Spin Fluctuation Theory for Unconventional Superconductivity in Antiferromagnetic Metals

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Spin Fluctuation Theory for Unconventional Superconductivity in Antiferromagnetic Metals
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1 online resource (124 p.)
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english
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Rowe, Wenya
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University of Florida
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Gainesville, Fla.
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Degree:
Doctorate ( Ph.D.)
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University of Florida
Degree Disciplines:
Physics
Committee Chair:
HIRSCHFELD,PETER J
Committee Co-Chair:
MUTTALIB,KHANDKER A
Committee Members:
KUMAR,PRADEEP
STEWART,GREGORY R
PHILLPOT,SIMON R

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Subjects / Keywords:
antiferromagnetism -- superconductivity
Physics -- Dissertations, Academic -- UF
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Physics thesis, Ph.D.
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theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
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Abstract:
The understanding of unconventional superconductivity is still a challenge for condensed matter physicists. To understand the interplay between antiferromagnetic order and superconductivity is crucial for the development of unconventional superconductivity theory, not only because the antiferromagnetic state coexists with superconductivity in many materials such as cuprates, iron-based pnictides, heavy fermions and organic superconductors, but also because spin fluctuations near a magnetically ordered phase have been proposed to possibly mediate superconductivity. In Chapter 1, we introduce the mechanism of spin fluctuations and review important categories of unconventional superconductors. In Chapter 2 we review elements of the relevant theory of magnetism. We discuss the differences between localized and itinerant approaches to study magnetism. We focus on cuprates which have a simple one-band Fermi surface. We use the Hubbard model to describe the band structure of the cuprates, and introduce the mean field phase diagram of electron-doped cuprates. In Chapter 3, we study the dynamic susceptibility of cuprates in the pure antiferromagnetic state and in the coexistence state of antiferromagnetism and superconductivity. We identify the key features of particle-hole spin excitations which are affected by the next-nearest neighbor hopping, $t^\prime$. We compare the different spin wave features between electron- and hole-doped cuprates. We conclude that the long range commensurate antiferromagnetic state is unstable on the hole doped side within the self-consistent-mean field theory due to the negative spin-stiffness. In the coexistence state, we see the spin resonance peak caused by superconductivity as well as the Goldstone mode in the spin excitation spectrum. Last we present the instability analysis of superconductivity from spin fluctuations in the antiferromagnetic state. We derive the superconducting pairing potentials and the gap equations for the spin singlet and triplet pairings. We separate the singlet potentials into longitudinal and transverse channels and expand the them around the pocket's center in the small pocket limit. Our result shows on the electron-doped side the leading symmetry is $d_{x^2-y^2}-$wave and on the hole-doped side it is $p-$wave. This implies that from the paramagnetic state to the antiferromagnetic state, the superconducting gap has a smooth transition on the electron-doped side whereas the gap has to change symmetry on the hole-doped side. We conjecture that this may account for the lack of bulk coexistence of antiferromagnetic and superconducting order on the hole-doped side of the cuprates phase diagram.
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Statement of Responsibility:
by Wenya Rowe.
Thesis:
Thesis (Ph.D.)--University of Florida, 2014.
Local:
Adviser: HIRSCHFELD,PETER J.
Local:
Co-adviser: MUTTALIB,KHANDKER A.

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SPINFLUCTUATIONTHEORYFORUNCONVENTIONALSUPERCONDUCTIVITYINANTIFERROMAGNETICMETALSByWENYAW.ROWEADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2014

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c2014WenyaW.Rowe 2

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ACKNOWLEDGMENTS IthankmyadvisorProfessorPeterJ.Hirschfeldforhiscontinuedsupportandencouragementwhichhavehelpmetogrow.Hispatience,humor,enthusiasm,andbroadknowledgehavebeeninvaluableformydoctoralstudy.Iamalsogratefulforthetimeandsupportfrommembersofmycommittee,Profes-sorsK.Muttalib,P.Kumar,G.Stewart,andS.Phillpot.IwouldliketothankProfessors.D.Maslov,K.Muttalib,andK.Ingersentfortheirlucidandrichlectures,theirpatienceandextraguidance.Myworkhasmostlybeendoneincollaborationwithresearchersinotherinstitutes.IexpressmyspecialappreciationtoProfessorIlyaEreminattheRuhrUniversityinBochumforhisguidanceovertheyears.ThankstoDr.J.Knollewhohelpedmewiththecalculationsofthespinsusceptibilityandthemeaneldenergy.ThankstoProfessorB.M.Andersenforhishelpwiththepotentialcalculations.AndthankstoA.Rmerforhermeticulouscomparisonoftheresults.IwouldliketothanktheRuhrUniversityinBochumfortheirhospitalityduringmyshortvisitsandduringthelastyearofmydoctoralstudyinGermany.IthankDrs.G.Boyd,A.Kemper,V.MishraandM.Korshunov,formermembersoftheHirschfeldgroup.Theyprovidedencouragementandinformativeadvicesduringthebeginningofmyresearchyears.IthankDr.A.Kreisel,Y.WangandP.Choubeyfortheirhelpfuldiscussionsaboutallmatters. 3

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 3 LISTOFTABLES ...................................... 6 LISTOFFIGURES ..................................... 7 ABSTRACT ......................................... 8 CHAPTER 1INTRODUCTION ................................... 10 1.1Spinuctuationmodels ............................ 10 1.1.1KondoModel .............................. 10 1.1.2Andersonmodel ............................ 12 1.1.3Hubbardmodel ............................. 13 1.2Spinuctuationsandsuperconductivity ................... 14 1.3Unconventionalsuperconductivity ...................... 16 1.3.1Cuprates ................................. 17 1.3.2Iron-pnictides .............................. 21 1.3.3Heavyfermions ............................. 22 1.3.4Organicandfullerenesuperconductors ................ 23 2ANTIFERROMAGNETICSTATE .......................... 24 2.1Ferro-andAntiferromagnetism ........................ 24 2.2Itinerantelectronmagnetism ......................... 26 2.3MeaneldphasediagramincludingAFandsuperconductivity ....... 28 3DYNAMICSPINSUSCEPTIBILITY ........................ 36 3.1Theoryandcalculations ............................ 36 3.1.1Neutronscattering ........................... 36 3.1.2Spinwaves ............................... 38 3.2Spinexcitationsinthepureantiferromagneticstate ............. 39 3.2.1Thedynamicspinsusceptibilityintheantiferromagneticstate ... 39 3.2.2Theeffectofnext-nearesthopping,t0onthespinexcitations .... 42 3.2.3Theeffectofthedopantsonspinexcitations ............. 44 3.3Spinexcitationsinthecoexistencestate ................... 48 4THEPAIRINGINTERACTIONARISINGFROMANTIFERROMAGNETICSPINFLUCTUATIONS ................................... 55 4.1Thepairinginteractionintheantiferromagneticbackground ........ 55 4.2Thepairingsymmetries ............................ 62 4.2.1Angulardependenceofthecoherencefactors ............ 63 4

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4.2.2Angulardependenceofthepairingpotentials ............ 67 4.2.2.1Chargeandlongitudinalinteraction ............ 67 4.2.2.2Transverseinteraction .................... 69 4.2.2.3Interbandinteractions .................... 70 4.2.3LAHAexpansionofgapequation ................... 71 4.2.4Comparisonwithnumericalevaluation ................ 74 5CONCLUSION .................................... 77 APPENDIX AMEANFIELDQUANTITIESINTHEPUREANTIFERROMAGNETICSTATE 81 A.1Antiferromagneticorderparameterequation:derivation .......... 81 A.2Theelectronlling:derivation ......................... 82 BDERIVATIONSINTHECOEXISTENCESTATEOFANTIFERROMAGNETISMANDSUPERCONDUCTIVITY ........................... 83 B.1Antiferromagneticorderparameterequationinthecoexistencestatewithsuperconductivity:derivation ......................... 83 B.2Fillinglevelofelectronsinthecoexistencestate:derivation ........ 83 B.3Meaneldenergyinthecoexistencestate:derivation ........... 85 CDERIVATTIONSOFDYNAMICSPINSUSCEPTIBILITYINTHEPUREAN-TIFERROMAGNETICSTATE ............................ 90 C.1Transversedynamicspinsusceptibilityintheantiferromagneticstate:derivation .................................... 90 C.2Umklapptermforthetransversedynamicspinsusceptibility ........ 92 C.3Thelongitudinaldynamicspinsusceptibility ................. 94 C.4ThelongitudinalUmklappsusceptibility ................... 96 C.5AnalyticprooffortheformationoftheGoldstonemode .......... 98 DDERIVATIONSOFDYNAMICSPINSUSCEPTIBILITYINTHECOEXISTENCESTATEOFANTIFERROMAGNETICANDSUPERCONDUCTIVITY ...... 99 D.1Derivationsoftransversedynamicspinsusceptibilityinthecoexistencestateofantiferromagnetismandsuperconductivity ............. 99 D.2TheUmklapptermforthetransversedynamicspinsusceptibility ..... 106 D.3Thelongitudinaldynamicspinsusceptibility ................. 110 REFERENCES ....................................... 115 BIOGRAPHICALSKETCH ................................ 124 5

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LISTOFTABLES Table page 4-1Coherencefactors,p2(k,k0)andn2(k,k0)expandedaroundholepocketsfork=( 2, 2)andk0=( 2, 2) ............................. 64 4-2Coherencefactors,p2(k,k0)andn2(k,k0)expandedaroundholepocketsfork=()]TJ /F8 7.97 Tf 10.5 4.71 Td[( 2, 2)andk0=( 2, 2) ............................ 65 4-3Coherencefactors,p2(k,k0)andn2(k,k0)expandedaroundelectronpockets 66 4-4Coherencefactors,p2(k,k0)andn2(k,k0)expandedaroundelectronandholepockets ........................................ 66 4-5Potentialsfromthecharge-andlongitudinalspin-uctuationcontributionex-pandedaroundholepockets ............................ 68 4-6Potentialsfromthetransversespin-uctuationcontribution,)]TJ /F6 11.955 Tf 9.29 0 Td[(2)]TJ /F9 7.97 Tf 13.06 -1.8 Td[(sexpandedaroundholepocketsinthelimitofkhF!0 ..................... 70 4-7Potentialsfromthecharge-andlongitudinalspin-uctuationinterbandcontri-butionexpandedbetweenelectronandholepockets ............... 71 4-8Angulardependenceofthes-waveanddx2)]TJ /F9 7.97 Tf 6.59 0 Td[(y2-wavesymmetriesontheholepockets ........................................ 72 4-9Angulardependenceofthes-waveanddx2)]TJ /F9 7.97 Tf 6.59 0 Td[(y2-wavesymmetriesontheelec-tronpockets. ..................................... 72 6

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LISTOFFIGURES Figure page 1-1TheFeynmandiagramintheBerk-Schriefferapproximationtotheeffectiveelectron-electroninteraction ............................. 14 1-2Therelativesignsofsuperconductinggaponacuprate-likeFermisurface ... 16 1-3Phasediagramsofhole-dopedandelectron-dopedcuprates .......... 18 1-4Crystalstructuresoftheelectron-dopedR2)]TJ /F9 7.97 Tf 6.59 0 Td[(xSrxCuO4andthehole-dopedLa2)]TJ /F9 7.97 Tf 6.59 0 Td[(xSrxCuO4 .................................... 19 1-5Crystalandspinstructuresoftheelectron-dopedR2)]TJ /F9 7.97 Tf 6.59 0 Td[(xSrxCuO4andthehole-dopedLa2)]TJ /F9 7.97 Tf 6.59 0 Td[(xSrxCuO4. ................................ 20 2-1TheFermisurfaceandthebandstructureofelectron-dopedcuprates ..... 35 2-2Thedoping-temperaturephasediagramofelectron-dopedcuprates ...... 35 3-1NeutronscatteringonPr1)]TJ /F9 7.97 Tf 6.58 0 Td[(xLaCexCuO(PLCCO) ................. 37 3-2Thebandstructuresandimaginarypartoftransversedynamicspinsuscepti-bilityathalf-lling ................................... 43 3-3ThreepossibletypesofFermisurfacetopologyintheantiferromagneticstateinlayeredcuprates .................................. 45 3-4Calculatedimaginarypartoftransverse+)]TJ /F9 7.97 Tf -6.59 -8.28 Td[(RPA(q,q,) ............... 46 3-5CalculatedImaginarypartofthetransverse+)]TJ /F9 7.97 Tf -6.59 -8.27 Td[(RPA(q,q,)spinexcitationspectra 50 3-6Calculatedimaginarypartofthelongitudinalsusceptibility,zzRPA(q,q,)spinexcitationspectra .................................. 52 4-1GeneralstructureoftheFermisurfaceoflayeredcuprates ............ 64 4-2Comparisonoftheanalyticalcalculationsupto(keF)2forthelongitudinalandtransversepairingpotentials ............................ 75 7

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophySPINFLUCTUATIONTHEORYFORUNCONVENTIONALSUPERCONDUCTIVITYINANTIFERROMAGNETICMETALSByWenyaW.RoweMay2014Chair:PeterJ.HirschfeldMajor:PhysicsTheunderstandingofunconventionalsuperconductivityisstillachallengeforcon-densedmatterphysicists.Tounderstandtheinterplaybetweenantiferromagneticorderandsuperconductivityiscrucialforthedevelopmentofunconventionalsuperconductivitytheory,notonlybecausetheantiferromagneticstatecoexistswithsuperconductivityinmanymaterialssuchascuprates,iron-basedpnictides,heavyfermionsandorganicsuperconductors,butalsobecausespinuctuationsnearamagneticallyorderedphasehavebeenproposedtopossiblymediatesuperconductivity.InChapter1,weintroducethemechanismofspinuctuationsandreviewimportantcategoriesofunconventionalsuperconductors.InChapter2wereviewelementsoftherelevanttheoryofmagnetism.Wediscussthedifferencesbetweenlocalizedanditinerantapproachestostudymag-netism.Wefocusoncuprateswhichhaveasimpleone-bandFermisurface.WeusetheHubbardmodeltodescribethebandstructureofthecuprates,andintroducethemeaneldphasediagramofelectron-dopedcuprates.InChapter3,westudythedynamicsusceptibilityofcupratesinthepureantiferromagneticstateandinthecoexistencestateofantiferromagnetismandsuperconductivity.Weidentifythekeyfeaturesofparticle-holespinexcitationswhichareaffectedbythenext-nearestneighborhopping,t0.Wecomparethedifferentspinwavefeaturesbetweenelectron-andhole-dopedcuprates.Weconcludethatthelongrangecommensurateantiferromagneticstateisunstableontheholedopedsidewithintheself-consistent-meaneldtheoryduetothenegative 8

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spin-stiffness.Inthecoexistencestate,weseethespinresonancepeakcausedbysuperconductivityaswellastheGoldstonemodeinthespinexcitationspectrum.Lastwepresenttheinstabilityanalysisofsuperconductivityfromspinuctuationsintheantiferromagneticstate.Wederivethesuperconductingpairingpotentialsandthegapequationsforthespinsingletandtripletpairings.Weseparatethesingletpotentialsintolongitudinalandtransversechannelsandexpandthethemaroundthepocket'scenterinthesmallpocketlimit.Ourresultshowsontheelectron-dopedsidetheleadingsymmetryisdx2)]TJ /F9 7.97 Tf 6.59 0 Td[(y2)]TJ /F1 11.955 Tf 9.3 0 Td[(waveandonthehole-dopedsideitisp)]TJ /F1 11.955 Tf 9.3 0 Td[(wave.Thisimpliesthatfromtheparamagneticstatetotheantiferromagneticstate,thesuperconductinggaphasasmoothtransitionontheelectron-dopedsidewhereasthegaphastochangesymmetryonthehole-dopedside.Weconjecturethatthismayaccountforthelackofbulkcoexistenceofantiferromagneticandsuperconductingorderonthehole-dopedsideofthecupratesphasediagram. 9

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CHAPTER1INTRODUCTION 1.1SpinuctuationmodelsForlocalizedspinsystems,magneticpropertiescanbefairlywelldescribedbyavarietyoftheoreticalapproaches[ 1 ].Forweaklyferromagneticsystems,whichareitinerant,predictionsarenotaccurate,sincestatisticaluctuationsofthechargeandscreeningofthespinmomenthavetobeincludedfortheitinerantelectronsystems.Boththermalandquantumuctuationscanbeimportant.Thermaluctuationsvanishwhentemperatureiszero,andthenincreasewithtemperature,butquantumuctuationsarepresentevenwhenthetemperatureiszero.HereweconsiderquantumspinuctuationtheorywhichhasbeendevelopedwithGreen'sfunctions.Spinuctuationtheoryisreallyacollectionofmethodsbasedonasmallsetofmodels[ 2 ],themainonesbeingtheKondo,AndersonandHubbardmodels,eachofthemhavingseveralvariations.ThetheoreticaltreatmentofmagnetisminmetalsbeganwithStonertheoryinthe1930s.Atthattime,peoplehaddoubtsaboutthepossibilityofdescribingspinwavesinitinerantsystems.TheRPAwaslaterdevelopedbyDoniachandEngelsberg[ 3 ],andappliedittoPdmetalwhichisnearlyferromagnetic.AndersonandBrinkman[ 4 ]usedthesametheorytounderstandthestabilityofthe3HeA-phaseliquid.BerkandSchrieffer[ 5 ]madetheimportantobservationthatspinuctuationswouldsuppresss-wavesuperconductivity.Inowbrieyreviewthemodelswhichhavebeendiscussedinthecontextofspinuctuations. 1.1.1KondoModelTheKondomodelhasbeenusedtodescribeisolatedimpuritiesinmetalsandquantumdotsystems.Themagnetismcomesfromthepartiallylledd)]TJ /F1 11.955 Tf 12.62 0 Td[(orf)]TJ /F1 11.955 Tf 12.62 0 Td[(shell, 10

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whichresultsinanuncompensatedmomentSj.TheclassicmaterialismagneticMnimpurityinCu.ThemodelisdescribedbythefollowingHamiltonian: H=Xk,"kcyk,ck,)]TJ /F3 11.955 Tf 11.84 8.09 Td[(J NXk,p,jeiRj(k)]TJ /F5 7.97 Tf 6.59 0 Td[(p)[(cyk,"cp,")]TJ /F3 11.955 Tf 11.96 0 Td[(cyk,#cp,#)S(z)j+cyk,"cp,#S()]TJ /F5 7.97 Tf 6.59 0 Td[()j+cyk,#cp,"S(+)j](1)Thersttermisthekineticenergyoftheconductionelectrons.TheothertermsarethescatteringsoftheelectronsfromthelocalspinatRj.J<0istheantiferromagneticexchangecoupling.Condensedmatterphysicistsobservedsingulareffectsofmagneticimpurityinnonmagneticmetals.Theelectronscatteringcontributiontotheresistivityistemperatureindependentforthenonmagneticimpurity.Butfornonmagneticmetalswithmagneticimpurity,thereisaminimuminresistivityatlowtemperature.TheminimumiscalledtheKondoeffect,representingasingularityinthemany-bodyscatteringamplitudeoftheelectronsfromthelocalizedspin.Therst-orderself-energyoftheelectronisindependentofwavevectororenergy.Thesecond-orderperturbationoftheGreen'sfunctionwhichhastobeevaluatedis G(2)(k,)=)]TJ /F6 11.955 Tf 10.5 8.09 Td[(1 2Z0d1Z0d2hTck,()^V(1)^V(2)cyk,(0)i (1) where^Vistheinteractionin 1 V=)]TJ /F3 11.955 Tf 11.84 8.09 Td[(J NXkpexp[iRj(k)]TJ /F7 11.955 Tf 11.95 0 Td[(p)]Scykcp. (1) Forcalculatingtheanomalousbehavioroftheresistivity,thethirdorderinperturbationtheoryhastobeconsidered.ThethirdorderintheGreen'sfunctionis G(3)(k,)=)]TJ /F3 11.955 Tf 10.49 8.09 Td[(cJ3 N2XpqZ0d1Z0d2Z0d3G(k,)]TJ /F10 11.955 Tf 11.95 0 Td[(1) (1) G(p,1)]TJ /F10 11.955 Tf 11.96 0 Td[(2)G(q,2)]TJ /F10 11.955 Tf 11.96 0 Td[(3)G(k,3)L(1,2,3) (1) 11

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whereGisthebareGreen'sfunctioninthepuresystemand L(1,2,3)=X()s()ss0()s0hTS()(1)S()(2)S()(3). (1) Inthirdorder,onendsalogdivergenceoftheGreen'sfunction.Kondoderivedthetemperaturedependenceofresistivityfromperturbationtheorytothirdorder[ 6 ],andgot (T)=(0)[1+4Jg(0)ln(kBT=W)]+bT5. (1) ThetermwithT5isduetothelatticevibrations.Thetermwithln(T)istheKondoeffect,whichcreatesaminimumatnitetemperature.Thisdivergencesignalsthebreakdownofperturbationtheoryatlowtemperatures,Theexactsolutiontotheproblem[ 7 8 ]aswellastherenormalizationgroups(RG)showthatthemoment~SisscreenedbelowatemperatureknownastheKondotemperature,whichisapproximately TK"Fp Jexp()]TJ /F6 11.955 Tf 9.3 0 Td[(1=N0J),(1)where"FistheFermienergy,JistheexchangeinteractionstrengthandNisthedensityofstates. 1.1.2AndersonmodelTheAndersonmodelisamodeldescribingalocalizedstatewhenthestateisfarbelowtheFermilevelinteractingwithconductionelectrons.TheHamiltonianis H=Xk,["kcyk,cyk,+Vk p N(cyk,f+fyck,)] (1) +"fXfyf+UX>nnwheren=fyf.fisthedestructionoperatorforthelocalstateandcisthedeconstruc-tionoperatorontheconductionstate.Vkisthehybridization.Uistheelectron-electroninteractionbetweenthelocalizedelectrons.Theelectroncanmovefromalocallevelwithenergy"ftoaconductionstateandviceversa.Inthelimitwhere"F<<)]TJ /F1 11.955 Tf 10.09 0 Td[(where 12

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=N0V2isthebarehybridizationwidth,themodelreducestoaKondoimpuritymodelwithJ=2V2="F[ 9 ]. 1.1.3HubbardmodelTheHubbardmodelhasaverysimpleform[ 10 ]: H=)]TJ /F12 11.955 Tf 11.29 11.36 Td[(Xhijitijcyjci+UXini"ni# (1) wherehijiarelatticesitesandhijimeansnearestneighborbonds.Therstpartisthehoppingtermbetweenlatticesites.Thesecondpartistheinteractionterm.Uistheon-siteCoulombinteraction.Anelectroncanhopfromsitetosite.Inthenon-interactinglimit,U<>t,theelectronshavetoolargeaCoulombinteractiontoovercome,andthesystembecomesaMottinsulator.WithcomparablevaluesofUandt,therewillbemetal-to-insulatortransition.Theone-dimensionalHubbardmodelhasbeensolvedexactly[ 11 ]butnoexactsolutionisavailableinhigherdimensions.ForthelargeU=tlimit,theHubbardmodelathalfllingcanbemappedontotheHeisenbergmodel,andshowstherelationJ=4t2=U.Awayfrom1/2-lling,Jistheantiferromagneticexchangecouplinginthet)]TJ /F3 11.955 Tf 11.96 0 Td[(Jmodelthatdescribesstronglycorrelatedelectronsystems.Theone-bandHubbardmodelhasbeengeneralizedtoaccommodatedifferentsystems.Thethree-bandHubbardmodeldescribes2)]TJ /F3 11.955 Tf 12.09 0 Td[(dCuO2layersincuprates.ThismodelhasthreedifferentU's-Cu(3dx2)]TJ /F9 7.97 Tf 6.58 0 Td[(y2)statewithUd,O(2px,y)statewithUpandthenearest-neighborinteractionwithUpd.TheHubbardHamiltonianinrealspacecanbetransformedintomomentumspace.TheHamiltonianis HXk,"kcyk,ck,+U 2Xk,k0,q,cyk,ck+q,cyk0,ck0+q(1) 13

