin -orbit torque, hence spin and magnetic dynamics shall not be neglected. We derived the general ex pression of a spin-orbit anisotropy field for the curved surfaces and provided explicit solutions in the special context s of the spherical, cylindrical and flat coordinates. Our expression s allow spin-orbit anisotropy field s and hence spin -orbit torque to be comp uted over the entire surface s of devices of any geometry . Corresponding author: † Seng Ghee Tan (Prof) Department of Optoelectric Physics, Chinese Culture University, 55, Hwa -Kang Road, Yang -Ming -Shan, Taipei, Taiwan 11114 ROC (Tel: 886 -02-2861 -0511, DID:25221) † Email: csy16@ulive.pccu.edu.tw ; tansengghee@gmail.com PACS: 2 2 INTRODUCTION The rise of spin physics in condensed matter and nanoscience is so prolific that spinor -based expressions is now a hallmark formulation in nearly all the modern research fields like the spintronics [1-5], graphene [6,7], topological insulators [8,9], topological Weyl and Dirac systems [10, 11], atomic and optical physics [12], quantum computation and information science [13, 14]. Central to the success of spin in science is the advancements in our understanding of the spin-orbit coupling. On the level of pure science, it is a Dirac relativistic phenomenon that exists in the non -relativistic limit . From the condensed matter point of view , spin-orbit coupling is a highly accessible physics that exists in bulk and structural forms in materials like the semiconductor , semimetal , as well as heterostructures comprising these materials . In applied science, it is simply an effective magnetic field that can be controlled via electrical mean s to perform electronic functions like in transistors and memory. In this article, we would focus on the spin orbit coupling in a specific area known as the spin -orbit torque. In magnetic memory, spin current is normally injected into magnetic materia ls to flip the magnetic moments via the force physics of spin transfer torque [15]. In nanoscience systems, spin torque is usually studied in the ferromagnetic materials, in which the net magnetization, reacts to the injection or the “creation” of an accum ulation of spin moments in the material. One example of such “creation” is via the spin -orbit effect. With spin-orbit effect identified for the new role, the force involved in this process would be known as the spin-orbit torque [16-22]. This paper is dedicated to providi ng a general formulation of the spin -orbit coupling and hence the spin-orbit torque in a 2D environment that comprises curved surfaces, with the 2D planar surfaces being a special case of the curve s. Such formulation is im portant because as device becomes smaller, the physics of local geometries on spin -orbit coupling and its torque shall not be neglected. In fact, as spin-orbit coupling is versatile in many modern fields, a general formulation is simply right for the occasion. For example, curve physics has recently been studied in Dirac -electronic transport due to the topological surface states [23-26]. Curves were studied for their effects on inducing topological transitions [27] in a Rashba system , as well as shift ing the band -inversion point [28] in the BHZ topological -insulator . In spintronics, the effect of curve on spin current [29-31] spin Hall [32], and spin Chern number [33] have also been studied. In this paper, we present a general formulation of the spin -orbit torque in a 2D condensed matter system. We will begin with a 3D infinitesimal bulk i n which a 2D surfa ce can be retrieved later by setting 𝑞3→0. Let the electric field penetrating the bulk be normal to the surface (to be denoted by 𝒆𝟑) as shown in Fig.1 below. The spin -orbit energy of a charged particle contained therein would be given by 𝐻𝑠𝑜𝑐=𝛼3 ℏ𝝈.(𝑷×𝒆𝟑) (1) where 𝝈 is the Pauli matrices , and 𝛼3 is a constant to characterize the spin -orbit strength . While Equation (1) is in the general form, constant 𝛼3 would reflect the material property and the type of spin-orbit coupling, e.g. Rashba, Dresselhaus, in bulk , heterostructures or topological surface states. The momenta and Pauli matrices on the 2D surface take on coordinates 𝒆𝟏,𝒆𝟐. And the y are related to the electric field direction by 𝒆𝟏×𝒆𝟐=\|𝑛3\| 𝒆𝟑. It can then be shown that \|𝑛3\|=√𝐺 𝐺33 , which leads to 𝐻𝑠𝑜𝑐=(𝛼3 ℏ)𝜎𝑣𝒆𝒗.[𝑃𝑘𝒆𝒌×(𝒆𝟏×𝒆𝟐)𝐺33 √𝐺] (2) where (𝑣,𝑘) runs over coordinates (𝑞1,𝑞2). 𝐺33=𝜕𝑹 𝜕𝑞3.𝜕𝑹 𝜕𝑞3 is the 3D metric of the system and 𝐺 is the determinant given by 𝐺=(𝐺11𝐺22−𝐺12𝐺21)𝐺33. Note that w ith 𝛼3 being contravariant , the upper and lower indices would balance out. The contravariant property of 𝛼3 will be discussed later. 3 3 FIG.1. An infinitesimal bulk can be shrunk along 𝒆𝟑 to a 2D surface embeded in a 3D space . This is tantamont to taking the limit of 𝑞3→0. Equation (2) is then wri tten as follows 𝐻𝑆𝑂𝐶=(𝛼3 ℏ) 𝑃𝑘𝜎𝑣 𝑅𝑘𝑣3 (3) where 𝑅𝑘𝑣3=𝐺33 √𝐺(𝐺𝑣1𝐺𝑘2−𝐺𝑣2𝐺𝑘1). Expanding the above with the electric field pre -determined along 𝒆𝟑, discussion is kept to the 2D spin -orbit effect, and one has the general expression of 𝐻𝑠𝑜𝑐=(𝛼3 ℏ) [𝜎1𝑃2−𝜎2𝑃1]√𝐺 (4) Note that (1,2,3) is the a bbrev iated form of general coordinates (𝑞1,𝑞2,𝑞3). A compact expression of the above is given by 𝐻𝑠𝑜𝑐=𝑃1 𝑚𝑒𝑆1+𝑃2 𝑚𝑒𝑆2 (5) where 𝑆1=−(𝛼3 ℏ)𝑚𝑒√𝐺 𝜎2 , 𝑆2=(𝛼3 ℏ)𝑚𝑒√𝐺 𝜎1. Note that (𝑃1,𝑃2) and (𝜎1,𝜎2) can b e found by transforming the ir Cartesian counterparts to the desired coordinates in a contravariant manner. When the general coordinates (𝑞1,𝑞2,𝑞3) take on the Cartesian (𝑥,𝑦,𝑧), the term √𝐺 goes to 1. The general Hamiltonian 𝐻𝑆𝑂𝐶 returns to the fami liar spin -orbit system in the Cartesia n frame, 𝐻𝑠𝑜𝑐=(𝛼3 ℏ)(−𝑃𝑥 𝜎𝑦+𝑃𝑦𝜎𝑥) (6) We will now examine the physical significance of 𝛼3, the spin-orbit constant that characterizes its strength. As 𝛼3 captures the material property pertaining directly to the strength of the electric field or the effective electric field in the case of the 2D spin -orbit effect e.g. the Rashba or the Dresellhaus , we will now explicitly express 𝛼3 in terms of i ts electric field as 𝛼3=𝛼′𝐸3. Eq.(1) can now be rewritten to better reflect the actual scenario whe re the presence of the electric field is explicit 𝐻𝑠𝑜𝑐=𝛼3 ℏ𝝈.(𝒑×𝒆𝟑) → 𝐻𝑠𝑜𝑐=𝛼′ ℏ𝝈.(𝒑×𝐸3𝒆𝟑) (7) Now 𝐸3 would be the strength of the electric field normal to surface i.e. along 𝒆𝟑. For better clarity, we use the simple spherical surface s for illustration in Fig.2 below . 𝒆𝟏 𝒆𝟐 𝒏=𝒆𝟏×𝒆𝟐=\|𝑛3\| 𝒆𝟑 𝑹 𝑶 𝒆𝟑 a 3D infinitesimal bulk 𝑞3 𝒓 4 4 (a) (b) FIG.2 . Illustration of the electric field orientations for the spherical surfaces. Solid lines are the actual electric fields , i.e. in (a) 𝑬=𝐸𝑧𝒆𝒛 , in (b) 𝑬=𝐸𝑅𝒆𝑹. Dotted lines denote the orientation normal to the surface, i.e. 𝒆𝟑= 𝒆𝑹. Consider the case of a spherical surface now where 𝒆𝟑=𝒆𝑹. In the event that the actual electric field is 𝑬=𝐸𝑧𝒆𝒛, the expression for the electric field normal to the surface is 𝐸S=(𝐸𝑧 𝒆𝒛) .𝒆𝑹 , and that would result in 𝐸𝑆=𝐸𝑧cos𝜃. In the case where actual electric field is given by 𝑬=𝐸𝑅𝒆𝑹 , one would obtain 𝐸𝑆=(𝐸𝑅 𝒆𝑹) .𝒆𝑹=𝐸𝑅. Therefore, in the event of a general surface marked by 𝒆𝟑 or 𝒆𝟑 and an electric field oriented along 𝒆𝟑, the electric field normal to the surface is 𝐸𝑆=(𝐸3 𝒆𝟑) .𝒆𝟑=𝐸3 (8) It thus becomes clear that 𝐻𝑠𝑜𝑐=𝛼3 ℏ𝝈.(𝒑×𝒆𝟑), with upper index 3 , generalizes the representation s for electric field s orient ed vertical to a surface 𝒆𝟑. The illustration above lends a clear physical meaning to the expressions of 𝑆1=−(𝛼3 ℏ)𝑚𝑒√𝐺 𝜎2 and 𝑆2=(𝛼3 ℏ)𝑚𝑒√𝐺 𝜎1. With 𝛼3 and hence (𝑆1,𝑆2) properly understood, a general formulation of the spin -orbit coupling on a 2D curved surface with electric field normal to the surface have thus been derived in Eq.(5). In the event that the electric field is not restricted to the normal surface, one could derive using Eq.(2) and letting (𝑣,𝑘) run over coordinates (𝑞1,𝑞2,𝑞3) and obtain 𝑆1=(𝛼2 ℏ)𝑚𝑒√𝐺 𝜎3−(𝛼3 ℏ)𝑚𝑒√𝐺 𝜎2 (9) 𝑆2=(𝛼3 ℏ)𝑚𝑒√𝐺 𝜎1−(𝛼1 ℏ)𝑚𝑒√𝐺 𝜎3 (10) 𝑆3=(𝛼1 ℏ)𝑚𝑒√𝐺 𝜎2−(𝛼2 ℏ)𝑚𝑒√𝐺 𝜎1 (11) One should , however, not take for granted that in the event of 𝛼2, 𝒆𝟐 would be normal to 𝒆𝟏 and 𝒆𝟑, and likewise for 𝛼1. The orthogonal property of the space metric continues to be enforced by 𝒆𝟏× 𝒆𝟐=√𝐺 𝐺33 𝒆𝟑 throughout and the metric of the system is [𝐺11𝐺12𝐺13 𝐺21𝐺22𝐺23 𝐺31𝐺32𝐺33]=[𝐺11𝐺120 𝐺21𝐺220 001] (12) 𝜃 𝑬=𝐸𝑧𝒆𝒛 𝑬=𝐸𝑅𝒆𝑹 5 5 The spin -orbit Hamiltonian would simply be 𝐻𝑠𝑜𝑐=𝑃1 𝑚𝑒𝑆1+𝑃2 𝑚𝑒𝑆2+𝑃3 𝑚𝑒𝑆3 (13) SPIN -ORBIT TORQUE In the context of the spin-orbit torque, the system under consideration would be a hetero structure that comprises the ferromagnet and the oxide layers (we cite the example of Ta \CoFeB \MgO [21]), with the f erromagnetic interface hosting a collective density of spin-orbit -induced spin moment denoted by 𝒔. At the same time, the fe rromagnetic material possesses a collective density of intrinsic moment known as 𝒎. Therefore, t he physics of spin -orbit torque arises due to the simultaneous presence of the kinetic energy, the spin -orbit energy and the magnetic energy in the system. The ful l Hamitonian, must now be presented to incorporate th ese energies as follows 𝐻=1 2𝑚𝑒𝑃𝑎𝑃𝑎+1 2𝑚𝑒(𝑃𝑎𝑆𝑎+𝑆𝑎𝑃𝑎)+𝜎𝑎𝑚𝑎 (14) For generality, we let 𝑎 runs over (𝑞1,𝑞2,𝑞3). In the above, the kinetic energy would be 1 2𝑚𝑒𝑃𝑎𝑃𝑎= −𝑖ℏ 2𝑚𝑒(𝜕𝑎+1 2𝜕𝑎ln𝐺)𝑃𝑎. This is because when 𝑃𝑎 is quantized, the covariant property of the operator has to be accounted for . Now, a minimal coupling form is presented, which reflect s intuitively the “forceful” physics of the Hamiltonian. Written in this way, 𝑆𝑎 relates directly to the non- Abelian ga uge that has been studied as an origin of spin forces and phases in many systems [5]. 𝐻=1 2𝑚𝑒(𝑃𝑎+𝑆𝑎)(𝑃𝑎+𝑆𝑎)+𝐺𝑎𝑏𝜎𝑎𝑚𝑏 (15) Expanding the above, 𝐻=−𝑖ℏ 2𝑚𝑒(𝜕𝑎+1 2𝜕𝑎ln𝐺)𝑃𝑎𝜓−𝑖ℏ 2𝑚𝑒(𝜕𝑎+1 2(𝜕𝑎ln𝐺))𝑆𝑎𝜓+1 2𝑚𝑒𝑆𝑎𝑃𝑎+𝐺𝑎𝑏𝜎𝑎𝑚𝑏 (16) One can now perform a local gauge transformation in the spin space suc h that the local frame would rotate to “some” axis – the choice of which would pretty much determine the physics to be revealed from those energies. The locally transformed system is given by 𝐻′=−𝑖ℏ 2𝑚𝑒(∇𝑎+𝑖𝑒 ℏ𝐴𝑎)(𝑃𝑎+𝑒𝐴𝑎)−𝑖ℏ 2𝑚𝑒(∇𝑎+𝑖𝑒 ℏ𝐴𝑎) 𝑈𝑆𝑎𝑈†−𝑈𝑆𝑎𝑈†𝑖ℏ 2𝑚𝑒(𝜕𝑎+𝑖𝑒 ℏ𝐴𝑎) +𝐺𝑎𝑏𝑈𝜎𝑎𝑚𝑏𝑈† (17) where ∇𝑎=𝜕𝑎+1 2𝜕𝑎ln𝐺≡𝜕𝑎+𝛤𝑎 and 𝐴𝑎=−𝑖ℏ 𝑒𝑈(𝜕𝑎𝑈†) is a gauge potential related to the magnetization and the curved geometry of the device. Note that the transformed Hamiltonian is still general in the sense that as of now, no decision has been taken as to whi ch axis the local frame should rotate to. Therefore, the physics that one would like to “see” lies in this important decision, i.e. the choice of the transformation operator 𝑈. And in this paper, s pin-orbit torque physics would become apparent in a frame rotation that rotates the 𝑍 axis to the magnetizatio n axis 𝒆𝒎 as follows 𝑈𝜂𝑚=𝜂𝑧 ,𝑈𝜎𝑚𝑈†=𝜎𝑧 (18) where 𝜂𝑚 and 𝜂𝑧 are respectively, the eigenstate s along 𝒆𝒎 and 𝒆𝒛. And 𝑈(𝜃𝑚,𝜙𝑚) operates in the space of the magnetic moment 𝒎. The frame rotation takes place in the presence of spin -orbit coupling . Upon transformation, local gauge potentials would appear in the Hamiltonian. Rewriting the Hamiltonian , one has 6 6 𝐻′=(1 2𝑚𝑒)(−𝑖ℏ∇𝑎+𝑒𝐴𝑎+𝑈𝑆𝑎𝑈†)(−𝑖ℏ𝜕𝑎+𝑒𝐴𝑎+𝑈𝑆𝑎𝑈†)+𝐺𝑎𝑏𝑈𝜎𝑎𝑚𝑏𝑈† (19) The physics of curve, spin -orbit coupling, and magnetism is now captured in the expression of a total gauge: ℚ𝑎=𝑒𝐴𝑎+𝑈𝑆𝑎𝑈†. In fact, the rotation gauge 𝑒𝐴𝑎 has previously been associate d with a form of adiabatic spin -tansfer torque [34] that w ould not be further elaborated in this article. As our interest lies in the spin -orbit torque , we will only focus on the transformed spin -orbit gauge also known henceforth as ℱ𝑎=𝑈𝑆𝑎𝑈†. Relevant to our study are the energy terms as follows : 𝐸=𝜓†(1 2𝑚𝑒)(−𝑖ℏ∇𝑎+ℱ𝑎)(−𝑖ℏ𝜕𝑎+ℱ𝑎)𝜓 (20) Dropping the second -order derivative kinetic energy terms, 𝐸𝑖𝑛𝑡=(−𝑖ℏ 2𝑚𝑒)𝜓†[∇𝑎 ℱ𝑎 +ℱ𝑎 𝜕𝑎] 𝜓 (21) Now, we will examine each energy density (∇𝑎 ℱ𝑎,ℱ𝑎 𝜕𝑎 ) term in details: ∇𝑎 ℱ𝑎 → (−𝑖ℏ 2𝑚𝑒)[∂𝑎(𝜓†ℱ𝑎𝜓)−(∂𝑎𝜓†)ℱ𝑎𝜓+𝛤𝑎 (𝜓† ℱ𝑎 𝜓)] (22) ℱ𝑎 𝜕𝑎 → (−𝑖ℏ 2𝑚𝑒)𝜓†ℱ𝑎 𝜕𝑎 𝜓 (23) The expression ∇𝑎 ℱ𝑎 compri ses three terms on the RHS. It is reduced to (−𝑖ℏ 2𝑚𝑒)[−(∂𝑎𝜓†)ℱ𝑎𝜓] when the first and the third term vanish as surface terms as the energy density is integrated over the entire device √𝐺∫[∂𝑎+𝛤𝑎](𝜓†ℱ𝑎𝜓) 𝑑𝑉=√𝐺∫(𝜓†ℱ𝑎𝜓).𝑑𝑆=0 (24) It suffices now to proceed with the remaining terms which combine to make physical sense in terms of 𝑗𝑎 taking on the physical meaning of current density as follows, (−𝑖ℏ 2𝑚)[ ℱ𝑎 𝜓†𝜕𝑎𝜓−(𝜕𝑎𝜓†) ℱ𝑎 𝜓 ] ↔ 𝑗𝑎ℱ𝑎 (25) The infinitesimal bulk is then compressed along 𝒆𝟑, i.e. taking the operation of 𝑞3→0 where (√𝐺)𝑞3→0=√𝑔. As the Z axis is rotated to the magnetic moment (𝒎), and s pin is assumed to align with 𝒎 in an adiabat ic manner and at all times (even as 𝒎 changes spatially) as it propagates, the eigenspinor of the electron would undergo 𝜂𝑚→𝜂𝑧. Refering to the formulation [35] for the volume and surface integrals, ∫𝜓† 𝜓 𝑑𝑉=∫𝜓† 𝜓 𝑑𝑞1 𝑑𝑞2 𝑑𝑞3 √𝐺 (26) With 𝑑𝑆=√𝑔 𝑑𝑞1𝑑𝑞2 ∫𝜓† 𝜓 𝑑𝑉=∫𝜓† 𝜓 𝑓 𝑑𝑆 𝑑𝑞3 (27) 7 7 From the above, 𝑑𝑉=𝑓 𝑑𝑆 𝑑𝑞3. Now, writing ∫𝜓† 𝜓 𝑑𝑉=∫ 𝜒†𝜒 𝑑𝑆 𝑑𝑞3 means the wave - functions are taken to be 𝜓=𝜒 𝜂𝑧 √𝑓 ,𝜓†=𝜂𝑧† 𝜒† √𝑓 and √𝑓=√𝐺 √𝑔. Note that 𝜒=𝜒𝑠𝜒𝑛 are separable scalar function s where 𝜒𝑠 is the surface wave -function, and 𝜒𝑛 is normal to the surface. The energy terms can now be written as follows : 𝜓†ℱ𝑎𝜕𝑎𝜓=(𝜂𝑧† 𝜒† 𝑓 ℱ𝑎 (𝜕𝑎𝜒) 𝜂𝑧+𝜂𝑧† 𝜒† √𝑓 ℱ𝑎 𝜒 𝜂𝑧 ( 𝜕𝑎 𝑓−1 2 )) (28) −(𝜕𝑎𝜓†) ℱ𝑎 𝜓=( −(𝜕𝑎 𝜂𝑧† 𝜒†) 𝑓 ℱ𝑎 𝜒 𝜂𝑧 −𝜂𝑧† 𝜒† √𝑓(𝜕𝑎 𝑓−1 2)ℱ𝑎 𝜒 𝜂𝑧) (29) Combining the above, 𝜓†ℱ𝑎𝜕𝑎𝜓−(𝜕𝑎𝜓†) ℱ𝑎 𝜓= 1 𝑓 ⟨𝜂𝑧\|ℱ𝑎 𝐽𝑎 \|𝜂𝑧⟩ (30) where 𝐽𝑎=𝜒†𝜕𝑎𝜒−(𝜕𝑎𝜒†) 𝜒 is now t he current density. Since ℱ𝑎 orginates from the spin -orbit coupling in a curved space, the entire term 𝑗𝑎ℱ𝑎 would be an interaction energy density (𝐸𝑖𝑛𝑡) related to the coupling of the current flow (𝑗𝑎) with the curved spin -orbit physics . As 𝜒=𝜒𝑠(𝑞1,𝑞2) 𝜒𝑛(𝑞3) , the term 𝑗𝑎ℱ𝑎 is decou pled into 𝑗𝑎ℱ𝑎=(𝑗1ℱ1+𝑗2ℱ2)+𝑗3ℱ3 (31) where 𝑗1ℱ1+𝑗2ℱ2 is the surface current den sity and 𝑗3ℱ3 the normal current density. Recall that ℱ𝑎=𝑈𝑆𝑎𝑈†, and 𝑆𝑎 is summarized below (𝑆1 𝑆2 𝑆3)=𝑚𝑒√𝐺 ℏ(0−𝛼3𝛼2 𝛼30−𝛼1 −𝛼2𝛼10)(𝜎1 𝜎2 𝜎3) (32) To confine the spin -orbit effect to the 2D which is prevalent in physical systems like the Rashba , 2D Dres sellhaus , topological surface states, 2D graphene and silicene, one takes t he limit of 𝑞3→0 and notes that 𝑓→1. It’s worth noting that the term s 𝜂𝑧† 𝜒† √𝑓 ( 𝜕𝑎 𝑓−1 2 )ℱ𝑎 𝜒 𝜂𝑧 would vanish for 𝑎=1,2 as limit 𝑞3→0 is taken. The surviving term 𝜂𝑧† 𝜒† √𝑓 ( 𝜕3 𝑓−1 2 )ℱ3 𝜒 𝜂𝑧 would be expected to capture the physics of the confinement effect . But as steps have been taken to ensure Hermiticty in spin-orbit coupling Eq.(14), the confinement terms of Eq.(28) and Eq.(29) cancel one another. This is an unexpected effect for the spin -orbit torque tha t eliminates a curved -surface confinement effect because of the symmetrization in the current density . With the limit taken, the normal current density is discarded and one should from now on ignore the effect of 𝑆3. We will proceed to the spin-orbit constant which is given by 𝛼𝑛=𝛼′𝐸𝑛. We will now let 𝐸1 and 𝐸2 approach zero, so that only 𝛼3 is retained . The matrix is reduced back the 2D formalism like in the beginning of the article, (𝑆1 𝑆2)=𝑚𝑒√𝑔 ℏ(0−𝛼3 𝛼30)(𝜎1 𝜎2) (33) It’s worth noting that there’s a caveat in the limit taking, i.e. we have let 𝑞3→0 first so that the surface confinement term 𝜂𝑧† 𝜒† √𝑓 ( 𝜕3 𝑓−1 2 )ℱ3 𝜒 𝜂𝑧 survived although it would vanish later because of 8 8 symmetrization. Had the limit of 𝛼1→0,𝛼2→0 been taken first, the confinement term would not have show n up. Therefore, t he general spin-orbit gauge potential s can now be expressed as follows: ℱ1=−𝑈(𝛼3 𝑚𝑒 ℏ√𝑔𝜎2)𝑈† , ℱ2=𝑈(𝛼3 𝑚𝑒 ℏ√𝑔𝜎1)𝑈† (34) Finally, the energy density is 𝐸𝑖𝑛𝑡=𝑗𝑎ℱ𝑎=(𝛼3𝑚 𝑒ℏ√𝑔)[−𝑗1 𝑈𝜎2𝑈†+𝑗2 𝑈𝜎1𝑈†] (35) The energy density in magnetic space is an important development for the study of magnetic torque and flipping via the concept of the anisotropy field. We consider a potential in the magnetic space to arise fr om the energy density 𝐸𝑖𝑛𝑡. The potential is essentially the effective anisotropy field as it is commonly underst ood in magnetic physics: 𝜇 𝑯=𝜕𝐸𝑖𝑛𝑡 𝜕𝒎=𝜕 𝜕𝒎 (𝑗𝑎ℱ𝑎 𝑒) (36) The explicit expression of 𝜇 𝑯 is therefore , given as 𝜇 𝑯=(𝛼3 𝑚𝑒 ℏ√𝑔)[ −𝑗1 𝑒(𝜕 𝜕𝒎𝑈𝜎2𝑈†)+𝑗2 𝑒(𝜕 𝜕𝒎 𝑈𝜎1𝑈†)] (37) The magnetic moment (𝒎) of the ferromagnetic material is superimposed on the curved surface as shown in Fig.3. below. The curved surface can be accessed at every point in space by 𝑹=𝒓(𝑞1,𝑞2)+ 𝑞3𝒆𝟑 that charcaterizes the surface on which 𝒎 is located. Vector 𝒎 will also be indexed by (𝑚1,𝑚2) to reflect its components in the general coordinates of the real space . In that way, the magnetic components will also track the structure of the space in which it is superimposed . The actual orientation of 𝒎 is described by (𝜃𝑚,𝜙𝑚,𝑚) with respect to the Cartesian coordinates , where superscripts of (𝜃𝑚,𝜙𝑚) merely show that these are coordinates for the magnetization. FIG.3 A 2D curve is embeded in a 3D space and characterized by parameters (𝑞1,𝑞2,𝑞3). The magnetization 𝒎 is superimposed on the curved surface. 𝒆𝟑 𝒆𝟐 𝒆𝟏 𝒎=𝑚𝒆𝒎 𝒆𝒙 𝒆𝒚 𝒆𝒛 𝜃 𝒓(𝑞1,𝑞2) 𝒆𝒙 𝒆𝒚 𝒆𝒛 𝜃𝑚 9 9 We will now look into the spin physics and focus our attention on the Pauli matrices. Originally expressed in the genera l corodinates, the Pauli matrices are now re-expressed in the Cartesian coordinates in which the adiabatic physics will be introduced at a later stage . In the physics of forces, and magnetism, the important physical quantities are: magnetic moment (𝒎), Pauli matrix (𝜎), current density (𝑗). There are frame of references under which the physical quantities most relevant to our studies are described, e.g. the Cartesian (𝑥,𝑦,𝑧), and the geneal coordinates (𝑞1,𝑞2,𝑞3). Physical quantities and their dimensions are expressed in both frames. For example, 𝒎=𝑚1𝒆𝟏+ 𝑚2𝒆𝟐=𝑚𝑣∂𝑣𝑎𝒆𝒂 , implies that a physical quantity remains unchanged when expressed under any frame of reference . Explicitly, it will look as follows 𝒎=𝑚𝑥𝒆𝒙+𝑚𝑦𝒆𝒚+𝑚𝑧𝒆𝒛=𝑚1𝒆𝟏+𝑚2𝒆𝟐+𝑚3𝒆𝟑 (38) The two are connected by the transformaton as follows 𝒎=(𝑚1∂1𝑥+𝑚2∂2𝑥+𝑚3∂3𝑥)𝒆𝒙+(𝑚1∂1𝑦+𝑚2∂2𝑦+𝑚3∂3𝑦)𝒆𝒚+(𝑚1∂1𝑧+𝑚2∂2𝑧+𝑚3∂3𝑧)𝒆𝒛 (39) The above provides an illustration of coordinate transformatio n. In the actual context, the same principle is applied to the Pauli matrices to reappear in the Cartesian frame. As a result, t he effective anisotropy field would now be, 𝜇 𝜹𝑯=(𝛼3𝑚𝑒 ℏ𝑚√𝑔)[−𝑗1 𝑒𝜕 𝜕𝒏(∂𝑣2𝑈𝜎𝑣𝑈†)+𝑗2 𝑒𝜕 𝜕𝒏(∂𝑣1𝑈𝜎𝑣𝑈†)] (40) where 𝑣 runs over the Cartesian (𝑥,𝑦,𝑧). In the real space, we note once again that the Pauli matrices in the general coordinates have been re -expressed i n the Cartesian frame whereby the curve effects would be reflected in the quantities of ∂𝑣2 and ∂𝑣1. On the other hand, t he unitary operator 𝑈 is now applied to rotate the magnetic axis (𝒆𝒎) to the (𝑥,𝑦,𝑧) frame , thereby providing the 𝒎 an indirect link in terms of orientation to the general coordinates. It is now possible to associate the Pauli matices with 𝒎 and perform the operation of 𝜕 𝜕𝒏 for the anisotropy field as required in Eq.(38 ). In the following, we provide an explicit demonstration of the 𝑈 operation. Recall that 𝑈 is parameteri zed by (𝜃𝑚,𝜙𝑚). Refer to Fig.4 below and observe that 𝑈 rotates the eigenstate along 𝒆𝒎 to the Z axis about axis 𝒆𝒎𝟐= −𝒆𝒚cos𝜙𝑚+𝒆𝒙sin𝜙𝑚. 𝑈=(cos𝜃𝑚 2sin𝜃𝑚 2𝑒−𝑖𝜙𝑚 −sin𝜃𝑚 2𝑒𝑖𝜙𝑚cos𝜃𝑚 2) (41) Rotation of the magnetic axis (𝒆𝒎) to the (𝑥,𝑦,𝑧) frame is given by 𝑈𝝈𝑈†=𝜎𝑧𝒆𝒎+𝜎𝑎𝒆𝒎𝟏+𝜎𝑚2𝒆𝒎𝟐 (42) Refer to Fig.4 for the d irections implied by the superscripts and subscripts in the equation above. Note that the 𝒎 physical quantities of (𝒆𝒎,𝒆𝒎𝟏,𝒆𝒎𝟐) and (𝜎𝑎,𝜎𝑚2) can all be related to the (𝑥,𝑦,𝑧) frame as follows 𝒆𝒎=𝒆𝒙sin𝜃𝑚cos𝜙𝑚+𝒆𝒚sin𝜃𝑚sin𝜙𝑚+𝒆𝒛cos𝜃𝑚 =𝑛𝑥𝒆𝒙+𝑛𝑦𝒆𝒚+𝑛𝑧𝒆𝒛 10 10 (43) 𝒆𝒎𝟏=−𝒆𝒚cos𝜃𝑚sin𝜙𝑚−𝒆𝒙cos𝜃𝑚cos𝜙𝑚+𝒆𝒛sin𝜃𝑚 (44) 𝒆𝒎𝟐=−𝒆𝒚cos𝜙𝑚+𝒆𝒙sin𝜙𝑚 (45) 𝜎𝑎=−𝜎𝑥cos𝜙𝑚−𝜎𝑦sin𝜙𝑚 (46) 𝜎𝑚2=𝜎𝑥sin𝜙𝑚−𝜎𝑦cos𝜙𝑚 (47) The following is a schematic showing the magnetic moment vectors in the Cartesian frame. FIG.4. Spin rotation about 𝒆𝒎𝟐 moves the Z axis to the magnetic moment axis 𝒆𝒎, and the 𝒆𝒂 axis to 𝒆𝒎𝟏 or vice versa. Following the mark of 𝑈𝜎𝑣𝑈† in Eq.(40 ) where 𝑣 runs over (𝑥,𝑦,𝑧), it is handy to write down the following for the sake of easy inspection. 𝑈𝝈𝑈†＝𝑈𝜎𝑥𝑈†𝒆𝒙+𝑈𝜎𝑦𝑈†𝒆𝒚+𝑈𝜎𝑧𝑈†𝒆𝒛 (48) Using E qs. (42) to (47), the process of linking the original magnetic axis through unitary rotation to the (𝑥,𝑦,𝑧) frame is complete, where 𝑈𝜎𝑥𝑈†=sin𝜃𝑚cos𝜙𝑚 𝜎𝑧+𝑓(𝜎𝑎,𝜎m2) (49) 𝑈𝜎𝑦𝑈†=sin𝜃𝑚sin𝜙𝑚 𝜎𝑧+𝑔(𝜎𝑎,𝜎m2) (50) 𝑈𝜎𝑧𝑈†=cos𝜃𝑚 𝜎𝑧+ℎ(𝜎𝑎) (51) Note that 𝑓,𝑔,ℎ are linear combination s of 𝜎𝑎,𝜎𝑚2 and these terms are non -issue s as they would vanish later . At this point, an important physical step is taken, i.e. the adiabatic approximation in which it is assumed that electron spin aligns along the new Z axis that has been rotated to 𝒆𝒎. Under the adiabatic spin alignment whereby the following is resulted: ⟨𝜂𝑧\|𝜎𝑥\|𝜂𝑧⟩=0 ,⟨𝜂𝑧\|𝜎𝑦\|𝜂𝑧⟩=0, ⟨𝜂𝑧\|𝜎𝑧\|𝜂𝑧⟩=1, functions 𝑓,𝑔,ℎ vanish while the following remains ⟨𝜂𝑧\|𝑈𝜎𝑥𝑈†\|𝜂𝑧⟩=𝑛𝑥 ,⟨𝜂𝑧\|𝑈𝜎𝑦𝑈†\|𝜂𝑧⟩=𝑛𝑦,⟨𝜂𝑧\|𝑈𝜎𝑧𝑈†\|𝜂𝑧⟩=𝑛𝑧 𝑥 𝑦 𝑧 𝒆𝒎𝟐=−𝒆𝒚cos𝜙𝑚+𝒆𝒙sin𝜙𝑚 𝜃𝑚 𝜙𝑚 𝒎 𝜙𝑚 𝒆𝒎 𝒆𝒎𝟏 𝒆𝒂 11 11 (52) It thus follows that ⟨𝜂𝑧\|𝜇 𝜹𝑯\|𝜂𝑧⟩=(𝛼3𝑚 ℏ𝑀𝑠√𝑔)[−𝑗1 𝑒𝜕 𝜕𝒏(∂𝑣2𝑛𝑣)+𝑗2 𝑒𝜕 𝜕𝒏(∂𝑣1𝑛𝑣)] =(𝛼3𝑚 𝑒ℏ𝑀𝑠√𝑔)[−𝑗1(∂𝑣2𝒆𝒗)+𝑗2(∂𝑣1𝒆𝒗)] (53) Note in the above that 𝑛𝑣 carries a contravariant index. But, 𝑛𝑎 of 𝜕 𝜕𝒏=𝜕 𝜕𝑛𝑎𝒆𝒂 is covariant . As a result, ⟨𝜂𝑧\|𝜇 𝜹𝑯\|𝜂𝑧⟩=(𝛼3𝑚 𝑒ℏ𝑀𝑠√𝑔)[−𝑗1𝒆𝟐+𝑗2𝒆𝟏] (54) Equation (54 ) is the general form of what we called the spin-orbit anisotropy field . This is another important result of this paper . Note that (𝑗1,𝑗2) and (𝒆𝟏,𝒆𝟐) can be found by transforming their Cartesian counterparts to the desired coordinates in a contravariant manner. Such formulation is important because as device becomes smaller, the physics of local geometries on spin -orbit effect and its torque shall not be neglected. The general formulation allow s spin and magnetic dynamic to be studied o n local curved surfaces. With the spin -orbit anisotropy field derived, the spin -orbit torque is a straightforward cross product with the 𝒎 𝝉=−𝒎×⟨𝑧\|𝜇 𝜹𝑯\|𝑧⟩ (55) As the formulation is provided in general coordinates, the spin -orbit effective field can be derived for any surfaces to be characterized by 𝑹=𝒓′(𝑞1,𝑞2)+𝑞3𝒆𝟑. For illustration, w e will now take the example s of the spherical, cylindrical and flat Cartesian surfaces. In the spherical system , the coordinates are 𝑹=𝒓′(𝜃,𝜙)+𝛿𝑟𝒆𝒓 and ⟨𝑧\|𝜇 𝜹𝑯\|𝑧⟩=(𝛼3𝑚 𝑒ℏ𝑀𝑠√𝑔)(−𝜕𝜃 𝜕𝑣𝜕𝜙 𝜕𝑣′+𝜕𝜙 𝜕𝑣𝜕𝜃 𝜕𝑣′) 𝑗𝑣𝒆𝒗′ (56) The current carrying carriers are c onstrained to the surface , i.e. 𝛿𝑟=0. The spherical surface is then accessed by 𝒓′=(𝑟sin𝜃cos𝜙,𝑟sin𝜃sin𝜙 ,𝑟cos𝜃) ( 𝒆𝒙 𝒆𝒚 𝒆𝒛)= ( 𝜕𝜃 𝜕𝑥𝜕𝜙 𝜕𝑥𝜕𝑟 𝜕𝑥 𝜕𝜃 𝜕𝑦𝜕𝜙 𝜕𝑦𝜕𝑟 𝜕𝑦 𝜕𝜃 𝜕𝑧𝜕𝜙 𝜕𝑧𝜕𝑟 𝜕𝑧) (𝒆𝜽 𝒆𝝓 𝒆𝑹)= ( cos𝜙cos𝜃 𝑟sin𝜙 −𝑟sin𝜃sin𝜃cos𝜙 sin𝜙cos𝜃 𝑟cos𝜙 𝑟sin𝜃sin𝜃sin𝜙 −sin𝜃 𝑟0 cos𝜃) (𝒆𝜽 𝒆𝝓 𝒆𝒓) (57) Note that 𝑣,𝑣′ run over (𝑥,𝑦,𝑧) and summation is only non -vanishing for 𝑣≠𝑣′. Meanwhile, 𝑔= √(𝑔𝜃𝜃𝑔𝜙𝜙−𝑔𝜃𝜙𝑔𝜙𝜃)𝑔𝑟𝑟=𝑟2sin𝜃. ⟨𝑧\|𝜇 𝜹𝑯\|𝑧⟩=(𝛼3𝑚 𝑒ℏ𝑀𝑠√𝑔)(1 𝑟2tan𝜃(−𝑗𝑥𝒆𝒚+𝑗𝑦𝒆𝒙)+sin𝜙 𝑟2(𝑗𝑥𝒆𝒛−𝑗𝑧𝒆𝒙)+cos𝜙 𝑟2(𝑗𝑧𝒆𝒚−𝑗𝑦𝒆𝒛)) 12 12 =(𝛼3𝑚 𝑒ℏ𝑀𝑠)(cos𝜃(−𝑗𝑥𝒆𝒚+𝑗𝑦𝒆𝒙)+sin𝜃sin𝜙(𝑗𝑥𝒆𝒛−𝑗𝑧𝒆𝒙)+sin𝜃cos𝜙(𝑗𝑧𝒆𝒚−𝑗𝑦𝒆𝒛)) (58) With the above, the spin -orbit anisotropy field can be estimated over the entire spherical surface provided (𝑗𝑥,𝑗𝑦,𝑗𝑧) is computed or measured over the surface. Likewise, with (𝑚𝑥,𝑚𝑦,𝑚𝑧) computed or measured over the surface , the spin -orbit torque can be determined at every point of the surface. Figure 5 illustrates how current flows into and out of the nanoscale structures. D evice s’ central region can be fabri cated to these structures and one can regard the current to flow from source to drain. (a) (b) FIG.5 (a) A spherical surface across which current flows from left to right as indicated by arrows. (b) A cylindrical surface across which current flows along the z direction as shown by the arrow. In the cylindrical s ystem , the coordinates are 𝑹=𝒓′(𝜙,𝑧)+𝛿𝑟𝒆𝒓. The current carrying carriers are constrained to the surface, i.e. 𝛿𝑟=0. The cylindrical surface is then accessed by 𝒓′= (𝑟cos𝜙,𝑟sin𝜙 ,𝑧). ( 𝒆𝒙 𝒆𝒚 𝒆𝒛)= ( 𝜕𝜙 𝜕𝑥𝜕𝑧 𝜕𝑥𝜕𝑟 𝜕𝑥 𝜕𝜙 𝜕𝑦𝜕𝑧 𝜕𝑦𝜕𝑟 𝜕𝑦 𝜕𝜙 𝜕𝑧𝜕𝑧 𝜕𝑧𝜕𝑟 𝜕𝑧) (𝒆𝝓 𝒆𝒛 𝒆𝒓)= ( −sin𝜙 𝑟0cos𝜙 +cos𝜙 𝑟0sin𝜙 010) (𝒆𝝓 𝒆𝒛 𝒆𝒓) (59) It follows that ⟨𝑧\|𝜇 𝜹𝑯\|𝑧⟩=(𝛼3𝑚 𝑒ℏ𝑀𝑠√𝑔)(sin𝜙 𝑟(𝑗𝑥𝒆𝒛−𝑗𝑧𝒆𝒙)+cos𝜙 𝑟(𝑗𝑧𝒆𝒚−𝑗𝑦𝒆𝒛)) (60) Note that 𝑣,𝑣′ run over (𝑥,𝑦,𝑧) and summation is only non -vanishing for 𝑣≠𝑣′. In the meantime , 𝑔=√(𝑔𝜙𝜙𝑔𝑧𝑧−𝑔𝜙𝑧𝑔𝑧𝜙)𝑔𝑟𝑟=𝑟. Thus, ⟨𝑧\|𝜇 𝜹𝑯\|𝑧⟩=(𝛼3𝑚 𝑒ℏ𝑀𝑠)(sin𝜙(𝑗𝑥𝒆𝒛−𝑗𝑧𝒆𝒙)+cos𝜙(𝑗𝑧𝒆𝒚−𝑗𝑦𝒆𝒛)) (61) 𝑥 𝑧 𝑦 𝜃 𝜙 𝑧 𝑥 𝑦 𝜙 𝑟 𝑧 13 13 Last, i n the Cartesian system , physical quantities are re-expressed in the x-y basis. The results are ⟨𝑧\|𝜇 𝜹𝑯\|𝑧⟩=(𝛼𝑧𝑚 𝑒ℏ𝑀𝑠)[−𝑗𝑥𝒆𝒚+𝑗𝑦𝒆𝒙] (62) The spin -orbit anisotropy field above when substituted in the standard expression for spin torque in a flat surface device , would lead to the to the well -known, experimentally proven [20, 21 ] spin-orbit torque , also known to the experimental community as the field -like spin -orbit torque. This is, however, not the same as another form of spin-orbit torque aka the anti -damping spin -orbit torque [22]. In other words, what we have derived is the general formulation for the field -like spin -orbit spin torque. CONCLUSION We have provided a general formulation of the spin -orbit coupling on a curved surface in Eq.(5) with 𝛼3 and hence (𝑆1,𝑆2) properly defined and understood. We had then consider ed a curved ferromagnetic and oxide heterostructure to give rise to a spin -orbit torque. A proper choice of the transformation operator 𝑈 that rotates the Z axis to the magnetization axis 𝒆𝒎 leads to the derivation of the spin -orbit anisotropy field and hence the spi n-orbit torque . We note that an unexpected effect that arises in the symmetriation of the current density actually eliminates the curved -surface confinement. Finally, with the adia batic approximation, we completed the general formluation of the spin-orbit anisotropy field in Eq.(54 ) and hence the spin-orbit torque that can be computed over the entire surface of devices of any shape . We provided examples in spherical, cylindrical and the Cartesian surfaces Acknowledgments We would like to thank the National Science and Technology Council of Taiwan for supporting this work under Grant No. 110 -2112 -M-034-001-MY3. ORCID iD Seng Ghee Tan https://orcid.org/0000 -0002 -3233 -9207 References [1] Igor Žutić, Jaroslav Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). [2] T. Fujita, M. B. A. Jalil, S. G. Tan, Shuichi Murakami, J. Appl. Phys. [Appl. Phys. Rev.] 110, 1213 01 (2011); S. G. Tan, M. B. A. Jalil, Xiong -Jun Liu, and T. Fujita, Phys. Rev. B 78, 245321 (2008). [3] Seng Ghee Tan, Mansoor B.A. Jalil, Introduction to the Physics of Nanoelectronics, Woodhead Publishing, Cambridge (Elsevier), U.K. (2012). [4] Gen Tatara , Physica E: Low -dimensi onal Systems and Nanostructures 106, 208 ( 2019 ). [5] Seng Ghee Tan, Son -Hsien Chen, Cong Son Ho, Che -Chun Huang, Mansoor B.A. Jalil, Ching Ray Chang, Shuichi Murakami, Physics Reports 882, Pages 1-36 (2020). [6] A. H. Castro Neto , F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009). [7] M.A.H. Vozmediano, M.I. Katsnelson, F. Guinea, Physics Reports 496, Pages 109 -148 (2010). [8] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010). [9] X.-L. Qi and S. -C. Zhang, Rev. Mod. Phys. 83 1057 (2011). [10] Shengyuan A. Yang, SPIN 6, 1640003 (2016) . [11] N.P. Armitage, E.J. Mele, and Ashvin Vishwanath , Rev. Mod. Phys. 90, 015001 (2018). 14 14 [12] Jean Dalibard, Fabrice Gerbier, Gediminas Juzeliūnas, and Patrik Öhberg, Rev. Mod. Phys. 83, 1523 (2011). [13] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki , Rev. Mod. Phys. 81, 865 (2009) . [14] Che -Chun Huang , Seng Ghee Tan , Ching -Ray Chang, New J. Phys. 25 063026 (2023). [15] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004). [16] S. G. Tan, M. B. A. Jalil, and Xiong -Jun Liu, arXiv:0705.3502, (2007). [17] K. Obata and G. Tatara, Phys. Rev. B 77, 214429 (2008). [18] A. Manchon, S. Zhang, Phys. Rev. B 78, 2124 05 (2008). [19] S. G. Tan, M. B. A. Jalil, T. Fujita, and X. J. Liu, Ann. Phys. (NY) 326, 207 (2011). [20] Ioan Mihai Miron, Gilles Gaudin, Stéphane Auffret, Bernard Rodmacq, Alain Schuh l, Stefania Pizzini, Jan Vogel, Pietro Gambardella Nature Materials 9, 230 (2010) . [21] Junyeon Kim, Jaivardhan Sinha, Masamitsu Hayashi, Michihiko Yamanouchi, Shunsuke Fukami, Tetsuhiro Suzuki, Seiji Mitani , Hideo Ohno , Nature Materials 12, 240 (2013) . [22] H. Kurebayashi, Jairo Sinova, D. Fang, A. C. Irvine, T. D. Skinner , J. Wunderlich, V. Novák, R. P. Campion, B. L. Gallagher, E. K. Vehstedt, L. P. Zârbo, K. Výborný, A. J. Ferguson & T. Jungwirth, Nature Nanotechnology 9, 211 (2014). [23] Yositake Takane, Ken -Ichiro Imura, J. Phys. Soc. Japan 82, 074712 (2013). [24] Ken -Ichiro Imura, Yositake Takane, and Akihiro Tanaka, Phys. Rev. B 84, 195406 (2011). [25] V. Parente, P. Lucignano, P. Vitale, A. Tagliacozzo, and F. Guinea, Phys. Rev. B 83, 075424 (2011). [26] Dung -Hai Lee, Phys. Rev. Letts. 103, 196804 (2009) . [27] Paola Gentile, Mario Cuoco, Carmine, Phys. Rev. Letts. 115, 256801 (2015). [28] Zhuo Bin Siu, Jian -Yuan Chang, Seng Ghee Tan, Mansoor B. A. Jalil, Ching -Ray Chang, Sci. Reps. 8, 16497 (2018). [29] Tai -Chung Cheng, Jiann -Yeu Chen, Ching -Ray Chang, Phys. R ev. B 84, 214423 (2011). [30] Ming -Hao Liu, Jhih -Sheng Wu, Son -Hsien Chen, and Ching -Ray Chang, Phys. Rev. B 84, 085307 (2011). [31] Z.B. Siu, M.B.A. Jalil, S G Tan, J. Appl. Phys. 115 17C513 (2014). [32] Zhuo Bin Siu, Mansoor B. A. Jalil, and Seng Ghee T an, J. Appl. Phys. 121, 233902 (201 4). [33] Zhuo B in Siu, Mansoor B . A. Jalil, and Seng Ghee Tan, Nanotechnology 33, 335703 (2022). [34] Ya. B. Bazaliy, B. A. Jones, Shou -Cheng Zhang, Phys. Rev. B 57, R3213 (1998). [35] R. C. T. da Costa, Phys. Rev. A 23, (1982).
0707.4493v1.Temperature_Dependence_of_Rashba_Spin_orbit_Coupling_in_Quantum_Wells.pdf	1 Temperature Dependence of Rashba Spin-orbit Coupling in Quantum Wells P.S. Eldridge1, W.J.H. Leyland2, P.G. Lagoudakis1, O.Z.Karimov1, M. Henini3, D.Taylor3, R.T. Phillips2 and R.T. Harley1 1School of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, UK. 2Cavendish Laboratory, Madingley Road, Cambridge, CB3 0HE, UK. 3School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 4RD, UK. Abstract We perform an all-optical spin-dynamic measurement of the Rashba spin- orbit interaction in (110)-oriented GaAs/AlGaAs quantum wells. The crystallographic direction of quantum confinement allows us to disentangle the contributions to spin- orbit coupling from the structural inversion asymmetry (Rashba term) and the bulk inversion asymmetry. We observe an unexpected temperature dependence of the Rashba spin-orbit interaction strength that signifies the importance of the usually neglected higher-order terms of the Rashba coupling. pdfMachine by Broadgun Software - a great PDF writer! - a great PDF creator! - http://www.pdfmachine.com http://www.broadgun.com 2 Intense worldwide interest is being focused on new semiconductor spintronic and spin- optronic quantum devices in which electronic spin replaces charge for data processing or is used to control optical polarisation. Indeed manipulation of electron spins has been signposted as the preferred route to quantum computing [1]. Progress towards realistic devices depends on engineering the spin-orbit interactions that result in effective magnetic fields seen by the electron spins under application of external electric fields. A classic example is the Datta and Das spin transistor, wherein the precession of spin polarised carriers confined in a plane is controlled by a gate voltage which tunes the Rashba or structural inversion asymmetry (SIA) component of spin-orbit coupling [2]. Whereas the ability to tune the Rashba (SIA) coupling strength originates in the field-induced spin-splitting of the electronic bands, other contributions to the spin-splitting, from bulk inversion asymmetry (BIA or Dresselhaus coupling) and natural interface asymmetry in heterostructures (NIA), complicate direct characterisation of the Rashba term. Disentangling and evaluating the contribution of the different spin-orbit coupling mechanisms to the spin-splitting of the electronic bands is of utmost importance for the engineering of spintronic devices. Most of the studies that focus on the characterisation of the Rashba coupling have either neglected the contribution of the Dresselhaus coupling or have measured the ratio of the strength of the two mechanisms [3,4]. Here we have designed and grown quantum well heterostructures that utilise crystal asymmetries in a way that allows us, through combined optical measurements of spin relaxation and of electron mobility, to separate the terms and measure directly the strength of the Rashba coupling. The values are in good quantitative agreement with k.p-theoretical estimations but unexpectedly, we observe a temperature dependence of the Rashba coefficient that signifies the importance of usually neglected higher order terms in the Rashba spin-orbit coupling [5]. The spin relaxation of a non-equilibrium population of electron spins in non- centrosymmetric semiconductors may involve several mechanisms [1], the dominant one in all except p-type material is that identified by Dyakonov, Perel and Kachorovskii (DPK)[6,7]. The driving force for spin reorientation and therefore loss of spin memory is the combination of inversion asymmetry and the spin-orbit interaction; as an electron propagates its spin tends to precess. The corresponding precession vector (k), which describes conduction band spin- splitting, varies in magnitude and direction according to the electrons wavevector k. Strong scattering of the electron wavevector randomises the precession and causes spin relaxation. The vector (k) is the sum of the three components described above and denoted BIA(k), SIA(k) and NIA(k) [1,8] . For a quantum well grown on a (110)-oriented substrate the interface component NIA(k) is zero [1,9]. Furthermore, since electron motion is confined to the (110) plane, BIA(k) is, by symmetry, normal to the plane, parallel or antiparallel to the growth axis [110], for all electron wavevectors. Therefore, the BIA term makes no contribution to spin relaxation along the growth axis [7]. Thus using this configuration we can measure directly the spin relaxation due to the structural inversion asymmetry alone and derive the strength of the Rashba coupling, SIA(k). The DPK mechanism gives the relaxation rate for the component of a spin population along a particular axis, i, as [1] ),1\|\|( ,21 , p pis (1) where <2> is the averaged square component of (k) perpendicular to the axis i taken over the spin-oriented population and p* is the momentum scattering time of an electron. Furthermore for a symmetrical quantum well, to first order, the SIA term has the form [8,10] SIA(k) = (e/ħ) Fk (2) 3 where is the Rashba coefficient and F the applied electric field. By applying F along the growth axis, SIA(k) lies in the quantum well plane and thus causes DPK spin relaxation along the growth axis. Squaring eq. 2 and taking the thermal average assuming kBT>> ħ SIA gives 2TBkm22F22e22k2F22e22)SIA( (3) where m is the electron effective mass and kB is Boltzmanns constant. Substitution eq.3 in eq. 1 shows that the spin relaxation rate should be linear in F2 and the Rashba coefficient becomes 21 * psB2 )Tkm2( eF (4) Thus we can obtain the Rashba coefficient from combined measurements of momentum relaxation time and of spin relaxation in applied electric field. Our sample consists of a GaAs/Al0.4Ga0.6As p+-i-n+ structure grown on a semi-insulating (110)-oriented GaAs substrate. The insulating portion of the structure comprises 100nm layers of undoped AlGaAs on each side of a stack of 20 7.5 nm undoped GaAs quantum wells with 12 nm undoped AlGaAs barriers. For the pump-probe measurements of the spin-relaxation time, s, a portion of the wafer is processed into a mesa device 400 microns in diameter with an annular metal contact to the top n+ layer to allow optical access to the quantum wells and a second contact to the lower p+ layer. The electric field is varied by means of applied bias voltage. The total applied electric field F is obtained as the sum of that due to the bias and that to the built-in electric field of the pin structure which we calculate to be 2.80 106 Vm-1. All measurements are made at temperatures above 80K in order to reveal effects of free electrons rather than excitons in the quantum wells [11]. Photocurrent measurements in the pin device as a function of reverse bias and of photon energy reveal evidence of resonant tunnelling between n=1 and n=2 confined electron states in adjacent wells above about 3 volts. In the measurements reported here we therefore concentrate on applied bias less than 3 volts equivalent to F 8 106 Vm-1. Under these conditions the electrons are resonantly excited into the n=1 confined state of a quantum well and can be considered to remain there until recombination, thermal excitation over the barriers being negligible. The momentum relaxation time, p, is obtained from measurement of the electron diffusion coefficient using a transient spin-grating method on an unprocessed portion of the same wafer. This gives p* at only one value of transverse field namely the built-in field, 2.80 106 Vm-1, and in our analysis we assume that it has no significant dependence on the field. The spin relaxation of the electrons is investigated using a picosecond-resolution polarised pump-probe reflection technique (see figure.1a) [12]. Wavelength-degenerate circularly polarised pump and delayed linearly polarised probe pulses from a mode-locked Ti-sapphire laser are focused at close to normal incidence on the sample and tuned to the n=1 heavy-hole to conduction band transition. The pulse duration is ~1.5 ps and repetition frequency is 75 MHz. Absorption of each pump pulse generates a photoexcited population of electrons spin-polarised along the growth axis. The time evolution of the photoexcited population and of the spin polarisation <Sz>(t) are monitored by measuring pump-induced changes of, respectively, probe reflection R and of probe polarisation rotation as functions of probe pulse delay. These measurements are combined to give the spin relaxation rate s-1. They also show that the recombination time of photoexcited carriers, r, is typically five times longer than s. The pump beam intensity is typically 0.5 mW focused to a 60-micron-diameter spot giving an estimated photoexcited spin-polarised electron density Nex~109 cm−2; the probe intensity is 25% of the pump. Figure 2 shows s-1 vs F2 for three different temperatures. For fields above ~3 106 V m-1 4 the relationship is linear within experimental uncertainty; the lines represent best fits to the experimental points from which the Rashba coefficient is obtained. For lower values of field, s-1 tends to a constant value which may be associated with imperfections of the interfaces [13] or Bir-Aronov-Pikus spin relaxation [1] due to accumulation of photoexcited carriers under forward bias of the pin structure. The spin-grating measurements (see figure 1b) [14, 15] are made using twin 0.5 mW pump beams from a 200 fs pulse-length mode-locked Ti-sapphire laser tuned to the n=1 valence- conduction band transition. The beams are linearly polarised at 90 degrees to one another and incident on the sample at 4.1 degrees to the normal. This produces interference fringes of polarisation but not intensity, resulting in a transient grating of spin population with a pitch ~ 5.7 microns. The focal spot size on the sample is again of order 60 microns giving excitation density Nex ~109 cm-2. The decay rate of the amplitude of such a grating is given by [14] 1 r1 s22 s4D (5) where Ds is the electron spin-diffusion coefficient. The decay is monitored by measuring first- order diffraction, in reflection geometry, using a delayed 0.25 mW linear polarised probe beam from the same laser, incident on the sample at normal incidence. Signal to noise is enhanced by use of an optical heterodyne detection scheme [15]; the decay rate of the diffracted intensity is 2. Since the sample is undoped we can equate the electron spin-diffusion coefficient to the diffusion coefficient De and obtain the electron mobility from the Einstein equation = (e/kBT)De. Figure 3 shows an example of a measured decay together with the extracted values of electron mobility as a function of temperature. The grating decay rate (and therefore De) is found to be insensitive to temperature so that ~ T-1 (see Fig.3). This temperature dependence is as expected for a non-degenerate two-dimensional electron system, that is with constant density of states, and dominant phonon scattering with probability ~kBT. From the mobility we obtain the ensemble momentum relaxation time p= m/e and since we are dealing with intrinsic material with negligible electron-electron scattering we can equate this to the momentum scattering time p [1,16] Figure 4 shows the values of the Rashba coefficient obtained by combining the two sets of measurements. The value is approximately 0.1 nm2 however there is a clear upward trend which is consistent with a linear increase of the Rashba coefficient with electron kinetic energy. The solid curve is based on the 8-band k.p treatment given by Winkler [10]. The spin splitting is given as v2 0g2 g222SIAF\|k\| )E(1 E1 3PeF\|k\|e\|\| (6) where Eg is the band gap of GaAs, 0 the spin-orbit splitting in the valence band and P is Kanes momentum matrix element [17]. Fv is the effective electric field in the valence band and is given by [10] F67.1 1FF cv v (7) v and c being the valence and conduction band offsets between GaAs and AlGaAs which we take to be in the ratio 3: 2. The weak temperature dependence of the curve results from the temperature dependence of Eg and recently identified temperature dependence of P [18]. This theoretical estimate is in satisfactory agreement with the magnitude of the experimental values of Rashba coefficient but it does not reproduce the observed temperature dependence. We note that extension of the k.p treatment to 14 bands, following [10,19], increases the calculated values by less than 2%. The dotted curve in Fig. 4 is the k.p theory plus an empirical term linear in temperature. Within the experimental uncertainties this reproduces the data reasonably 5 well. As the photoexcited electron gas is nondegenerate, the additional linear temperature dependence suggests that the Rashba coefficient has an additional approximately linear dependence on the electrons kinetic energy. This is reminiscent of the dependence of the Zeeman splitting on kinetic energy [20] which in turn gives rise to the observed increase of the effective electron g-factor with quantum confinement in GaAs/AlGaAs quantum wells [21]. The observed dependence signifies the importance of the usually neglected higher order terms in the Rashba Hamiltonian. In conclusion, by combined measurements of spin relaxation and of electron mobility in an undoped and nominally inversion symmetric (110)-oriented quantum wells in a pin structure, we have been able to investigate directly the electric field spin-splitting of the conduction band without interfering effects from bulk inversion asymmetry. The observed splittings are in qualitative agreement with a theoretical k.p calculation [10] but also reveal an unexpected significant temperature dependence. The observation of the temperature dependence of the Rashba coefficient is important for developing an understanding of the fundamental interactions in semiconductor nanostructures and for engineering spintronic devices. We acknowledge useful discussions with Roland Winkler and Xavier Cartoixa and financial support of the Engineering and Physical Sciences Research Council (EPSRC). References [1] For a reviews see: F.Meier and B.P.Zakharchenya (ed) 1984 Optical Orientation Modern Problems in Condensed Matter Science (Amsterdam: North-Holland); Semiconductor Spintronics and Quantum Computation ed D.D.Awschalom et al (Berlin: Springer); ibid chapter 4, Spin Dynamics in Semiconductors by M.E.Flatté, J.M.Byers and W.H.Lau 2002. [2] S.Datta and B.Das, App. Phys. Lett. 56, 665 (1990) [3] S.D.Ganichev, V.V.Bel'kov, L.E.Golub, E.L.Ivchenko, Petra Schneider, S. Giglberger, J.Eroms, J.De Boeck, G.Borghs, W.Wegscheider, D.Weiss, and W.Prettl, Phys. Rev. Lett. 92, 256601 (2004); S.Giglberger, L.E.Golub, V.V.Belkov, S.N.Danilov, D.Schuh, C.Gerl, F.Rohlfing, J.Stahl, W.Wegscheider, D.Weiss, W.Prettl, and S.D.Ganichev, Physical Review B75 035327 (2007) [4] Takaaki Koga, Junsaku Nitta, Tatsushi Akazaki, and Hideaki Takayanagi, Phys. Rev. Lett. 89, 046801 (2002) [5] X.Cartoixà, L.-W.Wang, D.Z.-Y.Ting, and Y.-C.Chang, Physical Review B73 205341 (2006) [6] M.I.Dyakonov and V.I.Perel 1971 Sov. Phys. JETP 33 1053 [7] M.I.Dyakonov and V.Yu. Kachorovskii 1986 Sov. Phys. Semicond. 20 110 [8] Optical Spectroscopy of Semiconductor Nanostructures by E.L.Ivchenko (Alpha Science 2005) [9] K.C.Hall, K.Gründoðdu, E.Altunkaya, W.H.Lau, M.E.Flatté, T.F.Boggess, J.J.Zinck, W.B.Barvosa-Carter and S.L.Skeith, Physical Review B68 115311 (2003) [10] R.Winkler Physica E22 450 (2004); Spin-orbit Coupling Effects in Two-Dimensional Electron and Hole Systems by R. Winkler (Springer 2003) [11] A.Malinowski, R.S.Britton, T.Grevatt, R.T.Harley, D.A.Ritchie, and M.Y.Symmonds, Phys.Rev. B62, 13034-39 (2000). [12] R.T.Harley, O.Z.Karimov, and M.Henini, J. Phys. D 36, 2198 (2003). [13] O.Z. Karimov, G.H.John, R.T.Harley W.H.Lau M.E.Flatté, M.Henini and R. Airey, Phys. Rev. Lett 91 (2003) 246601. [14] A.R.Cameron, P.Riblet and A.Miller, Phys. Rev. Lett. 76 4793 (1996) [15] W.J.H. Leyland, PhD Thesis (Cambridge, 2007) in preparation. 6 [16] W. J. H. Leyland, G. H. John, and R. T. Harley, M. M. Glazov, E. L. Ivchenko, D. A. Ritchie, I. Farrer,A. J. Shields, M. Henini, Physical Review B75 165309 (2007) and references therein. [17] E.O. Kane, J.Phys.Chem.Solids 1 249 (1957) [18] J. Hubner, S. Dohrmann, D. Hagele and M. Oesteich, arXiv:cond-mat/0608534 (2006) [19] W. Knap, C. Skierbiszewski, A. Zduniak, E. Litwin-Staszewska, D. Bertho, F. Kobbi, J. L. Robert, G. E. Pikus, F. G. Pikus, S. V. Iordanskii, V. Mosser, K. Zekentes, Yu. B. Lyanda- Geller, Phys. Rev. B53, 3912 (1996) [20] M. A. Hopkins, R. J. Nicholas, P. Pfeffer, W. Zawadzki, D. Gauthier, J. C. Portal and M. A. DiForte-Poisson, Semicond. Sci. Technol. 2, 568 (1987) [21] M. J. Snelling, G. P. Flinn, A. S. Plaut, R. T. Harley, A. C. Tropper, R. Eccleston and C. C. Phillips, Phys. Rev. B 44, 11345 (1991) 7 pump probe signal Nex<Sz> pump 1 pump 2 probe signal Nex<Sz> . (a) (b) Figure 1. Experimental configurations for measurements of (a) spin relaxation and (b) spin diffusion. Upper diagrams, incident pump and probe beams and their polarisations; lower diagrams, profile of focussed pump spot with (a) a spin polarised population and (b) spin grating with spacing . 8 0 2 4 605101520 (110)-oriented MQW 200K 170K 80KSpin relaxation rate (ns-1) (Electric field)2 (V2m-2x10-13) Figure 2. Electric field dependence of spin relaxation rate at three temperatures. The high field linear regions of the graphs extrapolate to the origin and the slopes are used to determine the Rashba coefficient 9 60 100 3005000.312Mobility (m2 V-1 s-1) Temperature (K)0 100 200 3000.010.11 Spin grating decay T=221K 2=20.4±0.4 ps-1Diffracted power (arb. units) Probe delay (ps)T-1 Figure 3. Logarithmic plot of the electron mobility, in a sample from the same wafer as data of Fig.2, determined by the spin-grating method. The electric field is the built-in field of the pin structure, 2.80 106 Vm-1. The T-1 temperature dependence is as expected for a non-degenerate two- dimensional electron system with dominant phonon scattering. Inset is a typical grating decay signal at 221 K. 10 0501001502002503000.000.050.100.15 k.p theory (k.p)+1.75x10-4TRashba coefft. (nm2) Temperature (K) Figure 4. Rashba coefficient obtained from combination of spin relaxation and mobility measurements. Solid curve is 8-band k.p calculation including temperature dependence of band edges and of interband momentum matrix element. Dotted curve is k.p theory plus an empirical linear-in-T term.
1204.2189v2.Spin_current_absorption_by_inhomogeneous_spin_orbit_coupling.pdf	arXiv:1204.2189v2 [cond-mat.mes-hall] 3 Oct 2012Spin-current absorption by inhomogeneous spin-orbit coup ling Kazuhiro Tsutsui,1,∗Kazuhiro Hosono,2and Takehito Yokoyama1 1Department of Physics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan 2International Center of Materials Nanoarchitectonics (WP I-MANA), Namiki 1-1, Tsukuba 305-0044, Japan (Dated: August 27, 2018) We investigate the spin-current absorption induced by an in homogeneous spin-orbit coupling due to impurities in metals. We consider the system with spin cur rents driven by the electric ﬁeld or the spin accumulation. The resulting diﬀusive spin current s, including the gradient of the spin-orbit coupling strength, indicate the spin-current absorption a t the interface, which is exempliﬁed with experimentally relevant setups. PACS numbers: 72.25.Ba, 72.25.Mk I. INTRODUCTION Spintronics aims to utilize not only the charge degree of freedom but also the spin degree of freedom1. Spin currents are an important notion in spintronics from the aspect of both basic and applied science. Spin currents are generated by a current injection into a ferromagnet or spin pumping using the magnetization precession2,3. Other methods to generate spin currents are the spin Hall eﬀect4–6or the excitation of the spin wave7. In the development of spintronic devices, techniques to de- tect spin currents eﬃciently are indispensable as well as the spin-current generation. For instance, spin currents have been successfully detected via the inverse spin Hall eﬀect8–10. Due to this eﬀect, spin currents are converted into electriccurrentsandthereforeareobservedasavolt- age drop. The electric voltage induced by the spin cur- rent has been observed in a lateral junction of a ferro- magnetic metal and a nonmagnetic metal with spin-orbit coupling (SOC) such as aluminum8or platinum9,10. In particular, platinum is a typical spin-currentdetector be- cause of its strong SOC due to impurities, which is used to absorb a spin current7. Kimura et al.demonstrated the spin-current detec- tion via the inverse spin Hall eﬀect and suggested that the absorption of the spin current occurs from the Cu cross into the Pt wire10. They explained the spin-current absorption using the resistance mismatch for the spin current. Namely, they deﬁned the spin resistance as Rs=λ/[σS(1−P2)] with the spin diﬀusion length λ, the spin polarization P, the conductivity σand the ef- fective cross-sectional area Sfor the spin current, and then considered that the spin current ﬂowing in a mate- rial is absorbed into the adjacent material with smaller spin resistance10,11. Since platinum is a strong spin-orbit coupled material while copper is a weak one, the spin resistance of Pt is smaller than that of Cu and hence the injected spin current in the Cu cross is partly ab- sorbed into the Pt wire. Here, the diﬀerence of the SOC strength, i.e., the inhomogeneous SOC is the key to the absorption of spin current. As shown above, the physics of the spin-current absorption induced by an inhomo-geneous SOC has been understood only phenomenolog- ically, in that previous theories involve the phenomeno- logicalparameterssuchasthe spin-diﬀusionlength orthe spin resistance12,13and microscopic theories of the spin- current absorption are missing. The spin Hall eﬀect14–16 and the spin-current generation17due to an inhomoge- neous Rashba SOC have been theoretically examined. Also, a spin current generation near an interface due to interfacial spin-orbit coupling has been investigated18. In the present work, we consider the spin-current ab- sorption as follows. In general, the spin-continuity equa- tion for spin-orbit coupled systems has a source term: ∂sα ∂t+∇·jα s=Tα. Here,sα,jα sandTαrepresent the spin density, the spin current and the spin torque (or source term) with αbeing a component in spin space, respectively. Thus, the divergence of spin currents in the steady state is non-zero and therefore leads to the generation of spin currents. We refer to a spin-current generation induced near an interface between diﬀerent spin-orbit coupled materials or due to an inhomogeneous SOC as the spin-current absorption. On the other hand, the spin-current absorption in previous phenomenologi- cal studies is based on the continuity of a spin current at an interface, and therefore absorbed spin currents are not generated ones at the interface. In this paper, we theoretically examine the spin- current generation induced by an inhomogeneous SOC due to impurities. We consider the spin-current absorp- tion in systems where spin currents are generated by an external electric ﬁeld in Sec. II. In Sec. III, we also investigate systems where spin currents are induced by spin accumulation. We present analytical expressions of the spin currents as a response to the gradient of the SOC strength. Then, we apply these results to the vicin- ity of the interface between metals with diﬀerent SOC strengths, verifying the absorption of spin currents. Sec- tion IV is devoted to the discussions. We summarize the paper in Sec. V.2 II. ABSORPTION OF SPIN CURRENT DRIVEN BY EXTERNAL FIELD A. Model and Formalism We consider the ferromagnetic conductor in the pres- ence of the spin-orbit scattering due to impurities19–22 and assume that the coupling constant of the spin-orbit scattering slowly varies in space. This spatial variation of the SOC has not been considered so far. In order to address the spin-current absorption, we describe the input spin current by the spin-polarized current ﬂowing in a ferromagnet under an external electric ﬁeld. The total Hamiltonian is thus composed of the free-electron part with the exchange coupling ( HFM), the impurity- scattering part (Himp), the SOC due to impurities with its minimal substitution ( H0 SO+HA SO), and the interac- tion between the vector potential and the electric current (Hem): HFM=/summationdisplay σ=±1/integraldisplay drc† σ(r,t)/parenleftbigg −/planckover2pi12 2m∇2−ǫFσ/parenrightbigg cσ(r,t), (1) Himp=/summationdisplay σ=±1/integraldisplay drU(r)c† σ(r,t)cσ(r,t), (2) H0 SO=/planckover2pi1 2i/summationdisplay σ,σ′=±1/integraldisplay drλSO(r) ×∇U(r)·c† σ(r,t)/parenleftBig← →∇×σσσ′/parenrightBig cσ′(r,t),(3) HA SO=e/summationdisplay σ,σ′=±1/integraldisplay drλSO(r) ×∇U(r)·c† σ(r,t)(A(r,t)×σσσ′)cσ′(r,t),(4) Hem=e/planckover2pi1 2mi/summationdisplay σ=±1/integraldisplay dr A(r,t)·c† σ(r,t)← →∇cσ(r,t).(5) Here,cσ(r,t) (c† σ(r,t)) represents the annihilation (cre- ation) operator of a conduction electron, σαβrepre- sents Pauli matrices, e(>0) is the electric charge and c† σ← →∇cσ≡c† σ∇cσ−(∇c† σ)cσ.σ= 1 and σ=−1 corre- spond to the up ( ↑) and down spins ( ↓), respectively. We setǫFσ≡ǫF+σMwithǫFthe Fermi energy and Mthe exchange energy. λSO(r) represents the SOC strength with spatial variation, averaged over the impurity posi- tions.A(r,t) is the vectorpotential for the external elec- tric ﬁeld and we adopt the ﬁxed gauge as Eem=−˙A. U(r) is the short ranged random impurity potential and averaging over the impurity positions is carried out as /angb∇acketleftU(r)U(r′)/angb∇acket∇ightimp=u2 0nimpδ(r−r′), where u0andnimp are the strength of the impurity potential and the im- purity concentration, respectively. Here, we assume that the exchange ﬁeld is much stronger than the stray ﬁeld and hence we neglect the eﬀect of the stray (magnetic) ﬁeld in our model. Note that, in the Hamiltonian H0 SO, the operator∇does not act on λSO(r).The quantum description of the spin current is ob- tained by identifying the Heisenberg equation of mo- tion for the spin density with the spin-continuity equation22,23. The spin current operator in the i- direction with α-spin polarization reads (ˆjα s)i≡1 2m/planckover2pi1 ic†(r,t)σα← →∂ic(r,t)−e mAiσαc†(r,t)c(r,t) +λSO/summationdisplay jǫαji∂jU(r)c†(r,t)c(r,t). (6) The spin current is thus given by (see also Fig. 1 ) (jα s)i=1 2m/planckover2pi12 i/summationdisplay k,k′ei(k−k′)·r(k+k′)itr[σαGk,k′(t)]< −e/planckover2pi1 m/integraldisplaydΩ 2πeiΩt/summationdisplay k,k′,qei(k−k′+q)·rAi(q,Ω)tr[σαGk,k′(t)]< +i/planckover2pi1/summationdisplay j/summationdisplay k,k′,u,pei(k−k′+u+p)·rǫαjipjλSO(u)U(p)tr[Gk,k′(t)]<. (7) Here,Gk,k′(t,t′)≡1 i/planckover2pi1/angb∇acketleftTc[ck(t)c† k′(t′)]/angb∇acket∇ightdenotes the time- ordered Green’s function (Keldysh Green’s function) of the total Hamiltonian and Gk,k′(t)≡limt′→tGk,k′(t,t′). [···]<means taking the lesser component of the Green’s functions and tr[···] means taking trace over spin. We perturbatively calculate the spin currents induced by the inhomogeneity of the SOC strength using the KeldyshGreen’sfunction formalism. We consider HFM+ Himpas the non-perturbative part and H0 SO,HA SOand Hemas perturbative parts. We remark that the non- perturbative Green’s function contains the self energy by the impurity scattering within the ﬁrst Born approxima- tion. Within the linear response with respect to the elec- tric ﬁeld, the second part of the spin currents in Eq.(7) correspondsto an equilibrium spin current, and therefore we ignore this contribution. The leading contributions of the spin current contain the ﬁrst order term with respect to the SOC strength λSO. Since we are interested in spin currents generated by the inhomogeneous SOC strength, we focus on the contribution with the spatial derivative of the SOC strength. The third part of the spin currents, on the other hand, does not involve the spatial deriva- tive of the SOC strength. Consequently, the ﬁrst part of Eq.(7) is relevant to our study. B. Local spin current We ﬁrst treat the local spin current, which is driven by the external electric ﬁeld locally. The diagrams of the local spin current are shown in Fig. 2. The inhomogene- ity of the external electric ﬁeld is required when we dis- cuss the leading eﬀect driven by an inhomogeneous SOC3 FIG. 1. Diagrams of the spin current for the total Hamilto- nian. Bold lines denote the total Green’s function. An wavy line denotes the vector potential ( A(q,Ω)). A broken line and double broken line represent the SOC strength ( λSO(u)) and the impurity potential ( U(p)), respectively. FIG. 2. Diagrams of the leading contribution of the local spi n current induced by the inhomogeneity of the SOC strength. Solid lines denote the free Green’s function. Broken lines denote the SOC strength ( λSO(u)) and double broken lines denote the impurity potential ( U(p)). Wavy lines represent the vector potential ( Al(q,Ω)). strength. Namely, the leading contribution of spin cur- rentsinvolvingthederivativeoftheSOCstrengthhasthe form∂λSO∂Eem. The reason is as follows. Spin current and electric ﬁeld are odd under spatial inversion, while λSOand other quantities are even with respect to spatial inversion. Therefore, the leading terms including spatial derivative of the SOC have the forms of ∂2λSOEemor ∂λSO∂Eem. We focus on the contributions ∂λSO∂Eemby assuming ∂2λSOis negligibly small in this section. Now, we present the resulting local spin currents. (For the de- tail of the calculation, refer to Appendix A.) Under the condition ǫFσ≫/planckover2pi1/2τσ(≡ησ) withτσ≡/planckover2pi1/2πu2 0nimpνσ the spin-dependent relaxation time and νσthe density of state with spin index σ=±1, we obtain three types of the spin currents induced by an inhomogeneous SOC (jα s)local i= (jα s)(1) i+(jα s)(2) i+(jα s)(3) i: (jα s)(1) i=C(α1eα+α2(eα×ez))·(∇λSO(r)×∂iEem(r)), (8) (jα s)(2) i=C(β1(ei×eα) +β2(δizeα−δiαez))·∇λSO(r)(∇·Eem(r)), (9) (jα s)(3) i=C(γ1eα+γ2(eα×ez))·(∇λSO(r)×∇Ei em(r)), (10)whereC≡πe/planckover2pi12 2m(ν↑+ν↓). The detailed expressions of αi,βi,andγi(i= 1,2) are given in Appendix A. For α=z, the second terms of Eqs. (8), (9) and (10) vanish. It should be noted that even when Mgoes to zero, the spin currents still have ﬁnite contributions and become isotropic in spin space. Therefore, the magnetization is not indispensable for the spin-current generation itself. C. Diﬀusive spin current +... +... FIG. 3. Diagrams of the leading contribution of the diﬀu- sive spin current induced by the inhomogeneity of the SOC strength. (a) Contributions with the vertex correction to t he spin-current operator. (b) Contributions with the vertex c or- rection to the vector potential. (c) The vertex correction i s deﬁned as the summation of the ladder diagram with respect to the normal impurity potential. We calculate the diﬀusive (or non-local) spin current, which can be obtained by including the vertex correc- tion to the local spin current. Here, the vertex correc- tion means the summation of the ladder diagram with respect to the normal impurity potential as shown in Fig. 3 (c). Since we focus on the diﬀusive behavior of the spin current induced by the exchange coupling, the ladder diagram does not involve the spin-orbit coupled impurities. We consider two types of the vertex correc- tion, i.e., that for the spin-current operator (Fig. 3 (a)) and that for the vector potential (Fig. 3 (b)). The vertex4 correction that cuts across the impurity-averaged line is negligible because it is of higher order in ησ22. For the diﬀusive contribution under the condition ǫFσ≫/planckover2pi1/2τσ,we obtain two types of the spin currents induced by the inhomogeneous SOC ( jα s)diﬀusive i= (jα s)(1′) i+(jα s)(2′) i: (jα s)(1′) i=/braceleftbiggC/summationtext σ(ℜ[ασeα·Ki (1),σ(r)]+ℑ[ασ(eα×ez)·Ki (2),σ(r)]),(α=x,y) C/summationtext σασez·Ki (2),σ(r),(α=z)(11) Ki (n),σ(r)≡/integraldisplay dr′/integraldisplay dr′′/integraldisplay dt′χ(n) σ(r−r′,r−r′′,t−t′)(∇λSO(r′)×∂iEem(r′′,t′)),(n= 1,2) (12) (jα s)(2′) i=C/summationdisplay σ((ei×eα)β1,σ+(δizeα−δiαez)β2,σ)·∇λSO(r)/integraldisplay dr′/integraldisplay dt′χσ(r−r′,t−t′)∇·Eem(r′,t′).(13) Here, χ(1) σ(r1,r2,t) =/integraldisplaydΩ 2π/summationdisplay u,qeiΩt+iu·r1+iq·r2 Fσ−iGσ,(14) χ(2) σ(r1,r2,t) =/integraldisplaydΩ 2π/summationdisplay u,qeiΩt+iu·r1+iq·r2 (Dσ(u+q)2−iΩ)τσ,(15) χσ(r,t) =/integraldisplaydΩ 2π/summationdisplay qeiΩt+iq·r (Dσq2−iΩ)τσ,(16) represent the spin-diﬀusion propagators. Dσ≡2ǫFτσ 3mde- notes the diﬀusion coeﬃcient. The detailed expressions ofασ,βi,σ,Fσ,andGσ(i= 1,2) are given in Appendix B. The vertex corrected spin current ( jα s)(1′) i((jα s)(2′) i) corresponds to the local spin current ( jα s)(1) i((jα s)(2) i). There does not exist the diﬀusive-spin current contribu- tion which corresponds to ( jα s)(3) i. (jα s)(2′) idiﬀers from (jα s)(1′) iin that only the divergence of the external ﬁeld propagates. D. Spin-current absorption Let us consider a concrete conﬁguration to show the absorption of the spin current by the inhomogeneous SOC. We suppose that an external electric ﬁeld is ap- plied to a ferromagnetic metal and therefore a spin- polarized current ﬂows in the y-direction as shown in Fig. 4. Here, we assume that the applied electric ﬁeld varies smoothly in space from the ferromagnet to the attached non-magnetic metal with SOC, which is modeled as Eem(r) =Ey1+tanh( −z/ξE) 2eywhereξEis of the order of the lattice constant. We also assume λSO(r) =λSOθ(z). In the perturbative calculation, we have considered a smoothly varying SOC strength. How- ever, this sharp spatial variation of the SOC would give a qualitatively correct results.22In the present conﬁgu- ration, a spin-polarized current induced by the external electric ﬁeld ﬂows in the direction parallel to the inter- face. Therefore, our setup is diﬀerent from that of the spin-current injection.The local spin current in the present conﬁguration is generated when λSOandEemvary in space, i.e., only near the interface between the ferromagnetic metal and the spin-orbit coupled metal. Hence, the local spin current is not generated in the bulk spin-orbit coupled metal. We thus focus on the non-local spin current as the spin-current absorption. Since only ( ∇λSO(r′)× ∂iEem(r′′))j=EyλSOδ(z′)(2ξEcosh2(z′′/ξE))−1δizδjx remains non-zero and ∇·Eem= 0, the contribution of (jα s)(1′) m(Eq.(11)) appears in the present conﬁguration. From Eq. (14), the spin-diﬀusive propagatorin the static limit (Ω→0) reduces to χ(1) σ(r1,r2) =π 2DστσAσe−√ Bσ/Dστσ\|r1\| \|r1\|δ(r2−r1), (17) where Aσ≡η2 ση2 +(η2 +−3M2) (M2+η2 +)3−iσMη2 ση+(3η2 +−M2) (M2+η2 +)3,(18) Bσ≡M(M2+η2 +)2 η2ση+(η2 +(η2 +−3M2)2+M2(3η2 +−M2)2) ×/parenleftbig 4Mη+(η2 +−M2)+iσ(6η2 +M2−M4−η4 +)/parenrightbig , (19) withη±≡η↑±η↓ 2. Therefore, we obtain the spin currents for each spin component from Eq. (11): jx s=π2C 2ξEEyλSO/summationdisplay σ1 Dστσℜ/bracketleftbiggασe−i\|z\|kσ s Aσ√Bσ/bracketrightbigg e−\|z\|/lσ sez, (20) jy s=π2C 2ξEEyλSO/summationdisplay σ1 Dστσℑ/bracketleftbiggασe−i\|z\|kσ s Aσ√Bσ/bracketrightbigg e−\|z\|/lσ sez, (21) jz s=0. (22) Here lσ s≡√Dστσ/vextendsingle/vextendsingle√Bσ/vextendsingle/vextendsinglecos(arg(√Bσ)), (23) kσ s≡sin(arg(/radicalbig Bσ)), (24)5 which represent the spin diﬀusion length and the wave number for each spin, respectively. Here, we used the assumption b≫lσ swithbthe diameter of the interface22. We ﬁnd that spin currents are induced perpendicularly to the interface between the ferromagnetic metal and the spin-orbit coupled metal, and then decay exponentially in an oscillatory fashion as shown in Fig. 4. This ex- ponential decay is due to the spin-ﬂip scattering by the magnetizationin the ferromagnetand also the oscillatory behaviororiginatesfromthediﬀerencebetweentheFermi wave numbers for each exchange spin-split bands. The above spin currents are spin-polarized perpendicularly to the spin polarization of the input spin-polarized current (namely, zcomponent). This reﬂects the violation of the conservation law of spin near the interface, and in other words, spin torque generated at the interface produces the present spin-current absorption. It is also found that the magnitudes of the resulting spin currents in Eqs. (20) and (21) at z= 0 are, respec- tively, proportionalto η1/2 +andη−η−1/2 +in the dirty limit (τσ→0(σ=↑,↓)), andη−η1/2 +andη3/2 +inthecleanlimit (τσ→∞(σ=↑,↓)). Namely, the magnitude of the ab- sorbed spin currents gets reduced in clean ferromagnets. This is due to the competition between the two eﬀects of impurity scattering. In general, a SOC due to impurities in clean ferromagnets is weak due to low impurity con- centration, leading to a small spin current. On the other hand, low impurity concentration leads to a large spin current since impurity scattering is suppressed. When the former eﬀect overcomes the latter, spin currents are expected to be suppressedin clean ferromagnets. Finally, we note that for junctions composed of a weak spin-orbit coupledferromagnetandastrongspin-orbitcouplednon- magnet (namely, λSO(r) =λ1 SOθ(z) +λ2 SOθ(−z) with λ1 SO> λ2 SO), spin current absorption predicted in this paper also occurs. III. ABSORPTION OF SPIN CURRENT DRIVEN BY SPIN ACCUMULATION A. Model In this section, we examine the spin-current absorp- tion originating from the variation of the SOC strength in systems with spin accumulation. These systems are modeled by the conducting electrons in the presence of thegradientofthespin-dependentchemicalpotentialand that of the SOC strength. Our Hamiltonian consists of FIG. 4. (Color online) Schematic illustration of the spin- current absorption by the input spin-polarized current. Th e external electric ﬁeld Eemis applied to a ferromagnetic metal and therefore a spin-polarized current jz sﬂows in the direc- tion ofyaxis. The non-magnetic metal attached to the fer- romagnetic metal has a strong SOC λSOdue to impurities. The diﬀusive spin currents jx,y sare induced perpendicularly to the interface and then decay exponentially in an oscillat ory fashion. H0,Hacc,HimpandH0 SO. Here, H0=/summationdisplay σ=±1/integraldisplay drc† σ(r,t)/parenleftbigg −/planckover2pi12 2m∇2/parenrightbigg cσ(r,t), (25) Hacc=/summationdisplay σ=±1/integraldisplay drµσ(r,t)c† σ(r,t)cσ(r,t) =/summationdisplay σ,σ′=±1/integraldisplay drc† σ(r,t)(¯µ(r,t)δσσ′+µs(r,t)σz σσ′)cσ′(r,t). (26) Here, wehavedeﬁned thespin accumulationasthediﬀer- ence between spin-up and spin-down chemical potentials, namely,µs(r) :=µ↑−µ↓ 2.12,13This spatial distribution of spin accumulation leads to a diﬀusive spin current. In the following, we will also assume the chemical potential ¯µ(r,t) to be constant. B. Spin-current We will calculate the spin current corresponding to the diagrams in Fig. 3 in a similar manner to the pre- vious section. Here, we have replaced the vector po- tentialA(q,Ω) with the spin accumulation µs(q,Ω) in the diagrams of Fig. 3. We consider H0as the non- perturbative part and Hacc,HimpandH0 SOas pertur- bative parts. In this section, we adopt the self energy including both the normal and the spin-orbit coupled impurity potentials22,24. We also treat only the vertex6 correction for the spin-current operator22. This correc- tion provides a ﬁnite decay length of the spin current determined by the SOC strength. After taking the lesser component of the Green’s functions and averaging over impurity positions, the spin current shown in Fig. 2 (a) and (b) reads (jα s)m=i/planckover2pi12 m/planckover2pi1/integraldisplaydΩ 2πΩeiΩt/summationdisplay i,j,k/summationdisplay k,k′,u,q ×ei(u+q)·rλSO(u)µs(q,Ω)/angb∇acketleftU(p)U(−p)/angb∇acket∇ightimp ×ǫijkℜ(χαijkm)tr/bracketleftbig σασkσz/bracketrightbig . (27) For the expression of χαijkm, refer to Appendix C. In the same manner as in the previous section, we expand the coeﬃcient χαijkmwith respect to uandq. Ac- cording to the inversion operation r→−r, it is found that only the contributions with an odd order of u andqremain. Now, we perform the vertex correction χαijkmtr/bracketleftbig σασkσz/bracketrightbig →˜χαijk/summationtext a,b,c,d,eσα daΓad,cbσk beσz ec with ˜χαijkbeing the vertex-corrected coeﬃcient and Γad,cbbeing the vertex correction corresponding to the ladder diagram including the normal and the spin or- bit coupled impurities22. The contributions from the vertex-corrected coeﬃcient with the ﬁrst order of uor qvanish, and hence the leading contribution in the dif- fusion regime is of the third order of uandq: ˜χαijk= ˜χ(2,1) αijk+ ˜χ(1,2) αijk+···, where the superscript ( i,j) denotes the order of uandq. The contributions from ˜ χ(3,0) αijkand ˜χ(0,3) αijkvanish exactly. From the contribution of ˜ χ(1,2) αijkm, we obtain the expres- sion of the spin current: (˜jαs)(1,2) m=ζ(1,2)/integraldisplay dr′/integraldisplay dr′′/integraldisplay dt′D(r−r′,r−r′′,t−t′) ×[∂αλSO(r′)∂z(∂mµs(r′′,t′)) −∂zλSO(r′)∂α(∂mµs(r′′,t′))], (28)whereζ(1,2)≡4π/planckover2pi12τν 3mwithτ≡/planckover2pi1/2πu2 0nimpνbeing the relaxation time and νbeing the density of state. D(r1,r2,t) :=/integraldisplaydΩ 2π/summationdisplay u,qeiΩt+iu·r1+iq·r2 ×Ω(1−κ)(1+3κ) Dτ((u+q)2+ξ2)+iΩτ(29) is the spin-diﬀusive propagator with κ≡λ2 SOk4 F/3 (kF is the Fermi wave number), D≡2ǫFτ/3mandξ≡/radicalbig 4κ/Dτ. As seen from Eq.(29), ξdetermines the de- cay length of the spin current. As for the other contribution ˜ χ(2,1) αijkm, we obtain the spin current of the form: (˜jαs)(2,1) m=ζ(2,1)/integraldisplay dr′/integraldisplay dr′′/integraldisplay dt′D(r−r′,r−r′′,t−t′) ×[∂αµs(r′′,t′)∂z(∂mλSO(r′)) −∂zµs(r′′,t′)∂α(∂mλSO(r′))], (30) whereζ(2,1)≡608π/planckover2pi12τν 45m/parenleftbigǫFτ /planckover2pi1/parenrightbig2. Here, we have focused on the diﬀusive spin currents, corresponding to contri- butions with the vertex correction. It is found that if the scales of spatial variations of λSO(r) andµs(r,t) are nearly the same, the magnitude of the spin current (˜jαs)(2,1) mis larger than that of ( ˜jαs)(1,2) mby the factor of ǫFτ//planckover2pi1. C. Spin-current absorption Let us consider a typical conﬁguration in order to in- vestigate the spin-current absorption. We assume that two non-magnetic metals with diﬀerent SOC strengths are connected at the z= 0 plane as shown in Fig. 5. In addition, there exists a gradient of the spin accumulation along the ydirection in one of the metals. In the vicinity of thez= 0 plane, we assume that λSO(r) =λSO(z) and µs(r,t) =µs(y,z,t), leading to (˜jxs)z= 0, (31) (˜jy s)z=/integraldisplay dr′/integraldisplay dr′′/integraldisplay dt′D(r−r′,r−r′′,t−t′) ×/bracketleftbig −ζ(1,2)∂z′′λSO(z′′)∂z′(∂y′µs(y′,z′,t′))+ζ(2,1)∂y′µs(y′,z′,t′)∂2 z′′λSO(z′′)/bracketrightbig , (32) (˜jzs)z= 0. (33) The ﬁrst term of the right hand side in Eq. (32) rep- resents the interplay between the variation of the SOC strength and that of the input spin current near the in- terface. Note that spin currentis induced by the gradient of the spin accumulation without SOC.13On the other hand, the second term comes from the variation of the SOC strength and uniform input spin current. Remark thataspincurrentpolarizedalongthe y-axisisgeneratedas a response to that polarized along the z-axis. Similar to the absorption of spin currents driven by an external electric ﬁeld, a spin current absorbed into the top metal is non-local, and diﬀusive spin current decays due to the spin-orbit coupled scattering as seen from Eq.(29).7 FIG. 5. (Color online) Schematic illustration of the absorp - tion of spin currents driven by the spin accumulation. The arrows in the bottom metal denote input spin currents ( jz s,in)y by the spin accumulation. In the top metal, the left arrows are normal diﬀusion of the input spin currents ( jz s,in)z, which is irrelevant to the variation of a SOC, whereas the middle and the right arrows are diﬀusion of spin currents ( jy s,abs.)z induced in the vicinity of the interface, relevant to the diﬀ er- ence of SOCs. IV. DISCUSSION In the present study, we have focused on diﬀusion of spin currents including the gradient of the SOC strength. However, there also exists a diﬀusion of spin currents ir- relevant to the diﬀerence of SOC strength as shown in Fig. 5. We can extract the former contribution by com- paring spin currents in the triple lateral spin valves us- ing middle junctions with diﬀerent SOC11. In fact, the absorption of non-local spin currents has been success- fully demonstrated in the spin-valve measurement con- sisting of double Py/Cu junctions and middle Au/Cu junction10,11. Since the argument based on the mismatch of the spin resistance by Kimura et al.12assumes the continuity of a spin current, the absorbed spin currents are not those generated at an interface. Consequently, the absorbedspin currents in our theory are of diﬀerent origin from those in the theory based on the spin-resistance mis- match. Hence, we cannot compare our results with pre- vious results directly. Both studies show the spin-current absorption with diﬀerent mechanisms. In asymmetric structures, conduction electrons at its interface in general feel a Rashba SOC25, leading to the spin polarization under an external electric ﬁeld or cur- rent injection26. In the conﬁgurations of Figs. 4 and 5, there would exist a Rashba SOC at their interfaces. The current-induced spin polarization due to the Rashba SOC may aﬀect the absorption of spin currents predicted in our models. V. SUMMARY Wehaveinvestigatedthegenerationofspincurrentsby an inhomogeneous SOC due to impurities, which have been applied to the interface system to show the spin- current absorption. Using the Keldysh Green’s function formalism, we have presented analytical expressions of the spin currents with the gradient of the SOC strength for two systems: the system with ﬁeld-driven spin cur- rents and one with spin-accumulation-driven spin cur- rents. The resulting spin current indicates the absorp- tionofthespin currentat theinterfacebetweenmaterials with diﬀerent SOC strengths. In the present study, we assumed a homogeneous mag- netization. The extension of our model to an inhomoge- neous magnetization is also an interesting future work. ACKNOWLEDGMENTS We thank Y. Tserkovnyak, G. Tatara and A. Takeuchi forhelpful discussions. K. T.thanksS. Murakamifordis- cussions. K. H. thanks Y. Nozaki for discussions. This work is supported by Grant-in-Aid for Young Scientists (B) (No. 23740236andNo. 24710153)andthe ”Topolog- ical Quantum Phenomena”(No. 23103505)Grant-in Aid forScientiﬁc Researchon InnovativeAreasfrom the Min- istry of Education, Culture, Sports, Science and Tech- nology (MEXT) of Japan. K. T. also acknowledges the ﬁnancial support from the Global Center of Excellence Program by MEXT, Japan, through the gNanoscience and Quantum Physicsh Project of the Tokyo Institute of Technology. ∗tsutsui@stat.phys.titech.ac.jp 1I. Zutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). 2R. H. Silsbee, A. Janossy, and P. Monod, Phys. Rev. B 19, 4382 (1979). 3Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002).4M. I. Dyakonov and V. I. Perel, Phys. Lett. A 35, 459 (1971). 5J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999). 6S. Murakami, N. Nagaosa, and S. -C. Zhang, Science 301, 1348 (2003). 7Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K.8 Takanashi, S. Maekawa, and E. Saitoh, Nature 464, 262 (2010). 8S.O.ValenzuelaandM. Tinkham, Nature 442, 176(2006). 9E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Lett. 88, 182509 (2006). 10T. Kimura, Y. Otani, T. Sato, S. Takahashi, and S. Maekawa, Phys. Rev. Lett. 98, 156601 (2007). 11T. Kimura, Y. Otani, and L. Vila, J. Appl. Phys. 103, 07F310 (2008). 12T. Kimura, J. Hamrle, and Y. Otani, Phys. Rev. B 72, 014461 (2005). 13S. Takahashi and S. Maekawa, Phys. Rev. B 67, 052409 (2003). 14Y. Tserkovnyak, B. I. Halperin, A. A. Kovalev, and A. Brataas, Phys. Rev. B 76, 085319 (2007). 15V. K. Dugaev, M. Inglot, E. Ya. Sherman, and J. Barna´ s, Phys. Rev. B 82, 121310(R) (2010). 16M. M. Glazov, E. Ya. Sherman, and V. K. Dugaev, Physica E42, 2157 (2010). 17K. Tsutsui, A. Takeuchi, G. Tatara, and S. Murakami, J. Phys. Soc. Jpn. 80, 084701 (2011). 18J. Linder and T. Yokoyama, Phys. Rev. Lett. 106, 237201(2011). 19W. -K. Tse and S. Das Sarma, Phys. Rev. Lett. 96, 056601 (2006). 20V. K. Dugaev, P. Bruno, and J. Barna´ s, Phys. Rev. B 64, 144423 (2001). 21V. K. Dugaev, P. Bruno, M. Taillefumier, B. Canals, and C. Lacroix, Phys. Rev. B 71, 224423 (2005). 22K. Hosono, A. Yamaguchi, Y. Nozaki, and G. Tatara, Phys. Rev. B 83, 144428 (2011). 23G. Tatara and P. Entel, Phys. Rev. B 78, 064429 (2008). 24S. Hikami, A. I. Larkin, and Y. Nagaoka, Prog. Theor. Phys.63, 707 (1980). 25E. I. Rashba, Sov. Phys. Solid State 2, 1109 (1960); Yu. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6039 (1984). 26V. M. Edelstein, Solid State Commun. 73, 233 (1990). 27N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and P. Ong, Rev. Mod. Phys. 82, 1539 (2010). 28H. Hung and A. P. Jauho, Quantum Kinetics in Trans- port and Optics of Semiconductors (Springer-Verlag, Hei- delberg, 1998). Appendix A: Calculation of local spin current We perturbatively calculate the spin currents induced by the inhomo geneity of the SOC strength using the Keldysh Green’s function formalism. We ﬁrst treat the local spin current, w hich is locally driven by the external electric ﬁeld. The leading contribution of the spin current involves the ﬁrst o rder with respect to the SOC strength, which is diagrammatically shown in Fig. 2. The local spin currents with α-component spin polarization ﬂowing in the m direction, i.e., ( jα s)local m= (jα s)sj m+(jα s)sk mthus read (jα s)sj m=i/planckover2pi12 2m/planckover2pi1e/planckover2pi1 2m/integraldisplaydΩ 2π/integraldisplaydω 2πΩeiΩt/summationdisplay i,j,k,l/summationdisplay k,k′,u,qei(u+q)·rλSO(u)/angb∇acketleftU(p)U(−p)/angb∇acket∇ightimpAl(q,Ω)ǫijk2ℜ(χαijklm),(A1) (jα s)sk m=i/planckover2pi12 2me/integraldisplaydΩ 2π/integraldisplaydω 2πΩeiΩt/summationdisplay i,j,k/summationdisplay k,k′,u,qei(u+q)·rλSO(u)/angb∇acketleftU(p)U(−p)/angb∇acket∇ightimpAj(q,Ω)ǫijk2ℜ(χαikm),(A2) where the superscripts ”sj” and ”sk” denote the side-jump and t he skew-scattering contributions, respectively,27and each coeﬃcients are given by χαijklm= +(k−u−k′)i(k+k′)j(2k−2u−q)l(2k−u−q)mtr/bracketleftbig σαgR kσkgR k′gR k−ugA k−u−q/bracketrightbig +(k′−k)i(k−u+k′)j(2k−2u−q)l(2k−u−q)mtr/bracketleftbig σαgR kgR k′σkgR k−ugA k−u−q/bracketrightbig +(k−u−k′)i(k+k′)j(2k′−q)l(2k−u−q)mtr/bracketleftbig σαgR kσkgR k′gA k′−qgA k−u−q/bracketrightbig , (A3) χαikm= (k−k′−u−q)i(2k−u−q)mtr/bracketleftbig σαgR kσkgA k′gA k−u−q/bracketrightbig . (A4) Here,gR(A) k≡gR(A) k,ω=0, andgR k,ωandgA k,ωdenote the non-perturbative retarded and advanced Green’s fu nc- tion, respectively, which are 2 ×2 matrices in spin space. We also use the Langreth’s method to obtain [gk1,ω···gki,ωgki+1,ω−Ω···gkn,ω−Ω]<≃ −Ωf′(ω)gR k1,ω···gR ki,ωgA ki+1,ω···gA kn,ωwithf(ω) the Fermi distribution function28. Here, we set temperature to zero, i.e., f′(ω)≃−δ(ω). Since we are interested in spin currents produced by the inhomogen eous SOC strength, we focus on the contribution which involves the spatial derivative of the SOC strength. After so me algebraic calculations, we obtain the above9 coeﬃcients represented as χαijklm≃2 3k2(δjluiqm+δjmuiql)/parenleftbig tr/bracketleftbig σαgR kσkgR k′gR kgA k/bracketrightbig +tr/bracketleftbig σαgR kgR k′σkgR kgA k/bracketrightbig/parenrightbig +4 15/planckover2pi12 mk4(δjluiqm+δjmuiql+δlmuiqj)/parenleftbig tr/bracketleftbig σαgR kσkgR k′gR k(gA k)2/bracketrightbig +tr/bracketleftbig σαgR kgR k′σkgR k(gA k)2/bracketrightbig/parenrightbig +2 3/parenleftbig k′2δjluiqm+k2δjmuiql/parenrightbig tr/bracketleftbig σαgR kσkgR k′gA k′gA k/bracketrightbig +8 9/planckover2pi12 mk2k′2δjluiqmtr/bracketleftbig σαgR kσkgR k′gA k′(gA k)2/bracketrightbig +4 9/planckover2pi12 mk2k′2δjmuiqltr/bracketleftbig σαgR kσkgR k′(gA k′)2gA k/bracketrightbig ,(A5) χαikm≃(uiqm+qium)/parenleftbigg tr/bracketleftbig σαgR kσkgA k′gA k/bracketrightbig +/planckover2pi12 mk2tr/bracketleftbig σαgR kσkgA k′(gA k)2/bracketrightbig/parenrightbigg . (A6) Here, weassumedtherotationalsymmetryofthesystem, i.e.,/summationtext kkikj=/summationtext kk2 3δijand/summationtext kkikjklkm=/summationtext kk4 15(δijδlm+ δilδjm+δimδjl). We remark that if we do not consider the inhomogeneity of the ext ernal electric ﬁeld, the present contributions vanish. The inhomogeneity of the external electric ﬁ eld is required to obtain the contributions from the inhomogeneous SOC strength. Next, we take traces over spin space in Eqs. (A5) and (A6) by using the formula tr[ σαAσkB] =/summationtext σ=±1(δαk− δαzδkz−iσǫαzk)AσB¯σ+/summationtext σ=±1δαzδkzAσBσwithA,BGreen’s functions, and then Eqs. (A1) and (A2) reduce to (jα s)sj m≃e/planckover2pi12 2mu2 0nimp/integraldisplaydΩ 2πΩeiΩt/summationdisplay u,q/summationdisplay i,j,k,lei(u+q)·rλSO(u)Al(q,Ω)ǫijk/summationdisplay σ((δαk−δαzδkz)ℜ+σǫαkℑ) ×/bracketleftbigg2 3δjluiqm/parenleftbigg (Iσ 01;00+Iσ 10;00)/parenleftbigg Qσ 11;01+4 5Sσ 11;02/parenrightbigg +Qσ 01;01Iσ 10;01+8 3Qσ 01;01Qσ 10;02/parenrightbigg +2 3δjmuiql/parenleftbigg/parenleftbig Iσ 01;00+Iσ 10;00/parenrightbig/parenleftbigg Qσ 11;01+4 5Sσ 11;02/parenrightbigg +Iσ 01;01Qσ 10;01+4 3Qσ 01;02Qσ 10;01/parenrightbigg +8 15δlmuiqj(Iσ 01;00+Iσ 10;00)Sσ 11;02/bracketrightbigg +e/planckover2pi12 2mu2 0nimp/integraldisplaydΩ 2πΩeiΩt/summationdisplay u,q/summationdisplay i,j,k,lei(u+q)·rλSO(u)Al(q,Ω)ǫijk/summationdisplay σδαzδkzℜ ×/bracketleftbigg2 3δjluiqm/parenleftbigg 2Iσ 10;00/parenleftbigg Qσ 20;10+4 5Sσ 20;20/parenrightbigg +Qσ 10;10Iσ 10;10+8 3Qσ 10;10Qσ 10;20/parenrightbigg +2 3δjmuiql/parenleftbigg 2Iσ 10;00/parenleftbigg Qσ 20;10+4 5Sσ 20;20/parenrightbigg +Iσ 10;10Qσ 10;10+4 3Qσ 10;20Qσ 10;10/parenrightbigg +16 15δlmuiqjIσ 10;00Sσ 20;20/bracketrightbigg , (A7) (jα s)sk m≃e/planckover2pi12 2mu2 0nimp/integraldisplaydΩ 2πΩeiΩt/summationdisplay i,j,k/summationdisplay u,qei(u+q)·rλSO(u)Aj(q,Ω)ǫijk/bracketleftBigg/summationdisplay σ(δαk−δαzδkz)ℜ+σǫαkℑ) ×uiqmIσ 00;01(Iσ 10;01+2Qσ 10;02)+/summationdisplay σδαzδkzuiqmℜIσ 00;10(Iσ 10;10+2Qσ 10;20)/bracketrightBigg . (A8) Here, we deﬁne integrals of the Green’s functions appearing in the a bove expressions as Iσ ab;cd≡/summationdisplay k(gR k,σ)a(gR k,¯σ)b(gA k,σ)c(gA k,¯σ)d, (A9) Qσ ab;cd≡/summationdisplay kǫk(gR k,σ)a(gR k,¯σ)b(gA k,σ)c(gA k,¯σ)d, (A10) Sσ ab;cd≡/summationdisplay kǫ2 k(gR k,σ)a(gR k,¯σ)b(gA k,σ)c(gA k,¯σ)d, (A11) wheregR k,σ= [ǫFσ−ǫk+iησ]−1withησ≡/planckover2pi1/2τσandgA k,σ= (gR k,σ)∗withǫk≡/planckover2pi12k2 2m. WesumupEqs. (A7)and(A8)and consider the dominant contribution from the integrals of the Green ’s functions under the condition ǫFσ≫/planckover2pi1/2τσ. By10 carrying out the integrals of the Green’s functions in Eqs.(A9-A11) and transforming to the real-space representation, we obtain the local spin current of the form (jα s)m≃πe/planckover2pi12 2m(ν↑+ν↓)[(α1eα+α2(eα×ez))·(∇λSO(r)×∂mEem(r)) +(β1(em×eα)+β2(δmzeα−δmαez))·∇λSO(r)(∇·Eem(r)) +(γ1eα+γ2(eα×ez))·(∇λSO(r)×∇Em em(r))]. (A12) Here, each coeﬃcients are dimensionless and given by αi≡γi+δi, (A13) βi≡γi+ǫi, (A14) δ1≡/braceleftBigg 16 9/parenleftBig ǫFM2η+ (M2+η2 +)2−1 41 ν↑+ν↓η+ M2+η2 +/summationtext σǫFσν¯σ/parenrightBig ,(α=x,y) −4 9ǫF η+,(α=z)(A15) δ2≡16 9/parenleftBigg ǫFM(M2−η2 +) (M2+η2 +)2−1 41 ν↑+ν↓M M2+η2 +/summationdisplay σǫFσν¯σ/parenrightBigg , (A16) ǫ1≡/braceleftBigg 8 9/parenleftBig M M2+η2 +/summationtext σσǫFσǫFσ ησ+2 ν↑+ν↓/summationtext σǫFσ ησν¯σ/parenrightBig ,(α=x,y) 16 9ǫFσ ησ,(α=z)(A17) ǫ2≡8 9η+ M2+η2 +/summationdisplay σσǫFσǫFσ ησ, (A18) γ1≡ 2 15/bracketleftbigg Mη+ M2+η2 +/summationtext σσ/parenleftBig ǫFσ ησ/parenrightBig2 +Mη− M2+η2 −/summationtext σ/parenleftBig ǫFσ ησ/parenrightBig2 −Mη2 + (M2+η2 +)2/summationtext σσǫFσǫFσ ησ−ǫFη+ M2+η2 +/bracketrightbigg ,(α=x,y) −2 15ǫF η+,(α=z)(A19) γ2≡−2 15/bracketleftBigg M2(η2 +−η2 −) (M2+η2 +)(M2+η2 −)/summationdisplay σσ/parenleftbiggǫFσ ησ/parenrightbigg2 +η+(M2−η2 +) (M2+η2 +)2/summationdisplay σσǫFσǫFσ ησ+ǫFM M2+η2 +/bracketrightBigg , (A20) andη±≡η↑±η↓ 2. Appendix B: Calculation of diﬀusive spin current We calculate the diﬀusive (or non-local) spin current, which is repres ented by the vertex correction to the local spin current. Thiscanbeperformedbyreplacing χαijklmandχαikminEqs. (A1)and(A2)with ˜ χαijklm= Γm u,qΠu,qχ(1) αijkl+ Γl qΠqχ(2) αijkmand ˜χαikm= Γm u,qΠu,qχαik, respectively, where Γm σ,¯σ(u) =u2 0nimp V/summationdisplay k(2k−u)mgR k,ω,σgA k−u,ω−Ω,¯σ, (B1) Πσ,¯σ(u) =∞/summationdisplay n=0/parenleftBigg u2 0nimp V/summationdisplay kgR k,ω,σgA k−u,ω−Ω,¯σ/parenrightBiggn , (B2) (B3) χ(1) αijkl= +(k−u−k′)i(k+k′)j(2k−2u−q)ltr/bracketleftbig σαgR kσkgR k′gR k−ugA k−u−q/bracketrightbig +(k′−k)i(k−u+k′)j(2k−2u−q)ltr/bracketleftbig σαgR kgR k′σkgR k−ugA k−u−q/bracketrightbig +(k−u−k′)i(k+k′)j(2k′−q)ltr/bracketleftbig σαgR kσkgR k′gA k′−qgA k−u−q/bracketrightbig , (B4) χ(2) αijkm= +(k−u−k′)i(k+k′)j(2k−u−q)mtr/bracketleftbig σαgR kσkgR k′gR k−ugA k−u−q/bracketrightbig +(k′−k)i(k−u+k′)j(2k−u−q)mtr/bracketleftbig σαgR kgR k′σkgR k−ugA k−u−q/bracketrightbig +(k−u−k′)i(k+k′)j(2k−u−q)mtr/bracketleftbig σαgR kσkgR k′gA k′−qgA k−u−q/bracketrightbig , (B5) χαik= (k−k′−u−q)itr/bracketleftbig σαgR kσkgA k′gA k−u−q/bracketrightbig . (B6)11 Expanding with respect to uorq, we obtain the leading contributions as follows χ(1) αijkl≃−2 3k2δjlui/parenleftbig tr/bracketleftbig σαgR kσkgR k′gR kgA k/bracketrightbig +tr/bracketleftbig σαgR kgR k′σkgR kgA k/bracketrightbig/parenrightbig −2 3k′2δjluitr/bracketleftbig σαgR kσkgR k′gA k′gA k/bracketrightbig −4 9/planckover2pi12 mk2k′2δjl(u+q)itr/bracketleftbig σαgR kσkgR k′gA k′(gA k)2/bracketrightbig , (B7) χ(2) αijkm≃−2 3k2δjmui/parenleftbig tr/bracketleftbig σαgR kσkgR k′gR kgA k/bracketrightbig +tr/bracketleftbig σαgR kgR k′σkgR kgA k/bracketrightbig/parenrightbig −2 3k2δjmuitr/bracketleftbig σαgR kσkgR k′gA k′gA k/bracketrightbig −4 9/planckover2pi12 mk2k′2δimqjtr/bracketleftbig σαgR kσkgR k′(gA k′)2gA k/bracketrightbig , (B8) χαik≃−(u+q)i/parenleftbigg tr/bracketleftbig σαgR kσkgA k′gA k/bracketrightbig +1 3/planckover2pi12 mk2tr/bracketleftbig σαgR kσkgA k′(gA k)2/bracketrightbig/parenrightbigg . (B9) Next, we take traces over spin space in Eqs. (B7), (B8) and (B9) in a similar manner to the local spin current, and then the diﬀusive spin currents are reduced to (jα s)sj m≃e/planckover2pi12 2mu2 0nimp/integraldisplaydΩ 2πΩeiΩt/summationdisplay u,q/summationdisplay i,j,k,lei(u+q)·rλSO(u)Al(q,Ω)ǫijk/summationdisplay σ((δαk−δαzδkz)ℜ+σǫαkℑ) ×/bracketleftbigg2 3δjluiqmΠσ,¯σ(u+q)u2 0nimpIσ 10;01/parenleftbigg (Iσ 01;00+Iσ 10;00)Qσ 11;01+Qσ 01;01Iσ 10;01+4 3Qσ 01;01Qσ 10;02/parenrightbigg +2 3δjmuiqlΠσ,σ(q)u2 0nimpIσ 10;10/parenleftbig (Iσ 01;00+Iσ 10;00)Qσ 11;01/parenrightbig +Iσ 01;01Qσ 10;01/bracketrightbigg +e/planckover2pi12 2mu2 0nimp/integraldisplaydΩ 2πΩeiΩt/summationdisplay u,q/summationdisplay i,j,k,lei(u+q)·rλSO(u)Al(q,Ω)ǫijk/summationdisplay σδαzδkzℜ ×/bracketleftbigg2 3δjluiqmΠσ,σ(u+q)u2 0nimpIσ 10;10/parenleftbigg 2Iσ 10;00Qσ 20;10+Qσ 10;10Iσ 10;10+4 3Qσ 10;10Qσ 10;20/parenrightbigg +2 3δjmuiqlΠσ,σ(q)u2 0nimpIσ 10;10/parenleftbig 2Iσ 10;00Qσ 20;10+Iσ 10;10Qσ 10;10/parenrightbig/bracketrightbigg , (B10) (jα s)sk m≃e/planckover2pi12 2mu2 0nimp/integraldisplaydΩ 2πΩeiΩt/summationdisplay i,j,k/summationdisplay u,qei(u+q)·rλSO(u)Aj(q,Ω)ǫijk /bracketleftBigg/summationdisplay σ(δαkℜ+σǫαkℑ)(uiqm+qium) Πσ,¯σ(u+q)u2 0nimpIσ 10;01/parenleftbigg Iσ 00;01Iσ 10;01+2 3Iσ 00;01Qσ 10;02/parenrightbigg +/summationdisplay σδαzδkzℜ(uiqm+qium) Πσ,σ(u+q)u2 0nimpIσ 10;10/parenleftbigg Iσ 00;10Iσ 10;10+2 3Iσ 00;10Qσ 10;20/parenrightbigg/bracketrightBigg . (B11) Here Πσ,¯σ(u)≃/bracketleftbigg 1−/parenleftbigg u2 0nimpIσ 10;01+/planckover2pi1Ωu2 0nimpIσ 10;02+4/planckover2pi12 2m1 3u2u2 0nimpQσ 10;03/parenrightbigg/bracketrightbigg−1 , (B12) Πσ,σ(u)≃/bracketleftbigg 1−/parenleftbigg u2 0nimpIσ 10;10+/planckover2pi1Ωu2 0nimpIσ 10;20+4/planckover2pi12 2m1 3u2u2 0nimpQσ 10;30/parenrightbigg/bracketrightbigg−1 . (B13) By carrying out the above integrals of the Green’s functions and tr ansforming to the real-space representation using uλSO(u) =−i/integraltext dr′∇r′λSO(r′)e−ir′·uand ΩqmA(q,Ω) =−/integraltext dt′/integraltext dr′′∂′′ m˙A(r′′,t′)e−ir′′·qe−iΩt′, we obtain the12 diﬀusive spin current of the form (jα s)m≃πe/planckover2pi12 2m(ν↑+ν↓) ×/bracketleftBigg (δαx+δαy)/summationdisplay σ(eαℜασ+(eα×ez)ℑασ)·/integraldisplay dr′/integraldisplay dr′′/integraldisplay dt′ /integraldisplaydΩ 2π/summationdisplay u,qeiΩ(t−t′)+iu·(r−r′)+iq·(r−r′′) Fσ−iGσ(∇λSO(r′)×∂mEem(r′′,t′)) +δαz/summationdisplay σασez·/integraldisplay dr′/integraldisplay dr′′/integraldisplay dt′ /integraldisplaydΩ 2π/summationdisplay u,qeiΩ(t−t′)+iu·(r−r′)+iq·(r−r′′) (Dσ(u+q)2−iΩ)τσ(∇λSO(r′)×∂mEem(r′′,t′)) +/summationdisplay σ(β1,σ(ei×eα)+β2,σ(δizeα−δiαez))·∇λSO(r)/integraldisplay dr′/integraldisplay dt′/integraldisplaydΩ 2π/summationdisplay qeiΩ(t−t′)+iq·(r−r′) (Dσq2−iΩ)τσ∇·Eem(r′,t′)/bracketrightBigg , (B14) where ασ≡/braceleftBigg 2 9Mη+((1+σ)η2 +−M2) (M2+η2 +)3ǫ2 Fσ+i2 9η2 +((1+σ)M2−η2 +) (M2+η2 +)3ǫ2 Fσ,(α=x,y) −1 9ǫFσ η+,(α=z)(B15) β1,σ≡/braceleftBigg −2 3ǫF¯στ¯σ /planckover2pi1M2(η2 +−η2 −) (M2+η2 +)(M2+η2 −)+π2τσνσ /planckover2pi1ν↑ν↓ǫFσ ησ,(α=x,y) π2τσνσ /planckover2pi1ν↑ν↓ǫFσ ησ,(α=z)(B16) β2,σ≡2 3ǫF¯στ¯σ /planckover2pi1M(η++ ¯ση−)(η+η−+ ¯σM2) (M2+η2 +)(M2+η2 −), (B17) Fσ≡M2 M2+η2 ++2σMηση2 + (M2+η2 +)2Ωτσ+η2 ση2 +(η2 +−3M2) (M2+η2 +)32ǫF¯στσ 3m(u+q)2τσ, (B18) Gσ≡σMη+ M2+η2 ++ηση+(η2 +−M2) (M2+η2 +)2Ωτσ+σMη2 ση+(3η2 +−M2) (M2+η2 +)32ǫF¯στσ 3m(u+q)2τσ, (B19) Dσ≡2ǫFτσ 3m. (B20) Appendix C: Calculation of spin current driven by spin accum ulation The spin current shown in Fig. 2 (a) and (b) reads (jα s)m=−/planckover2pi12 2m/planckover2pi1/integraldisplaydΩ 2πΩeiΩt/summationdisplay i,j,k/summationdisplay k,k′,u,qei(u+q)·rλSO(u)/angb∇acketleftU(p)U(−p)/angb∇acket∇ightimpµs(q,Ω)ǫijktr/bracketleftbig σασkσz/bracketrightbig 2ℜ(χαijkm) =−4i/planckover2pi12 2m/planckover2pi1u2 0nimp/integraldisplaydω 2π/summationdisplay i,j/summationdisplay k,k′,u,qei(u+q)·rλSO(u)µs(q,Ω)(δizδjα−δiαδjz)ℜ(χαijkm), (C1) χαijkm= (k−u−k′)i(k+k′)j(2k−u−q)mgR kgR k′gR k−ugA k−u−q+(k′−k)i(k−u+k′)j(2k−u−q)mgR kgR k′gR k−ugA k−u−q +(k−u−k′)i(k+k′)j(2k−u−q)mgR kgR k′gA k′−qgA k−u−q. (C2) As for the above coeﬃcient χαijkm, the contribution involving two q’s and one uis calculated as follows; χ(1,2) αijkm=−2 3/planckover2pi12 2m/bracketleftbig 2k2uiqmqjgR k′(gR k)2(gA k)2+k2uiqmqjgR kgR k′gA k′(gA k)2+k′2uiqjqmgR kgR k′(gA k′)2gA k/bracketrightbig −8 9/parenleftbigg/planckover2pi12 2m/parenrightbigg2 k2k′2qjumqigR k′(gA k′)2gR k(gA k)2. (C3)13 The resultant spin current reduces to (jα s)(1,2) m=−4/planckover2pi12 2m/planckover2pi1u2 0nimp/integraldisplaydΩ 2πΩeiΩt/summationdisplay u,qei(u+q)·rλSO(u)µs(q,Ω)(uαqmqz−uzqmqα)4 3ℜ/bracketleftbig Iσ 10;00Qσ 20;20+Iσ 10;10Qσ 10;20/bracketrightbig . (C4) Now we perform the vertex correction χαijkmtr/bracketleftbig σασkσz/bracketrightbig →˜χαijk/summationtext a,b,c,d,eσα daΓad,cbσk beσz ecwith Γ ad,cbbeing the vertexcorrectioncorrespondingto the ladderdiagramincluding th e normalimpurity and the SOC due to impurities22, and ˜χαijk= (k−u−k′)i(k+k′)jgR kgR k′gR k−ugA k−u−q+(k′−k)i(k−u+k′)jgR kgR k′gR k−ugA k−u−q +(k−u−k′)i(k+k′)jgR kgR k′gA k′−qgA k−u−q. (C5) According to Ref. 22, the vertex correction is calculated as Γ ad,cb(q,Ω) = Γ C(q,Ω)δadδcb+ΓS(q,Ω)/summationtext lσl adσl cbwith ΓC(q,Ω) =1+3κ 2(Dq2+iΩ)τ, (C6) ΓS(q,Ω) =(1−κ)(1+3κ) 2(4κ+(Dq2+iΩ)τ), (C7) leading to /summationdisplay a,b,c,d,eσα daΓad,cbσk beσz ec= ΓCtr[σαˆ1]tr[ˆ1σkσz]+/summationdisplay lΓStr[σασl]tr[σlσkσz] = 4iǫαkzΓS. (C8) Here,τ≡/planckover2pi1/2πu2 0nimpν,κ≡λ2 SOk4 F/3 (kFis the Fermi wave number), D≡2ǫFτ/3m, andξ≡/radicalbig 4κ/Dτ.ˆ1 is the identity matrix. Therefore, we obtain the spin current including the vertex correction of the form (˜jαs)(1,2) m≃4π/planckover2pi12τν 3m/integraldisplay dr′/integraldisplay dr′′/integraldisplay dt′/integraldisplaydΩ 2π/summationdisplay u,qeiΩ(t−t′)+iu·(r−r′)+iq·(r−r′′)Ω/τ(1−κ)(1+3κ) D((u+q)2+ξ2)+iΩ ×(∂αλSO(r′)∂z(∂mµs(r′′,t′))−∂zλSO(r′)∂α(∂mµs(r′′,t′))), (C9) withνbeing the density of state. We remark that the spin-diﬀusion propa gator is given by D(r1,r2,t) :=/integraldisplaydΩ 2π/summationdisplay u,qeiΩt+iu·r1+iq·r2Ω/τ(1−κ)(1+3κ) D((u+q)2+ξ2)+iΩ =−δ(r2−r1)/radicalbiggπ Dt/parenleftbigg Dξ2−\|r1\|2 4Dt2+1 2t/parenrightbigg exp/bracketleftbigg −/parenleftbigg Dξ2t+\|r1\|2 4Dt/parenrightbigg/bracketrightbigg . (C10) On the other hand, the contribution involving two u’s and one qis given as follows; χ(2,1) αijkm=−4 3/planckover2pi12 2mk2uiujqm[gR kgR k′gR k)2gA k−8 15/parenleftbigg/planckover2pi12 2m/parenrightbigg2 k4(uiujqm+uiuoqoδjm+uiumqj)gR kgR k′(gR k)2(gA k)2 −4 3/planckover2pi12 2mk2ui(umqj+qmuj)gR kgR k′gR k(gA k)2−1 3k2(uiumqj+uiujqm)gR kgR k′gA k′(gA k)2 −1 3k′2uiumqjgR kgR k′(gA k′)2gA k−4 9k2k′2uiumqjgR kgR k′(gA k′)2(gA k)2. (C11) The corresponding spin current reduces to (jα s)(2,1) m=−4/planckover2pi12 2m/planckover2pi1u2 0nimp/integraldisplaydΩ 2πΩeiΩt/summationdisplay u,qei(u+q)·rλSO(u)µs(q,Ω)/summationdisplay i,j,kǫijkǫαkz ×/parenleftbigg uiumqjℜ/bracketleftbigg −4 3Iσ 10;00Qσ 20;20−4 3Iσ 10;10Qσ 10;20−16 9(Qσ 10;20)2−16 15Iσ 10;00Sσ 30;20/bracketrightbigg +uiuoqoδjmℜ/bracketleftbigg −16 15Iσ 10;00Sσ 30;20/bracketrightbigg/parenrightbigg . (C12) By performing the vertex correction, we obtain the ﬁnal express ion of the spin current (˜jαs)(2,1) m≃152π/planckover2pi12τν 45m/parenleftbigg2ǫFτ /planckover2pi1/parenrightbigg2/integraldisplay dr′/integraldisplay dr′′/integraldisplay dt′/integraldisplaydΩ 2π/summationdisplay u,qeiΩ(t−t′)+iu·(r−r′)+iq·(r−r′′)Ω/τ(1−κ)(1+3κ) D((u+q)2+ξ2)+iΩ ×(∂αµs(r′,t′)∂z(∂mλSO(r′′))−∂zµs(r′,t′)∂α(∂mλSO(r′′))). (C13)
1202.1604v1.Role_of_spin_orbit_coupling_on_the_electronic_structure_and_properties_of_SrPtAs.pdf	arXiv:1202.1604v1 [cond-mat.supr-con] 8 Feb 2012Role of spin-orbit coupling on the electronic structure and properties of SrPtAs S. J. Youn1,2, S. H. Rhim2, D. F. Agterberg3, M. Weinert3, A. J. Freeman2 1Department of Physics Education and Research Institute of N atural Science, Gyeongsang National University, Jinju 660-701, Korea 2Department of Physics and Astronomy, Northwestern Univers ity, Evanston, Illinois, 60208-3112, USA and 3Department of Physics, University of Wisconsin-Milwaukee , Milwaukee, WI 53201-0413, USA (Dated: November 9, 2018) The eﬀect of spin-orbit coupling on the electronic structur e of the layered iron-free pnictide su- perconductor, SrPtAs, has been studied using the full poten tial linearized augmented plane wave method. The anisotropy in Fermi velocity, conductivity and plasma frequency stemming from the layered structure are found to be enhanced by spin-orbit cou pling. The relationship between spin- orbit interaction and the lack of two-dimensional inversio n in the PtAs layers is analyzed within a tight-binding Hamiltonian based on the ﬁrst-principles c alculations. Finally, the band structure suggests that electron doping could increase Tc. PACS numbers: 74.20.Pq,74.70.Xa,71.20.-b,71.18.+y I. INTRODUCTION The recent discovery of superconductivity in pnictides has attracted extensive attention owing to their surpris- ingly high Tc.1While the highest Tcso far is 56 K for GdFeAsO,2a consensus on the pairing mechanism has not yet been reached.3This class of materials share a common crystal structure, that is, the Fe square lat- tice. While most of the superconducting pnictides are Fe-based, pnictides without iron also exhibit supercon- ductivity, although Tcis drastically lower than with iron. Recently, another superconducting pnictide, SrPtAs, has been discovered, which is the ﬁrst non-Fe based super- conductor with a hexagonal lattice rather than square lattice.4Although Tc=2.4K is lower than those of Fe- based pnictides, it possesses interesting physics associ- ated with its hexagonal crystal structure. SrPtAs crystallizes in a hexagonal lattice of ZrBeSi type with space group P63/mmc(No.194, D4 6h) — the same(non-symmorphic)spacegroupasthe hcp structure — with two formula units per primitive cell. As depicted in Fig. 1(a), its structure resembles5that of MgB 2(with the symmorphic space group P6/mmm, No.191, D1 6h), with a double unit cell along the caxis: the boron layers of MgB 2are replaced by PtAs layers, rotated by 60◦in successivelayers(responsibleforthenon-symmorphicna- ture) and Mg is replaced by Sr. Although the crystal as a whole has inversion symmetry (as do the Sr atoms), the individual PtAs layers lack two-dimensional inversion. Thus, SrPtAs diﬀers from MgB 2in two signiﬁcant ways: (i) it exhibits strong spin-orbit coupling (SOC) at the Pt ions and ( ii) the PtAs layers individually break inversion symmetry, exhibiting only C3v(orD3h ifz-reﬂection is included) symmetry. These two prop- erties play an important role in determining the band structure and also aﬀect the superconducting state. As- suming that the superconductivity is largely determined by the two-dimensional PtAs layers, the lack of in- version symmetry in the individual layers (which we call “broken local inversion symmetry”) opens up thepossibility to see the unusual physics associated with non-centrosymmetric superconductors.6With large spin- orbit coupling, nominally s-wave non-centrosymmetric superconductors exhibit spin-singlet and spin-triplet mixing,7,8enhanced spin susceptibilities,7,9enhanced Pauli limiting ﬁelds,6non-trivial magnetoelectric eﬀects and Fulde-Ferrell-Larkin-Ovchinnikov(FFLO)-like states in magnetic ﬁelds,10–16and Majoranamodes.17The local inversion symmetry breaking, together with a SOC that is comparable to the c-axis coupling, suggests that SrP- tAs will provide an ideal model system to explore related eﬀects in centrosymmetric superconductors.18,19 InthispaperwediscusstheelectronicstructureofSrP- tAs, including spin-orbit coupling, which was neglected in a previous theoretical study.20In Sec. II, we describe details of the calculations. The eﬀects of SOC on the bands, the Fermi surface, density ofstates, and transport properties at the Fermi surface of SrPtAs are presented in Sec. III, along with a tight-binding analysis. Finally, we suggest an enhanced Tcmight be possible via doping. FIG. 1. (a) (Color online) Crystal structure of SrPtAs, wher e red, blue, and grey spheres denote Pt, As, and Sr atoms, respectively. (b) Brillouin zone of SrPtAs and high symmetr y kpoints.2 II. METHOD First-principles calculations are performed using the full-potential linearized augmented plane wave (FLAPW) method21,22and the local density approxi- mation (LDA) for the exchange-correlation functional of Hedin and Lundqvist.23Then, SOC is included by a second variational method.24Experimental lattice con- stants,a= 4.244˚A andc= 8.989˚A, are used.25Cutoﬀs used for wave function and potential representations are 196 eV and 1360 eV, respectively. Muﬃn-tin radii are 2.6, 2.4, and 2.1 a.u for Sr, Pt, and As, respectively. Semicore electrons such Sr 4 pand As 3 dare treated as valence electrons, which are explicitly orthogonalized to the core states.26Brillouin zone summations were done with 90kpoints in the Monkhorst-Pack scheme,27while the density of states are obtained by the tetrahedron method.28Although most of the calculations were per- formed using LDA, some results were also done with the GGA as well.29In order to calculate the Fermi velocity and plasma frequency for in-plane and out-of-plane con- tributions, eigenvalues from self-consistent calculations are ﬁtted by a spline method over the whole Brillouin zone.30–32 III. RESULTS The band structure of SrPtAs is presented in Fig. 2 along the symmetry lines shown in Fig. 1(b). Plots in the left (right) column are without (with) spin-orbit cou- pling, and plots in the upper and lowerrowsare the same but highlighted for As pand Ptdorbitals, respectively. Our energy bands and Fermi surfaces without SOC agree with those of Ref. [20]. The main band of Sr 5 sorigin is located far above the Fermi level ( EF), consistent with Zintl’s scheme33that Sr donates electrons to the PtAs layer and behaves almost like an inert Sr2+ion.Without SOC, bands on the zone boundary face, the kz=π/c plane(H−A−L−H), exhibit four fold degeneracy — two from spin and the other two from two diﬀerent PtAs layers — as a consequence of the non-symmorphic trans- lations along the c-axis, just as for the hcp structure. For thekz= 0 plane ( K−Γ−M−K), there is no such symmetry-dictated degeneracy due to the two equivalent layers, but instead the magnitude of splitting is propor- tional to the inter-layer coupling. With SOC, the bands change markedly: The four-fold degeneracy on the zone boundary face is reduced to a two fold pseudospin degen- eracy due to inversion symmetry, whereas bands along theA−Lline keeps the fourfold degeneracy as a conse- quence of time-reversal symmetry.34 In a simple atomic picture, the SOC Hamiltonian is Hsoc=δL·σ,whereδrepresentsthestrengthoftheSOC. Values of the SOC strength derived from the calculations areδPt=0.32 eV for the Pt dorbitals and δAs=0.23 eV for Asporbitals, which are used in laterdiscussions. The SOC splitting at the Apoint for Pt dxz,dyz(dxy,dx2−y2)orbitalsis0.59(0.54)eV,whereasforAs px,ythesplitting 0.28 eV. Togaininsightintothe eﬀectofspin-orbitcoupling, we consider a simple tight-binding theory for a single PtAs layer. A single PtAs layer lacks a center of inversion symmetry and therefore allows an anti-symmetric spin- orbit coupling of the form Hso=/summationdisplay k,s,s′gk·σss′c† kscks′(1) exists, where c† ks(cks) creates (annihilates) an elec- tron with momentum kand pseudo-spin s, andσde- note the Pauli matrices. Time-reversal symmetry im- posesgk=−g−k. Here we ﬁnd the form of gk through a consideration of the coupling between the Pt dx2−y2,dxyorbitals and the As px,pyorbitals. These orbitals give rise to the Fermi surfaces with cylindrical topology around the Γ– Aline. On the Pt sites, we deﬁne states\|d±,s >=\|dx2−y2,s >±i\|dxy,s >and on the As sites we deﬁne the states \|p±,s >=\|px,s >±i\|py,s >(s denotes spin). On a single Pt or As site, the spin-orbit coupling L·Shas only LzSzwith non-zero matrix ele- ments in these subspaces. This splits the local four-fold degeneracyinto twopairs so that forPt (As) \|d+,↑>and \|d−,↓>(\|p+,↑>and\|p−,↓>) form one degenerate pair while\|d+,↓>and\|d−,↑>(\|p−,↑>and\|p+,↓>) form the other degenerate pair. For both pairs, time-reversal symmetry is responsible for the degeneracy and we can (a) p ,p x ypzsEnergy (eV) -6-4-2024 H K ALHK ΓM (c) d , d xy x -y 2 2d , d xz yzdz2 sEnergy (eV) -6-4-2024 H K ALHK ΓM(b) Energy (eV) -6-4-2024 H K ALHK ΓM (d) Energy (eV) -6-4-2024 H K ALHK ΓM FIG. 2. (Color online) Band structure of SrPtAs (a),(c) with - out and (b),(d) with spin-orbit coupling. In (a) and (b), As px,yandpzorbitals are shown in red and green, respectively, and Assin blue. In (c) and (d), Pt ( dxy,dx2−y2), (dxz,dyz), anddz2orbitals are presented in red, green, and blue, respec- tively. Contribution from Pt sis shown in purple.3 label the two degenerate partners of each pair through a pseudo-spin index. We include a spin-independent near- est neighbor hopping between the As and Pt sites. Thisyields the following tight binding Hamiltonian in k-space H0=/summationdisplay k,sΨ† s(k)Hs(k)Ψs(k), (2) where Ψ s(k) = (ck,d+,s,ck,d−,s,ck,p+,s,ck,p−,s)T,sis the spin label ( s={↑,↓}). For spin up, H↑(k) is (to get H↓(k) change the sign of δPtandδAs) ǫd+δPt 0 t(1+ν∗e−ik·T3+νeik·T2)˜t(1+e−ik·T3+eik·T2) 0 ǫd−δPt˜t(1+e−ik·T3+eik·T2)t(1+νe−ik·T3+ν∗eik·T2) t(1+νeik·T3+ν∗e−ik·T2)˜t(1+eik·T3+e−ik·T2) ǫp+δAs 0 ˜t(1+eik·T3+e−ik·T2)t(1+ν∗eik·T3+νe−ik·T2) 0 ǫp−δAs (3) whereν=ei2π/3,T1=a(1,0),T2=a(−1/2,√ 3/2), T3=a(−1/2,−√ 3/2),tand˜tare the hopping param- eters between the As pand Ptdorbitals, and δPt(δAs) is the atomic SOC parameter for Pt d(Asp) orbitals. To ﬁnd an eﬀective Hamiltonian for the Pt d-orbitals, we as- sume that \|ǫd−ǫp\|is the largest energy scale and treat allotherparameters( t,˜t,δPt,δAs) asperturbations. This yields the following eﬀective Hamiltonian HPt=/summationdisplay k,sΨ† Pt,s(k)HPt,s(k)ΨPt,s(k),(4)where Ψ Pt,s(k) = (ck,d+,s,ck,d−,s)T,sis the spin label, and HPt,↑=/parenleftbigg ǫd+δPt+t1/summationtext icos(k·Ti)+t2/summationtext isin(k·Ti)t3[cos(k·T1)+νcos(k·T2)+ν∗cos(k·T3)] t3[cos(k·T1)+ν∗cos(k·T2)+νcos(k·T3)]ǫd−δPt+t1/summationtext icos(k·Ti)−t2/summationtext isin(k·Ti)/parenrightbigg (5) wheret1= 2[˜t2+ cos(2π/3)t2]/(ǫd−ǫp),t2= −2sin(2π/3)t2/(ǫd−ǫp), andt3=−2t˜t/(ǫd−ǫp), and the sum over iis over the three vectors T1,T2, andT3. The Hamiltonian HPt,↑yields the disper- sionsǫ(k) =t1(k)±/radicalbig (δPt+t2(k))2+\|t3(k)\|2where t1(k) =t1/summationtext icos(k·Ti),t2(k) =t2/summationtext isin(k·Ti), and t3(k) =t3[cos(k·T1)+νcos(k·T2)+ν∗cos(k·T3)]. The SOC contribution that lifts the four-fold degeneracy into two bands Ek,±=Ek±\|g(k)\|atkz=π/ccan be found to order δPt//radicalbig δ2 Pt+\|t2(k)\|2+\|t3(k)\|2and is given by g(k) =ˆzt2δPt/radicalbig δ2 Pt+\|t2(k)\|2+\|t3(k)\|2 ×[sin(k·T1)+sin(k·T2)+sin(k·T3)] (6) where we have also included the contribution from HPt,↓(k). Note that this gleads to pseudo-spin in- teraction in Eq. 1 denoted by σz. We emphasize that thisσzoperates on pseudo-spin, not actual spin, the up and down pseudo-spin states are related by time- reversal symmetry (for example, the local states \|d+,↑>and\|d−,↓>form a pseudo-spin pair). The expres- sion forgclearly reveals how the interplay between the atomic SOC ( δPt) and the broken inversion symmetry (g(k) =−g(−k)) of a single PtAs layer leads to the rel- evant band SOC. Note that this single-layer band SOC will be of opposite signs for the two inequivalent PtAs layers in the unit cell. Further, the band spin-orbit split- ting will have additional contributions from terms of or- derδPt/(ǫd−ǫp) andδAs/(ǫd−ǫp) that were neglected in the above derivation. However, these additional contri- butions do not qualitatively change the results. Similar considerations apply for the other Pt d-orbitals. ThepreviousparagraphconsideredasinglePtAslayer. To complete the description, the coupling between the two inequivalent layers must be included. For the Pt dx2−y2,dxyorbitals considered above, the nearest neigh- bor inter-layer hopping matrix between \|d+,s >states is (the same expression appears for \|d−,s >states) ǫc(k) =tccos(kzc/2)(1+e−ik·T3+eik·T2).(7) Including this inter-layer hopping leads to the following4 Hamiltonian H±,s=/summationdisplay kΨ† ±,s(k)/braceleftig [ǫ±(k)−µ]σ0τ0+g(k)·στz +Re[ǫc(k)]σ0τx+Im[ǫc(k)]σ0τy/bracerightig Ψ±(k,s′),(8) where Ψ ±,s(k) = (c±k↑1,s,c±k↓1,s,c±k↑2,s,c±k↓2,s)T, 1,2 denote the two inequivalent PtAs layers, σi(τi) are Pauli matrices that operate on the pseudo-spin (layer) space,ǫ±=t1(k)±/radicalbig (t2(k))2+\|t3(k)\|2, andg(k) is given in Eq. (6) (the τzmatrix describes the sign change ofgon the two layers). This Hamiltonian can be di- agonalized with resulting dispersion relations ǫ(k) = ǫ±(k)±/radicalbig \|ǫc(k)\|2+g2(k) and each state is 2-fold degen- erate due to time-reversal symmetry (Kramers degener- acy). Note that the tight binding theory described above suggests that eigenstates of Szare also eigenstates of the single electron Hamiltonian. However, inter-layer cou- pling termscan lead toadditional termsthat donot com- mute with Sz. The band structure suggests that these terms are not large for the states near the Fermi surface. The SOC found in in Eq. (6) has opposite sign for the diﬀerent layers as well as for the pseudo-spin direction. This is demonstrated in Fig. 3, where the band structure is resolved by layer and spin for H–L–H′, whereLis one ofthe time-reversalinvariantmomentum (TRIM) points, and the lines H–LandL–H′are related both by a mirror plane and, and morerelevant to the discussion, bya com- d , d xy x -y 2 2d , d xz yz d z 2(a) Energy (eV) -2.5-2-1.5-1-0.500.5 H H’ L (c) Energy (eV) H H’ L-2.5-2-1.5-1-0.500.5(b) Energy (eV) -2.5-2-1.5-1-0.500.5 H H’ L (d) Energy (eV) H H’ L-2.5-2-1.5-1-0.500.5 FIG. 3. (Color online) SOC splitting of bands. (a) and (b) [(c) and (d)] show components from Pt 5 dorbitals for up- per [lower] Pt atoms. Left column[(a) and (c)] is for spin-up components and right[(b) and (d)] is for spin-down compo- nents. Radius of circles is proportional to the magnitude of the components.bination of inversion and reciprocal lattice vectors. In Fig. 3, spin up[(a)] and down[(b)] components of the Pt dorbital from the upper PtAs layer are marked in diﬀer- ent colors, while components from the other PtAs layers are shown in (c) and (d). As expected from the tight- binding analysis, Figs. 3(a),(d) [and similarly for (b) and (c)] appear the same since they correspond to spatial in- version and time-reversal (opposite spin). Despite the presence of a global inversion center, the locally broken inversion symmetry in PtAs is evident in Fig. 3 where the spin degeneracy is broken in a single layer, for which the consequences are nontrivial.18(This result is not sur- prising or unexpected since in the limit that the coupling between PtAs layers vanishes, i.e., the separation goes to inﬁnity, the single layer result must be recovered.) This spin separation, however, does not result in magnetism because of spin compensation in each layer. These anti- symmetric splittings ( g(k) =−g(−k)) occur not only at theLpoint but also at the other TRIM points35— Γ,A, M, and the A-Lline — in the hcp structure. As expected from the layer structure, anisotropy oc- curs in the Fermi surfaces, conductivity, and plasma fre- quency. Two-dimensionalcrosssections of the Fermi sur- faces are shown in Fig. 4, both without [(a)-(d)] and with [(e)-(h)] SOC. Contours in the kz=π/c[(a),(e)] andkz= 0 [(d),(h)] planes clearly exhibit the conse- quence of the crystal symmetry: hexagonal symmetry around Γ and Aand trigonal symmetry around Kand H. All sheets exhibit almost two-dimensional cylindri- cal features except for a small pocket around H. This Α L LL L HH(a) 31 32333434 33 Γ MA L(b) 31 32 33 34 ΓA H KL M(c) 31 32 33 3434 333334 34 34 Γ M MM M KK(d) 3132333434 34 Α L LL L HH(e) 31323334 3433 Γ MA L(f) 31 32 33 34 ΓA H KL M(g) 31 32 33 34343334 Γ M MM M KK(h) 31323334 34 FIG. 4. Cross section of the Fermi surface (a)-(d): without and (e)-(h): with spin-orbit coupling. Cross section along the zone boundary face, kz=π/c[(a),(e)] and at the zone center, kz= 0 [(d),(h)]. (b),(c) are contours along the vertical plane shown with their corner points in the Brillouin zone, where (f),(g) are those with SOC included. Numbers next to Fermi surface sheets indicate band indices.5 anisotropy is further exempliﬁed by transport proper- ties which will be discussed later. Moreover, because of the symmetry-dictated degeneracy due to the non- symmorphic symmetry, there is no spin-orbit splitting along the time-reversal invariant direction, A-L, as seen in Fig. 4(e). Comparing Figs. 4(c) and (g), the SOC ap- pears to make the Fermi surfaces more cylindrical, conse- quently enhancing the two dimensional character of the Fermi surfaces. All the Fermi surfaces are hole-like after turning on SOC,in sharpcontrasttootherpnictide superconductors with two electron-like Fermi surfaces and two hole-like Fermi surfaces.36Instead of electron-like and hole-like Fermi surfaces, they are distinguished by orbital char- acter. Sheets around the Γ- Aline consist of σorbitals of the PtAs layer, As px,yand Ptdxy,x2−y2, while sheets around the K-Hline are from πorbitls, As pzand Pt dxz,yz. Two kinds of Fermi surfaces with diﬀerent orbital character might give rise to a two energy gap supercon- ductor in SrPtAs. The anisotropy due to the layered structure is further manifested in the average Fermi velocities and plasma frequencies. Neglecting SOC, the in-plane and out-of- plane Fermi velocities are /an}bracketle{tv2 x,y/an}bracketri}ht1/2=3.72×107cm/s and /an}bracketle{tv2 z/an}bracketri}ht1/2=1.02×107cm/s, respectively, and the plasma frequencies are Ω x,y=5.70 eV and Ω z=1.57 eV. The anisotropy ratio, deﬁned as a ratio of conductivities be- tween in-plane and out-of-plane components, is 13.3, as- suming an isotropic scattering rate. With SOC, the anisotropy is enhanced: /an}bracketle{tv2 x,y/an}bracketri}ht1/2=3.76×107cm/s and /an}bracketle{tv2 z/an}bracketri}ht1/2=6.78×106cm/s; Ω x,y=5.57eV, Ω z=1.00eV, and the anisotropy ratio increases to a much higher value of 30.8. The decrease of vzby 33% by SOC is consis- tent with the enhanced two dimensional character of the Fermi surfaces. Table I summarizes contributions from each Fermi surface. The largest contribution to the den- sity of states at the Fermi level, N(0), comes from the 34th band at around K, which is due to the low veloc- ity at the Fermi surface. The anisotropy ratio is usu- ally much larger than 1 except for the small hole pocket aroundH. TABLE I. Fermi surface properties with SOC included for each surface: Density of states, N(0) (states/eV/spin); av- erage velocities, /angbracketleftv2 x/angbracketright1/2,/angbracketleftv2 z/angbracketright1/2(107cm/s); anisotropy ratio, /angbracketleftv2 x/angbracketright//angbracketleftv2 z/angbracketright; and plasma frequencies, Ω x, Ωz(eV). The numbers 31,32,33, and 34 represent band indices and Γ and K in the parenthesis represent the locations of the Fermi surface. 31 32 33(Γ) 34(Γ) 33(K) 34(K) Total N(0) 0.085 0.107 0.209 0.346 0.267 0.943 1.898 /angbracketleftv2 x/angbracketright1/27.03 6.79 5.92 4.35 1.25 1.68 3.76 /angbracketleftv2 z/angbracketright1/21.03 0.67 0.47 1.04 1.37 0.41 0.678 /angbracketleftv2 x/angbracketright//angbracketleftv2 z/angbracketright46.9 103 159 17.3 0.83 16.9 30.8 Ωx2.20 2.39 2.91 2.74 0.69 1.75 5.57 Ωz0.32 0.24 0.23 0.66 0.76 0.43 1.00 0 10 20 30 (a)total 0 5 -0.200.2 0 0.1 0.2 DOS (States/eV)(b)Sr s Pt s As s 0 0.2 0.4 0.6 0.8 (c)As pxAs pz 0 1 2 3 -6-4-2024 Energy(eV)(d)Pt dxyPt dyzPt dz2 FIG. 5. (Color online) Density of states (DOS) of SrPtAs: (a) total DOS, (b) sorbitals, (c) As porbitals, and (d) Pt d orbitals. The total density of states (DOS) and orbital decom- posed partial DOS are presented in Fig. 5. The states aroundEFarise mainly from As pand Ptd. The non- bonding Pt dz2bands are located at around -2.2 eV, and becausetheyhybridizelittlewithotherorbitals,theygive riseto a peak in the DOS. In contrast, π-bonding orbitals such as As pzand Ptdxz,yzare located around the Fermi levelwith awide band width. In particular, the van Hove singularity(vHS) comingfrom asaddlepoint near Kjust aboveEF(inset of Fig. 5(a))exhibits two dimensional charactersince a 2D vHS gives rise to a singularity in the DOS while a 3D vHS gives only a discontinuity in slope. RaisingEFto vHS, which might be realized by electron doping via Sr layers, could potentially increase Tc: As- suming rigid bands, we estimate that electron doping by roughly 25% will lift EFto the vHS, enhancing the DOS at the Fermi level by 6%. Further, using a weak-coupling form for Tc(Tc= 1.14TDe−1/N(0)V), assuming that the pairing potential Vdoes not change by electron doping, and assuming Debye temperature of TD= 229K,37we estimate that Tccould be enhanced to as high as 3.8K by electron doping. ApplyinghydrostaticpressurecouldalsoraisetheDOS and could, as often happens, enhance Tc. Rh is a can- didate to supply a chemical pressure to SrPtAs since it has a similar electronegativity as Pt but a smaller ionic radius. However, the crystal structure is sensitive to the constituent atoms; for example, SrPtSb where As is re- placed by Sb has an AlB 2-type structure, while YPtAs, where Sr is replaced by Y with more electrons, has a6 hexagonalstructurewith fourslightlypuckeredPtAs lay- ers in a unit cell.5 Finally we consider the possibility of simple collinear magneticsolutions. Theantiferromagnetic(AFM)phase, where the moments in a PtAs layer are aligned, by an- tiparallel to those in adjacent layers, is favored by 0.23 meV (0.49 meV in GGA) per formula unit, than the non- magnetic phase; the ferromagnetic orientation converged to the non-magnetic solution. Magnetic moments are given in Table II. While the energy diﬀerences and cal- culated moments are too small to make deﬁnitive state- ments regardingmagnetic phasesin SrPtAs, the material appears to be near a magnetic instability. IV. SUMMARY First-principles calculations of the electronic structure of SrPtAs have been presented with SOC fully taken into account. The role of SOC on the electronic structure is manifested in the energy bands and Fermi surfaces. The important physics originates from two factors: strong SOC in Pt atoms and locally broken inversion symme- try in PtAs layers. We have constructed a tight-binding Hamiltonian based on the self-consistent electronicstruc- ture that provides insight into the SOC. Sheets of the Fermi surface are spatially well separated in the Bril- louin zone: cylindrical Fermi surfaces with σ-character at the zone center (around Γ- A) and two Fermi sur-faces,i.e.,a pocket and a cylinder, with π-character at the zone corner (around K-H). All the Fermi surfaces are hole-like which distinguishes this material from other pnictide superconductors. The transport properties are highly anisotropic between x,y−andz−directions. Rh is suggested for a positive pressure eﬀect to increase Tc. Furthermore, the van Hove singularity is shown in the DOS above EF. Assuming rigidity of bands, we predict thatTCincreases up 3.4 K with 25% doping, which may be achieved by chemical doping in place of the Sr atom. ACKNOWLEDGMENTS SJY and SHR are indebted to Hosub Jin for fruitful discussions. SJY acknowledges the sabbatical research Grant by Gyeongsang National University. SHR and AJF are supported by the Department of Energy (DE- FG02-88ER45382). DFA is supported by NSF grant DMR-0906655. MWissupportedbyNSFDMR-1105839. TABLE II. Spin µsand orbital µlmagnetic moments and total magnetic moment µtin unit of µB. µs µl µt Pt 0.014 0.064 0.078 As 0.006 -0.027 -0.021 1Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono, J. Am. Chem. Soc. 130, 3296 (2008). 2C. Wang, L. Li, S. Chi, Z. Zhu, Z. Ren, Y. Li, Y. Wang, X. Lin, Y. Luo, S. Jiang, et al., Europhys. Lett. 83, 67006 (2008). 3I. Mazin and J. Schmalian, Physica C 469, 614 (2009). 4Y. Nishikubo, K. Kudo, and M. Nohara, J. Phys. Soc. Jpn. 80, 055002 (2011). 5R.-D. Hoﬀmann and R. P¨ ottgen, Z. Kristallogra. 216, 127 (2001). 6P. A. Frigeri, D. F. Agterberg, A. Koga, and M. Sigrist, Phys. Rev. Lett. 92, 097001 (2004). 7L. Gor’kov and E. Rashba, Phys. Rev. Lett. 87, 037004 (2001). 8H. Yuan, D. Agterberg, N. Hayashi, P. Badica, D. Van- dervelde, K. Togano, M. Sigrist, and M. Salamon, Phys. Rev. Lett. 97, 017006 (2006). 9P. A. Frigeri, D. Agterberg, and M. Sigrist, New J. Phys. 6, 115 (2004). 10V. Barzykin and L. Gor’kov, Phys. Rev. Lett. 89, 227002 (2002). 11D. F. Agterberg, Physica C 387, 13 (2003). 12R. P. Kaur, D. F. Agterberg, and M. Sigrist, Phys. Rev. Lett.94, 137002 (2005). 13K. V. Samokhin, Phys. Rev. B 78, 224520 (2008). 14V. M. Edelstein, J. Phys.: Condens. Matter 8, 339 (1996). 15S. K. Yip, Phys. Rev. B 65, 144508 (2002). 16S. Fujimoto, Phys. Rev. B 72, 024515 (2005).17M. Sato and S. Fujimoto, Phys. Rev. B 79, 094504 (2009). 18S.J. Youn,M. H.Fischer, S.H.Rhim, M.Sigrist, andD.F. Agterberg, e-print arXiv:cond-mat/1111.5058 (2011). 19M. H. Fischer, F. Loder, and M. Sigrist, Phys. Rev. B 84, 184533 (2011). 20I. Shein and A. Ivanovskii, Physica C 471, 594 (2011). 21E. Wimmer, H. Krakauer, M. Weinert, and A. J. Freeman, Phys. Rev. B 24, 864 (1981). 22M. Weinert, E. Wimmer, and A. J. Freeman, Phys. Rev. B26, 4571 (1982), and references there in. 23L. Hedin and B. I. Lundqvist, J. Phys. C. 4, 2064 (1971). 24A. H. MacDonald, W. E. Pickett, and D. D. Koelling, J. Phys. C 13, 2675 (1980). 25G. Wenski and A. Mewis, Z. Anorg. Allg. Chem. 535, 110 (1986). 26M. Weinert, G. Schneider, R. Podloucky, and J. Redinger, J. Phys.: Cond. Matt. 21, 084201 (2009). 27H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188 (1976). 28P. E. Bl¨ ochl, O. Jepsen, and O. K. Andersen, Phys. Rev. B49, 16223 (1994). 29J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett.77, 3865 (1996). 30D. D. Koelling and J. H. Wood, J. Comp. Phys. 67, 253 (1986). 31W. E. Pickett, H. Krakauer, and P. B. Allen, Phys. Rev. B38, 2721 (1988).7 32G. K. H. Madsen and D. J. Singh, J. Comput. Phys. Comm.175, 67 (2006). 33H. Sch¨ afer, Ann. Rev. Mater. Sci. 15, 1 (1985). 34L. F. Mattheiss, Phys. Rev. 151, 450 (1966).35L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. 98, 106803 (2007). 36D. J. Singh and M.-H. Du, Phys. Rev. Lett. 100, 237003 (2008). 37K. Kudo (2011), private communication.
0808.2372v1.Dilution_effect_in_correlated_electron_system_with_orbital_degeneracy.pdf	arXiv:0808.2372v1 [cond-mat.str-el] 18 Aug 2008Dilutioneffect incorrelated electron systemwithorbital degeneracy Takayoshi Tanaka,∗and Sumio Ishihara Department of Physics, Tohoku University, Sendai 980-8578 , Japan. (Dated: November 29, 2018) Theory of dilution effect in orbital ordered system is prese nted. The egorbital model without spin degree of freedom and the spin-orbital coupled model in a three-dim ensional simple-cubic lattice are analyzed by the Monte-Carlo simulation and the cluster expansion method. I n theegorbital model without spin degree of free- dom, reductionof the orbitalordering temperature due todi lutionissteeper thanthat inthedilute magnet. This isattributedtoamodiﬁcationoftheorbitalwave-function aroundvacantsites. Inthespin-orbitalcoupledmodel, it is found that magnetic structure is changed from the A-typ e antiferromagnetic order into the ferromagnetic one. Orbital dependent exchange interaction and a sign chan ge of this interaction around vacant sites bring about thisnovel phenomena. Presentresultsexplaintherec ent experiments intransition-metalcompounds with orbitaldilution. PACS numbers: 71.10.-w,71.23.-k, 75.30.-m I. INTRODUCTION Impurity effect in correlated electron system is one of the attractive themes in recent solid state physics.1,2The well known example is doping of non-magnetic impurity in high Tc superconducting cuprates; a small amount of substitu- tion ofCu by Zn dramaticallydestroysthe superconductivit y. Non-magneticimpurityeffect in the low-dimensionalgappe d spin system is another example. A few percent doping of Zn or Mg, which does not have a magnetic moment, into two- leg ladder systems, e.g. SrCu 2O3, and spin-Peierls systems, e.g. CuGeO 3, induces long-range orders of antiferromag- netism (AFM).3,4,5,6,7Impurity effect in charge and orbital ordered state is also studied in the colossal magnetoresist ive manganites.8,9Itisreportedinaso-calledhalf-dopedmangan- ite La0.5Ca0.5MnO3that a few percent substitution of Mn by Crcollapsesthecharge/orbitalorderassociatedwiththeA FM oneandinducesaferromagneticmetallicstate. Becauseofn o egelectrons in Cr3+, unlike Mn3+with one egelectron, Cr is regardedasan impuritywithoutorbitaldegreeoffreedom. Recently, impurity dopingeffect in an orbital orderedstat e is examined experimentally in a more ideal material. Mu- rakamiet al.have studied substitution effect in an orbital ordered Mott insulator KCuF 3with the three dimensional (3D) Perovskite crystal structure.10A Cu2+ion in the cubic- crystalline ﬁeld shows the (t2g)6(eg)3electron conﬁguration where one hole has the orbital degree of freedom. The long- range orbital order (OO), where the dy2−z2- anddz2−x2-like orbitalsarealignedwithamomentum (π,π,π),wasobserved at roomtemperaturesbyseveralexperiments. SincetheAFM spin ordering temperature is 39K, which is much lower than the OO temperature ( >1200K), a substitution of Cu by Zn, whichhasanelectronconﬁguration (t2g)6(eg)4,isregardedas anorbitaldilution. Itwas revealedbytheresonantx-raysc at- tering experimentsin KCu 1−xZnxF3that the OO temperature decreaseswithdopingofZnmonotonicallyandthediffracti on intensityat (3/23/23/2)disappearsaround x=0.45. Atthe sameZnconcentration,thecrystalsymmetryischangedfrom the tetragonal to the cubic one. That is to say, the OO disap- pearsaround x=0.45. Indilutemagnets,e.g. KMn 1−xMgxF3, thexdependenceofthemagneticorderingtemperatureaswellas the critical concentration where the magnetic order van- ishesarewellexplainedbythepercolationtheory.11,12Onthe contrary, the critical concentration in KCu 1−xZnxF3, where the OO disappears, is much smaller than the site-percolatio n threshold in a 3D simple cubic lattice, xp=0.69. These ex- perimentalobservationsimply that the diluteOO maybelong toanewclassofdilutedsystemsbeyondtheconventionalper - colationtheory. Dilution effect in orbital ordered state was also examined experimentallyin a mothercompoundofthe colossal magne- toresisitivemanganites,LaMnO 3. Thelong-rangeOO,where thed3x2−r2- andd3y2−r2-like orbitals align with a momen- tum(π,π,0), appears below 780K. The A-type AFM order, where spins are aligned ferromagnetically in the xyplane and are antiferromagnetically along the zaxis, is realized at 140K. Substitution of Mn3+by Ga3+, which has a 3 d10 electron conﬁguration, corresponds to both the orbital and spin dilution.13,14,15,16,17,18From the x-ray diffractionand X- rayabsorptionnear-edgestructure(XANES)experiments,t he tetragonally distorted MnO 6octahedra become regular cubic ones around the Ga concentration x=0.6. That is, the OO disappear around x=0.6 which is smaller than the percola- tion threshold xp=0.69 for the simple cubic lattice. Differ- ence between LaMn 1−xGaxO3and KCu 1−xZnxF3is seen in the magnetic structure. Blasco et al.observed by the neu- tron diffraction experiments in LaMn 1−xGaxO3that the fer- romagnetic (FM) component appears by substitution by Ga and increases up to x=0.5. This change of the magnetic structurefromthe A-typeAFM to FM wasalso conﬁrmedby the magnetization measurements. This FM component can- not be attributed to the itinerant electrons through the dou ble exchange interaction, since the electrical resistivity in creases with increasing x. These phenomena are in contrast to the conventional dilute magnets where the ordering temperatur e is reduced, but the magnetic structure is not changed. Far- rellandGehringpresentedaphenomenologicaltheoryforth e magnetism in LaMn 1−xGaxO3.13They noticed that a volume in a GaO 6octahedron is smaller than that in a MnO 6. Un- deran assumptionthat the Mn 3 dorbitalsarounda dopedGa tend to be toward the Ga, the magnetic structure change was examined.2 Inthispaper,amicroscopictheoryofdilutioneffectsinth e egorbital degenerate system is presented. We study the dilu- tion effectsin the eg-orbitalHamiltonianwithout the spin de- gree of freedom, termed HT[see Eq. (10)], and the spin and egorbital coupledone, termed HST[see Eq. (9)]. The classi- cal Monte-Carlo (MC) method in a ﬁnite size cluster, as well as the cluster expansion (CE) method is utilized. It is known that, in the classical ground state of HTwithout impurity, a macroscopic number of orbital states are degenerated due to frustrated nature of the orbital interaction. We demonstra te numericallythatthisdegeneracyisliftedatﬁnitetempera ture. It is shown that the OO temperature decreases rapidly with increasing dilution. From the system size dependence of the orbital correlation function in the MC method, the OO is not realizedattheimpurityconcentration x=0.2. Theresultsob- tainedbytheCEmethodalsoshowrapidquenchingofOOby dilutionin comparisonwith dilute spin models. These resul ts areinterpretedthat orbitalsaroundimpuritysites arecha nged so as to gain the remaining bond energy. This is a conse- quence of the bond-direction dependent interaction betwee n the inter-site orbitals. In the analyses of the spin-orbita l cou- pledmodel,itisshownthattheA-typeAFMstructurerealize d inx=0ischangedintoFMonebydilution. Thisisexplained by changing a sign of the magnetic exchange interaction due totheorbitalmodiﬁcationaroundimpuritysites. Implicat ions ofthepresentmicroscopictheoryandtheexperimentalresu lts inKCu 1−xZnxF3andLaMn 1−xGaxO3are discussed. In Sect. II, the model Hamiltonianfor the egorbitaldegree of freedomin a cubic lattice and the spin-orbitalcoupledon e are introduced. In Sect. III, the classical MC simulation an d theCE methodarepresented. Resultsofthenumericalanaly- ses in HTandHSTare presented in Sects. IV and V, respec- tively. Section VI is devoted to summary and discussion. A partofthenumericalresultsforthe egorbitalmodelhavebeen brieﬂypresentedin Ref.19. II. MODEL Doubly degenerate egorbital degree of freedom is treated bythepseudo-spin(PS)operatorwithmagnitudeof1/2. This operatoris deﬁnedby Ti=1 2∑ sγγ′d† iγsσγγ′diγ′s, (1) wherediγsistheannihilationoperatorofanelectronwithspin s(=↑,↓)and orbital γ(=3z2−r2,x2−y2)at sitei, andσare thePaulimatrices. Occupiedorbitalisrepresentedbyanan gle θof PS. The eigen state of the z-component of PS with an angleθis \|θ/an}bracketri}ht=cos/parenleftbiggθ 2/parenrightbigg/vextendsingle/vextendsingled3z2−r2/angbracketrightbig +sin/parenleftbiggθ 2/parenrightbigg/vextendsingle/vextendsingle/vextendsingledx2−y2/angbracketrightBig .(2) For example, θ=0, 2π/3,and4π/3correspondto the states where the d3z2−r2,d3y2−r2, andd3x2−r2orbitals are occupied by an electron, respectively. It is convenient to introduce thelinearcombinationsofthePS operatorsdeﬁnedby τl i=cos/parenleftbigg2πnl 3/parenrightbigg Tz i−sin/parenleftbigg2πnl 3/parenrightbigg Tx i, (3) withl=(x,y,z)andanumericalfactor (nx,ny,nz)=(1,2,3). Thesearetheeigenoperatorsforthe d3l2−r2orbitals. It is known that dominantorbital interactionsin transitio n- metal compoundsare the electronic exchangeinteractionan d phononic one. The former is derived from the generalized Hubbard-typemodelwiththe doublydegenerate egorbitals; Hele=∑ /an}bracketle{tij/an}bracketri}htγγ′s/parenleftBig tγγ′ ijd† iγsdjγ′s+H.c./parenrightBig +U∑ iγniγ↑niγ↓ +1 2U′∑ iγ/ne}ationslash=γ′niγniγ′+1 2K∑ iγ/ne}ationslash=γ′ss′d† iγsd† iγ′s′diγs′diγ′s,(4) whereniγ=∑sniγs=∑sd† iγsdiγs. Wedeﬁnetheelectrontrans- fer integral tγγ′ ijbetween the a pair of the nearest neighbor- ing (NN) sites. The intra-orbital Coulomb interaction U, the inter-orbital one U′, and the Hund coupling K. Through the perturbational expansion with respect to the NN transfer in - tegral under the strong Coulomb interaction, the spin-orbi tal superexchangemodelisobtained.20,21Byassumingarelation U=U′+K,forsimplicity,it isgivenas Hexc=−2J1∑ /an}bracketle{tij/an}bracketri}ht/parenleftbigg3 4+Si·Sj/parenrightbigg/parenleftbigg1 4−τl iτl j/parenrightbigg −2J2∑ /an}bracketle{tij/an}bracketri}ht/parenleftbigg1 4−Si·Sj/parenrightbigg/parenleftbigg3 4+τl iτl j+τl i+τl j/parenrightbigg ,(5) whereSiis the spin operator at site iwith a mgnitude of 1/2, andlrepresents a bond direction connecting sites iand j. Amplitudes of the superexchangeinteractionsare given as J1[=t2/(U−3K)]andJ2(=t2/U)wheretis the transfer in- tegralbetweenthe d3z2−r2orbitalsalongthe zdirection. The phononic interaction between the orbitals is derived fromtheorbital-latticecoupledmodelgivenby HJT=−gJT∑ imQm iTm i +∑ kξωkξ 2/parenleftBig p∗ kξpkξ+q∗ kξqkξ/parenrightBig , (6) whereasubscript mtakesxandz. Theﬁrsttermrepresentsthe Jahn-Teller (JT) coupling with a coupling constant gJT. Two distortionmodesin a O 6octahedronwith the Egsymmetryis denotedby Qz iandQx i. The second term is for the JT phonon whereqkξandpkξarethephononcoordinateandmomentum, respectively, and ωkξis the phonon frequency. Subscripts k andξare the momentumand the phononmode, respectively. Here, the spring constant between the NN metal and oxygen ions are taken into account. The interaction between orbita ls andtheuniformstrainandthestrain-energy,whicharenece s- sary in study of the cooperative JT effect, are not shown, for simplicity,inthisequation. Forconvenience,theﬁrstand sec- ondtermsinEq.6 aredenotedby Horb−lattandHlatt,respec- tively. By introducing the canonical transformation deﬁne d3 by /tildewideqkξ=qkξ−2 √ωkξ∑ mg∗ kξmTm −k, (7) and neglectingthe non-commutabilitybetween Hlatand/tildewideqkξ, the orbital and lattice degrees of freedom are decoupled as22,23,24,25 HJT=2g∑ /an}bracketle{tij/an}bracketri}htτl iτl j+/tildewiderHlatt. (8) The ﬁrst term in this equation gives the inter-site orbital i n- teraction with a coupling constant g=g2 JT/(3KS)whereKS is a spring constant, and /tildewiderHlattis given by the second term in Eq. (6), i.e. Hlatt, where the phonon coordinate and momen- tum are replaced by /tildewideqkξand its canonical conjugate momen- tum/tildewidepkξ, respectively. The model Hamiltonian studied in the present paper is given by a sum of the above two contributions. Quenched impurity without spin and orbital degrees of freedom is de- noted by a parameter εiwhich takes zero (one), when site i is occupied(unoccupied)by an impurity. The Hamiltonianis givenas HST=−2J1∑ /an}bracketle{tij/an}bracketri}htεiεj/parenleftbigg3 4+Si·Sj/parenrightbigg/parenleftbigg1 4−τl iτl j/parenrightbigg −2J2∑ /an}bracketle{tij/an}bracketri}htεiεj/parenleftbigg1 4−Si·Sj/parenrightbigg/parenleftbigg3 4+τl iτl j+τl i+τl j/parenrightbigg +2g∑ /an}bracketle{tij/an}bracketri}htεiεjτl iτl j. (9) Numericalresultsin this Hamiltonianis presentedin Sect. V. Wealsostudydilutioneffectintheorbitalmodelwithoutsp in degree of freedom. This model is given by taking Si·Sjin Eq. (9) to be zero. This procedure may be justiﬁed in the diluted orbital system of KCu 1−xZnxF3where the N´ eel tem- perature (T N) is much below the OO temperature TOO. The explicit form of the egorbital model without spin degree of freedomis givenby HT=2J∑ /an}bracketle{tij/an}bracketri}htεiεjτl iτl j, (10) whereJ(=2g+3J1/4−J2/4)is the effective coupling con- stant. Numerical results of this model Hamiltonian are pre- sentedinSect. IV. III. METHOD In order to analyze the model Hamiltonian introduced above by using the unbiased method, we adopt mainly the classical MC simulation in ﬁnite size clusters. The orbital PS operator is treated as a classical vector deﬁned in the Tz−Tx plane, i.e. Tz i=(1/2)cosθiandTx i=(1/2)sinθiwhereθiis acontinuousvariable. AswellastheconventionalMetropol is algorithm,the Wang-Landau(WL) methodis utilized.26Thisis suitable for the present spin-orbital coupled model wher e the energy scales of the two degrees are much different with eachother. Inordertocalculatethedensityofstate, g(E),with high accuracy in the WL method, we take that the minimum energy edge Emining(E)is higher a little than the ground state energy EGS, and assume g(EGS<E<Emin) =0. As a result, the present MC simulation is valid above a character - istic temperature Tminwhich is determined by \|Emin−EGS\|. This situation will be discussed in Sect. IV in more detail. The simulations have been performed in L×L×Lcubic lat- tices (L=12∼18) with the periodic-boundarycondition. In theMetropolismethod,foreachsample,3 ×104−1×105MC steps are spent for measurement after 8 ×103−2×104MC stepsforthermalization. Physicalquantitiesareaverage dover 20−80samplesateachparameterset. IntheWLmethod,the ﬁnal modiﬁcationfactor26is set to be ffinal=exp(2−27). Af- tercalculatingthedensityofstates,2 ×107MCstepsarespent formeasurement. To supplement the classical MC simulation, the ordering temperatures are also calculated by utilizing the CE method . We apply the CE method proposed in Ref. 27 to the present orbital model. For a given impurities conﬁguration {ε}in a latticewith Nsites, theOO parameterisgivenas M{ε}=TrN∑ iεiTz iρN{ε}, (11) withthedensitymatrix ρN{ε}=e−βH{ε} TrNe−βH{ε}, (12) where Tr Nrepresents the trace over the PS operator at sites withεi=1 in a crystal lattice, and H{ε}is the Hamiltonian with impurity conﬁguration {ε}. The OO parameter per site is obtained by averaging about all possible impurity conﬁgu - ration{ε}as M=1 (1−x)N/angbracketleftbig M{ε}/angbracketrightbig {ε}, (13) wherexis the impurity concentration. In the CE method, a cluster consisting of msites, termed {m}, is considered, and the PS operatorswhich do not belong to {m}are replaced by stochastic variables σi. Here we take (Tx i,Tz i)=(0,σi). The effectiveHamiltonianthusobtainedisdenotedas H{ε}{m}{σ} where{σ}isasetof σi,andthecorrespondingdensitymatrix is ρ{ε}{m}{σ}=exp/parenleftbig −βH{ε}{m}{σ}/parenrightbig Tr{m}exp/parenleftbig −βH{ε}{m}{σ}/parenrightbig,(14) where Tr {m}represents the trace over the PS operators in a cluster{m}. We expand M{ε}intoa seriesofclusteraverages asfollows, M{ε}=N ∑ m=1∑ {m}m ∑ k=1∑ {k}(−1)k−m ×Tr{k}/bracketleftBigg/parenleftBigg ∑ i∈{k}εiTz i/parenrightBigg ∑ {σ}ρ{ε}{k}{σ}/bracketrightBigg ,(15)4 FIG. 1: (a) Scematic picture of the degenerate PS conﬁgurati ons termedthe type-(I) degeneracy, and (b) that of the type-(II )one. where∑{m}is taken overall possible clustersconsisting of m sites, and ∑{k}is taken over all subclusters of ksites belong- ing to a given {m}. The variable σitakes 1/2 or−1/2 by a probabilityof P(σi)=δσi,1 2/parenleftbigg1+2M 2/parenrightbigg +δσi,−1 2/parenleftbigg1−2M 2/parenrightbigg .(16) By solving Eqs. (13)-(16) self-consistently, the order par am- eter and the ordering temperature are obtained as a function of impurity concentration. In the present study, we adopt th e CE method in the two-site cluster approximation,i.e. m=2. It wasshownthat, evenin the two-site clusterapproximatio n, theobtainedresultsshowgoodaccuracyinthecaseofthefer - romagneticHeisenbergmodelinasimplecubiclattice;devi a- tions fromthe resultsby other reliable methodsare about 2% for the critical impurity concentration.27To compare the re- sultsintheclassical MCsimulation,theorderingtemperat ure is also calculated in the classical version of the CE method where the traces in Eqs. (14) and (15) are replaced by inte- grals with respect to the continuous variable Tz ibetween 1 /2 and−1/2. IV. DILUTIONIN THE egORBITALMODEL In this section, numerical results for dilution effects in t he egorbital Hamiltonian (10) are presented. First, we show the results in the MC simulation without impurity. It is known that there is a macroscopic degeneracy in the mean- ﬁeld(MF)groundstatein HTwithoutimpurity.28Thisdegen- eracy is classiﬁed into the following two types: (I) Conside r a staggered-type OO with two sublattices, termed A and B, andmomentum Q=(π,π,π). IntheMFgroundstate,thePS anglesinthesublatticesaregivenby( θA,θB)=(θ,θ+π)with anyvalueof θ. Suchcontinuousrotationalsymmetryisunex- pectedfromthe Hamiltonian HTwhereanycontinuoussym- metries do not exist. (II) Consider an OO with Q=(π,π,π) and(θA,θB)=(θ0,θ0+π),andfocusononedirectioninthree- dimensional simple-cubic lattice, e.g., the zdirection. The MF energy is preserved by changing all PS in each layer perpendicular to the zaxis independently as ( θ0,θ0+π)→ !"#$ % &' () + ,- . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B C D E F G H IJ KLM NOPQRS T U V W XY Z[ \ ] ^ _` a b FIG.2: (a)Systemsizedependenceoftheorbitalcorrelatio nfunction MOO(x=0), and (b) that of the orbital angle function Mang(x=0). Theminimumenergy EminintheWLmethodistakentobe0 .95EGS in(a)and 0 .98EGSin(b). (−θ0,−θ0−π). These are schematically shown in Fig. 1. Bothtypesofdegeneracyareunderstoodfromthemomentum representationoftheorbitalinteraction, HT=2J∑ kψ† kˆE(k)ψk, (17) withψk=[Tz k,Tx k]and the 2 ×2 matrix ˆE(k). By diagonaliz- ingˆE(k), weobtainthe eigenvalues E±(k) =cx+cy+cz ±/radicalBig c2x+c2y+c2z−cxcy−cycz−czcx,(18) wherecl=cosaklwith a lattice constant a. The lower eigen valueJ−(k)has its minima along (π,π,π)−(0,π,π)and other two-equivalent directions.29At the point Γ, the two eigenvalues E+(k)andE−(k)aredegenerate. Thatis,theor- bital states correspondingto these momenta are energetica lly degenerateintheMFlevel. Aliftingofthisdegeneracyinth e MF ground state has been examined from the view points of the order-by-disorder mechanism by utilizing the spin wave analyses.28,30,31 Herewedemonstratethedegeneracyliftingandappearance ofthelong-rageOObytheMCmethod. Weintroduce,forim- purity concentration x, the staggered orbital correlation func- tion MOO(x)=1 N(1−x)/angbracketleftBigg/braceleftbigg ∑ i(−1)iεiTi/bracerightbigg2/angbracketrightBigg1/2 ,(19)5 andtheanglecorrelationfunction Mang(x)=1 N(1−x)/angbracketleftBigg/braceleftbigg ∑ i(−1)iεicos3θi/bracerightbigg2/angbracketrightBigg1/2 ,(20) where/an}bracketle{t.../an}bracketri}htrepresents the MC average and N=L3. The or- bitalcorrelationatthemomentum Q=(π,π,π)isrepresented byMOO(x),andtheanglecorrelation Mang(x)takesone,when theorbitalPSangleis2 πn/3withanintegernumber n. There- fore,MOO(x)andMang(x)are utilized as monitors for lifting of the type-(II) and (I) degeneracies, respectively. Tempe r- ature dependences of MOO(x=0)for various Lare shown in Fig. 2(a). With decreasing temperature, calculated resu lts for allLshow a sharp increasing around T/J=0.35. This increasing becomes sharper with the system size L. Below T/J=0.08,MOO(x=0)takes a temperature-independent value of about 0 .47. This ﬂat behavior is attributed to the lowest energy edge Eminfor the density of state calculated in the WL method, as explained in Sect. III. An extrapolated value ofMOO(x=0)towardT=0 is close to 0.5 which in- dicatesthatthetype-(II)degeneracyislifted andthe OO wi th themomentum Q=(π,π,π)isrealized. Temperaturedepen- dences of Mang(x=0)presented in Fig. 2(b) increase mono- tonicallytowardoneinthelowtemperaturelimit. Almostno - size dependence is seen in Mang(x=0). Therefore, the type- (I) degeneracy is also lifted and the PS angle is ﬁxed. Both results indicate the long-range OO where the momentum is Q= (π,π,π), and the PS angles are (θA,θB) = (θ0,θ0+π) withθ0=2πn/3. The temperature at which MOO(x=0)andMang(x=0) change abruptly is around T/J=0.33 corresponding to the OO temperature TOO(x=0). In more detail, this tempera- tureisdeterminedbytheﬁnite-sizescalingforthecorrela tion length. Thisis calculatedbythesecond-momentmethod; ξ(x)=1 2sin(akmin/2)/radicalBigg MOO(x)2−Mkmin(x)2 Mkmin(x)2,(21) with Mkmin(x)=1 N(1−x)/angbracketleftBigg/braceleftBigg ∑ iei(Q−k)·riεiTi/bracerightBigg2/angbracketrightBigg1/2 ,(22) wherekmin=(2π/L,0,0). Thescalingrelationfor ξ(x)is ξ(x)=LF/bracketleftBig L1/ν{T−TOO(x)}/bracketrightBig , (23) whereνis the critical exponent for correlation length, and F isthescalingfunction. Thecorrelationlengths ξ(x=0)/Lfor various sizes cross with each other at TOO(x=0). In Fig. 3, we plotξ(x=0)/Las a function of L1/ν[T−TOO(x=0)]. The scaling analyses work quite well for L=10, 12, and 14. The OO temperature TOO(x=0)and the critical exponent ν are determined by the least-square ﬁtting for the polynomia l expansion. We obtain as TOO(x=0)/J=0.344±0.002 and ν=0.69−0.81,although statistical errors are not enough to obtainthe precisevalueof ν. c d e f g h i j k l m n o p q r s t u v w x y z {\| } ~ FIG. 3: Scaling plot of the correlation length ξ(x=0)for the staggered orbital correlation. Numerical data are obtaine d by the Metropolis algorithm. ¡ ¢ £ ¤ ¥ ¦ § ¨ © ª « ¬ ® ¯ ° ± ² ³´ µ ¶ ·¸¹º»¼ ½ ¾ ¿À Á Â ÃÄ Å ÆÇ È É ÊË Ì ÍÎ Ï ÐÑ Ò ÓFIG. 4: Impurity concentration dependence of the staggered orbital correlationfunction MOO(x). Systemsize istaken tobe L=18. Nu- merical data are obtained bythe Metropolis algorithm. Now, we examine impurity effect in the OO. In Fig. 4, we present the staggered orbital correlation function MOO(x) for several impurity concentration x. Numerical data are ob- tainedbytheMetropolisalgorithmintheclassicalMCmetho d and the system size is chosen to be L=18. First, we fo- cus on the region of x≤0.15. As shown above, MOO(x=0) abruptly increases at TOO(x=0)∼0.34Jand is saturated to 0.5 in the low temperature limit. By introducing impurity, MOO(x>0)does not reach 0 .5 even at T/J=0.01, and its saturated value in low temperatures gradually decreases wi th increasing x. Although the system sizes are not sufﬁcient to estimateMOO(x)in the thermodynamic limit, MOO(x>0)at zero temperature does not show the smooth convergence to 0.5 in contrast to the diluted spin models. Beyond x=0.15, results are different qualitatively; although MOO(x)starts to increase around a certain temperature (e.g. T/J∼0.24 at x=0.2), saturated values of MOO(x)in the low temperature limit are rather small. In order to compare the size depen- dencesof MOO(x),temperaturedependencesof MOO(x=0.1) andMOO(x=0.2)for several system sizes are presented in6Ô Õ Ö × Ø Ù Ú Û Ü Ý Þ ß à á â ã ä å æ ç è é êë ìí îïðñ òó ôõö ÷ ø ùú û ü ý þ ÿ ! " # $ %&' () + , FIG.5: (a)Systemsizedependenceoftheorbitalcorrelatio nfunction MOO(x)atx=0.1, and(b) that at x=0.2. Fig.5. InFig.5(a)for x=0.1,MOO(x)forseveralsizescross aroundT/J=0.25 below which MOO(x)increases with L. On the other hand, In Fig. 5(b) for x=0.2,MOO(x)mono- tonically decreases with Lin all temperature range. This dif- ference above and below x=0.15 is also seen in the results of the correlation length. In Fig. 6, a correlation length at x=0.15 andx=0.2 are compared. In x=0.15,ξ(x)for different sizes cross around T/J=0.25. As shown in the in- set of Fig. 6(a), the scaling analyses works well. From this analysesfor ξ(x), theOOtemperaturein x=0.15isobtained asTOO(x=0.15)/J=0.248±0.003. On the other hand, in x=0.2 [see Fig. 6(b)], ξ(x)for different sizes do not seem tocrosswitheachotheratacertaintemperature,andthesca l- ing analyses does not work. From the above numerical re- sults, it is thought that the long-range OO disappears aroun d 0.15<x<0.2. The impurity concentration xdependence of TOO(x)ob- tained by the MC and CE methods are presented in Fig. 7. Two kinds of the CE methods, where the PS operators are treatedasclassicalvectorsandquantumoperators,arecar ried out. These are termed the classical and quantum CE meth- ods,respectively. Inbothcases,weadoptthetwo-sizeclus ter. As a comparison,the N´ eel temperaturesin the 3D XY model obtained by the classical MC method, and those in the 3D Heisenberg model by the classical CE method are also plot- ted in the same ﬁgure. It is shownthat decrease of TOO(x)by the MC method is much steeper than that of TN(x)in the XY and Heisenberg models. As shown in the size dependences ofMOO(x)andξ(x)atx=0.2, it is thought that the long range OO is not realized at this impurity concentration. A - . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B C D EF G H I JKLMN OPQRSTU V WX Y Z[ \] ^ _ ` a b c d e f gh i j k l m n o p q r s t u v w x y z { \| } ~ ¡ ¢ £¤ ¥ ¦ § ¨© ª «¬®¯° FIG.6: (a) System size dependence of the correlation length ξ(x)/L atx=0.15, and (b) that at x=0.2. The inset of (a) is the scaling plot forξ(x)atx=0.1. The OOtemperature andthe critical exponent at x=0.15areobtainedtobe TOO(x)=0.248±0.003 andν=0.755± 0.085, respectively.±²³ ´ µ ¶ · ¸ ¹ º » ¼ ½ ¾ ¿ À Á Â Ã Ä Å Æ Ç È É Ê Ë Ì ÍÎ Ï ÐÑ ÒÓÔÕÖ ×Ø ÙÚÛÜÝÞ ß à á â ã ä å æç è é ê ë ì í î ï ð ñò ó ô õ ö ÷ ø ù ú û ü ý þ ÿ FIG.7: Impurityconcentration xdependence of the OOtemperature TOO(x). Filled circles are obtained by the MC method. Results by the quantum and classical CE method are shown by broken lines . For comparison, xdependence of the N´ eel temperature TN(x)in the 3D XY model obtained by the MC method and that in 3D Heisen- berg model by the classical CE one are presented by ﬁlled tria ngles and dotted line, respectively. Thick arrow indicates the pe rcolation threshold ina 3D simple cubic lattice.7 ! "# $ % & FIG.8: (a)AsnapshotintheMCsimulationforthePSconﬁgura tion atx=0.1, and (b)that at x=0.3. Filledcirclesindicate impurities. rapid decrease of TOO(x)in comparison with the spin order- ing temperaturesis also obtainedby the CE method. TheOO temperature monotonically decreases with x, and disappears aroundx=0.4inthequantumCEcalculation,andaround0 .5 in the classical CE one. The critical impurity concentratio ns obtained by the MC and CE methods are much smaller than the percolation threshold xp=0.69 in the 3D simple-cubic lattice. Let us explain the physical picture of the orbital dilution. Snapshots of the PS conﬁguration in the MC simulation are shown in Figs. 8(a) and (b) for x=0.1 and 0.3, respectively. The staggered-type OO with the orbital angle ( θA,θB)=(0,π) isseeninthebackgroundofFig.8(a). Attheneighboringsit es of the impuritiesindicated by the open circles, PS vectorst ilt from the angle of (0,π). This deviation of the PS angles is notonlyduetothethermalﬂuctuation. FocusontheNNsites alongthe xdirectionofan impuritywhichoccupiesthe down PS sublattice. Inalmost all thesesites, PS anglesare chang ed from0to a positiveangle δθ. Thiskindoftiltingfrom (0,π) becomes remarkableat x=0.3. Then, we explain the micro- scopicmechanismofthisPStiltingduetodilution[seeFig. 9]. Focus on a PS at a certain site termed i. The interaction act- ingonthis site isconsideredbythe MF approximationwhere weassumethestaggered-typeOOwiththePSangle (0,π)ex- ceptforthesite iandanimpuritysite. TheHamiltonianwhich ' ( ) +, -. / 0 1 2 3 4 5 6 78 FIG. 9: (a) A schematic PS conﬁguration without impurity, an d (b) that withanimpurity. Aﬁlledcirclerepresents animpurity . concernstheinteractionactingonthissite isgivenas H(i) T=2J∑ l=(x,y,z)/angbracketleftBig εi+ˆelτl i+ˆel+εi−ˆelτl i−ˆel/angbracketrightBig τl i =−∑ l=(x,y,z)hl·Ti, (24) where ˆelis a unit vector along lin the simple cubic lat- tice, and hl= (hx l,hz l)are the MF. In the case of no dilution [Fig. 9(a)], the mean-ﬁelds are given by hx= J(−√ 3/2,−1/2),hy=J(√ 3/2,−1/2)hz=J(0,−2), and theHamiltonianinEq.(24)isreducedto H(i) T=3JTz i. (25) This implies that the stable PS conﬁguration at the site iis θi=π. Then,introduceanimpurityatsite i−ˆexandconsider the PS at site i[Fig. 9(b)]. The x-componentofthe MF in the casewithoutimpurityischangedinto hx=J(−√ 3/4,−1/4), and others are not. The effective interaction in Eq. (24) is givenas H(i) T=J 4/parenleftBig 11Tz i−√ 3Tx i/parenrightBig , (26) implyingthatthestableorbitalangleatsite iisθi∼π−0.15. This PS tilting due to dilution is attributed to the fact that the orbital interaction explicitly depends on the bond dire c- tion and is the essence of the diluted orbital systems. This is highly in contrast to the dilute spin system where dilutio n doesnotcausespeciﬁcspintiltingaroundtheimpuritysite but simplyincreasesthermalspin ﬂuctuationsincenumberof th e interactingbondis reduced. V. DILUTION INTHE SPIN-ORBITALMODEL In this section, we examine the dilution effect in the spin- orbital coupled model described by HSTin Eq. (9). First, we brieﬂy introduce the MF calculation for the spin and or- bital structures at x=0. The two sublattice structures for both the spin and orbital ordered states are considered, and the PS angles in sublattices A and B are assumed to be (θA,θB)=(θ,−θ). Weobtaintheferromagneticspinorderin89 : ; < = > ? @ A B C D E F G HI J KL MNOP QRST UVWXYZ[ \ ] ^ _ ` a b c de f g h ij k l m n o p qr s tu v w x y z { \| } ~ FIG. 10: (a) Total energy E, and (b) the A-type AFM correlation functionMA−AF(x=0)calculatedforseveralvaluesoftheminimum energyEmininthe WLmethod. System size ischosen tobe L=10. the case of J1/J2≥3, and the A-type AFM one in J1/J2<3. In the A-type AFM state, the orbital PS angle is uniquelyde- termined as θ=cos−1{2J2/(5J1−J2+6g)}. By taking the MF results into account, for the following MC calculations, we choose the parameter set as (J1/J2,g/J2) = (2.9,5). In thesevalues,theOOappearsatmuchhighertemperaturethan the N´ eel one, and the A-type AFM is realized near the phase boundary between FM and A-type AFM. These are suitable to demonstratethe magnetic structurechange due to dilutio n. The MC simulation results in the realistic parameter set for LaMnO 3will be introducedin theSect.VI. IntheMCsimulation,weutilizetheWLmethodin L×L× Lsite cluster ( L=6−10)with the periodic-boundarycondi- tion. Thespinoperator SiintheHamiltonianistreatedasa3D classical vector with an amplitude of 1/2. In the simulation , 2×107MC steps are spent for measurement after calculating thehistogramforthe densityofstates. Physicalquantitie sare averaged over 10MC samples at each parameter set. We no- tice againthe lowest energyedge Eminin the densityof states which is introduced in Sec. III. In Fig. 10, we show the Emin dependence of the total energy, E, and A-type AFM correla- tionfunction, MA−AF(x)deﬁnedby MA−AF(x)=1 N(1−x)/angbracketleftBigg/braceleftbigg ∑ i,l(−1)ilεiSi/bracerightbigg2/angbracketrightBigg1/2 (27) whereilforl=(x,y,z)representsthe lcomponentof the co- ordinateat site i. Theresultsin Fig. 10(a)imply thatthe tem- perature below which the total energy is ﬂat is determined ¡ ¢ £¤ ¥¦ § ¨© ª «¬ ®¯ ° ± ² ³ ´ µ FIG.11: Temperature dependence of the orbital correlation function MOO(x=0),thePSanglefunction Mang(x=0),theA-typeAFMone MA−AF(x=0),andtheFMone MF(x=0). Systemsizeischosento beL=8. by an adopted value of Emin. This temperature is denoted asTminfrom now on. As shown in Fig. 10(b), in the case ofEmin=−9.75(−9.84), an obtained MA−AF(x=0)below Tminis about 55% (75%)of its maximum value of 1/2. That is,TminatEmin=−9.84islowerthantheN´ eeltemperatureof theA-typeAFM.Althoughasaturatedvalueof MA−AF(x=0) is less than 0.5, this result is enough to examine the orderin g temperature. We chose Emin=−9.84 in the following MC simulationandfocusonchangeofthemagneticorderingtem- peratureduetodilution. First, we show the results without impurities. In Fig. 11, calculated MOO(x=0),MA−AF(x=0),andtheferromagnetic correlationfunctiondeﬁnedby MF(x)=1 N(1−x)/angbracketleftBigg/parenleftbigg ∑ iεiSi/parenrightbigg2/angbracketrightBigg1/2 (28) are presented. The staggered-type orbital correlation fun c- tionMOO(x=0)abruptlyincreasesaround T/J2=2.5which corresponds to the OO temperature TOO(x=0). This value is consistent with the previous results obtained in the mode l Hamiltonian HT; the effective orbital interaction in the present Hamiltonian HSTwith paramagnetic state is Jorb= g+3J1/4−J2/4 whereSi·SjinHSTis replaced by zero. The obtained TOO(x=0) =2.5J2corresponds to 0 .3Jorbin the present parameter set. This value is consistent with TOO(x=0) =0.344Jobtained in Sect. IV [see Fig. 2(a)]. In Fig. 11, the angle correlation function Mang(x=0)starts to increase at TOO(x=0). With decreasing temperature, at aroundT/J2=0.5[≡TN(x=0)], the second transition oc- curs. The orbital correlation function MOO(x=0)decreases abruptly, and MA−AF(x=0)grows up. The ferromagnetic correlationfunction MF(x=0)showsa small humpstructure aroundTN(x=0). That is, TN(x=0)is the N´ eel tempera- ture of A-type AFM. The PS angle correlation Mang(x=0) decreases and almost becomes zero below TN(x=0). This result indicates that, due to the magnetic transition, the P S9¶ · ¸ ¹ º » ¼ ½ ¾ ¿ À Á Â Ã Ä ÅÆ Ç ÈÉ Ê ËÌ ÍÎÏÐÑÒ ÓÔÕÖ ×ØÙÚÛ Ü Ý Þ ß à á â ã ä å æ ç è é ê ë ì í î ï ð ñ ò ó ô õ ö÷øù ú û ü ýþ ÿ FIG. 12: (a) System size dependence of the orbital correlati on func- tionMOO(x=0),and(b)thatoftheA-typeAFMcorrelationfunction MA−AF(x=0). angle is changed into (θA,θB)∼(π/2,−π/2)which is con- sistent with the MF results. In Figs. 12, size dependences of MA−AF(x=0),MOO(x=0)are presented. With increasing L, changes of MOO(x=0)andMA−AF(x=0)atTN(x=0) become steep, although a saturated values of MA−AF(x=0) is still less than 0.5 due to a ﬁnite value of \|Emin−EGS\|as mentionedabove. Impurity concentration xdependences of MA−AF(x)and MF(x)are presented in Fig. 13. With increasing xfrom the x=0 case,MA−AF(x)decreases gradually and almost dis- appears around x=0.09. On the other hand, MF(x), which showsasmallhumpstructurearound T/J2=0.4atx=0,in- creases with x, and takes about 0.35 in the case of x>0.09. Thatistosay,themagneticstructureischangedfromA-AFM into FM by dilution. At x=0.06, bothMA−AF(x)andMF(x) coexistdownto the lowest temperaturein the presentsimula - tion. This is supposed to be a cant-type magnetic order or a magneticphaseseparationoftheFMandA-typeAFMphases. To clarify the mechanismof the magnetic structurechange due to dilution, the effective magnetic interaction and the PS conﬁguration are examined. Here, the AFM stacking in the A-type AFM structure is chosen to be parallel to the zaxis. The effective magnetic interaction Jl iis deﬁned such that the Hamiltonian HSTin Eq. (9) is rewritten as HST=∑/an}bracketle{tij/an}bracketri}htJl iSi· Sj. Theexplicitformoftheeffectiveinteractionisgivenas Jz i=2(J1+J2)Tz iTz j+2J2/parenleftBig Tz i+Tz j/parenrightBig +3 2J2−1 2J1,(29) where we consider a NN pair of sites iandj(=i+ˆez)along thezdirection, since we are interested in the magnetic struc- ture along z. A contour map of the effective interaction Jz i, ! " # $% & ' ( ) + , - . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B C D E F G H IJ K L MN OPQR S T UV WXYZ [ \ ] ^ _`a b cd ef ghi j kl mn o p qr s t FIG. 13: (a) Impurity concentration xdependence of the A-type AFM correlation function MA−AF(x), and (b) that of the FM cor- relationfunction MF(x). Systemsize is chosen tobe L=8. and a snapshot of the PS conﬁgurations in the same xyplane are presented in Fig. 14(a) and (b), respectively. Signs of Jz i inalmostallregionarepositive(antiferromagnetic),reﬂ ecting theA-AFMstructure. Attheneighboringsitesoftheimpurit y along the ydirection, Jz is are negative(ferromagnetic). Away from the impurity, PS are ordered as ±Txin the staggered- typeOO. Near theimpurity,PS tilt from ±Txand ﬁnite com- ponentsof Tzappears. Thistilting ofPS is seen in theresults ofHTas explained in Sec. IV. Based on these numerical simulation, we explain mechanism of the magnetic structure change due to dilution. Start from the staggered-type orbit al orderedstate of (Tx,Tz)=(±1/2,0). Introduceoneimpurity atasitei0whichbelongstothe Tx=1/2sublattice,andfocus onthePSconﬁgurationandtheeffectiveexchangeinteracti on at sitesi0+ˆemandi0+ˆem+ˆezform= (x,y)(see Fig. 15). As explained in Sect. IV, orbital dilution induces the PS til t- ing so as to gain the energiesof the bondswhere an impurity doesnotoccupy. Thus,PS at site i0+mˆetiltsfrom θ=3π/2 to 3π/2+δθ(−δθ)form=x(y)with a positive angle δθ. Since the orbital interaction is the staggered-type, the bi lin- ear termTz i0+ˆemTz i0+ˆem+ˆezin Eq. (29) are negative for both the m=xandycases. As for the linear term in Eq. (29), there is a relation (Tz i0+ˆex+Tz i+ˆex+ˆez) =−(Tz i0+ˆey+Tz i+ˆey+ˆez). That is, contributionof this linear term to the spin alignment along z, whichisdeterminedbyasumof Jz i0+ˆexandJz i0+ˆey,iscanceled out. Therefore,when the ﬁrst term in Eq. (29) overcomesthe positive constant 3 J2/2−J1/2,Jz ibecomes negative and the ferromagneticalignmentalongthe zdirectionisstablearound theimpuritysites.10uv w x y z { \|} ~ ¡ ¢ £¤ ¥ FIG. 14: (a) Contour map of Jz ideﬁned in Eq. (29), and (b) a snap- shot of the PS conﬁguration around an impurity in the xyplane ob- tainedintheMCmethod. Aﬁlledcirclerepresentsanimpurit y. Tem- perature is chosen tobe T/J2=0.3. VI. SUMMARYAND DISCUSSION In this section, we discuss implications of the present nu- merical calculations on the recent experimental results in the transition-metalcompounds. Firstwehaveremarksonthere - lationbetweenthecalculatedresultsof HTshowninSect.IV andtheexperimentsinKCu 1−xZnxF3.10AsshowninSectIV, TOO(x)rapidlydecreaseswithincreasing xincomparisonwith dilute magnets (see Fig. 7). Although the critical concentr a- tion (x∼0.2−0.5), where the OO disappears, depends on the calculationmethods,that is, MCand CE, these valuesare far belowthe percolationthreshold( xp=0.69). Thisresult is consistentqualitativelywiththeZnconcentrationdepend ence of the OO temperature in KCu 1−xZnxF3where OO vanishes aroundx=0.45. One of the discrepancies between the the- ory and the experiments are seen in their quantitative value s of the critical impurity concentration where OO disappears . Some of the reasons of this discrepancy may be attributed to the anharmonic JT coupling and the long-range PS interac- tions due to the spring constants beyond the NN ions and so on,bothofwhicharenottakenintoaccountinthepresentcal - culation. Theformersubject,i.e. theanharmonicJTcoupli ng, ¦§ ¨©ª «¬ ® ¯°± ²³´µ ¶· ¸ ¹ º »¼ ½¾¿ ÀÁ ÂÃ ÄÅÆ ÇÈÉÊ ËÌ Í Î Ï FIG.15: AschematicPSconﬁgurationaroundanimpurityatsi tei0. Aﬁlledcirclerepresents animpurity. induces the anisotropy in a bottom of the adiabatic potentia l oftheQx−Qzplane,andpreventsthePStiltingaroundimpu- rity sites. This effect on the reduction of TOO(x)was studied brieﬂy in Ref. 19. It was shown that, in the realistic parame- ter values, the reduction of TOO(x)becomes moderate by the anharmonic coupling, but it is still steeper than that in dil ute magnets. Anotherfactor which may explainsthe discrepancy betweenthetheoryandtheexperimentsisthequantumaspect for the orbital degree of freedom. In the results obtained by the quantum CE method as shown in Fig. 7, the critical xfor TOO(x) =0 is larger than the results by other two classical calculations for the orbital model and is close to the experi - mental value of x=0.45. This may be due to the fact that quantumﬂuctuationin low temperaturesweakensthe low di- mensionalcharacterintheOOstateandpreventsacollapseo f OOagainstdilution. Thiskindofquantumeffectsinthedilu te orbitalsystemwasexaminedbythepresentauthorsinthetwo dimensional quantum orbital model.32It was shown that the reduction of TOO(x)due to dilution is weaker than that in the classical orbitalmodel. We brieﬂy mention the orbital PS tilting due to dilution. Similar phenomenaare knownas a quadrupolarglass state in molecular crystals where different kind interactions betw een moleculeswithquadruplemomentcoexists.33Akindofglass stateintermsofthequadrupolemomentappearswithincreas - ing randomnessfor the interactions. We suggest a possibili ty thatthe presentobservedPS tiltingaccompaniedwith thela t- ticedistortionofligandionsisabletobedetectedexperim en- tally. One of the most adequate experimental techniques are the pair-distribution function method by the neutron diffr ac- tionexperiments,andX-rayabsorptionﬁnestructure(XAFS ) wheretheincidentx-rayenergyistunedattheabsorptioned ge of the impurity ions. This observation may work as a check forthe presentscenarioin thediluteorbitalsystem. Next we discuss implications of the calculated results in Sect. V to the experimental results in LaMn1−xGaxO3.13,14,15,16,17,18By analyzing the spin- orbital coupled Hamiltonian HST, we ﬁnd that the magnetic11Ð Ñ Ò Ó Ô Õ Ö × Ø Ù Ú Û Ü Ý Þ ß à á â ã ä å æ ç è é ê ë ìí îï ð ñ ò ó ôõö÷ ø ù ú û üý FIG. 16: Impurity concentration dependence of the A-type AF M transition temperature, TN(x), and the FM one, Tc(x), calculated in the realistic parameter values for LaMn 1−xGaxO3. The parameters and the system size are chosen to be (J1/J2,g/J2) = (2.5,5)and L=8, respectively. structure is changed from the A-AFM order into the FM one. This calculation qualitatively explains the experime ntal results in LaMn 1−xGaxO3from the macroscopic point of view. In Sect. V, the parameter set is chosen to be close to the values for the A-type AFM/FM phase boundary, in order to demonstrate clearly the magnetic structure change due to the orbital dilution. Here we brieﬂy introduce the numerica l results obtained in the realistic parametervalues. To eval uate the realistic values, we calculate the OO temperature, the N´ eel temperature by the MF approximation, and the spin wave stiffness by the spin wave approximation from HST, and compare the experimental results in LaMnO 3. Then, we set up the parameters as (J1/J2,g/J2) = (2.5,5). The xdependences of the magnetic transition temperatures are presented in Fig. 16. With increasing xfromx=0,TN(x) of the A-type AFM order gradually decreases, and around x=0.2, the A-type AFM is changed into the FM order which remains at least to x=0.4. In semi-quantitative sense,thisresultisconsistentwiththeexperimentalmagn etic phase diagram in LaMn 1−xGaxO3. However, one of the discrepancies is that the canted phase survives up to x=0.4in LaMn 1−xGaxO3. This difference between the theory and the experiments is supposed to be due to the t2gspins in Mn sites and the antiferromagnetic superexchange interactio n betweenthemwhicharenotincludedexplicitlyinthepresen t calculation. ThisinteractionstabilizestheA-typeAFMph ase in comparison with the FM one, and maintains the canted phaseupto ahigher xregion. Insummary,wepresentamicroscopictheoryofdilutionef- fectsinthe egorbitaldegeneratesystem. Weanalyzethedilu- tion effects in the eg-orbital Hamiltonian without spin degree offreedom, HT,andthespinand egorbitalcoupledHamilto- nian,HST. The classical MC simulation and the CE method are utilized. It is shown that the OO temperature decreases rapidlywithincreasingdilution. Fromthe systemsize depe n- dence of the orbital correlation function in the MC method, the OO is not realized at the impurity concentration x=0.2. Tilting of orbital PS around impurity is responsible for thi s characteristic reduction of TOO(x). This is consequence of thebonddependentinteractionbetweentheinter-siteorbi tals. In the analyses of the spin-orbital coupled model, the mag- neticstructureischangedfromtheA-typeAFMstructureint o the FM one by dilution. This is explained by changing of the magnetic interaction due to the orbital PS tilting aroun d the impurity. The present results explain microscopically the novel dilution effects in KCu 1−xZnxF3and LaMn 1−xGaxO3, and provide a uniﬁed picture for the dilution effect in the or - bitalorderedsystem. Acknowledgments The authors would like to thank Y. Murakami, M. Mat- sumoto, and H. Matsueda for their valuable discussions. The authors also thank T. Watanabe and J. Nasu for their crit- ical reading of the manuscript. This work was supported by JSPS KAKENHI (16104005),and TOKUTEI (18044001, 19052001, 19014003) from MEXT, NAREGI, and CREST. One of the authors (T.T,) thanks the ﬁnancial support from JSPS. 1M. Imada, A. Fujimori,andY. Tokura, Rev. Mod. Phys. 70, 1039 (1998). 2S.Maekawa, T.Tohyama,S.E.Barnes,S.Ishihara,W.Koshiba e, and G.Khaliullin, Physics ofTransitionMetalOxides , (Springer- Verlag, Berlin,2004). 3M. Azuma, Y. Fujishiro, M. Takano, M. Nohara, and H. Takagi Phys. Rev. B 55, R8658 (1997). 4M. Sigristand A.Furusaki, J.Phys. Soc.Jpn. 65, 2385 (1996). 5S. B. Oseroff, S-W. Cheong, B. Aktas, M. F. Hundley, Z. Fisk, and L.W.Rupp, Jr.,Phys. Rev. Lett. 74, 1450 (1995). 6M. Hase, K. Uchinokura, R. J. Birgeneau, K. Hirota, and G. Shi - rane, J. Phys.Soc. Jpn. 65, 1392 (1996). 7H. Fukuyama, N. Nagaosa, M. Saitoh, and T. Tanimoto, J. Phys. Soc. Jpn. 65, 2377 (1996). 8A. Barnabe, A. Maignan, M. Hervieu, F. Damay, C.MartinandB.Raveau, Appl. Phys.Lett. 713907 (1997). 9T. Kimura, Y. Tomioka, R. Kumai, Y. Okimoto, and Y. Tokura, Phys. Rev. Lett. 83, 3940 (1999). 10N. Tatami, Y. Ando, S. Niioka, H. Kira, M. Onodera, H. Nakao, K. Iwasa, Y. Murakami, T. Kakiuchi, Y. Wakabayashi, H. Sawa, S.Itoh,J.Mag.Mag.Mat. 310,787(2007),andN.Tatami,Master thesis inTohoku University(2004). 11R. J. Elliott, P. Leath, and J. A. Krumhansl, Rev. Mod. Phys. 46, 465 (1974). 12R. B. Stinchcombe, Phase Transitions and Critical Phenomen a, Vol. 7, edited by C. Domb and J. L. Lebowitz, (Academic Press, London, 1983). 13J. Farrell,and G.A.Gehring, NewJ. Phys. 6, 168 (2004). 14J. Blasco, J. Garc ´ia, J. Campo, M. C. S´ anchez, and G. Sub ´ias, Phys. Rev. B 66, 174431 (2002).12 15M. C. Sa´ nchez, G. Subi´ as, J. Garci´ a, and J. Blasco, Phys. R ev. B 69, 184415 (2004). 16J. B. Goodenough, A. Wold, R. J. Arnott, and N. Menyuk, Phys. Rev.124, 373 (1961). 17J.-S. Zhou, H. Q. Yin, and J. B. Goodenough, Phys. Rev. B 63, 184423 (2001). 18J.-S. Zhou and J. B. Goodenough, Phys. Rev. B 68, 144406 (2003),ibid.Phys.Rev. B 77, 172409 (2008). 19T. Tanaka, M. Matsumoto and S. Ishihara, Phys. Rev. Lett. 95, 267041 (2005). 20K. I.Kugel, andD. I.Khomskii, Sov. Phys.Usp. 25, 231 (1982). 21S. Ishihara, J. Inoue and S. Maekawa, Phys. Rev. B 55, 8280 (1997). 22S. Okamoto, S. Ishihara, and S. Maekawa, Phys. Rev. B 65, 144403 (2000). 23J. Kanamori, J.Appl. Phys. 31, 14S (1960). 24M. Kataoka andJ. Kanamori, J.Phys. Soc.Jpn. 32, 113 (1972).25A. J.Millis,Phys.Rev. B 53, 8434 (1996). 26F.Wang,andD.P.Landau, Phys.Rev.Lett. 86,2050(2001), ibid. Phys. Rev. E 64, 056101 (2001). 27H. Mano, Prog. Theor.Phys. 57, 1848 (1977). 28Z. Nussinov, M. Biskup, L. Chayes, and J. van den Brink, Euro. Phys. Lett. 67, 990 (2004). 29S.Ishihara, M. Yamanaka, andN.Nagaosa, Phys.Rev. B 56, 686 (1997). 30J. van den Brink, P. Horsch, and F. Mack, Phys. Rev. B 59, 6795 (1999). 31K. Kubo, J.Phys. Soc.Jpn. 71, 1308 (2002). 32T. Tanaka and S.Ishihara, Phys.Rev. Lett. 98, 256402 (2007). 33K. Binder andJ. D.Reger, Adv. Phys. 41, 547 (1992). ∗Present address: The Institute for Solid State Physics, UniversityofTokyo,Kashiwa,Chiba277-8581,Japan.
1201.1607v3.Spin_orbit_coupled_Fermi_liquid_theory_of_ultra_cold_magnetic_dipolar_fermions.pdf	arXiv:1201.1607v3 [cond-mat.quant-gas] 10 May 2012Spin-orbit coupled Fermi liquid theory of ultra-cold magne tic dipolar fermions Yi Li and Congjun Wu Department of Physics, University of California, San Diego , California 92093, USA We investigate Fermi liquid states of the ultra-cold magnet ic dipolar Fermi gases in the simplest two-component case including both thermodynamic instabil ities and collective excitations. The magnetic dipolar interaction is invariant under the simult aneous spin-orbit rotation, but not under either the spin or the orbit one. Therefore, the correspondi ng Fermi liquid theory is intrinsically spin-orbit coupled. This is a fundamental feature of magnet ic dipolar Fermi gases diﬀerent from electric dipolar ones. The Landau interaction matrix is cal culated and is diagonalized in terms of the spin-orbit coupled partial-wave channels of the tota l angular momentum J. The leading thermodynamic instabilities lie in the channels of ferroma gnetism hybridized with the ferronematic order with J= 1+and the spin-current mode with J= 1−, where + and −represent even and odd parities, respectively. An exotic propagating collect ive mode is identiﬁed as spin-orbit coupled Fermi surface oscillations in which spin distribution on th e Fermi surface exhibits a topologically nontrivial hedgehog conﬁguration. PACS numbers: 03.75.Ss,05.30.Fk,75.80.+q,71.10.Ay I. INTRODUCTION Recentexperimentalprogressofultracoldelectricdipo- lar heteronuclear molecules has become a major focus of ultracold atom physics1–3. Electric dipole moments are essentially classic polarization vectors induced by the ex- ternal electric ﬁeld. When they are aligned along the z axis, the electric dipolar interaction becomes anisotropic exhibiting the dr2−3z2-type anisotropy. In Fermi sys- tems, this anisotropy has important eﬀects on many- body physics including both single-particle and collective properties4–14. Fermi surfaces of polarized electric dipo- lar fermions exhibit quadrupolar distortion elongated along thezaxis4,5,7,13. Various Fermi surface instabil- ities have been investigated including the Pomeranchuk type nematic distortions6,7and stripelike orderings10,14. The collective excitations of the zero sound mode exhibit anisotropic dispersions: The sound velocity is largest if the propagation wavevector /vector qis along the zaxis, and the sound is damped if /vector qlies in the xyplane7,8. Un- der the dipolar anisotropy, the phenomenological Lan- dau interaction parameters become tridiagonal matrices, which are calculated at the Hartree-Focklevel6,7, and the anisotropicFermi liquid theoryfor such systemshas been systematically studied7. The magnetic dipolar gases are another type of dipolar system. Compared to the extensive research on electric dipolar Fermi systems, the study on magnetic dipolar ones is a new direction of research. On the experimen- tal side, laser cooling and trapping Fermi atoms with large magnetic dipole moments (e.g.,161Dy and163Dy withµ= 10µB)15–17have been achieved, which provides a new opportunity to study exotic many-body physics with magnetic dipolar interactions. There has also been a great amount of progress for realizing Bose-Einstein condensations of magnetic dipolar atoms17–21. Although the energy scale of the magnetic dipolar in- teraction is much weaker than that of the electric one, it is conceptually more interesting if magnetic dipoles arenot aligned by external ﬁelds. Magnetic dipole moments are proportional to the hyperﬁne spin up to a Lande fac- tor, thus, they are quantum-mechanical operators rather than the nonquantized classic vectors as electric dipole moments are. Furthermore, there is no need to use ex- ternal ﬁelds to induce magnetic dipole moments. In fact, the unpolarized magnetic dipolar systems are isotropic. The dipolar interaction does not conserve spin nor orbit angular momentum, but is invariant under simultaneous spin-orbit (SO) rotation. This is essentially a spin-orbit coupled interaction. Diﬀerent from the usual spin-orbit coupling of electrons in solids, this coupling appears at the interaction level but not at the kinetic-energy level. The study of many-body physics of magnetic dipolar Fermi gases is just at the beginning. For the Fermi liquid properties, although magnetic dipolar Fermi gases were studied early in Refs. [22] and [6], the magnetic dipoles arefrozen, thus, their behaviorisnot much diﬀerent from the electric ones. It is the spin-orbit coupled nature that distinguishes non-polarized magnetic dipolar Fermi gases from polarized electric ones. The study along this line was was pioneered by Fregoso and Fradkin23,24. They studied the coupling between ferromagnetic and ferrone- matic orders, thus, spin polarization distorts the spher- ical Fermi surfaces and leads to a spin-orbit coupling in the single-particle spectrum. Since Cooper pairing superﬂuidity is another impor- tant aspect of the many-body phase, we also brieﬂy sum- marizethecurrentprogressinelectricandmagneticdipo- lar systems. For the single-component electric dipolar gases, the simplest possible pairing lies in the p-wave channel because s-wave pairing is not allowed by the Pauli exclusion principle. The dipolar anisotropy se- lects thepz-channel pairing25–32. Interestingly, for the two-component case, the dipolar interaction still favors the triplet pairing in the pzchannel even though the s wave is also allowed. It provides a robust mechanism for the triplet pairing to the ﬁrst order in the interaction strength33–36. The mixing between the singlet and the2 tripletpairingsiswitharelativephase ±π 2,whichleadsto a novel time-reversal symmetry-breaking pairing state33. The investigation of the unconventional Cooper pairing symmetry in magnetic dipolar systems was studied by the authors37. We have found that it provides a robust mechanismforanovel p-wave(L= 1)spintriplet( S= 1) Cooper pairing to the ﬁrst order in interaction strength. It comesdirectly from the attractivepart ofthe magnetic dipolar interaction. In comparison, the triplet Cooper pairings in3He and solid-state systems come from spin ﬂuctuations, which is a second-order eﬀect in interaction strength38,39. Furthermore, that pairing symmetry was not studied in3He systems before in which orbital and spin angular momenta of the Cooper pair are entangled into the total angular momentum J= 1. In contrast, in the3He-Bphase40,LandSare combined as J= 0, and in the3He-Aphase,LandSare decoupled and Jis not well-deﬁned41,42. Fermi liquid theory is one of the most important paradigms in condensed matter physics on interacting fermions38,43. Despitethepioneeringpapers6,22–24, asys- tematic study of the Fermi liquid properties of magnetic dipolar fermions is still lacking in the literature. In par- ticular, Landau interactionmatrices havenot been calcu- lated, and a systematic analysis of the renormalizations from magnetic dipolar interactions to thermodynamic quantities has not been performed. Moreover, collective excitations in magnetic dipolar ultracold fermions have not been studied before. All these are essential parts of Fermi liquid theory. The experimental systems of161Dy and163Dy are with a very largehyperﬁne spin of F=21 2, thus the Fermi liquid theory taking into account of all the complicated spin structure should be very challeng- ing. We take the ﬁrst step by considering the simplest case of spin-1 2magnetic dipolar fermions which preserve the essential features of spin-orbit physics and address the above questions. In this paper, we systematically investigate the Fermi liquid theory of the magnetic dipolar systems includ- ing both the thermodynamic properties and the collec- tive excitations, focusing on the spin-orbit coupled ef- fect. The Landau interaction functions are calculated and are diagonalized in the spin-orbit coupled basis. Renormalizations for thermodynamic quantities and the Pomeranchuk-type Fermi surface instabilities are stud- ied. Furthermore, thecollectivemodesarealsospin-orbit coupled with a topologically non-trivial conﬁguration of the spin distribution in momentum space. Their disper- sion relation and conﬁgurations are analyzed. Uponthecompletionofthispaper, webecameawareof the nice work by Sogo et al.44. Reference 44 constructed the Landau interaction matrix for dipolar fermions with a general value of spin. The Pomeranchuk instabilities wereanalyzedforthespecialcaseofspin1 2, andcollective excitations were discussed. Our paper has some overlaps on the above topics with Ref. [44] but with a signiﬁ- cant diﬀerence, including the physical interpretation of the Pomeranchuk instability in the J= 1−channel andour discovery of an exotic propagating spin-orbit sound mode. The remaining part of this paper is organized as fol- lows. The magnetic dipolar interaction is introduced in Sec. II. The Landau interaction matrix is constructed at the Hartree-Fock level and is diagonalized in Sec. III. In Sec. IV, we present the study of the Fermi liquid renor- malization to thermodynamic properties from the mag- netic dipolar interaction. The leading Pomeranchuk in- stabilities areanalyzed. In Sec. V, the spin-orbitcoupled Boltzmann equation is constructed. We further perform the calculation of propagating spin-orbit coupled collec- tive modes. We summarize the paper in Sec. VI. II. MAGNETIC DIPOLAR HAMILTONIAN We introduce the magnetic dipolar interaction and the subtlety of its Fourier transform in this section. The magnetic dipolar interaction between two spin-1 2particles located at /vector r1,2reads Vαβ;β′α′(/vector r) =µ2 r3/bracketleftBig /vectorSαα′·/vectorSββ′−3(/vectorSαα′·ˆr)(/vectorSββ′·ˆr)/bracketrightBig ,(1) where/vectorS=1 2/vector σ;α,α′,β,β′take values of ↑and↓;/vector r= /vector r1−/vector r2and ˆr=/vector r/ris the unit vector along /vector r. The Fourier transform of Eq. (1) is Vαβ;β′α′(/vector q) =4πµ2 3/bracketleftBig 3(/vectorSαα′·ˆq)(/vectorSββ′·ˆq)−/vectorSαα′·/vectorSββ′/bracketrightBig ,(2) which depends on the direction along the momentum transfer but not its magnitude. It is singular as /vector q→0. More rigorously, Vαβ,β′α′(/vector q) should be further multiplied by a numeric factor7as g(q) = 3/parenleftBigj1(qǫ) qǫ−j1(qL) qL/parenrightBig , (3) whereǫis a short range scale cut oﬀ, and Lis the long distance cut oﬀ at the scale of sample size. The spher- ical Bessel function j1(x) shows the asymptotic behav- iorj1(x)→x 3atx→0, andj1(x)→1 xsin(x−π 2) as x→ ∞. In the long wavelength limit satisfying qǫ→0 andqL→ ∞,g(q)→1 and we recover Eq. (2). If /vector q is exactly zero, Vαβ;β′α′= 0, because the dipolar inter- action is neither purely repulsive nor attractive, and its spatial average is zero. The second quantization form for the magnetic dipolar interaction is expressed as Hint=1 2V/summationdisplay /vectork,/vectork′,/vector qψ† α(/vectork+/vector q)ψ† β(/vectork′)Vαβ;β′α′(/vector q) ×ψβ′(/vectork′+/vector q)ψα′(/vectork), (4) whereVis the volume of the system. The density of states of two-component Fermi gases at the Fermi energy isN0=mkf π2/planckover2pi12, and we deﬁne a dimensionless parameter3 λ=N0µ2.λdescribes the interaction strength, which equals the ratio between the average interaction energy and the Fermi energy up to a factor on the order of 1. III. SPIN-ORBIT COUPLED LANDAU INTERACTION Inthissection, wepresenttheLandauinteractionfunc- tions of the magnetic dipolar Fermi liquid, and perform the spin-orbit coupled partial wave decomposition. A. The Landau interaction function Interaction eﬀects in the Fermi liquid theory are cap- tured by the Landau interaction function. It describes the particle-hole channel forward-scattering amplitudes among quasiparticles on the Fermi surface. At the Hartree-Fock level, the Landau function is expressed as fαα′,ββ′(ˆk,ˆk′) =fH αα′,ββ′(ˆq)+fF αα′,ββ′(ˆk,ˆk′),(5) where/vectorkand/vectork′are at the Fermi surface with the mag- nitude ofkfand/vector qis the small momentum transfer in the forward scattering process in the particle-hole chan- nel.fH αα′,ββ′(/vector q) =Vαβ,β′α′(ˆq) is the direct Hartree in- teraction, and fF αα′,ββ′(/vectork;/vectork′) =−Vαβ,α′β′(/vectork−/vectork′) is the exchange Fock interaction. As /vector q→0,fHis singular, thus we need to keep its dependence on the direction of ˆq. More explicitly, fH αα′,ββ′(ˆq) =πµ2 3Mαα′,ββ′(ˆq), (6) fF αα′,ββ′(ˆk;ˆk′) =−πµ2 3Mαα′,ββ′(ˆm),(7) where the tensor is deﬁned as Mαα′,ββ′(ˆq) = 3(/vector σαα′· ˆq)(/vector σββ′·ˆq)−/vector σαα′·/vector σββ′and ˆmis the unit vector along the direction of the momentum transfer ˆ m=/vectork−/vectork′ \|/vectork−/vectork′\|. We have used the following identity: 3(/vector σαβ′·ˆm)(/vector σβα′·ˆm)−/vector σαβ′·/vector σβα′ = 3(/vector σαα′·ˆm)(/vector σββ′·ˆm)−/vector σαα′·/vector σββ′(8) to obtain Eq. (7). B. The spin-orbit coupled basis Due to the spin-orbit nature of the magnetic dipolar interaction, we introduce the spin-orbit coupled partial- wave basis for the quasiparticle distribution over the Fermi surface following the steps below. Theδnαα′(/vectork) is deﬁned as δnαα′(/vectork) =nαα′(/vectork)−δαα′n0(/vectork), (9)wherenαα′(/vectork) =∝an}bracketle{tψ† α(/vectork)ψα′(/vectork)∝an}bracketri}htis the Hermitian single- particle density matrix with momentum /vectorkand satisﬁes nαα′=n∗ α′αandn0(/vectork) is the zero-temperature equilib- rium Fermi distribution function n0(/vectork) = 1−θ(k−kf). δnαα′(/vectork)isexpandedintermsoftheparticle-holeangular momentum basis as δnαα′(/vectork) =/summationdisplay SszδnSsz(/vectork)χSsz,αα′ =/summationdisplay Sszδn∗ Ssz(/vectork)χ† Ssz,αα′,(10) whereχSsz,αα′are the bases for the particle-hole singlet (density) channel with S= 0 and triplet (spin) channel withS= 1, respectively. They are deﬁned as χ00,αα′=δαα′, χ10,αα′=σz,αα′, χ1±1,αα′=∓1√ 2(σx,αα′±iσy,αα′), (11) which satisfy the orthonormal condition tr( χ† SszχS′s′ z) = 2δSS′δszs′z. Since quasiparticles are only well deﬁned around the Fermi surface, we integrate out the radial direction and arrive at the angular distribution, δnαα′(ˆk) =/integraldisplayk2dk (2π)3δnαα′(/vectork). (12) Please note that angular integration is not performed in Eq. (12). We expand δnαα′(ˆk) in the spin-orbit decou- pled bases as δnαα′(ˆk) =/summationdisplay LmSs zδnLmSs zYLm(ˆk)χSsz,αα′, =/summationdisplay LmSs zδn∗ LmSs zY∗ Lm(ˆk)χ† Ssz,αα′,(13) whereYLm(ˆk) is the spherical harmonics satisfying the normalization condition/integraltext dˆkY∗ Lm(ˆk)YLm(ˆk) = 1. We can also deﬁne the spin-orbit coupled basis as YJJz;LS(ˆk,αα′) =/summationdisplay msz∝an}bracketle{tLmSs z\|JJz∝an}bracketri}htYLm(ˆk)χSsz,αα′, Y† JJz;LS(ˆk,αα′) =/summationdisplay msz∝an}bracketle{tLmSs z\|JJz∝an}bracketri}htY∗ Lm(ˆk)χ† Ssz,αα′, (14) where∝an}bracketle{tLmSs z\|JJz∝an}bracketri}htis the Clebsch-Gordon coeﬃcient andYJJz;LSsatisﬁes the orthonormal condition of /integraldisplay dˆktr[Y† JJz;LS(ˆk)YJ′J′z;L′S′(ˆk)] = 2δJJ′δJzJ′zδLL′δSS′. (15)4 Using the spin-orbit coupled basis, δnαα′(ˆk) is expanded as δnαα′(ˆk) =/summationdisplay JJz;LSδnJJz;LSYJJz;LS(ˆk,αα′) =/summationdisplay JJz;LSδn∗ JJz;LSY† JJz;LS(ˆk,αα′),(16) whereδnJJz;LS=/summationtext msz∝an}bracketle{tLmSs z\|JJz∝an}bracketri}htδnLmSs z. C. Partial-wave decomposition of the Landau function We are ready to perform the partial-wave decomposi- tion for Landau interaction functions. The tensor struc- tures in Eqs. (6) and (7) only depend on /vector σαα′and/vector σββ′,thus the magnetic dipolar interaction only contributes to the spin-channel Landau parameters, i.e., S= 1. In the spin-orbit decoupled basis, the Landau functions of the Hartree and Fock channels are expanded, respectively, as N0 4πfH,F αα′;ββ′(ˆk,ˆk′) =/summationdisplay Lmsz;L′m′s′zYLm(ˆk)χ1sz(αα′) ×TH,F Lm1sz;L′m′1s′zY∗ L′m′(ˆk′)χ† 1s′z(ββ′). (17) For later convenience, we have multiplied the density of statesN0and the factor of 1 /4πsuch thatTH,Fare di- mensionless matrices. Without loss of generality, in the Hartree channel, we choose ˆ q= ˆz. The matrix elements in Eq. (17) are presented below. In the Hartree channel, TH Lm1sz;L′m′1s′ z=πλ 3(2δsz,0−δsz,±1)δL,0δL′,0δm,0δm′,0δszs′ z; (18) and in the Fock channel, TF Lm1sz;L′m′1s′z=−πλ 2/parenleftBigδLL′ L(L+1)−δL+2,L′ 3(L+1)(L+2)−δL−2,L′ 3(L−1)L/parenrightBig ×/integraldisplay dΩr[δszs′ z−4πY1sz(Ωr)Y∗ 1s′ z(Ωr)]YLm(Ωr)Y∗ L′m′(Ωr). (19) The magnetic dipolar interaction is isotropic, thus the spin-orbit cou pled basis are the most convenient. In these basis, the Landau matrix is diagonal with respect to the total angu lar momentum Jand itsz-component Jzas N0 4πfαα′;ββ′(ˆk,ˆk′) =/summationdisplay JJzLL′YJJz;L1(ˆk,αα′)FJJzL1;JJzL′1Y† JJz;L′1(ˆk,ββ′). (20) The matrix kernel FJJzL1;JJzL′1reads as FJJzL1;JJzL′1=πλ 3δJ,1δL,0δL′,0(2δJz,0−δJz,±1)+/summationdisplay msz;m′s′z∝an}bracketle{tLm1sz\|JJz∝an}bracketri}ht∝an}bracketle{tL′m′1s′ z\|JJz∝an}bracketri}htTF Lm1sz;L′m′1s′z.(21) We foundthat up toapositivenumericfactor, the second term in Eq. (21) is the same as the partial-wave matri- ces in the particle-particle pairing channel, which was derived for the analysis of the Cooper pairing instability in magnetic dipolar systems37. However, the above matrix kernel FJJzL1;JJzL′1is not diagonal for channels with the same values of JJz but diﬀerent orbital angular momentum indices Land L′. Moreover, the conservation of parity requires that even and odd values of Ldo not mix. Consequently, FJJzL1;JJzL′1is either diagonalized or reduced into a small size of just 2 ×2. For later convenience of study- ing collective modes and thermodynamic instabilities, we present below the prominent Landau parameters in somelow partial-wave channels. Below, we use ( J±JzLS) to represent these channels in which ±represents even and odd parities, respectively. The parity odd channel of J= 0−only has one possi- bility of (0−011) in which F0−011;0−011=π 2λ. (22) There is another even parity density channel with J= 0+, i.e., (0+000), which receives contribution from short ranges-wave interaction but no contribution from the magnetic dipolar interaction at the Hartree-Fock level. The parity odd channel of J= 1−only comes from5 (1−Jz11) in which F1−Jz11;1−Jz11=−π 4λ. (23) Another channel of J= 1−, i.e., (1−Jz10), channel from thep-wave channel density interactions, which again re- ceives no contribution from magnetic dipolar interaction at the Hartree-Fock level. These two J= 1−modes are spin- and charge-current modes, respectively, and thus, do not mix due to their opposite symmetry properties under time-reversal transformation. We next consider the even parity channels. The J= 1+channels include two possibilities of ( JJzLS) = (1+Jz01),(1+Jz21). The former is the ferromagnetism channel, and the latter is denoted as the ferronematic channel in Refs. [6] and [24]. Due to the spin-orbit na- ture of the magnetic dipolar interaction, these two chan- nels are no longer independent but are coupled to each other. Because the Hartree term breaks the rotational symmetry, the hybridization matrices for Jz= 0,±1 are diﬀerent. For the case of Jz= 0, it is F1+0=/parenleftbigg F1001;1001F1001;1021 F1021;1001F1021;1021/parenrightbigg =πλ 12/parenleftbigg 8√ 2√ 2 1/parenrightbigg , (24) whose two eigenvalues and their associated eigenvectors are w1+0 1= 0.69πλ, ψ1+0 1= (0.98,0.19)T, w1+0 2= 0.06πλ, ψ1+0 2= (−0.19,0.98)T.(25) The hybridization is small. For the case of Jz=±1, the Landau matrices are the same as F1+1=/parenleftbigg F1101;1101F1101;1121 F1121;1101F1121;1121/parenrightbigg =πλ 12/parenleftbigg −4√ 2√ 2 1/parenrightbigg . (26) Again the hybridization is small as shown in the eigen- values and their associated eigenvectors w1+1 1=−0.37πλ, ψ1+1 1= (0.97,−0.25)T, w1+1 2= 0.12πλ, ψ1+1 2= (0.25,0.97)T.(27) Landau parameters, or, matrices, in other high partial- wave channels are neglected, because their magnitudes are signiﬁcantly smaller than those above. We need to be cautious on using Eqs. (24) and (26) in which the Hartree contribution of Eq. 6 is taken. How- ever, Eq. (6) is valid in the limit q≪kfbut should be much larger than the inverse of sample size 1 /L. It is valid to use Eqs. (24) and (26) when studying the collective spin excitations in Sec. V below. However, when studying thermodynamic properties, say, magnetic susceptibility, under the external magnetic-ﬁeld uniform at the scale of L, the induced magnetization is also uni- form. In this case, the Hartree contribution is suppressedto zero, thus the Landau matrices in the J= 1+channel are the same for all the values of Jzas F1+,thm(λ) =/parenleftbigg F1Jz01;1Jz01F1Jz01;1Jz21 F1Jz21;1Jz01F1Jz21;1Jz21/parenrightbigg thm =πλ 12/parenleftbigg 0√ 2√ 2 1/parenrightbigg . (28) In this case, the hybridization between these two chan- nels is quite signiﬁcant. The two eigenvalues and their associated eigenvectors are w1+ 1=−π 12λ, ψ1+ 1= (/radicalbigg 2 3,−/radicalbigg 1 3)T, w1+ 2=π 6λ, ψ1+ 2= (/radicalbigg 1 3,/radicalbigg 2 3)T. (29) IV. THERMODYNAMIC QUANTITIES In this section, we study the renormalizations for ther- modynamic properties by the magnetic dipolar interac- tion and investigatethe Pomeranchuk-typeFermi surface instabilities. A. Thermodynamics susceptibilities The change in the ground-state energy with respect to the variation in the Fermi distribution density matrix include the kinetic and interaction parts as δE V=δEkin V+δEint V. (30) The kinetic-energy variation is expressed in terms of the angular distribution of δnαα′(ˆk) as δEkin V=4π N0/summationdisplay αα′/integraldisplay dˆkδnαα′(ˆk)δnα′α(ˆk) =8π N0/summationdisplay LmSS zδn∗ LmSs zδnLmSs z,(31) where the units of δnSsz(ˆk) andδnLmSs zare the same as the inverse of the volume. The variation in the inter- action energy is δEint V=1 2/summationdisplay αα′ββ′/integraldisplay/integraldisplay dˆkdˆk′fαα′,ββ′(ˆk,ˆk′)δnα′α(ˆk)δnβ′β(ˆk′) = 2/summationdisplay LmszL′m′s′z;Sδn∗ LmSs zfLmSs z,L′m′Ss′zδn∗ L′m′Ss′z. (32) Adding them together and changing to the spin-orbit coupled basis, we arrive at δE V=8π N0/summationdisplay JJz;LL′;Sδn∗ JJz;LSMJJzLS;JJzL′SδnJJz;L′S,(33)6 where the matrix elements are MJJzLS;JJzL′S=δLL′+FJJzLS;JJzL′S.(34) In the presenceofthe externalﬁeld hJJzLS, the ground state energy becomes δE V= 16π/braceleftBig1 2χ0/summationdisplay JJzLL′Sδn∗ JJz;LSMJJzLS;JJzL′SδnJJz;L′S −/summationdisplay JJzLShJJzLSδnJJz;LS/bracerightBig , (35) whereχ0=N0is the Fermi liquid density of states. At the Hartree-Fock level, N0receives no renormalization from the magnetic dipolar interaction. The expectation value ofδnJJzLSis calculated as δnJJzLS=χ0/summationdisplay L′(M)−1 JJzLS;JJzL′ShJJzL′S.(36) For theJ= 1+channel,M−1≈I−F1+,thm(λ) up to ﬁrstorderof λinthecaseof λ≪1. Asaresult,theexter- nal magnetic ﬁeld /vectorhalong thezaxis not only induces the z-component spin polarization, but also induces a spin- nematic order in the channel of ( J+JzLS) = (1+021), which is an eﬀective spin-orbit coupling term as δH=√ 2 12πλh/summationdisplay kψ† α(/vectork)/braceleftBig/bracketleftbig (k2−3k2 z)σz −3kz(kxσx+kyσy)/bracketrightbig/bracerightBig ψβ(/vectork). (37) Apparently, this term breaks time-reversal symmetry, and thus cannot be induced by the relativistic spin-orbit couplinginsolidstates. Thismagneticﬁeldinducedspin- orbit coupling in magnetic dipolar systems was studied by Fregoso et al.6,24 B. Pomeranchuk instabilities Even in the absence of external ﬁelds, Fermi surfaces can be distorted spontaneously known as Pomeranchuk instabilities45. Intuitively, we can imagine the Fermi sur- face as the elastic membrane in momentum space. The instabilities occur if the surface tension in any of its partial-wavechannels becomes negative. In the magnetic dipolar Fermi liquid, the thermodynamic stability condi- tion is equivalent to the fact that all the eigenvalues of the matrix MJJzLS;JJzL′Sare positive. We next check the negative eigenvalues of the Landau matrix in each partial-wave channel. Due to the absence of external ﬁelds, the Pomeranchuk instabilities are al- lowed to occur as a density wave state with a long wave lengthq→0. For the case of J= 1+, it is clear that in the channel of Jz=±1, the eigenvalue w1+1 1in Eq. (27) is negative and the largest among all the channels. Thus the leading channel instability is in the ( JJz) = (1+±1)channel, which occurs at w1+1 1<−1, or, equivalently, λ > λc 1+1= 0.86. The corresponding eigenvector shows that it is mostly a ferromagnetism order parameter with small hybridization with the ferronematic channel. A re- pulsiveshort-range swavescattering, which weneglected above will enhance ferromagnetism and, thus, will drive λc 1+1to a smaller value. The wavevector /vector qof the spin polarization should be on the order of 1 /Lto minimize the energy cost of twisting spin, thus, essentially exhibit- ing a domain structure. The spatial conﬁguration of the spin distribution should be complicated by actual bound- ary conditions. In particular, the three-vector nature of spins implies the rich conﬁgurations of spin textures. An interesting result is that the external magnetic ﬁeld actu- ally weakens the ferromagnetism instability. If the spin polarization is aligned by the external ﬁeld, the Landau interaction matrix changes to Eq. (28). The magnitude of the negative eigenvalue is signiﬁcantly smaller than that of Eq. (26). As a result, an inﬁnitesimal external ﬁeld cannot align the spin polarization to be uniform but a ﬁnite amplitude is needed. For simplicity, we only consider ferromagnetism with a single plane wave vector /vector qalong thezaxis, then the spin polarization spirals in the xy-plane. Since q∼1/L, we can still treat a uniform spin polarization over a dis- tance large comparable to the microscopic length scale. Without loss of generality, we set the spin polariza- tion along the xaxis. As shown in Ref. 24, ferro- magnetism induces ferronematic ordering. The induced ferronematic ordering is also along the xaxis, whose spin-orbit coupling can be obtained based on Eq. (37) by a permutation among components of /vectorkasH′ so(/vectork)∝ (k2−3k2 x)σx−3kx(kyσy+kzσz). According to Eq. (27), ferromagnetism and ferronematic orders are not strongly hybridized, the energy scale of the ferronematic SO cou- pling is about 1 order smaller than that of ferromag- netism. An interesting point of this ferromagnetism is thatit distortsthe sphericalshapeofthe Fermisurfaceas pointed by Fregoso and Fradkin24. This anisotropy will also aﬀect the propagationof Goldstone modes. Further- more, spin waves couple to the oscillation of the shape of Fermi surfaces bringing Landau damping to spin waves. This may result in non-Fermi liquid behavior for fermion excitations, and will be studied in a later paper. This ef- fect in the nematic symmetry-breakingFermi liquid state has been extensively studied before in the literature46–51. The next subleading instability is in the J= 1−chan- nel withL= 1 andS= 1 as shown in Eq. (23), which is a spin-current channel. The generated order parame- ters are spin-orbit coupled. For the channel of Jz= 0, the generated SO coupling at the single-particle level ex- hibits the three-dimensional (3D) Rashba type as Hso,1−=\|nz\|/summationdisplay kψ† α(/vectork)(kxσy−kyσx)αβψβ(/vectork),(38) where\|nz\|is the magnitude of the order parameter. The same result was also obtained recently in Ref. 44. In the absence of spin-orbit coupling, the L=S= 1 chan-7 nel Pomeranchuk instability was studied in Refs. [52] and [53], which exhibits the unconventional magnetism with both isotropic and anisotropic versions. They are particle-hole channel analogies of the p-wave triplet Cooper pairings of3He isotropic Band anisotropic A phases, respectively. In the isotropic unconventional magnetic state, the total angular momentum of the or- der parameter is J= 0, which exhibits the /vectork·/vector σ-type spin-orbit coupling. This spin-orbit coupling is gener- ated from interactions through a phase transition and, thus, was denoted as the spontaneous generation of spin- orbit coupling. In Eq. (38), the spin-orbit coupling that appears at the mean-ﬁeld single-particle level cannot be denoted as spontaneous because the magnetic dipolar in- teraction possesses the spin-orbit nature. Interestingly, in the particle-particle channel, the dominant Cooper pairing channel has the same partial-wave property of L=S=J= 137. The instability in the J= 1−(spin current) channel is weaker than that in the 1+(ferromagnetism) channel because the magnitude of Landau parameters is larger in the former case. The 1−channel instability should occur after the appearance of ferromagnetism. Since spin-current instability breaks parity, whereas, ferromag- netism does not, this transition is a genuine phase tran- sition. For simplicity, we consider applying an external magnetic ﬁeld along the zaxis in the ferromagnetic state to remove the spin texture structure. Even though the J= 1+and 1−channels share the same property un-der rotation transformation, they do not couple at the quadratic level because of their diﬀerent parity proper- ties. The leading-order coupling occurs at the quartic order as δF=β1(/vector n·/vector n)(/vectorS·/vectorS)+β2\|/vector n×/vectorS\|2, (39) where/vector nand/vectorSrepresent the order parameters in the J= 1−and 1+channels, respectively. β1needs to be positive to keep the system stable. The sign of β2de- termines the relative orientation between /vector nand/vectorS. It cannot be determined purely from the symmetry analy- sis but depends on microscopic energetics. If β2>0, it favors/vector n∝bardbl/vectorS, and/vector n⊥/vectorSis favored at β2<0. V. THE SPIN-ORBIT COUPLED COLLECTIVE MODES In this section, we investigate another important fea- tureoftheFermiliquid, thecollectivemodes, whichagain exhibit the spin-orbit coupled nature. A. Spin-orbit coupled Boltzmann equation We employ the Boltzmann equation to investigate the collective modes in the Fermi liquid state43 ∂ ∂tn(/vector r,/vectork,t)−i /planckover2pi1[ǫ(/vector r,/vectork,t),n(/vector r,/vectork,t)]+1 2/summationdisplay i/braceleftBig∂ǫ(/vector r,/vectork,t) ∂ki,∂n(/vector r,/vectork,t) ∂ri/bracerightBig −1 2/summationdisplay i/braceleftBig∂ǫ(/vector r,/vectork,t) ∂ri,∂n(/vector r,/vectork,t) ∂ki/bracerightBig = 0,(40) wherenαα′(/vector r,/vectork,t)andǫαα′(/vector r,/vectork,t)arethedensityanden- ergy matrices for the coordinate ( /vector r,/vectork) in the phase space and [,] and{,}mean the commutator and anticommuta- tor, respectively. Under small variations in nαα′(/vector r,/vectork,t) andǫαα′(/vector r,/vectork,t), nαα′(/vector r,/vectork,t) =n0(k)δαα′+δnαα′(/vector r,/vectork,t), ǫαα′(/vector r,/vectork,t) =ǫ(k)δαα′+/integraldisplayd3k′ (2π)3fαα′,ββ′(ˆk,ˆk′) ×δnββ′(ˆk′). (41) the above Boltzmann equation can be linearized. Plug- ging the plane-wave solution of δnαα′(/vector r,/vectork,t) =/summationdisplay qδnαα′(/vectork)ei(/vector q·/vector r−ωt),(42)we arrive at δnαα′(ˆk)−1 2cosθk s−cosθk/summationdisplay ββ′/integraldisplay dΩk′N0 4πfαα′,ββ′(ˆk,ˆk′) ×δnββ′(ˆk′) = 0, (43) wheresis the dimensionless parameter ω/(vfq). The propagationdirectionofthe wavevector /vector qisdeﬁnedalong thez-direction. In the spin-orbit decoupled basis deﬁned as δnLmSs z in Sec. IIIB, the linearized Boltzmann equation becomes δnLmSs z+ ΩLL′;m(s)FL′m′Ssz;L′′m′′Ss′′zδnL′′m′′Ss′′z= 0, (44) where Ω LL′(s) is equivalent to the particle-hole channel Fermi bubble in the diagrammatic method as ΩLL′;m(s) =−/integraldisplay dΩˆkY∗ Lm(ˆk)YL′m(ˆk)cosθk s−cosθk.(45)8 For later convenience, we present Ω LL′;min several chan- nels ofLL′andmas follows Ω00;0(s) = 1−s 2ln\|1+s 1−s\|+iπ 2sΘ(s<1), Ω10;0(s) = Ω 01;0=√ 3sΩ00;0(s), Ω11;0(s) = 1+3s2Ω00;0(s), Ω11;1(s) = Ω 11;−1(s) =−1 2/bracketleftBig 1−3(1−s2)Ω00;0(s)/bracketrightBig . (46) Equation (44) can be further simpliﬁed by using the spin-orbit coupled basis δnJJz;LSdeﬁned in Sec. IIIB, δnJJz;LS+/summationdisplay J′;LL′KJJzLS;J′JzL′S(s)FJ′JzL′S;J′JzL′′S ×δnJ′JzL′′S= 0, (47) where the matrix kernel KJJzLS;J′JzL′Sreads KJJzLS;J′JzL′S(s) =/summationdisplay msz∝an}bracketle{tLmSs z\|JJz∝an}bracketri}ht∝an}bracketle{tL′mSsz\|J′Jz∝an}bracketri}ht ×ΩLL′;m(s). (48) B. The spin-orbit coupled sound modes Propagating collective modes exist if Landau parame- ters are positive. In these collective modes, interactions among quasiparticles rather than the hydrodynamic col- lisions provide the restoring force. Because only the spin channelreceivesrenormalizationfromthe magnetic dipo- lar interaction, we only consider spin channel collectivemodes. The largest Landau parameter is in the (1+001) channel in which the spin oscillates along the direction of/vector q. The mode in this channel is the longitudinal spin zero sound. On the other hand, due to the spin-orbit coupled nature, the Landau parameters are negative in the transverse spin channels of (1+±1 0±1), and thus no propagating collective modes exist in these channels. The hybridization between (1+001) and (1+021) is small as shown in Eq. (25), and the Landau parameter in the (1+021) channel is small, thus, this channel also is ne- glected below for simplicity. Because the propagation wave vector /vector qbreaks the par- ityand3Drotationsymmetries, the(1+001)channelcou- ples to other channels with the same Jz. As shown in Eq. (47), the coupling strengths depend on the magni- tudes of Landau parameters. We truncate Eq. (47) by keeping the orbital partial-wave channels of L= 0 and L= 1 because Landau parameters with orbital-partial wavesL≥2arenegligible. Therearethreechannelswith L=S= 1 as(0−011), (1−011), and (2−011). We further check the symmetry properties of these four modes un- der the reﬂection with respect to any plane containing /vector q. The mode of (1−011)is even and the other three are odd, thus it does not mix with them. The Landau parameter in the (2−011) channel is calculated asπ 20λ, which is 1 order smaller than those in (1+001) and (1−001), thus this channel is also neglected. We only keep these two coupled channels (1+001) and (1−001) in the study of collective spin excitations. The solution of the two coupled modes reduces to a 2×2 matrix linear equation as /parenleftbigg 1+Ω00;0(s)F1001;1001sΩ00;0(s)F0011;0011 sΩ00;0(s)F1001;1001 1+Ω00;0(s)F0011;0011/parenrightbigg/parenleftbigg δn1001 δn0011/parenrightbigg = 0, (49) where the following relations are used K1001;1001(s) = Ω 00;0(s) K1001;0011(s) =K0011;1001(s) =∝an}bracketle{t0010\|10∝an}bracketri}ht∝an}bracketle{t1010\|00∝an}bracketri}htΩ01;0(s) =sΩ00;0(s) K0011,0011(s) =/summationdisplay m\|∝an}bracketle{t1m1−m\|00∝an}bracketri}ht\|2Ω11;m(s) =1 3Ω11;0(s)+2 3Ω11;1(s) = Ω00;0(s). (50) The condition of the existence of nonzerosolutions of Eq. (49) becomes (1−s2)Ω2 00;0(s)+2Ω 00;0(s)F+ F2 ×+1 F2 ×= 0,(51) whereF+= (F1001:1001 +F0011;0011)/2 andF×=/radicalbig F1001:1001F0011;0011.Let us discuss several important analytical properties of its solutions. In order for collective modes to prop- agate in Fermi liquids, its sound velocity must satisfy s >1, otherwise it enters the particle-hole continuum and is damped, a mechanism called Landau damping.9 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48 /s52/s46/s53 /s53/s46/s48/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53/s52/s46/s48 /s32/s115 λ FIG. 1: (Color online) The sound velocity sin the unit of vfv.s. the dipolar coupling strength λ. At 0 < λ≪1, s(λ)≈1 + 0+. On the order of λ≫1,s(λ) becomes linear with the slope indicated in Eq. (56). We can solve Eq. (51) as Ω± 00;0(s) =F+±/radicalBig F2 ++(s2−1)F2 × (s2−1)F2 ×.(52) Only the expression of the Ω− 00;0(s) is consistent with s> 1 and is kept. The other branch has no solution of the propagating collective modes. Let us analytically check two limits with large and small values of λ, respectively. In the case of 0 <λ≪1 such thats→1+0+, Eq. (51) reduces to Ω00;0(sλ≪1)≈1−1 2ln2+1 2ln(s−1) =−1 2F+.(53) Its sound velocity solution is sλ≪1≈1+2e−2/parenleftbig 1+1 2F+/parenrightbig = 1+2e−2−12 7πλ.(54) The eigenvector can be easily obtained as1√ 2(1,1)T, which is an equal mixing between these two modes. On the other hand, in the case of λ≫1, we also expect s≫1, and thus Eq. (51) reduces to Ω00;0(sλ≫1)≈ −1 sF×=−1 3s2, (55) whose solution becomes sλ≫1≈F× 3=π 3√ 3λ. (56) In our case, F1001is larger than F0011but is on the same order. The eigenvector can be solved as 1√2F+(√F0011,√F1001)Tin which the weight of the (0011) channel is larger. The dispersion of the sound velocity swith respect to the dipolar interaction strength λis solved numerically as presented in Fig. 1. Collective sound excitations exist for all the interaction strengths with s >1. In both limits of 0 ≪λ≪1 andλ≫1, the numerical solutions 0 1 -1-1010 1-1 x F/k kk /ky F k /kz F FIG. 2: (Color online) The spin conﬁguration [Eq. (57)] of the zero-sound mode over the Fermi surface shows hedgehog- type topology at λ= 10. The common sign of u1andu2is chosen to be positive, which gives rise to the Pontryagin ind ex +1. Although the hedgehog conﬁguration is distorted in the zcomponent, its topology does not change for any values of λdescribing the interaction strength. agree with the above asymptotic analysis of Eqs. (54) and (56). In fact, the linear behavior of s(λ) already appears atλ∼1, and the slope is around 0 .6. For all the interaction strengths, the (1+001) and (0−011) modes are strongly hybridized. This mode is an oscillation of spin-orbit coupled Fermi surface distortions. The conﬁguration of the (0−011) mode exhibits an oscillating spin-orbit coupling of the /vectork·/vector σtype. This is the counterpart of the isotropic un- conventional magnetism, which spontaneously generates the/vectork·/vector σ-type coupling52,53. The diﬀerence is that, here, it is a collective excitation rather than an instability. It stronglyhybridizeswith the longitudinal spin mode. The spin conﬁguration over the Fermi surface can be repre- sented as /vector s(/vector r,/vectork,t) = u2sinθ/vectorkcosφ/vectork u2sinθ/vectorksinφ/vectork u2cosφ/vectork+u1 ei(/vector q·/vector r−sqvft),(57) where(u1,u2)Tisthe eigenvectorforthe collectivemode. We have checked that for all the values of λ,\|u2\|>\|u1\| is satisﬁed with no change in their relative sign, thus the spin conﬁguration as shown in Fig. 2 is topologically non-trivial with the Pontryagin index ±1 which periodi- cally ﬂips the sign with time and the spatial coordinate along the propagating direction. It can be considered as a topological zero sound. VI. CONCLUSIONS To summairze, we have presented a systematic study on the Fermi liquid theory with the magnetic dipolar interaction, emphasizing its intrinsic spin-orbit coupled nature. Although this spin-orbit coupling does not ex- hibit at the single-particle level, it manifests in various10 interaction properties. The Landau interaction function is calculated at the Hartree-Fock level and is diagonal- ized by the total angular momentum and parity quan- tum numbers. The Pomeranchuk instabilities occur at the strong magnetic dipolar interaction strength gener- ating eﬀective spin-orbit coupling in the single-particle spectrum. We have also investigated novel collective excitations in the magnetic dipolar Fermi liquid theory. The Boltz- mann transportequations aredecoupled in the spin-orbit coupled channels. We have found an exotic collective ex-citation, which exhibits spin-orbit coupled Fermi surface oscillations with a topologically nontrivial spin conﬁgu- ration, which can be considered as a topological zero- sound-like mode. Acknowledgments Y. L. and C. W. were supported by the AFOSR YIP program and NSF-DMR-1105945. 1S. Ospelkaus, K. K. Ni, M. H. G. de Miranda, B. Neyen- huis, D. Wang, S. Kotochigova, P. S. Julienne, D. S. Jin, and J. Ye, Faraday Discuss. 142, 351 (2009).. 2K. K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er, B. Neyenhuis, J. J. Zirbel, S. Kotochigova, P. S. Julienne, D. S. Jin, and J. Ye, Science 322, 231 (2008). 3A. Chotia, B. Neyenhuis, S. A. Moses, B. Yan, J. P. Covey, M. Foss-Feig, A. M. Rey, D. S. Jin, and J. Ye, Phys. Rev. Lett.108, 080405 (2012). 4T. Sogo, L. He, T. Miyakawa, S. Yi, H. Lu, and H. Pu, New J. Phys. 11, 055017 (2009). 5T. Miyakawa, T. Sogo, and H. Pu, Phys. Rev. A 77, 061603 (2008). 6B. M. Fregoso, K. Sun, E. Fradkin, and B. L. Lev, New J. Phys.11, 103003 (2009). 7C. K. Chan, C. Wu, W. C. Lee, S. Das Sarma, Phys. Rev. A81, 023602 (2010), 8S. Ronen and J. L. Bohn, Phys. Rev. A 81, 033601 (2010). 9C. Lin, E. Zhao, and W. V. Liu, Phys. Rev. B 81, 045115 (2010). 10Y. Yamaguchi, T. Sogo, T. Ito, T. Miyakawa, Phys. Rev. A82, 013643 (2010). 11Q. Li, E. H. Hwang, S. Das Sarma, Phys. Rev. B 82, 235126 (2010). 12C. M. Chang, W. C. Shen, C. Y. Lai, P. Chen, D. W. Wang, Phys. Rev. A 79, 053630 (2009). 13J. N. Zhang, S. Yi, Phys. Rev. A 81, 033617 (2010); J. N. Zhang, S. Yi, Phys. Rev. A 80, 053614 (2009). 14K. Sun, C. Wu, S. Das Sarma, Phys. Rev. B 82, 075105 (2010). 15S. H. Youn, M. Lu, U. Ray, and B. L. Lev, Phys. Rev. A 82, 043425 (2010). 16M. Lu, S. H. Youn, and B. L. Lev, Phys. Rev. Lett. 104, 063001 (2010). 17M. Lu, N. Q. Burdick, S. H. Youn, B. L. Lev, Phys. Rev. Lett.107, 190401 (2011). 18T. Koch, T. Lahaye, J. Metz, B. Frohlich, A. Griesmaier, and T. Pfau, Nature Physics 4, 218 (2008). 19T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and T. Pfau, Rep. Prog. Phys. 72, 126401 (2009). 20T. Lahaye, J. Metz, T. Koch, B. Frohlich, A. Griesmaier, and T. Pfau, Pushing the Frontiers of Atomic Physics, 1, 160 (2009). 21C. Menotti, M. Lewenstein, T. Lahaye, and T. Pfau, in Dipolar Interaction in Ultra-Cold Atomic Gases , edited by A. Campa, A. Giansanti, G. Morigi and F. S. Labini, AIP Conf. Proc. No. 970 (AIP, New York, 2008), p. 332.22J. Quintanilla, S. T. Carr, and J. J. Betouras, Phys. Rev. A79, 031601 (R) (2009). 23B. M. Fregoso, E. Fradkin, Phys. Rev. Lett. 103, 205301 (2009) 24B. M. Fregoso, E. Fradkin, Phys. Rev. B 81, 214443 (2010). 25L. You and M. Marinescu, Phys. Rev. A 60, 2324 (1999). 26M. A. Baranov, M. S. Mar’enko, V. S. Rychkov, and G. V. Shlyapnikov, Phys. Rev. A 66, 013606 (2002). 27M. A. Baranov, /suppress L. Dobrek, and M. Lewenstein, Phys. Rev. Lett.92, 250403 (2004). 28M. A. Baranov, Physics Reports 464, 71 (2008). 29G. M. Bruun, E. Taylor, Phys. Rev. Lett. 101, 245301 (2008). 30J. Levinsen, N. R. Cooper, and G. V. Shlyapnikov, Phys. Rev. A84, 013603 (2011). 31A. C. Potter, E. Berg, D. W. Wang, B. I. Halperin, and E. Demler, Phys. Rev. Lett. 105, 220406 (2010). 32R. M. Lutchyn, E. Rossi, and S. Das Sarma, Phys. Rev. A 82, 061604 (2010). 33C. Wu, and J. E. Hirsch, Phys. Rev. B 81, 020508 (R) (2010). 34K. V. Samokhin, and M. S. Mar’enko, Phys. Rev. Lett. 97, 197003 (2006). 35T. Shi, J. N. Zhang. C. P. Sun and S. Yi, Phys. Rev. A 82, 033623, (2010). 36B. Kain, Hong Y. Ling, Phys. Rev. A 83, 061603(R), (2011). 37Y. Li, C. Wu, Scientiﬁc Reports 2, 392 (2012). 38A. J. Leggett, Rev. Mod. Phys. 47, 331 (1975). 39G. E. Volovik, “The Universe in a Helium Droplet” , (Ox- ford University Press, Oxford 2009). 40R. Balian, N. R. Werthamer, Phys. Rev. 131, 1553 (1963). 41P. W. Anderson, P. Morel, Phys. Rev. 123, 1911 (1961). 42W. F. Brinkman, J. W. Serene, and P. W. Anderson, Phys. Rev. A10, 2386 (1974). 43W. Negele and H. Orland, Quantum Many-Particle Sys- tems, (Perseus Books, New York, 1988). 44T. Sogo, M. Urban, P. Schuck, T. Miyakawa, Phys. Rev. A 85, 031601(R) (2012). 45I. J. Pomeranchuk, Sov. Phys. JETP 8, 361 (1959). 46V. Oganesyan, S. A. Kivelson, and E. Fradkin, Phys. Rev. B64, 195109 (2001). 47M. Garst and A. V. Chubukov, Phys. Rev. B 81, 235105 (2010). 48M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, 075127 (2010) 49P. W¨ olﬂe, and A. Rosch, Journal of Low Temperature11 Physics, 147, 165 (2007). 50C. A. Lamas, D. C. Cabra, and N. Grandi, Phys. Rev. B 80, 075108 (2009). 51C. A. Lamas, D. C. Cabra, and N. E. Grandi, Int. J. of Mod. Phys. B 25, 3539 (2011).52C. Wu, and S. C. Zhang, Phys. Rev. Lett. 93, 036403(2004). 53C. Wu, K. Sun, E. Fradkin, and S. C. Zhang, Phys. Rev. B75, 115103 (2007).
1203.6684v1.Soliton_Magnetization_Dynamics_in_Spin_Orbit_Coupled_Bose_Einstein_Condensates.pdf	Soliton Magnetization Dynamics in Spin-Orbit Coupled Bose-Einstein Condensates O. Fialko,1J. Brand,1and U. Z ulicke2 1Centre for Theoretical Chemistry and Physics and New Zealand Institute for Advanced Study, Massey University, Private Bag 102904 NSMC, Auckland 0745, New Zealand 2School of Chemical and Physical Sciences and MacDiarmid Institute for Advanced Materials and Nanotechnology, Victoria University of Wellington, PO Box 600, Wellington 6140, New Zealand Ring-trapped Bose-Einstein condensates subject to spin-orbit coupling support localized dark soliton excitations that show periodic density dynamics in real space. In addition to the density feature, solitons also carry a localized pseudo-spin magnetization that exhibits a rich and tunable dynamics. Analytic results for Rashba-type spin-orbit coupling and spin-invariant interactions pre- dict a conserved magnitude and precessional motion for the soliton magnetization that allows for the simulation of spin-related geometric phases recently seen in electronic transport measurements. PACS numbers: 03.75.Lm, 67.85.Fg, 03.65.Vf, 71.70.Ej The recent realization of articial light-induced gauge potentials for neutral atoms [1] has added a power- ful new instrument to the atomic-physics simulation toolkit [2]. In particular, possibilities to induce Zeeman- like and spin-orbit-type couplings in (pseudo-)spinor atom gases [3] render them ideal laboratories to investi- gate the intriguing interplay of spin dynamics and quan- tum connement that has been the hallmark of semicon- ductor spintronics [4, 5]. At the same time, the unique aspects of Bose-Einstein-condensed atom gases [6] asso- ciated, e.g., with their intrinsically nonlinear dynamics, promise to give rise to novel behavior under the in uence of synthetic spin-orbit couplings [7{15]. One of the special properties resulting from nonlinear- ity in Bose-Einstein condensates (BECs) is the existence of solitary-wave excitations [16]. Basic types of these are distinguished by the shape of their localized density feature: dark (gray) solitons are associated with a full (partial) depletion of a uniform condensate density in a nite region of space, whereas bright solitons are local- ized density waves on an empty background. A further characteristic associated with solitons is the phase gra- dient of the condensate order parameter centered at the position of the density feature. In multi-component sys- tems, the dynamics of soliton excitations is found to be enriched by the additional degrees of freedom [17{20]. We have studied solitons in ring-trapped pseudo-spin- 1=2 condensates with spin-invariant repulsive atom-atom interactions subject to a Rashba-type [21, 22] spin-orbit coupling and nd that they exhibit a third feature: a pseudo-magnetization vector with conserved magnitude and rich dynamics that unfolds in tandem with the soli- ton's periodic propagation in real space. Figure 1 shows an example and also illustrates the interesting fact that the magnetization directions at the beginning and the end of a full cycle of the soliton's motion are generally not parallel. The appearance of such a geometric phase [23] and the precessional time evolution of the solitonic mag- netization is reminiscent of the spin dynamics of electrons traversing a mesoscopic semiconductor ring [24{28].In the following, we consider several soliton congura- tions and obtain analytical results for their density and magnetization proles as well as the magnetization dy- namics associated with their motion. We start by intro- ducing the basic theoretical description of our system of interest. Using the basis of a spatially varying local spin frame [26] for the condensate spinor, the nonlinear Gross- Pitaevskii equation [6] for the spin-orbit-coupled ring BEC turns out to be of Manakov-type [17], making it pos- sible to apply standard methods [18, 20] to nd solitary- wave solutions. Accounting for the presence of spin-orbit coupling adds an important twist: Spinors have to sat- isfy non-standard boundary conditions, which introduce background-density ows in the local spin frame that contribute to the nontrivial magnetization dynamics ex- hibited by the moving solitons in the lab frame. We consider a two-component BEC trapped in the xy plane and conned to a ring of radius R. The atoms are assumed to be in the lowest quasi-onedimensional FIG. 1. Time evolution of a gray-bright soliton's magnetiza- tion in a ring-trapped BEC with Rashba spin-orbit coupling. During a full cycle of the soliton's motion on the ring, the magnetization vector follows a trajectory on the surface of a sphere. The magnetization vectors at the beginning and the end of a cycle (indicated by arrows) dier by an angle #that is related to a spin-related geometric phase. Soliton parameters (see text):vs=c= 0:5, tan= 2,g= 100,=0:01.arXiv:1203.6684v1 [cond-mat.quant-gas] 29 Mar 20122 subband [29] and subject to a spin-orbit coupling of the familiar Rashba form [21, 22] R[x(i@y)y(i@x)] as well as a spin-rotationally invariant contact inter- action. (Here x;y;z are the spin-1/2 Pauli matrices.) The energy functional of such a system [30] is given by E[ ] =R d' y(H) , where 'is the azimuthal angle,the chemical potential, = ( "; #)Tthe two- component (pseudo-spin-1 =2) spinor order parameter in the representation where the ( z) direction perpendicular to the ring's plane is the spin-quantization axis, and H=E0 @2 '+g 2 y + tan +ei'+ei' i@'+z 2 :(1) We use(xiy)=2 to denote raising and lower- ing operators for spin-1/2 components, E0=~2=(2MR2) is the energy scale for quantum connement of atoms with massMin a ring of radius R,E0gis the two-body contact-interaction strength, and tan = 2MR R=~2is a dimensionless measure of the spin-orbit coupling. The eect of Rashba spin-orbit coupling in a ring ge- ometry can be elucidated by performing a suitable SU(2) transformation. Dening = UandHloc=U1HU, withU= ei'z=2eiy=2ei'z=(2 cos), we nd Hloc=E0 @2 '(tan)2 4+g 2y : (2) The transformation U1amounts to a '-dependent rota- tion of the pseudo-spin quantization axis [26], followed by a spin-dependent gauge transformation. We will refer to the original representation where the spin-quantization axis coincides with the axis of the ring as the lab frame , whereas the representation in which the Hamiltonian of the system is diagonal in pseudo-spin space [i.e., given byHlocof Eq. (2)] will be the local spin frame [26]. Note that the spinors in the lab frame are periodic func- tions of', whereas the spinors = (+;)Tfrom the local spin frame have to satisfy the boundary conditions (') =('+ 2)eiAwith a spin dependent phase twist originating from the spin-orbit coupling, where A=1 cos1 : (3) Knowledge of the local-spin-frame spinors enables the calculation of expectation values for any observables ac- cessible to measurement in the lab frame. The total densityn=j "j2+j #j2j+j2+jj2is obviously the same irrespective of which representation is chosen in spin space. The pseudo-spin-1/2 projections in the lab frame correspond to denite atomic states, hence their density proles n"(#)= y([1 + ()z]=2) are of in- terest. In addition, we will consider the magnetization- density vector s= y in the lab frame, with = (x;y;z) being the vector of Pauli matrices.We analyze the properties of localized excitations in spin-orbit-coupled ring-trapped BEC based on the time- dependent Gross-Pitaevskii equation [6] E[]= = i~@=@t. After rescaling to use the dimensionless time variable=tE0=~, it has the form i@ @=h @2 '+g j+j2+jj2n0i (4) for the two components of the spinor = (+;)T, wheren0= [+ (tan)2=4]=(gE0) is the uniform (back- ground) density consistent with the chemical potential . While the spin-orbit coupling has formally disap- peared from the nonlinear equation (4), it is still im- plicitly present via the boundary conditions that the in- dividual components (';) must satisfy. We have obtained several soliton solutions of Eqs. (4) using established techniques [17, 18, 20] and implemented the appropriate boundary conditions. Before giving fur- ther details, we like to summarize a few general features. The soliton spinors in the local-spin-frame representation turn out to be of the form (s) = (s) ('vs) eivb'=2iv2 b=4; (5) where (s) () are complex amplitude functions encoding the specic soliton-like density features, vsis the propa- gation speed of the soliton, and vbare background ow velocities of the individual spinor components that are necessary to implement the boundary conditions arising due to the presence of spin-orbit coupling. The density n(s) "(#)and magnetization density s(s)exhibit spatially lo- calized features. Subtracting s(s)from the magnetiza- tion density s(s) bof the condensate background yields the magnetization density that is associated with the soli- ton excitation only. Its integral S(s)=R d'[s(s) bs(s)] is the vector of total soliton magnetization, which is an additional property of localized excitations in multi- component BECs. For soliton solutions of the form (5), S(s)has constant magnitude. Its temporal evolution is most conveniently described by a set of four angles as dened in Fig. 2(c). While the tilt angles and0are time-independent, the angles and0vary linearly in time, signifying the precession of S(s)around tilted z0 axis with the universal result =; =vs+: (6) Thez0axis is tilted by the angle characterizing the spin-orbit coupling and it rotates around the zaxis with the same angular velocity vsthat characterizes the soli- ton propagation. The second tilt angle 0is found to de- pend only on the soliton prole ('), while the preces- sion frequency d0=dhas complicated dependences on the parameters of the soliton solutions. Figure 2 shows exemplary magnetization dynamics for gray-bright and3 FIG. 2. Magnetization dynamics of a gray-bright soliton [panel (a)] and a gray-gray soliton with zero background magnetization in the local spin frame [panel (b)]. Parameters used are g= 100 (100), tan = 0:2 (0:5),vs=c= 0:5 (0.2), and =0:5 for the gray-bright (gray-gray) case. (c) Angles used to describe the two-step precessional motion of S. The angle 0is measured with respect to an x0axis that is perpendicular to both the zandz0axes. gray-gray solitons. Interestingly, we nd that the mag- netization vector is usually not parallel to its initial di- rection after the soliton has completed a full cycle of its motion around the ring as, e.g. seen in gure 1. The an- gle#between the magnetization directions at the start and the end of a cycle turns out to be nite only as a consequence of spin-orbit coupling, as it depends promi- nently on the phase Agiven in Eq. (3) that also governs spin-dependent interference in mesoscopic ring conduc- tors [27]. In order to nd explicit soliton solutions, we introduce ='u, whereuis a velocity parameter, and initially look for solutions of the form (';) =p n() ei(). This allows us to rewrite Eq. (4) in the form u@n @+ 2@ @ n@ @ = 0;(7a) u@ @+1pn@2pn @2@ @2 g(nn0) = 0;(7b) wheren=n++n. Single component solutions for = ~are easily found by integration of Eqs. (7) to yield ~()/() and~= 0, with the well-known dark soliton solution on the innite line [6] () =pn0 iu c+ utanh u D : (8) Here 2 u= 1u2=c2,c2= 2gn0, 1=2 D=gn0=2. The soli- ton prole (8) is appropriate for suciently strong non- linearity, where D= u2[31]. However, ( ) does not satisfy the proper boundary condition since it has a phase step =2 arccos(u=c). To compensate for the phase step and ensure the correct phase shift associated with the gauge transformation U, we perform a Galilean transformation on (8), which yields SC ~(';) = (vb) eivb'=2iv2 b=4: (9) Herevb=(+ ~A)=is the background velocity imposed by the boundary condition. Thus the single- component soliton solution is of the form (5), with SC ~= and SC ~= 0, and propagation speed vs=u+vb.A straightforward calculation yields sSC= ~j(' vs)j2(sincos';sinsin';cos)Tfor the magne- tization density of the single-component soliton solution. In essence, the density depletion at the soliton's posi- tion gives rise to a reduction of the magnetization den- sitysSC b= ~n0(sincos';sinsin';cos)Tasso- ciated with the background. Thus sSC bsSCconsti- tutes the magnetization density associated with the soli- ton itself, as it is the change in the background mag- netization density due to the presence of the localized excitation. For the single-component soliton, this corre- sponds to a peak in magnetization density at the soli- ton's position. The total magnetization vector is ob- tained by integrating that peak in real space, which yields SSC= ~(sincos(vs);sinsin(vs);cos)T. This magnetization vector is precessing in a perfectly synchro- nized fashion with the soliton's motion around the ring [cf. Fig. 2(c) with Eq. (6) and 0= 0], i.e.,#SC= 0. We now consider a solution of Eqs. (7) that is a gray- bright (GB) soliton in the local spin frame. We assume that the densities approach constant values n+!n0+ (gray part) and n!0 (bright part) far away from the soliton's position. To decouple Eq. (7b), we use the ansatz [20] n=(n+n0+) with10. We apply a Galilean boost to both components to match the phase of the gray part only, hence they are of the form (5) with GB +() given by ( ) from Eq. (8) but with rescaledc2= 2gn0+(1 +), 1=2 D=gn0+(1 +)=2, and GB () =p n0+ ueiu=2 cosh( u=D): (10) Furthermore, vb=vb+andvs=u+vb+. Figure 3 shows the density proles [panel (a)] and magnetization- density prole [panel (c)] associated with a GB soliton. The vector SGBof total magnetization for a GB soli- ton precesses concomitantly with the soliton's motion; cf. Fig. 2(c) with Eq. (6) and tan 0= (pu)=[(1)c u], 0=vs(1 +A=). Figures 1(a) and 2(a) show exam- ples of possible time evolutions of the GB-soliton magne- tization. The magnitude of the magnetization vector is4 /LParen1a/RParen1 0.00.20.40.60.81.00.00.20.40.60.81.0 /CurlyPhi/Slash12Π /LParen1b/RParen1 0.00.20.40.60.81.00.00.20.40.60.81.0 /CurlyPhi/Slash12Π /LParen1c/RParen1 0.00.20.40.60.81.0/Minus0.50.00.51.0 /CurlyPhi/Slash12Π /LParen1d/RParen1 0.00.20.40.60.81.0/Minus0.50.00.51.0 /CurlyPhi/Slash12Π FIG. 3. Lab-frame spin densities [(a),(b)] and zcompo- nent of the magnetization density [(c),(d)] for a stationary gray-bright soliton [(a),(c)] and a stationary gray-gray soliton [(b),(d)]. Panels (a) and (b) show the total density (solid yel- low curve) and individual-spin (dashed blue = ", dotted red =#) densities normalized to the background-density value. In (c) and (d), the total density is plotted again for reference as the solid yellow curve, together with the magnetization pro- less(s) z(dashed blue curve), and the magnetization density s(s) bzs(s) zassociated with the soliton excitations only (dotted red curve). Parameters are tan = 1:0,g= 100, and (for the gray-bright soliton) =0:5. found to be S= 2(1)D u=cos0. For a GB soliton, the magnetization vector turns out to be notaligned with its initial direction after completion of a full cycle of its motion around the ring. A straightforward calculation yields sin(#GB=2) = sin0sinA. As0is a known func- tion of the soliton parameters, a measurement of #GBwill yield the spin-related geometric phase A. The solutions representing gray-gray (GG) solitons in the local spin frame are obtained by Hirota's method [18]. The spinor components are of the form (5) with GG () =pn0 iu c+ utanh(a) : (11) Here,a2= 2 u+=2 D++ 2 u=2 D,vs= 2au=c u+vb, c2 = 2gn0, 1=2 D=gn0=2. The back-ground ows are given by vb=(+A)=, where = 2 arccos(u=c). The independent parameters charac- terizing a GG soliton are the ratio n0+=n0(or, equiv- alently, the background magnetization in the local spin frame) and the speed vsof the soliton. All other param- eters can be found by solving transcendental equations given just after Eq. (11). For simplicity, we consider the case of a GG soliton with zero background magnetiza- tion in the local spin frame (i.e., n0+=n0n0=2). Figure 3 shows results for spinor-density [panel (b)] and magnetization-density [panel (d)] proles. The time evolution of the magnetization vector asso- ciated with a moving GG soliton is characterized by theangles dened in Fig. 2(c) with Eq. (6), 0==2, and 0=!+=2, where!=(vb+vb)vs=2 + (v2 b+ v2 b)=4(1+A=)vs. Figure 2(b) illustrates this dynam- ics which, for small , corresponds to a slow rotation of the magnetization vector in the ring's plane with super- imposed fast small-amplitude oscillations in the normal direction. As in the case of the GB soliton, the magneti- zation vector does not evolve back to its initial direction after a period of the soliton's ring revolution. The angle between magnetizations at the start and the end of the cycle is found to be #GG= 2j!j=vs!2AforA 1. Again, the dependence of #GGonAenables determina- tion of the latter by measuring the former. In conclusion, we have investigated the properties of soliton excitations in ring-trapped spin-orbit-coupled BECs. We nd that a magnetization degree of freedom is generally associated with a soliton, and that the magne- tization vector precesses around an axis that is rotating synchronously with the soliton's orbital motion around the ring. The magnetization direction at the end of a cycle of revolution does not coincide with the initial di- rection for the gray-bright and gray-gray cases, making it possible to measure a spin-orbit-related geometric phase. Our work opens up new avenues for the realization and manipulation of magnetic soliton excitations in BECs. It also creates the opportunity to study spin-dependent interference and scattering eects that, until now, were only accessible in semiconductor nanostructures. This work was supported by the Marsden fund (con- tract no. MAU0910) administered by the Royal Society of New Zealand. [1] Y.-J. Lin, R. L. Compton, K. Jimenez-Garcia, J. V. Porto, and I. B. Spielman, Nature 462, 628 (2009). [2] J. Dalibard, F. Gerbier, Juzeli unas, and P. Ohberg, Rev. Mod. Phys. 83, 1523 (2011). [3] Y.-J. Lin, K. Jimenez-Garcia, and I. B. Spielman, Nature (London) 471, 83 (2011). [4] D. D. Awschalom, D. Loss, and N. Samarth, eds., Semiconductor Spintronics and Quantum Computation (Springer, Berlin, 2002). [5] I. Zuti c, J. Fabian, and S. D. Sarma, Rev. Mod. Phys. 76, 323 (2004). [6] L. Pitaevskii and S. Stringari, Bose-Einstein Condensa- tion (Clarendon Press, Oxford, 2003). [7] T. D. Stanescu, B. Anderson, and V. Galitski, Phys. Rev. A 78, 023616 (2008). [8] M. Merkl, A. Jacob, F. E. Zimmer, P. Ohberg, and L. Santos, Phys. Rev. Lett. 104, 073603 (2010). [9] C. Wang, C. Gao, C.-M. Jian, and H. Zhai, Phys. Rev. Lett.105, 160403 (2010). [10] T.-L. Ho and S. Zhang, Phys. Rev. Lett. 107, 150403 (2011). [11] C.-J. Wu, I. Mondragon-Shem, and X.-F. Zhou, Chin. Phys. Lett. 28, 097102 (2011). [12] S.-K. Yip, Phys. Rev. A 83, 043616 (2011).5 [13] S. Sinha, R. Nath, and L. Santos, Phys. Rev. Lett. 107, 270401 (2011). [14] H. Hu, B. Ramachandhran, H. Pu, and X.-J. Liu, Phys. Rev. Lett. 108, 010402 (2012). [15] Y. Zhang, L. Mao, and C. Zhang, Phys. Rev. Lett. 108, 035302 (2012). [16] R. Carretero-Gonz aez, D. J. Frantzeskakis, and P. G. Kevrekidis, Nonlinearity 21, R139 (2008). [17] S. V. Manakov, Zh. Eksp. Teor. Fiz. 65, 505 (1973), [Sov. Phys. JETP 38, 248 (1974)]. [18] A. P. Sheppard and Y. S. Kivshar, Phys. Rev. E 55, 4773 (1997). [19] P. Ohberg and L. Santos, Phys. Rev. Lett. 86, 2918 (2001). [20] J. Smyrnakis, M. Magiropoulos, G. M. Kavoulakis, and A. D. Jackson, Phys. Rev. A 81, 063601 (2010). [21] E. I. Rashba, Fiz. Tverd. Tela (Leningrad) 2, 1224 (1960), [Sov. Phys. Solid State 2, 1109 (1960)]. [22] Y. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6039(1984). [23] A. Shapere and F. Wilczek, eds., Geometric Phases in Physics (World Scientic, Singapore, 1989). [24] D. Loss, P. Goldbart, and A. V. Balatsky, Phys. Rev. Lett.65, 1655 (1990). [25] A. G. Aronov and Y. B. Lyanda-Geller, Phys. Rev. Lett. 70, 343 (1993). [26] J. Splettstoesser, M. Governale, and U. Z ulicke, Phys. Rev. B 68, 165341 (2003). [27] D. Frustaglia and K. Richter, Phys. Rev. B 69, 235310 (2004). [28] F. Nagasawa, J. Takagi, Y. Kunihashi, M. Kohda, and J. Nitta, Phys. Rev. Lett. 108, 086801 (2012). [29] This limitation is not crucial, as nite-width eects and higher subbands could be treated straightforwardly. [30] M. Merkl, G. Juzeli unas, and P. Ohberg, Eur. Phys. J. D59, 257 (2010). [31] Otherwise periodic solutions involving elliptic functions have to be used. See, e.g., L. D. Carr, C. W. Clark, and W. P. Reinhardt, Phys. Rev. A 62, 063610 (2000).
2011.10483v1.Anisotropy_of_the_spin_orbit_coupling_driven_by_a_magnetic_field_in_InAs_nanowires.pdf	Anisotropy of the spin-orbit coupling driven by a magnetic eld in InAs nanowires Pawe l W ojcik,1,Andrea Bertoni,2,yand Guido Goldoni3, 2,z 1AGH University of Science and Technology, Faculty of Physics and Applied Computer Science, Al. Mickiewicza 30, 30-059 Krakow, Poland 2CNR-NANO S3, Istituto Nanoscienze, Via Campi 213/a, 41125 Modena, Italy 3Department of Physics, Informatics and Mathematics, University od Modena and Reggio Emilia, Italy (Dated: November 23, 2020) We use the kptheory and the envelope function approach to evaluate the Rashba spin-orbit coupling induced in a semiconductor nanowire by a magnetic eld at dierent orientations, taking explicitely into account the prismatic symmetry of typical nano-crystals. We make the case for the strongly spin-orbit-coupled InAs semiconductor nanowires and investigate the anisotropy of the spin- orbit constant with respect to the eld direction. At suciently high magnetic elds perpendicular to the nanowire, a 6-fold anisotropy results from the interplay between the orbital eect of eld and the prismatic symmetry of the nanowire. A back-gate potential, breaking the native symmetry of the nano-crystal, couples to the magnetic eld inducing a 2-fold anisotropy, with the spin-orbit coupling being maximized or minimized depending on the relative orientation of the two elds. We also investigate in-wire eld congurations, which shows a trivial 2-fold symmetry when the eld is rotated o the axis. However, isotropic spin-orbit coupling is restored if a suciently high gate potential is applied. Our calculations are shown to agree with recent experimental analysis of the vectorial character of the spin-orbit coupling for the same nanomaterial, providing a microscopic interpretation of the latter. I. INTRODUCTION The spin-orbit (SO) interaction, which couples the spin of electrons with their momentum, is the functioning principle of many spintronic applications, including spin transistor,1,2spin lters3{5or spin-orbit qubits.6,7Recent investigations focus towards semiconductor nanowires (NWs) with strong SO interaction8{16as host materials for topological quantum computing based on Majorana zero energy modes.17{21These exotic quasi-particles form at the ends of a NW as a result of the interplay between the SO coupling, Zeeman spin splitting and s-wave super- conductivity induced in the NW by the proximity eect from a superconducting shell.22{24 In general, a nite SO constant originates from the lack of the inversion symmetry. In semiconductors, this could either be an intrinsic feature of the crystallographic structure (Dresselhaus SO coupling25) or induced by the connement potential (Rashba SO coupling26,27). In zincblende NWs grown along the [111] direction, the crys- tal inversion symmetry is preserved and the Dresselhaus term vanishes.10On the other hand, for spintronic ap- plications the Rashba term has the essential advantage of being tunable by external elds, e.g., using external gates attached to the NW.28In general, external elds interplay with the overall NW geometry, which is typi- cally prismatic, and the value of the SO constant depends on the position with respect to the underlying substrate, the details of the dielectric conguration, as well as on the compositional details of the NW which determine the electronic states.13For example, we have recently discussed the additional possibilities to engineer the SO constant in core-shell NWs with respect to homogeneous samples.12Since the SO constant depends, in general, on the symmetry and localization of the electronic states, amagnetic eld may also induce a nite SO constant due to orbital eects. Despite the number of experiments with measurements of the Rashba SO constant in semiconductor NWs,8{10 the study of its anisotropy with respect to the magnetic eld orientation is limited. Recently, such a vectorial con- trol was reported for InAs NWs which were suspended in order to eliminate the SO contribution originating from the substrate.29In Ref. 29 the authors tracked the non- trivial evolution of the weak anti-localization (WAL) sig- nal and determined the SO length as a function of the magnetic eld intensity and direction. Interestingly, they observed that the average SO coupling is isotropic with respect to the magnetic eld orientation and does not reveal any hallmark of the prismatic symmetry. When applying a transverse electric eld by a gate, however, a 2-fold anisotropy appears, with the maximal SO length when Bis perpendicular the electric eld. Motivated by the availability of such experiments, we use the 88kpmethod to analyze the dependence of the Rashba SO constant on the magnetic eld intensity and orientation. The full vectorial character of the SO constant is taken into account by evaluating the SO cou- pling constants separately in dierent directions. While the magnetic eld perpendicular to the NW axis is able to generate a nite SO constant which turns out to be isotropic at low intensity (below 1 T), for larger elds the SO constant shows a slight 6-fold symmetry with re- spect to the eld orientation, due to the interplay be- tween the orbital eects of the eld and the prismatic symmetry of the NW. A back-gate potential couples to the magnetic eld, which maximizes or minimizes the SO coupling depending on the relative orientation, lead- ing to a 2-fold symmetry. We also investigate in-wire eld congurations. The trivial 2-fold symmetry whenarXiv:2011.10483v1 [cond-mat.mes-hall] 20 Nov 20202 the eld is rotated in a plane which contains the axis, is almost completely removed by a gate potential. Our re- sults are discussed in light of recent experiments reported in Ref. 29. The paper is organized as follows. In Sec. II the Rashba SO coecients are derived from the 8 8kpmodel within the envelope function approximation, including the orbital eects which originate from the magnetic eld. The eective Hamiltonian for the conduction electrons is derived in Sec. II A with details on the numerical method given in Sec. II B. Results of our calculations for homo- geneous InAs NWs are reported in Sec. III, with a dis- cussion of recent experiments. Sec. IV summaries our results. II. THEORETICAL MODEL We consider a homogeneous InAs NW with hexagonal cross-section, grown along the [111] direction for which the Dresselhaus contribution to the SO interaction can be neglected.30The NW is subjected to the external mag- netic eld B=B(cossin;sinsin;cos), with in- tensityBand the direction being dened by the angle formed with the NW axis along zand the angle formed with thexaxis, which connects two corners of the NW in thexyplane, see Fig. 1(a). We employ the gauge A(r) =B(ycos;0;ycossinxsinsin). A back- gate is directly attached to the bottom of the NW, along a facet, generating an electric eld parallel to the NW section, in the xyplane.10,13 Below we use the 8 8 Kane model to derive the Rashba SO constants in terms of a realistic description of the quantum states in a magnetic eld. This allows for quan- titative predictions of SO coecients as a function of the magnetic eld and the gate voltage for dierent electron concentrations.12,13 A. Eective SO Hamiltonian for conduction electrons Our theoretical model is based on the 8 8kp Kane Hamiltonian within the envelope function approx- imation. We neglect here the spin Zeeman splitting, to focus on the dominating orbital eects, that is the dis- tortion of the envelope function due to the eld. It is straightforward to add the Zeeman splitting to the elec- tron spin levels. The 8 8 Kane Hamiltonian reads31 H88=HcHcv Hy cvHv ; (1) whereHcis the Hamiltonian of conduction electrons cor- responding to the 6cband, while Hvis the Hamiltonian of the valence bands, 8v, 7v Hc=H6122; (2) Hv=H8144H7122: (3) FIG. 1. (a) Schematics of a NW with a bottom gate. In our simulations, anisotropy is evaluated with a magnetic eld B either perpendicular to the NW axis ( ==2) and rotated with an azimuthal angle , or with==2 and rotated in theyzplane. (b) Occupation of the lowest subband as a function of the wave vector kzat each chemical potential atB= 4 T. Asincreases, the occupation saturates to one at anykzbelow the Fermi energy. Of course, in general sev- eral subbands are occupied. The non-parabolic dispersion is clearly appreciated, with the eld inducing a seemingly Lan- dau level dispersion. Two vertical dashed lines mark values ofselected for the further analysis. In the above expressions H6=P2 2m0+Ec+V(r); (4) H8=Ec+V(r)E0; (5) H7=Ec+V(r)E00; (6) where P=peA(r),m0is the free electron mass, Ec is the conduction band edge, E0is the energy gap, 0 is the split-o gap and V(r) is the potential energy. In our target systems, the potential V(r) is the sum of the Hartee potential energy generated by the electron gas and the electrical potential induced by the bottom gate attached to NW, V(r) =VH(r) +Vg(r).3 The o-diagonal matrix Hcvin (1) reads Hcv=P0 h0 @P+p 2q 2 3PzPp 60Pzp 3Pp 3 0P+p 6q 2 3PzPp 2P+p 3Pzp 31 A; (7) whereP=PxiPyandP0=ihhSj^pxjXi=m0is the conduction-to-valence band coupling with jSi,jXibeing the Bloch functions at the point of Brillouin zone. Finally, the folding-down transformation31 H(E) =Hc+Hcv(HvE)1Hy cv: (8) reduces the 88 Hamiltonian (1) into the 2 2 eective Hamiltonian for the conduction band electrons. The in-plane vector potential is introduced into the nu- merical model through the Peierls substitution.32Note that the eld does not break translational invariance along the wire axis (the zdirection). Therefore, assuming n;kz(x;y;z ) = [ " n;kz(x;y); # n;kz(x;y)]Teikzzand ex- panding the on- and o-diagonal elements of the Hamil- tonian (8) to second order, we obtain H=P2 2D 2m+1 2m!2 c (ycosxsin) sinkzl2 B2 +Ec+V(x;y) 122+ (xx+yy)Pz h; (9) where P2 2D=P2 x+P2 y= (px+Bycos)2+p2 y,!c= eB=m,lB=p h=eB is the magnetic length, iare the Pauli matrices, mis the eective mass 1 m=1 m0+2P2 0 3h22 Eg+1 Eg+ 0 ; (10) andx,yare the SO coecients given by x(x;y)P2 0 31 (E0+ 0)21 E2 0@V(x;y) @y;(11) y(x;y)P2 0 31 (E0+ 0)21 E2 0@V(x;y) @x:(12) B. SO coupling constants calculations Representing the Hamiltonian (9) in the basis of the in- plane envelope functions n;kz(x;y), calculated without SO coupling, i.e., the diagonal part of (9), the matrix elements of the SO term are given by nm i(kz) =Z Z n;kz(x;y)i(x;y) m;kz(x;y)dxdy: (13) These coecients dene intra- ( n=m) and inter- subband (n6=m) SO constants whose magnetic eld- dependence is studied in Sec. III. Note that the 's co- ecients depend both on the envelope functions and the gradient of the potential.Calculations of the n;kz(x;y)'s is performed by the standard self-consistent Sch odinger-Poisson approach which includes electron-electron interaction at the mean- eld level. First, the in-plane envelope functions n;kz(x;y) are determined from the diagonal term of (9) P2 2D 2m+1 2m!2 c (ycosxsin) sinkzl2 B2 +Ec+V(x;y) n;kz(x;y) =En;kz n;kz(x;y):(14) In the presence of a magnetic eld, the subbands are not parabolic and n;kz(x;y) is explicitly kz-dependent. An example of the non-parabolic dispersion is shown in Fig. 1(b). Therefore, Eq. (14) is solved at selected kzon a uniform grid in [ kmax z;kmax z], withkmax zfairly above the Fermi wave vector. Then, the electron density is obtained by ne(x;y) = 2X nZkmax z kmaxz1 2j n;kz(x;y)j2f(En;k;T)dkz; (15) where the factor 2 accounts for spin degeneracy, Tis the temperature, is the chemical potential and f(En;k ;T) is the Fermi-Dirac distribution given by f(En;k;T) =1 1 + exp En;kz kBT: (16) Finally, for a given ne(x;y) we solve the Poisson equation r2 2DV(x;y) =ne(x;y) 0; (17) whereis the dielectric constant. Equations (14) and (17) are solved numerically on a triangular grid assuming Dirichlet boundary conditions. The symmetry of the discretization grid matching the symmetry of the hexagonal integration domain avoids nu- merical artifacts at the boundaries using smaller grid den- sities. The procedure of alternately solving Eqs. (14) and (17) is repeated until self-consistency is reached, which we consider to occur when the relative variation of the charge density between two consecutive iterations is lower than 0:001 at every point of the discretization domain. Then, the self-consistent potential energy prole V(x;y) and the corresponding envelope functions n;kz(x;y) are used to determine the SO constants nm ifrom Eq. (13). Further details concerning the self-consistent method for hexagonal NWs can be found in our previous papers.33,34 Calculations have been carried out for the material parameters corresponding to InAs:35E0= 0:42 eV, 0= 0:38 eV,m= 0:0265,EP= 2m0P2=h2= 21:5 eV, = 15:15,T= 4:2 K, and for the NW width W= 100 nm (facet-to-facet). In our calculations we x the chemi- cal potential. Results will be reported in the follow- ing section for = 0:3 eV and= 0:35 eV, which4 FIG. 2. (a-c) Intra- ( nn i,n= 1;2;3) and (d,e) inter-subband ( 1m i) selected Rashba SO coupling constants as a function of the magnetic eld Band the wave vector kz. (f)nn i(B;kz) atB= 4 T for the three lowest states. Results are shown for = 0:3 eV and a magnetic eld perpendicular to the NW axis and along the corner-corner direction ( ==2;= 0). are marked by vertical dashed lines in Fig. 1(b). For B= 0, these values correspond to the electron concen- trationne= 4:81016cm3andne= 1:361017cm3, respectively. Note, however, that an increasing perpen- dicular magnetic eld progressively depletes the NW.36 Therefore, in a transport experiment the chemical po- tential must be set to a suciently large value. In our calculations, the above two values of have been cho- sen suciently large as to provide an occupied ground state at the largest magnetic eld intensity used here, B= 4 T [Fig.1(b)]. For a given magnetic eld, dierent values ofcorrespond to dierent occupations, hence a dierent self-consistent potential and charge distribution within the section of the NW, which in turn aects the SO coupling. III. RESULTS We shall now discuss predictions of the SO constant as a function of the magnetic eld intensity and direc- tion. We shall put particular emphasis on the role of the eld-induced orbital eects and the interplay with the gate potential, which also in uences electronic states lo- calization and symmetry. We conclude this section by a discussion of the recent experiment.29A. Perpendicular magnetic eld with no backgate potential We rst show that a magnetic eld perpendicular to the NW axis induces a nite Rashba SO coecients even in the absence of any transverse electric eld ( Vg= 0). In this case only the Hartree term VHcontributes to the self-consistent potential. ForB= 0 the self-consistent potential, having the same hexagonal symmetry of the conning potential of the NW, is symmetric with respect to the xandydirec- tions. Hence, envelope functions have even or odd parity, leading tonn x=nn y= 0 for all electronic states, as im- plied by Eq. (9). Let us now consider a nite magnetic eld, directed along, e.g, the xaxis (==2;= 0). The eld gener- ates an eective parabolic potential along y, see Eq. (9), removing the symmetry of the Hamiltonian in this direc- tion. This, in turn, induces a nite potential gradient and akz-dependent displacement of the envelope function, hence, nite diagonal SO couplings nn x[see Eq. (11)], as shown in Fig. 2 (a-c) for selected subbands. For a constant Fermi energy, as assumed in our calculations, the number of occupied subbands changes with magnetic eld. AtB= 1 T,N= 8 subbands are occupied, while onlyN= 3 of them are populated at B= 4 T. The behavior of nn i(kz) (i=x;y) for all three subbands is both qualitatively and quantitatively similar, especially5 for the high magnetic eld, as presented in Fig. 2 (f). The maps of nm i(B;kz) in Fig. 2 (d,e) report selected SO o-diagonal couplings between the ground state and the two lowest excited states. Other coecients 1m iare four orders of magnitude lower than 11 xand are not reported here. Note that the suppression of these o- diagonal matrix elements occurs only for a magnetic eld along the corner-corner direction, = 0. For an arbitrary direction of the magnetic eld, no symmetry applies with respect to the specic xyreference frame, and all o- diagonal SO constants have comparable values at kz= 0. FIG. 3. Squared envelope functions of the three lowest mag- netic subbands, at kz= 0 and 0:2 nm1, with a transverse magnetic eld (red arrow) at intensities (a) B= 1 T and (b) B= 4 T. Right panels show the electron density neand the self-consistent potential prole Vat the corresponding eld intensities. The magnetic eld dependence of nm ican be traced to the envelope functions localization and ensuing self- consistent potential, as shown in Fig. 3. For B= 0 (not shown) the symmetry of the envelope functions naturally leads tonn i= 0.13However, the eld strongly changes the envelope function symmetry. The magnetic states of a NW have been thoroughly investigated in Ref. 36. In short, at kz= 0 these are localized by the eld in the two corners along the eld direction, where the verti- cal component of the eld is the strongest, in seemingly dispersionless Landau levels (see also Fig. 1(b)). There- fore, such states have the inversion symmetry and do not contribute to the SO coupling. At nite kzthe elec- tron states are localized at one of the facets in dispersive states, which are the analog of the traveling edge states in a Hall bar. Accordingly, the SO constant nn xis nite, it depends on kz, and changes sign at kz= 0, as shown in Fig. 2 (a-c). Note that kzstates have opposite localiza- tion alongy. Therefore, regardless of the magnetic eld intensity, the self-consistent potential, which is obtained by summing states up to the Fermi wavevector, has theinversion symmetry induced by the NW connement, as shown in the right panels of Fig. 3. For similar reasons, but with the opposite behavior due to symmetry, the inter-subband SO couplings 1n iare largest atkz= 0. Its exact value strongly depends on the eld intensity. Note that for the analyzed magnetic eld direction the symmetry around the y-axis is preserved, hencenn y= 0. FIG. 4. (a) The intra-subband SO constant 11 xas a function ofkzatB= 1 T and B= 4 T and at chemical potentials = 0:30 eV and = 0:35 eV. (b) The intra-subband SO constantnn x(kF n;z); n = 1;2;3 (left axis) calculated at kF n;z and number of occupied subbands (black line, right axis) as a function of the magnetic eld intensity, B. While a nite SO can be induced by a constant mag- netic eld due to the removal of the inversion symmetry, its magnitude also depends on the electric eld in the NW, see Eqs. (11),(12), which in turn depends on the electron concentration via the chemical potential . At suciently high electron density, the free charge moves to the corners of the NW to reduce the repulsive Coulomb energy.33The large gradient of the self-consistent poten- tial where the envelope function is large generates SO constantsnn iwhich increase with . As an example, in Fig. 4(a) we show the calculated 11 ias a function of the wavevector for = 0:30 eV and= 0:35 eV. Note that 11 iincreases rapidly with kz, but then saturates as the corresponding envelope functions are squeezed more and more to the NW edges. In a transport experiment, electrons are injected in one of the subbands of the NW with a well dened Fermi6 FIG. 5. Maps of nn xandnn yas a function of and wave vector kz. Results are shown for = 0:30 eV and magnetic elds (a)B= 1 T and (b) B= 4 T. FIG. 6. Maps of 11as a function of and wave vector kz. Results are shown for = 0:30 eV and magnetic elds (a) B= 1 T and (b) B= 4 T. Insets under the main panels zoom in the kzrange marked by dashed black rectangle of the corresponding panel. wave vector, kF n;z, which is a function of the magnetic eld intensity due to the eld induced charge depletion. In Fig. 4(b) we show nn i(Vg) at the Fermi wave vectorkF n;z. The strong localization of the electron charge at opposite NW edges gives rise to a strong susceptibility ofnn i(Vg) aroundB= 0, analogously to what happens when a gate potential is switched on, as we discussed in Ref. 13. On the other hand, nn isaturates for high magnetic elds due to the orbital eect which squeezes the envelope functions to NW edges. Slight oscillations ofnn x(B) correspond to changes in the self-consistent potential due to depopulation of subsequent subbands when increasing eld [see the black line in Fig. 4(b)]. We next analyze the anisotropy of the SO constant with respect to the transverse eld direction. Indeed, as a nitenn ioriginates from the connement induced by the eld, it is expected that the latter intertwines with the natural connement of the electron charge at the NW edges, as discussed above. Therefore, we expect a 6-fold anisotropy with respect to . The angular dependence of the intra-subband SO cou- plings is shown in Fig. 5 for the three lowest subbands and dierent magnetic eld intensities. Note these sub- bands exhaust the occupied states at B= 4 T, but they are only a subset of the N= 8 occupied subbands at B= 1 T [see also Fig. 4(b)]. Subbands with N > 3 are not shown here, however, as they do not add information. In Fig. 6 we show nn=q (nnx)2+ (nny)2calculated7 for the ground state n= 1. The SO coupling 11ap- pears isotropic and unaected by the magnetic eld ori- entation. However, a very weak dependence on can be observed in the bottom subpanels which zoom in the kzrange marked by the dashed rectangular at the main graph. A similar weak 6-fold anisotropy is shown by all the occupied states and corresponds to the hexagonal ge- ometry of NW. It is due to the slight reshaping of the en- velope functions which localize alternately on facets and corners as the magnetic eld is rotated around the NW (see Fig. 7). Interestingly, at B= 4 T the SO coupling shows a ower-like pattern around kz= 0 forn= 1;2, see Fig. 5(b). This behavior emerges in the low kzrange, where the eld drives the electron charge around the NW due to the parabolic well generated by the eld. However, a smallkz-dependent term slightly removes the symme- try, displacing the envelope function on one side and in- terplaying with the hexagonal potential. In Fig. 7(a) the = 28case is much more symmetric than the other two directions, due to the larger tunneling energy between the lobes, which makes the symmetric conguration more ro- bust. In Fig. 7(b), instead, the envelope function of the ground state for kz= 0:4 nm1is strongly localized by the eld near the edges. In this case the symmetry of the envelope function is strongly removed, regardless of the eld direction, and only a weak anisotropy is present thereof. FIG. 7. Squared envelope function of the ground state at selected angles atB= 4 T. (a) kz= 0:04 nm1(b)kz= 0:4 nm1. In Fig. 8 (a-c) we report polar diagrams of the intra- subband SO constant calculated at the Fermi wave vector kF n;zfor all occupied states ( N= 3) atB= 4 T. The x- andycomponents and the modulus nnare shown sep- arately. The value of SOC is the largest for the ground state, panel (a), which is almost isotropic. On the con- trary, other electronic bands have a smaller values but a stronger anisotropy. The total SOC, tot, averaged over all occupied subbands, panel (d), to be compared with the observed value in the magnetotransport experiment, FIG. 8. (a-c) Angular dependence of the x(blue) and y(red) components of intra-subband SO coupling constant (in units of meVnm) at kF n;ztogether with the modulus nn=p (nnx)2+ (nny)2(black) for the three occupied states. Panel (d) presents the total SO coupling constant, tot, averaged over all occupied states. Results for B= 4 T, = 0:3 eV. shows a slight 6-fold anisotropy, with the smaller value along the corner-corner direction and the larger value along the facet-facet direction. The total SOC for dierent Bandis shown in Fig. 9. At the lowest magnetic eld B= 0:1 T, panel (a), we do not observe any anisotropy. A slight 6-fold anisotropy can be appreciated at B= 1 T, in panel (b). In this case a dierent behaviour of the SOC as compared to that ob- tained atB= 4 T is due to the averanging over a larger number of subbands ( N= 8), including higher excited states whose angular dependence is a combined eect of the orbital eects and the envelope function symmetry. Although the orbital eects for these higher excited states are suppressed due to low kn;F, and therefore the contri- bution of them to the SOC is reduced, they cause a visible ripples of SOC, but still with the lowest SOC along the corner-corner line. The observed 6-fold anisotropy of SOC is actually expected. Due to external connement and the self- consistent eld arising from Coulomb interaction, the electron gas is strongly localized near the edges of NW for lowB. A weak magnetic eld cannot perturbate the symmetry of such strongly localized states. For higher magnetic eld the Coulomb interaction weakens due to the magnetically induced charge depletion (see Fig.4(b)). Therefore a suciently strong magnetic eld may squeezee the envelope functions to the surface in8 FIG. 9. The angular dependence of the total SO coupling constant (in units of meVnm), tot, averaged over all Noc- cupied subbands at kF n;z. (a)B= 0:1 T,= 0:3 eV (N= 8) and (b)B= 1 T,= 0:3 eV (N= 8). (c)11at= 0:3 eV (dashed line) and = 0:35 eV (solid line). a way which depends on the relative orientation of the surface and the eld. Note that the localization of the wave function at the surface is enhanced by the Coulomb repulsion at the high concentration regime. Indeed, as presented in Fig. 9(c), the 6-fold anisotropy of 11(for the ground state) is somewhat larger for higher . Our results qualitatively agree with experimental ev- idence in Ref. 29 where the SO coupling was measured to be isotropic in a suspended hexagonal InAs NW. This negative result is expected in the low magnetic eld used in the experiments ( B < 0:1 T). Evaluating the eld intensity at which anisotropy is exposed is a non trivial issue. The reason is that increasing the eld enhances the orbital eects on the charge density, which at zero eld tends to be localized near to the surface, but it also de- pletes the NW from free charge, which makes the charge to delocalize, due to the small Coulomb repulsion, and less sensitive to the anisotropy of the NW. B. Perpendicular magnetic eld with a nite backgate potential Next we consider the eect of a bottom gate attached to the NW (see Fig. 1). As in the previous section, the magnetic eld is perpendicular to the NW axis. We rst consider the = 0 (corner-to-corner) direction, hence the two elds are orthogonal to each other. The total intra-subband SO coupling totaveraged FIG. 10. The total intra-subband SO, tot, as a function of Vgat selected magnetic elds B= 0;1;4 T directed in the = 0 (corner-to-corner) direction. Results are shown for = 0:30 eV. over all occupied states at the Fermi wave vector kF n;z is shown in Fig. 10 as a function of the back-gate poten- tialVgat selected eld intensities. For the present elds conguration the symmetry around the y-axis is not bro- ken, hence nn y(Vg) = 0. Figure 10 shows that tot(Vg), which is nite due to the broken symmetry along x, in- creases with BforVg>0.tottakes o at a threshold Vgwhich moves toward negative gate voltages with in- creasing magnetic eld. The strong asymmetry shown in Fig. 10 between posi- tive and negative voltages is easily understood. For pos- itive voltages the electron charge is pulled toward the gates, where the self-consistent eld has the largest gra- dient. For negative voltages, instead, electrons are pulled far from the gate, where the potential is almost at.13 Note, however, the opposite eect of the magnetic eld. Here, the electric and magnetic elds are orthogonal, = 0. Therefore, for positive voltages both the gate po- tential and the magnetic eld push electrons toward the bottom edge, hence the magnetic eld reinforces the back gate eect, increasing the SO coupling. The opposite is true forVg<0; in this case, electric and magnetic eld push the electrons on opposite sides, and the magnetic eld weakens the SO coupling. Of course, the opposite situation takes place when the magnetic eld is directed at= 180. Therefore, for a xed Vg, we expect a strong anisotropy with respect to the magnetic eld orientation, as shown below. Figure 11 shows the polar plot of totaveraged over kF n;zforVg= 0:1 V together with 11 i. In the absence of a magnetic eld, the electronic charge is strongly local- ized by the electric eld at the edge of the NW, near to the backgate. At a small magnetic eld [ B= 0:1 T in panel (a)], the orbital eects are negligible, and the SO coupling is isotropic. If we increase the magnetic eld (panel (b)), however, tot(as well as11 x) shows a 2-fold anisotropy, as expected from the interplay between the9 FIG. 11. The angular dependence of the x(blue) and y component (red) of the intra-subband SO constant (in units of meVnm) 11 icalculated at the Fermi wave vector kF 1;zfor the lowest subband and the total SO constant averaged over all occupied states at kF n;z(magenta line). Insets in panel (b) show the squared envelope functions of the lowest subband at kF 1;zfor the magnetic eld with = 0 and= 180. Calcula- tions are performed with = 0:30 eV andVg= 0:1 eV. two elds. Note that at = 180, the SO coupling of the ground state is nearly zero as the orbital eects local- izes the electron wave function near the upper facet (see the inset), overcoming the gate eect. There, the electric eld is weak due to the distance from the gate, and the gradient is almost vanishing.13The nonzero value of tot in this case results from the other states which contribute to the total SOC. Further increasing the eld intensity B enhances the orbital eect enhancing the anisotropy due to suppressing nn xin a wide angular range, as shown in panel (c) for the ground state. A similar 2-fold anisotropy has been reported in Ref. 29 with a dierent gate conguration, but with the same symmetry. We postpone the detailed analysis of this ex- periment to Sec. III D. C. Axial magnetic eld We now consider the SO coupling constants under a magnetic eld with a component along the NW axis. This is the relevant conguration in the context of Majo- rana states engineering, which requires the axially mag- netic eld and the SO interaction to create Majorana zero energy modes at the ends of a NW. The question concerning the relative relationship between the SO cou-pling and the magnetic eld is still an open issue.37 FIG. 12. The total intra-subband SO constant totas a func- tion of the gate voltage Vgfor dierent axial magnetic elds. Inset: squared envelope functions of the lowest subband for dierent magnetic elds at Vg= 0. FIG. 13. Angular dependence of the x(blue) and y(red) component of the intra-subband SO (in units of meVnm) of the ground state 11 icalculated at kF 1;ztogether with the total SOCtot(magenta). The magnetic eld is rotated in the yz plane. Results for = 0:30 V,B= 1 T and (a) Vg= 0 and (b)Vg= 0:1 V. Figure 12 shows the calculated tot(Vg)vseld inten- sityBwith an axial eld ( = 0). Clearly, the axial mag- netic eld aects the SO coupling to a slight extent up to B= 16 T. This is in agreement with previous calculations within the Spin Density Functional formalism.38Indeed, in the axial eld conguration, the inversion symmetry is not removed (see Eq. 9), although the orbital eect is still visible in the inset of Fig. 12, where the envelope function is shown to localize further at the edges with the eld. There is almost no eld-induced depletion ef- fect here, which is only due to the part of the orbital eect related with the eld-induced quadratic terms in Eq. 9. Note the strong asymmetry with respect to the gate potential, which has the same explanation as the one in Fig. 10. Next, we consider a magnetic eld rotating in the y10 zplane, see Fig. 13, which shows a 2-fold anisotropy. However, the anisotropy is almost removed by the gate potential, with the SO constant being only slightly larger for the axially magnetic eld. The behaviour shown in Fig. 13 is easily traced to the wave function localization. At Vg= 0, SOC is trivially zero if the magnetic eld is in the axial direction (inver- sion symmetry holds), while it is at maximum with the eld in the orthogonal direction, ==2, as discussed in the previous paragraphs. If Vg= 0:1 V, instead, the wave function is localized near to the bottom edge, where the electric eld is the largest, and the SO coupling is large as well. AtB= 1 T the magnetic eld does not change the localization, although if the magnetic eld is perpendic- ular to NW the orbital eects squeezes the wave function to the side edges (either to the right or to the left) where the electric eld is lower, slightly lowering the SO cou- pling. Hence, a small gate potential restores the yz isotropy. D. Comparison with experiment [Ref.29] In Ref. 29 the authors used magnetotransport experi- ments to determine the SO coupling in suspended InAs NWs. Using a vectorial magnet, the non-trivial evolution of weak anti-localization (WAL) is tracked and the SO length is determined as a function of the magnetic eld intensity and direction. This study shows no anisotropy related to the geometrical connement in a low eld regime. The isotropy of SO coupling is however removed in the presence of an external electric eld induced by side gates. In this case, the SO coupling demonstrates a 2-fold periodic angular modulation when the magnetic eld is rotated in both the yzandxyplane. To simulate the experimental conditions, we consider a InAs NW attached to two side electrodes located 200 nm from the NW, see Fig. 14(a). Potentials applied to the gates generate an electric eld which is assumed to change linearly in the region between the electrodes. All parameters are taken from the experiment. We assume W= 100 nm (facet-facet) and ne= 21018cm3, which for the considered NW geometry, gives EF= 0:935 eV. In order to keep the electron density constant, the eld is induced by applying an asymmetric potential VSG1= gVSG2, wheregis determined separately for each Vg, as to keep the density constant. We consider only the case with the magnetic eld directed perpendicular to the NW and rotating in the xyplane, with B= 0:1 T as used in the experiment. Thexandycomponents of the intra-subband SO cou- pling for the ground state 11 icalculated at Vg= 0 is pre- sented in Fig. 14(b,c). The rapid switch between the two components results from the Coulomb interaction. At the considered high electron concentrations the electron- electron repulsion localizes the charge in quasi-1D chan- nels at the corners.33When the magnetic eld rotates the localization of the ground state suddenly moves betweenthe corners resulting in a step-like change between the xandycomponents which swap their intensities. The total SO coupling constant averaged over all oc- cupied states at kz n;Fis presented in panels (d) and (e) for two dierent gate voltages. The total SO cou- pling atVg= 0, panel (d), is nearly isotropic exhibiting slight oscillations with the 6-fold symmetry due the pris- matic symmetry of the NW which, in the considered high electron density regime, is more pronounced due to the strong localization of electrons at the six corners. Note that in Ref. 29 the authors reported full isotropic be- haviour of SOC at Vg= 0 without the oscillations. This inconsistency remains to be claried. It may be the re- sult of the specic extraction of the SO length used in Ref. 29 which includes the correction from the eective NW width. Alternatively, a low resolution of the magne- totransport measurement might not be able to capture small changes of SOC. Finally, we apply a potential Vg= 2 V, as in the ex- periments, to the side electrodes ( g= 0:96). In this conguration, the ycomponent of SO coupling becomes dominant and is barely aected by the magnetic eld ori- entation. For such a high gate potential the wave func- tion of the ground state is strongly localized in the right corner [see the inset Fig. 14(e)] and it is only slightly dis- turbed by the orbital eects originating from the weak magnetic eld used in the experiment ( B= 0:1 T). This results in the slight 2-fold anisotropy of SOC, shown in panel (e), similarly as reported in the experiment.29Note however that the experimental evidence shows a 2-fold anisotropy with respect to the magnetic eld orientation in theyzplane (although authors suggested its exis- tence also in the xymagnetic eld rotation) and its intensity is much stronger. Although we did not perform explicit calculations in this conguration for such a high electron density, which implies a very large number of subbands ( 100) and a correspondingly large numerical eort, results presented in Fig. 13 for a lower electron density and higher mag- netic eld agree with the experimental result and support the interpretation. Note however that at Vg= 0 and the axially directed magnetic eld, the inversion symmetry around either the xandyaxis is not broken, which re- sults intot= 0 as presented in Fig. 13(a). This sce- nario is however not supported by the experimental data which exhibit nonzero SOC even for the axially magnetic eld. This strongly suggests the presence in the samples of an intrinsic electric eld of an unknown origin, which is a source of SO coupling whose distortion by the weak magnetic eld used in the experiment ( B= 0:1 T) is not possible, resulting in the isotropic SOC. An intrin- sic electric eld would explain also the absence of the SO coupling angular oscillations [as in Fig. 14(a)] and the slightly lower value of SOC from the calculations, tot10 meVnm, as compared with the corresponding experimental value exp tot15 meVnm. Interestingly, it might also explain the observed unexplained phase shift in the magnetoconductance measurement [see Fig. 3(c,d)11 FIG. 14. (a) Schematic illustration of the experimental setup. (b,c) The xandycomponent of the intra-subband SO coupling11 ias a function of the angle and the wave vector kz. The magnetic eld is rotated in the xyplane. (d,e) The angular dependence of the total SO constant, tot. Results for B= 0:1 T and==2. in Ref. 29] in terms of the relative alignment between the magnetic eld and the resultant electric eld (sum of the non-collinear intrinsic and extrinsic electric eld) which changes depending on the applied voltage. IV. SUMMARY Based on the kptheory within the envelope function approximation, we have analyzed the orbital eects of a magnetic eld on the Rashba SO coupling in InAs homo- geneous semiconductor NWs. The full vectorial character of the SO constant has been studied under the magnetic eld magnitude and orientation. The Rashba SO interaction of conduction electrons in a NW is determined by the position and symmetry of the electron's wave function, which can be tuned by gate- induced electric elds as well as by the the orbital eects induced by a magnetic eld. Specically, when we ap- ply the magnetic eld perpendicular to NW the inver- sion symmetry of the envelope functions is broken and the wave functions is squeezed to the NW surface by a kz-dependent eective potential. This eect results in a nite SO coupling, which is also sensitive to the geomet- rical connement. As we have shown, at low magnetic eld (<1 T for the considered NW), when orbital eects are weak, the SO coupling is isotropic with respect tothe magnetic eld in the NW section. Interestingly, the slight 6-fold anisotropy appears at higher magnetic elds (or high electron concentration), when the wave function is squeezed to the NW edges to a larger extent. When a gate potential is applied in the direction or- thogonal to the magnetic eld, the two elds intertwin in a way which may enhance or suppress the SO coupling, depending on the relative direction, leading to a 2-fold anisotopy with respect to the magnetic eld rotation in both thexyplane. Finally, in light of our simulations, we have analyzed qualitatively recent experiments with suspended InAs NWs29and good agreement with the experimental data has been found. However, we suggest that an unintended electric eld is present in the sample, which would rec- oncile observations with our predictions. As a nal remark, we note that in real devices a dielec- tric spacer often separates the gate from the NW, which reduces the SO constant. However, a spacer layer could change the cancellation eect, as it only lowers the inter- nal electric eld. Importantly, our study has shown no signicant changes of the SO coupling with the axially magnetic eld.12 V. ACKNOWLEDGEMENT This work was supported by the AGH UST statutory tasks No.11.11.220.01/2 within subsidy of the Ministryof Science and Higher Education in part by PL-Grid In- frastructure. pawel.wojcik@s.agh.edu.pl yandrea.bertoni@nano.cnr.it zguido.goldoni@unimore.it 1H. C. Koo, J. H. Kwon, J. Eom, J. Chang, S. H. Han, and M. Johnson, Science 325, 1515 (2009). 2P. W ojcik, J. Adamowski, B. J. Spisak, and M. Wo loszyn, J. Appl. Phys. 115, 104310 (2014). 3P. W ojcik and J. Adamowski, Sci Rep 7, 45346 (2017). 4A. T. Ngo, P. Debray, and S. E. Ulloa, Phys. Rev. B 81, 115328 (2010). 5M. Kohda, S. Nakamura, Y. Nishihara, K. Kobayashi, T. Ono, J. O. Ohe, Y. Tokura, T. Mineno, and J. Nitta, Nature Communications 3, 1082 (2012). 6S. Nadj-Perge, S. M. Frolov, E. P. A. M. Bakkers, and L. P. Kouwenhoven, Nature 468, 1084 (2010). 7J. W. G. van den Berg, S. Nadj-Perge, V. S. Pribiag, S. R. Plissard, E. P. A. M. Bakkers, S. M. Frolov, and L. P. Kouwenhoven, Phys. Rev. Lett. 110, 066806 (2013). 8J. Kammhuber, M. C. Cassidy, F. Pei, M. P. Nowak, A. Vuik, D. Car, S. R. Plissard, E. P. A. M. Bakkers, M. Wimmer, and L. P. Kouwenhoven, Nat Commun. 8, 478 (2017). 9S. Heedt, N. Traverso Ziani, F. Cr epin, W. Prost, S. Trel- lenkamp, J. Schubert, D. Gr utzmacher, B. Trauzettel, and T. Sch apers, Nature Phys. 13, 563 (2017). 10I. van Weperen, B. Tarasinski, D. Eeltink, V. S. Pribiag, S. R. Plissard, E. P. A. M. Bakkers, L. P. Kouwenhoven, and M. Wimmer, Phys. Rev. B 91, 201413(R) (2015). 11T. Campos, P. E. Faria Junior, M. Gmitra, G. M. Sipahi, and J. Fabian, Phys. Rev. B 97, 245402 (2018). 12P. W ojcik, A. Bertoni, and G. Goldoni, Appl. Phys. Lett. 114, 073102 (2019). 13P. W ojcik, A. Bertoni, and G. Goldoni, Phys. Rev. B 97, 165401 (2018). 14A. Bringer, S. Heedt, and T. Sch apers, Phys. Rev. B 99, 085437 (2019). 15X. W. Zhang and J. B. Xia, Phys. Rev. B 74, 075304 (2006). 16K. Takase, K. Tateno, and S. Sasaki, Applied Physics Express 12, 117002 (2019). 17J. Alicea, Rep. Prog. Phys. 75, 076501 (2012). 18A. Kitaev, Ann. Phys. 303, 2 (2003). 19S. Das Sarma, M. Freedman, and C. Nayak, npj Quantum Information 1, 15001 (2015).20Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev. Lett. 105, 177002 (2010). 21K. O. Klausen, A. Sitek, S. I. Erlingsson, and A. Manolescu, Nanotechnology 31, 354001 (2020). 22V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A. M. Bakkers, and L. P. Kouwenhoven, Science 336, 1003 (2012). 23S. M. Albrecht, A. P. Higginbotham, M. Madsen, F. Kuem- meth, T. S. Jespersen, J. Nyg aard, P. Krogstrup, and C. M. Marcus, Nature 531, 206 (2016). 24J. D. Sau, S. Tewari, and S. Das Sarma, Phys. Rev. B 85, 1 (2012). 25G. Dresselhaus, Phys. Rev. 100, 580 (1955). 26E. I. Rashba, Sov. Phys. Solid State 2, 1109 (1960). 27A. Manchon, C. Koo, H, J. Nitta, M. Frolov, S, and A. Duine, R, Nature Materials 14, 871 (2015). 28S. Nadj-Perge, V. S. Pribiag, J. W. G. van den Berg, K. Zuo, S. R. Plissard, E. P. A. M. Bakkers, S. M. Frolov, and L. P. Kouwenhoven, Phys. Rev. Lett. 108, 166801 (2012). 29A. Iorio, M. Rocci, L. Bours, M. Carrega, V. Zannier, L. Sorba, S. Roddaro, F. Giazotto, and E. Strambini, Nano Lett. 19, 652 (2019). 30J.-W. Luo, L. Zhang, and A. Zunger, Phys. Rev. B 84, 121303 (2011). 31J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and I.Zuti c, Acta Physica Slovaca 57, 565 (2007). 32M. P. Nowak and P. W ojcik, Phys. Rev. B 97, 045419 (2018). 33A. Bertoni, M. Royo, F. Mahawish, and G. Goldoni, Phys. Rev. B 84, 205323 (2011). 34M. Royo, A. Bertoni, and G. Goldoni, Phys. Rev. B 89, 155416 (2014). 35I. Vurgaftman, R. Meyer, J, and R. Ram-Mohan, L, Appl. Phys. Lett. 89, 5815 (2001). 36M. Royo, A. Bertoni, and G. Goldoni, Phys. Rev. B 87, 115316 (2013). 37G. W. Winkler, D. Varjas, R. Skolasinski, A. A. Soluyanov, M. Troyer, and M. Wimmer, Phys. Rev. Lett. 119, 037701 (2017). 38M. Royo, C. Segarra, A. Bertoni, G. Goldoni, and J. Planelles, Phys. Rev. B 91, 115440 (2015).
1710.10579v2.Contact_theory_for_spin_orbit_coupled_Fermi_gases.pdf	arXiv:1710.10579v2 [cond-mat.quant-gas] 24 Dec 2017Contact theory for spin-orbit-coupled Fermi gases Shi-Guo Peng1, Cai-Xia Zhang1,4, Shina Tan2,3,∗and Kaijun Jiang1,3† 1State Key Laboratory of Magnetic Resonance and Atomic and Mo lecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academ y of Sciences, Wuhan 430071, China 2School of Physics, Georgia Institute of Technology, Atlant a, Georgia 30332, USA 3Center for Cold Atom Physics, Chinese Academy of Sciences, W uhan 430071, China and 4School of Physics, University of Chinese Academy of Science s, Beijing 100049, China (Dated: December 27, 2017) We develop the contact theory for spin-orbit-coupled Fermi gases. By using a perturbation method, we derive analytically the universal two-body beha vior at short distance, which does not depend on the short-range details of interatomic potential s. We ﬁnd that two new scattering pa- rameters need to be introduced because of spin-orbit coupli ng, besides the traditional s- andp-wave scattering length (volume) and eﬀective ranges. This is a ge neral and unique feature for spin- orbit-coupled systems. Consequently, two new adiabatic en ergy relations with respect to the new scattering parameters are obtained, in which a new contact i s involved because of spin-orbit cou- pling. In addition, we derive the asymptotic behavior of the large-momentum distribution, and ﬁnd that the subleading tail is corrected by the new contact. Thi s work paves the way for exploring the profound properties of spin-orbit-coupled many-body syst ems, according to two-body solutions. Introduction .—Universality, referring to observations independent of short-range details, is one of the most fascinating and intriguing phenomena in modern physics. In ultracold atoms, a set of universal relations, following from the short-range behavior of the two-body physics are discovered [1]. These relations are connected sim- ply by a universal contact parameter, which overarches between microscopic and macroscopic properties of a strongly interacting many-body system. Nowadays, the contact theory becomes signiﬁcantly important in ultra- cold atomic physics, and has systematically been veriﬁed and investigated both experimentally and theoretically [2–7]. Nevertheless, the contact theory for spin-orbit- coupled systems is still unexplored till now, even though the spin-orbit (SO) coupling was realized in cold atoms several years ago [8–10], and resulted unique phenomena have attracted a great deal of interest, such as topological insulators and superconductors [11–14]. In this letter, for the ﬁrst time, we generalize the contact theory to strongly interacting spin-orbit-couple d Fermi gases, and the single-particle Hamiltonian takes the form, ˆH1=/planckover2pi12ˆk2 1 2M+/planckover2pi12λ Mˆk1·ˆσ+/planckover2pi12λ2 2M, (1) whereˆk1=−i∇andˆσare respectively the single- particle momentum and spin operators, λ >0is the strength of SO coupling, Mis the atomic mass, and /planckover2pi1is the Planck’s constant divided by 2π. Here, the SO coupling is assumed to be isotropic for simplicity, and the possible scheme for the realization of the three- dimensional (3D) isotropic SO coupling is proposed in [15]. Because of SO coupling, the orbital angular momen- tum of the relative motion of two fermions is no longer conserved, and then all the partial-wave scatterings are coupled [16]. Fortunately, the total momentum Kof twofermions is still conserved as well as the total angular momentum J. Therefore, we may reasonably focus on the two-body problem in the subspace of K= 0and J= 0for simplicity, and then only s- andp-wave scatter- ings are coupled [16, 17]. Consequently, the two spin-half fermions in the subspace of K= 0andJ= 0is described by the following two-body Hamiltonian ˆH2=/planckover2pi12ˆk2 M+/planckover2pi12λ Mˆk·(ˆσ2−ˆσ1)+/planckover2pi12λ2 M+V(r),(2) whereˆkis the momentum operator for the relative mo- tionr=r2−r1,ˆσiis the spin operator of the ith atom, andV(r)is the interatomic potential with a ﬁnite range ǫ. Our theory may also be generalized to the case of K/ne}ationslash= 0andJ/ne}ationslash= 0, and then more partial waves should be involved. One of the most daunting challenges for establishing the contact theory is how to obtain the universal two- body behavior at short distance for a SO-coupled Fermi gas. Although the SO-coupled two-body problem was considered recently by using a spherical square-well po- tential [16–18], the general form of such universal behav- ior for any interatomic potential still remains elusive til l now. In this work, we develop a perturbation method to construct the short-range asymptotic form of the two- body wave function for a SO-coupled system. We ﬁnd that two new scattering parameters u,vneed to be in- troduced in the short-range behavior of two-body wave functions, besides the traditional scattering length (vol - ume) and eﬀective ranges. The obtained universal be- havior does not depend on the short-range details of the interatomic potentials, and thus is feasible for any inter- atomic potential with short range. Two new adiabatic energy relations are accordingly found with respect to2 the new scattering parameters, i.e., ∂E ∂u=/planckover2pi12λ 32π2M/parenleftbigg C(0) a−λPλ 2/parenrightbigg , (3) ∂E ∂v=λ3/planckover2pi12C(1) a 32π2M, (4) in which we hold all the other two-body parameters un- changed in the partial derivatives. Here, C(0) ais the well knowns-wave contact, C(1) ais thep-wave contact corre- sponding to the p-wave scattering volume [3, 6]. In addi- tion,Pλis the new contact introduced by SO coupling. Further, we derive the asymptotic behavior of the large- momentum distribution from the universal two-body be- havior at short distance, n(q) =C(1) a q2+/parenleftBig C(0) a+C(1) b+λPλ/parenrightBig1 q4+O/parenleftbig q−6/parenrightbig ,(5) in which C(1) bis thep-wave contact corresponding to the p-wave eﬀective range. We ﬁnd that the subleading tail (q−4) of the large-momentum distribution is amended by the new contact Pλbecause of SO coupling. Universal short-range behavior of two-body wave func- tions.—Let us consider the two-body problem of a SO- coupled system in the subspace of K= 0andJ= 0, and the corresponding Hamiltonian takes the form of Eq.(2). The subspace is spanned by two orthog- onal basis, i.e., Ω0(ˆr) =Y00(ˆr)\|S/an}b∇acket∇i}htandΩ1(ˆr) = −i[Y1−1(ˆr)\|↑↑/an}b∇acket∇i}ht+Y11(ˆr)\|↓↓/an}b∇acket∇i}ht−Y10(ˆr)\|T/an}b∇acket∇i}ht]/√ 3, where Ylm(ˆr)is the spherical harmonics, ˆrdenotes the an- gular part of the coordinate r, and\|S/an}b∇acket∇i}ht=(\|↑↓/angbracketright−\|↓↑/angbracketright )√ 2 and/braceleftBig \|↑↑/an}b∇acket∇i}ht,\|↓↓/an}b∇acket∇i}ht,\|T/an}b∇acket∇i}ht=\|↑↓/angbracketright+\|↓↑/angbracketright√ 2/bracerightBig are the spin-singlet and spin-triplet states with total spin 0and1, respec- tively. The two-body solution can formally be written in the basis of {Ω0(ˆr),Ω1(ˆr)}asΨ(r) =ψ0(r)Ω0(ˆr)+ ψ1(r)Ω1(ˆr)[16, 17]. Since the SO eﬀect exists even inside the interatomic potential, it should modify the short-range behavior of the two-body wave function dramatically [19]. However, in current experiments of ultracold atoms [20], the SO- coupling strength λis of the order µm−1, pretty small compared to the inverse of the range of interatomic po- tentialǫ−1(of the order nm−1). Moreover, the momen- tumk=/radicalbig ME//planckover2pi12is also much smaller than ǫ−1in the low-energy scattering limit. Therefore, when two fermions get as close as the range ǫ, we may deal with the SO coupling perturbatively as well as the energy, and assume that the form of the two-body solution has the following structure, Ψ(r)≈φ(r)+k2f(r)−λg(r) (6) asr∼ǫ. Here, we keep up to the ﬁrst-order terms of k2andλ. The advantage of this ansatz is that the func- tionsφ(r),f(r)andg(r)are all independent on theenergy and SO-coupling strength. Therefore, they are determined only by the short-range details of the inter- action, and characterize the intrinsic properties of the interatomic potential. We expect that the traditional scattering length or volume in the absence of SO cou- pling are included in the zero-order term φ(r), while the eﬀective ranges are involved in f(r), the coeﬃcient of the ﬁrst-order term of k2. Interestingly, new scattering parameters should appear in the ﬁrst-order term of λ (ing(r)), which are introduced by SO coupling. Conve- niently, more scattering parameters may be introduced if higher-order terms of k2andλare perturbatively con- sidered. Inserting the ansatz (6) into the Schrödinger equation, and comparing the corresponding coeﬃcients ofk2andλ, we obtain /bracketleftbigg −∇2+M /planckover2pi12V(r)/bracketrightbigg φ(r) = 0, (7) /bracketleftbigg −∇2+M /planckover2pi12V(r)/bracketrightbigg f(r) =φ(r), (8) /bracketleftbigg −∇2+M /planckover2pi12V(r)/bracketrightbigg g(r) =Q(r)φ(r), (9) whereQ(r) =−i∇ ·(ˆσ2−ˆσ1). These equations can analytically be solved for r>ǫ, and simply yield Ψ(r) =α0/bracketleftbigg1 r+/parenleftbigg −1 a0+b0 2k2+uλ/parenrightbigg −k2 2r/bracketrightbigg Ω0(ˆr) +α1/bracketleftbigg1 r2+/parenleftbiggk2 2+α0 α1λ/parenrightbigg +/parenleftbigg −1 3a1+b1 6k2+vλ/parenrightbigg r/bracketrightbigg Ω1(ˆr) +O/parenleftbig r2/parenrightbig ,(10) whereα0andα1are two complex superposition coeﬃ- cients. Apparently, a0,b0are thes-wave scattering length and eﬀective range, and a1,b1are thep-wave scattering volume and eﬀective range without SO coupling, respec- tively. For simplicity, we may only consider the case with b0≈0for broads-wave resonances throughout the paper. We can see that the s-wave component is hybridized in thep-wave channel by SO coupling as manifested as the termα0λ/α1. Interestingly, two new scattering parame- tersuandvas we anticipate are involved. They are the corrections from SO coupling to the short-range behav- ior of the two-body wave function in s- andp-wave chan- nels, respectively. If λ= 0, thes- andp-wave scatterings decouple, and the asymptotic form of Ψ(r)at small r, i.e., Eq.(10), simply reduces to the ordinary s- andp- wave short-range boundary conditions, respectively. The derivation above doesn’t depend on the short-range de- tails of the interaction, and thus is universal and appli- cable for all kinds of neutral fermionic atoms. In general, the s- andp-wave scatterings in diﬀerent spin channels should both be taken into account because of SO coupling. We may roughly estimate which partial wave is more important as follows. Without SO coupling, and away from any resonances—in the weak interacting3 limit, the two-body wave function should well behave as r→0asΨ(r)∼(α0/a0)Ω0(ˆr) + (α1r/3a1)Ω1(ˆr). If we assume that the atoms are initially prepared equally in the spin channels Ω0(ˆr)andΩ1(ˆr), we haveα0/α1∼ a0r/3a1. When interatomic interactions are turned on, the two-body wave function becomes divergent as r→ 0(>ǫ),α0r−1andα1r−2fors- andp-wave scatterings, re- spectively. This divergent behavior is unchanged even in the presence of SO coupling. Then the ratio between the strengths of s- andp-wave scatterings at small rbecomes/parenleftbig α0r−1/parenrightbig //parenleftbig α1r−2/parenrightbig ≈a0r2/3a1. Nears-wave resonances, we havea0∼k−1 f,a1∼ǫ3,r∼ǫ, wherekfis the Fermi wavenumber, and then this ratio is approximately of the order(kfǫ)−1≫1. Therefore, the s-wave interaction dominates the two-body scattering. By noticing Ω0(ˆr) = \|S/an}b∇acket∇i}ht/√ 4π, andΩ1(ˆr) =−i(ˆσ2−ˆσ1)·(r/r)\|S/an}b∇acket∇i}ht/√ 16π, and if thep-wave interaction could be ignored near broad s-wave resonances, Eq.(10) becomes (up to a prefactor α0/√ 4π), Ψ(r) =/parenleftbigg1 r−1 a0+uλ/parenrightbigg \|S/an}b∇acket∇i}ht−iλ 2(ˆσ2−ˆσ1)·r r\|S/an}b∇acket∇i}ht+O(r), (11) which exactly recovers the result of [19] (see Eq.(31) of [19]) witha−1 R=a−1 0−uλ. Nearp-wave resonances, for example, the p-wave Fes- hbach resonance at B0= 185.09G in6Li [21], we have a0∼ǫ,a1∼k−3 f,r∼ǫ, then the ratio between the strengths of s- andp-wave scatterings is roughly of the order(kfǫ)3≪1. In this case, the p-wave scattering becomes signiﬁcantly important. Large-momentum distribution. —For a many-body sys- tem withNspin-half fermions, if only two-body correla- tions are taken into account, the many-body wave func- tionΨNcan approximately be written as the form of Eq.(10), when fermions (i,j)get close while all the others are far away. In this case, r=ri−rj, and the arbitrary complex numbers α0andα1become the functions of the variableX, which involves both the center-of-mass (c.m.) coordinate of the pair being considered and the coordi- nates of all the other fermions. Further, α0andα1should be constrained by the normalization of the many-body wave function. Using the asymptotic form of the many- body wave function ΨNat small r, we can easily obtain the behavior of the tail of the single-particle momentum distribution at large q(but smaller than ǫ−1), which is deﬁned asn(q)≡/summationtextN i=1´/producttext j/negationslash=idrj/vextendsingle/vextendsingle´ driΨNe−iq·ri/vextendsingle/vextendsingle2. After straightforward algebra, we easily obtain the mo- mentum distribution n(q)taking the form of Eq.(5) at largeq/parenleftbig <ǫ−1/parenrightbig . Here, we are only interested in the de- pendence of the momentum distribution on the ampli- tude ofq, and have already integrated over the angular part ofq. We ﬁnd that C(ν) a= 32π2Nˆ dX\|αν(X)\|2,(ν= 0,1), (12)C(1) b=64π2MN /planckover2pi12ˆ dXα∗ 1(X)/bracketleftBig E−ˆT(X)/bracketrightBig α1(X)(13) are the conventional s- andp-wave contacts [6], where ˆT(X)denotes the operators of the c.m. motion of the pair (i,j)and all the other fermions, and N= N(N−1)/2is the number of all the possible ways to pair atoms. Besides, a new contacts Pλresulted from SO coupling appears, which is deﬁned as Pλ≡64π2Nˆ dXα∗ 0(X)α1(X)+c.c..(14) Obviously, this new contact describes the interplay of the s- andp-wave scatterings because of SO coupling. Since the momentum distribution at large qis only characterized by the short-range behavior of the two- body physics, we may roughly estimate the order of all the quantities in the large- qbehavior of the momentum distribution simply according to the two-body picture as before. Near s-wave resonances, if initially without SO coupling and away from any resonances, the atoms are prepared equally in the spin states Ω0(ˆr)andΩ1(ˆr), we haveα0/α1∼a0r/3a1, and then C(1) aq−2/C(0) aq−4≈ 9a2 1q2/a2 0r2, which is roughly of the order (kfǫ)4≪1. Besides, we may also ﬁnd C(1) b/C(0) a∼(kfǫ)4≪1. This means that the p-wave contribution to the tail of momen- tum distribution at large qmay reasonably be ignored, which is consistent with the discussion before. However, the SO-coupling correction is notable compared to the p-wave contact in the subleading tail of the momentum distribution, i.e., λPλ/C(1) b∼(kfǫ)−2≫1. Nearp-wave resonances, the leading tail q−2of the large-momentum distribution becomes important, be- causeC(1) aq−2/C(0) aq−4∼(kfǫ)−4≫1. In the sub- leading tail of q−4, we ﬁnd C(1) b/C(0) a∼(kfǫ)−4≫1, thus thes-wave contribution may be ignored. Conse- quently, the momentum distribution at large qbehaves asC(1) aq−2+/parenleftBig C(1) b+λPλ/parenrightBig q−4with a considerable cor- rection ofλPλin the subleading tail due to SO coupling, compared to the s-wave contribution, i.e. λPλ/C(0) a∼ (kfǫ)−2≫1. Adiabatic energy relations. —The thermodynamics of many-body systems, which is seemingly uncorrelated to the momentum distribution, is also characterized by the contacts deﬁned above. A set of adiabatic energy rela- tions describe how the energy of a many-body system changes as the two-body interaction is adiabatically ad- justed. Let us consider two many-body wave functions ΨNandΨ′ Ncorresponding to diﬀerent interatomic inter- action strengths. From the Schrödinger equations satis-4 ﬁed byΨNandΨ′ N, we easily obtain (E−E′)˚ Dǫdr1dr2···drNΨ′∗ NΨN= −/planckover2pi12N M" r=ǫI·ˆndΣ+/planckover2pi12λN 2πM" r=ǫF·ˆndΣ,(15) whereI≡Ψ′∗∇Ψ−(∇Ψ′∗)Ψ,F≡(ψ′∗ 1ψ0−ψ′∗ 0ψ1)ˆer with the unit radial vector ˆerofr, andΨN(X,r) = ψ0(X,r)Ω0(ˆr)+ψ1(X,r)Ω1(ˆr). Here, the domain Dǫis the set of all conﬁgurations (ri,rj)withr=\|ri−rj\|>ǫ, Σis the surface in which the distance between the two atoms in the pair (i,j)isǫ, andˆnis the direction normal toΣbut is opposite to the radial direction. Using the asymptotic form of the many-body wave function ΨNat smallr, we ﬁnd δE=−/planckover2pi12 32π2M/bracketleftbigg/parenleftbigg C(0) a−λPλ 2/parenrightbigg δa−1 0+C(1) aδa−1 1 −C(1) b 4δb1−λ/parenleftbigg C(0) a−λPλ 2/parenrightbigg δu−3λC(1) aδv/bracketrightBigg ,(16) which characterizes how the energy of the system varies as the scattering parameters adiabatically change. In the absence of SO coupling, Eq.(16) simply reduces to the or- dinary form of the adiabatic energy relations for s- and p-wave interactions [6, 22], with respect to the scattering length (or volume) as well as eﬀective range. Because of SO coupling, two new scattering parameters come into the problem, and then additional new adiabatic energy relations appear, i.e., Eqs.(3)-(4). These adiabatic en- ergy relations demonstrate how the macroscopic thermo- dynamics of SO-coupled many-body systems varies with microscopic two-body scattering parameters. Contacts in a two-body problem .—On behalf of the future experiments and calculations, we may explicitly evaluate the contacts deﬁned above for a two-body bound state, the wave function of which may be written as a col- umn vector in the basis of {Ω0(ˆr),Ω1(ˆr)}as [17] Ψb(r) =Bκ−/bracketleftBigg h(1) 0(κ−r) −h(1) 1(κ−r)/bracketrightBigg +Dκ+/bracketleftBigg h(1) 0(κ+r) h(1) 1(κ+r)/bracketrightBigg , (17) whereκ±=iκ±λ, andκ=/radicalbig −ME//planckover2pi12. The bind- ing energy Ecan be determined by expanding Ψb(r)at smallrand comparing with the short-range boundary condition (10), then the two-body contacts are easily ob- tained according to the adiabatic relations. Near s-wave resonances, we ﬁnd E=−/planckover2pi12 Ma2 0+2/planckover2pi12u Ma0λ+O/parenleftbig λ2/parenrightbig , (18) which simply reduces to the result E=−/planckover2pi12/Ma2 0in the absence of SO coupling. Then we immediately obtainC(0) a= 64π2/a0andPλ= 128π2uby using adiabatic relations. Near p-wave resonances, we ﬁnd E=2/planckover2pi12 Ma1b1−6/planckover2pi12v Mb1λ+O/parenleftbig λ2/parenrightbig , (19) which is consistent with that without SO coupling [3], and then it yields C(1) a=−64π2/b1andC(1) b= −256π2/parenleftbig a−1 1−3vλ/parenrightbig /b2 1. Grand canonical potential and pressure relation .—The adiabatic relations as well as the large-momentum dis- tribution we obtained above is valid for any pure energy eigenstate. Therefore, they should still hold for any in- coherent mixed state statistically at ﬁnite temperature. Then the energy, particle number density and contacts become their statistical average values. It should be in- teresting to discuss how the results presented above aﬀect the ﬁnite-temperature thermodynamics. To this end, let us look at the grand canonical potential, which is de- ﬁned asJ ≡ −PV[23], where Pis the pressure and Vis the volume of the system. According to straightforward dimensional analysis [24, 25], we can obtain J=−2 3E−/planckover2pi12 96π2Ma0/parenleftbigg C(0) a−λPλ 2/parenrightbigg −/planckover2pi12C(1) a 32π2Ma1+/planckover2pi12b1C(1) b 384π2M+λ/planckover2pi12vC(1) a 16π2M,(20) which alternatively yields the pressure relation by divid- ing both sides of Eq.(20) by −V. Conclusions. —We systematically study the contact theory for spin-orbit-coupled Fermi gases. The univer- sal two-body behavior at short distance is analytically derived, by introducing a perturbation method, which doesn’t depend on the short-range details of interatomic potentials. For simplicity, we focus on the s- andp-wave scatterings in the subspace of vanishing center-of-mass momentum and total angular momentum. Interestingly, two new microscopic scattering parameters appear in the short-range behavior of two-body wave functions because of spin-orbit coupling. We claim that this is a general and unique feature for spin-orbit-coupled systems, and thus the obtained universal short-range behavior of two- body wave functions is feasible for all kinds of neutral fermionic atoms. Consequently, a new contact is intro- duced originated from spin-orbit coupling, which, com- bining with conventional s- andp-wave contacts, char- acterizes the universal properties of spin-orbit-coupled many-body systems. In general, more partial-wave scat- terings should be taken into account for nonzero center- of-mass momentum and nonzero total angular momen- tum. Then more contacts should appear. Our method could conveniently be generalized to other kinds of spin- orbit couplings as well as to low dimensions. Besides, our method could also be applied to bosons. In the presence of spin-orbit coupling, we expect that additional contacts would be introduced for bosonic systems.5 S. G. P and K. J are supported by the NKRDP (Na- tional Key Research and Development Program) under Grant No. 2016YFA0301503 and NSFC under Grant No. 11474315, 11674358, 11434015, 91336106. S. T is sup- ported by the US National Science Foundation CAREE award Grant No. PHY-1352208. S. G. P and C. X. Z contributed equally to this work. ∗shina.tan@physics.gatech.edu †kjjiang@wipm.ac.cn [1] S. Tan, Ann. Phys. 323, 2952 (2008); S. Tan, Ann. Phys. 323, 2971 (2008); S. Tan, Ann. Phys. 323, 2987 (2008). [2] W. Zwerger, The BCS-BEC Crossover and the Unitary Fermi Gas , Col 836 of Lecture Notes in Physics (Springer, Berlin, 2011); Please see Chapter 6 for a brief review. [3] Z. H. Yu, J. H. Thywissen, and S. Z. Zhang, Phys. Rev. Lett. 115, 135304 (2015); Z. Yu, J. H. Thywissen, and S. Zhang, Phys. Rev. Lett. 117, 019901(E) (2016). [4] M. Y. He, S. L. Zhang, H. M. Chan, and Q. Zhou, Phys. Rev. Lett. 116, 045301 (2016). [5] C. Luciuk, S. Trotzky, S. Smale, Z. Yu, S. Zhang, and J. H. Thywissen, Nat. Phys. 12, 599 (2016). [6] S. G. Peng, X. J. Liu, and H. Hu, Phys. Rev. A 94, 063651 (2016). [7] R. J. Fletcher, R. Lopes, J. Man, N. Navon, R. P. Smith, M. W. Zwierlein, and Z. Hadzibabic, Science 355, 377 (2017). [8] Y. J. Lin, K. Jimenez-Garcia, and I. B. Spielman, Nature 471, 83 (2011). [9] P. J. Wang, Z. Q. Yu, Z. K. Fu, J. Miao, L. H. Huang, S. J. Chai, H. Zhai, and J. Zhang, Phys. Rev. Lett. 109, 095301 (2012). [10] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah, W. S. Bakr, and M. W. Zwierlein, Phys. Rev. Lett. 109, 095302 (2012). [11] X. L. Qi and S. C. Zhang, Phys. Today 63, 33 (2010). [12] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045(2010). [13] J. Dalibard, F. Gerbier, G. Juzeliunas, and P. Ohberg, Rev. Mod. Phys. 83, 1523 (2011). [14] J. Zhang, H. Hu, X.-J. Liu, and H. Pu, Annual Review of Cold Atoms and Molecules , Vol. 2, 81 (World Scieniﬁc, 2014). [15] B. M. Anderson, G. Juzeliunas, V. M. Galitski, and I. B. Spielman, Phys. Rev. Lett. 108, 235301 (2012). [16] X. L. Cui, Phys. Rev. A 85, 022705 (2012). [17] Y. X. Wu and Z. H. Yu, Phys. Rev. A 87, 032703 (2013). [18] X. L. Cui, Phys. Rev. A 95, 030701 (2017). [19] P. Zhang, L. Zhang, and Y. J. Deng, Phys. Rev. A 86, 053608 (2012). [20] P. J. Wang, Z. Q. Yu, Z. K. Fu, J. Miao, L. H. Huang, S. J. Chai, H. Zhai, and J. Zhang, Phys. Rev. Lett. 109, 095301 (2012); L. W. Cheuk, A. T. Sommer, Z. Hadz- ibabic, T. Yefsah, W. S. Bakr, and M. W. Zwierlein, Phys. Rev. Lett. 109, 095302 (2012). [21] J. Zhang, E. G. M. van Kempen, T. Bourdel, L. Khaykovich, J. Cubizolles, F. Chevy, M. Teichmann, L. Tarruell, S. J. J. M. F. Kokkelmans, and C. Salomon, Phys. Rev. A 70, 030702 (2004); C. H. Schunck, M. W. Zwierlein, C. A. Stan, S. M. F. Raupach, W. Ketterle, A. Simoni, E. Tiesinga, C. J. Williams, and P. S. Julienne, Phys. Rev. A 71, 045601 (2005). [22] For the s-wave interaction, there is a diﬀerence of the factor8πfrom the well-known form of adiabatic rela- tions (see Eq.(36)-(37) of [6]). This is because here we include the spherical harmonics Y00(ˆr) = 1/√ 4πin the s-partial wave function. Besides, an additional factor 1/2 is introduced in order to keep the deﬁnition of the con- tacts consistent with those in the tail of the momentum distribution at large q. [23] L. D. Landau, and E. M. Lifshitz, Statistical Physics, Part I (Elsevier, Singapore, Third Edition, 2007). [24] E. Braaten and L. Platter, Physical Review Letters 100, 205301 (2008); E. Braaten, D. Kang, and L. Platter, Physical Review A 78, 053606 (2008). [25] M. Barth and W. Zwerger, Annals of Physics 326, 2544 (2011).