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Figure1-1.TheFeynmandiagramintheBerk-Schriefferapproximationtotheeffectiveelectron-electroninteraction.Therearenon-spin-ipprocesses,(a)andspin-ipprocesses,(b)and(c) where"=)]TJ /F6 11.955 Tf 9.3 0 Td[(2t(coskx+cosky)+4t0coskxcosky)]TJ /F10 11.955 Tf 12.38 0 Td[(ifweonlyconsidernearest-andnext-nearest-hoppinginatwo-dimensionallattice.isthechemicalpotentialtodecidethedopingofthesystem.Thismodelhasbeenadaptedtodescribethehole-dopedcuprateswithantiferromagneticorder[ 12 14 ]. 1.2SpinuctuationsandsuperconductivitySpinuctuationssuppresssingletsuperconductivityinelectron-phononmediatedsuperconductors,butlateritwasshownthatspinuctuationsmaygiverisetop-wavepairinginthesuperuidphaseof3Heandalsosuperconductivityinheavyfermionsandotherhigh-Tcsuperconductors. 14

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Inspinuctuationtheory,wecansumoverasubclassofallFeynmandiagrams.Therandomphaseapproximation(RPA)showninFigure 1-1 retainsaninnitesubsetofthesediagrams.Itcontainsnon-spin-ipprocessesinFigure 1-1 (a)andspin-ipprocessesinFigure 1-1 (b)and(c).ThesummationofFigures 1-1 (a),(b)and(c)arewrittenas: )]TJ /F3 11.955 Tf 20.94 8.09 Td[(U320 1)]TJ /F3 11.955 Tf 11.96 0 Td[(U220=)]TJ /F6 11.955 Tf 13.15 8.09 Td[(1 2Uh1 1)]TJ /F3 11.955 Tf 11.95 0 Td[(U0)]TJ /F6 11.955 Tf 31.19 8.09 Td[(1 1+U0i (1a)U320 1)]TJ /F3 11.955 Tf 11.96 0 Td[(U220=1 2Uh1 1)]TJ /F3 11.955 Tf 11.96 0 Td[(U0+1 1+U0)]TJ /F6 11.955 Tf 11.96 0 Td[(2i (1b)U20 1)]TJ /F3 11.955 Tf 11.96 0 Td[(U0=Uh1 1)]TJ /F3 11.955 Tf 11.95 0 Td[(U0)]TJ /F6 11.955 Tf 11.96 0 Td[(1i (1c)where0(q,!)isthebaredynamicsusceptibilityofthemetal, 0(q,!)=Xkf("k+q))]TJ /F3 11.955 Tf 11.96 0 Td[(f("k) !)]TJ /F10 11.955 Tf 11.96 0 Td[("k+q+"k+i0+.(1)HereisanexamplefromScalapinoetal.[ 15 ]showinghowspinuctuationscangiverisetod)]TJ /F1 11.955 Tf 9.3 0 Td[(wavepairingsincuprates.Weseparatetheinteractionintosinglet(s)andtriplet(t)channelsandobtain Vs=U20 1)]TJ /F3 11.955 Tf 11.95 0 Td[(U0+U32 1)]TJ /F3 11.955 Tf 11.95 0 Td[(U220 (1a)Vt=)]TJ /F3 11.955 Tf 23.59 8.09 Td[(U20 1)]TJ /F3 11.955 Tf 11.96 0 Td[(U220. (1b)Inamodelband"k=)]TJ /F6 11.955 Tf 9.29 0 Td[(2t(coskx+cosky)+4t0coskxcosky,itmaybeshownthat=0=(1)]TJ /F3 11.955 Tf 12.71 0 Td[(U0)isstronglypeakedat(,).IfwesolvetheBCSgapequationatT=0, (k)=)]TJ /F12 11.955 Tf 11.29 11.36 Td[(XpV(k)]TJ /F7 11.955 Tf 11.96 0 Td[(p) 2Ep(p),(1)withEpthequasiparticleenergy,thelargestcontributionwouldbethewavevectorof(,)whichspanstheFermisurfaceatthesocalledhotspotasshowninFigure 1-2 .Ifwearesolvingthesuperconductinggapatanarbitrarypointk,thentheinteraction 15

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Figure1-2.Therelativesignsofsuperconductinggaponacuprate-likeFermisurface.ThebluelineistheFermisurface.Thebrowndashedlineisthepositionofgapnodes. couplestothesuperconductinggapatpointpwhichis(,)awayfrompointkasshowninFigure 1-2 .V(,)=Vs(,)andEparebothpositive.Soweneedthegaptochangesigninordertosatisfythegapequation.Thereforethesuperconductinggapatpointkandphaveoppositesigns.WiththisrelationandtheperiodicconditionsoftheFermisurface,wecangetad-wavesuperconductinggapasindicatedinFigure 1-2 foracuprate-likeFermisurface.Thisresultisfortheinteractionintheparamagneticstate. 1.3UnconventionalsuperconductivityAftertheemergenceofBCStheoryin1957,physiciststhoughtthemysteryofsuperconductorshadbeensolved.Howeverin1979,SteglichreportedthediscoveryofsuperconductivityinheavyfermionCeCu2Si2[ 16 ].Scientistscontinuediscoveringsuperconductivityinmaterialswheresuperconductivityidnotphononmediated.Theyarecategorizedasunconventionalsuperconductors[ 17 ]. 16

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Sincethentheoristsandexperimentalistsinvesttheireffortsstudyingthepairingsymmetriesofdifferentmaterials,andexpectitcanhelpusreachtheunderstandingofthepairingmechanism.Experimentalistshavevariouskindsofwaystoextractinfor-mationaboutthematerials.Themostcommonmethodsinclude:resistivity,magneticsusceptibility,andspecicheattoconrmaphasetransition;ARPES(Angle-ResolvedPhotoemissionSpectroscopy)tomapouttheFermisurfaceswhichplayanimportantroleinsuperconductivity,andthegapmagnitudeitself;phase-sensitiveexperimentslikeJosephsontunnelingtodistinguishonepairsymmetryfromanother;andelectro-magneticresponsepropertiessuchasopticalconductivityandRamanscatteringtodeterminethesuperconductinggapenergy.Fromtherstdiscoveredunconventionalsuperconductorstothemostrecent,wecangenerallygroupthemintoseveralmainbranches:heavy-fermionSC,organicSCandhighTcSC(cupratesandironpnictides). 1.3.1CupratesSuperconductivityincuprateswasdiscoveredbyBednorzandMuller[ 18 ].Thecupratessuperconductorshavethehighestcriticaltemperatures,whichmakesthecupratesthemostpopularmaterialsforhighertemperaturesuperconductorapplications.HgBa2Ca2Cu3O8hasthehighestTc,around150Kunderpressure.AllcuprateshavealayeredstructurewithCuO2planes.TheparentcompoundsareantiferromagneticMottinsulators.Upondoping,theantiferromagnetismisdestroyedabruptly,especiallyonthehole-dopedsideasinFigure 1-3 .Tounderstandthecuprates,weshouldtakeacloserlookatthedoping-temperaturephasediagram.Thephasediagramsofhole-dopedandelectron-dopedcupratesarequitedifferent,asshowninFigure 1-3 .Theantiferromagneticregiononthehole-dopedsideisalmostvetimesnarrowerthanontheelectron-dopedside(seegrayregionofFigure 1-3 ).Theelectron-dopedcuprates[ 19 ]havearobustcommensurateantiferromagneticphasebutamuchnarrowerSCrange.Itisbelievedthatthereisacoexistenceofsuperconductivityandantiferromagnetismontheelectron-dopedside. 17

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Figure1-3.Phasediagramsofhole-dopedandelectron-dopedcuprates.T?isthepseudogaptransitiontemperature.TNistheNeeltempetatureandTcisthesuperconductingtemperature.ReproducedwithpermissionfromArmitage,N.P.andFournier,P.andGreene,R.L.,Progressandperspectivesonelectron-dopedcuprates.Rev.Mod.Phys.,82(3):24212487,Sep2010(Page2422,Figure2).c(2010)byTheAmericanPhysicalSociety TheexperimentalevidenceincludesneutronscatteringexperimentonNd2)]TJ /F9 7.97 Tf 6.59 0 Td[(xCexCuO4whichshowscoexistenceregionuptotheoptimaldopinglevel[ 20 21 ].NMR(NuclearMagneticResonance)couldbeusedasaprobeindeterminingthecoexistencestateandsuperconductivityintheelectroncuprates.Buttherare-earthatomsinthespacerlayeroftheelectron-dopedcupratesgivealargemagneticresponse;thereforeitishardtointerpretthedata[ 19 ].Onthehole-dopedside,superconductivitycoexistswithonlystripedorglassydisorderinducedmagneticorderaccordingtoNMR(NuclearMagneticResonance)[ 22 ]andneutronscatteringmeasurements[ 23 ].Forareview,seeRef.[ 24 ].Oneaspectthatreectstheasymmetryinthephasediagramisthatthehole-dopedcupratesgenerallyhaveincommensurateantiferromagneticorderwhiletheelectron-dopedonesalwayshavecommensurateorderwithmomentumvector(,). 18

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Oneofthemostpuzzlingregionsofthephasediagramistheso-calledpseudogapphase,whichreprentsaregionwherethesystemdisplaysapartialgappingoflow-energyexcitations,althoughnoantiferromagneticorsuperconductinglong-rangeorderispresent.Longthoughttobeacrossoverphasewithnobrokensymmetries,thepseudogaptransitionThasrecentlybeenshowntocoincidewithaweakbreakingoftimereversalsymmetry,althoughthenatureofthephaseisstillnotclear[ 25 ]. Figure1-4.Crystalstructuresoftheelectron-dopedR2)]TJ /F9 7.97 Tf 6.58 0 Td[(xSrxCuO4andthehole-dopedLa2)]TJ /F9 7.97 Tf 6.59 0 Td[(xSrxCuO4.HereRisoneoftherare-earthions-Nd,Pr,SmorEu.ReproducedwithpermissionfromArmitage,N.P.andFournier,P.andGreene,R.L.,Progressandperspectivesonelectron-dopedcuprates.Rev.Mod.Phys.,82(3):24212487,Sep2010(Page2422,Figure1).c(2010)byTheAmericanPhysicalSociety. Understandingtheparticle-holeasymmetryinthephasediagrammaybefunda-mentaltoelucidatingthenatureofthecupratesuperconductorsandtheirrelationtotheMottinsulatingphaseathalf-lling.Theelectron-dopedandhole-dopedcuprateshaveslightlydifferentcrystalstructures.Figure 1-4 showsthecrystalstructuresof 19

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La2CuO4(LCO)andofanelectron-dopedNd2)]TJ /F9 7.97 Tf 6.58 0 Td[(xCexCuO4(NCCO).Theelectron-dopedcompoundshaveaT0crystalstructurewhichlacksoxygenatomintheapicalposition.Thesuperconductingcupratesgenerallyhaved-wavesymmetry.ThishasbeenconrmedbySTM(ScanningTunnelingMicroscopy),Londonpenetrationdepth,ARPESandtricrystal-grainboundaryexperiments[ 26 27 ].AlthoughbothelectronFigure1-5.Crystalandspinstructuresoftheelectron-dopedR2)]TJ /F9 7.97 Tf 6.59 0 Td[(xSrxCuO4andthehole-dopedLa2)]TJ /F9 7.97 Tf 6.58 0 Td[(xSrxCuO4.HereRisoneoftherare-earthions-Nd,Pr,SmorEu.ReproducedwithpermissionfromArmitage,N.P.andFournier,P.andGreene,R.L.,Progressandperspectivesonelectron-dopedcuprates.Rev.Mod.Phys.,82(3):24212487,Sep2010(Page2435,Figure16).c(2010)byTheAmericanPhysicalSociety. dopedandhole-dopedcuprateshavethesamemagneticperiodicity,themagneticmomentsforelectron-dopedcasepointalongtheCu-Obonddirections,whereasforhole-dopedcasetheypointatroughly45totheCu-ObonddirectionsasinFigure 1-5 [ 19 23 ].Cuprateshavebeenstudiedbyusingthet)]TJ /F3 11.955 Tf 12.24 0 Td[(JmodelduetotheantiferromagneticMottinsulatorparentcompounds[ 28 31 ].Athalf-lling(ornodoping),themagnetic 20

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momentsaretreatedaslocalizedspins.Upondoping,electronsareintroducedtothesystemwithantiferromagneticbackground.Thisisjustiedforthehole-dopedcuprateswhichexhibitmanypropertiesofdopedMottinsulators.Althoughcupratesareconsideredtobestronglycorrelatedsystems,experimentshaveshownthatintheelectron-dopedside,thesystemismetallicinpartoftheantiferromagneticstate.[ 19 ]FurthermoreDMFTcalculationsofundopedn-typecuprateswithT0crystalstructurehavearguedtheinsulatingpropertyisduetothepresenceofmagneticlong-rangeorderratherthantheMottchargetransfergapphysics[ 32 ].Thissuggeststhataweak-couplingapproachforthemagnetisminelectron-dopedcupratesmaybemoreappropriate. 1.3.2Iron-pnictidesKamiharaetal.[ 33 ]ofHosono'sgroupdiscoveredtherstiron-pnictidesuper-conductorLaFePOwithTc=6Kin2006.Later,withmoreandmorediscoveriesofmaterialsinthiscategory,peoplerealizedthatiron-basedsuperconductorscomeinmanyformsjustlikethecuprates.Theiron-basedsuperconductorscanbecategorizedasLaFeAsO(1111),MFe2As2(122),MFeAs(111),FeSe(11),Sr2MO3FePn(21311,M=Sc,V,Cr;Pn=P,As)andthedefectA0.8Fe1.6Se2structure(122*,A=K,Rb,Cs,Tl)[ 34 ].Thereisalsolongrangeantiferromagnetismintheiron-pnictides,foundtobesuppressedbyelectron-,hole-andisovalent-dopingandalsobyapplyingpressure.ThesuperconductivityappearsaroundthepointwhenAForderdisappears.Thecrystalstructurestypicallyundergoatetragonaltoorthorhombictransitionuponcooling.ThestructuralphasetransitiondoesnotnecessarilycoincidewiththeNeeltemperature,however.Fortheparentcompound,thesetwopointsareclosetogether.Otherwisetheantiferromagnetictransitionalwayshappensataslightlylowertemperature.Thetemperaturedifferencegetslargerwithlargerdopingormoreimpurities.Someiron-pnictidesexhibitcoexistencestateofsuperconductivityandantiferromagnetisminbothelectron-andhole-dopings[ 35 ].Thevariousformsofthecoexistencestatehave 21

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beenstudiedfordifferentsubcategoriesofiron-pnictidesbyusingNMR[ 36 38 ].Co-dopedBaFe2As2(122)andothermaterialsdisplaycoexistenceofsuperconductivityandantiferromagnetismattheatomicscale[ 36 ].ButRuisovalentdopingin122resultsinsuperconductingclustersinantiferromagneticbackground[ 37 ].Inthe245iron-selenideRbFeSe,thesystemexhibitsphaseseparationofthesetwostates[ 38 ].Unlikethecuprates,theFermisurfacesofiron-basedsuperconductorshavemulti-orbitalcharacter[ 39 ].Theyhavemorecomplicatedbandstructuresacrossdifferentsubcategoriesofmaterials.GenerallytheFermisurfacesofthevariousfamilieshaveholepocketsaround)]TJ /F1 11.955 Tf 10.1 0 Td[(pointandelectronpocketsaround(,0)or(0,).ExceptionsincludeKFe2As2,whichhasnoelectronpockets,andKFe2Se2andmonolayerFeSe,whichhavenoholepockets.Dopingcanaffectthenumberofpockets,ellipticity(nesting)ofthepocketsaswellasthequasi-twodimensionalnatureoftheFermisurface.Thequestionofwhetheriron-basedsuperconductorsaremorelocalizedoritinerantisunderdebate.Experimentshaveshownthatthespreadinthedegreeofcorrelationandlocalizationofmagneticstatesisprettywidefortheiron-basedsuperconductors[ 40 ]. 1.3.3HeavyfermionsTherstheavyfermionsuperconductortobefoundwasCeCu2Si2bySteglichetal.[ 16 ]in1978.Itwasalsotherstunconventionalsuperconductortobediscovered.Thesuperconductingtransitiontemperaturesforheavyfermionsareprettylow,onlyseveralKelvinmaximum.Thematerialsinthiscategorycontainelementsfromthelanthanideoractinideserieswhichhaveincompletefshells.ThissystemisinabalancebetweenthestrongCoulombinteractionwhichtendtoformlocalizedmomentsandthehybridizationwithextendedbandstatewhichtendtoinduceitinerantelectrons.Forsomeheavyfermions,dopinglevelorpressurecanchangenotonlythemagnitudesofsuperconductingandmagneticorders,butalsothesymmetriesoftheseorders. 22

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Antiferromagnetismiscommonamongheavyfermions.Howeverinsomematerials,thelowertemperaturephaseshowsweakmagneticmomentswhichisconrmedbymuonspinresonancemeasurementssuchasCeAl3andCeCu6[ 41 ].Thesupercon-ductingpairsinUPt3andUBe3havethepossibilityofbeingp)]TJ /F1 11.955 Tf 9.3 0 Td[(wavesymmetrywithparallelspinsfromthespecicheatmeasurements[ 42 43 ].Therearealsotransportandthermalmeasurementsshowingthatthesymmetrycouldbed)]TJ /F1 11.955 Tf 9.3 0 Td[(wave[ 44 ].Althoughheavy-fermionsweretherstunconventionalsuperconductorstobediscovered,withstronginteractionsandcomplexphasediagram,themicroscopicexplanationforsuper-conductivityremainsamystery.Thereareexperimental[ 42 45 ]andtheoretical[ 46 47 ]evidenceswhichsuggestsuperconductivitymediatedbyspinuctuations. 1.3.4OrganicandfullerenesuperconductorsLastly,anotherremarkablesetofunconventionalsuperconductorsistheorganicsu-perconductors.ThehighestTcatambientpressureis33Kinthealkali-dopedfullereneRbCs2C60.Thesuperconductivitycanbeinducedbypressureordoping.Organicsuper-conductorshaveveryrichphasediagrams.Inadditiontosuperconductivity,theyalsocontainmetal-insulatortransition,antiferromagneticorder,charge-,spin-density-wavephasesanddimensionalcrossover.Thereisalsocoexistencestateofsuperconductivityandspindensitywavesinorganicsuperconductorssuchasthequasi-one-dimensional(TMTSF)2PF6compounds[ 48 ].Withitsrichphasediagram,organicsuperconductorsrepresentagreattestinggroundforcompetingorderstudies. 23

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CHAPTER2ANTIFERROMAGNETICSTATESomeofthematerialIpresentedherehasappearedasSpinexcitationsinlayeredantiferromagneticmetals,W.Rowe,J.Knolle,I.Eremin,andP.J.Hirschfeld,Phy.Rev.B,86,134513(2012). 2.1Ferro-andAntiferromagnetismTheclassicaltheoryofmagnetismstartedfromalocalizedpictureofmag-neticmoments.Withthisconcept,LangevinexplainedtheCurie'slawofmagneticsusceptibility[ 49 ]whichwasrstdiscoveredbyPierreCurieexperimentally.Langevin'sresultforthemagneticsusceptibilityisinverselyproportionaltoT =N0m2=3kBT=C=T(2)whereN0isthenumberofatomsinthecrystal,misthemagneticmomentandCistheCurieconstant.LaterWeissintroducedinteractionbetweentheatomicmagneticmomentswhichisaveragedfromamoleculareldandthenaddedtotheexternaleldinLangevin'scalculation[ 50 ].HeobtainedtheresultabovetheCurietemperatureforthesusceptibilityas =C=(T)]TJ /F3 11.955 Tf 11.95 0 Td[(TC).(2)ThisiscalledtheCurie-Weisslaw.TherelationiscommonformostferromagnetsabovetheCurietemperature.TheexperimentaldatacanbecomparedwiththeCurie-Weisslawtoseewhetherthestudiedmaterialshavelocalizedspinsotherthanitinerantelectronsinthesystems.Fromthemicroscopicpointofview,thereexisttwochallengestothisclassicaldescription.Oneistheexplanationofthesourceofthemagneticmoment.TheotheristoexplaintheWeissmoleculareldwhichistoosmallifonecalculatestheaveragemagneticdipole-dipoleinteractioncomparingwiththevaluesfromtheobservedTc.Thesetwoproblemswerelateraddressedwiththehelpofquantummechanics. 24

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Accordingtoquantummechanics,theelectronstatecanbedescribedbytheorbitalangularmomentumLandspinS.ThemagneticmomentofanelectronisdescribedbyM=B(L+2S)withtheBohrmagneton,B=e~=2mc.ThephysicalvaluesintheLangevinequationcanbereplacedbytheirquantum-mechanicalcounterparts.Thenthemagneticsusceptibilitycanbere-writtenas =N0g2JBj(j+1)=3kBT(2)wheregJistheLandegfactor,jisthequantumnumberofthetotalangularmomentum.ThesusceptibilityhasthesametemperaturedependenceasintheclassicalresultandwehaveCurieconstant,C=N0g2JBj(j+1)=3kB.TheexplanationoftheWeissmoleculareldwasgivenbyHeisenberg[ 51 ].Hear-guedthattheoriginoftheeldwasfromthequantum-mechanicalexchangeinteractionsbetweentheatoms.Hedescribedthemagneticsystemwiththeatomicspinoperators: H=JXi,jSiSi(2)Withtheinteratomicexchangeinteractionconstant,J<0,thisHamiltoniandescribesaferromagneticstate,whereaswithJ>0,itdescribesanantiferromagneticstate.Intheantiferromagneticcase,inordertominimizetheenergy,jSi)]TJ /F7 11.955 Tf 12.03 0 Td[(Sjj=0,theadjacentspinshavetobeanti-parallel.Antiferromagnetismiscommonincorrelatedsystems.Theadjacentspinsliketoalignanti-paralleltolowertheenergyduetotheexchangecouplinginthesystems.Thisstate,similartotheferromagneticstate,occursatlowertemperatures.Aboveacertaincriticaltemperature,thespinorderingwouldbedestroyedbythermaluctuationsandthesystemturnsintoparamagneticstate.LouisNeelwasthersttoidentifythisantiferromagneticordering,thereforethiscriticaltemperatureisalsocalledtheNeeltemperature. 25

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BelowtheNeeltemperature,thenetmagnetizationisstillzeroasintheparamag-neticstate.Thereforeitisnoteasytodetect.TherstexperimentalconrmationofthisstatewasbyShullin1949withneutronscattering[ 52 ].Itcanalsobetestedbytheuni-formsusceptibility,(T)whichwouldshowaweakmaximumattheNeeltemperature,orbymeasurementofthestaggeredsusceptibility(q,T)whichshowsadivergencebyneutronscattering.TheNMRprobecanalsodeterminethestructurebythesymmetryanalysisofthehypernecouplingtensor[ 53 ].ThecommondescriptionofthisstateisbyassumingtwoferromagneticsublatticesAandBwhichhaveoppositespinorientation.ThetwomainmodelsareHeisenbergmodelforlocalizedspinsandtheHubbardmodelforitinerantelectrons. 2.2ItinerantelectronmagnetismTheHeisenbergmodelhassuccessindescribingferro-,antiferro-andferrimag-netismsinsystemswithlocalizedspins.Itledtothediscoveryofspinwavesandthedevelopofmagneticresonancetechniques.SincetheHeisenbergmodelassumeslocalizedspins,inprincipleitcanonlyexplainthephysicsofinsulatingmagnets.ButweknowthatmaterialsliketransitionmetalswhicharemagneticmetalssometimesdisplayweakmomentsandcannotbedescribedbytheHeisenbergmodel.BlochproposedthepossibilityofferromagnetismarisingfromelectrongaswiththehelpoftheHartree-Fock(HF)approximation[ 54 ].Asdiscussedintheintroduction,electrondopedcupratesap-peartodisplayitinerantantiferromagneticbehaviorclosetothesuperconductingphase.Wewouldliketounderstandthenatureofthismagneticstateinordertodescribeitsinuenceonsuperconductivity.Thebaresusceptibilityatwavevectorqis 0(q)=Xkf("k+q))]TJ /F3 11.955 Tf 11.95 0 Td[(f("k) "k)]TJ /F10 11.955 Tf 11.95 0 Td[("k+q.(2) 26

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WithintheRPAtheinteractingsusceptibilityisthen (q)=0 1)]TJ /F3 11.955 Tf 11.96 0 Td[(U0.(2)TheStonercriterionforthespindensitywaveinstabilityatqis U0(q)=1.(2)Wecanseethat0(q)issensitivetothetopologyoftheFermisurface,especiallyifthereisanesting.NestingmeansthatthereexistssegmentsoftheFermisurfacewhichcanbetranslatedbyavectorqandalignwiththeoriginalFermisurface.Thenestingoftenresultsinasingularbehaviorofthesusceptibilitywhichsignalsaninsta-bilitytoanewmagneticphase.Suchsystemsarecalleditinerantmagnets.Accordingtotheusualcriterionofbandtheory,ifoneormorebandsarepartiallylled,thesystemwouldbeametal.Localizedanditinerantmodelsrepresentoppositeapproaches.Theformerhastheelectronstateslocalizedinrealspacewhereasthelaterhasthestateslocalizedinmomentumspace.Themagnetisminalocalizedsystemduetostronginteractionsusuallymakethesysteminsulating.Andthemagneticmomentinanitinerantsystemcanmovefreely,thereforethesystemsremainmetallic.Theclassicationofmetalandinsulatorsaccordingtobandtheorywasquitesuccessfulbutfails,forinstanceinNiO,whichisaninsulatorbutissupposedtobeametalaccordingtoitscalculatedbandstructure.Mottlaterproposedanexplanationtosolvethepuzzle[ 55 ].TheMottinsulatorsareantiferromagnetic.ThestudyoftheMottinsulatorsathalf-llingandwithadditionalelectronsorholesdopinginthesystemisimportantforthesuperconductivitystudy,especiallyinsystemslikethehole-dopedcuprates. 27

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2.3MeaneldphasediagramincludingAFandsuperconductivityTheHubbardmodelinrealspaceisexpressedas H=Xij,tijcyi,cj,+U 2Xi,nini.(2)wherei,jarelatticesites,niisthenumberoperatoronsiteiwithspinwhichisoppositetospinandtijisthehoppingmatrixelementbetweensitesiandj.InFourierspace,wehave H=Xk,"kcyk,ck,+U 2Xk,k0,q,cyk,ck+q,cyk0,ck0+q(2)Weconsideratwo-dimensionalsystemwithnormalstatetight-bindingenergydispersion"k=)]TJ /F6 11.955 Tf 9.3 0 Td[(2t(coskx+cosky)+4t0coskxcosky)]TJ /F10 11.955 Tf 10.68 0 Td[(.AlthoughtheHubbardmodelisapparentlyverysimple,ithasnoexactsolutioninhigherdimensions.ItisthesimplestmodeltodisplaytheMottphenomenon,sincewhenUislarge,doubleoccupationisforbiddenandahalf-lledsystemcannotconduct.Intheantiferromagneticstate,theunitcellinrealspacedoubles,thereforetheunitcellinmomentumspaceishalfofthesizeofthatoftheparamagneticstate.ThesumofkoverthefullBrillouinzonethereforehastobefoldedintothereducedBrillouinzoneasinFigure 2-1 (a).Weuseamean-eldapproximationtodecouplethespindensitywavetermbydeningtheantiferromagneticorderparameteras W=U=2Xksgn().(2)withorderingmomentumQ=(,).WeapplytheBogoliubovtransformation, ck=ukk+vkkck+Q=sgn()()]TJ /F3 11.955 Tf 9.29 0 Td[(vkk+ukk)(2)totheHamiltoniananddeterminuk,vktodiagonalizeit.WegettheHamiltonianwiththe 28

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newbasisofkandkas H=Xk,0Ekyk,k,+Ekyk,k,(2)TheprimeoverthesummationindicatesthesumoverthereducedBrillouinzone.Thequasiparticleenergiesfortheelectronband,andtheholebandare: E,k="k+"k+Q 2r ("k)]TJ /F10 11.955 Tf 11.95 0 Td[("k+Q 2)2+W2(2)Thespindensitywaves(SDW)coherencefactorsareu2k=1 21+")]TJ /F5 7.97 Tf 0 -8.28 Td[(k p (")]TJ /F5 7.97 Tf 0 -8.28 Td[(k)2+W2 (2)v2k=1 21)]TJ /F10 11.955 Tf 43 8.09 Td[(")]TJ /F5 7.97 Tf 0 -8.27 Td[(k p (")]TJ /F5 7.97 Tf 0 -8.28 Td[(k)2+W2 (2)where"k=("k"k+Q)=2.TheantiferromagneticorderforcestheFermisurfacetoreconstruct.Itsplitsintoanelectronband(-band)andaholeband(-band)asinFigure 2-1 (b).InFigure 2-1 ,thegreenlineshowstheoriginalFermisurfaceinthenormalstatewithnoantiferromagneticorder.ThenintheantiferromagneticstatetheFermisurfacesplitsintoelectronpocketsaround(,0)and(0,)(redlines)andholepocketsaround(=2,=2)(bluelines).ThepositionofthedashlineinFigure 2-1 (b)atFermilevelcanbeadjustedbychangingforcertaindopinglevels.FromthedenitionofW,Equation 2 ,themagnitudeofthemagneticmomentforagivenUshouldbecalculatedself-consistentlyby W=UXk0W p ("k)]TJ /F10 11.955 Tf 11.95 0 Td[("k+Q)2+4W2htanhEk 2kBT)]TJ /F6 11.955 Tf 11.95 0 Td[(tanhEk 2kBTi(2)ThederivationisincludedinAppendix A.1 .Theelectronbandllingisdenedbyn=1+x=Pk,hcyk,ck,i.Toobtaincertaindopinglevels,wecanchangethechemicalpotential,suchthatitsatisesthefollowing 29

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equation: n=1+x=2)]TJ /F12 11.955 Tf 11.95 11.35 Td[(Xk0htanhEk 2kBT+tanhEk 2kBTi(2)ThederivationisincludedinAppendix A.2 .ThesizesoftheholepocketsandelectronpocketsdependonthemagnitudeofW.WhenWincreases,thesizeoftheholepocketshrinksandthesizeoftheelectronpocketexpandsontheelectron-dopedside.Inordertostudythesituationinthecoexistencestateofantiferromagneticandsuperconductingstates,weaddaphenomenologicalsuperconductingtermtotheHamiltonian.ThetotalHamiltonianiswrittenas: H=Xk"kcykck+Xk,k0,U 2cykck+Qcyk0+Qck0+Xk,p,q,Vqcyk+qcyp)]TJ /F5 7.97 Tf 6.58 0 Td[(qcpck (2) NowweperformameanelddecompositionontheVterm(superconductingterm)inEquation 2 ,assumingthespinsingletsuperconductingorderparameter k=Vhck"c)]TJ /F5 7.97 Tf 6.58 0 Td[(k#i(2)ThefullmeaneldHamiltonianthenbecomes H=Xk,0Ekyk,k,+Ekyk,k,)]TJ /F6 11.955 Tf 11.96 0 Td[(yk,y)]TJ /F5 7.97 Tf 6.58 0 Td[(k,)]TJ /F6 11.955 Tf 11.96 0 Td[(yk,y)]TJ /F5 7.97 Tf 6.59 0 Td[(k,(2)WeperformaBCSBogoliubovtransformationtofurtherdiagonalizethecoexistencestateofsuperconductivityandspindensitywaveswiththeBCStransformationinthespindensitywavestate: k"=ukk0+vkykly)]TJ /F5 7.97 Tf 6.59 0 Td[(k#=)]TJ /F6 11.955 Tf 9.58 0 Td[(vkk0+ukykl.(2) 30

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Fortheoperator,thetransformationisthesamewithexchangeofwith.TheBCScoherencefactorsare (uk)2=1 21+Ek k(vk)2=1 21)]TJ /F3 11.955 Tf 13.24 8.09 Td[(Ek k(2)inthecoexistencestate.Wenowgetnewquasiparticleenergydispersions: k=q (Ek)2+(k)2(=,).(2)TheSCgaps,karedeterminedself-consistentlyfromtwocoupledgapequations,derivedpreviouslybyIsmeretal[ 56 ].Thegapfunctionsare k=)]TJ /F12 11.955 Tf 11.96 11.35 Td[(Xp2R"(Vk)]TJ /F5 7.97 Tf 6.59 0 Td[(pFu,vk,p)]TJ /F3 11.955 Tf 11.95 0 Td[(Vk)]TJ /F5 7.97 Tf 6.59 0 Td[(p+QFv,uk,p)p 2p+(Vk)]TJ /F5 7.97 Tf 6.58 0 Td[(pNv,uk,p)]TJ /F3 11.955 Tf 11.96 0 Td[(Vk)]TJ /F5 7.97 Tf 6.59 0 Td[(p+QNu,vk,p)p 2p#k=)]TJ /F12 11.955 Tf 11.96 11.36 Td[(Xp2R"(Vk)]TJ /F5 7.97 Tf 6.59 0 Td[(pNv,uk,p)]TJ /F3 11.955 Tf 11.95 0 Td[(Vk)]TJ /F5 7.97 Tf 6.58 0 Td[(p+QNu,vk,p)p 2p+(Vk)]TJ /F5 7.97 Tf 6.58 0 Td[(pFu,vk,p)]TJ /F3 11.955 Tf 11.96 0 Td[(Vk)]TJ /F5 7.97 Tf 6.59 0 Td[(p+QFv,uk,p)p 2p#(2)whereNx,yk,p,Fx,yk,p=u2kx2p2ukvkupvp+v2ky2pandx,y=uorv.ThesumoverpislimitedtojEpj~!D,with~!DbeingtheDebyefrequencyonitsanaloginanelectronicpairingmodel.Forthedx2)]TJ /F9 7.97 Tf 6.58 0 Td[(y2)]TJ /F1 11.955 Tf 9.29 0 Td[(wavecase,wemaychooseVk)]TJ /F5 7.97 Tf 6.59 0 Td[(k0=Vd(coskx)]TJ /F6 11.955 Tf 12.65 0 Td[(cosky)(cosk0x)]TJ /F6 11.955 Tf -434.29 -23.91 Td[(cosk0y)=4=Vd'k'k0=4,andwendthatthesuperconductingorderparametertakestheform k='k(0+ukvk1).(2)Thecoexistenceofcommensurateantiferromagneticandsuperconductingstatescanthereforegenerateahigherharmoniccomponent1,withthedx2)]TJ /F9 7.97 Tf 6.58 0 Td[(y2-wavepotential[ 56 ].Thisharmonicisproportionaltothemagnitudeoftheantiferromagneticorderandsuper-conductinggaps.ItarisesduetoUmklappCooper-pairingtermslikehck,"c)]TJ /F5 7.97 Tf 6.59 0 Td[(k)]TJ /F5 7.97 Tf 6.59 0 Td[(Q,#i.TheseexpectationvaluesappearinthecoexistencephaseasthewavevectorQbecomesthenewreciprocalwavevectorofthelatticeintheantiferromagneticstate.Atthesametime, 31

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duetoadditionalbreakingofthespinrotationalsymmetryassociatedwiththeantifer-romagnetictransition,theUmklappCooper-pairingtermsformallybelongnowtothespin-tripletcomponentoftheCooper-pairwavefunctionwithmz=0aswasdiscussedpreviouslybyseveralauthors[ 57 59 ].Thisindicatesthattheappearanceofthe1isassociatedwithanadditionalphasetransitioninthecoexistencephasewithachangeoftheunderlyingsymmetryofthemean-eldHamiltonian.Theequationsforcalculatingantiferromagneticorderparameter,Wandthedopinglevel,xhavetobemodiedinthecoexistencestateaccordingtoboththeSDWandthesuperconductingcoherencefactors. W=UXk2RW 2p ")]TJ /F9 7.97 Tf 0 -8.28 Td[(k+W20hEk ktanh 2kBT)]TJ /F3 11.955 Tf 13.24 8.09 Td[(Ek ktanh 2kBTi(2)ThederivationofWinthecoexistencestateisincludedinAppendix B.1 n=1+x=Xk2)]TJ /F3 11.955 Tf 13.24 8.09 Td[(Ek ktanh(k 2T))]TJ /F3 11.955 Tf 13.24 8.09 Td[(Ek ktanh(k 2T)(2)ThederivationofnisincludedinAppendix B.2 .Withtheaboveequations,wecansolvetheorderparametersatdifferentdopingsandtemperaturesself-consistentlyontheelectron-dopedside.ThemeaneldphasediagramscanbeconstructedasinFigure 2-2 withadx2)]TJ /F9 7.97 Tf 6.59 0 Td[(y2)]TJ /F1 11.955 Tf 9.29 0 Td[(wavesuperconductingpairingpotential[ 56 ].Theantiferromagneticmomentdecreasesasthedopinglevelincreases.Thesuperconductivityorderparameterhasnon-zerosolutionsfromn=1to1.2.Themeaneldenergywascalculatedinordertodeterminethestablestatesolutions.Themeaneldenergyis: EMF=hHi=Xk0Ek)]TJ /F6 11.955 Tf 11.95 0 Td[(k+Ek)]TJ /F6 11.955 Tf 11.95 0 Td[(k+2k 2ktanh)]TJ /F6 11.955 Tf 7.44 -1.6 Td[(k 2T+2k 2ktanh)]TJ /F6 11.955 Tf 7.58 -1.6 Td[(k 2T+2kf(k)+2kf(k)+W2 U(2) 32

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ThederivationisincludedinAppendix B.3 .Thisresultsinarstordersuperconductingtransitionatdopingaroundn1.05inthephasediagraminFigure 2-2 .Forourphasediagram,weassumethesuperconductinggapappearswithanantiferromagneticbackground,i.e.inthelimitofTSCTNeel[ 61 ].Bycomparingwitha44transformation[ 58 ]whereantiferromagneticandsuperconductingordersareonequalfooting,wecanbesurethattheresultsarejustied.Thequasiparticlesarethesamebetweenourapproachandthe44oneswith1=0.For16=0theeigenenergiesagreeonlyuptoverysmalltermsoftheorderO(1)butstarttodifferforthehigherorderterms.Torecoverthesamequasiparticleenergy,oneneedstotakeintoaccounttheCooper-pairtermshyk,"y)]TJ /F5 7.97 Tf 6.59 0 Td[(k,#i.TheymayhavetobetakenintoaccountwhentheantiferromagneticorderbecomessmallandtheinterbandCooper-pairingmaybecomeimportant.Furthermore,theparticularformofthepairinginteraction(s)]TJ /F1 11.955 Tf 9.29 0 Td[(wave,d)]TJ /F1 11.955 Tf 9.29 0 Td[(waveorothers)inmomentumspacecanfurthermodifythestructureofthesuperconductinggapequationsinthecoexistencestate.InFigure 2-2 ,wepresentthemean-eldphasediagraminthecoexistencestateofcommensurateantiferromagnetismandd)]TJ /F1 11.955 Tf 9.3 0 Td[(wavesuperconductivitywith16=0.Althoughtherearesimilarresultsintheliterature[ 58 ],therearesomeimportantfeaturesthatareimportanttomention.Withinapureantiferromagneticphaseatnitedoping,thereisaLifshitztransition(bluecurve)separatingphaseswithadifferentFStopologywitheitheroneortwotypesofFSpockets.Athighertemperatures,bothelectronandholetypeofpocketsarepresentattheFS,whilebelowtheLifshitztransitiononlytheelectronpocketsarepresent.Theopencircleathalf-lling,n=1isthetransitionbetweenasemimetal(higherT)andaninsulator(lowerT) 33

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Anotherinterestingfeatureconcernsthecharacterofthephasetransitionintothecoexistingstateatlowtemperatures.Afteranalyzingthefreeenergy,wendthatthetransitionfromantiferromagnetismtothecoexistencestateisrstorderasafunctionofdoping,whileitbecomessecondorderasafunctionoftemperature.Furthermore,wenoticethatourtotalenergyanalysisshowsthatthestationarysolutionsforcoexistenceofantiferromagnetismandd)]TJ /F1 11.955 Tf 9.3 0 Td[(wavesuperconductivityintheelectron-dopedcuprateshavealwaysslightlylowerfreeenergyinthecase1=0,inotherwords,whenthetripletcomponentoftheCooper-pairingisabsent.Inourapproach,thesymmetryoftheproblemremainsSU(2)U(1)inthecoexistenceregime.WebelievethatitisconnectedtothefactthatweignoredthecontributionfromtheinterbandCooper-pairaverageshyk,"y)]TJ /F5 7.97 Tf 6.59 0 Td[(k,#iinthecoexistencephase.Althoughthesevaluesaresmall,theycouldchangethebalanceofthefreeenergytowardsthecoexistencestatewithnite'triplet'componentoftheCooper-pairing.Furthermore,amodicationofthemomentumdependenceoftheCooper-pairinginteractionmayalsochangethebalanceofthemean-eldstates.Oneapparentdisadvantageofthepredictionofthephasediagramshowsthatthesystemismetallicwiththeexceptionofhalf-llingattemperaturesbelowaboutT=0.05t,whileinexperimentstheparentcompoundcupratesarefoundtobeinsulatorsuptotheNeeltemperature.However,inelectron-dopedcupartes,thesystemappearstobemetallicfordopingsabovex0.125,whereasNeelorderdoesnotdisappearuntilaboutx0.15.Itisinpreciselythedopingrangewhereoptimalsuperconductivityoccurs. 34

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Figure2-1.Leftpanel:theFermisurfaceofelectron-dopedcupratesinnormalstate(greenlines),andintheantiferromagneticstatewithelectronpocket(redlines)andholepocket(bluelines).Rightpanel:thebandstructure. Figure2-2.Thedoping-temperaturephasediagramofelectron-dopedcuprateswiththesuperconductingtransition(redline)andNeeltemperaturs(blackline).ThereisaLifshitztransitionofbothpocketspresenttoonlyelectronpocketpresent(bluelines).Theopencircleathalf-lling,n=1isthetransitionbetweenasemimetal(higherT)andaninsulator(lowerT)[ 60 ].ReproducedwithpermissionfromW.Rowe,J.Knolle,I.Eremin,andP.J.Hirschfeld.Spinexcitationsinlayeredantiferromagneticmetalsandsuperconductors.Phys.Rev.B,86:134513,Oct2012(Page134513-7,Figure4).c(2012)byTheAmericanPhysicalSociety. 35

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CHAPTER3DYNAMICSPINSUSCEPTIBILITYSomeofthematerialIpresentherehasappearedasSpinexcitationsinlayeredantiferromagneticmetals,W.Rowe,J.Knolle,I.Eremin,andP.J.Hirschfeld,Phy.Rev.B,86,134513(2012). 3.1TheoryandcalculationsThesuperconductivityinlayeredoxides[ 62 63 ]andiron-pnictides[ 64 65 ]isproposedtoarisefromspinuctuations.Thereforeanunderstandingofthespinexcitationsisimportantforthestudyofunconventionalsuperconductivity.Thesymmetryofthesuperconductinggapaswellastheelectronicstructureofthematerialscaninuencethespinexcitations.Thespinexcitationscanbedetectedwithneutronscatteringexperiments[ 66 ]. 3.1.1NeutronscatteringTheneutronscatteringisimportantformeasuringspinuctuations.Itcandeterminethecrystalandmagneticstructureandalsothemotionoftheatoms.Theneutronscatteringexperimentfocusneutronbeamswithcertainmomentumandenergyonthematerial,thentheneutronsarescatteredfromthecrystal.Thedetectorscollecttheneutronsandthescattereddiffractionpatternshowsthepositionsoftheatomsinthematerial.Theneutronsmayalsocreatephononsormagnonswhentheypenetratethematerial.Theirenergychangesduetothisinelasticprocess.Thequantitywhichismeasuredbyinelasticneutronscatteringisthedynamicstructurefactor.Itisrelatedtothespinsusceptibilityviatherelation, S(q,!)=00(q,!) 1)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F15 7.97 Tf 6.59 0 Td[(~!=kBT.(3)Therehavebeenmanyneutronscatteringexperimentsonlayeredcuprates[ 20 ],manyofwhichexhibitafeaturecalledtheneutronresonanceorspinresonanceforT
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dopedcuprates[ 69 ].Figure 3-1 showsthespinresonanceofPr1)]TJ /F9 7.97 Tf 6.59 0 Td[(xLaCexCuO(PLCCO)inthesuperconductingstate.InFigure 3-1 (a),thetemperaturedifferencespectrumbetween2and30Ksuggestsaresonance-likeenhancementat11meV.InFigure 3-1 (b),blacksquaresshowtemperaturedependenceoftheneutronintensity(1hperpoint)at(1/2,1/2,0)and10meV.InFigure 3-1 (c),Q-scansat~!=10meVobtainedwithEf=14.7meVat2Kandat30KThespinresonancesinthesuperconducting Figure3-1.NeutronscatteringonPr1)]TJ /F9 7.97 Tf 6.58 0 Td[(xLaCexCuO(PLCCO).(a)Thetemperaturedifferencespectrumbetween2and30Ksuggestsaresonance-likeenhancementat11meV.(b)Blacksquaresshowtemperaturedependenceoftheneutronintensity(1hperpoint)at(1/2,1/2,0)and10meV.(c)Q-scansat~!=10meVobtainedwithEf=14.7meVat2Kandat30K.ReproducedwithpermissionfromS.D.Wilson,P.Dai,S.Li,S.Chi,H.J.Kang,andJ.W.Lynn,Resonanceintheelectron-dopedhigh-transition-temperaturesuperconductorPr0.88LaCe0.12CuO4)]TJ /F8 7.97 Tf 6.58 0 Td[(.Nature,442(7098):5962,072006.(Page61,Figure3(e)andFigure4(d,e)).c(2006)byNaturePublishingGroup 37

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statehavebeenstudiedinmaterialssuchasYBa2Cu3O7(YBCO)[ 70 ],Sr2RuO4[ 71 ]andiron-pnictides[ 66 72 ].Thesuperconductingresonancecanbetracedtothesignchangeinthed)]TJ /F1 11.955 Tf 9.3 0 Td[(wavesuperconductingorderparameterkundermomentumtransferk!k+qwithkQ=(,),duetoacoherencefactor(1)]TJ /F10 11.955 Tf 12.09 0 Td[(kk+q=EkEk+q)whichappearsinthedynamicalsusceptibility. 3.1.2SpinwavesTherearemainlytwoapproachesforthestudyofspinwavesincuprates.Oneisthestrong-couplingapproachrepresentingbythet)]TJ /F3 11.955 Tf 12.61 0 Td[(Jmodelandthet)]TJ /F3 11.955 Tf 12.61 0 Td[(J1)]TJ /F3 11.955 Tf 12.61 0 Td[(J2model.Thet)]TJ /F3 11.955 Tf 12.48 0 Td[(JmodelisderivedfromHubbardmodelinthelimitoflargeUandthet)]TJ /F3 11.955 Tf 12.38 0 Td[(J1)]TJ /F3 11.955 Tf 12.38 0 Td[(J2modelisanextensionofHeisenbergmodelincludinginteractionsbetweennext-nearest-neighborspins.Theotherapproachforstudyingspinwavesistheweak-couplingapproach,whichassumeselectronsareitinerant.Usingdifferentapproachestostudythespinexcitationsgivesusdifferentspinexcitationspectra.Thiscanhelpuscategorizewetherthespinsinaspecicmaterialaremorelocalizedormoreitinerantbycomparingthetheoreticalresultswiththeexperimentaldata.Insomelimits,thesetwoapproachescanyieldthesamegeneralfeatures.Forinstance,theHubbardmodelreducesinthelargeUlimittothet)]TJ /F3 11.955 Tf 12.33 0 Td[(Jmodel.Ideallywehopethereisawaytodescribethespinexcitationswithaccuracyandwithoutdependingonassumptionsofsensitiveparameters.Thepursuitofauniedmodelforgeneralmagneticsystemscontinuestobeanintriguingchallenge.Herewestartfromaitinerantapproach,usingtheHubbardmodel,whichwasintroducedinEquation 2 .Thederivationofthedynamicsusceptibilitystartswiththelinearresponsetheory[ 73 74 ]. 38