1807.05106v2.Spin_orbit_coupling_and_correlations_in_three_orbital_systems.pdf	Spin-orbit coupling and correlations in three-orbital systems Robert Triebl,1,Gernot J. Kraberger,1Jernej Mravlje,2and Markus Aichhorn1 1Institute of Theoretical and Computational Physics, Graz University of Technology, NAWI Graz, 8010 Graz, Austria 2Jozef Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia (Dated: November 30, 2018) We investigate the in uence of spin-orbit coupling in strongly-correlated multiorbital systems that we describe by a three-orbital Hubbard-Kanamori model on a Bethe lattice. We solve the problem at all integer llings Nwith the dynamical mean-eld theory using the continuous-time hybridization expansion Monte Carlo solver. We investigate how the quasiparticle renormalization Zvaries with the strength of spin-orbit coupling. The behavior can be understood for all llings exceptN= 2 in terms of the atomic Hamiltonian (the atomic charge gap) and the polarization in thej-basis due to spin-orbit induced changes of orbital degeneracies and the associated kinetic energy. At N= 2,increasesZat smallUbut suppresses it at large U, thus eliminating the characteristic Hund's metal tail in Z(U). We also compare the eects of the spin-orbit coupling to the eects of a tetragonal crystal eld. Although this crystal eld also lifts the orbital degeneracy, its eects are dierent, which can be understood in terms of the dierent form of the interaction Hamiltonian expressed in the respective diagonal single-particle basis. I. INTRODUCTION Strongly-correlated electronic systems with sizable spin-orbit coupling (SOC) are a subject of intense cur- rent interest. We stress a few aspects: (i) In the limit of strong interactions, the associated \spin" models are characterized by unusual exchange and are argued to lead to exotic phases such as spin-liquid ground states [1{12]. (ii) The electronic structure of layered iridate Sr 2IrO4, which features both SOC and sizable electronic repul- sion, is (at low energies) similar to the one of layered cuprates and is argued to lead to high-temperature su- perconductivity [13{20]. (iii) In Sr 2RuO 4, a compound in which the correlations are driven by the Hund's rule coupling, the SOC aects the Fermi surface [21, 22] and plays an important role in the ongoing discussion regard- ing the superconducting order parameter [23, 24]. (iv) Last, but not least, the development and improvement of multiorbital dynamical mean-eld theory (DMFT) tech- niques (also driven by the interest in multiorbital com- pounds following the discovery of superconductivity in iron-based superconductors) has lead to a detailed and to a large extent even quantitative understanding of sev- eral correlated multiorbital materials. Particular empha- sis has been put on the importance of the Hund's rule coupling for electronic correlations [25{27]. A question that is imminent in this respect is how this picture is aected by the SOC. Let us rst summarize the key results for the three- orbital models without SOC. The overall behavior was in part understood in terms of the atomic criterion, com- paring the atomic charge gap atto the kinetic energy. This criterion failed for an occupancy of N= 2, where the additional suppression of the coherence scale is im- portant [25{27]. This suppression coincides with the slowing down of the spin uctuations [28] and was ex- plained from the perspective of the impurity model that is in uenced by a reduction of the spin-spin Kondo cou-pling due to virtual uctuations to a high-spin multiplet at half lling [29{32]. The occurrence of strong corre- lations atN= 2 for moderate interactions was also in- terpreted (in the context of iron-based superconductors) as a consequence of the proximity to a half-lled (in our caseN= 3) Mott insulating state [33{36], for which the critical interaction is very small due to the Hund's rule coupling. The compounds characterized by the behavior discussed above were dubbed Hund's metals. In each case, the SOC modies all aspects of this pic- ture. First, the local Hamiltonian changes, and as a re- sult the atomic charge gap also changes. Second, the SOC reduces the ground-state degeneracy and hence the kinetic energy. Therefore, both the qualitative picture inferred from the atomic criterion, as well as quantita- tive results, can be expected to be strongly aected by the SOC. In this work, we use multiorbital DMFT to investi- gate the role of SOC in a three-orbital model with semi- circular noninteracting density of states and Kanamori interactions. We are particularly interested in the elec- tronic correlations and aim to establish the key properties that control their strength, similarly to what has been achieved for the materials without SOC in earlier works. For this purpose, we calculate the quasiparticle residue Z and investigate its behavior as a function of interaction parameters and SOC for dierent electron occupancies. We nd rich behavior, where, depending on the occu- pancy and the interaction strength, the SOC increases or suppresses Z. Partly, this is understood in terms of the in uence of the SOC on the atomic charge gap at and the associated changes of the critical interaction for the Mott transition [26]. In the Hund's metal regime, where the SOC leads to a disappearance of the charac- teristic Hund's metal tail, this criterion fails. Instead, we interpret the behavior in terms of the suppression of the half-lled Mott insulating state in the phase diagram. We discuss also the eects of the electronic correlationsarXiv:1807.05106v2 [cond-mat.str-el] 28 Nov 20182 on the SOC. Earlier DMFT work investigated some aspects of the SOC, for instance its in uence on the occurence of dif- ferent magnetic ground states at certain electron ll- ings [37{39]. Zhang et al. successfully applied DMFT to Sr 2RuO 4and pointed out an increase of the eective SOC by correlations [21], discussed also in LDA+U [40] and slave-boson/Gutzwiller approaches [41, 42]. Kim et al.also investigated Sr 2RuO 4and reconciled the Hund's metal picture with the presence of SOC in this com- pound [22, 43]. In an important work Kim et al. looked at the semicircular model [44], as in the present work but did not systematically investigate the evolution of the quasiparticle residue. The eects of the SOC were studied also with the rotationally invariant slave boson methods [45, 46]. Notably, Ref. [45] that studied a ve or- bital problem also found the disappearance of the Hund's metal tail due to the SOC. This paper is structured as follows. In Sec. II, we start by describing the model and the methods used. In Sec. III we give a qualitative discussion of the expected behavior in terms of the atomic problem. In Sec. IV we discuss the results of the DMFT calculations and put them into context of real materials. We end with our conclusions in Sec. V. In Appendix A we discuss the atomic Hamiltonian for small and large SOC, and in Appendix B we discuss the enhancement of the eects of SOC by electronic cor- relations in the large- and in the small-frequency limits. II. MODEL AND METHOD We consider a three-orbital problem with the (non- interacting) semicircular density of states () = 2 D2p D22. We use the half bandwidth Das the en- ergy unit. Such a density of states pertains to the Bethe lattice, for which the DMFT provides an exact solution. For real materials, however, this density of states, as well as the DMFT itself, is only an approximation. Neverthe- less, qualitative aspects of the results reported here can be expected to apply to real materials, see also Sec. IV F below. The eects of spin-orbit coupling are, in general, de- scribed by the one-particle operator H=ls (1) where landsare the orbital angular momentum and the spin of the respective electron. Our three-orbital model is motivated by cases where the eg-t2gcrystal-eld splitting within the dmanifold of a material is large. Therefore, one retains only the three t2gorbitalsdxy,dxz, anddyz. The matrix representations of the l= 2 operators lx,ly, andlzin the cubic basis within the t2gsubspace are up to a sign equal to the ones for the l= 1 operators in cubic basis, which is called TP correspondence [19, 47]. To be more precise, the dxyorbital corresponds to the pzorbital,dxztopy, anddyztopx. Therefore, the SOCoperator reads H=lt2gs=lps==2 (j2 el2 ps2);(2) where lpare the generators of the l= 1 orbital angu- lar momentum and jeis the eective total one-particle angular momentum je=lp+s. In order to keep the no- tation light, we will drop the index \e" in the following, and denote the total one-electron angular momentum by j. With the eigenvalues lp= 1 ands= 1=2 (h= 1),j can be 1=2 or 3=2 andmj=j;j+ 1;:::; +j. The eigenvalues of Hare thus=2 forj= 3=2 andfor j= 1=2, leading to a spin-orbit splitting of3 2. Note that in contrast to porbitals, the j= 3=2 band is lower in energy because of the minus sign in the TP correspon- dence. Therefore, the noninteracting electronic structure consists of four degenerate j= 3=2 bands and two de- generatej= 1=2 bands, the latter higher in energy. In the second-quantization formalism, the SOC Hamil- tonian reads H=X mm00hmjlt2gsjm00icy mcm00 =X mm00hmjlpjm0ihjsj0icy mcm00 =i 2X mm0m000mm0m00m00 0cy mcm00;(3) where we expressed the orbital state in the cubic t2gbasis, thuscy mcreates an electron in orbital m2fxy;xz;yzg with spin2f";#g. The matrix elements of the spin op- erators sare given by =2, where is the vector of Pauli matrices. The matrix elements of of the components of the orbital angular momentum operator are in case of theporbitalshmjlk pjm0i=ikmm0, wherek;m;m02 fx;y;zg. In case of t2gorbitals, this notation takes use of the TP correspondence fx;y;zgb=fyz;xz;xyg. The atomic interaction is described in terms of the Kanamori Hamiltonian, which reads in the second quan- tization formalism HI=X mUnm"nm#+U0X m6=m0nm"nm0# + (U0JH)X m<m0;nmnm0 +JHX m6=m0cy m"cy m0#cm#cm0" +JHX m6=m0cy m"cy m#cm0#cm0":(4) We setU0=U2JHto make the Hamiltonian ro- tationally invariant in orbital space. One can express HIin terms of the total number of electrons N=P mnm, the total spin S=P mP 0cy ms0cm0, and the total orbital isospin Lwith components Lk=3 P mm0hmjlk pjm0icy mcm0, HI= (U3JH)N(N1) 2+5 2JHN 2JHS2JH 2L2:(5) In thet2gbasis, again the generators of the por- bitals and the TP correspondence are used. The rst two Hund's rules are manifest in this form. The full problem is solved by the DMFT [48, 49], where the Hamiltonian is mapped self-consistently to an Ander- son impurity model. This impurity problem is solved by the continuous-time quantum Monte Carlo hybridization expansion method [50]. We performed the calculations using the TRIQS package [51, 52]. In the j-basis, which is dened to diagonalize the local Hamiltonian H, also the hybridization is diagonal, hence one can use real- valued imaginary-time Green's functions for the calcula- tions. This is convenient because it reduces the fermionic sign problem and makes the calculations feasible [37, 44]. However, the sign problem still remains a limiting factor for large Hund's couplings and small temperatures. All results reported in this paper were calculated at an in- verse temperature D= 80. All calculations are done in the paramagnetic state, as we focus on the eect of the SOC in the corre- lated metallic regime. Note that dierent kinds of in- sulating states occur because antiferromagnetic and ex- citonic order parameters do not vanish in some parameter regimes [9, 37, 38, 53{55]. III. CRYSTAL FIELD ANALOGY AND THE ATOMIC PROBLEM The ground-state energies and the atomic charge gaps for a Kanamori Hamiltonian with spin-orbit coupling have been already analyzed in the supplementary mate- rial of Ref. [44]. Here, we brie y recapitulate certain lim- its and compare them to the case of a tetragonal crystal- eld splitting. The SOC lowers the energy of the j= 3=2 bands by=2 and increases the energy of the j= 1=2 orbitals by . Therefore, the crystal-eld splitting pa- rameter cfis chosen such that it increases the on-site energy of one orbital by cfand that it lowers the en- ergy of the other two by cf=2 in accordance with the eect of. Physically, this crystal eld corresponds to a tetragonal tensile distortion in the zdirection. Both and cfare supposed to be positive; a negative sign would correspond to a particle-hole transformation. In Fig. 1 we illustrate the eects of the SOC and the tetrag- onal crystal eld on the energy levels and also include a real-space representation of the respective orbitals. Al- though the SOC and the considered crystal eld give an identical splitting of the single-electron energy levels, the corresponding orbitals and hence also the corresponding matrix elements are dierent, which has important con- sequences as discussed below. 3/2 ∆ cf 3/2λdxy dxz dyzj= 1/2 j= 3/2 HI=Unxz↑nxz↓+...HI=Unxy↑nxy↓HI=/parenleftbig U−4 3JH/parenrightbig n1 2,1 2n1 2,−1 2 HI=(U−JH)n3 2,3 2n3 2,−3 2+...crystal ﬁeld splitting spin-orbit coupling spin up spin down FIG. 1. Energy levels of the considered models. For both SOC and tetragonal crystal-eld splitting, the orbitally three- fold degenerate t2glevel splits into a twodfold degenerate and a onefold degenerate level. Each level has an additional spin degeneracy. In case of the crystal eld, the dxyorbital is higher in energy, whereas it is the j= 1=2 orbital in case of SOC. The respective orbitals are plotted left (crystal eld) and right (SOC) of the energy levels. The color denotes the spin. The fact that the interaction matrix elements in the j basis dier from the ones in the cubic t2gbasis is also indi- cated in the gure. Before discussing the issue of the interactions, let us brie y discuss the noninteracting case. Since both SOC and tetragonal crystal eld lift the orbital degeneracy, they change the kinetic energy in the system. Without in- teractions where SOC and crystal eld are equivalent, the kinetic energy can be readily calculated from the semi- circular density of states, EK=R ()f()d, withf() the Fermi function at T= 0. The SOC suppresses the noninteracting kinetic energy. In the large- limit we nd EK(0)=EK(!1 ) to be 1.13, 1.34, 1.96, and 1.73 for the casesN= 1, 2, 3, and 5, respectively (for N= 4, the large-limit corresponds to a band insulator with a van- ishing kinetic energy). The reduction of kinetic energy due to the SOC was discussed in the case of the N= 3 compound NaOsO 3, where even a somewhat larger re- duction of 2.3 was found in a realistic density-functional simulation [56]. We now turn to the atomic problem with interactions. It is instructive to rewrite the Kanamori Hamiltonian to thejbasis, HI=X abcdUabcdcy acy bcdcc=X ~U dy dy dd (6) with ~U =X abcdUabcdA aA bA cAd; (7) whereAis the unitary transformation between the cubic t2gand thejbasis [57]. The Latin indices are combined indices of orbital and spin; the Greek indices are com- bined indices of jandmj. As the Kanamori Hamilto- nian is invariant under this transformation for JH= 04 [seen easily from Eq. (5)], the result of the crystal-eld splitting and the SOC is identical in this case. On the other hand, for a nite Hund's coupling, the crystal eld and SOC lead to dierent results. The trans- formed Hamiltonian in the jbasis diers from its form in the cubic basis (4). We can split it into a pure j= 1=2 part, a pure j= 3=2 part, and a part that mixes the j= 1=2 andj= 3=2 parts, HI=Hj=1 2+Hj=3 2+Hmix: (8) The rst two terms read Hj=1 2= U4 3JH n1 2;1 2n1 2;1 2; (9) Hj=3 2= (UJH) n3 2;3 2n3 2;3 2+n3 2;1 2n3 2;1 2 + U7 3JH n3 2;3 2n3 2;1 2+n3 2;3 2n3 2;1 2 + U7 3JH n3 2;3 2n3 2;1 2+n3 2;3 2n3 2;1 2 +4 3JHdy 3 2;3 2dy 3 2;3 2d3 2;1 2d3 2;1 2 +4 3JHdy 3 2;1 2dy 3 2;1 2d3 2;3 2d3 2;3 2; (10) the density-density part of Hmixis Hmix, dd = U5 3JH n1 2;1 2n3 2;3 2+n1 2;1 2n3 2;3 2 + (U2JH) n1 2;1 2n3 2;1 2+n1 2;1 2n3 2;1 2 + U7 3JH n1 2;1 2n3 2;1 2+n1 2;1 2n3 2;1 2 + U8 3JH n1 2;1 2n3 2;3 2+n1 2;1 2n3 2;3 2 : The convention is that n1 2;1 2, for example, means nj=1 2;mj=1 2.Hmixcontains 30 more terms that are not shown here. Hj=1 2is a one-band Hubbard Hamiltonian with an ef- fective interaction Ue=U4=3JH. For the density- density part of Hj=3 2, one observes that the terms with the samejmjj's have prefactors UJH, whereas terms with dierentjmjj's have prefactors U7=3JH. If one usesjmjjas the orbital index and the sign of mjas the spin, the density-density part of this Hamiltonian is sim- ilar to the density-density part of a two-band Kanamori Hamiltonian, but with dierent prefactors. Importantly, there is only one kind of prefactor for interorbital interac- tions, namely U7=3JH, instead of U2JHandU3JH in Eq. (4). This in uences the electronic correlations, as we will see below in the case of N= 2. Following this interpretation of the mj's, the last two terms are pair-hopping-like expressions with an eective strengthof 4=3JH. A detailed analysis of this Hamiltonian can be found in Appendix A. It is useful to characterize the atomic Hamiltonian Hloc=HI+Hin terms of the atomic charge gap at=E0(N+ 1) +E0(N1)2E0(N); (11) whereE0(N) is the ground state of a system with Nelec- trons [27]. According to the Mott-Hubbard criterion, the metal-insulator transition takes place when atexceeds the kinetic energy. Hence, the proximity of interaction parameters to the associated critical value Uccan be used to anticipate the strength of electronic correlations. We start with a discussion of the crystal-eld split- ting [58{61]. For llings N= 1, 2, and 5, the ground state does not change with the crystal-eld splitting. For N= 3 andN= 4, there is a level crossing with a tran- sition from a high-spin to a low-spin state (e.g., from j";";"itoj"#;";0i), which is responsible for dierences in the atomic charge gap for small and large cf. The respective values for the charge gap in the limits of small and large cfare listed in Tables I and II. Note that in the large cflimit, the relevant Hamiltonian is a two- orbital one for llings N= 1;2;and 3, and a one-orbital one forN= 5. For the Kanamori Hamiltonian with orbitals, the charge gap depends on the relative lling; at half lling it is at=U+ (1)JH, otherwise U3JH. The lling N= 4 is special as an electron can only be added by paying additionally crystal-eld splitting en- ergy. We now turn to the discussion of SOC. Note that the limitsJHandJHcorrespond to the LSandjj coupling scheme, respectively. A look at Tables I and II reveals that practically all entries are dierent from the corresponding crystal-eld ones. The values for a large SOC can be obtained from the Hamiltonian expressed in thejbasis discussed above. For N= 5, where the eective model is a single-orbital model, the interaction parameter is U4 3JH, as seen from Eq. (9), in contrast to the crystal eld result, where one obtains simply U, instead. In the case of N= 2, it is interesting to note that the dependence of the charge gap on JHis dierent in sign for the SOC and the crystal eld. This follows from Eq. (10), which does not favor the alignment of the angular momenta jzof the respective orbitals (see also Appendix A). This opposite behavior is also re ected in the full DMFT solution, as we discuss below. We will see that forN= 2, there are parameter regimes, where the correlation strength increases with crystal-eld splitting, but it decreases with SOC. IV. DMFT RESULTS We now turn to the DMFT results. We focus on the interplay between the SOC and electronic correla- tions, which we follow by calculating the Matsubara self- energies. Due to the symmetry, the Green's functions5 TABLE I. Comparison of the atomic charge gap atobtained from a spin-orbit coupling or a tetragonal crystal-eld split- ting cfin the limit ;cfJH. N SOC crystal eld 1U3JH+ 1=2U3JH 2U3JH+ 1=2U3JH+ 3=2 cf 3U+ 2JH3=2U+ 2JH3=2 cf 4U3JH+ U3JH 5U3JH+U3JH+ 3=2 cf TABLE II. Comparison of the atomic charge gap atobtained from a spin-orbit coupling or a tetragonal crystal-eld split- ting cfin the limit ;cfJH. N SOC crystal eld 1U7=3JH U3JH 2UJH U+JH 3U7=3JH U3JH 4U3JH+ 3=2U5JH+ 3=2 cf 5U4=3JH U and the self-energies are diagonal in the jbasis with two independent components 1=2and 3=2. Figure 2 displays the calculated self-energies for the N= 1 case. One can see that due to the SOC jIm 3=2j is larger and its slope at low energies that determines the quasiparticle residue Z= lim i!n!0 1@Im(i!n) @i!n1 (12) is larger. The origin of that is discussed below, where we investigate the evolution of Zwithfor all integer occupancies, but let us rst discuss the other part of the interplay, namely the in uence of the electronic correla- tions on the SOC. A. In uence of electronic correlations on the SOC; eective SOC For this purpose it is convenient to introduce the av- erage self-energy a=2 33 2+1 31 2(13) and the dierence d= 1 23 2: (14) In terms of a,dthe self-energy matrix can be written in the form = a1+2 3dlt2gs; (15) 1.41.61.82.0ReΣ(a) j= 3/2 j= 1/2 λ= 0 0 1 2 3 ωn−0.6−0.5−0.4−0.3−0.2−0.10.0ImΣ (b)ﬁtj= 3/2 ﬁtj= 1/2 ﬁtλ= 0FIG. 2. Real (a) and imaginary (b) part of the self-energy for the parameters N= 1,= 0:1,U= 3, andJH= 0:1U. The green squares display the results without SOC for comparison. The lines show a polynomial t of degree four through the rst six Matsubara frequencies. which holds in any basis (see Appendix B). This form is also convenient as one can directly see that ddetermines the in uence of electronic correlations on the physics of SOC. In particular, because the Green's function is G(k;i!n) = [i!n+H0(k)(i!n)]1(16) withH0(k) the noninteracting Hamiltonian that includes the SOC, the real part of the self-energy can be used to dene the eective spin-orbit-coupling constant e=+2 3Re d(i!n!0): (17) For all cases we looked at (some data is shown in Ap- pendix B), we nd that the real part of d(i!n) is pos- itive for all !n(as long as the system is metallic) and its eect hence adds up to the bare SOC Hamiltonian so thate>, as found also in realistic studies [21, 22, 40]. Notice that there is also a further renormalization of the overall bandstructure due to the frequency dependence of the self-energy [22, 41]. The eects on the quasiparticle dispersions, for instance on the liftings of the quasipar-6 ticle degeneracies, can be phrased in terms of the quasi- particle SOC constant =Ze[22] with quasiparticle renormalization Z < 1, hencecan be smaller or larger than the bare . However, relative to the other features of the quasiparticle dispersions that are obviously renor- malized by Z, too, the SOC splittings are enhanced due to the eect of d. B. In uence of SOC on electronic correlations: One and ve electrons In the remainder of the paper we investigate how the SOC in uences the electronic correlations, which is fol- lowed by calculating the j-orbital occupations and the quasiparticle residues Z. These are calculated by tting six lowest frequency points of Matsubara self-energies to a fourth order polynomial, as shown in Fig. 2(b). Without SOC, one electron and one hole (ve elec- trons) in the system are equivalent due to the particle- hole symmetry, but the SOC breaks this symmetry. For large, only thej= 3=2 (j= 1=2) orbitals are partially occupied for N= 1 (N= 5). Hence, these are more in- teresting regarding electronic correlations. In Figs. 3(a) and 3(b), we show how the quasiparticle weights and the llings of these orbitals change when the SOC is in- creased. The corresponding atomic charge gap is also plotted, Fig. 3(c). The change in orbital polarization in uences the corre- lation strength. This is best seen for JH= 0, since then the eective repulsion is simply U, independent of the SOC. The quasiparticle weight of the relevant orbitals is reduced by the SOC as the polarization increases, which is shown in Fig. 3(b) for U= 3 (circles). The reduction is weak for N= 1 but strong for N= 5, which is due to the lower kinetic energy of one hole in one j= 1=2 orbital compared to the energy of one electron in two j= 3=2 orbitals. In the case of U= 3 andJH= 0, even a metal-insulator transition takes place. The Hund's coupling reduces the correlation strength (stars, crosses). This happens for two reasons: JHre- duces the polarization, and it decreases the atomic charge gap. The latter is expected for N= 1, where the eec- tive number of orbitals reduces with increasing from three to two. In this case, a nite exchange interaction JHleads to a reduction of the repulsion between electrons in dierent orbitals. Interestingly, JHalso decreases the strength of corre- lations for N= 5 in the limit of large , although the eective number of orbitals is one and interorbital eects are thus suppressed. However, the transformation from the cubic Kanamori Hamiltonian to its jbasis equivalent mixes inter- and intraorbital interactions, so that the ef- fectivej= 1=2 interaction strength is U4=3JH, as explained in Sec. III. In contrast, in the case of a large tetragonal crystal-eld splitting, the atomic charge gap is indeed simply given by UforN= 5. It is also interesting to compare the dependence of 0.00.20.40.60.81.0ZN= 5,j= 1/2 N= 1,j= 3/2(a)JH= 0.0U JH= 0.1UJH= 0.2U noninteracting 0.00.20.40.6n(b) 1.0 0 .5 0 0 .5 1 .0 λ1234∆at(c)FIG. 3. In uence of the spin-orbit coupling for a lling of N= 1 (right column) and N= 5 (left column) for U= 3. (a) Quasiparticle weight Zof thej= 3=2 orbitals (for N= 1) and of thej= 1=2 orbitals (for N= 5). (b) Electron density nof thej= 3=2 orbitals ( N= 1) and hole density of the j= 1=2 orbitals (N= 5) to allow for a better comparability. The green dotted line displays the respective noninteracting results. (c) Atomic charge gap at. the respective orbital occupation nwith the noninter- acting result [green dotted line in Fig. 3(b)]. One can see that the correlations increase the orbital polarization n3=2n1=2, in line of what one would expect from the enhancement of the SOC physics by electronic correla- tions discussed above. As shown below, we nd similar behavior also for other llings, but not for N= 3 when the Hund's coupling is large.7 1 2 3 4 U0.00.20.40.60.81.0Z3/2λ= 0.0 λ= 0.5 λ= 1.0 λ=∞ FIG. 4. Quasiparticle weight Z3=2of thej= 3=2 orbital as a function of UforJH= 0:1Uand a total lling of N= 3. C. Half lling In Fig. 4 we display the quasiparticle weight of the j= 3=2 orbitals (again, the j= 1=2 are emptied out with SOC and are therefore not discussed here) at N= 3 for several . One can see that strongly increases Uc and changes the behavior drastically. To understand why this occurs, rst recall that at = 0, Hund's coupling strongly reduces the kinetic energy since it enforces the high-spin ground state [25]. Hence, the Hund's coupling leads to a drastic reduction of the critical interaction strength [26]. This causes a steep descent of Zas a func- tion ofUwhen the critical Uis approached (see Fig. 4 for= 0 andJH= 0:1U). Asis large, this physics does not apply any more. The lling of the j= 3=2 orbitals increases to three electrons in two orbitals. Since the Hamiltonian of the j= 3=2 orbitals alone is particle-hole symmetric, this largelimit shows identical physics to the large limit in the case of N= 1. As described above in Sec. IV B, this!1 system is characterized by an increase of Zwith increasing JH. This is opposite to the half-lled N= 3 case at = 0, where Zdecreases with JH. In Figs. 5(a)-5(c) we show how the quasiparticle weight, the orbital polarization, and the atomic charge gap vary with , respectively. We nd that Zin- creases for physically relevant Hund's couplings (e.g., JH= 0:1U,JH= 0:2U). Furthermore, the qualitative dierence between the small and the large limits dis- cussed above results in crossings of the Z() curves for dierent Hund's couplings [see Fig. 5(a)]. These crossings are already expected from the atomic charge gap, which isU+ 2JHfor= 0 and drops to U7=3JHfor!1 , as shown in Tables I and II as well as in Fig. 5(c). The results in Fig. 5 show that SOC can strongly mod- ify the correlation strength. One needs to notice, though, that it takes a quite large for these changes to occur; for instance, full polarization is reached at 1, whereas it occurs at 0:3 in the case of N= 1 andU= 3 0.00.20.40.60.81.0Z3/2(a)JH= 0.0U JH= 0.1UJH= 0.2U noninteracting 0.40.50.60.70.8n3/2(b) 0.0 0 .5 1 .0 1 .5 2 .0 λ1.01.52.02.53.0∆at(c)FIG. 5. Quasiparticle weight Z3=2(a) and lling n3=2(b) of the electrons in the j= 3=2 orbitals as functions of forU= 2. The green dotted line displays the respective noninteracting results. (c) Atomic charge gap at. (compare Fig. 5 with Fig. 3). In this respect we notice also that in contrast to the N= 1 case, the electronic correlations increase the orbital polarization at N= 3 as compared to the noninteracting result only for small values ofJH. D. Two electrons We now discuss the interesting case of two electrons. In the absence of SOC, this is the case of a Hund's metal. Figure 6 shows the dependence of ZonUfor several values ofandJH=U= 0:2. The data at small ex- hibit a tail with small Z, which is characteristic for the8 0 1 2 3 4 5 6 U0.00.20.40.60.81.0Zλ= 0.0 λ= 0.5 λ=∞ ∆cf=∞ FIG. 6. Quasiparticle weight Zof thej= 3=2 orbital as a function of UforJH= 0:2UandN= 2. The dashed line shows the corresponding Zof thedxzorbital in the case of an innite tetragonal crystal-eld splitting. Hund's metal regime. The SOC has a drastic eect here; increasingsuppresses the Hund's metal behavior and leads to a featureless, almost linear, approach of Zto- wards 0 with increasing U. Interestingly, the in uence ofonZis opposite at small Uwhere increasing in- creasesZ, thus making the system less correlated, and at a highU, whereZdiminishes with and hence cor- relations become stronger. The latter behavior is easy to understand. A strong SOC reduces the number of relevant orbitals from three to two, and leads to the increase of the atomic charge gap fromU3JHtoUJH[see Fig. 8(c) and Sec. III]. Both the reduction of the kinetic energy due to the reduced degeneracy and the increase of the atomic charge gap withcontribute to a smaller critical U, which is indeed seen on the plot. We want to note here that the reduc- tion of the critical Uis even stronger for the crystal-eld case (shown as a dashed line in Fig. 6), since there the corresponding atomic gap is larger ( U+JH, see Sec. III). We turn now to the small- Uregime where the SOC reduces the electronic correlations. One can rationalize this from a scenario that pictures Hund's metals as doped Mott insulators at half lling [33{36]. Figure 7 presents the values of Uwhere a Mott insulator occurs. Let us rst discuss the case without SOC, i.e., the left panel of Fig. 7. In this picture of doped Mott insulators, the correlations for small interactions at N= 2 are due to proximity to a half-lled insulating state. For interac- tion parameters UandJHthat lead to a Mott insulator at half lling, doping with holes leads to a metallic state with low quasiparticle weight. This low- Zregion persists to doping concentrations of more than one hole per atom, as can be seen from Fig. 2 in Ref. [26]. As a result, for an interaction Uin between the critical values for two and three electrons Uc(N= 3)<U <U c(N= 2), the quasi- particle weight is small, but not zero. As one increases now, the critical UatN= 3 increases strongly, and 0 1 2 3 4 5 6 N0246810U (a)λ= 0 0 1 2 3 4 5 6 N0246810 (b)λ=∞FIG. 7. The Mott insulator occurs for values of Uindicated by bars for a Hund's coupling of JH= 0:2U. The left picture (a) shows the case without SOC, the right (b) with an innite SOC. Note that in the latter case no Mott insulator occurs for N= 4 since this case is a band insulator. The critical values for= 0 are taken from Ref. 26. The red crosses indicate the critical Uin the case where a tetragonal crystal eld is applied instead of the SOC. the insulating state appears only for large values of U, see the right panel of Fig. 7. Consequently, the N= 2 state cannot be viewed as a doped N= 3 Mott insulator any more. In fact, for a large SOC, the critical interaction strengthUcfor a Hund's coupling of JH=U= 0:2 is low- est forN= 2, as displayed in Fig. 7. As a consequence, the Hund's tail disappears (this was earlier noted also in a rotationally-invariant slave boson study of a ve orbital problem [45]), as highlighted in Fig. 6, and the quasipar- ticle weight increases with SOC in the case of a small Uand large Hund's couplings [see Fig. 8(a)]. In pass- ing we note that the DMFT self-consistency is essential to account for the increase of Zin the small Uregime. Calculations for an impurity model found a suppression of the Kondo temperature (and hence a suppression of Z) with increased [43], which is dierent from what we nd in the DMFT results here. Figure 8(b) shows the orbital occupancy as a function of. Like inN= 1,N= 5, and, for small enough JH, alsoN= 3, from a comparison with the noninteracting result one nds that the SOC usually leads to a larger orbital polarization when the interactions are present. Looking at the data more precisely, this ceases to hold in the large- regime. We actually nd this at other ll- ings, too. At values of where the noninteracting result is already fully polarized, the electronic correlations rein- troduce some charge in the empty/fully polarized orbital. In Fig. 8, we also compare the in uence of the SOC9 0.00.20.40.60.81.0Z(a)JH= 0.0U, spin-orbit coupling JH= 0.1U, spin-orbit coupling JH= 0.2U, spin-orbit coupling JH= 0.0U, crystal-ﬁeld splitting JH= 0.1U, crystal-ﬁeld splitting JH= 0.2U, crystal-ﬁeld splitting noninteracting 0.20.30.40.50.6n(b) 0.00 0 .25 0 .50 0 .75 1 .00 1 .25 1 .50 λ,∆cf0123∆at(c) FIG. 8. Quasiparticle weight of the electrons (a), lling (b), and the atomic charge gap (c) for N= 2 andU= 2. Solid lines correspond to the SOC case, and j= 3=2 quantities are plotted as functions of . Dashed lines are the results for a crystal-eld splitting, where we plot dxz=yz quantities as functions of cf. to that of a tetragonal crystal eld. One sees that the crystal eld always increases the correlation strength. To understand this it is convenient to recall that the atomic gaps are dierent, and as a result, also the critical U's are dierent. For an innite crystal eld, they are marked with crosses in Fig. 7(b). In particular, the critical inter- action atN= 2 in the case of an innite crystal eld is only slightly larger than the critical interaction at N= 3 without any splitting. Therefore, Hund's metals with in- teractions in the range Uc(N= 3)< U < U c(N= 2), becomes insulating, as the interaction driven Mott tran- sition atN= 2 is pushed to such small values of Uc 0 1 2 3 4 5 6 U0.00.20.40.60.81.0Z3/2 JH= 0.0U JH= 0.1U JH= 0.2U JH= 0.3U−2.5 0.0 2 .5 ω0.000.250.50AFIG. 9. Quasiparticle weight Z3=2of thej= 3=2 orbital as a function of Ufor!1 and a total lling of N= 2. The inset shows the respective impurity spectral functions forU= 3 andJH= 0 (blue) and JH= 0:2U(black). As the Hund's coupling JHincreases, the quasiparticle weight (= area of the quasiparticle peak) stays the same, whereas the position of the Hubbard bands changes due to dierent charge gaps. To obtain the spectral functions, imaginary-time data has been analytically continued using a maximum entropy method [62] with an alternative evidence approximation [63] and the preblur formalism [64]. by the large cf. Another dierence is the ground state degeneracy, which is three for the S= 1 ground state of the two-orbital Kanamori and ve in the case of the J= 2 ground state of Hj=3=2, see Appendix A, which also points to weaker correlations in the SOC case. Another interesting observation from Fig. 8(a) is that the quasiparticle weight is almost independent of Hund's coupling in the limit of large forU= 2. In Fig. 9, we show that the weak dependence on JHis also apparent for other values of U, and only becomes signicant when the Hund's coupling is exceeding JH>0:2U. However, since the atomic gap does depend on JH, the position of the Hubbard bands are dierent, even though Zis the same, as shown in the inset of Fig. 9. E. Four electrons The lling of four electrons is special because strong SOC leads to a band insulator with fully occupied j= 3=2 orbitals and empty j= 1=2 orbitals, with no renor- malization ( Z= 1) for both orbitals in the large regime. Figure 10(a) shows the quasiparticle renormalization of both orbitals in the metallic phase as a function of . One can see that Z3=2is hardly aected, and Z1=2increases only slightly for the given parameters U= 2 andJH= 0:2U, indicating that the orbital polarization aects only weakly the correlation strength, unless in close vicinity to the metal-insulator transition. A comparison to the crystal-eld results shows two ma-10 jor dierences: First, the orbital polarization, displayed in Fig. 10(b), is smaller in the case of the crystal eld, as compared to the SOC case, and a larger value of crystal- eld splitting is needed to reach a band insulator. The reason for this is a larger atomic gap in the SOC case [see Fig. 10(c) and Tables I and II]. Second, the quasi- particle renormalization of the less occupied (in the case of crystal eld dxy) orbital is lowest when its lling is around 1/2. This enhancement of correlation eects at half lling is absent for the j= 1=2 orbital. F. Discussion It is interesting to discuss our results in the context of real materials and to consider which parameter regimes are realized (see also Refs. 19 and 44). One can rst recall the atomic values for the SOC that roughly increase with the fourth power of the atomic number. It takes small values in 3 d(Mn: 0:04 eV, Co: 0 :07 eV), intermedi- ate values in 4 d(Ru: 0:13 eV, Rh: 0 :16 eV), and reaches considerable strength in 5 d(Os: 0:42 eV, Ir: 0 :48 eV) atoms [65]. These atomic values are representative also for the values of SOC found in corresponding oxides. Regarding interaction parameters, one can roughly take thatJH=U= 0:1 and values of Uthat diminish from 4 eV(in 3d), 3 eV(4d), 2 eV(5d). Finally, the bandwidth will vary from case to case, since it depends the most on structural details among all the microscopic parameters. As a rule of thumb, however, it increases with the prin- ciple quantum number, giving values of half bandwidth fromD=1 eV(3d), 1:5 eV(4d), 2 eV(5d). These all are of course only rough estimates, meant to indicate trends. The clear-cut case with strong in uence of SOC are 5 d oxides atN= 5. In iridates, =D ranges from 0.26 in Sr2IrO4up to 2.0 in Na 2IrO3due to the small bandwidth in this compound [44]. Inspecting now Fig. 3, one sees that the SOC leads to a strong orbital polarization and strongly aects the correlations at those values of =D. Actually, the sensitivity to SOC at N= 5 is so strong that one can expect signicant impact also in 4 d5com- pounds, like rhodates, too, although is by a factor of three smaller there. Indeed, the enhancement of corre- lations has been observed in a material-realistic DMFT study of Sr 2RhO 4[18, 19]. Rather small SOC leads also to a large polarization in the particle-hole transformed counterpart N= 1 (with potentially important conse- quences for the magnetic ordering [66]), but the increase of the quasiparticle renormalization is weak, see Fig. 3(a). Opposite to the N= 1 andN= 5 cases, the SOC at N= 3 makes the electronic correlations weaker. Also in contrast to the former two cases, the eect of SOC on polarization and quasiparticle renormalization becomes pronounced only at larger values of . From Fig. 5(b) we can infer that for full polarization =D > 0:5 is nec- essary. Large values of =D can be obtained in dou- ble perovskites based on 5 delements. In Sr 2ScOsO 6, for instance, quite a substantial reduction of correlations 0.00.20.40.60.81.0Z(a)j= 3/2, spin-orbit coupling j= 1/2, spin-orbit coupling dxz,dyz, crystal-ﬁeld splitting dxy, crystal-ﬁeld splitting noninteracting 0.00.20.40.60.81.0n(b) 0.00 0 .25 0 .50 0 .75 1 .00 1 .25 1 .50 λ,∆cf0123∆at(c)FIG. 10. Quasiparticle renormalization (a), lling (b), and atomic charge gap (c) of the orbitals as functions of spin-orbit coupling (full lines) and crystal-eld splitting (dashed lines) forN= 4,U= 2,JH= 0:2U. Full dots indicate insulating phases. In the case of SOC, all calculations with 0:7 are insulating, whereas in the case of a crystaleld only the last point shown ( cf= 1:5) is insulating. The green dotted lines shows the orbital llings in the noninteracting case. Then, crystal eld and SOC are equivalent. occurs with SOC [67]. In case of the single perovskite NaOsO 3, the SOC modies the band structure [68] too, which leads to an important suppression of kinetic en- ergy [56], as discussed also in Sec. III. In the case of 4 d elements, typically =D < 0:2; therefore we expect only small eects of the SOC on the correlation strength in these materials. For the lling N= 2, we show in Fig. 6(a) a system- atic suppression of the Janus-faced behavior with SOC,11 making the Hund's tail disappear. This eect is already sizable for =D0:5 and should, hence, be present in many 5dsystems. Indeed, it has been seen in calculations for the 5d2compound Sr 2MgOsO 6[67]. For a smaller SOC of=D0:1, which is a good estimate for many 4 d materials, we do not nd a substantial change of Z[see, for example, Fig. 8(a)]. Therefore, we think the SOC only weakly aects the correlation strength in materials with 4d2conguration, such as Sr 2MoO 4[69{71]. ForN= 4, our model calculations predict that the SOC aects the correlation strength only a little, pro- vided it is small enough such that the system remains in the metallic phase. If it exceeds a certain magnitude, though, a metal-insulator transition occurs. The critical decreases with increasing U. Examples for this behav- ior are on one hand Sr 2RuO 4(= 0:10 eV), where the quasiparticle renormalization hardly changes as the SOC is turned on [22], and, on the other hand, NaIrO 3(= 0:33 eV), where the interplay of SOC and Uleads to an insulating state [72]. V. CONCLUSION In this paper we investigated the in uence of the SOC on the quasiparticle renormalization Zin a three-orbital model on a Bethe lattice within DMFT. Depending on the lling of the orbitals (and for N= 2 also the inter- action strength), the SOC can decrease or increase the strength of correlations. The behavior can be understood in terms of the SOC-induced changes of the eective de- generacy, the llings of the relevant orbitals, and the in- teraction matrix elements in the low-energy subspace. The spin-orbital polarization leads to an increase of the correlation strength for N= 1 and 5, with particularly strong eect for N= 5, where a half-lled single-band problem is realized, relevant for iridate compounds. For the nominally half-lled case N= 3, the opposite trend is observed. Here, turning on SOC makes the system less correlated, and the critical interaction strength Uc for a Mott transition is increased. For the N= 2 Hund's metallic phase, the in uence of SOC is more involved. We nd that there are two regimes as a function of U with opposite eect of SOC. For small U, the inclusion of SOC increases Z, whereas for large Uit decreases Z, and in turn also the critical interaction Ucdecreases. As a result, the so-called Hund's tail with small quasiparticle renormalization for a large region of interaction values, disappears. We also considered the eects of the electronic cor- relations on SOC and found that in the cases where the system remains metallic, correlations always enhance the eective SOC.ACKNOWLEDGMENTS We thank Michele Fabrizio, Antoine Georges, Alen Horvat, Minjae Kim, Andrew Millis, and Hugo Strand for helpful discussion. We acknowledge nancial support from the Austrian Science Fund FWF, START program Y746. Calculations have been performed on the Vienna Scientic Cluster. J.M. acknowledges the support of the Slovenian Research Agency (ARRS) under Program P1- 0044. Appendix A: Atomic Hamiltonian in the limit of small and large spin-orbit couplings The full local Hamiltonian reads [see also Eq. (5)] Hloc=HI+H+H = (U3JH)N(N1) 2+5 2JH+ N 2JHS2JH 2L2+lt2gs;(A1) with an SOC and an on-site energy . Note that this Hamiltonian contains both two-particle terms like N2, L2, and S2, as well as one-particle terms like Nandlt2g s. For= 0, the total spin Sand the total orbital angular momentum Lare good quantum numbers and determine together with the total number of electrons Nthe eigenenergies. As is nite, the energy levels split according to their total angular momentum J. For example, the nine-fold degenerate S= 1,L= 1 ground state in the N= 2 sector splits into a J= 2, aJ= 1, and aJ= 0 sector. The respective degeneracies are 2 J+ 1. The total angular momentum Jis for all values of a good quantum number, in contrast to the total spin S and the total orbital angular momentum L. For a small SOC ( JH), one can use rst-order per- turbation theory in order to calculate the level splitting due to the SOC. In this approximation, the spin-orbit term is approximated by CLS. The constant Cde- pends on the number of electrons and is C= 1;1=2 for one and two electrons, and C=1;1=2 for one and two holes. For three electrons, L= 0, and the rst-order perturbation theory gives no energy correction. Since the total angular momentum is approximated by J=L+S, this regime is known as LScoupling regime. In the limit of large SOC ( JH), the spin-orbit term is the dominant term that is solved exactly, whereas S2 andL2may be treated perturbatively. The many-body eigenstates of the unperturbed system are then the Slater determinants of j= 1=2 andj= 3=2 one-electron states. Following Eq. (2), the matrix elements of lt2gsdepend in this unperturbed eigenbasis only on the number of elec- trons in the j= 3=2 and thej= 1=2 orbitals. The total angular momentum is J=P iji, therefore, this regime is thejjcoupling regime. For llings N4, only the j= 3=2 orbitals are occupied in the ground state. The12 TABLE III. Eigenenergies of the Hamiltonian Hj=3 2of the j= 3=2 orbitals, Eq. (10). NJEj=3=2 00 0 13/2 222+U7=3JH 202+U+ 1=3JH 33/23+ 3U17=3JH 404+ 6U34=3JH TABLE IV. Full list of quantum numbers and eigenenergies in the two-particle sector of a two-orbital system. We compare energiesEegof the ordinary Kanamori Hamiltonian for eg orbitals with energies Ej=3=2for the eective j= 3=2 Hamil- tonian stemming from a large SOC in t2gorbitals. NTTy~S~SzEegEj=3=2 2001-1U3JHU7=3JH 20010U3JHU7=3JH 20011U3JHU7=3JH 21-100UJHU7=3JH 21000U+JHU+ 1=3JH 21100UJHU7=3JH spin-orbit term is then proportional to the particle num- berNand can be absorbed in the one-electron energy . Calculating the matrix elements of S2andL2for Slater determinants with dierent NandJusing Clebsch- Gordan coecients, one can nd the eigenenergies of the Hamiltonian in the jjcoupling regime. This approach is equivalent to looking for the eigenvalues of Hj=3 2pre- sented in Eq. (10) in the main text, where all contribu- tions of the j= 1=2 orbitals are neglected. The eigenen- ergies ofHj=3 2, including an on-site energy , are shown in Table III. It is possible to bring the Hamiltonian Hj=3 2into a more symmetric form if one assigns the absolute value of mjas orbitals and its sign as spin, e.g., d3 2;1 27!c1"and d3 2;3 27!c2#. It reads then Hj=3 2= U5 3JHN(N1) 21 3JHN +4 3JH T22T2 y(A2) with a total spin ~S=1 2X mX 0cy m0cm0 (A3) and the two-orbital isospin T=1 2X X mm0cy mmm0cm0 (A4)Note that ~Sis not a physical spin, since it stems from mapping the sign of mjto an articial spin. Hamiltonian (A2) has the structure of a generalized Kanamori Hamiltonian, where the spin- ip and pair- hopping parameters JSFandJPHare not restricted to be equal to the Hund's coupling JHas in the ordinary Kanamori Hamiltonian (4). In terms of Tand ~S, the generalized Kanamori Hamiltonian reads [27] HGK= (U+U0JH+JSF)N(N1) 4 (UU0JH+ 3JSF)N 4 + (JSF+JPH)T2 x+ (JSFJPH)T2 y + (UU0)T2 z+ (JSFJH)~S2 z:(A5) In order that Hj=3 2ts into the structure of the gener- alized Hamiltonian, one has to replace the parameters of HGKbyU7!UJH,JH7!0,JSF7!0,JPH7!4 3JH, andU07!U7 3JH. Hamiltonian (A5) with the parameters of the usual Kanamori Hamiltonian, U0=U2JH,JSF=JPH= JH, is the symmetric form of the two-band Hamiltonian describingegbands [27] Heg= (UJH)N(N1) 2JHN + 2JH T2T2 y :(A6) WhileHj=3 2is the Hamiltonian relevant for the two j= 3=2 orbitals of a three orbital system with innite SOC,Hegis its counterpart describing the dxzanddxy orbitals when the tetragonal crystal-eld splitting is in- nite. The dierence between these two operators is thus responsible for the qualitative dierent behavior of crys- tal eld and SOC in the N= 2 case (see Sec. IV D). The operators (A2) and (A6) are of similar form, but have dierent prefactors. A complete set of commuting operators for both Hamiltonians is N,T2,Ty,~S2, and ~Sz. The full list of quantum numbers and the eigenenergies of the two oper- ators are shown in Table IV for N= 2. For the j= 3=2 orbitals, one sees that due to the prefactors, the ~S= 1 ground state is degenerate with two ~S= 0 states. This is related to the fact that spin- ip and Hund's coupling terms vanish in the related generalized Kanamori Hamil- tonian so that the relative orientation of pseudo-spins of two electrons in dierent orbitals has no in uence on the energy. The physical reason for this is that all ve states belong to the J= 2 ground state manifold that is found in the picture of jjcoupling and therefore have to be degenerate. As a consequence, charge uctuations to dif- ferent values of pseudospin ~Sare still possible for large Hund's couplings, in contrast to an ordinary Kanamori Hamiltonian, where JHsplits energy levels of dierent spins.13 Appendix B: Eective spin-orbit coupling The SOC (2) leads to o-diagonal elements in the non- interacting Hamiltonian in the cubic basis. If both inter- actions and SOC are present, the self-energy will have o- diagonal elements as well, changing the eective strength eof the SOC. The structure of the o-diagonal elements can be un- derstood in the case of our degenerate three-orbital model system using simple analytical considerations. In the j basis, both the local Hamiltonian and the hybridization function are diagonal, hence is diagonal as well, with dierent values for the j= 3=2 and thej= 1=2 orbitals. This diagonal matrix can be split into a term propor- tional to the unit matrix and a term proportional to the matrix representation of the lt2gsoperator, which is diagonal in the jbasis with elements 0:5 in the case of j= 3=2 and 1 in the case of j= 1=2. Therefore, = a1+2 3dlt2gs; (B1) with an average self-energy a=2 33 2+1 31 2(B2) and the dierence d= 1 23 2: (B3) The eective SOC can be dened as e=+2 3Re d(i!n!0): (B4) In the cubic basis, the diagonal elements of the self- energy are given by a, the o-diagonal elements up to a phase by 2 =3 d. Let us have a look now at the frequency dependence of the self-energy. For large frequencies, the values of d are given by the Hartree-Fock values. Using Eq. (8), the Hartree-Fock values in the jbasis are HF 1 2=* @HI @n 1 2;1 2+ = U4 3JH n1 2(B5) + 4U26 3JH n3 2(B6) HF 3 2=* @HI @n 3 2;3 2+ = 2U13 3JH n1 2(B7) + 3U17 3JH n3 2; (B8) hence d(!!1 ) = HF d= (U3JH) n3 2n1 2 :(B9) The eective SOC for large frequencies is therefore deter- mined by an eective correlation strength U3JHand 0 1 2 3 4 5 ωn−0.050.000.050.100.150.20Σd(a)ReΣd,JH=0.2U ImΣd,JH=0.2U 0 1 2 3 4 5 ωn−0.050.000.050.100.150.20Σd(b) ReΣd,JH=0.1U ImΣd,JH=0.1U 0.00 0 .05 0 .10 0 .15 0 .20 0 .25 0 .30 JH/U0.00.10.20.30.4ReΣd(c) ΣHF d Σd(iω0)FIG. 11. Dierence of the self-energies d= 1 23 2for N= 4,= 0:1, andU= 2. Subplots (a) and (b) show das a function of Matsubara frequencies !nfor Hund's couplings JH= 0:2UandJH= 0:1U, respectively. The dashed lines are the corresponding Hartree-Fock values. Subplot (c) shows Re d(i!0)Re d(i!n!0) (full line) and the Hartree-Fock values HF dequivalent to d(i!n!1 ) (dashed) as a function ofJH. While the Hartree-Fock value strongly decreases with JH, d(i!0) is hardly in uenced. the orbital polarization. Since the j= 3=2 orbital is lower in energy, its occupation is higher, and HF dis always pos- itive as long as the eective interaction is repulsive. As a consequence, the correlations usually enhance the SOC at large frequencies. At low frequencies and temperatures, assuming a metal, the values of are related to electronic occupan- cies, too. Namely, j= 1=2 andj= 3=2 problems are independent and the corresponding Fermi surface must,14 0.0 0 .2 0 .4 0 .6 0 .8 1 .0 λ0.00.10.20.30.40.50.60.70.8ReΣd(iω0)N= 1 N= 2 N= 3 N= 4 N= 5 FIG. 12. Increase of the rst Matsubara self-energy d(i!0)d(!= 0) with the SOC for U= 2,JH= 0:1U, and all integer llings. For N= 3 and<0:3, the system is a Mott insulator, and for N= 4 and>0:3 a band insulator. The data points are not shown for these parameters. by Luttinger theorem, contain the correct number of elec- trons. At the Fermi surface, +kRe = 0, which can be used to relate the dierence of kto the dierence of . Assuming that the electronic density of states is a constantindependent of energy (square shaped func- tion), the result is d(0) = 1= n3=2n1=2 3=2. In general, d(0) depends on the density of states, the SOC, and the orbital polarization, but not explicitly onthe interaction parameters UandJH. Since the Hartree- Fock value does depend on the interaction parameters, the large frequency and small frequency values of dcan be quite dierent, as shown in Fig. 11. In contrast to the Hartree-Fock value valid at large frequencies, d(!= 0) cannot be given in a closed form. However, for all metal- lic solutions we veried numerically that d(i!0) is pos- itive, hence the eective SOC is also increased for low frequencies [41]. The results for U= 2,JH= 0:1Uare shown in Fig. 12. In the case of Sr 2RuO 4, the DMFT work of Ref. [22] and Ref. [21] found that the real part of dwas to a good approximation a constant and the imaginary part nearly vanishing, which motivated the introduction of e. We reproduce this result in a DMFT calculation with param- etersN= 4,U= 2,JH= 0:2U, and= 0:1, which cor- respond approximately to the values in Sr 2RuO 4. How- ever, if the parameters are changed, for example to a Hund's coupling of JH= 0:1U, the o-diagonal elements of start to show a more pronounced frequency depen- dence, as shown in Fig. 11. The reason for this is the strong direct dependence of eon the interaction param- eters in the Hartree-Fock limit, which is not present at low frequencies. In Fig. 11(c), one sees that the Hartree- Fock value strongly decreases with the Hund's coupling, whereas the static value at != 0 only changes slightly. The stronger frequency dependence of dimplies that the accuracy of describing the eects of correlations on the SOC physics in terms of eis in general restricted to low energies only. robert.triebl@tugraz.at 1M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys. 70, 1039 (1998). 2Y. Zhou, K. Kanoda, and T.-K. Ng, Rev. Mod. Phys. 89, 025003 (2017). 3Z. Nussinov and J. van den Brink, Rev. Mod. Phys. 87, 1 (2015). 4G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 102, 017205 (2009). 5J. c. v. Chaloupka, G. Jackeli, and G. Khaliullin, Phys. Rev. Lett. 105, 027204 (2010). 6G. Chen, R. Pereira, and L. Balents, Phys. Rev. B 82, 174440 (2010). 7G. Chen and L. Balents, Phys. Rev. B 84, 094420 (2011). 8O. N. Meetei, W. S. Cole, M. Randeria, and N. Trivedi, Phys. Rev. B 91, 054412 (2015). 9G. Khaliullin, Phys. Rev. Lett. 111, 197201 (2013). 10J. c. v. Chaloupka and G. Khaliullin, Phys. Rev. B 92, 024413 (2015). 11K. W. Plumb, J. P. Clancy, L. J. Sandilands, V. V. Shankar, Y. F. Hu, K. S. Burch, H.-Y. Kee, and Y.-J. Kim, Phys. Rev. B 90, 041112 (2014). 12S. M. Winter, Y. Li, H. O. Jeschke, and R. Valent , Phys. Rev. B 93, 214431 (2016). 13B. J. Kim, H. Ohsumi, T. Komesu, S. Sakai, T. Morita, H. Takagi, and T. Arima, Science 323, 1329 (2009). 14F. Wang and T. Senthil, Phys. Rev. Lett. 106, 136402(2011). 15H. Zhang, K. Haule, and D. Vanderbilt, Phys. Rev. Lett. 111, 246402 (2013). 16Z. Y. Meng, Y. B. Kim, and H.-Y. Kee, Phys. Rev. Lett. 113, 177003 (2014). 17J. c. v. Chaloupka and G. Khaliullin, Phys. Rev. Lett. 116, 017203 (2016). 18C. Martins, M. Aichhorn, L. Vaugier, and S. Biermann, Phys. Rev. Lett. 107, 266404 (2011). 19C. Martins, M. Aichhorn, and S. Biermann, Journal of Physics: Condensed Matter 29, 263001 (2017). 20C. Martins, B. Lenz, L. Perfetti, V. Brouet, F. m. c. Bertran, and S. Biermann, Phys. Rev. Materials 2, 032001 (2018). 21G. Zhang, E. Gorelov, E. Sarvestani, and E. Pavarini, Phys. Rev. Lett. 116, 106402 (2016). 22M. Kim, J. Mravlje, M. Ferrero, O. Parcollet, and A. Georges, Phys. Rev. Lett. 120, 126401 (2018). 23B. Kim, S. Khmelevskyi, I. Mazin, D. Agterberg, and C. Franchini, npj Quantum Materials 2, 37 (2017). 24A. P. Mackenzie, T. Scadi, C. W. Hicks, and Y. Maeno, npj Quantum Materials 2, 40 (2017). 25K. Haule and G. Kotliar, New J. Phys. 11, 025021 (2009). 26L. de Medici, J. Mravlje, and A. Georges, Physical Review Letters 107, 256401 (2011). 27A. Georges, L. d. Medici, and J. Mravlje, Annual Review of Condensed Matter Physics 4, 137 (2013).15 28P. Werner, E. Gull, M. Troyer, and A. J. Millis, Phys. Rev. Lett. 101, 166405 (2008). 29Z. P. Yin, K. Haule, and G. Kotliar, Phys. Rev. B 86, 195141 (2012). 30C. Aron and G. Kotliar, Phys. Rev. B 91, 041110 (2015). 31K. M. Stadler, Z. P. Yin, J. von Delft, G. Kotliar, and A. Weichselbaum, Phys. Rev. Lett. 115, 136401 (2015). 32A. Horvat, R. Zitko, and J. Mravlje, Phys. Rev. B 94, 165140 (2016). 33H. Ishida and A. Liebsch, Phys. Rev. B 81, 054513 (2010). 34T. Misawa, K. Nakamura, and M. Imada, Phys. Rev. Lett. 108, 177007 (2012). 35L. de' Medici, G. Giovannetti, and M. Capone, Phys. Rev. Lett. 112, 177001 (2014). 36L. Fanfarillo and E. Bascones, Phys. Rev. B 92, 075136 (2015). 37T. Sato, T. Shirakawa, and S. Yunoki, Phys. Rev. B 91, 125122 (2015). 38T. Sato, T. Shirakawa, and S. Yunoki, arXiv preprint arXiv:1603.01800 (2016). 39M. Kim, Phys. Rev. B 97, 155141 (2018). 40G.-Q. Liu, V. N. Antonov, O. Jepsen, and O. K. Ander- sen., Phys. Rev. Lett. 101, 026408 (2008). 41J. B unemann, T. Linneweber, U. L ow, F. B. Anders, and F. Gebhard, Phys. Rev. B 94, 035116 (2016). 42M. Behrmann, C. Piefke, and F. Lechermann, Phys. Rev. B86, 045130 (2012). 43A. Horvat, R. Zitko, and J. Mravlje, Phys. Rev. B 96, 085122 (2017). 44A. J. Kim, H. O. Jeschke, P. Werner, and R. Valent , Phys. Rev. Lett. 118, 086401 (2017). 45C. Piefke and F. Lechermann, Phys. Rev. B 97, 125154 (2018). 46J. I. Facio, J. Mravlje, L. Pourovskii, P. S. Cornaglia, and V. Vildosola, Phys. Rev. B 98, 085121 (2018). 47S. Sugano, Y. Tanabe, and H. Kamimura, Multiplets of transition-metal ions in crystals , Pure and Applied Physics, Vol. 33 (Academic Press, New York, 1970). 48A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Rev. Mod. Phys. 68, 13 (1996). 49W. Metzner and D. Vollhardt, Phys. Rev. Lett. 62, 324 (1989). 50P. Werner and A. J. Millis, Phys. Rev. B 74, 155107 (2006). 51O. Parcollet, M. Ferrero, T. Ayral, H. Hafermann, I. Krivenko, L. Messio, and P. Seth, Computer Physics Communications 196, 398 (2015). 52P. Seth, I. Krivenko, M. Ferrero, and O. Parcollet, Com- put. Phys. Commun. 200, 274 (2016). 53N. Kaushal, J. Herbrych, A. Nocera, G. Alvarez, A. Moreo, F. A. Reboredo, and E. Dagotto, Phys. Rev. B 96, 155111(2017). 54J. Kunes, Journal of Physics: Condensed Matter 27, 333201 (2015). 55A. Akbari and G. Khaliullin, Phys. Rev. B 90, 035137 (2014). 56B. Kim, P. Liu, Z. Erg onenc, A. Toschi, S. Khmelevskyi, and C. Franchini, Phys. Rev. B 94, 241113 (2016). 57L. Du, L. Huang, and X. Dai, The European Physical Journal B 86, 94 (2013). 58P. Werner and A. J. Millis, Physical Review Letters 99, 126405 (2007). 59P. Werner, E. Gull, and A. J. Millis, Physical Review B 79, 115119 (2009). 60T. Kita, T. Ohashi, and N. Kawakami, Physical Review B84, 195130 (2011). 61T. Kita, T. Ohashi, and N. Kawakami, Journal of the Physical Society of Japan 80, SA142 (2011). 62M. Jarrell and J. Gubernatis, Physics Reports 269, 133 (1996). 63W. von der Linden, R. Preuss, and V. Dose, \The Prior- Predictive Value: A Paradigm of Nasty Multi-Dimensional Integrals," in Maximum Entropy and Bayesian Methods , edited by W. von der Linden, V. Dose, R. Fischer, and R. Preuss (Kluwer Academic Publishers, Dortrecht, 1999) pp. 319{326. 64J. Skilling, \Fundamentals of MaxEnt in data analysis," in Maximum Entropy in Action , edited by B. Buck and V. A. Macaulay (Clarendon Press, Oxford, 1991) pp. 19{40. 65M. Montalti, A. Credi, L. Prodi, and M. T. Gandol, Handbook of Photochemistry , 3rd ed. (CRC press, 2006). 66K.-W. Lee and W. E. Pickett, EPL (Europhysics Letters) 80, 37008 (2007). 67G. Giovannetti, arXiv preprint arXiv:1611.06482 (2016). 68Y. G. Shi, Y. F. Guo, S. Yu, M. Arai, A. A. Belik, A. Sato, K. Yamaura, E. Takayama-Muromachi, H. F. Tian, H. X. Yang, J. Q. Li, T. Varga, J. F. Mitchell, and S. Okamoto, Phys. Rev. B 80, 161104 (2009). 69S.-I. Ikeda, N. Shirakawa, H. Bando, and Y. Ootuka, Jour- nal of the Physical Society of Japan 69, 3162 (2000). 70I. Nagai, N. Shirakawa, S.-i. Ikeda, R. Iwasaki, H. Nishimura, and M. Kosaka, Applied Physics Letters 87, 024105 (2005). 71H. Wadati, J. Mravlje, K. Yoshimatsu, H. Kumigashira, M. Oshima, T. Sugiyama, E. Ikenaga, A. Fujimori, A. Georges, A. Radetinac, K. S. Takahashi, M. Kawasaki, and Y. Tokura, Phys. Rev. B 90, 205131 (2014). 72L. Du, X. Sheng, H. Weng, and X. Dai, EPL (Europhysics Letters) 101, 27003 (2013).
1003.1618v1.Spin_orbit_coupling_in_a_graphene_bilayer_and_in_graphite.pdf	Spin-orbit coupling in a graphene bilayer and in graphite F. Guinea1 1Instituto de Ciencia de Materiales de Madrid, CSIC, Sor Juana In es de la Cruz 3, E28049 Madrid, Spain The intrinsic spin-orbit interactions in bilayer graphene and in graphite are studied, using a tight binding model, and an intraatomic ~L~Scoupling. The spin-orbit interactions in bilayer graphene and graphite are larger, by about one order of magnitude, than the interactions in single layer graphene, due to the mixing of andbands by interlayer hopping. Their value is in the range 0 :11K. The spin-orbit coupling opens a gap in bilayer graphene, and it also gives rise to two edge modes. The spin-orbit couplings are largest, 14K, in orthorhombic graphite, which does not have a center of inversion. INTRODUCTION The isolation and control of the number of carriers in single and few layer graphene akes[1, 2] has lead to a large research activity exploring all aspects of these materials[3]. Among others, the application of graphene to spintronic devices[4{10] and to spin qubits[11{13] is being intensively studied. The understanding of these devices requires a knowledge of the electronic spin-orbit interaction. In principle, this interaction turns single layer graphene into a topological insulator[14], which shows a bulk gap and edge states at all boundaries. The magnitude of the spin-orbit coupling in single layer graphene has been studied[15{18]. The calculated cou- plings are small, typically below 0.1K. The observed spin relaxation[8, 19] suggests the existence of stronger mech- anisms which lead to the precession of the electron spins, like impurities or lattice deformations[20{22]. Bilayer graphene is interesting because, among other properties, a gap can be induced by electrostatic means, leading to new ways for the connement of electrons[23]. The spin-orbit interactions which exist in single layer graphene modulate the gap of a graphene bilayer[24]. The unit cell of bilayer graphene contains four carbon atoms, and there are more possible spin-orbit couplings than in single layer graphene. We analyze in the following the intrinsic and extrinsic spin-orbit couplings in bilayer graphene, using a tight binding model, and describing the relativistic eects responsible for the spin-orbit interaction by a ~L~Sin- traatomic coupling. We use the similarities between the electronic bands of a graphene bilayer and the bands of three dimensional graphite with Bernal stacking to gen- eralize the results to the latter. THE MODEL . We describe the electronic bands of a graphene bi- layer using a tight binding model, with four orbitals, the 2sand the three 2 porbitals, per carbon atom. We con- sider hoppings between nearest neighbors in the sameplane, and nearest neighbors and next nearest neighbors between adjacent layers, see[25]. The couplings between each pair of atoms is parametrized by four hoppings, Vss;Vsp;VppandVpp. The model includes also two in- traatomic levels, sandp, and the intraatomic spin-orbit coupling HsosoX i~Li~Si (1) The parameters used to describe the bands of graphite[26, 27], 0; 1; 2; 3; 4; 5and , can be de- rived from this set of parameters. We neglect the dier- ence between dierent hoppings between atoms which are next nearest neighbors in adjacent layers, which are re- sponsible for the dierence between the parameters 3 and 4. We also set the dierence in onsite energies between the two inequivalent atoms, to zero. The parameters 2and 5are related to hoppings between next nearest neighbor layers, and they do not play a role in the description of the bilayer. The total number of parameters is 15, although, without loss of generality, we setp= 0. We do not consider hoppings and spin orbit interactions which include dlevels, although they can contribute to the total magnitude of the spin-orbit couplings[18, 28]. The eects mediated by dorbitals do not change the order of magnitude of the couplings in single layer graphene, and their contribution to interlayer eects should be small. The main contribution to the eective spin-orbit at the Fermi level due to the interlayer coupling is due to the hoppings between porbitals in next nearest neigh- bor atoms in dierent layers. This interaction gives rise to the parameters 3and 4in the parametrization of the bands in graphite. For simplicity, we will neglect couplings between sandporbitals in neighboring layers. The non zero hoppings used in this work are listed in Table I. The hamiltonian can be written as a 32 32 matrix for each lattice wavevector. We dene an eective hamil- tonian acting on the , orpz, orbitals, by projecting out the rest of the orbitals: Heff H+H(!H)1H (2)arXiv:1003.1618v1 [cond-mat.mes-hall] 8 Mar 20102 s-7.3 t0 ss2.66 t0 sp4.98 t0 pp2.66 t0 pp-6.38 t1 pp 0.4 t2 pp 0.4 t2 pp-0.4 so0.02 TABLE I: Non zero tight binding parameters, in eV, used in the model. The hoppings are taken from[29, 30], and the spin- orbit coupling from[31]. Superindices 0,1, and 2 correspond to atoms in the same layer, nearest neighbors in dierent layers, and next nearest neighbors in dierent layers. A1,B2 B1A2 FIG. 1: (Color online). Unit cell of a graphene bilayer. La- bels A and B dene the two sublattices in each layer, while subscripts 1 and 2 dene the layers. We isolate the eect of the spin-orbit coupling by den- ing: Hso ~k Heff (so)Heff (so= 0) (3) Note thatHso depends on the energy, !. We analyzeHso at theKandK0points. The two matrices have a total of 16 entries, which can be labeled by specifying the sublattice, layer, spin, and valley. We dene operators which modify each of these degrees of freedom using the Pauli matrices ^ ;^;^s, and ^. The unit cell is described in Fig. 1. The hamiltonian has inversion and time reversal sym- metry, and it is also invariant under rotations by 120. -0.4-0.20.20.4EHeVL0.01400.01440.0148Èl1ÈHmeVL -0.4-0.20.20.4EHeVL0.0070.0080.009Èl2ÈHmeVL -0.4-0.20.00.20.4EHeVL0.00500.00550.00600.0065Èl3ÈHmeVL -0.4-0.20.20.4EHeVL0.480.500.52Èl4ÈHmeVLFIG. 2: (Color online). Dependence on energy of the spin- orbit couplings, as dened in eq. 5. These symmetries are dened by the operators: Ixxx T isyxK C120 1 2+ip 3 2sz! 1 2ip 3 2zz! 1 2+ip 3 2zz! (4) whereKis complex conjugation. The possible spin dependent terms which respect these symmetries were listed in[32], in connection with the equivalent problem of three dimensional Bernal graphite (see below). In the notation described above, they can be written as Hso =1zzsz+2zzsz+3z(ysxzxsy) + +4z(ysx+zxsy) (5) The rst term describes the intrinsic spin-orbit coupling in single layer graphene. The other three, which involve the matrices i, are specic to bilayer graphene. The term proportional to 3can be viewed as a Rashba cou- pling with opposite signs in the two layers. RESULTS . Bilayer graphene . The energy dependence of the four couplings in eq. 5 is shown in Fig. 2. The values of the couplings scale linearly with so. This dependence can be understood by treating the next nearest neighbor interlayer coupling and the intratomic spin-orbit coupling as a perturbation.3 -0.10.1EgHeVL0.01400.0144Èl1ÈHmeVL -0.10.1EgHeVL0.00760.0080Èl2ÈHmeVL -0.10.1EgHeVL0.005450.005460.00547Èl3ÈHmeVL -0.10.1EgHeVL0.480.490.50Èl4ÈHmeVL FIG. 3: (Color online). Dependence on interlayer gap, Eg, of the spin-orbit couplings, as dened in eq. 5. The spin-orbit coupling splits the spin up and spin down states of the bands in the two layers. The interlayer couplings couple the band in one layer to the band in the other layer. Their value is of order 3. Thestates are shifted by: 2 3 jj/so 3 02 (6) where0 is an average value of a level in the band. The model gives for the only intrinsic spin-orbit cou- pling in single layer graphene the value SLG 1= 0:0065meV (7) This coupling depends quadratically on so, 2 so=0 [15]. The band dispersion of bilayer graphene at low ener- gies, in the absence of spin-orbit couplings is given by four Dirac cones, because of trigonal warping eects as- sociated with 3[23]. Hence, we must to consider the couplings for wavevectors ~kslightly away from the K andK0points. We have checked that the dependence of the couplings ion momentum, in the range where trig- onal warping is relevant, is comparable to the changes with energy shown in Fig. 2. A gap,Eg, between the two layers breaks inversion symmetry, and can lead to new couplings. The calcula- tions show no new coupling greater than 106meV for gaps in the range 0:1eVEg0:1eV. The depen- dence of the couplings on the value of the gap is shown in Fig. 3. This calculation considers only the eect in the shift of the electrostatic potential between the two layers. The existence also of an electric eld will mix the pzands orbitals within each atom, leading to a Rashba term sim- ilar to the one induced in single layer graphene[15, 16]. The eect of 1is to open a gap of opposite sign in the two valleys, for each value of sz. The system will be- come a topological insulator[14, 33]. The number of edge states is two, that is, even. The spin Hall conductivity 02p 34p 32pkzc0.010.020.03Èl1ÈHmeVL 02p 34p 32pkzc0.010.020.03Èl2ÈHmeVL 02p 34p 32pkzc0.010.02Èl3ÈHmeVL 02p 34p 32pkzc0.40.8Èl4ÈHmeVLFIG. 4: (Color online). Dependence on momentum perpen- dicular to the layers in Bernal graphite of the spin-orbit cou- plings, as dened in eq. 5. is equal to two quantum units of conductance. A per- turbation which preserves time reversal invariance can hybridize the edge modes and open a gap. Such pertur- bation should be of the form xsy. The terms with 3and4describe spin ip hoppings which involve a site coupled to the other layer by the parameter 1. The amplitude of the wavefunctions at these sites is suppressed at low energies[23]. The shifts induced by 3and4in the low energy electronic levels will be of order 2 3= 1;2 4= 1. Bulk graphite . The hamiltonian of bulk graphite with Bernal stack- ing can be reduced to a set of bilayer hamiltonians with interlayer hoppings which depend on the momentum along the direction perpendicular to the layers, kz. We neglect in the following the (small) hoppings which de- scribe hoppings between next nearest neighbor layers, 2 and 5, and the energy shift between atoms in dierent sublattices. At the KandK0points of the three dimen- sional Brillouin Zone (2 kzc= 0, where cis the interlayer distance) the hamiltonian is that of a single bilayer where the value of all interlayer hoppings is doubled. At the H andH0points, where 2 kzc=, the hamiltonian reduces to two decoupled layers, and in the intermediate cases the interlayer couplings are multiplied by j2 cos(kzc)j. Carry- ing out the calculations described in the previous section, kzdependent eective couplings, i(kz), can be dened. These couplings are shown in Fig. 4. The results for bilayer graphene correspond to kzc= 2=3;4=3. The layers are decoupled for kzc=. In this case, the only coupling is1, which gives the coupling for a single layer, given in eq. 7. The signicant dispersion as function of momentum parallel to the layers shown in Fig. 4 implies the exis- tence of spin dependent hoppings between layers in dier- ent unit cells. This is consistent with the analysis which4 p4p 35p 3kya30.060.080.100.12Èl1ÈHmeVL p4p 35p 3kya30.200.25Èl2ÈHmeVL FIG. 5: (Color online). Dependence on wavevector, 2 ky, of the spin-orbit couplings for orthorhombic graphite, as dened in eq. 9. The point kx= 0;kyap 3 = 4=3 corresponds to the Kpoint (ais the distance between carbon atoms in the plane). showed that the spin-orbit coupling in a bilayer has a contribution from interlayer hopping, see eq. 6. The spin-orbit couplings can be larger in bulk graphite than in a graphene bilayer. The bands in Bernal graphite do not have electron-hole symmetry. The shift in the Fermi energy with respect to the Dirac energy is about EF20meV1;3[34]. Hence, the spin-orbit cou- pling is not strong enough to open a gap throughout the entire Fermi surface, and graphite will not become an insulator. A similar analysis applies to orthorhombic graphite, which is characterized by the stacking sequence ABCABC[35]. The electronic structure of this al- lotrope at low energies diers markedly from Bernal graphite[36, 37], and it can be a model for stacking defects[36{38]. If hoppings beyond nearest neighbor lay- ers are neglected, the hamiltonian can be reduced to an eective one layer hamiltonian where all sites are equiv- alent. The eective hamiltonian which describes the K andK0valleys contains eight entries, which can be de- scribed using the matrices i;si, andi. Orthorhombic graphene is not invariant under inversion, and a Rashba like spin-orbit coupling is allowed. The spin-orbit cou- pling takes the form: Hso orthoortho 1zszz+ortho 2 (ysxzxsy) (8) As in the case of Bernal stacking, the couplings have a signicant dependence on the momentum perpendicular to the layers, kz, and interlayer hopping terms are in- duced. For != 0;~k= 0 andkz= 0, we nd: ortho 1 = 0:134meV ortho 2 = 0:275meV (9) In orthorhombic graphite the Fermi level is away from theKandK0points, in the vicinity of a circle dened byj~kj= 1=vF[36, 37]. The variation of the couplings as function of wavevector is shown in Fig. 5. CONCLUSIONS We have studied the intrinsic spin-orbit interactions in a graphene bilayer and in graphite. We assume that theorigin of the couplings is the intraatomic ~L~Sinteraction, and we use a tight binding model which includes the 2 s and 2patomic orbitals. The intrinsic spin-orbit couplings in a graphene bilayer and in graphite are about one order of magnitude larger than in single layer graphene, due to mixing between theandbands by interlayer hoppings. Still, these couplings are typically of order 0 :010:1meV, that is, 0:11K. Bilayer graphene becomes an insulator with an even number of edge states. These states can be mixed by per- turbations which do not break time reversal symmetry. These perturbations can only arise from local impurities with strong spin-orbit coupling, as a spin ip process and intervalley scattering are required. The interplay of spin-orbit coupling and interlayer hop- ping leads to spin dependent hopping terms. The spin- orbit interactions are largest in orthorhombic graphite, which does not have inversion symmetry. ACKNOWLEDGEMENTS Funding from MICINN (Spain), through grants FIS2008-00124 and CONSOLIDER CSD2007-00010 is gratefully acknowledged. [1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, , and A. A. Firsov, Science 306, 666 (2004). [2] K. S. Novoselov, D. Jiang, F. Schedin, T. J. Booth, V. V. Khotkevich, S. V. Morozov, and A. K. Geim, Proc. Natl. Acad. Sci. U.S.A. 102, 10451 (2005). [3] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009). [4] E. W. Hill, A. K. Geim, K. S. Novoselov, F. Schedin, and P. Blake, IEEE Trans. Magn. 42, 2694 (2006). [5] S. J. Cho, Y.-F. Chen, and M. S. Fuhrer, Appl. Phys. Lett. 91, 123105 (2007). [6] M. Nishioka and A. M. Goldman, Appl. Phys. Lett. 90, 252505 (2007). [7] N. Tombros, C. J ozsa, M. Popinciuc, H. T. Jonkman, and B. J. van Wees, Nature 448, 571 (2007). [8] N. Tombros, S. Tanabe, A. Veligura, C. J ozsa, M. Popin- ciuc, H. T. Jonkman, and B. J. van Wees, Phys. Rev. Lett. 101, 046601 (2008). [9] W. H. Wang, K. Pi, Y. Li, Y. F. Chiang, P. Wei, J. Shi, and R. K. Kawakami, Phys. Rev. B 77, 020402 (2008). [10] M. Popinciuc, C. J. P. J. Zomer, N. Tombros, A. Veligura, H. T. Jonkman, and B. J. van Wees, Phys. Rev. B 80, 214427 (2009). [11] B. Trauzettel, D. V. Bulaev, D. Loss, and G. Burkard, Nature Phys. 3, 192 (2007). [12] J. Fischer, B. Trauzettel, and D. Loss, Phys. Rev. B 80, 155401 (2009).5 [13] W. L. Wang, O. V. Yazyev, S. Meng, and E. Kaxiras, Phys. Rev. Lett. 102, 157201 (2009). [14] C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005). [15] D. Huertas-Hernando, F. Guinea, and A. Brataas, Phys. Rev. B 74, 155426 (2006). [16] H.Min, J. E. Hill, N. Sinitsyn, B. Sahu, L. Kleinman, and A. MacDonald, Phys. Rev. B 74, 165310 (2006). [17] Y. Yao, F. Ye, X.-L. Qi, S.-C. Zhang, and Z. Fang, Phys. Rev. B 75, 041401 (2007). [18] M. Gmitra, S. Konschuh, C. Ertler, C. Ambrosch-Draxl, and J. Fabian, Phys. Rev. B 80, 235431 (2009). [19] C. J ozsa, T. Maassen, M. Popinciuc, P. J. Zomer, A. Veligura, H. T. Jonkman, and B. J. van Wees, Phys. Rev. B 80, 241403 (2009). [20] A. H. Castro Neto and F. Guinea, Phys. Rev. Lett. 103, 026804 (2009). [21] D. Huertas-Hernando, F. Guinea, and A. Brataas, Phys. Rev.Lett. 103, 146801 (2009). [22] C. Ertler, S. Konschuh, M. Gmitra, and J. Fabian, Phys. Rev. B 80, 041405 (2009). [23] E. McCann and V. I. Fal'ko, Phys. Rev. Lett. 96, 086805 (2006). [24] R. van Gelderen and C. M. Smith (2009), arXiv:0911.0857. [25] L. Chico, M. P. L opez-Sancho, and M. C. Mu~ noz, Phys.Rev. B 79, 235423 (2009). [26] J. W. McClure, Phys. Rev. 108, 612 (1957). [27] J. C. Slonczewski and P. R. Weiss, Phys. Rev. 109, 272 (1958). [28] J. W. McClure and Y. Yafet, in 5th Conference on Car- bon(Pergamon, University Park, Maryland, 1962). [29] D. Tom anek and S. G. Louie, Phys. Rev. B 37, 8327 (1987). [30] D. Tom anek and M. A. Schluter, Phys. Rev. Lett. 67, 2331 (1991). [31] J. Serrano, M. Cardona, and J. Ruf, Solid St. Commun. 113, 411 (2000). [32] G. Dresselhaus and M. S. Dresselhaus, Phys. Rev. 140, A401 (1965). [33] F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988). [34] M. S. Dresselhaus and J. Mavroides, IBM Journ. of Res. and Development 8, 262 (1964). [35] J. W. McClure, Carbon 7, 425 (1969). [36] F. Guinea, A. H. C. Neto, and N. M. R. Peres, Phys. Rev. B 73, 245426 (2006). [37] D. P. Arovas and F. Guinea, Phys. Rev. B 78, 245716 (2008). [38] N. B. Brandt, S. M. Chudinov, and Y. G. Ponomarev, in Semimetals I: Graphite and Its Compounds (North Hol- land, Amsterdam, 1988), vol. 20.1.
2210.01700v2.Spin_orbit_enhancement_in_Si_SiGe_heterostructures_with_oscillating_Ge_concentration.pdf	Spin-orbit enhancement in Si/SiGe heterostructures with oscillating Ge concentration Benjamin D. Woods,1M. A. Eriksson,1Robert Joynt,1and Mark Friesen1 1Department of Physics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA We show that Ge concentration oscillations within the quantum well region of a Si/SiGe het- erostructure can signiﬁcantly enhance the spin-orbit coupling of the low-energy conduction-band valleys. Speciﬁcally, we ﬁnd that for Ge oscillation wavelengths near = 1:57nm with an aver- age Ge concentration of nGe= 5%in the quantum well region, a Dresselhaus spin-orbit coupling is induced, at all physically relevant electric ﬁeld strengths, which is over an order of magnitude larger than what is found in conventional Si/SiGe heterostructures without Ge concentration oscil- lations. This enhancement is caused by the Ge concentration oscillations producing wave-function satellite peaks a distance 2=away in momentum space from each valley, which then couple to the opposite valley through Dresselhaus spin-orbit coupling. Our results indicate that the enhanced spin-orbit coupling can enable fast spin manipulation within Si quantum dots using electric dipole spin resonance in the absence of micromagnets. Indeed, our calculations yield a Rabi frequency Rabi=B> 500MHz/T near the optimal Ge oscillation wavelength = 1:57nm. I. INTRODUCTION Following the seminal work of Loss and DiVincenzo [1], quantum dots in semiconductors have emerged as a leading candidate platform for quantum computation [2– 5]. Gate-deﬁned quantum dots in silicon [6, 7] are par- ticularly attractive due to their compatibility with the microelectronics fabrication industry. Moreover, in con- trast with GaAs, for which coherence times are limited by unavoidable hyperﬁne interactions with nuclear spins [8], isotropic enrichment dramatically suppresses these interactions in Si, enabling long coherence times [9]. While recent progress in Si quantum dots has been quite promising, many of the leading qubit architectures rely on synthetic spin-orbit coupling arising from micro- magnets [10–13], leading to challenges for scaling up to systems with many dots. An alternative approach is to use intrinsic spin-orbit coupling for qubit manipulation, for example, through the electric dipole spin resonance (EDSR) mechanism [14, 15]. While this possibility has been considered for Ge and Si hole-spin qubits, where the degeneracyofthe p-orbital-dominatedvalencebandleads to strong spin-orbit coupling [16, 17], the weak spin-orbit coupling of the Si conduction band appears unfavorable for electron-spin qubits. In this work, we show how the spin-orbit coupling in Si/SiGe quantum well heterostructures can be enhanced by more than an order of magnitude by incorporating Ge concentration oscillations insidethe quantum well, leading to the possibility of exploiting intrinsic spin-orbit couplinginSiquantumdotsforfastgateoperations. Fig- ure1(a)showsaschematicofthesystemwhichconsistsof a Si-dominated quantum well region sandwiched between Si0:7Ge0:3barrier regions, where the growth direction is taken along the [001]crystallographic axis. In contrast to “conventional” Si/SiGe quantum wells, the quantum well region contains a small amount of Ge with concentration oscillations of wavelength , as shown in Fig. 1(b). For comparison, Fig. 1(c) shows the Ge concentration pro- ﬁle of a conventional Si/SiGe quantum well. Previous (a) Si0.7Ge0.3Si0.7Ge0.3 Si well with Ge oscillations 01530 nGe (%)growth direction, z(b) 01530 nGe (%)growth direction, z(c)FIG. 1. (a) Schematic of the Si/SiGe heterostructure consid- ered here, consisting of a Si-dominated quantum well sand- wiched between Si 0.7Ge0.3barrier regions. Note that the growth direction is along the [001]crystallographic axis. (b) Ge concentration proﬁle along the growth ( z) direction of a wiggle well. Ge concentration oscillations of wavelength in- side of the quantum well region lead to spin-orbit coupling enhancement for a proper choice of . (c) Ge concentration proﬁle of a “conventional” Si/SiGe quantum well, for compar- ison. works [18, 19] have studied such a structure, which has been named the wiggle well, and found that the periodic Ge concentration leads to an enhancement of the valley splitting. Here, we develop a theory of spin-orbit cou- pling within such structures and show that the periodic nature of the device, along with the underlying diamond crystal structure and degeneracy of the Si zvalleys, also gives rise to an enhancement in spin-orbit coupling. Im- portantly, we ﬁnd that the wavelength must satisfy a resonance condition to give rise to this spin-orbit cou- pling enhancement. As discussed in detail in Sec. IV, this involves a two-step process that can be summarized as follows. First, the periodic potential produced by the Ge concentration oscillations produces wave-function satellites a distance 2=away in momentum space from each valley. Then, a satellite of a given valley couples strongly to the opposite valley through Dresselhaus spin-arXiv:2210.01700v2 [cond-mat.mes-hall] 16 Jan 20232 orbit coupling, provided that the satellite-valley separa- tion distance in momentum space is 4=a, corresponding to the condition = 1:57nm. From the outset, it is important to remark that the spin-orbit coupling introduced by the Ge concentration oscillations is fundamentally distinct from the spin-orbit coupling of conventional Si/SiGe quantum wells. For a given subband of a conventional Si/SiGe quantum well immersed in a vertical electric ﬁeld, the C2vpoint group symmetry of the system allows for both Rashba and “Dresselhaus-type” linear- kkterms of the form [20, 21], HSO=(kyxkxy) +(kxxkyy);(1) where jare the Pauli matrices acting in (pseudo)spin space andandare the Rashba and Dresselhaus coef- ﬁcients, respectively, of the subband. The presence of Rasbha spin-orbit coupling is unsurprising due to the structural asymmetry provided by the electric ﬁeld [22], while the presence of the Dresselhaus-type term is ini- tially surprising since the diamond lattice of Si/SiGe quantum wells possesses bulk inversion symmetry [23]. However, these systems still support a Dresselhaus-type term of the same form (kxxkyy), due to the broken inversion symmetry caused by the quantum well inter- faces [20, 21, 24, 25]. This is in stark contrast to the true Dresselhaus spin-orbit coupling in III-V semicon- ductors, where the asymmetry of the anion and cation in the unit cell leads to bulk inversion asymmetry [23]. Importantly, we ﬁnd in Sec. IIIB that the spin-orbit cou- pling of the wiggle well does not rely upon the presence of an interface. Rather, it is an intrinsic property of a bulksystem with Ge concentration oscillations. In this sense, the spin-orbit coupling investigated here is more akin to the true Dresselhaus spin-orbit coupling of III- V semiconductors than the Dresselhaus-type spin-orbit coupling of conventional Si/SiGe quantum wells brought about by interfaces. Indeed, the only requirement for linear- kkDresselhaus spin-orbit coupling in a wiggle well with an appropriate is conﬁnement in the growth di- rection (even symmetric conﬁnement), to allow for the formation of subbands. For simplicity in the remainder of this work, we simply refer to this form of spin-orbit coupling as Dresselhaus. The rest of this paper is organized as follows. In Sec. II we describe our model used to study the quantum well heterostructure. Section III then presents our numerical results for the spin-orbit coeﬃcients. This also includes the calculation of the EDSR Rabi frequency and studies the impact of alloy disorder on the spin-orbit coeﬃcients. In Sec. IV we provide an extensive explanation of the mechanism behind the spin-orbit coupling enhancement. Finally, we conclude in Sec. V. II. MODEL In this section, we outline the model used to study our Si/SiGe heterostructure along with the methods usedto calculate the spin-orbit coeﬃcients. In Sec. IIA, we describe the tight binding model used to model generic SiGe alloy systems. Next, in Sec. IIB we employ a vir- tual crystal approximation to impart translation invari- ance in the plane of the quantum well, allowing us to reduce the problem to an eﬀective one-dimensional (1D) Hamiltonian parametrized by in-plane momentum kk. In Sec.IIC,weexpandthemodelaround kk= 0toseparate out the Hamiltonian components that give rise to Rashba and Dresselhaus spin-orbit coupling, respectively, and we explain the important diﬀerences between the two com- ponents. Finally, in Sec. IID, we transform the Hamil- tonian into the subband basis, which allows us to obtain expressions for the Rashba and Dresselhaus spin-orbit coeﬃcients in each subband. A. Model of SiGe alloys To study the spin-orbit physics of our system we use the empirical tight-binding method [26], where the elec- tronic wave function is written as a linear combination of atomic orbitals: j i=X n;j;;jnji nj: (2) Here,hrjnji=(rRn;j)jiis an atomic orbital centered at position Rn;j, corresponding to atom jof atomic layer nalong the growth direction [001],is a spatial orbital index, and jiis a two component spinor with=";#indicating the spin of the orbital. We use an sp3d5sbasis set with 20 orbitals per atom, on- site spin-orbit coupling, nearest-neighbor hopping, and strain. Note that nearest-neighbor sp3d5stight-binding models are well established for accurately describing the electronic structure of semiconductor materials over a wide energy range [27]. Explicitly, is a spatial orbital index from the set including s,s,pi(i=x;y;z), and di(i=xy;yz;zx;z2;x2y2) orbitals, which are meant to model the outer-shell orbitals of individual Si and Ge atoms that participate in chemical bonding. Addi- tionally, these orbitals possess certain spatial symmetries that, combined with the diamond crystal structure of the SiGe alloy, dictate the forms of the nearest neighbor cou- plings, as ﬁrst explained in the work of Slater and Koster [26]. The free parameters of the tight-binding model (in- cluding onsite orbital energies, nearest-neighbor hopping energies, strain parameters, etc.) are then chosen such that the band structure of the system agrees as well as possible with experimental and/or ab initio data. In this work, we use the tight binding model and parameters of Ref. [28], which allows for the modeling of strained, random SiGe alloys with any Ge concentration proﬁle. The Hamiltonian of an arbitrary SiGe alloy takes the3 form, Hmi;nj ; =nj mih 0 "(nj) +Vn +0 C(nj) +S(nj) ;0i +0 n+1 mT(n) i;j +n1 mT(m)y i;j ;(3) whereHmi;nj ;0=hmijHjnj0iandequals 1if its subscripts match its superscripts and 0otherwise. The ﬁrst line in Eq. (3) contains intra-atomic terms, where "(nj) is the onsite energy of orbital , for atom jin atomic layer n,Vnis the potential energy due to the ver- tical electric ﬁeld, C(nj) accounts for onsite energy shifts and couplings caused by strain, and S(nj) ;0accounts for spin-orbit coupling. The matrix C(nj) is determined by the deformation of the lattice due to strain as detailed in Ref. [28] and arises from changes in the onsite poten- tial of the atom due to the displacement of its neighbors. In addition, spin-orbit coupling S(nj)is an intra-atomic coupling between porbitals [29] and is the only term in Eq. (3) that does not conserve spin . (See Appendix B for the explicit form of S(nj).) Note that the superscripts (nj), which index the atoms, are needed here because the intra-atomic terms depend on whether an atom is Si or Ge, as well as the local strain environment. The second line in Eq. (3) contains inter-atomic terms describing the hopping between atoms on adjacent atomic layers, where T(n)is the hopping matrix from atomic layer nto atomic layern+1. Nearlyallelementsof T(n)arezero, withnon- zero hoppings occurring only between nearest-neighbor atoms. A non-zero hopping matrix element T(n) i;jthen depends on three things: (1) the orbital indices and , (2) the types of atoms involved, and (3) the direction andmagnitudeofthevector Rn+1;iRn;jconnectingthe atoms. We then use the Slater-Koster table in Ref. [26] along with the parameters of Ref. [28] to calculate T(n) i;j. We note that strain aﬀects the hopping elements by al- tering the direction and length of the nearest-neighbor vectors (i.e., the crystalline bonds) [28, 30]. In this work, we let the Ge concentration vary be- tween layers, as shown in Figs. 1(b) and 1(c), but as- sume it to be uniform within a given layer. Note that the large diﬀerence in Ge concentration between the bar- rier and well regions results in a large conduction band oﬀset that traps electrons inside the quantum well. This occurs naturally in the tight binding model of Eq. (3) because"(nj);C(nj), andT(n) ijare diﬀerent for Si and Ge atoms. Finally, we point the reader to Appendix A for a description of the lattice constant dependence on strain. B. Virtual crystal approximation and pseudospin transformation While the Hamiltonian in Eq. (3) provides an accurate description of SiGe alloys, it lacks translation invariance when alloy disorder is present. This makes the modelcomputationally expensive to solve, and it obscures the physics of the spin-orbit enhancement coming from the averaged eﬀects of the inhomogeneous Ge concentration proﬁle. We therefore employ a virtual crystal approxima- tion where the Hamiltonian matrix elements are replaced by their value averaged over all alloy realizations. Specif- ically, we deﬁne a virtual crystal Hamiltonian HVCwith elements (HVC)mi;nj ;0=D Hmi;nj ;0E , whereh:::iindicates anaverageoverallpossiblealloyrealizations. TheHamil- tonian is then translation invariant within the plane of the quantum well. In addition, it is useful to move be- yond the original orbital basis, where the spin is well- deﬁned, to a pseudospin basis deﬁned by jnji=X jnjiU(n) ;; (4) where are the hybridized orbital states, =;+are the pseudospins, and U(n)is the transformation matrix for layern. Full details of this basis transformation can be found in Appendix B. Here, we mention three important features of Eq. (4). First, the basis transformation diag- onalizes the onsite spin-orbit coupling, making the spin- orbit physics more transparent. Second, the pseudospin states represent linear combinations of orbitals including both spin"and spin#. Third, the transformation matrix U(n)can be shown to satisfy U(n)=( U(0); n2Zeven; U(1); n2Zodd:(5) This alternating structure for the transformation matrix is crucial to the results that follow, and results from the presence of two sublattices in the diamond crystal struc- ture of Si, as shown in Fig. 2(a). Making use of the virtual crystal approximation, we now convert our three-dimensional (3D) Hamiltonian into an eﬀective 1D Hamiltonian. To begin, we note that the virtual crystal Hamiltonian takes the simpliﬁed form, (HVC)mi;nj ;0=nj0 mih "(n) +Vn +C(n) i +n+1 mT(n) i;j0+n1 mT(m)y i;j0;(6) where (HVC)mi;nj ;0=hmijHVCjnj0i,"(n) is the on- site energy of pseudospin orbital , and C(n)and T(n) are the onsite strain and hopping matrices, respectively, transformed into the pseudospin basis and averaged over alloy realizations. Note that "(n)includes contributions from diagonalizing the onsite spin-orbit coupling. Impor- tantly, "(n)and C(n)maintain a dependence on the layer indexndue to the non-uniform Ge concentration pro- ﬁle. In contrast, the dependence of intra-atomic terms on theintra-layer atom index jhas vanished due to the virtual crystal approximation. Moreover, the translation invariance of the virtual crystal approximation implies that hopping matrix elements between any two layers only depend upon the relative position of atoms, i.e.,4 T(n) i;j0=T(n) ;0(Rn+1;iRn;j). We therefore intro- duce in-plane momentum kkas a good quantum number and Fourier transform our Hamiltonian. To do so, we deﬁne the basis state kkn =1pNkX jeikkRnjjnji;(7) where kk= (kx;ky)andNkis the number of atoms within each layer. The Hamiltonian has matrix elements eHmn ;0(kk) =n0 mh "(n) +Vn +C(n) i +n+1 meT(n) ;0 kk +n1 meT(m)y ;0 kk ;(8) whereeHmn ;0(kk) = kkmHVCkkn0 , and eT(n) ;0 kk is the Fourier-transformed hopping matrix given by eT(n) ;0 kk =2X l=1eikkr(n) lT(n) ;0(r(n) l);(9) where r(n) lis a nearest-neighbor vector from a reference atom in layer nto one of its nearest neighbors in layer n+ 1. For a diamond lattice, each atom has only has two such bonds, as indicated in Fig. 2(a). Note that the Hamiltonian matrix elements vanish between states with diﬀerent momenta due to translational invariance. Hence, we obtain an eﬀective 1D Hamiltonian as a func- tion of kk. An important feature of the Fourier-transformed hop- pingmatrixeT(n)(kk)isthatitdependsonthelayerindex nfor two reasons. First, the inhomogeneous Ge concen- tration along the growth axis causes the hopping param- eters to change slightly from layer to layer. Second, and more importantly, the diamond crystal structure is com- posed of two interleaving face-centered-cubic sublattices which each contribute an inequivalent atom to the prim- itive unit cell. This is illustrated in Fig. 2(a) where the atoms belonging to the two sublattices are colored red and blue, respectively. Indeed, the atoms for n2Zeven andn2Zoddbelong to sublattice 1 and 2, respectively, and have diﬀerent nearest neighbor vectors. It is there- fore useful to deﬁne eT(n) kk =(eT(n) + kk ; n2Zeven eT(n) kk ; n2Zodd(10) asthehoppingmatricesforthetwosublattices. Westress that the dependence of eT(n) +(kk)andeT(n) (kk)on the layer index nis due to the inhomogeneous Ge concentra- tion proﬁle, and that eT(n) +(kk)andeT(n) (kk)diﬀer due to the diamond crystal structure having two inequiva- lent atoms in its primitive unit cell. We can therefore visualize the system, for any given kk, as a 1D, multi- orbital tight binding chain, as shown in Fig. 2(b), where the hopping terms alternate in successive layers. (a) z (b) /tildewideT(0) +/parenleftbig k∥/parenrightbig/tildewideT(1) −/parenleftbig k∥/parenrightbig/tildewideT(2) +/parenleftbig k∥/parenrightbig/tildewideT(3) −/parenleftbig k∥/parenrightbigzFIG. 2. (a) Diamond crystal structure of silicon. The dashed lines outline the conventional unit cell of the face centered cubic lattice. Both red and blue atoms are silicon but be- long to diﬀerent sublattices. Notice that the vectors connect- ing an atom to its four nearest neighbors are fundamentally diﬀerent for the red and blue atoms, giving rise to the al- ternating hopping structure shown in (b). (b) Eﬀective 1D tight-binding chain, with hopping matrix terms alternating between eT(n) +(kk)and eT(n) (kk). Note that each site has 20 orbitals, and only the forward hopping terms are shown. On- site and backward hopping terms are not shown. For a SiGe alloy in the virtual crystal approximation, the two-sublattice structure is retained, but the atoms are replaced by virtual atoms, with averaged properties consistent with the Ge con- centration of a given layer. C. Expansion around kk= 0 Our goal is to understand the spin-orbit physics of low- energy conduction band states near the Fermi level. In strained Si/SiGe quantum wells, these derive from the two degenerate valleys near the Z-point of the strained Brillouin zone [31]. Therefore, the low-energy states have smalljkkj, and we can understand the spin-orbit physics by expanding the Fourier-transformed hopping matrices eT(n) kk to linear order. We ﬁnd that eT(n) (kk) =eT(n) 0+eT(n) R(kk)eT(n) D(kk) +O(k2 k);(11) whereeT(n) 0is the hopping matrix for kk= 0, andeT(n) R andeT(n) Dcontain the linear kkcorrections. These hop- ping matrix components are found to be eT(n) 0= (n)0; (12) eT(n) R(kk) = (n)(kyxkxy); (13) eT(n) D(kk) = (n)(kxxkyy); (14) where (n)and(n)are real-valued 1010matrices, andjarethePaulimatricesactingonpseudospinspace, withj= 0;x;y;z. There are several features to remark on in Eqs. (12)- (14). First, the momentum-spin structure of eT(n) Rand eT(n) Dhave the familiar forms of Rasbha and Dresselhaus spin-orbit coupling; hence, we apply the subscript labels5 RandD. Second, the hopping matrices for the two sub- lattices,eT(n) +andeT(n) , diﬀer by the sign in front of eT(n) D, while thesign of eT(n) Ris sublatticeindependent. Asnoted above, this alternating sign is a consequence of the two sublattices of the diamond crystal being inequivalent. As weshallshowinSec.IVC,thisalternatinghoppingstruc- ture ofeT(n) Dis key to explaining the mechanism behind the enhanced spin-orbit coupling. In Appendix C, we also provide a symmetry argument for why the system has this particular alternating hopping structure for the Rashba and Dresselhaus terms. Third, for the special case of kk= 0, the two sublattices become equivalent (i.e., there is no even/odd structure due to the vanishing ofeT(n) RandeT(n) Datkk= 0). Furthermore, the pseu- dospin sectors are uncoupled and equivalent at kk= 0, which implies that all eigenstates of our Hamiltonian are doubly degenerate at kk= 0as is expected since the sys- temhastime-reversalsymmetry. Fourth, thedependence of (n)and(n)on the layer index narises only from the inhomogeneous Ge concentration proﬁle. However, note that neither matrix vanishes if the Ge concentration is uniform. For interested readers, we provide expressions for (n)and(n)in Appendix D, for the case of a pure Si structure. Finally, we note that the diagonal elements of(n)all vanish. To linear order in kk, the Hamiltonian then takes the compact form H=H(z) 00+H(z) R(kyxkxy) +H(z) D(kxxkyy) +O(k2 k);(15) whereH(z) 0(which we call the subband Hamiltonian) de- scribes the physics at kk= 0, andH(z) RandH(z) Ddescribe the linear kkperturbations arising from eT(n) RandeT(n) D, respectively. These take the form hmjH(z) 0jni=n mh "(n) +Vn +C(n) i +n+1 m (n) +n1 m (m)T ;(16) hmjH(z) Rjni=n+1 m(n) +n1 m(m)T ; (17) hmjH(z) Djni=(1)n n+1 m(n) n1 m(m)T ;(18) where the superscript zindicates that only the orbital degrees of freedom in the zdirection (i.e., the growth direction) are acted upon. Hence, the momentum kkand pseudospin indices are both dropped in Eqs. (16)-(18). Also note that the alternating factor in front of eT(n) D in Eq. (11) is reﬂected in the (1)nfactor in Eq. (18). D. Transformation to the subband basis The largest term in Hamiltonian (15) (by far) is the subband Hamiltonian H(z) 0. The eigenstates of H(z) 0arereferred to as the orbital subbands of the quantum well, including two distinct valley states per subband. The subband and valley states, in turn, serve as a natural basis for representing the Hamiltonian, since the lateral conﬁnement associated with a quantum dot barely per- turbs this subband designation, although disorder may cause hybridization of the valley states. It is therefore the properties of the individual subbands that largely de- termine the properties of quantum dot states, including their spin-orbit behavior. To perform a subband basis transformation, we de- ﬁnej'`ias the`theigenstate of H(z) 0with energy E`. (Here, for convenience, we include both subband and val- ley states in the set f`g.) Generically we can write j'`i=X njniQn;`; (19) whereQis an orthogonal matrix, deﬁned such that H(z) 0j'`i=E`j'`i (20) for each`. Using these eigenstates as a basis, the Hamil- tonian can then be expressed as H=0+ (kyxkxy) +(kxxkyy) +O(k2 k);(21) where ,, and are real-symmetric matrices acting in subband space with elements ``0=``0E`; (22) ``0=h'`jH(z) Rj'`0i; (23) ``0=h'`jH(z) Dj'`0i: (24) The matrix elements ``0and ``0are referred to as the Rashba and Dresselhaus spin-orbit coupling coeﬃcients, respectively [16]. The diagonal elements are of particular importance since they determine the linear dispersion of a given subband near kk= 0. Indeed, the diagonal ele- ments ``and``themselves are often referred to in the literature as the Rasbha and Dresselhaus spin-orbit cou- pling coeﬃcients, respectively, and are typically denoted simply asand. Furthermore, we focus on the diag- onal elements of the ground ( `= 0) and excited ( `= 1) valleystatescorrespondingtothelowestorbitalsubband, which we henceforth refer to as simply the ground and excited valley states. These represent the lowest-energy conduction subbands, which are nearly degenerate due to the wide separation of the two degenerate zvalleys within the Brillouin zone of Si [32], thus playing a domi- nating role in the physics of Si spin qubits. In some cases, we may also be interested in the spin-orbit coupling be- tween the ground and excited valleys, often referred to as spin-valley coupling, since the valley states are much closer in energy than the orbitally excited subbands. We note that conﬁnement in the growth direction is a cru- cialingredientforobtainingnonzerovaluesof ``and``.6 While this latter fact is not obvious from the structure of the Hamiltonian, it can be shown to be true, using the fact that (n)has vanishing diagonal elements and the structure of the (n)and(n)matrices described in Ap- pendix D. Finally, we also mention that one can arrive at an eﬀective 2D theory, similar to previous SU (2)SU(2) approaches to spin-valley physics in Si [33], by projecting the Hamiltonian in Eq. (21) onto the subspace contain- ing the ground ( `= 0) and excited ( `= 1) valleys. E. Summary of calculation procedure To conclude this section, we present a brief summary of the procedure used in a typical calculation, like those reported in Sec. III. First, we specify a Ge concentra- tion proﬁle as a function of layer index n. Second, we construct the subband Hamiltonian H(z) 0using Eq. (16). Third, we diagonalize the subband Hamiltonian to ob- tain a set of eigenstates fj'`ig. Finally, we construct the H(z) RandH(z) Dmatrices in Eqs. (17) and (18) and cal- culate the matrix elements in Eqs. (23) and (24), which yields the spin-orbit coupling coeﬃcients. III. NUMERICAL RESULTS In this section, we present our numerical results. We ﬁrst present in Sec. IIIA spin-orbit coupling results for a “conventional” Si/SiGe quantum well system withoutGe oscillations included in the quantum well region. These results serve as a baseline for comparison. Next, we provide spin-orbit coupling results in Sec. IIIB for the Si/SiGe quantum well system withGe oscillations in- cluded in the quantum well region, namely, a wiggle well. In this case, we observe signiﬁcant enhancement of the Dresselhaus spin-orbit coupling for appropriate Ge con- centration oscillation wavelengths . In Sec. IIIC, we show that the enhanced spin-orbit coupling allows for fast Rabi oscillations using EDSR. Finally, we study in Sec. IIID the impact of alloy disorder on the spin-orbit coupling. A. Spin-orbit coupling in Si/SiGe quantum wells withoutGe concentration oscillations We ﬁrst calculate spin-orbit coupling for the conven- tional Si/SiGe quantum well shown in Fig. 1(c). We as- sume barrier regions with a uniform Ge concentration of nGe,bar = 30%and a quantum well width of Lz20nm, consisting of an even number of atomic layers; the lat- ter value was chosen to ensure that the wave functions have negligible weight at the bottom barrier except in the limit of very weak electric ﬁelds. In addition, the interfaces are given a nonzero width of Lint0:95nm (7atomic layers), in which the Ge concentration linearly interpolates between values appropriate for the barrier 25 025 (eV nm) (a) 0 2 4 6 8 10 Fz (mV/nm)4 2 0 (eV nm) (b)FIG. 3. Diagonal Dresselhaus ``(a) and Rashba ``(b) spin-orbit coupling coeﬃcients as a function of vertical elec- tric ﬁeldFz, for the ground (solid blue, 00and00) and ex- cited (dashed red, 11and11) valleys of the “conventional” Si/SiGe quantum well shown in Fig. 1(c). Notice that the Dresselhaus coeﬃcients ``are7times larger in magnitude than the Rashba coeﬃcients ``. and well regions. Such ﬁnite-width interfaces occur in realistic devices, and are known to signiﬁcantly impact important properties of the quantum well such as the valley splitting [34]. The diagonal Dresselhaus and Rashba spin-orbit cou- plingcoeﬃcientsarecalculatedfortheground(solidblue, 00and00) and excited (dashed red, 11and11) valley states, andareplottedinFigs.3(a)and3(b)asafunction of vertical electric ﬁeld Fz. At zero ﬁeld, Fz= 0, the di- agonal spin-orbit coupling coeﬃcients in Fig. 3 all vanish, as consistent with the system being inversion symmetric [20, 21]. By turning on the electric ﬁeld we break the structural inversion symmetry, and the resulting spin- orbit coeﬃcients vary linearly over the entire ﬁeld range considered here. Note again that Dresselhaus spin-orbit coupling in SiGe requires the presence of a broken struc- tural inversion symmetry [20, 21, 24, 25], in contrast with GaAs, which requires a broken bulk inversion symme- try [23]. As consistent with previous studies [20, 35–37], theDresselhauscoeﬃcientsofthegroundandexcitedval- leys are found to be approximately opposite in sign, with the Dresselhaus coeﬃcient of the excited valley being slightly smaller in magnitude. Additionally, the diagonal Rashba matrix elements are seen to be much smaller in magnitude than the Dresselhaus elements. Here, we ﬁnd thatj``j7j``jfor both low-energy valleys ( `= 0;1). Inaddition, themagnitudesofthespin-orbitelementsare found to quantitatively agree with Ref. [20], in the large electric ﬁeld regime. (The system studied in Ref. [20] assumed a narrower quantum well, resulting in diﬀerent behavior at low electric ﬁelds.) We note, however, that the Rashba coeﬃcients of the ground and excited valley stateswerefoundtohaveoppositesignsinRef.[20], while we ﬁnd them to have the same sign here. This diﬀerence in our results occurs because we have used a softened in- terface where the Ge concentration interpolates between7 400 0400 (eV nm) (a) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 (nm) 5 05 (eV nm) (b) 0.25 0.29 (nm) 600 300 0300 (eV nm) (c) 0.44 0.48 (nm) 10 010 (eV nm) (d) FIG. 4. Dresselhaus ``0(a) and Rashba ``0(b) spin-orbit matrix elements as a function of Ge oscillation wavelength for an electric ﬁeld of strength Fz= 10mV/nm. Coeﬃ- cients for the ground and excited valleys are shown in blue (solid, 00and 00) and red (dashed, 11and 11), respec- tively. The oﬀ-diagonal elements ( 01and01), often referred to as the spin-valley coeﬃcients, are shown as purple dashed- dotted lines. A wide bump centered at 1:57nm occurs in (a), corresponding to a dramatically enhanced Dresselhaus spin-orbit coupling. At the center of the bump, j``jis15 times larger than the “conventional” Si/SiGe system at the same electric ﬁeld [See Fig. 3(a).] Narrow bumps for the di- agonal Dresselhaus and Rashba coeﬃcients at small values areshownin(c)and(d). ThecorrespondingGeconcentration proﬁle is shown in Fig. 1(b), where the average Ge concentra- tion in the quantum well region is nGe= 5%. the barrier and well regions over a ﬁnite width, whereas Ref. [20] used a completely sharp interface. We have numerically conﬁrmed our results for the diagonal spin- orbit matrix elements using the computational scheme of Ref. [20], which is unrelated to our scheme, summarized in Eqs. (23) and (24). B. Spin-orbit coupling in Si/SiGe quantum wells withGe concentration oscillations We now calculate spin-orbit coeﬃcients for the wiggle well geometry shown in Fig. 1(b), with a sinusoidally varying Ge concentration ranging from nGe= 0%to a maximum amplitude of nGe= 10%, and an oscillation wavelength of . These parameters were chosen to match those of an experimental device reported in Ref. [18]. Here, the system parameters, nGe,bar,Lz, andLintare the same as before, and we apply an electric ﬁeld of Fz= 10mV/nm. The resulting diagonal Dresselhaus ``and Rashba `` spin-orbit coeﬃcients for the ground (solid blue, 00and 400 0400 (eV nm) (a) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 (nm) 5 05 (eV nm) (b) 0.25 0.29 (nm) 600 300 0300 (eV nm) (c) 0.44 0.48 (nm) 10 010 (eV nm) (d)FIG. 5. Same as Fig. 4 except with an electric ﬁeld strength ofFz= 2mV/nm. Notice that the magnitude of ``at the peaks of the bumps are nearly the same as in Fig. 4, but that the features are narrower than in Fig. 4. 00) and excited (dashed red, 11and11) valley states are shown in Figs. 4(a) and 4(b) as a function of the Ge oscillation wavelength . We ﬁnd that the spin-orbit co- eﬃcients vary nontrivially with the choice of oscillation wavelength , exhibiting bumps where the coeﬃcients are signiﬁcantly enhanced. In particular, we observe a broad bump in Fig. 4(a), centered at 1:57nm, in which the Dresselhaus spin-orbit coeﬃcient reaches the valuesj00j365eVnm for the ground valley and j11j357eVnm for the excited valley. Note that this peak location corresponds to the same wavelength as the long-period wiggle well predicted in Refs. [18, 19] to en- hance the valley splitting. (Note that a slightly diﬀerent value was predicted in Refs. [18, 19], due to the valley minima residing at slightly diﬀerent momenta k0com- pared to our model.) Similar to the conventional Si/SiGe quantum well shown in Fig. 4, the ground and excited valleys have approximately opposite Dresselhaus coeﬃ- cients for the range of values considered here, except at very small . At the peak of the bump, however, the diagonal Dresselhaus coeﬃcients ``for the wiggle well are15times larger than those of the conventional sys- tem at the same electric ﬁeld strength. [See Fig. 3(a) at Fz= 10mV/nm.] Moreover, the region of enhancement appears very broad, with a full-width at half-maximum of= 0:55nm. In contrast to the diagonal Dres- selhaus coeﬃcients ``, the diagonal Rashba coeﬃcients ``show no enhancement features except at small wave- lengths. Indeed, for &0:5nm, the Rashba coeﬃcients are essentially independent of with 003:7eVnm and113:3eVnm for the ground and excited val- leys, respectively. Comparing this to the results of the conventional system in Fig. 3(b) for the same electric8 ﬁeld, we see that the Rashba elements are nearly identi- cal in the two cases. Evidently, the Rashba coeﬃcients are unaﬀected by the Ge concentrations oscillations for all but the shortest values. We also plot the oﬀ-diagonal spin-orbit coupling el- ements (01and 01)as purple dashed-dotted lines in Figs. 4(a) and 4(b). Note that these quantities are of- ten referred to as spin-valley coupling. Interestingly, the oﬀ-diagonal Dresselhaus coeﬃcient 01vanishes at the center of the 1:57nm feature. Except at this partic- ular wavelength for the Dresselhaus spin-orbit coupling, the oﬀ-diagonal spin-orbit coeﬃcients are comparable in size to the diagonal coeﬃcients. To illustrate the features at small wavelengths, we plot the diagonal Dresselhaus and Rashba coeﬃcients over se- lected, narrow ranges of in Figs. 4(c) and 4(d). Here we observeanenhancementintheDresselhauscoeﬃcient `` centered at =a=20:27nm, whereais the size of the cubic unit cell in Fig. 2(a) with an amplitude about twice that of the = 1:57nm peak. This value corresponds to having a Ge concentration proﬁle that alternates on every other atomic layer, essentially transforming the di- amondlatticeofSiintoazinc-blendelattice. Inthiscase, the system can be thought of as a III-V semiconductor with the cation and anion corresponding to diﬀerent con- centrations of Ge. Interestingly, we ﬁnd that the ground and excited valleys have the same sign of ``here, in contrast to the bump centered at 1:57nm. We also observe a enhancement in the Rashba coeﬃcient ``near = 0:46nm as shown in Fig. 4(d). However, note that the magnitude of this peak is still more than an order of magnitude smaller than the ``peaks. Indeed, the magnitude of this peak is even smaller than the Dressel- haus coeﬃcients ``shown in Fig. 3(a) for the conven- tional Si/SiGe system at the same electric ﬁeld strength. Finally, we note that these small- bumps have a much narrower width than the ``bump at= 1:57nm, which has important consequences for practical applications. To study the tunability of the spin-orbit physics, we also calculate the spin-orbit coeﬃcients of a wiggle well for a weaker vertical electric ﬁeld. These results are shown in Fig. 5, which is the same system as Fig. 4, ex- cept withFz= 2mV/nm. The results are qualitatively similar to the stronger electric ﬁeld case. However, there exists an important quantitative similarity and diﬀerence between the two cases, which we address in the following two paragraphs. The remarkable similarity is that the Dresselhaus co- eﬃcients ``at the peaks of the bumps are nearly iden- tical in the two cases. Note that this is true for both the= 1:57nm and= 0:27nm bumps. Evidently, at the center of the bumps, the electric ﬁeld plays a minor role. This is in stark contrast with the Dresselhaus coef- ﬁcients for the conventional system, where the diagonal Dresselhaus coeﬃcients are proportional to Fzas shown inFig.3(a). Thishighlightstheimportantdiﬀerenceﬁrst discussedintheIntroductionbetweenspin-orbitcoupling in conventional Si/SiGe quantum wells and wiggle wells,namely, that the conventional system requires the pres- ence of an interface and structural asymmetry, while the wiggle well fundamentally does not. In the latter case, the spin-orbit coupling is an intrinsic property of the bulk system. Indeed, we have checked that the Dresselhaus spin-orbit coupling persists in a wiggle well of wavelength = 1:57nm, in the absence of an interface, by cal- culating the spin-orbit coeﬃcients for a system without barriers, but instead immersed in a harmonic potential, Vn=V0z2 n, obtaining similar results. The main diﬀerence between the two cases is that the features are narrower in space for the weak electric ﬁeld of Fig. 5, compared to the strong electric ﬁeld of Fig. 4. This represents an important advantage of strong electric ﬁelds for the wiggle well system, since it can be challenging to grow heterostructures with perfect oscil- lation periods. We conclude that stronger electric ﬁelds provide more reliable access to the enhanced spin-orbit coupling provided by Ge concentration oscillations, since the control of does not need to be as precise during the growth process. The reason for narrower features in weaker electric ﬁeld will be explained in Sec. IVC. Fi- nally, we note that stronger electric ﬁelds also provide larger valley splittings [38]. C. Electric dipole spin resonance Aside from being of interest from a purely scientiﬁc standpoint, the presence of spin-orbit coupling can be exploited to perform gate operations within a quantum computation context. In particular, electric dipole spin resonance (EDSR) is a powerful technique to manipu- late individual spins through all-electrical means [14, 15]. Here, we calculate the EDSR Rabi frequency for a single electron in a quantum dot embedded in a wiggle well. As shown in Ref. [15], applying an AC, in-plane electric ﬁeld of amplitude Fxwith frequency !dacross a quan- tum dot with spin-orbit coupling leads to an eﬀective AC in-plane magnetic ﬁeld. For valley `, the magnitude of this eﬀective AC magnetic ﬁeld is given by Beﬀ(t) =2eFx ~!!d !q 2 ``+ 2 `` gBsin (!dt);(25) where~!is the level spacing characteristic of the dot’s harmonicconﬁnementpotential, gisthe g-factor, and B is the Bohr magneton. Note that for j``jj ``j, as is the case for Si/SiGe systems, the eﬀective AC magnetic ﬁeld is parallel to the in-plane electric ﬁeld Fx. Apply- ing a static, out-of-plane magnetic ﬁeld B, the eﬀective Hamiltonianforthequantumdotrestrictedtotheorbital ground state is then Heﬀ(t) =1 2gB[Bz+Beﬀ(t)x]. For a system initialized in the spin- "ground valley state and driven at the resonance frequency !d=gBB=~set by the external magnetic ﬁeld, the probability of ﬁnd- ing the electron in the spin- #state is given by P#(t) = sin2( Rabit=2)[39], where the Rabi frequency is found to9 FIG.6. Electricdipolespinresonance(EDSR)Rabifrequency Rabias a function of magnetic ﬁeld Band Ge concentra- tion oscillation wavelength for a quantum dot with conﬁne- ment energy ~!= 1meV and an in-plane AC electric ﬁeld amplitudeFx= 102mV/nm. The vertical electric ﬁeld is Fz= 10mV/nm. be Rabi=eFxgBq 2 ``+ 2 `` ~(~!)2B: (26) The EDSR Rabi frequency Rabiof a quantum dot in a wiggle well is plotted in Fig. 6 as a function of mag- netic ﬁeld and Ge oscillation wavelength for the real- istic parameters of ~!= 1meV andFx= 102mV/nm. We see that for oscillation wavelengths near the peak of the the spin-orbit enhancement at 1:57nm, we can obtain Rabi frequencies of 100’s of MHz for moderate magnetic ﬁeld strengths. Indeed, a Rabi frequency of Rabi= 1GHz is achieved at the peak of the spin-orbit enhancement for a magnetic ﬁeld B1:5T. We there- foreconcludethatincludingGeconcentrationoscillations in Si/SiGe quantum wells enables a dramatic speedup of single qubit gates using EDSR. D. Impact of alloy disorder SiGe is a random alloy, and the resulting alloy disor- derisknowntohaveaneﬀectonquantumwellproperties such as valley splitting, both for conventional quantum wells [34] and wiggle wells [18, 19]. It is therefore impor- tant to explore the eﬀects of alloy disorder on spin-orbit coupling. Recall from Sec. IIB that we have employed a virtual crystal approximation in our model, which aver- ages over all possible alloy realizations. While providing tractability to our calculations, this approximation ig- nores the ﬂuctuations arising from the random nature of the Ge atom arrangements in the SiGe alloy. In this sec- tion, we explain how Ge concentration ﬂuctuations can be reintroduced into our model, to explore the eﬀects of alloy disorder. A full 3D calculation including alloy disorder is com- putationally expensive due to the loss of translation in- variance, and is beyond the scope of this work. We canstill, however, include the eﬀects of alloy disorder ap- proximately within our 1D eﬀective model by allowing for ﬂuctuations in the Ge concentration in each atomic layer. Thisisaccomplishedusingtheproceduredescribed in Ref. [34], which can be summarized as follows. First, we assume a dot of radius adot=p ~=mk!= 20nm in the plane of the quantum well, where mk= 0:19meis the in-plane eﬀective mass and ~!= 1meV is the orbital ex- citation energy characterizing the parabolic conﬁnement of the dot. We then calculate the eﬀective Ge concentra- tionneﬀ Ge;ninlayernofourdisorderedsystembycounting the number of Ge atoms withinour dot. Here, the prob- ability of any given atom in layer nbeing a Ge atom isnGe;n, wherenGe;nis the average germanium concen- tration throughout the entire layer n, and the number of atoms in our dot is Neﬀ= 4a2 dot=a217100, where a= 0:543nmisthecubiclatticeconstantofSi. Theeﬀec- tive Ge concentration neﬀ Ge;nin layerncan then be drawn from the distribution neﬀ Ge;n=N1 eﬀBinom (Neﬀ;nGe;n), where Binom (n;p)is the binomial distribution with n trials and probability of success p. In the limit of Neﬀ!1, the resulting, randomized eﬀective Ge con- centrationneﬀ Ge;napproaches the ideal Ge concentration nGe;n, but for smaller dots, ﬂuctuations from this ideal limit become more pronounced. We then calculate the spin-orbit coeﬃcients for our eﬀective 1D model as in previous sections but with the Ge concentration proﬁle given byneﬀ Ge;ninstead ofnGe;n. Note that this method of including alloy disorder in the 1D eﬀective model was shown in Ref. [34] to yield valley splitting distributions in good agreement with 3D calculations. We now calculate the diagonal Dresselhaus coeﬃcients ``for the same Si/SiGe system as Fig. 4, with Ge con- centration oscillations of wavelength = 1:57nm, but now with Ge concentration ﬂuctuations included. Note that thisvalue corresponds to the peak of the main spin-orbit enhancement bump in Fig. 4(a). The distribu- tion of the spin-orbit coeﬃcients is plotted in Fig. 7(a) for1000random-alloy realizations, for the ground (blue, 00) and excited (red, 11) valleys. Unsurprisingly, we see that the alloy ﬂuctuations aﬀect the spin-orbit co- eﬃcients, with the ground valley coeﬃcient spanning the range377<00<355eVnm. The distribu- tions are highly peaked, however, near the barevalues 00=365eVnm and 11= 357eVnm of the disorder-free system (see Fig. 4). Indeed, we ﬁnd that 86%of the alloy realizations have j00j>200eVnm. We therefore conclude that the spin-orbit enhancement arisingfromtheGeconcentrationoscillationswithinwig- gle wells is robust against alloy ﬂuctuations. We gain further insight into the eﬀects of the alloy ﬂuctuations by studying the inter-valley spin-orbit coef- ﬁcient 01. Figure 7(b) presents a scatter plot showing boththediagonal( 00and11)andoﬀ-diagonal( 01)co- eﬃcients of all 1000alloy realizations. Interestingly, all realizations yield coeﬃcients that land in a narrow semi- circular region of parameter space. This is an indication that the alloy disorder is essentially mixing the original10 400 200 0 200 400 (eV nm) 020040001 (eV nm) (b)0100200counts(a) 00 11 FIG. 7. (a) Distribution of the diagonal Dresselhaus coef- ﬁcients ``of the wiggle well when including alloy disor- der as described in the main text. System parameters are = 1:57nm andFz= 10mV/nm. The data includes 1000 alloy realizations. We see that the spin-orbit coupling en- hancement is robust against alloy ﬂuctuations with 86%of alloy realizations having j00j>200eVnm. (b) Scatter plot of the diagonal [ 00(blue) and 11(red)] and oﬀ-diagonal (01) spin-orbit coeﬃcients of all alloy realizations. This re- sult indicates that the main eﬀect of the alloy disorder is to mix the ground and excited valley states. ground and excited valley states of the system without disorder. Note that perturbations arising from higher orbital subbands are insigniﬁcant, except for inducing a small width to the distribution. This is not unexpected since the energy diﬀerence between the ground and ex- cited valley states is <1meV, while the lowest orbital excitation energy is &20meV. For completeness, we also calculate the distribution of Dresselhaus spin-orbit coeﬃcients in conventional Si/SiGe quantum wells that include alloy ﬂuctuations. These results are shown in Fig. 8(a) for a system with no Ge in the well region and electric ﬁeld Fz= 10mV/nm. Here, the Ge concentration proﬁle is shown in the in- set. Again, we obtain ``distributions centered at the same values as the disorder-free system. (See Fig. 3, withFz= 10mV/nm.) Here, however, the spread is signiﬁcantly narrower than the wiggle well results shown in Fig. 7(a), with the full-width-at-half-maximum being 10eVnm. This is because no alloy ﬂuctuations occur in the well region where the majority of the wave func- tion resides. We also consider a system with nGe= 5% distributed uniformly throughout the well region, tak- ing into account the eﬀects of random alloy ﬂuctuations. [Here, we do not include intentional Ge concentration os- cillations; the resulting Ge concentration proﬁle is shown in the inset of Fig. 8(b).] Note that this system con- tains the same amount of Ge in the well region as the wiggle well system shown in Fig. 7. Unsurprisingly, we ﬁnd that the distribution of the ``coeﬃcients spreads considerably, compared to the results in Fig. 8(a), due to the presence of alloy ﬂuctuations inside of the well. 0100200counts(a)00 11 40 20 0 20 40 (eV nm) 0100200counts(b) z030nGe (%)z030nGe (%)FIG. 8. (a) Distribution of the diagonal Dresselhaus coeﬃ- cients ``of a conventional Si/SiGe system with a pure Si quantum well when including alloy disorder as described in the main text. The electric ﬁeld is Fz= 10mV/nm. (b) Same as (a) except that nGe;n= 5%throughout the well re- gion. Insets show the nGe;nproﬁles. We see that inclusion of Ge inside of the well region widens the distribution of ``but does not on average increase its magnitude. Importantly, however, a uniform Ge concentration in the well region does not on average increase the spin-orbit coupling, in contrast to the eﬀect on valley splitting. In- deed, we ﬁnd that j00j = 26eVnm for the data in Fig. 8(a), while j00j = 22eVnm for the data in Fig. 8(b). This highlights that fact that in order to enhance spin-orbit coupling, it is not enough to simply include Ge in the well region, but rather the Ge concen- tration must oscillate with the appropriate wavelength . We note that this result is consistent with the experi- mentalobservationthatthe g-factormeasuredinuniform Si1xGexalloys is only slightly altered by changing the Ge concentration [40, 41]. IV. MECHANISM BEHIND THE SPIN-ORBIT ENHANCEMENT Having shown through simulations that the inclusion of Ge concentration oscillations of an appropriate wave- length can signiﬁcantly enhance the spin-orbit coupling, the natural question is what mechanism leads to this en- hancement? In this section, we provide an explanation of this mechanism. To begin, we describe in Sec. IVA a simpliﬁed version of our model that allows for easier un- derstanding of the spin-orbit enhancement mechanism. Next, we study in Sec. IVB the real-space representa- tion of the ground valley wave function in the absence and presence of Ge concentration oscillations. We ﬁnd that a real-space picture is inadequate in explaining the spin-orbit enhancement. We therefore study in Sec. IVC the structure of the wave function in momentum space. We show how the combination of the oscillating potential produced by the Ge concentration oscillations and the se- lection rules of Dresselhaus spin-orbit coupling leads to11 the spin-orbit coupling enhancement. A. Simpliﬁed Model To focus on the essential physics for the spin-orbit coupling enhancement, we use in this section a model for SiGe alloys that is slightly simpliﬁed compared to the model presented in Sec. II and used in our numeri- cal calculations in Sec. III. In this model, Ge atoms are assumed to be identical to Si atoms, except for their orbitals being shifted up in energy by a constant, i.e. "(Ge) = "(Si) +EGewhereEGe= 0:8eV is the extra en- ergy of every Ge orbital. This is meant to capture at the simplest level that inclusion of Ge increases the energy of the conduction band minima. In particular, the cho- sen value produces a band oﬀset of 0:24eV between a pure Si region and a barrier region with a nominal 30% Ge concentration. Note, however, that the precise value is not important since we are only using this simpliﬁed model to understand the spin-orbit enhancement mecha- nism, leaving quantitative questions to the more accurate model of Sec. II. In addition, we also neglect the eﬀects of strain, such that C(n)!0, since they are not crucial inunderstandingthespin-orbitenhancementmechanism. With these simpliﬁcations, the addition of Ge is equiv- alent to adding a term to the potential energy Vof a pure Si system. For simplicity, we deﬁne a new potential Vn=eFzzn+EGenGe;n, that includes both the electric ﬁeldFzand the energy shift from the Ge concentration nGe;nof the layer, and we let all orbitals energies take values appropriate for Si: "(n) !"(Si) . Here,zn=na=4 is thezcoordinate of atomic layer n, anda= 0:543nm is the cubic lattice constant of Si. Importantly, in this simpliﬁed model, the onsite orbital energies and hop- ping matrices all lose their dependence on the layer indexn,n "(n) ;eT(n) 0;eT(n) R(kk);eT(n) D(kk); (n);(n)o !n "(Si) ;eT0;eTR(kk);eTD(kk); ;o . The Hamiltonian com- ponents then take the simpliﬁed forms hmjH(z) 0jni=n m "(Si) +Vn +n+1 m +n1 m T ; (27) hmjH(z) Rjni=n+1 m+n1 mT ; (28) hmjH(z) Djni=(1)n n+1 mn1 mT :(29) Note that the numerical values of the andmatrices used in the above equations are speciﬁed in Appendix D. B. Real-space wave functions Let us now study the eﬀects of the Ge concentration oscillations on the wave functions of the subband Hamil- tonianH(z) 0. To do so, we ﬁrst calculate the ground valley wave function of a conventional Si/SiGe quantum \|\|2 (a) 0 5 10 15 z (nm)\|\|2 (b)01530 nGe (%) 01530 nGe (%)FIG. 9. Comparison of the ground valley wave function for a “conventional” Si/SiGe system with a pure Si well region (a) andawigglewell(b)withoscillationwavelength = 1:62nm. Blue lines show the Ge concentration proﬁles while red ﬁlled in curves are the wave functions j j2. Note that we sum over orbital indices. The electric ﬁeld is Fz= 5mV/nm in both (a) and (b). We see in (b) that the wave function is suppressed in regions of high Ge concentration, consistent with the fact thattheconductionbandminimaarehigherinenergyinthose regions. well that has the Ge concentration proﬁle shown as the blue line in Fig. 9(a). The ground valley wave func- tionj j2of the subband Hamiltonian H(z) 0is shown in red in Fig. 9(a), where the state is pushed up against the barrier-well interface by an electric ﬁeld of strength Fz= 5mV/nm. The wave function also exhibits fast oscillations characteristic of the superposition of the two valley minima [32]. We next consider the same system, except with Ge concentration oscillations of wavelength = 1:62nm included in the well region, as shown by the blue line in Fig. 9(b). Comparing the two cases, we see that the wave function is suppressed in regions of high Ge concentration, as consistent with the fact that the conduction-band minima (and thus the local potential energies) are higher in energy in those regions. However, these qualitative observations do not directly explain the enhancement of Dresselhaus spin-orbit coupling observed in Figs. 4 and 5. C. Momentum-space wave functions WegainaclearerunderstandingoftheeﬀectsoftheGe concentration oscillations by studying the momentum- space representation of the ground valley. To begin, let us ﬁrst investigate the band structure of Si at kk= 0, which is shown as the orange dashed lines in Fig. 10. A central feature of the band structure is the presence of two degenerate valleys near zero energy that give rise to the ground and excited valley states in a quantum well. In addition, the valley minima are located only a short distance of 0:17(2=a)away from the Brillouin zone edge at kz=2=a. Interestingly, we notice that atkz=2=a, there are no band anti-crossings (only12 4/a 2/a 02/a 4/a kz10 5 05E (eV) FIG. 10. The band structure of Si for kk= 0. The band structure for the conventional Brillouin zone extending to kz=2=ais shown by dashed orange lines. The presence of bandcrossings at the zone edge kz=2=aindicates that the Brillouin zone can be enlarged. Indeed, the equivalence of the two sublattices of Si for kk= 0allows us to extend the Brillouin zone to kz=4=a. The band structure for the extended zone is shown as solid black lines. See the main text for details on the green, blue, and red points. crossings). This is an indication that the Brillouin zone can be enlarged for kk= 0. Indeed, for the special case ofkk= 0, the two sublattice hopping matrices become equal,eT+(0) =eT(0), and the primitive unit cell of the 1D chain shown in Fig. 2(b) reduces from two sites to one. The Brillouin zone should therefore extend to kz= 4=ainstead ofkz=2=a. Here, we will refer to the Brillouin zone that extends to kz=4=aas the extended zone , while the zone extending only to kz= 2=aas theconventional zone . To calculate the band structure in the extended zone, we deﬁne a plane wave basis state with momentum kzas jkzi=1pNzX neikzznjni; (30) whereNzis the number of sites in the 1D chain and 4=a < kz4=a, wherekz= 8n=(Nza). The sub- band Hamiltonian H(z) 0then has matrix elements, hkzjH(z) 0jk0 zi=k0 z kz "(Si) +eikza 4 +eikza 4 T + eV(kzk0 z); (31) whereeV(kzk0 z)is the Fourier transform of the poten- tialVand is given by eV(qz) =N1 zP nexp(iqzzn)Vn. Notice that in the absence of the potential [ eV(qz)!0], kzis a good quantum number for H(z) 0, and Eq. (31) represents a Bloch Hamiltonian. The spectrum of H(z) 0 foreV(qz) = 0is shown as the solid black lines in Fig. 10, whichwerefertoastheextendedbandstructure. Within the conventional zone, we see that the extended band \|\| (a) 4/a 2/a 02/a 4/a kz \|\| 2 4/a (b)FIG. 11. Wave-function proﬁles of the ground valley in the plane-wave representation. (a), (b) Correspond to Ge proﬁles (a) and (b) in Fig. 9, respectively. Both states show main peaks centered at the valley minima, kz0:83(2=a), as shown in Fig. 10. As shown in (b), however, the Ge concen- trationoscillationsproducewave-functionsatellitesadistance 2=awayfromthemainpeaks. Thewavevectorfromamain peak to the opposite outer satellite is 4=afor this choice of . Since Dresselhaus spin-orbit coupling connects locations in reciprocal space separated by 4=a, spin-orbit enhancement occurs when a satellite peak is separated from a main peak by4=a. structure aligns perfectly with half of the conventional (orange) bands, while the other half of the bands have been removed from the conventional zone and instead re- side in the regions between kz=j2=ajandj4=aj. Fur- thermore, we observe that the conventional bands that do not match with the extended band structure can be made to match by shifting them by a reciprocal lattice vectorGz=4=aof the lattice containing two sites. For example, the green point at kz==ain Fig. 10 gets shifted to the blue point at kz= 3=a, which coin- cides with a band of the extended band structure. This is expected since the points kzandkz+Gzare equivalent from the point of view of the conventional zone [42]. We stress that the extended band structure contains more information than the conventional band structure. Indeed, the extended scheme clariﬁes which states can be coupled by a given Fourier component qzof the po- tentialeV(qz). For example, let us consider what Fourier component of the potential could couple the states of the conventional band structure marked by the green and red points at kz==aand1:3=a, respectively, in Fig. 10. Looking at the conventional band structure, one may initially believe the eV(0:3=a)Fourier component could couple the states. Looking at these states in the extended band structure (red and blue points in Fig. 10), however, we immediately see that these states are instead coupled by the eV(3:7=a)Fourier component. This ad- ditional information will be crucial in understanding the enhanced spin-orbit coupling mechanism below. Returning to our comparison of the systems with and without Ge concentration oscillations, we now plot the ground valley wave functions in the plane-wave repre-13 sentation. These are shown in Figs. 11(a) and 11(b) and correspond to the real-space wave functions shown in Figs. 9(a) and 9(b), respectively. That is, we plot (P jhkzj'`ij2)1=2as a function of kzwherej'`iis the ground valley wave function. As shown in Fig. 11(a), the ground valley wave function of the conventional Si/SiGe heterostructure consists of two peaks centered at kz 0:83(2=a). These coincide with the conduction-band minima in Fig. 10(a) as expected. Note that the posi- tions of these minima in the band structure of Si depends on the exact tight binding parameters used, and diﬀers slightly from other models in the literature. As shown in Fig. 11(b), the ground valley wave fuction in the pres- ence of the Ge oscillations still has its main peaks but also includes surrounding satellite features. The location of these satellites with respect to the central peaks is de- termined by the wavelength of the Ge oscillations, with a peak-satellite separation of 2=, as shown in Fig. 11(b). The second step in explaining the enhanced spin-orbit coupling requires a mechanism for coupling diﬀerent re- gions of the Brillouin zone. We ﬁrst express the ma- trix elements of the Rashba H(z) Rand Dresselhaus H(z) D Hamiltonian components in the plane wave basis, giving hkzjH(z) Rjk0 zi=k0 z kz eikza 4+eikza 4T ;(32) hkzjH(z) Djk0 zi=4=a jkzk0zj eikza 4eikza 4T : (33) The key features to notice here are the selection rules be- tweenkzandk0 z: whileH(z) Rconserveskz,H(z) Dcouples states with momenta diﬀering by 4=a. The latter result is obtained by Fourier transforming the (1)nfactor in Eq. (29), which itself is a manifestation of the alternat- ing sign in front of eT(n) Din Eq. (11). To our knowledge, these selection rules and their relation to Rashba and Dresselhaus spin-orbit couplings have not been noticed previously, although we speculate that they could be de- duced from group-theory methods, such as the method of invariants [16, 43]. We emphasize, however, that the extended zone scheme is key to obtaining such results, since in the conventional zone scheme, kzvalues diﬀering by4=aare equivalent, and cannot yield a selection rule like Eq. (33). We also note that Eqs. (32) and (33) apply to any system with a diamond crystal structure. The Dresselhaus momentum selection rule in Eq. (33) indicates that the spin-orbit coupling will be enhanced when wave-function peaks associated with two diﬀerent valleys are separated by 4=a. In Fig. 11(b), we see that this can only occur for coupling between a central valley peak and the outer satellite associated with the oppo- site valley. Since the central valley peaks are located at kz0:83(2=a), we see that the resonance condition for the oscillation wavelength to enhance the Dresselhaus spin-orbit is given by res= 2:94a= 1:59nm. Uncoinci- dentally, thiswavelengthvalueisveryclosetothepeakof thebumpcenteredat 1:57nmofournumericalresult in Fig. 4(a) where the Dresselhaus spin-orbit coupling issigniﬁcantly enhanced. Note that similar considerations apply to the excited valley. Our understanding of the spin-orbit enhancement mechanism also explains the narrower features observed in Fig. 5 for a weak vertical electric ﬁeld Fz= 2mV/nm, as compared to the wider features observed in Fig. 4 for a strong electric ﬁeld Fz= 10mV/nm. Essen- tially, a weaker electric ﬁeld produces wave functions with narrower features in the plane-wave representation than those from a strong electric ﬁeld. These thinner features make it more diﬃcult to satisfy the Dresselhaus resonance condition since the narrower peaks need to be situated more precisely in kzspace. Finally, we comment that the spin-orbit enhancement mechanism relies fundamentally on the degeneracy of the twozvalleys in the band structure of strained Si. Such a situation could not occur if, for example, Si was a direct bandgap semiconductor with a single non-degenerate val- ley at the point, since the key coupling in Fig. 11(b) occurs between the central peak of one valley and the outer satellite of the opposite valley. Interestingly, in this case the valley degeneracy of Si can be considered as ben- eﬁcial, while in other scenarios it is often considered to be problematic. In addition, the spin-orbit enhancement mechanism has similarities to holes in semiconductors, for which the enhancement arises due to degeneracy at the valence-band edge. In contrast to the valence-band case, however, which has degenerate bands at the same momentum, the degenerate zvalleys in Si require a peri- odicpotentialintheformofGeconcentrationoscillations to enhance the coupling strength. V. CONCLUSIONS We have shown that the inclusion of periodic Ge con- centrations oscillations within the quantum well region of a Si/SiGe heterostructure leads to enhanced spin-orbit coupling when the oscillation wavelength is properly chosen. Speciﬁcally, we ﬁnd that the Dresselhaus spin- orbit coupling coeﬃcient is enhanced by over an order of magnitude when 1:57nm, as shown in Figs. 4 and 5. We have provided a detailed explanation for this be- havior: the Ge concentration oscillations produce wave function satellites in momentum space which can cou- ple strongly to the valley minima through Dresselhaus spin-orbit coupling provided that the satellite-valley sep- aration is approximately 4=ain the extended Brillouin zone as shown in Fig. 11. Importantly, the region of en- hancement in Fig. 4 is quite wide in space, which has the important implication that the wiggle well structure should allow for rather large growth errors in the Ge con- centration proﬁle while maintaining the enhanced spin- orbit eﬀect. Additionally, our results indicate that the spin-orbit enhancement is robust against alloy disorder, as shown in Fig. 7. Enhancement of both the Dresselhaus and Rashba co- eﬃcients at smaller values have also been found in14 Figs. 4(c) and 4(d), although these bumps are much nar- rower in width than the 1:57nm bump, making such structures more challenging to fabricate. Assuming that the wiggle well with 0:27nm can be practi- cally realized, however, this period is quite attractive as it could provide the enhanced Dresselhaus spin-orbit cou- pling studied here along with a huge deterministic valley splitting [18, 19]. With regards to possible applications, the enhanced spin-orbit coupling of the wiggle well indicates that EDSR can be used for fast, electrically-driven manipula- tions of single-spin, Loss-DiVincenzo qubits without the use of micromagnets. Indeed, a fast, spin-orbit driven EDSR capability is one of the main attractive features of hole-spin qubits [17, 44–47], and has recently also attracted interest in Si electron-spin qubits [48]. This possibility is supported by our calculations in Sec. III, where it was shown that an EDSR Rabi frequency of Rabi=B > 500MHz/T can be obtained near the opti- mal Ge oscillation wavelength = 1:57nm. It is also possible that the enhanced spin-orbit coupling between the valleys may be used to drive fast singlet–triplet ro- tations near the valley-Zeeman hot spot [49]. Finally, we mention that the enhanced and spatially varying spin- orbit coupling may have interesting eﬀects on many-body physics in multi-electron dots [33, 50–53]. ACKNOWLEDGEMENTS Research was sponsored in part by the Army Research Oﬃce (ARO) under Award No. W911NF-17-1-0274 and No. W911NF-22-1-0090. The views, conclusions, and recommendationscontainedinthisdocumentarethoseof the authors and are not necessarily endorsed nor should they be interpreted as representing the oﬃcial policies, either expressed or implied, of the Army Research Of- ﬁce (ARO) or the U.S. Government. The U.S. Govern- ment is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein. Appendix A: Lattice constants in strained Si/SiGe quantum wells To determine the lattice constants of the strained Si/SiGe heterostructure, we use pseudomorphic bound- ary conditions [31, 54], where the in-plane lattice con- stantakthroughout the system is given by the relaxed lattice constant of the Si 0.7Ge0.3barrier regions. This then also sets the lattice spacing along the growth direc- tion as described below. We take the relaxed lattice constant of a Si 1-xGexalloy as ao(x) = (1x)aSi+xaGe; (A1)whereaSi= 0:5431nm andaGe= 0:5657nm are the relaxed lattice constants of Si and Ge, respectively. Our structure therefore has an in-plane lattice constant ak=ao(0:3) = 0:5499nm throughout the entire sys- tem. Atomic layers in our system with Ge concentration nGe;n6= 0:3are therefore strained. Explicitly, the in- plane strain k;nof layernis k;n=akao(nGe;n) ao(nGe;n): (A2) For a bulk Si1-x0Gex0alloy under biaxial stress perpen- dicular to the [001], we have strains xx=yy=kand zz=?that are related by [54] ?(x0) =2C12(x0) C11(x0)k; (A3) whereC12andC11are elastic constants that depend on the Ge concentration x0. For pure Si ( x= 0), we haveC11(0) = 165:8GPa andC12(0) = 63:9GPa, while for pure Ge ( x= 1), we have C11(1) = 131:8GPa and C12(1) = 48:3GPa [28]. For simplicity, we assume that the elastic constants vary linearly with the Ge concentra- tion For a well region composed of Si1-x0Gex0, we would then have the out-of-plane lattice constant, a?(x0) = 1 +?(x0) ao(x0): (A4) In our system with its inhomogeneous Ge concentration proﬁle, we take the lattice spacing between layers nand n+ 1with Ge concentrations nGe;nandnGe;n+1, respec- tively, asa(n+1;n) ?=4, where a(n+1;n) ?=1 2 a?(nGe;n+1) +a?(nGe;n) ;(A5) is the average of the out-of-plane lattice constants ex- pected for strained regions with Ge concentration nGe;n andnGe;n+1, respectively. Appendix B: Pseudospin basis transformation details In this appendix, we provide details of the pseudospin basis introduced in Sec. IIB of the main text. The pseu- dospin basis is deﬁned as kkn =X kkn U(n) ;; (B1) where andlabel the new orbitals with =;+being apseudospin label,andare indices of the original basis with=";#simply denoting spin, and U(n)is the transformation matrix of layer n. Our ﬁrst requirement of the new basis is that it diag- onalizes the onsite spin-orbit coupling. Following Chadi [29], spin-orbit coupling is taken to be an intra-atomic (onsite) coupling between porbitals and enters into the15 Hamiltonian as the matrix S. The explicit form of the Smatrix of atom jin atomic layer nin thep-orbital subspacefjpz"i;jpx#i;jpy#i;jpz#i;jpx"i;jpy"igis S(nj)=(nj) SO 30 BBBBB@01i0 0 0 1 0i0 0 0 ii0 0 0 0 0 0 0 0 1 i 0 0 0 1 0i 0 0 0i i 01 CCCCCA;(B2) where (nj) SOis the spin-orbit energy. All matrix elements involvings,s, anddorbitals are set to zero in Sand are not shown. Note that spin-orbit coupling, in prin- ciple, does exist between d-orbitals, but is much smaller than thep-orbital couplings and is typically neglected [27]. Also note that the spin-orbit energy depends on if the atom is Si or Ge, but the form of the spin-orbit coupling matrix is independent of atom type. It turns out that the Smatrix is diagonalized by the eigenstates of total angular momentum [16]. Within the p-orbital subspace, these states are given by jp1i=ip 2(jpx#iijpy#i); (B3) jp2i=1p 6(2jpz"ijpx#iijpy#i);(B4) jp3i=1p 3(jpz"i+jpx#i+ijpy#i);(B5) jp1+i=ip 2(jpx"i+ijpy"i); (B6) jp2+i=1p 6(2jpz#i+jpx"iijpy"i);(B7) jp3+i=1p 3(jpz#ijpx"i+ijpy"i);(B8) where;+arepseudospin labels. Weﬁndjp1iandjp2i both have an Smatrix eigenvalue of (nj) SO=3, whilejp3i has an eigenvalue of 2(nj) SO=3. We also deﬁne pseu- dospinsandsorbitals as jsi=js"i; (B9) jsi=js"i; (B10) js+i=js#i; (B11) js+i=js#i; (B12) and pseudospin d-orbitals as jd1i=ip 2(jdzx#iijdyz#i); (B13) jd2i=1p 6(2jdz2"ijdzx#iijdyz#i);(B14) jd3i=1p 3(jdz2"i+jdzx#i+ijdyz#i);(B15) jd4i=jdxy"i; (B16) jd5i=dx2y2" ; (B17)jd1+i=ip 2(jdzx"i+ijdyz"i); (B18) jd2+i=1p 6(2jdz2#i+jdzx"iijdyz"i);(B19) jd3+i=1p 3(jdz2#ijdzx"i+ijdyz"i)(B20) jd4+i=jdxy#i; (B21) jd5+i=dx2y2# : (B22) Note that all pseudospin s,s, andd-orbitals are triv- ially eigenstates of the spin-orbit matrix Swith eigen- value 0. In addition, note that for j= 1;2;3,jdjiand jdj+iare found fromjpjiandjpj+i, respectively, by lettingpx;py;pz!dzx;dyz;dz2. The other pseudospin dorbitals and sorbitals are trivially related to the orig- inal basis. Note that we adopt these altered d-orbital pseudospin states even though they possess no spin-orbit coupling such that we obtain the pseudospin structure of the hopping matrices in Eqs. (12 - 14) of the main text. Failure to adopt these d-orbital pseudospin states would result in coupling between the pseudospin sectors forkk= 0. Secondly, we require the pseudospin basis to transform the Hamiltonian in such a way that the minimum unit cell (in the absence of an inhomogenous potential Vn) decreases from two sites to one site for kk= 0. In other words, the Fourier-transformed hopping matrix, which is introduced in Eq. (9) of the main text, must become site independent for kk= 0. Naively adopting the orbitals deﬁned in Eqs. (B3 - B22) for every site does not fulﬁll this requirement. However, this requirement is fulﬁlled if we adopt the orbitals deﬁned in Eqs. (B3 - B22) if the jp1i,jp1+i,jd1i,jd1+i,jd4i, andjd4+i, orbitals are multiplied by (1)n, wherenis the site index in the 1D chain. In other words, we ﬂip the sign of these select orbitals on every other site. As stated in the main text, this alternating structure for the transformation matrix is due to the presence of two sublattices in the diamond crystal structure of Si as shown in Fig. 2 (a). For clarity, we now provide the explicit form of U(n). We writeU(n)as U(n)=0 B@U(n) s 0 0 0U(n) p 0 0 0U(n) d1 CA; (B23) whereU(n) s,U(n) p, andU(n) dare matrix blocks which de- scribehowthe s,p, anddorbitalstransform, respectively. Here,U(n) s=I44is just identity. The pblock is given16 by U(n) p=0 BBBBBBBBB@02p 61p 30 0 0 0 0 0i(1)n p 21p 61p 3 0 0 0(1)n+1 p 2ip 6ip 3 0 0 0 02p 61p 3 i(1)n p 21p 61p 30 0 0 (1)n p 2ip 6ip 30 0 01 CCCCCCCCCA;(B24) where the columns correspond to the pseudospin orbitals in the orderfp1;p2;p3;p1+;p2+;p3+g, and the rows correspond to the “standard” orbitals in the order fpz";px";py";pz#;px#;py#g. Finally, the dblock is U(n) d=0 BBBBBBBBBBBBBBBBB@02p 61p 30 0 0 0 0 0 0 0 0 0 0 0i(1)n p 21p 61p 30 0 0 0 0 0 0(1)n+1 p 2ip 6ip 30 0 0 0 0 (1)n0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 02p 61p 30 0 i(1)n p 21p 61p 30 0 0 0 0 0 0 (1)n p 2ip 6ip 30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ( 1)n0 0 0 0 0 0 0 0 0 0 11 CCCCCCCCCCCCCCCCCA; (B25) where the columns correspond to the pseudospin orbitals in the orderfd1;d2;d3;d4;d5;d1+;d2+;d3+ ;d4+;d5+g, and the rows correspond to the “standard” orbitals in the order fdz2";dzx";dyz";dxy";dx2y2" ;dz2#;dzx#;dyz#;dxy#;dx2y2#g. Notice the (1)n factors that ﬂips the signs of the jp1i,jp1+i,jd1i, jd1+i,jd4*i, andjd4+iorbitals on every odd site. This allows for a unit cell containing only one site for kk= 0 as described above. Appendix C: Symmetry argument for the hopping structure of the Rashba and Dresselhaus Hamiltonian components In Sec. IIC of the main text, we found the Fourier transformed hopping matrix eT(n) (kk)to have the form eT(n) (kk) =eT(n) o+eT(n) R(kk)eT(n) D(kk) +O(k2 k);(C1) whereeT(n) R(kk)andeT(n) D(kk)are the Rashba and Dres- selhaus hopping matrices, respectively, and are given by eT(n) R(kk) = (n)(kyxkxy);(C2) eT(n) D(kk) = (n)(kxxkyy);(C3)with (n)being a real-valued 1010matrix with van- ishing diagonal elemental. Importantly, the sign in front of the Rashba hopping matrix eT(n) Ris site independent, while the sign of the Dresselhaus hopping matrix eT(n) D changes sign between every site, as indicated by the in Eq. (C1). This can be understood as originating from the diamond crystal structure of the Si by the follow- ing symmetry argument; Let us consider the case of pure Si such thatn (n);eT(n) R;eT(n) Do !n ;eTR;eTDo all lose their dependence on the layer index. Next, note that under aC4rotation about the z-axis (growth axis), we havefkx;ky;x;yg!fky;kx;y;xg. This leaves invariant the Rashba term in Eq. (C2) and ﬂips the sign of the Dresselhaus term in Eq. (C3). Finally, performing the sameC4rotation on our diamond crystal structure in Fig. 2 (a) transforms red atoms into blue atoms and vice versa in the sense that the nearest neighbor vectors of even and odd atomic layers swap. In other words, the two sublattices of the Si lattice swap. This in turn, swaps eT+andeTwithin the tight binding chain. Clearly then, the Rashba hopping term eTR, being invariant under a C4rotation, should contain the part common to eT+and eT. Incontrast, theDresselhaushoppingterm eTDshould contain the part which is diﬀerent between eT+andeT,17 since it ﬂips sign under a C4rotation. This then explains thein front ofeT(n) Din Eq. (C1). Appendix D: and matrices In Sec. IIC of the main text, we introduced the 1010 matrices (n)and(n)as components of the hopping matrices in Eqs. (12 - 14). These matrices can be further decomposed into the block forms, (n)=2 64 (n) 00 (n) 01 0 (n)T 01 (n) 11 0 0 0 (n) 223 75; (D1) (n)=2 64(n) 00 (n) 01(n) 02 (n)T 01 (n) 11(n) 12 (n)T 02 (n)T 12 03 75;(D2) where the shapes of the diagonal blocks are 55, 44, and 11, respectively, and the diagonal block matrices satisfy (n)T ii = (n) iiand (n)T ii =(n) ii. Note that this implies that all diagonal elements of (n)are zero. Here, the orbital ordering used is fs;s;p1;d2;d3;p2;p3;d1;d4;d5g. Generically, these ma- trices depend on the layer index ndue to Ge concentra- tion changing from layer to layer. In the case of a uni- form Ge concentration, however, this layer dependence goes away, (n);(n) !f ;g. In the particular case of an unstrained Si system, the matrix blocks (in eV)are given by 00=0 BBB@3:732:78 0 0 0 2:789:03 0 0 0 0 00:731:71 2:42 0 01:71 1:63 2:03 0 0 2 :42 2:03 0:191 CCCA;(D3) 11=0 B@0:73 0 1 :712:42 0 0:732:421:71 1:712:42 1:24 0 2:421:71 0 1 :241 CA; (D4) 22= 3:06; (D5) 01=0 BBB@2:74 1:94 02:59 2:89 2:05 00:90 2:153:04 0:19 0 2:071:60 0:702:92 1:60 0:940:992:071 CCCA; (D6) and the matrix blocks (in eV Å) are given by 00=0 BBB@0 0 3 :23 1:432:03 0 0 3 :40 0:500:70 3:233:40 01:250:89 1:430:50 1:25 0 1 :72 2:03 0:70 0:891:72 01 CCCA;(D7) 11=0 B@0 3:570:150:10 3:57 00:1 0:15 0:150:1 01:16 0:100:15 1:16 01 CA; (D8) 02=0 0 2:66 1:72 2:43 ; (D9) 12=1:54 2:17 2:98 0 : (D10) Note that these andare precisely the matrices used in the simpliﬁed model of Sec. IV, where Ge atoms are treated as Si atoms with orbitals shifted up by a constant energyEGe. [1] D. Loss and D. P. DiVincenzo, Quantum computation with quantum dots, Phys. Rev. A 57, 120 (1998). [2] R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha, and L. M. K. Vandersypen, Spins in few-electron quan- tum dots, Rev. Mod. Phys. 79, 1217 (2007). [3] G. Burkard, T. D. Ladd, J. M. Nichol, A. Pan, and J.R.Petta,Semiconductorspinqubits,arXiv:2112.08863 (2021). [4] A.Chatterjee, P.Stevenson, S.D.Franceschi, A.Morello, N. P. de Leon, and F. Kuemmeth, Semiconductor qubits in practice, Nature Reviews Physics 3, 157 (2021). [5] X. Zhang, H.-O. Li, G. Cao, M. Xiao, G.-C. Guo, and G.- P. Guo, Semiconductor quantum computation, National Science Review 6, 32 (2019). [6] F. A. Zwanenburg, A. S. Dzurak, A. Morello, M. Y. Sim- mons, L. C. L. Hollenberg, G. Klimeck, S. Rogge, S. N. Coppersmith, and M. A. Eriksson, Silicon quantum elec- tronics, Rev. Mod. Phys. 85, 961 (2013).[7] M. F. Gyure, A. A. Kiselev, R. S. Ross, R. Rahman, and C. G. V. de Walle, Materials and device simulations for silicon qubit design and optimization, MRS Bulletin 46, 634 (2021). [8] J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, Coherent manipulation of coupled electron spins in semiconductor quantum dots, Science 309, 2180 (2005). [9] L. V. C. Assali, H. M. Petrilli, R. B. Capaz, B. Koiller, X. Hu, and S. Das Sarma, Hyperﬁne interactions in sili- con quantum dots, Phys. Rev. B 83, 165301 (2011). [10] M. Pioro-Ladrière, T. Obata, Y. Tokura, Y.-S. Shin, T.Kubo,K.Yoshida,T.Taniyama,andS.Tarucha,Elec- trically driven single-electron spin resonance in a slanting Zeeman ﬁeld, Nat. Phys. 4, 776 (2008). [11] X. Xue, M. Russ, N. Samkharadze, B. Undseth, A. Sam- mak, G.Scappucci,andL.M.K.Vandersypen,Quantum logicwithspinqubitscrossingthesurfacecodethreshold,18 Nature601, 343 (2022). [12] A.Noiri,K.Takeda,T.Nakajima,T.Kobayashi,A.Sam- mak, G. Scappucci, and S. Tarucha, Fast universal quan- tum gate above the fault-tolerance threshold in silicon, Nature601, 338 (2022). [13] A. R. Mills, C. R. Guinn, M. J. Gullans, A. J. Sigillito, M. M. Feldman, E. Nielsen, and J. R. Petta, Two-qubit silicon quantum processor with operation ﬁdelity exceed- ing 99%, Science Advances 8, eabn5130 (2022). [14] V.N.Golovach,M.Borhani,andD.Loss,Electric-dipole- induced spin resonance in quantum dots, Phys. Rev. B 74, 165319 (2006). [15] E. I. Rashba, Theory of electric dipole spin resonance in quantum dots: Mean ﬁeld theory with gaussian ﬂuctua- tions and beyond, Phys. Rev. B 78, 195302 (2008). [16] R. Winkler, Spin-Orbit Coupling Eﬀects in Two- Dimensional Electron and Hole Systems (Springer,2003). [17] L. A. Terrazos, E. Marcellina, Z. Wang, S. N. Copper- smith, M. Friesen, A. R. Hamilton, X. Hu, B. Koiller, A. L. Saraiva, D. Culcer, and R. B. Capaz, Theory of hole-spin qubits in strained germanium quantum dots, Physical Review B 103, 125201 (2021). [18] T. McJunkin, B. Harpt, Y. Feng, M. P. Losert, R. Rah- man, J. P. Dodson, M. A. Wolfe, D. E. Savage, M. G. Lagally, S. N. Coppersmith, M. Friesen, R. Joynt, and M. A. Eriksson, SiGe quantum wells with oscillating Ge concentrations for quantum dot qubits, Nature Commu- nications13, 7777 (2022). [19] Y. Feng and R. Joynt, Enhanced valley splitting in Si layers with oscillatory Ge concentration, Phys. Rev. B 106, 085304 (2022). [20] M. O. Nestoklon, E. L. Ivchenko, J.-M. Jancu, and P. Voisin, Electric ﬁeld eﬀect on electron spin splitting in SiGe=Siquantum wells, Phys. Rev. B 77, 155328 (2008). [21] M. Prada, G. Klimeck, and R. Joynt, Spin–orbit split- tings in Si/SiGe quantum wells: from ideal Si membranes to realistic heterostructures, New Journal of Physics 13, 013009 (2011). [22] Y. A. Bychkov and E. I. Rashba, Oscillatory eﬀects and the magnetic susceptibility of carriers in inversion layers, JournalofPhysicsC:SolidStatePhysics 17,6039(1984). [23] G. Dresselhaus, Spin-orbit coupling eﬀects in zinc blende structures, Phys. Rev. 100, 580 (1955). [24] E. L. Ivchenko, A. Y. Kaminski, and U. Rössler, Heavy- light hole mixing at zinc-blende (001) interfaces under normal incidence, Phys. Rev. B 54, 5852 (1996). [25] L. Vervoort, R. Ferreira, and P. Voisin, Eﬀects of in- terface asymmetry on hole subband degeneracies and spin-relaxation rates in quantum wells, Phys. Rev. B 56, R12744 (1997). [26] J. C. Slater and G. F. Koster, Simpliﬁed LCAO method for the periodic potential problem, Phys. Rev. 94, 1498 (1954). [27] J.-M. Jancu, R. Scholz, F. Beltram, and F. Bassani, Em- pirical spdstight-binding calculation for cubic semicon- ductors: Generalmethodandmaterialparameters,Phys. Rev. B57, 6493 (1998). [28] Y. M. Niquet, D. Rideau, C. Tavernier, H. Jaouen, and X. Blase, Onsite matrix elements of the tight-binding hamiltonian of a strained crystal: Application to silicon, germanium, and their alloys, Phys. Rev. B 79, 245201 (2009). [29] D. J. Chadi, Spin-orbit splitting in crystalline and com- positionally disordered semiconductors, Phys. Rev. B 16,790 (1977). [30] S. Y. Ren, J. D. Dow, and D. J. Wolford, Pressure de- pendence of deep levels in GaAs, Phys. Rev. B 25, 7661 (1982). [31] F. Schäﬄer, High-mobility Si and Ge structures, Semi- conductor Science and Technology 12, 1515 (1997). [32] M. Friesen, S. Chutia, C. Tahan, and S. N. Coppersmith, Valley splitting theory of SiGe=Si=SiGequantum wells, Phys. Rev. B 75, 115318 (2007). [33] C. Yannouleas and U. Landman, Valleytronic full conﬁguration-interaction approach: Application to the excitation spectra of Si double-dot qubits, Phys. Rev. B 106, 195306 (2022). [34] B. P. Wuetz, M. P. Losert, S. Koelling, L. E. A. Ste- houwer, A.-M. J. Zwerver, S. G. J. Philips, M. T. Mądzik, X. Xue, G. Zheng, M. Lodari, S. V. Amitonov, N. Samkharadze, A. Sammak, L. M. K. Vandersypen, R. Rahman, S. N. Coppersmith, O. Moutanabbir, M. Friesen, and G. Scappucci, Atomic ﬂuctuations lifting the energy degeneracy in si/sige quantum dots, Nature Communications 13, 7730 (2022). [35] M. O. Nestoklon, L. E. Golub, and E. L. Ivchenko, Spin and valley-orbit splittings in SiGe=Siheterostructures, Phys. Rev. B 73, 235334 (2006). [36] M. Veldhorst, R. Ruskov, C. H. Yang, J. C. C. Hwang, F. E. Hudson, M. E. Flatté, C. Tahan, K. M. Itoh, A. Morello, and A. S. Dzurak, Spin-orbit coupling and operation of multivalley spin qubits, Phys. Rev. B 92, 201401(R) (2015). [37] R. Ruskov, M. Veldhorst, A. S. Dzurak, and C. Tahan, Electrong-factor of valley states in realistic silicon quan- tum dots, Phys. Rev. B 98, 245424 (2018). [38] T. B. Boykin, G. Klimeck, M. A. Eriksson, M. Friesen, S. N. Coppersmith, P. von Allmen, F. Oyafuso, and S. Lee, Valley splitting in strained silicon quantum wells, Applied Physics Letters 84, 115 (2004). [39] J. S. Townsend, A Modern Approach to Quantum Me- chanics(University Science Books, 2012). [40] W. Jantsch, Z. Wilamowski, N. Sandersfeld, M. Mühlberger, and F. Schäﬄer, Spin lifetimes and g-factor tuning in Si/SiGe quantum wells, Physica E: Low-dimensional Systems and Nanostructures 13, 504 (2002). [41] H. Malissa, W. Jantsch, M. Mühlberger, F. Schäﬄer, Z. Wilamowski, M. Draxler, and P. Bauer, Anisotropy of g-factor and electron spin resonance linewidth in modu- lation doped sige quantum wells, Applied Physics Letters 85, 1739 (2004). [42] C. Kittel and P. McEuen, Introduction to solid state physics(John Wiley & Sons, 2019). [43] P. Li and H. Dery, Spin-orbit symmetries of conduction electrons in silicon, Phys. Rev. Lett. 107, 107203 (2011). [44] R. Maurand, X. Jehl, D. Kotekar-Patil, A. Corna, H. Bo- huslavskyi, R. Laviéville, L. Hutin, S. Barraud, M. Vinet, M. Sanquer, and S. D. Franceschi, A CMOS silicon spin qubit, Nature Communications 7, 13575 (2016). [45] H. Watzinger, J. Kukučka, L. Vukušić, F. Gao, T. Wang, F. Schäﬄer, J.-J. Zhang, and G. Katsaros, A germanium hole spin qubit, Nature Communications 9, 3902 (2018). [46] K. Wang, G. Xu, F. Gao, H. Liu, R.-L. Ma, X. Zhang, Z. Wang, G. Cao, T. Wang, J.-J. Zhang, D. Culcer, X. Hu, H.-W. Jiang, H.-O. Li, G.-C. Guo, and G.-P. Guo, Ultrafast coherent control of a hole spin qubit in a ger- manium quantum dot, Nature Communications 13, 20619 (2022). [47] C. Kloeﬀel and D. Loss, Prospects for spin-based quan- tum computing in quantum dots, Annual Review of Con- densed Matter Physics 4, 51 (2013). [48] W.Gilbert, T.Tanttu, W.H.Lim, M.Feng, J.Y.Huang, J. D. Cifuentes, S. Serrano, P. Y. Mai, R. C. C. Leon, C. C. Escott, K. M. Itoh, N. V. Abrosimov, H.-J. Pohl, M. L. W. Thewalt, F. E. Hudson, A. Morello, A. Laucht, C. H. Yang, A. Saraiva, and A. S. Dzurak, On-demand electricalcontrolofspinqubits,arXiv:2201.06679 (2022). [49] R. M. Jock, N. T. Jacobson, M. Rudolph, D. R. Ward, M.S.Carroll,andD.R.Luhman,Asiliconsinglet–triplet qubit driven by spin-valley coupling, Nature Communi- cations13, 641 (2022). [50] M.Governale,QuantumdotswithRashbaspin-orbitcou- pling, Phys. Rev. Lett. 89, 206802 (2002).[51] C. F. Destefani, S. E. Ulloa, and G. E. Marques, Spin- orbit coupling and intrinsic spin mixing in quantum dots, Phys. Rev. B 69, 125302 (2004). [52] J. Corrigan, J. P. Dodson, H. E. Ercan, J. C. Abadillo- Uriel, B. Thorgrimsson, T. J. Knapp, N. Holman, T. McJunkin, S. F. Neyens, E. R. MacQuarrie, R. H. Foote, L. F. Edge, M. Friesen, S. N. Coppersmith, and M. A. Eriksson, Coherent control and spectroscopy of a semiconductorquantumdotWignermolecule,Phys.Rev. Lett.127, 127701 (2021). [53] H. E. Ercan, S. N. Coppersmith, and M. Friesen, Strong electron-electron interactions in Si/SiGe quantum dots, Phys. Rev. B 104, 235302 (2021). [54] C. G. Van de Walle and R. M. Martin, Theoretical cal- culations of heterojunction discontinuities in the Si/Ge system, Phys. Rev. B 34, 5621 (1986).