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3.2Spinexcitationsinthepureantiferromagneticstate 3.2.1ThedynamicspinsusceptibilityintheantiferromagneticstateWeemploytheRPAformalismwithasingle-bandHubbardmodel.Thedynamicalspinsusceptibilityforthelongitudinal,zz,andthetransverse,+)]TJ /F1 11.955 Tf 7.09 -4.34 Td[(,aredenedas lm(q,q0,)=Zdti 2NhTSlq(t)Sm)]TJ /F5 7.97 Tf 6.59 0 Td[(q(0)ie!+it, (3) withlm=zzor+)]TJ /F1 11.955 Tf 9.3 0 Td[(.Thespinoperatorscanbewrittenintermsofraisingandloweringoperators, S+q()=Xkcyk+q"()ck#(),S)]TJ /F5 7.97 Tf -1.07 -7.89 Td[(q()=Xkcyk+q#()ck"(),Szq()=Xkcyk+q()ck().(3)TheantiferromagneticorderingatQ=(,)whichcorrespondingtothemagneticorderinthecupratesdoublestheunitcellandrequiresaccountingforthebreakingoftranslationalsymmetry[ 12 14 ].TheBrillouinzoneinmomentumspacewouldbehalfofthefullzone.Asaresult,thetotalsusceptibilityinthetransversechannelisa22matrixwithoff-diagonalUmklappterms ^+)]TJ /F5 7.97 Tf -6.59 -7.97 Td[(0=0B@+)]TJ /F6 11.955 Tf 7.08 -4.34 Td[((q,q,!)+)]TJ /F6 11.955 Tf 7.08 -4.34 Td[((q,q+Q,!)+)]TJ /F6 11.955 Tf 7.09 -4.34 Td[((q+Q,q,!)+)]TJ /F6 11.955 Tf 7.08 -4.34 Td[((q+Q,q+Q,!).1CA (3) ThespinsusceptibilitywouldbeenhancedifweconsidertheRPAproccess.BysolvingtheDysonequation,wegetthesusceptibility[ 12 ]as ^+)]TJ /F9 7.97 Tf -6.59 -8.28 Td[(RPA=^1)]TJ /F6 11.955 Tf 13.41 2.66 Td[(^U^+)]TJ /F5 7.97 Tf -6.58 -7.98 Td[(0)]TJ /F5 7.97 Tf 6.58 0 Td[(1^+)]TJ /F5 7.97 Tf -6.59 -7.98 Td[(0,(3) 39

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andthebarecomponentsaregivenby +)]TJ /F5 7.97 Tf -6.58 -7.98 Td[(0(q,q,!)=)]TJ /F6 11.955 Tf 10.49 8.09 Td[(1 2Xk,00B@1+")]TJ /F5 7.97 Tf 0 -8.28 Td[(k")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q)]TJ /F3 11.955 Tf 11.96 0 Td[(W2 q )]TJ /F10 11.955 Tf 5.48 -9.68 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k2+W2q )]TJ /F10 11.955 Tf 5.48 -9.68 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q2+W21CAf(Ek+q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(Ek) !+i)]TJ /F3 11.955 Tf 11.96 0 Td[(Ek+q+Ek)]TJ /F6 11.955 Tf 10.49 8.09 Td[(1 20Xk,6=00B@1)]TJ /F10 11.955 Tf 57.51 9.32 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q)]TJ /F3 11.955 Tf 11.96 0 Td[(W2 q )]TJ /F10 11.955 Tf 5.48 -9.69 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k2+W2q )]TJ /F10 11.955 Tf 5.48 -9.69 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q2+W21CAf(E0k+q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(Ek) !+i)]TJ /F3 11.955 Tf 11.95 0 Td[(E0k+q+Ek, (3) with=,.f(Ek)istheFermifunctionandtheprimereferstothesumoverthemagnetic(reduced)BrillouinZone.ThedetailedderivationisincludedinAppendix C.1 .Wecanusethesameequationforthecalculationof+)]TJ /F5 7.97 Tf -6.58 -7.98 Td[(0(q+Q,q+Q,!)FortheUmklappterm,thesusceptibilityis +)]TJ /F5 7.97 Tf -6.59 -7.97 Td[(0(q,q+Q,!)=W 2Xk00@1 q )]TJ /F10 11.955 Tf 5.48 -9.68 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q2+W2)]TJ /F6 11.955 Tf 46.74 8.09 Td[(1 q )]TJ /F10 11.955 Tf 5.48 -9.68 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k2+W21A f(Ek+q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(Ek) !+i)]TJ /F3 11.955 Tf 11.95 0 Td[(Ek+q+Ek)]TJ /F3 11.955 Tf 20.15 9.32 Td[(f(Ek+q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(Ek) !+i)]TJ /F3 11.955 Tf 11.95 0 Td[(Ek+q+Ek!)]TJ /F12 11.955 Tf 11.29 24.03 Td[(0@1 q )]TJ /F10 11.955 Tf 5.48 -9.69 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q2+W2+1 q )]TJ /F10 11.955 Tf 5.48 -9.69 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k2+W21A f(Ek+q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(Ek) !+i)]TJ /F3 11.955 Tf 11.95 0 Td[(Ek+q+Ek)]TJ /F3 11.955 Tf 20.14 9.1 Td[(f(Ek+q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(Ek) !+i)]TJ /F3 11.955 Tf 11.95 0 Td[(Ek+q+Ek!. (3) ThedetailedderivationisincludedinAppendix C.2 .Theotherelementinthesuscepti-bilitymatrixcanbeobtainedbytherelation,+)]TJ /F5 7.97 Tf -6.59 -7.98 Td[(0(q,q+Q)=+)]TJ /F5 7.97 Tf -6.58 -7.98 Td[(0(q+Q,q).Weclearlyseethatthistermhascoherencefactors(whicharethecoefcientsinthefrontofeachtermexplicitlydependingon")]TJ /F5 7.97 Tf 0 -8.28 Td[(kandW)proportionaltotheantiferromagneticorderparameterW.Ifthereisnomagneticorder,wewoundnothavetranslationalsymmetrybreakingandthistermwouldbezero.Forthelongitudinalpartofthespinsusceptibility,thecalculationissimilar.Sincethereisnobreakingofspinrotationalsymmetryinthexy)]TJ /F1 11.955 Tf 9.3 0 Td[(plane,wewouldgettheUmklappsusceptibilitytobezero.Thiscanalsobeprovedbydirectcalculationwhichis 40

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includedinAppendix C.4 .ThereforethecalculationoftheRPAsusceptibilityisasimpleequationwhichisgivenby zzRPA(q,q,!)=zz0(q,q,!) 1)]TJ /F3 11.955 Tf 11.96 0 Td[(Uzz0(q,q,!).(3)Thebarelongitudinalspinsusceptibilityis zz0(q,q,!)=)]TJ /F6 11.955 Tf 10.49 8.09 Td[(1 2Xk,00B@1+")]TJ /F5 7.97 Tf 0 -8.27 Td[(k")]TJ /F5 7.97 Tf 0 -8.27 Td[(k+q+W2 q )]TJ /F10 11.955 Tf 5.48 -9.68 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k2+W2q )]TJ /F10 11.955 Tf 5.48 -9.68 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q2+W21CAf(Ek+q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(Ek) !+i)]TJ /F3 11.955 Tf 11.96 0 Td[(Ek+q+Ek)]TJ /F6 11.955 Tf 10.49 8.09 Td[(1 20Xk,6=00B@1)]TJ /F10 11.955 Tf 57.51 9.32 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q+W2 q )]TJ /F10 11.955 Tf 5.48 -9.68 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k2+W2q )]TJ /F10 11.955 Tf 5.48 -9.68 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q2+W21CAf(Ek+q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(E0k) !+i)]TJ /F3 11.955 Tf 11.95 0 Td[(Ek+q+E0k. (3) ThedetailedderivationisincludedinAppendix C.3 .Notethatthecoherencefactorsaredifferent(oppositesigninfrontofW2)forlongitudinalandtransversespinsusceptibili-ties.ThestructureofthedynamicspinsusceptibilityintheantiferromagneticstatewithorderingmomentumQhasbeenstudied[ 12 14 ]inthecontextoft0=0andonthehole-dopedside.Theseworksdidnotincludetheeffectsofnon-zeronext-nearest-neighborhopping,t0andthecasewithelectrondopantsThesetwopoints,t06=0andelectrondopings,whichcaneffectthespinexcitationspectrumsignicantlywillbediscussedseparatelyinthisthesis.Themainfeaturesthatwealsoobservedarethebreakingofthespin-rotationalsymmetryintheanisotropyinthesusceptibility,+)]TJ /F2 11.955 Tf 12.11 -4.34 Td[(6=zzandthegaplessGoldstonemodeasexpected.TheimaginarypartofthetransversesusceptibilityisgaplessanddisplaystheGoldstonemodeattheorderingvectorQ=(,)and!!0intheantiferromagneticstate.TheGoldstonemodeisguaranteedbythefactthattheconditionofthepoleformationintheRPApartofthetransversespinsusceptibilitycoincideswiththemean-eldequationforWandisvalidforanydopinglevelassoonastheequationsarecalculatedself-consistently.Our 41

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analysisisincludedinAppendix C.5 .Forthelongitudinalpart,thesusceptibilityatQisgappedbytwicetheantiferromagneticgapmagnitude,W. 3.2.2Theeffectofnext-nearesthopping,t0onthespinexcitationsWhenthenext-nearest-neighborhoppingt0isturnedon,thebehaviorofspinexcitationsawayfromQislessknown.Athalf-lling(x=0),itispossibletohaveacompensatedmetalatnitetemperatures.Atzerotemperature,theFermisurfaceisgappedbyavaluewhichdependsonWandt0andisdeterminedbytheself-consistentcalculationofthechemicalpotential.Tostudytheeffectoft0onthesystem,weplottedthespinsusceptibilitiesofthetransversechannelinFigure 3-2 forthehalf-lledcase.TheexcitationsinthetransversechanneloftheStonerinsulatorarespinwaves-collectivespinmodesoftheantiferromagneticgroundstate[ 75 ].Thebandstructuresandimaginarypartoftransversedynamicspinsusceptibilityathalf-llingareshowninFigure 3-2 with(a),(b)t0=0.0t,(c),(d)t0=0.2tand(e),(f)t0=0.35t.AllpanelshaveU=2.80tandW=0.75t.TheFermisurfaceisfullygappedforthehalf-lledcaseatlowtemperature,asshowninFigure 3-2 (a),(c)and(e).Theparticle-holeStonerexcitationsandthespinwavesarethereforeseparatedinenergyandmayinteractonlyaroundtheparticle-holecontinuumfrequency,!p)]TJ /F9 7.97 Tf 6.58 0 Td[(h(q).Fort0=0,theonsetoftheparticle-holecontinuumisgappedatleastupto!p)]TJ /F9 7.97 Tf 6.59 0 Td[(h(Q)=2W.Thisisbecausethetopofthelower-bandandthebottomoftheupper-bandarelocatedatthereducedBrillouinzoneboundary,i.e.coskx+cosky=0,atenergies)]TJ /F3 11.955 Tf 9.3 0 Td[(Wand+W,respectivelyasshowninFigure 3-2 (a).Therefore,thereexistsadegeneratemanifoldofqwavevectorsforwhich!p)]TJ /F9 7.97 Tf 6.58 0 Td[(h(q)=2W.Asaresult,thespinwavesdonotinteractwiththeparticle-holecontinuumforsufcientlylargevaluesofWandlookidenticaltothoseobtainedwithinaHeisenbergmodeloflocalizedspinswhichinteractviaanantiferromagneticexchangebetweennearestneighbors,J1t2 U,seeFigure 3-2 (b).Withthenon-zerovaluesofnextnearesthopping,t0,therearenon-degeneratepositionsofthetopofthe)]TJ /F1 11.955 Tf 9.3 0 Td[(bandandbottomofthe-bandasclearlyseeninFigure 3-2 (c)and 42

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Figure3-2.Thebandstructuresandimaginarypartoftransversedynamicspinsusceptibilityathalf-lling.with(a),(b)t0=0.0t,(c),(d)t0=0.2tand(e),(f)t0=0.35t.AllpanelshaveU=2.80tandW=0.75t[ 60 ].ReproducedwithpermissionfromW.Rowe,J.Knolle,I.Eremin,andP.J.Hirschfeld.Spinexcitationsinlayeredantiferromagneticmetalsandsuperconductors.Phys.Rev.B,86:134513,Oct2012(Page134513-3,Figure2).c(2012)byTheAmericanPhysicalSociety (e).Thereforeitreducestheoverallmagnitudeoftheindirectgapintheparticle-holecontinuumandshiftsittolowerenergiesatthe( 2, 2)pointoftheBrillouinzone.Foranynon-zerot0,thebottomoftheupper-bandislocatedat(,0)and(0,)orYpointsoftheBrillouinzoneatenergy)]TJ /F6 11.955 Tf 9.3 0 Td[(4t0+W)]TJ /F10 11.955 Tf 13.21 0 Td[(>0,whereasthetopofthelower)]TJ /F1 11.955 Tf 9.3 0 Td[(bandislocatedat( 2, 2)pointsoftheBrillouinzoneatenergy)]TJ /F3 11.955 Tf 9.3 0 Td[(W)]TJ /F10 11.955 Tf 12.51 0 Td[(<0.Asaresult,thesmallestindirectgapbetweenthesetwobandswhichdeterminesalsothelowestpositionoftheparticle-holecontinuumoccursat 43

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!p)]TJ /F9 7.97 Tf 6.59 0 Td[(h(~q)=2W)]TJ /F6 11.955 Tf 13.17 0 Td[(4t0for~q=( 2, 2).Forincreasingt0=tratioandaconstantvalueofW,thespinwavesareboundedfromaboveatmomentum~q=( 2, 2)andformalocalminimumatenergiesbelow2W)]TJ /F6 11.955 Tf 12.84 0 Td[(4t0>0.Inparticular,inFigure 3-2 (d)itoccursbelow0.7twithW=0.75tandt0=0.2tandisshiftedtomuchlowerenergiesfort0=0.35tforaxedW=0.75tasshowninFigure 3-2 (f).Forzerodopingwealwaysndeitheraninsulatingantiferromagneticstateoranormalstatemetalatlowtemperature.Ifwehavethectitiouscaseofacompensatedmetalandhave2W)]TJ /F6 11.955 Tf 12.07 0 Td[(4t0<0inthebandstructure,therealpartoftransversesusceptibilitywillbecomenegativeatq=~qand!=0.Thiswouldbeanunstablesolution,thereforewenevergetaself-consistentsolutioninthiscase.Thelocalminimumforanitet0at~qisduetotheinteractionofspinwaveswiththeparticle-holecontinuum.Thisisafeatureofweak-couplingwhichallows2Wtobeofthesameorderas4t0.ThiseffectwouldnotoccurforthelocalizedmodelsuchHeisenbergorJ1)]TJ /F3 11.955 Tf 12.97 0 Td[(J2models,whereJ2referstotheantiferromagneticexchangebetweenthenext-nearestneighbors.J2onlylowersthepositionofthemaximumofthespinwavedispersionattheYpointoftheBrillouinzone,aneffectclearlyreproducedintheweak-couplingcalculationsaswell,comparingwithFigures 3-2 (d)and(f).Atthesametime,withinthelocalizedmodeltheparticle-holeexcitationsalwaysremaingappedbythelargevalueofUandW.Correspondinglythelocalminimuminthespinsusceptibilityat~qneverformsinthelocalizedpicture. 3.2.3TheeffectofthedopantsonspinexcitationsForthedopedcase,westudythreedifferentscenarios:hole-dopedwithonlyholepockets,electron-dopedwithonlyelectronpocketsandelectron-dopedwithbothelectronandholepockets.Thesethreecasescanberealizedbyusingdopinglevelatx=)]TJ /F6 11.955 Tf 9.3 0 Td[(0.05,0.10and0.12respectively.ThetopologyoftheFermisurfacesareshowninFigure 3-3 .Figure 3-3 (a)isthecaseforholedoping.(b)and(c)arebothelectrondoping.TheoriginalFermisurfaceinthenormalstateisshownasgreencurve.The 44

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Figure3-3.ThreepossibletypesofFermisurfacetopologyintheantiferromagneticstateinlayeredcuprates[ 60 ].(a)isthecaseforholedoping.(b)and(c)arebothelectrondoping.TheoriginalFermisurface(greencurve)inthenormalstate,Theholepockets(bluecurves)centeredaround( 2, 2)andelectronpockets(redcurves)centeredaround(,0)[(0,)]pointsoftheBrillouinzone.Forlargerdopingandsmallersizesoftheantiferromagneticgapbothtypesofthepocketscanbepresent.Asarguedinthetext,thecommensurateantiferromagneticorderbecomesunstableoncetheholepocketsappeararound( 2, 2)pointsoftheBrillouinzone.ReproducedwithpermissionfromW.Rowe,J.Knolle,I.Eremin,andP.J.Hirschfeld.Spinexcitationsinlayeredantiferromagneticmetalsandsuperconductors.Phys.Rev.B,86:134513,Oct2012(Page134513-2,Figure1).c(2012)byTheAmericanPhysicalSociety. holepockets(bluecurves)centeredaround( 2, 2)andelectronpockets(redcurves)centeredaround(,0)[(0,)]pointsoftheBrillouinzone.Forlargerdopingandsmallersizesoftheantiferromagneticgapbothtypesofthepocketscanbepresent.Thecommensurateantiferromagneticorderbecomesunstableoncetheholepocketsappeararound( 2, 2)pointsoftheBrillouinzone.Thespinwavedispersionissymmetricwithrespecttothe(0,0)and(,)points,whichreectsthefactthatbothareequivalentsymmetrypointsofthemagnetic(re-duced)Brillouinzone.Atthesametime,theabsoluteintensityofthespinwavesisdifferentandisdeterminedbytheantiferromagneticcoherencefactorswhicharesup-pressedaroundthe)]TJ /F1 11.955 Tf 6.77 0 Td[(-point.WecanseeclearlyfromEquation 3 thatatlowfrequency,thenon-vanishingcontributiontotheintensitycomesfromtheinterband(!and 45

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viceversa)transitionswhichareproportionaltotheantiferromagneticcoherencefactorcinterk,q=1)]TJ /F8 7.97 Tf 45.16 7.54 Td[(")]TJ /F5 5.978 Tf 0 -6.42 Td[(k")]TJ /F5 5.978 Tf 0 -6.42 Td[(k+q)]TJ /F9 7.97 Tf 6.59 0 Td[(W2 q (")]TJ /F5 5.978 Tf 0 -6.42 Td[(k)2+W2q (")]TJ /F5 5.978 Tf 0 -6.42 Td[(k+q)2+W2.ForqQonends")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+Q)]TJ /F10 11.955 Tf 22.8 0 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(kandcinterk,qQ2,whereasitiscinterk,q0/(2W2 (")]TJ /F5 5.978 Tf 0 -6.42 Td[(k)2+W2forq0.Thisapparentlyaffectstheintensityofthesusceptibilitygreatlyat(0,0)and(,)points,althoughtheyareconsideredthesamepointsinthemagneticmomentumspace. Figure3-4.Calculatedimaginarypartoftransverse+)]TJ /F9 7.97 Tf -6.58 -8.28 Td[(RPA(q,q,),(leftpanel)andlongitudinal,zzRPA(q,q,)(rightpanel)[ 60 ].ReproducedwithpermissionfromW.Rowe,J.Knolle,I.Eremin,andP.J.Hirschfeld.Spinexcitationsinlayeredantiferromagneticmetalsandsuperconductors.Phys.Rev.B,86:134513,Oct2012(Page134513-4,Figure3).c(2012)byTheAmericanPhysicalSociety. Thespinexcitationspectravs.qforthemetallicantiferromagneticstateareshowninFigure 3-4 witht0=t=0.35andU=1.3875t.(a),(d)refertotheholedoping,x=-0.05,W=0.61t,=)]TJ /F6 11.955 Tf 9.3 0 Td[(0.8819t,(b),(e)refertotheelectrondoping,x=0.10,W=0.4404t,=)]TJ /F6 11.955 Tf 9.3 0 Td[(0.4284t,and(c),(f)refertotheelectrondopingx=0.14, 46

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W=0.12t,=)]TJ /F6 11.955 Tf 9.3 0 Td[(0.302t.ThecorrespondingFermisurfacetopologyisshowninFigure 3-3 .Theintensityinstates=tisshownonalogscale.ThewhitearrowsinFigure 3-4 (a)denotetheincommensuratemomentum.Notethattheintensitymapsaredifferentbetweenleftandrightpanels.Theincommensuratemodeintheholedopedsideshowsthatourmeaneldassumptionofmomentumorderingat(,)isnotastablemagneticstructure.TheinstabilityisrelatedtotheappearanceofthesmallFSholepockets,andtothespinstiffnessofthecommensuratespinexcitationsatQ.Incontrasttotheundopedcase,thereisanadditionalcontributiontothespinstiffnesswhicharisesduetointraband)]TJ /F10 11.955 Tf 11.98 0 Td[(transitionswhicharenowgapless.Weexpandthedispersionofthelower)]TJ /F1 11.955 Tf 9.3 0 Td[(bandforU>>taround(=2,=2)pointswhichyieldsEk=)]TJ /F10 11.955 Tf 9.3 0 Td[()]TJ /F3 11.955 Tf 12.1 0 Td[(W)]TJ /F9 7.97 Tf 16.74 8.3 Td[(p2jj 2mjj)]TJ /F9 7.97 Tf 16.74 6.48 Td[(p2? 2m?,wherepjj=(kx)]TJ /F3 11.955 Tf 12.54 0 Td[(ky)=2,p?=(kx+ky)=2andmjj=(8t0))]TJ /F5 7.97 Tf 6.59 0 Td[(1,andm?=(16t2=W)]TJ /F6 11.955 Tf 12.53 0 Td[(8t0))]TJ /F5 7.97 Tf 6.59 0 Td[(1.FollowingtheanalysisofthedenominatorofthetransversespinsusceptibilityatQ,weseethatthespinstiffnesssacquiresanitecorrectioninthedopedantiferromagneticmetal[ 14 76 77 ]ass=0s(1)]TJ /F3 11.955 Tf 12.62 0 Td[(z)wherez=2Up m?mjj isproportionaltothePaulisusceptibilityofthe-band,and0sisthebarespinstiffnessintheundopedcase.Thecorrectionz>1forlargeUindicatesthatthecommensurateantiferromagneticorderisunstableuponholedoping.Onanotherhand,fortheoppositecasewithU<1.Therefore,theinstabilityofthecommensurateantiferromagneticorderforUtandholedopingoccursduetonegativecorrectionstothespinstiffness.Theinstabilityofthecommensuratemagneticstructuremaybeanexplanationofwhyinhole-dopedsidecupratesthelong-termmagneticorderisvulnerableupondopingwhileontheelectron-dopedsidethemagneticorderisrobust 47

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uptovetimesofthecriticaldopinglevelonthehole-sidewhentheantiferromagneticleveldisappears.Intheexperimentstheincommensuratemodehasbeenfoundonthelightlyhole-dopedLa2CuO4andcoexistswiththecommensuratemodeatcertaindopings[ 23 ].Thequestionofhowtodescribesuchaphenomenonwithatheoreticalmodelisintriguing. 3.3SpinexcitationsinthecoexistencestateForthecalculationofthedynamicspinsusceptibilityinthecoexistencestateofantiferromagnetismandsuperconductivity,inadditiontotheunitarytransformation,wealsoneedaBCSBogoliubovtransformationtogetthecorrectquasiparticles.ThedetailedderivationofthetransversespinsusceptibilitiesisincludedinAppendix D.1 .Thediagonaltransversespinsusceptibilityinthecoexistencestateis +)]TJ /F5 6.974 Tf -5.76 -7 Td[(0(q,q,!)=)]TJ /F12 9.963 Tf 12.28 9.47 Td[(Xk,=001 40BB@1+")]TJ /F5 6.974 Tf 0 -7.27 Td[(k")]TJ /F5 6.974 Tf 0 -7.27 Td[(k+q)]TJ /F3 9.963 Tf 9.97 0 Td[(W2 q )]TJ /F17 9.963 Tf 4.57 -8.07 Td[(")]TJ /F5 6.974 Tf 0 -7.27 Td[(k2+W2r ")]TJ /F5 6.974 Tf 0 -7.27 Td[(k+q2+W21CCA("1+EkEk+q+kk+q kk+q#f(k+q))]TJ /F3 9.963 Tf 9.97 0 Td[(f(k) !+i)]TJ /F6 9.963 Tf 9.97 0 Td[(k+q+k+1 2"1)]TJ /F3 9.963 Tf 11.16 8.05 Td[(EkEk+q+kk+q kk+q# f(k+q)+f(k))]TJ /F6 9.963 Tf 9.96 0 Td[(1 !+i+k+q+k+1)]TJ /F3 9.963 Tf 9.96 0 Td[(f(k+q))]TJ /F3 9.963 Tf 9.96 0 Td[(f(k) !+i)]TJ /F6 9.963 Tf 9.96 0 Td[(k+q)]TJ /F6 9.963 Tf 9.96 0 Td[(k!))]TJ /F12 9.963 Tf 12.28 9.46 Td[(Xk,6=001 40BB@1)]TJ /F17 9.963 Tf 49.91 8.05 Td[(")]TJ /F5 6.974 Tf 0 -7.27 Td[(k")]TJ /F5 6.974 Tf 0 -7.27 Td[(k+q)]TJ /F3 9.963 Tf 9.97 0 Td[(W2 q )]TJ /F17 9.963 Tf 4.57 -8.07 Td[(")]TJ /F5 6.974 Tf 0 -7.27 Td[(k2+W2r ")]TJ /F5 6.974 Tf 0 -7.27 Td[(k+q2+W21CCA("1+EkE0k+q+k0k+q k0k+q#f(0k+q))]TJ /F3 9.963 Tf 9.97 0 Td[(f(k) !+i+0k+q)]TJ /F6 9.963 Tf 9.96 0 Td[(k+1 2"1)]TJ /F3 9.963 Tf 11.16 8.05 Td[(EkE0k+q+k0k+q k0k+q# f(0k+q)+f(k))]TJ /F6 9.963 Tf 9.96 0 Td[(1 !+i+0k+q+k+1)]TJ /F3 9.963 Tf 9.96 0 Td[(f(0k+q))]TJ /F3 9.963 Tf 9.96 0 Td[(f(k) !)]TJ /F6 9.963 Tf 9.96 0 Td[(0k+q)]TJ /F6 9.963 Tf 9.97 0 Td[(k!). (3) Thesuperconductinggapskandkaredenedinchapter2.Weassumeadx2)]TJ /F9 7.97 Tf 6.59 0 Td[(y2-wavesuperconductinggapwhichwasobservedinmostcupratesexperimentalstudies[ 26 27 ]. 48

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TheUmklapptransversespinsusceptibilityis +)]TJ /F5 7.97 Tf -6.59 -7.97 Td[(0(q,q+Q,!)=W 4Xk,00@1 q )]TJ /F10 11.955 Tf 5.48 -9.69 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q2+W2)]TJ /F6 11.955 Tf 46.74 8.09 Td[(1 q )]TJ /F10 11.955 Tf 5.48 -9.68 Td[(")]TJ /F5 7.97 Tf 0 -8.27 Td[(k2+W21AEk+q !k+q+Ek kf(k+q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(k) !)]TJ /F6 11.955 Tf 11.95 0 Td[(k+q+kEk+q k+q)]TJ /F3 11.955 Tf 13.24 8.08 Td[(Ek k1)]TJ /F3 11.955 Tf 11.96 0 Td[(f(k+q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k) !)]TJ /F6 11.955 Tf 11.95 0 Td[(k+q)]TJ /F6 11.955 Tf 11.96 0 Td[(k+f(k+q)+f(k))]TJ /F6 11.955 Tf 11.96 0 Td[(1 !+k+q+k+W 4Xk,6=000@1 q )]TJ /F10 11.955 Tf 5.48 -9.69 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q2+W2+1 q )]TJ /F10 11.955 Tf 5.48 -9.68 Td[(")]TJ /F5 7.97 Tf 0 -8.27 Td[(k2+W21A( Ek+q k+q+E0k 0k!f(k+q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(0k) !)]TJ /F6 11.955 Tf 11.96 0 Td[(k+q+0k Ek+q k+q)]TJ /F3 11.955 Tf 13.24 8.09 Td[(E0k 0k! 1)]TJ /F3 11.955 Tf 11.96 0 Td[(f(k+q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(0k) !)]TJ /F6 11.955 Tf 11.95 0 Td[(k+q)]TJ /F6 11.955 Tf 11.95 0 Td[(0k+f(k+q)+f(0k))]TJ /F6 11.955 Tf 11.95 0 Td[(1 !+i+k+q+0k!). (3) Thesigncorrespondsto=and=respectively.Thelongitudinalspinsusceptibilityhastheform zz0(q,q,!)=Xk,01 40BB@1+")]TJ /F5 6.974 Tf 0 -7.27 Td[(k")]TJ /F5 6.974 Tf 0 -7.27 Td[(k+q+W2 q )]TJ /F17 9.963 Tf 4.56 -8.07 Td[(")]TJ /F5 6.974 Tf 0 -7.27 Td[(k2+W2r ")]TJ /F5 6.974 Tf 0 -7.27 Td[(k+q2+W21CCA("1+EkEk+q+kk+q kk+q#f(k+q))]TJ /F3 9.963 Tf 9.96 0 Td[(f(k) !+i)]TJ /F6 9.963 Tf 9.96 0 Td[(k+q+k+1 2"1)]TJ /F3 9.963 Tf 11.16 8.05 Td[(EkEk+q+kk+q kk+q# f(k+q)+f(k))]TJ /F6 9.963 Tf 9.96 0 Td[(1 !+i+k+q+k+1)]TJ /F3 9.963 Tf 9.96 0 Td[(f(k+q))]TJ /F3 9.963 Tf 9.97 0 Td[(f(k) !+i)]TJ /F6 9.963 Tf 9.96 0 Td[(k+q)]TJ /F6 9.963 Tf 9.96 0 Td[(k!)Xk,6=001 40BB@1)]TJ /F17 9.963 Tf 49.91 8.05 Td[(")]TJ /F5 6.974 Tf 0 -7.26 Td[(k")]TJ /F5 6.974 Tf 0 -7.26 Td[(k+q+W2 q )]TJ /F17 9.963 Tf 4.56 -8.07 Td[(")]TJ /F5 6.974 Tf 0 -7.27 Td[(k2+W2r ")]TJ /F5 6.974 Tf 0 -7.27 Td[(k+q2+W21CCA("1+EkE0k+q+k0k+q k0k+q#f(0k+q))]TJ /F3 9.963 Tf 9.96 0 Td[(f(k) !+i)]TJ /F6 9.963 Tf 9.96 0 Td[(0k+q+k+1 2"1)]TJ /F3 9.963 Tf 11.16 8.05 Td[(EkE0k+q+k0k+q k0k+q# f(0k+q)+f(k))]TJ /F6 9.963 Tf 9.96 0 Td[(1 !+i+0k+q+k+1)]TJ /F3 9.963 Tf 9.96 0 Td[(f(0k+q))]TJ /F3 9.963 Tf 9.97 0 Td[(f(k) !+i)]TJ /F6 9.963 Tf 9.96 0 Td[(0k+q)]TJ /F6 9.963 Tf 9.96 0 Td[(k!). (3) ThederivationsofthedynamicsusceptibilitiesareincludedinAppendix D .TheUmklapptermofthelongitudinalpartisagainzeroasinthepureantiferromag-neticcaseforthesamereasonthatthesymmetryisnotbrokenalongthezdirection.Ithasthesamecoefcientwhichiscomprisedofantiferromagneticcoherencefactorsasinthepureantiferromagneticstate,thereforeremainszero. 49

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Figure3-5.CalculatedImaginarypartofthetransverse+)]TJ /F9 7.97 Tf -6.58 -8.28 Td[(RPA(q,q,)spinexcitationspectraforthreedifferentelectrondopings,n=1.06,n=1.09,andn=1.12,(fromuppertolowerpanel)forthecoexistencestateand1=0(rightpanel).Forcomparisontheleftpanelshowstheresultsforthepureantiferromagneticstate.Thebluelinesdenote1=URe+)]TJ /F5 7.97 Tf -6.58 -7.98 Td[(0(q,)condition.Theintensityisshownonthelogscale.Thefollowingparametersareusedintheunitsoftfor(a)=)]TJ /F6 11.955 Tf 9.29 0 Td[(0.5362,W=0.5617,for(b)=)]TJ /F6 11.955 Tf 9.3 0 Td[(0.4573,W=0.4617,for(c)=)]TJ /F6 11.955 Tf 9.3 0 Td[(0.3694,W=0.3456,for(d)=)]TJ /F6 11.955 Tf 9.3 0 Td[(0.5349,W=0.5530,0=0.0727,for(e)=)]TJ /F6 11.955 Tf 9.3 0 Td[(0.4509,W=0.4555,0=0.0705,andfor(f)=)]TJ /F6 11.955 Tf 9.3 0 Td[(0.3624,W=0.3458,0=0.0618[ 60 ].ReproducedwithpermissionfromW.Rowe,J.Knolle,I.Eremin,andP.J.Hirschfeld.Spinexcitationsinlayeredantiferromagneticmetalsandsuperconductors.Phys.Rev.B,86:134513,Oct2012(Page134513-8,Figure5).c(2012)byTheAmericanPhysicalSociety. 50

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Thespinexcitationspectraforthreedifferentelectrondopingswithn=1.06,n=1.09,andn=1.12inFigure 3-5 (fromuppertolowerpanels)forthepureantifer-romagneticstate(leftpanels)andthecoexistencestatewith1=0(rightpanels).Forcomparisontheleftpanelsshowtheresultsforthepureantiferromagneticstate.Thebluelinesdenote1=URe+)]TJ /F5 7.97 Tf -6.58 -7.97 Td[(0(q,)condition.Theintensityisshownonthelogscale.Thefollowingparametersareusedintheunitsoftfor(a)=)]TJ /F6 11.955 Tf 9.3 0 Td[(0.5362,W=0.5617,for(b)=)]TJ /F6 11.955 Tf 9.3 0 Td[(0.4573,W=0.4617,for(c)=)]TJ /F6 11.955 Tf 9.3 0 Td[(0.3694,W=0.3456,for(d)=)]TJ /F6 11.955 Tf 9.3 0 Td[(0.5349,W=0.5530,0=0.0727,for(e)=)]TJ /F6 11.955 Tf 9.3 0 Td[(0.4509,W=0.4555,0=0.0705,andfor(f)=)]TJ /F6 11.955 Tf 9.3 0 Td[(0.3624,W=0.3458,0=0.0618.TheGoldstonemodeinthetransversechannelremainsrobustandgaplessinthecoexistenceregime.Thiscanbeprovedbyananalyticalcheckusingtheself-consistentequationforWandU,asinthecaseofthepureantiferromagneticstate.Theexcitationsinthetransversechannelaredominatedbytherenormalizedspectrumofthespinwaves.Atthesametime,wendthattheexcitationsinthelongitudinalchannelincludearesonancemodeatthecommensuratemomentumcloseto(,)duetothesupercon-ductinggap.Thespinvelocityisalsocalculatedinthecoexistencestate.Toevaluatethespinwavevelocity,weexpandthedenominatorofEquation 3 aroundqQwith!=0uptoquadraticorder.Thisprocedureleadstothespinwavevelocity,coftheform c2=yt2(1=U)]TJ /F3 11.955 Tf 11.95 0 Td[(W2z) W2x2+(v)(1=U)]TJ /F3 11.955 Tf 11.96 0 Td[(W2z)(3)where x=Xk01 q )]TJ /F10 11.955 Tf 5.48 -9.69 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k2+W2k+k2 Ek k)]TJ /F3 11.955 Tf 13.24 8.17 Td[(Ek k!(3) v=Xk01 k+k3 1)]TJ /F3 11.955 Tf 13.15 8.16 Td[(EkEk)]TJ /F6 11.955 Tf 11.95 0 Td[(2 kk!(3) 51

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Figure3-6.Calculatedimaginarypartofthelongitudinalsusceptibility,zzRPA(q,q,)spinexcitationspectravs.qforthreedifferentelectrondopings,n=1.06,n=1.09,andn=1.12(fromuppertolowerpanels).[ 60 ].ReproducedwithpermissionfromW.Rowe,J.Knolle,I.Eremin,andP.J.Hirschfeld.Spinexcitationsinlayeredantiferromagneticmetalsandsuperconductors.Phys.Rev.B,86:134513,Oct2012(Page134513-9,Figure6).c(2012)byTheAmericanPhysicalSociety. z=Xk01 )]TJ /F10 11.955 Tf 5.48 -9.68 Td[(")]TJ /F5 7.97 Tf 0 -8.27 Td[(k2+W2k+k 1)]TJ /F3 11.955 Tf 13.15 8.16 Td[(EkEk+2k kk!.(3)Thecoefcientyiscomprisedoftwoterms.Therstonearisesfromtheintrabandcontribution 52

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y1=Xk=,02k 2)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(k3264W2sin2kx )]TJ /F10 11.955 Tf 5.48 -9.68 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k2+W22)]TJ /F6 11.955 Tf 13.15 7.92 Td[((coskx+cosky)2 )]TJ /F10 11.955 Tf 5.48 -9.68 Td[(")]TJ /F5 7.97 Tf 0 -8.27 Td[(k2+W2375)]TJ /F6 11.955 Tf 60.67 8.16 Td[(3")]TJ /F5 7.97 Tf 0 -8.28 Td[(k2kEk )]TJ /F10 11.955 Tf 5.48 -9.69 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k2+W2(k)50@2sin2kxcosky")]TJ /F5 7.97 Tf 0 -8.28 Td[(ksin2kx q )]TJ /F10 11.955 Tf 5.48 -9.68 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k2+W21A. (3) andtheotheronefromtheinterbandcontributions y2=1 4Xk,6=002W2sin2kx )]TJ /F17 9.963 Tf 4.48 -7.96 Td[(")]TJ /F5 6.974 Tf 0 -7.23 Td[(k2+W22k+0k 1)]TJ /F3 9.963 Tf 11.16 6.88 Td[(EkE0k)]TJ /F6 9.963 Tf 9.96 0 Td[(2k k0k!)]TJ /F6 9.963 Tf 49.32 6.74 Td[(2 k0kk+0k2640B@t0 tcos(kx+ky))]TJ /F6 9.963 Tf 4.47 -7.96 Td[(cos2kx+coskxcosky q )]TJ /F17 9.963 Tf 4.48 -7.96 Td[(")]TJ /F5 6.974 Tf 0 -7.24 Td[(k2+W2W2sin2kx )]TJ /F17 9.963 Tf 4.48 -7.96 Td[(")]TJ /F5 6.974 Tf 0 -7.23 Td[(k2+W23=21CAE0k+2k 2t2375)]TJ /F6 9.963 Tf 56.82 6.74 Td[(1 )]TJ /F6 9.963 Tf 4.48 -7.96 Td[(k30kk+0k )]TJ /F3 9.963 Tf 4.47 -7.96 Td[(E0k2EkE0k)]TJ /F6 9.963 Tf 9.96 0 Td[(22k20sin2kx t2!+EkE0k+2k )]TJ /F6 9.963 Tf 4.48 -7.96 Td[(k50kk+0k )]TJ /F3 9.963 Tf 4.47 -7.96 Td[(E0k2)]TJ /F3 9.963 Tf 4.48 -7.96 Td[(Ek2+22k20sin2kx t2!)]TJ /F6 9.963 Tf 50.5 6.74 Td[(1 )]TJ /F6 9.963 Tf 4.48 -7.96 Td[(k2k+0k3 1)]TJ /F3 9.963 Tf 11.16 6.88 Td[(EkE0k)]TJ /F6 9.963 Tf 9.97 0 Td[(2k k0k! )]TJ /F3 9.963 Tf 4.48 -7.96 Td[(E0k2)]TJ /F3 9.963 Tf 4.48 -7.96 Td[(Ek2+22k20sin2kx t2!)]TJ /F6 9.963 Tf 59.01 6.74 Td[(1 )]TJ /F6 9.963 Tf 4.48 -7.95 Td[(k20kk+0k2 )]TJ /F3 9.963 Tf 4.47 -7.96 Td[(E0k2EkE0k)]TJ /F6 9.963 Tf 9.96 0 Td[(22k20sin2kx t2!+EkE0k)]TJ /F6 9.963 Tf 9.96 0 Td[(2k )]TJ /F6 9.963 Tf 4.48 -7.95 Td[(k40kk+0k2 )]TJ /F3 9.963 Tf 4.47 -7.96 Td[(E0k2)]TJ /F3 9.963 Tf 4.48 -7.96 Td[(Ek2+22k20sin2kx t2!)]TJ /F12 9.763 Tf 9.97 22.6 Td[(264EkE0k+2k )]TJ /F6 9.963 Tf 4.48 -7.95 Td[(k20kk+0k)]TJ /F12 9.763 Tf 9.97 16.75 Td[( 1)]TJ /F3 9.963 Tf 11.16 6.88 Td[(EkE0k)]TJ /F6 9.963 Tf 9.97 0 Td[(2k k0k!1 k+0k2375h2k)]TJ /F3 9.963 Tf 4.48 -7.95 Td[(E0k2+)]TJ /F3 9.963 Tf 4.48 -7.95 Td[(Ek220sin2kx=t2 )]TJ /F6 9.963 Tf 4.48 -7.96 Td[(k3)]TJ /F6 9.963 Tf 14.88 6.74 Td[(4 k2640B@t0 tcos(kx+ky))]TJ /F6 9.963 Tf 4.47 -7.96 Td[(cos2kx+coskxcosky q )]TJ /F17 9.963 Tf 4.48 -7.96 Td[(")]TJ /F5 6.974 Tf 0 -7.23 Td[(k2+W2W2sin2kx )]TJ /F17 9.963 Tf 4.48 -7.96 Td[(")]TJ /F5 6.974 Tf 0 -7.24 Td[(k2+W23=21CAEk+2k t2375i.(3) 53

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Thisresultcanbereducedtothespinwavevelocityinthepureantiferromagneticstateandbecomparedwiththespinvelocitycalculationsin[ 13 78 ].Thespinvelocityiscon-sistentwiththedispersionsinFigures 3-5 (a),(b)and(c)forthepureantiferromagneticstateandinFigures 3-5 (d),(e)and(f)forthecoexistencestate.Thespinwavesarestronglymodiedbytheresonancecreatedbysuperconductinggap.Theeffectispar-ticularlystrongaround20wherethespinwavesinthecoexistenceregionexhibitakinkstructureduetotheinteractionwiththeparticle-holecontinuumoftheintrabandtran-sitions.Theyareenhancedduetod)]TJ /F1 11.955 Tf 9.3 0 Td[(wavesymmetryofthesuperconductinggap,i.e.duetothefactthatonendsk=)]TJ /F6 11.955 Tf 9.29 0 Td[(k+qiforincommensuratemomentaqi>(0.8,0.8).Thisleadstoanenhancementoftheintrabandparticle-holecontinuumofbothbandsfor20.Astheelectronbandaround(,0)and(0,)pointsalwayscrossestheFermilevelintheantiferromagneticstate,theenhancementoftheparticle-holecontinuumofthisbandaround20isresponsibleforthekinkstructureseeninthespinwaves.Inotherwords,thedampingeffectsoftheparticle-holecontinuumonthespinwavesarepresentinbothpuremetallicantiferromagneticandcoexistencestates.However,inthecoexistenceregionthereisalsoaneffectofthestrongrenormalizationofthespinwaveduetothe20structureoftheparticle-holecontinuumoftheintrabandsusceptibility,whichthenyieldstherenormalizationofthespinwavevelocityaround20.Anotherinterestingfeatureisthatd)]TJ /F1 11.955 Tf 9.3 0 Td[(wavesuperconductivitystabilizesthecom-mensurateantiferromagneticstatebypartialgappingtheparticle-holecontinuuminthecoexistencestate.Observe,forexample,thatthespinwavescomputedforn=1.12inthepureantiferromagneticstateshowatendencytowardsincommensurability,whileinthecoexistencestatethespinexcitationsarestillcommensurate. 54

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CHAPTER4THEPAIRINGINTERACTIONARISINGFROMANTIFERROMAGNETICSPINFLUCTUATIONSThischapterpresentstheanalysisofthepairinginstabilitycreatedbythespinuctuationsintheantiferromagneticstate.ThestudyfollowsthetheoryproposedbySchrieffer,WenandZhang[ 12 ].SomeofthematerialIpresentedherehasappearedasDopingasymmetryofsuperconductivitycoexistingwithantiferromagnetisminspinuctuationtheory,W.Rowe,I.Eremin,A.Rmer,B.M.AndersenandP.J.Hirschfeld,arXiv:1312.1507. 4.1ThepairinginteractionintheantiferromagneticbackgroundTheeffectiveinteractionarisesfromthespincorrelationproposedbyBerkandSchrieffer[ 5 ].Theystudiedtheparamagnon-mediatedinteractioninanearlyferromag-neticFermiliquid.ThisapproachwasadaptedbyNakajimatostudyliquidHe3[ 79 ]andbyFayandAppel[ 80 ]tostudyp)]TJ /F1 11.955 Tf 9.3 0 Td[(statesuperconductivitywithitinerantferromagnetism.ThepairinginteractionarisesfromsummingovertheRPAdiagrams.TheeffectiveHamiltoniansobtainedfromsumoverallthepossibleRPAprocessesasinFigure 1-1 inthecharge-uctuationchannelis[ 5 ] Hc=1 4NXk,k0,qXs1,s2[2U)]TJ /F3 11.955 Tf 11.95 0 Td[(Vc(k)]TJ /F7 11.955 Tf 11.95 0 Td[(k0)]cyk0s1cy)]TJ /F5 7.97 Tf 6.59 0 Td[(k0+qs2c)]TJ /F5 7.97 Tf 6.59 0 Td[(k+qs2cks1, (4) inthelongitudinalspin-uctuationchannel Hz=)]TJ /F6 11.955 Tf 15.2 8.08 Td[(1 4NXk,k0,qXs1,s2,s3,s4Vz(k)]TJ /F7 11.955 Tf 11.96 0 Td[(k0)3s1,s23s3,s4cyk0s1cy)]TJ /F5 7.97 Tf 6.59 0 Td[(k0+qs3c)]TJ /F5 7.97 Tf 6.58 0 Td[(k+qs4cks2, (4) andinthetransversespin-uctuationchannel H+)]TJ /F6 11.955 Tf 10.4 1.79 Td[(=)]TJ /F6 11.955 Tf 15.2 8.09 Td[(1 4NXk,k0,qXs1,s2,s3,s4V+)]TJ /F6 11.955 Tf 7.09 1.79 Td[((k)]TJ /F7 11.955 Tf 11.95 0 Td[(k0)(+s1,s2)]TJ /F9 7.97 Tf -0.43 -7.9 Td[(s3,s4+)]TJ /F9 7.97 Tf -0.43 -7.9 Td[(s1,s2+s3,s4)cyk0s1cy)]TJ /F5 7.97 Tf 6.58 0 Td[(k0+qs3c)]TJ /F5 7.97 Tf 6.59 0 Td[(k+qs4cks2 (4) 55

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where Vc(q)=U2000(q) 1+U000(q), (4a)Vz(q)=U2zz0(q) 1)]TJ /F3 11.955 Tf 11.96 0 Td[(Uzz0(q), (4b)V+)]TJ /F6 11.955 Tf 7.08 1.79 Td[((q)=U2^+)]TJ /F5 7.97 Tf -6.58 -7.98 Td[(0(q) 1)]TJ /F3 11.955 Tf 11.96 0 Td[(U^+)]TJ /F5 7.97 Tf -6.59 -7.97 Td[(0(q), (4c)and000,zz0and^0+)]TJ /F1 11.955 Tf 10.4 -5.15 Td[(arebarestatic(!=0)susceptibilityforcharge,longitudinalspinandandtransversespin.TheseresultswereobtainedbySchriefferetal.[ 12 ].Intheparamagneticstate,spinrotationalinvarianceimplieszz0=1 2+)]TJ /F5 7.97 Tf -6.59 -7.97 Td[(0=000.However,intheantiferromagneticstatewithstaggeredmagnetizationalong^z-axis,zz06=1 2+)]TJ /F5 7.97 Tf -6.58 -7.97 Td[(0,andtheonlyremainingdegeneracyiszz0=000.Inthesuperconductingstate,allsymmetriesarebroken.Westudytheinstabilityintheantiferromagneticbackground,thereforeweconsiderthesusceptibilityinthepureantiferromagneticstate.TheexpressionsforthebaresusceptibilitiesareshowninEquations 3 3 and 3 .Therelativemagnitudeofthechargepotential,Vc,longitudinal,VzandtransverseV+)]TJ /F1 11.955 Tf 10.41 1.8 Td[(canbeestimated.Therenormalizationforthechargepotentialisnotstrongasforthelongitudinalspinone.WehavealwaysVz>Vcforthesamemomentum.V+)]TJ /F1 11.955 Tf 10.4 1.79 Td[(hasasingularityattheorderingvectorQthereforeitisadominantcontribution.Intheantiferromagneticstate,thespinuctuationsaremodiedduetothebreakingofrotationalsymmetry.WeapplytheunitarytransformationdescribedinEquation 2 56