1307.4668v1.Non_monotonic_spin_relaxation_and_decoherence_in_graphene_quantum_dots_with_spin_orbit_interactions.pdf	Non-monotonic spin relaxation and decoherence in graphene quantum dots with spin-orbit interactions Marco O. Hachiya,1Guido Burkard,2and J. Carlos Egues1 1Instituto de F´ ısica de S˜ ao Carlos, Universidade de S˜ ao Paulo, 13560-970 S˜ ao Carlos, S˜ ao Paulo, Brazil 2Department of Physics, University of Konstanz, D-78457 Konstanz, Germany (Dated: August 17, 2021) We investigate the spin relaxation and decoherence in a single-electron graphene quantum dot with Rashba and intrinsic spin-orbit interactions. We derive an e ective spin-phonon Hamiltonian via the Schrie er-Wol transformation in order to calculate the spin relaxation time T1and decoherence time T2within the framework of the Bloch-Redﬁeld theory. In this model, the emergence of a non-monotonic dependence of T1on the exter- nal magnetic ﬁeld is attributed to the Rashba spin-orbit coupling-induced anticrossing of opposite spin states. A rapid decrease of T1occurs when the spin and orbital relaxation rates become comparable in the vicinity of the spin-mixing energy-level anticrossing. By contrast, the intrinsic spin-orbit interaction leads to a monotonic magnetic ﬁeld dependence of the spin relaxation rate which is caused solely by the direct spin-phonon coupling mechanism. Within our model, we demonstrate that the decoherence time T2'2T1is dominated by relaxation processes for the electron-phonon coupling mechanisms in graphene up to leading order in the spin-orbit inter- action. Moreover, we show that the energy anticrossing also leads to a vanishing pure spin dephasing rate for these states for a super-Ohmic bath. PACS numbers: Valid PACS appear here I. INTRODUCTION Carbon-based materials such as graphene and carbon nanotubes are of recognized importance for their poten- tial spintronic and quantum computation applications. No- tably, single-layer graphene, a one-carbon-atom-thick layer arranged in a honeycomb crystal lattice, has attracted much interest in the last decade due to its unique electronic properties1. The electron spin degree of freedom in graphene quantum dots makes them promising candidates for univer- sal scalable quantum computing2,3, which would rely on spin relaxation and decoherence times much longer than the gate operation times4. Graphene has a relatively weak hyperﬁne interaction and spin-orbit (SO) couplings. A graphene sheet is composed naturally of 99% of12Cwith nuclear spin 0, and of 1%13Cwith nuclear spin 1 =2, leading us to long dephas- ing times in carbon-based quantum dots due to a weak hy- perﬁne interaction5. Thus graphene emerges as a good candi- date to host a spin qubit, in contrast to GaAs quantum dots, whose spin dynamics is strongly modiﬁed by the nuclear spin bath. Moreover, the weak SO couplings in graphene gener- ates a spin-splitting on the order of tens of eVdue to the low atomic weight of carbon atoms6,7. Long spin relaxation times are expected since the mechanisms that enable relax- ation channels arise as a combined e ect of non-piezo-electric electron-phonon interaction and weak SO coupling. Despite the lack of measurements of the spin relaxation and dephasing times in graphene quantum dots, experimen- tal results have already been reported in a two-electron13C nanotube double quantum dot8that has been isotopically- enriched. These results showed a non-monotonic magnetic ﬁeld dependence of the spin relaxation time near the energy anticrossing. In this case, the spin relaxation minimum is re- lated to the coupling between electron spin in the quantum dot and the nanotube deﬂection9,10. In this paper, we derive a spin-phonon Hamiltonian us- Figure 1: Schematic of a gate-tunable circular graphene quantum dot setup. An homogeneous magnetic ﬁeld is applied perpendicu- larly to the gapped graphene sheet. A metallic gate put on top of the graphene deﬁnes the conﬁnement potential for a single-electron. Figure not drawn to scale. ing the Schrie er-Wol transformation for all mechanisms of electron-phonon and spin-orbit interactions. This e ec- tive Hamiltonian captures the combined e ect of the SO interaction and electron-phonon-induced potential ﬂuctua- tions. Within the Bloch-Redﬁeld theory, we ﬁnd that a non- monotonic behavior of the spin relaxation time occurs as a function of the external magnetic ﬁeld around the spin mix- ing energy-level anticrossing by the Rashba SO coupling in combination with the deformation potential and bond-length change electron-phonon mechanisms. We predict that the mininum of the spin relaxation time T1could be experimen- tally observed in graphene quantum dots. This energy anti- crossing takes place between the ﬁrst two excited energy lev- els at the accidental degeneracy for a certain value Bof the external magnetic ﬁeld. We treat the accidental degeneracy mixed by the Rashba SO coupling using degenerate-state per-arXiv:1307.4668v1 [cond-mat.mes-hall] 17 Jul 20132 turbation theory. T1strongly increases at the energy anticross- ing, reaching the same order as the orbital relaxation time11–13. In contrast with carbon nanotubes, the intrinsic SO does not couple these states due to the selection rules in a circular quan- tum dot, exhibiting a monotonic magnetic ﬁeld dependence ofT1due to direct spin-phonon coupling (deﬂection coupling mechanism). We also demonstrate that pure spin dephasing rates vanish in the leading order of the electron-phonon inter- action and SO interactions causing a decoherence dominated by relaxation processes, i.e. T2=2T1. Moreover, we ﬁnd a vanishing spin dephasing rate for a super-Ohmic bath as a general property of the energy anticrossing spectrum. This paper is organized as follows: In Sec. II, we introduce the model to describe a circular graphene quantum dot. In Sec. III, we derive the e ective spin-phonon Hamiltonian. In Sec. IV, we present a calculation of the spin relaxation time T1within the Bloch-Redﬁeld theory. In Sec. V, we discuss the vanishing spin dephasing rate within our model. Finally, we summarize our results and draw our conclusions in Sec. VI. II. THE MODEL In this section, we introduce the model for a circular and gate-tunable graphene quantum dot. Within our model, we consider a gapped graphene taking into account electron- phonon coupling mechanisms and spin-orbit interactions. We also analyze the energy spectrum of the quantum dot and its energy-level degeneracy. The degenerate levels are mixed by the Rashba SO coupling, and the energy crossings are re- moved using the standard degenerate perturbation theory. A. Graphene quantum dots The low-energy e ective Hamiltonian for graphene is anal- ogous to the two-dimensional massless Dirac equation. The characteristic linear dispersion for massless fermions occurs at the two non-equivalent points KandK0(valleys), in the honeycomb lattice Brillouin zone. The graphene energy bands in the vicinity of these high-symmetry points consti- tute a solid-state realization of relativistic quantum mechan- ics. However, conﬁning electrons in graphene quantum dots is a di cult task, since the particles tend to escape from the electrostatic conﬁnement potential due to Klein tunnel- ing. This problem can be overcome by putting graphene on top of a substrate, such as SiC14and BN15,16, that induces a non-equivalent potential for each atom of the two carbon sub- lattices and adds a mass term to the Hamiltonian17. The sub- lattice A(B) will feel a potential parametrized by +()which breaks inversion symmetry, opening a gap 2 in the electron- hole energy spectrum. Combined with the mass term, an ex- ternal magnetic ﬁeld Bis necessary to break the time-reversal symmetry and lift the valley degeneracy. Thus it is reasonable to conﬁne a single electron in a quantum dot with the restric- tion of its being localized in a single valley. Consider then, a circular and gate-tunable graphene quan- tum dot in an external magnetic ﬁeld with SO interactionsand the electron-phonon interaction described by the follow- ing low-energy Hamiltonian for the Kvalley17, H=Hd+HZ+HSO+Hph+Heph; (1) with the quantum dot Hamiltonian Hdand the Zeeman term HZ, respectively, given by Hd=~vF+U(r)+ z;HZ=1 2gBBs;(2) where=peAis the canonical momentum. The vector potential is chosen such that B=rA=(0;0;B), i.e., per- pendicular to the graphene sheet. Here, vF=106m=s is the Fermi velocity, U(r)=U0(rR) is the circular-shaped elec- trostatic potential, with (x)=1 for x0 and (x)=0 for x<0. The operator acts on the pseudospin subspace (A,B sublattices), while sacts on the real spin. Both operators andsare represented by Pauli matrices. The SO Hamiltonian for the Kvalley reads18 HSO=Hi+HR=izsz+R(xsyysx); (3) whereHiandHRdenote the intrinsic and Rashba SO e ec- tive Hamiltonians6, respectively. The intrinsic SO coupling originates from the local atomic SO interaction. At ﬁrst, only the contribution from the orbital coupling was con- sidered, resulting in a second-order term to the intrinsic SO coupling strength i7. However, some dorbitals hybridize with pzforming a-band that gives a ﬁrst-order contribution which plays a major role in the spin-orbit-induced gap19. The Rashba SO coupling, also called the extrinsic contribution, arises when an electric ﬁeld is applied perpendicular to the graphene sheet. The major contribution of the SO coupling Rcomes from the hybridization7, in contrast with the intrinsic case. The Rashba SO could also be enhanced by cur- vature e ects in the graphene sheet20. The free phonon Hamiltonian is given by Hph=X q;~!q;by q;bq; (4) with the dispersion relation !q;=sjqjm, where sis the sound velocity and m=1;2 depending on the type of phonon branch. Finally, we have the electron-phonon interaction Heph. We consider long-wavelength acoustic phonons represented by two main mechanisms: the deformation potential and the bond-length change mechanism21. The former is an e ective potential generated by static distortions of the lattice. It is rep- resented in the sublattice space as a diagonal energy shift in the band structure. The latter are o -diagonal terms due to modiﬁcations of the bond-length between neighboring carbon atoms, which causes changes in the hopping amplitude. The electron-phonon interaction in the sublattice space is given by21 Heph=X q;qpA!q; g1a1g2a 2 g2a2g1a1! (eiqrby q;eiqrbq;); (5)3 where g1andg2are the deformation potential and bond-length change coupling constants. Here, Ais the area of the graphene layer andis the mass area density. The constants a1,a2 and the sound velocities sLA,sTAfor the longitudinal-acoustic (=LA) and transverse-acoustic ( =TA) modes are given in Table I. Both phonon branches have a linear dispersion re- lation given by !q;=sjqj. Optical phonons are not taken into account in this work, since their energies do not match the Zeeman splitting for typical laboratory ﬁelds. The out- of-plane phonons ( =ZA) will be discussed further below. Notice that the electron-phonon interaction is spin indepen- dent and can only cause a spin relaxation when assisted by the SO interaction. In the following subsection, we analyze the bare quantum dot spectrum and perform a perturbation theory calculation for degenerate levels treating the SO Hamiltonian as a pertur- bative term. B. Degenerate state perturbation theory In order to calculate T1andT2, we use the quantum dot eigenstates perturbed by the SO interaction. Before doing so, we have to get rid of the degeneracies in the quantum dot spec- trum by applying degenerate state perturbation theory. This procedure makes it clearer to deﬁne which states constitute our spin qubit and where the spin relaxation occurs. Due to the selection rules for the matrix elements of the SO interaction22, only the Rashba SO term couples states from the degenerate subspace. Thus we intend to ﬁnd a linear combina- tion of eigenstates from the degenerate subspace of the quan- tum dot such that these states are not coupled by the Rashba SO HamiltonianHR. Consider then, ﬁrst the bare quantum dot Hamiltonian in theKvalleyHd, withHdjj;;si=Ej;jj;;siand the quantum dot wave functions17 hr;jj;;si= j;;s(r;)=ei(j1=2) j;;s A(r) j;;s B(r)ei! : (6) The spinor components j;;s A;B(r) are proportional to the con- ﬂuent hypergeometric functions and are described by the set j;;s, where we introduce the angular ( j=1=2;3=2;:::), radial (=1;2;3;:::) and spin s=";#quantum numbers. Matching the spinors at r=Rresults in a transcenden- tal equation for the eigenvalues Ej;which can be obtained numerically17. Since we are going to calculate the spin relax- ation rates due to transitions between the lowest three energy Table I: Electron-phonon constants and sound velocities for longitu- dinal (LA) and transverse (TA) acoustic phonons. The phonon emis- sion angle is denoted by q. a1 a2 s(104m=s) LA i ie2iq 1.95a TA 0 e2iq 1.22a aFrom Ref. 31. 5.05.56.0B=6.4T7.07.58.00.00.020.040.060.080.10 B@TDEd Gg0¬g1Gg0¬g2 Eg0-Eg0Eg1-Eg0Eg2-Eg0 B0.0390.040Figure 2: Magnetic ﬁeld dependence of the energy di erence be- tween the perturbed three lowest energy levels and the ground state in a circular graphene quantum dot. Our spin qubit is composed by the ground state and the ﬁrst excited state with opposite spin orientation. Sequentially from bottom to top, E 0E 0(solid), E 1E 0(dashed) andE 2E 0(dot-dashed). The Rashba SO interaction-induced anti- crossing of the bare quantum dot states E1=2;1;"andE1=2;1;#, atB=B (solid lines in the inset). The spin relaxation rate takes place between the statesj 0iandj 1i(#"= 0 1) before the anticrossing, and between the states j 0iandj 2i(#"= 0 2) after the anticross- ing. Inset: Blowup of the energy levels in the vicinity of the crossing region. levels of the quantum dot, we restrict ourselvels to the anal- ysis of the subspace fj+1=2;1;#i;j1=2;1;"i;j1=2;1;#ig. In- cluding the Zeeman spin-splitting, it leads to a crossing of the energy levels E1=2;1;"andE1=2;1;#, for a certain magnetic ﬁeld Bdepending on the size of the quantum dot. The ground state j+1=2;1;#iis not degenerate for any value of B. The Rashba SO interactionHRcouples two of these states j+1=2;1;"iand j1=2;1;#idue to its selection rule for the angular quantum number j22, which is given by jjj0j=1. By contrast, the intrinsic SO interaction Hidoes not couple them since its se- lection rule isjjj0j=0. Now, we have to ﬁnd an appropriate linear combination of the states from the degenerate subspace j+1=2;1;"i;j1=2;1;#iin whichHRbecomes diagonal in or- der to remove the accidental energy level degeneracy from the denominator in the usual non degenerate perturbation theory. Then, performing standard degenerate state perturbation the- ory, we obtain the zero-order eigenstates for the three lowest energy levels are given by 26666666666666666664j 0i j 1i j 2i37777777777777777775=266666666666666666641 0 0 0 cos(#=2)eisin(#=2) 0 sin(#=2)eicos(#=2)3777777777777777777526666666666666666664j1=2;1;#i j1=2;1;"i j1=2;1;#i37777777777777777775;(7)4 with the associated ﬁrst-order eigenvalues E 0=E1=2;1~!Z 2;E 1; 2=+q 2 +jSOj2; (8) plotted in Fig. 2. We deﬁne +=(E1=2;1+E1=2;1)=2 and =(E1=2;1E1=2;1+~!Z)=2,~!Z=gBBis the Zeeman energy splitting. Here, SO=h1=2;1;"jH Rj1=2;1;#i= 4iRR drr1=2;1 A(r)1=2;1 B(r), tan#= SO=and tan= I[SO]=R[SO], where I[x] is the imaginary part and R[x] the real part of x. As a result, the Rashba SO induces an en- ergy gap 2 SOat the energy anticrossing ( =0), as shown in Fig. 2. We have two dominant spin components for j 1i andj 2idepending on whether the spin relaxation takes place before or after the energy anticrossing region. Before the en- ergy anticrossing SO=>0,j 1ij 1=2;1;"i+O(SO=) andj 2ij 1=2;1;#i+O(SO=). Increasing the magnetic ﬁeld we go through the energy anticrossing region such that #!=2 when=0. As a result, the states from the degen- erate subspace hybridize j 1i(j1=2;1;"ij 1=2;1;#i)=p 2 andj 2i (j1=2;1;"i+j1=2;1;#i)=p 2. After the energy anticrossing SO=<0,j 1ij 1=2;1;#i+O(SO=) and j 2ij 1=2;1;"i+O(SO=). Thus before the energy anti- crossing, the spin relaxation takes place between j 1i!j 0i and after the energy anticrossing between j 2i!j 0i. At the energy anticrossing, the spin up and down are equivalently mixed and the orbital relaxation rate dominates over the spin relaxation rate, since the latter is a higher-order process as- sisted by the SO interaction11–13. These results will be used to study the energy relaxation with spin-ﬂip between excited states and the ground state. III. EFFECTIVE SPIN-PHONON HAMILTONIAN The electron-phonon coupling allows for energy relaxation between the Zeeman levels via the admixed states with oppo- site spin due to the presence of the SO interaction. To study this admixture mechanism we derive an e ective Hamilto- nian describing the coupling of spin to potential ﬂuctuations generated by the electron-phonon coupling. We perform a Schrie er-Wol transformation in order to eliminate the SO interaction in leading order23,24, eH=eSHeS=Hd+HZ+Hph+Heph+h S;Hephi ;(9) where we have retained terms up to O(HSO)25. The oper- atorSobeys the commutatorHd+HZ;S=HSO, with SO (HSO). The termh S;Hephi represents the coupling of the electron spin to the charge ﬂuctuations induced by the electron-phonon interaction via the SO interaction (ad- mixture mechanism). The operator Scan be rewritten as S=(Ld+LZ)1HSOwhere ˆLiis the Liouvillian superoper- ator deﬁned as LiA=Hi;A, where A denotes an arbitrary operator. Here, we make the distinction S=SR+Si, where Si/iandSR/R.For the Rashba SO coupling, we have to consider the new basisfj 1i;j 2igcalculated in Sec. II B using perturbation the- ory for the degenerate levels. As explained in Sec. II B, we are interested in transitions from the excited states j kito the ground statej 0i. In this case, we calculate the matrix element of the e ective spin-phonon Hamiltonian h 0jHR sphj ki= h 0jHeph+h SR;Hephi j ki, where k= 1; 2. We ﬁnd that h 0jHR sphj ki=h 0jHephj ki (10) +X n;s, 0 1( 0;n; k) E 0En +X n;s,D 2( 0;n; k) E kEn; where the degenerate subspace is given by D = fj+1=2;1;"i;j1=2;1;#ig. Here, we have deﬁned the product of the matrix elements as 1( 0;n;s; k)=h 0jHRjn;sihn;sjHephj ki; (11) 2( 0;n;s; k)=h 0jHephjn;sihn;sjHRj ki: (12) The matrix elements of the Rashba SO coupling give the se- lection rulejjj0j=126. These transitions are compati- ble with the selection rules of the electron-phonon interaction mechanisms depending on the order of the dipole expansion considered in the term eiqr22. In this instance, the selection rules matchjjj0j=1 for the ﬁrst order and zero order of the dipole expansion of the deformation potential (LA) and bond-length change (LA, TA), respectively. For the intrinsic SO, the matrix element of the spin- phonon Hamiltonian is given by hn0;#jHi sphjn0;"i= hn0;#jh Si;Hephi jn0;"i, with the ground state set of angular and radial quantum numbers n0=(1=2;1), sinceHidoes not connect the quantum states related with the crossed energy levels. Explicitly, we have hn0;#jHi sphjn0;"i/X n0,n0j;j0 NAA n0n0NBB n0n0 ; (13) where NAA nn0=R drrn A(r)n0 A(r) and NBB nn0=R drrn B(r)n0 B(r). The selection rule of the intrinsic SO is jjj0j=0 which is compatible with the the zero order and ﬁrst order of the dipole expansion of the deformation potential (LA) and bond- length change (LA, TA), respectively. The functions n A(r) andn B(r) are respectively, purely real and purely imaginary. ThusHi sphcan be rewritten as proportional to hj;jj;0iwith ,0which is identically zero. Consequently, the admixture mechanism due to the intrinsic SO does not contribute to the spin relaxation and dephasing process within our model. In addition to the admixture mechanism, the spin relaxation can also take place due to the direct coupling of spin and local5 out-of-plane deformations of the graphene sheet (deﬂection coupling mechanism)10,22. Assuming small amplitudes for the displacement compared to the phonon wavelength, the normal vector to the graphene sheet is ˆ n(z)ˆz+ru(x;y). The dis- placement operator is given by uz=p1=A!q(eiqrbyeiqrb), where we consider linear and quadratic behaviors to the dis- persion relation ~!q=~sq+~q2, where=p =, with the bending rigidity =1:1 eV . The matrix element of the ef- fective Hamiltonian containing only the terms connecting the Zeeman levels of the ground state reads hn0;#jHZA sphjn0;"i=iipA!q qx+iqy (14) NAA n0n0+NBB n0n0 ; where sZA=0:25103m=s is the sound velocity. Here, only the lowest order of the dipole approximation gives a nonzero contribution. The spin-phonon terms presented here will be used to cal- culate the spin relaxation and dephasing rates in the following sections. IV . SPIN RELAXATION RATES In this section, we calculate the spin relaxation time using the eective spin-phonon Hamiltonian derived in the previous section. First, we introduce the Bloch-Redﬁeld theory28,29, which allows us to derive the general expression for the spin relaxation and decoherence times. Consider a general Hamiltonian given by H=HS+HB+ HS B, whereHSdescribes the system, HBa reservoir in ther- mal equilibrium (bath) and HS Bdescribes the interaction be- tween them. This general Hamiltonian His analogous to the one derived in Sec. III for all electron-phonon mecha- nisms and SO interactions via the mapping, HS!H d+HZ, HB!H phandHS B!H sph. The system and the bath are uncorrelated initially, i.e., their spin matrices can be sepa- rated as(0)=S(0)B(0). Nevertheless, as time goes by, the system and the bath become correlated via the interaction termHsph. This system dynamics is described by an equa- tion of motion for the density matrix in the interaction picture (ˆ=ei(Hd+HZ+Hph)t=~ei(Hd+HZ+Hph)t=~) with the bath variables traced out ˆS=TrBˆas d dtˆS(t)=i ~Zt 0dt0TrBhˆHsph(t);hˆHsph(t0);ˆS(t0)ˆB(0)ii (15) This equation of motion for the reduced density matrix is called the Nakajima-Zwanzig equation29. If we assume that the coupling system-bath is weak, this equation can be fur- ther simpliﬁed by neglecting terms up to O(H2 sph) in Eq. (15), which is equivalent to approximating the density matrix in the integral as(t)=S(t)B(0)+O(Hsph) (Born approximation). Considering a phonon bath, we assume that the time evolution of theS(t) depends only on its present value and not on itspast state (Markov approximation), i.e., ˆ (t0)!ˆ(t) in the integral of Eq. (15). Taking the matrix elements of Eq. (15) between the eigenstates of HS, we have that d dtˆS mn(t)=i ~!mnmnX k;lRnmklkl(t) (16) wheremn=hmjjniand!nm=!n!m. The term Rnmklis the Redﬁeld tensor Rnmkl=nmX r+ nrrk+nkX r lrrm+ lmnk lmnk;(17) where + lmnk=R1 0dtei!nkthljHsphjmihnjHsph(t)jki, with + lmnk= knml. Here, the overbar denotes the average over a phonon bath in thermal equilibrium at temperature T. Using Eq. (16) in the secular approximation where Rnmklis approximatedly given by a diagonal tensor and hdSz=dti= Tr[(d=dt)S], we can derive the di erential equation describ- ing time evolution of the average values of the spin compo- nents, also known as Bloch equations. The solution for the hSzicomponent with a magnetic ﬁeld applied along the same direction ishSzi(t)=S0 z(S0 zSz(0))et=T1, where S0 zis the equilibrium spin polarization (ensemble of spin-down elec- trons) and Sz(0) is the initial non-equilibrium spin alignment considered in the problem (ensemble of spin-up electrons). Explicitly, the spin relaxation rate is given by29 #"=1 T1=2R + 0 k k 0+ + k 0 0 k ; (18) Equation (18) can be simpliﬁed to 1 T1=2 ~X qh 0jHsphj ki2(~! 0 k~!q)coth ~! 0 k 2kbT! : (19) The spin relaxation rate is then calculated combining Eqs. (19) and (10). The contribution due to the deforma- tion potential (LA) combined with the Rashba SO coupling is given by g1:LA 0 k= 2g2 1 ~s2 LA E kE 0 ~sLA!4Z2 0dqh k i(Ag1)i2:(20) And those due to the bond-length change mechanism for = LA;TA, g2:LA;T A 0 k=2g2 2 ~s2 E kE 0 ~s!2Z2 0dqh k i(Ag2)i2;(21) where we imply summation over the repeated index i=1;2;3. In the above we have deﬁne k 1(Ag1;g2)=n 1h1=2;1;#jAg1;g2j1=2;1;"ik; (22)6 k 2(Ag1;g2)=X n,(1=2;1)n 2h1=2;1;#jAg1;g2jn;#i (23) hn;#jHRj1=2;1;"ik; k 3(Ag1;g2)=X n,(1=2;1)n 3h1=2;1;#jHRjn;"i (24) hn;"jAg1;g2j1=2;1;"ik; where Ag1=a1112x2,Ag2=g2 +a 2+a2 , with=(x iy)=2. Their respective matrix elements are given by hnjAg1jn0i=Mnn0 j;j0+1eiq+j;j01e+iq ;(25) with Mnn0=R drr2 n An0 A+n Bn0 B , and hnjAg2jn0i= g2a 2j;j0+1NAB nn0+g2a2j;j01NAB n0n ;(26) where NAB nn0=R drrn A(r)n0 B(r). Here, 1=sin(#=2), 1= cos(#=2) and 2=cos(#=2), 2=sin(#=2). The energy- dependent denominators are given by n 1=1,n 2=1=Ek En+gBB=2,n 3=1=E1=2;1EngBB=2. As stated in Sec. II B, the energy relaxation accompanied by a spin-ﬂip transition occurs between the states j 0iandj 1i before the energy anticrossing #"= 0 1, and between the statesj 0iandj 2iafter the energy anticrossing #"= 0 2, for all electron-phonon mechanisms R #"= g1:LA 0 k+ g2:LA 0 k+ g2:T A 0 k. The contribution from the out-of-plane ﬂexural phonons via the deﬂection coupling mechanism, calculated using Eq. (19) combined with Eq. (15), is ZA #"=42 2 i gBB1 Q(B) sZA+Q(B) 2!3 (27) Z dr rn A2n B22 ; where we deﬁne Q(B)=q s2 ZA+4(gBB=~), with sZA= 0:25103m=s. In the low magnetic ﬁeld limit, the term ZA #" simpliﬁes to ZA #"=422 i 1 s5 ZA(gBB)2Z dr rn A2n B22 :(28) The magnetic ﬁeld dependence of T1=(R #"+ ZA #")1with all the mechanisms considered in this work is evaluated numeri- cally and is presented in Fig. 3. It can be observed that at the energy anticrossing region, the spin relaxation time rapidly decreases, characterizing its non-monotonic behavior induced by an external electric ﬁeld via the Rashba SO interaction. Notice that if no external electric ﬁeld is applied, the spin re- laxation time is monotonic with contributions from only the intrinsic SO interaction via deﬂection coupling mechanism.The magnetic ﬁeld dependence of the spin relaxation rate for each electron-phonon coupling mechanism can be under- stood using the spectral density of the system-bath interaction J 0 k(!)=Z1 1dtei!th 0jHsph(0)j kih kjHsph(t)j 0i:(29) Further simpliﬁcations in Eq. 18 allow us to ﬁnd the following relation 1 =T1/J 0 k(! 0 k), where! 0 k/ !Z/gBB. In a general form, we have that 1 =T1/P qKqh 0jeiqrj kih kjHSOj 0i(!q! 0 k), where Kq= q=p!qsinceHeph/Kqeiqr. Also,P q/R dqqd1, where d=2 is the dimensionality of graphene. Each SO coupling deﬁnes the selection rule for the quantum number jand consequently, the order of the dipole expansion as ex- plained in Sec. III. We ﬁnd that for the Rashba SO coupling, J 0 k(! 0 k)/!s Zwith s=4 for the deformation potential (LA) and s=2 for the bond-length change mechanism (LA, TA). Also, for the intrinsic SO, s2 for the direct spin- phonon coupling (ZA). Therefore the spectral density of the system-bath interaction is super-Ohmic ( s>1) with a strong dependence with the bath frequency for all phonons consid- ered in graphene. V . SPIN DEPHASING RATES Next we evaluate the spin dephasing rates for all the electron-phonon mechanisms introduced in Sec. III. Within the Bloch-Redﬁeld theory, we can also solve the Bloch equa- tions for the spin components perpendicular to the magnetic ﬁeld, which are given by hSxi(t)=S0 xcos(!Zt)et=T2andD SyE (t)=S0 ysin(!Zt)et=T2, where S0 x;yare the initial spin po- larizations along the x;ydirections. The decoherence time Table II: Parameters for the numerical evaluation of the spin relax- ation rates. The electron-phonon coupling constants for the deforma- tion potential g1and for the bond-length change mechanism g2and the coupling strengths for Rashba Rfor an external electric ﬁeld E and the intrinsic iSO couplings. The graphene layer is character- ized by its mass area density . The quantum dot parameters are its radius R, potential height U0and the substrate-induced energy gap . The system is assumed to be in thermal equilibrium with the bath at temperature T. g1 30 eVa g2 1:5 eVa R 11eVb E 50 V/300 nmc i 12eVd 7:5107kg=m2e R 35nm U0= 260 meV T 100 mK aFrom Ref. 21. cFrom Ref. 6. cFrom Ref. 7. dFrom Ref. 19. eFrom Ref. 31.7 024B*=6.4T81010-1110-910-710-510-310-1 B@TDT1@sD 0.81.31.810-310-2 Figure 3: Magnetic ﬁeld dependence of the spin relaxation time. Parameters used in the numerical evaluation are given in Table II. Contributions from the deformation potential g 1: LA (dark, dotted), bond-length change mechanism g 2: LA (dark, dotted) and g 2: TA (light, dashed) and the out-of-plane phonons ZA (light, dot-dashed). Dark solid: the sum of all processes. The minimum in T1occurs at the energy-level anticrossing at B. Inset: Blowup of the low mag- netic ﬁeld regime. Competition between the two electron-phonon dominant mechanisms: deformation potential and ﬂexural phonons. The absence of Van Vleck cancellation22,30leads to a ﬁnite value for T1atB=0. can be separated into two contributions: the spin relaxation and the pure spin dephasing 1 =T2=1=2T1+1=T, where the pure spin dephasing rate is29 =1 T=R + 0 0 0 0+ + k k k k2+ 0 0 k k : (30) In the low-temperature limit, we ﬁnd that 1 T=lim !!0h 0jHsphj 0ih kjHsphj ki2(~!~!q)2kbT ~!: (31) The dephasing time can also be rewritten in terms of the spectral density of the system-bath interaction as 1 =T/ lim!!0J(!)coth (~!=2kbT)/lim!!0J(!)=!. As we have shown in Sec. III, the spectral function for all electron-phonon coupling mechanisms considered in this work are super-Ohmic. Thus the spin dephasing vanishes in all cases, since 1 =T/lim!!0!s=!!0, with s>1. In other words, there are no phonons available in leading order to cause dephasing in graphene quantum dots. The decoherence time T2is determined only by the relaxation contribution, i.e., T2=2T1. Notice that this relation is no longer necessarily true considering two-phonon processes since the combination of emission and absorption energies can fulﬁll the energy con- servation requirement.24 Aditionally, the spin dephasing rate could also vanish at the energy anticrossing for a super-Ohmic bath. Within the subspace spanned by the states fj+1=2;1;"i;j1=2;1;#ig, the HamiltonianHcan be rewritten as H= +(B)1+ (B)z, wherezdenote a Pauli matrix and =(E 3E 2)=2. This magnetic ﬁeld can be divided into two contributions B=B0+B(t): an external source B0and an internal con-tributionB(t) due to the bath. For small ﬂuctuation of B(t), theHis approximatedly given by H=((B0)+@B(B0)B(t))z; (32) where we have not included the term proportional to +112x2 since it does not cause spin dephasing. Calculating the spin dephasing rate within the Bloch-Redﬁeld theory using Eq. 30, we ﬁnd that 1 T= 2 ~@B(B0)!2 lim !!0RZ1 0dt0ei!thB(0)B(t0)i;(33) wherehA(t)iis the thermal equilibrium expectation value of the operator A(t) on the bath. Therefore the spin dephasing rate goes to zero at the energy anticrossing, since @B(B0)! 0. This condition is valid under the assumption that the ther- mal average of the ﬂuctuating magnetic ﬁeld does not diverge. Following the result given by Eq. (31), the spin dephasing rate still vanishes as long as the spectral density of the system-bath interaction is super-Ohmic, i.e., J(!)/!s, with s>1. VI. CONCLUSION In summary, we ﬁnd a minimum in the spin relaxation time as a function of the magnetic ﬁeld that is induced by the Rashba SO coupling and is controllable by an external electric ﬁeld. In larger quantum dots, the intrinsic SO dominates the spin relaxation over the Rashba SO contribution at low mag- netic ﬁelds. As the magnetic ﬁeld increases, the extrinsic con- tribution takes over, generating a non-monotonic behaviour of T1due to the Rashba SO interaction-induced level anticross- ing. We have also analyzed the spectral density of the system- bath interaction for the ﬁrst-order electron-phonon interaction and we have identiﬁed a vanishing contribution to the energy- conserving dephasing process. Therefore the phonon-induced pure spin dephasing rate is of the same order of magnitude as the spin relaxation rate, i.e., T2=2T1, in the leading order of the electron-phonon interaction. Other mechanisms such as nuclear spins from the13Catoms and charge noise com- bined with SO interaction could lead to a non-vanishing spin dephasing rate. Nevertheless, these mechanism are expected to be weak in graphene3,5. Moreover, we have shown that any super-Ohmic bath has a vanishing spin dephasing rate at the energy anticrossing. Acknowledgments We wish to acknowledge useful discussions with P. R. Struck and Peter Stano. Funding for this work was provided by CNPq, FAPESP and PRP /USP within the Research Sup- port Center initiative (NAP Q-NANO) (MOH and JCE) and DFG and ESF under grants FOR912, SPP1285, and Euro- Graphene (CONGRAN) (GB).8 1A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009). 2D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120 (1998). 3B. Trauzettel, D. V . Bulaev, D. Loss, and G. Burkard, Nature Physics 3, 192 (2007). 4D. P. DiVincenzo, Fortschr. Phys. 48, 771 (2000). 5J. Fischer, B. Trauzettel, and D. Loss. Phys. Rev. B 80, 155401 (2009). 6C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005). 7H. Min, J. E. Hill, N. A. Sinitsyn, B. R. Sahu, L. Kleinman, and A. H. MacDonald, Phys. Rev. B 74, 165310 (2006). 8H. O. H. Churchill, F. Kuemmeth, J. W. Harlow, A. J. Bestwick, E. I. Rashba, K. Flensberg, C. H. Stwertka, T. Taychatanapat, S. K. Watson, and C. M. Marcus, Phys. Rev. Lett. 102, 166802 (2009). 9D. V . Bulaev, B. Trauzettel, and D. Loss, Phys. Rev. B 77, 235301 (2008). 10M.S. Rudner and E.I. Rashba, Phys. Rev. B 81, 125426 (2010). 11D.V . Bulaev and D. Loss, Phys. Rev. B 71, 205324, (2005). 12P. Stano and J. Fabian, Phys. Rev. B 72, 155410, (2005). 13P. Stano and J. Fabian, Phys. Rev. B 74, 045320, (2005). 14S. Y . Zhou, G.-H. Gweon, A. V . Fedorov, P. N. First, W. A. de Heer, D.-H. Lee, F. Guinea, A. H. Castro Neto, and A. Lanzara, Nature Materials 6, 770 (2007). 15G. Giovannetti, P. A. Khomyakov, G. Brocks, P. J. Kelly, and J. van den Brink, Phys. Rev. B 76, 073103 (2007). 16J. Slawinska, I. Zasada, P. Kosiski, and Z. Klusek. Phys. Rev. B 82, 085431 (2010). 17P. Recher, J. Nilsson, G. Burkard, B. Trauzettel, Phys. Rev. B 79,085407 (2009). 18The tensor product notation is implicit, i.e., isji sj, for i;j=x;y;z. 19M. Gmitra, S. Konschuh, C. Ertler, C. Ambrosch-Draxl, J. Fabian, Phys. Rev. B 80, 235431 (2009). 20Huertas-Hernando, F. Guinea, and A. Brataas, Phys. Rev. B 74 155426 (2006). 21T. Ando, J. Phys. Soc. Jpn. 74, 777 (2005). 22P.R. Struck and Guido Burkard, Phys. Rev. B 82, 125401 (2010). 23R. Winkler, Spin-Orbit Coupling E ects in Two-Dimensional Electron and Hole Systems (Springer-Verlag, Berlin, 2003). 24V . N. Golovach, A. Khaetskii, and D. Loss, Phys. Rev. Lett. 93, 016601 (2004). 25We do not carry the termh S;Hphi since it does not con- tribute to the e ective spin-phonon Hamiltonian, i.e., h 0jH ph+h S;Hphi j ki=0. 26The matrix elements of the Rashba SO and all the electron-phonon coupling mechanisms are presented in Ref. [22]. 27A. V . Khaetskii and Yu. V . Nazarov, Phys. Rev. B 64, 125316, (2001). 28K. Blum, Density Matrix Theory and Applications (Springer, New York, 2012), 2nd ed. 29L. Chirolli and G. Burkard, Adv. Phys. 57, 225 (2008) 30J. H. van Vleck, Phys. Rev. 54, 426, (1940). 31L. A. Falkovsky, Phys. Lett. A 3725189 (2008); J. Exp. Theor. Phys. 105, 397 (2007).
1505.07937v1.Topological_Surface_States_Originated_Spin_Orbit_Torques_in_Bi2Se3.pdf	1 Topological Surface States Orig inated Spin-Orbit Torques in Bi 2Se3 Yi Wang,1 Praveen Deorani,1 Karan Banerjee,1 Nikesh Koirala,2 Matthew Brahlek,2 Seongshik Oh,2 and Hyunsoo Yang1,* 1Department of Electrical and Computer Engineering, National University of Singapore, 117576, Singapore 2Department of Physics & Astronomy, Rutgers Cent er for Emergent Materials, Institute for Advanced Materials, Devices and Nanotechnology, The State University of New Jersey, New Jersey 08854, USA Three dimensional topological insulator bismuth selenide (Bi 2Se3) is expected to possess strong spin-orbit coupling and spin-textured to pological surface states , and thus exhibit a high charge to spin current conversion efficiency. We evaluate spin-orbit torques in Bi 2Se3/Co 40Fe40B20 devices at different temperatures by spin torque ferromagnetic resonance measurements. As temperature decreases, the sp in-orbit torque ratio increases from ~ 0.047 at 300 K to ~ 0.42 below 50 K. Moreover, we observe a significant out-of-plane torque at low temperatures. Detailed analysis indicates that the origin of the observed spin-orbit torques is topological surface states in Bi 2Se3. Our results suggest that topological insulators with strong spin-orbit coupling could be promising candidates as highly efficient spin current sources for exploring next generation of spintronic applications. *eleyang@nus.edu.sg 2 The realization of functional devices su ch as the non-volatile memories and spin logic applications is of key importan ce in spintronic research [1]. The functions of these magnetic devices require highly effici ent magnetization manipulation in a ferromagnet (FM), which can be achieved by an external magnetic field or a spin polarized current by spin transfer torque (STT). Recent advances have demonstrat ed that pure spin currents resulting from charge currents via spin-orbit coupling in heavy metals, such as Pt [2-7], Ta [8-10], and W [11], can produce strong spin-orbit torques on the adjacent magnetic layers. The reported amplitude of spin Hall angles (i.e. efficiency of spin-orbit torques) in Pt and Ta is in the range of ~ 0.012 to ~ 0.15, and in W is ~ 0.33. The exploration for new materials exhibiting new physics and possessing an even higher conversion efficiency between the charge current density ( J c) and spin current density ( Js) is crucial to exploit next generation spintronic devices. The three dimensional (3D) topological insula tors (TI) are a new class of quantum state of materials that have an insulating bulk an d spin-momentum-locked metallic surface states [12-14]. They exhibit strong spin-orbit coupling and are expected to show a high charge to spin current conversion efficiency. So far, by extensively employing angle-resolved photoemission spectroscopy (ARPES) and spin-resolved ARPES, the Dirac cones and the helical spin polarized topological surface states (TSS) have been observed and the topological nature has been confirmed in TIs [15,16]. The surface state dominant conduction has also been confirmed by thickness dependent transport measurements in Bi 2Se3 [17]. The TSS in TI is immune to the nonmagnetic impurities due to the time reversal symmetry protection. Although a gap opening in the TSS dispersion was reported in Bi 2Se3 3 doped with Fe in the bulk [18], most recently repor ts have confirmed that the TSS is intact in Bi2Se3 covered with Fe [19,20] or Co [21] with in-plane magnetic anisotropy. The spin dependent transport is known to be significant near the Fermi level in the Bi 2Se3 surface states. However, limited spin dependent transport experiments have been focused on TI/FM heterostructures. Only recently, spin-orbit effects have been reported by spin pumping measurements [22-24] and magnetor esistance measurements [25,26]. Direct charge current induced spin-orbit torque on the FM layer has been demonstrated by spin torque ferromagnetic resonance (ST-FMR) measurement only at room temperature [27] and magnetization switching at cryogenic temperature [28]. It is known that for Bi 2Se3 the bulk channel provides an inevitable contribution to transport at room temperature and may diminish the signals of spin-orbit torques ar ising from surface states . At low temperatures, however, the surface contribution should become significant [17], and spin-orbit torques in TI/FM heterostructures should be enhanced [28]. In this work, we adopt extensively studied Bi 2Se3 as the TI layer and investigate the temperature dependence of charge-spin conversion efficiency, spin-orbit torque ratio ( \|\| = Js/Jc), by the ST-FMR technique in Bi 2Se3/Co 40Fe40B20 heterostructures. In this structure, the spin currents generated from ch arge currents flowing in Bi 2Se3 are injected into ferromagnetic Co 40Fe40B20 layer and exert torques on it. It must be pointed out that the spin-orbit torques could be attributed to either the spin Hall effect (SHE) in the Bi 2Se3 bulk, Rashba-split states at the interface [29-31], or Bi 2Se3 topological surface states [23,27,28,31]. We find that \|\| drastically increases when the temperature decreases to ~ 50 K. As the temperature decreases furthermore, \|\| reaches up to ~ 0.42, which is ~ 10 times larger than 4 that at 300 K. In addition, a significant out-of-plane torque is extracted at low temperatures. We argue that our observations could be correlated with the TSS in our Bi 2Se3/Co 40Fe40B20 heterostructures. 20 quintuple layer (QL, 1 QL 1 nm) of Bi 2Se3 films are grown on Al 2O3 (0001) substrates using a custom designed SVTA MOSV-2 molecular beam epitaxy (MBE) system with a base pressure < 3 × 10-10 Torr. The detailed procedures for Bi 2Se3 thin film growth can be found in previous reports [17,32]. The temperature dependent resistivity of Bi 2Se3 film is measured by four probe method. Figure 1(a) shows a typical characteristic of Bi 2Se3 that the sheet resistivity decreases as temperature decreases and then saturates at temperature < 30 K [17,33]. High resistivity Co 40Fe40B20 (CFB) is chosen as the FM layer in order to minimize the current shutting effect thru the FM layer. We have prepared five Bi 2Se3/CFB ( t) samples (thickness t = 1.5, 2, 3, 4 and 5 nm) and measured their magnetization response as a function of external magnetic field as plotted in Fig. 1(b). From the inset of Fig. 1(b), the CFB dead layer in Bi 2Se3/CFB samples is estimated to be 1.36 nm, similar to a recent report in which the Co dead layer at the interface of Bi 2Se3/Co is ~ 1.2 nm [34]. The ST-FMR devices are fabricated by the following process. First, a 5 nm CFB layer is sputtered onto the Bi 2Se3 film at room temperature with a base pressure of 3×10-9 Torr followed by a MgO (1 nm)/SiO 2 (3 nm) capping layer to prevent CFB from oxidation. Then the film is patterned into rectangular shaped mi crostrips (dotted blue line) with dimensions of L (130 µm) × W (10 20 µm) by photolithography and Ar ion milling as shown in Fig. 2(a). In the next step, coplanar waveguides (CPWs) are fabricated. Different gaps (10 55 µm) between ground (G) and signal (S) electrodes are designed to tune the device impedance ~ 50 5 Ω. A radio frequency (RF) current ( Irf) with frequencies from 7 to 10 GHz and a nominal power of 15 dBm from a signal generator (SG, Agilent E8257D) is applied to the Bi2Se3/CFB bilayer via a bias-tee, and the ST-FMR signal ( Vmix) is detected simultaneously by a lock-in amplifier. An in-plane external magnetic field ( Hext) is applied at a fixed angle (H) of 35º with respect to the microstrip length direction [6]. We present the data from three different devices, denoted as D1, D2 and D3. Figure 2(b) shows the measured ST-FMR signals from D1 at different temperatures ranging from 20 to 300 K. Vmix can be fitted by a sum of symmetric and antisymmetric Lorentzian functions, mix s sym ext a asym ext () () VV F H V F H [3,6,27]. From fitting, the symmetric component Vs (corresponding to in-plane torque τ\|\| on CFB) and antisymmetric component Va (corresponding to total out-of-plane torque τ) can be determined, simultaneously. The spin-orbit torque ratio from ST-FMR me asurements can be characterized by two methods. One is to obtain \|\| from the analysis of Vs/Va via 12 sa 0s e f f e x t(/) ( / ) [ 1 + ( 4 / ) ]/VV e M t d M H [3], where t and d represent the thickness of the CFB and Bi 2Se3 layer, respectively. Ms is the saturation magnetization of CFB and Meff is the effective magnetization. This method (denoted as ‘by Vs/Va’ hereafter) is to date widely used in ST-FMR measurements of heavy metals Pt (or Ta)/FM bilayers [3,6,8]. However, one assumption of this method is that the Va is only attributed to the Oersted field induced out-of-plane torque. However, in the case of a TI , the TSS in TI and/or Rashba-split states at the interface could also contribute to Va, therefore, we cannot estimate the actual \|\| value by Vs/Va. On the other hand, the second method can avoid such an issue by analyzing only the symmetric component Vs (denoted as ‘by Vs only’ hereafter) using the following equations:6 rf H sym ext Hs1()4I cos dRFHdV , ss s //JEM t E , and s/ [6,27], where Irf is the RF current flowing through the device, H/dR d is the angular dependent magnetoresistance at H = 35, is the linewidth of ST-FMR signal, Fsym (Hext) is a symmetric Lorentzian, τ\|\| is the in-plane spin-orbit torque on unit CFB moment at H = 0, s is the Bi 2Se3 spin Hall conductivity, is the Bi 2Se3 conductivity, and E is the microwave field across the device. The second met hod avoids the possible contamination to \|\| arising from Va, therefore we can extract the \|\| values in Bi 2Se3 by analyzing only Vs. At the same time, the total out-of-plane torque τ can be derived by using 12 rf H 0 eff ext asym ext Ha[1 ( / )]()4/I cos dR M HFdVH [27], where Fasym (Hext) is an antisymmetric Lorentzian. Figure 3(a-b) show the τ\|\| and τ as functions of temperature, respectively, using the 2nd method. Here, the τ\|\| (τ) represents the mean value for different RF frequencies. At 300 K, the τ\|\| is ~ 0.43 Oe for D1 (~ 0.84 Oe for D2 and ~ 0.48 Oe for D3). As the temperature decreases from 300 to 100 K, τ\|\| for all three devices gradually increases. At ~ 50 K, τ\|\| shows a steep increase and finally reaches ~ 5.25 Oe for D1 (~ 4.11 Oe for D2 and ~ 2.26 Oe for D3), which is ~ 10 times larger than that at 300 K. It is noteworthy that the observed drastic temperature dependent behavior of τ \|\| is different from the recently reported results in heavy metals such as Ta [10,35] as well as Pt [6,36,37], where the damping-like torque (equivalent to τ\|\| here), often argued to arise mainly from the SHE, shows a weak temperature dependence. This difference indicates the SH E mechanism may not account for the observed τ\|\| in our Bi 2Se3/CFB. Moreover, the τ shows a similar temperature dependent behavior as τ\|\| 7 shown in Fig. 3(b). It is worth noting that the difference in τ\|\| (and τ) among D1, D2 and D3 can be attributed to the slight variation of the Bi 2Se3/CFB interface during the fabrication process considering recent challenges in TI film growth and device fabrication. However, a qualitatively similar temperature dependence of torques is observed in all devices. The \|\| values as a function of temperature determined by above two methods have been shown in Fig. 3(c). From analysis by Vs only, \|\| is ~ 0.047 for D1 (~ 0.113 for D2 and ~ 0.072 for D3) at 300 K, and increases to ~ 0.158 for D1 (~ 0.225 for D2 and ~ 0.149 for D3) as temperature decreases to 100 K. In this temperature range (100 - 300 K), \|\| has similar amplitudes as the spin Hall angle in heavy metals such as Pt, Ta, and W [3,8,11,42-44]. However, \|\| increases sharply as temperature decreases to ~ 50 K and reaches maximum values of ~ 0.42 for D1 (~ 0.44 for D2 and ~ 0.30 for D3) at lower temperatures, respectively. Remarkably, \|\| increases ~ 10 times compared to that at 300 K for D1. Similarly, from the analysis by Vs/Va, \|\| also shows an abrupt increase as temperature decreases to ~ 50 K in Fig. 3(c). It is worth noting that we use the effective CFB thickness of t = 3.64 nm due to the dead layer for \|\| estimation by Vs/Va at different temperatures. Interestingly, as shown in Fig. 3(d), the ratio of [ \|\| (by Vs only) ‒ \|\| (by Vs/Va)]/\|\| (by Vs/Va) obtained by two different methods increases as temperature decreases and becomes more significant below ~ 50 K, as discussed later. In the context of spin Hall mechanism, the spin Hall angle ( sh) is found to be almost independent of temperature from Pt [6,36], Ta [45], Cu 99.5Bi0.5, and Ag 99Bi1 [46], which is attributed to the extrinsic mechanisms. In some cases, sh shows a gradual increase as the temperature decreases, which behaves as a typical intrinsic mechanism based on the 8 degeneracy of d-orbits by spin-orbit coupling [47,48]. In contrast, in our Bi 2Se3/CFB, the spin-orbit torque ratio ( \|\|) shows an abrupt and nonlinear increase as temperature decreases, especially below ~ 50 K. Therefore, the SHE from the Bi 2Se3 bulk is probably not the dominant mechanism for our observation of temperature dependent spin-orbit torque (ratio) in Bi 2Se3/CFB. From the measured ST-FMR signals as shown in Fig. 2(b), we also find that the Rashba-split state at the Bi 2Se3/CFB interface is not the main mechanism for our observations, since the Rashba-split states lead to opposite direction (and sign) of charge current-induced spin polarization (and \|\|) on the basis of the spin structure [27,31]. Instead, we ascertain that the direction of in-plane spin polarization to the electron momentum in our Bi2Se3/CFB is consistent with expectations of the TSS of TIs (spin-momentum locking) [12-14,27,31,37]. From further analysis [37], we have found that in our devices a large portion of the charge current flows through the TSS in Bi 2Se3. The effective \|\| attributed to only TSS is in the range from ~ 1.62 ± 0.18 to ~ 2.1 ± 0.39. As mentioned before, the temperature dependent \|\| obtained from the above two methods shown in Fig. 3(c) should not show any difference, if Va is attributed to only the charge current induced Oersted field. Therefore, the observed difference implies the existence of other contributions to Va (i.e. to τ). For the Bi 2Se3/CFB system, the difference can be attributed to the TSS in Bi 2Se3 [23,27,28,31] and/or Rashba-split states at the Bi2Se3/CFB interface [29-31]. We analyze τ = τ τOe as the other contributions to the out-of-plane torque, where τ is the total out-of-plane torque as shown in Fig. 3(b), and τOe is a partial out-of-plane torque fro m charge current (flowing in Bi 2Se3) induced Oersted field. By using the measured \|\| by Vs only, we can deduce τOe and thus τ by 9 12 sa 0s e f f e x t(/ ) ( / ) [ 1 + ( 4 / ) ]/VV e M t d M H , and 12 rf H 0 eff ext Oe asy e t Ha mx[1 ( / )]()4/Ic o s d R MHFHdV [3,27], where Va is the equivalent antisymmetric component only due to the current induced Oersted field ( τOe). As shown in Fig. 4(a), the out-of-plane torque ( τ) in all three devices becomes much larger at low temperatures < 50 K, compared to the τ at high temperatures (100 – 300 K). Consequently, we can obtain the out-of-plane spin-orbit torque ratio ( ) as a function of temperature by using the same method by which we deduce \|\| from τ\|\| above. As shown in Fig. 4(b), we find that in all three devices also becomes more significant at low temperatures (< 50 K). More interestingly, the almost has the same order of magnitude compared to \|\|. We now discuss the origin of the out-of-plane torque. As has been reported recently, a Rashba-split surface state in two dimensional electron gas (2DEG) coexists with TSS in the Bi2Se3 surface due to the band bending and structural inversion asymmetry [29,30,49-52]. The Rashba effective magnetic field can be written as TR /( z )ˆ H k [49-51], where zˆ is a unit vector normal to film plane, k is the average electron Fermi wavevector, and αR is a characteristic parameter of the strength of Rashba splitting in 2DEG. Since the electron Fermi wavevector can be assumed to show a weak temperature dependence and the αR decrease as temperature decreases in a typical 2DEG [53,54], HT is expected to decrease as temperature decreases in these semiconductor sy stems. In addition, the similar temperature dependent behavior of HT has been recently reported in Ta/CoFeB heterostructures, where HT decreases and eventually almost reaches to zero at low temperatures [10,35]. However, the observed τ (equivalent to HT) in our Bi 2Se3/CFB presents the opposite temperature 10 dependent behavior which is not in line with the reports about Rashba induced torques. Therefore, we conclude that the Rashba-split surface state in 2DEG of Bi 2Se3 is not the main mechanism for the out-of-plane torque ( τ). On the other hand, a possible out-of-plane spin polarization in the TSS has been theoretically predicted [55,56] and experimentally observed in Bi 2Se3 [57,58], which is attributed to the hexagonal warping effect in the Fermi surface [55,59]. This out-of-plane spin polarization in the TSS can account for the observed τ especially in the low temperature range (< 50 K) and the τ adds to the τOe [27,31]. Moreover, as shown in Fig. 3(a) and 4(a), the out-of-plane torque ( τ) has the same order of magnitude comparable to in-plane torque ( τ\|\|) below 50 K ( τ/τ\|\| ~ 60%) [37], which is in agreement with the behavior of hexagonal TSS in TI [55,56]. With the analysis from different aspects, our findings especially in the low temperature range (< 50 K) indicate a TSS origin of spin-orbit torques in Bi 2Se3/CFB. In summary, we have studied the temperature dependence of spin-orbit torques in Bi2Se3/CoFeB heterostructures. As temperature decreases, the spin-orbit torque ratio increases drastically and eventually reaches a maximum value of ~ 0.42, which is almost 10 times larger than that at 300 K. A significant out-of-plane torque ( τ), in addition to charge current induced Oersted field torque ( τOe), can be observed below 50 K. The observed spin-orbit torques are attributed to the topological surface states in Bi 2Se3. Our results suggest that topological insulators with strong spin-orbit coupling and spin-momentum locking are promising spin current sources for next generation of spintronic devices. 11 This work was partially supported by the National Research Foundation (NRF), Prime Minister’s Office, Singapore, under its Competitive Research Programme (CRP award no. NRFCRP12-2013-01), the Ministry of Educatio n-Singapore Academic Research Fund Tier 1 (R-263-000-A75-750) & Tier 2 (R-263-00 0-B10-112), Office of Naval Research (N000141210456), and by the Gordon and Betty Moore Foundation’s EPiQS Initiative through Grant GBMF4418. 12 References [1]I.Zutic,J.Fabian,andD.Sarma,Rev.Mod.Phys.76,323(2004). [2]I.M.Mironetal.,Nature476,189(2011). [3]L.Q.Liu,T.Moriyama, D.C.Ralph,andR.A.Buhrman, Phys.Rev.Lett.106,036601(2011). [4]L.Q.Liu,O.J.Lee,T.J.Gudmundsen, D.C.Ralph,andR.A.Buhrman, Phys.Rev.Lett.109,096602(2012). [5]T.D.Skinner,M.Wang,A.T.Hindmarch, A.W.Rushforth, A.C.Irvine,D.Heiss,H.Kurebayashi, andA.J. Ferguson, Appl.Phys.Lett.104,062401(2014). [6]Y.Wang,P.Deorani,X.P.Qiu,J.H.Kwon,andH.Yang,Appl.Phys.Lett.105,152412(2014). [7]X.Qiu,K.Narayanapillai, Y.Wu,P.Deorani,D.H.Yang,W.S.Noh,J.H.Park,K.J.Lee,H.W.Lee,andH. Yang,NatureNanotechnol. 10,333(2015). [8]L.Q.Liu,C.F.Pai,Y.Li,H.W.Tseng,D.C.Ralph,andR.A.Buhrman, Science336,555(2012). [9]J.Kim,J.Sinha,M.Hayashi,M.Yamanouchi, S.Fukami,T.Suzuki,S.Mitani,andH.Ohno,NatureMater.12, 240(2013). [10]X.Qiu,P.Deorani,K.Narayanapillai, K.S.Lee,K.J.Lee,H.W.Lee,andH.Yang,Sci.Rep.4,4491(2014). [11]C.‐F.Pai,L.Liu,Y.Li,H.W.Tseng,D.C.Ralph,andR.A.Buhrman, Appl.Phys.Lett.101,122404(2012). [12]J.E.Moore,Nature464,194(2010). [13]M.Z.HasanandC.L.Kane,Rev.Mod.Phys.82,3045(2010). [14]X.‐L.QiandS.‐C.Zhang,Rev.Mod.Phys.83,1057(2011). [15]D.Hsiehetal.,Science323,919(2009). [16]A.Nishideetal.,Phys.Rev.B81,041309(2010). [17]N.Bansal,Y.S.Kim,M.Brahlek,E.Edrey,andS.Oh,Phys.Rev.Lett.109,116804(2012). [18]Y.L.Chenetal.,Science329,659(2010). [19]M.R.Scholz,J.Sánchez‐Barriga,D.Marchenko, A.Varykhalov, A.Volykhov, L.V.Yashina,andO.Rader, Phys.Rev.Lett.108,256810(2012). [20]J.Honolkaetal.,Phys.Rev.Lett.108,256811(2012). [21]M.Yeetal.,Phys.Rev.B85,205317(2012). [22]P.Deorani,J.Son,K.Banerjee, N.Koirala,M.Brahlek,S.Oh,andH.Yang,Phys.Rev.B90,094403(2014). [23]Y.Shiomi,K.Nomura,Y.Kajiwara,K.Eto,M.Novak,K.Segawa,Y.Ando,andE.Saitoh,Phys.Rev.Lett.113, 196601(2014). [24]M.Jamali,J.S.Lee,Y.Lv,Z.Y.Zhao,N.Samarth,andJ.P.Wang,arXiv:1407.7940. [25]C.H.Li,O.M.J.van'tErve,J.T.Robinson, Y.Liu,L.Li,andB.T.Jonker,NatureNanotechnol. 9,218(2014). [26]Y.Ando,T.Hamasaki, T.Kurokawa, K.Ichiba,F.Yang,M.Novak,S.Sasaki,K.Segawa,andM.Shiraishi, NanoLett.14,6226(2014). [27]A.R.Mellniketal.,Nature511,449(2014). [28]Y.Fanetal.,NatureMater.13,699(2014). [29]P.D.C.Kingetal.,Phys.Rev.Lett.107,096802(2011). [30]H.M.Benia,A.Yaresko,A.P.Schnyder, J.Henk,C.T.Lin,K.Kern,andC.R.Ast,Phys.Rev.B88,081103(R) (2013). [31]M.H.Fischer,A.Vaezi,A.Manchon, andE.‐A.Kim,arXiv:1305.1328. [32]N.Bansaletal.,ThinSolidFilms520,224(2011). [33]J.Son,K.Banerjee, M.Brahlek,N.Koirala,S.‐K.Lee,J.‐H.Ahn,S.Oh,andH.Yang,Appl.Phys.Lett.103, 213114(2013). [34]J.Li,Z.Y.Wang,A.Tan,P.A.Glans,E.Arenholz, C.Hwang,J.Shi,andZ.Q.Qiu,Phys.Rev.B86,054430 (2012).13 [35]J.Kim,J.Sinha,S.Mitani,M.Hayashi,S.Takahashi, S.Maekawa, M.Yamanouchi, andH.Ohno,Phys.Rev. B89,174424(2014). [36]L.Vila,T.Kimura,andY.Otani,Phys.Rev.Lett.99,226604(2007). [37]SeeSupplemental MaterialatURLfordetailsonspin‐momentum locking,additional dataaboutspin‐orbit torquesandspin‐orbittorqueratiosandanalysis,whichincludesRefs.[38‐41]. [38]J.Linder,T.Yokoyama, andA.Sudbø,Phys.Rev.B80,205401(2009). [39]L.A.Wray,Su‐YangXu,Y.Xia,D.Hsieh,A.V.Fedorov,Y.S.Hor,R.J.Cava,A.Bansil,H.Lin,andM.Z.Hasan, NaturePhys.7,32(2011). [40]T.Valla,Z.‐HPan,D.Gardner,Y.S.Lee,andS.Chu,Phys.Rev.Lett.108,117601(2012). [41]E.Wang,P.Tang,G.Wan,A.V.Fedorov,I.Miltkowski, Y.P.Chen,W.Duan,andS.Zhou,NanoLett.15,2031 (2015). [42]A.Hoffmann, IEEETrans.Magn.49,5172(2013). [43]L.Bai,P.Hyde,Y.S.Gui,C.M.Hu,V.Vlaminck, J.E.Pearson,S.D.Bader,andA.Hoffmann, Phys.Rev.Lett. 111,217602(2013). [44]S.Kasai,K.Kondou,H.Sukegawa, S.Mitani,K.Tsukagoshi, andY.Otani,Appl.Phys.Lett.104,092408 (2014). [45]P.Deorani,Y.Wang,andH.Yang,International Magnetics Conference (INTERMAG), GE‐11,Beijing,May 2015. [46]Y.Niimi,H.Suzuki,Y.Kawanishi, Y.Omori,T.Valet,A.Fert,andY.Otani,Phys.Rev.B89,054401(2014). [47]G.Y.Guo,S.Murakami, T.W.Chen,andN.Nagaosa,Phys.Rev.Lett.100,096401(2008). [48]T.Tanaka,H.Kontani,M.Naito,T.Naito,D.Hirashima, K.Yamada,andJ.Inoue,Phys.Rev.B77,165117 (2008). [49]Yu.A.BychkovandE.I.Rashba,J.Exp.Theor.Phys.Lett.39,78(1984). [50]A.Manchon andS.Zhang,Phys.Rev.B78,212405(2008). [51]I.M.Miron,G.Gaudin,S.Auffret,B.Rodmacq, A.Schuhl,S.Pizzini,J.Vogel,andP.Gambardella, Nature Mater.9,230(2010). [52]M.Bianchi,D.Guan,S.Bao,J.Mi,B.B.Iversen,P.D.King,andP.Hofmann, NatureCommun. 1,128 (2010). [53]P.Eldridge,W.Leyland,P.Lagoudakis, O.Karimov,M.Henini,D.Taylor,R.Phillips,andR.Harley,Phys.Rev. B77,125344(2008). [54]M.A.Leontiadou etal.,J.Phys.:Condens. Matter23,035801(2011). [55]L.Fu,Phys.Rev.Lett.103,266801(2009). [56]O.V.Yazyev,J.E.Moore,andS.G.Louie,Phys.Rev.Lett.105,266806(2010). [57]Y.H.Wang,D.Hsieh,D.Pilon,L.Fu,D.R.Gardner,Y.S.Lee,andN.Gedik,Phys.Rev.Lett.107,207602 (2011). [58]M.Nomura,S.Souma,A.Takayama, T.Sato,T.Takahashi, K.Eto,K.Segawa,andY.Ando,Phys.Rev.B89, 045134(2014). [59]K.Kurodaetal.,Phys.Rev.Lett.105,076802(2010). 14 Figure captions FIG. 1. (a) Temperature dependent sheet resistivity of Bi 2Se3 films (20 QL). (b) The magnetization versus field ( H) for Bi 2Se3 (20 QL)/CFB ( t) (nominal thickness t =1.5, 2, 3, 4 and 5 nm) at room temperature. The inset show s the magnetization per unit area versus CFB thickness. FIG. 2. (a) The schematic diagram of the ST-FMR measurement, illustrating a bias-tee, lock-in amplifier, RF signal generator (SG), and ST-FMR device with a Bi 2Se3/CFB (5 nm). Micro-strip is denoted by a dashed blue rectangle. (b) The measured ST-FMR signals from a Bi2Se3/CFB (5 nm) device (D1) at different temperatures. FIG. 3. Temperature dependence of (a) τ\|\|, (b) τ, (c) \|\|, and (d) [ \|\| (by Vs only) ‒ \|\| (by Vs/Va)]/[\|\| (by Vs/Va)] in Bi 2Se3/CFB (5 nm) for D1, D2, and D3. The \|\| is analyzed by two different methods, by ‘ Vs only’ and by ‘ Vs/Va’. FIG. 4. (a) Temperature dependent out-of-plane torque ( τ = τ τOe) and (b) out-of-plane torque ratio ( ) in Bi 2Se3/CFB (5 nm) devices. 15 FIG. 1 -200 0 200 400-400-2000200400 CFB 5 nm CFB 4 nm CFB 3 nm CFB 2 nm CFB 1.5 nm M/area (emu/cm2) H (Oe)123450100200300400 M/area CFB (nm)MDL = 1.36 nm 1 10 100200300400 T(K)Rxx () Bi2Se3 20 QL(a) (b)16 FIG. 2 (a) (b) -1000 0 1000-4-20246 50 K 20 K f = 8 GHz 300 K 200 K 100 KV (V) H (Oe) Bias-Tee 100 µmSGH xy Hext GS Ref Signal Lock-in17 FIG. 3 0 50 100 150 200 250 3000.00.40.81.21.6 (,Vs-,Vs/Va)/,Vs/Va D 1 D 2 D 3 T (K)(a) (b) (c) (d) 0 50 100 150 200 250 3000.00.10.20.30.4 D 1 D 2 D 3By Vs Only D 1 D 2 D 3 T (K)By Vs/Va0 50 100 150 200 250 300012345\|\| (Oe) T (K) D 1 D 2 D 3 0 50 100 150 200 250 300012345 (Oe) T (K) D 1 D 2 D 318 FIG. 4 0 50 100 150 200 250 3000123 T (K) (Oe) D 1 D 2 D 3(a) (b) 0 50 100 150 200 250 3000.00.10.20.30.4 T (K) D 1 D 2 D 3
1908.02232v1.Spectral_properties_of_spin_orbital_polarons_as_a_fingerprint_of_orbital_order.pdf	Spectral properties of spin-orbital polarons as a ﬁngerprint of orbital order Krzysztof Bieniasz,1, 2, 3,∗Mona Berciu,2, 3and Andrzej M. Ole´ s4, 1,† 1Marian Smoluchowski Institute of Physics, Jagiellonian University, Prof. S. /suppress Lojasiewicza 11, PL-30348 Krak´ ow, Poland 2Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1 3Quantum Matter Institute, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z4 4Max Planck Institute for Solid State Research, Heisenbergstraße 1, D-70569 Stuttgart, Germany, (Dated: 15 May 2019) Transition metal oxides are a rich group of materials with very interesting physical properties that arise from the interplay of the charge, spin, orbital, and lattice degrees of freedom. One interesting consequence of this, encountered in systems with orbital degeneracy, is the coexistence of long range magnetic and orbital order, and the coupling between them. In this paper we develop and study an eﬀective spin-orbital superexchange model for e3 gsystems and use it to investigate the spectral properties of a charge (hole) injected into the system, which is relevant for photoemission spectroscopy. Using an accurate, semi-analytical, magnon expansion method, we gain insight into various physical aspects of these systems and demonstrate a number of subtle eﬀects, such as orbital to magnetic polaron crossover, the coupling between orbital and magnetic order, as well as the orbital order driving the system towards one-dimensional quantum spin liquid behavior. Our calculations also suggest a potentially simple experimental veriﬁcation of the character of the orbital order in the system, something that is not easily accessible through most experimental techniques. I. INTRODUCTION It is a well established fact that the ground state and ex- citations of a Hubbard-like model in the regime of strong Coulomb interactions are faithfully reproduced by an ef- fective model, derived using second order perturbation theory, which describes almost localized electrons with suppressed charge ﬂuctuations. The simplest and the most extensively studied of such models is the t-Jmodel [ 1], which describes an antiferromagnetic (AF) Heisenberg ex- change interaction between localized spins. Doping away from half-ﬁlling generates an electron (or hole) hopping in the subspace without double occupancies, a formidable many-body problem. Notably, this model predicts that a charge added to the system will produce a string of misaligned spins, when the N´ eel AF state is considered, that would trap it in a linear string potential [ 2–5], while on the other hand it allows for coherent charge propa- gation by means of spin ﬂuctuations [ 6–8] which remove the spin excitations produced by the charge. As such, this is a simple demonstration of a quasiparticle (QP), in which the charge can only move freely if it couples to the magnetic background of the system. In systems with active orbital degrees of freedom, such a low-energy eﬀective model includes superexchange interac- tions between spins and orbitals [ 9,10]. The development of multiorbital Hubbard models [ 11,12], most commonly employed in the description of transition metal oxides withdorbital degeneracy, led to the derivation of spin- orbital superexchange models [ 9], which are t-J-like model generalizations which accommodate the orbital degrees of freedom on equal footing with electron spins [ 13–26]. Such ∗krzysztof.t.bieniasz@gmail.com †a.m.oles@fkf.mpi.de, corresponding authormodels are composed of products of a spin term, charac- terized by the common SU(2) symmetry, and the orbital pseudospin part of a lower symmetry [ 20], reﬂecting the or- bitals’ spatial extent and their interdependence on lattice symmetry. These models allow not only spin but also or- bital long range order in the system, and predict coherent orbital excitations (orbitons) akin to magnons, to which a charge can couple in a similar fashion [ 27–29]. However, the unusual properties of orbitons and their interaction with the spin degree of freedom make this problem even more challenging than the one described above. It is for this reason that these models have remained a challenge that requires novel theoretical approaches. Here we are primarily interested in egsystems, which realize a pseudospin T=1/2interactions and are thus the closest analogue of the t-Jmodel with S=1/2spins. However, due to non-conservation of the orbital quantum number, free propagation of charge will be permitted by the kinetic Hamiltonian, and the interaction with orbitons will primarily make the resulting QP heavier, especially in view of the much smaller role played by orbital ﬂuctu- ations. It was nonetheless suggested that the importance of the ﬂuctuations increases with the dimensionality of theegproblem in the case of ferromagnetic (FM) spin order, with one-dimensional (1D) alternating orbital (AO) systems being Ising-like [30]. On the other hand, for an AF system hole dynamics is dominated by orbital excitations which leads to quasi- localization when AF and AO order coexist [ 31,32]. Here we shall address the interesting complementary question of what happens in an intermediate state where AF and AO orders exist simultaneously, but in orthogonal di- rections, such that the system can be decomposed into 1D AF chains and orthogonal two-dimensional (2D) AO planes. Such a situation occurs in numerous real three- dimensional (3D) systems, in particular in copper-ﬂuoride perovskite KCuF 3[33], and in the perovskite manganitearXiv:1908.02232v1 [cond-mat.str-el] 6 Aug 20192 LaMnO 3[34,35]. Both of these systems are of high inter- est either from the point of view of basic research, or novel phenomena triggered by spin-orbital interplay. KCuF 3is a rare example of a nearly perfect 1D spin liquid [ 36,37], while LaMnO 3has almost perfect orbital order and ap- plications stemming from the colossal magnetoresistance are found in doped La 1−xSrxMnO 3[38, 39]. It is the type of orbital order in spin-orbital systems which is very intriguing. The orbitals occupied by elec- trons in LaMnO 3are tuned by the tetrahedral ﬁeld which splits theegorbitals [ 16,40]. It has been realized long ago that the photoemission spectra in LaMnO 3strongly de- pend on the type of orbital order in the ground state [ 41], but there is no systematic method to measure this order experimentally. Resonance Raman spectroscopy [ 42] and optical properties [ 43,44] were proposed to investigate the orbital order but one has to realize that the orbitals couple rather strongly to spins [ 45] and it is thus chal- lenging to investigate the hole coupling to spin-orbital excitations in a systematic way. In the regime of interme- diate coupling, the spectral functions could be obtained using the generalized gradient approximation with dy- namical mean-ﬁeld theory (GGA+DMFT) [ 46]. Below we use the strong coupling approach and show that the spectral functions of spin-orbital polarons, obtained from the respective Green’s function, may be used to identify the orbitals occupied in the ground state. The remainder of this paper is organized as follows. We introduce the spin-orbital model with egdegrees of freedom in Sec. II. The variational momentum average method used to generate the spectra with increasing num- ber of excitations is described in Sec. III. In Sec. IV we present and discuss the numerical results obtained for two representative types of orbital order in the intermediate phase with AF/AO order. The paper is summarized with main conclusions in Sec. V. Finally, we present the de- tails of the derivation of the mean ﬁeld phase diagram in Appendix A, and some of the more involved steps of the derivation of the fermion-boson polaronic model in Appendix B. II. THE SPIN-ORBITAL MODEL KCuF 3is a tetragonal system (pseudo-cubic to ﬁrst approximation), with Cu( d9) ions placed in octahedral cages of ﬂuorides. The crystal-ﬁeld splitting splits the 3 d orbitals into the low-lying t2gﬁlled states and the active eg states. Thus, the copper conﬁguration can be equivalently described as e3 gin terms of electron occupation, or e1 gin terms of hole occupation. The kinetic part of the Hamiltonian includes the electron hopping tbetween two directional orbitals \|zγ/angbracketright= (3z2 γ−r2)/√ 6, located on nearest neighbor (NN) Cu(3d9) sites, where zγ≡x/y/z is parallel to the main cubic directions a/b/c of the system [ 47]. The comple- mentary orbitals \|¯zγ/angbracketright= (x2 γ−y2 γ)/√ 2do not contribute because they are orthogonal to the intermediary ligandF(2p6) orbitals. The above deﬁnition of the hopping is not practical, however, due to the orbital basis changing with the hopping direction. Transforming all terms into the{\|z/angbracketright,\|¯z/angbracketright}basis we ﬁnd: Ht=−t 4/summationdisplay /angbracketleftij/angbracketright⊥c/parenleftBig d† izσ∓√ 3d† i¯zσ/parenrightBig/parenleftBig djzσ∓√ 3dj¯zσ/parenrightBig −t/summationdisplay /angbracketleftij/angbracketright/bardblcd† izσdjzσ+ H.c., (1) where the upper/lower sign corresponds to the in-plane directionsa/b, respectively. Here, d† izσandd† i¯zσcreate electrons with spin σin the\|z/angbracketrightor the\|¯z/angbracketrightorbital, respec- tively, at site i. The electron interactions are described using a multi- orbital Hubbard-like model, including on-site Coulomb repulsionUand Hund’s exchange interaction JHwhich drives the site towards maximal spin. We are interested in the strongly correlated limit U/greatermucht, which, when con- sidering virtual excitations, e3 ge3 g e2 ge4 g, leads to an eﬀective superexchange model [ 9]. Due to the proximity of degeneracy of the egorbitals, one needs to consider the multiplet structure of the e2 gion. The spectrum of these excitations has four eigenenergies U−3JH,U−JH(dou- ble), andU+JH[48]. Taking all this into consideration leads to the following superexchange Hamiltonian: Hγ 1=−2Jr1/summationdisplay /angbracketleftij/angbracketright/bardblγ/parenleftbigg Si·Sj+3 4/parenrightbigg/parenleftbigg1 4−τγ iτγ j/parenrightbigg ,(2a) Hγ 2= 2Jr2/summationdisplay /angbracketleftij/angbracketright/bardblγ/parenleftbigg Si·Sj−1 4/parenrightbigg/parenleftbigg1 4−τγ iτγ j/parenrightbigg , (2b) Hγ 3= 2Jr3/summationdisplay /angbracketleftij/angbracketright/bardblγ/parenleftbigg Si·Sj−1 4/parenrightbigg/parenleftbigg1 2−τγ i/parenrightbigg/parenleftbigg1 2−τγ j/parenrightbigg , (2c) Hγ 4= 2Jr4/summationdisplay /angbracketleftij/angbracketright/bardblγ/parenleftbigg Si·Sj−1 4/parenrightbigg/parenleftbigg1 2−τγ i/parenrightbigg/parenleftbigg1 2−τγ j/parenrightbigg , (2d) where the{ri}coeﬃcients serve to impose the multiplet structure at ﬁnite Hund’s exchange JH>0, r1=1 1−3η, r 2=r3=1 1−η, r 4=1 1 +η,(3) with η=JH/U, (4) whileτγ iare bond-direction-dependent orbital operators for the principal cubic axes, which can be expressed using the pseudospin operators in the following way: τa/b i=−1 2/parenleftBig Tz i∓√ 3Tx i/parenrightBig , τc i=Tz i, (5) under the standard convention, \|¯z/angbracketright≡\|↑/angbracketright,\|z/angbracketright≡\|↓/angbracketright. (6)3 It can be shown that assuming a FM spin state in the abplanes and under a purely octahedral crystal ﬁeld, the orbital order preferred by the superexchange Hamiltonian is AO, with the \|±/angbracketright= (\|¯z/angbracketright±\|z/angbracketright)/√ 2 (7) states occupied. However, this need not be the case for other magnetic orders. In the general case, the occupied orbitals are given by rotation of the basis, which is most conveniently parametrized with an angle ±(π/2 +φ), where the sign depends on the orbital sublattice, with φ= 0 corresponding to the {\|+/angbracketright,\|−/angbracketright} reference basis (7). For further convenience, we also introduce an orbital crystal ﬁeld into the Hamiltonian, which serves to remove the orbital degeneracy of the system [ 16], and to make the model more realistic [49]: Hz=−Ez/summationdisplay iTz i. (8) This term simulates an axial pressure along the caxis, and for large values of \|Ez\|it supports ferro-orbital (FO) order, with occupied states either \|¯z/angbracketright(forEz>0) or\|z/angbracketright (forEz<0). Tuning the orbital ﬁeld thus allows one to drive the system from AO all the way to FO order in a continuous manner, although we will not be interested in this extreme limitof the superexchange terms are similar to the crystal ﬁeld in that they are linear in the τγoperators, and thus when these are active ( i.e., when the magnetic order is not assumed to be FM) there is an internal orbital ﬁeld already present in the superexchange Hamiltonian. Thus, the external ﬁeld will work either to counter or to enhance these terms, in turn aﬀecting the magnetic order. In this way the system incorporates spin-orbit coupling through indirect means, allowing for the magnetic and orbital orders to aﬀect each other and, furthermore, to be controlled through external parameters, such as an axial pressure. In order to derive an eﬀective polaronic Hamiltonian for a single charge doped into the system, we need to perform a series of rather involved steps: (i) determine the classi- cal ground state by calculating the mean ﬁeld energy and minimizing it with respect to the crystal ﬁeld Ez, for more details see Appendix A; (ii) transform the kinetic part of the Hamiltonian (1)to the orbital basis correspond- ing to the classical ground state; (iii) introduce magnons and orbitons (to represent magnetic and orbital excita- tions above the classical ground state) as slave bosons by means of a Holstein-Primakoﬀ transformation. As these operations are rather tedious and unlikely to be of much interest to the general audience, we relegate this derivation of the polaronic Hamiltonian to Appendix B. It is only important to notice that from this point onward we will be mostly relying on the outlined formalism, and thus we will be referring to magnetic and orbital excita- tions as magnons (denoted with the operators b† i) and orbitons (denoted as a† i), respectively, and treating them as well-deﬁned, spinless bosons, while the charge degree of freedom will be represented by the spinless fermion f† i.The ﬁnal Hamiltonian consists of the exchange term HJ, and the kinetic term Ht. It is important for the understanding of the paper what physical processes are realized by each of those terms. The exchange term, HJ≡H I+HII, is of course responsible for the spin- orbital order in the presence of the crystal ﬁeld; here we have conveniently divided it into the terms quadratic in (pseudo)spin operators, included in HI, and the linear (crystal ﬁeld like) terms included in HII, see the Ap- pendix B. After the Holstein-Primakoﬀ transformation these terms are purely bosonic operators, and include the Ising terms which only serve to “count” the bosonic en- ergy, and the ﬂuctuation terms which create and destroy the various bosons without involving the doped charge, similar to the spin polaron in the t-Jmodel [7]. The kinetic Hamiltonian, Ht≡T +V⊥ t+V/bardbl t, on the other hand, contains all of the charge dynamics, as shown in the Appendix B. The free hopping term Tis restricted to the FMabplanes due to spin conservation—any hop- ping out of plane necessarily produces magnons. The Vt term includes all the processes responsible for the electron- boson coupling and constitute the actual interaction in our model. Because of the in-plane FM order, the per- pendicular term, V⊥ t, can only produce orbitons, while its inﬂuence on magnons is limited to a fermion-magnon swap term. Finally, the out of plane term V/bardbl tdescribes hole dynamics by the coupling to both magnons and or- bitons at the same time. Altogether, these terms represent all the fermion-boson coupling processes possible in this system and include terms as complicated as ﬁve particle interactions. Our variational technique, which we will brieﬂy describe in the next section, allows us to include all of those terms, something that would not be possible to do in more standard polaronic methods relying on the linear spin wave (LSW) approximation. It needs to be emphasized, however, that the present model employs a number of idealizations ( e.g., we neglect the intermediary oxygen orbitals and proper Jahn-Teller interactions, and ignore any resulting structural transi- tions that might occur in the system) and is not intended to produce a realistic low energy excitation spectrum, but rather to study the eﬀects of spin and orbital excitations on the charge dynamics in systems with the A-AF/C-AO ground state, as encountered in KCuF 3and LaMnO 3. The results presented here are therefore not meant to directly address the experimental results, although some of the observed qualitative eﬀects could be relevant to interpret or guide the experiment. III. THE MOMENTUM AVERAGE METHOD We use the well-established momentum average (MA) variational method [ 52–55] to determine the one- electron Green’s function, G(k,ω) =/angbracketleftk\|G(ω)\|k/angbracketright, where G(ω) = [ω+iη−H]−1is the resolvent operator and \|k/angbracketright=f† k\|0/angbracketrightis the Bloch state for an electron injected4 FIG. 1. The mean-ﬁeld phase diagram of the 3D Kugel- Khomskii model. We focus on the A-AF/C-AO spin-orbital order for which we determine the spectral function (10) occurs between two AF phases with FO order (white areas), AF z (left) and AF ¯z(right). The color scale indicates the detuning angleφin degrees. The values of φ= 0 andφ=π/6, found atη= 0.16 (red dashed line), used to investigate the spectral functions in the present study, are indicated by ×and +, respectively. Note that a more complete mean-ﬁeld phase diagram including possible phases described by variational wave functions with short-range order was presented before in Ref. [58]. into the undoped, semiclassical ground state \|0/angbracketright. The HamiltonianHis divided intoH0=T+Hz J, whereHz Jis the Ising part of the exchange terms in Eq. (A2) (usually, the quantum ﬂuctuations are of little importance and can be ignored, see also Ref. [ 56]), and the interaction, V=V⊥ t+V/bardbl t, which might also be extended to include the spin ﬂuctuation terms of the exchange Hamiltonian. The variational MA method uses Dyson’s identity, G(ω) =G0(ω) +G(ω)VG0(ω), (9) to generate the equations of motion (EOMs) for the Green’s functions, within a chosen variational space. Speciﬁcally, evaluation of V\|k/angbracketrightin real space links to gener- alized propagators that involve various bosons beside the fermion; the variational expansion controls which such conﬁgurations are included in the calculation. The EOMs for these generalized Green’s functions are then obtained using the same procedure and the process is continued until all the variational conﬁgurations are exhausted, at which point this hierarchy of coupled EOMs automatically truncates. The validity and accuracy of the approxima- tion is determined by how appropriate is the choice of the variational space; this is usually based on some physically- motivated criterion restricting the spatial spread of the bosonic cloud, as exempliﬁed below. The accuracy of the results can be systematically improved by increasing the variational space until convergence is achieved. In this way we generate analytical EOMs that easilyallow for exact implementation of the local constraints (i.e., charge and bosons are forbidden from being at the same site simply by removing from the variational space the conﬁgurations which violate this constraint). Once generated, the EOMs form an inhomogeneous system of linearly coupled equations, which is solved numerically to yield all the Green’s functions, and in particular G(k,ω) from which we determine the spectral function, A(k,ω) =−1 π/IfracturG(k,ω). (10) This quantity is directly measured through angle resolved photoemission spectroscopy for LaMnO 3, or inverse pho- toemission for KCuF 3. We shall be interested in the spectral function obtained for theA-AF/C-AO spin-orbital order phase where both magnon and orbiton excitations may couple to the moving charge. The mean ﬁeld analysis of this phase includes the energy minimization to select the optimal value of the detuning angle φ, as described in Appendix A. We investigate two ground states with φ= 0 andφ=π/6 found atη= 0.16, shown by the respective symbols in Fig. 1, and take t≡1.0 as the energy unit. Our method, while highly accurate and versatile, does not come without its limitations. The most important stems from the very basis of the expansion, namely the cut-oﬀ criterion being implemented in real space. As a consequence, only local processes can be treated exactly, while other interactions have to be approximated in a way compatible with this methodology. As such, this method is especially well-suited to polaronic problems, where a charge couples to bosonic excitations either on-site or on the nearest-neighboring site, such as in this paper. The most common obstacle here is the treatment of quantum ﬂuctuations, which are not tied to the itinerant charge and are therefore completely non-local. These are gen- erally treated by being included only in the immediate neighborhood of the electron, the logic behind this being that only then will they aﬀect the properties of the arising QP. This works as long as the classical ground state is not too diﬀerent from the true quantum ground state, i.e., the classical state is a good starting point for the expansion. This would make our method tricky to use in 1D, but any higher dimensional problem is easily treatable. An- other limitation comes from the use of real space Green’s functions, which are hard to calculate already for a single electron. Treatment of multi-electron problems is an on- going, highly challenging eﬀort, although this is certainly true of all semi-analytical Green’s function methods. Here we only focus on single-electron spectral functions, which are relevant for photoemission spectroscopies. IV. RESULTS AND DISCUSSION We carry out the MA calculation in the variational space deﬁned by conﬁgurations with up to 4 bosons present. Because the calculation is done for a 3D system5 FIG. 2. The spectral functions A(k,ω) (shown by intensity of brown/yellow color) in partial and full variational space for φ= 0 andJ= 0.1 (left) and J= 0.5 (right). The dashed blue line indicates the free charge dispersion, /epsilon1kφ. The numbers in the upper-left corner indicate the maximal number of magnons and orbitons, respectively. The number in the upper-right corner gives the size of the variational space. The high-symmetry points are: Γ = (0 ,0),X= (π,0),Y= (0,π),S= (π/2,π/2), and M= (π,π). Parameter: η= 0.16. with full treatment of the charge coupling to bosonic de- grees of freedom, the branching factor for the EOMs is far too great to allow us to include more conﬁgurations. Nevertheless, based on our previous research within simi- lar models [ 56,57], we expect this choice to be suﬃcient for the ground state convergence to be satisfactory. In order to distinguish the physical eﬀects arising due to the coupling to magnons and orbitons, we have performed the calculation not only in the variational space with up to four bosons of any kind, but also in subspaces where we further restrict the number of individual bosonic ﬂavors (e.g., up to three orbitons and up to one magnon). This allows us, to some extent, to trace the evolution of the spectral function depending on the bosonic content of the QP’s cloud in its ground state. By comparing these subspace projections to the full calculation, we can infer which bosons dominate the QP dynamics. The spectral functions (10) were obtained for two rep- resentative mixing angles φwith coexisting A-AF/C-AO spin-orbital order, φ= 0 andφ=π/6. They occur at ﬁnite Hund’s exchange η>0 near the orbital degeneracy, Ez≈0. We have selected η= 0.16 which is represen- tative for the AO order in KCuF 3considered here and close to what is reported in earlier studies [ 20,58–62]. This value ensures that both the φ= 0 andφ=π/6 A-AF/C-AO phases appear as the actual ground states within the range of variation of the crystal ﬁeld [58]. The ﬁrstA-AF/C-AO spin-orbital phase is obtained forEz>0 close to the boundary between A-AF and AF ¯z phases, see Fig. 1. It is characterised by symmetric and antisymmetric linear combinations of the basis orbitals, {\|¯z/angbracketright,\|z/angbracketright}; a ﬁnite value of Ez>0 is needed because of the spin order which is AF in the ( a,b) planes and FM along thecaxis. The second spin-orbital phase discussed belowhas the orbital angle φ=π/6 (A1), which is obtained for Ez<0, see Fig. 1. It corresponds to the other extreme characterized by the external orbital ﬁeld favoring the Kugel-Khomskii orbitals. We start by analyzing the spectral functions for the φ= 0 phase in the Ising limit, see Fig. 2. The occupied \|±/angbracketrightorbitals (7) form an AO state shown in Fig. 3.The Ising limit used here is deﬁned by neglecting both spin and orbital ﬂuctuations, i.e., discarding all terms containing operators other than SzorTz. Note that the spectral function density maps are presented in a nonlinear ∝tanh scale which allows us to highlight the low amplitude states that would otherwise not be visible. The results are shown for two values of the superexchange constant, J= 0.1 (canonical value, note that the deﬁnition of J≡t2/Udoes not include here the factor of 4, conventionally present in the standard t-Jmodel) and J= 0.5 (weak interaction regime, this is not a physically relevant limit but it is FIG. 3. The in-plane orbital arrangement of the φ= 0 phase.6 FIG. 4. The extracted QP ground state energies E(k) (left) and spectral weights Z(k) for the full fourth order expansion (right), and the respective subspace expansions for the ex- change constants J= 0.1 (upper panels) and J= 0.5 (lower panels). Parameter: η= 0.16. Labeling conventions are the same as in Fig. 2. useful for exploring the interdependence between orbitons and magnons in the system, and its eﬀect on the polaronic physics). Each panel in Fig. 2 is marked in the upper-left cor- ner with the maximal number of magnons and orbitons, respectively, allowed in a given subspace, and in the upper-right corner with the size of the variational Hilbert space. The lower-right panel marked with the word “Full” presents the full expansion for up to 4 bosons (without further specifying individual bosonic ﬂavors). The dashed blue line indicates the free charge dispersion /epsilon1kφ, and serves as a reference energy for the QP state. As ex- pected, the dressing with bosons creates a QP which is energetically more stable than the free particle, how- ever this comes at the cost of an increased eﬀective mass and decreased mobility. Note that this is all consistent with standard polaronic physics. The renormalization is much smaller for the large Jlimit. This can be easily understood because the cost of creating any boson is pro- portional to J, so the bigger Jis, the more expensive it is to create a big bosonic cloud. Thus, for large Jthere will be fewer bosons in the cloud, resulting in smaller renormalization of physical properties. Remarkably, by comparing the full results against the FIG. 5. The full and partial spectral functions A(k,ω) for the φ=π/6 phase. Parameters: J= 0.1 andη= 0.16. Notation and conventions are the same as in Fig. 2. partial results, we can see that in the strong interaction case (J= 0.1) the QP behaves predominantly like in the orbiton rich cases (1,3) and (2,2). To highlight this eﬀect we extract the ground state energy and spectral weight for all these solutions and plot them against each other, see Fig. 4. As is evident, the full solution tends to include more orbitons and fewer magnons. Having said that though, a cloud consisting of only orbitons would not be suﬃcient to achieve the optimal QP energy, either. Thus, we can already see that this is an intrinsically spin- orbital system, where the interaction of alldegrees of freedom (charge, spin, and orbital) is crucial to achieve the complete understanding of underlying physics. Even more interestingly, if we now make the same comparison for the weak interaction limit ( J= 0.5), we see that this time the QP band behaves most like the magnon- rich solutions (3,1) and (2,2). This suggests a crossover, controlled by the exchange parameter J, between orbiton- rich and magnon-rich QP clouds. This happens because magnons have lower energy and are cheaper to create than orbitons. In the large Jlimit, only very few bosons are created and they are more likely to be magnons, which therefore dominate the dynamics of the resulting QP. In contrast, for small Jall bosons are cheap(er) and orbitons dominate by means of geometric eﬀects, i.e., the fact that the charge can couple to them by moving in any of the three principal cubic directions, in contrast to magnons which couple only when the particle moves along the single AFcdirection [57]. Figure 5 shows the spectral functions for φ=π/6 with J= 0.1. The orbital order itself is depicted in Fig. 6.The ﬁrst striking observation is that the bands show hardly any dispersion at all, except for the purely orbitonic solution (0,4). This is easily understood if we look at the free charge dispersion /epsilon1kφ, which vanishes for φ=π/6, as illustrated by the ﬂat dashed blue reference line in7 FIG. 6. The in-plane orbital arrangement of the φ=π/6 Kugel-Khomskii phase. Fig. 5. In other words,the unrenormalized particle is completely localized, and the coupling to bosons does not change that in any substantial way. The tiny dispersion observed in the orbitonic solution is due to Trugman loops [ 2], which require a 2D AO order, just like we have in this system, and the existence of at least three- boson clouds, hence its appearance in the purely orbitonic solution. In fact, a very tiny dispersion can also be seen in the (1,3) panel, however there the interference between orbitons and magnons clearly suppresses the Trugman processes [ 2], again underlining the crucial role of orbiton- magnon interplay in the physics of these systems. The lack of dispersion in this orbital phase is a straight- forward consequence of a special symmetry of the orbital order in the Kugel-Khomskii state. Namely, as evident from Fig. 6, the φ=π/6 detuning corresponds to the occupation of AO y2−z2/z2−x2, so the hopping pro- cess would require the charge to move from a lobe of one such orbital to the nodal point of the neighboring orbital, which is forbidden by symmetry of the wave function. The results presented thus far point to an interesting experimental possibility. Namely, the orbital order should be discernible from a spectral experiment: the ﬂatter the QP band, the closer the occupied orbitals should be to the φ=π/6 phase. Naturally, determining the exact phase might not be simple, however, verifying the validity of the φ=π/6 case to which most local density approximation (LDA) studies seem to point [ 60–64] should be possible owing to the dispersionless character of this phase. Having said that, the issue of an insulating sample and thus strong charging during an angle resolved photoemission spectroscopy (ARPES) experiment might pose a barrier even to this veriﬁcation. There is another possibility, however, owing to the QP mass renormalization. Going back to Fig. 2 and comparing the QP vs.the free charge dispersion, we see that not only is there a diﬀerence in bandwidth between the two cases, but also the symmetry between the Γ andMpoints is signiﬁcantly suppressed, with the QP band at the Mpoint being much ﬂatter and having a FIG. 7. Comparison of the density of states for the two major orbital phases discussed in this paper, φ= 0 andφ=π/6; Parameters: J= 0.1 andη= 0.16. greatly reduced spectral weight. If we would now integrate the spectrum to produce the density of states (DOS) for this system, we would see that the QP DOS for a dispersive phase should be highly asymmetric, whereas the dispersionless phase should be characterized by a sharp and completely symmetric QP DOS, as veriﬁed in Fig. 7. Thus, the orbital phase could be inferred, even if only approximately, from the shape and asymmetry of the QP DOS. In turn, the DOS can be obtained from a scanning tunneling microscope experiment for which sample charging might be less problematic. In all of the above we have assumed an Ising interac- tion, or to put it diﬀerently, that the eﬀect of quantum ﬂuctuations is negligible. This is a reasonable assumption because ﬂuctuations generally are less important in higher dimensions and here we are dealing with an ostensibly 3D system. Having said that, however, the AO order can at the same time be thought as 2D and the AF or- der as 1D. While it has been established that the role of ﬂuctuations for egorbital pseudospin in a 2D planar subsystem is indeed negligible [ 56], the same assumption seems less justiﬁed for magnetic excitations. Apart from the dimensionality of the corresponding order, another argument is that the relative lack of importance of or- bitonic ﬂuctuations comes from the fact that the orbiton spectrum is gapped, which is not the case for magnons. This is why it is reasonable to neglect the orbital ﬂuctua- tions while including magnetic ﬂuctuations, in order to explicitly establish whether they are relevant or not. Fortunately, magnetic ﬂuctuations may be fairly easily included within MA by allowing arbitrary ﬂuctuations but only in the vicinity of the propagating charge (the variational space cutoﬀ is controlled with exactly the same cloud spatial criteria as before), since these are the only ones which will aﬀect the QP dynamics. Fluctuations occurring far from the charge will instead only aﬀect the8 FIG. 8. Comparison of the spectral functions in the Ising approximation (ising) and the one including full magnetic ﬂuctuations (mﬂuct), and for the two orbital phases, φ= 0 and φ=π/6. To highlight the eﬀect we take the weak interaction regimeJ= 0.5. Notation and conventions are the same as in Fig. 2. Parameter: η= 0.16. nature of undoped regions far from the particle, aﬀecting the overall energy. However, as long as the classical ground state is close enough to the true quantum state realized for a given set of parameters, this would only be reﬂected by a constant shift of the entire spectrum, which is not a physically signiﬁcant eﬀect. To illustrate the role of ﬂuctuations, we present a com- parison between the Ising solution and the one including local ﬂuctuations for both angles, φ= 0 andφ=π/6, see Fig. 8. As their eﬀect proves to be rather elusive, we focus on the weak interaction limit J= 0.5, where the changes can be more readily observed. One immedi- ately sees the huge diﬀerence in the size of the variational spaces, even though only magnetic ﬂuctuations (albeit in all three cubic directions) are considered here. This, however, has surprisingly little overall eﬀect on the QP dispersion. While their eﬀect seems more considerable for the excited states, these are likely not fully converged any- way, so that part of the spectra is not suﬃciently reliable for comparison. A tiny dispersive eﬀect can also be ob- served, most readily visible in the φ=π/6 phase, however it is much too small to be of any practical importance. There is however one interesting feature, namely the pure gain in energy experienced by the QP ground state, indicative of a stronger binding of the QP, which however does not aﬀect its dynamical properties. In particular,while the gain for the φ= 0 phase is relatively small, the one observed for φ=π/6 is considerable. This diﬀerence could be indicative of a subtle quantum eﬀect arising from the spin-orbital coupling in the system. Clearly, the importance of the magnetic ﬂuctuations strongly depends on the orbital phase in the system, and in particular, the ﬂuctuations in the φ=π/6 phase grow particularly strong. This indicates that the magnetic order becomes less classical in character, which is likely caused by the system decoupling into 1D AF chains. There is ample evidence of the actual KCuF 3exhibiting a 1D quantum AF character [ 36], so this would seem to point to the actual orbital phase in that system being close to φ=π/6, something that was long proposed based on electronic structure calculations using LDA. Here we were able to arrive at similar conclusions through indirect means and by a completely diﬀerent methodology. V. SUMMARY AND CONCLUSIONS We have developed an eﬀective spin-orbital superex- change model for an e3 gsystem, and computed the single- polaron spectrum resulting when a single charge is doped in the system by using the semi-analytical, variational momentum average method for calculating Green’s func- tions. This allowed us not only to obtain the relevant spectral functions, but also to gain insight into the nature of the magnetic and orbital order in the system. Thus we were able to demonstrate a number of subtle quantum eﬀects arising from the interaction between the charge, orbital, and magnetic degrees of freedom. One such eﬀects is the change of the character of the polaronic quasiparticle cloud from being dominated by orbitons to being dominated by magnons; this is controlled by the strength of the superexchange interaction J. This behavior, although only a theoretical prediction due to the impossibility of experimentally tuning the parameter Jin such a wide range of values, nonetheless points towards a strong interplay between orbital and magnetic degrees of freedom in this model. It should come as no surprise, then, that their intermingling should also crop up in other properties of the system, some of which might be more readily accessible to experiment. One possible experimental consequence lies in the quasi- particle dispersion being strongly dependent on the orbital order in the system which, coupled with the polaronic sup- pression of the symmetry between the Γ and Mpoints of the Brillouin zone, suggests that the quasiparticle density of states should be particularly sensitive to the orbital order in these systems. In turn, this would point towards Scanning Tunneling Microscopy as a promising tool for an experimental probe of the orbital order type. We thus propose that the orbital order could be inferred by inves- tigating the amplitude to width ratio and the asymmetry of the density of states peaks. Finally, we point out that the orbital order around the detuning angle φ=π/6 seems to drive the magnetic sys-9 tem closer towards the 1D AF chain. Indeed, the already available results of neutron scattering experiments [ 36] demonstrate a nearly ideal 1D spin liquid behavior. We suggest that this is a strong indication that the orbital order in KCuF 3is likely to be close to φ=π/6. ACKNOWLEDGMENTS K. B. and A. M. O. kindly acknowledge support by UBC Stewart Blusson Quantum Matter Insti- tute (SBQMI), by Natural Sciences and Engineering Research Council of Canada (NSERC), and by Naro- dowe Centrum Nauki (NCN, Poland) under Projects Nos. 2016/23/B/ST3/00839 and 2015/16/T/ST3/00503. M. B. acknowledges support from SBQMI and NSERC. A. M. Ole´ s is grateful for the Alexander von Humboldt Foundation Fellowship (Humboldt-Forschungspreis). Appendix A: The mean ﬁeld ground state In this Section we provide the more technical details concerning the derivation of the eﬀective polaronic spin- orbital model that is the basis of our calculation. In order to ﬁnd the classical orbital ground state, we parametrizethe orbital basis in terms of a standard rotation of the {¯z,z}basis. However, since the reference, ﬁeld-free order is composed of alternating \|±/angbracketrightstates (7), the rotation is most conveniently parametrized with an angle ±(π/2+φ), where the sign depends on the orbital sublattice, \|φA/angbracketright= cos/parenleftbiggπ 4+φ 2/parenrightbigg \|¯z/angbracketright+ sin/parenleftbiggπ 4+φ 2/parenrightbigg \|z/angbracketright, \|φB/angbracketright= cos/parenleftbiggπ 4+φ 2/parenrightbigg \|¯z/angbracketright−sin/parenleftbiggπ 4+φ 2/parenrightbigg \|z/angbracketright.(A1) This choice also serves to transform the underlying AO order to an FO order, eﬀectively eliminating the bipartite division of the lattice in the reference state. It should be stressed that this operation does not aﬀect the ac- tual ground state, it merely changes its representation to one that is more convenient—it spares us the trouble of distinguishing between bosons on diﬀerent sublattices (cf.Ref.7).Nowφwill indicate a detuning from the ﬁeld- free orbital order, and the angle between the occupied orbitals on the two sublattices will be 2 φ(i.e., in general, the bases on diﬀerent sublattices will not be mutually orthogonal). To ﬁnd the relation between the orbital ﬁeld Ezand the detuning angle φin Eqs. (A1), we start by writing out the superexchange Hamiltonian in the new basis: H⊥ I=J/summationdisplay /angbracketleftij/angbracketright⊥c/parenleftbig AηSi·Sj+1 4Bη/parenrightbig/braceleftBig 2Cη Aη−1−(2 cos 2φ+ 1)Tz iTz j−(2 cos 2φ−1)Tx iTx j +2eiQRisin 2φ(Tz iTx j−Tx iTz j)∓√ 3 (Tx iTz j+Tz iTx j)/bracerightBig , (A2a) H/bardbl I= 2J/summationdisplay /angbracketleftij/angbracketright/bardblc/parenleftbig AηSi·Sj+1 4Bη/parenrightbig/braceleftBig Cη Aη−1 2+ 2 sin2φTz iTz j+ 2 cos2φTx iTx j−eiQRisin 2φ(Tx iTz j+Tz iTx j)/bracerightBig ,(A2b) H⊥ II=−JCη/summationdisplay /angbracketleftij/angbracketright⊥c/parenleftbig Si·Sj−1 4/parenrightbig/braceleftBig/bracketleftBig sinφ(Tz i+Tz j)∓√ 3eiQRicosφ/bracketrightBig (Tz i−Tz j) −eiQRicosφ(Tx i−Tx j)∓√ 3 sinφ(Tx i+Tx j)/bracerightBig , (A2c) H/bardbl II= 2JCη/summationdisplay /angbracketleftij/angbracketright/bardblc/parenleftBig Si·Sj−1 4+Ez 4JCη/parenrightBig/bracketleftbig sinφ(Tz i+Tz j)−eiQRicosφ(Tx i+Tx j)/bracketrightbig , (A2d) where the last term incorporates the orbital ﬁeld Hz. Here, Q= (π,π,0) is the ordering vector for the C-AO state, and the resulting phase factor encodes the alternating nature of the orbital order. The symbol ⊥//bardblrefers to the cubic directions with respect to the c-axis. Note that the various superexchange terms of Eq. (2)have been split into terms quadratic in {Tz i}operators (HI) and linear in{Tz i}operators (HII). The Hund’s exchange (4) is now encoded in the three prefactors (if η= 0, one ﬁndsA0=B0=C0= 1): Aη=1−η (1 +η)(1−3η), (A3a) Bη=1 + 3η (1 +η)(1−3η), (A3b) Cη=1 1−η2, (A3c) which themselves result from various combinations of therimultiplet parameters listed above. Note that the exchange Hamiltonian has also been shifted in energy so10 that the Ising energy for the ground state of the system is set to zero. This is done merely for reasons of convenience, so that the excitation energies are easier to track once we start considering excitations in the system. Next we evaluate the mean ﬁeld energy assuming the classical ground state to be A-AF/ C-AO, as is known to be the case in KCuF 3. We ﬁnd: EMF=1 4J(Bη−Aη) sin2φ−1 8J(Aη+Bη)(2 cos 2φ+1) −J/parenleftbigg Cη−Ez 2J/parenrightbigg sinφ.(A4) This expression is then minimized with respect to the detuning angle φ, yielding the relation Ez=J[2Cη−(Aη+ 3Bη) sinφ]. (A5) This identity can now be used to eliminate Ezfrom the Hamiltonian by replacing it with the detuning angle φ. Note that if we now set φ= 0, we will, seemingly para- doxically, get Ez= 2JCη,i.e., a ﬁnite orbital ﬁeld corre- sponding to the ﬁeld-free case. This is due to the fact that the superexchange Hamiltonian already includes terms linear in pseudospin operators which behave like an or- bital ﬁeld, and the external ﬁeld works to compensate these terms. In other words, the exchange Hamiltonian breaks cubic symmetry by itself, and the above estimate is the orbital ﬁeld needed to restore it. On the other hand, the caseEz= 0 corresponds to φ=π/6 in the 2D orbital model [ 50], which is the Kugel-Khomskii state composed of alternating y2−z2/z2−x2states. These two limits are commonly cited as the extreme possibilities for the orbital order in this system. The actual orbital order realized in the system will be bounded by these two extremes, and in fact could be, to some extent, tuned by means of an axial pressure∝Ezapplied along the caxis. Appendix B: Fermion-boson polaronic model To go beyond the mean ﬁeld ground state, we derive the eﬀective Hamiltonian transforming the physics of a single charge doped into a spin-orbital model to a fermion- boson many-body problem. To that end, we transform the kinetic Hamiltonian to the same basis as the one considered in the last Section, so that the entire model isexpressed in compatible representations. This leads to H⊥ t=−t 4/summationdisplay /angbracketleftij/angbracketright⊥c,σ/braceleftBig/bracketleftBig (1−2 sinφ)d† iσ0djσ0 −2eiQRicosφ(d† iσ0djσ1−d† iσ1djσ0) ∓√ 3 (d† iσ0djσ1+d† iσ1djσ0) −(1 + 2 sinφ)d† iσ1djσ1/bracketrightBig + H.c./bracerightBig , (B1a) H/bardbl t=−t 2/summationdisplay /angbracketleftij/angbracketright/bardblc,σ/braceleftBig/bracketleftBig (1 + sinφ)d† iσ0dj¯σ0 −eiQRicosφ(d† iσ0dj¯σ1+d† iσ1dj¯σ0) −(1−sinφ)d† iσ1dj¯σ1/bracketrightBig + H.c./bracerightBig , (B1b) where the 0 (1) indices denote the ground (excited) orbital states, respectively. Finally, following Mart´ ınez and Horsch [ 7], we repre- sent the spin and orbital degrees of freedom using a slave boson representation. This is achieved by expanding the (pseudo)spin operators around the assumed mean ﬁeld ground state by means of a Holstein-Primakoﬀ transfor- mation, d† i↑0=f† i, d† i↓0=f† ibi,d† i↑1=f† iai, d† i↓1=f† iaibi,(B2) whereb† icreates a spin excitation at site i,a† icreates an orbital excitation, and f† icreates a spinless fermion which represents the charge degree of freedom, where 0 indicates the siteiin the ground state, while 1 means that the respective site hosts an excited state. Thus, a charge can be added to the system only if it is locally in its (classical) ground state, otherwise if a boson occupied the considered site, ﬁrst it has to be removed before the charge can be added. Also note that the on-site bosonic Hilbert space is restricted to (2 S+ 1) states, and since both the spin and the pseudospin have length 1/2, each site can host not more than one boson of each kind. This local constraint applies to every site and is fully taken into account in our calculations through the variational technique employed. The exchange Hamiltonian also has to be transformed into its bosonic representation, which is done by means of the Holstein-Primakoﬀ transformation of the spin op- erators, Sz i=1 2−b† ibi, S+ i=/radicalBig 1−b† ibibi, S− i=b† i/radicalBig 1−b† ibi, (B3) and similarly for the pseudospin operators, but in terms of orbiton operators/braceleftBig ai,a† i/bracerightBig . There are two issues that are still worth pointing out concerning this transforma- tion. Firstly, the Sz ioperators are the ones that most readily introduce higher order terms into the Hamilto- nian, and thus are principally responsible for inter-bosonic interactions, which can have profound eﬀects for low- dimensional physics [ 51] but which nonetheless are all11 too often neglected in techniques reetalying on the LSW approximation. It is therefore worth mentioning that in our variational method this part of the transformation is not strictly necessary, as the Sz ioperators merely “count” the Ising energy of the bosons and thus their eﬀect can be discerned directly from the conﬁguration of the system. Or, to put it diﬀerently, in our method it would actu- ally be more cumbersome (although possible) to calculate the energy in the LSW approximation than it is to do it exactly. Secondly, the square root factors in the ﬂuctuation operatorsS± iare conventionally treated by Taylor ex- pansion and truncation at second order, to be consistent with the LSW approximation. However, here again, one should realize that these factors merely serve to impose the restriction of a single boson per site, and thus thisconstraint can be taken into account by excluding from the variational space the conﬁgurations which violate it. Therefore, with our variational technique we can bosonize the exchange interactions without the need to abandon any of the inter-bosonic interactions or constraints. Applying these transformations decouples the original fermions into their constituent charge, spin, and orbital degrees of freedom. The free charge propagation Ht, as well as its coupling to the bosonic degrees of freedom, will now be described by the kinetic Hamiltonian, which after the above transformations reads, Ht=T+V⊥ t+V/bardbl t, (B4) where: T=−t 4/summationdisplay /angbracketleftij/angbracketright⊥c(1−2 sinφ)/parenleftBig f† ifj+ H.c./parenrightBig =/summationdisplay k/epsilon1kφf† kfk, (B5a) V⊥ t=t 4/summationdisplay /angbracketleftij/angbracketright⊥c/braceleftBig/bracketleftBig 2eiQRicosφ(a† j−ai)±√ 3(a† j+ai) + (1 + 2 sin φ)a† jai/bracketrightBig (1 +bib† j)f† ifj+ H.c./bracerightBig −t 4/summationdisplay /angbracketleftij/angbracketright⊥c/bracketleftBig (1−2 sinφ)bib† jf† ifj+ H.c./bracketrightBig , (B5b) V/bardbl t=−t 2/summationdisplay /angbracketleftij/angbracketright/bardblc/braceleftBig/bracketleftBig (1 + sinφ)−eiQRicosφ(ai+a† j)−(1−sinφ)aia† j/bracketrightBig (bi+b† j)f† ifj+ H.c./bracerightBig , (B5c) and/epsilon1kφ=−1 2t(1−2sinφ)(coskx+cosky) is the free electron dispersion. We emphasize that it depends on the orbital order (A1) through the angle φ, and vanishes when φ=π/6. The free charge hopping term Tis restricted to theabplanes because of the magnetic order alternating in the perpendicular direction c, which thus requires creation or annihilation of a magnon when the charge hops in that direction.The remaining terms couple the charge to the bosonic degrees of freedom, and some of them are high-order many-body interactions, sometimes coupling the charge to multiple bosons at the same time. Treating these interactions with a technique relying on the LSW approx- imation would be impossible, whereas this can be done within the variational momentum average approach, as presented in Sec. IV. [1]K. A. Chao, J. Spa/suppress lek, and A. M. Ole´ s, J. Phys. C 10, L271 (1977). [2] S. A. Trugman, Phys. Rev. B 37, 1597 (1988). [3]Z. Liu and E. Manousakis, Phys. Rev. B 45, 2425 (1992). [4] E. Manousakis, Phys. Rev. B 75, 035106 (2007). [5]F. Grusdt, M. K´ anasz-Nagy, A. Bohrdt, C. S. Chiu, G. Ji, M. Greiner, D. Greif, and E. Demler, Phys. Rev. X 8, 011046 (2018). [6]C. L. Kane, P. A. Lee, and N. Read, Phys. Rev. B 39, 6880 (1989). [7]G. Mart´ ınez and P. Horsch, Phys. Rev. B 44, 317 (1991). [8]P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys. 78, 17 (2006). [9]K. I. Kugel and D. I. Khomskii, Sov. Phys. Usp. 25, 231(1982). [10] Y. Tokura and N. Nagaosa, Science 288, 462 (2000). [11] A. M. Ole´ s, Phys. Rev. B 28, 327 (1983). [12]S. Hoshino and P. Werner, Phys. Rev. B 93, 155161 (2016). [13]L. F. Feiner, A. M. Ole´ s, and J. Zaanen, Phys. Rev. Lett. 78, 2799 (1997). [14]S. Ishihara, J. Inoue, and S. Maekawa, Phys. Rev. B 55, 8280 (1997). [15]L. F. Feiner and A. M. Ole´ s, Phys. Rev. B 59, 3295 (1999). [16]M. Snamina and A. M. Ole´ s, Phys. Rev. B 97, 104417 (2018). [17]G. Khaliullin and S. Maekawa, Phys. Rev. Lett. 85, 3950 (2000).12 [18]G. Khaliullin, P. Horsch, and A. M. Ole´ s, Phys. Rev. Lett. 86, 3879 (2001); Phys. Rev. B 70, 195103 (2004). [19]G. Khaliullin, Prog. Theor. Phys. Suppl. 160, 155 (2005). [20]A. M. Ole´ s, G. Khaliullin, P. Horsch, and L. F. Feiner, Phys. Rev. B 72, 214431 (2005). [21]J. Chaloupka and G. Khaliullin, Phys. Rev. Lett. 100, 016404 (2008). [22]P. Horsch, A. M. Ole´ s, L. F. Feiner, and G. Khaliullin, Phys. Rev. Lett. 100, 167205 (2008). [23]B. Normand and A. M. Ole´ s, Phys. Rev. B 78, 094427 (2008); B. Normand, Phys. Rev. B 83, 064413 (2011); J. Chaloupka and A. M. Ole´ s, Phys. Rev. B 83, 094406 (2011). [24]J. Sirker, A. Herzog, A. M. Ole´ s, and P. Horsch, Phys. Rev. Lett. 101, 157204 (2008); A. Herzog, P. Horsch, A. M. Ole´ s, and J. Sirker, Phys. Rev. B 83, 245130 (2011). [25]W. Brzezicki, A. M. Ole´ s, and M. Cuoco, Phys. Rev. X 5, 011037 (2015). [26] W. Brzezicki, arXiv:1904.11772 (2019). [27]J. van den Brink, P. Horsch, and A. M. Ole´ s, Phys. Rev. Lett. 85, 5174 (2000). [28]S. Ishihara, Y. Murakami, T. Inami, K. Ishii, J. Mizuki, K. Hirota, S. Maekawa, and Y. Endoh, New J. Phys. 7, 119 (2005). [29] S. Ishihara, Phys. Rev. Lett. 94, 156408 (2005). [30]M. Daghofer, A. M. Ole´ s, and W. von der Linden, Phys. Rev. B 70, 184430 (2004). [31]K. Wohlfeld, A. M. Ole´ s, and P. Horsch, Phys. Rev. B 79, 224433 (2009). [32] M. Berciu, Physics 2, 55 (2009). [33]A. Okazaki and Y. Suemune, J. Phys. Soc. Japan 16, 176 (1961). [34]J.-S. Zhou and J. B. Goodenough, Phys. Rev. Lett. 96, 247202 (2006). [35]T. Kimura, S. Ishihara, H. Shintani, T. Arima, K. T. Takahashi, K. Ishizaka, and Y. Tokura, Phys. Rev. B 68, 060403 (2003). [36]B. Lake, D. A. Tennant, C. D. Frost, and S. E. Nagler, Nature Mat. 4, 329 (2005). [37]B. Lake, D. A. Tennant, and S. E. Nagler, Phys. Rev. B 71, 134412 (2005). [38]G. H. Jonker and J. H. van Santen, Physica 16, 337 (1950). [39]Y. Tokura, Reports on Progress in Physics 69, 797 (2006). [40]K. Ro´ sciszewski and A. M. Ole´ s, Phys. Rev. B 99, 155108 (2019). [41]J. Ba/suppress la, G. A. Sawatzky, A. M. Ole´ s, and A. Macridin,Phys. Rev. Lett. 87, 067204 (2001). [42]R. Kr¨ uger, B. Schulz, S. Naler, R. Rauer, D. Budelmann, J. B¨ ackstr¨ om, K. H. Kim, S.-W. Cheong, V. Perebeinos, and M. R¨ ubhausen, Phys. Rev. Lett. 92, 097203 (2004). [43]M. W. Kim, J. H. Jung, K. H. Kim, H. J. Lee, J. Yu, T. W. Noh, and Y. Moritomo, Phys. Rev. Lett. 89, 016403 (2002). [44]N. N. Kovaleva, A. M. Ole´ s, A. M. Balbashov, A. Maljuk, D. N. Argyriou, G. Khaliullin, and B. Keimer, Phys. Rev. B81, 235130 (2010). [45]M. Snamina and A. M. Ole´ s, New Journal of Physics 21, 023018 (2019). [46]J. Kuneˇ s, I. Leonov, M. Kollar, K. Byczuk, V. Anisimov, and D. Vollhardt, Eur. Phys. J. Special Topics 180, 5 (2010). [47]L. F. Feiner and A. M. Ole´ s, Phys. Rev. B 71, 144422 (2005). [48]A. M. Ole´ s, L. F. Feiner, and J. Zaanen, Phys. Rev. B 61, 6257 (2000). [49]W. Brzezicki, J. Dziarmaga, and A. M. Ole´ s, Phys. Rev. Lett. 109, 237201 (2012). [50]P. Czarnik, J. Dziarmaga, and A. M. Ole´ s, Phys. Rev. B 96, 014420 (2017). [51]K. Bieniasz, P. Wrzosek, A. M. Ole´ s, and K. Wohlfeld, arXiv:1809.07120 (2018). [52] M. Berciu, Phys. Rev. Lett. 97, 036402 (2006). [53]D. J. J. Marchand, G. De Filippis, V. Cataudella, M. Berciu, N. Nagaosa, N. V. Prokof’ev, A. S. Mishchenko, and P. C. E. Stamp, Phys. Rev. Lett. 105, 266605 (2010). [54]M. Berciu and H. Fehske, Phys. Rev. B 84, 165104 (2011). [55]H. Ebrahimnejad, G. A. Sawatzky, and M. Berciu, J. Phys.: Cond. Mat. 28, 105603 (2016). [56]K. Bieniasz, M. Berciu, M. Daghofer, and A. M. Ole´ s, Phys. Rev. B 94, 085117 (2016). [57]K. Bieniasz, M. Berciu, and A. M. Ole´ s, Phys. Rev. B 95, 235153 (2017). [58]W. Brzezicki, J. Dziarmaga, and A. M. Ole´ s, Phys. Rev. B87, 064407 (2013). [59]A. I. Liechtenstein, V. I. Anisimov, and J. Zaanen, Phys. Rev. B 52, R5467 (1995). [60] M. Kataoka, J. Phys. Soc. Jpn. 73, 1326 (2004). [61]E. Pavarini, E. Koch, and A. I. Lichtenstein, Phys. Rev. Lett. 101, 266405 (2008). [62]I. Leonov, D. Korotin, N. Binggeli, V. I. Anisimov, and D. Vollhardt, Phys. Rev. B 81, 075109 (2010). [63]N. Binggeli and M. Altarelli, Phys. Rev. B 70, 085117 (2004). [64]E. Pavarini and E. Koch, Phys. Rev. Lett. 104, 086402 (2010).
1912.05181v2.Selective_tuning_of_spin_orbital_Kondo_contributions_in_parallel_coupled_quantum_dots.pdf	Selective tuning of spin-orbital Kondo contributions in parallel-coupled quantum dots Heidi Potts,1,Martin Leijnse,1Adam Burke,1Malin Nilsson,1 Sebastian Lehmann,1Kimberly A. Dick,1, 2and Claes Thelander1,y 1Division of Solid State Physics and NanoLund, Lund University, SE-221 00 Lund, Sweden 2Centre for Analysis and Synthesis, Lund University, SE-221 00 Lund, Sweden (Dated: March 18, 2020) We use co-tunneling spectroscopy to investigate spin-, orbital-, and spin-orbital Kondo transport in a strongly conned system of InAs double quantum dots (QDs) parallel-coupled to source and drain. In the one-electron transport regime, the higher symmetry spin-orbital Kondo eect manifests at orbital degeneracy and no external magnetic eld. We then proceed to show that the individual Kondo contributions can be isolated and studied separately; either by orbital detuning in the case of spin-Kondo transport, or by spin splitting in the case of orbital Kondo transport. By varying the inter-dot tunnel coupling, we show that lifting of the spin degeneracy is key to conrming the presence of an orbital degeneracy, and to detecting a small orbital hybridization gap. Finally, in the two-electron regime, we show that the presence of a spin-triplet ground state results in spin-Kondo transport at zero magnetic eld. I. INTRODUCTION The Kondo eect is a widely studied many-body phenomenon that has increased the understanding of strongly correlated electron systems. Experimentally it can be investigated using quantum dots (QDs) with highly transparent tunnel barriers, and manifests as a zero-bias conductance resonance. Most studies focus on the spin-1/2 Kondo eect, where an unpaired spin is screened in the absence of a magnetic eld1{3. Two- electron spin states represent another common system for Kondo studies, where resonances arise from a vanish- ing singlet-triplet exchange energy4{6, a magnetic eld induced singlet-triplet crossing7{12, or a triplet ground state13{15. More recently, the orbital Kondo eect, which relies on two degenerate orbitals, has received consider- able attention. In particular, by combining spin and or- bital degrees of freedom, the SU(4) Kondo eect can be studied16. The presence of such a higher symmetry can be identied by dierent temperature scaling17, and an expected enhancement in the shot-noise properties18{20. In single QDs, orbital and spin-orbital Kondo trans- port has been observed in carbon nanotubes18{23, and silicon based devices24,25. However, in such systems, the orbital degeneracy is an inherent material property and the tunability of the orbital alignment is therefore limited. An alternative approach is to fabricate two parallel-coupled QDs for which the orbitals can be tuned independently. Typically, this is realized in two- dimensional electron gases17,26{30, but at the expense of limited Zeeman splitting of spin states. In this work, we use parallel-coupled quantum dots in indium-arsenide (InAs) nanowires to investigate the spin- and orbital Kondo eect. We show that each degeneracy here can be induced and lifted selectively, which makes it an ideal system for studies of higher Kondo symmetries. We start by discussing Kondo res- onances in the one-electron regime, and present resultson the spin-, orbital-, and combined spin-orbital Kondo eect. The individual contributions to the transport are isolated by electric eld-induced orbital detuning, and magnetic eld-induced spin splitting. Furthermore, by controlling the inter-dot tunnel coupling, t, we demonstrate that the formation of hybridized states inhibits the orbital Kondo eect. If tis small, the energy gap of the avoided crossing can only be resolved if the spin-degeneracy is lifted. This underlines the importance of isolating dierent Kondo mechanisms when studying their contribution to Kondo eects of higher symmetry. Finally, we investigate the Kondo eect in the two-electron regime when each of the QDs contains one unpaired electron, and a nite tunnel coupling between the QDs gives rise to two-electron states. A Kondo resonance at zero magnetic eld is here observed due to a spin-triplet ground state. II. EXPERIMENT Our study is based on InAs nanowires where a QD is dened by a thin zinc-blende (ZB) section between two wurtzite (WZ) barriers31. Figure 1(a) shows a schematic representation of the nanowire and a scanning electron microscopy (SEM) image of a representative de- vice. Metal contacts (Ni/Au 25 nm/75 nm) are placed on the outer ZB sections as source and drain32. Trans- port measurements are performed in a dilution refriger- ator with an electron temperature Te70 mK. If ap- plied, the magnetic eld, B, is aligned perpendicular to the substrate plane in this study. Previous studies have shown that two sidegates ( VL,VR) and a global backgate (VBG) allow to split the QD into two parallel-coupled QDs, and the electron population on the two QDs can be controlled independently. The system is highly tun- able, and the small eective electron mass provides large intra-dot orbital separations32{34. In recent works, we have furthermore demonstrated the formation of ring-likearXiv:1912.05181v2 [cond-mat.mes-hall] 17 Mar 20202 ZB WZ QDsEc VR (V)3 4 5 6 7(0,0)(1,0) (0,1)VL (V)-5.0 -5.5 -6.0 -6.5-7.0 -7.5 00.2 0.1 500 nm(a) (b) sourcedrainVL VRQDs G (e2/h) (1,1) Vsd = 25 μV B = 1 T FIG. 1. (a) Top: Schematic of the nanowire and conduction band alignment. Bottom: 45tilted SEM image of a represen- tative device. (b) Conductance ( G) as a function of sidegate voltages at B= 1 T. The electron populations on the left and right QD (Nleft,Nright) after subtracting 2 Nelectrons are in- dicated. Important gate vectors are shown in red, green, and orange. states when coupling the QDs in two points, resulting in a vanishing one-electron hybridization energy35, and a spin-triplet ground state when each QD contains an un- paired electron. III. RESULTS AND DISCUSSION In this article, we investigate transport in a regime where two orbitals (one from each QD) cross in energy and interact, as shown in Fig. 1(b). The electron population of the left and right dot ( Nleft,Nright) is indicated. Here, 2 Nelectrons were subtracted on both QDs, as the contribution of lled orbitals can be neglected due to strong connement (a large-range overview measurement can be found in the supporting information). In such an orbital crossing, Kondo transport due to both spin and orbital degeneracies is possible. The spin-1/2 Kondo eect is expected when one of the QDs contains an unpaired electron, which is the case for transport involving the (1,0) and (0,1) orbitals. Assuming a vanishing hybridization energy for these two orbitals, we additionally expect an orbital Kondo eect where they cross in energy. Additionally, in the (1,1) regime, two-electron spin states provide degeneracies that can also result in Kondo transport. In order to study these dierent types of Kondo origins, we will focus on transport along the gate vectors indicated in green, red, and orange in Fig. 1(b). A. Spin-orbital Kondo eect Figure 2(a) shows a measurement of dierential conductance, d I/dVsd, versus the source-drain voltage, Vsd, along the green gate vector indicated in Fig. 1(b). The gate vector is chosen such that the (1,0) and (0,1) orbitals are approximately degenerate along thisvector in the one-electron regime. A zero-bias peak in the dierential conductance can be observed within the outlined Coulomb diamonds corresponding to the one-electron (1e), two-electron (2e), and three-electron (3e) regimes. Next, we investigate the 1e regime in more detail by detuning the orbitals of the left and right QD with respect to each other (red gate vector). Due to the large interdot Coulomb energy U1;23 meV, the system holds one electron along the whole vector, and all transport features are related to co-tunneling events. The transport measurement at B= 0 T in Fig. 2(b) shows a zero-bias peak36, as well as a step in dierential conductance that approaches zero bias at zero detuning. Comparing with a schematic representation of the states and resulting onset of co-tunneling transport (Fig. 2(c)), we can distinguish two transport processes: 1) the spin-1/2 Kondo eect on the populated QD, and 2) co-tunneling involving both QDs, which we will refer to as orbital co-tunneling. For the latter, an electron tunnels out of the left (right) QD and is replaced by an electron on the right (left) QD (with or without a spin- ip), and the energy cost of this process corresponds to the detuning of the two QDs. At zero detuning of the QDs ( Eorb= 0), the onset of orbital co-tunneling crosses zero bias. However without further investigation it is unclear whether both the spin and the orbital degeneracy contribute to the Kondo resonance. Transport measurements at dierent temperatures were performed in order to extract the Kondo temper- atures in dierent locations in the honeycomb diagram. The intensity of a Kondo zero-bias peak shows a charac- teristic decay with temperature, which can be described by the phenomenological expression1,37 G(T) =G0 1 + (21=s1)T TKns +G1(1) whereTKis the Kondo temperature, G0is the Kondo conductance at T= 0 K, and G1corresponds to a constant background conductance. The parameters s andndepend on the symmetry of the Kondo state. For the spin-1/2 Kondo eect s= 0:22 andn= 2 is commonly used (SU(2) parameters), while Keller et al. have introduced s= 0:2 andn= 3 for the spin-orbital Kondo eect with SU(4) symmetry17. Figure 2(d) shows the temperature dependence of the conductance at Vsd= 0 and Eorb= 0 in the 1e regime. Assuming that both the spin and orbital de- generacies contribute to the Kondo resonance, we use the SU(4) tting parameters, and extract TK= 610 mK (G0= 0:25e2/h,G1= 0:15e2/h) using Eq. 1. However, also the standard parameters for the SU(2) Kondo eect provide a good t, and would give a Kondo temperature ofTK= 950 mK ( G0= 0:33e2/h,G1= 0:08e2/h), thus making it impossible to conclude the symmetry of the Kondo eect by the temperature dependence alone.3 SU(2) TK = 350 mK0.050.100.15 SU(2) TK = 950 mK SU(4) TK = 610 mK(1,0) (0,1) (d) (e)(b)(d) (e) (1,0) (0,1)SU(2) TK = 470 mK 0.20.30.40.5 0.1 T (K)0.01 0.1 1(g) (i)-6.8, 5.4-6.3,4.0 -6.7,5.2-6.2,4.0Vsd (mV)1.0 0 -1.0 0.51.5 -0.5 -1.5(0,0) (1,1) (2,2) 1e 3eB = 0 dI/dVsd (e2/h) 0 0.4 0.2Vsd (mV)1.0 0 -1.00.51.5 -0.5 -1.5 0 0.4 0.2(a) (f)VL,R (V)-6.8, 4.2-5.6,5.6 -6.8,4.3-5.5,5.40.05 0.4 B = 0, t = 0 (1,0) (0,1)E E-EGS (0,1) (1,0)(1,0) (0,1) spin-flip spin-flip detuning, ΔEorb(c) (1,0) (0,1) B = 1 T, t = 0 (1,0)(0,1) (1,0)(0,1) spin-flip spin-flip(0,1) (1,0)EZ (1,0)(0,1)(h) (1,0) (0,1)B = 0 -0.40.4 0Vsd (mV)1 K VR (V)VL (V) (1,0) (0,1)70 mK 1 K VR (V)VL,R (V) VL,R (V) VL,R (V)0.20.30.40.5 0.1dI/dVsd (e2/h) T (K)0.01 0.1 1 T (K)0.01 0.1 1 70 mK 300 mK(j) (k)B = 1 T B = 1 T 0.05 0.3 E E-EGSdI/dVsd (e2/h) dI/dVsd (e2/h) detuning, ΔEorb FIG. 2. Spin-orbital and isolated orbital Kondo transport in the one-electron regime. (a) Measurement of d I/dVsdversusVsd along the green gate vector indicated in Fig. 1(b) for B= 0. A zero-bias peak can be observed in the 1e, 2e, and 3e regimes. (b) Corresponding measurement in the 1e regime, when detuning the orbitals along the red gate vector. The dashed lines represent the sidegate voltages where the temperature sweeps are performed. (c) Schematic representation of the electron energy levels and resulting onset of co-tunneling when detuning the QD orbitals (assuming t= 0). (d) Temperature dependence of the zero-bias conductance peak at orbital degeneracy. Blue and dashed black lines show the t using Eq. 1 and the parameters for SU(2) and SU(4) Kondo scaling, respectively. (e) Temperature dependence and SU(2) Kondo t of the zero-bias conductance peak in the (0,1) state. (f-i) Corresponding measurements and schematic for B= 1T. The spin degeneracy is lifted, but the orbital degeneracy at zero detuning in the 1e regime remains. (j) Same as (f) but for dierent temperatures (only 1e regime). (k) Conductance as a function of sidegate voltages ( Vsd= 25V) for 70 mK and 1 K (1e regime). B. Isolating the spin-1/2 Kondo eect Next, we isolate spin-Kondo transport in the right QD by electrostatically detuning the system to the (0,1) regime, and study its temperature dependence (Fig. 2(e)). For the chosen sidegate voltages, TK= 350 mK can be extracted using Eq. 1 and standard SU(2) pa- rameters (G0= 0:06e2/h,G1= 0:06e2/h). The fact thatG02e2/hcan be explained by an asymmetry in the tunnel couplings to source and drain (Appendix D). In our QD system, the asymmetry can be due to a dierence in barrier thickness and shape of the QDs. C. Isolating the orbital Kondo eect In order to unambiguously verify the orbital contribu- tion to the Kondo eect at Eorb= 0, we isolate the ef- fect from spin-Kondo transport using Zeeman spin split- ting atB= 1 T. The large eective g-factor (g) of InAs (-14.7 in bulk) facilitates lifting of the spin degeneracy by the Zeeman energy EZ=gBB, whereBis the Bohr magneton. In Fig. 2(f) a resulting gap is observed in the2e and 3e regimes, where spin- ip transport by inelastic co-tunneling is possible when the source-drain voltage is larger than the gap energy. Preliminarily, the absence of zero-bias peaks in the 2e and 3e regimes indicates that the observed Kondo peaks in Figs. 2(a) were due to a spin-related Kondo eect. However, in the 1e region the zero-bias peak remains, indicating that a dierent degen- eracy is still present38. Detuning the orbitals along the red gate vector allows to distinguish the Zeeman gap of the single QD orbitals from the onset of orbital co-tunneling (Figs. 2(g,h)). In the (1,0) and (0,1) regimes, we observe EZ0:5 meV, corresponding to g9. The step in dierential conduc- tance, originating from the onset of orbital co-tunneling transport, crosses in the Zeeman gap at Eorb= 0. A zero-bias peak can be observed, which corresponds to the orbital Kondo eect. Temperature dependent con- ductance data at this point (Fig. 2(i)) can be tted with Eq. 1 and standard SU(2) parameters, resulting in TK= 470 mK ( G0= 0:28e2/h,G1= 0:08e2/h). Com- paring the Kondo temperature for the spin-orbital Kondo resonance with that of the pure spin-1/2 or the pure or- bital Kondo eect, we nd a higher TKandG0when both degeneracies are present, which is in agreement with a4 B = 1 T, t > 0 (2,1)(1,2) (2,1)(1,2) spin-flip spin-flip(1,2) (2,1)EZ (2,1)(1,2)Vsd (mV)0.8 0 -0.80.4 -0.4Vsd (mV)0.8 0 -0.80.4 -0.4 VR (V)VL (V) -2.5-2.0-1.5-1.0-0.5 -9 -8 -7 -6(1,1)(2,1) (2,2) (0,0)dI/dVsd (e2/h) 0.05 0.10.10.2 (a) (b) (c) (d)-1.5, -6.6-1.3,-7.2 -1.5,-6.6-1.3,-7.2 (2,1) (1,2)B = 0 B = 1 TVL,R (V) VL,R (V)00.10.20.3 G (e2/h) Vsd = 25 μV B = 1 T E E-EGSdI/dVsd (e2/h) detuning, ΔEorb(1,2) FIG. 3. The eect of a nite tunnel coupling between the QDs in another orbital crossing. (a) Overview measurement with the electron number on the left and right QD ( Nleft,Nright) indicated. (b,c) Measurement of d I/dVsdversusVsdrecorded along the pink gate vector at B= 0 T, and B= 1 T. (d) Schematic representation of the energy levels and resulting onset of co-tunneling when detuning the QD orbitals, assum- ing nite tunnel coupling between the two QD levels. higher symmetry. In Figs. 2(j-k), measurements at dierent tempera- tures are presented for B= 1 T. The bias-dependent measurements of d I/dVsdalong the green gate vector in the 1e regime (Fig. 2(j)) show that the zero-bias peak disappears with increasing temperature, while transport corresponding to sequential tunneling only exhibits a weak temperature dependence. Accordingly, the con- ductance line at orbital degeneracy strongly decays with temperature in Fig. 2(k). We note that the presence of the orbital Kondo eect implies that the orbital quantum number is also a good quantum number in the leads (similar to spin). In this work, the results are discussed in the picture of coupled double QDs, where the orbital degeneracy is commonly attributed to zero tunnel coupling between the two QDs. An orbital Kondo resonance therefore requires two transport channels in the leads, each of them coupling predominantly only to one of the QDs. However, an orbital degeneracy would also be present if the two QDs are coupled into ring-like states (c.f.35), if the hybridiza- tion gap due to spin-orbit interaction and backscattering is smaller than the Kondo temperature18,21,39.D. Eect of orbital hybridization In the following, we will discuss the eect of inter- dot tunnel coupling, based on data from the same device but from a dierent crossing of QD orbitals (Fig. 3(a)). When detuning the orbitals along the pink gate vector forB= 0, both a zero-bias peak, and a sloped step in dierential conductance due to orbital co-tunneling can be observed (Fig. 3(b)). From this data it appears as if the onset of orbital co-tunneling crosses at zero bias for Eorb= 0, similar to what has been shown in Fig. 2(b). However, when the spin degeneracy is lifted ( B= 1 T, Fig. 3(c)), a nite gap at zero detuning is visible instead of an orbital Kondo resonance. This can be understood considering an avoided crossing due to inter-dot tunnel coupling, as schematically presented in Fig. 3(d). In this case the hybridization gap is 70eV, corresponding to a temperature of 800 mK. Since the gap is comparable to the observed Kondo temperatures, it can only be detected by rst lifting the spin-degeneracy. A similar behavior but with a much larger avoided crossing is found for the 3e regime of Fig. 1(b) (see Appendix Fig. 5). E. Spin-1 Kondo transport Finally, we investigate Kondo transport in the 2e regime by detuning the orbitals along the orange gate vector in Fig. 1(b). A zero-bias peak can be observed in the (1,1) regime at B= 0 T (Fig. 4(a)), indicating a Kondo resonance due to a degenerate ground state when each QD contains one electron. At B= 1 T (Fig. 4(b)) this degeneracy is lifted, resulting in a gap of 800eV in the (1,1) regime. A second excited state can be observed 350eV higher in energy than the rst excited state, which implies the existence of two-electron states due to hybridization, rather than non-interacting orbitals on each QD. To study the two-electron states in more detail, bias-dependent transport as function of magnetic eld was performed in the center of (1,1), as shown in Fig. 4(c). The zero-bias peak Zeeman splits withg7, which is slightly dierent compared to g9 in the 1e regime, likely due to the interaction with another orbital40. The second onset of co-tunneling evolves parallel to the rst excited state. These observa- tions are in agreement with a spin-triplet ground state and a spin-singlet excited state, with an exchange energy ofJ350eV, as schematically depicted in Fig. 4(d). The spin-triplet ground state could be explained by the formation of a quantum ring41, which can form when two quantum dots couple in two points, as previously shown for this material system35. We nd that the intensity of the zero-bias peak monotonically decreases both with B-eld (Fig. 4(c)) and temperature (Fig. 4(d)), and can therefore be interpreted as an underscreened spin-1 Kondo eect15,42,43. The temperature scaling can be5 Vsd (mV)1.0 0 -1.00.51.5 -0.5 -1.5 Vsd (mV)0.8 0 -0.80.4 -0.4 B (T)00.4 0.8(2,0) (1,1) (0,2)(a) (b) (c) 0.100.150.20 SU(2) TK = 780 mK(e) 0.10.20.3 -7.0, 7.1-5.7,3.5-7.1,7.4-5.5,3.0B = 0 B = 1 T VL,R (V) VL,R (V) T (K)0.01 0.1 1dI/dVsd (e2/h)dI/dVsd (e2/h)E E-EGS BS T0 T +T-J T+ T0T+ S 0.060.100.14dI/dVsd (e2/h)(d) FIG. 4. Kondo transport in the two-electron regime. (a) Measurement of d I/dVsdversusVsdrecorded along the or- ange gate vector at B= 0 T, showing a Kondo resonance in (1,1). (b) Corresponding measurement at B= 1 T. The ground state degeneracy is lifted, and an additional onset of co-tunneling is observed with an energy of 350eV higher than the rst excited state. (c) Magnetic eld sweep in the center of (1,1). (d) Schematic representation of the 2e state energies and resulting onset of co-tunneling as a function of B-eld. The triplet GS Zeeman splits into T +, T0and T. At niteB-eld, T +is the GS, and we observe transitions to T0and to the singlet state, S. (e) Temperature dependence of the zero-bias conductance peak in the center of (1,1) at B= 0 T. described by Eq. 1, and corresponds to TK= 780 mK (G0= 0:07e2/h,G1= 0:08e2/h) if using standard SU(2) parameters. IV. SUMMARY AND CONCLUSION In summary, we have studied Kondo transport in parallel-coupled QDs in InAs nanowires. In the 1e regime we observe the spin-1/2 Kondo eect, the combined spin-orbital Kondo eect, and the orbital Kondo eect when the spin degeneracy is lifted. We demonstrate that nite inter-dot tunnel coupling inhibits the orbital Kondo eect by hybridizing the orbitals, and emphasize that the presence of a small energy gap can only be detected when the spin degeneracy is selectively lifted. The 2e regime exhibits a triplet ground state likely due to the formation of ring-like states, leading to a Kondo resonance at B= 0. The possibility to isolate the dierent degeneracies makes this an ideal material system for studies of higher symmetry Kondo eects.ACKNOWLEDGMENTS The authors thank I-J. Chen and J. Paaske for fruit- ful discussions. This work was carried out with nan- cial support from the Swedish Research Council (VR), NanoLund, the Knut and Alice Wallenberg Foundation (KAW), and the Crafoord Foundation. H.P. thank- fully acknowledges funding from the Swiss National Sci- ence Foundation (SNSF) via Early PostDoc Mobility P2ELP2 178221. Appendix A: Overview conductance measurement Figure 5(a) shows conductance ( G) for a wide range of sidegate voltages ( VL,VR). The overview measurement shows that honeycombs with no discernible hybridiza- tion can be observed for specic orbital crossings. We explain this by the formation of ring-like states when the orbitals are coupled in two points, as investigated in detail in Ref.35. The orbital crossing showing the spin- orbital Kondo eect (c.f. Fig. 2 of the main article) is highlighted with a red square, while the additional cross- ing where the orbital Kondo eect is suppressed due to a small hybridization gap (c.f. Fig. 3 of the main ar- ticle) is highlighted with a green square. We note that the data presented in Fig. 3 was obtained at a backgate voltage ofVBG=0:8 V, while the overview here was obtained at VBG=1 V. In this supplementary mate- rial, we present additional measurements for the orbital crossing highlighted in red. The gate vectors which will be discussed in the following are indicated in the conduc- tance measurement in Fig. 5(b) (same as Fig. 1(b) of the main article). Appendix B: Extraction of TKfrom the voltage dependence In the main article, the red gate vector where the or- bitals are detuned from the (1,0) to the (0,1) regime is discussed in great detail, and the Kondo temperature (TK) is extracted from the temperature dependence of the zero-bias conductance peak. Alternatively, TKcan be determined from the source-drain voltage dependence of the zero-bias peak. Figure 1(c) shows dierential con- ductance (d I/dVsd) versus the source-drain voltage ( Vsd) along the red gate vector at a magnetic eld of B= 1 T (same as Fig. 2(g) of the main article). Dierential conductance as a function of applied voltage at the or- bital degeneracy point (highlighted with a dashed line) is presented in Fig. 5(d). Note that the x-axis corre- sponds to the voltage applied across the device ( VDUT), after subtracting the voltage drop across the series re- sistance of the amplier and cryostat wiring and lters (Rs= 16:5 k ). For the spin-1/2 Kondo eect, Ple- tyukhov et. al. have introduced that the voltage depen- dence of the dierential conductance can be described6 G (e2/h) 01VL (V) VR (V)-12-8-44 -15 -10 -5 00VBG = -1 V 5(a) VR (V)3 4 5 6 7(0,0)(2,2) (1,0) (0,1)Vsd = 25 μV B = 1 TVL (V)-5.0 -5.5 -6.0 -6.5 -7.0 -7.5 00.3(b) G (e2/h) (1,1) VRVL (0,0) (0,1)(1,1)(1,0)(2,1) (2,2) (1,2)seq. tunneling left QD seq.tunneling right QDspin-orbital Kondo spin-1/2 Kondo spin-1 Kondo(h) TK = 800 mK 0.10.20.3 VDUT (mV)-0.2 -0.1 00.1 0.20.4(d) (1,0) (0,1)(c) -6.7, 5.2-6.2,4.0V L,R (V)B = 1 T Vsd (mV)1.0 0 -1.00.51.5 -0.5 -1.5B = 0 B = 1 T0.050.3 (e) (f) (2,1) (1,2) -6.6 6.9-5.8,4.6V L,R (V)-6.4, 6.5-5.7,4.8V L,R (V) Vsd (mV)1.0 0 -1.00.51.5 -0.5 -1.5 0.050.3dI/dVsd (e2/h) dI/dVsd (e2/h) dI/dVsd (e2/h) VL,R (V)-6.8, 4.2-5.6,5.6GVsd=0 (e2/h) 0.10.20.30.4 0(g) (0,0) (1,1)1e 3eB = 0 FIG. 5. (a) Conductance as a function of sidegate voltages. The red and green square indicate the orbital crossings which are studied. (b) Magnied conductance plot of the red orbital crossing, where relevant gate vectors are indicated. (c) Dierential conductance versus source-drain voltage recorded along the red gate vector at B= 1 T. (d) Dierential conductance as a function of voltage across the device measured at the sidegate voltages corresponding to the dashed line in (c). The data is tted with Eq. B144. (e)-(f) Transport recorded in the 3e regime along the cyan gate vector at B= 0 T, and B= 1 T, respectively. (g) Conductance Gat zero bias along the green gate vector. (h) Schematic representation of the orbital crossing and the dierent transport mechanisms at B= 0 T. by dI=dVsd(Vsd) = dI=dVsd(0) 1 +2(21=s11) (1b+bs2)s1 (B1) withs1= 0:32,b= 0:05,s2= 1:26 and=e(Vsd V0)=(kBT K). The dierential conductance at zero bias (dI/dVsd(0)), the Kondo temperature ( T K), andV0are tting parameters44. Using this equation to t our exper- imental data for the orbital Kondo resonance, and after substituting T K= 1:8TK, as suggested by Pletyukhov et. al.44, we extract a Kondo temperature of TK= 800 mK. We note that the Kondo temperature extracted from the voltage dependence is slightly higher compared to the one obtained by the temperature dependent data. Appendix C: Transport in the 3e regime In the main article, we showed that the orbital Kondo eect is absent if tunnel coupling between the two QDs leads to an avoided level crossing at Eorb= 0 (c.f. Fig. 3 of the main article). In order to illustrate that lift- ing the spin degeneracy can be key to detecting a small hybridization gap, we presented data from a dierent or- bital crossing (highlighted with a green square in Fig. 5(a)). Here, we show transport measurements for the 3e regime of the orbital crossing highlighted with a redsquare in Fig. 5(a). Figures 5(e-f) show d I/dVsdas a function of VsdforB= 0 T and B= 1 T, when de- tuning the orbitals along the cyan gate vector. We ob- serve a relatively large hybridization gap which can be seen both with and without magnetic eld. This nding shows that adding electrons to the QD orbitals slightly alters the tunnel coupling between them. In the 1e regime the gate voltages were tuned in order to have the orbital Kondo eect (which requires the absence of a hybridiza- tion gap). In the 2e regime the two electrons strongly interact, resulting in two-electron states (c.f. Fig 4 of the main article). Finally, in the 3e regime the electrons also strongly interact, which we interpret as a gradual in- crease in tunnel coupling strength with particle number. Figure 5(g) shows the conductance at zero bias along the green gate vector (linecut at Vsd= 0 in Fig. 2(a) of the main article). The single electron peaks are clearly visi- ble, indicating that the sample is not in the mixed valence regime for sidegate voltages in the center of the valleys23. This will be conrmed by estimating the tunnel coupling in Appendix D. Based on our observations, the dierent mechanisms leading to transport through this particular orbital cross- ing are summarized in Fig. 5(h): The red and green lines correspond to sequential tunneling through the left and right QD, respectively. The spin-1/2 Kondo eect is ob- served in the (0,1), (1,0), (1,2), and (2,1) regimes (yellow shading). In the 1e regime, the tunnel coupling between7 Vsd (mV)1.0 0 -1.00.51.5 -0.5 -1.5 -5.0, 4.8-5.6,3.6B = 0 V L,R (V)-5.5, 3.4-6.3,3.0V L,R (V)-4.5, 3.4-6.2,2.2V L,R (V)B = 0 B = 1 T Vgate (meV)0.10.2 0 1.0 2.0 0.5 1.5Left QD Right QD Right QD Left QD Left QD Right QD Vgate (meV)G (e2/h) 0.10.20.30.5 0 1 2 3FWHM = 0.8 meV FWHM = 1.0 meV(a)(b) (c)(d) (e) 00.3 (0,0) (1,0) (2,0) (2,0) (2,1) (2,2) (0,0) (1,0) (2,0) (2,1) (2,2)dI/dVsd (e2/h)0.4 FIG. 6. (a) Dierential conductance versus source drain voltage as a function of gate vector through the unperturbed left and right QD levels (solid and dashed orange gate vector in Fig. 5(b)) for B= 0 T. (c) Corresponding measurement along the yellow gate vector for B= 1 T. (d)-(e) Conductance as a function of gate voltage for sequential tunneling through the left and right QD (purple and pink gate vector in Fig. 5(b)). The data is tted using Eq. D1 to extract . the two QDs is zero, leading to the spin-orbital Kondo ef- fect at orbital degeneracy (blue line). In the (1,1) regime, a Kondo resonance is found due to a degenerate triplet ground state (orange shading). No orbital Kondo eect is found between the (2,0)-(1,1)-(0,2) regimes, explained by a small spin-orbit-induced mixing (see small avoided crossings in Fig. 4(b)). Appendix D: Extraction of the leverarms and tunnel coupling In Fig. 6(a,b) we present d I/dVsdversusVsdfor gate vectors through the unperturbed levels of the left and right QD (corresponding to the solid/dashed orange vec- tors in Fig. 5(b)) at B= 0 T. A zero-bias peak can be observed due to the spin-1/2 Kondo eect in the (1,0) and (2,1) regimes, when the electron population on one of the QDs is odd. At B= 1 T (Fig. 6(c)) the spin de- generacy is lifted, resulting in a Zeeman gap. From the height of the Coulomb diamonds in Figs. 6(a)-(c), the charging energy Ecof the QDs can be extracted. The width of the Coulomb diamonds is then used to calculate the leverarms of the QDs with respect to each of the two sidegate voltages. We note that the Coulomb diamonds were recorded by sweeping both sidegates simultaneously (as indicated on the x-axis of Figs. 6(a)-(c)), and we used the slope of conductance lines in Fig. 5(b) in order to calculate an eective gate vector which depends only on one sidegate voltage. The relevant parameters for both QDs are summarized in Table I. Here, VL=Ldenotes the leverarm of the left QD (L) with respect to the left sidegate voltage VL, and the other leverarms are labeled correspondingly. Next, we extract the tunneling coupling () to the two QDs. This is relevant, since strictly speaking the em- pirical equation to calculate TKfrom the temperature dependence is only valid as long as 0=<0:5, where 0is the energy dierence of the nearest lower energy state relative to the Fermi level of the contacts1.EcVL=LVR=LVL=RVR=R (meV) (meV/V) (meV/V) (meV/V) (meV/V) Left QD 6 14 4 - - Right QD 8 - - 14 8 TABLE I. Approximate values of the charging energy and the QD leverarms. can be estimated from the conductance due to se- quential tunneling through a QD level. At low source- drain voltage and low temperature ( kBT), the con- ductanceGas a function of gate voltage ( Vgate) can be described by a Lorentzian G=Gmax 1 + (2Vgate=)2+c (D1) whereGmaxis the peak conductance, Vgate=Vgate Vgate;0is the detuning from the QD resonance centered atVgate;0, is the full width half maximum (FWHM) of the Lorentzian, and cis a constant oset. The FWHM corresponds to the sum of the tunnel couplings to source and drain = S+ D. Figures 6(d,e) show linecuts through an orbital of the left and right QD, correspond- ing to the purple and pink gate vector in Fig. 5(b). We note that the gate vectors were chosen such that they are perpendicular to the conductance lines of the QD levels, after having converted the sidegate voltages into energies using the leverarms. Using equation D1, 0:8 meV and 1 meV can be extracted for the left and right QD, respectively. We note that the se- quential tunneling lines were measured at Vsd= 25eV, leading to an additional broadening of the peak. The cal- culated therefore can be considered an upper bound. For the orbital Kondo eect, the measurement was done at orbital degeneracy in the 1e in the middle of the two charge state resonances (see Fig. 2(a)). We therefore have0=0:5U1;2=1:5 meV, where U1;2= 3 meV is the inter-dot Coulomb energy. With 0= =1:5<0:5 transport is in the Kondo regime, and the empirical equa-8 tion to extract the Kondo temperature from the temper- ature dependence is therefore valid. Finally, we extract the asymmetry S=Dfor the two QDs based on the sequential tunneling conductance in Figs. 6(d)-(e). Using Gmax= (4 SD)=(S+ D)2e2/h, we extract S=D20 and S=D8 for the left andright QD, respectively. We note that an estimation of the asymmetry based on G0in the Kondo regime would be less accurate since the electron temperature is com- parable toTK45. The asymmetry of the tunnel couplings of each QD also depends on the sidegate voltages due to a change of connement potential. heidi.potts@ftf.lth.se yclaes.thelander@ftf.lth.se 1D. Goldhaber-Gordon, J. G ores, M. A. Kastner, H. Shtrik- man, D. Mahalu, and U. Meirav, Phys. Rev. Lett. 81, 5225 (1998). 2S. M. Cronenwett, T. H. Oosterkamp, and L. P. Kouwen- hoven, Science 281, 540 (1998). 3J. Schmid, J. Weis, K. Eberl, and K. v. Klitzing, Physica B: Condensed Matter 256-258 , 182 (1998). 4J. Schmid, J. Weis, K. Eberl, and K. v. Klitzing, Phys. Rev. Lett. 84, 5824 (2000). 5A. Kogan, G. Granger, M. A. Kastner, D. Goldhaber- Gordon, and H. Shtrikman, Phys. Rev. B 67, 113309 (2003). 6J. Paaske, A. Rosch, P. Woel e, N. Mason, C. M. Marcus, and J. Nygard, Nature Physics 2, 460 (2006). 7S. Tarucha, D. G. Austing, Y. Tokura, W. G. van der Wiel, and L. P. Kouwenhoven, Phys. Rev. Lett. 84, 2485 (2000). 8M. Pustilnik, Y. Avishai, and K. Kikoin, Phys. Rev. Lett. 84, 1756 (2000). 9S. Sasaki, S. De Franceschi, J. Elzerman, W. van der Wiel, M. Eto, S. Tarucha, and L. Kouwenhoven, Nature 405, 764 (2000). 10D. Giuliano, B. Jouault, and A. Tagliacozzo, Phys. Rev. B63, 125318 (2001). 11W. G. van der Wiel, S. De Franceschi, J. M. Elzerman, S. Tarucha, L. P. Kouwenhoven, J. Motohisa, F. Nakajima, and T. Fukui, Phys. Rev. Lett. 88, 126803 (2002). 12S. J. Chorley, M. R. Galpin, F. W. Jayatilaka, C. G. Smith, D. E. Logan, and M. R. Buitelaar, Phys. Rev. Lett. 109, 156804 (2012). 13W. Liang, M. Bockrath, and H. Park, Phys. Rev. Lett. 88, 126801 (2002). 14C. H. L. Quay, J. Cumings, S. J. Gamble, R. dePicciotto, H. Kataura, and D. Goldhaber-Gordon, Phys. Rev. B 76, 245311 (2007). 15N. Roch, S. Florens, T. A. Costi, W. Wernsdorfer, and F. Balestro, Phys. Rev. Lett. 103, 197202 (2009). 16D. Krychowski and S. Lipi nski, Phys. Rev. B 93, 075416 (2016). 17A. J. Keller, S. Amasha, I. Weymann, C. P. Moca, I. G. Rau, J. A. Katine, H. Shtrikman, G. Zar and, and D. Goldhaber-Gordon, Nature Physics 10, 145 (2013), 1306.6326. 18T. Delattre, L. G. Herrmann, P. Morn, J. Berroir, G. F eve, B. Pla cais, D. C. Glattli, M. Choi, C. Mora, and T. Kontos, Nature Physics 5, 208 (2009). 19M. Ferrier, T. Arakawa, T. Hata, R. Fujiwara, R. De- lagrange, R. Deblock, R. Sakano, A. Oguri, and K. Kobayashi, Nature Physics 12, 230 (2015). 20M. Ferrier, T. Arakawa, T. Hata, R. Fujiwara, R. Dela- grange, R. Deblock, Y. Teratani, R. Sakano, A. Oguri, and K. Kobayashi, Phys. Rev. Lett. 118, 196803 (2017).21P. Jarillo-Herrero, J. Kong, H. S. J. v. d. Zant, and C. Dekker, Nature 434, 484 (2005). 22A. Makarovski, A. Zhukov, J. Liu, and G. Finkelstein, Phys. Rev. B 75, 241407(R) (2007). 23A. Makarovski, J. Liu, and G. Finkelstein, Phys. Rev. Lett. 99, 066801 (2007). 24G. P. Lansbergen, G. C. Tettamanzi, J. Verduijn, N. Col- laert, S. Biesemans, M. Blaauboer, and S. Rogge, Nano Letters 10, 455 (2010). 25G. C. Tettamanzi, J. Verduijn, G. P. Lansbergen, M. Blaauboer, M. J. Calder on, R. Aguado, and S. Rogge, Phys. Rev. Lett. 108, 046803 (2012). 26U. Wilhelm, J. Schmid, J. Weis, and K. Klitzing, Physica E: Low-dimensional Systems and Nanostructures 14, 385 (2002). 27A. W. Holleitner, A. Chudnovskiy, D. Pfannkuche, K. Eberl, and R. H. Blick, Phys. Rev. B 70, 075204 (2004). 28S. Sasaki, S. Amaha, N. Asakawa, M. Eto, and S. Tarucha, Phys. Rev. Lett. 93, 017205 (2004). 29S. Amasha, A. J. Keller, I. G. Rau, A. Carmi, J. A. Katine, H. Shtrikman, Y. Oreg, and D. Goldhaber-Gordon, Phys. Rev. Lett. 110, 046604 (2013). 30Y. Okazaki, S. Sasaki, and K. Muraki, Phys. Rev. B 84, 161305(R) (2011). 31M. Nilsson, L. Namazi, S. Lehmann, M. Leijnse, K. A. Dick, and C. Thelander, Phys. Rev. B 93, 195422 (2016). 32M. Nilsson, I.-J. Chen, S. Lehmann, V. Maulerova, K. A. Dick, and C. Thelander, Nano Letters 17, 7847 (2017). 33M. Nilsson, F. Vi~ nas Bostr om, S. Lehmann, K. A. Dick, M. Leijnse, and C. Thelander, Phys. Rev. Lett. 121, 156802 (2018). 34C. Thelander, M. Nilsson, F. Vi~ nas Bostr om, A. Burke, S. Lehmann, K. A. Dick, and M. Leijnse, Phys. Rev. B 98, 245305 (2018). 35H. Potts, I.-J. Chen, A. Tsintzis, M. Nilsson, S. Lehmann, K. A. Dick, M. Leijnse, and C. Thelander, Nature Com- munications 10, 5740 (2019). 36(2019), we also note a nite splitting of the spin-1/2 Kondo resonance in the (1,0) regime. This could imply spin- polarization of the electrons in the InAs nanowire segments that act as source and drain due to the Ni contacts, similar to observations in carbon nanotubes with Ni contacts46. 37T. A. Costi, A. C. Hewson, and V. Zlatic, Journal of Physics: Condensed Matter 6, 2519 (1994). 38(2019), we note that the small gap in the zero-bias peak in the 1e regime can be explained by the gate vector being not exactly aligned with the degeneracy of the QD levels. 39J. Paaske, A. Andersen, and K. Flensberg, Phys. Rev. B 82, 081309(R) (2010). 40S. Csonka, L. Hofstetter, F. Freitag, S. Oberholzer, C. Schnenberger, T. S. Jespersen, M. Aagesen, and J. Ny- grd, Nano Letters 8, 3932 (2008). 41B. Baxevanis and D. Pfannkuche, \Coulomb interaction in9 nite-width quantum rings," in Physics of Quantum Rings (Springer Berlin Heidelberg, 2014) Chap. 15, pp. 381{408. 42M. Pustilnik and L. I. Glazman, Phys. Rev. Lett. 87, 216601 (2001). 43A. Posazhennikova and P. Coleman, Phys. Rev. Lett. 94, 036802 (2005).44M. Pletyukhov and H. Schoeller, Phys. Rev. Lett. 108, 260601 (2012). 45M. Pustilnik and L. Glazman, Journal of Physics: Con- densed Matter 16, R513 (2004). 46J. R. Hauptmann, J. Paaske, and P. E. Lindelof, Nature Physics 4, 373 (2008).
1311.0965v1.Spin_accumulation_detection_of_FMR_driven_spin_pumping_in_silicon_based_metal_oxide_semiconductor_heterostructures.pdf	1 Spin accumulation detection of FMR driven spin pumping in silicon -based metal -oxide -semiconductor hetero structures Y. P u1, P. M. Odenthal2, R. Adur1, J. Beardsley1, A. G. Swartz2, D. V. Pelekhov1, R. K. Kawakami2, J. Pelz1, P. C. Hammel1, E. Johnston -Halperin1 1Department of Physics, The Ohio State University , Columbus, Ohio 43210 2Department of Physics and Astronomy, University of California, Riverside, California 92521 The use of the spin Hall effect and its inverse to electrically detect and manipulate dynamic spin currents generated via ferromagnetic resonance (FMR) driven spin pumping has enabled the investigation of these dynamically injected currents across a wide variety of ferromagnetic materials. However, while this approach has proven to be an invaluable diagnostic for exploring the spin pumping process it requires strong spin -orbit coupling , thus substantially limit ing the materials basis available for the detector/ch annel material (primarily Pt, W and Ta) . Here , we re port FMR driven spin pumping into a weak spin - orbit channel through the measurement of a spin accumulation voltage in a Si-based metal - oxide -semiconductor (MOS) heterostructure . This alternate experimental approach enables the investigation of dynamic spin pumping in a broad class of materials with weak spin-orbit coupling and long spin lifetime while providing additional information regarding the phase evolution of the injected s pin ensemble via Hanle -based measurements of the effective spin lifetime . 2 The creation and manipulation of non -equilibrium spin populations in non -magnetic materials (NM) is one of the cornerstones of modern spintronics. These excitations have to date relied primarily on charge based phenomena, either via direct electrical injection from a ferromagnet (FM)1-7 or through the exploitation of the spin -orbit interaction8-10. Ferromagnetic resonance (FMR) driven spin pumping11-22 is an emerging method to dynam ically inject pure spin current into a NM with no need for an accompanying charge current, implying substantial potential impacts on low energy cost , high efficiency spintronics . However, while the creation of these non -equilibrium spin currents does not require a charge current , previous studies of transport -detected spin pumping do rely on a strong spin -orbit interaction in the NM to convert the spin current into a charge current in the detector via the inverse spin -Hall effect (ISHE) 12-20. This approach has proven to be an instrumental diagnostic , but it does carry with it several limitations; specifically, the ISHE measures spin current not spin density , is only sensitive to a single component of the ful l spin vector and is only effective in materials with strong spin -orbit coupling . Here we demonstrate an alternate detection geometry relying on the measurement of a spin accumulation voltage using a ferromagnetic electrode, similar to the three -terminal geometry pioneered for electrically -driven spin injection4,23-28. This approach dramatically expands the materials basis for FMR driven spin pumping, allows for the direct measurement of spin accumulation in the channel and enables the phase -sensitive meas urement of the injected spin population. Our study is performed in a silicon -based metal -oxide -semiconductor (MOS) structure compatible with current semiconductor logic technologies. Using a Fe/MgO/p -Si tunnel diode we achieve spin pumping into a semiconductor across an insulat ing dielectric. This approach allows voltage -based detection of the spin accumulation under the electrode4-6,23-29. Further, we 3 demonstrate sensitivity to the phase of the injected spin via the observation of Hanle dephasing in the presence of an out -of-plane magnetic field . These results establish a bridge between the pure spin currents generated by FMR driven spin pumping and traditional charge -based spin injection, laying the foundation for a new class of experimental probes and promising the development of novel spin -based devices compatible with current CMOS technologies. Tunne l diodes are fabricated from Fe(10nm)/ MgO( 1.3nm)/ Si(100) heterostructures grown by mo lecular beam epitaxy (MBE). The p -type Si substrates are semiconductor on insulator (SOI) wafers with a 3 μm thick Si device layer containing 5×1018 cm-3 boron dopants , producing a room temperature resistivity of 2× 10-2 Ωcm. The device is patterned by conv entional photolithography technique s into a Fe/MgO/Si tunnel contact of 500 μm × 500 μm lateral size, placed 1 mm away from Au reference contacts for voltage measurements. Spin pumping and FMR measurements are performed in the center of a radio frequency (RF) microwave cavity with f = 9.85GHz with a DC magnetic field , DCH , applied along the x-axis, as sketched in Fig. 1a. On resonance , a pure sp in current is injected into the silicon channel via coupling between the precessing magnetization of the ferromagnet, M, and the conduction electrons in the silicon. This spin current induces an imbalance in the spin -resolved electrochemical potential and consequent spin accumulation given by S , where and are the chemical potentials of up and down spin s, respectively. Using a standard electrical spin detection technique4-6,23-29 the spin accumulation can be detected electrically via the relationship : )1(2eP VS S where SV is the spin -resolved voltage between Fe and Si, P is the spin polarization of Fe , is the spin detection efficiency , and S is assumed to be proportional to the component of the net 4 spin polarization parallel to M30. Figure 1c shows the magnetic field dependence of the FMR intensity (upper panel) and the spin accumulation induced voltage VS (lower panel), clearly demonstrating spin accumulation at the ferromagnetic resonance . Figure 2a show s the RF power , PRF, dependence of the FMR intensity (upper panel) and SV (lower panel) on resonance ; the former is proportional to the square root of RFP and latter is linear with RFP , consistent with ISHE detected spin pumping18-20. As shown in Fig.2b, SV is constant when M reverses, consistent with our local detection geometry wherein the injected spin is always parallel to the magnetization of the FM electrode . Note that this is in contrast to the magnetization dependence of ISHE detection, wherein the sign of the ISHE gives a measure of the spin orientation relative to the detection electrode . As a result, o ur technique distinguishes the spin accumulation signal from artifacts due to magneto -transport or spin transport, such as the anomalous Hall effect, ISHE or spin Seebeck effect , which depend on the direction of M. In addition, the current -voltage characteristic ( I-V) of the tunnel contact is linear at room temperature (see Supplementary Information ), implying at best a weak rectification of any RF - induced pickup currents. This expectation is confirmed by the small offset (below 10 V) observed in spin accumulation voltage measurement s. Our technique also rules out potential spurious signal s due to magneto -electric transport such as tunneling anisotropic magneto - resistance (TAMR) that would require a rectified bias voltage of at least mV scale to give the observed V scale signal observed at resonance. In order to probe the dynamics of the observed spin accumulation, the applied magnetic field is rotated towards the sample normal (within the xz-plane) by an angl e , introducing an out-of-plane component, zH. Due to the strong demagnetization field of our thin -film geometry (~2.2 T) the orientation of the magnetization lags the orientation of the applied field, remaining 5 almost entirely in -plane ( the maximum estimated deviation is 2°). As a result , the injected spins (parallel to M) precess due to the applied field. As the magnitude of zH increases this precession will lead to a dephasing of the spin ensemble and consequent decrease in its net magnetization , (the Hanle effect3-6,23-29). Figure 3a shows the FMR spectrum for different angles ; the resonant field FMRH changes from 250G to 400G as changes from 0 to 40 degrees. This increase is consistent with the fact that the in -plane component of H primarily determines the resonance condition, so as increases a larger total applied field is therefore required to drive FMR (see Supplementa ry Information ). Figure 3b shows SV vs. DCH over the same angular range. The peak position of SV shifts in parallel with the FMR spectrum, but the peak value decreases with increasing zH , as expected for Hanle -induced dephasing in an ensemble of injected spins3-6,23-29. For an isotropic ensemble of spins precessing in a uniform field perpendicular to M the effect of this dephasing on SV can be described by a simple Lorentzian function: )2()(1)(20 SHS Vz x S S where is the Larmor frequency given by /z B effH g , effg is the effective Land é g-factor, B is the Bohr magneton , is the Plank’s constant , and is the spin lifetime. In the more general case that H is not perpendicular to M, as is the case here, then Eq. (2) should be replaced by the more general function: )3() (11 2 22 2 22 0 total totalz y totalx x S S S V where ),, (/ zyxi gHB effi i 23,27. If H is in the xz-plane, this reduces to 6 )4() (11sin cos22 2 0 totalx S S S V This general behavior has been observed in previous studies of three terminal electrical spin injection4,23-28 (Fig. 4a). However, it has been widely reported in electrically detected spin injection experiments that spatially varying local fields due to the magnetic electrodes, coupling to interfacial spin states and other non -idealities generate spin dynamics that are not well described by this simple model. As a result is generally understo od to represent an effective spin lifetime, eff , and while there are some initial efforts to more quantitatively account for the real sample environment, such as the so -called “inver ted-Hanle” measurement23,26,27, a detailed model of these interactions is currently lacking. We explore the functional dependence of the dephasing of our FMR driven spin current by plotting the peak spin accumulation voltage, peak SV , as a function of zH (Fig. 4b, solid circles). The suppression of the spin accumulation at high magnetic fields seen in Fig. 3 is a clear indication of the dephasing of the injected spin ensemble; however, attempts to fit this behavior to Eq. (4) reveal that this simple, isotropic model fails to accurately reproduce our data (Fig. 4b, black dashed lines). In particular, it is a feature of the isotropic model that for a magnetic field that is not parallel to z that the spin polarization along M, and therefore the measured accumulation voltage, does not go to zero at high field even for infinite spin lifetime. This discrepancy likely arise s from contributions due to the various non -idealities discussed above ; in particular, as we discuss below, we believe that the coupling to localized states and the impact of bulk spin diffusion may play a more central role in this experimental geometry. We note that our data is well described by a simple Lorentzian (Eq. 2), though the relationship between the effective spin lifetime extracted from this fit (0.6 ns) , which we label 7 FMR, to the eff defined in Eq. (4) is not clear. For comparison, the intensity of the FMR signal is found to be constant to within roughly 10% (solid red triangles) , suggesting that the spin current is roughly constant and indicating that FMR -driven heating19,20, if present, does not contribute significantly to the field -induced suppression of peak SV seen i n Fig. 4b. A key advantage of our experimental geometry is that it allows direct comparison of this dephasing with the more traditional three -terminal electrical injectio n within the same device26. Figure 4b (open purple circles) shows the spin accumulation voltage measured in the three - terminal geometry as a function of a perpendicular applied field, zH . The dephasing in this geometry is clearly much slower than in FMR driven s pin pumping. This observation is supported by the Lorentzian fit to Eq. (2) indicated by the solid purple line, yielding an effective lifetime of 0.11 ns, consistent with previous reports by our group and others4,23-27. In considering the origin of this discrepancy in observed lifetime a natural suspicion falls on the different experimental methodologies . Specifically , for the spin pumping case the field is applied at an angle , resulting in both in -plane and perpendicular components to the field, while for the DC current injection only the perpendicular component is present. The consequence of this vector magnetic field is twofold: first, it will rotate the precession axis of the injected spins away from the perpendicular case implicit in the simple Hanle model as described above, and second, it will generate an “inver ted” Hanle effect that has been proposed to derive from the interplay between an in -plane applied magnetic fiel d and some finite inhomogeneity in the local fields due to the magnetization of the electrode. In Fig. 4c w e explore this behavior in a control sample wherein we perform both traditional Hanle (i.e. wherein the only applied magnetic field is zH ) and rotating Hanle measurements (i.e. wherein the magnetic field is rotated by an angle , yielding both in -plane and perpendicular components to 8 H), see Supplementary Information. The field values for the rotating Hanle experiment are chosen to correspond to the values o f zH from Fig. 4b. As expected, the traditional Hanle measurement again yields an effective lifetime of roughly 0.1 4 ns (open purple circles) . While the rotating Hanle geometry does yield a slightly shorter lifetime of 0.09 ns (solid blue squares) , this variation is too small (and in the wrong direction) to account for the longer effective lifetime observed for FMR driven spin pumping. We therefore conclude that th e enhanced dephasing rate observed in Fig. 4b indicates the FMR -driven and electrically -driven spin injection processes differ . While the origin of this discrepancy is still an active area of investigation, we note that the observed FMR of 0.6 ns is consistent with previous measurements of the spin lifetime in the bulk silicon channel at these doping level4,23-25. Further, the requirement that there be no net charge flow during FMR driven spin injection implies that any forward propagati ng tunneling process be balanced by an equal and opposite back tunneling process. As shown by the band diagram of spin pumping in Fig. 4a this opens up a potential pathway for coupling from the bulk Si states into the intermediate states that dominate the three -terminal accumulation voltage26,28. The band diagram of electrical spin injection (upper panel of Fig. 4a) shows that this process is strongly suppressed in electrical spin injection due to the finite bias ( 5 mV in this case) present across the tunnel junction . If we assume for the sake of argument that the spin pumping measurement is in fact sensitive to the bulk spin polarization in silicon, we can calculate the spin current using the relation sf S SeJ ; according to Eq . (1) with P=0.4 for Fe and assuming =0.5, the spin current density is SJ ~ 2 × 105 Am-2. This spin current is roughly one order of magnitude smaller than previous reports of spin pumping from conventional ferromagnets into metals. 9 In summary, we demonstrate spin pumping into a semiconductor through an insulat ing dielectric . This approach allows observation of precession of the dynamically injected spins an d characteriz ation of the effective spin lifetime . Our results directly probe the coherence and phase of the dynamically injected spins and the spin manipulation of that spin ense mble via spin precession, lay ing the foundation for novel spin pumping based spintronic applications . Acknowledgements This work is supported by the Center for Emergent Materials at the Ohio State University, a NSF Materials Research Science and Engineering Center (DMR -0820414) (YP, PO, JB, AS, RK, JP and EJH) and by the Department of Energy through grant DE -FG02 -03ER46054 (RA and PCH). Technical support is provided by the NanoSystems Laboratory at the Ohio State University. The aut hors thank Andrew Berger and Steven Tjung for discussion s and assistance. Reference and notes 1. Žutić, I. , Fabian , J. & Das Sarma , S. Spintronics: Fundamentals and applications. Rev. Mod. Phys. 76, 323 (2004). 2. Awschalom , D. D. & Flatté , M. E. Challenges for semiconductor spintronics. Nature Phys. 3, 153 (2007). 3. Appelbaum , I., Huang , B., & Monsma, D. J. Electronic measurement and control of spin transport in silicon . Nature 447, 295 (2007). 4. Dash, S. P. , Sharma, S., Patel, R. S. , de Jong, M. P. & Jansen, R. Electrical creation of spin polarization in silicon at room temperature . Nature 462, 491 (2009). 10 5. Jedema, F. J., Heersche, H. B., Filip, A. T., Baselmans, J. J. A. & van Wees, B. J. Electrical detection of spin precession in a metallic mesoscopic spin valve. Nature 416, 713 (2002). 6. Lou, X. et al. Electrical detection of spin transport in lateral ferromagnet semiconductor devices. Nature Phys. 3, 197 (2007). 7. Jonker, B. T. , Kioscog lou, G., Hanbicki, A. T., Li, C. H., & Thompson, P. E. Electrical spin - injection into silicon from a ferromagnetic metal /tunnel barrier contact . Nature Phys. 3, 542 (2007). 8. Kato , Y. K., Myers , R. C., Gossard , A. C. & Awschalom , D. D. Observation of the Spin Hall Effect in Semiconductors . Science 306, 1910 (2004). 9. Valenzuela , S. O. & Tinkham , M. Direct electronic measurement of the spin Hall effect . Nature 442, 176 (2006). 10. Liu, L. et al. Spin-Torque Switching with the Giant Spin Hall Effect of Tantalum. Science 336, 555 (2012). 11. Tserkovnyak Y., Brataas, A. & Bauer, G. E. W. Enhanced Gilbert damping in thin ferromagnetic Films. Phys. Rev. Lett . 88, 117601 (2002). 12. Saitoh, E., Ueda, M., Miyajima, H. & Tatara, G. Conversion of spin current into charge current at room temperature: Inverse spin -Hall effect. Appl. Phys. Lett. 88, 182509 (2006). 13. Tserkovnyak Y., Brataas, A. & Bauer, G. E.W. & Halperin, B. I. Nonlocal magnetization dynamics in ferromagnetic heterostructures. Rev. Mod. Phys. 77, 1375 (2005). 14. Kajiwara , Y. et al . Transmission of electrical signals by spin -wave interco nversion in a magnetic insulator . Nature 464, 262 (2010). 15. Kurebayashi , H. et al . Controlled enhancement of spin -current emission by three -magnon splitting . Nature Materials 10, 660 (2011). 11 16. Sandweg, C. W . et al. Spin Pumping by Parametrically Excited Exchange Magnons . Phys. Rev. Lett. 106, 216601 (2011). 17. Czeschka , F. D. et al. Scaling behavior of the spin pumping effect in ferromagnet -platinum bilayers . Phys. Rev. Lett. 107, 046601 (2011). 18. Ando , K. et al. Inverse spin -Hall effect induced by spin pumping in metallic system . J. Appl. Phys. 109, 103913 (2011). 19. Ando , K. et al . Electrically tunable spin injector free from the impedance mismatch problem . Nature Materials 10, 655 (2011). 20. Ando , K. & Saitoh, E. Observation of the inverse spin Hall effect i n silicon. Nature Comm. 3, 629 (2012 ). 21. Costache, M. V. et al. Electrical detection of spin pumping due to the precessing magnetization of a single ferromagnet. Phys. Rev. Lett. 97, 216603 (2006). 22. Heinrich , B. et al . Spin pumping at the magnetic insulator (YIG)/normal metal (Au) interfaces . Phys. Rev. Lett. 107, 066604 (2011) . 23. Dash, S. P. et al. Spin precession and inverted Hanle effect in a semiconductor near a finite - roughness ferromagnetic interface . Phys. Rev. B 84, 054410 (2011). 24. Li, C. H., van‘t Erve , O. & Jonker , B. T. Electrical injection and detection of spin accumulation in silicon at 500K with magnetic metal/silicon dioxide contacts . Nature Comm. 2, 245 (2011 ). 25. Gray , N. W. & Tiwaria , A. Room temperature electrical injection and detection of spin polarized carriers in silicon using MgO tunnel barrier . Appl. Phys. Lett. 98, 102112 (2011 ). 26. Pu, Y . et al. Correlation of electrical spin injection and non -linear charge -transport in Fe/MgO/Si . Appl. Phys. Lett. 103, 012402 (2013 ). 12 27. Jeon, K. et al. Electrical investigation of the oblique Hanle effect in ferromagnet / oxide / semiconductor c ontacts . arXiv 1211.3486 (2013). 28. Tran, M. et al. Enhancement of the spin accumulation at the i nterface between a spin- polarized tunnel junction and a semiconductor . Phys. Rev. Lett. 102, 036601 (2009 ). 29. Sasaki, T., Oikawa, T., Suzuki, T., Shiraishi, M., Suzuki, Y. & Noguchi, K. Comparison of spin signals in silicon between nonlocal four -terminal and three -terminal methods. Appl. Phys. Lett. 98, 012508 (2011). 30. As discussed in Ref. 26, Eq. (1) is derived assuming a linear tunneling model, and might underestimate the actual value of S at higher bias if the current depends super -linearly on applied bias. 13 Figure legends Figure 1 \| Experimental setup (a) Schematic of experimental setup. ( b) Diagram of spin accumulation and spin -resolved voltage SV. (c) FMR intensity (upper panel; arrows indicate state of the Fe magnetiza tion) and spin-resolved voltage SV (lower panel) as a function of DCH . Figure 2 \| RF power - and magnetic field - dependence (a) FMR intensity (upper panel) and spin-resolved voltage (lower panel) as a function of RF power; ( b) Solid symbols: SV vs. FMR DC H H when DCH is parallel or anti -parallel with the x- axis; open symbols indicate the voltage between two Au/Si reference contacts; all data is measured under the same experimental conditions. A background offset of ~2 V has been subtracted from all data. Figure 3 \| Experiments with increasing Hz (a) FMR intensity spectra at various magnetic field orientations as described in the text; ( b) SV vs. DCH measured at the same set of magnetic field orientations. The shift in FMR center frequency tracks the expected magnetization anisotropy of the Fe thin film, see text. Figure 4 \| Hanle effect measurements 14 (a) Schematics of experimental setup and band diagram for three -terminal electrical spin injection (upper panel) and spin pumping (lower panel); ( b) Hanle effect as a function of zH for three -terminal (open circles) and spin pumping (solid circles), solid lines are Lorentzian fits yielding 0.11 ns and 0.6 ns, respectively; solid triangles are FMR absorption as a function of H z, the red dashed line is a guide to the eye and the scale bar represents 10% variation; Black dashed lines are simulated using Eq. (4) with , xH = 248G ( dash dot ) and 310G ( dash), respectively , see Supplementary Information . (c) Hanle effect measured in a control sample by the three -terminal method, open circles are measured with magnetic field applied out of plane, solid squares are obtained using same magnetic field configuration as for FMR driven spin pumping; lines are Lorentzian fits yielding 0.14 ns and 0.09 ns, respectively. 15 Figures Figure 1 Y. Pu et al. 16 Figure 2 Y. Pu et al. 17 Figure 3 Y. Pu et al. 18 Figure 4 Y. Pu et al. 19 Supplementary Information Spin accumulation detection of FMR driven spin pumping in silicon -based metal -oxide -semiconductor heterostructures Y. Pu1, P. M. Odenthal2, R. Adur1, J. Beardsley1, A. G. Swartz2, D. V. Pelekhov1, R. K. Kawakami2, J. Pelz1, P. C. Hammel1, E. Johnston -Halperin1 1Department of Physics, The Ohio State University, Columbus, Ohio 43210 2Department of Physics and Astronomy, University of California, Riverside, California 92521 A. Linear I -V of Fe/MgO/Si contact at room temperature The I-V characteristic of the Fe/MgO/p -Si contact is linear at room temperature , as shown in Fig. S1, indicating that in this regime the contact resistance is domin ated by the MgO insulating barrier and contribution from the Schottky barrier is negligible. The linear fit gives 40.9 resistance with about 25m uncertainty. B. Impact of interface roughness and in -plane external magnetic field In contrast to traditional three -terminal spin accumulation measurements, for the FMR driven measurements described in Figs. 3 and 4 it is necessary to apply an external magnetic field both parallel and perpendicular to the magnetization. As described in the main text, the parallel component of the field is necessary to satisfy the conditions of magnetic resonance and 20 the perpendicular component contributes to the decay of spin accumulation, allowing for a Hanle -style measurement of the effective spin life time. However, in consulting the literature23,27 it becomes evident that this geometry potentially raises an additional concern regarding the interpretation of this data. Specifically, the in plane component of the magnetic field will itself induce an “inv erted -Hanle” effect wherein the measured spin accumulation rises with the magnitude of the parallel component of the magnetic field (Fig. S2a). The accepted interpretation of this effect is an annealing of fluctuations in the magnetization induced by surfa ce roughness of the magnetic layer in large external fields23,27. We measure the spin accumulation on a control sample via 3T electrical spin injection, as shown in Fig. S2a, where magnetic field is applied in xz -plane with orientation ranging from in - plane ( H//x, 0 degree) to out of plane (H//z, 90 degree). Using the 3T data of ) (FMR S H HV , where FMRH for given magnetic field orientation is obtained by spin pumping experiment as shown in Figure 3, we can first directly compare electrical spin injection and spin pumping under the same experimental configuration, as discussed in main text. To get a quantitative understanding of the angular dependence shown in Fig. S2a, one can start with a general formula: SSωSSDt (S1) where is the Larmor frequency , D is the spin diffusion constant and is the spin lifetime. In the 3T geometry the impact of spin diffusion is usually believed to be negligible23,27, from this one can obtain an analytical solution under arbitrary applied magnetic field: 21 2 22 2 22 0) (11 total totalz y totalx xS S (S2) where ),, (/ zyxi gHB effi i , representing each component of the applied magnetic field. Figure S2b shows the simulation according to Equation (S2) with = 0.14ns as obtained by Lorentzian fit. Clearly the model fails to explain the data shown in Fig. S2a, especially the observation that at certain orientations the measured spin accumulation rises with the applied magnetic field increasing. As pointed out by previous studies23,27 stray magnetic fields from the injector due to interface roughness should strongly impact on the Hanle -style measurements. The total magnetic field should be taken as ),, ( zyxi H H Hms iext i i , which re presents the contribution from external and magnetic -stray fields. The stray field strength is taken to have a spatial variation )/2cos()0( )( x Hx Hms ims i , where is the typical length scale (~20nm) of the surface roughness23,27. Assuming the spin diffusion length is much longer than we average the total magnetic field over a full period of , i.e. 2 2 2) () (ms iext i i H H H , we therefore have a formula: 2 2 2 22 2 22 2 0)/ (11 ) () () () ( total B totalext zms totalext xms x xH g HH H HH HS S (S3) where ms xH and msH represent the averaged stray field parallel or perpendicular to the injected spins, respectively. Figure S2c shows the simulation according to Equation (S3) with parameter s = 0.9ns, ms xH 270G and msH 440G. The simulation qualitatively agrees with the experiment, but shows some systematic deviations especially in the low -field regime. 22 Although the Equation (S1) is generally accepted, and in principle rigorous analysis can be done with spin precessi on, spin diffusion and spin flip involved, a well -established approach to determine the intrinsic spin lifetime using the local spin detection geometry is still lacking. A precise determination of the intrinsic spin lifetime in our sample is beyond the sco pe of this report; we treat the lifetime obtained by the simple Lorentzian fit as an effective spin lifetime or spin dephasing time, which represents the decay rate of average spin accumulation under applied perpendicular magnetic field. C. Simulation on the Hanle effect under FMR condition As indicated by the FMR spectrum, at FMR there are varying x- and z- components of the applied magnetic field with different field orientation . The table below is a summary: (deg.) 0 10 20 25 30 40 60 Hx (G) 250 276 287 295 310 302 248 Hz (G) 0 49 105 138 179 253 430 As shown in the table, zH increases monotonically with and xH is in range of 248 – 310 G (roughly constant to maintain the conditions for magnetic resonance), both should impact on the Hanle effect. The black dashed curves shown in Fig. 4(b) are simulated using Eq. (4) with and xH = 248G, 310G respectively. In the situation that Eq. (4) is valid, finite values of the spin lifetime should give a weaker H z-dependence than the simulation curve. 23 Figure S1: Plot of current vs. voltage of the Fe/MgO/p -Si contact at room temperature, symbol s are data and the solid line is a linear fit. 24 Figure S 2: (a) Spin accumulation from 3T electrical spin injection, magnetic field is applied in different orientations, ranging from in -plane ( H//x, 0 degree, top curve) to out of plane ( H//z, 90 degree, bottom curve); (b) Simu lations according to Equation (S2) with = 0.14ns, assuming no stray field; ( c) Simulations using Equation (S3 ) with parameters = 0.9ns, ms xH 270G and msH 440G .