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totheeffectiveHamiltonian,andget Hc=1 4NXk,k0Xs1,s2,s3,s4f[2U)]TJ /F3 11.955 Tf 11.96 0 Td[(Vc(k)]TJ /F7 11.955 Tf 11.95 0 Td[(k0)]l2(k,k0)s1,s2s3,s4+[2U)]TJ /F3 11.955 Tf 11.96 0 Td[(Vc(k)]TJ /F7 11.955 Tf 11.96 0 Td[(k0+Q)]m2(k,k0)3s1,s23s3,s4g[yk0s1y)]TJ /F5 7.97 Tf 6.58 0 Td[(k0s3)]TJ /F5 7.97 Tf 6.59 0 Td[(ks4ks2+yk0s1y)]TJ /F5 7.97 Tf 6.59 0 Td[(k0s3)]TJ /F5 7.97 Tf 6.59 0 Td[(ks4ks2]+f[2U)]TJ /F3 11.955 Tf 11.95 0 Td[(Vc(k)]TJ /F7 11.955 Tf 11.95 0 Td[(k0)]p2(k,k0)s1,s2s3,s4+[2U)]TJ /F3 11.955 Tf 11.96 0 Td[(Vc(k)]TJ /F7 11.955 Tf 11.95 0 Td[(k0+Q)]n2(k,k0)3s1,s23s3,s4g[yk0s1y)]TJ /F5 7.97 Tf 6.58 0 Td[(k0s3)]TJ /F5 7.97 Tf 6.59 0 Td[(ks4ks2+yk0s1y)]TJ /F5 7.97 Tf 6.59 0 Td[(k0s3)]TJ /F5 7.97 Tf 6.59 0 Td[(ks4ks4],(4) Hz=)]TJ /F6 11.955 Tf 17.85 8.08 Td[(1 4NXk,k0Xs1,s2,s3,s4f[Vz(k)]TJ /F7 11.955 Tf 11.95 0 Td[(k0)]l2(k,k0)3s1,s23s3,s4+[Vz(k)]TJ /F7 11.955 Tf 11.95 0 Td[(k0+Q)]m2(k,k0)s1,s2s3,s4g[yk0s1y)]TJ /F5 7.97 Tf 6.58 0 Td[(k0s3)]TJ /F5 7.97 Tf 6.59 0 Td[(ks4ks2+yk0s1y)]TJ /F5 7.97 Tf 6.59 0 Td[(k0s3)]TJ /F5 7.97 Tf 6.59 0 Td[(ks4ks2]+f[Vz(k)]TJ /F7 11.955 Tf 11.95 0 Td[(k0)]p2(k,k0)3s1,s23s3,s4+[Vz(k)]TJ /F7 11.955 Tf 11.96 0 Td[(k0+Q)]n2(k,k0)s1,s2s3,s4g[yk0s1y)]TJ /F5 7.97 Tf 6.58 0 Td[(k0s3)]TJ /F5 7.97 Tf 6.59 0 Td[(ks4ks2+yk0s1y)]TJ /F5 7.97 Tf 6.59 0 Td[(k0s3)]TJ /F5 7.97 Tf 6.59 0 Td[(ks4ks2],(4)and H+)]TJ /F6 11.955 Tf 10.4 1.79 Td[(=)]TJ /F6 11.955 Tf 17.85 8.09 Td[(1 4NXk,k0Xs1,s2,s3,s4f[V+)]TJ /F6 11.955 Tf 7.08 1.79 Td[((k)]TJ /F7 11.955 Tf 11.96 0 Td[(k0)]n2(k,k0))]TJ /F6 11.955 Tf 11.96 0 Td[([V+)]TJ /F6 11.955 Tf 7.09 1.79 Td[((k)]TJ /F7 11.955 Tf 11.95 0 Td[(k0+Q)]p2(k,k0)g(+s1,s2)]TJ /F9 7.97 Tf -0.42 -7.89 Td[(s3,s4+)]TJ /F9 7.97 Tf -0.43 -7.89 Td[(s1,s2+s3,s4)[yk0s1y)]TJ /F5 7.97 Tf 6.59 0 Td[(k0s3)]TJ /F5 7.97 Tf 6.58 0 Td[(ks4ks2+yk0s1y)]TJ /F5 7.97 Tf 6.59 0 Td[(k0s3)]TJ /F5 7.97 Tf 6.58 0 Td[(ks4ks2]+f[V+)]TJ /F6 11.955 Tf 7.08 1.8 Td[((k)]TJ /F7 11.955 Tf 11.96 0 Td[(k0)]m2(k,k0))]TJ /F6 11.955 Tf 11.95 0 Td[([V+)]TJ /F6 11.955 Tf 7.08 1.8 Td[((k)]TJ /F7 11.955 Tf 11.96 0 Td[(k0+Q)]l2(k,k0)g(+s1,s2)]TJ /F9 7.97 Tf -0.42 -7.9 Td[(s3,s4+)]TJ /F9 7.97 Tf -0.43 -7.9 Td[(s1,s2+s3,s4)[yk0s1y)]TJ /F5 7.97 Tf 6.59 0 Td[(k0s3)]TJ /F5 7.97 Tf 6.59 0 Td[(ks4ks2+yk0s1y)]TJ /F5 7.97 Tf 6.58 0 Td[(k0s3)]TJ /F5 7.97 Tf 6.59 0 Td[(ks4ks2],(4) 57

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withthecoherencefactors m(k,k0)=ukv0k+vku0k (4a)l(k,k0)=uku0k+vkv0k (4b)p(k,k0)=ukv0k)]TJ /F3 11.955 Tf 11.95 0 Td[(vku0k (4c)n(k,k0)=uku0k)]TJ /F3 11.955 Tf 11.95 0 Td[(vkv0k. (4d)InSchrieffer,WenandZhang'sstudy[ 12 ],theyconsideredthehole-doped-casewithonlyholepocketsandignoredallthetermsexcepttheonesonlywiththeholebandoperators,.HereoureffectiveHamiltonianisgeneralbutreducestotheirresultinthislimit.ThetotalmeaneldHamiltoniancorrespondstotheeffectiveHamiltonianinclud-ingthekineticterm,HubbardtermandthesuperconductingeffectiveinteractioninEquations 4 4 and 4 canbeexpressedintermsoftheSDWquasiparticles, H=XkEykk)]TJ /F12 11.955 Tf 14.27 11.36 Td[(Xk00)]TJ /F5 7.97 Tf 6.58 0 Td[(kk0)]TJ /F12 11.955 Tf 14.26 11.36 Td[(Xk00yky)]TJ /F5 7.97 Tf 6.59 0 Td[(k0. (4) Here=,aretheindicesofthebands.Thegapfunction0(k)isamatrixinspinspace, ^(k)=0B@""(k)"#(k)#"(k)##(k)1CA=0B@)]TJ /F3 11.955 Tf 9.3 0 Td[(dx(k)+idy(k)s(k)+dz(k))]TJ /F6 11.955 Tf 9.3 0 Td[(s(k)+dz(k)dx(k)+idy(k).1CA (4) Weobtainthediagonaltermofthematrixinspinspaceas (k)=)]TJ /F6 11.955 Tf 15.19 8.09 Td[(1 4NXk0h)]TJ /F8 7.97 Tf 6.78 -1.79 Td[((k,k0)+)]TJ /F9 7.97 Tf 26.26 4.94 Td[(zs(k,k0)ih)]TJ /F5 7.97 Tf 6.58 0 Td[(k"k"i (4) +h)]TJ /F4 7.97 Tf 6.78 4.94 Td[(0(k,k0)+)]TJ /F9 7.97 Tf 26.27 4.94 Td[(z0s(k,k0)ih0)]TJ /F5 7.97 Tf 6.59 0 Td[(k"0k"i, 58

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whereand0areoppositebands,andtheoff-diagonaltermsas (k)=)]TJ /F6 11.955 Tf 15.19 8.09 Td[(1 4NXk0h)]TJ /F8 7.97 Tf 6.78 -1.79 Td[((k,k0))]TJ /F6 11.955 Tf 11.96 0 Td[()]TJ /F9 7.97 Tf 6.77 4.94 Td[(zs(k,k0)ih)]TJ /F5 7.97 Tf 6.59 0 Td[(kki+h)]TJ /F4 7.97 Tf 6.77 4.94 Td[(0(k,k0))]TJ /F6 11.955 Tf 11.95 0 Td[()]TJ /F9 7.97 Tf 6.78 4.94 Td[(z0s(k,k0)ih0)]TJ /F5 7.97 Tf 6.58 0 Td[(k0ki+2h)]TJ /F4 7.97 Tf 6.77 4.94 Td[(?s(k,k0)ih)]TJ /F5 7.97 Tf 6.58 0 Td[(kki+2h)]TJ /F4 7.97 Tf 6.77 4.94 Td[(?0s(k,k0)ih0)]TJ /F5 7.97 Tf 6.59 0 Td[(k0ki. (4) The)]TJ /F1 11.955 Tf 6.78 0 Td[(saredenedas )]TJ /F8 7.97 Tf 6.77 -1.79 Td[((k,k0)=[2U)]TJ /F3 11.955 Tf 11.95 0 Td[(Vc(k)]TJ /F7 11.955 Tf 11.96 0 Td[(k0)]l2(k,k0))]TJ /F6 11.955 Tf 11.96 0 Td[([Vz(k)]TJ /F7 11.955 Tf 11.95 0 Td[(k0+Q)]m2(k,k0), (4a))]TJ /F4 7.97 Tf 6.77 4.93 Td[(0(k,k0)=[2U)]TJ /F3 11.955 Tf 11.95 0 Td[(Vc(k)]TJ /F7 11.955 Tf 11.96 0 Td[(k0)]p2(k,k0))]TJ /F6 11.955 Tf 11.95 0 Td[([Vz(k)]TJ /F7 11.955 Tf 11.96 0 Td[(k0+Q)]n2(k,k0), (4b))]TJ /F9 7.97 Tf 6.78 4.93 Td[(zs(k,k0)=[2U)]TJ /F3 11.955 Tf 11.95 0 Td[(Vc(k)]TJ /F7 11.955 Tf 11.96 0 Td[(k0+Q)]m2(k,k0))]TJ /F6 11.955 Tf 11.95 0 Td[([Vz(k)]TJ /F7 11.955 Tf 11.95 0 Td[(k0)]l2(k,k0), (4c))]TJ /F9 7.97 Tf 6.77 4.94 Td[(z0s(k,k0)=[2U)]TJ /F3 11.955 Tf 11.95 0 Td[(Vc(k)]TJ /F7 11.955 Tf 11.96 0 Td[(k0+Q)]n2(k,k0))]TJ /F6 11.955 Tf 11.96 0 Td[([Vz(k)]TJ /F7 11.955 Tf 11.95 0 Td[(k0)]p2(k,k0), (4d))]TJ /F4 7.97 Tf 6.77 4.93 Td[(?s(k,k0)=)]TJ /F6 11.955 Tf 9.3 0 Td[([V+)]TJ /F6 11.955 Tf 7.08 1.8 Td[((k)]TJ /F7 11.955 Tf 11.96 0 Td[(k0)]n2(k,k0)+[V+)]TJ /F6 11.955 Tf 7.09 1.8 Td[((k)]TJ /F7 11.955 Tf 11.95 0 Td[(k0+Q)]p2(k,k0), (4e))]TJ /F4 7.97 Tf 6.78 4.94 Td[(?0s(k,k0)=)]TJ /F6 11.955 Tf 9.3 0 Td[([V+)]TJ /F6 11.955 Tf 7.08 1.79 Td[((k)]TJ /F7 11.955 Tf 11.96 0 Td[(k0)]m2(k,k0)+[V+)]TJ /F6 11.955 Tf 7.08 1.79 Td[((k)]TJ /F7 11.955 Tf 11.96 0 Td[(k0+Q)]l2(k,k0), (4f)wheretheprimedverticesindicatetheinter-bandinteractions.Toseparatethegapfunctionsintosingletchannelandtripletchannels,wefollowthestandarddenition, ^(k)=0B@)]TJ /F3 11.955 Tf 9.3 0 Td[(dx(k)+idy(k)d0(k)+dz(k))]TJ /F3 11.955 Tf 9.3 0 Td[(d0(k)+dz(k)dx(k)+idy(k),1CA (4) andweget d0(k)=1 2("#)]TJ /F6 11.955 Tf 11.95 0 Td[(#") (4a)dx(k)=1 2()]TJ /F6 11.955 Tf 9.29 0 Td[(""+##) (4b)dy(k)=)]TJ /F3 11.955 Tf 9.3 0 Td[(i 2(""+##) (4c)dz(k)=1 2("#+#"). (4d) 59

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Thegapequationsforthetripletgapsare dx=y(k)=)]TJ /F6 11.955 Tf 15.2 8.09 Td[(1 4NXk0h)]TJ /F8 7.97 Tf 6.78 -1.79 Td[((k,k0)+)]TJ /F9 7.97 Tf 26.27 4.94 Td[(zs(k,k0)idi(k0) 2k0tanh)]TJ /F6 11.955 Tf 6.75 -1.6 Td[(k0 2T (4) +h)]TJ /F4 7.97 Tf 6.77 4.94 Td[(0(k,k0)+)]TJ /F9 7.97 Tf 26.27 4.94 Td[(z0s(k,k0)id0i(k0) 20k0tanh)]TJ /F6 11.955 Tf 6.67 -1.6 Td[(0k0 2T (4) and dz(k)=)]TJ /F6 11.955 Tf 15.2 8.09 Td[(1 4NXk0h)]TJ /F8 7.97 Tf 6.77 -1.79 Td[((k,k0))]TJ /F6 11.955 Tf 11.95 0 Td[()]TJ /F9 7.97 Tf 6.78 4.94 Td[(zs(k,k0)+2)]TJ /F4 7.97 Tf 32.55 4.94 Td[(?s(k,k0)i0(k0) 2k0tanh)]TJ /F6 11.955 Tf 6.76 -1.6 Td[(k0 2T+h)]TJ /F4 7.97 Tf 6.77 4.94 Td[(0(k,k0))]TJ /F6 11.955 Tf 11.95 0 Td[()]TJ /F9 7.97 Tf 6.78 4.94 Td[(z0s(k,k0)+2)]TJ /F4 7.97 Tf 32.54 4.94 Td[(?0s(k,k0)i00(k0) 20k0tanh)]TJ /F6 11.955 Tf 6.67 -1.6 Td[(0k0 2T, (4) andforthesingletgapsare s(k)d0(k)=)]TJ /F6 11.955 Tf 17.85 8.08 Td[(1 4NXk0h)]TJ /F8 7.97 Tf 6.78 -1.8 Td[((k,k0))]TJ /F6 11.955 Tf 11.96 0 Td[()]TJ /F9 7.97 Tf 6.77 4.93 Td[(zs(k,k0))]TJ /F6 11.955 Tf 11.96 0 Td[(2)]TJ /F4 7.97 Tf 13.05 4.93 Td[(?s(k,k0)is(k0) 2k0tanh)]TJ /F6 11.955 Tf 6.76 -1.6 Td[(k0 2T+h)]TJ /F4 7.97 Tf 6.77 4.93 Td[(0(k,k0))]TJ /F6 11.955 Tf 11.96 0 Td[()]TJ /F9 7.97 Tf 6.77 4.93 Td[(z0s(k,k0))]TJ /F6 11.955 Tf 11.96 0 Td[(2)]TJ /F4 7.97 Tf 13.05 4.93 Td[(?0s(k,k0)i0s(k0) 20k0tanh)]TJ /F6 11.955 Tf 6.68 -1.59 Td[(0k0 2T.Fromthedenitionofthecoherencefactors,m2,l2,p2andn2,Equations 4 ,wecanwritedownthegeneralexpressions, m2(k,k0)=1 2h1)]TJ /F10 11.955 Tf 56.32 8.09 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k")]TJ /F5 7.97 Tf 0 -8.34 Td[(k0)]TJ /F3 11.955 Tf 11.95 0 Td[(W2 p (")]TJ /F5 7.97 Tf 0 -8.28 Td[(k)2+W2p (")]TJ /F5 7.97 Tf 0 -8.34 Td[(k0)2+W2i, (4a)l2(k,k0)=1 2h1+")]TJ /F5 7.97 Tf 0 -8.28 Td[(k")]TJ /F5 7.97 Tf 0 -8.33 Td[(k0+W2 p (")]TJ /F5 7.97 Tf 0 -8.27 Td[(k)2+W2p (")]TJ /F5 7.97 Tf 0 -8.33 Td[(k0)2+W2i, (4b)p2(k,k0)=1 2h1)]TJ /F10 11.955 Tf 56.32 8.09 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k")]TJ /F5 7.97 Tf 0 -8.33 Td[(k0+W2 p (")]TJ /F5 7.97 Tf 0 -8.28 Td[(k)2+W2p (")]TJ /F5 7.97 Tf 0 -8.33 Td[(k0)2+W2i, (4c)n2(k,k0)=1 2h1+")]TJ /F5 7.97 Tf 0 -8.28 Td[(k")]TJ /F5 7.97 Tf 0 -8.34 Td[(k0)]TJ /F3 11.955 Tf 11.95 0 Td[(W2 p (")]TJ /F5 7.97 Tf 0 -8.28 Td[(k)2+W2p (")]TJ /F5 7.97 Tf 0 -8.34 Td[(k0)2+W2i. (4d) 60

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Byusingthecondition")]TJ /F5 7.97 Tf 0 -8.27 Td[(k+Q=)]TJ /F10 11.955 Tf 9.3 0 Td[(")]TJ /F5 7.97 Tf 0 -8.27 Td[(k,wehavetherelationsm2(k+Q,k0)=m2(k,k0+Q)=l2(k,k0)andp2(k+Q,k0)=p(k,k0+Q)=n2(k,k0).Thisresultsinaperiodicityconditionforthepotentials.Wehavem2(k+Q,k0=m2(k,k0+Q)=l2(k,k0).Thisimplies)]TJ /F8 7.97 Tf 6.78 -1.79 Td[((k,k0+Q)=)]TJ /F9 7.97 Tf 29.87 4.34 Td[(zs(k,k0)and)]TJ /F4 7.97 Tf 6.77 4.34 Td[(?s(k,k0+Q)=)]TJ /F6 11.955 Tf 9.29 0 Td[()]TJ /F4 7.97 Tf 6.78 4.34 Td[(?s(k,k0).Theserelationsguaranteetheantiperiodicconditionof Vs(q+Q)=)]TJ /F3 11.955 Tf 9.3 0 Td[(Vs(q),(4)whereVsisthesingletgappotentialVs(k)]TJ /F7 11.955 Tf 10.83 0 Td[(k0)=)]TJ /F8 7.97 Tf 27.59 -1.8 Td[((k,k0))]TJ /F6 11.955 Tf 10.83 0 Td[()]TJ /F9 7.97 Tf 6.77 4.34 Td[(zs(k,k0))]TJ /F6 11.955 Tf 10.83 0 Td[(2)]TJ /F4 7.97 Tf 13.05 4.34 Td[(?s(k,k0)[ 12 ].Theantiperiodicityofthepotentialleadstotheantiperiodicityofthesuperconductinggap, s(q+Q)=)]TJ /F6 11.955 Tf 9.29 0 Td[(s(q)(4)inordertofulllthegapequation.Wecancheckthatthegapequationintheantiferromagneticstatereducestothegapequationintheparamagneticsuperconductingstate.Westartwithsingletgapfunction s(k)=)]TJ /F6 11.955 Tf 15.19 8.09 Td[(1 4NXk00h2U(l2)]TJ /F3 11.955 Tf 11.95 0 Td[(m2))]TJ /F6 11.955 Tf 11.95 0 Td[([Vc(k)]TJ /F7 11.955 Tf 11.96 0 Td[(k0))]TJ /F3 11.955 Tf 11.95 0 Td[(Vz(k)]TJ /F7 11.955 Tf 11.96 0 Td[(k0)]l2+[Vc(k)]TJ /F7 11.955 Tf 11.96 0 Td[(k0+Q))]TJ /F3 11.955 Tf 11.96 0 Td[(Vz(k)]TJ /F7 11.955 Tf 11.96 0 Td[(k0+Q)]m2+2V+)]TJ /F6 11.955 Tf 7.09 1.79 Td[((k)]TJ /F7 11.955 Tf 11.95 0 Td[(k0)n2)]TJ /F6 11.955 Tf 11.96 0 Td[(2V+)]TJ /F6 11.955 Tf 7.08 1.79 Td[((k)]TJ /F7 11.955 Tf 11.96 0 Td[(k0+Q)p2is(k0) 2k0tanh)]TJ /F6 11.955 Tf 6.75 -1.6 Td[(k0 2T+h2U(p2)]TJ /F3 11.955 Tf 11.96 0 Td[(n2))]TJ /F12 11.955 Tf 11.95 9.68 Td[(Vc(k)]TJ /F7 11.955 Tf 11.96 0 Td[(k0))]TJ /F3 11.955 Tf 11.95 0 Td[(Vz(k)]TJ /F7 11.955 Tf 11.96 0 Td[(k0)]p2+[Vc(k)]TJ /F7 11.955 Tf 11.95 0 Td[(k0+Q))]TJ /F3 11.955 Tf 11.95 0 Td[(Vz(k)]TJ /F7 11.955 Tf 11.95 0 Td[(k0+Q)]n2+2V+)]TJ /F6 11.955 Tf 7.09 1.79 Td[((k)]TJ /F7 11.955 Tf 11.95 0 Td[(k0)m2)]TJ /F6 11.955 Tf 11.95 0 Td[(2V+)]TJ /F6 11.955 Tf 7.08 1.79 Td[((k)]TJ /F7 11.955 Tf 11.96 0 Td[(k0+Q)l2is(k0) 2k0tanh)]TJ /F6 11.955 Tf 6.76 -1.6 Td[(k0 2T.(4)Intheparamagneticstate,weonlyhaveoneband,thereforethesuperconductinggaphasonlyoneform=!s.Bycombiningtheinter-andintra-bandinteractionsin 61

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theoriginalantiferromagneticstate,theexpressionreducesto s!)]TJ /F5 7.97 Tf 29.07 4.71 Td[(1 4NPk00h)]TJ /F6 11.955 Tf 11.95 0 Td[([Vc(k)]TJ /F7 11.955 Tf 11.96 0 Td[(k0))]TJ /F3 11.955 Tf 11.95 0 Td[(Vz(k)]TJ /F7 11.955 Tf 11.96 0 Td[(k0)]+[Vc(k)]TJ /F7 11.955 Tf 11.96 0 Td[(k0+Q))]TJ /F3 11.955 Tf 11.96 0 Td[(Vz(k)]TJ /F7 11.955 Tf 11.95 0 Td[(k0+Q)]+2V+)]TJ /F6 11.955 Tf 7.08 1.8 Td[((k)]TJ /F7 11.955 Tf 11.96 0 Td[(k0))]TJ /F6 11.955 Tf 11.96 0 Td[(2V+)]TJ /F6 11.955 Tf 7.09 1.8 Td[((k)]TJ /F7 11.955 Tf 11.95 0 Td[(k0+Q)is(k0) 2Ek0tanh)]TJ /F9 7.97 Tf 6.69 -3.69 Td[(Ek0 2T. (4) Hereweusetheantiperiodicitys(k)=)]TJ /F6 11.955 Tf 9.29 0 Td[(s(k+Q)inthegapandget s(k)=)]TJ /F6 11.955 Tf 15.2 8.08 Td[(1 4NXk00h)]TJ /F6 11.955 Tf 11.96 0 Td[([Vc(k)]TJ /F7 11.955 Tf 11.95 0 Td[(k0))]TJ /F3 11.955 Tf 11.95 0 Td[(Vz(k)]TJ /F7 11.955 Tf 11.95 0 Td[(k0)]+2V+)]TJ /F6 11.955 Tf 7.08 1.8 Td[((k)]TJ /F7 11.955 Tf 11.96 0 Td[(k0)is(k0) 2Ek0tanh)]TJ /F3 11.955 Tf 7.55 -1.6 Td[(Ek0 2T+[Vc(k)]TJ /F7 11.955 Tf 11.96 0 Td[(k0+Q))]TJ /F3 11.955 Tf 11.96 0 Td[(Vz(k)]TJ /F7 11.955 Tf 11.96 0 Td[(k0+Q)])]TJ /F6 11.955 Tf 11.96 0 Td[(2V+)]TJ /F6 11.955 Tf 7.08 1.79 Td[((k)]TJ /F7 11.955 Tf 11.95 0 Td[(k0+Q)i)]TJ /F6 11.955 Tf 9.3 0 Td[(s(k0+Q) 2Ek0+Qtanh)]TJ /F3 11.955 Tf 6.68 -1.6 Td[(Ek0+Q 2T. (4) UsingtheconditionthatVz(k,k0)=V+)]TJ /F6 11.955 Tf 7.09 1.8 Td[((k,k0)Vs(k,k0)intheparamagneticstate,andrestrictingthesumtothefullBrillouinzone,wehave (k)=)]TJ /F6 11.955 Tf 15.2 8.09 Td[(1 2NXk00h3 2Vs(k)]TJ /F3 11.955 Tf 11.95 0 Td[(k0))]TJ /F6 11.955 Tf 13.15 8.09 Td[(1 2Vc(k)]TJ /F3 11.955 Tf 11.95 0 Td[(k0)is(k0) 2Ek0tanh)]TJ /F3 11.955 Tf 7.2 -1.6 Td[(Ek0 2T+3 2[Vs(k)]TJ /F3 11.955 Tf 11.95 0 Td[(k0+Q))]TJ /F6 11.955 Tf 13.15 8.09 Td[(1 2Vc(k)]TJ /F3 11.955 Tf 11.96 0 Td[(k0+Q)]is(k0+Q) 2Ek0+Qtanh)]TJ /F3 11.955 Tf 6.68 -1.59 Td[(Ek0+Q 2T=)]TJ /F6 11.955 Tf 15.2 8.09 Td[(1 2NXk0h3 2Vs(k)]TJ /F3 11.955 Tf 11.95 0 Td[(k0))]TJ /F6 11.955 Tf 13.15 8.09 Td[(1 2Vc(k)]TJ /F3 11.955 Tf 11.95 0 Td[(k0)is(k0) 2Ek0tanh)]TJ /F3 11.955 Tf 7.2 -1.6 Td[(Ek0 2T.(4)Thisgapequationisequivalenttotheresultinstudiesofsuperconductivityintheparamagneticstate[ 15 ].Notethatthesignoftheoverallinteraction3 2Vs)]TJ /F5 7.97 Tf 14.4 4.71 Td[(1 2Vcisrepulsive,i.e.>0. 4.2ThepairingsymmetriesThesingletsuperconductinggapsaretwonon-linearcoupledequationsfortheandbands.Intheweakcouplinglimit,wecanassumethattheinteractiononlyhappensaroundtheFermisurface.Thenwecansolvethemself-consistentlywithnumericalmethods.Thepairingpotentialisacomplicatedfunctionofk)]TJ /F7 11.955 Tf 12.61 0 Td[(k0.Itisnotpossibletoanalyzethesymmetryofthefullpotentialwithoutfurtherapproximation.Althoughweknowtheantiperiodicityofthepotential,thisisnotenoughtodeterminethesymmetryofthegaps.Withouttheknowledgeofthesymmetryofthegaps,the 62

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calculationwouldbeexpensive.TherearealsonumericaldifcultiesassociatedwiththedescriptionofthesingularityinthetransverseRPAsusceptibility.Theseobstaclestoacompletesolutioncanbeovercome,butitisusefultohaveananalyticalsolutioninawell-denedlimittoguidethecalculation.HerewetakethelimitofsmallpocketsizewhichappearsundersmalldopingwithlargeantiferromagneticorderW.WeestimatetheleadingsymmetryofthegapsbyexpandingthecoherencefactorsandtheRPAsusceptibilitiesaroundthecentersoftheelectronpockets,k=(,0),(0,)andtheholepockets,k=(=2,=2)assumingsmallcircularpocketsize.Theellipticityofthepocketsizeoranydeviationfromperfectcircularityshouldonlychangetheweightsoftheharmonicsbutnottheoverallsymmetryofthegap.Thenwecomparetheexpandedpotentialwiththeprojectedgapsymmetriestodeterminetheleadingsymmetryofthesuperconductinggaps. 4.2.1AngulardependenceofthecoherencefactorsFirst,inordertoexpandthecoherencefactorsaroundholepockets,wetakeEquation 4 ,andexpandkandk0aroundthepocketscenters.Hereweusetheexampleofanintrabandexpansionaroundk=k0=(=2,=2)toexplaintheprocess.Weassumesmallquantities~kand~k0suchthatk=( 2, 2)+~kandk0=( 2, 2)+~k0.WeuseMathematicatoexpandthecoherencefactorsinEquation 4 .Weget,forinstance,theleadingterms m2(k,k0)1)]TJ /F3 11.955 Tf 17.17 8.08 Td[(t2 W2(~kx+~ky)]TJ /F10 11.955 Tf 11.95 0 Td[(~k0x)]TJ /F10 11.955 Tf 11.95 0 Td[(~k0y)2(4)Thenwereplace~kand~k0withtheangulardependentexpressionalongthesmallpockets,~kx+~ky=p 2khFcosand~kx)]TJ /F10 11.955 Tf 12.8 0 Td[(~ky=p 2khFsinwherekhFistheFermimomentumorpocketradiusontheholepockets.Theanglesforholepocketsisandforelectronpocketis,asshowninFigure 4-1 .Thepresenceoftheholepocketscenteredaround(=2,=2)pointsandtheelectronpocketsaround(,0)and(0,)pointsoftheBrillouinzonedependonthetypes(electronorhole)andthe 63

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Figure4-1.GeneralstructureoftheFermisurfaceoflayeredcupratesinthecommensurateantiferromagneticstateforelectronorholedoping. amountofdoping.Notethatthedenitionoftheholeangleatzerodegreesis45fromthex)]TJ /F1 11.955 Tf 9.29 0 Td[(axis.Thisisforsimplifyingtheexpressionontheholepockets.Theexpansionsofthecoherencefactorsp2(k,k0)andn2(k,k0)aroundsmallholepocketswiththeleadingtermsarepresentedinTable 4-1 withk=( 2, 2)andTable 4-2 withk=()]TJ /F8 7.97 Tf 10.49 4.71 Td[( 2, 2).Theothertwocoherencefactorscanbeobtainedfromtherelations Table4-1.Coherencefactors,p2(k,k0)andn2(k,k0)expandedaroundholepocketsfork=( 2, 2)andk0=( 2, 2).Thecoherencefactorsl2andm2canbeobtainedfroml2(k,k0)=1)]TJ /F3 11.955 Tf 11.95 0 Td[(p2(k,k0)andm2(k,k0)=1)]TJ /F3 11.955 Tf 11.96 0 Td[(n2(k,k0). k0( 2, 2)()]TJ /F8 7.97 Tf 10.5 4.7 Td[( 2, 2) p2(k,k0)t2(khF)2 W2(2)]TJ /F6 11.955 Tf 11.95 0 Td[(4coscos0+cos2+cos20)t2(khF)2 W2(2)]TJ /F6 11.955 Tf 11.96 0 Td[(4cossin0+cos2)]TJ /F6 11.955 Tf 11.95 0 Td[(cos20)n2(k,k0)t2(khF)2 W2(2+4coscos0+cos2+cos20)t2(khF)2 W2(2+4cossin0+cos2)]TJ /F6 11.955 Tf 11.95 0 Td[(cos20) l2(k,k0)=1)]TJ /F3 11.955 Tf 12.9 0 Td[(p2(k,k0)andm2(k,k0)=1)]TJ /F3 11.955 Tf 12.9 0 Td[(n2(k,k0).Theexpansionofk0around 64

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Table4-2.Coherencefactors,p2(k,k0)andn2(k,k0)expandedaroundholepocketsfork=()]TJ /F8 7.97 Tf 10.5 4.7 Td[( 2, 2)andk0=( 2, 2).Thecoherencefactorsl2andm2canbeobtainedfroml2(k,k0)=1)]TJ /F3 11.955 Tf 11.95 0 Td[(p2(k,k0)andm2(k,k0)=1)]TJ /F3 11.955 Tf 11.96 0 Td[(n2(k,k0). k0( 2, 2)()]TJ /F8 7.97 Tf 10.49 4.7 Td[( 2, 2) p2(k,k0)t2(khF)2 W2(2)]TJ /F6 11.955 Tf 11.95 0 Td[(4sincos0)]TJ /F6 11.955 Tf 11.95 0 Td[(cos2+cos20)t2(khF)2 W2(2)]TJ /F6 11.955 Tf 11.96 0 Td[(4sinsin0)]TJ /F6 11.955 Tf 11.3 0 Td[(cos2)]TJ /F6 11.955 Tf 11.95 0 Td[(cos20)n2(k,k0)t2(khF)2 W2(2+4sincos0)]TJ /F6 11.955 Tf 11.95 0 Td[(cos2+cos20)t2(khF)2 W2(2+4sinsin0)]TJ /F6 11.955 Tf 11.3 0 Td[(cos2)]TJ /F6 11.955 Tf 11.95 0 Td[(cos20) ()]TJ /F8 7.97 Tf 10.5 4.71 Td[( 2,)]TJ /F8 7.97 Tf 10.49 4.71 Td[( 2)and( 2,)]TJ /F8 7.97 Tf 10.49 4.71 Td[( 2)canbeobtainedbyusingtherelationp2(k,k0)=n2(k,k0+Q).whereQ=(,).Fortheexpansionsofthecoherencefactorsaroundelectronpockets,wehavetheresultsshowninTable 4-3 .Weonlyhaveevenorderharmonicsintheleadingtermsfortheelectronpockets.Andwecanseeclearly,therelationp2(k,k0)=n2(k,k0+Q)issatisedinTable 4-3 .Theinterbandinteractioninthegapequationsmaybeimportantwhenwehavebothpocketspresent.ThecoherencefactorsoftheinterbandexpansionsbetweenelectronandholepocketsareshowninTable 4-4 65

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Table4-3.Coherencefactors,p2(k,k0)andn2(k,k0)expandedaroundelectronpocketswithk=(,0)andk0=(,0)and(0,).Thecoherencefactorsl2andm2canbeobtainedfroml2(k,k0)=1)]TJ /F3 11.955 Tf 11.96 0 Td[(p2(k,k0)andm2(k,k0)=1)]TJ /F3 11.955 Tf 11.95 0 Td[(n2(k,k0). k0=(,0)(0,) p2(k,k0)t2(keF)4 4W21)]TJ /F6 11.955 Tf 11.95 0 Td[(2cos2cos20+1 2(cos4+cos40)t2(keF)4 4W21+2cos2cos20+1 2(cos4+cos40)n2(k,k0)t2(keF)4 4W21+2cos2cos20+1 2(cos4+cos40)t2(keF)4 4W21)]TJ /F6 11.955 Tf 11.96 0 Td[(2cos2cos20+1 2(cos4+cos40) Table4-4.Coherencefactors,p2(k,k0)andn2(k,k0)expandedaroundelectronandholepocketswithk=(,0)andk0around()]TJ /F8 7.97 Tf 10.5 4.71 Td[( 2,)]TJ /F8 7.97 Tf 10.49 4.71 Td[( 2)and( 2,)]TJ /F8 7.97 Tf 10.49 4.71 Td[( 2).Thecoherencefactorsl2andm2canbeobtainedfroml2(k,k0)=1)]TJ /F3 11.955 Tf 11.95 0 Td[(p2(k,k0)andm2(k,k0)=1)]TJ /F3 11.955 Tf 11.96 0 Td[(n2(k,k0). k0=( 2, 2)()]TJ /F8 7.97 Tf 10.49 4.71 Td[( 2, 2) p2(k,k0)t2 W2h(khF)2(1+cos20))]TJ 11.96 9.95 Td[(p 2khF(keF)2(cos2cos0)it2 W2h(khF)2(1)]TJ /F6 11.955 Tf 11.96 0 Td[(cos20)+p 2khF(keF)2(cos2sin0)in2(k,k0)t2 W2h(khF)2(1+cos20)+p 2khF(keF)2(cos2cos0)it2 W2h(khF)2(1)]TJ /F6 11.955 Tf 11.96 0 Td[(cos20))]TJ 11.95 9.95 Td[(p 2khF(keF)2(cos2sin0)i 66

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4.2.2AngulardependenceofthepairingpotentialsNowwehavetheleadingtermexpansionofthecoherencefactors.Inordertogettheapproximatedpotentials,wereplacethefullcoherencefactorswiththeirangulardependentapproximations.InSchriefferelal.'sstudy[ 12 ],theyignorethecontribu-tionsfromthetransversesusceptibility,appealingtotheAdlerprinciple[ 31 81 ],thesuppressionofthedivergentpairinginteractionattheorderingwavevectorbyver-texcorrections.FrenkelandHanke[ 81 ]andothers[ 14 77 ]showed,howeverthatthetransversecontributions(spinwaves)totheinteractionswereofthesameorderasthelongitudinalexcitations.Herewewouldliketoinvestigatetherelativeimportancebetweentransverseandlongitudinalchannels.Weseparatethesingletpairingpotentialintothecharge-andlongitudinalspin-uctuationpartandthetransversespin-uctuationpart.Wediscussthesituationwithbothelectronandholepocketspresentbelow.Tosimplifytheanalysis,weassumeonlytheintrabandinteractionforthemoment. 4.2.2.1ChargeandlongitudinalinteractionThecharge-andlongitudinalspin-uctuationcontributiontothepotentialforsingletchannelis)]TJ /F8 7.97 Tf 6.77 -1.79 Td[((k,k0))]TJ /F6 11.955 Tf 11.95 0 Td[()]TJ /F9 7.97 Tf 6.78 4.94 Td[(zzs(k,k0)=2U(l2)]TJ /F3 11.955 Tf 11.95 0 Td[(m2))]TJ /F12 11.955 Tf 11.95 13.27 Td[(hVc(k)]TJ /F7 11.955 Tf 11.95 0 Td[(k0))]TJ /F3 11.955 Tf 11.96 0 Td[(Vz(k)]TJ /F7 11.955 Tf 11.95 0 Td[(k0)il2+hVc(k)]TJ /F7 11.955 Tf 11.95 0 Td[(k0+Q))]TJ /F3 11.955 Tf 11.95 0 Td[(Vz(k)]TJ /F7 11.955 Tf 11.95 0 Td[(k0+Q)im2. (4)Withinthischannel,wecanseetheantiperiodicityisstillvalid.ReplacingthecoherencefactorswiththeapproximatedangularexpressionsinTables 4-1 and 4-2 ,wegetthelongitudinalpotentialasinTable 4-5 .ThesymbolsinthetableareV=Vc(Q))]TJ /F3 11.955 Tf 11.71 0 Td[(Vc(0)+Vz(0))]TJ /F3 11.955 Tf 12.19 0 Td[(Vz(Q)andV0=Vz(Q))]TJ /F3 11.955 Tf 12.2 0 Td[(Vc(Q)+Vz(0))]TJ /F3 11.955 Tf 12.2 0 Td[(Vc(0).NotethatthebarepotentialsareallpositiveandthelargestvaluewouldbeVz(Q).ThereforewehaveV<0andV0>0.Theexpansionsaroundk=ork0=()]TJ /F8 7.97 Tf 10.49 4.71 Td[( 2,)]TJ /F8 7.97 Tf 10.49 4.71 Td[( 2),( 2,)]TJ /F8 7.97 Tf 10.49 4.71 Td[( 2)canbeobtainedby)]TJ /F8 7.97 Tf 6.77 -1.8 Td[((k,k0))]TJ /F6 11.955 Tf 12.09 0 Td[()]TJ /F9 7.97 Tf 6.77 4.33 Td[(zzs(k,k0)=)]TJ /F6 11.955 Tf 9.3 0 Td[()]TJ /F8 7.97 Tf 6.78 -1.8 Td[((k+Q,k0)+)]TJ /F9 7.97 Tf 26.53 4.33 Td[(zzs(k+Q,k0)=)]TJ /F6 11.955 Tf 9.29 0 Td[()]TJ /F8 7.97 Tf 6.78 -1.8 Td[((k,k0+Q)+)]TJ /F9 7.97 Tf 26.53 4.33 Td[(zzs(k,k0+Q). 67

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Table4-5.Potentialsfromthecharge-andlongitudinalspin-uctuationcontribution,)]TJ /F8 7.97 Tf 6.77 -1.8 Td[((k,k0))]TJ /F6 11.955 Tf 11.95 0 Td[()]TJ /F9 7.97 Tf 6.78 4.33 Td[(zzs(k,k0)expandedaroundholepockets. HHHHHHkk0( 2, 2)()]TJ /F8 7.97 Tf 10.49 4.71 Td[( 2, 2) ( 2, 2)V1)]TJ /F5 7.97 Tf 13.15 6.48 Td[(2t2(khF)2 W22Ut2(khF)2 W28cossin0+4t2(khF)2 W2(4U+V0)coscos0)]TJ /F6 11.955 Tf 9.29 0 Td[(8[Vc(,0))]TJ /F3 11.955 Tf 11.95 0 Td[(Vz(,0)])]TJ /F6 11.955 Tf 11.16 2.65 Td[(Vt2(khF)2 W2(cos2+cos20)t2(khF)2 W2cossin0()]TJ /F8 7.97 Tf 10.49 4.71 Td[( 2, 2)2Ut2(khF)2 W28sincos0V1)]TJ /F5 7.97 Tf 13.15 6.48 Td[(2t2(khF)2 W2)]TJ /F6 11.955 Tf 9.29 0 Td[(8[Vc(,0))]TJ /F3 11.955 Tf 11.95 0 Td[(Vz(,0)]t2(khF)2 W2sincos0+4t2(khF)2 W2(4U+V0)sinsin0)]TJ /F6 11.955 Tf 11.16 2.65 Td[(Vt2(khF)2 W2(cos2+cos20) ThescatteringbetweenthepocketitselfortothepocketshiftedbyQismuchstrongerthanthescatteringwhichisshiftedby(,0)or(0,).Forthescatteringwithinthesameelectronpocket,thepotentialfromthecharge-andlongitudinalspin-uctuationcontributioninEquation 4 isthenapproximately )]TJ /F8 7.97 Tf 6.78 -1.79 Td[((k,k0))]TJ /F6 11.955 Tf 11.96 0 Td[()]TJ /F9 7.97 Tf 6.77 4.94 Td[(zzs(k,k0)V)]TJ /F6 11.955 Tf 11.96 0 Td[((V)]TJ /F6 11.955 Tf 11.96 0 Td[(8U)t2(keF)4 4W2+V0t2(keF)4 2W2(cos2cos20)+Vt2(keF)4 8W2(cos4+cos40).(4)Forscatteringbetweendifferentelectronpockets,weusetheantiperiodicrelation)]TJ /F8 7.97 Tf 6.77 -1.79 Td[((k,k0))]TJ /F6 11.955 Tf 12.57 0 Td[()]TJ /F9 7.97 Tf 6.78 4.34 Td[(zzs(k,k0)=)]TJ /F6 11.955 Tf 9.3 0 Td[()]TJ /F8 7.97 Tf 6.78 -1.79 Td[((k,k0+Q)+)]TJ /F9 7.97 Tf 27.5 4.34 Td[(zzs(k,k0+Q).Thisrelationcansimplifytheelectronmulti-pocketproblemintoaoneelectronpocketproblem.Laterwewillseethatwhensolvingthesuperconductinggapequation,wecanusethesesymmetriestoallowustoconsideronlyonepocketfortheelectronandtwopocketsfortheholecase.Althoughthecharge-andlongitudinalspin-potentialdoesnotdivergeandcanbeviewedasaconstantintheexpansions,forconsistencytoorderkF,wealsoexpandthesetwotermsaroundsmallq.Wehave V+)]TJ /F6 11.955 Tf 7.08 1.79 Td[((q)V+)]TJ /F6 11.955 Tf 7.09 1.79 Td[((0)+1 2(UV+)]TJ /F6 11.955 Tf 7.08 1.79 Td[((0))U(zz000(0) 1Uzz0(0)q2(4) 68

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and V+)]TJ /F6 11.955 Tf 7.08 1.8 Td[((q+Q)V+)]TJ /F6 11.955 Tf 7.09 1.8 Td[((Q)+1 2(UV+)]TJ /F6 11.955 Tf 7.08 1.8 Td[((Q))U(zz000(Q) 1Uzz0(Q)q2.(4)whereV+andV)]TJ /F1 11.955 Tf 10.41 1.79 Td[(correspondtoVcandVz,respectivelyand000(x)isthesecondderivativeofwithrespecttoqandevaluatedatx. 4.2.2.2TransverseinteractionThecontributionfromthetransversespin-uctuationsis)]TJ /F6 11.955 Tf 9.3 0 Td[(2)]TJ /F4 7.97 Tf 13.05 4.94 Td[(?s(k,k0)=2[V+)]TJ /F6 11.955 Tf 7.08 1.79 Td[((k)]TJ /F7 11.955 Tf 11.96 0 Td[(k0)]n2)]TJ /F6 11.955 Tf 11.95 0 Td[(2[V+)]TJ /F6 11.955 Tf 7.08 1.79 Td[((k)]TJ /F7 11.955 Tf 11.96 0 Td[(k0+Q)]p2. (4)Herewehavesingularitieswhenk=k0intheevaluationofV+)]TJ /F6 11.955 Tf 7.08 1.79 Td[((k)]TJ /F7 11.955 Tf 12.49 0 Td[(k0+Q)orwhenk=k0+Q.Butthecoherencefactorsp2(k,k0)andn2(k,k0)arezerowhenk=k0andk=k0+Q,respectively.Thereforewehavetoexpandthetransversesusceptibilityandthecoherencefactorstoobtainthelimitatthesingularpoint.WefollowFrenkelandHanke'sstudy[ 29 31 77 81 82 ],andget V+)]TJ /F6 11.955 Tf 7.08 1.8 Td[((Q)'1 t2yq2=1 t2y(keF)2(2)]TJ /F6 11.955 Tf 11.95 0 Td[(2(coscos0+sinsin0))]TJ /F5 7.97 Tf 6.58 0 Td[(1,(4)wherey=2t2P0ksin2kx(1)]TJ /F5 7.97 Tf 6.59 0 Td[(3(")]TJ /F5 5.978 Tf 0 -6.42 Td[(Q)2=2((")]TJ /F5 5.978 Tf 0 -6.42 Td[(Q)2+W2)))]TJ /F5 5.978 Tf 7.79 3.26 Td[(1 2cos2kx)]TJ /F5 5.978 Tf 7.78 3.26 Td[(1 2coskxcosky ((")]TJ /F5 5.978 Tf 0 -6.41 Td[(Q)2+W2)3=2)]TJ /F6 11.955 Tf 11.96 0 Td[(4t02P0ksin2kxcos2ky ((")]TJ /F5 5.978 Tf 0 -6.41 Td[(Q)2+W2)3=2.WeusetheaboveexpressionforV+)]TJ /F6 11.955 Tf 7.08 1.79 Td[((Q)togetherwiththeexpansionfortheco-herencefactorsandgettheangularexpansionofthetransverseinteractionaroundholepocketsinTable 4-6 .Thelargecontributionstillcomesfromintra-pocketscattering.Thetransversesusceptibilityexpansionintermsofanglesontheelectronpocketsaroundq=Qis V+)]TJ /F6 11.955 Tf 7.08 1.79 Td[((Q)'1 t2yq2=1 t2y(keF)2(2)]TJ /F6 11.955 Tf 11.96 0 Td[(2(coscos0+sinsin0))]TJ /F5 7.97 Tf 6.59 0 Td[(1.(4) 69