1907.11433v1.Universal_relations_for_spin_orbit_coupled_Fermi_gases_in_two_and_three_dimensions.pdf	arXiv:1907.11433v1 [cond-mat.quant-gas] 26 Jul 2019Universal relations for spin-orbit-coupled Fermi gases in two and three dimensions Cai-Xia Zhang Guangdong Provincial Key Laboratory of Quantum Engineerin g and Quantum Materials, GPETR Center for Quantum Precision Measurement and SPTE, South China Normal University, Guangzhou 510006, China Shi-Guo Peng∗and Kaijun Jiang† State Key Laboratory of Magnetic Resonance and Atomic and Mo lecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academ y of Sciences, Wuhan 430071, China and Center for Cold Atom Physics, Chinese Academy of Sciences, W uhan 430071, China (Dated: July 29, 2019) We present a comprehensive derivation of a set of universal r elations for spin-orbit-coupled Fermi gases in three or two dimension, which follow from the short- range behavior of the two-body physics. Besides the adiabatic energy relations, the large-momentu m distribution, the grand canonical po- tential and pressure relation derived in our previous work f or three-dimensional systems [Phys. Rev. Lett. 120, 060408 (2018)], we further derive high-frequenc y tail of the radio-frequency spectroscopy and the short-range behavior of the pair correlation functi on. In addition, we also extend the deriva- tion to two-dimensional systems with Rashba type of spin-or bit coupling. To simply demonstrate how the spin-orbit-coupling eﬀect modiﬁes the two-body sho rt-range behavior, we solve the two- body problem in the sub-Hilbert space of zero center-of-mas s momentum and zero total angular momentum, and perturbatively take the spin-orbit-couplin g eﬀect into account at short distance, since the strength of the spin-orbit coupling should be much smaller than the corresponding scale of the ﬁnite range of interatomic interactions. The univers al asymptotic forms of the two-body wave function at short distance are then derived, which do no t depend on the short-range details of interatomic potentials. We ﬁnd that new scattering param eters need to be introduced because of spin-orbit coupling, besides the traditional s- andp-wave scattering length (volume) and eﬀective ranges. This is a general and unique feature for spin-orbit- coupled systems. We show how these two- body parameters characterize the universal relations in th e presence of spin-orbit coupling. This work probably shed light for understanding the profound pro perties of the many-body quantum systems in the presence of the spin-orbit coupling. I. INTRODUCTION Understanding strongly-interacting many-body sys- tems is one of the most daunting challenges in modern physics. Owing to the development of the experimental technique, ultracold atomic gases acquire a high degree of control and tunability in interatomic interaction, ge- ometry, purity, atomic species, and lattice constant (of optical lattices) [1–5]. To date, ultracold quantum gases have emerged as a versatile platform for exploring a broad variety of many-body phenomena as well as oﬀering nu- merous examples of interesting many-body states [6–8]. Unlike conventional electric gases in condensed matters, atomic quantum gases are extremely dilute, and the mean distance between atoms is usually very large (on the or- der ofµm), while the range of interatomic interactions is several orders smaller (on the order of several tens of nm). Therefore, the two-body correlations characterize the key properties of such many-body systems near scat- tering resonances, where the two-body interactions are simply described by the scattering length and become irrelevant to the speciﬁc form of interatomic potentials. ∗pengshiguo@gmail.com †kjjiang@wipm.ac.cnA set of universal relations, following from the short- range behavior of the two-body physics, govern some cru- cial features of ultracold atomic gases, and provide pow- erful constraints on the behavior of the system. Many of these relations were ﬁrst derived by Shina Tan, such as the adiabatic energy relation, energy theorem, general virial theorem and pressure relation [9–11]. Afterwards, more universal behaviors were obtained by others, such as the radio-frequency (rf) spectroscopy, photoassocia- tion, static structure factors and so on [12]. All these relations are characterized by the only universal quan- tity named contact , and therefore known as the contact theory. During past few years, the concept of contact theory was further generalized to higher-partial-wave in- teractions [13–20] as well as to low dimensions [21–29], and more contacts appear when additional two-body pa- rameters are involved. The reason why the contact theory is signiﬁcantly im- portant in ultracold atoms is attributed to its direct connection to the experimental measurements. Some of the universal relations were experimentally conﬁrmed, in- volving various measurements of the contact itself. For two-component Fermi gases with s-wave interactions, D. S. Jin’s group measured the contact according to three diﬀerent methods, i.e., the momentum distribution, pho- toemission spectroscopy, and rf spectroscopy, and tested the adiabatic energy relation when the interatomic in-2 teraction was adiabatically swept [30]. The asymptotic behavior of the static structure factor at large momen- tum was conﬁrmed by C. J. Vale’s group, by using Bragg spectroscopy technique [31, 32]. Recently, the tempera- ture evolution of the contact was resolved independently by M. Zwierlein’s group and C. J. Vale’s group, espe- cially the characteristic behavior of the contact across the superﬂuid transition [33, 34]. For single-component Fermi gases with p-wave interactions, the feasibility of generalizing the contact theory for higher-partial-wave scatterings was conﬁrmed experimentally by Thywissen’s group [35], in which the anisotropic p-wave interaction was tuned according to the magnetic vector [36]. Nowa- days, the contact gradually becomes one of fundamental concepts in ultracold atomic physics both theoretically and experimentally. In the past decade, the realizations of the spin-orbit (SO) coupling in ultracold neutral atoms have sparked a great deal of interest [37–44]. It provides an ideal plat- form on which to study novel quantum phenomena re- sulted from SO coupling in a highly controllable and tun- able way, such as topological insulators and superconduc- tors [6, 7], and (spin) Hall eﬀect [45–47]. Nevertheless, it is still challenging to theoretically deal with the many- body correlations for SO-coupled systems. Unlike the situation in condensed matters, the intrinsic short-range feature of interatomic potentials is unchanged for neutral atoms even in the presence of SO coupling. The natu- ral question may be raised, from the point of view of the contact theory, as to whether the two-body physics could provide crucial constraints on many-body behaviors of SO-coupled atomic systems. In addition, it was pointed out that although the short-range feature remains, the SO-coupling eﬀect does modify the short-range behav- ior of the two-body wave function [48]. Therefore, the existence and exact forms of universal relations for SO- coupled atomic systems attract a great deal of attention. In [49], we preliminarily discussed some of the universal relations for three-dimensional (3D) Fermi gases in the presence of 3D isotropic SO coupling. We proposed a simple way to construct the short-range wave function, in which the SO coupling eﬀect could be taken into ac- count perturbatively. Since SO-coupling in general cou- ples diﬀerent partial waves of the two-body scatterings, additional contact parameters appear in universal rela- tions. Before long, our theory was veriﬁed by diﬀerent groups near s-wave resonances [50, 51]. So far, the generalization of the contact theory in the presence of SO-coupling is mostly discussed in 3D, while the derivation of these universal relations is still elusiv e in two-dimensional (2D) systems. The short-range behavior of the two-body physics in 2D is diﬀerent from that in 3D: the two-body wave function in 3D is power-law divergent, while one has to deal with the logarithmic divergence in 2D. From the point of view of the contact theory, diﬀer- ent short-range correlations in two-body physics result in diﬀerent forms of universal relations. Therefore, it re- quires a direct extension to 2D in the similar manner asin 3D in the presence of SO coupling. The purpose of this article is to present a compre- hensive derivation of universal relations for SO-coupled Fermi gases. Besides the adiabatic energy relations, the large-momentum distribution, the grand canonical po- tential and pressure relation derived in our previous work for 3D systems [49], we further derive high-frequency tail of the rf spectroscopy and the short-range behavior of the pair correlation function. Then we generalize the deriva- tion of universal relations for 3D systems to 2D case with Rashba SO coupling in a similar way. For the convenience of the presentation, we still construct the short-range be- havior of the two-body wave function in the sub-Hilbert space of zero center-of-mass (c.m.) momentum and zero total angular momentum as before, and then only s- and p-wave scatterings are coupled [49, 52, 53]. Our results show that the SO coupling introduces a new contact and modiﬁes the universal relations of many-body systems. The remainder of this paper is organized as follows. In the next section, we present the derivations of the short-range behavior of two-body wave functions for SO- coupled Fermi gases in three and two dimensions, respec- tively. Subsequently, with the short-range behavior of the two-body wave functions in hands, we derive a set of universal relations for a 3D SO-coupled Fermi gases in Sec. III, and then generalize them to 2D SO-coupled Fermi gases in Sec. IV, including adiabatic energy rela- tions, asymptotic behavior of the large-momentum dis- tribution, the high-frequency behavior of the rf response, short-range behavior of the pair correlation function, grand canonical potential and pressure relation. Finally, the main results are summarized in Sec. V. II. UNIVERSAL SHORT-RANGE BEHAVIOR OF TWO-BODY WAVE FUNCTIONS The ultracold atomic gases are dilute, while the range of interatomic potentials is extremely small. When two fermions get close enough to interact with each other, they usually far away from the others. If only these two-body correlations are taken into account, some key properties of many-body systems are characterized by the short-range two-body physics, which is the basic idea of the contact theory. In this section, we are going to discuss the short-range behavior of two-body wave functions for 3D Fermi gases in the presence of 3D SO coupling and 2D Fermi gases in the presence of 2D SO coupling, respec- tively. Let us consider spin-half SO-coupled Fermi gases, and the Hamiltonian of a single fermion is modeled as ˆH1=/planckover2pi12ˆk2 1 2M+/planckover2pi12λ Mˆχ+/planckover2pi12λ2 2M, (1) whereˆk1=−i∇is the single-particle momentum oper- ator,Mis the atomic mass, /planckover2pi1is the Planck’s constant divided by 2π. Here, the SO coupling is described by the term /planckover2pi12λˆχ/M with the strength λ>0, andˆχtakes3 the isotropic form of ˆk1·ˆσin 3D or the Rashba form of ˆσ×ˆk1·ˆ nin 2D [54], where ˆσis the Pauli operator, and ˆ nis the unit vector perpendicular to the ( x−y) plane. Because of SO coupling, the orbital angular momen- tum of the relative motion of two fermions is no longer conserved, and then all the partial-wave scatterings are coupled [52]. Fortunately, the c.m. momentum Kof two fermions is still conserved as well as the total angular momentum J. For simplicity, we may reasonably focus on the two-body problem in the subspace of K= 0and J= 0, and then only s- andp-wave scatterings are in- volved [52, 53]. Consequently, the Hamiltonian of two spin-half fermions can be written as ˆH2=/planckover2pi12ˆk2 M+/planckover2pi12λ MˆQ(r)+/planckover2pi12λ2 M+V(r), (2) whereˆk=/parenleftBig ˆk2−ˆk1/parenrightBig /2is the momentum operator for the relative motion r=r2−r1,V(r)is the short-range interatomic interaction with a ﬁnite range ǫ,ˆQ(r) = (ˆσ2−ˆσ1)·ˆkin 3D or ˆQ(r) = (ˆσ2−ˆσ1)×ˆk·ˆ nin 2D, andˆσiis the spin operator of the ith atom. In the fol- lows, let us consider the two-body problems in the 3D systems with 3D SO coupling and 2D systems with 2D SO coupling, respectively. A. For 3D systems with 3D SO coupling In the subspace of K= 0andJ= 0, we may choose the common eigenstates of the total Hamiltonian ˆH2and total angular momentum J(= 0) as the basis of Hilbert space, which take the forms of Ω0(ˆ r) =Y00(ˆ r)\|S/an}b∇acket∇i}ht, (3) Ω1(ˆ r) =−i√ 3[Y1−1(ˆ r)\|↑↑/an}b∇acket∇i}ht +Y11(ˆ r)\|↓↓/an}b∇acket∇i}ht−Y10(ˆ r)\|T/an}b∇acket∇i}ht], (4) whereYlm(ˆ r)is the spherical harmonics, ˆ r≡ (θ,ϕ)denotes the angular degree of freedom of the coordinate r, and\|S/an}b∇acket∇i}ht= (\|↑↓/an}b∇acket∇i}ht−\|↓↑/an}b∇acket∇i}ht )/√ 2and/braceleftbig \|↑↑/an}b∇acket∇i}ht,\|↓↓/an}b∇acket∇i}ht,\|T/an}b∇acket∇i}ht= (\|↑↓/an}b∇acket∇i}ht+\|↓↑/an}b∇acket∇i}ht)/√ 2/bracerightbig are the spin-singlet and spin-triplet states with total spin S= 0and1, re- spectively. Then the two-body wave function can for- mally be written in the basis of {Ω0(ˆ r),Ω1(ˆ r)}as Ψ(r) =ψ0(r)Ω0(ˆ r)+ψ1(r)Ω1(ˆ r), (5) whereψi(r) (i= 0,1)is the radial part of the wave func- tion. Note that we here consider an isotropic p-wave in- teraction and the radial wave function ψ1(r)is identical for three scattering channels, i.e., m= 0,±1. Typically, the SO coupling strength (of the order µm−1) is pretty small compared to the inverse of theinteraction range (of the order nm−1) [38, 39], i.e., λ≪ ǫ−1. Moreover, in the low-energy scattering limit, the relative momentum k=/radicalbig ME//planckover2pi12is also much smaller thanǫ−1. Thus, when two fermions get as close as the range of the interaction, i.e., r∼ǫ, the SO coupling can be treated as perturbation as well as the energy. We as- sume that the two-body wave function may take the form of the following ansatz [49] Ψ(r)≈φ(r)+k2f(r)−λg(r), (6) as the distance of two fermions approaches ǫ. Here, we keep up to the ﬁrst-order terms of the energy ( k2) and SO coupling strength ( λ). The advantage of this ansatz is that the functions φ(r),f(r), andg(r)are all indepen- dent onk2andλ. These functions are determined only by the short-range details of the interaction, and thus characterize the intrinsic properties of the interatomic potential. We expect that in the absence of SO coupling the conventional scattering length or volume is included in the zero-order term φ(r), while the eﬀective range is included in f(r), the coeﬃcient of the ﬁrst-order term of k2. Interestingly, we may anticipate that new scattering parameters resulted from SO coupling appear in the ﬁrst- order term of λ[ing(r)]. Conveniently, more scattering parameters may be introduced if higher-order terms of k2andλare perturbatively considered. Inserting the ansatz (6) into the Schrödinger equation ˆH2Ψ(r) =EΨ(r), and comparing the corresponding co- eﬃcients of k2andλ, we ﬁnd /bracketleftbigg −∇2+MV(r) /planckover2pi12/bracketrightbigg φ(r) = 0, (7) /bracketleftbigg −∇2+MV(r) /planckover2pi12/bracketrightbigg f(r) =φ(r), (8) /bracketleftbigg −∇2+MV(r) /planckover2pi12/bracketrightbigg g(r) =ˆQ(r)φ(r). (9) These coupled equations can easily be solved for r > ǫ, and we obtain φ(r) =α0/parenleftbigg1 r−1 a0/parenrightbigg Ω0(ˆr) +α1/parenleftbigg1 r2−1 3a1r/parenrightbigg Ω1(ˆr)+O/parenleftbig r2/parenrightbig , (10) f(r) =α0/parenleftbigg1 2b0−1 2r/parenrightbigg Ω0(ˆr) +α1/parenleftbigg1 2+b1 6r/parenrightbigg Ω1(ˆr)+O/parenleftbig r2/parenrightbig , (11) g(r) =−α1uΩ0(ˆr)−α0(1+vr)Ω1(ˆr)+O/parenleftbig r2/parenrightbig ,(12) whereα0andα1are two complex superposition coeﬃ- cients,aiandbiares-wave scattering length and eﬀec- tive range for i= 0, andp-wave scattering volume and eﬀective range for i= 1, respectively. Interestingly, two4 new scattering parameters uandvare involved as we anticipate. They are corrections from SO coupling to the short-range behavior of the two-body wave function in s- andp-wave channels, respectively. In the absence of SO coupling, if atoms are initially prepared near an s-wave resonance, the contribution from thep-wave channel could be ignored, and we have α1≈0. Naturally, the two-body wave function Ψ(r)reduces to the known s-wave form of (up to a constant α0) Ψ(r) =/parenleftbigg1 r−1 a0+b0k2 2−k2 2r/parenrightbigg Ω0(ˆr)+O/parenleftbig r2/parenrightbig (13) at short distance r/apprgeǫ. Subsequently, when SO coupling is switched on near the s-wave resonance, a considerable p-wave contribution is involved, and the two-body wave function becomes Ψs(r) =/parenleftbigg1 r−1 a0+b0k2 2−k2 2r/parenrightbigg Ω0(ˆr) +(1+vr)λΩ1(ˆr)+O/parenleftbig r2/parenrightbig ,(14) which recovers the modiﬁed Bethe-Peierls boundary con- dition of [48] by noticing Ω0(ˆr) =\|S/an}b∇acket∇i}ht/√ 4πandΩ1(ˆr) =−i(ˆσ2−ˆσ1)·(r/r)\|S/an}b∇acket∇i}ht/√ 16π. We can see that the pa- rametervcharacterizes the hybridization of the p-wave component into the s-wave scattering due to SO coupling. If atoms are initially prepared near a p-wave resonance without SO coupling, the s-wave scattering could be ig- nored, then we have α0≈0. The two-body wave function Ψ(r)takes the known p-wave form at short distance, i.e., Ψ(r) =/parenleftbigg1 r2−1 3a1r+k2 2+b1k2 6r/parenrightbigg Ω1(ˆr)+O/parenleftbig r2/parenrightbig . (15) In the presence of SO coupling near the p-wave resonance, ans-wave component is introduced, and the two-body wave function becomes Ψp(r) =/bracketleftbigg1 r2+k2 2+/parenleftbigg −1 3a1+b1k2 6/parenrightbigg r/bracketrightbigg Ω1(ˆr) +uλΩ0(ˆr)+O/parenleftbig r2/parenrightbig (16) at short distance. We can see that the parameter ude- scribes the hybridization of the s-wave component into thep-wave scattering due to SO coupling. In general, boths- andp-wave scatterings exist between atoms in the absence of SO coupling. Therefore, when SO coupling is introduced, the two-body wave function is generally the arbitrary superposition of Eqs.(14) and (16), and can be written as Ψ3D(r) =α0/parenleftbigg1 r−1 a0+b0k2 2+α1 α0uλ−k2 2r/parenrightbigg Ω0(ˆr) +α1/bracketleftbigg1 r2+k2 2+α0 α1λ+/parenleftbigg −1 3a1+b1k2 6+α0 α1vλ/parenrightbigg r/bracketrightbigg Ω1(ˆr)+O/parenleftbig r2/parenrightbig (17) at short distance r/apprgeǫ. Eq.(17) can be treated as the short-range boundary condition for two-body wave func- tions in 3D in the presence of 3D SO coupling, when both s- andp-wave interactions are considered. B. For 2D systems with 2D SO coupling Let us consider two spin-half fermions scattering in the x−yplane. We easily ﬁnd that the total angular momen- tumJperpendicular to the x−yplane is conserved as well as the c.m. momentum K. Therefore, we may still focus on the two-body problem in the subspace of K=0 andJ=0, which is spanned by the following three or- thogonal basisΩ0(ϕ) =1√ 2π\|S/an}b∇acket∇i}ht, (18) Ω−1(ϕ) =e−iϕ √ 2π\|↑↑/an}b∇acket∇i}ht, (19) Ω1(ϕ) =eiϕ √ 2π\|↓↓/an}b∇acket∇i}ht, (20) whereϕis the azimuthal angle of the relative coordinate r. Then the two-body wave function can formally be expanded as Ψ(r) =/summationdisplay m=0,±1ψm(r)Ωm(ϕ), (21) andψm(r)is the radial wave function. Analogously, the strength of SO coupling as well as the energy can be taken into account perturbatively at short distance. We assume that the two-body wave function has the form5 of the ansatz (6), and the corresponding functions to be determined can easily be solved out from the Schrödinger equation outside the range of the interatomic potential, i.e.,r/apprgeǫ. After straightforward algebra, we obtain φ(r) =α0/parenleftbigg lnr 2a0+γ/parenrightbigg Ω0(ϕ) +/parenleftbigg1 r−π 4a1r/parenrightbigg/summationdisplay m=±1αmΩm(ϕ)+O/parenleftbig r2/parenrightbig ,(22) f(r) =−α0/parenleftbiggπ 4b0+1 4r2lnr 2a0/parenrightbigg Ω0(ϕ) +/parenleftbigg1−2γ 4r−1 2rlnr 2b1/parenrightbigg/summationdisplay m=±1αmΩm(ϕ)+O/parenleftbig r2/parenrightbig , (23)g(r) =−/parenleftBigg/summationdisplay m=±1αm/parenrightBigg uΩ0(ϕ) −α0/parenleftbigg vr+r√ 2lnr 2b1/parenrightbigg/summationdisplay m=±1Ωm(ϕ)+O/parenleftbig r2/parenrightbig (24) forr/apprgeǫ, whereγis Euler’s constant, αm(m= 0,±1) is complex superposition coeﬃcients, amandbmares- wave scattering length and eﬀective range for m= 0, and p-wave scattering area and eﬀective range for \|m\|= 1, respectively. Here, we have assumed that the p-wave in- teraction is isotropic and thus is the same in m=±1 channels, and applied the p-wave eﬀective-range expan- sion of the scattering phase shift, i.e., k2cotδ1=−1/a1+ 2k2ln(kb1)/π[55]. We ﬁnd that two new scattering pa- rameters are similarly introduced, and they demonstrate the hybridization of s- andp-wave scattering in the pres- ence of Rashba SOC in 2D. Finally, the asymptotic form of the two-body wave function at short distance can be written as Ψ2D(r) =α0/bracketleftBigg lnr 2a0+γ−π 4b0k2+/parenleftBigg/summationdisplay m=±1αm α0/parenrightBigg uλ−k2 4r2lnr 2a0/bracketrightBigg Ω0(ϕ) +/summationdisplay m=±1αm/bracketleftbigg1 r+/parenleftbigg −π 4a1+1−2γ 4k2+α0 αmvλ/parenrightbigg r+/parenleftbigg −k2 2+α0 αmλ√ 2/parenrightbigg rlnr 2b1/bracketrightbigg Ωm(ϕ)+O/parenleftbig r2/parenrightbig (25) forr/apprgeǫ. It is apparent that Ψ2D(r)naturally decouples to thes- andp-wave short-range boundary conditions in the absence of SO coupling. However, Rashba SO coupling mixes the s- andp-wave scatterings, and two new scattering parameters uandvare introduced. We should note that the short-range behaviors of the two- body wave function, i.e., Eqs.(17) and (25), are universal and does not depend on the speciﬁc form of interatomic potentials. III. UNIVERSAL RELATIONS IN THE PRESENCE OF ISOTROPIC 3D SO COUPLING In the previous section, we have discussed the two- body problem in the presence of SO coupling, and ob- tained the short-range behaviors of the two-body wave functions. Then, we are ready to consider Tan’s univer- sal relations of SO-coupled many-body systems, if only two-body correlations are taken into account. Owing to the short-range property of interactions between neutral atoms, when two fermions ( iandj) get as close as the range of interatomic potentials, all the other atoms are usually far away. In this case, the many-body wave func- tions approximately take the forms of Eq.(17) in 3D sys- tems with 3D SO coupling, when the fermions iandjapproach to each other. We need to pay attention that the arbitrary superposition coeﬃcient αm(X)then be- comes the functions of the c.m. coordinates of the pair (i,j)as well as those of the rest of the fermions, which we include into the variable X. In the follows, we de- rive a set of universal relations for SO-coupled many- body systems by using Eqs.(17) for 3D SO-coupled Fermi gases. These relations include adiabatic energy relations , the large-momentum behavior of the momentum distri- bution, the high-frequency tail of the rf spectroscopy, the short-range behavior of the pair correlation function, the grand canonical potential and pressure relation. Let us consider a strongly interacting two-component Fermi gases with total atom number N. For simplicity, we con- sider the case with b0≈0for broads-wave resonances in the follows. A. Adiabatic energy relations In order to investigate how the energy varies with the two-body interaction, let us consider two many- body wave functions ΨandΨ′, corresponding to diﬀer- ent interatomic interaction strengths. They satisfy the Schrödinger equation with diﬀerent energies6 N/summationdisplay i=1ˆH(i) 1Ψ =EΨ, (26) N/summationdisplay i=1ˆH(i) 1Ψ′=E′Ψ′, (27) if there is not any pair of atoms within the range of the interaction, where ˆH(i) 1denotes the single-atom Hamilto- nian (1) for the ith fermion. By subtracting [27]∗×Ψ fromΨ′∗×[26], and integrating over the domain Dǫ, the set of all conﬁgurations (ri,rj)in whichr=\|ri−rj\|>ǫ, we arrive at (E−E′)ˆ DǫN/productdisplay i=1driΨ′∗Ψ = −/planckover2pi12 MNˆ r>ǫdXdr/bracketleftbig Ψ′∗∇2 rΨ−/parenleftbig ∇2 rΨ′∗/parenrightbig Ψ/bracketrightbig +/planckover2pi12λ MNˆ r>ǫdXdr/bracketleftBig Ψ′∗/parenleftBig ˆQΨ/parenrightBig −/parenleftBig ˆQΨ′/parenrightBig∗ Ψ/bracketrightBig ,(28) whereN=N(N−1)/2is the number of all the possible ways to pair atom. Using the Gauss’ theorem, the ﬁrst term on the right-hand side (RHS) can be written as −/planckover2pi12 MNˆ r>ǫdXdr/bracketleftbig Ψ′∗∇2 rΨ−/parenleftbig ∇2 rΨ′∗/parenrightbig Ψ/bracketrightbig =−/planckover2pi12 MN" r=ǫ[Ψ′∗∇rΨ−(∇rΨ′∗)Ψ]·ˆndS =/planckover2pi12ǫ2 MN1/summationdisplay i=0ˆ dX/parenleftbigg ψ′∗ i∂ ∂rψi−ψi∂ ∂rψ′∗ i/parenrightbigg r=ǫ,(29) whereSis the surface in which the distance between the two atoms in the pair ( i,j) isǫwith,ˆnis the direction normal to Sbut opposite to the radial direction, and ψ0 (ψ1) is thes-wave (p-wave) component of the radial two- body wave function. In addition, for the second term on the RHS of Eq.(28), we have ˆQ(r)Ψ =−2 r2∂ ∂r/parenleftbig r2ψ1/parenrightbig Ω0(ˆ r)+2∂ψ0 ∂rΩ1(ˆ r),(30) then it becomes /planckover2pi12λ MNˆ r>ǫdXdr/bracketleftBig Ψ′∗/parenleftBig ˆQ(r)Ψ/parenrightBig −/parenleftBig ˆQ(r)Ψ′/parenrightBig∗ Ψ/bracketrightBig =2λ/planckover2pi12ǫ2 MNˆ dX(ψ′∗ 0ψ1−ψ′∗ 1ψ0)r=ǫ.(31)Combining Eqs.(28), (29) and (31), we obtain (E−E′)ˆ DǫN/productdisplay i=1driΨ′∗Ψ =/planckover2pi12ǫ2 MN1/summationdisplay i=0ˆ dX/parenleftbigg ψ′∗ i∂ ∂rψi−ψi∂ ∂rψ′∗ i/parenrightbigg r=ǫ +2λ/planckover2pi12ǫ2 MNˆ dX(ψ′∗ 0ψ1−ψ′∗ 1ψ0)r=ǫ.(32) Inserting the asymptotic form of the many-body wave function Eq.(17) into Eq.(32), and letting E′→Eand Ψ′→Ψ, we ﬁnd δE·ˆ DǫN/productdisplay i=1dri\|Ψ\|2=−/planckover2pi12 M/parenleftBig I(0) a−λIλ/parenrightBig δa−1 0 −/planckover2pi12I(1) a Mδa−1 1+E1 2δb1+3λ/planckover2pi12 2MIλδv −λ/planckover2pi12 M/parenleftbigg 2λI(1) a−1 2Iλ/parenrightbigg δu+/parenleftbigg1 ǫ+b1 2/parenrightbigg I(1) aδE, (33) where I(m) a=Nˆ dX\|αm(X)\|2, (34) Em=Nˆ dXα∗ m(X)/bracketleftBig E−ˆT(X)/bracketrightBig αm(X),(35) Iλ=Nˆ dXα∗ 0(X)α1(X)+c.c., (36) Eλ=Nˆ dXα∗ 0(X)/bracketleftBig E−ˆT(X)/bracketrightBig α1(X)+c.c.(37) form= 0,1, andˆT(X)is the kinetic operator including the c.m. motion of the pair (i,j)and those of all the rest fermions. Using the normalization of the many-body wave function (see appendix A) ˆ DǫN/productdisplay i=1dri\|Ψ\|2= 1 +/parenleftbigg1 ǫ+b1 2/parenrightbigg I(1) a,(38) we can further simplify Eq. (33) as δE=−/planckover2pi12 M/parenleftBig I(0) a−λIλ/parenrightBig δa−1 0−/planckover2pi12I(1) a Mδa−1 1 +E1 2δb1+3λ/planckover2pi12Iλ 2Mδv+λ/planckover2pi12 2M/parenleftBig Iλ−4λI(1) a/parenrightBig δu,(39) which yields the following set of adiabatic energy rela- tions7 ∂E ∂a−1 0=−/planckover2pi12 M/parenleftBig I(0) a−λIλ/parenrightBig , (40) ∂E ∂a−1 1=−/planckover2pi12I(1) a M, (41) ∂E ∂b1=E1 2, (42) ∂E ∂u=λ/planckover2pi12 2M/parenleftBig Iλ−4λI(1) a/parenrightBig , (43) ∂E ∂v=3λ/planckover2pi12Iλ 2M. (44) Interestingly, two additional new adiabatic energy re- lations appear, i.e. Eqs. (43) and (44), which origi- nate from new scattering parameters introduced by SO coupling. These relations demonstrate how the macro- scopic internal energy of an SO-coupled many-body sys- tem varies with microscopic two-body scattering param- eters. B. Tail of the momentum distribution at large q Let us then study the asymptotic behavior of the large momentum distribution for a many-body system with N fermions. The momentum distribution of the ith fermion is deﬁned as ni(q) =ˆ/productdisplay t/negationslash=idrt/vextendsingle/vextendsingle/vextendsingle˜Ψi(q)/vextendsingle/vextendsingle/vextendsingle2 , (45) where˜Ψi(q)≡´driΨ3De−iq·ri, and then the to- tal momentum distribution can be written as n(q) =/summationtextN i=1ni(q). When two fermions (i,j)get close while all the other fermions are far away, we may write the many- body function Ψ3Datr=\|ri−rj\| ≈0as the following ansatz Ψ3D(X,r) =/bracketleftbiggα0(X) r+B0(X)+C0(X)r/bracketrightbigg Ω0(ˆr) +/bracketleftbiggα1(X) r2+B1(X)+C1(X)r/bracketrightbigg Ω1(ˆr)+O/parenleftbig r2/parenrightbig ,(46) whereαm,BmandCm(m= 0,1) are all regular func- tions. Comparing Eq. (17) with (46) at small r, we ﬁnd B0(X) =−α0 a0+α1uλ, (47) B1(X) =α1k2 2+α0λ, (48) C0(X) =−α0k2 2, (49) C1(X) =−α1 3a1+α1b1k2 6+α0vλ. (50)The asymptotic form of the momentum distribution at largeqbut still smaller than ǫ−1is determined by the asymptotic behavior at short distance with respect to the two interacting fermions, then we have ˜Ψi(q)≈ q→∞ˆ drΨ3D(X,r∼0)e−iq·r.(51) Using∇2/parenleftbig r−1/parenrightbig =−4πδ(r), we have the identity f(q)≡ˆ dre−iq·r r=4π q2, (52) so that ˆ drα0(X) rΩ0(ˆr)e−iq·r=4π q2α0(X)Ω0(ˆq),(53) ˆ drB0(X)Ω0(ˆr)e−iq·r= 0, (54) ˆ drC0(X)rΩ0(ˆr)e−iq·r=−8π q4C0(X)Ω0(ˆq),(55) ˆ drα1(X) r2Ω1(ˆr)e−iq·r=−i4π qα1(X)Ω1(ˆq),(56) ˆ drB1(X)Ω1(ˆr)e−iq·r=−i8π q3B1(X)Ω1(ˆq),(57) ˆ drC1(X)rΩ1(ˆr)e−iq·r= 0. (58) Inserting Eqs. (53)-(58) into (51), and then into Eq. (45), we ﬁnd that the total momentum distribution n(q) at largeqtakes the form n3D(q)≈ Nˆ dX32π2α1α∗ 1Ω1(ˆq)Ω∗ 1(ˆq) q2 +i32π2[α0α∗ 1Ω0(ˆq)Ω∗ 1(ˆq)−α∗ 0α1Ω∗ 0(ˆq)Ω1(ˆq)] q3 +/bracketleftbig 32π2α0α∗ 0Ω0(ˆq)Ω∗ 0(ˆq)+64π2k2α1α∗ 1Ω1(ˆq)Ω∗ 1(ˆq) +64π2λ(α0α∗ 1+α∗ 0α1)Ω1(ˆq)Ω∗ 1(ˆq)/bracketrightbig1 q4+O/parenleftbig q−5/parenrightbig . (59) If we are only interested in the dependence of the momen- tum distribution on the amplitude of q, we may integrate over the direction of q, and then we ﬁnd all the odd-order terms ofq−1vanish. Finally, we obtain n3D(q) =C(1) a q2+/parenleftBig C(0) a+C(1) b+λPλ/parenrightBig1 q4+O/parenleftbig q−6/parenrightbig , (60) where the contacts are deﬁned as C(m) a= 32π2I(m) a,(m= 0,1), (61) C(1) b=64π2M /planckover2pi12E1, (62) Pλ= 64π2Iλ. (63)8 With these deﬁnitions in hands, the adiabatic energy re- lations (40)-(44) can alternatively be written as ∂E ∂a−1 0=−/planckover2pi12C(0) a 32π2M+λ/planckover2pi12Pλ 64π2M, (64) ∂E ∂a−1 1=−/planckover2pi12C(1) a 32π2M, (65) ∂E ∂b1=/planckover2pi12C(1) b 128π2M, (66) ∂E ∂u=λ/bracketleftBigg /planckover2pi12Pλ 128π2M−λ/planckover2pi12C(1) a 16π2M/bracketrightBigg , (67) ∂E ∂v=3λ/planckover2pi12Pλ 128π2M. (68) In the absence of SO coupling, Eqs. (64), (65) and (66) simply reduce to the ordinary form of the adiabatic en- ergy relations for s- andp-wave interactions [10, 17], with respect to the scattering length (or volume) as well as ef- fective range. We should note that for the s-wave inter- action, there is a diﬀerence of the factor 8πfrom the well- known form of adiabatic energy relations. This is because we include the spherical harmonics Y00(ˆ r) = 1/√ 4πin thes-partial wave function. Besides, an additional fac- tor1/2is introduced in order to keep the deﬁnition of contacts consistent with those in the tail of the momen- tum distribution at large q. In the presence of SO cou- pling, two additional adiabatic energy relations appear, i.e., Eqs. (67) and (68), and a new contact Pλis intro- duced. C. The high-frequency tail of the rf spectroscopy Next, we discuss the asymptotic behavior of the rf spectroscopy at high frequency. The basic ideal of the rf transition is as follows. For an atomic Fermi gas with two hyperﬁne states, denoted as \|↑/an}b∇acket∇i}htand\|↓/an}b∇acket∇i}ht, the rf ﬁeld drives transitions between one of the hyperﬁne states ( i.e.\|↓/an}b∇acket∇i}ht) and an empty hyperﬁne state \|3/an}b∇acket∇i}htwith a bare atomic hyperﬁne energy diﬀerence /planckover2pi1ω3↓due to the mag- netic ﬁeld splitting [56, 57]. The universal scaling behav- ior at high frequency of the rf response of the system is governed by contacts. In this subsection, we are going to show how the contacts deﬁned by the adiabatic energy relations characterize such high-frequency scalings of th e rf transition in 3D Fermi gases with 3D SO coupling. Here, we will present a two-body derivation ﬁrst, which may avoid complicated notations as much as possible, and the results can easily be generalized to many-body systems later. The rf ﬁeld driving the spin-down particle to the state \|3/an}b∇acket∇i}htis described by Hrf=γrf/summationdisplay k/parenleftBig e−iωtc† 3kc↓k+eiωtc† ↓kc3k/parenrightBig , (69)whereγrfis the strength of the rf drive, ωis the rf fre- quency, and c† σkandcσkare respectively the creation and annihilation operators for fermions with the momentum kin the spin states \|σ/an}b∇acket∇i}ht. For any two-body state \|Ψ2b/an}b∇acket∇i}ht, we may write it in the momentum space as \|Ψ2b/an}b∇acket∇i}ht=/summationdisplay σ1σ2/summationdisplay k1k2˜φσ1σ2(k1,k2)c† σ1k1c† σ2k2\|0/an}b∇acket∇i}ht,(70) where˜φσ1σ2(k1,k2)is the Fourier transform of φσ1σ2(r1,r2)≡ /an}b∇acketle{tr1,r2;σ1,σ2\|Ψ2b/an}b∇acket∇i}ht, i.e., ˜φσ1σ2(k1,k2) =ˆ dr1dr2φσ1σ2(r1,r2)e−ik1·r1e−ik2·r2, (71) andσi=↑,↓denotes the spin of the ith particle. The spe- ciﬁc form of ˜φσ1σ2(k1,k2)can easily be obtained by using that of the two-body wave function /an}b∇acketle{tr1,r2;σ1,σ2\|Ψ2b/an}b∇acket∇i}htin the coordinate space, i.e., Eq.(5). Acting Eq.(69) onto (70), we obtain the two-body wave function after the rf transition, Hrf\|Ψ2b/an}b∇acket∇i}ht=γrfe−iωt× /summationdisplay k1k2/bracketleftBig ˜φ↓↑(k1,k2)c† 3k1c† ↑k2−˜φ↑↓(k1,k2)c† 3k2c† ↑k1 +˜φ↓↓(k1,k2)/parenleftBig c† 3k1c† ↓k2−c† 3k2c† ↓k1/parenrightBig/bracketrightBig \|0/an}b∇acket∇i}ht.(72) The physical meaning of Eq.(72) is apparent: after the rf transition, the atom with initial spin state \|↓/an}b∇acket∇i}htis driven to the empty spin state \|3/an}b∇acket∇i}ht, while the other one remains in the spin state \|↑/an}b∇acket∇i}ht. Therefore, there are totally four possi- ble ﬁnal two-body states with, respectively, possibilitie s of/vextendsingle/vextendsingle/vextendsingle˜φ↓↑/vextendsingle/vextendsingle/vextendsingle2 ,/vextendsingle/vextendsingle/vextendsingle˜φ↑↓/vextendsingle/vextendsingle/vextendsingle2 ,/vextendsingle/vextendsingle/vextendsingle˜φ↓↓/vextendsingle/vextendsingle/vextendsingle2 , and/vextendsingle/vextendsingle/vextendsingle˜φ↓↓/vextendsingle/vextendsingle/vextendsingle2 . Taking all these ﬁnal states into account, and according to the Fermi’s golden rule [29], the two-body rf transition rate is therefore give n by the Franck-Condon factor, Γ2(ω) =2πγ2 rf /planckover2pi1× /summationdisplay k1k2/parenleftbigg/vextendsingle/vextendsingle/vextendsingle˜φ↓↑/vextendsingle/vextendsingle/vextendsingle2 +/vextendsingle/vextendsingle/vextendsingle˜φ↑↓/vextendsingle/vextendsingle/vextendsingle2 +2/vextendsingle/vextendsingle/vextendsingle˜φ↓↓/vextendsingle/vextendsingle/vextendsingle2/parenrightbigg δ(/planckover2pi1ω−△E),(73) where△Eis the energy diﬀerence between the ﬁnal and initial states, and takes the form of ∆E=/planckover2pi12k2 M−/planckover2pi12q2 M+/planckover2pi1ω3↓, (74) wherek= (k1−k2)/2,/planckover2pi12q2/Mis the relative energy of two fermions in the initial state, and ω3↓≡ω3−ω↓is the bare hyperﬁne splitting between the spin states \|3/an}b∇acket∇i}htand \|↓/an}b∇acket∇i}ht, and can be set to 0without loss of generality. Now, we are interested in the asymptotic form of Γ2(ω)at large9 ωbut still small compared to /planckover2pi1/Mǫ2, which is determined by the short-range behavior when two fermions get as close asǫ. Combining Eqs.(71) and (73), as well as the asymptotic form of the two-body wave function (17) at r=r1−r2∼0, we ﬁnally obtain the asymptotic behavior of the rf response of 3D SO-coupled Fermi gases at large ω, Γ2(ω) =Mγ2 rf 16π2/planckover2pi13/bracketleftBigg c(1) a (Mω//planckover2pi1)1/2+ c(0) a+3c(1) b/4+λpλ (Mω//planckover2pi1)3/2/bracketrightBigg ,(75) wherec(0) a,c(1) a,c(1) bandpλare contacts for a two-body system with N= 1in the deﬁnitions (61)-(63). For many-body systems, all possible N= N(N−1)/2pairs may contribute to the high-frequency tail of the rf spectroscopy, while high-order contribution s from more than two fermions are ignored. Then we can generalize the above two-body picture to many-body systems by simply redeﬁning the constant Ninto the contacts, and then obtain ΓN(ω) =Mγ2 rf 16π2/planckover2pi13/bracketleftBigg C(1) a (Mω//planckover2pi1)1/2+ C(0) a+3C(1) b/4+λPλ (Mω//planckover2pi1)3/2/bracketrightBigg ,(76) whereC(0) a,C(1) a,C(1) bandPλare corresponding contacts for many-body systems. In the absence of SO coupling, Eq. (76) simply reduces to the ordinary asymptotic be- haviors of the rf response for s- andp-wave interactions, respectively [13, 58]. D. Pair correlation function at short distances The pair correlation function g2(s1,s2)gives the prob- ability of ﬁnding two fermions with one at position s1 and the other one at position s2simultaneously, i.e., g2(s1,s2)≡ /an}b∇acketle{tˆρ(s1) ˆρ(s2)/an}b∇acket∇i}ht, where ˆρ(s) =/summationtext iδ(s−ri) is the density operator at the position s. For a pure many-body state \|Ψ/an}b∇acket∇i}htofNfermions, we have [29] g2(s1,s2) =ˆ dr1dr2···drN/an}b∇acketle{tΨ\|ˆρ(s1) ˆρ(s2)\|Ψ/an}b∇acket∇i}ht =N(N−1)ˆ dX′\|Ψ(X,r)\|2, (77) wherer=s1−s2is relative coordinates of the pair fermions at positions s1ands2, andX′denotes the de- grees of freedom of all the other fermions. If we further integrate over the c.m. coordinate of the pair, we candeﬁne the spatially integrated pair correlation function as G2(r)≡N(N−1)ˆ dX\|Ψ(X,r)\|2, (78) andXincludes the c.m. coordinate R= (s1+s2)/2of the pair besides X′. Inserting the short-range form of many-body wave functions for SO coupled Fermi gases, i.e. Eq.(17) into Eq. (78), we ﬁnd G2(r)≈N(N−1)ˆ dX/braceleftbiggα1α∗ 1Ω1Ω∗ 1 r4 +α∗ 0α1Ω∗ 0Ω1+α0α∗ 1Ω0Ω∗ 1 r3+[α0α∗ 0Ω0Ω∗ 0 +k2α1α∗ 1Ω1Ω∗ 1+λ(α∗ 0α1+α0α∗ 1)Ω∗ 1Ω1 +λuα∗ 1α1(Ω∗ 0Ω1+Ω0Ω∗ 1) −α∗ 0α1Ω∗ 0Ω1+α0α∗ 1Ω0Ω∗ 1 a0/bracketrightbigg1 r2+O/parenleftbig r−1/parenrightbig/bracerightbigg . (79) Further, if we are only care about the dependence of G2(r)on the amplitude of r=\|r\|, we can integrate over the direction of r, and use the deﬁnitions of contacts (61)-(63), then it yields G2(r)≈1 16π2/bracketleftBigg C(1) a r4+/parenleftBigg C(0) a+C(1) b 2+λPλ 2/parenrightBigg 1 r2 +/parenleftBigg −2C(0) a a0−2C(1) a 3a1+b1C(1) b 6+λ(u+v)Pλ 2/parenrightBigg 1 r/bracketrightBigg . (80) which reduces to the results in the absence of the SO coupling for s- andp-wave interactions, respectively[9, 13, 31, 59, 60]. E. Grand canonical potential and pressure relation The adiabatic energy relations as well as the large- momentum distribution we obtained is valid for any pure energy eigenstate. Therefore, they should still hold for any incoherent mixed state statistically at ﬁnite tempera- ture. Then the energy Eand contacts then become their statistical average values. Now, let us look at the grand thermodynamic potential Jfor a homogeneous system, which is deﬁned as [61] J ≡ −PV=E−TS−µN, (81) whereP,V,T,S,µ,Nare, respectively, the pressure, volume, temperature, entropy, chemical potential, and total particle number. The grand canonical potential J10 is the function of V,T,S, and takes the following diﬀer- ential form dJ=−PdV−SdT−Ndµ. (82) For the two-body microscopic parameters, we may evalu- ate their dimensions as a0∼Length1,a1∼Length3,b1∼ Length−1,u∼Length−1, andv∼Length−1. There- fore, there are basically following energy scales in the grand thermodynamic potential, i.e., kBT,µ,/planckover2pi12/MV2/3, /planckover2pi12/Ma2 0,/planckover2pi12/Ma2/3 1,/planckover2pi12b2 1/M,/planckover2pi12u2/M,/planckover2pi12v2/M. Then we may express the thermodynamic potential Jin the terms of a dimensionless function ¯Jas [21, 62] J(V,T,µ,a 0,a1,b1,u,v) =kBT¯J/parenleftbigg/planckover2pi12/MV2/3 kBT,µ kBT,/planckover2pi12/Ma2 0 kBT, /planckover2pi12/Ma2/3 1 kBT,/planckover2pi12b2 1/M kBT,/planckover2pi12u2/M kBT,/planckover2pi12v2/M kBT/parenrightBigg .(83) Consequently, one can deduce the simple scaling law J/parenleftBig γ−3/2V,γT,γµ,γ−1/2a0,γ−3/2a1,γ1/2b1,γ1/2u,γ1/2v/parenrightBig =γJ(V,T,µ,a 0,a1,b1,u,v).(84) The derivative of Eq.(84) with respect to γatγ= 1 simply yields /parenleftbigg −3V 2∂ ∂V+T∂ ∂T+µ∂ ∂µ−a0 2∂ ∂a0 −3a1 2∂ ∂a1+b1 2∂ ∂b1+u 2∂ ∂u+v 2∂ ∂v/parenrightbigg J=J,(85) where all the partial derivatives are to be understood as leaving all other system variables constant. Since J −T∂J ∂T−µ∂J ∂µ=J+TS+µN=E, (86) and the variation of the grand thermodynamic potential δJwith respect to the two-body parameters at ﬁxed vol- umeV, temperature Tand chemical potential µis equal to that of the energy δEat ﬁxed volume V, entropyS and particle number N, i.e.,(δJ)V,T,µ= (δE)V,S,Nand V∂J ∂V=J, we easily obtain from Eqs.(85) and (86) −3 2J −a0 2∂E ∂a0−3a1 2∂E ∂a1+b1 2∂E ∂b1+u 2∂E ∂u+v 2∂E ∂v=E, (87) Further by using adiabatic energy relations, Eq.(87) be- comesJ=−2 3E−/planckover2pi12 96π2Ma0/parenleftbigg C(0) a−λPλ 2/parenrightbigg −/planckover2pi12C(1) a 32π2Ma1+/planckover2pi12b1C(1) b 384π2M −λu/planckover2pi12 48π2M/parenleftbigg λC(1) a−Pλ 8/parenrightbigg +λvPλ/planckover2pi12 128π2M,(88) or the pressure relation by dividing both sides of Eq. (88) by−V, which respectively reduces to the well-known results in the absence of the spin-orbit coupling P=2E 3V+/planckover2pi12C(0) a 96π2MVa0(89) fors-wave interactions, which is consistent with the result of Ref.[11, 60, 63], and P=2E 3V+/planckover2pi12C(1) a 32π2MVa1−b1/planckover2pi12C(1) b 384π2MV(90) forp-wave interactions, which is consistent with the re- sult of Ref.[13]. IV. UNIVERSAL RELATIONS IN 2D SYSTEMS WITH RASHBA SO COUPLING The derivation of the universal relations for 3D Fermi gases with 3D SO coupling can directly be generalized to those for 2D systems with 2D SO coupling. In this section, with the short-range form of the two-body wave function for 2D systems with 2D SO coupling in hands, i.e., Eq.(25), we are going to discuss Tan’s universal rela- tions for 2D Fermi gases with 2D SO coupling , by taking into account only two-body correlations. A. Adiabatic energy relations Let us consider how the energy of the SO-coupled sys- tem varies with the two-body interaction in 2D systems with 2D SO coupling. The two wave functions of a many- body system Ψ(r)andΨ′(r), corresponding to diﬀerent interatomic interaction strengths, satisfy the Schröding er equation with diﬀerent energies, i.e. formally as Eqs. (26) and (27). Analogously, by subtracting [27]∗×Ψ fromΨ′∗×[26], and integrating over the domain Dǫ, the set of all conﬁgurations (ri,rj)in whichr=\|ri−rj\|>ǫ, we obtain11 (E−E′)ˆ DǫN/productdisplay i=1driΨ′∗Ψ = −/planckover2pi12 MNˆ r>ǫdXdr/bracketleftbig Ψ′∗∇2 rΨ−/parenleftbig ∇2 rΨ′∗/parenrightbig Ψ/bracketrightbig +/planckover2pi12λ MNˆ r>ǫdXdr/bracketleftBig Ψ′∗/parenleftBig ˆQΨ/parenrightBig −/parenleftBig ˆQΨ′/parenrightBig∗ Ψ/bracketrightBig ,(91) whereN=N(N−1)/2is again the number of all the possible ways to pair atom. Using the Gauss’ theorem, the ﬁrst term on the right-hand side (RHS) can be writ- ten as −/planckover2pi12 MNˆ r>ǫdXdr/bracketleftbig Ψ′∗∇2 rΨ−/parenleftbig ∇2 rΨ′∗/parenrightbig Ψ/bracketrightbig =−/planckover2pi12 MN˛ r=ǫ[Ψ′∗∇rΨ−(∇rΨ′∗)Ψ]·ˆndS, =/planckover2pi12ǫ MNˆ dX/summationdisplay m=0,±1/parenleftbigg ψ′∗ m∂ ∂rψm−ψm∂ ∂rψ′∗ m/parenrightbigg r=ǫ, (92) whereSis the boundary of Dǫthat the distance between the two fermions in the pair (i,j)isǫ,ˆnis the direction normal to S, but is opposite to the radial direction, and ψ0(ψ±1) is thes-wave (p-wave) component of the two- body wave function as deﬁned in Eq.(21). Since ˆQ(r)Ψ =/summationdisplay m=±1/bracketleftBigg −√ 2 r∂ ∂r(rψm)Ω0(ˆ r) +√ 2∂ψ0 ∂rΩm(ˆ r)/bracketrightbigg ,(93) we ﬁnd that the second term on the right-hand side (RHS) of Eq. (91) can be written as /planckover2pi12λ MNˆ r>ǫdXdr/bracketleftBig Ψ′∗/parenleftBig ˆQ(r)Ψ/parenrightBig −/parenleftBig ˆQ(r)Ψ′/parenrightBig∗ Ψ/bracketrightBig =√ 2λ/planckover2pi12ǫ MNˆ dX/summationdisplay m=±1(ψ′∗ 0ψm−ψ′∗ mψ0)r=ǫ.(94) Combining Eqs.(91), (92) and (94), we have (E−E′)ˆ DǫN/productdisplay i=1driΨ′∗Ψ =/planckover2pi12ǫ MNˆ dX/summationdisplay m=0,±1/parenleftbigg ψ′∗ m∂ ∂rψm−ψm∂ ∂rψ′∗ m/parenrightbigg r=ǫ +√ 2λ/planckover2pi12ǫ MNˆ dX/summationdisplay m=±1(ψ′∗ 0ψm−ψ′∗ mψ0)r=ǫ.(95)Inserting the asymptotic form of the many-body wave function Eq.(25) into Eq.(95), and letting E′→Eand Ψ′→Ψ, we arrive at δE·ˆ DǫN/productdisplay i=1dri\|Ψ\|2 =/planckover2pi12 M/parenleftBigg I(0) a+/summationdisplay m=±1√ 2 2λI(m) λ/parenrightBigg δlna0 +/summationdisplay m=±1/braceleftBigg −π/planckover2pi12I(m) a 2Mδa−1 1+/parenleftBigg Em−λ/planckover2pi12I(m) λ√ 2M/parenrightBigg δlnb1 −/planckover2pi12 M/bracketleftBigg/parenleftbigg√ 2λ2I(m) a+λ 2I(m) λ/parenrightbigg +√ 2λ2 2Ip/bracketrightBigg δu +λ/planckover2pi12 MI(m) λδv−/parenleftbigg lnǫ 2b1+γ/parenrightbigg I(m) aδE/bracerightbigg ,(96) where I(m) a=Nˆ dX\|αm\|2, (97) Em=Nˆ dXα∗ m/parenleftBig E−ˆT/parenrightBig αm (98) form= 0,±1, I(±1) λ=Nˆ dXα∗ 0α±1+c.c, (99) E(±1) λ=Nˆ dXα∗ 0/parenleftBig E−ˆT/parenrightBig α±1+c.c, (100) Ip=Nˆ dXα∗ −1α1+c.c., (101) andˆT(X)is the kinetic operator including the c.m. mo- tion of the pair as well as those of all the rest fermions. Using the normalization of the wave function (see ap- pendix B) ˆ DǫN/productdisplay i=1dri\|Ψ\|2= 1−/summationdisplay m=±1/parenleftbigg lnǫ 2b1+γ/parenrightbigg I(m) a,(102) we can further simplify Eq. (96) as δE=/planckover2pi12 M/parenleftBigg I(0) a+/summationdisplay m=±1λI(m) λ√ 2/parenrightBigg δlna0 +/summationdisplay m=±1/braceleftBigg −π/planckover2pi12I(m) a 2Mδa−1 1+/parenleftBigg Em−λ/planckover2pi12I(m) λ√ 2M/parenrightBigg δlnb1 −λ/planckover2pi12 M/bracketleftBigg √ 2λI(m) a+I(m) λ 2+λIp√ 2/bracketrightBigg δu+λ/planckover2pi12 MI(m) λδv/bracerightbigg , (103)12 which characterizes how the energy of a 2D system with 2D SO coupling varies as the scattering parameters adia- batically change, and yields the following set of adiabatic energy relations ∂E ∂lna0=/planckover2pi12 M/parenleftBigg I(0) a+λ√ 2/summationdisplay m=±1I(m) λ/parenrightBigg , (104) ∂E ∂a−1 1=−π/planckover2pi12 2M/summationdisplay m=±1I(m) a, (105) ∂E ∂lnb1=/summationdisplay m=±1/parenleftBigg Em−λ/planckover2pi12I(m) λ√ 2M/parenrightBigg , (106) ∂E ∂u=−/planckover2pi12λ√ 2M/summationdisplay m=±1/bracketleftBigg I(m) λ√ 2+λ/parenleftBig 2I(m) a+Ip/parenrightBig/bracketrightBigg ,(107) ∂E ∂v=/planckover2pi12λ M/summationdisplay m=±1I(m) λ. (108) Obviously, there are additional two new adiabatic energy relations appear, i.e. Eqs. (107) and (108), which orig- inate from new scattering parameters introduced by SO coupling. B. Tail of the momentum distribution at large q In general, the momentum distribution at large qis de- termined by the short-range behavior of the many-body wave function when the fermions iandjare close. Simi- larly as in the 3D case, we can formally write the many- body wave function Ψ2Datr≈0as the following ansatz Ψ2D(X,r) =/bracketleftbig α0lnr+B0+C0r2lnr/bracketrightbig Ω0(ˆr) +/summationdisplay m/bracketleftBigαm r+Bmrlnr+Cmr/bracketrightBig Ωm(ˆr)+O/parenleftbig r2/parenrightbig ,(109) whereαj,BjandCj(j= 0,±1) are all regular functions ofX. Comparing Eqs. (25) and (109) at small r, we ﬁnd that B0(X) =α0(γ−ln2a0)+/summationdisplay m=±1αmλu, (110) Bm(X) =−αmk2 2+λα0√ 2, (111) C0(X) =−α0k2 4, (112) Cm(X) =αm/parenleftbigg −π 4a1+1−2γ 4k2/parenrightbigg +α0λv+/parenleftbiggαmk2 2−λα0√ 2/parenrightbigg ln2b1.(113) In the follows, we derive the momentum distribution at largeqbut still smaller than ǫ−1. With the help of the plane-wave expansioneiq·r=√ 2π∞/summationdisplay m=0/summationdisplay σ=±ηmimJm(qr)e−iσmϕ qΩ(σ) m(ϕ), (114) whereηm= 1/2form= 0, andηm= 1form≥1, and ϕqis the azimuthal angle of q, we have ˆ drα0lnrΩ0(ˆr)e−iq·r=−2π q2α0Ω0(ˆq), (115) ˆ drB0Ω0(ˆr)e−iq·r= 0, (116) ˆ drC0r2lnrΩ0(ˆr)e−iq·r=8π q4C0Ω0(ˆq), (117) ˆ drαm rΩm(ˆr)e−iq·r=−i2π qαmΩm(ˆq),(118) ˆ drBmrlnrΩm(ˆr)e−iq·r=i4π q3BmΩm(ˆq),(119) ˆ drCmrΩm(ˆr)e−iq·r= 0, (120) whereˆqis the angular part of q. Inserting Eqs. (115)- (120) into (45), we ﬁnd that the total momentum distri- butionn2D(q)at largeqtakes the form of n2D(q)≈ Nˆ dX/summationdisplay m,m′αmα∗ m′Ωm(ˆq)Ω∗ m′(ˆq)8π2 q2 +i/summationdisplay m[α∗ 0αmΩ∗ 0(ˆq)Ωm(ˆq)−α0α∗ mΩ0(ˆq)Ω∗ m(ˆq)]8π2 q3 + α0α∗ 0Ω0(ˆq)Ω∗ 0(ˆq)+/summationdisplay m,m′/bracketleftBig −√ 2λ(α0α∗ mΩm′(ˆq)Ω∗ m(ˆq) +α∗ 0αmΩm(ˆq)Ω∗ m′(ˆq))+2k2αmα∗ m′Ωm(ˆq)Ω∗ m′(ˆq)/bracketrightbig/bracerightbig ×8π2 q4+O/parenleftbig q−5/parenrightbig (121) and the summations are over m,m′=±1. If we are only interested in the dependence of n2D(q)on the amplitude ofq, the expression can further be simpliﬁed by integrat- ingn2D(q)over the direction of q, and all the odd-order terms ofq−1vanish. Finally, we arrive at n2D(q) =/summationtext m=±1C(m) a q2 +/bracketleftBigg C(0) a+/summationdisplay m=±1/parenleftBig C(m) b−λP(m) λ/parenrightBig/bracketrightBigg 1 q4+O/parenleftbig q−6/parenrightbig , (122) where the contacts are deﬁned as C(j) a= 8π2I(j) a (123)13 forj= 0,±1, and C(m) b=16π2M /planckover2pi12Em, (124) P(m) λ= 8√ 2π2I(m) λ(125) form=±1. With these deﬁnitions in hands, the adi- abatic energy relations (104)-(108) can alternatively be written as ∂E ∂lna0=/planckover2pi12 8π2M/parenleftBigg C(0) a+λ 2/summationdisplay m=±1P(m) λ/parenrightBigg ,(126) ∂E ∂a−1 1=−/planckover2pi12 16πM/summationdisplay m=±1C(m) a, (127) ∂E ∂lnb1=/planckover2pi12 16π2M/summationdisplay m=±1/parenleftBig C(m) b−λP(m) λ/parenrightBig ,(128) ∂E ∂u=−/planckover2pi12λ 16√ 2π2M/summationdisplay m=±1P(m) λ, (129) ∂E ∂v=/planckover2pi12λ 8√ 2π2M/summationdisplay m=±1P(m) λ. (130) In the absence of SO coupling, Eqs. (126), (127) and (128) simply reduce to the ordinary form of the adiabatic energy relations for s- andp-wave interactions [24, 29], with respect to the scattering length (or area) as well as eﬀective range. And for the s-wave interaction, there is a diﬀerence of the factor 2πfrom the Ref.[24], which is because we include the angular part 1/√ 2πin thes- partial wave function. In addition, two additional new adiabatic energy relations, i.e., Eqs. (129) and (130), and new contacts P(m) λappear, due to SO coupling. C. The high-frequency tail of the rf spectroscopy We may carry out the analogous procedure as that in 3D systems with 3D SO coupling, and the two-body rf transition rate takes the form Γ2(ω) =2πγ2 rf /planckover2pi1× /summationdisplay k1k2/parenleftbigg/vextendsingle/vextendsingle/vextendsingle˜φ↑↓/vextendsingle/vextendsingle/vextendsingle2 +/vextendsingle/vextendsingle/vextendsingle˜φ↓↑/vextendsingle/vextendsingle/vextendsingle2 +2/vextendsingle/vextendsingle/vextendsingle˜φ↓↓/vextendsingle/vextendsingle/vextendsingle2/parenrightbigg δ(/planckover2pi1ω−∆E),(131) where ˜φσ1σ2(k1,k2) =ˆ dr1dr2φσ1σ2(r1,r2)e−ik1·r1e−ik2·r2. (132) If we are only interested in the high-frequency tail of the transition rate, we can use the asymptotic behavior ofthe two-body wave function for a 2D system with 2D SO coupling, i.e. Eq. (25). Combining with Eqs.(131) and (132), we obtain the two-body rf transition rate Γ2(ω) as Γ2(ω) =Mγ2 rf 4π/planckover2pi13/bracketleftBigg c(1) a Mω//planckover2pi1 +c(0) a/2+c(1) b/2−λp(1) λ (Mω//planckover2pi1)2/bracketrightBigg ,(133) wherec(0) a,c(1) a,c(1) bandp(1) λare contacts for a two-body system with N= 1in the deﬁnitions (123)-(125). For many-body systems, all N=N(N−1)/2pairs contribute to the transition rate. Similarly, we can re- deﬁning the constant Ninto the contacts, and then ob- tain ΓN(ω) =Mγ2 rf 4π/planckover2pi13/bracketleftBigg C(1) a Mω//planckover2pi1 +C(0) a/2+C(1) b/2−λP(1) λ (Mω//planckover2pi1)2/bracketrightBigg ,(134) whereC(0) a,C(1) a,C(1) bandP(1) λare corresponding con- tacts for many-body systems. In the absence of SO cou- pling, Eq. (134) simply reduces to the ordinary results fors- andp-wave interactions, respectively [29, 64]. D. Pair correlation function at short distances Let us then discuss the short-distance behavior of the pair correlation function for a 2D Fermi gases with 2D SO coupling. Inserting the asymptotic form of the many- body wave function at short distance, i.e. Eq. (25) into the Eq. (78), we easily obtain spatially integrated pair correlation function G2(r). If we are only interested in the dependence of G2(r)on the amplitude of r=\|r\|, we may integrate over the direction of r, and obtain G2(r)≈1 4π2 /summationtext m=±1C(m) a r2+C(0) a/parenleftbigg lnr 2a0/parenrightbigg2 +/parenleftBigg 2γC(0) a+λu√ 2/summationdisplay m=±1P(m) λ/parenrightBigg lnr 2a0 +/summationdisplay m=±11 2/parenleftBig −C(m) b+λP(m) λ/parenrightBig lnr 2b1/bracketrightBigg .(135) In the absence of SO coupling, Eq. (135) simply reduces to the ordinary results for s- andp-wave interactions, respectively [24, 29].14 E. Grand canonical potential and pressure relation Similarly, according to the dimension analysis, we eas- ily obtain −J −a0 2∂E ∂a0−a1∂E ∂a1−b1 2∂E ∂b1=E. (136) Further by using adiabatic energy relations, Eq.(136) be- comes J=−E−/planckover2pi12 16π2M/parenleftBigg C(0) a+λ 2/summationdisplay m=±1P(m) λ/parenrightBigg −/planckover2pi12 16πM/summationdisplay m=±1/bracketleftBigg C(m) a a1+1 2π/parenleftBig C(m) b−λP(m) λ/parenrightBig/bracketrightBigg .(137) The pressure relation can be obtained by dividing both sides of Eq.(137) by −V, which respectively reduces to the results in the absence of SO coupling P=E V+/planckover2pi12C(0) a 16π2MV(138) fors-wave interactions, which is consistent with the result of Ref.[22], and P=E V+/summationdisplay m=±1/planckover2pi12 16πMV/parenleftBigg C(m) a a1+C(m) b 2π/parenrightBigg (139) forp-wave interactions, which is consistent with the re- sult of Ref.[29]. V. CONCLUSIONS In conclusion, we systematically study a set of univer- sal relations for spin-orbit-coupled Fermi gases in three o r two dimension, respectively. The universal short-range forms of two-body wave functions are analytically de- rived, by using a perturbation method, in the sub-Hilbert space of zero center-of-mass momentum and zero total angular momentum of pairs. The obtained short-range behaviors of two-body wave functions do not depend on the short-range details of interatomic potentials. We ﬁnd that two new microscopic scattering parameters appear because of spin-orbit coupling, and then new contacts need to be introduced in both three- and two-dimensional systems. However, due to diﬀerent short-range behav- iors of two-body wave functions for three- and two- dimensional systems, the speciﬁc forms of universal re- lations are distinct in diﬀerent dimensions. As we antic- ipate, the universal relations for spin-orbit-coupled sys - tems, such as the adiabatic energy relations, the large- momentum distributions, the high-frequency behavior ofthe radio-frequency responses, short-range behaviors of the pair correlation functions, grand canonical poten- tials, and pressure relations, are fully captured by the contacts deﬁned. In general, more partial-wave scatter- ings should be taken into account for nonzero center-of- mass momentum and nonzero total angular momentum of pairs. Consequently, we may expect more contacts to appear. Our results may shed some light for understand- ing the profound properties of the few- and many-body spin-orbit-coupled quantum gases. ACKNOWLEDGMENTS This work has been supported by the NKRDP (National Key Research and Development Program) under Grant No. 2016YFA0301503, NSFC (Grant No.11674358, 11434015, 11474315) and CAS under Grant No. YJKYYQ20170025. APPENDIX A: NORMALIZATION OF THE WAVE FUNCTION FOR 3D SYSTEMS WITH 3D SO COUPLING In this section of Appendix A, we are going to derive ´ DǫN/producttext i=1dri\|Ψ\|2for 3D many-body systems with 3D SO coupling. Let us consider two many-body wave functions Ψ′andΨ, corresponding to diﬀerent energies /planckover2pi12k′2/M and/planckover2pi12k2/M, respectively. They should be orthogonal, i.e.,´ DǫN/producttext i=1driΨ′∗Ψ = 0 , and therefore we have ˆ r<ǫN/productdisplay i=1driΨ′∗Ψ =−ˆ r>ǫN/productdisplay i=1driΨ′∗Ψ. (140) From the Schrödinger equation satisﬁed by Ψ′andΨ outside the interaction potential, i.e., r > ǫ , we easily obtain ˆ r>ǫN/productdisplay i=1driΨ′∗Ψ =ǫ2 k2−k′2Nˆ dXˆ r=ǫdˆ r /bracketleftbigg/parenleftbigg Ψ′∗∂ ∂rΨ−Ψ∂ ∂rΨ′∗/parenrightbigg +λ 2π(ψ′∗ 0ψ1−ψ′∗ 1ψ0)/bracketrightbigg .(141) In the presence of SO coupling, only s- andp-wave scat- terings are involved in the subspace K= 0andJ= 0, and the wave function at short distance takes the form of Eq. (17). Using the asymptotic behavior of the wave function, we easily evaluate15 ˆ r<ǫN/productdisplay i=1dri\|Ψ\|2 =−lim k′→k1 2/parenleftBiggˆ r>ǫN/productdisplay i=1driΨ′∗Ψ+ˆ r>ǫN/productdisplay i=1driΨ′Ψ∗/parenrightBigg =−Nˆ dX/braceleftBigg \|α1\|2 ǫ+\|α1\|2b1 2/bracerightBigg =−/parenleftbigg1 ǫ+b1 2/parenrightbigg I(1) a, (142) which in turn yields ˆ DǫN/productdisplay i=1dri\|Ψ\|2= 1 +/parenleftbigg1 ǫ+b1 2/parenrightbigg I(1) a.(143) APPENDIX B: NORMALIZATION OF THE WAVE FUNCTION FOR 2D SYSTEM WITH 2D SO COUPLING In this section of Appendix B, we are going to derive ´ DǫN/producttext i=1dri\|Ψ\|2for 2D many-body systems with 2D SO coupling. Let us consider two many-body wave functions Ψ′andΨ, corresponding to diﬀerent energies /planckover2pi12k′2/M and/planckover2pi12k2/M, respectively. They should be orthogonal, i.e.,´ DǫN/producttext i=1driΨ′∗Ψ = 0 , and therefore we have ˆ r<ǫN/productdisplay i=1driΨ′∗Ψ =−ˆ r>ǫN/productdisplay i=1driΨ′∗Ψ. (144) From the Schrödinger equation satisﬁed by Ψ′andΨ outside the interaction potential, i.e., r > ǫ , we easilyobtain ˆ r>ǫN/productdisplay i=1driΨ′∗Ψ =ǫ k2−k′2Nˆ dXˆ r=ǫdˆ r /bracketleftBigg/parenleftbigg Ψ′∗∂ ∂rΨ−Ψ∂ ∂rΨ′∗/parenrightbigg +/summationdisplay m=±1λ√ 2π(ψ′∗ 0ψm−ψ′∗ mψ0)/bracketrightBigg . (145) In the presence of SO coupling, only s- andp-wave scat- terings are involved in the subspace K= 0andJ= 0, and the wave function at short distance takes the form of Eq. (25). Using the asymptotic behavior of the wave function, we easily evaluate ˆ r<ǫN/productdisplay i=1dri\|Ψ\|2 =−lim k′→k1 2/parenleftBiggˆ r>ǫN/productdisplay i=1driΨ′∗Ψ+ˆ r>ǫN/productdisplay i=1driΨ′Ψ∗/parenrightBigg =Nˆ dX/summationdisplay m=±1/parenleftbigg lnǫ 2b1+γ/parenrightbigg \|αm\|2 =/summationdisplay m=±1/parenleftbigg lnǫ 2b1+γ/parenrightbigg I(m) a, (146) which in turn yields ˆ DǫN/productdisplay i=1dri\|Ψ\|2= 1−/summationdisplay m=±1/parenleftbigg lnǫ 2b1+γ/parenrightbigg I(m) a.(147) [1] I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Rev. Mod. Phys. 80, 885 (2008). [2] S. Giorgini, L. P. Pitaevskii, and S. Stringari, Theory o f ultracold atomic Fermi gases, Rev. Mod. Phys. 80, 1215 (2008). [3] T. K ¨ ohler, K. G ´ oral, and P. S. Julienne, Production of cold molecules via magnetically tunable Feshbach reso- nances, Rev. Mod. Phys. 78, 1311 (2006). [4] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Fesh- bach resonances in ultracold gases, Rev. Mod. Phys. 82, 1225 (2010). [5] C. Gross and I. Bloch, Quantum simulations with ultra- cold atoms in optical lattices, Science 357, 995 (2017). [6] X. L. Qi and S. C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83, 1057 (2011).[7] J. Dalibard, F. Gerbier, G. Juzeli ¯ unas, and P. ¨Ohberg, Colloquium: Artiﬁcial gauge potentials for neutral atoms, Rev. Mod. Phys. 83, 1523 (2011). [8] D. A. Abanin, E. Altman, I. Bloch, M. Serbyn, Collo- quium: Many-body localization, thermalization, and en- tanglement, Rev. Mod. Phys. 91, 021001 (2019). [9] S. Tan, Energetics of a strongly correlated Fermi gas, Ann. Phys. 323, 2952 (2008). [10] S. Tan, Large momentum part of a strongly correlated Fermi gas, Ann. Phys. 323, 2971 (2008). [11] S. Tan, Generalized virial theorem and pressure relati on for a strongly correlated Fermi gas, Ann. Phys. 323, 2987 (2008). [12] W. Zwerger, The BCS-BEC crossover and the unitary Fermi gas , Lecture Notes in Physics Vol. 836 (Springer, Berlin, 2011), see Chap. 6 for a brief review.16 [13] Z. H. Yu, J. H. Thywissen, and S. Z. Zhang, Universal relations for a Fermi gas close to a p-wave interaction resonance, Phys. Rev. Lett. 115, 135304 (2015); Phys. Rev. Lett. 117, 019901(E) (2016). [14] S. M. Yoshida and M. Ueda, Universal high-momentum asymptote and thermodynamic relations in a spinless Fermi gas with a resonant p-wave interaction, Phys. Rev. Lett.115, 135303 (2015). [15] M. Y. He, S. L. Zhang, H. M. Chan, and Q. Zhou, Con- cept of a contact spectrum and its applications in atomic quantum Hall states, Phys. Rev. Lett. 116, 045301 (2016). [16] S. M. Yoshida and M. Ueda, p-wave contact tensor: Uni- versal properties of axisymmetry-broken p-wave Fermi gases, Phys. Rev. A 94, 033611 (2016). [17] S. G. Peng, X. J. Liu, and H. Hu, Large-momentum dis- tribution of a polarized Fermi gas and p-wave contacts, Phys. Rev. A 94, 063651 (2016). [18] F. Qin, X. L. Cui, and W. Yi, Universal relations and normal phase of an ultracold Fermi gas with coexisting s- and p-wave interactions, Phys. Rev. A 94, 063616 (2016). [19] P. F. Zhang, S. Z. Zhang, and Z. H. Yu, Eﬀective theory and universal relations for Fermi gases near a d-wave- interaction resonance, Phys. Rev. A 95, 043609 (2017). [20] S. L. Zhang, M. Y. He, and Q. Zhou, Contact matrix in dilute quantum systems, Phys. Rev. A 95, 062702 (2017). [21] M. Barth and W. Zwerger, Tan relations in one dimen- sion, Ann. Phys. 326, 2544 (2011). [22] M. Valiente, N. T. Zinner, and K. M /diameterlmer, Universal relations for the two-dimensional spin-1/2 Fermi gas with contact interactions, Phys. Rev. A 84, 063626 (2011). [23] F. Werner and Y. Castin, General relations for quan- tum gases in two and three dimensions: Two-component fermions, Phys. Rev. A 86, 013626 (2012). [24] F. Werner and Y. Castin, General relations for quantum gases in two and three dimensions. II. Bosons and mix- tures, Phys. Rev. A 86, 053633 (2012). [25] X. L. Cui, Universal one-dimensional atomic gases near odd-wave resonance, Phys. Rev. A 94, 043636 (2016). [26] X. L. Cui and H. F. Dong, High-momentum distribu- tion with a subleading k−3tail in odd-wave interacting one-dimensional Fermi gases, Phys. Rev. A 94, 063650 (2016). [27] Y.-C. Zhang and S. Zhang, Strongly interacting p-wave Fermi gas in two dimensions: Universal relations and breathing mode, Phys. Rev. A 95, 023603 (2017). [28] X. G. Yin, X. W. Guan, Y. B. Zhang, H. B. Su, and S. Z. Zhang, Momentum distribution and contacts of one- dimensional spinless Fermi gases with an attractive p- wave interaction, Phys. Rev. A 98, 023605 (2018). [29] S. G. Peng, Universal relations for a spin-polarized Fe rmi gas in two dimensions, J. Phys. A: Math. Theor. 52, 245302 (2019). [30] J. T. Stewart, J. P. Gaebler, T. E. Drake, and D. S. Jin, Veriﬁcation of universal relations in a strongly interacti ng Fermi gas, Phys. Rev. Lett. 104, 235301 (2010). [31] E. D. Kuhnle, H. Hu, X. J. Liu, P. Dyke, M. Mark, P. D. Drummond, P. Hannaford, and C. J. Vale, Universal be- havior of pair correlations in a strongly interacting Fermi gas, Phys. Rev. Lett. 105, 070402 (2010). [32] S. Hoinka, M. Lingham, K. Fenech, H. Hu, C. J. Vale, J. E. Drut, and S. Gandolﬁ, Precise determination of the structure factor and contact in a unitary Fermi gas, Phys. Rev. Lett. 110, 055305 (2013).[33] B. Mukherjee, P. B. Patel, Z. Yan, R. J. Fletcher, J. Struck, and M. W. Zwierlein, Spectral response and con- tact of the unitary Fermi gas, Phys. Rev. Lett. 122, 203402 (2019). [34] C. Carcy, S. Hoinka, M. G. Lingham, P. Dyke, C. C. N. Kuhn, H. Hu, and C. J. Vale, Contact and sum rules in a near-uniform Fermi gas at unitarity, Phys. Rev. Lett. 122, 203401 (2019). [35] C. Luciuk, S. Trotzky, S. Smale, Z. Yu, S. Zhang, and J. H. Thywissen, Evidence for universal relations describing a gas with p-wave interactions, Nat. Phys. 12, 599 (2016). [36] S.-G. Peng, S. Tan, and K. Jiang, Manipulation of p- wave scattering of cold atoms in low dimensions using the magnetic ﬁeld vector, Phys. Rev. Lett. 112, 250401 (2014). [37] Y. J. Lin, K. Jim ´ enez-Garcia, and I. B. Spielman, Spin- orbit-coupled Bose-Einstein condensates, Nature 471, 83 (2011). [38] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah, W. S. Bakr, and M. W. Zwierlein, Spin-injection spec- troscopy of a spin-orbit coupled Fermi gas, Phys. Rev. Lett.109, 095302 (2012). [39] P. J. Wang, Z. Q. Yu, Z. K. Fu, J. Miao, L. H. Huang, S. J. Chai, H. Zhai, and J. Zhang, Spin-orbit coupled degenerate Fermi gases, Phys. Rev. Lett. 109, 095301 (2012). [40] Z. Wu, L. Zhang, W. Sun, X. T. Xu, B. Z. Wang, S. C. Ji, Y. Deng, S. Chen, X. J. Liu, and J. W. Pan, Realization of two-dimensional spin-orbit coupling for Bose-Einstein condensates, Science 354, 83 (2016). [41] L. H. Huang, Z. M. Meng, P. J. Wang, P. Peng, S. L. Zhang, L. C. Chen, D. H. Li, Q. Zhou, and J. Zhang, Ex- perimental realization of two-dimensional synthetic spin - orbit coupling in ultracold Fermi gases, Nat. Phys. 12, 540 (2016). [42] H. R. Chen, K. Y. Lin, P. K. Chen, N. C. Chiu, J. B. Wang, C. A. Chen, P. P. Huang, S. K. Yip, Y. Kawaguchi, and Y. J. Lin, Spin-orbital-angular-momentum coupled Bose-Einstein condensates, Phys. Rev. Lett. 121, 113204 (2018). [43] P. K. Chen, L. R. Liu, M. J. Tsai, N. C. Chiu, Y. Kawaguchi, S. K. Yip, M. S. Chang, and Y. J. Lin, Rotating atomic quantum gases with light-induced az- imuthal gauge potentials and the observation of the Hess- Fairbank eﬀect, Phys. Rev. Lett. 121, 250401 (2018). [44] D. Zhang, T. Gao, P. Zou, L. Kong, R. Li, X. Shen, X. L. Chen, S. G. Peng, M. Zhan, H. Pu, and K. Jiang, Ground-state phase diagram of a spin-orbital- angular-momentum coupled Bose-Einstein condensate, Phys. Rev. Lett. 122, 110402 (2019). [45] S. L. Zhu, H. Fu, C. J. Wu, S. C. Zhang, and L. M. Duan, Spin Hall eﬀects for cold atoms in a light-induced gauge potential, Phys. Rev. Lett. 97, 240401 (2006). [46] M. C. Beeler, R. A. Williams, K. Jim ´ enez-Garcia, L. J. LeBlanc, A. R. Perry, and I. B. Spielman, The spin Hall eﬀect in a quantum gas, Nature 498, 201 (2013). [47] J. Y. Choi, S. Kang, S. W. Seo, W. J. Kwon, and Y. I. Shin, Observation of a geometric Hall eﬀect in a spinor Bose-Einstein condensate with a skyrmion spin texture, Phys. Rev. Lett. 111, 245301 (2013). [48] P. Zhang, L. Zhang, and Y. J. Deng, Modiﬁed Bethe- Peierls boundary condition for ultracold atoms with spin- orbit coupling, Phys. Rev. A 86, 053608 (2012).17 [49] S.-G. Peng, C.-X. Zhang, S. Tan, and K. Jiang, Contact theory for spin-orbit-coupled Fermi gases, Phys. Rev. Lett.120, 060408 (2018). [50] P. Zhang and N. Sun, Universal relations for spin-orbit - coupled Fermi gas near an s-wave resonance, Phys. Rev. A97, 040701(R) (2018). [51] J. Jie, R. Qi, and P. Zhang, Universal relations of an ultracold Fermi gas with arbitrary spin-orbit coupling, Phys. Rev. A 97, 053602 (2018). [52] X. L. Cui, Mixed-partial-wave scattering with spin-or bit coupling and validity of pseudopotentials, Phys. Rev. A 85, 022705 (2012). [53] Y. X. Wu and Z. H. Yu, Short-range asymptotic behavior of the wave functions of interacting spin-1/2 fermionic atoms with spin-orbit coupling: A model study, Phys. Rev. A87, 032703 (2013). [54] Y. A. Bychkov and E. I. Rashba, Oscillatory eﬀects and the magnetic susceptibility of carriers in inversion layer s, J. Phys. C: Solid State Phys. 17, 6039 (1984). [55] C.-X. Zhang, S.-G. Peng, and K. Jiang, High- temperature thermodynamics of spin-polarized Fermi gases in two-dimensional harmonic traps, Phys. Rev. A 98, 043619 (2018). [56] H. Hu, H. Pu, J. Zhang, S.-G. Peng, and X.-J. Liu, Radio-frequency spectroscopy of weakly bound moleculesin spin-orbit-coupled atomic Fermi gases, Phys. Rev. A 86, 053627 (2012). [57] S.-G. Peng, X.-J. Liu, H. Hu, and K. Jiang, Momentum- resolved radio-frequency spectroscopy of a spin-orbit- coupled atomic Fermi gas near a Feshbach resonance in harmonic traps, Phys. Rev. A 86, 063610 (2012). [58] E. Braaten, D. Kang, and L. Platter, Short-time operato r product expansion for rf spectroscopy of a strongly inter- acting Fermi gas, Phys. Rev. Lett. 104, 223004 (2010). [59] S. Gandolﬁ, K. E. Schmidt, and J. Carlson, BEC-BCS crossover and universal relations in unitary Fermi gases, Phys. Rev. A 83, 041601(R) (2011). [60] N. Sakumichi, Y. Nishida, and M. Ueda, Lee-Yang cluster expansion approach to the BCS-BEC crossover: BCS and BEC limits, Phys. Rev. A 89, 033622 (2014). [61] L. D. Landau and E. M. Lifshitz, Statistical Physics , 3rd ed. (Elsevier, Singapore, 2007), Part I. [62] E. Braaten, D. Kang, and L. Platter, Universal relation s for a strongly interacting Fermi gas near a Feshbach res- onance, Phys. Rev. A 78, 053606 (2008). [63] M. Iskin, Isothermal-sweep theorems for ultracold qua n- tum gases in a canonical ensemble, Phys. Rev. A 83, 033613 (2011). [64] M. Barth, and J. Hofmann, Pairing eﬀects in the nonde- generate limit of the two-dimensional Fermi gas, Phys. Rev. A89, 013614 (2014).
2211.04809v1.Giant_efficiency_of_long_range_orbital_torque_in_Co_Nb_bilayers.pdf	Giant efficiency of long-range orbital torque in Co/Nb bilayers Fufu Liu1, Bokai Liang1, Jie Xu1, Cheng long Jia1,2†, Changjun Jiang1,2* 1 Key Laboratory for Magnetism and Magnetic Materials, Ministry of Education, Lanzhou University, Lanzhou 730000, China 2 Lanzhou Center for Theoretical Physics & Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou University, Lanzhou 730000, China Corresponding author. E -mail address: †cljia@lzu.edu.cn, *jiangchj@lzu.edu.cn ABSTRACT We report unambiguous ly experimental evidence of a strong orbital current in Nb films with weak spin -orbit coupling via the spin-torque ferromagnetic resonance (ST -FMR) spectrum for Fe/Nb and Co/Nb bilayers. The sign change of the damping -like torque in Co/Nb demonstrates a large spin -orbit correlation and thus great efficiency of orbital torque in Co/Nb. By studying the efficiency as a function of the thickness of Nb sublayer, we reveal a long orbital diffusion length (~ 3.1 nm) of Nb. Further planar Hall resistance (PHE) measurements at positive and negative apply ing current confirm the nonlocal orbital transport in ferromagnetic -metal/Nb heterostructures. Spin-related torque s induc ed by spin current s have drawn considerable interest in spintronic over the decade [1-3]. Especially, spin Hall effect (SHE) prevail s in heavy metal (HM) with strong spin -orbit coupling , where the resulting spin -orbit torque (SOT) has served as an efficient manipulation of magnetization in HM/ferromagnet (HM/FM) heter ostructures [1, 4-7]. Recently, orbital current based on the flowing orbital angular momentum (OAM) [8] via orbital Hall effect (OHE) receive s quite attention [9-12]. Like SHE, orbital current is generated along the perpendicular direction to the charge current. However, there is distinctive features of OHE compared to SHE. Theoretically, the OHE roots in orbital textures in momentum -space , which means that the OHE allows for the absence of spin-orbit coupling (SOC) due to the direct action of charge current on orbital degree of freedom [13], where charge current directly acts on spin degree of freedom through strong SOC that results in SHE. Moreover , orbital Hall conductivity (OHC) is much larger than spin Hall conductivity (SHC) in many transition metals , which could be estimated that the orbital torque (OT) efficiency can be comparable to or larger than spin torque efficiency [10]. Thus, the OT promote s the birth of orbitronics which holds great potential for future highly efficient magnetic devices and a complement to spintronics . Many alternative materials can be regarded as the orbital source to induce OT , and theoretically found in multiorbital centrosymmetric systems such as transition metals [10], graphene [14], semiconductors [11] as well as two-dimensional transition metal dichalcogenides [15-17]. Experimentally, the OT was investigated in various systems, especially in thulium iron garnet TmIG/Pt/CuO x [18], FM/Cu/Al 2O3 [19], Py/CuO x [20- 21], etc. which all originates from the orbital Rashba effect (ORE) and modulates OT via layer design [22]. In these systems, Pt or oxidized Cu was utilized as a n orbital source , it undoubtedly complicate s the discussion of OT. Moreover, the intrinsic OHE was observed in Ta/Ni by evaluating the sign of damping -like torque similar to the sign of OHC in Ta [13]. However, the relatively larger SHC for Ta leads to slightly smaller OT efficiency in Ta/Ni system. Thus, materials selection without strong SOC for large OT efficiency is extremely necessary. The transition metal niobium (Nb) is characterized by weak SOC as well as large enough OHC (𝜎𝑂𝐻𝑁𝑏) relative to SHC (𝜎𝑆𝐻𝑁𝑏) and opposite sign of them [10,], which implies highly possible to observe obvious OHE via sign relation and generate expected OT acting on magnetization of FM [23]. Moreover, the large enough OHC is expected to be accompanied by a large OT efficiency . Herein , we present experimental evidence confirm ing the existence of OHE in FM/Nb bilayers by the sign relation of damping -like torque according with OHC (or SHC) in Nb via spin -torque ferromagnetic resonance (ST -FMR) . After fitting the FM dependence of ST -FMR spectra, the variety between the signs of damping -like torque for Fe/Nb and Co/Nb bilayers unambiguous ly proves the emergence of orbital torque. Furthermore, we support our conclusion that the sign of damping -like torque is examined by utilizing the method of planar Hall resistance (PHE) at positive and negative measuring current, which result coincides wi th ST -FMR and the theoretical calculations. Generally, it is difficult to disentangle OT and ST due to identical properties. Thus, the total torque usually is ascribed to the combined effect of ST and OT, which includes following three cases: The first case corresponds to the same sign of 𝜎𝑆𝐻 and 𝜎𝑂𝐻 as shown in Figure 1(a). The total torque is enhanced by a synergistic effect of the OT and ST. In the second case, the sign of 𝜎𝑆𝐻 varies from 𝜎𝑂𝐻 as shown in the top of Figure 1(b), while the magnitude of spin Hall contribution ( ASH) is larger than orbital Hall contribution ( AOH), the total torque is dominated by ST. In the final case, the sign of 𝜎𝑆𝐻 varies from 𝜎𝑂𝐻 in the bottom of Figure 1(b), and the ASH is smaller than AOH that provides a great chance to disentangl e the OT from the ST . Given all this, the 4d transition metal Nb is an excellent candidate to observe OT where 𝜎𝑂𝐻 is opposite sign and much larger than 𝜎𝑆𝐻 [10]. Specifically speaking , as shown in Figure 1(c), the orbital current and spin current are generated by OHE and SHE and injected into FM, respectively. Next the injected orbital current is converted to spin via SOC of the FM. A crucial conversion ratio 𝜂𝐹𝑀 (spin -orbit correlation) [ 24] is used to describe how much spin in FM is induced by the orbital current injected from the Nb. For most FMs, the sign of 𝜂𝐹𝑀 is positive, such as Fe, Co, Ni, etc. However, the small 𝜂𝐹𝑀 tends to induce ST , while the large 𝜂𝐹𝑀 leads to OT , as demonstrated in Figure 1(c). Based on the previous theoretical[13], we summary the total torque of FM/Nb systems in Figure 1(d). Obviously, the Co/Nb bilayer is a good candidate to clarify the OT, given that the sign of the effective Hall conductivity ( 𝜎𝑆𝐻𝑁𝑏+𝜂𝐹𝑀𝜎𝑂𝐻𝑁𝑏) obtained from torque coincides with the sign of 𝜎𝑂𝐻𝑁𝑏 . The FM/Nb thin films were deposited on MgO substrate by magnetron sputtering at room temperature. In order to clarify the physical mechanism of OT in experiments, f our different types of bilayers were prepared: Fe(tFe)/Nb(6 nm), Co(tCo)/Nb(6 nm). Fe(8 nm)/Nb( tNb) and Co(8 nm)/Nb( tNb). For comparison , the samples Pt(6 nm)/Co( tCo) and Pt(6 nm)/Fe( tFe) were grown on MgO substrate as well . Moreover, Fe(2 nm)/Nb(8 nm) and Co(2 nm)/Nb(8 nm) are prepared to conduct the PHE measurements. To measure the OT efficiency, we exploit the spin -torque ferromagnetic resonance (ST-FMR) measurement as shown in Figure 2(a), the ST -FMR technique is employed because it is a well-established method to determine the charge -spin conversion efficiency as well as the type, number and efficiency of torques [1, 25-28]. In this measurement, a radio frequency (RF) microwave current is applied along the x - direction , this RF current will induce the resulting torque and cause the FM magnetization process, which contributes to the ST -FMR resonance signal. The typical ST-FMR signal for Fe(8 nm)/Nb(6 nm) is shown in Figure 2(b), which could be fitted by symmetric (Vs) and antisymmetric component (Va). Generally, the Vs and Va corresponds to different torques, where the Vs arises solely from the damping -like torque ( 𝝉𝐃𝐋∝𝒎×(𝒚×𝒎)) and Va originates jointly from the field-like torque ( 𝝉𝐅𝐋∝ 𝒎×𝒚 and current -induced Oersted field , respectively [13]. Furthermore, the quantitative conversion efficiency ξ can be determined by the ratio of Vs and Va based on the following equation [1, 13, 29-30]: 𝜉=𝑉𝑠 𝑉𝑎𝑒𝜇0𝑀𝑠𝑡𝐹𝑀𝑑𝑁𝑏 ℏ[1+(𝑀𝑠𝐻𝑟⁄ )]12⁄, (1) where tFM and dNb are the thickness of FM and Nb sublayer s, respectively. Hr is the resonance field . Ms is the effective saturation magnetization that can be obtained by Kittel equation: (2𝜋𝑓)/𝛾=√[𝐻𝑟(𝐻𝑟+𝑀𝑠)] with γ being the gyromagnetic ratio. Analogous to torque components of SOT, the 𝝉𝐃𝐋 and 𝝉𝐅𝐋 components of OT can be generated in adjacent magnetic layer and separately determined via ST -FMR. In the present study, we are more interested in the sign of 𝝉𝐃𝐋, which could reflect the sign of SHC and /or OHC. As shown in Figure 2(b), a negative Vs is observed for Fe(8 nm)/Nb(6 nm), which mean s the existence of 𝝉𝐃𝐋. Obviously, the sign of Vs is consistent with the expected sign of 𝜎𝑆𝐻𝑁𝑏 and opposite t o the 𝜎𝑂𝐻𝑁𝑏. The sign of 𝝉𝐃𝐋 for Fe/Nb could be further determined by the FM layer thickness dependence of the ST-FMR resonance signal Vdc [24] as shown in Figure 2( c). The experimentally measured conversion efficiency ξ as a function of tFM provides a route to separately evaluate the damping - like ( 𝜉𝐷𝐿) and field-like (𝜉𝐹𝐿) torque efficiencies as [13, 20 ] 1 𝜉=1 𝜉𝐷𝐿(1+ℏ 𝑒𝜉𝐹𝐿 𝜇0𝑀𝑠𝑡𝐹𝑀𝑑𝑁𝑏) (2) where the intercept implies the sign of 𝜉𝐷𝐿. As shown in Figure 2(c), the sign of the intercept is in accord with the symmetric component Vs in Figure 2(b). Hence , it is apparent that the sign of 𝜉𝐷𝐿 of Fe/Nb bilayer is consistent with the sign of 𝜎𝑆𝐻𝑁𝑏 . However, we have a sign reversal of the symmetric component Vs in ST-FMR signal for Co(8 nm)/Nb(6 nm) , i.e., a positive Vs as shown in Figure 2( d). Consequently , the FM layer thickness dependence of efficiency for Co/Nb bilayers [24] gives a positive 𝜉𝐷𝐿 as plotted in Figure 2(e) , which coincides with the expected sign of 𝜎𝑂𝐻𝑁𝑏 but opposite to the sign of 𝜎𝑆𝐻𝑁𝑏. On the other hand , for the Pt/Fe and Pt/Co control sample as shown in Figures 3, the heavy metal Pt, is acknowledged as a strong spin source material [1]. From the experimental result, the positive sign of Vs is consistent with the sign of 𝜎𝑆𝐻𝑃𝑡 , and further accords w ith the sign of 𝜉𝐷𝐿 . According to schematic illustration for the mechanism of the orbital torque in Figure 1( c) and theoretical calculations [24], the to tal torque in Co/Nb includes conventional SOT arising from spin Hall contribution ( 𝜎𝑆𝐻𝑁𝑏 ) and OT attributing to orbital Hall contribution ( 𝜎𝑂𝐻𝑁𝑏 ). When the spin Hall contribution is much less than orbital Hall contribution, the sign of the total torque is same as the sign of 𝜎𝑂𝐻𝑁𝑏. In this case, as for Co/Nb bilayer, the sign of OT depends on the 𝜎𝑂𝐻𝑁𝑏 and spin -orbit correlation ˂L·S˃Co [13]. The fact from theoretical calculation is that the orbital Hall effect contribution is larger than the spin Hall one in magnitude of Co/Nb , the sign of total torque is in accord with the expected sign of 𝜎𝑂𝐻𝑁𝑏, i.e., the orbital torque . Furthermore, to verify the source of OT, we perform the ST -FMR measurement for Co(8 nm)/Cu(t)/Nb(6 nm) , Figure S4 shows the typical ST -FMR spectra. The positive Vs means the existence of OT even if inserting the Cu layer. Cu layer thickness dependence of ef ficiency ξ is plotted in Figure 4(a). Enough efficiency exclude s the interface effect at Co/Nb interface and bulk OHE is responsible for the OT . Moreover, to obtain the orbital diffusion length of Nb, a series of ST-FMR samples with different Nb thickness es of Co (and Fe, t = 8 nm) /Nb(d = 6-15 nm) was fabricated, and the conversion efficiency ξ was characterized by Eq. (1) and summarized in Figure 3(a), the Nb thickness dependence of ξ match with 𝜉=𝜉∞[1−sech (𝑑/𝜆𝑠)] [31], where ξ is the measured conversion efficiency at different Nb thickness and ξꝏ is the conversion efficiency at infinite Nb thickness. From the fitting, λs of Nb is quantitatively determined to be 3.1 nm. This point sets it apart from conventional heavy metal, such as Pt, where spin Hall contribution dominates and shows a slightly small spin diffusion length (~ 1.5 nm) [31]. Furthermore, to verify the sign relation of damping -like torque in FM/Nb bila yers, we perform the method of PHE at positive and negative measuring current, which has been regarded as a reliable and effective method to measure charge -to-spin conversion [32-34]. In this measurement, the damping -like torque induces the out -of-plane effective field Hp, while the field -like torque induces the in -plane effective field HT as shown in Figure 4(a). Here we are just concerned with the sign of Hp. Figure 4(b) and (c) exhibits the typical magnetic angle dependence of planar Hall resistance RH as well as the corresponding RDH curve at I = ± 0.9 mA for Fe(2 nm)/N b (8 nm) , and I = ± 1.0 mA for Co (2 nm)/Nb(8 nm), respectively (see Supplemental Material for other currents [24]). Note that the effective fields Hp and HT can be extracted from the resistance difference RDH and charactered by following equation [ 32-34]: 𝑅𝐷𝐻(𝐼,𝜑)=2𝑅𝐻(𝐻𝑇+𝐻𝑂𝑒) 𝐻𝑒𝑥𝑡(𝑐𝑜𝑠𝜑 +𝑐𝑜𝑠3𝜑)+2𝑑𝑅𝐴𝐻𝐸 𝑑𝐻𝑝𝑒𝑟𝑝𝐻𝑃𝑐𝑜𝑠𝜑 +𝐶 (3) where C is the resistance offset, HOe is the Oersted effective field, φ is defined in Figure 4(a), Hperp is the applied out -of-plane magnetic field, 𝑑𝑅𝐴𝐻𝐸 𝑑𝐻𝑝𝑒𝑟𝑝 is the slope of RAHE vs Hperp [24]. After determining relevant parameters in Eq. (3), the out -of-plane effective field Hp is obtained by fitting the φ dependence of RDH. Figure 4(d) shows the linear fitting Hp relative to the current for Fe/Nb and Co/Nb , respectively . The sign of effective conversion efficiency can be quantitatively determined by [ 32]:𝜉~𝐻𝑝 𝐽𝑁𝑀, where JNM is the density of current. Obviously, the sign of 𝜉 depends on the slope of Hp versus current I. Any difference in sign of slope for Hp vs I between Fe/Nb and Co/Nb would suggest an existence of orbital torque, which is consistent with ST -FMR result and the theoretical calculations. In conclusion, the generation of orbital current and orbital torque is experimentally confirmed based on orbital Hall effect present in FM/Nb systems measured by spin - torque ferromagnetic resonance. After fitting the FM thicknesses dependence of the ST - FMR reso nance sign al Vdc, we characterize the sign of 𝜉𝐷𝐿 for Fe/Nb and Co/Nb is positive, which is consistent with the spin Hall conductivity and orbital Hall conductivity of Nb from theoretical calculations , respectively . This result can be further verif ied by the method of planar Hall resistance . This strategy of generating orbital current and orbital torque by experimental method provide a route of enabling applications in spintronic and orbitronic devices by orbital engineering. REFERENCES [1] L. Q. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett. 106, 036601 (2010). [2] Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Science 306, 1910 – 1913 (2004). [3] M. H. Nguyen, C. F. Pai, K. X. Nguyen, D. A. Muller, D. C. Ralph, and R. A. Buhrman, Appl. Phys. Lett. 106, 222402 (2015). [4] W. Zhang, W. Han, X. Jiang, S .-H. Yang, and S. S. P. Parkin, Nat. Phys. 11, 496 (2015). [5] L. Liu, C. F. Pai, Y . Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Science 336, 555 (2012). [6] F. F. Liu, C. Zhou, R. J. Tang, G. Zh. Chai, and Ch. J. Jiang, Phys. Status Solidi RRL. 15, 2100342 (2021). [7] K. F. Huang, D. S. Wang, H. H. Lin, and C. H. Lai, Appl. Phys. Lett. 107, 232407 (2015). [8] P. M. Haney and M. D. Stiles , Phys. Rev. Lett. 105, 126602 (2010). [9] H. Kontani, T. Tanaka, D. S. Hirashima, K. Yamada, and J. Inoue , Phys. Rev. Lett. 102, 016601 (2009). [10] T. Tanaka, H. Kontani, M. Naito, T. Naito, D. S. Hirashima, K. Yamada, and J. Inoue , Phys. Rev. B 77, 165117 (2008). [11] B. A . Bernevig, T . L. Hughes, and Sh . Ch. Zhang , Phys. Rev. Lett. 95, 066601 (2005). [12] D. Go, D . Jo, Ch . Y . Kim, and H . W. Lee, Phys. Rev. Lett. 121, 086602 (2018). [13] D. Lee, D . Go, H . J Park, W . Jeong, H . W. Ko, D . Yun, D . Jo, S . Lee, G . Go, J . H. Oh, K . J. Kim, B . G. Park, B . Ch. Min, H . Ch. Koo, H . W. Lee, O . Lee and K. J. Lee, Nat. Commun . 12, 6710 (2021). [14] S. Bhowal and G . Vignale , Phys. Rev. B 103, 195309 (2021). [15] S. Bhowal and S. Satpathy , Phys. Rev. B 101, 121112(R) (2020). [16] L. M. Canonico, T . P. Cysne, A . M. Sanchez, R. B. Muniz, and T . G. Rappoport , Phys. Rev. B 101, 161409(R) (2020). [17] T. P. Cysne, M . Costa, L . M. Canonico, M. B . Nardelli, R.B. Muniz, and Tatiana G. Rappoport , Phys. Rev. Lett. 126, 056601 (2021). [18] Sh. L. Ding, A . Ross, D . W. Go, L . Baldrati, Z . Y . Ren, F . Freimuth, S . Becker, F . Kammerbauer, J . B. Yang, G . Jakob, Y . Mokrousov, and M . Kläui , Phys. Rev. Lett. 125, 177201 (2020). [19] J. Kim, D . W. Go, H . Sh. Tsai, D . Jo, K . Kondou, H . W. Lee, and Y . Ch. Otani , Phys. Rev. B 103, L020407 (2021). [20] H. Y . An, Y . Kageyama, Y . Kanno, N . Enishi and K. Ando , Nat. Commun. 7, 13069 (2016). [21] Y . Kageyama, Y . Tazaki, H. An, T. Harumoto, T. Gao, J. Shi, and K. Ando, Sci. Adv. 5, eaax4278 (2019). [22] D. W. Go, D . Jo, T . H. Gao, K . Ando, S . Blügel, H . W. Lee, and Y . Mokrousov , Phys. Rev. B 103, L121113 (2021). [23] S. Dutta and A. A. Tulapurkar, Phys. Rev. B 106, 184406 (2022) [24] See Supplemental Material for details of the theoretical analysis, FM-thickness dependence of ST -FMR spectra , anomalous Hall resistance measurements and p lanar Hall resistance measurements . [25] A. R. Mellnik, J. S. Lee, A. Richardella, J. L. Grab, P. J. Mintun, M. H. Fischer, A. Vaezi, A. Manchon, E. A. Kim, N. Samarth, D. C. Ralph, Nature 511, 449 -451 (2014). [26] D. MacNeill, G. M. Stiehl, M. H. D. Guimaraes, R. A. Buhrman, J. Park and D. C. Ralph, Nat. Phys. 13, 300 –305 (2017). [27] F. F. Liu, Y . Jin, Y . B. Zhao, Ch . L. Jia and Ch. J. Jiang , Appl. Phys. Lett. 121, 022404 (2022) . [28] T. Nan, C. X. Quintela, J. Irwin, G. Gurung, D. F. Shao, J. Gibbons, N. Campbell, K. Song, S. Y . Choi, L. Guo, R. D. Johnson, P. Manuel, R. V . Chopdekar, I. Hallsteinsen, T. Tybell, P. J. Ryan, J. W. Kim, Y . Choi, P. G. Radaelli, D. C. Ralph, E. Y . Tsymb al, M. S. Rzchowski, C. B. Eom, Nat. Commun. 11, 4671 (2020). [29] W. F. Zhang, W . Han, X . Jiang, S . H. Yang and S . S. P. Parkin , Nat. Phys. 11, 496 – 502 (2015) . [30] C. F. Pai, Y . X. Ou, L . H.e Vilela -Leão, D. C. Ralph, and R. A. Buhrman , Phys. Rev. B 92, 064426 (2015). [31] Y. Wang, P . Deorani, X . P. Qiu, J . H. Kwon, and H . Yang , Appl. Phys. Lett. 105, 152412 (2014) . [32] Q. Lu, P . Li, Zh . X. Guo, G . H. Dong, B . Peng, X . Zha, T . Min, Z . Y . Zhou and M. Liu, Nat. Commun . 13, 1650 (2022). [33] M. DC, R . Grassi, J . Y. Chen, M . Jamali, D . R. Hickey, D . L. Zhang, Zh . Y . Zhao, H. Sh. Li, P. Quarterman, Y . Lv, Mo Li, A . Manchon, K. A . Mkhoyan, T . Low and J. P. Wang , Nat. Mater . 17, 800 –807 (2018). [34] M. Kawaguchi, K . Shimamura, S . Fukami, F . Matsukura, H . Ohno, T . Moriyama, D. Chiba and T . Ono, Appl. Phys. Express 6, 113002 (2013). Figure 1 . (a) The same sign for 𝜎𝑆𝐻 and 𝜎𝑂𝐻, where the total torque includes the sum of OT and ST. (b) The opposite sign of 𝜎𝑆𝐻 and 𝜎𝑂𝐻. The total torque could be either ST or OT. (c) Schematic illustration for the mechanism of the orbital torque in FM/N b. The orbital current generated by OHE in the Nb is injected into FM. Next the injected orbital current is converted to spin through SOC o f the FM. The resulting torque exerted by t his spin exerts and acts on the magnetization of FM, which is referred to as orbital torque (OT). (d) Theoretical calculation for 𝜎𝑆𝐻𝑁𝑏+𝜂𝐹𝑀𝜎𝑂𝐻𝑁𝑏 of FM/Nb bilayers ( FM = Fe and Co ). Where 𝜎𝑆𝐻𝑁𝑏, 𝜎𝑂𝐻𝑁𝑏 and 𝜂𝐹𝑀 are spin Hall conductivity, orbital Hall conductivity of Nb and orbital -to-spin conversion efficiency of FM, respectively. -80-60-40-20020 +5 Co/NbNb SH + FMNb OH(ћ/e -1cm-1) Fe/Nb-68(c) (d)(a) (b) Figure 2. (a) Schematic of the circuit for ST -FMR measurements. ST-FMR voltage signal Vdc for (b) MgO/Fe (8 nm)/Nb(6 nm) at 8 GHz , (d) MgO/Co (8 nm)/Nb(6 nm) at 7 GHz . The inverse of the conversion efficiency 1/ ξ as a function of 1/ tFM for (c) Fe/Nb bilayer and (e) Co/Nb bilayer. The solid line s are the linear fit to the experimental data (grids) . 0.00 0.05 0.10 0.15 0.20-6061218 DL = - 0.22 Fe/Nb Linear1/ 1/t Fe (nm-1) 0 200 400 600 800-24-120122436 Data Fitting Va Vs Vdc (V) H (Oe)7 GHzCo(8nm)/Nb(6nm) 0 300 600 900-2-1012 Data Fitting Va Vs Vdc (V) H (Oe)8 GHzFe(8nm)/Nb(6nm)(b) (c) (d) (e)(a) 0.00 0.07 0.14 0.21 0.28-12-606 DL = + 0.16 Co/Nb Linear1/ 1/t Co (nm-1) Figure 3. ST-FMR voltage signal Vdc for (a) MgO/Pt (6 nm)/Fe(8 nm) at 7 GHz , (c) MgO/Pt (6 nm)/Co(8 nm) at 7 GHz . The inverse of the conversion efficiency 1/ ξ as a function of 1/ tFM for (b) Pt/Fe bilayer and (d) Pt/Co bilayer. The girds are the experimental data, and the solid line is the linear fit to the data. 0.00 0.07 0.14 0.21 0.28061218 DL = + 0.06 Pt/Co Linear1/ 1/t Co (nm-1) 0.00 0.07 0.14 0.21 0.28071421DL = + 0.05 Pt/Fe Linear1/ 1/t Fe (nm-1) 0 300 600 900-12-6061218 Data Fitting Va Vs Vdc (V) H (Oe)7 GHzPt(6nm)/Fe(6nm) 0 300 600 900 1200-10-5051015 Data Fitting Va Vs Vdc (V) H (Oe)7 GHzPt(6nm)/Co(6nm)(a) (b) (c) (d) Figure 4. (a) Cu-layer thickness tCu dependence of efficiency for Co(8 nm)/Cu( tCu)/Nb(6 nm). (b) The conversion efficiency ξ (orange squares) as a function of Nb thickness for Co (8 nm)/Nb(t) as well as Fe (8 nm)/Nb(t) systems and a fit (solid orange line). 0 1 2 3 4 50.000.060.120.180.240.30 tCu (nm)Co(8nm)/Cu (t Cu)/Nb(6nm) 0 4 8 12 16-0.26-0.130.000.130.26 Co/Nb Fitting Fe/Nb Fitting tNb(nm)s= 3.1nm(b) (a) Figure 5. (a) Current -induced effective fields and schematic illustration of the planar Hall resistance measurement srtup. The external magnetic field angle φ dependence of planar Hall resistance RH of (b) Fe(2 nm)/Nb(8 nm) and ( c) Co(2 nm)/Nb(8 nm). The right axis of (b) and ( c) represents the difference of the Hall resistances RDH at positive and negative currents. (d) The variety of Hp with the applied current for Fe(2 nm)/Nb(8 nm) and Co(2 nm)/Nb(8 nm) . (a) 0 90 180 270 360-0.2-0.10.00.10.2 RDH () (°) +1 mA -1 mA DifferenceRH () -0.18-0.090.000.090.18 Co/Nb 0 90 180 270 360-0.2-0.10.00.10.2 RH () (°) Fe/Nb -0.14-0.070.000.070.14 RDH () +0.9 mA -0.9 mA Difference 0.0 0.3 0.6 0.9-4004080 Co/Nb Fe/NbHP (Oe) Current (mA) (b) (c) (d) HP τDLτFL HTz yx y x zφIHext V
1701.00786v2.Universal_Absence_of_Walker_Breakdown_and_Linear_Current_Velocity_Relation_via_Spin_Orbit_Torques_in_Coupled_and_Single_Domain_Wall_Motion.pdf	arXiv:1701.00786v2 [cond-mat.mes-hall] 18 Apr 2017Universal Absence of Walker Breakdown and Linear Current–V elocity Relation via Spin–Orbit Torques in Coupled and Single Domain Wall Mot ion Vetle Risinggård∗and Jacob Linder Department of Physics, NTNU, Norwegian University of Scien ce and Technology, N-7491 Trondheim, Norway (Dated: June 29, 2018) We consider theoretically domain wall motion driven by spin –orbit and spin Hall torques. We ﬁnd that it is possible to achieve universal absence of Walker breakdow n for all spin–orbit torques using experimentally relevant spin–orbit coupling strengths. For spin–orbit to rques other than the pure Rashba spin–orbit torque, this gives a linear current–velocity relation instead of a satur ation of the velocity at high current densities. The effect is very robust and is found in both soft and hard magnet ic materials, as well as in the presence of the Dzyaloshinskii–Moriya interaction and in coupled domain w alls in synthetic antiferromagnets, where it leads to very high domain wall velocities. Moreover, recent exper iments have demonstrated that the switching of a synthetic antiferromagnet does not obey the usual spin Hall angle-dependence, but that domain expansion and contraction can be selectively controlled toggling only th e applied in-plane magnetic ﬁeld magnitude and not its sign. We show for the ﬁrst time that the combination of spi n Hall torques and interlayer exchange coupling produces the necessary relative velocities for this switch ing to occur. I. INTRODUCTION Domain wall motion in ferromagnetic strips is a central theme in magnetization dynamics and has recently been in- strumental to the discovery of several new current-induced effects.1–6The attainable velocity of a domain wall driven by conventional spin-transfer torques (STTs)7–9is limited by the Walker breakdown,10upon which the domain wall deforms, resulting in a reduction of its velocity. Current-induced torques derived from spin–orbit effects (SOTs) such as the spin Hall effect4–6,11or an interfacial Rashba spin–orbit coupling12–14have enabled large domain wall velocities. We here consider the dependence of the do- main wall velocity on the current and ﬁnd that regardless of the relative importance of the reactive and dissipative com - ponents of the torque it is possible to achieve universal ab- sence of Walker breakdown for all current densities for ex- perimentally relevant spin–orbit coupling strengths. For spin– orbit torques other than the pure Rashba SOTs, such as the spin Hall torques, the velocity will not saturate as a func- tion of current, but will increase linearly as long as a conve n- tional spin-transfer torque is present. This behavior is ro bust against the presence of an interfacial Dzyaloshinskii–Mor iya interaction15–17and is found both in perpendicular anisotropy ferromagnets, in shape anisotropy-dominated strips and in synthetic antiferromagnets (SAFs),18–22where it enables very high domain wall velocites for relatively small current den si- ties. Moreover, the combination of SOTs with the interlayer exchange torque was recently shown experimentally to pro- duce novel switching behavior that circumvents the usual sp in Hall angle-dependence.22We show that the combination of spin Hall torques and interlayer exchange produces the re- quired dependence of the domain wall velocity on the topolog - ical charge to qualitatively reproduce the experimental da ta.II. UNIVERSAL ABSENCE OF WALKER BREAKDOWN We consider an ultrathin ferromagnet with a heavy metal underlayer as shown in Figure 1 . We describe the dynamics of the magnetization m(r,t)using the Landau–Lifshitz–Gilbert (LLG) equation,23 ∂tm=γm×H−α mm×∂tm+τ, (1) whereγ < 0is the gyromagnetic ratio, mis the saturation magnetization,α<0is the Gilbert damping, H=−δF/δm is the effective ﬁeld acting on the magnetization and τis the current-induced torques. The free energy Fof the ferromag- net is a sum, F=/uni222B.dsp dr(fZ+fex+fDM+fa), (2) of the Zeeman energy due to applied magnetic ﬁelds, the isotropic exchange, the interfacial Dzyaloshinskii–Mori ya in- teraction and the magnetic anisotropy. The Zeeman energy and the isotropic exchange can be written respectively as fZ=−H0·m, whereH0is the ap- plied magnetic ﬁeld, and fex=(A/m2)[(∇mx)2+(∇my)2+ (∇mz)2], where Ais the exchange stiffness.23Inversion sym- metry breaking at the interface between the heavy metal and the ferromagnet gives rise to an anisotropic contribution t o the exchange known as the Dzyaloshinskii–Moriya interac- tion, which favors a canting of the spins.15–17The resulting Figure 1. Ultrathin ferromagnet with a heavy metal underlay er. We consider transverse domain wall motion along the xaxis. r,σland σsdenote the three nontrivial operations of the symmetry grou pC2v.2 contribution to the free energy is fDM=(D/m2)[mz(∇ ·m)− (m· ∇)mz], where Dis the magnitude of the Dzyaloshinskii– Moriya vector. Ultrathin magnetic ﬁlms are prone to exhibit perpendicular magnetization due to interface contributio ns to the magnetic anisotropy.24Consequently, we write the mag- netic anisotropy energy as fa=−Kzm2 z+Kym2 y, correspond- ing to an easy axis in the zdirection and a hard axis in the y direction. A. Current-Induced Torques The current-induced torques τare conventionally divided into spin-transfer torques and spin–orbit torques. The spi n- transfer torques can be written as7–9 τSTT=u∂xm−βu mm×∂xm, (3) where u=µBP j/[em(1+β2)]andjis the electric current, Pis its spin polarization, µBis the Bohr magneton, eis the electric charge andβis the nonadiabacity parameter. The spin–orbit torques can be written as4–6,11–14 τR=γm×HRey−γm×/parenleftbigg m×βHRey m/parenrightbigg , (4) τSH=γm×/parenleftbigg m×HSHey m/parenrightbigg +γm×βSHHSHey,(5) where HR=αRP j/[2µBm(1+β2)]andαRis the Rashba pa- rameter and where HSH=/planckover2pi1θSHj/(2emt)andθSHis the spin Hall angle and tis the magnet thickness. Since the spin Hall effect changes sign upon time-reversal, the principal spin Hall torque term is dissipative instead of reactive, in contrast to the principal term of the STTs and the Rashba SOTs. In fact, assuming that the stack can be described using the C2vsymmetry group (see Figure 1 ) it can be shown that these torques exhaust the number of possible torque components. Hals and Brataas25describe spin–orbit torques and general- ized spin-transfer torques in terms of a tensor expansion. A s- suming the lowest orders are sufﬁcient to describe the essen - tial dynamics, the reactive and dissipative spin–orbit tor ques are described by, respectively, an axial second-rank tenso r and a polar third-rank tensor while the generalized spin-trans fer torques are described using a polar fourth-rank tensor and a n axial ﬁfth-rank tensor. The torques that arise in a given str uc- ture are limited by the requirement that the tensors must be invariant under the symmetry operations fulﬁlled by the str uc- ture. We have assumed that the physical systems we con- sider are described by C2vsymmetry. Combined with the fact that the current is applied in the xdirection only and that∂ym=0and∂zm=0, this implies that there is only one relevant nonzero element in the axial second-rank tenso r, two elements in the polar third-rank tensor, three elements in the polar fourth-rank tensor and six elements in the axial ﬁf th- rank tensor.26 The three relevant nonzero elements of the second- and third-rank tensors give rise to three spin–orbit torques. A de- tailed analysis shows that these torque components are cap- tured by the Rashba and spin Hall torques in equations ( 4) and(5). As an aside, we note that although the Rashba and spin Hall effects may not necessarily capture all of the relevant mi- croscopic physics27–29these torques can still be used to model the dynamics because they contain three ‘free’ parameters, αR, θSHandβSH. As has been shown in Ref. 25, the generalized spin-transfer torques reduce to the ordinary STTs in the nonrelativistic l imit. Thus, by using the ordinary STTs we neglect possible spin– orbit coupling corrections to these higher-order terms. B. The Collective Coordinate Model The magnetization is conveniently parametrized in spher- ical coordinates as m/m=cosφsinθex+sinφsinθey+ cosθez. Using the assumption that there is no magnetic tex- ture along the yand the zaxes,∇=∂xex, we can ﬁnd the domain wall proﬁle by minimizing the free energy. The re- sulting Euler–Lagrange equations are A(θ′′cscθsecθ−φ′2)−Dφ′sinφtanθ=(Kz+Kysin2φ) and A(φ′′+2θ′φ′cotθ)+Dθ′sinφ=Kycosφsinφ. One solution of these differential equations is the Néel wal l solutionφ=nπandθ=2 arctan exp [Q(x−X)/λ], where Q is the topological charge of the wall,30Xis the wall position andλ=/radicalbig A/Kzis the domain wall width. nis even if D<0 andQ=+1, and nis odd if D<0andQ=−1. This domain wall proﬁle is known as the Walker proﬁle.10To be sure that φ=nπis really the global minimum, we solve the full LLG equation ( 1) for a single magnetic layer and let the solution re- lax without any applied currents or ﬁelds. The angle φ(x)can then be calculated as φ(x)=arctan[my(x)/mx(x)]. However, φ(x)is ill deﬁned in the domains where θ→0orπ. Conse- quently, we consider φonly inside the domain wall. As shown inFigure 2 (a), the solutionφ=0works very well. Substitution of the Walker proﬁle into the full LLG equa- tion ( 1) usingH0=HxexandQ=+1gives the collective coordinate equations, for the wall position Xand tiltφ α∝dotaccX λ−∝dotaccφ=+π 2γ/parenleftBig HSH−βHR/parenrightBig cosφ+βu λ, (6) (1+α2)∝dotaccφ=−αγKy msin 2φ+παγ(D−Hxmλ) 2mλsinφ(7) −u(α+β) λ−π 2γ/bracketleftBig HSH(1−αβSH)−HR(α+β)/bracketrightBig cosφ. By doing this substitution, we are assuming that the domain wall moves as a rigid object described by two collective coor - dinates X(t)andφ(t)(Ref. 30). In particular, we are neglect- ing any position dependence in the domain wall tilt φ. The col- lective coordinate model, or one-dimensional model, has be en used previously to explain the qualitative behavior of both spin-transfer and spin–orbit torques.4,5,7,10,18–20,28,30,31How- ever, it is important to remember that the model will always3 −2 0 2−0.02−0.010.000.010.02 positionx/λtilt angle φ(x)[rad] −2 0 20.000.250.500.751.00 magnetization mx(x)/m(a) 468 10−0.02−0.010.000.010.02 positionx/λtilt angle φ(x)[rad] 468 100.000.250.500.751.00 magnetization mx(x)/m(b) 20 22 24 260.000.010.020.030.04 positionx/λtilt angle φ(x)[rad] 20 22 24 260.000.250.500.751.00 magnetization mx(x)/m(c) 2 4 68−0.02−0.010.000.010.02 positionx/λtilt angle φ(x)[rad] 2 4 680.000.250.500.751.00 magnetization mx(x)/m(d) Figure 2. Position dependence of the domain wall tilt φ. In each panel, the orange curve mx(x)shows the extension of the domain wall while the black solid curve shows the domain wall tilt φ(x)ob- tained by solving the full LLG equation ( 1) and the black dashed line shows the prediction of the collective coordinate model. (a ) Equi- librium solution. (b) Spin-transfer torque dynamics. (c) S pin Hall torque dynamics. (d) Rashba spin–orbit torque dynamics. (a )–(d) We use the material parameters supplied in the ﬁrst column of Table I with j=5 MA/cm2except that J=0. be an approximation, and we cannot necessarily expect quant i- tative agreement between experimental results and model pr e- dictions nor can we completely exclude the possibility of dy - namics that is not captured by the one-dimensional model.31 We can nevertheless test the adequacy of the collective coor - dinate model by calculating φ(x)from a solution of the full LLG equation for a single magnetic layer, just as we did for the static case. As shown in Figure 2 (b) the xdependence of φis negligible for spin-transfer torques. The xdependence of φis larger for spin Hall [ Figure 2 (c)] and Rashba spin–orbit torques [ Figure 2 (d)]. Nonetheless, the ability of the collec- tive coordinate model to consistently qualitatively repro duce experimental behavior indicates that it captures the gener ality, if not all, of the physics in the system. Equations ( 6) and ( 7) can be simpliﬁed by introducing a j= π 2γ(HSH−βHR),bj=βu/λ,c=−2αγKy/m,d=παγ(D− Hxmλ)/(2mλ),ej=−π 2γ[HSH(1−αβSH) −HR(α+β)]and f j=−u(α+β)/λ. Walker breakdown is absent when the time derivative ∝dotaccφvanishes, resulting in the condition 0=csinφcosφ+dsinφ+j(ecosφ+f). (8) Provided that the transverse domain wall is not transformed into for instance a vortex wall,31Walker breakdown will be universally absent if e>fbecause this equation always has a solution forφregardless of the value of j. For increasing j,φwill level off to a value cosφ=−f/e. For realistic material values e>fcorresponds to a Rashba parameter αR>4µ2 B/(πeγλ)=1to6 meV nm (pure Rashba SOTs) or a spin Hall angleθSH>4µBPt/(π/planckover2pi1γλ)=0.05to0.09(pure spin Hall torques). To the best of our knowledge, the absenceof Walker breakdown for spin Hall torques has not been noted previously, whereas absence of Walker breakdown for sufﬁ- ciently strong Rashba spin–orbit coupling was pointed out i n Ref. 32, and can also be noted in Refs 13and33–35. Let us writeξ=cosφandη=sinφ, so thatξ2+η2=1. Solving equation ( 8) forηto getη=−j(eξ+f)/(cξ+d), this relation gives a quartic equation c2ξ4+2cdξ3+[(ej)2+d2−c2]ξ2+2(e f j2−cd)ξ=d2−(f j)2. The exact solutions of the quartic are hopelessly complicat ed. However, they all have the same series expansion around j= 0andj→ ∞ . We consider ﬁrst the asymptotic expansion, ξ=−f e+S1 j+O/parenleftBig j−2/parenrightBig , (9) where S1represents the solutions of the quadratic equation e6ζ2=d2e4+c2f4+(c2−d2)f2e2+2cde f(f2−e2). Using equation ( 6), the wall velocity is then α∝dotaccX λ=/parenleftbigg b−a f e/parenrightbigg j+aS1+aO/parenleftBig j−1/parenrightBig . (10) Back substitution of the abbreviations a,b,eand fshows that for pure Rashba SOTs the coefﬁcient of the linear term reduces to zero because the ratio of the reactive to the dissi - pative torque is the same for the STTs and the Rashba SOTs. Thus, for large jthe domain wall velocity approaches a con- stant. For pure spin Hall torques we get instead the linear te rm −uα(1+ββSH)/[λ(1−αβSH)]. This means that for large j the velocity is actually independent of the sign of the spin Hall angle and increases linearly with j. Note the importance of including the STTs—which are always present—in these con- siderations: in the absence of STTs ( u→0) both band fgo to zero and the velocity levels off to a constant for large jfor any combination of SOTs. For completeness, we also consider the series expansion about j=0, which gives ξ=−1+(e−f)2 2(c−d)2j2+O/parenleftBig j4/parenrightBig (11) and α∝dotaccX λ=(b−a)j+a(e−f)2 2(c−d)2j3+aO/parenleftBig j5/parenrightBig . (12) The key observation here is that in this regime the velocity does depend on the sign of the spin Hall angle ( a∝θSHfor pure spin Hall torques) and increases with the cube of j. Com- bined with the spin Hall angle-independence of the velocity in the j→ ∞ limit, this implies that even in the absence of Walker breakdown a nonmonotonic current–velocity rela- tion is possible. Figure 3 (a) shows a numerical solution of the coupled equations ( 6) and ( 7) as a function of jfor pure Rashba SOTs and for pure spin Hall torques both in the cases ofθSH>0andθSH<0together with the analytical solutions close to j=0and for large jfor parameters that are typical for a standard cobalt–nickel multilayer. We see that our ana - lytical results successfully approximate the full solutio n in the4 0 2 4 6 ·1013−2−10 currentj/bracketleftBig A/m2/bracketrightBigvelocity ˙X[km/s](a) (b) 0 3 69 ·1012−0.6−0.300.3 currentj/bracketleftBig A/m2/bracketrightBigvelocity ˙X[km/s] spin Hall, θSH>0 spin Hall, θSH<0 Rashba SOTs Figure 3. Current–velocity relation for three different SO Ts in the ab- sence of Walker breakdown. The Rashba SOTs level off to a cons tant velocity at large currents, whereas the spin Hall torques as ymptoti- cally approach a linear current–velocity relation. Dashed lines show the asymptotic expansion and dotted curves show the series a bout j=0. We use the material parameters supplied in the (a) ﬁrst and (b) second column of Table I except that J=0. expected ranges of validity indicating the absence of Walke r breakdown in the numerical solution. The in-plane hard axis included in the magnetic anisotropy is appropriate for narrow ferromagnetic strips, which gene r- ally host Néel walls. Wider strips give Bloch walls,24and by making the necessary modiﬁcations to the above calcula- tions, we ﬁnd that in this case the domain wall velocity re- tains the qualitative features elucidated above. This is al so true for shape anisotropy-dominated strips, which host hea d- to-head walls. This shows that universal absence of Walker breakdown is a robust effect that does not depend on the de- tails of the ferromagnetic material , unlike other SOT effects studied previously.36This fact is also illustrated by the numer- ics. In Figure 3 (b) we present numerical results obtained for a Néel wall in a PMA ferromagnet with anisotropies weaker byan order of magnitude, weaker magnetic damping and much larger Rashba spin–orbit coupling and spin Hall angle in the adjacent heavy metal. The results are qualitatively simila r to those obtained in Figure 3 (a). III. COUPLED DOMAIN WALLS IN A SAF STRUCTURE We consider next an asymmetric stack of two ultrathin ferromagnets separated by an insulating spacer as shown in Figure 4 (a). We describe the dynamics of each of the ferro- magnets using separate LLG equations, but add to the free energy a coupling term, FIEC=/uni222B.dspdr1 m(1)/uni222B.dspdr2 m(2)J(r1−r2)/bracketleftBig m(1)(r1)·m(2)(r2)/bracketrightBig ,(13) representing the interlayer exchange (IEC). We assume that the IEC is local in the plane, J(r1−r2)=Jδ(x1−x2)δ(y1− y2). Equation ( 13) then represent the lowest order coupling proposed by Bruno.37 Following the same procedure as in the previous section we may now derive four coupled collective coordinate equa- tions. With an antiferromagnetic coupling the walls will have opposite topological charges, Q2=−Q1. Since a lo- cal IEC can only affect the chiralities, and not the proﬁles o f the walls, we can use the static solution derived previously , θ=2 arctan exp [Q(x−X)/λ], whereλ=/radicalbig A/Kzis the do- main wall width and Qis the topological charge. For a single wall the azimuthal angle φis given byφ=nπ.nis even if D<0andQ=+1, and nis odd if D<0andQ=−1. To limit the scope of the treatment, we consider only the case where D1andD2have the same sign, D1<0andD2<0. Then the DMI and the IEC cooperate to give the static solu- tionφ1=0(Q1=+1) andφ2=π(Q2=−1). Substituting this static solution into the LLG equations us - ingH0=Hxexgives the collective coordinate equations (1+α2)∝dotaccX1 λ=−γKy msin 2φ1+πγ(D1−Hxmλ) 2mλsinφ1+γJt2 2m/bracketleftBig αU(s)cos(φ1−φ2)+αW(s)+V(s)sin(φ1−φ2)/bracketrightBig −u(1−αβ) λ+π 2γ/bracketleftBig H(1) SH/parenleftBig α+β(1) SH/parenrightBig +H(1) R(1−αβ)/bracketrightBig cosφ1,(14) (1+α2)∝dotaccX2 λ=+γKy msin 2φ2+πγ(D2+Hxmλ) 2mλsinφ2−γJt1 2m/bracketleftBig αU(s)cos(φ1−φ2)+αW(s)−V(s)sin(φ1−φ2)/bracketrightBig −u(1−αβ) λ+π 2γ/bracketleftBig H(2) SH/parenleftBig α+β(2) SH/parenrightBig +H(2) R(1−αβ)/bracketrightBig cosφ2,(15) (1+α2)∝dotaccφ1=−αγKy msin 2φ1+παγ(D1−Hxmλ) 2mλsinφ1−γJt2 2m/bracketleftBig U(s)cos(φ1−φ2)+W(s)−αV(s)sin(φ1−φ2)/bracketrightBig −u(α+β) λ−π 2αγ/bracketleftBig H(1) SH/parenleftBig 1−αβ(1) SH/parenrightBig −H(1) R(α+β)/bracketrightBig cosφ1,(16) (1+α2)∝dotaccφ2=−αγKy msin 2φ2−παγ(D2+Hxmλ) 2mλsinφ2−γJt1 2m/bracketleftBig U(s)cos(φ1−φ2)+W(s)+αV(s)sin(φ1−φ2)/bracketrightBig +u(α+β) λ+π 2αγ/bracketleftBig H(2) SH/parenleftBig 1−αβ(2) SH/parenrightBig −H(2) R(α+β)/bracketrightBig cosφ2.(17)5 Figure 4. (a) Two ultrathin ferromagnets separated by an ins ulating spacer with heavy metal over- and underlayers. The ferromag nets are identical except for their thicknesses, but the different h eavy metals induce different DMIs and SOTs. (b) Dependence of the IEC ter ms V(s),U(s)andW(s)on the wall separation. where we have assumed that the bulk parameters of the two ferromagnets are equal and where sis the separation between the two walls, s=(X1−X2)/λ. The IEC terms are expressed using the three functions V(s),U(s)andW(s); V(s)=2scsch s, U(s)=2 csch s−2scoth scsch s, W(s)=2 coth s−2scsch2s. These functions are plotted in Figure 4 (b). Equations ( 14) and ( 16) reduce to equations ( 6) and ( 7) when J→0. To solve equations ( 14)–(17) numerically, we rescale the equations to obtain dimensionless variables. T he dimension of equations ( 14)–(17) isHz. A convenient scal- ing factor with the same dimensions is µ0γm. By dividing equations ( 14)–(17) byµ0γmwe get the rescaled variables ˜t=tµ0γm,˜Xi=Xi/λ,˜Hx=Hx/µ0m,˜Ky=Ky/µ0m2, ˜Di=Di/µ0m2λ,˜ti=ti/λ,˜J=Jλ/µ0m2and˜u=u/µ0γmλ. We solve the equations using an explicit fourth order Runge– Kutta scheme with adaptive stepsize control, implemented a s a Dormand–Prince pair.38 A. Universal Absence of Walker Breakdown in SAF structures For parameter values representative of a standard cobalt– nickel multilayer we obtain the current–velocity and curre nt– tilt relations shown in Figure 5 (a) and (b) for t1/t2=1in the case where only STTs are present and in the case where spin Hall torques are additionally present. We see that the prese nce of the IEC delays Walker breakdown when the wall is driven by ordinary STTs, but the subcritical differential velocit y re- mains unaffected. This can also be shown analytically by sol v- ing for the tilt angle of the wall as a function of current. Suc h a calculation shows that the tilt angle is suppressed by the IE C00.61.2 1.8 ·1014−6−303 currentj/bracketleftBig A/m2/bracketrightBigvelocity ˙X[km/s](a) STT, single STT, coupledSH, single SH, coupled 00.61.2 1.8 ·1014−2−1012 currentj/bracketleftBig A/m2/bracketrightBigtilt angle φ1[rad](b) 0.5 1 1.5 20.511.522.5 thickness ratio t1/t2velocity ˙X[km/s](c) 0.5 1 1.5 2024 thickness ratio t1/t2tiltφ[rad] 12.22.42.62.8 tiltφ2−φ1[rad]φ2 φ1(d) 0612 18 ·1013−20−10010 currentj/bracketleftBig A/m2/bracketrightBigvelocity ˙X[km/s](e) STT, single STT, coupledSH, single SH, coupled 0 0.30.60.9 ·1013−0.8−0.400.4 currentj/bracketleftBig A/m2/bracketrightBigvelocity ˙X[km/s](f) 0612 18 ·1013−2−1012 currentj/bracketleftBig A/m2/bracketrightBigtilt angle φ1[rad](g) 0.25 0.5 1 2 402468 thickness ratio t1/t2velocity ˙X[km/s](h) Figure 5. Domain wall dynamics in interlayer exchange coupl ed ferromagnets. (a) and (b) The IEC delays Walker breakdown fo r STT driving, but the subcritical differential velocity rem ains unaf- fected. With spin Hall torques the tilt angle stabilizes at a ﬁnite value, indicating universal absence of Walker breakdown. T he tilt an- gle approaches its limiting value more slowly in the presenc e of IEC. (c) and (d) The IEC gives the velocity a nonmonotonic thickne ss- dependence resulting in a peak close to t1/t2=1. [j=3 GA/cm2, corresponding to the dashed vertical line in (a).] We use the material parameters supplied in the ﬁrst column of Table I . (e)–(h) These re- sults are robust against a change in parameters to those in th e second column of Table I . (but the breakdown angle is still π/4). Back-substitution of this angle into the torque acting on the wall shows that this torque is independent of J, explaining why there is no change in the differential velocity. When spin Hall torques are included, the domain wall tilt levels off to a ﬁnite value and the current–velocity relatio n is6 Table I. Parameters used for the numerical solution of equat ions ( 14)– (17) and for analytical estimates in the text. parameter Co–Ni strong SOC Bi et al. unit gyromagnetic ratio γ−0.19−0.19−0.19 THz /T domain wall width λ 4 16 2 nm hard axis anisotropy Ky200 20 2 kJ /m3 saturation magn. m 1 1 1 .1 MA/m DM constant D −1.4−1.0−0.1 mJ/m2 Gilbert dampingα −0.25−0.1−0.5 spin-polarization P 0.5 0.5 0.5 nonadiabacity param. β 0.5 0.4 2 Rashba parameterαR 6.3 75 meVnm spin Hall angleθSH 0.1 0.2 0.12 spin Hallβ-termβSH 0.02 0.02 0.02 interlayer exchange Jt1t25 5 1 .5 mJ/m2 thickness t1 1.2 1.2 0.6 nm thickness t2 1.2 1.2 1.7 nm linear in the j→ ∞ limit. This shows that universal absence of Walker breakdown is also found in SAF structures. The effect of the IEC can be understood simply as a rescaling of the constant S1and the higher order constants S2,S3,... in the expansion ( 9), making the tilt angle approach its limiting value more slowly. Thus, the effect of the IEC on both the STT and spin Hall results is to suppress the domain wall tilt, as shown in Figure 5 (b). We note that the combination of spin Hall torques and IEC produces much higher domain wall ve- locities than in single ferromagnets for comparatively sma ll current densities.20 In a single ferromagnet the velocity of a wall driven by spin Hall torques decreases with tas1/t. When changing t2from t2=t1/2tot2=2t1in a SAF structure, we ﬁnd that the veloc- ity peaks close to t1/t2≈1, which maximizes the IEC torque [seeFigure 5 (c); the deviation from 1 is due to the DMI]. This can be understood by considering Figure 5 (d); at t1/t2≈1 the magnetizations in both layers are tilted in the ydirection. Increasing (decreasing) t2tot2=2t1(t2=t1/2) reduces (in- creases) H(2) SHand increases (reduces) H(1) IEC, thus(φ2−φ1)ap- proachesπand the IEC torque is reduced. Just as for the single ferromagnetic layer the results for th e coupled walls are robust against a change of parameters, as shown in Figure 5 (e)–(h). B. Novel Switching Behavior in SAF Structures Biet al.22have very recently demonstrated completely novel switching behavior in SAF structures. In single ferro - magnets, domain walls with one topological charge will trav el faster than those with the opposite topological charge if an in-plane magnetic ﬁeld is applied.39If the relative velocity is large enough the favored domains can overcome the destabi- lizing action of the current (see Refs 40–43) and merge.44–46 The favored magnetization direction is uniquely determine d by the spin Hall angle and the applied magnetic ﬁeld for a ﬁxed direction of the current. Biet al. observed this behavior−2−1 0 1 21416 velocity ˙X[m/s] −2−1 0 1 2−16016 in-plane magnetic ﬁeld Hx[T]rel. vel.[m/s] (+,−) (−,+) Figure 6. Qualitative reproduction of the experimental res ults of Bi et al.22The sign of the relative velocity of walls with (Q1,Q2)= (+1,−1)and(Q1,Q2)=(−1,+1)can be toggled only by changing the magnitude of the applied ﬁeld. We use the material parame ters supplied in the third column of Table I . in SAF structures for small in-plane ﬁelds, but by toggling b e- tween large and small values of the in-plane ﬁeld (same sign) , they were able to toggle the sign of the relative velocity of t he walls and thereby the favored magnetization direction. Usi ng material parameters that approximate the samples of Biet al. , our model is the ﬁrst to qualitatively reproduce this behavi or, as shown in Figure 6 . Under an in-plane ﬁeld in the range 0.3 Tto1.4 T, walls with (Q1,Q2)=(+1,−1)travel faster than walls with (Q1,Q2)=(−1,+1)and ‘up’ magnetization is favored. If the ﬁeld is increased beyond 1.4 T, the relative velocity changes sign, and ‘down’ magnetization is favored . (The offset from zero is due to the DMI.) IV . CONCLUSION We have shown that complete suppression of Walker break- down is possible in a wide range of domain wall systems driven by spin–orbit torques, including head-to-head wall s in soft magnets, Bloch and Néel walls in perpendicular anisotropy magnets, in the presence of the Dzyaloshinskii– Moriya interaction and in coupled domain walls in syn- thetic antiferromagnets. For spin–orbit torques other tha n pure Rashba spin–orbit torques this leads to a linear curren t– velocity relation instead of a saturation of the velocity fo r large currents. In combination with interlayer exchange couplin g, spin–orbit torque driven domain wall motion in synthetic an ti- ferromagnets gives rise to novel switching behavior and ver y high domain wall velocities. ACKNOWLEDGMENTS Funding via the “Outstanding Academic Fellows” program at NTNU, the COST Action MP-1201, the NV Faculty, and the Research Council of Norway Grants No. 216700 and No. 240806, is gratefully acknowledged. We thank Morten Amundsen for very useful discussions of the numerics.7 ∗vetle.k.risinggard@ntnu.no 1I. M. Miron, P.-J. Zermatten, G. Gaudin, S. Auffret, B. Rodma cq, and A. Schuhl, Phys. Rev. Lett. 102, 137202 (2009) . 2I. M. Miron, G. Gaudin, S. Auffret, B. Rodmacq, A. Schuhl, S. Pizzini, J. V ogel, and P. Gambardella, Nat. Mater. 9, 230 (2010) . 3I. M. Miron, T. Moore, H. Szambolics, L. D. Buda-Prejbeanu, S. Auffret, B. Rodmacq, S. Pizzini, J. V ogel, M. Bonﬁm, A. Schuhl, and G. Gaudin, Nat. Mater. 10, 419 (2011) . 4S. Emori, U. Bauer, S.-M. Ahn, E. Martinez, and G. S. D. Beach, Nat. Mater. 12, 611 (2013) . 5K.-S. Ryu, L. Thomas, S.-H. Yang, and S. Parkin, Nat. Nanotechnol. 8, 527 (2013) . 6P. P. J. Haazen, E. Murè, J. H. Franken, R. Lavrijsen, H. J. M. Swagten, and B. Koopmans, Nat. Mater. 12, 299 (2013) . 7S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004) . 8D. C. Ralph and M. D. Stiles, J. Magn. Magn. Mater. 320, 1190 (2008) . 9A. Brataas, A. D. Kent, and H. Ohno, Nat. Mater. 11, 372 (2012) . 10N. L. Schryer and L. R. Walker, J. Appl. Phys. 45, 5406 (1974) . 11K. Ando, S. Takahashi, K. Harii, K. Sasage, J. Ieda, S. Maekaw a, and E. Saitoh, Phys. Rev. Lett. 101, 036601 (2008) . 12A. Manchon and S. Zhang, Phys. Rev. B 78, 212405 (2008) . 13K.-W. Kim, S.-M. Seo, J. Ryu, K.-J. Lee, and H.-W. Lee, Phys. Rev. B 85, 180404 (2012) . 14A. Chernyshov, M. Overby, X. Liu, J. K. Furdyna, Y . Lyanda- Geller, and L. P. Rokhinson, Nat. Phys. 5, 656 (2009) . 15I. E. Dzyaloshinskii, J. Exp. Theor. Phys. 5, 1259 (1957) . 16T. Moriya, Phys. Rev. Lett. 4, 228 (1960) ; Phys. Rev. 120, 91 (1960) . 17A. Fert, Mater. Sci. Forum 59–60 , 439 (1990) . 18H. Saarikoski, H. Kohno, C. H. Marrows, and G. Tatara, Phys. Rev. B 90, 094411 (2014) . 19S. Lepadatu, H. Saarikoski, R. Beacham, M. J. Benitez, T. A. Moore, G. Burnell, S. Sugimoto, D. Yesudas, M. C. Wheeler, J. Miguel, S. S. Dhesi, D. McGrouther, S. McVitie, G. Tatara, and C. H. Marrows, arXiv:1604.07992 . 20S.-H. Yang, K.-S. Ryu, and S. Parkin, Nat. Nanotechnol. 10, 221 (2015) . 21R. Tomasello, V . Puliaﬁto, E. Martinez, A. Manchon, M. Ricci , M. Carpentieri, and G. Finocchio, arXiv:1610.00894 . 22C. Bi, H. Almasi, K. Price, T. Newhouse-Illige, M. Xu, S. R. Allen, X. Fan, and W. Wang, Phys. Rev. B 95, 104434 (2017) . 23L. D. Landau and E. M. Lifshitz, Phys. Zeitschrift der Sowje- tunion 8, 153 (1935); Ukr. J. Phys. 53, 14 (2008) ; T. Gilbert, IEEE Trans. Magn. 40, 3443 (2004) .24R. C. O’Handley, Modern magnetic materials. Principles and ap- plications (John Wiley & Sons, 2000). 25K. M. D. Hals and A. Brataas, Phys. Rev. B 88, 085423 (2013) ; Phys. Rev. B 91, 214401 (2015) . 26R. R. Birss, Symmetry and Magnetism , 1st ed., edited by E. P. Wohlfarth, Selected Topics in Solid State Physics, V ol. 3 (N orth- Holland Publising Company, 1964). 27J. Kim, J. Sinha, M. Hayashi, M. Yamanouchi, S. Fukami, T. Suzuki, S. Mitani, and H. Ohno, Nat. Mater. 12, 240 (2012) . 28K.-S. Ryu, S.-H. Yang, L. Thomas, and S. S. P. Parkin, Nat. Commun. 5, 3910 (2014) . 29X. Fan, H. Celik, J. Wu, C. Ni, K.-J. Lee, V . O. Lorenz, and J. Q. Xiao, Nat. Commun. 5, 3042 (2014) . 30J. Shibata, G. Tatara, and H. Kohno, J. Phys. D. Appl. Phys. 44, 384004 (2011) . 31G. Beach, M. Tsoi, and J. Erskine, J. Magn. Magn. Mater. 320, 1272 (2008) . 32J. Linder and M. Alidoust, Phys. Rev. B 88, 064420 (2013) . 33M. Stier, M. Creutzburg, and M. Thorwart, Phys. Rev. B 90, 014433 (2014) . 34O. Boulle, L. D. Buda-Prejbeanu, E. Jué, I. M. Miron, and G. Gaudin, J. Appl. Phys. 115, 17D502 (2014) . 35P.-B. He, H. Yan, M.-Q. Cai, and Z.-D. Li, Europhys. Lett. 114, 67001 (2016) . 36A. V . Khvalkovskiy, V . Cros, D. Apalkov, V . Nikitin, M. Kroun bi, K. A. Zvezdin, A. Anane, J. Grollier, and A. Fert, Phys. Rev. B 87, 020402 (2013) . 37P. Bruno, Phys. Rev. B 52, 411 (1995) . 38J. R. Dormand and P. J. Prince, J. Comput. Appl. Math. 6, 19 (1980) . 39S.-G. Je, D.-H. Kim, S.-C. Yoo, B.-C. Min, K.-J. Lee, and S.-B . Choe, Phys. Rev. B 88, 214401 (2013) . 40J. Shibata, G. Tatara, and H. Kohno, Phys. Rev. Lett. 94, 076601 (2005) . 41Y . Nakatani, J. Shibata, G. Tatara, H. Kohno, A. Thiaville, a nd J. Miltat, Phys. Rev. B 77, 014439 (2008) . 42J. Torrejon, F. Garcia-Sanchez, T. Taniguchi, J. Sinha, S. M itani, J.-V . Kim, and M. Hayashi, Phys. Rev. B 91, 214434 (2015) . 43T. Taniguchi, S. Mitani, and M. Hayashi, Phys. Rev. B 92, 024428 (2015) . 44G. Yu, P. Upadhyaya, K. L. Wong, W. Jiang, J. G. Alzate, J. Tang , P. K. Amiri, and K. L. Wang, Phys. Rev. B 89, 104421 (2014) . 45K. Garello, C. O. Avci, I. M. Miron, M. Baumgartner, A. Ghosh, S. Auffret, O. Boulle, G. Gaudin, and P. Gambardella, Appl. Phys. Lett. 105, 212402 (2014) . 46J.-C. Rojas-Sánchez, P. Laczkowski, J. Sampaio, S. Collin, K. Bouzehouane, N. Reyren, H. Jaffrès, A. Mougin, and J.-M. George, Appl. Phys. Lett. 108, 082406 (2016) .
1807.11806v1.Spin_absorption_at_ferromagnetic_metal_platinum_oxide_interface.pdf	arXiv:1807.11806v1 [cond-mat.mes-hall] 31 Jul 2018Spin absorption at ferromagnetic-metal/platinum-oxide i nterface Akio Asami,1Hongyu An,1Akira Musha,1Makoto Kuroda,1and Kazuya Ando1,∗ 1Department of Applied Physics and Physico-Informatics, Ke io University, Yokohama 223-8522, Japan (Dated: August 1, 2018) We investigate the absorption of a spin current at a ferromag netic-metal/Pt-oxide interface by measuring current-induced ferromagnetic resonance. The s pin absorption was characterized by the magnetic damping of the heterostructure. We show that the ma gnetic damping of a Ni 81Fe19ﬁlm is clearly enhanced by attaching Pt-oxide on the Ni 81Fe19ﬁlm. The damping enhancement is disappeared by inserting an ultrathin Cu layer between the N i81Fe19and Pt-oxide layers. These results demonstrate an essential role of the direct contact between the Ni 81Fe19and Pt-oxide to induce sizable interface spin-orbit coupling. Furthermor e, the spin-absorption parameter of the Ni81Fe19/Pt-oxide interface is comparable to that of intensively st udied heterostructures with strong spin-orbit coupling, such as an oxide interface, topologic al insulators, metallic junctions with Rashba spin-orbit coupling. This result illustrates strong spin- orbit coupling at the ferromagnetic-metal/Pt- oxide interface, providing an important piece of informati on for quantitative understanding the spin absorption and spin-charge conversion at the ferromagneti c-metal/metallic-oxide interface. I. INTRODUCTION An emerging direction in spintronics aims at discover- ing novel phenomena and functionalities originatingfrom spin-orbit coupling (SOC)1. An important aspect of the SOC is the ability to convert between charge and spin currents. The charge-spin conversion results in the gen- eration of spin-orbit torques in heterostructures with a ferromagnetic layer, enabling manipulation of magneti- zation2–4. Recent studies have revealed that the oxida- tion of the heterostructure strongly inﬂuences the gen- eration of the spin-orbit torques. The oxidation of the ferromagnetic layer alters the spin-orbit torques, which cannot be attributed to the bulk spin Hall mechanism5–7. The oxidation of a nonmagnetic layer in the heterostruc- ture also oﬀers a route to engineer the spin-orbit devices. Demasius et al. reported a signiﬁcant enhancement of the spin-torque generation by incorporating oxygen into tungsten, which is attributed to the interfacial eﬀect8. The spin-torque generation eﬃciency was found to be signiﬁcantly enhanced by manipulating the oxidation of Cu, enablingto turn the light metal into aneﬃcient spin- torque generator, comparable to Pt9. We also reported that the oxidation of Pt turns the heavy metal into an electrically insulating generatorof the spin-orbit torques, which enables the electrical switching of perpendicular magnetization in a ferrimagnet sandwiched by insulating oxides10. These studies have provided valuable insights into the oxide spin-orbitronics and shown a promising way to develop energy-eﬃcient spintronics devices based on metal oxides. The SOC in solids is responsible for the relaxation of spins, as well as the conversion between charge and spin currents. The spin relaxation due to the bulk SOC of metals and semiconductors has been studied both ex- perimentally and theoretically11–14. The inﬂuence of the SOC at interfaces on spin-dependent transport has also been recognized in the study of giant magnetoresistance (GMR). The GMR in Cu/Pt multilayers in the current-perpendicular-to-plane geometry indicated that there must be a signiﬁcant spin-memory loss due to the SOC at the Cu/Pt interfaces15. The interface SOC also plays a crucial role in recent experiments on spin pumping. The spin pumping refers to the phenomenon in which precessing magnetization emits a spin current to the sur- rounding nonmagnetic layers12. When the pumped spin current is absorbed in the nonmagnetic layer due to the bulk SOC or the ferromagnetic/nonmagnetic interface due to the interface SOC, the magnetization damping of the ferromagnetic layer is enhanced because the spin- current absorption deprives the magnetization of the an- gularmomentum16. Althoughthedampingenhancement induced by the spin pumping has been mainly associated with the spin absorption in the bulk of the nonmagnetic layer, recent experimental and theoretical studies have demonstrated that the spin-current absorption at inter- faces also provides a dominant contribution to the damp- ing enhancement17. Since the absorption of a spin cur- rent at interfaces originates from the SOC, quantifying the damping enhancement provides an important infor- mation for fundamental understanding of the spin-orbit physics. In this work, we investigate the absorption of a spin current at a ferromagnetic-metal/Pt-oxide interface. We show that the magnetic damping of a Ni 81Fe19(Py) ﬁlm is clearly enhanced by attaching Pt-oxide, Pt(O), despite the absence of the absorption of the spin current in the bulk of the Pt(O) layer. The damping enhancement dis- appears by inserting an ultrathin Cu layer between the Pyand Pt(O)layers. This resultindicates that the direct contact between the ferromagnetic metal and Pt oxide is essential to induce the sizable spin-current absorption, or the interface SOC. We further show that the strength of the damping enhancement observedfor the Py/Pt(O) bi- layer is comparable with that reported for other systems with strong SOC, such as two-dimensional electron gas (2DEG) at an oxide interface and topological insulators.2 II. EXPERIMENTAL METHODS Three sets of samples, Au/SiO 2/Py, Au/SiO 2/Py/Pt(O) and Au/SiO 2/Py/Cu/Pt(O), were deposited on thermally oxidized Si substrates (SiO2) by RF magnetron sputtering at room tempera- ture. To avoid the oxidation of the Py or Cu layer, we ﬁrst deposited the Pt(O) layer on the SiO 2substrate in a mixed argon and oxygen atmosphere. After the Pt(O) deposition, the chamber was evacuated to 1 ×10−6Pa, and then the Py or Cu layer was deposited on the top of the Pt(O) layer in a pure argon atmosphere. For the Pt(O) deposition, the amount of oxygen gas in the mixture was ﬁxed as 30%, in which the ﬂow rates of argon and oxygen were set as 7.0 and 3.0 standard cubic centimeters per minute (sccm), respectively. The SiO 2 layer was deposited from a SiO 2target in the pure argon atmosphere. The ﬁlm thickness was controlled by the deposition time with a precalibrated deposition rate. We measured the magnetic damping using current- induced ferromagnetic resonance (FMR). For the fabri- cation of the devices used in the FMR experiment, the photolithography and lift-oﬀ technique were used to pat- terntheﬁlmsintoa10 µm×40µmrectangularshape. A blanket Pt(O) ﬁlm on a 1 cm ×1 cm SiO 2substrate was fabricatedforthecompositionconﬁrmationbyx-raypho- toelectron spectroscopy (XPS). We also fabricated Pt(O) single layer and SiO 2/Py/Pt(O) multilayer ﬁlms with a Hall bar shape to determine the resistivity of the Pt(O) and Py using the four-probe method. The resistivity of Pt(O) (6.3 ×106µΩ cm) is much larger than that of Py (106µΩ cm). Because of the semi-insulating nature of the Pt(O) layer, we neglect the injection of a spin cur- rent into the Pt(O) layer from the Py layer; only the Py/Pt(O) interface can absorb a spin current emitted from the Py layer. Transmission electron microscopy (TEM) was used to directly observe the interface and multilayer structure of the SiO 2/Py/Pt(O) ﬁlm. All the measurements were conducted at room temperature. III. RESULTS AND DISCUSSION Figure 1(a) exhibits the XPS spectrum of the Pt(O) ﬁlm. Previous XPS studies on Pt(O) show that bind- ing energies of the Pt 4 f7/2peak for Pt, PtO and PtO 2 are around 71.3, 72.3 and 74.0 eV, respectively18. Thus, the Pt 4 f7/2peak at 72.3 eV in our Pt(O) ﬁlm indi- cates the formation of PtO. By further ﬁtting the XPS spectra, we conﬁrm that the Pt(O) ﬁlm is composed of a dominant structure of PtO with a minor portion of PtO2. Figure 1(b) shows the cross-sectional TEM image of the SiO 2(4 nm)/Py(8 nm)/Pt(O)(10 nm) ﬁlm. As can be seen, continuous layer morphology with smooth and distinct interfaces is formed in the multilayer ﬁlm. Al- though we deposited the Py layer on the Pt(O) layer to avoid the oxidation of the Py, it might still be possible that the Py layer is oxidized by the Pt(O) layer. There- FIG. 1. (a) The XPS spectrum of the Pt(O) ﬁlm. The gray curve is the experimental data, and the red ﬁttingcurve is th e merged PtO and PtO 2peaks. (b) The cross-sectional TEM image of the SiO 2(4 nm)/Ni 81Fe19(8 nm)/Pt(O)(10 nm) ﬁlm. fore, we measured the resistance of the Au/SiO 2/Py and Au/SiO 2/Py/Pt(O) samples used in the FMR experi- ment. The resistance of both samples show the same value (60 Ω). Furthermore, as described in the follow- ing section, the saturation magnetization for each device was obtained by using Kittel formula (0.746 T and 0.753 T for the Au/SiO 2/Py and Au/SiO 2/Py/Pt(O), respec- tively). The only 1% diﬀerence indicates that the minor oxidationofthePylayerdue tothe presenceofthe Pt(O) layer can be neglected. Next, we conduct the FMR experiment to investigate the absorption and relaxation of spin currents induced by the spin pumping. Figure 2(a) shows a schematic of the experimental setup for the current-induced FMR. We applied an RF current to the device, and an in-plane external magnetic ﬁeld µ0Hwas swept with an angle of 45ofrom the longitudinal direction. The RF charge currentﬂowingintheAulayergeneratesanOerstedﬁeld, which drives magnetization precession in the Py layer at the FMR condition. The magnetization precession induces an oscillation of the resistance of the device due to the anisotropic magnetoresistance (AMR) of the Py layer. We measured DC voltage generated by the mixing of the RF current and the oscillating resistance using a bias tee. Figures 2(b), 2(c) and 2(d) show the FMR spec- tra for the Au/SiO 2/Py, Au/SiO 2/Py/Pt(O) and Au/SiO 2/Py/Cu/Pt(O) ﬁlms, respectively. For the FMRmeasurement, asmallRFcurrentpower P= 5mW was applied. Around P= 5 mW, the FMR linewidth is independent of the RF power as shown in the inset to Fig. 3(a). This conﬁrms that the measured linewidth is unaﬀected by additional linewidth broadening due to nonlinear damping mechanisms and Joule heating. As shown in Fig. 2, clear FMR signals with low noise are obtained, allowing us to precisely ﬁt the spectra and ex- tract the magnetization damping for the three samples. Here, the mixing voltage due to the FMR, Vmix, is ex-3 FIG. 2. Schematic illustration of the experimental setup for the current-induced FMR. Mis the magnetization in the Py layer. The FMR spectra of the (b) Py(9 nm), (c) Py(9 nm)/Pt(O)(7.3 nm), and (d) Py(9 nm)/Cu(3.6 nm)/Pt(O)(7.3 nm) ﬁlms by changing the RF current fre- quency from 4 to 10 GHz. All the ﬁlms are capped with 3 nm-thick SiO 2and 10 nm-thick Au layers. The RF current power was set as 5 mW. The schematic illustrations of the corresponding ﬁlms are also shown. pressed as Vmix=Vsym(µ0∆H)2 (µ0H−µ0HR)2+(µ0∆H)2 +Vasyµ0∆H(µ0H−µ0HR) (µ0H−µ0HR)2+(µ0∆H)2,(1) whereµ0∆Handµ0HRare the spectral width and res- onance ﬁeld, respectively19.VsymandVasymare the magnitudes of the symmetric and antisymmetric com- ponents. The symmetric and antisymmetric components arise from the spin-orbit torques and Oersted ﬁeld. In the devices used in the present study, the Oersted ﬁeld created by the top Au layer dominates the RF eﬀective ﬁelds acting on the magnetization in the Py layer [see also Fig. 2(a)]. The large Oersted ﬁeld enables the elec- tric measurement of the FMR even in the absence of the spin-orbit torques in the Au/SiO 2/Py ﬁlm. The damping constant αof the Py layer in the Au/SiO 2/Py, Au/SiO 2/Py/Pt(O) and Au/SiO 2/Py/Cu/Pt(O) ﬁlms can be quantiﬁed byﬁtting the RF current frequency fdependence of the FMR spectral width µ0∆Husing µ0∆H=µ0∆Hext+2πα γf, (2) where ∆ Hextandγare the inhomogeneous linewidth broadening of the extrinsic contribution and gyromag- netic ratio, respectively19,20. Figure 3(a) shows the f dependence of the FMR linewidth µ0∆H, determined by ﬁtting the spectra shown in Fig. 2 using Eq. (1). As shown in Fig. 3(a), the frequency dependence of the linewidth is well ﬁtted by Eq. (2). Importantly, the slope of thefdependence of µ0∆Hfor the Py/Pt(O) ﬁlm is clearly larger than that for the Py and Py/Cu/Pt(O) ﬁlms. This indicates larger magnetic damping in the Py/Pt(O) ﬁlm. By using Eq. (2), we determined the damping constant αas 0.0126, 0.0169 and 0.0124 for the Py, Py/Pt(O) and Py/Cu/Pt(O)ﬁlms, respectively. The diﬀerence in αbetween the Py and Py/Cu/Pt(O)ﬁlms is vanishingly small, which is within an experimental error. In contrast, the damping of the Py/Pt(O) ﬁlm is clearly larger than that of the other ﬁlms, indicating an essen- tial role of the Py/Pt(O) interface on the magnetization damping. The larger magnetic damping in the Py/Pt(O) ﬁlm demonstrates an important role of the direct contact be- tween the Py and Pt(O) layers in the spin-current ab- sorption. If the bottom layers inﬂuence the magnetic properties of the Py layer, the diﬀerence in the mag- netic properties can also result in the diﬀerent magnetic damping in the Au/SiO 2/Py, Au/SiO 2/Py/Pt(O) and Au/SiO 2/Py/Cu/Pt(O) ﬁlms. However, we have con- ﬁrmed that the diﬀerence in the magnetic damping is not caused by diﬀerent magnetic properties of the Py layer. In Fig. 3(b), we plot the RF current frequency fdepen- dence of the resonance ﬁeld µ0HR. As can be seen, the fdependence of µ0HRis almost identical for the diﬀer- ent samples, indicating the minor change of the magnetic propertiesofthe Pylayerdueto the diﬀerent bottomlay- ers. In fact, by ﬁtting the experimental data using Kittel formula21, 2πf/γ=/radicalbig µ0HR(µ0HR+µ0Ms), the satura- tion magnetizationis obtainedto be µ0Ms= 0.746, 0.753 and 0.777 T for the Py, Py/Pt(O) and Py/Cu/Pt(O) ﬁlms, respectively. The minor diﬀerence ( <5%) in the saturation magnetization indicates that the larger damp- ing of the Py/Pt(O) ﬁlm cannot be attributed to possi- ble diﬀerent magnetic properties of the Py layer. Thus, the larger magnetic damping of the Py/Pt(O) ﬁlm can only be attributed to the eﬃcient absorption of the spin current at the interface. Notable is that the additional damping due to the spin-current absorption disappears by inserting the 3.6 nm-thick Cu layer between the Py and Pt(O) layers. Here, the thickness of the Cu layer is muchthinnerthanitsspin-diﬀusionlength( ∼500nm)22, allowing us to neglect the relaxation of the spin current in the Cu layer. This indicates that the directcontactbe- tween the Py and Pt(O) layersis essential for the absorp- tion of the spin current at the interface, or the interface SOC.4 0∆H (mT) µ 3 6 9 12 0246 f (GHz) (a) (b) 8 610 0 0HR (mT) µf (GHz) 4 100150 50 Py/Cu/Pt(O) Py/Pt(O)Py Py/Cu/Pt(O) Py/Pt(O) Py 036 10 010 110 2 P (mW) 0∆H (mT) µ FIG. 3. (a) The RF current frequency fdependence of the half-width at half-maximum µ0∆Hfor the Py, Py/Pt(O) and Py/Cu/Pt(O) samples. The solid lines are the linear ﬁt to the experimental data. The inset shows RF current power Pdependence of µ0∆Hfor the Py ﬁlm at f= 7 GHz. (b) The RF currentfrequency fdependenceof the resonance ﬁeld µ0HRfor the three samples. The solid curves are the ﬁtting result using the Kittel formula. TABLE I. The summarized spin-absorption parameter Γ 0η in diﬀerent material systems. In order to directly compare this work with previous works, we used International Sys- tem of Units. We used the magnetic permeability in vac- uumµ0= 4π×10−7H/m. ∆ αand Γ 0ηfor the Sn 0.02- Bi1.08Sb0.9Te2S/Ni81Fe19is the values at T <100 K. Heterostructure ∆ αΓ0η[1/m2] Ref. Bi/Ag/Ni 80Fe20 0.015 8.7 ×1018[25] Bi2O3/Cu/Ni 80Fe20 0.0045 1.5 ×1018[26] SrTiO 3/LaAlO 3/Ni81Fe19 0.0013 2.3 ×1018[27] Pt(O)/Ni 81Fe19 0.0044 2.3 ×1018This work α-Sn/Ag/Fe 0.022 1.2 ×1019[28] Sn0.02-Bi1.08Sb0.9Te2S/Ni81Fe190.013 1.4 ×1019[29] Bi2Se3/Ni81Fe19 0.0013 2.5 ×1018[30] To quantitatively discuss the spin absorption at the Py/Pt(O)interfaceand comparewith othermaterialsys- tems, we calculate the spin absorption parameters. In a model of the spin pumping where the interface SOC is taken into account, the additional damping constant is expressed as23 ∆α=gµBΓ0 µ0Msd/parenleftbigg1+6ηξ 1+ξ+η 2(1+ξ)2/parenrightbigg .(3) Here,g= 2.11 is the gfactor24,µB= 9.27×10−24Am2 is the Bohr magneton, dis the thickness of the Py layer, andΓ0isthemixingconductanceattheinterface. ξisthe back ﬂow factor; no backﬂow refers to ξ= 0 and ξ=∞indicates that the entire spin current pumped into the bulk ﬂows back across the interface. ηis the parameter that characterizes the interface SOC. For the Py/Pt(O) ﬁlm,ξcan be approximated to be ∞because of the spin pumping into the bulk of the semi-insulating Pt(O) layer can be neglected. Thus, Eq. (3) can be simpliﬁed as ∆α=6gµBΓ0η µ0Msd. (4) Here, 6Γ 0ηcorresponds to the eﬀective spin mixing conductance g↑↓ eﬀ. From the enhancement of magnetic damping ∆ α, we obtain Γ 0η= 2.3×1018m−2for the Py/Pt(O) ﬁlm. We further compared this value with Γ0ηfor other systems where eﬃcient interface charge- spin conversion has been reported. As shown in Table I, the spin-absorption parameter Γ 0ηof the Py/Pt(O) ﬁlm is comparable with that of the heterostructures with the strong SOC, such as the 2DEG at an oxide interface, topological insulators, as well as metal/oxide and metal- lic junctions with the Rashba SOC. This result therefore demonstrates the strong SOC at the Py/Pt(O) interface. IV. CONCLUSIONS In summary, we have investigated the spin-current ab- sorption and relaxation at the ferromagnetic-metal/Pt- oxide interface. By measuring the magnetic damping for the Py, Py/Pt(O)and Py/Cu/Pt(O)structures, we show that the direct contact between Py and Pt(O) is essential for the absorption of the spin current, or the sizable in- terface SOC. Furthermore, we found that the strength of the spin-absorption parameter at the Py/Pt(O) interface is comparable to the value for intensively studied het- erostructureswithstrongSOC,suchas2DEGatanoxide interface, topological insulators, metallic junction with Rashba SOC. The comparable value with these material systems illustrates the strong SOC at the ferromagnetic- metal/Pt-oxide interface. This indicates that the oxida- tion of heavy metals provides a novel approach for the development of the energy-eﬃcient spintronics devices based the SOC. ACKNOWLEDGMENTS This work was supported by JSPS KAKENHI Grant Numbers 26220604, 26103004, the Asahi Glass Founda- tion, JGC-SScholarshipFoundation, andSpintronicsRe- search Network of Japan (Spin-RNJ). H.A. is JSPS In- ternational Research Fellow (No. P17066) and acknowl- edges the support from the JSPS Fellowship (Grant No. 17F17066). ∗ando@appi.keio.ac.jp1A. Soumyanarayanan, N. Reyren, A. Fert, and C. Panagopoulos, Nature (London) 539, 509 (2016).5 2P. Gambardella and I. M. Miron, Philos. Trans. R. Soc. London A 369, 3175 (2011). 3X.Qiu, Z.Shi, W.Fan, S.Zhou, andH.Yang,Adv.Mater. 30, 1705699 (2018). 4A. Brataas, A. D. Kent, and H. Ohno, Nat. Mater. 11, 372 (2012). 5X. Qiu, K. Narayanapillai, Y. Wu, P. Deorani, D.-H. Yang, W.-S.Noh, J.-H. Park, K.-J.Lee, H.-W.Lee, andH.Yang, Nat. Nanotechnol. 10, 333 (2015). 6Y. Hibino, T. Hirai, K. Hasegawa, T. Koyama, and D. Chiba, Appl. Phys. Lett. 111, 132404 (2017). 7T. Gao, A. Qaiumzadeh, H. An, A. Musha, Y. Kageyama, J. Shi, and K. Ando, Phys. Rev. Lett. 121, 017202 (2018). 8K.-U. Demasius, T. Phung, W. Zhang, B. P. Hughes, S.-H. Yang, A. Kellock, W. Han, A. Pushp, and S. S. Parkin, Nat. Commun. 7, 10644 (2016). 9H. An, Y. Kageyama, Y. Kanno, N. Enishi, and K. Ando, Nat. Commun. 7, 13069 (2016). 10H. An, T. Ohno, Y. Kanno, Y. Kageyama, Y. Monnai, H. Maki, J. Shi, and K. Ando, Sci. Adv. 4, eaar2250 (2018). 11S. Mizukami, Y. Ando, and T. Miyazaki, Phys. Rev. B 66, 104413 (2002). 12Y. Tserkovnyak, A. Brataas, and G. E. Bauer, Phys. Rev. B66, 224403 (2002). 13L. Chen, F. Matsukura, and H. Ohno, Nat. Commun. 4, 2055 (2013). 14H. Nakayama, K. Ando, K. Harii, T. Yoshino, R. Taka- hashi, Y. Kajiwara, K.-i. Uchida, Y. Fujikawa, and E. Saitoh, Phys. Rev. B 85, 144408 (2012). 15H. Kurt, R. Loloee, K. Eid, W. Pratt Jr, and J. Bass, Appl. Phys. Lett. 81, 4787 (2002).16Y. Tserkovnyak, A. Brataas, and G. E. Bauer, Phys. Rev. Lett.88, 117601 (2002). 17J. R. S´ anchez, L. Vila, G. Desfonds, S. Gambarelli, J. At- tan´ e, J. De Teresa, C. Mag´ en, and A. Fert, Nat. Commun. 4, 2944 (2013). 18Q. Wang, X. Lu, Q. Xin, and G. Sun, Chin. J. of Catal. 35, 1394 (2014). 19D.Fang, H.Kurebayashi, J. Wunderlich,K.V` yborn` y,L.P. Zˆ arbo, R. Campion, A. Casiraghi, B. Gallagher, T. Jung- wirth, and A. Ferguson, Nat. Nanotechnol. 6, 413 (2011). 20B. Heinrich, J. Cochran, and R. Hasegawa, J. Appl. Phys. 57, 3690 (1985). 21C. Kittel, Physical Review 73, 155 (1948). 22H. Wang, C. Du, Y. Pu, R. Adur, P. C. Hammel, and F. Yang, Phys. Rev. Lett. 112, 197201 (2014). 23K. Chen and S. Zhang, Phys. Rev. Lett. 114, 126602 (2015). 24J. M. Shaw, H. T. Nembach, T. J. Silva, and C. T. Boone, J. Appl. Phys. 114, 243906 (2013). 25M. Jungﬂeisch, W. Zhang, J. Sklenar, W. Jiang, J. Pear- son, J. Ketterson, and A. Hoﬀmann, Phys. Rev. B 93, 224419 (2016). 26S. Karube, K. Kondou, and Y. Otani, Appl. Phys. Express 9, 033001 (2016). 27E. Lesne, Y. Fu, S. Oyarzun, J. Rojas-S´ anchez, D. Vaz, H. Naganuma, G. Sicoli, J.-P. Attan´ e, M. Jamet, E. Jacquet, et al., Nat. Mater. 15, 1261 (2016). 28J.-C. Rojas-S´ anchez, S. Oyarz´ un, Y. Fu, A. Marty, C. Vergnaud, S. Gambarelli, L. Vila, M. Jamet, Y. Oht- subo, A. Taleb-Ibrahimi, et al., Phys. Rev. Lett. 116, 096602 (2016). 29A. Nomura, N. Nasaka, T. Tashiro, T. Sasagawa, and K. Ando, Phys. Rev. B 96, 214440 (2017). 30P. Deorani, J. Son, K. Banerjee, N. Koirala, M. Brahlek, S. Oh, and H. Yang, Phys. Rev. B 90, 094403 (2014).
0904.1185v2.Curvature_enhanced_spin_orbit_coupling_in_a_carbon_nanotube.pdf	arXiv:0904.1185v2 [cond-mat.mes-hall] 3 Aug 2009Curvature-enhanced spin-orbit coupling in a carbon nanotu be Jae-Seung Jeong and Hyun-Woo Lee PCTP and Department of Physics, Pohang University of Scienc e and Technology, Pohang, 790-784, Korea (Dated: November 1, 2018) Structure of the spin-orbit coupling varies from material t o material and thus ﬁnding the correct spin-orbit coupling structure is an important step towards advanced spintronic applications. We show theoretically that the curvature in a carbon nanotube g enerates two types of the spin-orbit coupling, one of which was not recognized before. In additio n to the topological phase-related contribution of the spin-orbit coupling, which appears in t he oﬀ-diagonal part of the eﬀective Dirac Hamiltonian of carbon nanotubes, there is another contribu tion that appears in the diagonal part. The existence of thediagonal term can modify spin-orbitcou pling eﬀects qualitatively, an example of whichistheelectron-hole asymmetricspinsplittingobser vedrecently, andgeneratefourqualitatively diﬀerent behavior of energy level dependence on parallel ma gnetic ﬁeld. It is demonstrated that the diagonal term applies to a curved graphene as well. This resu lt should be valuable for spintronic applications of graphitic materials. PACS numbers: I. INTRODUCTION Graphitic materials such as carbon nanotubes (CNTs) and graphenes are promising materials for spintronic ap- plications. Various types of spintronic devices are re- ported such as CNT-based three terminal magnetic tun- nel junctions1, spin diodes2, and graphene-based spin valves3. Graphitic materials are believed to be excellent spin conductors4. The hyperﬁne interaction of electron spins with nuclear spins is strongly suppressed since12C atoms do not carry nuclear spins. It is estimated that the spin relaxation time in a CNT5and a graphene6is limited by the spin-orbit coupling (SOC). Carbon atoms are subject to the atomic SOC Hamil- tonianHso. In an ideal ﬂat graphene, the energy shift caused byHsois predicted to be ∼10−3meV7,8. Re- cently it is predicted8,9that the geometric curvature can enhance the eﬀective strength of the SOC by orders of magnitude. This mechanism applies to a CNT and also to a graphene which, in many experimental situa- tions, exhibits nanometer-scale corrugations10. There is also a suggestion?that artiﬁcial curved structures of a graphene may facilitate device applications. A recent experiment12on ultra-clean CNTs measured directly the energy shifts caused by the SOC, which provides an ideal opportunity to test theories of the curvature-enhanced SOC in graphitic materials. The measured shifts are in order-of-magnitude agreement with the theoretical predictions8,9, conﬁrming that the curvature indeed enhances the eﬀective SOC strength. The experiment revealed discrepancies as well; While ex- isting theories predict the same strength of the SOC for electrons and holes, which is natural considering that both the conduction and valence bands originate from the sameπorbital, the experiment found considerable asymmetry in the SOC strength between electrons and holes. Thiselectron-holeasymmetryimpliesthatexisting theoriesoftheSOCingraphiticmaterialsareincomplete. In this paper, we show theoretically that in additionto eﬀective SOC in the oﬀ-diagonal part of the eﬀective Dirac Hamiltonian, which was reported in the existing theories8,9, there exists an additional type of the SOC that appears in the diagonal part both in CNTs and curved graphenes. It is demonstrated that the combined action ofthe twotypes ofthe SOC producesthe electron- hole asymmetry observed in the CNT experiment12and gives rise to four qualitatively diﬀerent behavior of en- ergy level dependence on magnetic ﬁeld parallel to the CNT axis. This paper is organized as follows. In Sec. II, we show analytical expressions of two types of the eﬀective SOC in a CNT and then explain how the electron-hole asym- metric spin splitting can be generated in semiconducting CNTs generically. Section III describes the second-order perturbation theory that is used to calculate the eﬀective SOC, and tight-binding models of the atomic SOC and geometric curvature. Section IV reports four distinct en- ergy level dependence on magnetic ﬁeld parallel to the CNT axis. We conclude in Sec. V with implications of our theory on curved graphenes and a brief summary. II. EFFECTIVE SPIN-ORBIT COUPLING IN A CNT We begin our discussion by presenting the ﬁrst main result for a CNT with the radius Rand the chiral angle θ(0≤θ≤π/6, 0(π/6) for zigzag (armchair) CNTs). We ﬁnd that when the two sublattices AandBof the CNT are used as bases, the curvature-enhanced eﬀective SOC Hamiltonian Hsocnear the K point with Bloch momen- tumKbecomes HK soc=/parenleftbigg (δ′ K/R)σy(δK/R)σy (δ∗ K/R)σy(δ′ K/R)σy/parenrightbigg , (1) whereσyrepresents the real spin Pauli matrix along the CNT axis. The pseudospin is deﬁned to be up (down) when an electron is in the sublattice A(B). Here the oﬀ- diagonal term that can be described by a spin-dependent2 topological phase are reported in Refs.8,9but the diago- nal term was not recognized before. Expressions for the parameters δKandδ′ Kare given by13 δK R=λsoa(εs−εp)(Vπ pp+Vσ pp) 12√ 3Vσsp2e−iθ R(2) and δ′ K R=λsoaVπ pp 2√ 3(Vσpp−Vπpp)cos3θ R, (3) whereλso∼12meV14is the atomic SOC constant, ais the lattice constant 2 .49˚A, andεs(p)is the atomic energy for thes(p) orbital. Here, Vσ spandVπ(σ) pprepresent the coupling strengths in the absence of the curvature for the σcoupling between nearest neighbor sandporbitals and theπ(σ) coupling between nearest neighbor porbitals, respectively. Note that the \|δ′ K\|has theθ-dependence, whose implication on the CNT energy spectrum is ad- dressed in Sec. IV. For K′point with K′=−K,HK′ socis given by Eq. (1) with δKandδ′ Kreplaced by δK′=−δ∗ K andδ′ K′=−δ′ K, respectively. Implications of the diagonal term of the SOC be- come evident when Eq. (1) is combined with the two- dimensional Dirac Hamiltonian HDiracof the CNT. For astate nearthe Kpoint with the Blochmomentum K+k [k= (kx,ky),\|k\| ≪ \|K\|],HDiracbecomes15 HK Dirac=/planckover2pi1vF/parenleftbigg 0e−iθ(kx−iky) e+iθ(kx+iky) 0/parenrightbigg ,(4) wherevFis the Fermi velocity and the momentum com- ponentkxalong the circumference direction satisﬁes the quantization condition kx= (1/3R)νfor a (n,m) CNT withn−m= 3q+ν(q∈Zandν=±1,0) andθ= tan−1[√ 3m/(2n+m)]. For a semiconducting ( ν=±1) E E ky ky ky0 0 0E (a) (b) (c) 2δ′ K/R−2νRe[δKeiθ]/R−2δ′ K/R−2νRe[δKeiθ]/R −2νRe[δKeiθ]/R −2νRe[δKeiθ]/R FIG. 1: (Color online) Schematic diagram of the lowest con- duction (red, E >0) and highest valence (blue, E <0) band positions of asemiconducting CNT predictedby HK Dirac+HK soc for (a)δK=δ′ K=0, (b)δK/negationslash=0,δ′ K=0, and (c) δK/negationslash=0,δ′ K/negationslash=0. In (c), the conduction or valence band has larger spin split- ting depending on the sign of ν. Arrows (green) show the spin direction along the CNT. The expressions for the energy level spacing are also provided. When they are negative, the positions of the two spin-split bands should be swapped.CNT, thediagonalizationof HK Dirac+HK socresultsindiﬀer- ent spin splittings [Fig. 1(c)] of −2δ′ K/R−2νRe[δKeiθ]/R and 2δ′ K/R−2νRe[δKeiθ]/Rfor the conduction and va- lence bands, respectively. This explains the electron-hole asymmetryobservedin the recent experiment12. Here we remark that neither the oﬀ-diagonal ( δK) nor the diago- nal(δ′ K) termofthe SOCalonecangeneratetheelectron- hole asymmetrysince the twospin splittings can diﬀer by sign at best, which actually implies the same magnitude of the spin splitting (see Fig. 1 for the sign convention). Thus the interplay of the two types is crucial for the asymmetry. III. THEORY AND MODEL We calculate the δKandδ′ Kanalytically using degener- atesecond-orderperturbationtheoryandtreatingatomic SOC and geometric curvature as perturbation. For sim- plicity, we evaluate δKandδ′ Kin the limit k= 0. Al- though this limit is not strictly valid since k=0 does not generally satisfy the quantization condition on kx, one may still take this limit since the dependence of δKand δ′ Konkis weak. An electron at the K point is described by the total Hamiltonian HK,(0)+Hso+Hc, whereHc describes the curvature eﬀects and HK,(0)describes the πandσbands in the absence of both HsoandHc. The πband eigenstates of HK,(0)are given by \|ΨK,(0) ↑(↓)/angbracketright=1√ 2/parenleftbig νe−iθ/vextendsingle/vextendsingleψK A/angbracketrightbig ±/vextendsingle/vextendsingleψK B/angbracketrightbig/parenrightbig χ↑(↓)(5) with the corresponding eigenvalues EK,(0) ↑(↓)≡E(0)= 0. Here\|ΨK,(0) ↑(↓)/angbracketrightwith the upper (lower) sign amounts to the k= 0 limit of the eigenstate at the the conduction band bottom (valenceband top). χ↑(↓)denotes the eigenspinor ofσy.\|ψK A(B)/angbracketright=1√ N/summationtext r=rA(B)eiK·r\|pr z/angbracketrightis the orbital projection of \|ΨK,(0) ↑(↓)/angbracketrightinto the sublattice A(B),\|pr z/angbracketrightrep- resents the pzorbital at the atomic position r, and thez axis is perpendicular to the CNT surface. WhenHsoandHcare treated as weak perturbations, the ﬁrst order contribution Hsoto the eﬀective SOC van- ishes since it causes the inter-band transition (Fig. 3) to theσband8. The next leading order contribution to the eﬀective SOC comes from the following second order perturbation Hamiltonian HK,(2)16, HK,(2)=HcPK E(0)−HK,(0)Hso+H.c.,(6) where the projection operator PKis deﬁned by PK≡1−/summationtext α=↑,↓\|ΨK,(0) α/angbracketright/angbracketleftΨK,(0) α\|. Another spin-dependent second order term Hso[PK/(E(0)−HK,(0))]Hso17is smaller than Eq. (6) (by two orders of magnitude for a CNT with R∼2.5nm), and thus ignored. Then the second order3 energy shift EK,(2) ↑(↓)is given by18 EK,(2) ↑=/angbracketleftbig ψK A/vextendsingle/vextendsingleHK,(2)/vextendsingle/vextendsingleψK A/angbracketrightbig ±νRe/bracketleftBig /angbracketleftψK A\|HK,(2)\|ψK B/angbracketrighteiθ/bracketrightBig EK,(2) ↓=−EK,(2) ↑, (7) where the upper (lower) sign applies to the energy shift of the conduction band bottom (valence band top) and /angbracketleftψK A\|HK,(2)\|ψK A/angbracketright=/angbracketleftψK B\|HK,(2)\|ψK B/angbracketrightis used. Then by com- paringEK,(2) ↑(↓)with Fig. 1, one ﬁnds δK R=/angbracketleftbig ψK A/vextendsingle/vextendsingleHK,(2)/vextendsingle/vextendsingleψK B/angbracketrightbig ,δ′ K R=/angbracketleftbig ψK A/vextendsingle/vextendsingleHK,(2)/vextendsingle/vextendsingleψK A/angbracketrightbig .(8) Note thatδKandδ′ Kare related to pseudospin-ﬂipping and pseudospin-conserving processes, respectively. To evaluate Eq. (8), one needs explicit expressions for Hso,Hc, andHK,(0).Hsois given by λso/summationtext rLr·Sr7, whereLrandSrare respectively the atomic orbital and spin angular momentum of an electron at a carbon atom r. The tight-binding Hamiltonian of the Hsocan be written8asHso= (λso/2)/summationtext r=rA/B(cz† r−cx r+−cz† r+cx r−+ icz† r+cy r−+icz† r−cy r++icy† r+cx r+−icy† r−cx r−) + H.c., where cx r+(−),cy r+(−), andcz r+(−)denote the annihilation opera- tors for\|pr x/angbracketrightχ+(−),\|pr y/angbracketrightχ+(−), and\|pr z/angbracketrightχ+(−). Hereχ+(−) denotes the eigenspinor of σz(+/−for outward/inward). For later convenience, we express χ+(−)in term ofχ↑(↓) to obtain a expression for Hso, Hso=λso 2/summationdisplay r=rA/B/bracketleftBig i/parenleftBig cz† r↓cx r↓−cz† r↑cx r↑/parenrightBig +/parenleftBig e−iϕcz† r↑cy r↓−eiϕcz† r↓cy r↑/parenrightBig +i/parenleftBig e−iϕcy† r↑cx r↓+eiϕcy† r↓cx r↑/parenrightBig/bracketrightBig +H.c..(9) For the curvature Hamiltonian Hc, we retain only the leading order term in the expansion in terms of a/R. Up y xy′ ω3 ω2ω1 a B3AB1 B2a2 a1 θϕzxz′ x x′ FIG. 2: (Color online) Two-dimensional honeycomb lattice structure. x(y) is the coordinate around (along) the CNT with chiral vector na1+ma2≡(n,m) and chiral angle θ. ωj(j=1,2,3), the length between yaxis passing Aatom and its parallel (red dashed) line is related with ξjbyξj≈ωj/(2R) [Eq. (10)]. The coordinates for the CNT is illustrated on the right. Here, x=ϕR.to the ﬁrst order in a/R,Hcreduces toHπσ c, Hπσ c=/summationdisplay rA3/summationdisplay j=1/summationdisplay α=↑,↓/bracketleftBig Sj/parenleftBig cz† rAαcs Bjα+cs† rAαcz Bjα/parenrightBig +Xj/parenleftBig cz† rAαcx Bjα−cx† rAαcz Bjα/parenrightBig (10) +Yj/parenleftBig cz† rAαcy Bjα−cy† rAαcz Bjα/parenrightBig/bracketrightBig +H.c., whererAis a lattice site in the sublattice Aand its three nearest neighbor sites in the sublattice Bare rep- resented by Bj(j=1,2,3) (Fig. 2). Here Sj,Xj,Yjare proportional to a/Rand denote the curvature-induced coupling strengths of s,px,pyorbitals with a nearest neighborpzorbital. Their precise expressions that can be determined purely from geometric considerations, are given bySj=ξj˜Sj,Xj=ξj˜Xj, andYj=ξj˜Yjwith ξ1≈a/(2√ 3R)sinθ,ξ2≈a/(2√ 3R)sin(π/3−θ), and ξ3≈a/(2√ 3R)sin(π/3+θ)(Fig. 2). Here ˜S1=Vσ spsinθ, ˜S2=Vσ spcos/parenleftBigπ 6+θ/parenrightBig , ˜S3=Vσ spcos/parenleftBigπ 6−θ/parenrightBig , ˜X1=−Vσ ppsin2θ−Vπ pp−Vπ ppcos2θ, ˜X2=−Vσ ppsin2/parenleftBigπ 3−θ/parenrightBig −Vπ pp−Vπ ppcos2/parenleftBigπ 3−θ/parenrightBig , ˜X3=Vσ ppsin2/parenleftBigπ 6−θ/parenrightBig +Vπ pp+Vπ ppcos2/parenleftBigπ 6−θ/parenrightBig , ˜Y1= sin(2θ)Vπ pp−Vσ pp 2, ˜Y2= sin/parenleftbigg 2θ−2π 3/parenrightbiggVπ pp−Vσ pp 2, ˜Y3= sin/parenleftBig 2θ−π 3/parenrightBigVπ pp−Vσ pp 2. (11) Lastly, for the factor HK,(0), we use the Slater-Koster parametrization19for nearest-neighbor hopping. In σ band calculation, s,px, andpyorbitals are used as basis. Combined eﬀects of the three factors Hso,PK/(E(0)− HK,(0)),Hcare illustrated in Fig. 3. The real spin de- pendence arises solely from Hso, which generates the factorσy20. For the pseudospin, the combined eﬀect ofHsoandHcis to ﬂip the pseudospin. When they are combined with the pseudospin conserving part of PK/(E(0)−HK,(0)), one obtains the pseudospin ﬂip- ping process [Eq. (8)] determining δK. In addition, PK/(E(0)−HK,(0)) contains the pseudospin ﬂipping part, which is natural since states localized in one particular sublattice are not eigenstates of HK,(0). When the pseu- dospin ﬂipping part of PK/(E(0)−HK,(0)) is combined withHsoandHc, one obtains the pseudospin conserving process [Eq. (8)] determining δ′ K. The signs of δKeiθandδ′ K/cos3θare negative. We ﬁnd\|(δ′ K/cos3θ)/δK\|= 4.5 for tight-binding parame- ters in Ref.21. Thusδ′ Kis of the same order as δK22,4 HsoHπσ c Hso πbandHπσ cσbandA B ABPK E(0)−HK,(0)PK E(0)−HK,(0)PK E(0)−HK,(0) FIG. 3: (Color online) Schematic diagram ofthe second order transition process generated by HK,(2)[Eq. (6)]. Pseudospin transitions (between the sublattices AandB) and interband transitions (between πandσbands) are illustrated. which is understandable since pseudospin ﬂipping terms inE(0)−HK,(0)(with amplitudes Vσ pp,Vσ sp) are compara- ble in magnitude to pseudospin conserving terms (with amplitudes E(0)−εs(p)). IV. BEHAVIOR IN A MAGNETIC FIELD Next we examine further implications of our result in view of the experiment12, where the conduction band bottom and valence band top positions of semiconduct- ing CNTs ( ν=±1) are measured as a function of the magnetic ﬁeld Bparallel to the CNT axis. We ﬁnd that theθdependence [Eq. (3)] of δ′ Khas interesting impli- cations. When cos3 θis suﬃciently close to 0 (close to armchair-type), \|δ′ K\|is smaller than \|δKeiθ\|. The predic- tion of our theory in this situation is shown in Figs. 4(a) and (b). Note that the spin splitting of both the conduc- tion and valence bands becomes smaller as the energy Eincreases. On the other hand, when cos3 θis suﬃ- ciently close to 1 (close to zigzag-type), \|δ′ K\|is larger than\|δKeiθ\|. In this situation [Figs. 4(c) and (d)], the energy dependence of either valence or conduction band is inverted; For ν=+1(−1), the spin splitting of the va- lence (conduction) band becomes largerasthe the energy increases. Combined with the electron-prevailing [Figs. 4(a) and (c) forν=+1] vs. hole-prevailing [Figs. 4(b) and (d) for ν=−1] asymmetries in the zero-ﬁeld splitting, one then ﬁnds that there exist four distinct patterns of Evs.B diagram, which is the second main result of this paper. Among these 4 patterns, only the pattern in Fig. 4(a) is observed in the experiment12, which measured two CNT samples. We propose further experiments to test the ex- istence of the other three patterns. Here we remark that although Eqs. (1), (2), (3) are demonstrated so far for semiconducting CNTs, they hold for metallic CNTs ( ν=0) as well. For armchair CNTs with cos3θ=0,δ′ Kbecomes zero and the spin splitting isdetermined purely by δK. For metallic but non-armchair CNTs, ﬁnding implications of Eq. (1) is somewhat tech- nical since the curvature-induced minigap appears near the Fermi level23. Our calculation for (37 ,34)(cos3θ≈0) and (60,0)(cos3θ=1) CNTs including the minigap eﬀect indicates that they show behaviors similar to Fig. 4(b) and (d), respectively. Thus nominally metallic CNTs ex- hibit spin splitting patterns of ν=−1 CNTs. -0.0810-0.08050.08050.0810 -0.5 0.0 0.5E ( eV ) B ( T ) (a) ∆so ∆so= 0.09 meV = 0.03 meVν = +1 K K′ K K′ -0.5 0.0 0.5-0.0820-0.08150.08150.0820 E ( eV ) B ( T ) (b) ∆so ∆so= 0.03 meV = 0.09 meVν = −1 -0.083-0.0820.0820.083 -1.0 0.0 1.0E ( eV ) B ( T ) (c) ∆so ∆so= 0.34 meV = 0.22 meVν = +1 K K′ K K′ -1.0 0.0 1.0-0.082-0.0810.0810.082 E ( eV ) B ( T ) (d) ∆so ∆so= 0.21 meV = 0.33 meVν = −1 FIG. 4: (Color online) Calculated energy spectrum of the conduction band bottom (red, E >0) and valence band top (blue,E <0) near K (solid lines) and K′(dashed lines) points in semiconducting CNTs with R≈2.5nm as a function of magnetic ﬁeld Bparallel to the CNT axis. The chiral vectors for each CNT are (a) (38,34), (b) (39,34), (c) (61,0), and (d) (62,0), respectively. Arrows (green) show spin direction a long theCNTaxis and∆ sodenotesthezero-ﬁeld splitting. Assum- ingky=0, the energy Eincluding the SOC, the Aharonov- Bohm ﬂux15φAB=BπR2, and the Zeeman coupling eﬀects is, E=±/planckover2pi1vFp (kx+(1/R)(φAB/φ0))2+EK(K′),(2) ↑(↓)+(g/2)µBτ/bardblB, with upper (lower) sign applying to the conduction (valence ) band.φ0=hc/\|e\|,τ/bardbl=+1(−1) forχ↑(↓),vF=−aVπ pp√ 3/2, andg= 212. For estimation of EK(K′),(2) ↑(↓), we use tight- binding parameters in Ref.21;Vσ ss=−4.76eV,Vσ sp=4.33eV, Vσ pp=4.37eV,Vπ pp=−2.77eV,εs=−6.0eV, andεp=022.5 V. DISCUSSION AND SUMMARY Lastly we discuss brieﬂy the eﬀective SOC in a curved graphene10. Unlike CNTs, there can be both convex- shaped and concave-shaped curvatures in a graphene. We ﬁrst address the convex-shaped curvatures. When the local structure of a curved graphene has two princi- pal curvatures, 1 /R1and 1/R2with the corresponding binormal unit vectors n1andn2, each principal curva- ture 1/Ri(i=1,2) generates the eﬀective SOC, Eq. (1), withσyreplaced by σ·niandRbyRi. The correspond- ingδiandδ′ ivalues are given by Eqs. (2) and (3) with θ replaced by θi, whereθiis the chiral angle with respect toni. Thus the diagonal term of the eﬀective SOC is again comparable in magnitude to the oﬀ-diagonal term. For the concave-shaped curvatures, we ﬁnd that the two types of the SOC become −δiand−δ′ iwithθi, respec- tively. We expect that this result may be relevant for the estimation of the spin relaxation length in graphenes6 and may provide insights into unexplained experimen- tal data in graphene-based spintronic systems24. We also remark that the eﬀective SOC in a graphene may be spatially inhomogeneous since the local curvature of the nanometer-scale corrugations10is not homogeneous, whose implications go beyond the scope of this paper. In summary, we have demonstrated that the interplayofthe atomic SOC and the curvature generatestwotypes of the eﬀective SOC in a CNT, one of which was not recognized before. Combined eﬀects of the two types of the SOC in CNTs explain recently observedelectron-hole asymmetric spin splitting12and generates four qualita- tively diﬀerent types of energy level dependence on the parallel magnetic ﬁeld. Our result may have interesting implications for graphenes as well. Note added.– While we werepreparingour manuscript, we became aware of a related paper25. However the ef- fective Hamiltonian [Eq. (1)] for the SOC and the four distinct types of the magnetic ﬁeld dependence (Fig. 4) are not reported in the work. Acknowledgments We appreciate Philp Kim for his comment for the curved graphenes. We acknowledge the hospitality of Hyunsoo Yang and Young Jun Shin at National Uni- versity of Singapore, where parts of this work were per- formed. We thank Seung-Hoon Jhi, Woojoo Sim, Seon- Myeong Choi and Dong-Keun Ki for helpful conversa- tions. This work was supported by the KOSEF (Ba- sic Research Program No. R01-2007-000-20281-0) and BK21. 1S. Sahoo, T. Kontos, J. Furer, C. Hoﬀmann, M. Gr¨ aber, A. Cuttet, and C. Sch¨ onenberger, Nat. Phys. 1, 99 (2005). 2C. A. Merchant and N. Markovi´ c, Phys. Rev. Lett. 100, 156601 (2008). 3N. Tombros, C. Jozsa, M. Popinciuc, H. T. Jonkman, and B. J. van Wees, Nature (London) 448, 571 (2007); S. Cho, Y.-F. Chen, and M. S. Fuhrer, Appl. Phys. Lett. 91, 123105 (2007). 4K. Tsukagoshi, B. W. Alphenaar, and H. Ago, Nature (London) 401, 572 (1999). 5D. V. Bulaev, B. Trauzettel, and D. Loss, Phys. Rev. B 77, 235301 (2008). 6D. Huertas-Hernando, F. Guinea, and A. Brataas, arXiv:0812.1921 (unpublished). 7H. Min, J. E. Hill, N. A.Sinitsyn, B. R.Sahu, L. Kleinman, and A. H. MacDonald, Phys. Rev. B 74, 165310 (2006); Y. Yao, F. Ye, X.-L. Qi, S.-C. Zhang, and Z. Fang, ibid. 75, 041401(R) (2007). 8D. Huertas-Hernando, F. Guinea, and A. Brataas, Phys. Rev. B74, 155426 (2006). 9T. Ando, J. Phys. Soc. Jpn. 69, 1757 (2000); A. De- Martino, R. Egger, K. Hallberg, and C. A. Balseiro, Phys. Rev. Lett. 88, 206402 (2002). 10J. C. Meyer, A. K. Geim, M. I. Katsnelson, K. S. Novoselov, T. J. Booth, and S. Roth, Nature (London) 446, 60 (2007); E. Stolyarova, K. T. Rim, S. Ryu, J. Maultzsch, P. Kim, L. E. Brus, T. F. Heinz, M. S. Hy- bertsen, and G. W. Flynn, Proc. Natl. Acad. Sci. U.S.A. 104, 9209 (2007); V. Geringer, M. Liebmann, T. Echter- meyer, S. Runte, M. Schmidt, R. R¨ uckamp, M. C. Lemme, and M. Morgenstern, Phys. Rev. Lett. 102, 076102 (2009);A. K. Geim, Science 324, 1530 (2009). 11V. M. Pereira and A. H. Castro Neto, Phys. Rev. Lett. 103, 046801 (2009). 12F. Kuemmeth, S. Ilani, D. C. Ralph, and P. L. McEuen, Nature (London) 452, 448 (2008). 13The corresponding expression in Ref.8is slightly diﬀerent from Eq. (2) since the σband is treated in diﬀerent ways. When a few minor mistakes in Ref.8are corrected, the two expressions result in similar numerical values. 14J. Serrano, M. Cardona, and T. Ruf, Solid State Commun. 113, 411 (2000). 15J. Ajiki and T. Ando, J. Phys. Soc. Jpn. 62, 1255 (1993). 16Leonard I. Schiﬀ, Quantum Mechanics (McGraw-Hill, New York, 1968). 17Hc[PK/(E(0)−HK,(0))]Hcis spin-independent and thus ig- nored [see Eq. (10)]. 18For the K′point,EK′,(2) ↑=/angbracketleftψK′ A\|HK′,(2)\|ψK′ A/angbracketright±νRe [/angbracketleftψK′ A \|HK′,(2)\|ψK′ B/angbracketrighte−iθ],EK′,(2) ↓=−EK′,(2) ↑with upper (lower) sign for the conduction band bottom (valence band top). 19J. C. Slater and G. F. Koster, Phys. Rev. 94, 1498 (1954). 20The last four terms of Hsoin Eq. (9), which do not com- mute withσy, do not contribute to the eﬀective SOC near the Fermi energy due to the factor e±iϕ8. 21J. W. Mintmire and C. T. White, Carbon 33, 893 (1995). 22Using other sets of tight-binding parameters [D. Tom´ anek and M. A. Schluter, Phys. Rev. Lett. 67, 2331 (1991); R. Saito, M. Fujita, G. Dresselhaus, and M. S. Dresselhaus, Phys. Rev. B 46, 1804 (1992)] does not change results qualitatively. 23C. L. Kane and E. J. Mele, Phys. Rev. Lett. 78, 19326 (1997); L.YangandJ.Han, ibid.85, 154(2000); A.Kleiner andS. Eggert, Phys. Rev.B 63, 073408 (2001); J.-C. Char- lier, X. Blase, and S. Roche, Rev. Mod. Phys. 79, 677 (2007). 24See for instance, W. Han, W. H. Wang, K. Pi, K. M. Mc-Creary, W. Bao, Y. Li, F. Miao, C. N. Lau, and R. K. Kawakami, Phys. Rev. Lett. 102, 137205 (2009). 25L. Chico, M. P. L´ opez-Sancho, and M. C. Mu˜ noz, Phys. Rev. B79, 235423 (2009).
1302.1063v3.Inertial_effect_on_spin_orbit_coupling_and_spin_transport.pdf	Inertial eect on spin orbit coupling and spin transport B. Basuand Debashree Chowdhuryy1 1Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B.T.Road, Kolkata 700 108, India We theoretically study the renormalization of inertial eects on the spin dependent transport of conduction electrons in a semiconductor by taking into account the interband mixing on the basis of ~k:~ pperturbation theory. In our analysis, for the generation of spin current we have used the extended Drude model where the spin orbit coupling plays an important role. We predict enhancement of the spin current resulting from the rerormalized spin orbit coupling eective in our model in cubic and non cubic crystal. Attention has been paid to clarify the importance of gauge elds in the spin transport of this inertial system. A theoretical proposition of a perfect spin lter has been done through the Aharonov-Casher like phase corresponding to this inertial system. For a time dependent acceleration, eect of ~k:~ pperturbation on the spin current and spin polarization has also been addressed. Furthermore, achievement of a tunable source of polarized spin current through the non uniformity of the inertial spin orbit coupling strength has also been discussed. PACS numbers: 72.25.-b, 85.75.-d, 71.70.Ej I. INTRODUCTION In semiconductor band structure spin-orbit coupling (SOC), which originates from the relativistic coupling of spin and orbital motion of electrons, plays a very important role from the perspective of spin Hall eect. Understanding the eect of SOI is indispensable in the study of spincurrent , a ow of spins. Although spin current can be induced easily, detection and control of spin current is a challenging research area both for theoretical and experimental physicists and has attracted a lot of attention in the eld of spintronics [1{3]. Spintronics aims to use the spin properties of electrons along with the charge degrees of freedom and has emerged as the most pursued area in condensed matter physics and nanotechnology. In this regard, the theoretical prediction of the spin Hall eect (SHE) [4] and its application to spintronics has seen considerable advancement. This eect is observed experimentally in semiconductors [5, 6] and metals[7]. Though studies on the inertial eect of electrons has a long standing history [8{11] but the contribution of the spin-orbit interaction (SOI) in accelerating frames has not much been addressed in the literature [12, 13]. Recently, a theory has been proposed [14, 15] describing the direct coupling of mechanical rotation and spin where the generation of spin current arising from rotational motion has been predicted. Inclusion of the inertial eects in semiconductors can open up some fascinating phenomena, yet not addressed. So it is appealing to investigate how the inertial eect aects some aspects of spin transport in semiconductors. In addition, the role of SOI in connection to spin Hall eect may inspire one to study the gauge theory of this inertial spin orbit Hamiltonian. The spin dynamics of the semiconductor is in uenced by the ~k:~ pperturbation theory as the band structure of a semiconductor in the vicinity of the band edges can be very well described by the ~k:~ pmethod. On the basis of ~k:~ p perturbation theory, by taking into account the interband mixing, one can reveal many characteristic features related to spin dynamics. In this paper, we theoretically investigate the generation of spin current in a solid on the basis of ~k:~ pperturbation [6] with a generalized spin orbit Hamiltonian which includes the inertial eect due to acceleration. The generation of spin current is studied in the extended Drude model framework, where the spin orbit coupling has played an important role. It is shown in our present paper that spin current appearing due to the combined action of the external electric eld, crystal eld and the induced inertial electric eld via the total eective spin-orbit interaction is enhanced by the interband mixing of the conduction and valence band states. We have also studied the Aharonov-Casher like phase which corresponds to the eective SOI present in the model. Through the interplay of Aharonov-Bohm phase ( AB) and Aharonov-Casher ( AC) phases, we are able to propose a perfect spin lter for the accelerating system. Also by taking into consideration of a special prole of the acceleration in a trilayer system, we can set up a tunable spin lter. Renormalization of the spin current and spin polarization for the time dependent acceleration has also been investigated. Here we consider the ~k:~ pperturbation in the 8 8 Kane model and write the total Hamiltonian including the inertial eect due to acceleration. Electronic address: sribbasu@gmail.com yElectronic address:debashreephys@gmail.comarXiv:1302.1063v3 [cond-mat.mes-hall] 20 May 20132 The paper is organized as follows. In Section II we write the total Hamiltonian of the 8 8 Kane model with ~k:~ pperturbation including the eect of acceleration. Section III deals with the generation of the spin current and conductivity in the semiconductors with cubic and non-cubic symmetry. The eect of time dependent acceleration on spin current and conductivity is discussed in section IV. The details on the gauge theory of our model, particularly theACphase, perfect spin lter and tunable spin lter is narrated in section V. Finally we conclude with section VI. II. INERTIAL SPIN ORBIT HAMILTONIAN AND ~k:~ pMETHOD We start with the Dirac Hamiltonian for a particle with charge eand massmin an arbitrary non-inertial frame with constant linear acceleration ~ aand without rotation which is given by [11], HI=mc2+c :(~ pe~A c)! +1 2c" (~ a:~ r)((~ pe~A c):~ )((~ pe~A c):~ )(~ a:~ r)# +m(~ a:~ r) +eV(~ r); (1) where the subscript Iin Hamiltonian (1) is due to the eect of inertia. Applying a series of Foldy-Wouthuysen (FW) transformations [16, 17] on the Hamiltonian(1 ) we can write the Pauli-Schrodinger Hamiltonian for the two component electron wave function in the low energy limit as HFW= mc2+(~ pe~A c)2 2m! +eV(~ r) +m(~ a:~ r)e~ 2mc~ :~Be~ 4m2c2~ :(~E~ p) +~ 4mc2~ :(~ a~ p) (2) where~Eand~Bare the external electric and magnetic eld respectively. In the right hand side of Hamiltonian (2), the third term is an inertial potential term arising due to the acceleration ~ a. This potential V~ a(~ r) =m e~ a:~ rinduces an electric eld ~E~ a=m e~ a[12, 14]. The induced electric eld ~E~ aproduces an inertial SOI term (sixth term in the right hand side of (2)) apart from the SOI term due to the external electric eld (fth term in the right hand side of (2)). The Hamiltonian (2) thus can be rewritten in terms of ~E~ aandV~ a(~ r) as HFW= mc2+(~ pe~A c)2 2m! +e(V(~ r)Va(~ r))e~ 2mc~ :~Be~ 4m2c2~ :((~EE~ a)~ p): (3) In the above calculations we have neglected the1 c4terms and the terms due to red shift eect of kinetic energy. The generalized spin-orbit interactione~ 4m2c2~ : (~E~E~ a)~ p eective in our inertial system plays a signicant role in our analysis. We are interested in an eective Hamiltonian describing the motion of electrons in a solid incorporating the inertial eect due to acceleration. It is known that the physical parameters present in any Hamiltonian in vacuum are renormalized when considered in a solid. In an inertial frame, such renormalization eects in a crystalline solid is generally studied in the framework of ~k:~ pperturbation theory using the Bloch eigenstates. One can renormalize the eect of acceleration on the basis of ~k:~ pperturbation and the 8 8 Kane model [18]. The basic idea of the Kane model is that the band edge eigenstates constitute a complete basis and to obtain the eigenstates away from the band edge the wave function is expanded in the band edge states, which gives rise to an 88 band Hamiltonian. Bands that are far away in energy can be neglected. In presence of magnetic eld the crystal momentum is given by ~~k=~ pq~A:The~k:~ pmethod leads to high-dimensional Hamiltonians, for example, an 8 8 matrix for the Kane model [6]. To this end, we start with a Hamiltonian of the well known 8 8 Kane model which takes into account the ~k:~ p coupling between the 6conduction band and 8and 7valance bands which is given by H88=0 @H6c6cH6c8vH6c7v H8v6cH8v8vH8v7v H7v6cH7v8vH7v7v1 A (4) =0 B@(Ec+eVtot)I2p 3P~T:~k Pp 3~ :~kp 3P~Ty:~k (Ev+eVtot)I4 0 Pp 3~ :~k 0 (Ev4 0+eVtot)I21 CA (5)3 Here,Vtot=V(~ r)Va(~ r); EcandEvare the energies at the conduction and valence band edges respectively. 40 is the spin orbit gap, P is the Kane momentum matrix element which couples slike conduction bands with plike valence bands. This Kane Momentum matrix is almost constant for group III to V semiconductors, whereas 40and EG=EcEvvaries with materials. The ~Tmatrices are given as Tx=1 3p 2 p 3 0 1 0 01 0p 3 ; Ty=i 3p 2p 3 0 1 0 0 1 0p 3 ;Tz=p 2 3 0 1 0 0 0 0 1 0 (6) andI2;I4are unit matrices of size 2 and 4 respectively. It may be noted here that as the eect of rotation [14] is not considered, the crystal momentum used in ~k:~ p perturbation is not modied in our model. The eect of acceleration changes the electric potential V(~ r) as well as the electric eld ~E. In our framework, the total potential and total electric eld have been modied as Vtot(~ r) and ~Etotrespectively. The Hamiltonian (5) can now be reduced to an eective Hamiltonian of the conduction band electron states [6] in presence of acceleration as Hkp=P2 32 EG+1 EG+40 ~k2+eVtot(~ r)P2 31 EG1 (EG+40)ie ~~ :(~k~k)+eP2 31 E2 G1 (EG+40)2 ~ :(~k~Etot) (7) In our analysis the derivation of ~k:~ pperturbed Hamiltonian of the accelerated system is carried out by using ~Etotand Vtot:The total Hamiltonian for the conduction band electrons including the eect of acceleration is then given by, Htot=~2~k2 2m+eVtot(~ r) + (1 +g 2)B~ :~B+e(+)~ :(~k~Etot); (8) where1 m=1 m+2P2 3~2 2 EG+1 EG+40 is the eective mass and ~Etot=~rVtot(~ r) =~E~Ea;is the eective total electric eld of the inertial system and =~2 4m2c2is the spin orbit coupling strength as considered in vacuum. Furthermore, the perturbation parameters gandare given by g=4m ~2P2 31 EG1 EG+40 = +P2 31 E2 G1 (EG+40)2 (9) Specically, the parameter gis related to the renormalized Zeeman coupling strength, whereas is responsible for the renormalization of spin orbit coupling. Now one can rewrite the Hamiltonian as Htot=~2k2 2m+eVtot+ (1 +g 2)B~ :~B+eeff~ :(~k~Etot); (10) whereeff=+is the eective SO coupling. We shall note in due course that the parameter eff= (+), which comes into play due to the interband mixing on the basis of ~k:~ pperturbation theory, is responsible for the enhancement of the spin current. III. SPIN HALL CURRENT AND CONDUCTIVITIES FOR CUBIC AND NONCUBIC CRYSTAL We are interested in the generation of spin current through the eective SOI and therefore consider only the relevant part of the Hamiltonian (for the positive energy solution) of spin1 2electron for zero external magnetic eld as H=~ p2 2m+eVtot(~ r)effe ~~ :(~Etot~ p) (11) The semiclassical equation of motion of electron can be dened as ~F=1 i~h m~_r;Hi +m@~_r @t; (12)4 with~_r=1 i~[~ r;H]:Thus from (11) ~_r=~ p meffe ~ ~ ~Etot (13) Finally, the force ~F=m~ r=e~rVtot(~ r) +effem ~_~ r~r(~ ~Etot) (14) is the spin Lorentz force with an eective magnetic eld ~r(~ ~Etot):Explicitly, the vector potential is given by ~A(~ ) =effmc ~(~ ~Etot) (15) Later we shall discuss about this spin dependent gauge ~A(~ ) which is closely related to the ACphase and show how the~k:~ pperturbation modies the corresponding ACphase. The spin dependent eective Lorentz force noted in eqn.(14) is responsible for the spin transport of the electrons in the system, and hence responsible for the spin Hall eect of this inertial system. It is clear from (9) that the expression in (14) i.e the Lorentz force is enhanced due to ~k:~ pperturbation in comparison to the inertial spin force studied in [12]. From the expression of _~ rin (13) we can write the linear velocity in a linearly accelerating frame with ~k:~ pperturbation as _~ r=~ p m+~ v~ ; ~ a (16) where ~ v~ ; ~ a=effe ~(~ ~Etot) (17) is the spin dependent anomalous velocity term. The anomalous velocity term is related to the spin current as ji s=enTri~ v~ ; ~ a:One should note that the velocity depends on i.e on the spin orbit gap and the band gap energy of the crystal considered. The expression shows for a non zero spin orbit gap, the spin dependent velocity changes with the energy gap. For vanishing spin-orbit gap, there is no extra contribution to the anomalous velocity for the ~k:~ pperturbation. The spin current and spin Hall conductivity in an accelerated frame of a semiconductor can now be derived by taking resort to the method of averaging [12, 19]. We proceed with equation(14) as ~F=~F0+~F~ (18) where~F0and~F~ are respectively the spin independent and the spin dependent parts of the total spin force. With the help of eqn. (15) the Hamiltonian (11) can be written as H=1 2m(~ pe c~A(~ ))2+eVtot(~ r) (19) whereVtot(~ r) =V(~ r)Va(~ r) andV(~ r) is the sum of the external electric potential V0(~ r) and the lattice electric potentialVl(~ r). In this calculation we have neglected the terms of O(~A2(~ )):Breaking into dierent parts, the solution of equation (14) can be written as _~ r=_~ r0+_~ r~ [19]. If the relaxation time is independent of ~ and for the constant total electric eld ~Etot, following [12, 19] we can write, h_~ r0i= m@Vtot @r =e m~Eeff; (20) and D _~ r(~ )E =effe22 m~~Eeff@ @r(~ @Vl @r) +effe22 m~~Eeff@ @r(~ @V~ a @r) : (21)5 where~Eeff=e~r(V0(~ r)V~ a(~ r)):We can now derive the spin current by the evaluation of the averages in equation (21) with dierent symmetry. Semiconductors with cubic symmetry For the case of semiconductors with cubic symmetry and constant acceleration [12] D _~ r(~ )E =2e22 m~eff(~ ~Eeff) (22) as in this case the only non zero contribution permitted by symmetry [12, 19] is @2Vl @ri@rj =ij; (23) withbeing a system dependent constant. The total spin current of this inertial system with ~k:~ pperturbation can now be obtained as ~jkp=eD s~_rE =~jo;~ a kp+~js;~ a kp(~ ) (24) The charge component of this current in our accelerated system is ~jo;~ a kp=e2 m(~E0~E~ a): (25) Let us introduce the density matrix for the charge carriers as s=1 2(1 +~ n:~ ); whereis the total charge concentration and ~ n=h~ iis the spin polarization vector. Within the ~k:~ pframework, due to the interband mixing the spin current of this inertial system is given by ~js;~ a kp(~ ) =2e32 m~~2 4(m)2c2+P2 31 E2 G1 (EG+40)2 ~ n(~E0~E~ a (26) or ~js;~ a kp(~ ) =m m 1 +4m2c2P2 3~21 E2 G1 (EG+40)2 ~js;~ a(~ ) (27) =m m(1 + )~js;~ a(~ ) (28) where~js;~ a(~ ) =~e32 2m3c2 ~ n(~E0~E~ a is the spin current in an accelerating frame[12] without ~k:~ pperturbation. With~E0= (0;0;Ez^z) and~ a= (0;0;az^z) we can explicitly derive the spin currents in the xandydirections. The ratio of spin current in an accelerating system with and without ~k:~ pperturbation is given by j~js;~ a(~ )kpj j~js;~ a(~ )j=m m(1 + ): (29) The coupling constant has dierent values for dierent materials and , the coupling parameter in the vacuum has a constant value 3 :7106A2:We tabulate the ratio of spin currents for dierent semiconductors with cubic symmetry as EG(eV)40(eV)P(eVA)(A2)j~js;~ a(~ )kpj j~js;~ a(~ )j GaAs = 1.519 0.341 10.493 5.3 2:154107 AlAs = 3.13 0.300 8.97 0.318 5:748105 InSb = 0.237 0.810 9.641 523.33 1:01751010 InAs = 0.418 0.380 9.197 120 1:41109 The table reveals that in a linearly accelerating frame, how ~k:~ pmethod is useful for the generation of large spin current6 GaAsInAsInSb 0 10 20 30 40az02/Multiply1094/Multiply1096/Multiply1098/Multiply1091/Multiply1010A/LBracketBar1jx/RBracketBar1 FIG. 1: (Color online) Variation of spin current with acceleration for three dierent semiconductors, where A =2m2c2 ~e22. in semiconductors. In gure 1 we have plotted the variation of spin current( xdirection) with acceleration( zdirection) using equ. (31) and the values of the Kane model parameters in the table, for three dierent semiconductors. Now if we switch o the external electric eld, we have the following expression of spin current in our system ~js;~ a kp(~ ) =s;a H;kp ~ n~E~ a ; (30) wheres;a H;kp=2e32 m~effis the spin Hall conductivity. If we now consider the acceleration along zdirection, the spin current in the xdirection becomes j~js;~ a x;kp(~ )j=s;a H;kp(nyEa;z) (31) One can notice that even if the external electric eld is zero, still we can achieve huge spin current by the application of acceleration only. For dierent semiconductors we get dierent spin current for non zero spin orbit gap. It is interesting to point out that for a critical value ~E0=~Eawe see no spin current in the system. Though the acceleration under which we get the result is very high [12], still it elicits that we can control spin current by adjusting acceleration. The corresponding charge and spin Hall conductivities in a linearly accelerating frame from the expressions of the currents (25), (26) can be readily obtained as ~ a H;kp =e2 m s;~ a H;kp=2e32 m~eff (32) As expected [12], both the charge and spin conductivities are not aected by the inertial eect of acceleration but the spin conductivity is renormalized by the ~k:~ pperturbation. The ratio of spin and charge Hall conductivity is s;~ a H;kp ~ a H;kp=2e ~eff (33) =2e ~~2 4(m)2c2+P2 31 E2 G1 (EG+40)2 (34) This spin to charge ratio is independent of the concentration of charge carriers but depends on the relaxation time, the constant term , and also on the Kane model parameters P,EGand40, i.e. the ratio varies with the material considered. For a non accelerating system with zero spin orbit gap parameter, the ratio in (33) is the same as found in [19]. The condition under which the spin conductivity becomes exactly equal to charge conductivity is eff=~ 2e: The comparison of the spin Hall conductivity in our system with that as obtained in [12] without ~k:~ pperturbation shows an enhancement due to the presence of the term which can be observed from the relation js;~ a H;kpj js;~ a Hj=m m(1 + ) (35) in any crystalline solid with a non zero spin orbit gap 40: Semiconductors with non-cubic symmetry7 There are semiconductors which do not have cubic symmetry, but are examples of producing spin Hall eect. Our objective now is to study those systems and derive the expressions for the spin current. We can consider orthorhombic crystals and can choose the axes of the coordinate frame along the crystal axes [20]. Instead of eqn (23), for the non cubic symmetry we can now write @2V @ri@rj =iij; (36) wherex6=y6=zare the factors of order unity. Using this, eqn (22) becomes, D _~ r(~ )iE =effe22 m~[x+y+zi](~ ~Eeff)i (37) which shows that the spin dependent velocity is not uniform in all directions. Following the same procedure [19], the spin current in ~ xdirection is attained as ~js;~ a x;kp(~ ) =effe32 m~ (y+z) ~ n(~E0~E~ a) x(38) Hence, the spin Hall conductivity is s;~ a x;kp=effe32 m~ (y+z) (39) The charge conductivity remains the same as in the cubic case i.e ~ a H;kp =e2 m(40) For an orthorhombic crystal in the ~ xdirection we can nd out the ratio of the spin to charge conductivity as s;~ a H;kp ~ a H;kp=effe(y+z) ~(41) =e(y+z) ~(42) ~2 4m2c2+P2 31 E2 G1 (EG+40)2 : (43) The charge to spin conductivity ratio does not depend upon the concentration of charge carriers, rather it depends on the Kane model parameters and the values of xandy:It can be readily observed that we return back to the result of the cubic case for x=y=z. IV. SPIN CURRENT AND SPIN POLARIZATION WITH TIME DEPENDENT ACCELERATION Let us now analyze the case for a time dependent acceleration. As an example of time dependent acceleration we consider [12, 14] ~ a=u!2 aexp(i!~ at)~ ex; (44) where the acceleration is induced by harmonic oscillation with frequency !~ aand amplitude u:The time dependent acceleration ~ ainduces a time dependent electric eld ~E~ aas ~E~ a=mu!2 a eexp(i!~ at)~ ex: (45) For the external electric eld ~E= 0, from (26), the spin current for a semiconductor with cubic symmetry is then given by ~js;~ a kp(~ ;t) =mu!2 a2e22 m~~2 4(m)2c2+P2 31 E2 G1 (EG+40)2 (~ nex)exp(i!~ at) (46)8 Ω = 10 GHz GaAs 2 4 6 8 10u2.0/Multiply1084.0/Multiply1086.0/Multiply1088.0/Multiply1081.0/Multiply1091.2/Multiply1091.4/Multiply109A/LBracketBar1jz/RBracketBar1 u = 10 nm GaAs 2 4 6 8 10Ωa2.0/Multiply1084.0/Multiply1086.0/Multiply1088.0/Multiply1081.0/Multiply1091.2/Multiply1091.4/Multiply109A/LBracketBar1jz/RBracketBar1 FIG. 2: (Color online) Left: Variation of Ajjzjwith ufor!a= 10GHz . Right: Variation of Ajjzjwith !aforu= 10 nm for GaAs semiconductor, where A=2mmc2 e22~. Thezpolarized spin current along y direction is ~js;~ a z;kp(~ ;t) =mu!2 a2e22 m~~2 4(m)2c2+P2 31 E2 G1 (EG+40)2 exp(i!~ at)~ ey (47) The absolute value of the current is then given by jjs;~ a z;kp(~ )j=1 Au!2 a(1 + );whereA=2mmc2 e22~. Equation (47) demonstrates that if the semiconductor sample is attached to a mechanical resonator and vibrated in xdirection with !a= 10GHz andu= 10nm, we get the zpolarized ac spin current along ydirection. Application of ~k:~ pperturbation enhances the ac spin current in semiconductor [14]. In gure 2 we plot the variation of spin current with amplitude uand frequency !afor theGaAs semiconductor. Now we move towards the evaluation of the out of plane spin polarization. The constant acceleration of the inertial system cannot explain the out-of-plane transverse spin current and in what follows we consider a time dependent acceleration within the ~k:~ pperturbation formalism. From (11) we write the time dependent Hamiltonian for the time dependent acceleration with the choice of ~ a(t) = (0;0;az^z(t)), which subsequently results ~E~ a(t) = (0;0;Ea;z^z(t));[12] and H(t) =~2~k2 2m+eff(kx;tyky;tx); (48) where we use the fact that, for electrons moving through a lattice, the electric eld ~Eis Lorentz transformed to an eective magnetic eld ( ~k~E)~B(~k) in the rest frame of the electron. Hamiltonian (48) resembles to the well known Rashba Hamiltonian and eff;the spin orbit coupling strength depends on the acceleration of the system as well as on the material parameters. This Rashba like coupling parameter has signicant importance in the understanding of the spin transport with inertial eects. The SOC in semiconductor causes electron to experience an eective momentum dependent magnetic eld ~B~ a(~k);which breaks the spin degeneracy of electron. Time dependence of the spin orbit Hamiltonian will generate an additional component [21] ~B?= (_~n~ a~ n~ a);in addition to the eective magnetic eld ~Ba(~k);where the unit vector ~ na=~Ba(~k) j~Ba(~k)j:Let us now assume the eective electric eld due to acceleration is in the x direction such that ~E~ a=E~ a;x^x[12, 21]. As ~B~ a(~k) is in the x-y plane, the term ~B?completely represents an eective out of plane magnetic eld component along zdirection. With ~B=~B~ a(~k) +~B?;the total contribution of magnetic eld, the classical spin vector in the zdirection can be written as sz=1 j~Bj~ 2(_~n~ a~ n~ a):^z (49) Now if we make a choice for the unit vector along ~Ba(~k) as~ n~ a=p1(py;px;0);we get _~ na=p1(0;e~E~ a;x;0):Here represents the spin aligned parallel and anti-parallel to ~B:In the adiabatic limit, where ~B~ a(~k)~B?[12],~B9 0 5 10 15 20 0 0.5 1 1.5 2 0 250 500 750 1000A\|sz\| az py/p3A\|sz\| 0 20 40 60 80 100 0 2 4 6 8 10 0 25 50 75 100A\|sz\| az uA\|sz\| 0 5 10 15 20 0 2 4 6 8 10 0 10 20 30A\|sz\| azωaA\|sz\| FIG. 3: (Color online) (i) Variation of Ajszjwith azandpy=p3for!a= 10 GHz, u= 10 nm. (ii) Variation of Ajszjwith az andufor!a= 10 GHz andpy p3=const . (iii) Variation of Ajszjwith azand!aforu= 10 nm andpy p3=const where A=4eff e~3 . approaches ~B~ a(~k) and the out of plane spin polarization can be derived as sz;kp 1 j~B~ a(~k)j~ 2(_~n~ a~ n~ a):^z =~2 2effp~ 2 1 p2eEa;xpy =e~3pyEa;x 4effp3: (50) Substituting the value of Ea;xfrom (45) in (50), we obtain the absolute value of sz;kpas jsz;kpj=e~3pyu!2 a 4effazp3: (51) This shows how the Kane model parameters modify the spin polarization vector in an accelerated system [12]. The dependence of spin polarization on acceleration and Kane model parameters is clear from (51). In gure 3 we show the variation of the spin polarization with respect to acceleration azand any one of the parameters, u;!a;py p3keeping the two other parameters xed. Here A=4eff e~3, depends on the solid considered. As spin polarization is a measurable quantity, from the experimentally obtained values of szusing (51) we can have an insight for the experimental verication of the parameters of the Kane model. V. GAUGE FIELD THEORY OF INERTIAL SOC AND SPIN FILTER The study of gauge elds in spintronics has become a topic of recent interest [22]. In the context of SOI, the importance of Berry phase [23] was realized following the discovery of the intrinsic spin Hall eect[24]. The Berry phase results from cyclic, adiabatic transport of quantum states with respect to parameter space (e.g. real space ~ r;momentum space ~k):In this regard, analysis of the Aharonov-Casher phase through the spin dependent gauge potential also has remarkable importance. Our next goal is to explore the conditions of the Berry curvature and study their consequences in spin transport. In this section we consider the Dirac Hamiltonian in a linearly accelerating frame without any external electric eld and consider the physical consequences appearing as a result of the the induced inertial electric eld due to acceleration. A. Spin orbit coupling, spin dependent phase and perfect spin lter There are lots of attempt to describe spin lter in dierent systems [25], but as far as our knowledge goes, proposal of a perfect lter through an inertial system is not noted in the literature. As the name suggests, the function of a spin lter is to spin polarize the injected charge current. In this subsection we consider the induced spin orbit Hamiltonian (11) in the presence of the external magnetic eld and study the gauge theory of the inertial spin orbit interaction.10 In this regard, let us consider the SO Hamiltonian in (48)(time independent) in the presence of external magnetic eld~B, as H=~2 2m+eff ~(xyyx); (52) where~ =~ pe~A(~ r);and~B=r~A(~ r):The Hamiltonian in (52) can also be rewritten in the following form H=1 2m ~ pe c~A(~ r)q c~A0(~ r;~ )2 ; (53) where the spin dependent real space gauge eld, ~A0(~ r;) is ~A0(~ r;) =c 2(y;x;0): (54) Here we neglect the second order of ~A0in deriving eqn. (53). The new constant term q=2meff ~can be regarded as charge and effrepresents a Rashba [5] like spin orbit coupling strength [12]. The expression of the spin dependent gauge indicates that it is non-Abelian in nature, whereas the gauge due to external magnetic eld provides an Abelian contribution. The equation (52) can be written in terms of the total gauge eld~~A0(~ r;) acting on the system as H=1 2m ~ p~e c~~A2 (55) where~~A=e~A(~ r) +q~A(~ r;~ );is the total gauge eective in the system and ~ eis a coupling constant which is set to be 1 for future convenience. From the eld theoretical point of view, the physical eld generated due to the presence of the total gauge~~Ais given by = =@~A@~Ai~e c~h ~A;~Ai (56) The eld in the zdirection is then z= @x~Ay@y~Ax i~e c~h ~Ax;~Ayi : (57) As the commutators of dierent components of the spin gauge ~A(~ r;~ ) exists, zin our case boils down to the following form z=eBz+q2c 2~z: (58) The equn (58) can be expressed in terms of the ux generated through area Sas z=eB S+qI S; (59) whereB=SBz; ux due to external magnetic eld and I=Sqc 2~zis the physical eld generated due to inertial spin orbit coupling eect. The rst term on the right hand side(rhs) of (59) is the contribution due to the external magnetic eld, which causes the ABphase, whereas the second term on the rhs of (59) is the ux due to the physical eld, is actually responsible for a AClike phase. This AClike phase is generated when a spin circulates an electric ux. The second term in the expression of zin (58), actually represents a magnetic eld in zdirection, with opposite sign for spin polarized along + zdirection orzdirection. As the spin up and spin down electrons experience equal but opposite vertical magnetic elds, they will subsequently carry equal and opposite AClike phase [26] which can be obtained from AC=I d~ r:~A(~ r;); (60) whereas the ABphase appears due to the rst term in (58) is the same for both up and down electrons. Interestingly, one can take advantage of this acceleration induced spin orbit Hamiltonian for the proposition of a perfect spin lter. In a semiconductor the interplay between this ACphase due to the induced spin dependent gauge11 in presence of acceleration and the ABphase due to an external magnetic eld can be used to achieve a spin lter [27]. If a spatial circuit can be realized such that the up (down) spin electrons acquire an ACphase of=2 (-=2) and nite magnetic vector potential in the interior of the circuit makes both the up and down electrons attain an AB phase of=2, then the output consists of only spin down electrons and a perfect spin lter is set up. The reversal of the direction of the applied magnetic eld may switch the polarity of the lter and the output may consist of only spin down electrons. Thus we can propose theoretically a perfect spin lter without any external electric eld. The beauty of our result is that without the application of any external electric eld, only through the acceleration of the carriers and external magnetic eld we can at least theoretically propose a perfect spin lter for our system. B. Spatially non uniform SOC and tunable spin lter Spin transport through the magnetic barriers is a topic of recent interest, which naturally gives us an idea of spin lter [28]. Theoretically, spin ltering through the formation of magnetic barrier was rst investigated in a trilayer system constructed using the FM stripes and 2DEG [29] . In this subsection we consider a trilayer structure, within which a semiconducting (SC) channel is sandwiched between two metallic contacts. The SC channel is assumed to be in an accelerated frame with SO coupling strength eff, while within metallic parts have eff= 0 i.e we are considering a spatial discontinuity of the inertial spin orbit coupling strength. This sharp discontinuity in the SO coupling, in turn gives a highly localized eective magnetic eld barriers at the interfaces. The Hamiltonian with spatially non uniform spin orbit coupling can be obtained as H=~ p2 2m+eff(~ r)(kyxkxy); (61) which can be written in the following form H=1 2m ~ pe c~A(~ r;)2 (62) where ~A(~ r;) =eff(~ r)mc ~(y;x;0); (63) is the spin gauge. The trilayer structure consists of metals at x= 0 andx=Land in between there exists semiconducting channel [30]. Let us now consider the spatial prole of effto be a step function as eff(x) =0[(x)(xL)]; (64) where (x) is the unit step function and Lis the length of the semiconductor channel. Thus we can write the curvature eld using (64) as z(~ r) =mc ~(@xeff(x)x@yeff(y)y) +2 eff2(m)2ec ~3z: (65) As the spin orbit coupling is non uniform, the rst term in (65), which is vanishing for an uniform coupling, exists in our case. Finally we can write the curvature in the zdirection as z(~ r) =0mc ~[(x)(xL)]x+ (eff)22(m)2ec ~3z; (66) where0and(x) are the inertial spin orbit coupling at the barrier of the sample and the Dirac delta function respectively. The rst term on the rhs of eqn. (66) appears as the consequence of the spatial discontinuity of effand actually gives narrow spikes of magnetic elds at the interfaces. This term is interesting as it gives a Dirac function centered at the interfaces of the trilayer structure. The second term is a physical eld dierent from the eective magnetic eld, generated due to the SO coupling. This narrow magnetic elds are spin dependent as x=1:One should notice here that if the mixing of spin states are not considered, i.e if we take the length of the channel large compared to the spin precession length, we can write the non- Abelian gauge in (63) as an Abelian gauge as A= (0;A;y;0) = (0;effmc ~;0); (67) wheredenotes two states corresponding to x=1:This structure is useful as a tunable source of spin current, which is very important concept in spintronics applications.12 VI. CONCLUSION In this paper we have theoretically investigated the generation of spin Hall current in a linearly accelerating semi- conductor system in presence of electromagnetic elds with the help of well known Kane model by taking into account the interband mixing on the basis of ~k:~ pperturbation theory. The explicit form of inertial spin Hall current and conductivity is derived for both cubic and noncubic crystals. We have shown how the interband mixing explains the spin current in an inertial system and also show the dependence of conductivity on the ~k:~ pperturbation parameters. In the case of time dependent acceleration with ~k:~ pmethod we show the explicit expression of spin current and spin polarization. From the gauge theoretical point of view, next we have investigated the real space Berry curvature appearing in an inertial system with ~k:~ pmethod. Lastly based on the gauge theoretical aspects we have discussed a perfect spin lter and a tunable spin lter in an inertial frame. [1] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnar, M. L. Roukes, A.Y. Chtchelkanova, and D. M. Treger, Science 294, 1488 (2001). [2] I. Zutic, J. Fabian, and S. D. Sarma, Rev. Mod. Phys. 76, 323 (2004). [3] J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999). [4] M. I. Dyakonov and V. I. Perel, Sov. Phys. JETP Lett. 13: 467 (1971). [5] E.I Rashba Sov. Phys. Solid State 2, 1224 (1960); Y. Bychkov and E.I Rashba JETP Lett. 39, 78 (1984). [6] R. Winkler, Spin-orbit Coupling Eects in Two-Dimentional Electron and Hole Systems (Springer-Verlag, Berlin, 2003) [7] Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Science 306, 1910 (2004); J. Wunderlich, B. Kaestner, J. Sinova, T. Jungwirth, Phys. Rev. Lett. 94, 047204 (2005); S. O. Valenzuela and M. Tinkham, Nature 442, 176 (2006), D. Culcer, J. Sinova, N. A. Sinitsyn, T. Jungwirth and A. H. MacDonald and Q. Niu, Phys. Rev. Lett. 93, 046602 (2004); J. Shi, P. Zhang, D. Xiao and Q. Niu, Phys. Rev. Lett. 96, 076604 (2006). [8] S. J. Barnett, Phys. Rev. 6, 239 (1915). [9] A. Einstein and W. J. de Haas, Verh. Dtsch. Phys. Ges. 17, 152 (1915). [10] R. T. Tolman and T. Stewart, Phys. Rev. 8, 97 (1916). [11] F. W. Hehl and Wei Tou Ni, Phys. Rev. D 42, 2045 (1990). [12] Debashree Chowdhury and B. Basu, Annals of Physics 329166 (2013). [13] B Basu, D Chowdhury, S Ghosh, arXiv:1212.4625. [14] M. Matsuo et.al., Physical Review B 84, 104410 (2011). [15] M. Matsuo , J Ieda, S Maekawa, Phys. Rev. B 87, 115301 (2013). [16] L.L Foldy and S.Wouthuysen, Phys Rev 78, 29 (1950). [17] W. Greiner, Relativistic Quantum Mechanics:Wave Equation (Springer-Verlag, Berlin, 2000), p.277. [18] E. O. Kane, J. Phys. Chem. Solids 1, 249 (1957). [19] E. M. Chudnovsky, Phys. Rev. Lett. 99, 206601 (2007). [20] E. M. Chudnovsky, Phys. Rev. B 80.153105 (2009). [21] T Fujita, M B A Jalil and S G Tan, New Journal of Physics 12, 013016 (2010). [22] K.Yu. Bliokh, Yu.P. Bliokh, Annals of Physics 319, 13 (2005). [23] M.V. Berry, Proc. R. Soc. London A 392, 45 (1984). [24] S. Q. Shen, Phys. Rev. B 70, 081311(R)(2004). [25] Y. Z. Xu and Z. Shi, Appl. Phys. Lett., 81, 691 (2002), Y. Jiang, M. B. A. Jalil, and T. S. Low, Appl. Phys. Lett., 80, 1673, (2002), M. B. A. Jalil, S. G. Tan, T. Liew, K. L. Teo, and T. C. Chong, J. Appl. Phys., 95, 7321, (2004). [26] Y. Aharonov and A. Casher, Phys. Rev. lett. 53, 319 (1984). [27] N. Hatano, R. Shirasaki and H. Nakamura, Phys. Rev. A 75, 032107 (2007). [28] X. Hao, J. Moodera and R. Meservey, Phys. Rev. B 42, 8235 (1990). [29] A. Majumdar, Phys. Rev. B 54, 11911, (1996). [30] T. Fujita, M. B. A. Jalil and S. G. Tan, IEEE Transactions of Magnetics, 46, 6 (2010).
1709.07042v1.Superfluid_transition_temperature_of_spin_orbit_and_Rabi_coupled_fermions_with_tunable_interactions.pdf	Super uid transition temperature of spin-orbit and Rabi coupled fermions with tunable interactions Philip D. Powell,1, 2Gordon Baym,2and C. A. R. S a de Melo3 1Lawrence Livermore National Laboratory, 7000 East Ave., Livermore, CA 94550, USA 2Department of Physics, University of Illinois at Urbana-Champaign, 1110 W. Green Street, Urbana, Illinois 61801, USA 3School of Physics, Georgia Institute of Technology, 837 State Street, Atlanta, Georgia 30332, USA (Dated: October 12, 2021) We obtain the super uid transition temperature of equal Rashba-Dresselhaus spin-orbit and Rabi coupled Fermi super uids, from the Bardeen-Cooper-Schrieer (BCS) to Bose-Einstein condensate (BEC) regimes in three dimensions. Spin-orbit coupling enhances the critical temperature in the BEC limit, and can convert a rst order phase transition in the presence of Rabi coupling into second order, as a function of the Rabi coupling for xed interactions. We derive the Ginzburg-Landau equation to sixth power in the super uid order parameter to describe both rst and second order transitions as a function of spin-orbit and Rabi couplings. PACS numbers: 67.85.Lm, 03.75.Ss, 47.37.+q, 74.25.Uv, 75.30.Kz The ability to simulate magnetic elds in cold atoms systems opens the possibility of exploring new physics unachievable elsewhere. In addition to articial Abelian magnetic elds [1{3], one can also generate non-Abelian elds in both bosonic and fermionic systems [4{12]. The latter will eventually lead to the possibility of simulat- ing quantum chromodynamics lattice gauge theory [13{ 16]. Present experiments on three dimensional spin-orbit coupled Fermi gases are still at too high a temperature for these systems to undergo Bardeen-Cooper-Schrieer (BCS) pairing, because the current Raman scheme causes heating. In contrast, theory has concentrated at zero temperature [17{22]. Once such fermionic systems can be cooled below the super uid transition temperature, the spin-orbit coupling is expected to reveal new states with non-conventional pairing. Even weak spin orbit coupling will produce an admixture of s-wave and p-wave pairing. In this Letter, we investigate the transition tempera- ture of Fermi super uids with an equal mixture of Rashba and Dresselhaus spin-orbit coupling as a function of the Rabi coupling, throughout the entire BCS (Bardeen- Cooper-Schrieer)-to-BEC (Bose-Einstein condensation) evolution in three dimensions; the single particle Hamil- tonian matrix is Hso(^p) =(^pxy)2 2m+^p2 y 2m+^p2 z 2m 2z; (1) the Pauli sigma matrices operate in the two-level space, ^pis the momentum, is the Rabi frequency, and is the momentum transfer to the atoms in a two-photon Ra- man process [7]. This problem bears a close relation to spin-orbit coupling in solids, where the coupling pij is intrinsic, and where the role of the Rabi frequency is played by an external Zeeman magnetic eld. While a mean eld treatment describes well the evolution from the BCS to the BEC regime at zero temperature [23], this order of approximation fails to describe the correctcritical temperature of the system in the BEC regime, because the physics of two-body bound states (Feshbach molecules) [24] is not captured when the pairing order pa- rameter goes to zero. To remedy this problem, we include eects of order-parameter uctuations in the thermody- namic potential. We stress that our present results are applicable to both neutral cold atomic and charged condensed matter systems. We nd that the spin-orbit coupling can en- hance the critical temperature of the super uid in the BEC regime and that it can convert a discontinuous rst order phase transition in the presence of Rabi coupling into a continuous second order transition, as a function of the Rabi frequency (or Zeeman eld in solids) for xed in- teractions. We analyze the nature of the phase transition in terms of the Ginzburg-Landau free energy, calculating it to six powers of the super uid order parameter to al- low for the description of continuous and discontinuous transitions as a function of the spin-orbit coupling, Rabi frequency, and interactions. To describe three dimensional Fermi super uids in the presence of spin-orbit and Zeeman elds, we start from the Hamiltonian density H(r) =Hso(r) +HI(r); (2) and use units ~=kB= 1. The rst term in Eq. (2) is the independent-particle contribution including spin- orbit coupling, Hso(r) =X ss0 y s(r) [Hso(^p)]ss0 s0(r); (3) The second term describes the two-body s-wave contact interaction HI(r) =g y "(r) y #(r) #(r) "(r); (4) where the arrows indicate the pseudospins of the fermions, which we refer to simply as \spins". Here g>0arXiv:1709.07042v1 [cond-mat.quant-gas] 20 Sep 20172 corresponds to a constant attraction between opposite spins. The pairing eld ( r;) =gh #(r;) "(r;)ide- scribes the formation of pairs of two fermions with oppo- site spins, where =itis the imaginary time. Standard manipulations lead to the Lagrangian density L(r;) =1 2 y(r;)G1(r;) (r;) +j(r;)j2 g +K(r)(rr0); (5) where = ( " # y " y #)Tis the Nambu spinor, and Kr2=2mis the kinetic energy operator measured with respect to the fermion chemical potential . We note that the denition of already includes the overall positive shift 2=2min the single particle kinetic energies due to spin-orbit coupling, that is, is measured with respect to2=2m. The inverse Green's function appearing in Eq. (5) is G1 k() =0 BB@@K"ikx=m 0 ikx=m @K# 0 0 @+K"ikx=m 0ikx=m @+K#1 CCA; (6) whereK"=K =2;andK#=K+ =2 are the ki- netic energy terms shifted by the Rabi coupling. As men- tioned above, the mean eld treatment fails to describe the correct critical temperature of the system in the BEC regime. To incorporate the physics of two-body bound states, we must include eects of order-parameter uctu- ations in the thermodynamic potential. To obtain the transition temperature to the super uid state, we analyze the partition function Zas a functional integralR DDR D D yeSfor the Fermi super uid, whereS=R 0R d3rL(r;) is the full action of the sys- tem. Upon integration over the fermion elds, the ther- modynamic potential = TlnZcontains two terms = 0+ F, where 0=TlnZ0=TS0is the sad- dle point contribution, at which point ( r;) = 0, and F=TlnZFis the uctuation part. The subscript 0 denotes quantities calculated in mean eld. The mean-eld (saddle-point) term in the thermody- namic potential is 0=Vj0j2 gT 2X k;jlnh 1 +eEj(k)i +X kk;(7) wherek="k,"k=k2=2m, and theEj(k) are the eigenvalues of the Nambu Hamiltonian matrix H0(k) = 1@G1 k(), withj=f1;2;3;4g. The rst set of eigen- values E1;2(k) =r E2 0;k+h2 k2q E2 0;k+h2 kj0j2(kx=m)2; (8)describe quasiparticle excitations, and the second set of eigenvalues E3;4(k) =E1;2(k) correspond to quasiholes. Here E0;k=p 2 k+j0j2;andhkp (kx=m)2+ 2=4 is the magnitude of the combined spin-orbit and Rabi couplings. The order parameter equation is found from the saddle point condition 0= 0jT;V; = 0, leading to m 4as=1 2VX k1 "kA+h2 z khkA : (9) Here, we write the interaction gin terms of the renor- malizeds-wave scattering length asvia the relation 1=g=m=4as+(1=V)P k1=2"k[25{27], and for short writeA= (12nk;1)=2E1(12nk;2)=2E2;with nk;i= 1=(eEk;i+1) the Fermi function. In addition, the particle number at the saddle point N0=@ 0=@jT;V; is given by N0=X k 1k A++(kx=m)2 khkA : (10) The saddle point transition temperature T0is deter- mined by solving Eq. (9) for given . The correspond- ing number of particles is given by Eq. (10). This mean eld treatment leads to a transition temperature grow- ing ase1=kFasforkFas!0+. To nd the physically correct transition temperature we must, in construct- ing the thermodynamic potential, include the physics of two-body bound states near the transition via the two- particle t-matrix [28, 29]. With all the two particle chan- nels taken into account, the t-matrix calculation leads to a two-particle scattering amplitude, , where 1(q;z) =m 4as1 2VX k1 "k+2X i;j=1ijWij ; (11) herezis the (complex) frequency, Wij= (1nk;i nk+q;j)=(zEi(k)Ej(k+q)):In the limit that the order parameter goes to 0, the single particle eigenvalues reduce toE1;2(k) =khk[30]. The coecients 11= 22=jukuk+qvkv k+qj2and12=21=jukvk+q+ uk+qvkj2are weighting functions of the amplitudes uk=s 1 2 1 + 2hk ; v k=is 1 2 1 2hk :(12) As the fermion chemical potential becomes large and negative, the system becomes non-degenerate and 1(q;z) = 0 becomes the exact eigenvalue equation for the two-body bound state in the presence of spin-orbit and Rabi coupling [31]. The solution is z=Ebs(q)2, whereEbs(q) is the two-body bound state energy. The uctuation correction to the thermodynamic potential is then F=TP q;iqnln [(q;iqn)=V]:3 From Fwe obtain the uctuation contribution to the particle number NF=@ F=@jT;V=Nsc+Nb. Here, Nsc=X qZ1 !tp(q)d! nB(!)@(q;!) @@(q;0) @ V;T (13) is the number of particles in scattering states, where the phase shift (q;!) is dened via the relation ( q;! i) =j(q;!)jei(q;!)and!tp(q) is the two-particle continuum threshold corresponding to the branch point of 1(q;z) [28, 32]. Also, Nb= 2X qnB(Ebs(q)); (14) is the number of fermions in bound states with nB(!) = 1=(e!1) the Bose distribution function. The total number of fermions, as a function of , becomes N=N0+NF; (15) whereN0is given in Eq. (10) and NFis the sum the two contributions NscandNbdiscussed above [24, 28]. DeningkFto be the Fermi momentum of the atomic gas with total density n=k3 F=32, we obtain Tcas a function of the scattering parameter 1 =kFasby solving simultaneously the order parameter and number equa- tions (9) and (15). Figure 1 shows the eects of spin- orbit and Rabi couplings on the transition temperature Tc. The solutions correspond to minima of the free en- ergyF= +N. In Fig. 1, we scale energies and temperatures by the Fermi energy "F=k2 F=2m. The solid (black) line in Fig. 1a shows the transi- tion temperature Tcbetween the normal and super uid state versus the scattering parameter 1 =kFasfor zero Rabi coupling ( = 0) and zero one-dimensional Rashba- Dresselhaus (ERD) [33{35] spin-orbit coupling ( = 0). If = 0, the spin-orbit coupling can be removed by a simple gauge transformation, and thus plays no role. In this situation, the pairing is purely s-wave. The dashed (blue) line shows Tcfor 6= 0, with vanishing ERD spin- orbit coupling. We see that for xed interaction strength, the pair-breaking eect of the Rabi coupling (as a Zee- man eld breaks pairs in a superconductor) suppresses super uidity, compared with = 0. With both ERD spin-orbit and Rabi coupling present, the pairing is no longer pure s-wave, but has a triplet p-wave component (and higher) mixed into the super uid order parameter; the admixture stabilizes the super uid phase, as shown by the dotted (green) line. The latter curve shows that in the BEC regime with large positive 1 =kFas, the tran- sition temperature Tcis larger with spin-orbit and Rabi couplings than in their absence, as a consequence of the reduction of the bosonic eective mass in the x-direction below 2m. However, with suciently large , the ge- ometric mean bosonic mass MBincreases and Tcde- creases [36]. Temperature -1/k f a 0No SOC No SOC/uni03A9 = 0NORMAL PAIRED BEC molecules BCS pairs/uni03A9 = fSOC ε/uni03A9 = fεa) FIG. 1: (Color online) a) The transition temperature Tcfor ERD spin-orbit coupling for two Rabi coupling strengths, = 0 and "F. For = 0, solid (black) curve, Tcis that for zero spin-orbit coupling, since the ERD eld can be gauged away. The dashed (blue) line shows Tcfor zero spin-orbit coupling, with = "F, while the dotted (green) line shows Tcfor = 0 and "F, and = 0:5kF. In b)Tcis drawn at unitarity, 1 =kFas= 0, and in the inset at 1 =kFas=2:0, as a function of e = ="F. The solid (red) curves are for ~ = 0, and the dashed (blue) curves are for ~= 0:5. Across the dotted (red) curves below the solid (red) curves, the phase transition is rst order. Figure 1b shows Tcversus for xed 1 =kFas, without and with ERD spin-orbit coupling at = 0:5kF. When and the temperature are zero, super uidity is destroyed at a critical value of corresponding to the Clogston limit [37]. At low temperature the phase transition to the normal state is rst order, because the Rabi coupling (Zeeman eld) is suciently large to break singlet Cooper pairs. However, at higher temperatures the singlet s-wave super uid starts to become polarized due to thermally excited quasiparticles that produce a paramagnetic re- sponse. Therefore above the characteristic temperature indicated by the large (red) dots, the transition becomes second order, as pointed out by Sarma [38]. The critical temperature for 6= 0 vanishes only asymptotically in the limit of large . We note that for = EFand= 0 the transition from the super uid to the normal state is continuous at unitarity, but very close to a discontinuous transition. In the range 1 :05. =EF.1:10 numerical4 uncertainties as !0 prevent us from predicting ex- actly whether the transition at unitarity is continuous or discontinuous. To understand further the eects of uctuations on the order of the transition to the super uid phase and to as- sess the impact of spin-orbit and Rabi couplings near the critical temperature, we now derive the Ginzburg-Landau description of the free energy near the transition, where the actionSFcan be expanded in powers of the order parameter ( q), beyond Gaussian order. The expansion ofSFto quartic power is sucient to describe the con- tinuous (second order) transition in Tcversus 1=kFasin the absence of an external Zeeman eld [24]. However, to describe correctly the rst order transition [37, 38] at low temperature (Fig. 1), it is necessary to expand the free energy to sixth order in . The quadratic (Gaussian order) term in the action is SG=VX qjqj2 (q;z): (16) For an order parameter varying slowly in space and time, we may expand 1(q;z) =a+ciq2 i 2md0z+; (17) with the sum over i=x;y;z implicit. The full result, as a functional of ( r;), has the form SF=Z 0dZ d3r d0@ @ +ajj2 +cijrij2 2m+b 2jj4+f 3jj6 : (18) The full time-dependent Ginzburg-Landau action de- scribes systems in and near equilibrium, e.g., with col- lective modes. The imaginary part of d0measures the non-conservation of jj2in time. We are interested here in systems at thermodynamic equilibrium where the order parameter is independent of time. Then minimizing the free energy TSFwith respect to , we obtain the Ginzburg-Landau equation ci 2mr2 i+bjj2+fjj4+a = 0: (19) Forbpositive the system undergoes a continuous phase transition when achanges sign. However, when bis nega- tive the system is unstable in the absence of f. Forb<0 anda > 0, a rst order phase transition occurs when 3b2= 16af. Positivefstabilizes the system even when b<0. In the BEC regime, we dene an eective bosonic wave- function =pd0 to recast Eq. (19) in the form of the Gross-Pitaevskii equation for a dilute Bose gas r2 i 2Mi+U2j (r)j2+U3j (r)j4B (r) = 0: (20)Here,B=a=d 0is the bosonic chemical potential, the Mi=m(d0=ci) are the anisotropic bosonic masses, and U2=b=d2 0andU3=f=d3 0represent contact interactions of two and three bosons. In the BEC regime these terms are always positive, thus leading to a system consisting of a dilute gas of stable bosons. The boson chemical potentialBis2+Eb<0, whereEbis the two- body bound state energy in the presence of spin-orbit coupling and Rabi frequency, obtained from the condition 1(q;E2) = 0, discussed earlier. The anisotropy of the eective bosonic masses, Mx6= My=MzM?stems from the anisotropy of the ERD spin-orbit coupling, which together with the Rabi cou- pling modies the dispersion of the constituent fermions along thexdirection. In the limit kFas1 the many- body eective masses reduce to those obtained by ex- panding the two-body binding energy Ebs(q)Eb+ q2 i=2Mi;and agree with known results [31]. However, for 1=kFas<2, many-body and thermal eects produce deviations from the two-body result. In the absence of two and three-body boson-boson in- teractions ( U2andU3), we directly obtain the analytic expression for Tcin the Bose limit from Eq. (14), Tc=2 MBnB (3=2)2=3 ; (21) withMB= (MxM2 ?)1=3, by noting that !p(q) =Ebs(q) and using the condition that nB'n=2 (with corrections exponentially small in (1 =kFas)2), wherenBis the den- sity of bosons and nis the density of fermions. In the BEC regime, the results shown in Fig. 1 include the ef- fects of the mass anisotropy, but do not include eects of boson-boson interactions. To account for boson-boson interactions, we use the Hamiltonian of Eq. (20) with U26= 0, but with U3= 0, and apply the method developed in Ref. [39] to show that these interactions further increase TBEC to Tc(aB) = (1 + )TBEC; (22) where =n1=3 BaB. Here,aBis the s-wave boson- boson scattering length, is a dimensionless constant 1, and we used the relation U2= 4aB=MB. Since nB=k3 F=62and the boson-boson scattering length is aB=U2MB=4, we have =~fMBeU2;wherefMB= MB=2m;eU2=U2k3 F="F;and~==4(65)1=3=50: For xed 1=kFas,Tcis enhanced both by a spin-orbit and the dependent decrease in the eective boson mass MB (10-15%), as well as a stabilizing boson-boson repulsion U2(2-3%), for the parameters used in Fig. 1. In summary, we have analysed the nite temperature phase diagram of three dimensional Fermi super uids in the presence ERD spin-orbit coupling, Rabi coupling, and tunable s-wave interactions. Furthermore, we de- veloped the Ginzburg-Landau theory up to sixth power5 in the amplitude of the order parameter to show the ori- gin of discontinuous (rst order) phase transitions when the Rabi frequency is suciently large for vanishing spin- orbit coupling. The research of author PDP was supported in part by NSF Grant PHY1305891 and that of GB by NSF Grants PHY1305891 and PHY1714042. Both GB and CARSdM thank the Aspen Center for Physics, supported by NSF Grants PHY1066292 and PHY1607611, where part of this work was done. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE- AC52- 07NA27344. [1] Y. Lin, R. Compton, K. Jimin ez-Garc a, J. Porto, and I. Spielman, Nature (London) 462, 628 (2009). [2] Y. Lin, R. Compton, A. Perry, W. Phillips, J. Porto, and I. Spielman, Phys. Rev. Lett. 102, 130401 (2009). [3] C. J. Kennedy, W. C. Burton, W. C. Chung and W. Ketterle, Nature Physics, 11, 859 (2015). [4] Y.-J. Lin, K. Jim enez-Garc a and I. B. Spielman, Nature 471, 83 (2011). [5] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah, W. S. Bakr, and M. W. Zwierlein, Phys. Rev. Lett. 109, 095302 (2012). [6] P. Wang, Z.-Q. Yu, Z, Fu, J, Miao, L, Huang, S. Chai, H, Zhai, and J. Zhang Phys. Rev. Lett. 109, 095301 (2012). [7] R. A. Williams, M. C. Beeler, L. J. LeBlanc, K. Jimin ez- Garc a and I. B. Spielman, Phys. Rev. Lett. 111, 095301 (2013). [8] V. Galitski and I. B. Spielman, Nature 494, 49 (2013). [9] Z. Fu, L. Huang, Z. Meng, P. Wang, L. Zhang, S. Zhang, H. Zhai, P. Zhang, and J. Zhang, Nat. Phys. 10, 110 (2014). [10] M. Mancini, G. Pagano, G. Cappellini, L. Livi, M. Rider, J. Catani, C. Sias P. Zoller, M. Inguscio, M. Dalmonte, and L. Fallani, Science 349, 1510 (2015). [11] L. Huang, Z. Meng, P. Wang, P. Peng, S.-L. Zhang, L. Chen, D. Li, Q. Zhou, and J. Zhang, Nature Physics 12, 540 (2016). [12] Z. Wu, L.Zhang, W. Sun, X.-T. Xu, B.-Z. Wang, S.-C. Ji, Y. Deng, S. Chen, X.-J. Liu, and J.-W. Pan, Science 354, 83-88 (2016). [13] J. I. Cirac and P. Zoller, Nature Phys. 8, 264 (2012). [14] U.-J. Wiese, Ultracold Quantum Gases and Lattice Sys- tems: Quantum Simulation of Lattice Gauge Theories, Annalen der Physik 525, 777 (2013). [15] E. Zohar, J. I. Cirac, and B. Reznik, Rep. Prog. Phys. 79, 1 (2015). [16] M. Dalmonte and S. Montangero, Contemp. Phys. 57388 (2016); arXiv:1602.03776. [17] M. Gong, S. Tewari, and C. Zhang, Phys. Rev. Lett. 107, 195303 (2011). [18] Z.-Q. Yu and H. Zhai, Phys. Rev. Lett. 107, 195305 (2011).[19] H. Hu, L. Jiang, X.-J. Liu, and H. Pu, Phys. Rev. Lett. 107, 195304 (2011). [20] L. Han and C. A. R. S a de Melo, Phys. Rev. A 85, 011606(R) (2012). [21] K. Seo, L. Han, and C. A. R. S a de Melo, Phys. Rev. Lett. 109, 105303 (2012). [22] K. Seo, L. Han, and C. A. R. S a de Melo, Phys. Rev. A 85, 033601 (2012). [23] A. J. Leggett in Modern Trends in the Theory of Con- densed Matter edited by A. Pekalski and R. Przystawa, Springer-Verlag, Berlin (1980). [24] C. A. R. S a de Melo, M. Randeria, and J. Engelbrecht, Phys. Rev. Lett. 71, 3202 (1993). [25] Note that asis thes-wave scattering length in the ab- sence of spin-orbit and Zeeman elds. It is, of course, possible to express all relations obtained in terms of a scattering length which is renormalized by the presence of the spin-orbit and Zeeman elds [26, 27]. However, in addition to complicating our already cumbersome ex- pressions, it would make reference to a quantity that is more dicult to measure experimentally and that would hide the explicit dependence of the properties that we analyzed in terms of the spin-orbit and Zeeman elds, so we do not consider such complications here. [26] S. Gopalakrishnan, A. Lamacraft, and P. M. Goldbart, Phys. Rev. A 84, 061604 (2011). [27] T. Ozawa, Ph.D. Thesis, University of Illinois (2012). [28] P. Nozi eres and S. Schmitt-Rink, J. Low Temp. Phys., 59, 195 (1985). [29] Z. Yu and G. Baym, Phys. Rev. A 73, 063601 (2006). [30] Note that setting 0= 0 in the general eigenvalue ex- pressions yields E1;2=jjkjhkj. It is straightforward to show that neglecting the absolute values does not re- sult in any change in either the mean eld order pa- rameter or number equation. Thus, for simplicity we let E1;2=khk. [31] D. M. Kurkcuoglu and C. A. R. S a de Melo, Phys. Rev. A93, 023611 (2016). See also arXiv:1306.1964v1 (2013). [32] Z. Yu, G. Baym, and C. J. Pethick, J. Phys. B: At. Mol. Opt. Phys. 44, 195207 (2011). [33] G. Dresselhaus, Phys. Rev. 100, 580 (1955). [34] E. Rashba, Fizika Tverdogo Tela 2, 1244 (1960). [35] This form is equivalent to another common form of the Rashba-Dresselhaus coupling found in the litera- ture: hso=hR+hDwhere hR=vR(kx^yky^x) andhD=vD(kx^y+ky^x). The two forms are related via a momentum-space rotation and the correspondences =m(vR+vD) and= (vRvD)=(vR+vD). The Equal-Rashba-Dresselhaus limit (ERD) corresponds to vR=vD=v, leading to = 0 and= 2mv. [36] This renormalization of the mass of the bosons can be traced back to the actual change in the energy disper- sion of the fermions when both spin-orbit coupling and Zeeman elds are present. [37] A. M. Clogston, Phys. Rev. Lett. 9, 266 (1962). [38] G. Sarma, J. Phys. Chem. Sol. 24, 1029 (1963). [39] G. Baym, J.-P. Blaizot, M. Holzmann, F. Lalo e and D. Vautherin, Phys. Rev. Lett. 83, 1703 (1999).
2401.16738v1.A_novel_non_adiabatic_spin_relaxation_mechanism_in_molecular_qubits.pdf	A novel non-adiabatic spin relaxation mechanism in molecular qubits Philip Shushkova) Department of Chemistry, Indiana University, Bloomington, Indiana, 47405 (Dated: 31 January 2024) The interaction of electronic spin and molecular vibrations mediated by spin-orbit coupling governs spin relaxation in molecular qubits. I derive an extended molecular spin Hamiltonian that includes both adiabatic and non-adiabatic spin- dependent interactions, and I implement the computation of its matrix elements using state-of-the-art density functional theory. The new molecular spin Hamiltonian contains a novel spin-vibrational orbit interaction with non-adiabatic origin together with the traditional molecular Zeeman and zero-field splitting interactions with adiabatic origin. The spin-vibrational orbit interaction represents a non-Abelian Berry curvature on the ground-state electronic manifold and corresponds to an effective magnetic field in the electronic spin dynamics. I further develop a spin relaxation rate model that estimates the spin relaxation time via the two-phonon Raman process. An application of the extended molecular spin Hamiltonian together with the spin relaxation rate model to Cu(II) porphyrin, a prototypical S=1/2 molecular qubit, demonstrates that the spin relaxation time at elevated temperatures is dominated by the non-adiabatic spin-vibrational orbit interaction. The computed spin relaxation rate and its magnetic field orientation dependence are in excellent agreement with experimental measurements. I. INTRODUCTION Molecular qubits, paramagnetic systems that exhibit long spin-coherence times, are emerging as a promising platform for the implementation of quantum information processing.1–4 The molecular electronic spin is an excellent candidate to encode quantum information because of protection by time- reversal symmetry. Taken together with the unprecedented accuracy of chemical methods to synthesize and assem- ble molecular systems, there has been a recent surge in efforts to engineer molecular qubits that are suitable to advance the much sought-after room-temperature quantum technologies.5–18 A major limiting factor to the room-temperature quantum information storage in molecular qubits is the spin relax- ation time T1, called also spin-lattice or longitudinal relax- ation time,19that characterizes the timescale of thermal equi- libration of the electronic spin by the molecular vibrational motion.5–7,9The spin relaxation theory dates back to the pio- neering work of Van Vleck,20Mattuck and Strandberg,21and Orbach,22who focused on the relaxation dynamics of para- magnetic impurities in crystals. The physical picture that emerged from their seminal contributions is that both adia- batic and non-adiabatic processes can dominate the spin re- laxation time depending on the specifics of the electronic structure of the paramagnetic impurities. Furthermore, they showed that one-phonon processes contribute to the the spin relaxation dynamics only at very low temperatures and at el- evated temperatures spin relaxation is driven by two-phonon absorption-emission processes. Density functional theory and multi-reference wavefunc- tion approaches have provided a firm theoretical basis to pre- dict molecular spin interactions.23–28Fundamental to this suc- cess is the concept of the static spin Hamiltonian, an effective Hamiltonian defined at the equilibrium nuclear geometry that a)phgshush@iu.eduincorporates the influence of the molecular electronic struc- ture in spin-dependent interactions.29,30The dynamical exten- sion of the static spin Hamiltonian by accounting for the nu- clear geometry-dependence of the traditional spin-dependent interactions has underscored recent efforts to simulate the spin relaxation dynamics of molecular qubits.31–38This approach to spin relaxation, however, does not account for the contribu- tion of the non-adiabatic interactions. The goal of the paper is to derive a molecular spin Hamil- tonian that includes both adiabatic and non-adiabatic spin- dependent interactions and to implement the computation of the matrix elements of this Hamiltonian using density func- tional theory. I achieve this goal by applying the Born- Oppenheimer approximation in the adiabatic representation for the electronic wavefunctions,39,40circumventing the need for diabatization, and allowing seamless integration with linear response density functional theory. Similar to the static spin Hamiltonian, I use unitary degenerate perturbation theory41,42to derive the molecular spin Hamiltonian, which limits its applicability to molecular systems with weak spin- orbit interaction and orbitally non-degenerate ground elec- tronic states.30The derived molecular spin Hamiltonian con- tains a novel, non-adiabatic spin-vibrational orbit interaction together with the traditional molecular Zeeman and zero-field splitting interactions29that have an adiabatic origin. I fur- ther develop a rate model to estimate the contribution to the spin relaxation time of the interactions in the molecular spin Hamiltonian. The rate model is specialized to elevated tem- peratures and evaluates the spin relaxation time via the two- phonon Raman process20using density functional calcula- tions on the isolated paramagnetic molecule. A pilot appli- cation of the molecular spin Hamiltonian together with the spin relaxation rate model to Cu(II) porphyrin,43a prototyp- icalS=1/2 molecular qubit,44,45demonstrates that the two- phonon spin relaxation time is dominated by the non-adiabatic spin-vibrational orbit interaction with doubly degenerate nor- mal modes being the major vibrational relaxation channel. The paper is organized as follows: I present the derivation of the molecular spin Hamiltonian and of the spin relaxationarXiv:2401.16738v1 [physics.chem-ph] 30 Jan 20242 rate model in Sec. II. I outline the numerical evaluation of the molecular spin Hamiltonian matrix elements using density functional theory in Sec. III. I present the results of the appli- cation of the new approach to Cu(II) porphyrin in Sec. IV, and I conclude the paper with an outline of future directions in Sec. V. II. THEORY Spin-vibrational interactions involve the coupling of the electronic spin with the molecular vibrations that is mediated by the electronic orbital motion. I start Sec. II by specifying the molecular Hamiltonian and establishing the notation used in the paper. I then present an overview of unitary degenerate perturbation theory42that I apply to derive the molecular spin Hamiltonian in Sec. II A and proceed with the application of the Born-Oppenheimer approximation. In Sec. II A, I derive the expression for the Berry connection46–48on the ground- state spin manifold and arrive at a molecular spin Hamilto- nian that contains a novel spin-vibronic vector potential. In Sec. II B, I apply a unitary gauge transformation40to a sym- metrized gauge for the spin-vibronic vector potential and de- rive the spin-vibrational orbit interaction. Harmonic expan- sion of the resulting molecular spin Hamiltonian in Sec. II C gives one- and two-phonon spin-vibrational interactions with non-adiabatic and adiabatic origins. In Sec. II D, I present a rate model for the computation of the spin relaxation time of S=1/2 molecular qubits at elevated temperature and at weak to moderate magnetic field intensities using the two-phonon spin-vibrational interactions of Sec. II C. A. Molecular Spin Hamiltonian I write the molecular Hamiltonian,29including relativistic interactions and the interaction of the molecule with an exter- nal magnetic field as: ˆHmolec=ˆTR+ˆHNR R+ˆHSOZ R. (1) ˆTRin Eq. (1) is the nuclear kinetic energy, ˆHNR R- the non- relativistic electronic Hamiltonian together with the spin- independent, scalar relativistic corrections, and ˆHSOZ R- the spin-dependent relativistic interactions that include the spin- orbit, the electronic Zeeman, and the electronic spin-spin in- teractions. Rstands for the collection of Cartesian nuclear co- ordinates, and the subscript denotes the explicit dependence of the Hamiltonian terms on the nuclear configuration. In this paper, I do not include the interactions of the electronic spin with the nuclear magnetic moments, which give rise to hyper- fine splitting of the electronic energy levels. I define the zero-order adiabatic electronic wavefunctions,K(0) RE , and the zero-order adiabatic electronic potential en- ergy surfaces, E(0) K,R, as the eigenvectors and the eigenvalues of the non-relativistic electronic Hamiltonian: ˆHNR RK(0) RE =E(0) K,RK(0) RE . (2)The eigenfunctions and eigenvalues are zero order with re- spect to the spin-dependent interactions, in which case the zero-order electronic states form multiplets of degenerate states that correspond to different projection of the electronic spin angular momentum. To account for the degeneracy of the zero-order electronic spectrum, the multi-index Kstands for a collection of quantum numbers K={kSM S}that specify the orbital quantum number k, the spin quantum number S, and the associated spin projection MS. I assume that the state de- generacy in the multiplets is only of spin origin and exclude degeneracy in the multiplets that results from spatial symme- try. The adiabatic eigenfunctions in Eq. (2) are parametrically dependent on the nuclear configuration, R, and form an or- thonormal set at each nuclear geometry. The spin-dependent interactions in Eq. (1) couple the elec- tronic states of the non-relativistic Hamiltonian and remove partially or fully the degeneracy of the electronic spectrum. I apply unitary degenerate perturbation theory42to diagonal- ize the inter-multiplet interactions due to the spin-dependent Hamiltonian ˆHSOZ R. The perturbative treatment is justified for molecules that consist of atoms of light elements, including early transition metal and main group elements, because for them the spin-dependent multiplet splittings are significantly smaller than the energy differences between the zero-order electronic multiplets. The perturbed wavefunctions, \|KR⟩, in unitary perturbation theory are obtained by a unitary transfor- mation of the zero-order wavefunctions: \|KR⟩=eˆGRK(0) RE , (3) where ˆGis the generator of the transformation and satisfies ˆG†=−ˆGto ensure the unitarity of the transformation. The generator is obtained perturbatively from the requirement of vanishing inter-multiplet interactions in the unitarily trans- formed Hamiltonian, and the perturbed eigenvalues derive from the intra-multiplet blocks of the transformed Hamilto- nian: D J(0) Re−ˆGRˆHNR R+ˆHSOZ R eˆGRK(0) RE =δJKEK′K,R.(4) Multi-indeces J,K,..refer to general multiplets, and I reserve the multi-index Ifor the ground-state multiplet. HIJstands for the matrix block between two different multiplets IandJ, whereas HI′Istands for the matrix block of the multiplet I.δJK stands for a multi-index Kronecker delta symbol. Use of the Baker-Campbell-Hausdorff formula in Eq. (4) and collection of terms according to the order of perturbation provides the equation for the first-order generator: h ˆHNR R,ˆG(1) Ri JK+HSOZ JK,R=0, (5) which is solved for the inter-multiplet blocks of the genera- torG(1) JKand is supplemented with the additional condition of vanishing intra-multiplet blocks G(1) K′K=0. The perturbation expansion in Eq. (4) gives also the perturbed eigenvalues up to second order in the perturbation: EK′K,R=E(0) K,RδK′K+HSOZ K′K,R+1 2h ˆHSOZ R,ˆG(1) Ri K′K.(6)3 Eq. (5) gives the ground-state electronic wavefunction to first order in the spin-orbit perturbation as: \|IR⟩=I(0) RE +ˆG(1) RI(0) RE =I(0) RE −∑ J̸=IJ(0) RE HSOI JI,R E(0) J,R−E(0) I,R.(7) Components of multiplets with different projections of spin angular momentum are non-interacting at zero order; coupling between them arises in the first-order contribution to the elec- tronic wavefunction, the second term in Eq. (7), from the spin- orbit interaction (SOI) in ˆHSOZ R. The perturbative expansion of the ground-state multiplet energies from Eq. (6) is given by: EI′I,R=E(0) I,RδI′I+E(1) I′I,R+E(2) I′I,R =E(0) I,RδI′I+HSOZ I′I,R+∑ J̸=IHSOZ I′J,RHSOZ JI,R E(0) I,R−E(0) J,R,(8) where EI′I,Ris a matrix within the ground-state manifold, EM′M,R. The first-order and second-order adiabatic energy contributions, the second and third term in Eq. (8), give rise to the traditional molecular spin Hamiltonian with the molec- ular Zeeman interaction, µB⃗BgR⃗S, characterized by the elec- tronic g-tensor and µBthe Bohr magneton, and the zero-field splitting interaction, ⃗SDR⃗S, characterized by the electronic D- tensor. Both the g-tensor and D-tensor depend parametrically on the nuclear positions, and the existence of the molecular spin Hamiltonian requires an orbitally-non-degenerate ground state, which is the case for all systems that I consider in the paper. I establish the connection with the traditional spin Hamiltonian and the spin-component structure of the adia- batic wavefunctions in Eq. (7) using the Wigner-Eckart the- orem in Appendix A. I apply the Born-Oppenheimer approximation39,40toun- couple the electron-nuclear dynamics on different electronic multiplets and preserve the coupled electron-nuclear dynam- icswithin an electronic multiplet. This treatment of the electron-nuclear dynamics is justified when the energy dif- ferences between electronic multiplets are significantly larger than the vibrational quanta involved in the spin relaxation dy- namics, whereas the energy differences within electronic mul- tiplets are of similar magnitude or smaller than the vibrational quanta of the nuclear dynamics. This regime holds for the S=1/2 systems considered in this work. The case where two or more electronic multiplets come close in energy and interact non-adiabatically as it occurs in conical intersections is outside the scope of the present treatment. In the Born- Oppenheimer approximation, the vibronic wavefunction for the ground-state multiplet is expressed as the sum of prod- ucts of the electronic wavefunction of a multiplet component, \|iSM,R⟩, and the associated nuclear wavefunction, χiSM(R): ΨBO I =∑ M,M′iSM′,R cM′M,R0χiSM(R) =\|IR⟩χI(R).(9) The electronic wavefunctions in Eq. (9) are the perturbed wavefunctions of Eqs. (3) and (7), and the last equality impliesa sum over the multiplet components. I choose a diabatic rep- resentation within the multiplet manifold that conserves the character of the multiplet components, the dominant projec- tion of spin angular momentum M, and the matrix cM′M,R0 in Eq. (9) allows for a specific choice of the spin quantiza- tion axis at a fixed nuclear configuration. I use this matrix to specify the reference spin quantization axis for the spin re- laxation dynamics at the equilibrium nuclear configuration. If the matrix cM′M,Rwere allowed to vary with nuclear po- sition and at each nuclear configuration were chosen as the eigenvectors of the matrix, EI′I,R, in Eq. (8), then the result- ing wavefunctions, ∑M,M′\|iSM′,R⟩cM′M,R, would be the adi- abatic electronic wavefunctions of the multiplet components and the associated eigenvalues would be the adiabatic poten- tial energy surfaces of the multiplet components. To derive the Born-Oppenheimer Hamiltonian for the ground-state multiplet, I apply the nuclear kinetic energy op- erator to the Born-Oppenheimer wavefunction in Eq. (9) and obtain: ˆTRΨBO I =\|IR⟩ˆTRχI(R) −i∂µ\|IR⟩ˆpµχI(R)+χI(R)ˆTR\|IR⟩.(10) The notation in Eq. (10) implies a sum over the components of the electronic multiplet. I use the Einstein convention for an unrestricted summation over repeated indeces, and I ex- plicitly include the sum symbol when there are restrictions on the sum indeces and when there is a summation without in- dex repetition. µis the index of the mass-weighted nuclear Cartesian coordinates, and ˆ pµis the nuclear momentum oper- ator of coordinate µ.iis the imaginary unit, and I use atomic units, such that ¯h=1. The action of the nuclear kinetic en- ergy operator gives terms with nuclear derivatives acting only on the nuclear, first term, and the electronic wavefunctions, third term, as well as a mixed term with nuclear derivatives acting on both the electronic and nuclear wavefunctions. I project the molecular Hamiltonian, Eq. (1), in the ground- state multiplet, Eq. (9), and use Eqs. (4) and (10) to derive the Born-Oppenheimer molecular Hamiltonian: ˆHBO I′I=δI′IˆTR−iAI′Iµ,Rˆpµ+DI′I,R+EI′I,R, (11) The Hamiltonian in Eq. (11) is a matrix within the ground- state electronic multiplet space and an operator in nuclear Cartesian coordinate space. The first term is the nuclear ki- netic energy, and the last term is the adiabatic potential en- ergy matrix of the ground-state multiplet, Eq. (8). The second and third terms are non-adiabatic interactions and contain the first-derivative non-adiabatic coupling matrix, AI′Iµ,R= I′ R∂µ\|IR⟩, (12) and the second-derivative non-adiabatic coupling matrix, DI′I,R= I′ RˆTR\|IR⟩. (13) AI′Iµ,Ris the Berry connection,40,46,48,49which in this case has a non-Abelian algebraic structure and arises as a result of non-adiabatic interactions with excited electronic multiplets.4 All inter-multiplet non-adiabatic coupling matrices vanish be- cause of the application of the Born-Oppenheimer approxima- tion. I expand the Berry connection in the spin-orbit interaction to derive explicit expressions for the non-adiabatic coupling matrix elements between multiplet components. I use the per- turbative expansion of the electronic wavefunctions, Eq. (3), and the Baker-Campbell-Hausdorff formula in Eq. (12) and obtain to first order in the SOI: AI′Iµ,R=D I′(0) Re−ˆGR∂µeˆGRI(0) RE =D I′(0) R∂µI(0) RE +D I′(0) Rh ∂µ,ˆG(1) RiI(0) RE .(14) The zero-order term of the Berry connection A(0) I′Iµ,R, the first term in the last equality in Eq. (14), is diagonal in the ground- state multiplet space and is the generalization of the traditional Berry connection that arises from conical intersections.46,47 Because I consider systems away from conical intersections in this paper, A(0) I′Iµ,Rvanishes. The first-order term in the spin- orbit interaction, A(1) I′Iµ,R, the second term in Eq. (14), is the leading contribution to the Berry connection on the ground- state manifold and it couples different multiplet components as a result of the interplay of spin-orbit and non-adiabatic interactions with excited electronic states. Expansion of the commutator in Eq. (14) gives an explicit formula for the first- order Berry connection in terms of zero-order adiabatic elec- tronic wavefunctions: A(1) I′Iµ,R=∑ J̸=IHSOI I′J,R E(0) I′,R−E(0) J,RD J(0) R∂µI(0) RE −∑ J̸=ID I′(0) R∂µJ(0) RE HSOI JI,R E(0) J,R−E(0) I,R.(15) The non-Abelian algebraic structure of A(1) I′Iµ,Rfollows from an application of the Wigner-Eckart theorem50to Eq. (15), which gives an expression for A(1) I′Iµ,Rin terms of the set of non-commuting spin matrices, SM′M,α, with dimension equal to the dimensionality of the ground-state multiplet space: A(1) I′Iµ,R=iSM′M,αa(1) αµ,R, a(1) αµ,R=2∑ j̸=ihSOI i jα,R E(0) i,R−E(0) j,RD j(0) R∂µi(0) RE .(16) The sum over the Cartesian index α=x,y,zis implicit in Eq. (16), and hSOI i jα,Ris the imaginary part of the one- electron, mean-field spin-orbit coupling operator as given in Appendix A. The functions a(1) αµ,Rare the three Cartesian com- ponents of a vector potential associated with each nuclear de- gree of freedom µ.a(1) αµ,Rare real functions of the nuclear positions, which ensures that A(1) I′Iµ,Ris an anti-Hermitian ma- trix. Note that with the definition of the Berry connection in Eq. (12), the product −i¯hA(1) I′Iµ,Ris a Hermitian matrix. Thelast equality of Eq. (16) expresses the first-order vector po- tential, a(1) αµ,R, in terms of matrix elements of zero-order adia- batic electronic wavefunctions,{jSS}(0) RE =j(0) RE , with spin quantum number Sequal to the spin quantum number of the ground-state multiplet and maximum projection of the spin angular momentum MS=S.i(0) RE is the maximum-spin- projection ground-state electronic wavefunction and the sum in Eq. (16) runs over the maximum-spin-projection excited- state electronic wavefunctions. The benefit of the Wigner- Eckart theorem is to separate the exact spin-component struc- ture from the electronic matrix elements and to express all rel- evant quantities in terms of electronic wavefunctions of fixed projection of spin angular momentum. I implement Eq. (16) together with state-of-the-art density functional theory ap- proximations to evaluate the first-order Berry connection in Sec. III. Similar expansion of the second-derivative non-adiabatic coupling matrix, Eq. (13), in the spin-orbit interaction gives: DI′I,R=D I′(0) RˆTRI(0) RE +D I′(0) Rh ˆTR,ˆG(1) RiI(0) RE =D(0) I′I,R+D(1) I′I,R.(17) The zero-order term of the second-derivative non-adiabatic coupling matrix, D(0) I′I,R, is similarly diagonal in the ground- state multiplet space and vanishes away from conical inter- sections. The first-order contribution, D(1) I′I,R, can be entirely expressed in terms of the first-order Berry connection: D(1) I′I,R=−∑ µ1 2D I′(0) R∂2 µˆG(1) RI(0) RE −∑ µD I′(0) R∂µˆG(1) R∂µI(0) RE =−1 2∑ µ∂µA(1) I′Iµ,R.(18) The first line of Eq. (18) follows from the expansion of the commutator with the nuclear kinetic energy in Eq. (17), and in the second line I apply an identity proved in Appendix B together with the definition of the first-order Berry connection in Eq. (14). Collecting the results from the perturbative treatment of the non-adiabatic coupling matrices, Eqs. (14) and (18), and the adiabatic potential energy matrix, Eq. (8), gives an expansion of the Born-Oppenheimer molecular Hamiltonian, Eq.(11), in the spin-orbit interaction: ˆHBO I′I=δI′IˆTR−iA(1) I′Iµ,Rˆpµ−1 2∑ µ∂µA(1) I′Iµ,R +δI′IE(0) I,R+E(1) I′I,R+E(2) I′I,R.(19) Uniting the Berry connection terms with the nuclear kinetic energy gives a covariant form of this Hamiltonian: ˆHBO I′I=−1 2∑ µ ∂µ+A(1) I′Iµ,R2 +δI′IE(0) I,R+E(1) I′I,R+E(2) I′I,R.(20)5 Eq. (20) and Eq. (19) agree to first-order in the Berry connec- tion upon expansion of the kinetic energy term. I use Eq. (16) and Eq. (8) in Eq. (20) to derive the final form of the new molecular spin Hamiltonian in terms of spin operators, ˆSα, acting in the ground-state multiplet space: ˆHspin=−1 2∑ µ ∂µ+iˆSαa(1) αµ,R2 +E(0) R+µBBαg(2) αβ,RˆSβ+ˆSαD(2) αβ,RˆSβ.(21) Eq. (21), the extended molecular spin Hamiltonian, is a main result of this paper. It contains a novel, spin-vibronic vec- tor potential that derives from the first-order Berry connec- tion, as well as the traditional spin Hamiltonian terms that derive from the adiabatic potential energy matrix. The spin- vibronic vector potential gives rise to an effective magnetic field on the ground-state multiplet, and I explore its effects on the spin-vibrational dynamics in the rest of this paper. The spin-orbit interaction makes a leading first-order contribution to the vector potential, unlike the adiabatic molecular Zeeman and zero-field splitting terms, which are of second order in the spin-orbit and orbital Zeeman interactions. I expect that the spin-vibronic vector potential dominates the spin-vibrational dynamics in weak external magnetic fields, and I demonstrate its fundamental role for the spin relaxation dynamics of a pro- totypical molecular qubit. The external magnetic field also makes a contribution to the Berry connection: at first order the contribution to the Berry connection is diagonal in the ground-state multiplet space and does not couple to the spin- vibrational dynamics. B. Gauge transformation I showed in Sec. II A that the spin-vibrational dynamics in the ground-state multiplet is governed by the molecular spin Hamiltonian in Eq. (21), which contains a novel, spin-vibronic vector potential together with the traditional spin Hamiltonian interactions. In Sec. II B, I transform the representation of the Hamiltonian to bring the molecular spin Hamiltonian to a symmetrized form that is convenient for application in time- dependent perturbation theory. Gauge transformations40,48are nuclear configuration- dependent single-valued unitary transformations of the elec- tronic multiplet wavefunctions: ˜IR =\|IR⟩UI′I,R=∑ M′iSM′,R UM′M,R, (22) which preserve the dynamics generated by the Born- Oppenheimer molecular Hamiltonian, Eq. (11). As a result of the gauge transformation in Eq. (22), the spin-vibronic vec- tor potential, Eq. (12), transforms as a non-Abelian gauge potential,40,48 ˜AI′Iµ,R=U† I′I′′,RAI′′I′′′µ,RUI′′′I,R+U† I′I′′,R∂µUI′′I,R,(23) and the Born-Oppenheimer Hamiltonian undergoes a unitary transformation, ˜HBO I′I,R=U† I′I′′,RHBO I′′I′′′,RUI′′′I,R. (24)Physical observables are independent of the specific choice of electronic representation; they are gauge-covariant quantities, which transform upon gauge transformations like the Hamil- tonian in Eq. (24). For instance, the gauge-covariant nuclear momentum, called kinematic momentum ˆΠI′Iµ, differs from the canonical momentum ˆ pµby the vector potential: ˆΠI′Iµ=δI′Iˆpµ−iAI′Iµ,R, (25) and neither ˆ pµnorAI′Iµ,Rare separately gauge-covariant. The vector potential in Eq. (25) results in the non-commutativity of the kinematic momenta: ˆΠµ,ˆΠν I′I=−FI′Iµν,R, (26) unlike the well-known commutation relation of the canonical momenta, [ˆpµ,ˆpν] =δµν. In Eq. (26), bold symbols denote matrices in the ground-state multiplet space, and FI′Iµν,Ris the field tensor (also called Berry curvature46): FI′Iµν,R=∂µAI′Iν,R−∂νAI′Iµ,R+ Aµ,R,Aν,R I′I,(27) which is a gauge-covariant measure of the strength of the in- duced effective magnetic field on the ground-state manifold. The gauge-covariance of the field tensor requires that UI′IR is both unitary and single-valued. Expansion of the spin- vibronic vector potential to first order in the spin-orbit interac- tion as in Sec. II A gives the first-order spin-vibronic magnetic field tensor: F(1) I′Iµν,R=∂µA(1) I′Iν,R−∂νA(1) I′Iµ,R=iSM′M,αf(1) αµν,R,(28) with f(1) αµν,R=∂µa(1) αν,R−∂νa(1) αµ,R=∂µ∧a(1) αν,R. (29) In Eq. (29), ∧denotes the antisymmetric product, the gener- alization of the cross-product ×to multiple dimensions. With the first-order spin-vibronic vector potential, the kinematic momentum becomes: ˆΠM′Mµ=δM′Mˆpµ+SM′M,αa(1) αµν,R. (30) The nuclear dynamics depends only on gauge-covariant quan- tities, and I derive the Heisenberg equation of motion for the kinematic momentum Eq. (30) with the Hamiltonian Eq. (21) to leading order in the spin-orbit interaction: d2ˆRµ dt2=ˆSα1 2dˆRν dtf(1) ανµ,R−f(1) αµν,RdˆRν dt −∂µE(0) R−µBBα∂µg(2) αβ,RˆSβ−ˆSα∂µD(2) αβ,RˆSβ.(31) The derivation of Eq. (31) uses the equation of motion for the nuclear position δM′MdˆRµ/dt=ˆΠM′Mµwith the veloc- ity operator dˆRµ/dt. The first term on the right-hand side of Eq. (31) is a quantum Lorentz force, the quantum equivalent to the classical Lorentz force51FL=v×Bfor a unit-charge par- ticle moving in three dimensions in a magnetic field B, where Brelates to the electromagnetic tensor Fi jasεi jkBk=Fjiwith6 the completely antisymmetric tensor εi jk. Because of the non- Abelian algebraic structure of the spin-vibronic magnetic field tensor, there is a separate magnetic field for each generator of the associated non-Abelian group SO(3) or SU(2). The rest of the terms on the right-hand side are the forces that originate from the adiabatic potential energy matrix. I use the freedom of gauge transformation to explicitly de- rive the spin-vibronic magnetic field tensor contribution to the spin Hamiltonian in Eq. (21). With the hindsight of the har- monic approximation, I first expand the spin-vibronic vector potential to linear order in the deviations uµfrom a fixed nu- clear configuration R0, which is equivalent to the multipole expansion of the electromagnetic vector potential50: A(1) I′Iµ,R=A(1) I′Iµ,R0+∂νA(1) I′Iµ,R0uν. (32) I construct a single-valued unitary gauge transformation using the generator approach UI′I,R=eΛI′I,Rwith the anti-Hermitian generator function: ΛI′I,R=A(1) I′Iµ,R0uν+1 2∂νA(1) I′Iµ,R0uνuµ, (33) that transforms the molecular spin Hamiltonian to a symmet- ric gauge. I use this gauge transformation in Eq. (23) to obtain the first-order, symmetrized spin-vibronic vector potential: ˜A(1) I′Iµ,R=1 2 ∂νA(1) I′Iµ,R0uν−∂µA(1) I′Iν,R0uν . (34) Gauge transforming Eq. (21) together with the vector potential in Eq. (34) gives to leading order in the spin-orbit interaction the symmetrized molecular spin Hamiltonian: ˆHspin=ˆH(0)+1 4f(1) ανµ,R0ˆSα uν∧ˆpµ +µBBαg(2) αβ,RˆSβ+ˆSαD(2) αβ,RˆSβ,(35) with ˆH(0)=−1 2∑ µ∂2 µ+E(0) R and summation over repeated indeces. The significance of the spin-vibronic vector potential is particularly revealing in the symmetric gauge where it gives rise to a spin-vibrational orbit interaction, second term in Eq. (35), which involves the effective magnetic field f(1) ανµ,R0ˆlνµinduced by the vi- brational angular motion with vibrational angular momen- tum ˆlνµ=uν∧ˆpµ. The spin-vibrational orbit interaction is analogous to the spin-electronic orbit interaction (SOI) and it is similarly symmetric upon both time-reversal and par- ity transformations. Note that the time-reversal invariance requires that the first-order magnetic field tensor f(1) ανµ,Ris real. This implies that in the absence of an external magnetic field, the spin-vibrational orbit interaction preserves Kramers’ theorem50and guarantees that half-integer spin systems have doubly degenerate electronic states.C. Harmonic approximation I carried out a multipole expansion of the spin-vibronic vec- tor potential followed by a gauge transformation to a sym- metric gauge in Sec. II B to derive the spin-vibrational orbit interaction in the molecular spin Hamiltonian Eq. (35). In Sec. II C, I derive a harmonic approximation to the molec- ular spin Hamiltonian in Eq. (35), which I use to obtain an expression for the spin relaxation time in Sec. II D using time- dependent perturbation theory. I start with the case of har- monic vibrational dynamics of molecules in Sec. II C 1, which I extend to molecular crystals in Sec. II C 2. 1. Vibrations of molecules The harmonic approximation to the vibrational dynamics of molecules52is based on the second-order expansion of the ground-state potential energy surface E(0) RinˆH(0)around the equilibrium nuclear geometry R0: ˆHvib=−1 2∑ µ∂2 µ+1 2∂µ∂νE(0) R0uµuν =1 4∑ iωi P2 i+Q2 i(36) I introduce in Eq. (36) the dimensionless normal mode coor- dinates Qiand momenta Piof vibrations with frequencies ωi. The mass-weighted Cartesian displacements and momenta re- late to the normal mode coordinates and momenta by the lin- ear transformation: uµ=∑ iCµ,ir 1 2ωiQi,Qi=b† i+bi ˆpµ=∑ iCµ,ir ωi 2Pi,Pi=i b† i−bi with Cµ,ithe eigenvectors of the mass-weighted Hessian ma- trix. The equations also give the expressions for the quanti- zation of QiandPiin terms of the ladder operators biandb† i. The second-order expansion of the spin-dependent terms in the Hamiltonian Eq. (35) gives: (i) the traditional static spin Hamiltonian at the equilibrium nuclear configuration, which I call the reference spin Hamiltonian: ˆHspin-ref=µBBαg(2) αβ,R0ˆSβ+ˆSαD(2) αβ,R0ˆSβ; (37) (ii) one-phonon (1P) spin-vibrational coupling Hamiltonian with adiabatic (AD) origin, deriving from the adiabatic po- tential energy matrix in Eq. (35): ˆH1P-AD=µB∂νg(2) αβ,R0BαˆSβuν+∂νD(2) αβ,R0ˆSαˆSβuν;(38) (iii) two-phonon (2P) spin-vibrational coupling Hamiltonian with non-adiabatic (NA) origin, deriving from the spin- vibrational orbit interaction in Eq. (35), as well as the adia-7 batic two-phonon spin-vibrational coupling Hamiltonian: ˆH2P-NA=1 4f(1) ανµ,R0ˆSα uν∧ˆpµ , ˆH2P-AD=µB 2∂ν∂µg(2) αβ,R0BαˆSβuνuµ+ 1 2∂ν∂µD(2) αβ,R0ˆSαˆSβuνuµ.(39) Transforming the spin-vibrational coupling Hamiltonians in Eqs. (38) and (39) to normal mode representation gives: (i) the one-phonon spin-vibrational interactions: ˆH1P-AD=µB∂ig(2) αβBαˆSref βQi+∂iD(2) αβˆSref αˆSref βQi; (40) (ii) the two-phonon spin-vibrational interactions: ˆH2P-NA=1 4f(1) αjiˆSref α(Qj∧Pi), ˆH2P-AD=µB 2∂i∂jg(2) αβBαˆSref βQiQj+ 1 2∂i∂jD(2) αβˆSref αˆSref βQiQj.(41) In Eqs. (40) and (41), indeces iandjrefer to the dimension- less normal modes, and fαji=ωi∂jaαi−ωj∂iaαj. 2. Vibrations of molecular crystals The generalization of the harmonic approximation in Sec. II C 1 to molecular crystals requires a straightforward change of notation to account for the wavevector qdepen- dence of the normal mode coordinates Qiq, momenta Piq, and frequencies ωiq:53 ˆHvib=1 4∑ i,qωiq P2 iq+Q2 iq . (42) The sum in Eq. (42) goes over all normal mode branches iand all values of the wavevector qin the first Brillouin zone. The mass-weighted Cartesian displacements uµLand momenta ˆ pµLof atomic coordinate µin the cell centered at Lin terms of the normal mode coordinates and momenta be- come: uµL=∑ i,qCµ,iqs 1 2Nωiqeiq·LQiq,Qiq=b† iq+bi−q ˆpµL=∑ i,qCµ,iqr ωiq 2Neiq·LPiq,Piq=i b† iq−bi−q with Cµ,iqthe eigenvectors of the mass-weighted dynamical matrix. The equations give also the quantization of the crys- tal vibrations in terms of the ladder operators biqandb† iq. The spin relaxation rate model in Sec. II D 3 assumes that the para- magnetic molecular center, a paramagnetic molecule in a crys- tal environment of diamagnetic molecular counterparts, is lo- calized in the unit cell at the origin L=0 and relies only on thedynamics of the optical vibrational modes, for which I adopt an approximation that neglects the dependence of the opti- cal mode eigenvectors Cµ,iqon the wavevector q. Within this approximation, the paramagnetic molecular center displace- ments and momenta in terms of the optical normal modes are given by: uµ=∑ i,qr 1 2NωiCµ,iQiq, ˆpµ=∑ i,qr ωi 2NCµ,iPiq.(43) Substituting Eq. (43) in Eq. (39) gives the expression for the two-phonon spin-vibrational coupling Hamiltonian of the paramagnetic molecular center, ˆH2P-NA=1 N∑ iq,jk1 4f(1) αjiˆSref α Qjk∧Piq ˆH2P-AD=1 N∑ iq,jkµB 2∂i∂jg(2) αβBαˆSref βQjkQiq+ 1 N∑ iq,jk1 2∂i∂jD(2) αβˆSref αˆSref βQjkQiq.(44) Eq. (44) is the extension of the isolated molecule two-phonon spin-vibrational interactions Eq. (41) to a single paramagnetic molecular spin center in a diamagnetic crystal environment. I apply Eq. (44) in Sec. II D to compute the two-phonon Raman process contribution to the spin relaxation time. D. Spin relaxation rate model 1. Rate model assumptions The rate model in this paper focuses on the simulation of the spin relaxation dynamics of S=1/2 molecular qubits near room temperature and at weak to moderate magnetic field in- tensities. The rate model is motivated by experimental mea- surements of the temperature and magnetic field dependence of the spin-lattice relaxation time ( T1time) of molecular qubit crystals under diamagnetic dilution.5–9,11,14,43,44The assump- tions of the rate model are: • the model applies to single paramagnetic molecular centers which are embedded in the crystal environment of diamagnetic molecular homologs. • the model targets early-transition-metal-based molec- ular qubits, allowing the perturbative treatment of the spin-orbit interaction of Sec. II A. It further focuses onS=1/2 molecular qubits, for which the zero-field interaction is unnecessary, leaving the spin-vibronic magnetic-field and molecular Zeeman interactions as the only spin-dependent contributions to the Hamilto- nian in Eq. (35).8 • the model assumes that the spin relaxation dynam- ics near room temperature is determined by the two- phonon spin-vibrational interactions in Eq.(44), which involve a broad range of vibrational modes in the re- laxation dynamics by the virtual absorption-emission of phonons, the two-phonon Raman process. • the model includes only the intra-molecular, optical vibrational modes in the approximation of Eq. (43) as they are more efficient in modulating the intra- molecular spin-dependent interactions. • the model approximates the density-of-states of an op- tical vibrational mode by a Gaussian function centered at the isolated molecule normal mode frequency. The width of the Gaussian function is treated as an empiri- cal parameter. Subsections II D 2 and II D 3 develop the rate model based on time-dependent perturbation theory,20–22the molecular spin Hamiltonian of Sec. II C, and the model assumptions. 2. Reference state The application of time-dependent perturbation theory to evaluate the two-phonon Raman process rate of the model of Sec. II D 1 requires a definition of the initial and final states of the relaxation dynamics.21I assume that as a result of thermal equilibration the initial and final states are the eigenvectors of the reference spin Hamiltonian Eq. (37) at the equilibrium nuclear configuration: ˆU†ˆHspin-ref ˆU=EM ˆSref α=ˆU†ˆSαˆU.(45) This definition of reference states can be straightforwardly im- plemented in the molecular spin Hamiltonian by rotating the quantization axis of the reference spin operators ˆSref αas written in the second line of Eq. (45), where ˆUis the unitary matrix diagonalizing the reference spin Hamiltonian. With this defi- nition, the quantization axis of ˆSref αclosely follows the direc- tion of the external magnetic field for S=1/2 paramagnetic molecular centers with a near isotropic g-tensor. 3. Two-phonon Raman process rate The rate kb→afor the spin-flip transition from the initial state bto the final state avia the two-phonon Raman process, where a phonon in mode jkis absorbed and a phonon in mode iqis emitted, is given by time-dependent perturbation theory as: kb→a=2π ¯h∑ iq,niq∑ jk,njkρniqρnjkδ ¯hωiq−¯hωjk−∆ba a,niq+1,njk−1ˆH2Pb,niq,njk2.(46)In Eq. (46), the sum ranges over all optical vibrational modes according to the rate model assumptions in Sec. II D 1, their associated wavevectors and occupation numbers niq.ρniqis the thermal weight of mode iq,ωiq- the frequency of the mode, and ∆ba- the energy gap between the initial and final states. δis the Dirac delta function that imposes energy con- servation. The virtual absorption-emission process in Eq. (46) allows optical modes with close frequencies to match ener- getically the small energy gap in the spin-flip transition, jus- tifying the leading role of the two-phonon Raman process in the near-room-temperature spin relaxation dynamics. Evalu- ation of the two-phonon spin-vibrational Hamiltonian matrix elements using Eq. (44) gives: (i) for the non-adiabatic two- phonon coupling matrix elements a,niq+1,njk−1ˆH2P-NAb,niq,njk = =i f(1) αjiSref α,ab1 N√njkp niq+1 =H2P-NA ji,ab1 N√njkp niq+1,(47) with the definition of H2P-NA ji,abin the last line of Eq. (47); (ii) for the adiabatic two-phonon coupling matrix elements a,niq+1,njk−1ˆH2P-ADb,niq,njk = =µB∂i∂jg(2) αβBαSref β,ab1 N√njkp niq+1 =H2P-AD ji,ab1 N√njkp niq+1(48) with the definition of H2P-AD ji,abin the last line of Eq. (48). I include only the g-tensor contribution in Eq. (48) because I consider only S=1/2 systems, for which the D-tensor term vanishes. With the model for the optical vibrational modes in Eq. (43), the Hamiltonian matrix elements are independent of the mode wavevector, resulting in the simplified expression for the Raman process rate: kb→a=2π ¯h∑ i,jH2P-NA ji,ab+H2P-AD ji,ab2 1 N2∑ q,kδ ¯hωiq−¯hωjk−∆ba (¯n(ωiq)+1)¯n(ωjk).(49) I carry out the sum over the thermal weight in Eq. (49) to ob- tain the average thermal occupation of the vibrational mode ¯n(ωiq). Furthermore, I carry out the sum over the mode wavevectors in Eq. (49) using the vibrational mode density of states gi(ω) =1 N∑qδ(ω−ωiq)to obtain the final expression for the Raman process rate: kb→a=2π ¯h2∑ i,jH2P-NA ji,ab+H2P-AD ji,ab2 Z dωgi(ω−ωba)gj(ω)(¯n(ω−ωba)+1)¯n(ω).(50) The spin-lattice ( T1) relaxation time characterizes the timescale of thermal equilibration of the spin system as a re- sult of the spin-vibrational interactions with the phonon bath.9 For the case of an S=1/2 spin system that interacts with a phonon bath of reciprocal temperature βwith rate constants kb→aandka→bsatisfying detailed balance, the T1time is given by: 1 T1=kb→a+ka→b≈2k. (51) The last equality in Eq. (51) follows from the smallness of ωbacompared to the vibrational mode frequencies, and kis defined as the expression obtained from Eq. (50) for vanishing ωba. Substitution of Eq. (50) in Eq. (51) gives the expression for the contribution of the two-phonon Raman process to the T1time: 1 T1=4π ¯h2∑ i,jH2P-NA ji,ab+H2P-AD ji,ab2 Z dωgi(ω)gj(ω)(¯n(ω)+1)¯n(ω).(52) The implementation of Eq. (52) requires the vibrational mode density-of-states, for which I adopt the Gaussian approxima- tion from Sec. II D 1 with mode frequency ωiand width σi. With the Gaussian approximation, I arrive at the final ex- pression for the T1spin relaxation time of the rate model of Sec. II D 1: 1 T1=4π ¯h2∑ i,jH2P-NA ji,ab+H2P-AD ji,ab2 1q 2π(σ2 i+σ2 j)e−1 2(ωi−ωj)2 σ2 i+σ2 je−β¯hωji 1−e−β¯hωji2,(53) where ωji=σ2 jωi+σ2 iωj σ2 i+σ2 jis the center of the product of the mode density-of-states Gaussians. To obtain Eq. (53), I assumed that the thermal probabilities change slowly at the scale of the mode density-of-states, allowing their evaluation at the center of the Gaussian product. Section III presents the computa- tion of the spin-vibrational Hamiltonian matrix elements in Eq. (53) based on density functional theory calculations on the isolated paramagnetic molecule. The only empirical pa- rameters that enter in the implementation of Eq. (53) are the widths of the mode density-of-states, for which I use a simple model with mode widths: σi=10cm−1forωi<100cm−1, σi=5cm−1forωi<200cm−1, and σi=1cm−1forωi> 200cm−1. The ab initio estimation of the mode density-of- states requires a computational model of the molecular crystal, which is outside of the scope of the present work. I present re- sults in Sec. IV for the two-phonon Raman contribution to the T1time as a function of temperature and magnetic field orien- tation for the prototypical molecular qubit Cu(II) porphyrin. III. IMPLEMENTATION I derived in Sec. II an extended molecular spin Hamilto- nian that contains a novel, non-adiabatic contribution fromthe spin-vibronic vector potential. In Sec. III, I present the numerical evaluation of the spin-vibronic vector potential via state-of-the-art density functional theory. The starting point of the derivation is Eq. (16), which I eval- uate using linear response unrestricted density functional the- ory. As derived in Refs.23,54, the first-order perturbed Kohn- Sham electronic wavefunctionsψ(1) 0,αE with respect to the spin-orbit interaction are given by: ψ(1) 0,αE =∑ a,i,σUSOI aiσ,α\|ψaσ iσ⟩,(54) where I use the convention that i,j,..denote occupied molec- ular orbitals (MOs) and a,b,..- unoccupied (virtual) MOs in the ground-state determinant, and σis the MO spin projec- tion. The sum in Eq. (54) ranges over all single excitations from the occupied to the unoccupied MOs of both spin pro- jections. The coefficients USOI aiσsatisfy the coupled-perturbed self-consistent-field Kohn-Sham equations: ∑ a,iMσσ b j,aiUSOI aiσ,α=−hSOI bσjσ,α,(55) with the magnetic electronic Hessian Mσ′σ b j,aigiven by: Mσ′σ b j,ai=(εaσ−εiσ)δb j,aiδσσ′+ cHF( aσj′ σiσb′ σ − aσb′ σiσj′ σ ).(56) In Eq. (56), εiσare the Kohn-Sham orbital energies, cHFis the percentage of exact Hartree-Fock exchange in the density functional approximation, and ⟨aσj′ σ\|iσb′ σ⟩are the electron repulsion integrals. hSOI bσjσ,αin Eq. (55) are the matrix ele- ments of the spin-orbit coupling operator as defined in Ap- pendix A. With Eq. (54) the spin-vibronic vector potential in Eq. (16) becomes: a(1) αµ,R=−2∑ a,i,σUSOC iaσ,αR aσR∂µiσR .(57) With generalized-gradient density functional approximations, the coupled-perturbed Kohn-Sham equations have a non- iterative solution, and Eq. (57) gives: a(1) αµ,R=2∑ a,i,σhSOI iσaσ,αR εiσ,R−εaσ,R aσR∂µiσR , (58) which is the result of Eq. (57) for electronic states repre- sented as single Kohn-Sham Slater determinants. To com- plete the evaluation of the spin-vibronic vector potential, I ob- tain the last term in the sum in Eq. (57) from the solution of the coupled-perturbed Kohn-Sham equations for the nu- clear displacements55,56. These equations are conventionally solved when computing analytical second derivatives of the ground-state energy, and using the results of analytical gra- dient theory,55,56I write the nuclear derivatives of the Kohn- Sham MOs as: ∂µiσR =∑ p,iUG piσ,µ\|pσR⟩+∂µ˜iσR ,(59)10 i j ωi ωj\|fi j\| PBE/SV\|fi j\| PBE/TZ\|fi j\| PBE0/SV\|∂i jg\| PBE/SV\|∂i jg\| PBE/TZ\|∂i jg\| PBE0/SV 10 10 126.8 126.8 0.000 0.000 0.000 0.042 0.043 0.061 10 11 126.8 126.8 13.500 16.189 7.995 0.018 0.018 0.027 11 11 126.8 126.8 0.000 0.000 0.000 0.042 0.043 0.061 13 13 204.3 204.3 0.000 0.000 0.000 0.029 0.030 0.039 13 14 204.3 204.4 3.356 2.885 4.824 0.007 0.007 0.009 14 14 204.4 204.4 0.000 0.000 0.000 0.029 0.030 0.039 12 16 196.5 233.6 0.000 0.000 0.000 0.071 0.076 0.117 15 17 212.9 240.6 4.211 4.727 5.345 0.002 0.002 0.002 18 19 288.9 288.9 13.961 14.213 15.205 0.003 0.003 0.012 22 23 387.1 389.1 3.532 3.306 3.092 0.001 0.001 0.002 23 25 389.1 433.4 8.129 8.826 9.726 0.001 0.001 0.002 25 26 433.4 437.8 4.292 4.033 4.351 0.001 0.001 0.002 28 29 454.3 454.3 7.686 8.606 2.158 0.000 0.000 0.000 TABLE I. Two-phonon spin-vibrational Hamiltonian matrix elements for the leading pairs of normal modes. iandjare the indeces of the modes, and ωiandωjare the mode frequencies in cm−1.\|fi j\|2=∑αf2 αi jdenotes the magnitude of the spin-vibronic magnetic field tensor for modes iandjin 10−3cm−1.\|∂i jg\|2=µB∑α̸=β∂i∂jg2 αβi jdenotes the magnitude of the off-diagonal portion of the second derivative g-tensor for modes iandjin 10−3cm−1/T. The matrix elements are computed by: PBE density functional and def2-SVP basis set (PBE/SV), PBE density functional and def2-TZVP basis set (PBE/TZ), and PBE0 density functional and def2-SVP basis set (PBE0/SV). FIG. 1. Major normal modes contributing to the two-phonon spin relaxation time T1for the prototypical Cu(II) porphyrin S=1/2 molecular qubit. The four normal modes are doubly degenerate with normal mode frequencies 126 .8cm−1(upper pair) and 288 .9cm−1 (lower pair). The mode at 126 .8cm−1brings the molecule out of the symmetry plane, whereas the mode at 288 .9cm−1develops within the plane of Cu(II) porphyrin. Arrows portray the relative magnitude and direction of the atomic displacements. Color code: Carbon is dark gray, Nitrogen - blue, Copper - orange, and Hydrogen - light gray. where UG piσ,µare the solutions of the coupled-perturbed Kohn- Sham equations for the nuclear displacement µ, and˜iσR de- notes MOs with frozen orbital coefficients, such that the nu- clear derivative operates only on the atom-centered basis func- tions. Using Eq. (59) in Eq. (57), I arrive at the final formulafor the evaluation of the spin-vibronic vector potential: a(1) αµ,R=−2∑ a,i,σUSOC iaσ,αR UG aiσ,µR+Saiσ,µR ,(60) where Saiσ,µRis the right-hand derivative of the basis overlap matrix in the MO basis. The numerical evaluation of Eq. (60) requires two coupled-perturbed Kohn-Sham response calcu- lations: one response calculation for the spin-orbit interac- tion perturbation USOI iaσ,αR, and a second response calculation for the nuclear displacement perturbations UG aiσ,µR. I imple- ment Eq. (60) in an in-house version of the quantum chem- istry software package ORCA57, for which I adapted already existing highly optimized algorithms for molecular g-tensor calculations23and analytical second derivatives55. I applied in all calculations a widely-used mean-field approximation for the spin-orbit coupling operator26that is already available in the ORCA suite. The implementation of the spin-vibronic vector potential Eq. (60) is consistent with the numerical evaluation of the g-tensor23and the D-tensor54in linear response unrestricted density functional theory. The molecular g-tensor, for in- stance, is expressed as: g(2) αβ,R=−∑ a,i,σUSOI iaσ,αRIm(Laiσ,βR),(61) where Im (Laiσ,βR)is the imaginary part of the electronic or- bital angular momentum operator in the MO basis. Both the spin-vibronic vector potential Eq. (60) and the molecular g- tensor Eq. (61) expressions contain the self-consistent pertur- bation of the Kohn-Sham orbitals with respect to the spin-orbit interaction; the crucial difference is the second perturbing in- teraction: for the spin-vibronic vector potential, these are the nuclear displacement perturbations, whereas for the molecu- lar g-tensor this is the orbital Zeeman interaction. This dif- ference determines the different orders of the two effective11 103105107 PBE/SVP PBE/TZVP PBE0/SVP1031051071/T1, s-1Non-adiabatic rate 10-2100102 50 150 250 T, KTotal rate Adiabatic rateCuPc CuTTP FIG. 2. Two-phonon spin relaxation rate 1 /T1as a function of tem- perature Tin K for Cu(II) porphyrin. Rate is in s−1and the y-axis is a logarithmic scale (base 10). The rate is uniformly averaged over the magnetic field orientation. Magnetic field intensity is B=330mT. Upper panel is the total rate, middle panel - the non-adiabatic con- tribution to the rate, and lower panel - the adiabatic contribution to the rate. Panels share the same temperature range. Black circles rep- resent experimentally measured T1times for Cu(II)-phthalocyanine (CuPc) from Ref.14. Black diamond is an experimetally measured T1time for Cu(II) tetratolylporphyrin (CuTPP) from Ref.43. Color code: orange - PBE/def2-SVP, red - PBE/def2-TZVP, and blue - PBE0/def2-SVP. Note the difference in scale of the adiabatic and the non-adiabatic rate.i j ωi ωj DoS Th. Pr.1 T1 10 11 126.8 126.8 0.056 2.588 1.758 13 14 204.4 204.4 0.282 0.949 1.172 18 19 288.9 289.0 0.282 0.438 5.385 22 23 387.1 389.2 0.102 0.215 0.039 28 29 454.3 454.3 0.282 0.141 0.035 33 34 662.9 662.9 0.282 0.044 0.036 TABLE II. Two-phonon spin relaxation rate contributions for room temperature T=298K, magnetic field intensity B=330mT, and magnetic field orientation θ=90orelative to the molecular axis. iand jare the indeces of the modes, and ωiandωjare the mode frequencies in cm−1. DoS denotes the mode density-of-states contri- bution to the rate using the Gaussian model in Sec. II D. Th. Pr. is the thermal probability contribution to the rate. 1 /T1is the two-phonon spin relaxation rate in MHz. The spin-vibronic matrix elements are calculated at PBE0/def2-SVP level of theory. interactions: the spin-vibronic vector potential is first order in the spin-orbit interaction, whereas the molecular g-tensor is first order in the spin-orbit interaction but also first order in the small orbital Zeeman interaction, resulting in an overall second order expression. The matrix elements that enter in the calculation of the spin relaxation time Eq. (44) require geometric derivatives of both the spin-vibronic vector potential and the molecular g-tensor. I calculate these derivatives by numerical differen- tiation using a central difference approximation and normal mode displacements. I apply a dimensionless mode displace- ment (displacement in units of the zero-point amplitude) of 0.5. The calculation protocol starts with a gas-phase opti- mization of the molecular geometry using the selected ba- sis set and density functional together with: resolution-of- identity approximation for the Coulomb portion of the Fock matrix,58fine grid for the exchange-correlation functional in- tegration, very tight criteria for the self-consistent-field con- vergence, and tight criteria for the geometry convergence, all per ORCA 3.0.0 definitions.57A calculation of the analyti- cal second derivatives follows using the same convergence criteria as applicable. In this work, I computed the molec- ular Hessian using the Perdew-Burke-Ernzerhof (PBE) den- sity functional59and the Ahlrichs def2-TZVP basis set.60–62 I calculated the spin-vibronic magnetic field tensor and the molecular g-tensor second geometrical derivatives using three density functional and basis set combinations: PBE functional and def2-SVP basis set, PBE functional and def2-TZVP basis set, and the hybrid PBE0 functional63,64and def2-SVP ba- sis set. The results of these calculations and the associated spin-lattice relaxation times for the Cu(II) porphyrin molecu- lar qubit are presented in Sec. IV. IV. RESULTS I apply the new molecular spin Hamiltonian to the spin re- laxation dynamics of Cu(II) porphyrin, a prototypical S=1/2 molecular qubit that represents the common core of a homol-12 i j f jix fjiy fjiz1 T1 10 11 0.000 0.000 1.999 1.758 13 14 0.000 -0.003 -1.206 1.172 18 19 0.000 0.000 3.801 5.385 22 23 0.000 0.000 -0.773 0.039 28 29 0.004 0.000 0.539 0.035 33 34 0.000 0.000 0.971 0.036 TABLE III. Spin-vibronic magnetic-field tensor components for the leading contributions to the two-phonon spin relaxation rate for room temperature T=298K, magnetic field intensity B=330mT, and magnetic field orientation θ=90orelative to the molecular symme- try axis. iandjare the indeces of the modes. fjixdenotes the value of the spin-vibronic tensor x-component in 10−3cm−1. 1/T1is the two-phonon spin relaxation rate in MHz. The spin-vibronic matrix elements are calculated at PBE0/def2-SVP level of theory. ogous series of Cu(II)-based molecular qubits. I present the results for the two-phonon spin relaxation time using the spin rate model of Sec. II D, and I calculate the two-phonon spin- vibrational Hamiltonian matrix elements according to the im- plementation in Sec. III using three levels of density func- tional theory to investigate the density functional and basis set dependence of the computed matrix elements. Table I presents the magnitudes of the spin-vibronic mag- netic field tensor and the off-diagonal second derivative g- tensor for the leading pairs of normal modes. The striking result is that the spin-vibronic magnetic field matrix elements are two to three orders of magnitude larger than the deriva- tive g-tensor matrix elements, demonstrating that the domi- nating spin-vibrational coupling mechanism in this class of molecular qubits is of non-adiabatic character. The results in Table I affirm that this conclusion is independent of both the basis set size and the density functional approximation. The basis set dependence of the matrix elements is mild, show- ing that computationally affordable basis sets of double zeta size can be employed to compute spin-vibronic coupling ma- trix elements. The amount of Hartree-Fock exchange in the density functional (the PBE0 density functional has 25% ex- act exchange compared to 0% for the PBE functional) leads to larger deviations in the magnitudes of the matrix elements, but the values are in quantitative agreement with one another, allowing the use of generalized gradient density functionals for initial investigations. The data in Table I further demonstrate that degenerate normal modes have a disproportionately large contribution to the spin-vibrational coupling mechanism in Cu(II) porphyrin. Figure 1 depicts the two normal modes that are the major con- tributors to the spin-vibrational coupling in the qubit. The mode at 126 .8cm−1brings the molecule out of the molec- ular plane, whereas the mode at 288 .9cm−1develops in the molecular plane. Table I shows that the spin-vibronic matrix elements of the 288 .9cm−1in-plane mode increase with the amount of Hartree-Fock exchange in the density functional and this mode becomes the major spin-vibrational coupling channel at PBE0/def2-SVP level of theory. The emerging physical picture from the results in Table I and Fig. 1 is that fitNon-adiabatic rate 0 45 90 θ, deg1/T1, s-1 Adiabatic rate PBE/SVP PBE/TZVP PBE0/SVP10-410-2100102104106FIG. 3. Two-phonon spin relaxation rate 1 /T1as a function of the angle θin degrees relative to the C4axis of Cu(II) porphyrin. Rate is in s−1and the y-axis is a logarithmic scale (base 10). The rate is uni- formly averaged over the polar angle φin the plane of the molecule. Temperature is 100K and magnetic field intensity is 330 mT. Upper panel is the the non-adiabatic contribution to the rate and lower panel - the adiabatic contribution to the rate. Panels share the same angle range. The non-adiabatic rate is practically equal to the total rate - note the difference in the scales of the two panels. Color codes the level of theory: orange - PBE/def2-SVP, red - PBE/def2-TZVP, and blue - PBE0/def2-SVP. The dashed black line is a Asin2(θ)fit to the PBE0/def2-SVP data. The fit to the non-adiabatic contribution coin- cides with the computed results. nearly degenerate normal modes linearly superpose to give vi- brational rotational modes, which strongly interact with the molecular spin via the spin-vibrational orbit interaction (see end of Sec. II B). The spin-vibrational orbit interaction is of non-adiabatic origin and results from the non-Abelian Berry curvature on the ground-state electronic spin multiplet. Figure 2 presents the magnetic-field-orientation averaged two-phonon spin relaxation rate 1 /T1, resulting from the cou- pling to the optical normal modes of the molecular qubit com- puted by the spin relaxation rate model of Sec. II D. The data in Figure 2 confirm that the total two-phonon relaxation rate is determined by the non-adiabatic spin-vibrational orbit interac- tion and the adiabatic contribution to the relaxation rate is five13 orders of magnitude smaller. Similar to the spin-vibronic cou- pling, this conclusion holds true independently of the basis set size and the density functional approximation. The calculated rates are in excellent agreement with experimentally measured T1times for both a powder of Cu(II) phthalocyanine14and sin- gle crystals of Cu(II) tetratolylporphirin43. The theory con- firms that the leading spin relaxation mechanism in Cu(II)- based molecular qubits is the spin-vibrational orbit interac- tion. The rates in Fig. 2 display a characteristic activated dynam- ics with increasing temperature, which results from the suc- cessive thermal population of the doubly degenerate normal modes. Table II presents the major normal mode contribu- tions to the two-phonon spin relaxation rate for T=298K and B=330mT. The data demonstrate that the doubly degener- ate normal modes at 288 .9cm−1, 126.8cm−1, and 204 .4cm−1 are the major channels for two-phonon spin relaxation in the system, and the three normal mode pairs fully determine the order of magnitude of the rate. This conclusion suggests an appealing route to control the spin relaxation rate of S=1/2 molecular qubits by decreasing the molecular symmetry via removal of symmetry axes greater than second order. Figure 3 plots the two-phonon spin relaxation rate as a function of the angle θbetween the molecular C4axis and the external magnetic field for T=100K and B=330mT. The relaxation rates are averaged over the angle φthat de- scribes the rotation of the magnetic field vector around the C4 axis. The results in Figure 3 demonstrate the strong magnetic- field orientation dependence of the two-phonon relaxation rate with distinct functional dependence of the rate contributions onθ: the non-adiabatic contribution, which entirely domi- nates the total rate, shows a clear sin2(θ)dependence (dashed line), whereas the adiabatic contribution goes through a max- imum at 45oand is described poorly by the sin2(θ)func- tion. This orientational dependence of the two-phonon rate allows to distinguish theoretically and experimentally the non- adiabatic and the adiabatic mechanisms of spin relaxation. The computed sin2(θ)dependence of the total rate is in com- plete agreement with the observed sin2(θ)trend of the exper- imentally measured T1times43, which presents an indepen- dent confirmation of the non-adiabatic spin relaxation mech- anism. Furthermore, as a consequence of the orientational dependence, the two-phonon spin relaxation rate for in-plane orientation θ=90oof the magnetic field is several orders of magnitude larger than the rate for perpendicular orientation θ=0o. This result leads to a greater than one T1anisotropy, defined as the ratio of the in-plane to the perpendicular relax- ation rate, that similarly parallels experimental observations. The dependence of the two-phonon rate on the orienta- tion of the external magnetic field can be rationalized based on the data in Table III, which presents the components of the spin-vibronic magnetic field tensor for the leading normal mode pairs in the spin relaxation dynamics at T=298Kand B=330mT. Table III shows that the largest component of the spin-vibronic magnetic field tensor is along the C4axis of Cu(II) porphyrin and the in-plane components are smaller by several orders of magnitude. As a result, the induced effective magnetic field fαi jˆli jby the spin-vibrational orbit interactionis entirely directed along the C4axis of the molecule, and the most efficient relaxation occurs when the molecular spin is oriented perpendicularly to the C4axis in the molecular plane. Provided that the molecular g-tensor at the equilibrium geom- etry is very nearly axial, the molecular spin closely follows the orientation of the external magnetic field, such that the in- plane molecular spin component varies as sin (θ)withθthe angle between the magnetic field vector and the C4axis. The relaxation rate depends on the squared modulus of the spin- vibrational orbit interaction, resulting in the observed sin2(θ) dependence in Fig. 3. V. CONCLUSIONS I derive in the present paper an extended molecular spin Hamiltonian that takes into account both the traditional adia- batic spin-dependent interactions and includes a novel, non- adiabatic spin-vibrational interaction. The derivation employs the Born-Oppenheimer approximation to decouple the dy- namics on different electronic spin manifolds, which induces a non-Abelian Berry connection on the ground-state electronic multiplet. The resulting non-Abelian Berry curvature is the spin-vibrational orbit interaction, which couples the vibra- tional angular momentum to the electronic spin in complete analogy to the spin-electronic orbit interaction. Thus, the vi- brational angular motion induces an effective magnetic field in the spin dynamics on the electronic ground state and vice versa the electronic spin exerts a quantum Lorentz force on the nuclear motion. The spin-vibronic magnetic field tensor that quantifies the strength of coupling between the vibrational an- gular motion and the electronic spin is first order in the spin- orbit interaction and first order in the non-adiabatic interac- tions between the electronic multiplets. As such, this previ- ously unappreciated interaction is expected to dominate the spin relaxation dynamics at weak magnetic field intensities and elevated temperatures. I implement the computation of the matrix elements of the molecular spin Hamiltonian using linear response density functional theory, which allows the ab initio prediction of the strength of the different interactions and the unbiased compar- ison between them. The density-functional implementation requires two different kinds of coupled-perturbed calculations for computing the response of the electronic wavefunctions to the spin-orbit interaction and to the nuclear displacements. I further develop a rate model to estimate the spin relaxation time using the molecular spin Hamiltonian computed for the isolated paramagnetic molecule. The rate model is special- ized to isolated paramagnetic molecular centers in molecular crystals and to elevated temperatures and estimates the spin relaxation time via the two-phonon Raman process with the sole participation of the optical crystal vibrations. The only empirical parameters of the model are the widths of the mode density-of-states, for which I employ a Gaussian approxima- tion. I apply the molecular spin Hamiltonian together with the spin relaxation rate model to the prototypical S=1/2 molec- ular qubit Cu(II) porphyrin. The striking result is that the14 spin relaxation rate near room temperature is determined by the new spin-vibrational orbit interaction, and that the com- puted spin relaxation rate is in remarkable agreement with experimental measurements. This result is independent of both basis set size and density functional approximation. The physical picture that emerges for spin relaxation in Cu(II)- based molecular qubits is that the vibrational angular motion of nearly degenerate normal modes strongly couples to the electronic spin dynamics and is responsible for spin relaxation in the system. Furthermore, the spin relaxation time shows a strong dependence on the orientation of the magnetic field to theC4symmetry axis of the molecule and is distinct for the non-adiabatic and the adiabatic relaxation mechanisms, which allows to distinguish them both theoretically and experimen- tally. The current work lays the foundation for a further investiga- tion of a much broader set of S=1/2 qubits. Furthermore, the approach can be successfully applied to S>1/2 systems that include the emerging class of optically addressable molecular qubits. Taken together, the new molecular spin Hamiltonian formalism together with density functional theory is expected to provide a much-needed theoretical approach to simulate and understand spin relaxation dynamics in molecular qubits. ACKNOWLEDGMENTS I would like to acknowledge start-up funds from Indiana University, Bloomington and Tufts University. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. Appendix A SPIN STRUCTURE VIA THE WIGNER-ECKART THEOREM In Appendix A, I apply the Wigner-Eckart theorem to estab- lish the spin-operator equivalents of the effective interactions in the molecular spin Hamiltonian. The spin-dependent rel- ativistic interactions ˆHSOZ, including the interaction with the weak external magnetic field, consists of: (i) the spin-orbit interaction: ˆHSOI=∑ i,m(−1)mˆF(1) −m(i)ˆS(1) m(i), (62) which is a one-electron operator in the mean-field approxi- mation and contains an orbital-dependent part ˆF(i), a func- tion of the position and momentum of electron i, and a spin- dependent part ˆS(i), the spin operator of electron i; (ii) the Zeeman interaction: ˆHZ=µB∑ i,m(−1)m geB(1) −mˆS(1) m(i)+B(1) −mˆL(1) m(i) ,(63)which includes the interaction of the magnetic field Bwith the electron spin, first term, and the interaction with the orbital motion of the electrons, second term. The sum over iruns over all electrons, and the sum over mruns over all spher- ical components of the spherical tensors of rank (1).geis the anomalous g-factor of the electron. Application of the Wigner-Eckart theorem to the matrix elements of the spin- orbit interaction gives: HSOI JI= jSM′ˆHSOI\|jSM⟩ =∑ m(−1)m S1;MmS1;SM′ ⟨jS\|\|∑iˆF(1) −m(i)ˆS(1)(i)\|\|iS⟩√ 2S+1 =∑ m(−1)mS(1) M′Mm⟨jSS\|∑iˆF(1) −m(i)ˆS(1) 0(i)\|iSS⟩ S =i∑ αSM′MαIm⟨jSS\|∑iˆFα(i)ˆSz(i)\|iSS⟩ S =i∑ αSM′MαhSOI jiα.(64) The second line of Eq. (64) results directly from the Wigner- Eckart theorem, where ⟨S1;Mm\|S1;SM′⟩is a Clebsch-Gordan coefficient in the nomenclature of Sakurai50and⟨jS\|\|·\|\|iS⟩ denotes a reduced matrix element. In the third line, I express the Clebsch-Gordan coefficient using the matrix elements of the electron spin operator, and in the fourth line, I account for the purely imaginary nature of the resulting orbital-dependent operator. The last line of Eq. (64) provides the definition of hSOI jiα. The sum over mis a sum over the spherical compo- nents of the rank-1 spherical tensors as before, and the sum overαis the equivalent sum over the Cartesian components of the vector operators. Similar application of the Wigner-Eckart theorem to the Zeeman interaction reveals the following spin- component structure: ˆHZ JI= jSM′ˆHZ\|jSM⟩ =µBge∑ m(−1)mB(1) −mS(1) M′Mmδji +µB∑ m(−1)mB(1) −m⟨jSS\|ˆL(1) m\|iSS⟩δM′M.(65) Use of Eqs. (64) and (65) in the second order term of Eq. (8), and keeping only the combination terms between the orbital Zeeman and spin-orbit interactions gives the second order contribution to the molecular g-tensor: ∆g(2) αβ=1 S2∑ j̸=0Im⟨0SS\|ˆLα\|jSS⟩Im⟨jSS\|ˆFβˆSz\|0SS⟩ E(0) j−E(0) i +1 S2∑ j̸=0Im⟨0SS\|ˆFαˆSz\|jSS⟩Im⟨jSS\|ˆLβ\|0SS⟩ E(0) j−E(0) i.(66) Equation (66) is the sum-over-states expression of the sec- ond order molecular g-tensor and demonstrates the relation between adiabatic terms in the molecular spin Hamiltonian15 and the traditional static spin Hamiltonian interactions. Simi- lar account of the pure spin-orbit interaction terms in Eq. (8) gives the second-order sum-over-states expression for the molecular D-tensor, which also involves matrix elements of the spin-orbit interaction between spin manifolds that differ by one unit of angular momentum. Finally, the same-spin-manifold matrix elements of the first-order spin-orbit perturbing operator can be written using the Wigner-Eckart theorem as: G(1) JI= jSM′ˆG(1)\|iSM⟩=−∑ J̸=IHSOI JI E(0) J−E(0) I =−iSM′Mα∑ j̸=ihSOI jiα E(0) j−E(0) i=iSM′Mαg(1) jiα,(67) where in the last line I define the Cartesian components of the first-order perturbing operator gjiα.gjiαare purely real func- tions and are symmetric matrices with respect to the orbital indeces. Appendix B PROOF OF IDENTITY IN EQ. 18 I prove in Appendix B an identity that is needed to trans- form the second derivative non-adiabatic coupling matrix in Eq. (18): D I′(0) R∂µˆG(1) R∂µI(0) RE = =∑ J̸=I∂µGI′J J∂µI +∑ J̸=IGI′J ∂µJ∂µI + ∑ K,J,J̸=K I′∂µK GKJ J∂µI =iSM′Mα(∑ j̸=i∂µgi jα j∂µi +∑ j̸=igi jα ∂µj∂µi + ∑ k,j,j̸=k i∂µk gk jα j∂µi ) =iSM′Mα(∑ j̸=i ∂µij ∂µgjiα+∑ j̸=i ∂µi∂µj gjiα+ ∑ k,j,j̸=k ∂µij gjkα ∂µki ) =D ∂µI′(0) R∂µˆG(1) RI(0) RE(68) The first line follows from the expansion of the first-order per- turbing operator, the second line employs Eq. (67), and the third line uses the symmetry of gjiαand of the derivative ma- trix elements in the case of real zero-order wavefunctions. 1A. Gaita-Ariño, F. Luis, S. Hill, and E. Coronado, “Molecular spins for quantum computation,” Nature chemistry 11, 301–309 (2019). 2M. S. Fataftah and D. E. Freedman, “Progress towards creating optically ad- dressable molecular qubits,” Chemical Communications 54, 13773–13781 (2018). 3M. Atzori and R. Sessoli, “The second quantum revolution: role and chal- lenges of molecular chemistry,” Journal of the American Chemical Society 141, 11339–11352 (2019). 4M. J. Graham, J. M. Zadrozny, M. S. Fataftah, and D. E. Freedman, “Forg- ing solid-state qubit design principles in a molecular furnace,” Chemistry of Materials 29, 1885–1897 (2017).5N. P. Kazmierczak, K. M. Luedecke, E. T. Gallmeier, and R. G. Hadt, “T 1 anisotropy elucidates spin relaxation mechanisms in an s= 1 cr (iv) optically addressable molecular qubit,” The Journal of Physical Chemistry Letters 14, 7658–7664 (2023). 6N. P. Kazmierczak and R. G. Hadt, “Illuminating ligand field contributions to molecular qubit spin relaxation via t 1 anisotropy,” Journal of the Amer- ican Chemical Society 144, 20804–20814 (2022). 7N. P. Kazmierczak, R. Mirzoyan, and R. G. Hadt, “The impact of ligand field symmetry on molecular qubit coherence,” Journal of the American Chemical Society 143, 17305–17315 (2021). 8J. M. Zadrozny, J. Niklas, O. G. Poluektov, and D. E. Freedman, “Mil- lisecond coherence time in a tunable molecular electronic spin qubit,” ACS central science 1, 488–492 (2015). 9M. J. Amdur, K. R. Mullin, M. J. Waters, D. Puggioni, M. K. Wojnar, M. Gu, L. Sun, P. H. Oyala, J. M. Rondinelli, and D. E. Freedman, “Chemi- cal control of spin–lattice relaxation to discover a room temperature molec- ular qubit,” Chemical Science 13, 7034–7045 (2022). 10C.-J. Yu, S. V on Kugelgen, M. D. Krzyaniak, W. Ji, W. R. Dichtel, M. R. Wasielewski, and D. E. Freedman, “Spin and phonon design in modular ar- rays of molecular qubits,” Chemistry of Materials 32, 10200–10206 (2020). 11M. S. Fataftah, M. D. Krzyaniak, B. Vlaisavljevich, M. R. Wasielewski, J. M. Zadrozny, and D. E. Freedman, “Metal–ligand covalency enables room temperature molecular qubit candidates,” Chemical Science 10, 6707–6714 (2019). 12M. S. Fataftah, S. L. Bayliss, D. W. Laorenza, X. Wang, B. T. Phelan, C. B. Wilson, P. J. Mintun, B. D. Kovos, M. R. Wasielewski, S. Han, et al. , “Trig- onal bipyramidal v3+ complex as an optically addressable molecular qubit candidate,” Journal of the American Chemical Society 142, 20400–20408 (2020). 13M. J. Graham, J. M. Zadrozny, M. Shiddiq, J. S. Anderson, M. S. Fataftah, S. Hill, and D. E. Freedman, “Influence of electronic spin and spin–orbit coupling on decoherence in mononuclear transition metal complexes,” Jour- nal of the American Chemical Society 136, 7623–7626 (2014). 14A. H. Follmer, R. D. Ribson, P. H. Oyala, G. Y . Chen, and R. G. Hadt, “Understanding covalent versus spin–orbit coupling contributions to temperature-dependent electron spin relaxation in cupric and vanadyl phthalocyanines,” The Journal of Physical Chemistry A 124, 9252–9260 (2020). 15L. Tesi, E. Lucaccini, I. Cimatti, M. Perfetti, M. Mannini, M. Atzori, E. Morra, M. Chiesa, A. Caneschi, L. Sorace, et al. , “Quantum coherence in a processable vanadyl complex: new tools for the search of molecular spin qubits,” Chemical Science 7, 2074–2083 (2016). 16M. Atzori, L. Tesi, E. Morra, M. Chiesa, L. Sorace, and R. Sessoli, “Room- temperature quantum coherence and rabi oscillations in vanadyl phthalo- cyanine: toward multifunctional molecular spin qubits,” Journal of the American Chemical Society 138, 2154–2157 (2016). 17M. Atzori, S. Benci, E. Morra, L. Tesi, M. Chiesa, R. Torre, L. Sorace, and R. Sessoli, “Structural effects on the spin dynamics of potential molecular qubits,” Inorganic chemistry 57, 731–740 (2018). 18M. Atzori, L. Tesi, S. Benci, A. Lunghi, R. Righini, A. Taschin, R. Torre, L. Sorace, and R. Sessoli, “Spin dynamics and low energy vibrations: in- sights from vanadyl-based potential molecular qubits,” Journal of the Amer- ican Chemical Society 139, 4338–4341 (2017). 19M. Goldman, “Formal theory of spin–lattice relaxation,” Journal of Mag- netic Resonance 149, 160–187 (2001). 20J. Van Vleck, “Paramagnetic relaxation times for titanium and chrome alum,” Physical Review 57, 426 (1940). 21R. D. Mattuck and M. Strandberg, “Spin-phonon interaction in paramag- netic crystals,” Physical Review 119, 1204 (1960). 22R. Orbach, “Spin-lattice relaxation in rare-earth salts,” Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 264, 458–484 (1961). 23F. Neese, “Prediction of electron paramagnetic resonance g values using coupled perturbed hartree–fock and kohn–sham theory,” The Journal of Chemical Physics 115, 11080–11096 (2001). 24F. Neese and E. I. Solomon, “Calculation of zero-field splittings, g-values, and the relativistic nephelauxetic effect in transition metal complexes. ap- plication to high-spin ferric complexes,” Inorganic chemistry 37, 6568– 6582 (1998).16 25F. Neese, “Importance of direct spin- spin coupling and spin-flip excitations for the zero-field splittings of transition metal complexes: A case study,” Journal of the American Chemical Society 128, 10213–10222 (2006). 26F. Neese, “Efficient and accurate approximations to the molecular spin-orbit coupling operator and their use in molecular g-tensor calculations,” The Journal of chemical physics 122(2005). 27S. K. Singh, M. Atanasov, and F. Neese, “Challenges in multireference per- turbation theory for the calculations of the g-tensor of first-row transition- metal complexes,” Journal of Chemical Theory and Computation 14, 4662– 4677 (2018). 28J. Gauss, M. Kallay, and F. Neese, “Calculation of electronic g-tensors using coupled cluster theory,” The Journal of Physical Chemistry A 113, 11541–11549 (2009). 29R. McWeeny, “On the origin of spin-hamiltonian parameters,” The Journal of Chemical Physics 42, 1717–1725 (1965). 30M. Pryce, “A modified perturbation procedure for a problem in paramag- netism,” Proceedings of the Physical Society. Section A 63, 25 (1950). 31A. Lunghi and S. Sanvito, “The limit of spin lifetime in solid-state elec- tronic spins,” The Journal of Physical Chemistry Letters 11, 6273–6278 (2020). 32A. Lunghi and S. Sanvito, “Multiple spin–phonon relaxation pathways in a kramer single-ion magnet,” The Journal of Chemical Physics 153(2020). 33A. Lunghi, “Toward exact predictions of spin-phonon relaxation times: An ab initio implementation of open quantum systems theory,” Science Ad- vances 8, eabn7880 (2022). 34M. Briganti, F. Santanni, L. Tesi, F. Totti, R. Sessoli, and A. Lunghi, “A complete ab initio view of orbach and raman spin–lattice relaxation in a dysprosium coordination compound,” Journal of the American Chemical Society 143, 13633–13645 (2021). 35E. Garlatti, A. Albino, S. Chicco, V . Nguyen, F. Santanni, L. Paolasini, C. Mazzoli, R. Caciuffo, F. Totti, P. Santini, et al. , “The critical role of ultra-low-energy vibrations in the relaxation dynamics of molecular qubits,” Nature Communications 14, 1653 (2023). 36A. Albino, S. Benci, L. Tesi, M. Atzori, R. Torre, S. Sanvito, R. Ses- soli, and A. Lunghi, “First-principles investigation of spin–phonon coupling in vanadium-based molecular spin quantum bits,” Inorganic chemistry 58, 10260–10268 (2019). 37D. Reta, J. G. Kragskow, and N. F. Chilton, “Ab initio prediction of high- temperature magnetic relaxation rates in single-molecule magnets,” Journal of the American Chemical Society 143, 5943–5950 (2021). 38J. G. Kragskow, A. Mattioni, J. K. Staab, D. Reta, J. M. Skelton, and N. F. Chilton, “Spin–phonon coupling and magnetic relaxation in single- molecule magnets,” Chemical Society Reviews (2023). 39D. R. Yarkony, “Diabolical conical intersections,” Reviews of Modern Physics 68, 985 (1996). 40C. A. Mead, “The geometric phase in molecular systems,” Reviews of mod- ern physics 64, 51 (1992). 41J. Van Vleck, “On σ-type doubling and electron spin in the spectra of di- atomic molecules,” Physical Review 33, 467 (1929). 42H. Primas, “Generalized perturbation theory in operator form,” Reviews of Modern Physics 35, 710 (1963). 43J.-L. Du, G. R. Eaton, and S. S. Eaton, “Electron spin relaxation in vanadyl, copper (ii), and silver (ii) porphyrins in glassy solvents and doped solids,” Journal of Magnetic Resonance, Series A 119, 240–246 (1996). 44C.-J. Yu, M. D. Krzyaniak, M. S. Fataftah, M. R. Wasielewski, and D. E. Freedman, “A concentrated array of copper porphyrin candidate qubits,”Chemical science 10, 1702–1708 (2019). 45C. M. Moisanu, R. M. Jacobberger, L. P. Skala, C. L. Stern, M. R. Wasielewski, and W. R. Dichtel, “Crystalline arrays of copper porphyrin qubits based on ion-paired frameworks,” Journal of the American Chemi- cal Society 145, 18447–18454 (2023). 46M. V . Berry, “Quantal phase factors accompanying adiabatic changes,” Pro- ceedings of the Royal Society of London. A. Mathematical and Physical Sciences 392, 45–57 (1984). 47C. A. Mead and D. G. Truhlar, “On the determination of born–oppenheimer nuclear motion wave functions including complications due to conical inter- sections and identical nuclei,” The Journal of Chemical Physics 70, 2284– 2296 (1979). 48A. Shapere and F. Wilczek, Geometric phases in physics , V ol. 5 (World scientific, 1989). 49C. A. Mead, “Molecular kramers degeneracy and non-abelian adiabatic phase factors,” Physical review letters 59, 161 (1987). 50J. Sakurai and J. Napolitano, “Modern quantum mechanics. 2-nd edition,” Person New International edition , 39 (2014). 51J. D. Jackson, “Classical electrodynamics,” (1999). 52E. B. Wilson, J. C. Decius, and P. C. Cross, Molecular vibrations: the theory of infrared and Raman vibrational spectra (Courier Corporation, 1980). 53N. W. Ashcroft and N. D. Mermin, Solid state physics (Cengage Learning, 2022). 54F. Neese, “Calculation of the zero-field splitting tensor on the basis of hy- brid density functional and hartree-fock theory,” The Journal of chemical physics 127(2007). 55D. Bykov, T. Petrenko, R. Izsák, S. Kossmann, U. Becker, E. Valeev, and F. Neese, “Efficient implementation of the analytic second derivatives of hartree–fock and hybrid dft energies: a detailed analysis of different ap- proximations,” Molecular Physics 113, 1961–1977 (2015). 56J. Pople, R. Krishnan, H. Schlegel, and J. S. Binkley, “Derivative studies in hartree-fock and møller-plesset theories,” International Journal of Quantum Chemistry 16, 225–241 (1979). 57F. Neese, “The orca program system,” Wiley Interdisciplinary Reviews: Computational Molecular Science 2, 73–78 (2012). 58K. Eichkorn, F. Weigend, O. Treutler, and R. Ahlrichs, “Auxiliary basis sets for main row atoms and transition metals and their use to approximate coulomb potentials,” Theoretical Chemistry Accounts 97, 119–124 (1997). 59J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized gradient approxi- mation made simple,” Physical review letters 77, 3865 (1996). 60A. Schäfer, H. Horn, and R. Ahlrichs, “Fully optimized contracted gaussian basis sets for atoms li to kr,” The Journal of chemical physics 97, 2571– 2577 (1992). 61A. Schäfer, C. Huber, and R. Ahlrichs, “Fully optimized contracted gaus- sian basis sets of triple zeta valence quality for atoms li to kr,” The Journal of chemical physics 100, 5829–5835 (1994). 62F. Weigend and R. Ahlrichs, “Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for h to rn: Design and assessment of accuracy,” Physical Chemistry Chemical Physics 7, 3297– 3305 (2005). 63C. Adamo and V . Barone, “Toward reliable density functional methods without adjustable parameters: The pbe0 model,” The Journal of chemi- cal physics 110, 6158–6170 (1999). 64M. Ernzerhof and G. E. Scuseria, “Assessment of the perdew–burke– ernzerhof exchange-correlation functional,” The Journal of chemical physics 110, 5029–5036 (1999).
2403.14408v1.Spin_orbit_interaction_with_large_spin_in_the_semi_classical_regime.pdf	arXiv:2403.14408v1 [math-ph] 21 Mar 2024Spin-orbit interaction with large spin in the semi-classical regime Didier Robert∗ Abstract We consider the time dependent Schr¨ odinger equation with a cou- pling spin-orbit in the semi-classical regime /planckover2pi1ց0 and large spin num- bers→+∞suchthat /planckover2pi1δs=cwherec>0andδ>0areconstant. The initial state Ψ(0) is a product of an orbital coherent state in L2(Rd) and a spin coherent state in a spin irreducible representation space H2s+1. Forδ<1, at the leading order in /planckover2pi1, the time evolution Ψ( t) of Ψ(0) is well approximated by the product of an orbital and a spin co- herent state. Nevertheless for 1 /2<δ <1 the quantum orbital leaves the classical orbital. For δ= 1 we prove that this last claim is no more true when the interaction depends on the orbital variables. For th e Dicke model, we prove that the orbital partial trace of the projec tor on Ψ(t) is a mixed state in L2(R) for smallt>0. 1 Introduction We consider here the Schr¨ odinger equation for a system of pa rticles with large spin number when the spin and the position variables ar e coupled: i/planckover2pi1∂tΨ(t) =ˆH(t)Ψ(t), Ψ(0) =ϕz0⊗ψn0,inL2(Rd)⊗H2s+1(1.1) whereϕz0,ψn0are respectively Schr¨ odinger, spin coherent states and ˆH(t) =ˆH0(t)+/planckover2pi1/hatwideC(t)·S. (1.2) ˆH0(t) has a scalar /planckover2pi1-Weyl symbol as well as ˆCj(t) andS= (S1,S2,S3) are spin matrices representation in an irreducible space H2s+1of dimension 2s+1, wheresis the total spin number. A particular case is the Pauli equation for the electron ( s=1/2). ˆH=1 2(/planckover2pi1D−A)2+ˆV+/planckover2pi1B·σ, ∗Laboratoire de Math´ ematiques Jean Leray, Universit´ e de N antes, 2 rue de la Houssini` ere, BP 92208, 44322 Nantes Cedex 3, France Email:didier.robert@univ-nantes.fr 1whereD=i−1∇x,x∈R3,A= (A1,A2,A3) is a magnetic potential, Van electric potential, B= (B1,B1,B1) the magnetic ﬁeld in R3;σ= (σ1,σ2,σ3) are the Pauli matrices: σ1=/parenleftbigg0 1 1 0/parenrightbigg , σ2=/parenleftbigg0−i i0/parenrightbigg , σ3=/parenleftbigg1 0 0−1/parenrightbigg . Fors=1/2we haveSk=σk 2. More generally the Weyl symbol of the interaction is /planckover2pi1H1(t,X) where H1(t,X) =/summationdisplay 1≤k≤3Ck(t,X)Sk=C(t,X)·S. If the spin sis ﬁxed, in the semi-classical regime, /planckover2pi1ց0,H1(t,X) is the sub- principal symbol of ˆH(t). Hence we can get a semiclassical approximation at any order for Ψ( t) using generalized coherent states as it will be recalled later. For details see [3], Chapter 14 and [4] for matrix prin cipal symbol H0(t). The spin matrices Skare realized as hermitian matrices in the Hermitian spaceH2s+1such thatSk=dD(s)(σk/2) are deﬁned by the derivative at I of the irreducible representation D(s)inH2s+1. In particular from the Lie algebrasu(2) we have the commutation relations [Sk,Sℓ] =iǫk,ℓ,mSm. (1.3) Ck(t,X) are real scalar symbols in X∈R2d. So the full Weyl symbol of ˆH(t) is the matrix H(t,X) =H0(t,X)I+/planckover2pi1C(t,X)·S, with the spin opera- torS= (S1,S2,S3). For simplicity we shall assume in all this paper that the Weyl symbols H0(t) andC(t,X) are subquadratic [3](p.409). It follows that the quan- tum Hamiltonian ˆH(t) generates a propagator U(t,t0) in the Hilbert space L2(Rd,H2s+1). (see Proposition 123 of [3] easily extended to systems). It is known (see [3] ( Ch.1 and Chap.7) that the Schr¨ odinger c oherent states are labelled by the phase space T∗(Rd) and the spin (or atomic) coherent states are labelled by the sphere S2. So it is natural to study the semi-classical limit of the propagator with the phase space T∗(Rd)×S2. In particular we can reformulate the propagation for coherent statesϕz⊗ψn labelled by ( z,n)∈R2d×S2whereϕzis a Schr¨ odinger coherent state and ψ(s) n=ψnis a spin coherent state. Let us recall our notations. •Heisenberg translations in L2(Rd): ˆT(z) = exp/parenleftbiggi /planckover2pi1p·ˆx−q·ˆp/parenrightbigg wherez= (q,p)∈Rd×Rd, ˆxis mutiplication by xand ˆp=/planckover2pi1 i∇x. 2•Schr¨ odinger coherent states : ϕz=ˆT(z)ϕ0, ϕ0(x) = (π/planckover2pi1)−d/4exp/parenleftbigg\|x\|2 2/planckover2pi1/parenrightbigg •SU(2) and the sphere S2: let n= (sinθcosϕ,sinθsinϕ,cosθ),0≤θ<π,0≤ϕ<2π; To anyn∈S2we associate the transformation in SU(2), g=gn= exp/parenleftbigg iθ 2((sinϕ)σ1−(cosϕ)σ2)/parenrightbigg (1.4) •irreducible representations and spin coherent states : letsbe an half integer and Dsthe irreducible representation in the Hilbert space H2s+1of dimension 2 s+1. Ton∈S2is associated the spin coherent states1 ψn=Ds(gn)ψ0, whereψ0is a unit eigenvector of S3inH2s+1with minimal eigenvalue −s. Let us consider uncorrelated initial states Ψ z0,n0=ϕz0⊗ψ(s) n0, where ϕz0is a Schr¨ odinger (orbital) coherent state and ψ(s) n0a spin coherent state. When the spin number sis ﬁxed we have the following result proved in [2] and revisited in [3]. Theorem 1.1. [2] For the initial state Ψz0,n0=ϕΓ0z0⊗ψ(s) n0we have U(t,t0)Ψz0,n0= ei /planckover2pi1S(t,t0)+isα(t)ϕΓ(t) z(t)⊗ψ(s) n(t)+O(√ /planckover2pi1) (1.5) wherez0/ma√sto→z(t)is the classical ﬂow for the Hamiltonian for H0(t),S(t,t0) is the classical action, the covariance matrix Γ(t)is computed from the dy- namics generated by the linearized ﬂow of H0(t)andα(t)is a real phase computed from the spin motion n(t)which satisﬁes the Landau-Lifshitz [7] equation ∂tn(t) =C(t,z(t))∧n(t). (1.6) Remark 1.2. In Theorem 1.1 the motion of the spin depends on the motion along the orbit but the orbit motion is independent of the spi n. Remark 1.3. Theorem 1.1 is a particular case of propagation of coherent states for systems with a subprincipal term of order /planckover2pi1and a principal term H0(t)(here scalar) without crossing eigenvalues. Moreover one c an get a complete asymptotic expansion in power of√ /planckover2pi1modO(/planckover2pi1∞). For smooth crossings some non-adiabatic results are proved in [4]. 1this construction follows from the computation of the isotr opy subgroups of the SU(2) action: we get roughly SU(2)/ISO≈S2(see [3] for details). 3The ﬁrst new result in this paper is a generalization of Theor em 1.1 for s→+∞and/planckover2pi1ց0 such that s/planckover2pi1δ=cwhere 0< δ <1 andc >0 are constants. Let us denote κ:=/planckover2pi1swhich is here a small positive parameter forδ<1. Theorem 1.4. Let us assume that 0<δ<1. For the initial state Ψz0,n0= ϕz0⊗ψ(s) n0, the solution of (1.1)for the Hamiltonian (1.2)satisﬁes: U(t,t0)Ψz0,n0= ei /planckover2pi1S(t)+isα(t)ϕΓ(t) z(t)⊗ψ(s) n(t)+O(√ /planckover2pi1+κ)),(1.7) where the dynamics of the coherent states satisﬁes the follo wing system of equations: ˙q=∂pH0(t,q,p)+κ∂pC(t,q,p)·n(t) ˙p=−∂qH0(t,q,p)−κ∂qC(t,q,p)·n(t) ˙n(t) =C(t,q,p)∧n(t).(1.8) MoreoverS(t)is the action along the trajectory z(t)andα(t)is the action along the trajectory n(t)of the classical spin, in the time interval [t0,t]. Let be (zκ t,nκ t) the ﬂow satisfying (1.8) with zκ 0=z0,nκ 0=n0. Corollary 1.5. Let be1/2>ε>0small. Then we have •ifs≤c/planckover2pi1−1/2+εthen the Theorem 1.4 is valid by taking κ= 0in(1.8), with the error term O(/planckover2pi1ε). So in this case the quantum orbital follows the classical trajectory for H0(t)at the leading order in /planckover2pi1. •Ifs≈c/planckover2pi1−1/2−εthen the quantum orbital motion depends on the spin motion: in general we cannot take κ= 0in(1.8). If∇XC(t0,z0)/\e}atio\slash= 0 the orbital Gaussian ϕΓ(t) z(t)follows the new trajectory zκ tand notzκ t. Moreover there exist τ >0,c>0, such that \|zκ t−z0 t\| ≥c\|t−t0\|/planckover2pi11/2−ε,for\|t−t0\| ≤τ. (1.9) (see Lemma A.1) Remark 1.6. Notice that the coherent state ϕΓ z(x)is localized in any disc {\|X−z\| ≤c/planckover2pi11/2−ε},c>0,ε>0, in the phase space R2d. More precisely its Wigner function WϕΓz(X)satisﬁes for some µ>0,C >0, \|WϕΓz(X)\| ≤(π/planckover2pi1)−de−µ /planckover2pi1\|X−z\|2. And its Husimi function HϕΓz(X)satisﬁes, for some C′>0andµ′>0, 0≤ HϕΓz(X)≤C′/planckover2pi1−de−µ′ /planckover2pi1\|X−z\|2, (for more properties of the Husimi functions see [3], sectio n 2.5). Hence from (1.9)we get that HϕΓ(t) z(t)(X)isO(/planckover2pi1∞)in a neighborhood of size /planckover2pi11/2−ε/2ofz0 tfor0<t1<t<t+τ. 4Let us consider now the case κ>0 ﬁxed and /planckover2pi1ց0. We shall see that Theorem1.4 cannot besatisﬁed ingeneral. Inparticular for theDicke model considered by Hepp-Lieb (for /planckover2pi1= 1) [5] (See also [11] for a recent review). ˆHDic=/planckover2pi1ωca†a+ω3ˆS3+λ√ N(a†+a)ˆS1, In quantum optics this model describes the interaction betw een light and matter where the light (photons) is a ﬁeld operator. Here we c onsider a toy model with an quantum harmonic oscillator for photons and Natoms of matter being in two spin levels. ωc>0,ω3≥0,N= 2s+1is the dimension of the spin states space, λ>0 a coupling constant, a=1√ 2/planckover2pi1(x+/planckover2pi1∂x),a†=1√ 2/planckover2pi1(x−/planckover2pi1∂x) are the usual creation and annihilation operators (satisfying [ a,a†] = 1) and ˆSj=/planckover2pi1Sj. Here the Fock space is simply L2(R). A more explicite expression for ˆHDicconsidered here is the following ˆHDic=ωc 2(−/planckover2pi12∂2 x+x2)+ω3ˆS3+2λ√ N/planckover2pi1xˆS1. (1.10) With this elementary model we get a contradiction with Theor em 4.7 of [2]. In the regime /planckover2pi1≈s−1,sր ∞the orbital trajectory blows up into a mixed state fort>0 (”decoherence of the orbital state”). Remark 1.7. Why to consider large spin quantum systems? Notice that for Natoms of spin 1/2 the spin Hilbert space should be of dimen- sion2Ncorresponding to the tensor product ⊗ND1/2whereD1/2is the rep- resentation of degree 2 of SU(2)for the spin 1/2. Using the Clebsch-Gordon formula we see that the representation ⊗ND1/2contains the irreducible rep- resentation DN/2ofSU(2)of maximal degree N+ 1corresponding to an eﬀective spin s=N 2. So that in this settting the large spin limit is also the large number of atoms limit (thermodynamic limit). Theorem 1.8. Let be0<κ=/planckover2pi1s,Ψ(0) =ϕz0(x)ψn0andΨ(t) = e−it /planckover2pi1ˆHDicΨ(0) withΨ(0) =ϕz0⊗ψn0. Then there exists c0>0such that for any 0<µ<1/2and/planckover2pi1ց0, we have tr[trH2s+1ΠΨ(t)]2≤(1+c0κt2)−1/2+O(/planckover2pi1µ), (1.11) whereΠΨis the projection on the pure state Ψ∈L2(R)⊗H2s+1,trH2s+1ΠΨ is the partial trace of ΠΨinH2s+1and the last trace in (6.7)is computed for operators in the Hilbert space L2(R,C). Remark 1.9. The previous result shows that for a spin sof order /planckover2pi1−1the orbital evolution is transformed into a mixed state as the sp in interaction is 5switched on and /planckover2pi1is small enough. For a density matrix ˆρ,P[ˆρ] := tr[ˆρ2]is called the purity of the density matrixˆρ. Here it is applied for ˆρs(t) := tr H2s+1ΠΨ(t). This is related with the von-Neumann entropy which is deﬁned as SvN[ˆρ] =−tr[ˆρlog ˆρ]. So we have easily SvN[ˆρ]≥1−P[ˆρ]. For a pure state SvN[ˆρ] = 0andP[ˆρ] = 1. Then from (6.7)we get that the density matrix trH2s+1ΠΨ(t)has a positive von Neumann entropy for t>0and/planckover2pi1>0small enough. In the analysis of the Dicke model one consider that the orbita l part is an open sub-system of the closed total system (orbital+spin). For other open systems and time evolution of coherent states (like the ”Sch r¨ odinger cat”) we refer to [3], Chapter 13 for more details. 2 Preliminaries 2.1 Reduction to the interaction propagator It is convenient to annihilate the scalar part H0(X) by considering the inter- action representation for the propagator U(t,t0) of the Hamiltonian ˆH(t). Hence we have U(t,t0) =U0(t−t0)V(t,t0), where U0(t−t0) = e−it−t0 /planckover2pi1ˆH0and the propagator V(t,t0) must satisfy i/planckover2pi1∂tV(t,t0) = (/hatwideCI(t)·S)V(t,t0). For 1≤k≤3 the Weyl symbols CI,k(t,X) is computed by the Egorov Theorem. In particular its principal term is given by CI,k(t,X) =Ck(t,Φt−t0 0(X)). So for simplicity, in what follows we shall assume that H0≡0 and consider the simpler interaction spin-orbit Hamiltonian ˆHint(t) =/planckover2pi1/hatwideC(t)·S. For the Dicke model H0is the harmonic oscillator so we have: Φt 0(x,ξ) = (cos(ωct)x(0)+sinωctξ(0),−sin(ωct)x(0)+cos(ωct)ξ(0), henceHint(t,x,ξ) = (cos(ωct)x+sin(ωct)ξ)S1+/planckover2pi1ω3S3. 2.2 A realization of spin- srepresentation Because our aim is to perform explicit computations for the s pin side we choose to work with a concrete representation (see for examp le [3], chap.7). We assume here that V(s)=H2s+1, the complex linear space of homogenous polynomials of degree 2 sin two variables ( z1,z2)∈C2.H2s+1is an Hermi- tian space for the scalar product deﬁned such that the monomi als (named Dicke states): D(s) m=zs+m 1zs−m 2/radicalbig (s+m)!(s−m)!, 6where−s≤m≤sandmis an integer or half an integer according sis. InH2sthe spin operators are realized as S3=1 2(z1∂z1−z2∂z2), S+=z1∂z2, S−=z2∂z1 S1=S++S− 2, S2=S+−S− 2i, (2.1) with the commutation relations of the Lie algebra su(2) of the group SU(2): [S3,S±] =±S±. Recall that g∈SU(2) ifg=/parenleftbiggα−¯β β α/parenrightbigg ,α,β∈C,\|α\|2+\|β\|2= 1. Spin coherent states are deﬁned by the action of SU(2) on the Dicke state of minimal weight: D(s) −s. As known, to get a good parametrization of coherent states we choose a family of transformations in SU(2) indices by the sphere S2. This is obtained by computing the isotropy subgroup ISO of t heSU(2) action:g∈ISO if and only if g=/parenleftbiggα0 0 ¯α/parenrightbigg ,α= eiψ,ψ∈R. Then we get that the space of orbits SU(2)\ISO can be identiﬁed with the sphere S2(for details see for example [3]). Let us denote n= (sinθcosϕ,sinθsinϕ,cosθ),0≤θ<π,0≤ϕ<2π; To anyn∈S2we associate the transformation in SU(2), g=gn= exp/parenleftbigg iθ 2(sinϕσ1−cosϕσ2)/parenrightbigg (2.2) whereσ1, σ2are the Pauli matrices which satisfy the commutation relati ons [σk,σl] = 2iǫk,ℓ,mσm (2.3) Fors=1 2we haveSk=σk 2. So ifg/ma√sto→ D(s)gis the representation of SU(2) inH2s+1we have D(s)gn= exp/parenleftbiggθ 2(eiϕS−−e−iϕS+)/parenrightbigg (2.4) Deﬁnition 2.1. The coherent states of SU(2)are deﬁned in the represen- tation space H2s+1as follows: \|n/a\}b∇acket∇i}ht=Ds(gn)D(s) −s:=ψ(s) n. (2.5) 7It is some time convenient to consider a complex parametriza tion of the sphereS2using the stereographic projection n/ma√sto→η, on the complex plane and an identiﬁcation of the coherent states \|n/a\}b∇acket∇i}htwith the state \|η/a\}b∇acket∇i}ht:=ψη deﬁned as follows: \|η/a\}b∇acket∇i}ht= (1+\|η\|2)−jexp(ηS+)\|s,−s/a\}b∇acket∇i}ht, More precisely we denote \|η/a\}b∇acket∇i}ht=\|n/a\}b∇acket∇i}htwith the following correspondence: n= (sinθcosϕ,sinθsinϕ,cosθ), η=−tanθ 2e−iϕ The geometrical interpretation is that −¯ηis the stereographic projection of n. Recall the following expression of gn gn=/parenleftbiggcosθ 2−sinθ 2e−iϕ sinθ 2eiϕcosθ 2/parenrightbigg (2.6) For further investigations we shall need two results: 1) compute the derivative of t/ma√sto→T(gnt) for aC1path onS2. 2) compute the adjoint action of T(gn) onSk, 1≤k≤3. From (2.4) we get ∂ϕT(gn) =i 2T(gn)/parenleftbig sinθ(e−iϕS++eiϕS−)+(1−cosθ)S3/parenrightbig (2.7) i∂θT(gn) =i 2T(gn)(eiϕS−−e−iϕS+) (2.8) Lemma 2.2. Let be aC1path onS2:t/ma√sto→(θt,ϕt). Then we have ∂tT(gn) =i 2T(gn)/parenleftbig sinθ(e−iϕS++eiϕS−)+(1−cosθ)S3/parenrightbig ˙ϕt +1 2T(gn)/parenleftbig eiϕS−−e−iϕS+/parenrightbig˙θt. (2.9) To compute ˙θ, ˙ϕwe use the Riemann model for S2. Let be −¯η∈Cthe stereographic projection of non the equatorial plane, from the south pole nso. The coordinates of nare given by n1=−η+ ¯η 1+\|η\|2, n2=η−¯η i(1+\|η\|2), n3=1−\|η\|2 1+\|η\|2. So we have η=−tan/parenleftbiggθ 2/parenrightbigg e−iϕ,cosθ=1−\|η\|2 1+\|η\|2,sinθeiϕ=−2¯η 1+\|η\|2.(2.10) Then we get easily ˙θ=η˙¯η+ ¯η˙η \|η\|(1+\|η\|2),˙ϕ=i 2/parenleftbigg˙η η−˙¯η ¯η/parenrightbigg . (2.11) 8Lemma 2.3. −i∂tT(gnt) =iT(gnt)(A˙θ+B˙ϕ) with (A˙θ+B˙ϕ) =−1 1+\|η\|2(˙ηS+−˙¯ηS−)+\|η\|2 1+\|η\|2/parenleftbigg˙η η−˙¯η ¯η/parenrightbigg S3 Proof. Standard computations. /square Let us denote Sk(n) =Ds(gn)∗SkDs(gn),k= +,−,3 ork= 1,2,3. In the following Lemma similar results are stated in [6], sec tion 2.6, and proved by a diﬀerent method. Lemma 2.4. If(θ,ϕ)are the coordinates of nonS2, we have, S3(θ,ϕ) = cosθ.S3−sinθ/parenleftbiggeiϕS−+e−iϕS+ 2/parenrightbigg (2.12) S+(θ,ϕ) = eiϕsinθ.S3+cosθ+1 2S++cosθ−1 2e2iϕS−(2.13) S−(θ,ϕ) = e−iϕsinθ.S3+cosθ+1 2S−+cosθ−1 2e−2iϕS+(2.14) S1(θ,ϕ) =/parenleftbigcosθ+1 2+cosθ−1 2cos2ϕ/parenrightbig S1+cosθ−1 2sin2ϕS2 +cosϕsinθS3 (2.15) S2(θ,ϕ) =cosθ−1 2sin2ϕS1+/parenleftbigcosθ+1 2−cosθ−1 2cos2ϕ/parenrightbig S2 +sinθsinϕS3. (2.16) Proof. We write Ds(gn) = eθ 2LwhereL= eiϕS−−e−iϕS+. Let beSone of the spin operator, we have ∂θS(θ,ϕ) =1 2e−θ 2L[S,L]eθ 2L and the commutation relations [L,S3] = eiϕS++e−iϕS+,[L,S−] =−2e−iϕS3,[L,S+] =−2eiϕS3. So we ﬁnd ∂θS3(θ,ϕ) =−1 2(eiϕS−(θ,ϕ)+e−iϕS+(θ,ϕ)) (2.17) ∂θS+(θ,ϕ) = eiϕS3(θ,ϕ) (2.18) ∂θS−(θ,ϕ) = e−iϕS3(θ,ϕ). (2.19) 9hence ∂2 θS3(θ,ϕ) =−S3(θ,ϕ). (2.20) Solving the diﬀerential equation (2.20) we get (2.12). /square Now we come to the spin-coherent states. Le be n0the north pole on S2corresponding to the Dicke state D(s) −s:=ψn0. The following Lemma is basic for our next computations. Lemma 2.5. For any n∈S2we have Sψn=−snψn+/radicalbiggs 2v(n)ψ1,n whereψ1,n=D(s)(gn)D(s) 1−sandv(n) = (v1,v2,v3)∈C3is deﬁned as v1(n) =cosθ+1 2+cosθ−1 2e−2iϕ(2.21) v2(n) =cosθ+1 2i−cosθ−1 2ie−2iϕ(2.22) v3(n) =−sinθe−iϕ(2.23) In particular we have n·v(n) = 0. Proof. The formulas are proved from elementary computations usin g Lemma 2.4 and that S3ψn0=−sψn0,S+ψn0=√ 2sψ1,n0,S−ψn0= 0. In particular we recover here a well known property of the spin c oherent states: n·Sψn=−sψn. /square Remark 2.6. In Lemma 2.5 the formulas have only two terms, a leading term of order sand a second term of order√s. In Lemma 4.5 of [2] the authors claims that the expansion is an asymptotic power ser ie ins−1. This contradicts our computations. It is also convenient to write down the evolution of the spin m atrices on the Riemann sphere, denoting Sk(η) =Sk(n) whereη∈Cis the complex coordinate of n∈S2. Lemma 2.7. S3(η) =1−\|η\|2 1+\|η\|2S3+ηS++ ¯ηS− 1+\|η\|2, S+(η) =−2¯η 1+\|η\|2S3+1 1+\|η\|2S+−η2 1+\|η\|2S− S−(η) =−2η 1+\|η\|2S3+1 1+\|η\|2S−−¯η2 1+\|η\|2S+. 102.3 The classical spin space Let us recall that the sphere S2has a natural symplectic form: σn(u,v) = (u∧v)·n= det(u,v,n), where n∈S2,u,v∈Tn(S2). Let beH:R3→Ra smooth function. Its resriction to S2deﬁned an Hamiltonian vector ﬁeld XHonS2satisfyingdH(Y) =σn(Y,XH),∀Y∈ Tn(S2). So we get the Hamilton equation (named in this context the L andau equation): ˙n=∇H∧n. Incomplexcoordinates(2.10)thecovariant symbol Hc(t,η,¯η) =/a\}b∇acketle{tψη,ˆH(t)ψη/a\}b∇acket∇i}ht of the Hamiltonian ˆH(t) =/planckover2pi1C(t)·S, becomes Hc(t,η,¯η) = (1+ \|η\|2)−1(C3(t)(1−\|η\|2)−(C−(t)−¯η+C−(t)+η)) whereC±=C1±iC2. Recall that the symplectic form on the Riemann sphereˆCisσc= 2i(1 +\|η\|2)−2dη∧d¯η. So the Hamilton equation in ˆC becomes ˙η=(1+\|η\|2)2 2i∂¯ηHc(t,η,¯η). (2.24) This is the Landau-Lifschitz equation (1.6) in complex coor dinates. Following [10] the symplectic two form satisﬁes dθc=σcwhere the one-form isθc=i−1(∂¯ηKd¯η−∂ηKdη) andK(η,¯η) = 2log(1 + η¯η) is the K¨ ahler potential for ˆC. In particular the action Γ for Hcis the one form in ˆCη×Rtsatisfying dΓc=θc−Hcdt. So, along an Hamiltonian path in time interval [0 ,T], the action is given by (see Appendix): γ(T) =/integraldisplayT 0/parenleftbiggℑ(ηt˙¯ηt) 2(1+\|ηt\|2)−Hc(ηt,¯ηt)/parenrightbigg dt (2.25) In the same Appendix it is proved that α(t) =γ(t) whereα(t) is the phase given in Theorem 1.4. 2.4 The Schr¨ odinger coherent states We shall use some well known formulas concerning the Heisenb erg transla- tions operators ˆT(z) and coherent states. Lemma 2.8. Le bet/ma√sto→zt= (qt,pt)aC1path in the phase space R2d. Then we have i/planckover2pi1∂tˆT(zt) =ˆT(zt)/parenleftbigg1 2σ(zt,˙zt)+ ˙qt·/planckover2pi1∇x−˙pt·x/parenrightbigg .(2.26) Lemma 2.9. [3, chap.2] Assume that Aa sub-polynomial symbol. Then for everyN≥1, we have ˆAϕz=/summationdisplay \|γ\|≤N/planckover2pi1\|γ\| 2∂γA(z) γ!Ψγ,z+O(/planckover2pi1(N+1)/2), (2.27) 11the estimate of the remainder is uniform in L2(Rn)forzin every bounded set of the phase space and Ψγ,z=ˆT(z)Λ/planckover2pi1Opw 1(zγ)g. (2.28) whereg(x) =π−n/4e−\|x\|2/2,Opw 1(zγ)is the 1-Weyl quantization of the monomial : (x,ξ)γ=xγ′ξγ′′,γ= (γ′,γ′′)∈N2d. In particular Opw 1(zγ)g=Pγgwhere Pγis a polynomial of the same parity as \|γ\|. 3 The Schr¨ odinger equation for the spin-orbit in- teraction Recall our reduced Hamiltonian ˆH(t) =/planckover2pi1/hatwideC(t)·Sand the Schr¨ odinger equa- tion (i/planckover2pi1∂t−ˆH(t))Ψ(t) = 0,Ψ(0) =ϕz0⊗ψn0. (3.1) Let us consider the following ansatz Ψapp(t) = ei /planckover2pi1γ(t)ϕΓ(t) z(t)⊗ψ(s) n(t) such that for /planckover2pi1δs=κ>0and someµ>0 we have: (i/planckover2pi1∂t−ˆH(t))Ψapp(t) =O(/planckover2pi11+µ) for/planckover2pi1→0 (3.2) hence from Duhamel formula we should have /ba∇dblΨ(t)−Ψapp(t)/ba∇dbl=O(/planckover2pi1µ). IfC(t) depends only on time tthen the ansatz (3.2) gives an exact solution for someγ(t),nt[2]. Proposition 3.1. Let us assume that ˆH(t) =/planckover2pi1C(t)·S, whereC(t)depends only on time. Then the ansatz (3.2)is exact for any s. More explicitly we have Ψapp(t) = eisα(t)ϕz0⊗ψ(s) n(t)(3.3) The classical spin motion n(t)follows the Landau-Lifshitz equation ˙nt=C(t)∧nt. (3.4) and the phase α(t)is the classical spin action computed in (2.25). Proof.This is well known (see a proof in Appendix B). 124 The regime s=c/planckover2pi1−δ,0< δ <1 InthissectionwegiveaproofofTheorem1.4. Wehavetwosmal lparameters /planckover2pi1andκ:=s/planckover2pi1=c/planckover2pi11−δ. In this regime we shall prove that the formula (1.5) is still v alid with the error estimate O(√ /planckover2pi1+κ). Let us consider the ansatz Ψ♯(t) = ei /planckover2pi1S(t,t0)+isα(t)ϕΓ(t) z(t)⊗ψ(s) n(t). (4.1) As for the scalar Schr¨ odinger equation (see [3], Chap. 4) we shall compute a patht/ma√sto→(z(t),n(t)), a covariance matrix Γ( t) and phases S(t,t0),α(t,t0) such that (1.5) is satisﬁed. It is enough to prove the result for the interaction propagat orV(t,t0) such that the orbital Hamiltonian H0(t) is absent. The full result is obtained by applying the scalar propagator of the orbital motion. It is enough to prove, for the norm in L2(Rd,H2s+1), we have i/planckover2pi1∂tΨ♯(t) =ˆH(t)Ψ♯(t)+O(/planckover2pi1(√ /planckover2pi1+κ)). By standard computations we get asymptotic expansions in /planckover2pi1for each term i/planckover2pi1∂tΨ♯(t) andˆH(t)Ψ♯(t) mod an error O(/planckover2pi1(√ /planckover2pi1+κ)). Recall the notations ϕΓ z=ˆT(z)Λ/planckover2pi1gΓ 0, g0(x) = (π)−d/4cΓei<x,Γx>,Λ/planckover2pi1f(x) =/planckover2pi1−d/4f(x/√ /planckover2pi1) and ψn=T(gn)ψn0. (4.2) Using (2.26), (2.2), Lemma 2.3, Lemma 2.9, and Taylor formul a aroundzt at the order 3, the Schr¨ odinger equation for the ansatz (3.2 ) is transformed as follows. /parenleftbigg1 2σ(zt,˙zt)−∂tS(t)−s/planckover2pi1˙α(t)+√ /planckover2pi1(˙ qt·∇x−˙ pt·x)/parenrightbigg g0⊗ψn0+ (4.3) g0⊗/planckover2pi1(A˙θ+B˙ϕ)ψn0 =/summationdisplay 1≤k≤3/parenleftBig Ck(t,zt)+√ /planckover2pi1∇XCk(t,zt)Opw 1(X)/parenrightBig g0⊗/planckover2pi1Sk(nt)ψn0+O(/planckover2pi1(√ /planckover2pi1+κ)). But we have (˙qt·∇x−˙pt·x)g0= (i˙q−˙p)·xg0,and Opw 1(a·X)g0= (α+iβ)·xg0 wherea·X=α·x+β·ξ. From Lemma 2.3 we have A˙θ+B˙ϕ) =−√ 2s˙ηt 1+\|ηt\|2ψ1,n0+s\|ηt\|2 1+\|ηt\|2/parenleftbigg˙ηt ηt−˙¯ηt ¯ηt/parenrightbigg ψn0.(4.4) 13From Lemma 2.7 we have also S3(η)ψn0=−s1−\|η\|2 1+\|η\|2ψn0+√ 2sη 1+\|η\|2ψ1,n0, S+(η)ψn0= 2s¯η 1+\|η\|2ψn0+√ 2s1 1+\|η\|2ψ1,n0 S−(η)ψn0= 2sη 1+\|η\|2ψn0−√ 2s¯η2 1+\|η\|2ψ1,n0 Now we get the equations to compute the ansatz by identifying the co- eﬃcients of ( /planckover2pi1,κ) in (4.3). Easy computations give the following results. •Projection on g0⊗ψn0: the coeﬃcient of /planckover2pi10gives the classical action S(t,t0) andthecoeﬃcient of κgives thespinaction α(t). Inparticular. we get ˙α(t) =ℑ(ηt˙¯ηt) 2(1+\|ηt\|2)−Hc(ηt,¯ηt), (4.5) whereHc(η,¯η) =/a\}b∇acketle{tψη,C(t)·Sψη/a\}b∇acket∇i}ht. •Projection on xg0⊗ψn0determinestheHamilton equation fortheorbit zt. •projection on xg0⊗ψ1,n0determines the Landau-Lifshitz equation for n(t). Notice that for the interaction dynamics the covariant matr ix Γ is con- stant, itdependsonlyontheorbitaldynamicsfor H0(t)notintheinteraction with the spin, contrary to the orbital motion when κ≫√ /planckover2pi1. Using the same method as in the scalar case considered in [3], Chapter 4 we can complete the proof of Theorem 1.4. 5 The regime /planckover2pi1s=κ=constant The ansatz 3.2 is very natural when considering the classica l analogue of (3.1). We assume /planckover2pi1s=κ>0and we keep /planckover2pi1as our semi-classical parameter. For simplicity assume that κ=1 2. The symbol of ˆH(t) =/hatwideC(t)·ˆS, isH(t,q,p;n) =−κC(t,q,p)·n, deﬁned on T∗(R)×S2, (recall that the covariant symbol of Sis−sn, see [3], prop. 90) we get the classical system of equations ˙q=−1/2∂pC(t,q,p)·n, ˙p= 1/2∂qC(t,q,p)·n, ˙n=C(t,q,p)∧n. (5.1) WhenC(t) depends only on time there is no orbit interaction with the s pin and using section 2.2 we can compute the phase α(t) such that (3.2) is the 14exact solution of (3.1). /square But in the following computations we shall see that (3.2)is not possible if ∇XC(t,X)/\e}atio\slash= 0. For simplicity we assume here that for 1 ≤k≤3,Ck(t) is a linear form on R2d,Ck(t,X) =ak(t)·X=αk(t)·x+βk(t)·ξ,X= (x,ξ). Let us revisit the computations of Section 4 in the particula r case. Denote C±=C1±iC2. Let us denote (0) Land (0)Rthe coeﬃcient of /planckover2pi10on left and right side of (4.3). In the same way we introduce (1 /2)L,Rand (1)L,R. for the coeﬃcients of/planckover2pi11/2and/planckover2pi11. Just compute to get (0)L=1 2σ(zt,˙zt)−˙γ(t)−1 2\|ηt\|2 1+\|ηt\|2/parenleftbigg˙ηt η−˙¯ηt ¯ηt/parenrightbigg (5.2) (0)R=−1−\|ηt\|2 1+\|ηt\|2C3(t,zt)−¯η2 1+\|η\|2C−(t,zt)−¯η2 1+\|η\|2C+(t,zt) The equation (0) L= (0)Rdetermineγ(t) whenzt,ηtare known. In the linear case consider here ∇XC(t) is independent on X. So we have: (1/2)L= (−i˙q−˙p)xg0⊗ψn0−˙ηt 1+\|ηt\|2g0⊗ψ1,n0 (1/2)R=−/parenleftbigg1−\|ηt\|2 1+\|ηt\|2(α3(t)+iβ3(t))+ηt 1+\|ηt\|2(α+(t)+iβ+(t)) +¯ηt 1+\|ηt\|2(α−(t)+iβ−(t))/parenrightbigg ·xg0⊗ψ0 +/parenleftbigg C3(t)ηt 1+\|ηt\|2+C−(t)1 1+\|ηt\|2 −C+(t)¯η2 t 1+\|ηt\|2/parenrightbigg g0⊗ψ1,n0 (5.3) We denote C(t) :=C(t,zt). From the equation (1 /2)L= (1/2)R, using that the statesxg0⊗ψ0andg0⊗ψ1,n0are orthogonal in H2s, we obtain a system of coupled equations which determines a trajectory ( zt,ηt) inR2d×S2. In particular we get again for ntthe Landau equation (3.4) for C(t,zt) but here the time dependent equation for ztdepends on nt. Now let us consider (1) L,R. First we have (1) L= 0. Let us compute (1) R which is a term supported by the the mod xg0⊗ψ1,n0. (1)R=−/parenleftbiggηt 1+\|ηt\|2(α3(t)+iβ3(t))+1 1+\|ηt\|2(α+(t)+iβ+(t)) +1 1+\|ηt\|2(α−(t)+iβ−(t))/parenrightbigg ·xg0⊗ψ1,n0. Then we get that (1) R/\e}atio\slash= 0 if∇XC(t)/\e}atio\slash= 0. 15Remark 5.1. The section 4.2 of [2] use in a fondamental way their Lemma 4.5 concerning the action of the spin operators on spin coher ent states. But the conclusion of this Lemma is false as we have shown in (4.5). In particu- lar the√sterm in our Lemma 2.5 is at the origin of the wrong ansatz (3.2) in the regime /planckover2pi1s=κ. 6 More on the Dicke model 6.1 Preliminary computations and reductions A simpler form of the interaction Hamiltonian for the Dicke m odel is ˆHDint(t) = ((cost)x+(sint)/planckover2pi1Dx)/planckover2pi1S1. (6.1) More generally consider the Hamiltonian ˆH(t) = (α(t)x+β(t)Dx))A, where Ais a bounded Hermitian operator in the Hilbert space H. Let be U(t) the propagator for the Schr¨ odinger equation with initial data att= 0. i∂tΨ(t) =ˆH(t)Ψ(t) ThenwehavethefollowinglemmarelatedwithCampbell-Haus dorﬀformula. Lemma 6.1. There exist two scalar functions c(t)and˜c(t)such that U(t) = e−i 2c(t)A2e−ib(t)DxAe−ia(t)xA(6.2) wherec(t) =/integraltextt 0α(τ)b(τ)dτ,a(t) =/integraltextt 0α(τ)dτ,b(t) =/integraltextt 0β(τ)dτ, and U(t) = e−i 2˜c(t)A2e−i(b(t)Dx+a(t)x)A(6.3) where˜c(t) =c(t)−a(t)b(t). Proof.Letusﬁrstremarkthat (6.3)isadirectconsequenceof theCa mpbell- Hausdorﬀformula. Soitisenoughtoprove(6.2). Let V(t) = e−ib(t)DxAe−ia(t)xA. By a direct computation and a commutation we get that that i∂tV(t) = (˙a(t)xA+˙b(t)Dx)V(t)+a(t)b(t)A2. So we get (6.2) with ˙ c(t) =α(t)b(t). Now from the Campbell-Hausdorﬀ formula, we get (6.3) with ˜ c(t) =c(t)−a(t)b(t). /square. Let us assume that Ψ(0 ,x) =ϕz0(x)ψn0, the product of a Schr¨ odinger coherent state and a spin coherent state. Let us assume ﬁrst t hatβ= 0. Then the time evolution is given by Ψ( t,x) =ϕz0(x)e−ia(t)xS1ψn0. We have seen (Proposition 3.1) that e−ia(t)xS1ψn0= eisγ(t,x)ψn(t,x). 16The classical spin n(t,x) is given here by n1(t,x) =n0,1= cosθ0, n2(t,x) = sinθ0cos(θ(t,x)), n3(t,x) = sinθ0sin(θ(t,x)), whereθ(t,x) =θ0−a(t)x. The phase α(t,x) is given by ˙α(t,x) =α(t)x 1+\|η\|2/parenleftbiggn1(t,x) 1+n3(t,x)+\|η\|2n2(t,x) 1+n3(t,x)/parenrightbigg (6.4) withγ(t0,x) = 0 and \|η\|2=1−n2 3 (1+n3)2. So we have the formula Ψ(t,x) =ϕz0(x)eisγ(t,x)ψn(t,x). Ifsis frozen ( /planckover2pi1-independent ) we can use the Taylor formula and we get /ba∇dblΨ(t)−ϕz0eisγ(t,q0)ψn(t,q0)/ba∇dblL2(R,Hs)=O(√ /planckover2pi1) which is a compatible with the ansatz (3.2) for µ= 1/2. Fors/planckover2pi1=κ>0we also consider Taylor expansion. γ(t,x) =γ(t,q0)+∂xγ(t,q0)(x−q0)+∂2 xγ(t,q0)(x−q0)2 2+O(\|x−q0\|3) Hence ϕz0(x)eisγ(t,x)=ϕz0(x)eis(γ(t,q0)+∂xγ(t,q0)(x−q0)+∂2 xγ(t,q0)(x−q0)2 2)+O(κ√ /planckover2pi1) in this formula we see that the spin motion add a momentum to th e orbit motion (the linear term in ( x−q0) and a quadratic contribution to the Gaussian. But the Taylor argument to eliminate the x-dependence does not work for the spin part ψn(t,x). The reason is the following. From explicit formulas [3], we k now that \|/a\}b∇acketle{tψn,ψm/a\}b∇acket∇i}ht\|= eslog(1−\|n−m\|2 4), (6.5) hence we have /ba∇dblψn−ψm/ba∇dbl2≈2(1−e−s\|n−m\|2 4) With the localization by ϕz0forxin a neighborhood of q0we have\|n(t,x)− n(t,q0)\|of order/planckover2pi1. Assume that 0 <θ0<π/2 andf= 1. Then there exist c1>,c2>0 such that for t>0 we have /ba∇dblψn(t,x)−ψn(t,q0)/ba∇dbl2≥c1(1−e−c2st2\|x−q0\|2). 17Using that s=κ /planckover2pi1, we get with c3>0, fort0>0 small enough, /ba∇dblϕz0(x)(ψn(t,q0)−ψn(t,x))/ba∇dbl2 L2(Rx,H2s+1)≥c3κt2,∀t∈]0,t0]. This shows that the classical spin n(t,x) has to depend not only on the orbit but also on the position xon the orbit which is not compatible with the ansatz (3.2). Another and more accurate way to understand the last computa tions is related with an entanglement-decoherence phenomenon for t he time evolu- tion of the initial state Ψ(0) = ϕz0⊗ψn0, when the interaction with a large spin system is switched on. We shall see that for t∈[0,t0],κ >0 ands=κ /planckover2pi1, then Ψ(t,x) is anentan- gled state which means that it is not possible to have a decomposition li ke Ψ(t) =ϕ(t)⊗ψ(t) withϕ(t)∈L2(R) andψ(t)∈ H2s+1. WeusethepartialtracesintheHilbertspace L2(R,C)⊗H2s+1=L2(R,H2s+1). Recall that in a tensor product of Hilbert spaces H=H1⊗H2, for a trace class operator AinH, the partial trace of AonH2is the unique trace class operator in H1, denoted tr H2(A), such that the following Fubini identity is satisﬁed for any bounded operator BonH1, trH1(trH2(A)B) = trH(A(B⊗IH2)). We shall use the following invariance property: if Ukare invertible operators inHk, andU=U1⊗U2, then we have trH2(U−1AU) =U−1 1trH2(A)U1. Notice that if H2=Cthen we have H1=H1⊗Cand trC(A) =A. Let Ψ =ψ1⊗ψ2∈ H1⊗H2. and denote by Π Ψthe orthogonal projector on Ψ. Then we have tr H2ΠΨ= Πψ1. Physical interpretation : ifAis a density matrix in H(non negative operator with trace 1) then tr H2(A) is also a density matrix in H1which represents the state of the sub-system H1. Supposethat thetotal system satisﬁes aSchr¨ odingerequat ion withaninitial state Ψ 0=ψ1⊗ψ2(pure state in H). For time t >0 the density matrix ΠΨ(t)of the total system is pure but the density matrix tr H2(ΠΨ(t)) of the subsystem in H1is not necessary a pure state because it is not isolated (decoherence ). When the rank of tr H2(ΠΨ(t)) is≥2 the sub-system (1) can occupy at least two orthogonal pure states with probabiliti es in ]0,1[, like for the Schr¨ odinger cat. Let be Ψ ∈L2(R,C)⊗ H2s+1)≃L2(R,H2s+1). A simple computation gives trH2s+1(ΠΨ)f(x) =/integraldisplay Rf(y)/a\}b∇acketle{tΨ(y),Ψ(x)/a\}b∇acket∇i}htH2s+1dy,∀f∈L2(R). 18In the same way we have also trL2(R)u=/integraldisplay R/a\}b∇acketle{tΨ(x),u/a\}b∇acket∇i}htH2s+1Ψ(x)dx,∀u∈ H2s+1. The orbital density matrix ρO(t) := tr H2s+1(ΠΨ(t)) has the integral kernel K(t,x,y) = eis(γ(t,x)−γ(t,y)˜K(t,x,y) where ˜K(t,x,y) =ϕzt(x)ϕzt(y)/a\}b∇acketle{tψn(t,x),ψn(t,y)/a\}b∇acket∇i}htH2s+1. (6.6) So we have tr(ρO(t)2) =/integraldisplay R2\|˜K(t,x,y)\|2dxdy. Recall that ˆH(t) =α(t)xS1. Lemma 6.2. There exists c0>0such that for any 0<µ<1/2we have /integraldisplay R2\|˜K(t,x,y)\|2dxdy≤(1+c0κt2)−1/2+O(/planckover2pi1µ) (6.7) Notice that ˆK(t) is a non negative operator of trace 1. So let be λj(t) the eigenvalues of ρO(t) (in decreasing order with multiplicities). So if t>0, κ>0 and/planckover2pi1small enough, we get from the Lemma that /summationdisplay jλ2 j(t)</summationdisplay jλj(t) = 1 henceˆK(t) has at least two eigenvalues <1 when the spin interaction is switched on (recall that ˆK(0) is the projector on a pure state). Proofof Lemma 6.2. We use (6.5) to compute the Schwartz kernel (6.6) and Lemma 6. 3 So we get for any µ>0, /integraldisplay R2\|˜K(t,x,y)\|2dxdy≤(π/planckover2pi1)−1/integraldisplay {u2+v2≤r0}e−1 /planckover2pi1(u2+v2+c1κt2(u−v)2)dudv Then a direct computation gives the estimate: /integraldisplay R2\|˜K(t,x,y)\|2dxdy≤(1+2c1κt2)−1/2+O(/planckover2pi1µ) /square Let usnow consider thefull interaction Hamiltonian for the Dicke model: ˆHDint(t) = ((cost)x+(sint)/planckover2pi1Dx)/planckover2pi1S1. By a symplectic transformthis Hamil- tonian is conjugate to the previous one. So Lemma 6.2 is also t rue in this case. 196.2 Proof of Theorem 1.8 Strategy: 1) Reduce to the interaction picture with the propagator V(t,t0): Denote Ψ I(t) =V(t,t0)Ψ0. Then we have Π Ψ(t)=U0(t−t0)ΠΨI(t)U0(t0−t). Hence we have trH2s+1ΠΨ(t)=U0(t−t0)trH2s+1ΠΨI(t)U0(t0−t) ButU0(t−t0) is a unitary operator in L2(R,C) so it is enough to prove the result for Ψ I(t). 2) Proof of the result if ω3= 0 LetusconsiderthetimedependentHamiltonian ˆH(t) = (α(t)x+β(t)Dx))S1. Using (6.3), to compute partial trace on H2s+1it is enough the consider the propagator: W(t) := e−i(b(t)/planckover2pi1Dx+a(t)x)S1 Finally by a symplectic rotation R(t) we get a metaplectic transformation ˆR(t) inL2(R,C) such that ˆR(t)W(t)ˆR∗(t) = e˜α(t)xS1:=˜W(t). We havealready proved abovethedecoherencefor theevoluti on of theinitial state Ψ(0,x) =ϕz0(x)ψn0by˜W(t). So the proof for ω3= 0 is achieved. 3) The case ω3/\e}atio\slash= 0. Now the interaction is computed from the decoupled Hamilton ian ˆK0=ˆH0+/planckover2pi1ω3S3. So we have U(t) =U0(t)V(t), where U0(t) = e−it /planckover2pi1ˆK0and the propagator V(t) must satisfy i/planckover2pi1∂tV(t) =ˆKI(t)V(t),V(0) =I, where ˆKI(t) = (α(t)x+β(t)/planckover2pi1Dx)/parenleftBig cos(ω3t)ˆS1+sin(ω3t)ˆS2/parenrightBig . Like in the case ω3= 0, to compute the partial trace it is enough to consider the caseβ(t) = 0. Hence we can conclude, like for ω3= 0, using Proposition 3.1, (6.5) and the following Lemma Lemma 6.3. Let us consider the Landau equation depending on the param- eterx∈R, ∂tn(t) =C(t,x)∧n(t). Assume that C3≡0,∇xC1(t,x0)/\e}atio\slash= 0andn3(0)/\e}atio\slash= 0. Then there exists c0>0,t0>0,r0>0such that \|n(t,x)−n(t,y)\| ≥c0t\|x−y\|,if\|x0−y\|+\|x0−x\| ≤r0,0≤t≤t0.(6.8) 20Proof.From the Landau-Lifshitz equation we get \|∂s∂xn2(s,u)\| ≥c1>0 fromsandu−x0small enough. Then (6.8) follows. Remark 6.4. It seems possible that these results coud be extended to more general spin-orbit interaction like /hatwideC1S1such that ∇XC1(X)/\e}atio\slash= 0(principal type condition) by constructing a unitary Fourier-integra l operatorUsuch thatU/hatwideC1U∗≈ˆx. A Classical perturbations of Hamiltonians Our aim here is to analyze the consequence on the trajectorie s of the pertur- bation of order κin (1.8) for the critical regime s≈/planckover2pi1−1/2. More generally let us consider times dependent smooth vector ﬁelds in Rm,F(t,X),G(t,X) and the diﬀerential equations ˙Y(t) =F(t,Y(t)),˙X(t) =F(t,X(t))+κG(t,X(t)), X(0 =Y(0) =X0. Lemma A.1. There exists t0>0,κ0>0such that for we have /ba∇dblX(t)−Y(t)/ba∇dbl=κt/ba∇dblG(0,X0)/ba∇dbl+O(κt2),for 0<t<t 0,0<κ<κ 0. In particular if G(0,X0)/\e}atio\slash= 0we have, for c2>0, /ba∇dblX(t)−Y(t)/ba∇dbl ≥c2κt,for 0<t<t 0. Proof.The following argument is standard to compare solutions of d iﬀeren- tial equations. In a ﬁrst step, we get for some c1>0,t0>0, that we have /ba∇dblX(t)−Y(t)/ba∇dbl ≤c1κ,for 0<t<t 0. Thenwe usethevariation equation with the linearized equat ion around Y(t) to get X(t)−Y(t) =κ/integraldisplayt 0R(s,t)G(s,Y(s))ds+O(κ2t2). So we get the Lemma with t0>0 small enough. B Proof of Proposition 3.1 Here we can assume /planckover2pi1= 1. WeareusingcomplexcoordinatesonthespherefortheHamilt onian(Section 2.3): Hc(η,¯η) = (1+ \|η\|2)−1(C3(1−\|η\|2)−(C−¯η+C+η)) Recall that η=in2−n1 1+n3forn= (n1,n2,n3) (η(0,0,−1) =∞) Let us compute the time derivative of the ansatz (3.3), like i n Section 5. 21The classical dynamics of the spin is given by (2.24). By a straightforward computation we get the spin phase α: α(t) =/integraldisplayt 0/parenleftbiggℑ(˙η¯η) 2(1+\|η\|2)−Hc(η,¯η)/parenrightbigg dτ=γ(t). References [1] F.T Arecchi and E. Courtens and R. Gilmore and H. Thomas. Atomic Coherent States in Quantum Optics Physical Review A, Vol.6, No.6, p.2211-2237, December 1972. [2] J. Bolte and R. Glaser. Semi-classical propagation of coherent states with spin-orbit interaction Annales Henri Poincar´ e 6, p. 625-656, 2004. [3] M. Combescure and D. Robert. Coherent states and applications in mathematical physics .Theoretical and Mathematical Physics . Springer, 2nd edition 2021. [4] C. Fermanian-Kammerer, C. Lasser and D. Robert. Propagation of wave packets for systems presenting codimension one cros sings. Comm. Math. Phys., 385(3), p. 1685-1739 (2021). [5] K. Hepp and E. Lieb. On the superradiant phase transition for molecules in a quantized radiation ﬁeld: the Dicke maser mod el. An- nals of Physics. 76(2):p.360-404 (1973). [6] J. Fr¨ ohlich, A. Knowles and E. Lenzmann. Semi-Classical Dynamics in Quantum Spin Systems . Letters in Mathematical Physics, 82(2), p. 275-296 (2007). [7] L.LandauandE.Lifshitz. On the theory of dispersion of magnetic per- meability in ferromagnetic bodies . Phys. Zeitsch. der Sow., 8:153–169, (1935). [8] Y. Kosmann-Schwarzbach. Groupes et sym´ etries ´Editions de l’ ´Ecole Polytechnique (2006) [9] E. Lieb. A proof of an entropy conjecture of Wehrl . Communications in Mathematical Physics, 62p. 1-13, (1978). [10] E. Onofri A note on coherent state representations of Lie groups Jour- nal of Mathematical Physics. Vol. 16, No. 5 p. 1087-1089. (19 75) [11] M. Roses and G. Dalla Torre Dicke model PLOS, https://doi.org/10.1371/journal.pone.0235197 Published: September 4, ’2020). 22
2202.04797v1.Local_breaking_of_the_spin_degeneracy_in_the_vortex_states_of_Ising_superconductors__Induced_antiphase_ferromagnetic_order.pdf	arXiv:2202.04797v1 [cond-mat.supr-con] 10 Feb 2022Local breaking of the spin degeneracy in the vortex states of Ising superconductors: Induced antiphase ferromagnetic order Hong-Min Jiang1and Xiao-Yin Pan2 1School of Science, Zhejiang University of Science and Techno logy, Hangzhou 310023, China 2Department of Physics, Ningbo University, Ningbo 315211,C hina (Dated: February 11, 2022) Ising spin-orbital coupling is usually easy to identify in t he Ising superconductors via an in-plane critical ﬁeld enhancement, but we show that the Ising spin-o rbital coupling also manifests in the vortex physics for perpendicular magnetic ﬁelds. By self-c onsistently solving the Bogoliubov-de Gennes equations of a model Hamiltonian built on the honeyco mb lattice with the Ising spin-orbital coupling pertinent to the transition metal dichalcogenide s, we numerically investigate the local breaking of the spin and sublattice degeneracies in the pres ence of a perpendicular magnetic ﬁeld. Itisrevealedthattheferromagnetic ordersareinducedins idethevortexcoreregion bytheIsingspin- orbital coupling. The induced magnetic orders are antiphas e in terms of their opposite polarizations inside the two nearest-neighbor vortices with one of the two polarizations coming dominantly from one sublattice sites, implying the local breaking of the spi n and sublattice degeneracies. The ﬁnite- energy peaks of the local-density-of-states for spin-up an d spin-down in-gap states are split and shifted oppositely by the Ising spin-orbital coupling, and the relative shifts of them on sublattices AandBare also of opposite algebraic sign. The calculated results and the proposed scenario may not only serve as experimental signatures for identifying t he Ising spin-orbital coupling in the Ising superconductors, but also be prospective in manipulation o f electron spins in motion through the orbital eﬀect in the superconducting vortex states. PACS numbers: 74.20.Mn, 74.25.Ha, 74.62.En, 74.25.nj I. INTRODUCTION The superconductivity uncovered in atomically thin two-dimensional (2D) forms of layered transition metal dichalcogenides (TMDs) have recently attracted remark- able scientiﬁc and technical interests1–14. Although these superconductors belong to the conventional s-wave su- perconductivity with low transition temperature1–14, the uniqueness of the TMDs makes them alluring to the re- searchers. On one hand, similar to graphene, these ma- terials have a honeycomb lattice structure, and exhibit a valley degree of freedom with minima/maxima of con- duction/valence bands at the corners Kand−Kof the Brillouin zone. On the other hand, unlike graphene, the in-plane mirror symmetry is broken in the TMDs, lead- ing to a strong atomic Ising type spin-orbital coupling (ISOC)3,4,7–9. The ISOC strongly pins the electron spins to the out-of-plane directions and have opposite direc- tions in opposite valleys ( Kand−K)3,4,7–9,12,14–16, so that it preserves time-reversal symmetry and is compat- ible with superconductivity. Due to the strong pinning of electron spins in the out-of-plane directions, external in-plane magnetic ﬁelds are much less eﬀective in align- ing electron spins, and lead to the in-plane upper critical ﬁeldHc2ofthe system severaltimes largerthan the Pauli limit10,12. Nevertheless, an out-of-plane magnetic ﬁeld will gen- erate the magnetic ﬂux in conductors due to the domi- nating orbital eﬀect over the Zeeman splitting. It is well known that the superconductors expel the magnetic ﬂux from their interior, the so called Meissner eﬀect. While some superconductors expel the magnetic ﬁeld globally(they are called type I superconductors), a type II super- conductorwill only keep the whole magnetic ﬁeld out un- til a ﬁrst critical ﬁeld Hc1is reached. Then vortices start toappear. Avortexisalocalmagneticﬂuxquantumthat penetrates the superconductor, where the superconduct- ing (SC) order parameter drops to zero to save the rest of the SC state in metal from being destroyed. While the ISOC exempliﬁes itself as the spin-valley locking in the momentum space, it acts as coupling between spins and the orbital derived eﬀectively periodic spin and sublat- tice dependent ﬂuxes in real space with the quantization axis along the out-of-plane direction. This is to say the spins, sublattices and the eﬀectively periodic ﬂuxes are bound together by the ISOC in real space. Thus, the lo- cal breaking of the spin and sublattice degeneracies may be expected if the ﬂuxes are altered locally, and the spin orders in real space may also be expected to emerge. In this paper, we numerically demonstrate that the spin and sublattice degeneracies break locally with an induced ferromagnetic order inside the vortex core of the Ising superconductors, as a result of the contrast- ing variation of the eﬀectively periodic ﬂuxes for sublat- ticesAandBcaused by the out-of-plane magnetic ﬁeld. By self-consistently solving the Bogoliubov-de Gennes (BdG) equations of the Hamiltonian, it is shown that there is no magnetic order induced inside the vortex core when the ISOC is zero. Accordingly, the curves of the local-density-of-states (LDOS) for the spin-up and spin- down in-gap states are almost identical, forming a series of discrete energy peaks inside the core region. The in- clusion of the ISOC induces a ferromagnetic order inside the vortex core, where the SC order parameter is sup-2 pressed. The induced magnetic orders are antiphase in terms of their opposite polarizations inside two nearest- neighbor (NN) vortices with one of the two polarizations coming dominantly from one sublattice sites. The ﬁnite- energy peaks of the LDOS for spin-up and spin-down in-gap states are shifted oppositely by the ISOC, and the sign of the relative shifts of them depends on which sublattices the site is belonging to. Based on a scenario of local breaking of the spin and sublattice degeneracies due to the interactionofthe ISOCderived eﬀective ﬂuxes with the local magnetic ﬂux inside the vortex core, we give an explanation to the unusual phenomena regarding the polarization of the induced magnetic orders and the energy shifts of the ﬁnite-energy in-gap peaks. The cal- culated results may not only serve as experimental sig- natures for identifying the ISOC proposed in the Ising superconductors, but also put forward eﬀective thinking- ways in manipulation of electron spins in motion through the orbital eﬀect in the SC vortex states. The remainder of the paper is organized as follows. In Sec. II, we introduce the model Hamiltonian and carry out analytical calculations. In Sec. III, we present nu- merical calculations and discuss the results. In Sec. IV, we make a conclusion. II. THEORY AND METHOD The eﬀective electron hoppings between the NN sites iandi+τjon a honeycomb lattice can be described by the following tight-binding Hamiltonian, H0=−/summationdisplay i,τj,σ(ti,i+τja† i,σbi+τj,σ+h.c.)−µ(/summationdisplay i∈A,σa† i,σai,σ +/summationdisplay i∈B,σb† i,σbi,σ), (1) whereti,i+τjisthehoppingintegralbetweentheNNsites. τjdenotes the three NN vectors with τ0=a(√ 3 2,1 2),τ1= a(−√ 3 2,1 2) andτ2=a(0,−1) as deﬁned in Fig. 1(a) with abeing the lattice constant. a† i,σ(b† i,σ) is the electron creation operator in sublattice A(B) ifi∈sublattice A (B), andµthe chemical potential. For the free hopping case with ti,i+τj=t, the Hamiltonian H0can be written in the momentum space, H0(k) =/summationdisplay k,σ[ξka† k,σbk,σ+ξ∗ kb† k,σak,σ−µ(a† k,σak,σ +b† k,σbk,σ)], (2) where ξk=−t2/summationdisplay j=0eik·τj. (3)One can readily ﬁnd the energy bands for this Hamilto- nian as17, ε± k=±t[3+2cos(√ 3kx)+4cos(√ 3kx/2)cos(3ky/2)]1 2 −µ. (4) with + ( −) indexing the conduction (valence) band. We focus on systems which have been doped such that the chemical potential µlies in the upper conduction bands, and produce six spin degenerate pockets at the corners of the hexagonal Brillouin zone when ε+ k= 0, as shown in Fig. 1(b). FIG. 1: (a) Honeycomb lattice structure of the Ising super- conductor, made out of two sublattices A(blue dots) and B (red dots). τ0,τ1andτ2are the nearest-neighbor vectors, andτ′ 1-τ′ 6the next-nearest-neighbor vectors. (b) The Bril- louin zone (dashed line) and the six spin degenerate Fermi pockets (solid lines) of the Ising superconductor. The red a nd blue colors indicate the opposite sign of the eﬀective Zeema n ﬁelds between adjacent Fermi pockets located at Kand−K. The positive phase hopping directions for spin-up electron s depicted by HISOCin Eq. (6) (c1), and by HKMin Eq. (15) (c2), respectively. The arrows in bothﬁguresindicate thep os- itive phase hopping directions. (d) The ISOC dependencies o f the maximum of the absolute value for the induced magnetic order\|S\|maxand the magnitude of the relative energy shifts \|δ\|between the spin-up and spin-down in-gap state peaks on the vortex core center [reference to text and Fig. 4(b)]. The ISOC acts as strong eﬀective Zeeman ﬁelds, which polarize electron spins oppositely to the out-of-plane di- rection at opposite valleys, that is, at the Kand−K points in Fig. 1(b). If we choose the out-of-plane direc- tion as the z-axis, the ISOC term has the form18 HISOC(k) =β/summationdisplay k,σ,σ′gk·ˆσσσ′(a† k,σak,σ′+b† k,σbk,σ′),(5) whereβis the ISOC strength, and ˆ σdenotes the Pauli matricesactinginthespinspace. TheISOCrequiresthat3 the form factor gkalternates its sign between adjacent Fermipocketslocatedat Kand−K[seeFig.1(b)], which should be the form gk= ˆzFkwithFk= 2sin(√ 3kx)− 4cos(3ky/2)sin(√ 3kx/2) =−F−ksatisfying the time- reversal symmetry. In this way, the spins are bound to the orbitals in the momentum space and accordingly exhibit various valley dependent behaviors such as val- ley spintronics in these materials19–23. By making the Fouriertransformationof Fk, theISOCterminrealspace can be reached as24, HISOC=iβ/summationdisplay i,τ′ j,σ,σ′ˆσz σσ′(−1)j(a† i,σai+τ′ j,σ′+b† i,σbi+τ′ j,σ′),(6) where the vectors τ′ jconnecting the six next-nearest- neighbor (NNN) sites are located at τ′ 1=−τ′ 4=√ 3a(1,0),τ′ 2=−τ′ 5=√ 3a(1 2,√ 3 2) andτ′ 3=−τ′ 6=√ 3a(−1 2,√ 3 2), as indicated by the dashed arrows in Fig. 1(a). We will see later that the ISOC in real space depicted by Eq. (6) plays the role of the coupling be- tween spins and the eﬀectively periodic ﬂuxes with the quantization axis along the out-of-plane direction. Then, the Hamiltonian including both the free hoppings and the ISOC term is reached, in real space as, HTMD=H0+HISOC. (7) The SC pairing is assumed to be derived from the ef- fective attraction between electrons, HP=V0 2/summationdisplay i,σni,σni,¯σ. (8) Here, we consider the on-site interactions with V0denot- ing the eﬀective interaction potential15,16. By making the mean-ﬁeld decoupling, HPcan be rewritten in terms of the SC pairings as, HP=/summationdisplay i∈A(∆Aa† i,↑a† i,↓+h.c.)+/summationdisplay i∈B(∆Bb† i,↑b† i,↓ +h.c.), (9) where ∆ A=−V0/angbracketleftai,↑ai,↓/angbracketright(∆B=−V0/angbracketleftbi,↑bi,↓/angbracketright) deﬁnes the on-site spin-singlet s-wave SC pairing. Then the total Hamiltonian is arrived as follows, H=HTMD+Hpair. (10) Based on the Bogoliubov transformation, the diagonal- ization of the Hamiltonian Hcan be achieved by solving the following discrete BdG equations, /summationdisplay j −µδijHij,↑↑∆Aδij0 H∗ ij,↑↑−µδij0 ∆ Bδij ∆∗ Aδij0µδij−H∗ ij,↓↓ 0 ∆∗ Bδij−Hij,↓↓µδij uA,n,j,↑ uB,n,j,↑ vA,n,j,↓ vB,n,j,↓ = En uA,n,i,↑ uB,n,i,↑ vA,n,i,↓ vB,n,i,↓ ,(11)where, Hij,↑↑=−tijδi+τj,j+iβσz ↑↑(−1)jδi+τ′ j,j, Hij,↓↓=−tijδi+τj,j+iβσz ↓↓(−1)jδi+τ′ j,j,(12) withuA,n,j,↑(uB,n,j,↑) andvA,n,j,↓(vB,n,j,↓) being the Bogoliubovquasiparticleamplitudeson the j-th site with corresponding eigenvalues En. The SC pairing ampli- tudes satisfy the following self-consistent conditions, ∆A=−V0 2/summationdisplay nuA,n,i,↑v∗ A,n,i,↓tanh(En 2kBT), ∆B=−V0 2/summationdisplay nuB,n,i,↑v∗ B,n,i,↓tanh(En 2kBT).(13) The spin dependent electron density nA(B),i,σand the local magnetic orders SA(B),i,zare determined respec- tively by, nA(B),i,↑=/summationdisplay n\|uA(B),n,i,↑\|2f(En), nA(B),i,↓=/summationdisplay n\|vA(B),n,i,↓\|2f(En), SA(B),i,z=1 2[nA(B),i,↑−nA(B),i,↓]. (14) III. RESULTS AND DISCUSSION In numerical calculations, we choosethe zero ﬁeld hop- ping integral t= 200meVasthe energyunit, andﬁx tem- perature T= 1×10−5, unless otherwise speciﬁed. The ﬁlling factor n=/summationtext i,σni,σ/N= 1.08 (Ndenotes the number of total lattice sites) such that the chemical po- tentialµlies in the upper conduction band and gives rise to the Fermi surfaces in Fig. 1(b). In the presence of a perpendicularmagneticﬁeld, the orbitaleﬀect dominates over the the Zeeman splitting, so we neglect the Zeeman term of the external magnetic ﬁeld in the following cal- culations. In this case, the hopping terms are described by the Peierls substitution. For the NN hopping between sitesiandi+τj, one has ti,i+τj=teiϕi,i+τj, and for the NNN hopping between iandi+τ′ jone should have β→βeiϕi,i+τ′ j, whereϕi,i+τj(τ′ j)=π Φ0/integraltextri ri+τj(τ′ j)A(r)·dr with Φ 0=hc 2ebeing the SC ﬂux quanta. We consider a system with a parallelogram vortex unit cell as shown in Fig. 1(a), where two vortices are accommodated. The vortex unit cell with size of 24 a1×48a2is adopted in the calculations, unless otherwise stated. The vector poten- tialA(r) = (0,Bx,0) is chosen in the Landau gauge to give rise to the magnetic ﬁeld Balong the z-direction. In this study, we have no ambition to explore the SC mechanism underlying the Ising superconductors. In- stead, we assume a phenomenological pairing potential V0to give rise to the SC pairing. Within the BCS the- ory,the coherencelengthisgivenby ξ0=/planckover2pi1vF/π∆, where4 vFis the Fermi velocity, linking the coherence length to the inverse size of the SC gap ∆. The coherence length of NbSe 2is about 10nm as obtained from Hc2(T) measurement25,26. The estimated vortex core size is of ξV∼30nm26. A system contains two such vortex cores would be larger than the size of 60nm ×120nm, which roughly amounts to a parallelogram sample with the size largerthan 200 a1×400a2. Such a large size is far beyond the computational capability. However, it is still capable of mimicking the vortex physics on a relative small size of sample by artiﬁcially enlarging the SC gap ∆. In the self-consistent calculations, the length scale of the sam- ple with size 24 a1×48a2is about one order smaller than the actual size. Thus, we need to choose a large V0= 1.6 in the self-consistent calculations to give rise to a bulk value of ∆ ≈0.09∼18meV, a value about one order larger than the actual measurements26, so as to meet the requirement. FIG. 2: The spatial distributions of the SC and magnetic order parameters in the vortex states for β= 0.04 are shown in (a) and (b), respectively. The spatial distributions of t he magnetic order in the vortex states for β= 0.04 on sublattice A(c), and on sublattice B(d), respectively. A. The induced antiphase magnetic orders inside vortex cores Under a perpendicular magnetic ﬁeld, the vanishment of the screening current density at the vortex center drives the system into the vortexstates with the suppres- sionoftheSCorderparameteraroundthevortexcore. In the absence of the ISOC interaction, we ﬁnd that except for the suppression of the SC order around the vortex core region there is no other order to be induced. On the other hand, when the ISOC is present, a ferromagnetic order can emerge inside the vortex core region with its maximum appearing at the vortex core center. The max- imum of the absolute value for the magnetic order \|S\|max exhibits roughly linear increasing trend with βin a widerange of ISOC, and ﬁnally reaches a saturated value at large ISOC, as displayed in Fig. 1(d). Typical results on thevortexstructurewith β= 0.04areshowninFigs.2(a) and 2(b) for the spatial distributions of SC and magnetic orders, respectively. As shown in Fig. 2(a), each vortex unit cell accommodates two SC vortices each carrying a ﬂux quantum Φ 0. The SC order parameter \|∆A/B\|van- ishes at the vortex corecenter where the maximum of the induced magnetic order appears. It is interesting to note that the magnetic order parameters have opposite polar directions around two NN vortices along the long side of the parallelogram vortex unit cell, as shown in Fig. 2(b). The most unusual aspect of the spatial distribution of the magnetic order parameters SA(B),i,zappears when we replot in Figs. 2(c) and 2(d) the magnetic orders sep- arately on the sublattices AandB. Speciﬁcally, the pos- itive magnetic order alone z-axis inside one vortex comes dominantly from the Asublattice while the negative one inside another vortex comes dominantly from the Bsub- lattice. In ordertounderstandthe originaswellasthe unusual distributions of the induced magnetic order, we should note the fact that there is no magnetic order induced when ISOC is zero. In real space, the ISOC depicted by Eq. (6) plays the role of coupling between spins and the eﬀectively periodic ﬂuxes with the quantization axis along the out-of-plane direction. Following the ISOC term in Eq. (6), we display the positive phase [noting thati=eiπ/2] hopping directions of ISOC in Fig. 1(c1) by arrows on NNN bonds for spin-up electrons at sub- latticesAandB, from which the eﬀective spin ﬂuxes are generated. If the positive phase hoppings on NNN bonds for spin-up electrons on sublattice Agenerate spin ﬂux pointing to z-direction, then they generate spin ﬂux pointing to −z-direction on sublattice B, and contrary is true for spin-down electrons. That is, the NNN hoppings have opposite chirality, for sublattices AandB. Since the spins, sublattices and the eﬀectively periodic ﬂuxes are bound together in real space, local breaking of the spin and sublattice degeneracies may be expected if the eﬀective ﬂuxes for sublattices AandBare contrastively altered by an out-of-plane magnetic ﬁeld, and thus the spin ordersin realspacemay alsobe expected. Neverthe- less, we can not expect the appearance of magnetic order in the normal state under an out-of-plane magnetic ﬁeld. This is due to the fact that the energy scale of the hop- ping integral toverwhelms the ISOC strength β, inter- changing the electrons between sites of sublattices Aand Bleading to the suppression of the local orders. How- ever, the situation is totally diﬀerent in the vortex state, where the localized electrons in the vortex core, which come from the breaking of the Cooper pairs, contribute to the magnetic order. If one vortex core resides on the Asublattice site, the blue site shown in Fig. 1(a), the positive phase hoppings on NNN bonds bound to spin- up electrons on sublattice Agenerate eﬀective spin ﬂux pointing to z-direction [noting the negative charge of the electrons], which is in the same direction as the mag-5 netic ﬁeld. On the contrary, the spin-down electrons on sublattice Agenerate eﬀective spin ﬂux in the opposite direction of the magnetic ﬁeld. Thus, the spin degener- acy breaks locally to two branches with a lower energy for the spin-up electrons, leading to the positive mag- netic order around one vortex as shown in Fig. 2(c) on sublattice A. In principle, the pairing breaking from the spin-singlet SC pairings due to the orbital eﬀect of the magnetic ﬁeld results in equal numbers of spin-up and spin-down electrons, so the total spins should be zero as a global. The excess of spin-down electrons accumulate into the NN vortex to give rise to the negative magnetic ordershowninFig.2(d)onsublattice B, wherebyitsaves the energy as the eﬀective spin ﬂux generated by spin- down electrons being in compliance with the direction of the magnetic ﬁeld. Two situations could lend support to the above sce- nario. Firstly, we consider the case with a reversal of the directionofthemagneticﬁeld, i.e., amagneticﬁeld inthe −z-direction. From the above argument, the polariza- tions of the induced magnetic orders should be reversed if the magnetic ﬁeld reverses its direction. It is exactly the case as evidenced in Figs. 3(a) and 3(b), where the results are obtained with an out-of-plane magnetic ﬁeld in the−z-directionwhile keepother parametersthe same as that in Fig. 2. Secondly, we should make a comparison with the spin-orbital coupling (SOC) term in Kane-Mele model27, which has the form HKM=iβ/summationdisplay i,τ′ j,σ,σ′ˆσz σσ′(−1)j(a† i,σai+τ′ j,σ′−b† i,σbi+τ′ j,σ′).(15) BothHISOCandHKMpreserve time-reversal symme- try, so the spins remain degenerate in both cases. The only diﬀerence lies that HISOCpreserves the sublattice symmetry but HKMbreaks it. As a result, the NNN hopping phases carried by the same spins in HKMwould have same chirality for sublattices AandB, as denoted by arrowsin Fig. 2(c2). According to the above scenario, we deduce that the induced magnetic orders should be in the same direction for the two adjacent vortices. This is also veriﬁed in Figs. 3(c) and 3(d), where the results for the spatial distribution of the induced magnetic or- ders are calculated by replacing HISOCwithHKMwhile other parameters remain unchanged. B. The splitting and shift of the ﬁnite-energy peaks for the spin-resolved LDOS Next, we examine the energy dependence of the LDOS in the vortex states on the honeycomb lattice. The LDOS is deﬁned as N(Ri,E) =N↑(Ri,E)+N↓(Ri,E) withN↑(Ri,E) =−/summationtext n\|uA(B),n,i,↑\|2f′(En−E) and N↓(Ri,E) =−\|vA(B),n,i,↓\|2f′(En+E) being the spin- resolved LDOS for spin-up and spin-down states, respec- tively. In order to reduce the ﬁnite size eﬀect, the calcu- lations of the LDOS are carried out on a periodic latticeFIG. 3: The spatial distributions of the induced magnetic or - ders in the vortex states with the magnetic ﬁeld along the −z direction for β= 0.04 on sublattice A(a), and on sublattice B(b), respectively. (c) and (d) show the calculated results fot the spatial distributions of the induced magnetic order s by replacing HISOCwithHKM[see text for details]. which consists of 16 ×8 parallelogram vortex unit cells, with each vortex unit cell being the size of 24 a1×48a2. In Fig. 4, we plot a series of the spin-resolved LDOS as a function of energy at sites along the zigzag direction moving away from the vortex center for β= 0.0 and β= 0.04, respectively. For comparison, we have also displayed the LDOS at the midpoint between the two NN vortices, which resembles the U-shaped full gap fea- ture for the bulk system. In the absence of ISOC, the states of spin-up and spin-down are nearly equal occupa- tion and empty in the vortex core, as shown in Fig. 4(a), in accordance with the empty cores without the induced magnetic orders. Besides the almost identical LDOS line shapes for the spin-up and spin-down states, the LDOS shown in Fig. 4(a) exhibits another two prominent fea- tures within the SC gap edges. On one hand, the LDOS shows the pronounced discrete energy peaks inside the coreregionwithonelocatednearthezeroenergyandoth- ers located at ﬁnite energies, as indicated by the dashed vertical lines in the ﬁgure. Here, the asymmetric line shape of the LDOS with respect to zero energy reﬂects the lack of particle-hole symmetry as the chemical po- tentialµdeviates from zero for the ﬁlling factor nbeing greater than the half ﬁlling ( n >1). Due to the particle- hole asymmetry, the ﬁnite-energy bound states at the core site only appear on the E >0 side28(There are also weak peaks at ﬁnite energies on the E <0 side when moving away from the core center.). The existence of the zero-energy vortex core sates in the Dirac fermion system have been predicted analytically by Jackiw and Rossi in terms of the zero-energy solutions of relativistic ﬁeld theory29. Although these zero-energysolutions were6 subsequently demonstrated that the existence of these zero-energy solutions is connected to an index theorem30 and the zero modes were shown to exist in the Dirac con- tinuum theory of the honeycomb lattice at half ﬁlling31, the zero-energy levels split when adopting a honeycomb lattice model description by setting the size of the vortex coretobezero32. Itisalsofoundthattheenergysplitting decreases with the vortex size and leads to the near-zero- energy states in the circumstance of ﬁnite core size32. While the notion of the zero-energy vortex core states presents an important subject of study being worthy of further research, we identify the near-zero-energy vortex core states here in a self-consistent manner by employing the honeycomb lattice model, where the band structure hasthe Dirac-typedispersionnearthe halfﬁlling. Onthe other hand, though the peaks’ intensities are suppressed as the site departing from the core center, the energy lev- els of these peaks are almost independent of positions. It is worth while to note that a dispersionless zero-energy conductance peak has been recently observed inside the SC vortex core by Chen’s group33in the kagome super- conductor CsV 3Sb5, which shares the lattice structure with component of hexagonal honeycomb and the elec- tronic structure with Dirac points in a manner similar to those in honeycomb lattices. How the calculated results withnear-zero-energypeaksinthepresentstudyrelateto the experimental observations, and whether these near- zero-energy vortex core states have a common underly- ing symmetrical cause, constituting another fascinating questions deserving further studies. In the presence of the ISOC, the local breaking of the spin and sublattice degeneracies in the vortex states is also reﬂected in the energy dependence of the LDOS. Figs. 4(b), 4(c) and 4(d) present the typical results of the spin-resolved LDOS for β= 0.04. As can be seen from Fig. 4(b), while the energy level of the near-zero- energy peaks remain virtually unchanged for both spins, the energy levels of the ﬁnite-energy peaks are shifted diﬀerently by the ISOC for diﬀerent spins and at diﬀer- ent sublattice sites, as compared with the case of β= 0. Speciﬁcally, for the LDOS on the same site within the core region, the ﬁnite-energy peaks for the spin-up and spin-down bound states shift oppositely, as indicated by the arrows in the ﬁgures, depicting a picture of local breaking of the spin degeneracy. At the same time, for the bound states with the same spin, the ﬁnite-energy peaks on the sites belonging to diﬀerent sublattices also have the opposite shifts, indicating the local breaking of the sublattice degeneracy. Since there are induced mag- netic orders in the vortex cores as well as the similar ISOC dependencies of the magnitudes of the magnetic orders\|S\|maxand the relative energy shifts \|δ\|as shown in Fig. 1(d), it is natural to suspect whether the spin splitting for the LDOS is derived from the Zeeman eﬀect of the induced local magnetic order interacting with the electrons34, or from the above scenario where the spin degree of freedom is manipulated by the orbital eﬀect of magnetic ﬁeld via the ISOC. Several aspects render theFIG. 4: The energy dependence of the spin-resolved LDOS on a series of sites for β= 0.0 (a), and for β= 0.04 (b), (c) and (d). (a), (b) and (d) are the results for a magnetic ﬁeld along thez-direction, while (c) the results for a magnetic ﬁeld along the−z-direction. (b) and (c) show the LDOS inside the same vortex core, and (d) the LDOS inside another vortex core. In each panel from top to bottom, the curves stand for the LDOS at sites along the zigzag direction moving away from the core center. The curves are vertically shifted for clari ty. The three dashed vertical lines in each panel denote the thre e low energy peak positions for β= 0. The arrows in (b), (c) and (d) indicate the peak position shift with respect to that ofβ= 0. The magnitude of the relative energy shifts \|δ\|is shown in (b). Zeemaneﬀect mechanism impossible. As hasbeen shown in Fig. 2(b), the magnetic orders polarize oppositely in- side two NN vortices. If the Zeeman eﬀect mechanism runs, the energy level shifts of the peaks should behave the opposite way on the sites located respectively at the two NN vortices. Nevertheless, as displayed in Figs. 4(b) and 4(d), the consistency of the peaks’ shifts on the sites located at diﬀerent vortices while belonging to the same sublattice rules out the Zeeman eﬀect mechanism. The second thing we notice about the energy level shifts of the peaks is that they occur only for the ones with ﬁnite energy, while the near-zero-energy peaks almost stay the same, being at odds with the Zeeman eﬀect mechanism. Finally, if we reverse the direction of the out-of-plane magnetic ﬁeld, as shown in Fig. 4(c), the peaks’ shifts behave exactly the opposite way as compared with that in Fig. 4(b). It is thus conﬁrmed that the local spin split- ting and the local breaking of sublattice degeneracy are conformed with the above scenario where the spin de- gree of freedom is manipulated by the orbital eﬀect of7 magnetic ﬁeld via the ISOC in the SC vortex states. FIG. 5: Temperature (a), and magnetic ﬁeld (b) evolutions of the maximum of the absolute value for the induced magnetic orders at vortex cores with β= 0.04. C. The eﬀects of temperature and magnetic ﬁeld strength on the induced orders Due to the 2D nature of the Ising superconductors, the thermal eﬀect on the induced magnetic orders con- stitutes an inevitable issue from both theoretical per- spective and experimental realization. Although the sys- tem under study is 2D, the induced magnetic orders are formed under the combined actions of the magnetic ﬁeld, the ISOC and the SC order, so they are not spontaneous ones. Meanwhile, the induced magnetic orders are local- ized inside the vortex core regions, and thus they are lo- cal ones. As a result, one may expect a diﬀerent manner of the thermal eﬀect on the induced magnetic orders as compared with the Mermin-Wagner theorem35. To see the thermal eﬀect on the induced magnetic orders, we calculate the temperature dependence of the magnitude of the magnetic orders. Fig. 5(a) shows the temperature dependence of the maximum of the absolute value for the induced magnetic order at the vortex core, where T is rescaled by Tc≈0.05. As can be seen from the ﬁgure, the magnitude of the magnetic orders remains approxi- mately constantat lowtemperature T≤0.02Tc[seeinset of Fig. 5(a)] as a result of the small thermal excitations andthealmostunchangedvortexcoresizeatthistemper- ature regime36. After then it shows a steady decreasing trend with increasing temperature, and ﬁnally reaches a tiny value at T∼0.5Tc. The decreasing trend is mainly ascribed to the enlarging vortex core size with temper- ature36, the so-called Kramer-Pesch Eﬀect37. The en- largedvortexcorewould involvemorediﬀerent sublattice sites into the vortex core center, resulting in the reduc- tion of the induced magnetic orders. Though the mag- netic orders reduce their magnitude upon the increasing of the temperature, they sustain to a ﬁnite temperature. Therefore, one may expect to observe the induced mag- netic orders under temperatures well below the SC criti- cal temperature. Anotherimportantfactortobeconsideredinobserving theinducedmagneticordersishowthestrengthoftheex- ternal magnetic ﬁeld aﬀects the induced magnetic orders.Since one vortex unit cell accommodates two vortices in the calculations, we have B= 2Φ0/A∼1/NwithAand Nbeing the area and the site number of the vortex unit cell. Fig. 5(b) displays the variation of the maximum of the absolute value for the induced magnetic orders with respect to diﬀerent strengths of the magnetic ﬁeld, which are realized in the calculations by varying the size of the parallelogram vortex unit cell. In the weak to moderate magnetic ﬁeld region, there is little interference between the vortex cores due to the large inter-vortex spacing d. The increase of the magnetic ﬁeld leads to more broken Cooper pairs inside the vortex cores to contribute to the formation of the magnetic orders, so the magnitude of the induced magnetic orders increases with the magnetic ﬁeld strength, as evidenced in Fig. 5(b). However, as the magnetic ﬁeld increasing further, the adjacent vortex cores with opposite polarizations of the induced mag- netic order would get close enough (with a length scale being less than two times of the penetration depth λ) to interfere with one another, leading to the reduction of the magnitude of the magnetic order. This suggests the induced magnetic orders will be altered in an Abrikosov vortexlattice38. Ononehand, the formationofthe Bloch wave39or the interactions among vortices40in the vortex lattice will suppress the induced magnetic orders. On the other hand, since there are many vortices in the sample instead of just two, the polarizationofthe induced orders is not necessarily opposite for two adjacent vortex cores. Nevertheless, the result also means the induced magnetic orders would survive in the vortex lattice under a weak to moderate magnetic ﬁeld as long as d≫λ, i.e., the inter-vortex spacing is much larger than the penetration depth. IV. REMARKS AND CONCLUSION The local magnetic orders induced in the SC vor- tex states have been extensively investigated on the cuprates superconductors34,41–45, where the emergence of the magnetic orders inside the core region was gener- ally believed to be originated from the electrons’ correla- tions. These correlationsusually comefrom the Coulomb interactionsbetween electronsandthat the induced mag- netic orders have nothing to do with the chirality of the electrons. However, the induced local magnetic or- ders inside the SC vortex core by the ISOC has a di- rect bearing on what the electrons’ chirality is. As has been demonstrated that the amplitudes of the induced magnetic orders and the unusual energy shifts of the in- gap state peaks present here are related to the ISOC strength β, while their directions are determined by the direction of the magnetic ﬁeld. The amplitude and the diﬀerent polar direction of the induced local magnetic or- ders could be measured by the muon spin rotation ( µSR) spectroscopy and the nuclear magnetic resonance experi- ments, and the energy shifts of the in-gap state peaks on diﬀerent sublattice for diﬀerent spins could be observed8 in the spin-polarized scanning tunneling microscopy ex- periments. Both of these observations may be served as signatures to characterize the ISOC proposed for the Ising superconductors. In the meantime, since the in- duced magnetic orders are derived from the ISOC, the breaking of the spin degeneracy and the energy shifts of the in-gap state peaks are selectively occurred for the electrons which possess ﬁnite momentum with respect to the vortex center. The scenario proposed here may also provide a possibility in manipulation of electron spins in motion via the orbital eﬀect in the SC vortex states. In conclusion, we have numerically investigated the vortex states of the Ising superconductors, with the em- phasis on the local breaking of the spin and sublattice degeneracies as a result of the interaction between the ISOC derived eﬀective ﬂuxes and the local magnetic ﬂux inside the vortex core. In the absence of the ISOC, there was no magnetic order induced inside the vortex core, and the almost identical line shapes of the LDOS for the spin-upandspin-downin-gapstateswereshownupinside the core region, forming a series of discrete energy peaks within the gap edges. The inclusion of the ISOC induced the ferromagnetic orders inside the vortex core region,where the magnetic orders polarized oppositely for the two NN vortices with one of the two polarizations com- ing dominantly from one specie of the two sublattices. Accordingly, the ﬁnite-energy peaks of the LDOS on the same site for spin-up and spin-down in-gap states were shifted oppositely by the ISOC, and the relative shifts of them on sublattices AandBwere also of opposite alge- braic sign. The calculated results might serve as exper- imental signatures for identifying the ISOC in the Ising superconductors, and the scenario proposed here might also be prospective in manipulation of electron spins in motion through the orbital eﬀect in the SC vortex states. V. ACKNOWLEDGEMENT This work was supported by the National Natural Sci- ence Foundation of China (Grant Nos. 11574069 and 61504035) and the Natural Science Foundation of Zhe- jiang Province (No. LY16A040010). This work was also supported by K. C. Wong Magna Foundation in Ningbo University. 1R. A. Bromley, R. B. Murray, and A. D. Yoﬀe, J. Phys. C 5, 759 (1972). 2Th. B¨ oker, R. Severin, A. M¨ uller, C. Janowitz, R. Manzke, D. Voß, P. Kr¨ uger, A. Mazur, and J. Pollmann, Phys. Rev. B64, 235305 (2001). 3Z. Y. Zhu, Y. C. Cheng, and U. Schwingenschl¨ ogl, Phys. Rev. B84, 153402 (2011). 4D. Xiao, G.-B. Liu, W. Feng, X. Xu, and W. Yao, Phys. Rev. Lett. 108, 196802 (2012). 5J. T. Ye, Y. J. Zhang, R. Akashi, M. S. Bahramy, R. Arita, and Y. Iwasa, Science 338, 1193 (2012). 6K.Taniguchi, A.Matsumoto, H.Shimotani, andH.Takagi, Appl. Phys. Lett. 101, 042603 (2012). 7A. Korm´ anyos, V. Z´ olyomi, N. D. Drummond, P. Rakyta, G. Burkard, and V. I. Fal’ko, Phys. Rev. B 88, 045416 (2013). 8F. Zahid, L. Liu, Y. Zhu, J. Wang, and H. Guo, AIP Adv. 3, 052111 (2013). 9E. Cappelluti, R. Rold´ an, J. A. Silva-Guill´ en, P. Ordej´ o n, and F. Guinea, Phys. Rev. B 88, 075409 (2013). 10X. Xi, L. Zhao, Z. Wang, H. Berger, L. Forr´ o, J. Shan, and K. F. Mak, Nat. Nanotechnol. 10, 765 (2015). 11W. Shi, J. T. Ye, Y. Zhang, R. Suzuki, M. Yoshida, J. Miyazaki, N. Inoue, Y. Saito, and Y. Iwasa, Sci. Rep. 5, 12534 (2015). 12J. M. Lu, O. Zheliuk, I. Leermakers, N. F. Q. Yuan, U. Zeitler, K. T. Law, and J. T. Ye, Science 350, 1353 (2015). 13Y. Saito, Y. Nakamura, M. S. Bahramy, Y. Kohama, J. Ye, Y. Kasahara, Y. Nakagawa, M. Onga, M. Tokunaga, T. Nojima, Y. Yanase, and Y. Iwasa, Nat. Phys. 12, 144 (2016). 14X. Xi, Z. Wang, W. Zhao, J.-H. Park, K. T. Law, H. Berger, L. Forr´ o, J. Shan, and K. F. Mak, Nat. Phys. 12, 139 (2016).15B. T. Zhou, N. F. Q. Yuan, H.-L. Jiang, and K. T. Law, Phys. Rev. B 93, 180501(R) (2016). 16G. Sharma and S. Tewari, Phys. Rev. B 94, 094515 (2016). 17A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009). 18P. A. Frigeri, D. F. Agterberg, A. Koga, and M. Sigrist, Phys. Rev. Lett. 92097001 (2004). 19B. Radisavljevic, A. Radenovic, J. Brivio, V. Giacometti, and A. Kis, Nat. Nanotechnol. 6, 147 (2011). 20Y. J. Zhang, J. T. Ye, Y. Matsuhashi, and Y. Iwasa, Nano Lett.12, 1136 (2012). 21Q. H. Wang, K. Kalantar-Zadeh, A. Kis, J. N. Coleman, and M. S. Strano, Nat. Nanotechnol. 7, 699 (2012). 22W. Bao, X. Cai, D. Kim, K. Sridhara, and M. S. Fuhrer, Appl. Phys. Lett. 102, 042104 (2013). 23J. Lee, K. F. Mak, and J. Shan, Nat. Nanotechnol. 11, 421 (2016). 24H.-M. Jiang, J.-L. Shang, andL.-Z.Hu, J. Phys.: Condens. Matter31, 295602 (2019). 25V. G. Kogan and N. V. Zhelezina, Phys. Rev. B 71, 134505 (2005). 26A. Fente, E. Herrera, I. Guillam´ on, H. Suderow, S. Ma˜ nas- Valero, M. Galbiati, E. Coronado, and V. G. Kogan, Phys. Rev. B94, 014517 (2016). 27C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005). 28N. Hayashi, T. Isoshima, M. Ichioka, and K. Machida, Phys. Rev. Lett. 80, 2921 (1998). 29R. Jackiw and P. Rossi, Nucl. Phys. B 190, 681 (1981). 30E. J. Weinberg, Phys. Rev. D 24, 2669 (1981). 31P. Ghaemi and F. Wilczek, Phys. Scr. T146, 014019 (2012). 32D. L. Bergman and K. Le Hur, Phys. Rev. B, 79, 1845209 (2009). 33Z. Liang, X. Hou, F. Zhang, W. Ma, P. Wu, Z. Zhang, F. Yu, J.-J. Ying, K. Jiang, L. Shan, Z. Wang, and X.-H. Chen, Phys. Rev. X 11, 031026 (2021). 34J.-X. Zhu and C. S. Ting, Phys. Rev. Lett. 87, 147002 (2001). 35N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 (1966). 36R. I. Miller, R. F. Kieﬂ, J. H. Brewer, J. Chakhalian, S. Dunsiger, G. D. Morris, J. E. Sonier, and W. A. MacFar- lane, Phys. Rev. Lett. 85, 1540 (2000). 37L. Kramer and W. Pesch, Z. Phys. 269, 59 (1974). 38A. A. Abrikosov, Rev. Mod. Phys, 76, 975 (2004). 39M. Franz and Z. Teˇ sanovi´ c, Phys. Rev. Lett. 84, 554(2000). 40G. Blatter and V. Geshkenbein, Phys. Rev. Lett. 77, 4958 (1996). 41M. Ogata, Int. J. Mod. Phys. B 13, 3560 (1999). 42J.-X. Zhu, I. Martin, and A. R. Bishop, Phys. Rev. Lett. 89, 067003 (2002). 43Y. Chen, Z. D. Wang, J.-X. Zhu, and C. S. Ting, Phys. Rev. Lett. 89, 217001 (2002). 44M. Takigawa, M. Ichioka, and K. Machida, Phys. Rev. Lett.90, 047001 (2003). 45H. Tsuchiura, M. Ogata, Y. Tanaka, and S. Kashiwaya, Phys. Rev. B 68, 012509 (2003).
2304.12928v1.Designing_Valley_Dependent_Spin_Orbit_Interaction_by_Curvature.pdf	Designing Valley-Dependent Spin-Orbit Interaction by Curvature Ai Yamakage,1,∗T. Sato,2,†R. Okuyama,3T. Funato,4, 5W. Izumida,6K. Sato,7T. Kato,2and M. Matsuo5, 8, 9, 10 1Department of Physics, Nagoya University, Nagoya 464-8602, Japan 2Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan 3Department of Physics, Meiji University, Kawasaki 214-8571, Japan 4Center for Spintronics Research Network, Keio University, Yokohama 223-8522, Japan 5Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China. 6Department of Physics, Tohoku University, Sendai 980-8578, Japan 7National Institute of Technology, Sendai College, Sendai 989-3128, Japan 8CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China 9Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, 319-1195, Japan 10RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan We construct a general theoretical framework for describing curvature-induced spin-orbit interac- tions on the basis of group theory. Our theory can systematically determine the emergence of spin splitting in the band structure according to symmetry in the wavenumber space and the bending direction of the material. As illustrative examples, we derive the curvature-induced spin-orbit cou- pling for carbon and silicon nanotubes. Our theory oﬀers a strategy for designing valley-contrasting spin-orbit coupled materials by tuning their curvatures. Introduction.— The spin-orbit interaction (SOI) in solids lies at the heart of the the ability to manipu- late electron spins in spintronics. It has been utilized through the spin Hall eﬀect, a charge-to-spin conversion phenomenon, to generate, manipulate, and detect spin currents by various electrical means [1]. It also enables control of the magnetization by transferring spin as a torque from materials such as heavy metals, antiferro- magnets, as well as oxide, topological [2, 3], and chiral materials [4]. In addition, valleytronics, which utilizes the valley degree of freedom in materials, has stimulated researchers’ interest because the interplay between spin and valley plays a crucial role in quantum transport phe- nomena [5–7]. While the design of materials for a suitable SOI is an important subject in modern spintronics, the choices of elements and crystal structures are limited. As another strategy, researchers have extensively studied tuning the strength of the SOI by using the mechanical proper- ties of the materials, such as strain. Strain-induced SOI has been studied in semiconductor quantum wells with static [8–14] and dynamical [15, 16] strains. However, the application of strain engineering has been restricted to inherently strong SOI materials, because strain does not directly couple to spins. To overcome the previous limitations, we propose an alternate route for designing valley-dependent SOI by using the curvature of the ma- terial. In particular, we propose a novel strategy to realize valley-dependent SOI for electrons by breaking the crys- talline symmetry through the application of a ﬁnite cur- vature to a material as shown in Fig. 1. Our theoreti- cal framework provides a powerful method for determin- ing the emergence of spin splitting in the band structure depending on high-symmetry points in the wavenumber FIG. 1. Schematic diagram of valley-dependent spin splitting induced by the curvature of the material. space and the bending direction of the material. Our pro- cedure based on group theory generally describes valley- dependent spin splitting induced by the curvature. We consider a single-wall carbon nanotubes (CNTs) and sil- icon as prominent examples. For CNTs, we derive an eﬀective spin- and valley-dependent Hamiltonian, with an SOI allowed by the group theoretic approach, that is consistent with microscopic theory [17]. For silicon, we derive an eﬀective model for conduction electrons near six valleys that clariﬁes the valley-dependent SOI induced by the curvature. Our theory shares the group-theoretic ap- proach of Ref. [18], in which the curvature-induced spin- phonon coupling in graphene was discussed. It provides a general strategy for inducing SOI in materials and de- signing valley contrasting spintronics via tuning of the bending directions of material. General procedure.— By using group theory, we con- struct an eﬀective Hamiltonian with curvature as follows. First, we identify the space group of the crystal struc-arXiv:2304.12928v1 [cond-mat.mes-hall] 25 Apr 20232 ture without curvature. In particular, the magnetic little groupGis determined for a given valley. We can ﬁnd a symmetry of wavefunctions for a target band as an irre- ducible representation (irrep) of G. Then, an irreducible decomposition is performed for signiﬁcant physical quan- tities, such as momentum k, spins, and pseudospin σ, as well as the curvature which is the second derivative of the displacement ﬁeld. The products of these quanti- ties appear in the eﬀective Hamiltonian if they belong to the totally symmetric irrep, as the Hamiltonian is invari- ant to any symmetry operation in G. The speciﬁc forms and product rules of irreps are provided online [19–22]. With this procedure, we can systematically construct an eﬀective Hamiltonian with a valley-dependent SOI. Carbon nanotube.— A carbon nanotube (CNT) is rolled-up graphene forming a cylindrical surface. We will start by considering ﬂat graphene with the space group P6/mmm (No. 191). The electronic states nearby the Fermi energy are around the Kpoint. The Kpoint has the point symmetry of ¯62m(D3h) and the corresponding magnetic little group is 6/prime/mmm/prime(D3h×{I,PT}with PTbeing the parity-time transformation). We consider theπbands, which are doubly degenerate at the Fermi energy and belong to the K6(E/prime/prime) irrep ofD3h(deﬁned in the Supplemental Material [23]). The theory contains the degrees of freedom of the momentum kmeasured from theKpoint, spins, and pseudospin for sublat- ticeσ. For the irreducible decomposition, we examine representations for the generators of the magnetic little group: the threefold rotation along the z-axisC3, twofold rotation along the y-axisC/prime 2(y), and the horizontal mir- rorσh, are represented by DK(C3) =ei2πσz/3e−iπsz/3, DK(C/prime 2(y)) =−iσysy, andDK(σh) =−isz, respectively. DK(PT) =σxsyK, whereKdenotes complex conjuga- tion. An operator O(=σiorsi,i=x,y,z ) transforms by the symmetry operation gasO→DK(g)−1ODK(g), whereask→gk. Then, we ﬁnd the conversion rules for k,s, andσand decompose these quantities (as well as the possible products of sandσ) as shown in Table I. An eﬀective Hamiltonian for πelectrons is constructed from this table as a totally symmetric irrep, namely, a PT-even irrep of A/prime 1. For example, such an irrep can be obtained as a product of the two PT-even E/primeirreps: H(0) K(k) =/planckover2pi1vF(kxσx+kyσy). (1) This is nothing but the well-known Dirac Hamiltonian for graphene. The curvature is expressed in terms of the displacement ﬁeldu(x), wherex= (x,y,0) is the coordinates of the tube surface. For the tube geometry shown in Fig. 2, the displacement ﬁeld is given by u(x) = (0,0,uz(x·n)), (2) using the unit vector of the bending direction n= (cosϕ,sinϕ,0), whereϕ=π/6−θandθis the chiral (c) (b) B1B2 B3A(a) KK' AB2 B3B1 FIG. 2. (a) Atomic structure of a cylindrical surface. (b) Coordinates of CNT in real space. (c) Coordinates of CNT in momentum space. angle (see Fig. 2 (b)). The second derivative ∂i∂juzis proportional to the ﬁrst order of the curvature 1 /R: (∂2 x+∂2 y)uz∝1 R, (∂2 x−∂2 y)uz∝cos 2ϕ R,2∂x∂yuz∝sin 2ϕ R. (3) The irreducible decomposition of the ﬁrst- and second- order curvatures is also shown in Table I. The cou- pling terms in the Hamiltonian between the curva- ture and other quantities are obtained as PT-even ir- reps ofA/prime 1: (σxsy−σysx)/R, (sxcos 2ϕ−sysin 2ϕ)/R, [cos 2ϕ(σxsy+σysx)−sin 2ϕ(σxsx−σysy)]/R, (σxsin 2ϕ+ σycos 2ϕ)/R2, and (σxsin 4ϕ−σycos 4ϕ)/R2. To sim- plify the Hamiltonian, we introduce the tube coordinates (kc,kt)T=ˆR(−ϕ)(kx,ky)T, withkcandktbeing mo- menta along the circumference and axis directions, re- spectively. ˆR(ϑ) is a two-dimensional rotation matrix, ˆR(ϑ) =/parenleftBigg cosϑ−sinϑ sinϑcosϑ/parenrightBigg . (4) σandsare also represented in this basis as ( σc,σt)T= ˆR(−ϕ)(σx,σy)Tand (sc,st)T=ˆR(−ϕ)(sx,sy)T. Then, we can construct the eﬀective Hamiltonian in the pres- ence of the curvature-induced SOI as HK(k) by collect- ing the terms with PT-even irrep of A/prime 1. The eﬀective Hamiltonian for the opposite valley, K/prime, is obtained by requiring the Hamiltonian to be even under time re- versal:THK(k)T−1=HK/prime(−k). Here, we have used T=iτxσzsyK, withτbeing the valley-pseudospin opera- tor [24]. Finally, we obtain the valley-dependent eﬀective Hamiltonian as, HτzK(k) =/planckover2pi1vF(kcσc+τzktσt) −/epsilon1soτz(sccos 3ϕ−stsin 3ϕ) −/planckover2pi1vF∆kso(σcst−τzσtsc)−/planckover2pi1vF∆k/prime so(σcst+τzσtsc) −/planckover2pi1vF(τzσc∆kcsin 3ϕ+σt∆ktcos 3ϕ). (5) τzKrepresentsKandK/primeforτz=±1. In the curvature- induced SOI terms, /epsilon1so, ∆kso, and ∆k/prime soare proportional3 TABLE I. Irreducible decomposition under D3hof momentum k, spins, pseudospin for sublattice σ, and∂i∂juzof the order of 1/R(see Eq. (3)). The twofold axis of C/prime 2is set toy. The momentum kis PT-even and the curvature ∂i∂jukis PT-odd. Irrep PT-even PT-odd 1 /R(PT-odd) 1 /R2(PT-even) A/prime 1σzsz 1/R2 A/prime 2 σz,sz E/prime(kx,ky), (σx,σy) (σysx,−σxsz) (sin 2 ϕ,cos 2ϕ)/R2, (sin 4ϕ,−cos 4ϕ)/R2 A/prime/prime 1 σxsx+σysy A/prime/prime 2 σxsy−σysx 1/R E/prime/prime(σzsy,−σzsx) (sx,sy), (cos 2 ϕ,−sin 2ϕ)/R (σxsy+σysx,σxsx−σysy) to 1/R. The spin-independent shift (∆ kc, ∆kt) of Dirac points is proportional to 1 /R2. The obtained Hamil- tonian is consistent with the previous study [25]. We should note that the chiral-angle dependence of the CNT is also fully reproduced. This means that the 3 ϕdepen- dence is ascribable to crystalline symmetry. In conclu- sion, the curvature breaks the spatial-inversion symme- try of graphene and induces the antisymmetric SOI which leads to valley-dependent spin-split energy bands. Silicon.— Next, we consider curvature-induced SOI in silicon, which is a fundamental material used in semi- conductor technology. We focus on the lowest conduc- tion bands, whose edges are located at ∆ points near the zone boundary along the Γ-X symmetry lines and discuss the expected eﬀect of curvature on, e.g., silicon nanotubes [26, 27]. Three-dimensional silicon crystallizes into a diamond structure whose symmetry is characterized by the space group Fd¯3m(No. 227). The conduction elec- trons are located at the six valleys at ∆ points, (±k0,0,0),(0,±k0,0), and (0,0,±k0). Their magnetic little group is 4 /m/primemm (C4v×{I,PT}). The orbital wavefunctions for the conduction band minima belong to ∆ 1irrep [20, 28]. For ( ±k0,0,0)-valley, the mo- mentumk(measured from the band bottom), spin s, and curvature ∂i∂jukdecompose into the irreps summa- rized in Table II. The irreducible decomposition for the other valleys can be obtained by the threefold rotation, (x,y,z )→(y,z,x )→(z,x,y ). An eﬀective Hamiltonian for the valley located at k0 is a totally symmetric irrep of the little group (PT-even irrep ofA1). In the absence of curvature, we obtain quadratic kinetic terms with anisotropic eﬀective masses, H(0) k0(k) =/planckover2pi12k2 /lscript 2m/lscript+/planckover2pi12k2 t 2mt, (6) wherek/lscriptandktare longitudinal (parallel to k0) and transversal (perpendicular to k0) momenta, respectively. Hereafter, we consider curved silicon expressed with the displacement vector u= (0,0,uz(x·n)) withn= (cosϕ,sinϕ,0), for simplicity. The curvature is repre- sented by the second-order derivative u/prime/prime z(x·n)∝1/RTABLE II. Irreducible decomposition of momentum k, spin s, and∂i∂jukof the order of 1 /Runder the point group C4v, which is the little group of ( ±k0,0,0) from the space group Fd¯3m. Irrep PT-even PT-odd ∇∇u(PT-odd) A1kx/parenleftbig ∂2 y+∂2 z/parenrightbig ux,∂2 xux,∂x(∂yuy+∂zuz) A2 sx∂x(∂yuz−∂zuy) B1/parenleftbig ∂2 y−∂2 z/parenrightbig ux,∂x(∂yuy−∂zuz) B2 ∂y∂zux,∂x(∂yuz+∂zuy) E(ky,kz) (sz,−sy)∂2 x(uy,uz), ∂y∂z(uy,uz), (∂y,∂z)∂xux,/parenleftbig ∂2 yuy,∂2 zuz/parenrightbig ,/parenleftbig ∂2 zuy,∂2 yuz/parenrightbig as in the discussion on the CNT (see Eq. (3)). For (±k0,0,0)-valley, as shown in Table II [29], we have a non-vanishing term of order 1 /R,∂x(∂yuz−∂zuy)∝ sin 2ϕ/R, as anA2irrep. This can be coupled to the same irrep, sx, to be a totally symmetric irrep. Similarly, anEirrep (sz,−sy) can be coupled to ( ∂2 xuy,∂2 xuz)∝ (0,cos2ϕ)/Rand (∂2 zuy,∂2 yuz)∝(0,sin2ϕ)/R. Accord- ingly, the curvature-induced SOI yields H/prime (±k0,0,0) =±/planckover2pi1v1 Rsxsin 2ϕ±/planckover2pi1v2 Rsycos2ϕ±/planckover2pi1v3 Rsysin2ϕ. (7) Here, the double-sign is in the same order, and the signs are determined to satisfy time-reversal symme- try,syH(k0,0,0)(k)∗sy=H(−k0,0,0)(−k). For (0,±k0,0)- valley, the irreducible decomposition can be carried out by making the threefold rotation for momentum, spin, and curvature in Table II. The SOI reads H/prime (0,±k0,0) =∓/planckover2pi1v1 Rsysin 2ϕ∓/planckover2pi1v2 Rsxsin2ϕ∓/planckover2pi1v3 Rsxcos2ϕ.(8) For the (0 ,0,±k0)-valley, in contrast, no curvature- induced term appears up to 1 /R. We have a nonzero term, (∂2 x+∂2 y)uz∝1/R, as anA1irrep, but this can- not be coupled to a same irrep, e.g., kz, due to the PT symmetry.4 In order to estimate the magnitude of the spin split- ting, we calculated the band structure of the curved sil- icon by using the tight binding model, taking the ef- fect of the curvature up to order 1 /Rinto account [23]. Let us consider curved silicon as schematically shown in Fig. 3 (a), where the [010] axis is kept unchanged, cor- responding to ϕ= 0. Figure 3 (b) shows the calculated conduction bands near ( k0,0,0), (0,k0,0), and (0,0,k0) with a curvature of R= 50 nm. At the valleys of ( k0,0,0) and (0,k0,0), the electron spin is fully polarized along the yandxdirections, respectively. The spin splitting of the conduction band is estimated to be ∼2µeV at the val- ley of (k0,0,0) and∼90µeV at the valley of (0 ,k0,0). Note that the spin splitting at the valley of (0 ,0,±k0) is negligibly small. These results agree with the above group-theoretic prediction. We can also calculate the band structures when ϕ=π/4 [23]. By carefully compar- ing these numerical results with Eqs. (7) and (8), the val- ues ofv1,v2,v3can be estimated to be /planckover2pi1v1/R/similarequal30µeV, /planckover2pi1v2/R/similarequal1µeV, and /planckover2pi1v3/R/similarequal50µeV forR= 50 nm. We can show that v2is induced by interband mixing, while v1 andv3are induced by mixing between subbands formed by the circumferential boundary condition [23]. The spin splitting obtained here should be able to be detected with resonance microwave measurements [30]. Discussion.— Our theory can be utilized as a conve- nient method to produce valley-dependent SOI in various systems. Nanotubes produced by rolled-up atomic layer materials such as transition-metal dichalcogenides would be an interesting example [31]. Our method can also be used to control the SOI in rolled-up semiconductor struc- tures with diﬀerent elastic properties [32, 33]. While our theory was applied to homogeneous systems, it is also applicable to curvature-induced SOI by using spatially inhomogeneous structures such as corrugations of atomic layers [34]. Moreover, the curvature eﬀect de- scribed by the second derivative of the displacement has a similar aspect to the spin-vorticity coupling [35–37], Hsv=−1 2s·ω, whereωi=/epsilon1ijk∂j∂tukmeans vorticity, dynamical anti-symmetric lattice distortion. The vor- ticity can be regarded as a kind of curvature that ex- tends the second-order derivative of the lattice displace- ments in space to those in space-time. In this sense, our group-theoretic approach can be extended to include spin-vorticity theory in a uniﬁed way. We should note that the SOI induced in curved mate- rials has been discussed in the literature in terms of the technique of thin-ﬁlm quantization [38, 39]. We clariﬁed that this method cannot be applied to curved materi- als in usual situations because the conﬁnement potential has to be made unphysically large (see the Supplemental Material [23] for details). Conclusion.— We proposed a group-theoretical method to describe the valley-dependent spin-orbit in- teraction induced in curved materials. This method sys- tematically determines an eﬀective Hamiltonian from the (eV) (/μm) (a) (b) (eV) (/μm) (eV) (/μm) FIG. 3. (a) Crystal structure of silicon and schematic diagram of curved silicon whose bending direction is n= (1,0,0). (b) Schematic diagram of the valleys of silicon in the wavenumber space (lower left panel) and the conduction bands at the val- leys of [100] (lower right panel), [010] (upper right panel) and [001] (upper left panel) along the kydirection. The red and blue lines indicate band dispersions for diﬀerent spin polariza- tions. The solid lines indicate the lowest subbands, while the dashed line indicates the next lowest subband. Here, the sub- bands are formed by imposing the boundary condition along the circumferential direction. The curvature radius is taken to beR= 50 nm and the energy is measured from the con- duction band bottom for unbent silicon. symmetries of the crystals, momenta, and spins. The method succeeded in reproducing the eﬀective Hamilto- nian of CNTs with sublattice- and valley-dependent SOI induced by curvature. Furthermore, we derived an ef- fective Hamiltonian for curved silicon and revealed that the curvature activates SOI in a valley-selective man- ner. Combining this method with a tight-binding calcu- lation, we demonstrated signiﬁcant SOI splittings with nontrivial bending-direction dependences. Our method will provide a general strategy for curvature engineering of valley-dependent SOI in nanomaterials. We acknowledge JSPS KAKENHI for providing Grants (No. JP18H04282, No. JP19K14637, No. JP20K03831, No. JP20H01863, No. JP20K03835, No. JP20K05258, No. JP21K20356, and No. JP21K03414) and the Sumit-5 omo Foundation (190228). M. M. is partially supported by the Priority Program of the Chinese Academy of Sci- ences, Grant No. XDB28000000. T.S. was supported by the Japan Society for the Promotion of Science through the Program for Leading Graduate Schools (MERIT). ∗ai@st.phys.nagoya-u.ac.jp †sato-tetsuya163@g.ecc.u-tokyo.ac.jp [1] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. Back, and T. Jungwirth, Spin Hall eﬀects, Rev. Mod. Phys. 87, 1213 (2015). [2] A. Manchon, H. C. Koo, J. Nitta, S. Frolov, and R. Duine, New perspectives for rashba spin–orbit cou- pling, Nat. Mater. 14, 871 (2015). [3] A. Manchon, J. ˇZelezn` y, I. M. Miron, T. Jungwirth, J. Sinova, A. Thiaville, K. Garello, and P. Gambardella, Current-induced spin-orbit torques in ferromagnetic and antiferromagnetic systems, Rev. Mod. Phys. 91, 035004 (2019). [4] S.-H. Yang, R. Naaman, Y. Paltiel, and S. S. Parkin, Chiral spintronics, Nature Reviews Physics 3, 328 (2021). [5] S. A. Vitale, D. Nezich, J. O. Varghese, P. Kim, N. Gedik, P. Jarillo-Herrero, D. Xiao, and M. Rothschild, Val- leytronics: Opportunities, challenges, and paths forward, Small 14, 1801483 (2018). [6] A. Ciarrocchi, F. Tagarelli, A. Avsar, and A. Kis, Exci- tonic devices with van der waals heterostructures: val- leytronics meets twistronics, Nat. Rev. Mater. 7, 449 (2022). [7] J. F. Sierra, J. Fabian, R. K. Kawakami, S. Roche, and S. O. Valenzuela, Van der waals heterostructures for spin- tronics and opto-spintronics, Nature Nanotechnology 16, 856 (2021). [8] B. A. Bernevig and S.-C. Zhang, Spin splitting and spin current in strained bulk semiconductors, Physical Review B72, 115204 (2005). [9] M. Hruˇ ska, ˇS. Kos, S. A. Crooker, A. Saxena, and D. L. Smith, Eﬀects of strain, electric, and magnetic ﬁelds on lateral electron-spin transport in semiconductor epilay- ers, Phys. Rev. B 73, 075306 (2006). [10] Y. Li and Y. Q. Li, Strain-assisted spin manipulation in a quantum well, Eur. Phys. J. B 63, 493 (2008). [11] B. Norman, C. Trowbridge, J. Stephens, A. Gossard, D. Awschalom, and V. Sih, Mapping spin-orbit splitting in strained (in, ga) as epilayers, Physical Review B 82, 081304 (2010). [12] G. Liu, Y. Chen, C. Jia, G.-D. Hao, and Z. Wang, Spin splitting modulated by uniaxial stress in InAs nanowires, J. Phys.: Condens. Matter 23, 015801 (2011). [13] T. Matsuda and K. Yoh, Eﬀect of strain on the Dres- selhaus eﬀect of InAs-based heterostructures, Journal of Crystal Growth 323, 52 (2011). [14] B. M. Norman, C. J. Trowbridge, D. D. Awschalom, and V. Sih, Current-Induced Spin Polarization in Anisotropic Spin-Orbit Fields, Phys. Rev. Lett. 112, 056601 (2014). [15] T. Sogawa, P. V. Santos, S. K. Zhang, S. Eshlaghi, A. D. Wieck, and K. H. Ploog, Transport and Lifetime En- hancement of Photoexcited Spins in GaAs by Surface Acoustic Waves, Phys. Rev. Lett. 87, 276601 (2001). [16] H. Sanada, T. Sogawa, H. Gotoh, K. Onomitsu, M. Ko-hda, J. Nitta, and P. Santos, Acoustically induced spin- orbit interactions revealed by two-dimensional imaging of spin transport in gaas, Physical review letters 106, 216602 (2011). [17] W. Izumida, K. Sato, and R. Saito, Spin–Orbit Inter- action in Single Wall Carbon Nanotubes: Symmetry Adapted Tight-Binding Calculation and Eﬀective Model Analysis, J. Phys. Soc. Jpn. 78, 074707 (2009). [18] H. Ochoa, A. H. Castro Neto, V. I. Fal’ko, and F. Guinea, Spin-orbit coupling assisted by ﬂexural phonons in graphene, Phys. Rev. B 86, 245411 (2012). [19] M. I. Aroyo, J. M. Perez-Mato, C. Capillas, E. Kroumova, S. Ivantchev, G. Madariaga, A. Kirov, and H. Won- dratschek, Bilbao Crystallographic Server: I. Databases and crystallographic computing programs, Z. Kristallogr. Cryst. Mater. 221, 15 (2006). [20] M. I. Aroyo, A. Kirov, C. Capillasm, J. M. Perez-Mato, and H. Wondratschek, Bilbao Crystallographic Server. II. Representations of crystallographic point groups and space groups, Acta Cryst. A62, 115 (2006). [21] Y. Xu, L. Elcoro, Z.-D. Song, B. J. Wieder, M. G. Vergniory, N. Regnault, Y. Chen, C. Felser, and B. A. Bernevig, High-throughput calculations of mag- netic topological materials, Nature 586, 702 (2020). [22] L. Elcoro, B. J. Wieder, Z. Song, Y. Xu, B. Bradlyn, and B. A. Bernevig, Magnetic topological quantum chemistry, Nat. Commun. 12, 5965 (2021). [23] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevLett.*.**** for detailed theoretical formulation,... [24] The time reversal operator Tfor our basis (same as the one used in Ref. [17]) is derived from the well-known expression, T=−isyK, for the position basis. [25] For the perturbative calculation given in Ref. [17], we have ∆k/prime SO= 0. This is due to the oversimpliﬁcation in the model, i.e., the artiﬁcial electron-hole symmetry for π andσorbitals. ∆k/prime SO/negationslash= 0 is obtained from more realistic analyses. [26] X. Huang, R. Gonzalez-Rodriguez, R. Rich, Z. Gryczyn- ski, and J. L. Coﬀer, Fabrication and size dependent properties of porous silicon nanotube arrays, Chem. Commun. 49, 5760 (2013). [27] M. Taghinejad, H. Taghinejad, M. Abdolahad, and S. Mohajerzadeh, A nickel–gold bilayer catalyst engineer- ing technique for self-assembled growth of highly ordered silicon nanotubes, Nano Lett. 13, 889 (2013). [28] Unlike K6irrep (πelectrons) for graphene, ∆ 1irrep is non-degenerate, and therefore orbital degrees of freedom, such asσfor graphene, do not appear in this case. [29] Non-vanishing irreps of the curvature as functions of 1 /R andϕfor each valley are listed in Table IV of the Sup- plemental Material [23]. [30] J. Sichau, M. Prada, T. Anlauf, T. J. Lyon, B. Bosn- jak, L. Tiemann, and R. H. Blick, Resonance microwave measurements of an intrinsic spin-orbit coupling gap in graphene: A possible indication of a topological state, Phys. Rev. Lett. 122, 046403 (2019). [31] J. L. Musfeldt, Y. Iwasa, and R. Tenne, Nanotubes from layered transition metal dichalcogenides, Physics Today 73, 42 (2020). [32] V. Y. Prinz, V. A. Seleznev, A. K. Gutakovsky, A. V. Chehovskiy, V. V. Preobrazhenskii, M. A. Putyato, and T. A. Gavrilova, Free-standing and overgrown in- gaas/gaas nanotubes, nanohelices and their arrays, Phys-6 ica E: Low-dimensional Systems and Nanostructures 6, 828 (2000). [33] Y. Zhang, F. Zhang, Z. Yan, Q. Ma, X. Li, Y. Huang, and J. A. Rogers, Printing, folding and assembly meth- ods for forming 3D mesostructures in advanced materials, Nature Reviews Materials 2, 17019 (2017). [34] A. Avsar, H. Ochoa, F. Guinea, B. ¨Ozyilmaz, B. van Wees, and I. Vera-Marun, Colloquium: Spintronics in graphene and other two-dimensional materials, Reviews of Modern Physics 92, 021003 (2020). [35] M. Matsuo, J. Ieda, K. Harii, E. Saitoh, and S. Maekawa, Mechanical generation of spin current by spin-rotation coupling, Phys. Rev. B 87, 180402 (2013). [36] D. Kobayashi, T. Yoshikawa, M. Matsuo, R. Iguchi, S. Maekawa, E. Saitoh, and Y. Nozaki, Spin Current Gen- eration Using a Surface Acoustic Wave Generated via Spin-Rotation Coupling, Phys. Rev. Lett. 119, 077202 (2017). [37] S. Tateno, G. Okano, M. Matsuo, and Y. Nozaki, Electri- cal evaluation of the alternating spin current generated via spin-vorticity coupling, Phys. Rev. B 102, 104406 (2020). [38] R. C. T. da Costa, Quantum mechanics of a constrained particle, Phys. Rev. A 23, 1982 (1981). [39] S. Matsutani, Berry phase of dirac particle in thin rod, J. Phys. Soc. Jpn 61, 3825 (1992).Supplementary Information: Designing Valley-Dependent Spin-Orbit Interaction by Curvature Ai Yamakage,1,∗T. Sato,2,†R. Okuyama,3T. Funato,4, 5W. Izumida,6K. Sato,7T. Kato,2and M. Matsuo5, 8, 9, 10 1Department of Physics, Nagoya University, Nagoya 464-8602, Japan 2Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan 3Department of Physics, Meiji University, Kawasaki 214-8571, Japan 4Center for Spintronics Research Network, Keio University, Yokohama 223-8522, Japan 5Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China. 6Department of Physics, Tohoku University, Sendai 980-8578, Japan 7National Institute of Technology, Sendai College, Sendai 989-3128, Japan 8CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China 9Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, 319-1195, Japan 10RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan I. EFFECTIVE HAMILTONIAN IN THE PRESENCE OF CURVATURE This section shows a procedure to generate an eﬀective Hamiltonian induced by curvature that can be applied to arbitrary systems including ones with multiple internal degrees of freedom (spin, orbital, and sublattice). The outline of the procedure is: 1. Find the space group, kpoints, and irreducible rep- resentations (irreps) of the low-lying excitations in the ﬂat (uncurved) system. 2. Set the irreps of the symmetry operations. 3. Compute the irreducible decomposition of the ma- trices (operators) Oi. 4. Compute the irreducible decomposition of the mo- mentumki, strains∂iujand curvatures ∂i∂juk. 5. Construct the products of the matrix, momentum, strains, and curvatures. Decompose them into ir- reps. The totally symmetric representations can appear in the eﬀective Hamiltonian. The other irreps correspond to physical observables such as electric/spin currents. As a representative example, we consider a carbon nan- otube (CNT). 1. Irrep of low-energy states For nonmagnetic graphene, which is the ﬂat (unrolled) system of CNT, characterized by the magnetic space groupP6/mmm 1/prime(No. 191.234), the low-lying excita- tions are governed by the Dirac cones around the Kand ∗ai@st.phys.nagoya-u.ac.jp †sato-tetsuya163@g.ecc.u-tokyo.ac.jpK/primepoints, on which the magnetic little cogroup is given by 6/prime/mmm/primeand its unitary maximum subgroup ¯62m (D3h) [1, 2]. They come from the pzorbitals to form a double degeneracy, single-valued two-dimensional odd- parity irrep K6. 2. Symmetry operation TheK6irreps of the generators, i.e., the 3+threefold rotation along the /angbracketleft001/angbracketright(z) direction, 2 100twofold rota- tion along the/angbracketleft100/angbracketright(y) direction, and m001horizontal mirror, are listed as follows [1–4]: DK(3+) =eiσz2π/3, (1) DK(2100) =σy, (2) DK(m001) =−σ0, (3) whereσ0andσi(i=x,y,z ) denote the identity and Pauli matrices acting in the sublattice space, respectively. Ad- ditionally, we have the parity-time (PT) symmetry oper- ation, O→σxO∗σx. (4) Taking the spin degrees of freedom into account, the ir- reps are given by ¯DK(3+) =eiσz2π/3e−iszπ/3, (5) ¯DK(2100) =σy(−isy), (6) ¯DK(m001) =−σ0isz, (7) and the PT symmetry operation by O→σxsyO∗syσx. (8) 3. Irreps of matrices The theory on the Kpoint has the sublattice σand spinsdegrees of freedom, represented by 4 ×4 matrixarXiv:2304.12928v1 [cond-mat.mes-hall] 25 Apr 20232 σµsν. This matrix transforms as O→D†OD. They are decomposed into irreps under the point group ¯62m (D3h). Furthermore, they are decomposed into PT–even and odd irreps. The results are summarized in Table I. Note that we will use the following convention for the components of the two-dimensional irreps, E/primeand E/prime/prime. Their components, ( E(1),E(2)), are transformed for g∈D3has if they are components of in-plane geometric vectors: ¯DK(3+)†/parenleftbigg E(1) E(2)/parenrightbigg ¯DK(3+) =ˆR(2π/3)/parenleftbigg E(1) E(2)/parenrightbigg ,(9) ¯DK(2100)†/parenleftbigg E(1) E(2)/parenrightbigg ¯DK(2100) =/parenleftbigg −E(1) E(2)/parenrightbigg , (10) where ˆR(ϑ) is a two-dimensional rotation matrix, ˆR(ϑ) =/parenleftbigg cosϑ−sinϑ sinϑcosϑ/parenrightbigg . (11) By using this convention, the product rules involving E/prime andE/prime/primeirreps are simpliﬁed as shown in Table III. These results can be easily interpreted geometrically: A/prime 1and A/prime 2components of E/prime×E/primeare inner and outer products of the two in-plane vectors, respectively. 4. Irreps of momentum, strain, and curvature A three-dimensional displacement u= (ux,uy,uz) is transformed using a symmetry operation gof the little group as 3+: ux uy uz → −1/2−√ 3/2 0√ 3/2−1/2 0 0 0 1 ux uy uz ,(12) 2100: ux uy uz → −1 0 0 0 1 0 0 0−1 ux uy uz , (13) m001: ux uy uz → 1 0 0 0 1 0 0 0−1 ux uy uz . (14) Two-dimensional vectors, momentum k= (kx,ky) and gradient ∇= (∂x,∂y), transform in a similar way. Note that momentum is PT–even, while the gradient and dis- placement are PT–odd. The irreducible decomposition of strains and linear and quadratic polynomials of mo- mentum is shown in Table I. 5. Hamiltonian Now we are in a position to obtain any observable oper- ator, including the Hamiltonian. The Hamiltonian must belong to the totally symmetric irrep, A/prime 1. Of the orderofk0and on the ﬂat space, the Hamiltonian is given by a linear combination of σ0s0andσzsz, the latter of which corresponds to spin-orbit coupling in graphene leading to a quantum spin Hall insulator [5] and is ignored in the main text as it is quite small. Since both ( kx,ky) and (σx,σy) belong to the PT–even E/primeirrep, the Hamilto- nian of the ﬁrst order of kis given by∝kxσx+kyσy, which is nothing but a Dirac cone. Strain-induced chiral gauge ﬁeld. Similarly, strain ∂iujcan appear in the Hamiltonian when the product of ∂iujand matrices are in the PT–even A/prime 1irrep. The A/prime 2 strain,∂xuy−∂yux, is not allowed in the Hamiltonian because there is no PT–even A/prime 2matrix. The E/primestrains, on the other hand, can appear as ∝(∂xuy+∂yux)σx+ (∂xux−∂yuy)σy, (15) reproducing the chiral gauge ﬁeld in strained graphene [6]. TheE/prime/primestrain, (∂y,−∂x)uz, also appears in the Hamil- tonian to couple with PT–even spin-dependent E/prime/primema- trixσz(sy,−sx): ∝∂yuzσzsy+∂xuzσzsx, (16) which is the mass term for the Dirac cone with spin de- pendence. Curvature-induced spin-orbit interaction. The next-order terms, ∂i∂juk, including curvature, are cou- pled with PT–odd matrices in the Hamiltonian, result- ing in spin-split eigenstates. For instance, the A/prime/prime 2andE/prime/prime terms are included in the Hamiltonian in the form of ∝∇2uz(σxsy−σysx),/parenleftbig ∂2 y−∂2 x/parenrightbig uzsx+ 2∂x∂yuzsy, /parenleftbig ∂2 y−∂2 x/parenrightbig uz(σxsy+σysx) + 2∂x∂yuz(σxsx−σysy), (17) which is a kind of Rashba-like antisymmetric spin-orbit interaction (SOI) induced by curvature ∂i∂juz, as dis- cussed in the main text. Valley-asymmetric velocities. Similarly, we can re- produce the curvature-induced valley-asymmetric Fermi velocities of electrons that are microscopically derived in Ref. [7]. These are determined by terms proportional to k in the eﬀective Hamiltonian. For the Kvalley, we decom- pose the products of sublattice pseudospin, momentum, and derivatives of uby neglecting tiny spin dependence. We obtain the eﬀective Hamiltonian as the PT–even ir- reps ofA/prime 1, where the leading order in curvature is 1 /R2. By requiring the time-reversal symmetry, the additional terms for the Hamiltonian for τzKvalley yield, H/prime τzK(k) =1 R2/bracketleftBig g1σckc+g2τzσtkt+g3τzkcsin 3ϕ +g4τzktcos 3ϕ+g5(σc,−τzσt)ˆR(−6ϕ)/parenleftBigg kc kt/parenrightBigg/bracketrightBig .(18) Here,gj’s are constants independent of both Randϕ. kcandktare momenta along the circumference and3 TABLE I. Irreps of matrices under ¯62m(D3h) point group. Irreps of momenta and strains are also shown. Irrep 1 3+2100m001 PT–even PT–odd k kk ∇u A/prime 1 1 1 1 1 1, σzsz k2 x+k2 y ∂xux+∂yuy A/prime 2 1 1−1 1 σz,sz ∂xuy−∂yux E/prime2−1 0 2 ( σx,σy) ( σy,−σx)sz (kx,ky) (2kxky,k2 x−k2 y) (∂xuy+∂yux,∂xux−∂yuy) A/prime/prime 1 1 1 1−1 σxsx+σysy A/prime/prime 2 1 1−1−1 σxsy−σysx E/prime/prime2−1 0−2σz(sy,−sx) ( sx,sy), ( ∂y,−∂x)uz (σxsy+σysx,σxsx−σysy) TABLE II. Irreps of curvatures. Irrep ∇∇u A/prime 1 (∂2 x−∂2 y)uy+ 2∂x∂yux A/prime 2 (∂2 x−∂2 y)ux−2∂x∂yuy E/prime∇2(ux,uy),/parenleftbig 2∂x∂yuy+ (∂2 x−∂2 y)ux,2∂x∂yux−(∂2 x−∂2 y)uy/parenrightbig A/prime/prime 1 A/prime/prime 2 ∇2uz E/prime/prime(∂2 y−∂2 x,2∂x∂y)uz TABLE III. Product rules for two-dimensional irreps of D3h. XindicatesX/primeorX/prime/primein the table ( X=A1,A2,orE).X/prime× Y/prime=X/prime/prime×Y/prime/prime=Z/primeandX/prime×Y/prime/prime=X/prime/prime×Y/prime=Z/prime/prime. Product Components A1×E=E A 1(E(1),E(2)) A2×E=E A 2(E(2),−E(1)) E×E=A1E(1)E(1) +E(2)E(2) +A2E(1)E(2)−E(2)E(1) +E(E(1)E(2) +E(2)E(1),E(1)E(1)−E(2)E(2)) axis directions, respectively: ( kc,kt)T=ˆR(−ϕ)(kx,ky)T. Equation (18) is nothing but the Hamiltonian derived in Ref. [7]. Note that some coupling constants, which are independent in Ref. [7], satisfy the relation −c3=c4= 4g5//planckover2pi1in this analysis. This should be because the sym- metries of the system may not be fully taken into account in the previous calculation based on perturbation theory. Actually,c3andc4obtained from a ﬁtting with the nu- merical calculations satisfy −c3/similarequalc4in Ref. [7]. II. IRREDUCIBLE DECOMPOSITION FOR CURVED SILICON In this section, we provide the irreducible decomposi- tion for curved silicon, for each conduction band minima. The curvature is expressed in terms of the displacementﬁeld as, u= (0,0,uz(x·n)), (19) n= (cosϕ,sinϕ,0). (20) As shown in Eq. (3) in the main text, the second deriva- tive ofuis proportional to 1 /R. While the irreducible decomposition is only shown for valleys along [100]- direction in the forms of derivative in the main material, we provide them for all the six valleys in Table IV as a function of 1 /Randϕ. III. NUMERICAL ESTIMATE OF CURVATURE-INDUCED SPIN SPLITTING In this section, we summarize how to estimate the curvature-induced spin splitting for curved thin ﬁlms. In the theoretical work for CNTs [8], the spin splitting induced by the curvature was calculated by considering inclinations of the atomic orbitals at each site, which en- ables hopping between, e.g., pzandsorbitals that are forbidden in graphene. Here, we employ an alternative method to simplify the implementation of the numerical simulation. We consider a general coordinate transforma- tion to make a curved thin ﬁlm ﬂat in new coordinates. In the new coordinates, the hopping energy between neigh- boring sites is modiﬁed due to the curvature of the co- ordinates. This method can treat not only atomic-layer materials such as CNTs but also thin curved ﬁlms of three-dimensional materials. The coordinate transformation is schematically illus- trated in Fig. 1. The gray area in the left panel indi- cates a curved thin material with curvature R, which is assumed to be much larger than the thickness of the material. The Cartesian coordinates at an arbitrary position are expressed in the cylindrical coordinates as (x,y,z ) = (rsinφ,y,r cosφ−R), whereris the dis- tance from the center of the curved material and φis the azimuth angle measured from the z-axis (see Fig. 1). We consider a general transformation from the origi- nal Cartesian coordinates ( x,y,z ) to curved coordinates (x/prime,y/prime,z/prime), in which the curved thin material is mapped to a ﬂat thin material, keeping the ycoordinate unchanged (y=y/prime). The new coordinates are written with the cylin-4 TABLE IV. Irreducible decomposition of momentum k, spins, and curvature of order 1 /R, under the point group C4v, which is the little group of six conduction valley minima, (0 ,0,±k0), (±k0,0,0), and (0,±k0,0), from the space group Fd¯3m. We assume that curvature is expressed as u= (0,0,uz(x·n)) withn= (cosϕ,sinϕ,0). (0,0,±k0) (±k0,0,0) (0,±k0,0) s k 1/Rs k 1/R s k 1/R Irrep (PT-odd) (PT-even) (PT-odd) A1 kz 1/R kx ky A2sz sx sin 2ϕ/R sy −sin 2ϕ/R B1 cos 2ϕ/R B2 sin 2ϕ/R sin 2ϕ/R sin 2ϕ/R E(sy,−sx) (kx,ky) (sz,−sy) (ky,kz) (0,cos2ϕ/R),(0,sin2ϕ/R)(sx,−sz) (kz,kx) (sin2ϕ/R, 0),(cos2ϕ/R, 0) φ φ φ φ FIG. 1. Schematic diagram of the coordinate transformation. The gray region indicates a thin material. Through this coor- dinate transformation, the curved material in the laboratory coordinates ( x,y,z ) is mapped to a ﬂat one in the new coor- dinates (x/prime,y/prime,z/prime). drical coordinates as ( x/prime,y/prime,z/prime) = (Rφ,y,r−R), which leads to the relation, x= (z/prime+R) sin(x/prime/R), y=y/prime, z= (z/prime+R) cos(x/prime/R)−R.(21) For this coordinate transformation, the Jacobian is r/R (= (R+z/prime)/R). Therefore, the volume integral in the old coordinates can be rewritten as /integraldisplay dxdydz (···) =/integraldisplay dx/primedy/primedz/primer R(···). (22) Next, we consider a modulation of a potential energy by this coordinate transformation. For simplicity, we as- sume that the potential of an ion is the Coulomb poten- tial, i.e.,V(x,y,z ) =k/(x2+y2+z2)1/2, wherekis the Coulomb constant. Here, we take the Coulomb constant to bek=Z/planckover2pi1c/137/epsilon1r, wherecis the velocity of light, Zis the eﬀective ionic charge, and /epsilon1ris the relative electrical permittivity. We set /epsilon1r= 3 for carbon nanotubes [9] and /epsilon1r= 11.7 for silicon [10], while we set Z= 4 for both systems. If the curvature Ris much larger than r, the potential energy of an ion located at the origin can bewritten in the new coordinates V(x/prime,y/prime,z/prime) as V(x/prime,y/prime,z/prime) =k/radicalbig R2+r2−2rRcosφ+y2 =k/radicalbig x/prime2+y/prime2+z/prime2 ×/parenleftbigg 1−x/prime2z/prime 2R(x/prime2+y/prime2+z/prime2)+O(1/R2)/parenrightbigg ≡V0(x/prime,y/prime,z/prime) +1 RVcur(x/prime,y/prime,z/prime) +O(1/R2).(23) In addition to the spherical Coulomb potential V0(x/prime,y/prime,z/prime), we also obtain the curvature-induced anisotropic correction Vcur(x/prime,y/prime,z/prime)/R. The present coordinate transformation also modiﬁes the kinetic energy of electrons. The Laplace operator is expressed in cylindrical coordinates as ∆=1 r∂r(r∂r) +1 r2∂2 φ+∂2 y. (24) Using (x/prime,y/prime,z/prime) = (Rφ,y,r−R), the spatial derivatives in the new coordinates are written in terms of r,φ, and yas ∂x/prime=1 R∂φ, ∂y/prime=∂y, ∂z/prime=∂r. (25) Combining Eqs. (24) and (25), the Laplace operator in the new coordinates becomes ∆/similarequal∂2 x/prime+∂2 y/prime+∂2 z/prime+1 R∂z/prime−2z/prime R∂2 x/prime ≡∆0+1 R∆cur, (26) where∆0=∂2 x/prime+∂2 y/prime+∂2 z/primeis the Laplace operator in the new coordinates and ∆curis a correction term induced by the coordinate transformation. The wave functions are also modiﬁed by the coordinate transformation. For instance, the wave function of the pz5 orbital in the original frame is transformed as ψpz(r) =rcosφ−R 4/radicalbig 2πa5ze−√ (rcosφ−R)2+r2sin2φ+y2/2az /similarequal1 4/radicalbig 2πa5z/bracketleftBigg z/prime+1 R/parenleftBigg −x/prime2 2−x/prime2z/prime2 4az/radicalbig x/prime2+y/prime2+z/prime2/parenrightBigg/bracketrightBigg ×e−√ x/prime2+y/prime2+z/prime2/2az ≡ψ(0) pz(r/prime) +1 Rψ(1) pz(r/prime), (27) whereaz=a0/Z,a0is the Bohr radius, ψ(0) pz(r/prime) = ψpz(r/prime) is the original wavefunction, and ψ(1) pz(r/prime)/Risa correction term induced by the coordinate transforma- tion. In the new coordinates, the crystal structure has no curvature and its electronic structure can be obtained using the standard tight-binding calculation. This helps to reduce the computational cost compared with direct calculation for the curved crystal structure. In return, we have to consider the correction terms in the kinetic and potential energies induced by curvature carefully. We can evaluate these corrections using perturbation theory. In our tight-binding calculation, we employ the non- orthogonal Slater-Koster two-center parameters [11, 12] for silicon and CNTs and take the atomic SOIs into ac- count through one-site terms. The correction of the hop- ping integral, which is proportional to 1 /R, is written as /integraldisplay d3r/prime/braceleftbigg ψ(0)∗ n(r/prime+di/2)/parenleftbigg −/planckover2pi12 2m∆cur+Vcur(r/prime+di/2) +Vcur(r/prime−di/2)/parenrightbigg ψ(0) m(r/prime−di/2) (28) +ψ(1)∗ n(r/prime+di/2)/parenleftbigg −/planckover2pi12 2m∆0+V0(r/prime+di/2) +V0(r/prime−di/2)/parenrightbigg ψ(0) m(r/prime−di/2) +ψ(0)∗ n(r/prime+di/2)/parenleftbigg −/planckover2pi12 2m∆0+V0(r/prime+di/2) +V0(r/prime−di/2)/parenrightbigg ψ(1) m(r/prime−di/2) +z/primeψ(0)∗ n(r/prime+di/2)/parenleftbigg −/planckover2pi12 2m∆0+V0(r/prime+di/2) +V0(r/prime−di/2)/parenrightbigg ψ(0) m(r/prime−di/2)/bracerightbigg , FIG. 2. Cylindrical coordinates on a nanotube. where the displacement vectors between two atomic or- bitals are denoted with di(i= 1,···,M). Note that the last term is due to the correction in the Jacobian. The atomic SOI is deﬁned in cylindrical coordinates, which is (xc,xt,xn) in Fig. 2 and is given as λs·l, where s= (sc,st,sn) is the spin operator and l= (lc,lt,ln) is the orbital angular momentum operator in cylindrical coordinates. Since the spin operator is related to the spin of an itinerant electron, shas to be rewritten in the laboratory frame ( x,y,z ) (see Fig. 2). When the axis ofthe nanotube is taken to be in the y(=y/prime) direction, the spin operator in cylindrical coordinates, s, can be rewritten as [8] sc= (˜s+µ−−˜s−µ+)/i, (29) st= ˜sy, (30) sn= ˜s+µ−+ ˜s−µ+, (31) where ˜s= (˜sx,˜sy,˜sz) is the spin operator in the labo- ratory frame, ˜ s±= ˜sz±i˜sxis the spin ladder operator, andµ±is the operator which changes the momentum of the circumference direction kcby±δkc=±1/R. Using these relations, the atomic SOI takes the form of λs·l=λ(˜syly+ ˜s+µ−l−+ ˜s−µ+l+). (32) This spin-orbit interaction hybridizes a up-spin state with wavenumber ( kc,kt,kn) with a down-spin state with wavenumber ( kc±δkc,kt,kn). This transfer between kc and ˜syis ascribable to the conservation of the total angu- lar momentum in the ydirection in the laboratory frame, as we will discuss soon. For example, let us consider a band calculation of silicon using eight atomic orbitals, i.e., four atomic orbitals (3 s, 3px, 3py, and 3pz) per sub- lattice. Here, we need to consider a 16 ×16 matrix for the HamiltonianH(k) by taking the spin degree of freedom into account, in which the eight up-spin orbitals with6 (b) (d)2.0 -2.00 2.0 -2.00(meV) 000(eV) (meV) (eV)-1 1246 -246248 -248-1 1 0-2 21 0 -12.0 -2.00 -1 1(eV) 2.0 -2.00 0(eV) -2 20(a) (c) 5 -5(/nm) (/nm) (/nm)(/nm) (/nm) (/nm) FIG. 3. Band structures for (a-b) (9 ,0) zigzag nanotube and (c-d) (6 ,6) armchair nanotube. k0/similarequal1.3225 nm−1andktis a momentum along the axis direction. (a)(c) Band structures in the absence of curvature, which are obtained from that of graphene on the cutting lines, reﬂecting the boundary condition in the circumferential direction. (b)(d) Band structures in the presence of curvature. A band gap opens up at the Dirac point in both the zigzag and armchair nanotubes. The gap for the zigzag nanotube ( ∼500 meV) is much larger than that for the armchair ( ∼0.4 meV). For the zigzag nanotube, spin splitting is observed in both the conduction band ( ∼0.2 meV) and the valence band ( ∼1.1 meV). On the other hand, for the armchair nanotube, spin splitting is negligibly small in both bands. These results are consistent with Ref. [8]. wavenumber ( kc,kt,kn) couple to the eight downspin or- bitals with shifted wavenumber ( kc+δkc,kt,kn). Since scandsnchange the (quantized) momentum in the cir- cumference direction, kc, it is no longer a good quantum number. Instead, the total angular momentum in the di- rection of the y-axis,Jy=Rkc+sy/2, is conserved, due to the axial symmetry in the laboratory frame. Thus, the eﬀect of the curvature is taken into account through the modiﬁcation of the hopping integral and the SOI. We can obtain the band structure by numerically diagonalizing the Hamiltonian H(k) for ﬁxedk. For nanotube structures, we should also take into ac- count the boundary condition for the circumference di- rection. In the absence of the SOI, kcis discretized in order to satisfy the boundary condition. Therefore, the original Brillouin zone for an unbent material is quan- tized into line segments [13], called cutting lines. A cut- ting line is nothing but a quasi-one-dimensional subband, labeled bykc. In the presence of the SOI, Jyshould be used instead of kcto specify a subband [8], as kcis no longer a good quantum number.A. Carbon nanotubes To discuss the eﬀect of curvature on graphene, we will focus on (9 ,0) (zigzag) and (6 ,6) (armchair) nan- otubes. Fig. 3 (a) and (c) show the band structures for the low-lying cutting lines in the absence of curvature for zigzag and armchair nanotubes, respectively. When the curvature is non-zero, the band structures of the zigzag and armchair nanotubes change into those in Fig. 3 (b) and (d), respectively. Here, the radii of the (9 ,0) and (6,6) nanotubes are R/similarequal0.35 nm and R/similarequal0.41 nm. The band gap between the conduction and valence bands is estimated to be 500 meV for the (9 ,0) nanotube and 0.4 meV for the (6 ,6) nanotube. The spin splitting ener- gies for the conduction and valence bands are estimated to be 0.2 meV and 1 .1 meV for the (9 ,0) nanotube, while they become much smaller for the (6 ,6) nanotube. The band gap opens exactly at the Kpoint for (9 ,0) nan- otube, while its position is shifted from the Kpoint by ktT/2π∼0.08 for (6,6) nanotube, where ktis the mo- mentum along the axis direction, T= 2√ 3πR/dRis the length of the unit cell of the nanotube, and dRis the greatest common divisor of 2 n+mand 2m+nfor the (n,m) nanotube [13].7 (eV) FIG. 4. Energy dispersion at the valley of (0 ,k0,0) forϕ= 0 as a function of angular momentum along the axis direction, Jy. These results semi-quantitatively agree with the previ- ous theoretical work. In fact, the band gap is opened by the curvature in the same way as in Ref. [8]. The spin splitting of the (9 ,0) nanotube and the gap of the (6 ,6) nanotube, which are proportional to 1 /R, have similar values to those in Ref. [8]. However, the gap of the (9 ,0) nanotube, which is proportional to 1 /R2, is several times larger than the value reported in Ref. [8]. This deviation is consistent with the fact that our calculation neglected the contribution of 1 /R2; the present tight-binding cal- culation leads to a reasonable estimate only up to 1 /R. B. Silicon Next, we consider spin-orbit coupling in curved silicon. We assume that the ydirection is unchanged, while the xdirection is maximally curved. This situation corre- sponds toϕ= 0 in Eq. (7) in the main text. Here, the spinsyand the momentum kcare not conserved, because they do not commute with the eﬀective Hamiltonian near the valley. Instead, the total angular momentum along the axis direction, Jy=Rkc+sy/2, becomes a good quantum number (see Eq. (32) and the subsequent ex- planation), and therefore the cutting lines are labeled by Jy. Figure 4 depicts the energy dispersion at the valley of (0,k0,0) as a continuous function of Jy. Note that Jy is actually quantized by the boundary condition; Jy= 0 (Jy=±1) yields the lowest (next-lowest) cutting line, for example. All the band structures in Fig. 3(b) in the main text were obtained with this procedure. The band structure can be obtained in the case of ϕ= π/4 by changing the directions of xt,xc, andxnwith carefully treating the direction of a crystal structure.IV. THIN-FILM QUANTIZATION The thin-ﬁlm quantization is a method to describe a particle conﬁned in a one- or two-dimensional curved space embedded in three-dimensional space [14, 15]. In this method, the spatial constraint on a particle can be represented by the geometric potentials depending on the curvatures of the low-dimensional space. In this section, we brieﬂy explain that the thin-ﬁlm quantization proce- dure does not lead to an appropriate curvature-induced SOI by considering the CNT as an illustrative example. Let us derive an eﬀective Hamiltonian for the CNT by using the method given in Refs. [14, 15]. For an electron conﬁned in a curved thin ﬁlm such as the CNT, one starts with a Dirac equation in 3+1 dimensional curved space- time with the geometric potentials of a cylindrical surface which is considered to be the CNT. Following Ref. [15], we use the cylindrical coordinates shown in Fig. 2. Note thatxn= 0 when an electron is on the nanotube. We start with the Dirac equation: i/planckover2pi1∂ ∂tψ(x)/radicalbig 1 +xn/R = [−i/planckover2pi1cαˆiej ˆi∂j+βmc2+V(xn)]ψ(x)/radicalbig 1 +xn/R,(33) whereαˆiis the alpha matrix, V(xn) =vcx2 nis the con- ﬁnement potential to the nanotube with strength param- etervc, andej ˆiis the triad ﬁeld, whose indices ˆiandj respectively label the Cartesian coordinates ( x,y,z ) and cylindrical coordinates ( xc,xt,xn): ec ˆi=1/radicalbig 1 +xn/R/parenleftBig −sinxc Rδˆiz+ cosxc Rδˆix/parenrightBig ,(34) et ˆi=δˆiy, (35) en ˆi= cosxc Rδˆiz+ sinxc Rδˆix, (36) whereδˆiˆjis Kronecker’s delta. By taking the conﬁnement potential to be inﬁnitely large, i.e., vc→∞ , we obtain the SOI modulated by the geometric potential in the non- relativistic limit as follows: Htfq so=/planckover2pi1stpc 2mR. (37) Next, we estimate the SOI modulation derived by the thin-ﬁlm quantization on a CNT (see Fig. 5). The ma- trix elements of the SOI modulation between the nearest- neighbor sites on the CNT are given by /angbracketleftRA\|Htfq so\|RA+dj/angbracketright=2/planckover2pi1vF 3ist∆ktfq socosφj (38) wherej= 1,2,3 are integers, vF/similarequal8.32×105m/s is the Fermi velocity, and φj=−ϕ+2π 3(j−1) is the angle be- tween the displacement vector djand thexc-axis.∆ktfq so characterizes the energy gap between the conduction and8 FIG. 5. Crystalline structure of a CNT. a1anda2are primi- tive lattice vectors, and ∆j(j= 1,2,3) are vectors connecting the nearest-neighbor sites. θis the angle between a1and the xc-axis. valence bands around KandK/primepoints due to the SOI modulation: it is estimated as ∆ktfq so=−v 2vFR/similarequal−0.82×1 2R, (39)where we have deﬁned v=/planckover2pi1κ/m, and 2κ/3 denotes the matrix elements of the momentum operator, given by [16, 17] 2 3κ=/angbracketleftRA\|px\|RA+dj/angbracketright/similarequal0.22 a.u. (40) Our estimate of the curvature-induced SOI by the thin- ﬁlm quantization is three orders of magnitude larger than the previous results, ∆kso/similarequal5.3×10−4/2R, and its sign is opposite to that of the SOI estimated by a ﬁrst-principles calculation [8]. This overestimation is caused by the con- straint imposed by the inﬁnitely large conﬁnement poten- tialV(xn), which is larger than the energy of the electron mass. This situation is not suitable to the consideration of electronic states conﬁned in a CNT. [1] M. I. Aroyo, J. M. Perez-Mato, C. Capillas, E. Kroumova, S. Ivantchev, G. Madariaga, A. Kirov, and H. Won- dratschek, Bilbao Crystallographic Server: I. Databases and crystallographic computing programs, Z. Kristallogr. Cryst. Mater. 221, 15 (2006). [2] M. I. Aroyo, A. Kirov, C. Capillasm, J. M. Perez-Mato, and H. Wondratschek, Bilbao Crystallographic Server. II. Representations of crystallographic point groups and space groups, Acta Cryst. A62, 115 (2006). [3] Y. Xu, L. Elcoro, Z.-D. Song, B. J. Wieder, M. G. Vergniory, N. Regnault, Y. Chen, C. Felser, and B. A. Bernevig, High-throughput calculations of mag- netic topological materials, Nature 586, 702 (2020). [4] L. Elcoro, B. J. Wieder, Z. Song, Y. Xu, B. Bradlyn, and B. A. Bernevig, Magnetic topological quantum chemistry, Nat. Commun. 12, 5965 (2021). [5] C. L. Kane and E. J. Mele, Quantum spin Hall eﬀect in graphene, Phys. Rev. Lett. 95, 226801 (2005). [6] M. A. H. Vozmediano, M. I. Katsnelson, and F. Guinea, Gauge ﬁelds in graphene, Phys. Rep. 496, 109 (2010). [7] W. Izumida, A. Vikstr¨ om, and R. Saito, Asymmetric ve- locities of dirac particles and vernier spectrum in metallic single-wall carbon nanotubes, Phys. Rev. B 85, 165430 (2012). [8] W. Izumida, K. Sato, and R. Saito, Spin–Orbit Inter- action in Single Wall Carbon Nanotubes: Symmetry Adapted Tight-Binding Calculation and Eﬀective Model Analysis, J. Phys. Soc. Jpn. 78, 074707 (2009). [9] Y. Wang, V. W. Brar, A. V. Shytov, Q. Wu, W. Regan, H.-Z. Tsai, A. Zettl, L. S. Levitov, and M. F. Crommie, Mapping Dirac quasiparticles near a single Coulomb im- purity on graphene, Nature Physics 8, 653 (2012). [10] W. C. Dunlap and R. L. Watters, Direct measurement of the dielectric constants of silicon and germanium, Phys. Rev.92, 1396 (1953).[11] H. Min, J. E. Hill, N. A. Sinitsyn, B. R. Sahu, L. Klein- man, and A. H. MacDonald, Intrinsic and rashba spin- orbit interactions in graphene sheets, Phys. Rev. B 74, 165310 (2006). [12] D. A. Papaconstantopoulos, Handbook of the Band Struc- ture of Elemental Solids (Springer New York, NY, 2015) p. 344. [13] R. Saito, G. Dresselhaus, and M. S. Dresselhaus, Physical Properties of Carbon Nanotubes (Imperial College Press, London, 1998). [14] R. C. T. da Costa, Quantum mechanics of a constrained particle, Phys. Rev. A 23, 1982 (1981). [15] S. Matsutani, Berry phase of dirac particle in thin rod, J. Phys. Soc. Jpn 61, 3825 (1992). [16] A. Gr¨ uneis, R. Saito, G. G. Samsonidze, T. Kimura, M. A. Pimenta, A. Jorio, A. G. S. Filho, G. Dressel- haus, and M. S. Dresselhaus, Inhomogeneous optical ab- sorption around the k point in graphite and carbon nan- otubes, Phys. Rev. B 67, 165402 (2003). [17] A. Gr¨ uneis, Resonance Raman spectroscopy of single wall carbon nanotubes , Ph.D. thesis, Tohoku University (2004).
1503.06835v2.Critical_Temperature_and_Tunneling_Spectroscopy_of_Superconductor_Ferromagnet_Hybrids_with_Intrinsic_Rashba_Dresselhaus_Spin_Orbit_Coupling.pdf	Critical Temperature and Tunneling Spectroscopy of Superconductor/Ferromagnet Hybrids with Intrinsic Rashba–Dresselhaus Spin-Orbit Coupling Sol H. Jacobsen,1Jabir Ali Ouassou,1and Jacob Linder1 1Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway These authors contributed equally to this work. We investigate theoretically how the proximity effect in superconductor/ferromagnet hybrid structures with in- trinsic spin-orbit coupling manifests in two measurable quantities, namely the density of states and critical tem- perature. To describe a general scenario, we allow for both Rashba and Dresselhaus type spin-orbit coupling. Our results are obtained via the quasiclassical theory of superconductivity, extended to include spin-orbit cou- pling in the Usadel equation and Kupriyanov–Lukichev boundary conditions. Unlike previous works, we have derived a Riccati parametrization of the Usadel equation with spin-orbit coupling which allows us to address the full proximity regime and not only the linearized weak proximity regime. First, we consider the density of states in both SF bilayers and SFS trilayers, where the spectroscopic features in the latter case are sensitive to the phase difference between the two superconductors. We ﬁnd that the presence of spin-orbit coupling leaves clear spectroscopic ﬁngerprints in the density of states due to its role in creating spin-triplet Cooper pairs. Unlike SF and SFS structures without spin-orbit coupling, the density of states in the present case depends strongly on the direction of magnetization. Moreover, we show that the spin-orbit coupling can stabilize spin-singlet superconductivity even in the presence of a strong exchange ﬁeld hD. This leads to the possibility of a mag- netically tunable minigap: changing the direction of the exchange ﬁeld opens and closes the minigap. We also determine how the critical temperature Tcof an SF bilayer is affected by spin-orbit coupling and, interestingly, demonstrate that one can achieve a spin-valve effect with a single ferromagnet. We ﬁnd that Tcdisplays highly non-monotonic behavior both as a function of the magnetization direction and the type and direction of the spin-orbit coupling, offering a new way to exert control over the superconductivity of proximity structures. I. INTRODUCTION Material interfaces in hybrid structures give rise to proxim- ity effects, whereby the properties of one material can “leak” into the adjacent material, creating a region with properties derived from both materials. In superconductor/ferromagnet (SF) hybrid structures1, the proximity effect causes supercon- ducting correlations to penetrate into the ferromagnetic re- gion and vice versa. These correlations typically decay over short distances, which in diffusive systems is of the orderp D=h, where Dis the diffusion coefﬁcient of the ferromag- net and his the strength of the exchange ﬁeld. However, for certain ﬁeld conﬁgurations, the singlet correlations from the superconductor may be converted into so-called long-range triplets (LRTs)2. These triplet components have spin projec- tion parallel to the exchange ﬁeld, and decay over much longer distances. This results in physical quantities like supercur- rents decaying over the length scale xN=p D=T, which is usually much larger than the ferromagnetic coherence length xF=p D=h, where Tis the temperature. This distance is independent of h, and at low temperatures it becomes increas- ingly large, which allows the condensate to penetrate deep into the ferromagnet. The isolation and enhancement of this fea- ture has attracted much attention in recent years as it gives rise to novel physics and possible low-temperature applications by merging spintronics and superconductivity3. It is by now well-known that the conversion from singlet to long-range triplet components of the superconducting state can happen in the presence of magnetic inhomogeneities4,5, i.e.a spatially varying exchange ﬁeld, and until recently such inhomogeneities were believed to be the primary source of this conversion6–15, although other proposals using e.g.non- equilibrium distribution functions and intrinsic triplet super-conductors also exist16–19. However, it has recently been es- tablished that another possible source of LRT correlations is the presence of a ﬁnite spin-orbit (SO) coupling, either in the superconducting region20or on the ferromagnetic side21,22. In fact, it can be shown that an SF structure where the magnetic inhomogeneity is due to a Bloch domain wall, as considered ine.g. Refs. 23–25, is gauge equivalent to one where the ferromagnet has a homogeneous exchange ﬁeld and intrinsic SO coupling21. It is known that SO scattering can be caused by impurities26, but this type of scattering results in purely isotropic spin-relaxation, and so does not permit the desired singlet-LRT conversion. To achieve such a conversion, one needs a rotation of the spin pair into the direction of the ex- change ﬁeld27. This can be achieved by using materials with an intrinsic SO coupling, either due to the crystal structure in the case of noncentrosymmetric materials28, or due to inter- faces in thin-ﬁlm hybrids29, where the latter also modiﬁes the fundamental process of Andreev reﬂection30,31. The role of SO coupling with respect to the supercurrent in ballistic hy- brid structures has also been studied recently32. In this paper, we establish how the presence of spin-orbit coupling in SF structures manifests in two important exper- imental observables: the density of states D(e)probed via tunneling spectroscopy (or conductance measurements), and the critical temperature Tc. A common consequence for both of these quantities is that neither becomes independent of the magnetization direction. This is in contrast to the case with- out SO coupling in conventional monodomain ferromagnets, where the results are invariant with respect to rotations of the magnetic exchange ﬁeld. This symmetry is now lifted due to SO coupling: depending on the magnetization direction, LRT Cooper pairs are created in the system which leave clear ﬁn- gerprints both spectroscopically and in terms of the Tcbehav-arXiv:1503.06835v2 [cond-mat.supr-con] 6 Jul 20152 ior. On the technical side, we will present in this work for the ﬁrst time a Riccati parametrization of the Usadel equation and its corresponding boundary conditions that include SO cou- pling. This is an important advance in terms of exploring the full physics of triplet pairing due to SO coupling as it allows for a solution of the quasiclassical equations without any as- sumption of a weak proximity effect. We will also demon- strate that the SO coupling can actually protect the singlet su- perconducting correlations even in the presence of a strong exchange ﬁeld, leading to the possibility of a minigap that is magnetically tunable via the orientation of the exchange ﬁeld. The remainder of the article will be organised as follows: In Section II, we introduce the relevant theory and notation, starting from the quasiclassical Usadel equation, which de- scribes the diffusion of the superconducting condensate into the ferromagnet. We also motivate our choice of intrinsic SO coupling in this section, and propose a new notation for de- scribing Rashba–Dresselhaus couplings. The section goes on to discuss key analytic features of the equations in the limit of weak proximity, symmetries of the density of states at zero energy, and analytical results needed to calculate the critical temperature of hybrid systems. We then present detailed nu- merical results in Section III: we analyze the density of states of an SF bilayer in III A [see Fig. 1(a)], with the case of pure Rashba coupling considered in Section III B, and we study the SFS Josephson junction in III C [see Fig. 1(b)]. We con- sider different orientations and strengths of the exchange ﬁeld and SO coupling, and in the case of the Josephson junction, the effect of altering the phase difference between the con- densates. Then, in Section III D, we continue our treatment of the SF bilayer in the full proximity regime by including a self-consistent solution in the superconducting layer, and fo- cus on how the presence of SO coupling affects the critical temperature of the system. We discover that the SO coupling allows for spin-valve functionality with a single ferromagnetic layer, meaning that rotating the magnetic ﬁeld by p=2 induces a large change in Tc. Finally, we conclude in Section IV with a summary of the main results, a discussion of some additional consequences of the choices made in-text, as well as possibil- ities for further work. II. THEORY A. Fundamental concepts The diffusion of the superconducting condensate into the ferromagnet can be described by the Usadel equation, which is a second-order partial differential equation for the Green’s function of the system33. Together with appropriate bound- ary conditions, the Usadel equation establishes a system of coupled differential equations that can be solved in one di- mension. We will consider the case of diffusive equilibrium, where the retarded component ˆ gRof the Green’s function is sufﬁcient to describe the behaviour of the system34,35. We start by examining the superconducting correlations in the fer- romagnet, and use the standard Bardeen–Cooper–Schrieffer (BCS) bulk solution for the superconductors. In particular,we will clarify the spectroscopic consequences of having SO coupling in the ferromagnetic layer. In the absence of SO coupling, the Usadel equation33in the ferromagnet reads DFÑ(ˆgRÑˆgR)+i eˆr3+ˆM;ˆgR =0; (1) where the matrix ˆr3=diag(1;1), andeis the quasiparticle energy. The magnetization matrix ˆMin the above equation is ˆM= hs 0 0(hs) ; where h= (hx;hy;hz)is the ferromagnetic exchange ﬁeld, () denotes complex conjugation, s= (sx;sy;sz)is the Pauli vector, and skare the usual Pauli matrices. The corresponding Kupriyanov–Lukichev boundary conditions are36 2LjzjˆgR jÑˆgR j= [ˆgR 1;ˆgR 2]; (2) where the subscripts refer to the different regions of the hybrid structure; in the case of an SF bilayer as depicted in Fig. 1(a), j=1 denotes the superconductor, and j=2 the ferromagnet, while Ñdenotes the derivative along the junction 1 !2. The respective lengths of the materials are denoted Lj, and the in- terface parameters zj=RB=Rjdescribe the ratio of the barrier resistance RBto the bulk resistance Rjof each material. z = 0(a) Bilayer (b) Josephson junctionS F FS SSO SOz = ‒L S z = L F z = ‒L F/2 z = L F/2 FIG. 1: (Color online) (a) The SF bilayer in III A, III B and III D. We take the thin-ﬁlm layering direction along the z- axis, and assume an xy-plane Rashba–Dresselhaus coupling in the ferromagnetic layer. (b) The SFS trilayer in III C. We will use the Riccati parameterisation37for the quasi- classical Green’s function ˆ gR, ˆgR= N(1+g˜g) 2Ng 2˜N˜g˜N(1+˜gg) ; (3)3 where the normalisation matrices are N= (1g˜g)1and ˜N= (1˜gg)1. The tilde operation denotes a combination of complex conjugation i! iand energy e!e, with g!˜g, N!˜N. The Riccati parameterisation is particularly useful for numerical computation because the parameters are bounded [0;1], contrary to the multi-valued q-parameterisation34. In practice, this means that for certain parameter choices the nu- merical routines will only converge in the Riccati formulation. Appendix A contains some further details on this parameteri- sation. To include intrinsic SO coupling in the Usadel equation, we simply have to replace all the derivatives in Eq. (1) with their gauge covariant counterparts:21,38 Ñ()7!˜Ñ()Ñ()i[ˆA;]: (4) This is valid for any SO coupling linear in momentum. We consider the leading contribution; higher order terms, e.g. those responsible for the SU(2) Lorentz force, are neglected here. Such higher order terms are required to produce so- called j0junctions which have attracted interest of late39, and consequently we will see no signature of the j0effect in the systems considered herein. The object ˆAhas both a vector structure in geometric space, and a 4 4 matrix structure in Spin–Nambu space, and can be written as ˆA=diag(A;A) in terms of the SO ﬁeld A= (Ax;Ay;Az), which will be dis- cussed in more detail in the next subsection. SO coupling in the context of quasiclassical theory has also been discussed in Refs. 38,40. When we include the SO coupling as shown above, we derive the following form for the Usadel equation (see Appendix A): DF ¶2 kg+2(¶kg)˜N˜g(¶kg) =2iegih(sggs) +DF AAggAA+2(Ag+gA)˜N(A+˜gAg) +2iDF (¶kg)˜N(A k+˜gAkg)+(Ak+gA k˜g)N(¶kg) ;(5) where the index kindicates an arbitrary choice of direction in Cartesian coordinates. The corresponding equation for ˜gis found by taking the tilde conjugate of Eq. (5). Similarly, the boundary conditions in Eq. (2) become: ¶kg1=1 L1z1(1g1˜g2)N2(g2g1)+iAkg1+ig1A k; ¶kg2=1 L2z2(1g2˜g1)N1(g2g1)+iAkg2+ig2A k;(6) and the ˜gcounterparts are found in the same way as before. For the details of these derivations, see Appendix A. We will now discuss the deﬁnition of current in the presence of spin-orbit interactions. Since the Hamiltonian including SO coupling contains terms linear in momentum (see below), the velocity operator vj=¶H=¶kjis affected. We stated above that the Kupriyanov-Lukichev boundary conditions are sim- ply modiﬁed by replacing the derivative with its gauge covari- ant counterpart including the SO interaction. To make sure that current conservation is still satisﬁed, we must carefully examine the Usadel equation. In the absence of SO coupling,the quasiclassical expression for electric current is given by Ie=I0Z¥ ¥deTrfr3(ˇgÑˇg)Kg; (7) where ˇ gis the 88 Green’s function matrix in Keldysh space ˇg=ˆgRˆgK ˆ0 ˆgA ; (8) andI0is a constant that is not important for this discussion. Current conservation can now be proven from the Usadel equation itself. We show this for the case of equilibrium, which is relevant for the case of supercurrents in Josephson junctions. In this case ˆ gK= (ˆgRˆgA)tanh(e=2T)and we get Ie=I0Z¥ ¥deTrfr3(ˆgRÑˆgRˆgAÑˆgA)gtanh(e=2T):(9) Performing the operation Tr fr3g on the Usadel equation, we obtain DÑTrfr3(ˆgRÑˆgR)+iTrfr3[er3+ˆM;ˆgR]g=0: (10) Now, inserting the most general deﬁnition of the Green’s func- tion ˆgR, one ﬁnds that the second term in the above equation is always zero. Thus, we are left with ÑTrfr3(ˆgRÑˆgR)g=0; (11) which expresses precisely current conservation since the same analysis can be done for ˆ gA. Now, let us include the SO cou- pling. The current should then be given by Ie=I0Z¥ ¥deTrfr3(ˇg˜Ñˇg)Kg; (12) so that the expression for the charge current is modiﬁed by the presence of SO coupling, as is known. The question is now if this current is conserved, as it has to be physically. We can prove that it is from the Usadel equation by rewriting it as DÑ(ˆgR˜ÑˆgR) =D[A;ˆgRÑˆgR]+D[A;[A;ˆgR]]i[er3+ˆM;ˆgR];(13) and then performing the operation Tr fr3g, one ﬁnds: DÑTrfr3(ˆgR˜ÑˆgR)g=0; (14) so we recover the standard current conservation law ÑIe=0. B. Spin-orbit ﬁeld The precise form of the generic SO ﬁeld Ais imposed by the experimental requirements and limitations. As the name suggests, spin-orbit coupling couples a particle’s spin with its motion, and more speciﬁcally its momentum. As mentioned in the Introduction, the SO coupling in solids can originate from a lack of inversion symmetry in the crystal structure.4 Such spin-orbit coupling can be of both Rashba and Dressel- haus type and is determined by the point group symmetry of the crystal41,42. It is also known that the lack of inversion sym- metry due to surfaces, either in the form of interfaces to other materials or to vacuum, will give rise to antisymmetric spin- orbit coupling of the Rashba type. For sufﬁciently thin struc- tures, the SO coupling generated in this way can permeate the entire structure, but the question of precisely how far into ad- jacent materials such surface-SO coupling may penetrate ap- pears to be an open question in general. Intrinsic inversion asymmetry arises naturally due to interfaces between materi- als in thin-ﬁlm hybrid structures such as the ones considered herein. Noncentrosymmetric crystalline structures provide an alternative source for intrinsic asymmetry, and are considered in Ref. 43. In thin-ﬁlm hybrids, the Rashba spin splitting de- rives from the cross product of the Pauli vector swith the momentum k, HR=a m(sk)ˆz; (15) where ais called the Rashba coefﬁcient, and we have chosen a coordinate system with ˆ zas the layering direction. Another well-known type of SO coupling is the Dresselhaus spin split- ting, which can occur when the crystal structure lacks an in- version centre. For a two-dimensional electron gas (quantum well) conﬁned in the ˆ z-direction, then to ﬁrst order hkzi=0, so the Dresselhaus splitting becomes HD=b m(sykysxkx); (16) where bis called the Dresselhaus coefﬁcient. In our struc- ture, we consider a thin-ﬁlm geometry with the conﬁnement being strongest in the z-direction. Although there may cer- tainly be other terms contributing to the Dresselhaus SO cou- pling in such a structure, since real thin-ﬁlm structures will have three-dimensional quasiparticle diffusion and we use a 2Dform of the SO coupling here, we consider the standard form Eq. (16) as an approximation that captures the main physics in the problem. This is a commonly used model in the literature to explore the effects originating from SO cou- pling in a system. When we combine both interactions, we obtain the Hamiltonian for a general Rashba–Dresselhaus SO coupling, HRD=kx m(asybsx)ky m(asxbsy): (17) In this work, we will restrict ourselves to this form of SO cou- pling. It should be noted that our setup may also be viewed as a simpliﬁed model for a scenario where the SO coupling and ferromagnetism exist in separate, thin layers, in which case we expect qualitatively similar results to the ones reported in this manuscript. As explained in Ref. 21, the SO coupling acts as a back- ground SU(2) ﬁeld, i.e.an object with both a vector structure in geometric space, and a 2 2 matrix structure in spin space. We can therefore identify the interaction above with an effec- tive vector potential Awhich we will call the SO ﬁeld , HRDkA=m; (18)from which we derive that A= (bsxasy;asxbsy;0): (19) At this point, it is convenient to introduce a new notation for describing Rashba–Dresselhaus couplings, which will let us distinguish between the physical effects that derive from the strength of the coupling, and those that derive from the geometry. For this purpose, we employ polar notation deﬁned by the relations aasinc; bacosc; (20) where we will refer to aas the SO strength , andcas the SO angle . Rewritten in the polar notation, Eq. (19) takes the form: A=a(sxcosc+sysinc)ˆxa(sxsinc+sycosc)ˆy:(21) From the deﬁnition, we can immediately conclude that c=0 corresponds to a pure Dresselhaus coupling, while c=p=2 results in a pure Rashba coupling, with the geometric interpre- tation of cillustrated in Fig. 2. Note that A2 x=A2 y=a2, which means that A2=2a2. Another useful property is that we can switch the components Ax$Ayby letting c!3p=2c. sxcosc+sysinc kxcky sxsinc+sycosc c FIG. 2: Geometric interpretation of the SO ﬁeld (21) in polar coordinates: the Hamiltonian couples the momentum com- ponent kxto the spin component (sxcosc+sysinc)with a coefﬁcient +a=m, and the momentum component kyto the spin component ( sxsinc+sycosc) with a coefﬁcient a=m. Thus, adetermines the magnitude of the coupling, and cthe angle between the coupled momentum and spin components. The appearance of LRTs in the system depends on the inter- play between SO coupling and the direction of the exchange ﬁeld. Recall that the LRT components are deﬁned as hav- ing spin projections parallel to the exchange ﬁeld, as opposed to the short-ranged triplet (SRT) component which appears as long as there is exchange splitting44but has spin projec- tion perpendicular to the ﬁeld and is therefore subject to the same pair-breaking effect as the singlets3,27, penetrating only a very short distance into strong ferromagnets. Thus if we have an SO ﬁeld component along the layering direction, e.g. if we had Az6=0 in Figs. 1(a) and 1(b), achievable with a non- centrosymmetric crystal or in a nanowire setup, then a non- vanishing commutator [A;hs]creates the LRT. However, we will from now only consider systems where Az=0, in which5 case the criterion for LRT is21that[A;[A;hs]]must not be parallel to the exchange ﬁeld hs. Expanding, we have [A;[A;hs]] =4a2(hs+hzsz) 4a2(hxsy+hysx)sin2c; (22) from which it is clear that no LRTs can be generated for a pure Dresselhaus coupling c=0 or Rashba coupling c=p=2 when the exchange ﬁeld is in-plane. However, the effect of SO coupling becomes increasingly signiﬁcant for angles close to p=4 (see Fig. 4 in Section III A). We also see that no LRTs can be generated for in-plane magnetization in the special case hx=hyandhz=0, since hxsy+hysxcan then be rewritten as hxsx+hysy, which is parallel to h. There is no LRT genera- tion for the case hx=hy=0 and hz6=0 for similar reasons. In general however, the LRT will appear for an in-plane magne- tization as long as hx6=hyand the SO coupling is not of pure Dresselhaus or pure Rashba type. It is also important to note that the LRT can be created even for pure Rashba type SO coupling if the magnetization has both in- and out-of-plane magnetization components. We will discuss precisely this sit- uation in Sec. III B. Once the condition for long-range triplet generation is sat- isﬁed, increasing the corresponding exchange ﬁeld will also increase the proportion of long-range triplets compared with short-range triplets. Whether or not the presence of long- range triplets can be observed in the system, i.e.if they retain a clear signature in measurable quantities such as the density of states when the criteria for their existence is fulﬁlled, depends on other aspects such as the strength of the spin-orbit coupling and will be discussed later in this paper. Thus, a main moti- vation for this work is to take a step further than discussing their existence21and instead make predictions for when long- ranged triplet Cooper pairs can actually be observed via spec- troscopic or T cmeasurements in SF structures with spin-orbit coupling. However, we will also demonstrate that the pres- ence of SO coupling offers additional opportunities besides the creation of LRT Cooper pairs. We will show both ana- lytically and numerically that the SO coupling can protect the singlet component even in the presence of an exchange ﬁeld, which normally would suppress it. This provides the possi- bility of tuning the well-known minigap magnetically , both in bilayer and Josephson junctions, simply by altering the direc- tion of the magnetization. C. Weak proximity effect In order to establish a better analytical understanding of the role played by SO coupling in the system before presenting the spectroscopy and Tcresults, we will now consider the limit of weak proximity effect, which means that jgi jj1,N1 in the ferromagnet. The anomalous Green’s function in gen- eral is given by the upper-right block of Eq. (3), f=2Ng, which we see reduces to f=2gin this limit. It will also prove prudent to express the anomalous Green’s function using a singlet/triplet decomposition, where the singlet component is described by a scalar function fs, and the triplet componentsencapsulated in the so-called d-vector45,46, f= (fs+ds)isy: (23) Combining the above with the weak proximity identity f=2g, we see that the components of gcan be rewritten as: g=1 2 idydxdz+fs dzfsidy+dx! : (24) Under spin rotations, the singlet component fswill then transform as a scalar, while the triplet component d= (dx;dy;dz)transforms as an ordinary vector. Another useful feature of this notation is that it becomes almost trivial to dis- tinguish between short-range and long-range triplet compo- nents; the projection d=dˆhalong the exchange ﬁeld corre- sponds to the SRTs, while the perpendicular part d?=jdˆhj describes the LRTs, where ˆhhere denotes the unit vector of the exchange ﬁeld. For a concrete example, if the exchange ﬁeld is oriented along the z-axis, then dzwill be the short- range component, while both dxanddyare long-ranged com- ponents. In the coming sections, we will demonstrate that the LRT component can be identiﬁed from its density of states signature, as measurable by tunneling spectroscopy. In the limit of weak proximity effect, we may linearize both the Usadel equation and Kupriyanov–Lukichev boundary con- ditions. Using the singlet/triplet decomposition in Eq. (24), and the Rashba–Dresselhaus coupling in Eq. (19), the lin- earized version of the Usadel equation can be written: i 2DF¶2 zfs=efs+hd; (25) i 2DF¶2 zd=ed+hfs+2iDFa2W(c)d; (26) where we for brevity have deﬁned an SO interaction matrix W(c) =0 @1sin2c 0 sin2c 1 0 0 0 21 A: (27) We have now condensed the Usadel equation down to two coupled differential equations for fsandd, where the cou- pling is proportional to the exchange ﬁeld and the SO interac- tion term. The latter has been written as a product of a factor 2iDFa2, depending on the strength a, and a factor W(c)d, de- pending on the angle cin the polar notation. The matrix W(c) becomes diagonal for a Dresselhaus coupling with c=0 or a Rashba coupling with c=p=2, which implies that there is no triplet mixing for such systems. In contrast, the off- diagonal terms are maximal for c=p=4, which suggests that the triplet mixing is maximal when the Rashba and Dres- selhaus coefﬁcients have the same magnitude. In addition to the off-diagonal triplet mixing terms, we see that the diagonal terms of W(c)essentially result in imaginary energy contri- butions 2 iDFa2. As we will see later, this can in some cases result in a suppression of all the triplet components in the fer- romagnet. We will now consider exchange ﬁelds in the xy-plane, h=hcosqˆx+hsinqˆy: (28)6 Since the linearized Usadel equations show that the presence of a singlet component fsonly results in the generation of triplet components along h, and the SO interaction term only mixes the triplet components in the xy-plane, the only nonzero triplet components will in this case be dxanddy. The SRT amplitude dand LRT amplitude d?can therefore be written: d=dxcosq+dysinq; (29) d?=dxsinq+dycosq: (30) By projecting the linearized Usadel equation for dalong the unit vectors (cosq;sinq;0)and(sinq;cosq;0), respectively, then we obtain coupled equations for the SRTs and LRTs: i 2DF¶2 zfs=efs+hd; (31) i 2DF¶2 zd=[e+2iDFa2(1sin2qsin2c)]d 2iDFa2cos2qsin2cd?+h fs; (32) i 2DF¶2 zd?=[e+2iDFa2(1+sin2qsin2c)]d? 2iDFa2cos2qsin2cd: (33) These equations clearly show the interplay between the singlet component fs, SRT component d, and LRT component d?. If we start with only a singlet component fs, then the presence of an exchange ﬁeld hresults in the generation of the SRT component d. The presence of an SO ﬁeld can then result in the generation of the LRT component d?, where the mixing term is proportional to a2cos2qsin2c. This implies that in the weak proximity limit, LRT mixing is absent for an exchange ﬁeld direction q=p=4, corresponding to hx=hy, while it is maximized if q=f0;p=2;pgand at the same time c=p=4. In other words, the requirement for maximal LRT mixing is therefore that the exchange ﬁeld is aligned along either the x- axis or y-axis, while the Rashba and Dresselhaus coefﬁcients should have the same magnitude. It is important to note here that although the mixing between the triplet components is maximal at q=f0;p=2;pg, this does not necessarily mean that the signature of the triplets in physical quantities is most clearly seen for these angles, as we shall discuss in detail later. Moreover, these equations show another interesting conse- quence of having an SO ﬁeld in the ferromagnet, which is unrelated to the LRT generation. Note that the effective quasi- particle energies coupling to the SRTs and LRTs become E=e+2iDFa2(1sin2qsin2c); (34) E?=e+2iDFa2(1+sin2qsin2c): (35) When q=c=p=4, then the SRTs are entirely unaffected by the presence of SO coupling; the triplet mixing term van- ishes for these parameters, and Eis also clearly independent ofa. However, when q=c=p=4, the situation is dras- tically different. There is still no possibility for LRT genera- tion, however the SRT energy E=e+4iDFa2will now ob- tain an imaginary energy contribution which destabilizes the SRTs. In fact, numerical simulations show that this energy shift destroys the SRT components as aincreases. As we willsee in Section III D, this effect results in an increase in the critical temperature of the bilayer. Thus, switching between q=p=4 in a system with c'p=4 may suggest a novel method for creating a triplet spin valve. When c=p=4 but q6=p=4, the triplet mixing term proportional to cos2 qsin2cwill no longer vanish, so we get LRT generation in the system. We can then see from the ef- fective triplet energies that as q!sgn(c)p=4, the imaginary part of Evanishes, while the imaginary part of E?increases. This leads to a relative increase in the amount of SRTs com- pared to the amount of LRTs in the system. In contrast, as q! sgn(c)p=4, the imaginary part of E?vanishes, and the imaginary part of Eincreases. This means that we would expect a larger LRT generation for these parameters, up until the point where the triplet mixing term cos2 qsin2cbecomes so small that almost no LRTs are generated at all. The ratio of effective energies coupling to the triplet component at the Fermi level e=0 can be written as E?(0) E(0)=1+sin2qsin2c 1sin2qsin2c: (36) D. Density of states The density of states D(e)containing all spin components can be written in terms of the Riccati matrices as D(e) =Tr[N(1+g˜g)]=2; (37) which for the case of zero energy can be written concisely in terms of the singlet component fsand triplet components d, D(0) =1jfs(0)j2=2+jd(0)j2=2: (38) The singlet and triplet components are therefore directly com- peting to lower and raise the density of states47. Furthermore, since we are primarily interested in the proximity effect in the ferromagnetic ﬁlm, we will begin by using the known BCS bulk solution in the superconductor, ˆgBCS= cosh(q) sinh(q)isyeif sinh(q)isyeifcosh(q) ; (39) where q=atanh(D=e), and fis the superconducting phase. Using Eq. (24) and the deﬁnition of the tilde operation, and comparing ˆ gRin Eq. (3) with its standard expression in a bulk superconductor Eq. (39), we can see that at zero energy the singlet component fs(0)must be purely imaginary and the asymmetric triplet dz(0)must be purely real if the supercon- ducting phase is f=0. By inspection of Eq. (26), we can see that a transformation hx$hyalong with dx$dyleaves the equations invariant. The density of states will therefore be unaffected by such per- mutations, D[h= (a;b;0)] = D[h= (b;a;0)]; (40) while in general D[h= (a;0;b)]6=D[h= (b;0;a)]: (41)7 However, whenever one component of the planar ﬁeld is ex- actly twice the value of the other component, one can conﬁrm that the linearized equations remain invariant under a rotation of the exchange ﬁeld h= (a;2a;0)!h= (a;0;2a); (42) with associated invariance in the density of states. E. Critical temperature When superconducting correlations leak from a supercon- ductor and into a ferromagnet in a hybrid structure, there will also be an inverse effect, where the ferromagnet effectively drains the superconductor of its superconducting properties due to tunneling of Cooper pairs. Physically, this effect is observable in the form of a reduction in the superconducting gapD(z)near the interface at all temperatures. Furthermore, if the temperature of the hybrid structure is somewhat close to the bulk critical temperature Tcsof the superconductor, this inverse proximity effect can be strong enough to make the su- perconducting correlations vanish entirely throughout the sys- tem. Thus, proximity-coupled hybrid structures will in prac- tice always have a critical temperature Tcthat is lower than the critical temperature Tcsof a bulk superconductor. Depending on the exact parameters of the hybrid system, Tccan some- times be signiﬁcantly smaller than Tcs, and in some cases it may even vanish ( Tc!0). To quantify this effect, it is no longer sufﬁcient to solve the Usadel equation in the ferromagnet only. We will now also have to solve the Usadel equation in the superconductor, DS¶2 zg=2iegD(sygsyg)2(¶zg)˜N˜g(¶zg); (43) along with a self-consistency equation for the gap D(z), D(z) =N0lD0cosh(1=N0l)Z 0deReffs(z;e)gtanhp 2ege=D0 T=Tcs ;(44) where N0is the density of states per spin at the Fermi level, andl>0 is the electron-electron coupling constant in the BCS theory of superconductivity. For a derivation of the gap equation, see Appendix B. To study the effects of the SO coupling on the critical tem- perature of an SF structure, we therefore have to ﬁnd a self- consistent solution to Eq. (5) in the ferromagnet, Eq. (6) at the interface, and Eqs. (43) and (44) in the superconductor. In practice, this is done by successively solving one of the equations at a time numerically, and continuing the procedure until the system converges towards a self-consistent solution. To obtain accurate results, we typically have to solve the Us- adel equation for 100–150 positions in each material, around 500 energies in the range (0;2D0), and 100 more energies in the range (2D0;wc), where the Debye cutoff wc76D0for the superconductors considered herein. This procedure will then have to be repeated up to several hundred times before we obtain a self-consistent solution for any given temperatureof the system. Furthermore, if we perform a conventional lin- ear search for the critical temperature Tc=Tcsin the range (0;1) with a precision of 0.0001, it may require up to 10,000 such it- erations to complete, which may take several days depending on the available hardware and efﬁciency of the implementa- tion. The speed of this procedure may, however, be signiﬁ- cantly increased by performing a binary search instead. Using this strategy, the critical temperature can be determined to a precision of 1 =212+10:0001 after only 12 iterations, which is a signiﬁcant improvement. The convergence can be fur- ther accelerated by exploiting the fact that D(z)from iteration to iteration should decrease monotonically to zero if T>Tc; however, the details will not be further discussed in this paper. III. RESULTS We consider the proximity effect in an SF bilayer in III A, using the BCS bulk solution for the superconductors. The case of pure Rashba coupling is discussed in III B, and the SFS Josephson junction is treated in III C. We take the thin- ﬁlm layering direction to be oriented in the z-direction and ﬁx the spin-orbit coupling to Rashba–Dresselhaus type in the xy-plane as given by Eq. (19). We set LF=xS=0:5. The co- herence length for a diffusive bulk superconductor typically lies in the range 10 30 nm. We solve the equations using MATLAB with the boundary value differential equation pack- agebvp6c and examine the density of states D(e)for en- ergies normalised to the superconducting gap D. For brevity of notation, we include the normalization factor in the coefﬁ- cients aandbin these sections. This normalization is taken to be the length of the ferromagnetic region LF, so that for in- stance a=1 in the ﬁgure legends means aLF=1. Finally, in Section III D, we calculate the dependence of the critical tem- perature of an SF bilayer as a function of the different system parameters. A. SF Bilayer Consider the SF bilayer depicted in Fig. 1(a). In section II B we introduced the conditions for the LRT component to appear, and from Eq. (22) it is clear that no LRTs will be gen- erated if the exchange ﬁeld is aligned with the layering direc- tion, i.e. hkˆz, since Eq. (22) will be parallel to the exchange ﬁeld. Conversely, the general condition for LRT generation with in-plane magnetisation is both that hx6=hyand that the SO coupling is not of pure Rashba or pure Dresselhaus form. However, it became clear in Section II C that the triplet mix- ing was maximal for equal Rashba and Dresselhaus coupling strengths, and in fact the spectroscopic signature is quite sen- sitive to deviations from this. In Ref. 50, the density of states for an SF bilayer was shown to display oscillatory behavior as a function of distance pene- trated into the ferromagnet. The physical origin of this stems from the non-monotonic dependence of the superconducting order parameter inside the F layer, which oscillates and leads to an alternation of dominant singlet and dominant triplet cor-8 relations as a function of distance from the interface. When the triplet ones dominate, the proximity-induced change in the density of states is inverted compared to SN structures, giving rise to an enhancement of the density of states at low-energies in this so-called p-phase where the proximity-induced super- conducting order parameter is negative. For SF bilayers without SO coupling and a homogeneous exchange ﬁeld, one expects to see a spectroscopic mini- gap whenever the Thouless energy is much greater than the strength of the exchange ﬁeld. The minigap in SF structures closes when the resonant condition hEgis fulﬁlled, where Egis the minigap occuring without an exchange ﬁeld, and a zero-energy peak emerges instead48. The minigap Egdepends on both the Thouless energy and the resistance of the junc- tion. For stronger ﬁelds we will have an essentially feature- less density of states (see e.g.Ref. 49 and references therein). This is indeed what we observe for a=b=0 in Fig. 5. With purely out-of-plane magnetisation hkˆz, the effect of SO cou- pling is irrespective of type: Rashba, Dresselhaus or both will always create a minigap. With in-plane magnetisation how- ever, the observation of a minigap above the SO-free resonant condition h>Egindicates that dominant Rashba or domi- nant Dresselhaus coupling is present. The same is true for SFS trilayers, and thus to observe a signature of long-range triplets the Rashba–Dresselhaus coefﬁcients must be similar in magnitude, and in the following we shall primarily focus on this regime. To clarify quantitatively how much the Rashba and Dresselhaus coefﬁcients can deviate from each other be- fore destroying the low-energy enhancement of the density of states, which is the signature of triplet Cooper pairs in this sys- tem, we have plotted in Fig. 3 the density of states at the Fermi level ( e=0) as a function of the spin-orbit angle cand the magnetization direction q. For purely Rashba or Dresselhaus coupling (c=f0;p=2g), the deviation from the normal- state value is small. However, as soon as both components are present a highly non-monotonic behavior is observed. This is particularly pronounced for c!p=4, although the con- version from dominant triplets to dominant singlets as one ro- tates the ﬁeld by changing qis seen to occur even away from c=p=4. With either h=hˆx6=0, or equivalently h=hˆy6=0, LRTs are generated provided ab6=0, and in Fig. 6 we can see that the addition of SO coupling introduces a peak in the density of states at zero energy, which saturates for a certain cou- pling strength. This peak manifests as sharper around e=0 than the zero-energy peak associated with weak ﬁeld strengths of the order of the gap ( i.e. as evident from a=b=0 in Fig. 6), which occurs regardless of magnetisation direction or texture48,49. By analysing the real components of the triplets, for a gauge where the superconducting phase is zero, we can conﬁrm that this zero-energy peak is due to the LRT compo- nent, in this case dx, also depicted in Fig. 6, in agreement with the predictions for textured magnetisation without SO coupling49. However, it is also evident from Fig. 6 that in- creasing the ﬁeld strength rapidly suppresses the density of states towards that of the normal metal, making the effect more difﬁcult to detect experimentally. The way to amelio- rate this situation is to remember that the introduction of SOcoupling means the direction of the exchange ﬁeld is crucially important, as we see in Fig. 4, and this allows for a dramatic spectroscopic signature for ﬁelds without full alignment with thex- ory-axes. FIG. 3: Zero-energy density of states D(0)as a function of the spin-orbit angle cand magnetization angle q. We have used a ferromagnet of length LF=xS=0:5 with an exchange ﬁeld h=D=3 and a spin-orbit magnitude axS=2. Fig. 4 shows how the density of states at zero energy varies with the angle qbetween hxand hyat zero energy; with q=0 the ﬁeld is aligned with hx, and with q=p=2 it is aligned with hy. We see that the inclusion of SO coupling introduces a nonmonotonic angular dependance in the den- sity of states, with increasingly sharp features as the SO cou- pling strength increases, although the optimal angle at ap- proximately q=7p=32 and q=9p=32 varies minimally with increasing SO coupling. Clearly the ability to extract max- imum LRT conversion from the inclusion of SO coupling is highly sensitive to the rotation angle, with near step-function behaviour delineating the regions of optimal peak in the den- sity of states and an energy gap for strong SO coupling. It is remarkable to see how D(0)vs.qformally bears a strong re- semblance to the evolution of a fully gapped BCS64density of states D(e)vs.eto a ﬂat density of states as the SO coupling decreases. These results can again be explained physically by the lin- earized equations (31)–(33). Since the case a=bcorresponds toc=p=4 in the notation developed in the preceding sec- tions, Eq. (36) implies that E?(0)>E(0)when q<0, while E?(0)<E(0)when q>0. In other words, for negative q, the SO coupling suppresses the LRT components, and the ex- change ﬁeld suppresses the other components. Since the sin- glet and SRT components have opposite sign in Eq. (38), this renders the density of states essentially featureless. However, for positive q, both the SO coupling and the exchange ﬁeld suppress the SRT components, meaning that LRT generation is energetically favoured. Note that E?=E!¥asq!+p=4, which explains why the LRT generation is maximized in this regime. Since the triplet mixing term in Eq. (33) is propor-9 tional to (cos2qsin2c), the LRT component vanishes when the value of qgets too close to +p=4. Furthermore, since Ehas a large imaginary energy contribution in this case, the SRTs are also suppressed at q= +p=4. Thus, despite LRTs being most energetically favored at this exact point, we end up with a system dominated by singlets due to the SRT suppres- sion and lack of LRT production pathway. Nevertheless, one would conventionally expect that exchange ﬁelds of a mag- nitude hDas depicted in Fig.4 would suppress any fea- tures in the density of states, while we observe an obvious minigap. Thus, the singlet correlations become much more resilient against the pair-breaking effect of the exchange ﬁeld when spin-orbit coupling is present. To identify the physical origin of this effect, we solve the linearized equations (31)–(33) along with their corresponding boundary conditions for the speciﬁc case e=0,q=c= p=4. We consider a bulk superconductor occupying the space x<0 while the ferromagnet length LFis so large that one in practice only needs to keep the decaying parts of the anoma- lous Green’s function. We then ﬁnd the following expression for the singlet component at the SF interface in the absence of SO coupling: f0 s=sinh(arctanh (D=e)) 2zLFr DF h: (45) With increasing h, the singlet correlations are suppressed in the conventional manner. However, we now incorporate SO coupling in the problem. For more transparent analytical re- sults, we focus on the case 2 (ax)2h=D. This condition can be rewritten as 2 DFa2h. In this case, a similar calculation gives the singlet component at the SF interface in the presence of SO coupling: fs=f0 sr DFa2 2h: (46) Clearly, the SO coupling enhances the singlet component in spite the presence of an exchange ﬁeld sincep DFa2=h1. This explains the presence of the conventional zero energy gap for large SO coupling even with a strong exchange ﬁeld. A consequence of this observation is that SO coupling in fact provides a route to a magnetically tunable minigap . Fig. 4 shows that when both an exchange ﬁeld and SO coupling is present, the direction of the ﬁeld determines when a minigap appears. This holds even for strong exchange ﬁelds hDas long as the SO coupling is sufﬁciently large as well. We recall that the LRT Cooper pairs, deﬁned as the com- ponents of dperpendicular to h, may be characterized by a quantity d?which is deﬁned by the cross product of the two vectors: d?=jdˆhj. We saw above that the spectroscopic signature of LRT generation is strongly dependent on the an- gle of the ﬁeld, and this angle is a tunable parameter for sufﬁ- ciently weak magnetic anisotropy. In Fig. 7 we see an example of the effect this rotation can have on the spectroscopic signa- ture of LRT generation: when the exchange ﬁeld is changed from h= (6D;3D;0)!(6D;5D;0),i.e.changing the direction of the ﬁeld, we see that a strong zero-energy peak emerges due to the presence of LRT in the system. This large peakemerges despite the stronger exchange ﬁeld that would ordi- narily reduce the density of states towards the normal state, i.e.as in Fig. 6 for h=Dˆy!3Dˆy. If one were to remove the SO coupling, the low-energy density of states would thus have no trace of any superconducting proximity effect, which demonstrates the important role played by the SO interactions here. Finally, for completeness we include an example of the effect of rotating the ﬁeld to have a component along the junc- tion in Fig. 8. Comparing the case of h= (0;3D;6D)in Fig. 8 with h= (6D;3D;0)in Fig. 7, we see that the two cases are identical, as predicted in the limit of weak proximity effect, and increasing the magnitude of the out-of-plane zcomponent of the ﬁeld has no effect on the height of the zero-energy peak, which is instead governed by the in-plane ycomponent. π/4 π/2 0D(0) θα = β = 0 α = β = 0.5 α = β = 1 α = β = 5α = β = 2 h=6Δ(cos(θ), sin(θ), 0) 00.250.50.7511.251.51.75 -π/4 -π/2 FIG. 4: The dependence of the density of states of the SF bilayer at zero energy on the angle qbetween the xand ycomponents of the magnetisation exchange ﬁeld h=D= 6(cos(q);sin(q);0)for increasing SO coupling. As the strength of the SO coupling increases we see increasingly sharp variations in the density of states from an optimal peak at around q7p=32 and q9p=32 to a gap around q=p=4.10 -1.5 -1 -0.5 0 0.5 1 1.500.511.522.5 -1.5 -1 -0.5 0 0.5 1 1.500.511.522.5D(ε) ε/Δ ε/Δh = (0, 0, 3Δ) -1.5 -1 -0.5 0 0.5 1 1.500.511.522.5 ε/Δh = (0, 3Δ, 0) α = β = 0 α =β = 0.5 α = β = 1 α = β = 2α = β = 0 α = 0.1, β = 0.5 α = 0.1, β = 1 α = 0.1, β = 2 FIG. 5: Density of states D(e)for the SF bilayer with energies normalised to the superconducting gap Dand SO coupling normalised to the inverse ferromagnet length 1 =LF. The table shows the spectroscopic effect of increasing SO coupling witha=bwhen the magnetisation h=3Dˆz,i.e.with the ﬁeld perpendicular to the interface, and the effect of increasing difference between the Rashba and Dresselhaus coefﬁcients for both h=3Dˆzandh=3Dˆy. Although the conditions for LRT generation are fulﬁlled in the latter case, it is clear that no spectroscopic signature of this is present. 0.511.522.53 -1.5 -1 -0.5 0 0.5 1 1.50.60.811.2 ε/Δε/ΔD(ε)(0,Δ,0) (0,3Δ,0)\|Re(d y)\| α = β = 0 α = β = 0.1 α = β = 0.5 α = β = 2D(ε)h D(ε) \|Re(d x)\| 00.511.5\|Re(d y)\| \|Re(d y)\| \|Re(d x)\| \|Re(d x)\| -1.5 -1 -0.5 0 0.5 1 1.50.050.20.350.5 ε/Δε/Δ ε/Δ 00.40.81.21.4 -1.5 -1 -0.5 0 0.5 1 1.500.10.20.30.4 FIG. 6: Density of states D(e)for the SF bilayer with energies normalised to the superconducting gap Dand SO coupling normalised to the inverse ferromagnet length 1 =LF. The table shows the spectroscopic effect of equal Rashba–Dresselhaus coefﬁcients when the magnetisation is oriented entirely in the y-direction, and also the correlation between the SO-induced zero-energy peak with the long-range triplet component jRe(dx)jRe(d?). It is clear that the predominant effect of the LRT component, which appears only when the SO coupling is included, is to increase the peak at zero energies. Increasing the ﬁeld strength rapidly suppresses the density of states towards that of the normal metal.11 -1.5 -1 -0.5 0 0.5 1 1.50.60.811.2 -1.5 -1 -0.5 0 0.5 1 1.50.511.522.5 -1.5 -1 -0.5 0 0.5 1 1.500.511.52 ε/Δ ε/Δ ε/ΔD(ε)h = (6Δ, 3Δ, 0) h = (6Δ, 5Δ, 0) h = (6Δ, 5Δ, 0) Re(d ⊥) for α = β = 0α = β = 0.5 α = β = 2 α = β = 1 α = β = 5Re(d ⊥) FIG. 7: Density of states D(e)in the SF bilayer for energies normalised to the superconducting gap Dand SO coupling normalised to the inverse ferromagnet length 1 =LF. The table shows the spectroscopic features of the SF bilayer with rotated exchange ﬁeld in the xy-plane. Again we see a peak in the density of states at zero energy due to the LRT component, i.e.the component of dperpendicular to h,d?. The height of this zero-energy peak is strongly dependent on the angle of the ﬁeld vector in the plane, as shown in Fig. 4. For near-optimal ﬁeld orientations increasing the SO coupling leads to a dramatic increase in the peak of the density of states at zero energy. -1.5 -1 -0.5 0 0.5 1 1.50.60.811.2 -1.5 -1 -0.5 0 0.5 1 1.50.60.811.2 -1.5 -1 -0.5 0 0.5 1 1.50.60.811.2 ε/Δ ε/Δ ε/ΔD(ε)h = (0, 3Δ, 3Δ) h= (0, 3Δ, 6Δ) h = (0, 6Δ, 3Δ) α = β = 0 α = β = 0.5 α = β = 2 α = β = 1 α = β = 0.1 FIG. 8: Density of states D(e)in the SF bilayer for energies normalised to the superconducting gap Dand SO coupling normalised to the inverse ferromagnet length 1 =LF. The table shows the spectroscopic features of the SF bilayer with a rotated exchange ﬁeld in the xzyz-plane. Note that when the ﬁeld component along the junction is twice the component in the y-direction, here h= (0;3D;6D), the density of states is equivalent to the case h= (6D;3D;0)illustrated in Fig. 7, as predicted in the limit of weak proximity effect.12 B. SF bilayer with pure Rashba coupling There exists another experimentally viable setup where the LRT can be created. In the case where pure Rashba SO cou- pling is present, originating e.g. from interfacial asymmetry, the condition for the existence of LRT is that the exchange ﬁeld has a component both in-plane and out-of-plane. Al- though the LRT formally is non-zero, it is desirable to clarify if and how it can be detected through spectroscopic signatures. From an experimental point of view, it is known that PdNi and CuNi11can in general feature a canted magnetization orientation relative to the ﬁlm-plane due to the competition between shape anisotropy and magnetocrystalline anisotropy. This is precisely the situation required in order to have an ex- change ﬁeld with both an in-plane ( xy-plane in our notation) and out-of-plane ( z-direction) component. In our model, the ferromagnetism coexists with the Rashba SO coupling, which may be taken as a simpliﬁed model of two separate layers where the SO coupling is induced e.g. by a very thin heavy metal and PdNi or CuNi is deposited on top of it. To determine how the low-energy density of states is inﬂu- enced by the triplet pairing, we plot in Fig. 9(a) D(0)as a function of the misalignment angle jbetween the ﬁlm-plane and its perpendicular axis [see inset of Fig. 9(b) for junc- tion geometry]. In order to correlate the spectroscopic features with the LRT, we plot in Fig. 9(b) the LRT Green’s function jd?j. It is clear that the LRT vanishes when j=0 orj=p=2. This is consistent with the fact that for pure Rashba coupling, purely in-plane or out-of-plane direction of the exchange ﬁeld gives d?=0 according to our previous analysis. However, forj2(0;p=2)the LRT exists. Its inﬂuence on D(0)is seen in Fig. 9(a): an enhancement of the zero-energy density of states. For any particular set of junction parameters there is an optimal value of the SO coupling, and in approaching this value the density of states is correlated with Re fd?g. Beyond this optimal value, they are anticorrelated, as evident from Fig. 9 as the SO coupling increases, but the angular correlation re- mains. We note that the magnitude of the enhancement of the density of states is substantially smaller than what we ob- tained with both Rashba and Dresselhaus coupling. At the same time, the magnitude of the enhancement is of precisely the same order as previous experimental works that have mea- sured the density of states in S/F structures50,51. Note that it is only the angle between the plane and the tun- neling direction which is of importance: the density of states is invariant under a rotation in the ﬁlm-plane of the exchange ﬁeld. The SO-induced enhancement of the zero-energy den- sity of states reaches an optimal peak before further increases in the magnitude of the Rashba coupling results in a suppres- sion of both the short- and long-ranged triplet components, causing the low-energy density of states enhancement to van- ish. The correlation with the LRT component jd?jcorre- spondingly changes to anticorrelation, evident in Fig. 9. Nev- ertheless, the strong angular variation with D(0)remains al- though D(0)<1 for all j[see inset of Fig. 9(a)]. Increasing the exchange ﬁeld hfurther suppressed the proximity effect overall. The main effect of the SO coupling is that D(0)depends onthe exchange ﬁeld direction. As seen for the case of a=0 in Fig. 9(a), there is no directional dependence without SO coupling. Thus, depending on the exchange ﬁeld angle be- tween the in-plane and out-of plane direction, measuring an enhanced D(0)at low-energies is a signature of the presence of LRT Cooper pairs in the ferromagnet. More generally, mea- suring a dependence on the exchange ﬁeld direction jwould be a direct consequence of the presence of SO coupling in the system, even in the regime of e.g.moderate to strong Rashba coupling where the triplets are suppressed. (a) (b) FIG. 9: (Color online) (a) Plot of the zero-energy density of states D(0)in an S/F structure with pure Rashba spin- orbit coupling. We have set h=D=4 and L=xS=0:5. Inset: stronger SO coupling a=1:5, demonstrating that the angu- lar variation of D(0)remains, although the enhancement due to triplets is absent. (b) Plot of the magnitude of the LRT anomalous Green’s function jd?jate=0. As seen, its en- hancement correlates with an accompanying increase in the density of states for the same angle j, and beyond an optimal SO coupling value there is anticorrelation between the density of states peak and jd?j. The only angle of importance is the angle jbetween the out-of-plane and in-plane component of the exchange ﬁeld, shown in the inset. C. Josephson junction By adding a superconducting region to the right interface of the SF bilayer we form an SFS Josephson junction. It is well known that the phase difference between the supercon- ducting regions governs how much current can ﬂow through the junction52, and the density of states for a diffusive SNS junction has been measured experimentally with extremely high precision53. Here we consider such a transversal junc- tion structure as depicted in Fig.1(b), again with intrinsic SO coupling in the xy-plane (Eq. 19) in the ferromagnet and with BCS bulk values for each superconductor. In III C 1 we con- sider single orientations along the principal axes of the system (x;y;z)of the uniform exchange ﬁeld and in III C 2 we con- sider a rotated ﬁeld. Experimentally, the density of states can be probed at the superconductor/ferromagnet interface if one13 of the superconductors is a superconducting island, and the scanning tunneling microscope approaches from the top, next to this superconductor island. Let us ﬁrst recapitulate some known results. We saw in Sec- tion II that the spin-singlet, SRT and LRT components com- pete to raise and lower the density of states at low energies. Their relative magnitude is affected by the magnitude and di- rection of both the exchange ﬁeld and SO coupling and results in three distinctive qualitative proﬁles: the zero-energy peak from the LRTs, the singlet-dominated regime with a minigap, and the ﬂat, featureless proﬁle in the absence of superconduct- ing correlations. In the Josephson junction, the spectroscopic features are in addition sensitive to the phase difference fbe- tween the superconductors. In junctions with an interstitial normal metal, the gap decreases as f=0!p, closing entirely atf=psuch that the density of states is that of the isolated normal metal; identically one53,54. Without an exchange ﬁeld the density of states is unaffected by the SO coupling. This is because without an exchange ﬁeld the equations governing the singlet and triplet components are decoupled and thus no singlet-triplet conversion can occur. From a symmetry point of view, it is reasonable that the time-reversal invariant spin- orbit coupling does not alter the singlet correlations. Without SO coupling and as long as the exchange ﬁeld is not too large, changing the phase difference can qualitatively alter the density of states from minigap to peak at zero en- ergy (see Fig. 10), a useful feature permitting external con- trol of the quasiparticle current ﬂowing through the junction. The underlying reason is that the phase difference controls the relative ratio of the singlet and triplet correlations: when the singlets dominate, a minigap is induced which mirrors their origin in the bulk superconductor. As in the bilayer case, there is a resonant condition48,49indicating an exchange ﬁeld strength be- yond which the minigap can no longer be sustained and increasing the phase difference simply lowers the density of states towards that of the normal metal. Amongst the features we outline in the following subsections, one of the effects of adding SO coupling is to make this useful gap-to-peak effect accessible with stronger exchange ﬁelds, i.e. for a greater range of materials. At the same time, the SO coupling cannot betoo strong since the triplet correlations are suppressed in this regime leaving only the minigap and destroying the capability for qualitative change in the spectroscopic features. 1. Josephson junction with uniform exchange ﬁeld in single direction Consider ﬁrst the case in which the exchange ﬁeld is aligned in a single direction, meaning that we only consider an exchange ﬁeld purely along the principal fx;y;zgaxes of thesystem. If we again restrict the form of the SO-vector to (19), aligning hin the z-direction will not result in any LRTs. In this case the spectroscopic effect of the SO coupling is dictated by the singlet and short-range triplet features, much as in the SF bilayer case (Fig. 5). This is demonstrated in Fig. 10, where again we see a qualitative change in the density of states as the exchange ﬁeld increases, with the regions of minigap and zero-energy-peak separated by the resonant condition hEg without SO coupling. We will now examine the effect of increasing the exchange ﬁeld aligned in the x- or, equivalently, the y-direction. In this case, we have generation of LRT Cooper pairs. If his sufﬁ- ciently weak to sustain a gap independently of SO coupling, introducing weak SO coupling will increase the gap at zero phase difference while maintaining a peak at zero energy for a phase difference of 0 :75p(see Fig. 10). Increasing the SO coupling increases this peak at zero energy up to a saturation point. As the exchange ﬁeld increases sufﬁciently beyond the resonant condition to keep the gap closed, increasing the SO coupling increases the zero-energy peak at all phases, again due to the LRT component, eventually also reaching a satu- ration point. As the phase difference f=0!p, the density of states reduces towards that of the normal metal, closing en- tirely at f=pas expected43,54,55. As the value of the density of states at zero energy saturates for increasing SO coupling, ﬁxed phase differences yield the same drop at zero energy re- gardless of the strength of SO coupling. We note in passing that when the SO coupling ﬁeld has a component along the junction direction (z), it can qualitatively inﬂuence the nature of the superconducting proximity effect. As very recently shown in Ref. 43, a giant triplet proximity effect develops at f=pin this case, in complete contrast to the normal scenario of a vanishing proximity effect in p-biased junctions. 2. Josephson junction with rotated exchange ﬁeld With two components of the ﬁeld h,e.g.from rotation, it is again useful to separate the cases with and without a compo- nent along the junction direction. When the exchange ﬁeld lies in-plane (the xy-plane), and provided we satisfy the conditions hx6=hyandab6=0, increasing the SO coupling drastically in- creases the zero energy peak as shown in Fig. 11, again due to the LRT component. This is consistent with the bilayer behav- ior, where the maximal generation of LRT Cooper pairs occurs at an angle 0 <q<p=4. As the phase difference approaches p, the proximity-induced features are suppressed in the centre of the junction. This can be understood intuitively as a con- sequence of the order parameter averaging to zero since it is positive in one superconductor and negative in the other.14 01234 01234 01234 01234 01234 01234 01234 -1.5 -1 -0.5 0 0.5 1 1.501234 01234 -1.5 -1 -0.5 0 0.5 1 1.501234 -1.5 -1 -0.5 0 0.5 1 1.500.511.522.2 00.511.52 00.511.52 00.511.52 00.511.52 -1.5 -1 -0.5 0 0.5 1 1.500.511.52 00.511.52 00.511.52 00.511.52 00.511.52 ε/Δ ε/Δ ε/Δ ε/ΔD(ε) D(ε) D(ε) D(ε) D(ε)h = (0, 0, Δ) h = (0, Δ, 0) h = (0, 0, 3Δ) h = (0, 3Δ, 0) α, β 0 0.1 0.5 1 2 φ = 0 φ =0.25π φ = 0.5π φ = 0.75π φ = π FIG. 10: The table shows the density of states D(e)in the SFS junction with increasing SO coupling and exchange ﬁeld in a single direction, with D(e)normalised to the superconducting gap Dand SO coupling normalised to the inverse ferromagnet length 1 =LF. With no SO coupling and very weak exchange ﬁeld we see a phase-dictated gap-to-peak qualitative change in the density of states at zero energy. When the ﬁeld is strong enough to destroy this gap, i.e.above the resonant condition, increasing the phase difference simply lowers the density of states towards that of the normal metal, which is achieved at a phase difference of f=p. With the addition of SO coupling we see a clear difference in the density of states due to the long range triplet component, which is present when the ﬁeld is oriented in ybut not in z. When LRTs are present with weak exchange ﬁelds, a phase-dictated gap-to-peak feature is retained and increased as the strength of SO coupling increases the gap, with the peak shown here at a phase difference of 0 :75p. For stronger exchange ﬁelds, increasing the SO coupling produces the minigap when there is no LRT component, whereas the existence of an LRT component again introduces an increasing peak at zero energy when no minigap is present.15 0.50.7511.251.5 0.50.7511.251.5 0.7511.251.51.75 0.511.52 -1.5 -1 -0.5 0 0.5 1 1.500.511.522.5 -1.5 -1 -0.5 0 0.5 1 1.500.511.522.5 0.511.52 0.7511.251.51.75 0.50.7511.251.5 0.50.7511.251.5 -1.5 -1 -0.5 0 0.5 1 1.50.50.7511.251.5 0.50.7511.251.5 0.50.7511.251.5 0.50.7511.251.5 0.50.7511.251.5h = (1.5Δ, 3.5Δ, 0) h = (0, 1.5Δ, 3.5Δ) h = (0, 3.5Δ, 5.5Δ) α, β 0 0.1 0.5 1 2 D(ε) D(ε) D(ε) D(ε) D(ε) ε/Δ ε/Δ ε/Δ φ = 0 φ =0.25π φ = 0.5π φ = 0.75π φ = π FIG. 11: Density of states D(e)in the SFS junction for energies normalised to the superconducting gap Dand SO coupling normalised to the inverse ferromagnet length 1 =LF. The table shows the spectroscopic effects of increasing SO coupling in SFS with rotated exchange ﬁeld. In the absence of SO coupling, the density of states is ﬂat and featureless at low energies. Increasing the SO coupling again leads to a strong increase in the peak of the density of states at zero energy, while increasing the phase difference reduces the peak and shifts the density of states weight toward the gap edge for higher SO coupling strengths. With a component of the ﬁeld in the junction direction a qualitative change in the density of states from strongly suppressed to enhanced at zero energy can be achieved by altering the phase difference between the superconductors. This change can occur in the presence of stronger exchange ﬁelds when SO coupling is included. Increasing the exchange ﬁeld destroys the ability to maintain a gap in the density of states and the LRT component of the SO coupling increases the zero-energy peak as it did in the bilayer case.16 The 2D plots in this paper of the local density of states are given for the centre of the junction ( z=0), where one natu- rally expects the relative proportion of LRTs to be greatest. However, it is interesting to note that the large peak at zero energy – the signature of the LRTs – is maintained through- out the ferromagnet. This is shown in Fig. 12, for the case a=b=1 and h= (1:5D;3:5D;0), where the maximal peak forf=0 is almost twice the normal-state value. In compar- ison, the depletion of this peak is surprisingly small at the superconductor interfaces. FIG. 12: Spatial distribution of the density of states D(e) throughout the ferromagnet of an SFS junction with phase dif- ference f=0, spin-orbit coupling a=b=1 and magnetisa- tionh= (1:5D;3:5D;0). With one component of the exchange ﬁeld along the junc- tion and another along either xory, a phase-dictated gap-to- peak transition at zero energy is possible with stronger ﬁelds than with the ﬁeld aligned in a single direction, as shown in Fig. 11. Notice that in this case increasing the phase differ- encef=0!0:5pgives an increase in the peak at zero energy before reducing towards the normal metal state. For higher ﬁeld strengths we ﬁnd once again that increasing the SO cou- pling increases the peak at zero energy, up to a system-speciﬁc threshold, and increasing phase difference reduces the density of states towards that of the normal metal. It is also useful to consider how the zero-energy density of states depends simultaneously on the phase-difference and magnetization orientation. To this end, we show in Fig. 13 a contour plot of the density of states at the Fermi level (e=0) as a function of the superconducting phase difference facross the junction and the magnetization direction q. The proxim- ity effect vanishes in the centre of the junction at f=pfor any value of the exchange ﬁeld orientation, giving the normal- state value. Just as in the bilayer case (Fig. 3), we see that the proximity effect is strongly suppressed for the range of an- glesq>0. When rotating the ﬁeld in the opposite direction, q<0, strongly non-monotonic behavior emerges. For zero phase-difference, the physics is qualitatively similar to the bi- layer situation. In this case, we proved analytically that the LRT is not produced at all when q=p=4. Accordingly, Fig. 13 shows a full minigap there.Whether or not a clear zero-energy peak can be seen due to the LRT depends on the relative strength of the Rashba and Dresselhaus coupling. In the top panel, we have dominant Dresselhaus coupling in which case the low-energy density of states show either normal-state behavior or a minigap. In- terestingly, we see that the same opportunity appears in the present case of a Josephson setup as in the bilayer case: a magnetically tunable minigap appears. This effect exists as long as the phase difference is not too close to p, in which case the minigap closes. In the bottom panel correspond- ing to equal magnitude of Rashba and Dresselhaus, however, a strong zero-energy enhancement due to long-range triplets emerges as one moves away from q=p=4. With increasing phase difference, the singlets are seen to be more strongly sup- pressed than the triplet correlations since the minigap region (dark blue) vanishes shortly after f=p'0:6 while the peaks due to triplets remain for larger phase differences. FIG. 13: Zero-energy density of states D(0)as a function of the phase-difference fand magnetization angle q, both tun- able parameters experimentally. The other parameters used areLF=xS=0:5,h=D0=3,axS=2. In the top panel, we have dominant Dresselhaus strength ( c=0:15p) while in the bottom panel we have equal magnitude of Rashba and Dres- selhaus ( c=p=4).17 D. Critical temperature In this section, we present numerical results for the criti- cal temperature Tcof an SF bilayer. The theory behind these investigations is summarized in Section II E, and discussed in more detail in Appendix B. An overview of the physical system is given in Fig. 1(a). In all of the simulations we per- formed, we used the material parameter N0l=0:2 for the su- perconductor, the exchange ﬁeld h=10D0for the ferromag- net, and the interface parameter z=3 for both materials. The other physical parameters are expressed in a dimensionless form, with lengths measured relative to the superconducting correlation length xS, energies measured relative to the bulk zero-temperature gap D0, and temperatures measured relative to the bulk critical temperature Tcs. This includes the SO cou- pling strength a, which is expressed in the dimensionless form axS. The plots presented in this subsection were generated from 12–36 data points per curve, where each data point has a numerical precision of 0.0001 in Tc=Tcs. The results were smoothed with a LOESS algorithm. Before we present the results with SO coupling, we will brieﬂy investigate the effects of the ferromagnet length LF and superconductor length LSon the critical temperature, in order to identify the interesting parameter regimes. The criti- cal temperature as a function of the size of the superconductor is shown in Fig. 14. LF/ξS= LF/ξS= 0.25 LF/ξS= 0.50 = 1.00 0.000.250.500.751.00 0.5 0.6 0.7 0.8 0.9 1.0 LSξSTcTcs FIG. 14: Plot of the critical temperature Tc=Tcsas a function of the length LS=xSof the superconductor for axS=0. Be- low a critical length LS, superconductivity can no longer be sustained and Tcbecomes zero. For larger thicknesses of the superconducting layer, Tcreverts back to its bulk value. First of all, we see that the critical temperature drops to zero when LS=xS0:5. This observation is hardly surprising; since the superconducting correlation length is xS, the criti- cal temperature is rapidly suppressed once the length of the junction goes below xS. After this, the critical temperature in- creases quickly, already reaching nearly 50% of the bulk value when LS=xS=0:6, demonstrating that the superconductivity of the system is clearly very sensitive to small changes in pa- rameters for this region.The next step is then to observe how the behaviour of the system varies with the size of the ferromagnet, and these re- sults are presented in Fig. 15. LS/ξS = 0.575 LS/ξS = 0.525 LS/ξS = 0.550 0.000.250.500.751.00 0.00 0.25 0.50 0.75 1.00 LFξSTcTcs FIG. 15: Plot of the critical temperature Tc=Tcsas a function of the ferromagnet length LF=xSforaxS=0. Increasing the thickness of the ferromagnet gradually suppresses the Tcof the superconductor, causing a stronger inverse proximity effect. We again observe that the critical temperature increases with the size of the superconductor, and decreases with the size of the ferromagnet. The critical temperature for a superconduc- tor with LS=xS=0:525 drops to zero at LF=xS0:6, and stays that way as the size of the ferromagnet increases. Thus we do not observe any strongly nonmonotonic behaviour, such as reentrant superconductivity, for our choice of parameters. This is consistent with the results of Fominov et al. , who only reported such behaviour for systems where either the interface parameter or the exchange ﬁeld is drastically smaller than for the bilayers considered herein56. We now turn to the effects of the antisymmetric SO cou- pling on the critical temperature, which has not been studied before. Figs. 16 and 17 show plots of the critical temperature as a function of the SO angle cfor an exchange ﬁeld in the z- direction. The critical temperature is here independent of the SO angle c. This result is reasonable, since the SO coupling is in the xy-plane, which is perpendicular to the exchange ﬁeld for this geometry. We also observe a noticeable increase in critical temperature for larger values of a. This behaviour can be explained using the linearized Usadel equation. Accord- ing to Eq. (26), the effective energy Ezcoupling to the triplet component in the z-direction becomes Ez=e+4iDFa2; (47) so in other words, the SRTs obtain an imaginary energy shift proportional to a2. However, as shown in Eq. (25), there is no corresponding shift in the energy of the singlet component. This effect reduces the triplet components relative to the sin- glet component in the ferromagnet, and as the triplet prox- imity channel is suppressed the critical temperature becomes restored to higher values.18 aξS= 0 aξS= 2 aξS= 6 0.8250.8500.8750.9000.925 -0.50 -0.25 0.00 0.25 0.50 χπTcTcs FIG. 16: Plot of the critical temperature Tc=Tcsas a function of the SO angle c, when LS=xS=1:00,LF=xS=0:2, and hkˆz. Increasing the SO coupling causes Tcto move closer to its bulk value, since the triplet proximity effect channel becomes suppressed. aξS= 0 aξS= 2 aξS= 6 0.30.40.50.60.70.8 -0.50 -0.25 0.00 0.25 0.50 χπTcTcs FIG. 17: Plot of the critical temperature Tc=Tcsas a function of the SO angle c, when LS=xS=0:55,LF=xS=0:2, and hkˆz. The same situation for an exchange ﬁeld along the x-axis is shown in Figs. 18 and 19. For this geometry, we observe a somewhat smaller critical temperature for all a>0 and all c compared to Figs. 16 and 17. This can again be explained by considering the linearized Usadel equation in the ferromagnet, which suggests that the effective energy Excoupling to the x- component of the triplet vector should be Ex=e+2iDFa2; (48) which has a smaller imaginary part than the corresponding equation for Ez. Furthermore, note the drop in critical temper- ature as c!p=4. Since the linearized equations contain a triplet mixing term proportional to sin2 c, which is maximal precisely when c=p=4, these are also the geometries for which we expect a maximal LRT generation. Thus, this de- crease in critical temperature near c=p=4 can be explained by a net conversion of singlet components to LRTs in the sys- tem, which has an adverse effect on the singlet amplitude in the superconductor, and therefore the critical temperature. aξS= 0 aξS= 2 aξS= 6 0.8250.8500.8750.9000.925 -0.50 -0.25 0.00 0.25 0.50 χπTcTcsFIG. 18: Plot of the critical temperature Tc=Tcsas a function of the SO angle c, when LS=xS=1:00,LF=xS=0:2, and hkˆx. The critical temperature depends on the relative weight of the Rashba and Dresselhaus coefﬁcients. aξS= 0 aξS= 2 aξS= 6 0.30.40.50.60.70.8 -0.50 -0.25 0.00 0.25 0.50 χπTcTcs FIG. 19: Plot of the critical temperature Tc=Tcsas a function of the SO angle c, when LS=xS=0:55,LF=xS=0:2, and hkˆx. In Figs. 20 and 21 we present the results for a varying exchange ﬁeld hcosqˆx+sinqˆyin the xy-plane. In this case, we observe particularly interesting behaviour: the crit- ical temperature has extrema at jcj=jqj=p=4, where the extremum is a maximum if qandchave the same sign, and a minimum if they have opposite signs. Since q=p=4 is precisely the geometries for which we do not expect any LRT generation, triplet mixing cannot be the source of this behaviour. For the choice of physical parameters chosen in Fig. 21, this effect results in a difference between the minimal and maximal critical temperature of nearly 60% as the mag- netization direction is varied. As shown in Fig. 20, the effect persists qualitatively in larger structures as well, but is then weaker.19 χ = 0 χ = +π/4 χ = -π/4 0.830.840.850.860.87 -0.50 -0.25 0.00 0.25 0.50 θπTcTcs FIG. 20: Plot of critical temperature Tc=Tcsas a function of the exchange ﬁeld angle q, when LS=xS=1:00,LF=xS=0:2, andaxS=2. In contrast to ferromagnets without SO coupling, Tcnow depends strongly on the magnetization direction. This gives rise to a spin-valve like functionality with a single fer- romagnet featuring SO coupling. χ = 0 χ = +π/4 χ = -π/4 0.350.400.450.500.55 -0.50 -0.25 0.00 0.25 0.50 θπTcTcs FIG. 21: Plot of critical temperature Tc=Tcsas a function of the exchange ﬁeld angle q, when LS=xS=0:55,LF=xS=0:2, andaxS=2. Instead, these observations may be explained using the theory developed in Section II. When we have a general exchange ﬁeld and SO ﬁeld in the xy-plane, Eq. (34) reveals that the effective energy of the SRT component is E=e+2iDFa2(1sin2qsin2c): (49) Since the factor (1sin2qsin2c)vanishes for q=c=p=4, we get E=efor this case. This geometry is also one where we do not expect any LRT generation, since the triplet mix- ing factor cos2 qsin2c=0, so the conclusion is that the SO coupling has no effect on the behaviour of SRTs for these parameters—at least according to the linearized equations. However, since 1sin2qsin2c=2 forq=c=p=4, the situation is now dramatically different. The SRT effective en- ergy is now E=e+4iDFa2, with an imaginary contribution which again destabilizes the SRTs, and increases the critical temperature of the system. We emphasize that the variation ofTcwith the magnetization direction is present when c6=p=4 as well, albeit with a magnitude of the variation that gradually decreases as one approaches pure Rashba or pure Dresselhais coupling. E. Triplet spin-valve effect with a single ferromagnet The results discussed in the previous section show that the critical temperature can be controlled via the magnetization direction of one single ferromagnetic layer. This is a new re- sult originating from the presence of SO coupling. In conven- tional SF structures, Tcis independent of the magnetization orientation of the F layer. By using a spin-valve setup such as FSF57–61, it has been shown that the relative magnetization conﬁguration between the ferromagnetic layers will tune the Tcof the system. In contrast, in our case such a spin-valve effect can be obtained with a single ferromagnet (see Figs. 20 and 21): by rotating the magnetization an angle p=2,Tcgoes from a maximum to a minimum. The fact that only a single ferromagnet is required to achieve this effect is of practical importance since it can be challenging to control the relative magnetization orientation in magnetic multilayered structures. IV . SUMMARY AND DISCUSSION It was pointed out in Ref. 21 that for the case of transver- sal structures as depicted in Fig. 1(b), pure Rashba or pure Dresselhaus coupling and arbitrary magnetisation direction are insufﬁcient for long range triplets to exist. However, al- though these layered structures are more restrictive in their conditions for LRT generation than lateral junctions they are nevertheless one of the most relevant for current experimental setups10,11,50, and herein we consider the corresponding ex- perimentally accessible effects of SO coupling as a comple- ment to the ﬁndings of Ref. 21. We have provided a detailed exposition of the density of states and critical temperature for both the SF bilayer and SFS junction with SO coupling, high- lighting in particular the signature of long range triplets. We saw that the spectroscopic signature depends nonmono- tonically on the angle of the magnetic exchange ﬁeld, and that the LRT component can induce a strong peak in the density of states at zero energy for a range of magnetization direc- tions. In addition to the large enhancement at zero energy, we see that by carefully choosing the SO coupling and exchange ﬁeld strengths in the Josephson junction it is again possible to control the qualitative features of the density of states by al- tering the phase difference between the two superconductors e.g.with a loop geometry53. The intrinsic SO coupling present in the structures con- sidered herein derives from their lack of inversion symme- try due to the e.g. junction interfaces, so-called interfacial asymmetry, and we restricted the form of this coupling to the experimentally common and, in some cases, tunable Rashba- Dresselhaus form. A lack of inversion symmetry can also de- rive from intrinsic noncentrosymmmetry of a crystal. This could in principle be utilised to provide a component of the20 SO-ﬁeld in the junction direction, but to date we are not aware of such materials having been explored in experiments with SF hybrid materials. However, analytic and numerical data suggest that such materials could have signiﬁcant impor- tance for spintronic applications making use of a large triplet Cooper pair population43. It is also worth considering the possibility of separating the spin-orbit coupling and the ferromagnetic layer, which would arguably be easier to fabricate, and we are currently pursuing this line of investigation. In this case, we would expect similar conclusions regarding when the long-range triplets leave clear spectroscopic signatures and also regarding the spin-valve ef- fect with a single ferromagnet, as found when the SO cou- pling and exchange ﬁeld coexist in the same material. One way to practically achieve such a setup would be to deposit a very thin layer of a heavy normal metal such as Au or Pt between a superconductor and a conventional homogeneous ferromagnet. The combination of the large atomic number Z and the broken structural inversion symmetry at the interface region would then provide the required SO coupling. With a very thin normal metal layer (of the order of a couple of nm), the proximity effect would be signiﬁcantly stronger, and thus analysis of this regime is only possible with the full Usadel equations in the Riccati parameterisation developed herein. The current analysis pertains to thin ﬁlm ferromagnets. Upon increasing the length of ferromagnetic ﬁlm one will in- crease the relative proportions of long-range to short-range triplets in the middle of the ferromagnet. For strong ferro- magnets where the exchange ﬁeld is a signiﬁcant fraction of the Fermi energy, the quasiclassical Usadel formalism may no longer describe the system behaviour appropriately, since it assumes that the impurity scattering rate is much larger than the other energy scales involved, and the Eilenberger equation should be used instead62. In the previous section, we also observed that the presence of SO coupling will in many cases increase the critical tem- perature of a hybrid structure. This effect is explained through an increase in the effective energy coupled to the triplet com- ponent in the Usadel equation, which destabilizes the triplet pairs and closes that proximity channel. However, for the special geometry q=c=p=4, the linearized equations suggest that the SRTs are unaffected by the presence of SO coupling, and this is consistent with the numerical results. We also note that for the geometries with a large LRT generation, such as q=0 and c=p=4, the LRT generation reduces the critical temperature again. Thus, for the physical parameters considered herein, we see that there is a very slight increase in critical temperature for these geometries, but not as large as for the geometries without LRT generation. One particularly striking result from the critical temper- ature calculations is that when the Rashba and Dresselhaus contribution to the SO coupling is of similar magnitude, one observes that the critical temperature can change by as much as 60% upon changing q=p=4 toq= +p=4,i.e.by a 90 rotation of the magnetic ﬁeld. This implies that it is possible to create a novel kind of triplet spin valve using an SF bilayer, where the ferromagnet has a homogeneous exchange ﬁeld and Rashba–Dresselhaus coupling. This is in contrast to previoussuggestions for triplet spin valves, such as the one described by Fominov et al. , which have required trilayers with differ- ent homogeneous ferromagnets63. The construction of such a device is likely to have possible applications in the emerging ﬁeld of superconducting spintronics3. Acknowledgments The authors thank Angelo di Bernardo, Matthias Eschrig, Camilla Espedal, and Iryna Kulagina for useful discussions and gratefully acknowledge support from the ‘Outstanding Academic Fellows’ programme at NTNU and COST Action MP-1201’ Novel Functionalities through Optimized Conﬁne- ment of Condensate and Fields’. J.L. was supported by the Research Council of Norway, Grant No. 205591 (FRINAT) and Grant No. 216700. Appendix A: Riccati parametrization of the Usadel equation and Kupriyanov–Lukichev boundary conditions The 44 components of the retarded Green’s function ˆ g are not entirely independent, but can be expressed as ˆg(z;e) = g(z;+e) f(z;+e) f(z;e)g(z;e) ; (A1) which suggests that the notation can be simpliﬁed by intro- ducing the tilde conjugation ˜g(z;+e)g(z;e): (A2) Moreover, the normalization condition ˆ g2=1 further con- strains the possible form of ˆ gby relating the gcomponents to the fcomponents, ggf˜f=1; g ff˜g=0: (A3) Remarkably, if we pick a suitable parametrization of ˆ g, which automatically satisﬁes the symmetry and normalization re- quirements above, then both the Usadel equation and the Kupriyanov–Lukichev boundary conditions can be reduced from 44 to 22 matrix equations. In this paper, we em- ploy the so-called Riccati parametrization for this purpose, which is deﬁned by ˆg= N0 0˜N 1+g˜g2g 2˜g1+˜gg ; (A4) where the normalization matrices are N(1g˜g)1and ˜N (1˜gg)1. Solving the Riccati parametrized equations for the function g(z;e)in spin space is then sufﬁcient to uniquely construct the whole Green’s function ˆ g(z;e). It is noteworthy that ˆg!1 when g!0, while the elements of ˆ gdiverge to inﬁnity when g!1; so we see that a ﬁnite range of variation ingparametrizes an inﬁnite range of variation in ˆ g. We begin by deriving some basic identities, starting with the inverses of the two matrix products Ngandg˜N: (Ng)1=g1N1=g1(1g˜g) =g1˜g; (A5) (g˜N)1=˜N1g1= (1˜gg)g1=g1˜g: (A6)21 By comparison of the results above, we see that Ng=g˜N. Similar calculations for other combinations of the Riccati ma- trices reveal that we can always move normalization matrices past gamma matrices if we also perform a tilde conjugation in the process: Ng=g˜N;˜Ng=gN;N˜g=˜g˜N;˜N˜g=˜gN: (A7) Since we intend to parametrize a differential equation, we should also try to relate the derivatives of the Riccati matri- ces. This can be done by differentiating the deﬁnition of N using the matrix version of the chain rule: ¶zN=¶z(1g˜g)1 =(1g˜g)1[¶z(1g˜g)](1g˜g)1 = (1g˜g)1[(¶zg)˜g+g(¶z˜g)](1g˜g)1 =N[(¶zg)˜g+g(¶z˜g)]N: (A8) Performing a tilde conjugation of the equation above, we get a similar result for ¶z˜N. Thus, the derivatives of the normal- ization matrices satisfy the following identities: ¶zN=N[(¶zg)˜g+g(¶z˜g)]N; (A9) ¶z˜N=˜N[(¶z˜g)g+˜g(¶zg)]˜N: (A10) In addition to the identities derived above, one should note that the deﬁnition of the normalization matrix N= (1g˜g)1can be rewritten in many forms which may be of use when sim- plifying Riccati parametrized expressions; examples of this include g˜g=1N1and 1 =NNg˜g. Now that the basic identities are in place, it is time to parametrize the Usadel equation in the ferromagnet, DF˜Ñ(ˆg˜Ñˆg)+i eˆr3+ˆM;ˆg =0; (A11) where we for simplicity will let DF=1 in this appendix. We begin by expanding the gauge covariant derivative ˜Ñ(ˆg˜Ñˆg), and then simplify the result using the normalization condition ˆg2=1 and its derivative fˆg;¶zˆgg=0, which yields the result ˜Ñ(ˆg˜Ñˆg) =¶z(ˆg¶zˆg)i¶z(ˆgˆAzˆg) i[ˆAz;ˆg¶zˆg][ˆA;ˆgˆAˆg]:(A12) We then write ˆ gin component form using Eq. (A1), and also write ˆAin the same form using ˆA=diag(A;A). In the rest of this appendix, we will for simplicity assume that Ais real, so that ˆA=diag(A;A); in practice, this implies that Acan only depend on the spin projections sxandsz. The derivation for the more general case of a complex ˆAis almost identical.The four terms in Eq. (A12) may then be written as follows: ¶z(ˆg¶zˆg) = ¶z(g¶zgf¶z˜f)¶z(g¶zff¶z˜g) ¶z(˜g¶z˜f˜f¶zg)¶z(˜g¶z˜g˜f¶zf) ; (A13) ¶z(ˆgˆAˆg) = ¶z(gAg+f A˜f)¶z(gA f+f A˜g) ¶z(˜gA˜f+˜f Ag)¶z(˜gA˜g+˜f A f) ; (A14) [ˆA;ˆg¶zˆg] = [A;g¶zgf¶z˜f]fA;g¶zff¶z˜gg fA;˜g¶z˜f˜f¶zgg [A;˜g¶z˜g˜f¶zf] ; (A15) [ˆA;ˆgˆAˆg] = [A;gAg+f A˜f]fA;gA f+f A˜gg fA;˜gA˜f+˜f Agg[A;˜gA˜g+˜f A f] : (A16) Substituting these results back into Eq. (A12), we can ﬁnd the upper blocks of the covariant derivative ˜Ñ(ˆg˜Ñˆg), [˜Ñ(ˆg˜Ñˆg)](1;1) =¶z(g¶zgf¶z˜f)i¶z(gAzg+f Az˜f) i[Az;g¶zgf¶z˜f][A;gAg+f A˜f]; (A17) [˜Ñ(ˆg˜Ñˆg)](1;2) =¶z(g¶zff¶z˜g)i¶z(gAzf+f Az˜g) ifAz;g¶zff¶z˜ggf A;gAf+f A˜gg: (A18) In this context, the notation ˆM(n;m)refers to the n’th row and m’th column in Nambu space. Since the Green’s function ˆ g and background ﬁeld ˆAalso have a structure in spin space, the (1;1)element in Nambu space is the upper-left 2 2 block of the matrix, and the (1;2)element is the upper-right one. There are two kinds of expressions that recur in the equa- tions above, namely the components of ˆ g¶zˆg, and the compo- nents of ˆ gˆAˆg. After we substitute in the Riccati parametriza- tiong=2N1 and f=2Ng, these components take the form: [ˆg¶zˆg](1;1)=g¶zgf¶z˜f =2N[(¶zg)˜gg(¶z˜g)]N; (A19) [ˆg¶zˆg](1;2)=g¶zff¶z˜g =2N[(¶zg)g(¶z˜g)g]˜N; (A20) [ˆgˆAˆg](1;1)=gAg+f A˜f =4N(A+gA˜g)N2fA;Ng+A; (A21) [ˆgˆAˆg](1;2)=gAf+f A˜g =4N(Ag+gA)˜N2fA;Ngg: (A22) If we explicitly calculate the commutators of ˆAwith the two22 matrices ˆ g¶zˆgand ˆgˆAˆg, then we ﬁnd: [ˆA;ˆg¶zˆg](1;1)= [A;g¶zgf¶z˜f] =2N(1g˜g)AN[(¶zg)˜gg(¶z˜g)]N 2N[(¶zg)˜gg(¶z˜g)]NA(1g˜g)N; (A23) [ˆA;ˆg¶zˆg](1;2)=fA;g¶zff¶z˜gg =2N(1g˜g)AN[(¶zg)g(¶z˜g)g]˜N +2N[(¶zg)g(¶z˜g)g]˜NA(1˜gg)˜N; (A24) [ˆA;ˆgˆAˆg](1;1)= [A;gAg+f A˜f] =4AN(A+gA˜g)N 4N(A+gA˜g)NA 2[A2;N]; (A25) [ˆA;ˆgˆAˆg](1;2)=fA;gAf+f A˜gg =4AN(Ag+gA)˜N +4N(Ag+gA)˜NA 4ANgA2fA2;Ngg: (A26) If we instead differentiate the aforementioned matrices with respect to z, we obtain: [¶z(ˆg¶zˆg)](1;1)=¶z(g¶zgf¶z˜f) =2N[(¶2 zg)+2(¶zg)˜N˜g(¶zg)]˜gN 2Ng[(¶2 z˜g)+2(¶z˜g)Ng(¶z˜g)]N; (A27) [¶z(ˆg¶zˆg)](1;2)=¶z(g¶zff¶z˜g) =2N[(¶2 zg)+2(¶zg)˜N˜g(¶zg)]˜N 2Ng[(¶2 z˜g)+2(¶z˜g)Ng(¶z˜g)]g˜N; (A28) [¶z(ˆgAˆg)](1;1)=¶z(gAg+f A˜f) =2N(1+g˜g)AN[g(¶z˜g)+(¶zg)˜g]N +2N[g(¶z˜g)+(¶zg)˜g]NA(1+g˜g)N +4NgA˜N[(¶z˜g)+˜g(¶zg)˜g]N +4N[(¶zg)+g(¶z˜g)g]˜NA˜gN; (A29) [¶z(ˆgAˆg)](1;2)=¶z(gAf+f A˜g) =2N(1+g˜g)AN[(¶zg)+g(¶z˜g)g]˜N +2N[(¶zg)+g(¶z˜g)g]˜NA(1+˜gg)˜N +4NgA˜N[˜g(¶zg)+(¶z˜g)g]˜N +4N[g(¶z˜g)+(¶zg)˜g]NA˜g˜N: (A30) Combining all of the equations above, we can express Eqs. (A17) and (A18) using Riccati matrices. In order to iso- late the second-order derivative ¶2 zgfrom these, the trick is to multiply Eq. (A17) by gfrom the right, and subsequentlysubtract the result from Eq. (A18): 1 2N1 [˜Ñ(ˆg˜Ñˆg)](1;2)[˜Ñ(ˆg˜Ñˆg)](1;1)g =¶2 zg+2(¶zg)˜N˜g(¶zg) 2i(Az+gAz˜g)N(¶zg)2i(¶zg)˜N(Az+˜gAzg) 2(Ag+gA)˜N(A+˜gAg)A2g+gA2: (A31) If we ﬁnally rewrite [˜Ñ(ˆg˜Ñˆg)](1;1)and[˜Ñ(ˆg˜Ñˆg)](1;2)in the equation above by substituting in the Usadel equation (A11), then we obtain the following equation for the Riccati matrix g: ¶2 zg=2iegih(sggs)2(¶zg)˜N˜g(¶zg) +2i(Az+gAz˜g)N(¶zg)+2i(¶zg)˜N(Az+˜gAzg) +2(Ag+gA)˜N(A+˜gAg)+A2ggA2: (A32) The corresponding equation for ˜gcan be found by tilde con- jugation of the above. After restoring the diffusion coefﬁ- cient DF, and generalizing the derivation to a complex SO ﬁeld A, the above result takes the form shown in Eq. (5). After parametrizing the Usadel equation, the next step is to do the same to the Kupriyanov–Lukichev boundary condi- tions. The gauge covariant version of Eq. (2) may be written 2Lnznˆgn˜Ñˆgn= [ˆg1;ˆg2]; (A33) which upon expanding the covariant derivative ˆ g˜Ñˆgbecomes ˆgn¶zˆgn=1 2Wn[ˆg1;ˆg2]+iˆgn[ˆAz;ˆgn]; (A34) where we have introduced the notation Wn1=Lnznfor the interface parameter. We will now restrict our attention to the (1,1) and (1,2) components of the above, gn¶zgnfn¶z˜fn=1 2Wn(g1g2g2g1f1˜f2+f2˜f1) +ign[Az;gn]+i fnfAz;˜fng; (A35) gn¶zfnfn¶z˜gn=1 2Wn(g1f2g2f1f1˜g2+f2˜g1) +ignfAz;fng+i fn[Az;˜gn]: (A36) Substituting the Riccati parametrizations gn=2Nn1 and fn=2Nngnin the above, we then obtain: Nn[(¶zgn)˜gngn(¶z˜gn)]Nn=WnN1(1g1˜g2)N2 WnN2(1g2˜g1)N1 iNn(1gn˜gn)ANn iNnA(1gn˜gn)Nn +2iNn(A+gnA˜gn)Nn;(A37) Nn[(¶zgn)gn(¶z˜gn)gn]˜Nn=WnN1(1g1˜g2)g2˜N2 WnN2(1g2˜g1)g1˜N1 +iNn(1+gn˜gn)Agn˜Nn +iNngnA(1+˜gngn)˜Nn:(A38)23 If we multiply Eq. (A37) by gnfrom the right, subtract this from Eq. (A38), and divide by Nnfrom the left, then we obtain the following boundary condition for gn: ¶zgn=Wn(1g1˜g2)N2(g2gn) +Wn(1g2˜g1)N1(gng1) +ifAz;gng: (A39) When we evaluate the above for n=1 and n=2, then it sim- pliﬁes to the following: ¶zg1=W1(1g1˜g2)N2(g2g1)+ifAz;g1g; (A40) ¶zg2=W2(1g2˜g1)N1(g2g1)+ifAz;g2g: (A41) The boundary conditions for ¶z˜g1and¶z˜g2are found by tilde conjugating the above. If we generalize the derivation to a complex SO ﬁeld A, and substitute back Wn1=Lnznin the result, then we arrive at Eq. (6). Appendix B: Derivation of the self-consistency equation for D For completeness, we present here a detailed derivation of the self-consistency equation for the BCS order parameter64 in a quasiclassical framework. Similar derivations can also be found in Refs. 52,65–68. In this paper, we follow the conven- tion where the Keldysh component of the anomalous Green’s function is deﬁned as FK ss0(r;t;r0;t0)ih[ys(r;t);ys0(r;t)]i; (B1) where ys(r;t)is the spin-dependent fermion annihilation op- erator, and the superconducting gap is deﬁned as D(r;t)lhy"(r;t)y#(r;t)i; (B2) where l>0 is the electron–electron coupling constant in the BCS theory. For the rest of this appendix, we will also assume that we work in an electromagnetic gauge where Dis a purely real quantity. Comparing Eqs. (B1) and (B2), and using the fermionic anticommutation relation y"(r;t)y#(r;t) =y#(r;t)y"(r;t); (B3) we see that the superconducting gap D(r;t)can be expressed in terms of the Green’s functions in two different ways, D(r;t) =il 2FK "#(r;t;r;t); (B4) D(r;t) =il 2FK #"(r;t;r;t): (B5) We may then perform a quasiclassical approximation by ﬁrst switching to Wigner mixed coordinates, then Fourier trans- forming the relative coordinates, then integrating out the en- ergy dependence, and ﬁnally averaging the result over the Fermi surface to obtain the isotropic part. The resulting equa- tions for the superconducting gap are D(r;t) =1 4N0lZ defK "#(r;t;e); (B6) D(r;t) =1 4N0lZ defK #"(r;t;e); (B7)where fK ss0is the quasiclassical counterpart to FK ss0,eis the quasiparticle energy, and N0is the density of states per spin at the Fermi level. In the equilibrium case, the Keldysh component ˆ gKcan be expressed in terms of the retarded and advanced components of the Green’s function, ˆgK= (ˆgRˆgA)tanh(e=2T); (B8) and the advanced Green’s function may again be expressed in terms of the retarded one, ˆgA=r3ˆgR†r3; (B9) which implies that the Keldysh component can be expressed entirely in terms of the retarded component, ˆgK= (ˆgRr3ˆgR†r3)tanh(e=2T): (B10) If we extract the relevant anomalous components fK "#and fK #" from the above, we obtain the results fK "#= [fR "#(r;+e)+fR #"(r;e)]tanh(e=2T); (B11) fK #"= [fR #"(r;+e)+fR "#(r;e)]tanh(e=2T): (B12) We then switch to a singlet/triplet-decomposition of the retarded component fR, where the singlet component is de- scribed by a scalar function fs, and the triplet component by the so-called d-vector (dx;dy;dz). This parametrization is de- ﬁned by the matrix equation fR= (fs+ds)isy; (B13) or in component form, fR ""fR "# fR #"fR ##! = idydxdz+fs dzfsidy+dx! : (B14) Parametrizing Eqs. (B11) and (B12) according to Eq. (B14), we obtain fK "#(r;e) = [ dz(r;+e)+fs(r;+e) +dz(r;e)fs(r;e)]tanh(e=2T); (B15) fK "#(r;e) = [ dz(r;+e)fs(r;+e) +dz(r;e)+fs(r;e)]tanh(e=2T): (B16) The triplet component dzcan clearly be eliminated from the above equations by subtracting Eq. (B15) from Eq. (B16), fK "#fK #"=2[fs(r;e)fs(r;e)]tanh(e=2T); (B17) and a matching expression for the superconducting gap can be acquired by adding Eqs. (B6) and (B7), 2D(r) =1 4N0lZ de[fK "#(r;e)fK "#(r;e)]tanh(e=2T):(B18) By comparing the two results above, we ﬁnally arrive at an equation for the superconducting gap which only depends on the singlet component of the quasiclassical Green’s function: D(r) =1 4N0lZ de[fs(r;e)fs(r;e)]tanh(e=2T):(B19)24 If the integral above is performed for all real values of e, it turns out to be logarithmically divergent e.g.for a bulk su- perconductor. However, physically, the range of energies that should be integrated over is restricted by the energy spectra of the phonons that mediate the attractive electron–electron interactions in the superconductor. This issue may therefore be resolved by introducing a Debye cutoff wc, such that we only integrate over the region where jej<wc. Including the integration range, the gap equation is therefore D(r) =1 4N0lwcZ wcde[fs(r;e)fs(r;e)]tanh(e=2T):(B20) The equation above can however be simpliﬁed even further. First of all, both fs(e)fs(e)and tanh (e=2T)are clearly antisymmetric functions of e, which means that the product is a symmetric function, and so it is sufﬁcient to perform an integral over positive values of e, D(r) =1 2N0lwcZ 0de[fs(r;e)fs(r;e)]tanh(e=2T):(B21) However, because of the term fs(r;e), we still need to know the Green’s function for negative values of ebefore we can calculate the gap. On the other hand, the singlet component of the quasiclassical Green’s functions also has a symmetry when the superconducting gauge is chosen as real fs(r;e) =f s(r;e); (B22) which implies that fs(r;e)fs(r;e) =2Reffs(r;e)g: (B23) Substituting Eq. (B23) into Eq. (B21), the gap equation takes a particularly simple form, which only depends on the real part of the singlet component fs(r;e)for positive energies e: D(r) =N0lwcZ 0deReffs(r;e)gtanh(e=2T): (B24) Let us now consider the case of a BCS bulk superconductor, which has a singlet component given by the equation fs(e) =Dp e2D2; (B25)so that the gap equation may be written as D=N0lwcZ 0deReDp e2D2 tanh(e=2T): (B26) The part in the curly braces is only real when jejD, which means that the equation can be simpliﬁed by changing the lower integration limit to D. After also dividing the equation byDN0l, we then obtain the self-consistency equation 1 N0l=wcZ Ddetanh(e=2T)p e2D2: (B27) For the zero-temperature case, where T!0 and D!D0, per- forming the above integral and reordering the result yields wc=D0cosh(1=N0l): (B28) Using the above equation for wc, and the well-known result D0 Tc=p eg; (B29) where g0:57722 is the Euler–Mascheroni constant, we can ﬁnally rewrite Eq. (B24) as: D(r) =N0lD0cosh(1=N0l)Z 0deReffs(r;e)gtanhp 2ege=D0 T=Tc :(B30) This version of the gap equation is particularly well-suited for numerical simulations. One advantage is that we only need to know the Green’s function for positive energies, which halves the number of energies that we need to solve the Usadel equa- tion for. The equation also takes a particularly simple form if we use energy units where D0=1 and temperature units where Tc=1, which is common practice in such simulations. 1A. I. Buzdin, Rev. Mod. Phys. 77, 935 (2005). 2F.S. Bergeret, A.F. V olkov and K.B. Efetov, Rev. Mod. Phys. 77, 1321 (2005). 3J. Linder and J. W. A. Robinson, Nature Physics 11, 307 (2015). 4F. Bergeret, A. V olkov, and K. Efetov, Phys. Rev. Lett. 86, 4096 (2001). 5M. Eschrig, J. Kopu, J. C. Cuevas, and Gerd Sch ¨on, Phys. Rev. Lett. 90, 137003 (2003).6K. Halterman, P. H. Barsic, and O. T. Valls, Phys. Rev. Lett. 99, 127002 (2007). 7A. Cottet, Phys. Rev. Lett. 107, 177001 (2011). 8Y . Asano, Y . Sawa, Y . Tanaka, and A. A. Golubov, Phys. Rev. B 76, 224525 (2007). 9Y . Kalcheim et al. , Phys. Rev. B 89, 180506(R) (2014). 10J. W. A. Robinson, J. D. Witt, and M. Blamire, Science 329, 59 (2010).25 11T. S. Khaire et al. , Phys. Rev. Lett. 104, 137002 (2010). 12M. Alidoust and J. Linder, Phys. Rev. B 82, 224504 (2010); I. B. Sperstad et al. , Phys. Rev. B 78, 104509 (2008). 13L. Trifunovic and Z. Radovi ´c, Phys. Rev. B 82, 020505(R) (2010). 14I. Sosnin, H. Cho, V . T. Petrashov, and A. F. V olkov, Phys. Rev. Lett. 96, 157002 (2006). 15Z. Shomali, M. Zareyan, W. Belzig, New J. Phys. 13, 083033 (2012). 16G. Annunziata et al. , Phys. Rev. B 83, 060508(R) (2011). 17I. V . Bobkova and A. M. Bobkov, Phys. Rev. Lett. 108, 197002 (2012). 18F. Konschelle, J. Cayssol, and A. Buzdin, Phys. Rev. B 82, 180509(R) (2010). 19M. Houzet, Phys. Rev. Lett. 101, 057009 (2008). 20G. Annunziata, D. Manske, and J. Linder, Phys. Rev. B 86, 174514 (2012). 21F.S. Bergeret and I.V . Tokatly, Phys. Rev. B. 89, 134517 (2014). 22F.S. Bergeret and I.V . Tokatly, Phys. Rev. Lett. 110, 117003 (2013). 23M. Eschrig, J. Kopu, A. Konstandin, J.C. Cuevas, M. Fogelstr ¨om, G. Sch ¨on, Advances in Solid State Physics V ol. 44, pp. 533-546 (2004). 24I. S. Burmistrov and N. M. Chtchelkatchev, Phys. Rev. B 72, 144520 (2005). 25J. Linder and K. Halterman, Phys. Rev. B 90, 104502 (2014). 26A.A. Abrikosov and L.P. Gor’kov, Sov. Phys. JETP 15, 752 (1962). 27M. Eschrig, Physics Today 64(1), 43 (2011). 28K.V . Samokhin, Ann. Phys. 324, 2385 (2009). 29V .M. Edelstein, Phys. Rev. B. 67, 020505 (2003). 30J. Linder and T. Yokoyama, Phys. Rev. Lett. 106, 237201 (2011). 31K. Sun and N. Shah, Phys. Rev. B 91, 144508 (2015). 32X. Liu, J. K. Jain, and C.-X. Liu, Phys. Rev. Lett. 113, 227002 (2014). 33K. Usadel, Phys. Rev. Lett. 25, 507 (1970). 34W. Belzig et al. , Superlattices and Microstructures, 5, 25 (1999). 35V . Chandrasekhar, arXiv:0312507. 36M.Y . Kupriyanov and V .F. Lukichev, Sov. Phys. JETP 67, 1163 (1988). 37N. Schopohl and K. Maki, Phys. Rev. B. 52, 490 (1995); N. Schopohl, arXiv:cond-mat/9804064. 38C. Gorini, P. Schwab, R. Raimondi, and A. L. Shelankov, Phys. Rev. B 82, 195316 (2010). 39A.I. Buzdin, Phys. Rev. Lett. 101, 107005 (2008); A.A. Raynoso et al. , Phys. Rev. B. 86, 214519 (2012); I. Kulagina and J. Linder, Phys. Rev. B 90, 054504 (2014); F.S. Bergeret and I.V . Tokatly,preprint, arXiv:1409.4563. 40F. Konschelle, Eur. Phys. J. B 87, 119 (2014). 41G. Dresselhaus, Phys. Rev. 100, 580 (1955); E. Rashba, Fiz. Tverd. Tela (Leningrad) 2, 1224 (1960), [Sov. Phys. Solid State 2, 1109 (1960)]. 42E. Bauer, M. Sigrist. Non-Centrosymmetric Superconductors: In- troduction and overview. Springer (2012). 43S.H. Jacobsen and J. Linder, Phys. Rev. B 92, 024501 (2015) 44P. Fulde and R.A. Ferrell, Phys. Rev. 135, A550 (1964); A.I. Larkin and Y .N. Ovchinnikov, Sov. Phys. JETP 20762 (1965). 45R. Balian and N.R. Werthamer, Phys. Rev. 131, 1553 (1963). 46A.P. Mackenzie and Y . Maeno, Rev. Mod. Phys. 75, 657 (2003). 47Y . Tanaka and A. Golubov, Phys. Rev. Lett. 98, 037003 (2007). 48T. Yokoyama, Y . Tanaka and A.A. Golubov, Phys. Rev. B 72, 052512 (2005);T. Yokoyama, Y . Tanaka and A.A. Golubov, Phys. Rev. B 73, 094501 (2006). 49S. Kawabata et al. , J. Phys. Soc. Jpn. 82, 124702 (2013). 50T. Kontos et al. , Phys. Rev. Lett. 86, 304 (2001); V .V . Ryazanov et al. , Phys. Rev. Lett. 86, 2427 (2001). 51P. SanGiorgio et al. , Phys. Rev. Lett. 100, 237002 (2008). 52M. Tinkham, Introduction to Superconductivity , (McGraw-Hill, 1996), 2nd Edition. 53H. le Sueur et al, Phys. Rev. Lett. 100, 197002 (2008). 54F. Zhou, P. Charlat, and B. Pannetier, J. Low Temp. Phys. 110, 841 (1998). 55J. C. Hammer, J. C. Cuevas, F. S. Bergeret, and W. Belzig, Phys. Rev. B 76, 064514 (2007). 56Y . Fominov et al. , Phys. Rev. B. 66, 014507 (2002). 57J. Y . Gu et al. , Phys. Rev. Lett. 89, 267001 (2002). 58I. Moraru, W. P. Pratt Jr., and N. Birge, Phys. Rev. Lett. 96, 037004 (2006). 59J. Zhu, I. N. Krivorotov, K. Halterman, and O. T. Valls, Phys. Rev. Lett. 105, 207002 (2010). 60P. V . Leksin et al. , Phys. Rev. Lett. 109, 057005 (2012). 61N. Banerjee et al. , Nature Commun. 5, 3048 (2014). 62G. Eilenberger, Sov. Phys. JETP 214, 195 (1968). 63Y . Fominov et al. , JETP Lett., 91, 308 (2010). 64J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 108, 1175 (1957). 65N. Kopnin. Theory of Nonequilibrium Superconductivity (2001). 66A. A. Abrikosov, L.P. Gorkov, I.E. Dzyaloshinski. Quantum Field Theoretical Methods in Statistical Physics (V olume 4 of Interna- tional Series of Monographs in Natural Philosophy) (1965). 67J. W. Serene and D. Rainer, Phys. Rep. 101, 4 (1983). 68P. G. deGennes. Superconductivity Of Metals And Alloys . (1999).
1401.0575v1.Spin_Orbit_Coupled_Bose_Gases_at_Finite_Temperatures.pdf	arXiv:1401.0575v1 [cond-mat.quant-gas] 3 Jan 2014Spin-Orbit Coupled Bose Gases at Finite Temperatures Renyuan Liao1, Oleksandr Fialko2 1College of Physics and Energy, Fujian Normal University, Fu zhou 350108, China and 2Centre for Theoretical Chemistry and Physics, Massey Unive rsity, Auckland 0632, New Zealand (Dated: June 24, 2021) Spin-orbit couplingis predictedto havedramatic eﬀects on thermal properties of a two-component atomic Bose gas. We show that in three spatial dimensions it l owers the critical temperature of condensation and enhances thermal depletion of the condens ate fraction. In two dimensions we show that spin-orbit coupling destroys superﬂuidity at any ﬁnit e temperature, modifying dramatically the cerebrated Berezinskii-Kosterlitz-Thouless scenario. W e explain this by the increase of the number of low energy states induced by spin-orbit coupling, enhanc ing the role of quantum ﬂuctuations. PACS numbers: 67.85.Fg, 03.75.Mn, 05.30.Jp, 67.85.Jk There are numerous phenomena in a wide range of quantum systems, ranging from condensed matter to atomic and nuclear physics, where spin-orbit coupling (SOC) plays an important role. Recently discovered new class of topological insulators, quantum spin Hall eﬀect [1] and Majorana fermions [2] rely on SOC and are expected to retain their quantum properties up to room temperature. However, the electronic systems can not be easily controlled and the details of SOC are usually not known. Therefore, it is a diﬃcult task to manip- ulate such systems. In contrast, ultracold atoms have been demonstrated to be a remarkable platform for em- ulation of various condensed matter phenomena due to their ability to be easily manipulated at will [3]. The pio- neering experimental realization of synthetic gauge ﬁelds and SOC [4–8] is deﬁning a new dimension in explor- ing quantum many-body systems with ultracold atomic gases. The engineered SOC (with equal Rashba and Dresselhaus strength) in a neutral atomic Bose-Einstein condensate was achieved by dressing two atomic spin states with a pair of lasers. It allows to study the rich physics of SOC eﬀects in bosonic systems [9–13], which havenotbeenexploredbefore. Recently, methodstogen- erate pure Rashba type SOC have been suggested [14]. Its realization will make it possible to study rich ground state physics proposed in fermionic [15] and bosonic sys- tems [10, 16], of which many properties have no con- densed matter analogues. SOC leads to a huge degeneracy of the ground state of a single particle [17]. This enhances the role of quan- tum ﬂuctuations making condensation of non-interacting bosons not possible [17, 18]. However, it has been shown that interactions among atoms stabilize conden- sation [19–21]. The role of quantum ﬂuctuations is espe- cially essential in two dimensions destroying condensa- tion but not necessary superﬂuidity. This yields, in par- ticular, the celebrated Berezinskii-Kosterlitz-Thouless (BKT) phase transition in two dimensions with a criti- caltemperatureseparatingsuperﬂuidandnormalphases. How do the quantum ﬂuctuations in the presence of SOC aﬀect the BKT phenomenon? What is the eﬀect of SOCon thermal properties of a Bose condensate? These ques- tions shall be addressed in this Letter. Previous theo- retical studies have been focused mainly on the ground state properties of interacting SOC quantum gases, leav- ing the experimentally relevant physics at ﬁnite tempera- tures intact. In the light of the recent experiment [22] on a ﬁnite-temperature phase diagram of SOC Bose gases, our present study is an interesting and urgent task. According to the Mermin-Wagner theorem long-range order at ﬁnite temperature does not exist in one spatial dimension. In this Letter, we shall present studies of a SOC two-component atomic Bose gas at ﬁnite temper- ature in two and three spatial dimensions. The inter- play between quantum and thermal ﬂuctuations in the presence of SOC is shown to yield dramatic modiﬁca- tions of the familiar physics. First, by resorting to the Popov approximation, we develop a formalism suitable for treating the system at ﬁnite temperature in three di- mensions. Within this formalism, we ﬁnd that the SOC greatly suppresses the critical temperature of condensa- tion and enhances thermal depletion of the condensate. We then derive an eﬀective theory suitable to study the celebrated BKT phase transition in two dimensions. The BKT transition temperature, in contrast to the previous case, is shown to drop to zero in the presence of SOC. We consider a three-dimensional homogeneous two- component Bose gas with an isotropic in-plane (x-y plane) Rashba spin-orbit coupling, described by the fol- lowing grand canonical Hamiltonian in real space: H=/integraldisplay dr/bracketleftigg/summationdisplay σψ† σ/parenleftbigg −/planckover2pi12∇2 2m−µ/parenrightbigg ψσ+(ψ† ↑ˆRψ↓+h.c.) +/summationdisplay σgσσ 2(ψ† σψσ)2+g↑↓ψ† ↑ψ↑ψ† ↓ψ↓/bracketrightigg . (1) Here,ψσis a Bose ﬁeld satisfying the usual commuta- tion relation [ ψσ(r),ψ† σ′(r′)] =i/planckover2pi1δσσ′δ(r−r′), the spin indexσ=↑,↓denotes two pseudo-spin states of the Bose gas with atomic mass m. The inter-particle interac- tiongσσ′is related to the two-body scattering length aσσ′asgσσ′= 4π/planckover2pi12aσσ′/m. For simplicity, we shall2 assume that the interactions between like-spin particles are the same, g↑↑=g↓↓=g. The chemical potential µis introduced to ﬁx the total particle number den- sity. The Rashba spin-orbit coupling is described by the operator ˆR=λ(ˆpx−iˆpy) withλbeing the cou- pling strength. Throughout the rest of this paper, we set/planckover2pi1= 2m=kB= 1 and deﬁne n1/3as a momentum scale, while n2/3as an energy scale. For the system to be weakly interacting, we set g= 0.1n−1/3. The Hamiltonian of a non-interacting system is diag- onalized in the helicity basis after the Fourier transform of the ﬁelds ψσ(r) = 1/√ L3/summationtext qψσ(q)exp(−iqr). The gas is assumed to be in a box with size L. This results in two branches of spectrum E± q=q2−µ±λq⊥, where q⊥is the magnitude of the in-plane momentum. The lowest energy state is therefore inﬁnitely degenerate, sit- ting on the circle q⊥=λ/2 in the plane qz= 0. This increases the low energy density of states with dramatic implications on the thermal properties of the Bose gas to be explored below. For an interacting system, earlier mean-ﬁeld study [10] found that there exist the plane wave phase (PW) for g≥g↑↓and the striped phase for g < g ↑↓. The PW phase is the result of condensation at a single ﬁnite momentum state breaking explicitly the rotational symmetry, while the striped phase represents a coherent superposition of two condensates at two op- posite momenta. Within the framework of the functional ﬁeld in- tegral, the partition function of the system is [23] Z=/integraltext D[ψ∗,ψ]exp(−S[ψ∗,ψ]) with the action S=/integraltextβ 0dτ[/summationtext σψ∗ σ∂τψσ+H(ψ∗,ψ)], whereβ= 1/Tis the inverse temperature. Here for simplicity we restrict our- self to study the PW phase, as analogous treatment of the striped phase is more involved and will be addressed elsewhere. Our choice of the PW phase is also motivated by the fact that the striped phase has not been realized experimentally yet. We further assume that the conden- sation occurs at momentum κ= (λ/2,0,0). Without loss of generality, the condensate wavefunction can bechosen as ( φ0↑,φ0↓) =√n0(1,−1)eiλx/2, withn0being the condensate density for either species. We split the Bose ﬁeld into the mean-ﬁeld part φ0σand the ﬂuctuat- ing partφqσasψqσ=φ0σδqκ+φqσ. After substitution, the action can be formally written as S=S0+Sf, where S0=βL3/bracketleftbig −2(κ2+µ)n0+(g+g↑↓)n2 0/bracketrightbig is the mean-ﬁeld contribution and Sfdenotes a contribution from ﬂuctu- ating ﬁelds. At this point, the action is exact. However, it contains terms of cubic and quartic orders in ﬂuctu- ating ﬁelds. To deal with such an action, one must re- sort to some sort of approximation. The celebrated Bo- goliubov approximation for BEC is valid strictly at zero temperature. At ﬁnite temperatures, the self-consistent Hartree-Fock-Bogoliubov (HFB) approximation gives a gapped spectrum [24], violating the Hugenholtz-Pines theorem [25] and the Goldstone theorem [26], which re- sults from the spontaneous symmetry breaking of U(1) gauge symmetry. We choose the Popov theory [27] yield- ing a gapless spectrum and therefore it is more suitable for treating ﬁnite-temperature Bose gases. Under the Popov approximation, which takes into ac- count interactions between excitations [28], the terms with three and four ﬂuctuating ﬁelds in the action are approximated as follows (neglecting anomalous average): φ∗ σφσφσ≈2∝angb∇acketleftφ∗ σφσ∝angb∇acket∇ightφσ, (φ∗ σφσ)2≈4∝angb∇acketleftφ∗ σφσ∝angb∇acket∇ightφ∗ σφσand \|φ↑φ↓\|2≈ ∝angb∇acketleftφ∗ ↑φ↑∝angb∇acket∇ightφ∗ ↓φ↓+∝angb∇acketleftφ∗ ↓φ↓∝angb∇acket∇ightφ∗ ↑φ↑. Within the Popov approximation we also require the ﬁrst order term to vanish, which ﬁxes the chemical potential as µ=µp= −κ2+ (g+g↑↓)n0+ (2g+g↑↓)nf, wherenf≡ ∝angb∇acketleftφ∗ σφσ∝angb∇acket∇ight is the density of particles of either component excited out of the condensate. We deﬁne a four-dimensional col- umn vector Φ qn= (φκ+q,n↑,φκ+q,n↓,φ∗ κ−q,n↑,φ∗ κ−q,n↓), whose components are deﬁned through the Fouriertrans- formφσ(r,τ) = 1//radicalbig L3β/summationtext q,nφq,nσexp[i(q·r−wnτ)], wherewn= 2nπ/βis the bosonic Matsubara frequen- cies. Retaining terms of zeroth and quadratic orders in the ﬂuctuating ﬁelds we can then bring the ﬂuctu- ating part of the action into the compact form Sf≈/summationtext q,n1 2Φ∗ qnG−1(q,iwn)ΦT qn−β/summationtext qǫ−q, whereǫq= (q+ κ)2−µ+ (2g+g↑↓)(n0+nf) and the inverse Green’s function G−1(q,iwn) reads G−1(q,iwn) = −iwn+ǫqRκ+q−g↑↓n0gn0 −g↑↓n0 R∗ κ+q−g↑↓n0−iwn+ǫq−g↑↓n0gn0 gn0 −g↑↓n0iwn+ǫ−qR∗ κ−q−g↑↓n0 −g↑↓n0gn0Rκ−q−g↑↓n0iwn+ǫ−q , (2) whereRq=λ(qx−iqy) is the Fourier transformed Rashba operator. The poles of the Green’s function give the spectrum of elementary excitations ωqs. It can be found by replacing iωnwithωqsand solving the secu- lar equation det G−1(q,ωqs) = 0. The index sdenotesdiﬀerent solutions of the secular equation. We retain only two positive solutions in the following. The spec- trum is used to study thermodynamic properties of the system. The Gaussian action permits direct integration over the ﬂuctuating ﬁelds to yield the grand potential3 /s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s48/s46/s55/s53/s49/s46/s48/s48/s84 /s67/s47/s84/s48 /s67 /s47/s110/s49/s47/s51 FIG. 1. (color online). The critical temperature of conden- sation as a function of the SOC strength. SOC decreases the critical temperature due to increase of the low energy densi ty of states on the circle q⊥=λ/2. HereT0 cis the critical tem- perature for a non-interacting Bose system without SOC. We have set g↑↓=g. −lnZ/β=S0/β+Ωf, where the second term Ωf=/summationdisplay q,s/bracketleftigg ln(1−e−βωqs) 2β+ωqs−ǫq 2+(g2+g2 ↑↓)n2 0 2q2/bracketrightigg (3) is the result of the Gaussian integration. The density of particles excited out of the condensates for either species can be easily evaluated as nf= 1/(2L3)(∂Ωf/∂µ)µ=µp. At the critical temperature the condensed density n0= 0 andnfis equal to the density of the total number of particlesn. In this special case, it is straightforward to calculate analytically the poles of the Green’s function and obtain the following secular equation determining the critical temperature n=1 2L3/summationdisplay q,s=±1 e[q2z+(q⊥+sλ/2)2]/Tc−1.(4) IntheabsenceofSOC,thetwotermsareidenticalandwe get the usual number equation to determine the critical temperature of condensation. The biggest contribution to the sum over the momentum is from the vicinity of the point \|q\|= 0. SOC changes the situation. Now the biggest contribution to the sum comes from the circle qz= 0,q⊥=λ/2 with much higher weight than in the previous case. The critical temperature Tcmust be de- creasedforﬁxeddensity. Morequantitatively, the critical temperature in the thermodynamic limit is estimated as Tc≈T0 cg3/2(1) g3/2(z). (5)/s48/s46/s48/s48 /s48/s46/s50/s53 /s48/s46/s53/s48 /s48/s46/s55/s53 /s49/s46/s48/s48/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s48/s46/s55/s53/s49/s46/s48/s48/s110 /s48/s47/s110 /s84/s47/s84 /s67/s32 /s61/s48 /s32 /s110/s49/s47/s51 /s32 /s110/s49/s47/s51 FIG. 2. (color online). The condensate fraction as a func- tion of temperature for various spin-orbit coupling streng ths λ. The eﬀect of SOC on the thermal depletion is more pro- nouncedathighertemperatures, sincethethermalcomponen t beneﬁts more from the low energy states than the condensed one. We have set g↑↓=g. Here,T0 c= 4π[n/g3/2(1)]2/3is the critical tempera- ture for a non-interacting Bose gas without SOC, z= exp[−λ2/(4Tc)], andgp(z) =/summationtext∞ l=1zl/lp[29]. As shown in Fig. 1, the critical temperature Tcindeed decreases rapidly as the SOC strength increases. The situation is more complicated for temperatures smaller than the critical temperature. The secular equa- tionshouldbesupplementedwiththeequationfortheto- tal densityn0+nf=n. By solving the two equations we obtain the condensate density n0for a ﬁxed total density ofatomsn. BeinganintrinsicpropertyofaBose-Einstein condensate, the condensate fraction n0/nprovides key information about the robustness of the superﬂuid state. It is shown in Fig. 2, where three typical SOC couplings (λ= 0,λ/n1/3= 0.2, andλ/n1/3= 0.4) are chosen for comparison. In the absence of SOC, the condensate fraction decreases gradually to zero as the temperature reachesT0 c, since under the Popov approximation, the critical temperature of a weakly interacting Bose gas is identical to that of a non-interacting one [30, 31]. The eﬀect of SOC on the condensate fraction is more pro- nounced when the temperature gets closer to the tran- sition temperature Tc. We can explain this by a similar argument leading to the reduction of the critical temper- ature: At a ﬁxed temperature the thermal component beneﬁts mainly from the low energy states living on the circleq⊥=λ/2, while the condensed component mainly from a single momentum at qx=λ/2. Therefore, SOC aﬀects the thermal component more than the condensed one, leading to the more pronounced thermal depletion at higher temperatures.4 We turn now to study the eﬀects of SOC on the sys- tem in two spatial dimensions. In the absence of SOC there is a quasi-long range order in two dimensions at suﬃciently low temperatures with correlation function decaying according to some power law. At higher tem- peratures the correlation function decays exponentially, where the algebraic order is destroyed by proliferation of vortices. The BKT separates these two regimes [32]. Since there is no condensation now, the Popov approx- imation is not applicable. Here we derive a low energy eﬀective theory suitable to probe the correlation func- tion. Anticipating that phase degrees of freedom play an essential role now, we adopt the density-phase represen- tation of the ﬁeld operators, ψσ=ρσeiθσ. The ﬁelds are then separated again into a mean-ﬁeld part and a ﬂuctu- ating part as ρσ=ρ0+δρσandθσ=θ0σ+δθσ. Here, ρ0is usually called a quasi-condensate density and the explicit breaking of the rotational symmetry by SOC re- sults inθ0↑=θ0↓−π=λx/2. Retaining terms of zeroth and quadratic order in the ﬂuctuating ﬁelds, the action is split into two parts S≈S0+Sf, where similarly to the previous case S0=βL2/bracketleftbig −2(λ2/4+µ)ρ0+(g+g↑↓)ρ2 0/bracketrightbig is the mean-ﬁeld contribution. The Gaussian action Sf contains the ﬂuctuating ﬁelds up to the second order. This allows us to integrate the density ﬂuctuations δρσ out yieding Sf=/integraldisplay dτ/integraldisplay dr/braceleftbigg(∂τθ+)2 g+g↑↓+(∂τθ−+i2λ∂xθ−)2 g−g↑↓+λ2/(2ρ0) +ρ0 2/bracketleftbig (∇θ+)2+(∇θ−)2/bracketrightbig +ρ0λθ−∂yθ++ρ0 2λ2θ2 −/bracerightig ,(6) whereθ±=δθ↑±δθ↓. In the absence of SOC, the above action describes two decoupled quantum XY mod- els representing two branches of phase ﬂuctuations with linear spectrum in momentum space yielding the quasi- long range order. The corresponding correlation func- tionsC±(r) =∝angb∇acketlefte[θ±(r)−θ±(0)]∝angb∇acket∇ight ∝r−T/ρ0(g±g↑↓)decay ac- cording to the power law rather than exponentially [32]. In the presence of SOC, we notice that the ﬁeld θ−de- scribes a massive ﬂuctuation as the action Sgcontains the termρ0λ2θ2 −/2. At low energy, we can integrate out the massive ﬁeld θ−by assuming homogeneous temporal ﬂuctuations to get an eﬀective action solely in terms of the collective variable θ+(for simplicity we set g≈g↑↓): Sf≈1 2/integraldisplay dτ/integraldisplay dr/bracketleftbigg1 g(∂τθ+)2+ρ0(∂xθ+)2+ρ0 λ2(∂2 yθ+)2/bracketrightbigg . (7) The ﬁrst two terms in the action are the part of the fa- miliar XY model. The third term is the manifestation of the broken rotational symmetry and it adds a twist. To see this we calculate the spectrum of elementary excita- tions. We ﬁrst Fourier transform the action for it to be- comeSf= 1/2/summationtext q,n(w2 n/g+ρ0q2 x+ρ0q4 y/λ2)\|θ+(q,wn)\|2. The phase ﬂuctuations in the Fourier space can be eas- ily evaluated, ∝angb∇acketleft\|θ+(q,wn)\|2∝angb∇acket∇ight=g(w2 n+w2 q)−1. The polesof this expression gives the low energy spectrum of the symmetric phase wq=/radicalig gρ0(q2x+q4y/λ2). It is clear that now the energy carried by elementary excitations of the system does not scale linearly as in the XY model case. As a result, less energy is required to excite low energy modes as compared to the XY model case, en- hancing the role of quantum ﬂuctuations. The Lan- dau criterion for the superﬂuid critical velocity gives minq(wq/q) = 0, implying SOC destroys superﬂuidity in two dimensions. This is substantiated by the form of the correlation function, which reads C+(r)≈e−Iwith I=/summationtext q,n∝angb∇acketleft\|θ+(q,wn)\|2∝angb∇acket∇ight(1−eiqr)/2β. The asymptotic behavior of it for large separations along the x-direction isI≈T/radicalbig λ\|x\|/(2πgρ0) and along the y-direction is I≈Tλ\|y\|/(2gρ0). Hence, at any ﬁnite temperature, the correlation function decays exponentially along both x- andy- directions, in stark contrast to the power law ex- hibited by the conventional XY model in the low tem- perature phase. The superﬂuid order in two dimensions is predicted to be disordered by SOC. Summary. – We predict dramatic implications of SOC on the thermal properties of atomic Bose gases: In three dimensions SOC reduces the critical temperature of con- densation and enhances the thermal depletion of the con- densate, while in two dimensions it destroys superﬂuid order at any ﬁnite temperature. This is explained by the role of quantum ﬂuctuations ampliﬁed by SOC. We hope thatcurrentworkwillstimulateexperimentstoverifyour predictions and will add to the overall understanding of the thermal quantum ﬂuids. We would like to thank Zhi-Gao Huang, Xiu-Min Lin and Joachim Brand for useful discussions. RL was sup- ported by the NBPRC under Grant No. 2011CBA00200, the NSFC under Grants No. 11274064, and by NCET. OF was supported by the Marsden Fund Council from Government funding (contract No. MAU1205), adminis- trated by the Royal Society of New Zealand. [1] D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82, 1959 (2010), M. Z. Hasan and C. L. Kane, ibid. 82, 3045 (2010); X. L. Qi and S. C. Zhang, ibid. 83, 1057 (2011). [2] V. Mourik, K. Zuo, S. M. Frlov, S. R. Plissard, E.P.A.M.Bakkers, and L. P. Kouwenhoven, Science 336, 1003 (2012) [3] D. Jaksch and P. Zoller, Annals of Physics 315, 52 (2005) [4] Y.-J. Lin, R. L. Compton, A. R. Perry, W. D. Phillips, J. V. Porto, and I. B. Spielman, Phys. Rev. Lett. 102, 130401 (2009); Y.-J. Lin and R. L. Compton and K. Jim´ enez-Garcia and J. V. Porto and I. B. Spielman, Na- ture462, 628 (2009); Y.-J. Lin and K. Jim´ enez-Garcia and I. B. Spielman, Nature 471, 83 (2011). [5] M. Aidelsburger, M. Atala, S. Nascimb´ ene, S. Trotzky, Y.-A. Chen, and I. Bloch, Phys. Rev. Lett. 107, 255301 (2011) [6] J.Zhang, S.Ji, Z.Chen, L.Zhang, Z.Du,B.Yan, G.Pan,5 B. Zhao, Y. Deng, H. Zhai, S. Chen, and J. Pan, arxiv: 1201.6018 [7] P. Wang, Z.-Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H. Zhai, and J. Zhang, Phys. Rev. Lett. 109, 095301 (2012) [8] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah, W. S. Bakr, and M. W. Zwierlein, Phys. Rev. Lett. 109, 095302 (2012) [9] T. D. Stanescu, B. Anderson, and V. Galitski, Phys. Rev. A78, 023616 (2008) [10] C. Wang, C. Gao, C. M. Jian, and H. Zhai, Phys. Rev. Lett.105, 160403 (2010) [11] C.-M. Jian and H. Zhai, Phys. Rev. B 84, 060508 (2011) [12] T.-L. Ho and S. Zhang, Phys. Rev. Lett. 107, 150403 (2011) [13] Y. Li, L. P. Pitaevskii, and S. Stringari, Phys. Rev. Let t. 108,225301(2012); Y.Li, G.I.Martone, L.P.Pitaevskii, and S. Stringari, ibid. 110, 235302 (2013). [14] B. M. Anderson, G. Juzeli¯ unas, V. M. Galitski, and I. B. Spielman, Phys. Rev. Lett. 108, 235301 (2012); B. M. Anderson, I. B. Spielman and G. Juzeli¯ unas, Phys. Rev. Lett.111, 125301 (2013). [15] M. Iskin and A. L. Subasi, Phys. Rev. Lett. 107, 050402 (2011), M. Gong, S. Tewari, and C. Zhang, ibid. 107, 195303 (2011); H. Hu, L. Jiang, X.-J. Liu, and H. Pu, ibid.107, 195304 (2011); Z.-Q. Yu and H. Zhai, ibid. 107,195305 (2011); R. Liao, Y. Yi-Xiang and W-M. Liu, ibid.108, 080406(2012). [16] S. Sinha, R. Nath, and L. Santos, Phys. Rev. Lett. 107, 270401 (2011); H. Hu, B. Ramachandhran, H. Pu, and X.-J. Liu, Phys. Rev. Lett. 108,010402 (2012); T.Kawakami, T. Mizushima, M. Nitta, and K. Machida, Phys. Rev. Lett. 109,015301 (2012); Z. F. Xu, Y. Kawaguchi, L. You, and M. Ueda, Phys. Rev. A 86, 033628 (2012). [17] H. Hu and X.-J. Liu, Phys. Rev. A 85, 013619 (2012) [18] Q. Zhou and X. Cui, Phys. Rev. Lett. 110, 140407 (2013) [19] R. Barnett, S. Powell, T. Grass, M. Lewenstein, and S. DasSarma, Phys. Rev. A 85, 023615 (2012) [20] X. Cui and Q. Zhou, Phys. Rev. A 87, 031604 (2013) [21] T. Ozawa and G. Baym, Phys. Rev. Lett. 109, 025301 (2012) [22] J.-Y. Zhang, S.-C. Ji, L. Zhang, Z.-D. Du, W. Zheng, Y.- J. Deng, H.Zhai, S.Chen,andJ.-W. Panarxiv:1305.7054 [23] A. Altland and B. Simons, Condensed Matter Field The- ory(Cambridge University Press, Cambridge, 2006) [24] A. Griﬃn, Phys. Rev. B 53, 9341 (1996) [25] N. Hugenholtz and D. Pines, Phys. Rev. 116, 489 (1959) [26] J. Goldstone, Phys. Rev. 127, 2 (1962) [27] V. Popov, Functional integrals in quantum ﬁeld theory and stastical physics (Reidel, Dordrecht, 2001) [28] J. O. Anderson, Rev. Mod. Phys. 76, 599 (2004) [29] L. Pitaevskii and S. Stringari, Bose-Einstein Condensa- tion(Oxford, New York, 2003) page 17 [30] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. String ari, Rev. Mod. Phys. 71, 463 (1999) [31] C. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases (Cambridge University Press, Cambridge, 2008) [32] N. Nagaosa, Quantum Field Theory in Condensed Matter Physics (Springer, Berlin, 1999)
1812.07949v2.Manifestations_of_spin_orbit_coupling_in_a_cuprate_superconductor.pdf	arXiv:1812.07949v2 [cond-mat.supr-con] 12 Dec 2019Manifestations of spin-orbit coupling in a cuprate superco nductor Zachary M. Raines,∗Andrew A. Allocca, and Victor M. Galitski Joint Quantum Institute and Condensed Matter Theory Center , Department of Physics, University of Maryland, College Park, Maryland 20742-4111 , U.S.A. Exciting new work on Bi2212 shows the presence of non-trivia l spin-orbit coupling eﬀects as seen in spin resolved ARPES data [Gotlieb et al.,Science,362, 1271-1275 (2018)]. Motivated by these observations we consider how the picture of spin-orbit coup ling through local inversion symmetry breaking might be observed in cuprate superconductors. Fur thermore, we examine two spin-orbit driveneﬀects, thespin-Hall eﬀect andtheEdelstein eﬀect, focusing onthe details of theirrealizations within both the normal and superconducting states. I. INTRODUCTION Since their discovery three decades ago, the cuprate family of superconductors has been a focus of intense research interest.1To this day they maintain the high- est superconducting transition temperature at ambient pressure.2Despite many years of active investigation the cuprates continue to generate new discoveries and new puzzles. Discussion continues on phenomena rang- ing from the exact nature of the pairing mechanism, to the origin of the pseudogap,3–5and the various charge- ordered states now being observed.6–9 While there has been some work on the consequences of spin-orbit coupling in the cuprates, it is generally be- lieved that such eﬀects are weak10–14and they are often ignored. However, recent spin-resolved ARPES measure- ments have shown striking evidence of spin textures in the Brillouin zone.15In particular, the observed behav- ior can be explained by a model of spin-orbit coupling which is opposite on the two layers of the BSCCO unit cell. Such a model preserves the inversion symmetry of system but can still host non-trivial eﬀects arising from the spin-orbit coupling. There is precedent for supercon- ductors with such a staggered noncentrosymmetry,16but the consequences for the cuprate superconductors have not yet been investigated. It is well established both theoretically17–21and experimentally22–24that systems with spin-orbit cou- pling may display novel transport properties linking spin and chargedegrees of freedom. These are typically called spintronic eﬀects, and provide a potential means to ma- nipulate spins with charges and vice versa.25,26One of themostcommonlyconsideredoftheseeﬀectsisthe spin- Hall eﬀect, in which a net spin polarization accumulates at the boundaries of a sample in response to an electric current. As in the caseof the Hall eﬀect, the spin-Hall ef- fect can also be related to a transversecurrent associated with the accumulated quantity, although in the case of a superconductor the relation between the two viewpoints is moresubtle. Another notable phenomenon is the Edel- stein or inverse spin-galvanic eﬀect, which relates a spin polarization throughout the bulk of a sample to the ﬂow of a charge current.10In light of the observations of spin- orbit coupling in cuprate superconductors, the question naturally arises as to how such eﬀects manifest in thesematerials,particularlywithinthesuperconductingphase. The structure of this paper is as follows. In Section II we introduce the model of spin-orbit coupling in BSCCO and discuss some of its properties. In Section III we re- view the theory of superconductivity in spin-orbit cou- pled materials and construct a general Bogoliubov-de Gennes Hamiltonian describing d-wave superconductiv- ity in this model. In Sections IV and V we then use this model to calculate spintronic eﬀects in both the normal and superconducting states. In Section VI we review and discuss our results. II. MODEL In this work we use a model which was introduced to explain the experimentally observed spin-texture in Bi2212 under spin-resolved ARPES.27The model is a tight-binding description of the copper sites in a bilayer of CuO 2planes. It is given by H0=/summationdisplay kc† k/parenleftbig ξ(k)+t⊥(k)τx+λ(k)·στz/parenrightbig ck,(1) whereξ(k) includes hopping terms up to third-nearest neighbor, leading to a hole-like Fermi surface, λ(k) = λ(sinky,−sinkx,0) is the spin-orbit coupling vector, and t⊥(k) =t⊥(coskx−cosky)2is the interlayer hopping term.28,29Hereτandσrepresent two sets of Pauli ma- trices, with the τmatrices operating in the layer space and theσmatrices operating in spin space, and ckis the 4-component vector of electron annihilation operators in the spin and layerspaces. The form ofthe spin-orbitcou- pling can be ascribed to local inversion symmetry break- ing between the layers; ﬁeld eﬀects in between the layers lead to inversion symmetry breaking with opposite sense in the top and bottom layer, such that the system as a whole retains inversion symmetry. This system contains two Kramers degenerate bands with energies ǫb(k) =ξ(k)+bA(k), (2) whereb=±andA(k) =/radicalBig/vextendsingle/vextendsingleλ(k)/vextendsingle/vextendsingle2+t⊥(k)2. It should be noted that at the level of the electronic dispersion,2 FIG. 1. The Fermi surfaces for the two bands with spin aligned along the spin-orbit coupling vector λ(k). For the opposite helicity the two bands are exchanged. The spin- texture for the band parallel to the spin-orbit vector is sho wn by the arrows. spin-orbit coupling enters in the same manner as inter- layer coupling and it is diﬃcult to disentangle the two. The eigenstates of this model can be expressed as tensor products of spin and layer space states as \|b↑/angb∇acket∇ight=\|↑/angb∇acket∇ightσ⊗\|b/angb∇acket∇ightτ \|b↓/angb∇acket∇ight=\|↓/angb∇acket∇ightσ⊗τx\|b/angb∇acket∇ightτ. (3) The states \|↑/angb∇acket∇ightσand\|↓/angb∇acket∇ightσare deﬁned as the spin states pointing parallel or anti-parallel to λ(k), i.e. helicity states, while \|b=±/angb∇acket∇ightτare the states with layer pseudo- spin parallel or anti-parallel to ( t⊥,0,\|λ\|). They can be expressed as /angb∇acketleftσ\|h/angb∇acket∇ightσ=1√ 2/parenleftBig 1he−iφ(k)/parenrightBigT /angb∇acketleftτ\|+/angb∇acket∇ightτ=/parenleftbigw(k)z(k)/parenrightbigT /angb∇acketleftτ\|−/angb∇acket∇ightτ=/parenleftbig−z(k)w(k)/parenrightbigT(4) whereλcos/parenleftbig φ(k)/parenrightbig =λ(k)·ˆxand w(k), z(k) =/radicalBigg 1 2/parenleftbigg 1±λ(k) A(k)/parenrightbigg (5) which implicitly deﬁnes the change of basis ψk=ˇUck to eigenstate operators. The structure of the eigenstates leads to two Kramers degenerate Fermi surfaces, split from each other by the spin-orbit and interlayer coupling as depicted in Fig. 1. Each Kramers doublet consists of states with helical winding of the electronic spin about the Brillouin zone center in opposite senses. For the purposes of calculating response functions in Sections IV and V below it is convenient to re-expressthe Hamiltonian in the following manner. We deﬁne the matrices ˇΣ0=σzτx,ˇΣ1=σxτy, ˇΣ2=σyτy,ˇΣ3=σzτ0.(6) These matrices along with the products ˇΣ0ˇΣiare closed under the commutators /bracketleftBig ˇΣ0,ˇΣi/bracketrightBig = 0,/bracketleftBig ˇΣi,ˇΣj/bracketrightBig = 2iǫijkˇΣk(7) and generatethe algebra so(4)⊕u(1) with the ˇΣiforming ansu(2) sub-algebra. In terms of these matrices along with the the modiﬁed adjoint ¯ c≡c†ˇΣ0, the Hamiltonian is simply H0=/summationdisplay k¯ck/parenleftBig ξkˇΣ0+d(k)·ˇΣ/parenrightBig ck (8) where we have deﬁned the vector d(k)≡A(k)n(k)≡(λsinkx,λsinky,t⊥(k))T,(9) with/vextendsingle/vextendsingled(k)/vextendsingle/vextendsingle≡A(k) and unit vector n(k). For all calculations in this work we use intralayer hop- ping strengths t1= 1,t2=−0.32,t3= 0.16, inter- layer hopping strength t⊥= 0.08, and chemical potential µ=−1.18. III. BOGOLIUBOV-DE GENNES HAMILTONIAN When writing the Bogoliubov-de Gennes Hamiltonian for the superconducting state of this model, we impose several constraints in order to match empirical details of superconductivity in this system. The order parameter in cuprates is known to belong to the B1grepresentation (dx2−y2), which we enforce by hand. We additionally re- strict pairing to be between degenerate bands; pairing between bands with diﬀerent energies would either re- quire pairing at ﬁnite momentum, which is not observed, or pairing of excitations away from the Fermi surface, which is energeticallydisfavored. Finally, we impose that the system remains inversion symmetric. With these restrictions we can now write the BdG Hamiltonian for this system as HBdG=/summationdisplay kbh′Ψ† khb/bracketleftbigg ǫb(k) ∆bf(k) ∆∗ bf(k)−ǫb(k)/bracketrightbigg Ψkhb.(10) Here ∆ bis the order parameter for pairing in each band andf(k) = coskx−coskyis thed-wave form factor. The Nambu spinor Ψ =/parenleftBig ψk˜ψ† k/parenrightBigT is deﬁned in terms of the eigenstate operators ψbhassociated with the states in Eq. (3) and their time-reverse ˜ψ= ΘψΘ−1, where Θ is the time-reversal operator. We do this because this sys- tem has non-trivial properties under time-reversal owing3 to the presence of spin-orbit coupling. The notation/summationtext′ indicates that we sum overonly half the Brillouin zone to avoid double counting states that would naturally arise in this framework. We have written our Nambu spinors in this form be- cause the usual procedure of considering pairing be- tween states of opposite spin and momentum would haveunfavorableconsequencesin systemswith inversion- symmetrybreakingorspin-orbitcoupling,namelythesu- perconducting gap function would no longer transform- ing as a representation of the point group of the system. The order parameter would acquire an extra phase under group operations that could not be removed by a gauge transformation, and so provides an obstruction to clas- sifying its symmetry. By writing the BdG Hamiltonian explicitly in terms of operators and their time-reversal, however, this problem is avoided.30–32Additionally, one recovers the notion of separation into singlet and triplet gaps, where now this distinction is with respect to helic- ity instead of spin along a ﬁxed quantization axis. The global inversion symmetry of this system enforces that the gap is singlet in helicity space, analogously to the case of a system without spin-orbit coupling. Note that unlikein the caseofan inversionsymmetry-breaking superconductor, our quasiparticle bands remain doubly degenerate. The diﬀerence of the gap magnitude in the two bands depends on the strength of the interaction in the respective channels, which we do not focus on here. Onecan diagonalizethe Hamiltonian Eq.(10) asasum of normal BdG Hamiltonians, and the Bogoliubov quasi- particle dispersions are found to be Eb(k) =/radicalBig ǫb(k)2+\|∆b\|2f(k)2. (11) Because of the singlet nature of the gap, Bogoliubov quasiparticles are superpositions of a quasi-electron state andaquasi-holeinthecorrespondingtime-reversedstate. These two states will have the same spin and so as a di- rect consequence, the Bogoliubov quasiparticles inherit the spin-texture of the normal state bands. This means that near the nodes, there are gapless spin-orbit-coupled excitations. Finally, we note that the parametrization introduced in Eq. (9) can be straightforwardly extended to the BdG Hamiltonian Eq. (10), allowing the response functions to be neatly expressed in terms of the vector dand its derivatives (for more details see Appendix A). IV. SPIN-HALL EFFECT One of the most commonly studied spin transport ef- fects in spin-orbit coupled materials is the spin-Hall ef- fect (SHE), in which spin accumulates on the boundaries of a material parallel to the electrical current ﬂowing through it, with the projection of the spin being opposite on opposing boundaries, as depicted in Fig. 2. A quan- tity often considered in the context of the SHE is the FIG. 2. (Color online) A schematic depiction of the spin-Hal l eﬀect. The charge current jis split into spin-up (red) and spin-down (blue) components, which accumulate on oppos- ing boundaries. This can be interpreted as a charge current inducing a transverse spin current, jz S. spin-hall conductivity, which describes the ﬂow of a spin- current perpendicular to an applied electric current, with the projection of the carried spin being perpendicular to the plane deﬁned by the currents themselves.26Here we focus on the spin-hall conductivity as a hallmark of the SHE, but note that the two are not necessarily simply related, as will be discussed further below. In particular, we are interested in the dc intrinsic spin- Hall conductivity σz xy= lim ω→0lim q→0Πint xy(q,iωm→ω+i0+) iω,(12) written here in terms of the intrinsic contribution to the spin-Hall response, Πint xy(q,ωm) =/angbracketleftBig jz S,x(q,ωm)jy(−q,−ωm)/angbracketrightBig ,(13) jz S,x(q,ωm) =/summationdisplay k,ǫnc† k−q 21 2/braceleftbig vx(k),σz/bracerightbig ck+q 2,(14) jy(q,ωm) =e/summationdisplay k,ǫnc† k−q 2vy(k)ck+q 2.(15) Herek= (k,ǫn) stands for the momentum and Mat- subara frequency, respectively, e=−\|e\|is the electron charge,vi(k) =∂H0(k)/∂kiis the velocity operator, andc† k,ckare the electron creation and annihilation op- erators. For our analysis these operators create quasi- particles of deﬁnite spin and layer index. Equation (14) gives a common convention for the spin-current, and the superscript zindicates the polarization of the spin- current.33Explicit calculation leads to σz xy= 4e/summationdisplay k′∂ξ ∂kx/parenleftBigg ∂d ∂ky×n/parenrightBigg ·ˆz ×/bracketleftBigg/parenleftbiggǫ+ E++ǫ− E−/parenrightbiggπN E+−E− +/parenleftbiggǫ+ E+−ǫ− E−/parenrightbiggπS E++E−/bracketrightBigg (16)4 0.0 0.5 1.0 1.5 2.0 T/T c0.860.880.900.920.940.960.981.00σz xy/σz xy ;N 0.01000.01880.02750.03620.04500.05380.06250.07130.0800 FIG. 3. (Color online) The total DC spin-Hall conductivity σz xyas a function of temperature for diﬀerent values of the spin-orbit coupling strength λ. The conductivity is roughly constant in the normal phase. We therefore normalize all results by the value of the spin-hall conductivity in the nor - mal phase for the corresponding choice of λ. The spin hall conductivity however exhibits a decrease with the onset of s u- perconductivity. This can be attributed to a ﬁnite energy fo r reorienting spins associated with the formation of singlet -like Cooper pairs in the superconducting phase. In these calcula - tions, we have chosen Tc= 0.016t1. where we have deﬁnes the normal- and superﬂuid-like bubbles πN/S(k) =n(E−(k))−n(±E+(k)) E+(k)∓E−(k)(17) withnthe quasiparticle occupation function, and nand dthe vectors introduced in the parametrization Eq. (9). We have suppressed the dependence on the momentum k. Spin-orbit coupling will in general lead to a non-zero spin-Hall conductivity. There are notable exceptions, however, where the spin-Hall conductivity exactly van- ishes, and the exact conditions under which it remains ﬁnite, particularly for Rashba spin-orbit coupling, have been a subject of lively discussion.20,34–41These argu- ments for a vanishing of the spin-Hall conductivity do not hold in our model, however, and we ﬁnd a non-zero result. One of the main issues regarding the consideration of the spin-Hall conductivityis the diﬃculty in relatingit to experimental measurements of the spin-Hall eﬀect. Since spin is not a conserved quantity in a system with spin- orbit coupling, it does not obey a continuity equation, so a spin-current cannot be consistently and rigorously deﬁned; Eq. (14), which we use in our calculation, is a common choice, but is not uniquely determined. There- fore, the accumulation of spin at the boundaries of the system is not directly related to the spin-Hall conductiv- ityinthe samewaythataccumulationofchargeisrelated to electrical Hall conductivity. This can be most easily demonstrated by considering FIG. 4. (Color online) A schematic depiction of the Edelstei n eﬀect. The charge current jinduces a uniform transverse spin polarization s= ˆαEEjthroughout the bulk of the material. the transformation properties of spin and spin-current undertimereversal. AspointedoutbyRashba,42thedef- inition of the spin current used in deﬁning the spin-Hall conductivity is even under time reversal, while magneti- zation measured at sample boundaries is odd under the same operation. Consequently, there must be some addi- tional time-reversal-symmetry breaking process that re- lates spin-current to spin accumulation at sample bound- aries, which is not included in calculations of the spin- Hall conductivity. Indeed, we note that we ﬁnd a total spin-Hall conductivity, despite the fact that the system does not break global inversion symmetry. Furthermore, the sense of the spin-Hall conductivity does not depend on the sign of the spin-orbit coupling. This is a strongin- dication that the spin-Hall conductivity as traditionally calculated is not directly related to an observable. Nonetheless, we have included this calculation of the spin-Hall conductivity for completeness. For the afore- mentioned reasons it is not clear how to relate this result to a precise experimental prediction, except to note that a non-zerospin-Hall conductivity typically indicates that aspin-Halleﬀect canbe observed. Becausethe spin-orbit coupling is seen to change sense between layers, we may expectwhateverspin-Hallresponsethereisinexperiment to be layer-staggered as well. V. EDELSTEIN EFFECT Another frequently considered spintronic eﬀect is the Edelstein eﬀect, also called the inverse spin-galvanic ef- fect (ISGE). In the Edelstein eﬀect a charge current gen- erates a uniform spin polarizationthroughout the bulk of a SOC material.10Traditionally the eﬀect is discussed in the context ofapplyingan electricﬁeld andobservingthe resultant spin polarization, and such behavior has been experimentally observed.22,43 The eﬀect can typically be quantiﬁed through the dc Edelstein conductivity σEEgiven by χEE(q,iωm) =/angbracketleftbig sx(q,iωm)jy(−q,−iωm)/angbracketrightbig σEE= lim ω→0lim q→0χEE(q,iωm→ω+i0+) iω,(18)5 which deﬁnes the linear response relationship sx= σEEEy. Thisrelationis, however,problematicinthe con- text of a superconductor. As stated above, the Edelstein eﬀect is, properly, the relationship between spin polar- ization and current ﬂow. In the normal state the current is directly related to an applied electric ﬁeld through the dc electrical conductivity ji=σEi, and soσEEis an accurate proxy for the strength of the eﬀect. However, in a superconductor (or the pathological case of a metal without disorder), the electrical conductivity σis inﬁ- nite, so application of an electric ﬁeld does not result in a steady state current, and the dc Edelstein conductivity in Eq. (18) is not a meaningful quantity. Instead, in order to consider the Edelstein eﬀect in a superconductor we need to directly relate the spin po- larization and the supercurrent as sx=αEEjy, with αEEnow being the quantity of primary interest. Con- sidering the ﬁnite frequency response of the system, we can relateαEE(ω) to the Edelstein susceptibility χEE(ω) above and the electromagnetic response function Π( ω) from which one obtains the dc electrical conductivity as σ= limω→0Π(ω)/(iω). We have sx(ω) =αEE(ω)Π(ω) iωEy(ω) /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright =jy(ω)=χEE(ω) iωEy(ω) =⇒αEE(ω) =χEE(ω) Π(ω),(19) givingαEE(ω), forwhichthedclimit ω→0isﬁnite. This result will also be useful in evaluating the normal state response since our model contains no disorder eﬀects. We can understand, at a heuristic level, how the cur- rent and spin can be related through the following argu- ment. Let us consider the case where this model contains a supercurrent. The Cooper pairs then acquire ﬁnite mo- mentum Q. We can absorb this momentum by making the gauge transformation A→A+Q/(2e). To lowest order in Qthe action is shifted to S−j·Q/(2e). Com- puting the spin expectation value of the system, we ﬁnd to lowest order in Q /angb∇acketleftsx/angb∇acket∇ight=−Q 2e/angbracketleftbig sxjy/angbracketrightbig =−Q 2eχEE(0,ω→0).(20) From Ginzburg-Landau theory we have that the super- current is jy=−e mnsQ, (21) wherensis the density of superﬂuid electrons, and we assumethat quasiparticlesdonot signiﬁcantlycontribute tothecurrent,sothisapproximatelyrepresentstheentire charge current of the system. We thus have that in the superconducting state /angb∇acketleftsx/angb∇acket∇ight=mχEE 2e2nsjy. (22) So in the case of a uniform supercurrent the non- vanishing of the Edelstein susceptibility χEEimpliesa relationship between the supercurrent and spin- polarization, as expected from Eq. (19). Such a relation was studied in the case of an s-wave pairing previously.12 We note, however, that because our model is inversion symmetric, there can be no total spin polarization in the system,sx= 0. However, the layer-staggered analogs of Eqs. (19) and (22), corresponding to the response of the layer-staggered spin ˜sx≡sxτz, behave in much the same manner. This response ˜ sx=αEEjS ycan obtained be via the analog of (19), replacing the normal spin with layer- staggered spin. Some calculation allows us to ﬁnd the general formulae χ(ω→0) = 8e/summationdisplay k′A(k)∂ny(k) ∂ky ×/parenleftBig l(k)2πN(k)+p(k)2πS(k)/parenrightBig (23) and Π(ω→0) = 2e2/summationdisplay k′ × 2A(k)2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂n(k) ∂ky/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/parenleftBig l(k)2πN(k)+p(k)2πS(k)/parenrightBig +/summationdisplay b ∂2ǫb(k) ∂k2y−bA(k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂n(k) ∂ky/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 ǫb(k) 2Eb(k)tanhEb(k) 2T , (24) where we have deﬁned the coherence functions l(k)2,p(k)2= 1±ǫ+(k)ǫ−(k)+∆+∆−f2 k E+(k)E−(k),(25) nis again the unit vector introduced in Eq. (9), and πN/Sare the quantities deﬁned in Eq. (17). The cor- responding expressions for the normal state are found as the ∆ →0 limit of these. The quantity αEE, ob- tained as lim ω→0χ(ω)/Π(ω), is plotted in Fig. 5 as a function oftemperature acrossthe superconductingtran- sition. Whereas the eﬀect is only weakly dependent on temperature above Tc, in the superconducting phase the magnitude of the eﬀect can change dramatically with temperature, especially for weaker spin-orbit coupling. Theresultbecomesmorecomplicatedinthecasewhere the current is non-uniform or if the two gaps have dif- ferent phase structures. To obtain insight into this case, as well as the above linear response calculation, we now derive a Ginzburg-Landau-like generating func- tional for the layer-staggeredspin-density. We start with the Bogoliubov-deGennes (BdG) action SBdGalongwith the associated Hubbard-Stratonovich terms. The ﬁrst modiﬁcation is to give the order parameter a spatially varying phase in order to describe a supercurrent carry- ing state. The ∆ terms now connect states of momentum6 0.0 0.5 1.0 1.5 2.0 T/T c0.8250.8500.8750.9000.9250.9500.9751.000αEE/αEE ;N 0.01000.02750.04500.06250.0800 FIG. 5. (Color online) The Edelstein susceptibility αEErelat- ing the layer-staggered polarization to the transverse cur rent via ˜sx=αEEjyas depicted schematically in Fig. 4. The coef- ﬁcientαis plotted for a range of spin-orbit coupling strengths λwith each line normalized by the (roughly) constant normal state Edelstein response. There is a marked change in the magnitude of the eﬀect coinciding with the onset of super- conductivity. k+Q/2 andk−Q/2 where Qis the Cooper pair mo- mentum. Secondly we will add a source ﬁeld for layer staggered spin-density, which takes the form B¯ψ˜sxψ. In- tegrating out the Fermions we obtain a generating func- tional forthe staggeredspin density Z[∆,Q,B]such that /angb∇acketleft˜sx/angb∇acket∇ight=−Td dBlnZ\|B=0. Our next step is to approximate the generating func- tional within a Ginzburg-Landau approximation Z=e−βF≈e−βFGL(26) where FGL=/integraldisplay dr/parenleftBig αb\|∆b\|2+Kbb′D∆∗ bD∆b′ +βb\|∆b\|4+BKxy bb′∆∗ b(−iD)∆b′/parenrightBig (27) andD=∇−2ieAis the covariant derivative, with sums over repeated indices. To do so we start with the stan- dard Hubbard-Stratonovichdecoupled form of the mean- ﬁeld problem, including now the layer-staggered source term S=/summationdisplay b1 gb\|∆b\|2+SBdG[∆b]+/summationdisplay k′B¯ψ˜sxψ.(28) We then integrate out the gaussian fermionic theory to obtain S=/summationdisplay b1 gb\|∆b\|2−Trln/bracketleftBig ˆG−1 0−∆bˆV∆;b−BˆVB/bracketrightBig .(29) Performinga4thordergradientexpansionin∆andkeep- ing only to lowest order in Bwe obtain the explicit formof the Ginzburg-Landau coeﬃcients44 αb=1 gb−/summationdisplay k′f(k)2tanh/parenleftbigǫb 2T/parenrightbig ǫb βb=/summationdisplay k′f(k)4 2ǫ2 bc(ǫb) Kxy bb=b/summationdisplay k′f(k)2 ǫb/bracketleftBigg ny∂ǫb ∂ky/parenleftbigg n′′(ǫb)+c(ǫb) ǫb/parenrightbigg +∂ny ∂ky/parenleftbig c(ǫb)+ǫ+ǫ−S(ǫ+,ǫ−)/parenrightbig/bracketrightBigg Kxy b,−b=−2/summationdisplay k′f(k)2A∂ny ∂kyS(ǫ+,ǫ−) Kbb=/summationdisplay k′f(k)2 4ǫb/bracketleftBigg ∂2ǫb ∂k2xc(ǫb)+/parenleftbigg∂ǫb ∂kx/parenrightbigg2 n′′(ǫb) +2A2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂n ∂ky/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 ǫ+ǫ− ǫb−ǫ−bS(ǫ+ǫ−) Kb,−b=−1 4/summationdisplay k′f(k)2A2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂n ∂ky/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 S(ǫ+,ǫ−)(30) where c(x) =tanh/parenleftbigx 2T/parenrightbig 2x+n′(x), S(x,y) =1 xtanhx 2T−1 ytanhy 2T x2−y2,(31) nis the Fermi function, and we recall that/summationtext′indicates summation over half the Brillouin zone. The interest- ing magneto-electro eﬀects are due to the the presence of theKxyterms, sometimes called Lifshitz invariants, al- lowed by the breaking of inversion symmetry within each layer45, which arise from the Tr/bracketleftBig (ˆGˆV∆)3ˆGˆVB/bracketrightBig term of the above expansion. In our case, instead of coupling to the magnetic ﬁeld, these terms couple to the generating ﬁeld of layer-staggered spin. With this we can now see how the Edelstein eﬀect arises from Ginzburg-Landau theory. Suppose we have a uniform supercurrent in the ydirection. From the Ginzburg-Landau theory, we have that jy= 2e/summationdisplay bb′Kbb′∆∗ b∆b′Qy. (32) Recall that since Fis a generating functional, we also have /angb∇acketleft˜sx/angb∇acket∇ight=/summationdisplay bb′Kxy bb′∆∗ b∆b′Qy, (33) and we can then write /angb∇acketleft˜sx/angb∇acket∇ight=/summationtext bb′Kxy bb′∆∗ b∆b′ 2e/summationtext bb′Kbb′∆∗ b∆b′jy, (34)7 giving the Ginzburg-Landau theory equivalent of the quantityαEEcalculated above from linear response. In the more general case, the expression becomes /angb∇acketleft˜sx/angb∇acket∇ight=/summationtext bb′Kxy bb′∆∗ b∂y∆b′ 2e/summationtext bb′Kbb′∆∗ b∂y∆b′jy, (35) Similar eﬀects have been derived from the Ginzburg- Landau free energies of inversion-symmetry breaking superconductors12,46, but the layer-staggered polariza- tion predicted here is novel in the cuprates. One might expect from the above that since an ana- log of the ISGE exists in bilayer cuprates, so might an analog of the spin Galvanic eﬀect, which would allow a supercurrent to be driven by application of a static mag- netic ﬁeld. In inversion asymmetric systems such behav- ior is generally preempted by the transition to a helical superconducting phase with zero net supercurrent.47–50 It remains to be investigated whether similar reasoning rules out the spin Galvanic eﬀect in this system. VI. DISCUSSION AND CONCLUSIONS In this work we have considered a model of spin-orbit coupling the superconducting state of a bilayer cuprate superconductor. We have shown that the inversion- symmetry-preserving spin-orbit coupling posited to be present in Bi221215should lead to non-trivial layer po- larized spin-orbit eﬀects. In particular, wecalculate the layerpolarizedspin-Hall conductivity, which was found to be non-trivial. While this quantity cannot be directly related to a measurable quantity, such as the accumulation of spin at sample boundaries, we nonetheless expect that a layer-staggered spin-Hall eﬀect should be present. More interestingly, the presence of the new coupling term leads to a layer-staggered analog of the Edelstein (or inverse spin-galvanic) eﬀect, an in-plane spin polar- ization in the presence of an applied supercurrent. This should be visible through optical methods such as Fara- day rotation22,51or by measuring the degree of circular polarization in photoluminescence.43Furthermore, one could attempt ARPES measurements in a supercurrent- carrying state to directly see that canting of the in-plane spins due to this eﬀect.52 There are still interesting eﬀects to consider beyond what we have looked at in this work. In particular, spin- resolved ARPES observes a non-trivial variation of spin texture within the Brillouin zone, most notably including a reversal of the spin texture as a function from the zone center.15This suggests a more complicated form of the spin orbit coupling which could lead to further eﬀects. Additionally, for dx2−y2superconductivity a gradient in thed-wave order parameter can admix an s-wave pair- ing term through a coupling of the gradients.53–55Re- gardless, the presence of spin-orbit coupling in BSCCO should lead to the presence of a multitude of fascinatingspin-orbit driven eﬀects, including the spin-Hall eﬀect, Edelstein eﬀect, and more. ACKNOWLEDGMENTS We would like to thank A. Lanzara, K. Gotlieb, C.-Y Lin, M. Serbyn, I. Appelbaum, V. Yakovenko, and V. Stanev for enlightening discussions. This work was sup- ported by DOE-BES (DESC0001911) and the Simons Foundation (V.G. and A.A.) and NSF DMR-1613029 (Z.R.). Appendix A: Parametrization of the BdG Hamiltonian For purposes of calculation, it is convenient to parametrize the BdG Hamiltonian, and associated Nambu Green’s function in terms of the vector dand ˇΣ matrices introduced in Eq. (9). To do so we note that the matrices ˇP±≡/parenleftBig ˇΣ0±n·ˇΣ/parenrightBig /2 satisfy the the rela- tion ˇPbn·ˇΣˇPb′=bδbb′ˇPb, (A1) meaning that they act as projectors on the degenerate eigenspaces of the normal state Hamiltonian, with the somewhat non-standard properties ˇPbˇPb′=ˇΣ0ˇPbδbb′, n·ˇΣˇPb=bˇΣ0ˇPb.(A2) Using these matrices we can compactly express the BdG Hamiltonian HBdG=/summationdisplay k′¯Ψk/summationdisplay b/bracketleftbig ǫb(k)ρ3+∆bf(k)ρ1/bracketrightbigˇPbΨk(A3) where we have introduced the adjoint Nambu spinor ¯Ψ = Ψ†ˇΣ0and theρiare Pauli matrices in Nambu space. The corresponding Nambu Green’s function is found, completely analogous to the usual expresion, to be ˇG(iǫn,k) =/summationdisplay bˇPbiǫnρ0+ǫb(k)ρ3+∆bf(k)ρ1 (iǫn)2−E2 b(A4) with the energies Ebas deﬁned in Eq. (11). Appendix B: Spin Susceptibility and Knight Shift As the presence of spin-orbit coupling induces a triplet component to the Gor’kov anomalous Green’s function one might wonder why this is not in general seen in ex- perimental signatures such as the Knight shift, where the observed behavior is seen to be consistent with sin- glet pairing.56,57In general, the telltale sign of triplet8 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 T/T c0.00.20.40.60.81.0χij(T)/χN xx xx yy zzxy FIG. 6. (Color online) The components of the static spin sus- ceptiblity χij(0,0) as a function of temperature for the layer spin-orbit coupled model. The diagonal in-plane component s have a noticeable decrease below Tcbut do not go exactly to 0 at zero temperature. This behavior is consistent with the de - crease normally attributed to spin-singlet superconducti vity in the cuprates.pairing in the Knight shift is that the spin-susceptibility remains constant across the superconducting transition. On the other hand, for the case of singlet pairing there is a noticeable decrease.58,59This is, however, consistent with this model as the singlet component of the order pa- rameter still leads to qualitative behavior similar to the typical singlet case i.e. the Knight shift should rapidly decrease below Tc. However, unlike the pure singlet case as can be seen in Fig. 6, the spin-susceptibility does not go exactly to zero at T= 0, a fact which was already appreciated by Gor’kov and Rashba in 2001.60 ∗raineszm@gmail.com 1J. G. Bednorz and K. A. Mller, Z. Fr Phys. B Condens. Matter 64, 189 (1986). 2A. Schilling, M. Cantoni, J. D. Guo, and H. R. Ott, Nature363, 56 (1993). 3P. A. Lee, Phys. Rev. X 4, 031017 (2014). 4S.Badoux, W. Tabis, F. Lalibert, G.Grissonnanche, B. Vi- gnolle, D. Vignolles, J. Bard, D. A. Bonn, W. N. Hardy, R. Liang, N. Doiron-Leyraud, L. Taillefer, and C. Proust, Nature531, 210 (2016). 5S. Chatterjee and S. Sachdev, Phys. Rev. B 94, 205117 (2016). 6J. Chang, E. Blackburn, A. T. Holmes, N. B. Christensen, J. Larsen, J. Mesot, R. Liang, D. A. Bonn, W. N. Hardy, A. Watenphul, M. V. Zimmermann, E. M. Forgan, and S. M. Hayden, Nat Phys 8, 871 (2012). 7G. Ghiringhelli, M. Le Tacon, M. Minola, S. Blanco- Canosa, C. Mazzoli, N. B. Brookes, G. M. De Luca, A. Frano, D. G. Hawthorn, F. He, T. Loew, M. M. Sala, D. C. Peets, M. Salluzzo, E. Schierle, R. Sutarto, G. A. Sawatzky, E. Weschke, B. Keimer, and L. Braicovich, Science337, 821 (2012). 8K. Fujita, M. H. Hamidian, S. D. Edkins, C. K. Kim, Y. Kohsaka, M. Azuma, M. Takano, H. Tak- agi, H. Eisaki, S.-i. Uchida, A. Allais, M. J. Lawler, E.-A. Kim, S. Sachdev, and J. C. S. Davis, Proc. Natl. Acad. Sci. 111, E3026 (2014). 9R. Comin, R. Sutarto, F. He, E. H. da Silva Neto, L. Chauviere, A. Frao, R. Liang, W. N. Hardy, D. A. Bonn, Y. Yoshida, H. Eisaki, A. J. Achkar, D. G. Hawthorn, B. Keimer, G. A. Sawatzky, and A. Damas- celli, Nat Mater 14, 796 (2015). 10V.M.Edelstein,Solid State Communications 73, 233 (1990). 11W. Koshibae, Y. Ohta, and S. Maekawa, Phys. Rev. B 47, 3391 (1993).12V. M. Edelstein, Phys. Rev. Lett. 75, 2004 (1995). 13N. Harrison, B. J. Ramshaw, and A. Shekhter, Sci Rep 5(2015), 10.1038/srep10914. 14C. Wu, J. Zaanen, and S.-C. Zhang, Phys. Rev. Lett. 95, 247007 (2005). 15K. Gotlieb, C.-Y. Lin, M. Serbyn, W. Zhang, C. L. Small- wood, C. Jozwiak, H. Eisaki, Z. Hussain, A. Vishwanath, and A. Lanzara, Science 362, 12711275 (2018). 16M. Sigrist, D. F. Agterberg, M. H. Fischer, J. Goryo, F. Loder, S.-H. Rhim, D. Maruyama, Y. Yanase, T. Yoshida, and S. J. Youn, J. Phys. Soc. Jpn. 83, 061014 (2014). 17M. Dyakonov and V. Perel, Physics Letters A 35, 459 (1971). 18J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999). 19S. Murakami, N. Nagaosa, and S.- C. Zhang, Science 301, 1348 (2003), http://science.sciencemag.org/content/301/5638/1348 .full.pdf. 20J. Sinova, D. Culcer, Q. Niu, N. A. Sinit- syn, T. Jungwirth, and A. H. MacDonald, Phys. Rev. Lett. 92(2004), 10.1103/PhysRevLett.92.126603. 21V. V. Bel’kov and S. D. Ganichev, Semiconductor Science and Technology 23, 114003 (2008). 22Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Science 306, 1910 (2004), http://science.sciencemag.org/content/306/5703/1910 .full.pdf. 23J. Wunderlich, B. Kaestner, J. Sinova, and T. Jungwirth, Phys. Rev. Lett. 94, 047204 (2005). 24H. Zhao, E. J. Loren, H. M. van Driel, and A. L. Smirl, Phys. Rev. Lett. 96, 246601 (2006). 25A.Hoﬀmann,IEEE Transactions on Magnetics 49, 5172 (2013). 26J. Sinova, S. O. Valenzuela, J. Wunderlich, C. Back, and T. Jungwirth, Rev. Mod. Phys. 87, 1213 (2015). 27See the Supplemental material for Ref. 15.9 28T. Xiang and W. N. Hardy, Phys. Rev. B 63, 024506 (2000). 29R. S. Markiewicz, S. Sahrakorpi, M. Lindroos, H. Lin, and A. Bansil, Phys. Rev. B 72, 054519 (2005). 30I. A. Sergienko and S. H. Curnoe, Phys. Rev. B 70, 214510 (2004). 31K. V. Samokhin, Phys. Rev. B 92, 174517 (2015). 32X. Gong, M. Kargarian, A. Stern, D. Yue, H. Zhou, X. Jin, V. M. Galitski, V. M. Yakovenko, and J. Xia, Sci. Adv. 3, e1602579 (2017). 33The presence of spin-orbit coupling means that there is no longer a natural deﬁnition of spin-current in terms of a conserved Noether charge. 34J.-i. Inoue, G. E. W. Bauer, and L. W. Molenkamp, Phys. Rev. B 70, 041303 (2004). 35E. G. Mishchenko, A. V. Shytov, and B. I. Halperin, Phys. Rev. Lett. 93, 226602 (2004). 36O. V. Dimitrova, ArXivcond-Mat0405339 (2004), arXiv:cond-mat/0405339. 37O. V. Dimitrova, Phys. Rev. B 71, 245327 (2005). 38P. L. Krotkov and S. Das Sarma, Phys. Rev. B 73, 195307 (2006). 39W.-K. Tse and S. Das Sarma, Phys. Rev. Lett. 96, 056601 (2006). 40V. M. Galitski, A. A. Burkov, and S. Das Sarma, Phys. Rev. B 74, 115331 (2006). 41O.Bleibaum, Phys. Rev. B 74(2006), 10.1103/PhysRevB.74.113309. 42E.I.Rashba,Phys. E Low-Dimens. Syst. Nanostructures 34, 31 (2006). 43A. Y. Silov, P. A. Blajnov, J. H. Wolter, R. Hey, K. H. Ploog, and N. S. Averkiev, Appl. Phys. Lett. 85, 5929 (2004). 44Wenotethatthecoeﬃcientsobtainedherediﬀerfromwhat one would obtain by expanding Eqs. (23) and (24). This is due to diﬀerent order of limits between the two expres- sions. Here we take the limit ω→0 before q→0 reﬂecting the fact that we are considering a supercurrent carrying system in thermal equilibrium. In contrast, the general ex-pressions derived above consider the case where the system is responding to the presence of a supercurrent which has been “switched on”. The general expressions above can be derived in the opposite limit without much extra complica- tion leading toadditional intra-bandterms in theresponse . 45V. P. Mineev and K. V. Samokhin, Phys. Rev. B 78, 144503 (2008). 46F. Konschelle, I. V. Tokatly, and F. S. Bergeret, Phys. Rev. B 92, 125443 (2015). 47D.F.Agterberg,Physica C: Superconductivity Proceedings of the 3rd Polish-US Workshop on Superconductivity and Magn etism of Advanced Materials, 387, 13 (2003). 48O. Dimitrova and M. V. Feigelman, Phys. Rev. B 76, 014522 (2007). 49D.F.Agterberg,in Non-Centrosymmetric Superconductors , Vol. 847, edited by E. Bauer and M. Sigrist (Springer Berlin Heidelberg, Berlin, Heidelberg, 2012) pp. 155–170. 50M. Smidman, M. B. Salamon, H. Q. Yuan, and D. F. Agterberg, Rep. Prog. Phys. 80, 036501 (2017). 51Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Phys. Rev. Lett. 93, 176601 (2004). 52G. Boyd, S. Takei, and V. Galitski, Solid State Commu- nications 189, 6367 (2014). 53Q. P. Li, B. E. C. Koltenbah, and R. Joynt, Phys. Rev. B 48, 437 (1993). 54A. J. Berlinsky, A. L. Fetter, M. Franz, C. Kallin, and P. I. Soininen, Phys. Rev. Lett. 75, 2200 (1995). 55D. L. Feder and C. Kallin, Phys. Rev. B 55, 559 (1997). 56M. Takigawa, P. C. Hammel, R. H. Heﬀner, and Z. Fisk, Physical Review B 39, 7371 (1989). 57S. E. Barrett, D. J. Durand, C. H. Pennington, C. P. Slichter, T. A. Friedmann, J. P. Rice, and D. M. Gins- berg, Physical Review B 41, 6283 (1990). 58K. Yosida, Physical Review 110, 769 (1958). 59J. R. Schrieﬀer, Theory of Superconductivity , Advanced book classics (Advanced Book Program, Perseus Books, Reading, Mass, 1999). 60L. P. Gor’kov and E. I. Rashba, Phys. Rev. Lett. 87, 037004 (2001).
2106.01046v1.Enhanced_spin_orbit_coupling_and_orbital_moment_in_ferromagnets_by_electron_correlations.pdf	arXiv:2106.01046v1 [cond-mat.str-el] 2 Jun 2021Enhanced spin-orbit coupling and orbital moment in ferroma gnets by electron correlations Ze Liu,1Jing-Yang You,2Bo Gu,1,3,∗Sadamichi Maekawa,4,1and Gang Su1,3,5,† 1Kavli Institute for Theoretical Sciences, and CAS Center fo r Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China 2Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117551 3Physical Science Laboratory, Huairou National Comprehens ive Science Center, Beijing 101400, China 4Center for Emergent Matter Science, RIKEN, Walo 351-0198, J apan 5School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China In atomic physics, the Hund rule says that the largest spin an d orbital state is realized due to the interplay of the spin-orbit coupling (SOC) and the Coulo mb interactions. Here, we show that in ferromagnetic solids the eﬀective SOC and the orbital mag netic moment can be dramatically enhanced bya factor of 1 /[1−(2U′−U−JH)ρ0], whereUandU′are the on-site Coulomb interaction within the same oribtals and between diﬀerent orbitals, res pectively, JHis the Hund coupling, and ρ0is the average density of states. This factor is obtained by u sing the two-orbital as well as ﬁve-orbital Hubbard models with SOC. We also ﬁnd that the spi n polarization is more favorable than the orbital polarization, being consistent with exper imental observations. This present work provides a fundamental basis for understanding the enhance ments of SOC and orbital moment by Coulomb interactions in ferromagnets, which would have wid e applications in spintronics. Introduction —The Hund’s rule in atomic physics says that the state with both the largest spin moment and the largest orbital moment is realized in an atom, required by the minimum of the Coulomb repulsive energy. The similar picture was obtained in the magnetic impurity systems. In the Anderson impurity model, the spin mag- netic moment of impurities is developed due to the large on-site Coulomb interaction U [ 1]. In 1964, the extended Anderson impurity model with degenerate orbitals has been studied, where the role of Uand the Hund coupling JHhas been addressed [ 2,3]. Forty years ago, Yafet also studied the Anderson impurity model within Hartree- Fock approximation and found that the on-site Coulomb interaction of impurities can enhance the eﬀective SOC in the spin-ﬂip cross section [ 4]. Later, Fert and Jaoul applied this result to study the anomalousHall eﬀect due to magnetic impurities [ 5]. The relation between the on- site Coulomb interaction Uand the eﬀective spin-orbit coupling (SOC) in magnetic impurity systems has also been discussed by the density functional theory (DFT) calculations [ 6] and the quantum Monte Carlo simula- tions [7]. In these years, one of the fast developing areas in con- densed matter physics is spintronics [ 8,9]. It aims to manipulate the spin rather than the charge degree of freedom of electrons to design the next-generation elec- tronic devices with small size, faster calculating abil- ity, and lower energy consumption. SOC, as one of the key ingredients in spintronics, is related to many sig- niﬁcant physical phenomena and novel matter [ 10]. In addition to the magnetic anisotropy [ 8,11], SOC plays an important role in the phenomena such as anomalous Hall eﬀect [ 12], spin Hall eﬀect associated with the spin- charge conversion [ 13–16], topological insulators [ 17–21], skymions [ 22–24] and so on. To design better spintronicdevices, a large SOC is usually required. As SOC is a relativistic eﬀect in quantum mechanics, it is often small in many materials. A key issue is what factors can aﬀect the magnitude of the SOC in solids. On the other hand, the orbital moment in the FeCo nanogranules was experimentally shown to be about three times larger than that in bulk FeCo, as a result of the enhanced Coulomb interaction in the FeCo/insulator interface [ 25], because the Coulomb interaction in the FeCo/insulator interface is expected to be larger than that in the ferromagnetic FeCo bulk. In addition, a large Coulomb interaction up to 10 eV was discussed in Fe thin ﬁlms in the experiment [ 26]. The spin polarization in the Hubbard model with Rashba SOC can also be enhanced by the on-site Coulomb interaction U [ 27]. Re- cently, in the two-dimensional magnetic topological insu- lators PdBr 3and PtBr 3, the DFT calculations show that the band gap and the SOC can be strongly enhanced by the Coulomb interaction [ 28]. Inspired by recent experimental and numerical results on the enhanced SOC due to the Coulomb interaction in strongly correlated electronic systems, here we develop a theory on the relation between SOC and Coulomb inter- action in ferromagnets. By a two-orbital Hubbard model with SOC, we ﬁnd that the eﬀective SOC and orbital magnetic moment in ferromagnets can be enhanced by a factorof1 /[1−(2U′−U−JH)ρ0], whereUandU′arethe on-site Coulomb interaction within the same oribtalsand between diﬀerent orbitals, respectively, JHis the Hund coupling, and ρ0is the average density of states. The same factor has also been obtained for the ﬁve-orbital Hubbard model with degenerate bands. Our theory can be viewed as the realization of Hund’s rule in ferromag- nets. Two-orbital Hubbard model with SOC —Let us consider2 a two-orbital Hubbard model, where only a pair of or- bitals with opposite orbital magnetic quantum numbers m(-1 and 1, or -2 and 2) are considered. Thus, the Hamiltonian can be written as H=/summationdisplay k,m,σǫkmσnkmσ+U/summationdisplay i,mnim↑nim↓ +U′/summationdisplay i,σ,σ′nimσni¯mσ′−JH/summationdisplay i,σnimσni¯mσ,(1) whereǫkmσis the energy of electron with wave vector k, orbitalm, and spin σ(↑,↓) [29],UandU′are the on- site Cuolomb repulsion within the orbital mand between diﬀerent orbitals mandm′, respectively, JHis the Hund coupling, and nkmσ(nimσ)representstheparticlenumber with wave vector k(site index i), orbital mand spin σ. For simplicity, we consider four degenerate energy bands, which are lifted by an external magnetic ﬁeld hand the Ising-type SOC [ 5]: ǫkmσ=ǫk−σµBh−1 2σλsom, (2) whereλsois the SOC constant, ǫkis the electron en- ergy without external magnetic ﬁled and SOC. Using the Hartree-Fock approximation, we have nimσnim′σ′≈ /angbracketleftnimσ/angbracketrightnim′σ′+/angbracketleftnim′σ′/angbracketrightnimσ−/angbracketleftnimσ/angbracketright/angbracketleftnim′σ′/angbracketright. Assum- ing the system is homogeneous, the occupation number nimσisindependentoflatticesite i:/angbracketleftnimσ/angbracketright ≈ /angbracketleftnmσ/angbracketright, and through Fourier transformation/summationtext inimσ=/summationtext knkmσ, the Halmiltonian in Eq.( 1) can be diagonalized as: H≈/summationdisplay k,m,σ˜ǫkmσnkmσ, (3) with ˜ǫkmσ=ǫk−σµBh−1 2σλsom+U/angbracketleftnm¯σ/angbracketright +U′(/angbracketleftn¯mσ/angbracketright+/angbracketleftn¯m¯σ/angbracketright)−JH/angbracketleftn¯mσ/angbracketright.(4) We deﬁne the spin polarization per site as sz= µB(/angbracketleftnm↑/angbracketright−/angbracketleftnm↓/angbracketright+/angbracketleftn¯m↑/angbracketright−/angbracketleftn¯m↓/angbracketright), and the orbital po- larization per site as lz=mµB(/angbracketleftnm↑/angbracketright−/angbracketleftn¯m↑/angbracketright+/angbracketleftnm↓/angbracketright− /angbracketleftn¯m↓/angbracketright). Here we should remark that the so-deﬁned or- bital polarization from itinerant electrons on diﬀerent or- bitals with SOC diﬀers from the conventionalorbital mo- ments of atoms that are usually quenched owing to the presence of the crystal ﬁelds in transition metal ferro- magnets. Introduce the particle numbers of the parallel (np) and antiparallel ( nap) states of the spin σand or- bitalm:np=/angbracketleftnm↑/angbracketright+/angbracketleftn¯m↓/angbracketright,nap=/angbracketleftn¯m↑/angbracketright+/angbracketleftnm↓/angbracketright. Then the energy ˜Ekmσcan be written as ˜ǫkmσ=¯ǫ−σµB/parenleftbigg h+U+JH 4µ2 Bsz/parenrightbigg −1 2m/parenleftbigg σλso−U−2U′+JH 2µBm2lz/parenrightbigg .(5)When spin σand orbital mare antiparallel (parallel) the energy ¯ǫ= (ǫk+1 2Unap(p)+1 2U′nap(p)+1 2U′np(ap)− 1 2JHnap(ap)). Spin polarization — It is noted that without external magneticﬁeld handSOC λso, thefourenergybandswith spinσ(↑and↓) and orbital m(for example 1 and −1) aredegenerate,andtheoccupationnumbers nap=np. In termsofthe translationalsymmetryofthelatticesystem: /angbracketleftnmσ/angbracketright=1 N/summationtext i/angbracketleftnimσ/angbracketright=1 N/summationtext k/angbracketleftnkmσ/angbracketright=1 N/summationtext kf(˜ǫkmσ), wherefistheFermidistributionfunction, thespinpolar- izationcanbewrittenas sz=µB N/summationtext k[f(˜ǫkm↑)−f(˜ǫkm↓)+ f(˜ǫk¯m↑)−f(˜ǫk¯m↓)]. For the system with a paramagnetic (PM) state ( h= 0),f(˜ǫkmσ) can be expanded according toh, which is a small value compared to Fermi energy, andnap=np,sz=µB/summationtext k[f(˜ǫPM,km ↑)−f(˜ǫPM,km ↓) + f(˜ǫPM,k¯m↑)−f(˜ǫPM,k¯m↓)] = 0. Up to the linear order of h, the spin polarization becomes sz=4µ2 Bρ0 1−(U+JH)ρ0h, (6) whereρ0=1 4/integraltext∞ 0[−∂f(E) ∂E][ρm↑(E) +ρ¯m↑(E) +ρm↓(E) + ρ¯m↓(E)]dEis the average density of states of the four energy bands. The instability condition of the spin po- larization is (U+JH)ρ0>1. (7) This condition canbe takenas an extensionofStoner cri- terion in the presence of SOC in itinerant ferromagnets. Orbital polarization — Similarly, the orbital polariza- tion can be expressed as lz=µBm(/angbracketleftnm↑/angbracketright − /angbracketleftn¯m↑/angbracketright+ /angbracketleftnm↓/angbracketright−/angbracketleftn¯m↓/angbracketright) =µBm N/summationtext k[f(˜ǫkm↑)−f(˜ǫk¯m↑)+f(˜ǫkm↓)− f(˜ǫk¯m↓)]. Forthe ferromagnetic(FM) state, the SOCcan be regarded as a small value [ 5], sof(˜ǫkmσ) can be ex- panded according to λso, and when λso= 0,nap=np, the zero-order term is zero. To the linear order of λso, the orbital polarization gives lz=m2µBρs 1−(2U′−U−JH)ρ0λso, (8) whereρs=1 2/integraltext∞ 0[−∂f(E) ∂E][ρm↑(E) +ρ¯m↑(E)−ρm↓(E)− ρ¯m↓(E)]dEisthe averagespin polarizeddensityofstates. Then Eq.( 8) can be rewritten as lz=µBm2ρsλeﬀ so, where the eﬀective SOC λeﬀ sois λeﬀ so=λso 1−(2U′−U−JH)ρ0. (9) Onemaynotethattheorbitalpolarizationdiscussedhere [Eq. (8)] is totally induced by the SOC, which can be enhanced by increasing U′or decreasing UandJH, we willdiscussthisindetail. IntheabsenceoftheSOC,such an orbital polarization is absent according to Eq. ( 8). The instability condition of orbital polarization would be: (2U′−U−JH)ρ0>1. (10)3 TABLE I. Comparison of the theoretical results among the And erson impurity model, the one-orbital Hubbard model (Stone r model), and our two- and ﬁve-orbital Hubbard models with the spin-orbit coupling (SOC). szandlzare the spin and orbital polarization, respectively. The instability conditions ( IC) ofszandlzin these models are listed. λeﬀ sois the eﬀective SOC aﬀected by atomic SOC λso, the electron correlations U,U′andJH, and the electron density of state ρ. The equations of ﬁve-orbital Hubbard model can be found in the Supplementary Information. Anderson impurity modelOne-orbital Hubbard model (Stoner)Two-orbital Hubbard model with SOC( m=±1 orm=±2)Five-orbital Hubbard model with SOC ( m= 0,±1,±2) sz –2µ2 Bρ(EF) 1−Uρ(EF)h[30]4µ2 Bρ0 1−(U+JH)ρ0h[Eq.(6)]10µ2 Bρ0 1−(U+4JH)ρ0h[Eq.(63)] lz – –m2µBρs 1−(2U′−U−JH)ρ0λso[Eq.(8)]µB(ρ1s+4ρ2s) 1−(2U′−U−JH)ρ0λso[Eq.(78)] IC ofsz(U+4JH)ρ(EF)>1[2,3]Uρ(EF)>1 [30] ( U+JH)ρ0>1 [Eq.(7)] ( U+4JH)ρ0>1 [Eq.(65)] IC oflz – – (2 U′−U−JH)ρ0>1 [Eq.(10)] λeﬀ soλat 1−(U−JH)ρ(EF)[4] –λso 1−(2U′−U−JH)ρ0[Eq.(9)] The detailed derivation is given in the Supplementary Information. Five-orbital Hubbard model with SOC —Ourtheorycan beeasilyextendedtotheﬁve-orbitalHubbardmodelwith degenerate bands, and the detailed derivation is given in theSupplementaryInformation. Fortheﬁve-orbitalcase, the instability condition of the spin polarization becomes as (U+4JH)ρ0>1. The same expression has been ob- tained for the presence of localized spin moment in the Anderson impurity model with degenerate orbitals [ 2,3]. The obtained instability condition of the orbital polar- ization is (2 U′−U−JH)ρ0>1, which is the same as Eq.(10) for the two-orbital case. In the ﬁve-orbital case, the eﬀective SOC and the orbital magnetic moment can alsobe enhanced by a factorof1 /[(2U′−U−JH)ρ0], that isthesameenhancementfactorasin thetwo-orbitalcase. Discussion —The comparison between our theory, the Stoner model and the Anderson impurity model is shown in Table I. It is interesting to note that the instabil- ity conditions of szbetween our ﬁve-orbital Hubbard model with SOC and the Anderson impurity model are the same, while the obtained eﬀective SOC λeﬀ sobetween the two models are diﬀerent. Comparing Eqs.( 7) and (10), which are the spin and orbital instability condi- tions of the two-orbital model in Table I, one may note that the condition of the orbital spontaneous polariza- tion is more stringent than that of the spin spontaneous polarization. The phase diagram of the spin and orbital spontaneous polarizations as a function of the inverse of average density of state 1 /ρ0and the Coulomb interac- tion U obtained with Eqs. ( 7) and (10) is depicted in Fig.1. Considering the relation U=U′+2JHand the reasonable values of U= 4∼7 eV in the 3d transitional metal oxides [ 31], for 3d electrons, JH= 1,U′= 5, U= 7 eV are a set of reasonable values, for simplicity we keep the ratio U:U′:JH= 7 : 5 : 1 in Eq.( 9), and the shaded area with blue (red) solid lines indicatesUnpolarized Stoner 1ρ/g3674 (eV)(eV) Spin polarization (Eq. 7)Orbital polarization (Eq.10) FIG. 1. The phase diagram of spin and orbital spontaneous polarization as a function of the inverse average density of states and the Coulomb interaction U. The shaded area with blue solid lines represents the spin spontaneous polarizat ion determined by Eq.( 7). The shaded area with red solid lines representsthe orbital spontaneous polarization determin edby Eq.(10). The black dotted line indicates the Stoner criterion of the spin spontaneous polarization, which is obtained by t he single orbital Hubbard model. the spin (orbital) spontaneous polarization. The Stoner criterion of the spin spontaneous polarization based on the single orbital Hubbard model is also plotted in Fig. 1for a comparison. The results show that the area of or- bital spontaneous polarization is enclosed in the area of spin spontaneous polarization. In other words, it is more stringent to have the orbital spontaneous polarization, which is consistent with the fact that the orbital sponta- neous polarization is rarely observed in experiments. The relation between the electron correlations U,U′ andJHand the spin polarization szin Eq. (6) and the4 orbital polarization lzin Eq. (8) can be understood by theenergytermsinEq. ( 4). Foragivenstatewithorbital mand spin ↑, according to the principle of minimum energy, in order to compensate the Coulomb interaction U, the occupancy number /angbracketleftnm↓/angbracketrightwill decrease, which will increaseszand decrease lz. To compensate the Coulomb interaction U′, the occupancy numbers /angbracketleftn¯m↑/angbracketrightand/angbracketleftn¯m↓/angbracketright will equally decrease, which will have no eﬀect on szand increase lz. To compensate the Hund interaction JH, the occupancy number /angbracketleftn¯m↑/angbracketrightwill increase, which will increase szand decrease lz. The above argument by Eq. (4) is consistent with the obtained enhancement factor 1/[1−(2U′−U−JH)]ρ0forlzin Eq. ( 8), where U andJHwill decrease lzandU′will increase lz. The same argument by Eq. ( 4) is also consistent with the calculated enhancement factor 1 /[1−(U+JH)]ρ0forsz in Eq. (6), where UandJHwill increase szandU′has no eﬀect on sz. Application —Equation ( 9) shows that Coulomb inter- actions can enhance the eﬀective SOC. Recently, for magnetic topological insulators PdBr 3and PtBr 3, it is found that the energy gap increases with the increase of Coulomb interaction [ 28]. In these topological materi- als, the energy gap is opened due to the SOC, and the enhancement of SOC by the Coulomb interaction can be naturally obtained by Eq.( 9). In addition, since the magnetic optical Kerr eﬀect (MOKE) and the Faraday eﬀect aredetermined by the SOC, the experimentally ob- servedlargeFaradayeﬀectinmetalﬂuoridenanogranular ﬁlms [32] and the predicted large MOKE at Fe/insulator interfaces [ 33] can also be understood by the eﬀect of Coulomb interaction as revealed by Eq.( 9), because the Coulomb interaction becomes important with the de- creased screening eﬀect at the interfaces. It is noted that the Hubbard model with SOC has been extensively stud- ied, where the SOC can induce the Dzyaloshinski-Moriya interaction and the pesudo-dipolar interaction [ 29,34– 45]. The orbital magnetic moment can also be enhanced by Coulomb interaction, as given by Eq.( 8). In the re- cent experiment, the orbital magnetic moment in FeCo nanogranules is observed to be three times larger than that of FeCo bulk [ 25]. Using Eq.( 8), the ratio of the or- bital magnetic moment without the Coulomb interaction lz(U= 0) to the orbital magnetic moment with ﬁnite U lz(U) can be approximately written as: lz(U) lz(0)=1 1−(2U′−U−JH)ρ0. (11) As the Coulomb interactions can be approximately ne- glected in the metal bulk, and become important in the metal/insulator interfaces, lz(0) andlz(U) can represent the orbital moment of FeCo bulk and nanogranules, re- spectively. Toreproducethe experimentalratiooforbital magnetic moment between FeCo nanogranules and bulk wemaytake, lz(U)/lz(0) = 3, theﬁtted value U= 4.4eVis obtained by Eq.( 11), which is reasonable for 3d tran- sition metals. In the ﬁtting, we use the approximation in the DFT calculation, to keep JH= 0 eV, U=U′. ρ0= 0.15 (1/eV) is obtained by DFT calculation for the FeCo interface, where ρ0is approximately estimated as the density of states at Fermi level. Therefore, Eq.( 11) can be used to qualitatively explain the enhancement of orbital magnetic moment for the FeCo nanogranules in the experiment. Conclusion — Atwo-orbitalHubbardmodelwithSOC, we show that the orbital polarization and the eﬀec- tive SOC in ferromagnets are enhanced by a factor of 1/[1−(2U′−U−JH)ρ0], whereUandU′are the on-site Coulomb interaction within the same orbitals and be- tweendiﬀerentorbitals,respectively, JHistheHund cou- pling, and ρ0is the average density of states. The same factor is obtained for the ﬁve-orbital Hubbard model with degenerate bands. Our theory can be viewed as the realization of Hund’s rule in ferromagnets. The the- ory can be applied to understand the enhanced band gap due to SOC in magnetic topological insulators, and the enhanced orbital magnetic moment in ferromagnetic nanogranules in a recent experiment. In addition, our results reveal that it is more stringent to have the or- bital spontaneouspolarizationthan the spin spontaneous polarization, which is consistent with experimental ob- servations. As the electronic interaction in some two- dimensional (2D) systems can be controlled experimen- tally [46], according to our theory, the enhanced SOC, spin and orbital magnetic moments are highly expected to be observed in these 2D systems. This present work not only provides a fundamental basis for understanding the enhancements of SOC in some magnetic materials, but also sheds light on how to get a large SOC through hybrid spintronic structures. The authors acknowledge Q. B. Yan, Z. G. Zhu, and Z. C. Wang for many valuable discussions. This work is supported in part by the National Key R&D Program of China (Grant No. 2018YFA0305800), the Strategic Pri- ority Research Program of the Chinese Academy of Sci- ences (Grant No. XDB28000000), the National Natural Science Foundation of China (Grant No. 11834014), and Beijing Municipal Science and Technology Commission (Grant No. Z191100007219013). B.G. is also supported by the National Natural Science Foundation of China (Grants No. Y81Z01A1A9 and No. 12074378), the Chi- nese Academy of Sciences (Grants No. Y929013EA2and No. E0EG4301X2), the University of Chinese Academy of Sciences (Grant No. 110200M208), the Strategic Pri- ority Research Program of Chinese Academy of Sciences (Grant No. XDB33000000), and the Beijing Natural Science Foundation (Grant No. Z190011). SM is sup- ported by JST CREST Grant (No. JPMJCR19J4 No. JPMJCR1874 and No. JPMJCR20C1) and JSPS KAK- ENHI (Nos. 17H02927 and 20H01865) from MEXT, Japan.5 ∗gubo@ucas.ac.cn †gsu@ucas.ac.cn [1] P. W. Anderson, Localized magnetic states in metals, Phys. Rev. 124, 41 (1961) . [2] T. Moriya, Ferro- and antiferromagnetism of transition metals and alloys, Prog. Theor. Phys. 33, 157 (1965) . [3] K. Yosida, A. Okiji, and S. Chikazumi, Mag- netic anisotropy of localized state in metals, Prog. Theor. Phys. 33, 559 (1965) . [4] Y.Yafet,Spin-orbitinducedspin-ﬂipscatteringbyalo cal moment, J. Appl. Phys. 42, 1564 (1971) . [5] A. Fert and O. Jaoul, Left-right asymmetry in the scat- tering of electrons by magnetic impurities, and a hall eﬀect,Phys. Rev. Lett. 28, 303 (1972) . [6] G.-Y. Guo, S. Maekawa, and N. Nagaosa, En- hanced spin hall eﬀect by resonant skew scat- tering in the orbital-dependent kondo eﬀect, Phys. Rev. Lett. 102, 036401 (2009) . [7] B. Gu, J.-Y. Gan, N. Bulut, T. Ziman, G.-Y. Guo, N. Na- gaosa, and S. Maekawa, Quantum renormalization of the spin hall eﬀect, Phys. Rev. Lett. 105, 086401 (2010) . [8] I.ˇZuti´ c, J. Fabian, and S. D. Sarma, Spin- tronics: Fundamentals and applications, Rev. Mod. Phys. 76, 323 (2004) . [9] S. Maekawa, ed., Concepts in Spin Electronics (Oxford University Press, 2006). [10] A. Soumyanarayanan, N. Reyren, A. Fert, and C. Panagopoulos, Emergent phenomena induced by spin–orbit coupling at surfaces and interfaces, Nature539, 509 (2016) . [11] J.-Y. You, Z. Zhang, X.-J. Dong, B. Gu, and G. Su, Two- dimensional magnetic semiconductors with room curie temperatures, Phys. Rev. Research 2, 013002 (2020) . [12] N. Nagaosa, J. Sinova, S. Onoda, A. H. Mac- Donald, and N. P. Ong, Anomalous hall eﬀect, Rev. Mod. Phys. 82, 1539 (2010) . [13] M. Dyakonov and V. Perel, Current-induced spin orientation of electrons in semiconductors, Phys. Lett. A 35, 459 (1971) . [14] J. E. Hirsch, Spin hall eﬀect, Phys. Rev. Lett. 83, 1834 (1999) . [15] Y. K. Kato, Observation of the spin hall eﬀect in semi- conductors, Science306, 1910 (2004) . [16] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, Spin hall eﬀects, Rev. Mod. Phys. 87, 1213 (2015) . [17] L. Fu, C. L. Kane, and E. J. Mele, Topological insulators in three dimensions, Phys. Rev. Lett. 98, 106803 (2007) . [18] J. E. Moore and L. Balents, Topological invari- ants of time-reversal-invariant band structures, Phys. Rev. B 75, 121306(R) (2007) . [19] M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys. 82, 3045 (2010) . [20] X.-L. Qi and S.-C. Zhang, Topological insulators and su - perconductors, Rev. Mod. Phys. 83, 1057 (2011) . [21] J.-Y. You, C. Chen, Z. Zhang, X.-L. Sheng, S. A. Yang, and G. Su, Two-dimensional weyl half- semimetal and tunable quantum anomalous hall eﬀect, Phys. Rev. B 100, 064408 (2019) . [22] S. Muhlbauer, B. Binz, F. Jonietz, C. Pﬂei- derer, A. Rosch, A. Neubauer, R. Georgii, andP. Boni, Skyrmion lattice in a chiral magnet, Science323, 915 (2009) . [23] X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han, Y. Matsui, N. Nagaosa, and Y. Tokura, Real- space observation of a two-dimensional skyrmion crystal, Nature465, 901 (2010) . [24] N. Nagaosa and Y. Tokura, Topological prop- erties and dynamics of magnetic skyrmions, Nat. Nanotechnol. 8, 899 (2013) . [25] Y. Ogata, H. Chudo, B. Gu, N. Kobayashi, M. Ono, K. Harii, M. Matsuo, E. Saitoh, and S. Maekawa, Enhanced orbital magnetic moment in FeCo nanogranules observed by barnett eﬀect, J. Magn. Magn. Mater. 442, 329 (2017) . [26] R. Gotter, A. Verna, M. Sbroscia, R. Moroni, F. Bi- sio, S. Iacobucci, F. Oﬃ, S. R. Vaidya, A. Ruocco, and G. Stefani, Unexpectedly large electron correlation measured in auger spectra of ferromagnetic iron thin ﬁlms: Orbital-selected coulomb and exchange contribu- tions,Phys. Rev. Lett. 125, 067202 (2020) . [27] J. A. Riera, Spin polarization in the hubbard model with rashba spin-orbit coupling on a ladder, Phys. Rev. B 88, 045102 (2013) . [28] J.-Y. You, Z. Zhang, B. Gu, and G. Su, Two- dimensional room-temperature ferromagnetic semi- conductors with quantum anomalous hall eﬀect, Phys. Rev. Appl 12, 024063 (2019) . [29] T.A.Kaplan,Single-bandhubbardmodelwithspin-orbi t coupling, Z. Phys. B 49, 313 (1983) . [30] Collective electron ferronmagnetism, Proc. Math. Phys. Eng. Sci. 165, 372 (1938) . [31] S. Maekawa, T. Tohyama, S. E. Barnes, S. Ishihara, W. Koshibae, and G. Khaliullin, Physics of Transition Metal Oxides (Springer Berlin Heidelberg, 2004). [32] N. Kobayashi, K. Ikeda, B. Gu, S. Takahashi, H. Masumoto, and S. Maekawa, Giant fara- day rotation in metal-ﬂuoride nanogranular ﬁlms, Sci. Rep. 8, 4978 (2018) . [33] B. Gu, S. Takahashi, and S. Maekawa, Enhanced magneto-optical kerr eﬀect at fe/insulator interfaces, Phys. Rev. B 96, 214423 (2017) . [34] D. Coﬀey, T. M. Rice, and F. C. Zhang, Dzyaloshinskii-moriya interaction in the cuprates, Phys. Rev. B 44, 10112 (1991) . [35] L. Shekhtman, O. Entin-Wohlman, and A. Aharony, Moriya’s anisotropic superexchange interaction, frus- tration, and dzyaloshinsky’s weak ferromagnetism, Phys. Rev. Lett. 69, 836 (1992) . [36] L. Shekhtman, A. Aharony, and O. Entin- Wohlman, Bond-dependent symmetric and anti- symmetric superexchange interactions in La 2CuO4, Phys. Rev. B 47, 174 (1993) . [37] N. E. Bonesteel, Theory of anisotropic superexchange i n insulating cuprates, Phys. Rev. B 47, 11302 (1993) . [38] W. Koshibae, Y. Ohta, and S. Maekawa, Electronic and magnetic structures of cuprates with spin-orbit interac- tion,Phys. Rev. B 47, 3391 (1993) . [39] W. Koshibae, Y. Ohta, and S. Maekawa, Comment on “moriya’s anisotropic superexchange interaction, frus- tration, and dzyaloshinsky’s weak ferromagnetism”, Phys. Rev. Lett. 71, 467 (1993) . [40] H. E. Vierti¨ o and N. E. Bonesteel, Interplanar couplin g andtheweakferromagnetic transitioninLa 2−xNdxCuO4,6 Phys. Rev. B 49, 6088 (1994) . [41] T. Yildirim, A. B. Harris, A. Aharony, and O. Entin- Wohlman, Anisotropic spin hamiltonians due to spin-orbit and coulomb exchange interactions, Phys. Rev. B 52, 10239 (1995) . [42] S. G. Tabrizi, A. V. Arbuznikov, and M. Kaupp, Hub- bard trimer with spin–orbit coupling: Hartree–fock so- lutions, (non)collinearity, and anisotropic spin hamilto - nian,J. Phys. Chem. A 123, 2361 (2019) . [43] A. N. Kocharian, G. W. Fernando, K. Fang, K. Palandage, and A. V. Balatsky, Spin-orbit coupling, electron transport and pairing insta-bilities in two-dimensional square structures, AIP Advances 6, 055711 (2016) . [44] M. Laubach, J. Reuther, R. Thomale, and S. Rachel, Rashba spin-orbit coupling in the kane-mele-hubbard model,Phys. Rev. B 90, 165136 (2014) . [45] W. Koshibae, Y. Ohta, and S. Maekawa, Theory of dzyaloshinski-moriya antiferromagnetism in distorted CuO2and NiO 2planes,Phys. Rev. B 50, 3767 (1994) . [46] X. Liu, Z. Wang, K. Watanabe, T. Taniguchi, O. Vafek, and J. I. A. Li, Tuning electron correlation in magic- angle twisted bilayer graphene using coulomb screening, Science371, 1261 (2021) .