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Table4-6.Potentialsfromthetransversespin-uctuationcontribution,)]TJ /F6 11.955 Tf 9.3 0 Td[(2)]TJ /F9 7.97 Tf 13.05 -1.79 Td[(sexpandedaroundholepocketsinthelimitofkhF!0 HHHHHHkk0( 2, 2)()]TJ /F8 7.97 Tf 10.5 4.71 Td[( 2, 2) )]TJ /F5 7.97 Tf 16.88 4.71 Td[(2 yW2(1)]TJ /F6 11.955 Tf 11.95 0 Td[(coscos0+sinsin0)( 2, 2)+2V+)]TJ /F6 11.955 Tf 7.08 1.8 Td[((0)t2(khF)2 W2(2+4coscos016V+)]TJ /F6 11.955 Tf 7.09 1.8 Td[((,0)t2(khF)2 W2cossin0+cos2+cos20))]TJ /F5 7.97 Tf 16.88 4.7 Td[(2 yW2(1+coscos0)]TJ /F6 11.955 Tf 11.96 0 Td[(sinsin0)()]TJ /F8 7.97 Tf 10.49 4.71 Td[( 2, 2)16V+)]TJ /F6 11.955 Tf 7.09 1.79 Td[((,0)t2(khF)2 W2sincos0+2V+)]TJ /F6 11.955 Tf 7.09 1.79 Td[((0)t2(khF)2 W2(2+4sinsin0)]TJ /F6 11.955 Tf 11.29 0 Td[(cos2)]TJ /F6 11.955 Tf 11.95 0 Td[(cos20) Thetransverseinteractionexpansionaroundtheelectronpocketsisthen )]TJ /F6 11.955 Tf 9.3 0 Td[(2)]TJ /F4 7.97 Tf 13.05 4.94 Td[(?s(k,k0)=2V+)]TJ /F6 11.955 Tf 7.08 1.79 Td[((0)t2(keF)4 4W2h1+2cos2cos20+1 2(cos4+cos40)i)]TJ /F6 11.955 Tf 14.68 8.09 Td[((keF)2 2yW2h1+coscos0+sinsin0)]TJ /F6 11.955 Tf 13.15 8.09 Td[(1 2(cos3cos0)]TJ /F6 11.955 Tf 11.96 0 Td[(sin3sin0+coscos30)]TJ /F6 11.955 Tf 11.96 0 Td[(sinsin30))]TJ /F6 11.955 Tf 11.95 0 Td[(cos2cos20+sin2sin20i.(4) 4.2.2.3InterbandinteractionsTheinteractionbetweenelectronandholepocketsisexpectedtobesmallduetothefactthattheconnectingvectorisawayfromQandmostofthetimeweonlyhaveonekindofpocketpresent.Intherarecasewhenwehavebothpockets,wecanstillestimatethecontributionfrominterbandscattering.Tocalculatethepotentialforthescatteringbetweenholeandelectronpockets,wehavetoconsideradifferentinterbandpotential.Thechargeandthelongitudinalinterband(primed)potentialis)]TJ /F4 7.97 Tf 6.77 4.93 Td[(0(k,k0))]TJ /F6 11.955 Tf 11.95 0 Td[()]TJ /F9 7.97 Tf 6.78 4.93 Td[(zz0s(k,k0)=2U(p2)]TJ /F3 11.955 Tf 11.96 0 Td[(n2))]TJ /F12 11.955 Tf 11.95 13.27 Td[(hVc(k)]TJ /F7 11.955 Tf 11.95 0 Td[(k0))]TJ /F3 11.955 Tf 11.96 0 Td[(Vz(k)]TJ /F7 11.955 Tf 11.95 0 Td[(k0)ip2)]TJ /F12 11.955 Tf 11.96 13.27 Td[(hVz(k)]TJ /F7 11.955 Tf 11.96 0 Td[(k0+Q))]TJ /F3 11.955 Tf 11.95 0 Td[(Vc(k)]TJ /F7 11.955 Tf 11.96 0 Td[(k0+Q)in2. (4) 70

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Thetransverseinterbandpotentialis)]TJ /F6 11.955 Tf 9.3 0 Td[(2)]TJ /F4 7.97 Tf 13.05 4.94 Td[(?0s(k,k0)=+2V+)]TJ /F6 11.955 Tf 7.08 1.79 Td[((k)]TJ /F7 11.955 Tf 11.96 0 Td[(k0)m2)]TJ /F6 11.955 Tf 11.96 0 Td[(2V+)]TJ /F6 11.955 Tf 7.09 1.79 Td[((k)]TJ /F7 11.955 Tf 11.95 0 Td[(k0+Q)l2. (4)TheapproximatedinterbandpotentialisshowninTable 4-7 ThereisafourTable4-7.Potentialsfromthecharge-andlongitudinalspin-uctuationinterbandcontribution,)]TJ /F4 7.97 Tf 6.78 4.34 Td[(0(k,k0))]TJ /F6 11.955 Tf 11.96 0 Td[()]TJ /F4 7.97 Tf 6.77 4.34 Td[(0zzs(k,k0))]TJ /F6 11.955 Tf 11.96 0 Td[(2)]TJ /F4 7.97 Tf 13.05 4.34 Td[(?0s(k,k0)expandedbetweenelectronandholepockets PPPPPPPPPk=k0=(,0) ( 2, 2))]TJ /F6 11.955 Tf 9.3 0 Td[(2p 22U)]TJ /F3 11.955 Tf 11.96 0 Td[(Vc( 2, 2)+Vz( 2, 2)+2V+)]TJ /F6 11.955 Tf 7.08 1.8 Td[(( 2, 2)khF(keF)2coscos2()]TJ /F8 7.97 Tf 10.5 4.71 Td[( 2, 2)2p 22U)]TJ /F3 11.955 Tf 11.96 0 Td[(Vc( 2, 2)+Vz( 2, 2)+2V+)]TJ /F6 11.955 Tf 7.09 1.79 Td[(( 2, 2)khF(keF)2sincos2 foldsymmetricalpropertyofV(=2,=2)=V()]TJ /F10 11.955 Tf 9.3 0 Td[(=2,)]TJ /F10 11.955 Tf 9.3 0 Td[(=2)=V(=2,)]TJ /F10 11.955 Tf 9.3 0 Td[(=2)=V()]TJ /F10 11.955 Tf 9.3 0 Td[(=2,=2).WecanseethattheinterbandpotentialisquitesmalluptothirdpowerofthepocketsizekhF(keF)2. 4.2.3LAHAexpansionofgapequationInordertodeterminethegapsymmetry,weusetheleadingangularharmonicsapproximation(LAHA)method[ 83 ].TheLAHAmethodisasimpliedversionoftheBCSgapequationinEquation 1 .Thatbecomesexactinthelimitofsmallpocketsize.Thesimpliedform XjZ20d0 2NF,j)]TJ /F9 7.97 Tf 6.77 -1.79 Td[(i,j(,0),j(0)L=)]TJ /F10 11.955 Tf 9.3 0 Td[(,i()(4)isaeigenvalueproblemwhereiandjarethebandororbitalindices,isthesymmetryofthegap,)]TJ /F9 7.97 Tf 6.78 -1.79 Td[(ijistheinteraction,NF,jisthedensityofstateatFermisurfaceandLisaconstantproportionaltolnEF=TC,=,representsanglesoneitherholeorelectronpockets.Theinteractionofeachchannelisrestrictedwithintheleadingharmonics.FromtheantiperiodicityofthepotentialsV(q)=)]TJ /F3 11.955 Tf 9.3 0 Td[(V(q+Q),possibilitiesincludeextendeds)]TJ /F1 11.955 Tf 9.3 0 Td[(wave,whichforcesthesuperconductinggaptochangesignonthereducedBrillouinzoneboundary;dx2)]TJ /F9 7.97 Tf 6.59 0 Td[(y2-wave,whichdoesnotchangesignontheboundarybut 71

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whichhasnodesalongthe110directions;andalsoodd-parityp)]TJ /F1 11.955 Tf 9.3 0 Td[(wavesymmetry[ 12 ].dxy-wavepairingisexcludedbecauseitdoesnotfullltheantiperiodicitycondition.Hereweexpandthes-waveanddx2)]TJ /F9 7.97 Tf 6.59 0.01 Td[(y2-wavesymmetriesontheholepocketstogettheangulardependenceofthegapsasinTable 4-8 Table4-8.Angulardependenceofthes-waveanddx2)]TJ /F9 7.97 Tf 6.59 0 Td[(y2-wavesymmetriesontheholepockets. (k)coskx+cosky(extendeds-wave)coskx)]TJ /F6 11.955 Tf 11.95 0 Td[(cosky(dx2)]TJ /F9 7.97 Tf 6.59 0 Td[(y2-wave) ( 2, 2))]TJ 9.29 9.96 Td[(p 2khFcosp 2khFsin+p 2 36(khF)3(5cos)]TJ /F6 11.955 Tf 11.96 0 Td[(2cos3)+p 2 36(khF)3(5sin+2sin3)()]TJ /F8 7.97 Tf 10.49 4.71 Td[( 2, 2))]TJ 9.29 9.96 Td[(p 2khFsinp 2khFcos)]TJ /F4 7.97 Tf 10.49 11.35 Td[(p 2 36(khF)3(5sin+2sin3))]TJ /F4 7.97 Tf 10.5 11.35 Td[(p 2 36(khF)3(5cos)]TJ /F6 11.955 Tf 11.96 0 Td[(2cos3) TheleadingtermangulardependentgapontherstandsecondholepocketsaslabeledinFigure 4-1 canbewrittenas sh1()=shcos,sh2()=shsin (4) fortheextendeds)]TJ /F1 11.955 Tf 9.3 0 Td[(wavesymmetry,and dh1()=dhsin,dh2()=dhcos (4) forthedx2)]TJ /F9 7.97 Tf 6.59 0 Td[(y2-wavesymmetry. Table4-9.Angulardependenceofthes-waveanddx2)]TJ /F9 7.97 Tf 6.59 0 Td[(y2-wavesymmetriesontheelectronpockets. (k)coskx+cosky(extendeds-wave)coskx)]TJ /F6 11.955 Tf 11.96 0 Td[(cosky(dx2)]TJ /F9 7.97 Tf 6.58 0 Td[(y2-wave) (,0)(keF)2 2cos2)]TJ /F5 7.97 Tf 13.15 6.48 Td[((keF)4 24cos4)]TJ /F6 11.955 Tf 9.3 0 Td[(2+(keF)2 2)]TJ /F5 7.97 Tf 13.15 6.48 Td[((keF)4 96(3+cos4) Inaddition,wehavetheleadinggapsymmetryAnsatzefortheelectronpockets, se()=secos2,(4) 72

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de()=de(1+decos4).(4)Wealsohavethepossibilityofp)]TJ /F1 11.955 Tf 9.3 0 Td[(wavesymmetry[ 12 ].Oneachtypeofpockets,thegapshavetheangulardependenceasph1()=ph(1+phcos2),ph2()=ph(1+phcos2). (4)Thesignreferstotwodistinctp-wavestates,withsigns++\000or+)-258()]TJ /F6 11.955 Tf 21.68 0 Td[(+onholepocketsh1,...,h4.Bycomparingthegapangulardependencewiththeinteractions,)]TJ /F1 11.955 Tf 6.78 0 Td[(s,wecanndtheleadingsymmetry.FromTable 4-5 andEquation 4 ,wecangetthechargeandlongitudinalcontributionofthepotentials,V`)]TJ /F8 7.97 Tf 6.77 -1.79 Td[()]TJ /F6 11.955 Tf 12.24 0 Td[()]TJ /F9 7.97 Tf 6.78 4.34 Td[(zs.Theexpansionsuptoorderk2Foneachtypeofpocketsforthecharge-andlongitudinal-potentialsareexpressedas Vlh1h1(,0)ch+ahcoscos0+bhcoscos0+ch(cos2+cos20), (4a)Vlh2h2(,0)ch+ahsinsin0+bhcoscos0+ch(cos2+cos20), (4b)Vlee(,0)ce+de(coscos0+sinsin0), (4c)wherechV+h~V)]TJ /F5 7.97 Tf 13.25 4.71 Td[(2t2V W2ikhF2,ahh)]TJ /F6 11.955 Tf 13.92 2.66 Td[(~V+4t2V W2ikhF2,bh=~VkhF2,ceV+~VkeF2,de~VkeF2,andY(x)=4U00zz(x)Vz(x) 1+Uzz(x)1+Vz(x)2U)]TJ /F5 5.978 Tf 5.76 0 Td[(2 (1+Uzz(x))2,V[Vz(0))]TJ /F3 11.955 Tf 11.96 0 Td[(Vc(0)+(Vc(Q))]TJ /F3 11.955 Tf 11.96 0 Td[(Vz(Q))],and~VY(0))]TJ /F3 11.955 Tf 12.17 0 Td[(Y(Q).VispositivesinceVz(Q)isthedominantterminthedenition.Thisimpliesthattheleadingcontributiontotheintra-pocketinteractionsareattractiveforbothelectronandholecases.Butduetotheantiperiodicity,itdoesnotgiverisetoaconventionals)]TJ /F1 11.955 Tf 9.3 0 Td[(wavegap.Comparingtheelectronpotential,Vlee(,0)withtheexpansionofdx2)]TJ /F9 7.97 Tf 6.58 0 Td[(y2)]TJ /F1 11.955 Tf 12.62 0 Td[(gap,Equation 4 ,wecanseebothexpressionshaveleading 73

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termsbeingconstantand.Thisleadstoaleadingd)]TJ /F1 11.955 Tf 9.29 0 Td[(waveinstabilityforanelectron-dopedsystemwithantiferromagneticorder.Fortheholepocketcase,ontheotherhand,onlyp)]TJ /F1 11.955 Tf 12.63 0 Td[(wavehasaconstantleadingattractiveinteraction.Thereforethecharge-andlongitudinalspin-potentialgivesrisetoap)]TJ /F1 11.955 Tf 12.62 0 Td[(wavesymmetryinthesuperconductinggap.Thereisasub-dominantcontributionfromthes)]TJ /F1 11.955 Tf 9.3 0 Td[(wavesymmetry.Thes)]TJ /F1 11.955 Tf 9.3 0 Td[(wavecontributionisquitesmallandscaledwith(khF)2.TheexpansionforVtr)]TJ /F6 11.955 Tf 21.92 0 Td[(2)]TJ /F4 7.97 Tf 13.05 4.34 Td[(?sthenhasthefollowingform: Vtrh1h1(,0)Ah(1)]TJ /F6 11.955 Tf 11.96 0 Td[(coscos0+sinsin0)+Bh(2+4coscos0+cos2+cos20), (4a)Vtrh2h2(,0)Ah(1+coscos0)]TJ /F6 11.955 Tf 11.96 0 Td[(sinsin0)+Bh(2+4sinsin0)]TJ /F6 11.955 Tf 11.96 0 Td[(cos2)]TJ /F6 11.955 Tf 11.95 0 Td[(cos20), (4b)Vtree(,0)Aeh1+coscos0+sinsin0)]TJ /F6 11.955 Tf 13.15 8.09 Td[(1 2(cos3cos0)]TJ /F6 11.955 Tf 11.96 0 Td[(sin3sin0+coscos30)]TJ /F6 11.955 Tf 11.95 0 Td[(sinsin30))]TJ /F6 11.955 Tf 11.96 0 Td[(cos2cos20+sin2sin20i, (4c)whereAh)]TJ /F5 7.97 Tf 30.67 4.7 Td[(2 yW2,BhV(0)tkhF W2,andAe)]TJ /F9 7.97 Tf 28.58 6.47 Td[(keF2 2yW2.Forthetransversechannel,theholepocketshaveastrongerpairingpotentialcomparedwiththeelectronpockets.Thepotentialfortheholepockethasaleadingconstanttermwhiletheleadingtermforelectronpocketsscaleswith(keF)2.Fortheholepockets,thetransversepotentialstillsupportsp)]TJ /F1 11.955 Tf 9.29 0 Td[(wavepairingwhereasfortheelectronpockets,itsupportsdx2)]TJ /F9 7.97 Tf 6.59 0 Td[(y2)]TJ /F1 11.955 Tf 9.3 0 Td[(wavepairing. 4.2.4ComparisonwithnumericalevaluationWecomparethefullexpressionofthepotentialswiththeapproximateexpansionofthepotentialsbyplottingthesetwoalongtheelectronpocketshowninFigure 4-2 withdoping,x=0.03.Fortheholepocketswewillhavetoovercometheproblemof 74

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Figure4-2.Comparisonoftheanalyticalcalculationsupto(keF)2forthelongitudinal(leftpanel)andtransverse(rightpanel)pairingpotentials,VleeandVtreeontheelectronpocketsforthedopinglevelofn=1.03(blackcurves),togetherwiththefullnumericalevaluationof)]TJ /F8 7.97 Tf 6.78 -1.79 Td[()]TJ /F6 11.955 Tf 11.96 0 Td[()]TJ /F9 7.97 Tf 6.77 4.34 Td[(zs,and)]TJ /F6 11.955 Tf 9.3 0 Td[()]TJ /F4 7.97 Tf 6.78 4.34 Td[(?s(bluepoints). theincommensuratemodebymodifyingtheinterbandcontributioninthetransversesusceptibility.Thisissavedforfutureinvestigation.Thebluedotsarecalculatedwith0=0accordingtothecharge-andlongitudinalspin-uctuationsinEquation 4 inFigure 4-2 (a)andtothetransversespin-uctuationsinEquation 4 inFigure 4-2 (b).Thenthedotsarettedbytheleadingharmonics, F(,0)=a+bcoscos0+ccos2cos20+dcos3cos30,(4)witha,b,canddastparameters,givingtheredcurvesinFigure 4-2 .TheapproximatepotentialsgiveninEquations 4c and 4c areplottedasblackcurves.Wendce=)]TJ /F6 11.955 Tf 9.3 0 Td[(1.534,de=)]TJ /F6 11.955 Tf 9.29 0 Td[(0.611andAe=)]TJ /F6 11.955 Tf 9.29 0 Td[(0.308(inunitsoft).Theredcurvesdenotethetwhence,de,andAearenotcomputedanalyticallybutttedtothenumericalresultswiththeleastsquaremethod(ce=)]TJ /F6 11.955 Tf 9.3 0 Td[(1.457,de=)]TJ /F6 11.955 Tf 9.3 0 Td[(0.547andAe=)]TJ /F6 11.955 Tf 9.3 0 Td[(0.339(inunitsoft)).Here,weuseU=2.775twhichgivesW=0.6537t.Itisevidentthattheapproximatedpotentialsagreewiththefullexpressiononthegeneralsymmetryandthemagnitude.Theerrorsmaycomefromrestrictionoftheniteorderoftheexpansion,theevaluation 75

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ofthevalueofywhichisderivedassuminghalf-llinginthetransversechannel,andthenitesizeoftheelectronpockets.Theimplicationofdx2)]TJ /F9 7.97 Tf 6.59 0.01 Td[(y2)]TJ /F1 11.955 Tf 9.3 0 Td[(wavesymmetryontheelectronpocketsinthecoexistencestateandp)]TJ /F1 11.955 Tf 9.3 0 Td[(waveontheholepocketsprovidesanaturalexplanationofwhywedonothaverobustcoexistencestateonthehole-dopedsideofthecupratephasediagrams.Iniswell-knownthatonbothsideofthecuprateswehavedx2)]TJ /F9 7.97 Tf 6.58 0 Td[(y2)]TJ /F1 11.955 Tf 12.62 0 Td[(superconductinggaps.Inthecrossoverfrompuresuperconductingstatetothecoexistencestate,theelectron-dopedcupratesdonotneedtochangesymmetrywhereasthehole-dopedcuprateshavetochangesymmetrytoanodelesssingletp)]TJ /F1 11.955 Tf 9.3 0 Td[(wavetoavoidnodesonthepocket.Notethatthetripletp)]TJ /F1 11.955 Tf 9.3 0 Td[(wavegapgivesrisetoanodalstructureontheholepockets,thereforeislessfavored.Recentexperimentshaveshownafuliygappedsuperconductinggapinthedeeplyunderdopedcuprates[ 84 88 ],towhichourworkmayapply.Forfuturework,anewphasediagramcouldbegeneratedbasedonthesamemodel.ThepreviouscalculationofspinexcitationsinthecoexistencestateofAFandsuperconductivitycanbeincludedtocreateaself-consistencepairinginteractioncalculation.Thereforewedonotneedtheassumptionofphenomenologicalorderparameterforsuperconductivity.ItwouldbeinterestingtoseethesymmetryofthesuperconductinggapacrossthephasediagramandtocomparethegapstructurewhetherwewillseeasinterbandpairingcontributionwhichariseduetotheUmklappprocessesintheAFstate[ 56 ]. 76

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CHAPTER5CONCLUSIONThemeaneldphasediagramintheelectron-dopedcuprateshasbeenstudiedwithaone-bandsquarelatticeHubbardmodel.Intheantiferromagneticstate,theFermisurfaceisreconstructedandtheenergydispersionsplitsintoan(electron)bandanda(hole)band.ThiscreateselectronandholepocketsontheFermisurface.Wederivedtheself-consistentequationsforthecalculationoftheantiferromagneticorderparameterwithagivenon-siteCoulombinteractionU.WechangedthethechemicalpotentialtoadjustthedopinglevelandaddedaphenomenologicalsuperconductingHamiltoniantostudythecoexistencestateofantiferromagnetismandsuperconductivity.Themeaneldenergywasalsocalculatedtodeterminethefavorablestateforagivendopingandtemperature.Fromthehalf-llingregion,wehaveasuperconductingphasestartingataroundx=0.05doping.Thistransitionacrossthedopingisofrstorderduetothemean-eldenergy,unlikethetransitionacrosstemperaturewhichissecondorder.Forthiscalculation,weassumedadx2)]TJ /F9 7.97 Tf 6.59 0 Td[(y2)]TJ /F1 11.955 Tf 9.3 0 Td[(wavepairinginthesuperconductingstate.ThesuperconductinggapinthecoexistencestatehasthepossibilityofatripletSz=0termwhichisahigherharmoniccorrectionandisproportionaltoW.Withinthemeaneldcalculation,thephasewithouttripletcorrectionshaslowerenergy.Wenextstudiedthespindynamicsusceptibilityinthehalf-lledantiferromagneticstatewithdifferentnext-nearesthoppingt0.Fort0=0,thespinwavehasnosofteningat( 2, 2)whichisconsistentwiththestrongcouplingresults.Butwithincreasedt0,wefoundasofteningat( 2, 2).Thedentingofthedispersionisrelatedtothenon-degeneratepointsintroducedbyt0.Athalf-lling,witht0=0thebandandthebandarewellseparatedby2W.Butwithnitet0,theindirectgap(thedistancebetweenlowestbandthethehighestoftheband)become2W)]TJ /F6 11.955 Tf 13.11 0 Td[(4t0.Theminimumatq( 2, 2)isduethetheinteractionbetweenspinwavesandtheparticle-holespectrum.Thisisafeatureoftheweak-couplingapproachwhichallowsWtobethesameorderas 77

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t0.Ifwecontinuetoincreaset0and2W)]TJ /F6 11.955 Tf 12.1 0 Td[(4t0<0,therealpartoftheRPAsusceptibilitybecomesnegative.Thismakestheorderedsystemunstableanditreturnstothenormalstate.Withnitedopingsandt0=t=0.35inthepureantiferromagneticstate,westudiedthecasesofhole-dopedsystemswithonlyholepockets,electron-dopedsystemswithonlyelectronpocketsandelectron-dopedsystemswithbothpockets.WerecoveredtheGoldstonemodeinthetransversespinsusceptibility.Thelongitudinalspinsusceptibilityisgappedatq=(,).Inthecaseswithholepockets,thereisanincommensuratemode,indicatingthebreakdownofthemeaneldcommensurateassumption.Withthecalculationofthespin-stiffnessandtherealpartoftheRPAsusceptibility,wefoundthattheassumptionoflongrangecommensurateantiferromagneticorderisnotsuitableforthehole-dopedcase.Thismayalsoexplainwhytheantiferromagneticorderonthehole-dopedsideislessrobustthanontheelectron-dopedside.Wealsostudiedthedynamicspinsusceptibilityinthecoexistencestateofan-tiferromagneticandsuperconductivity.Weassumedadx2)]TJ /F9 7.97 Tf 6.59 0 Td[(y2)]TJ /F1 11.955 Tf 9.3 0 Td[(wavepairinginthesuperconductingstate.TheGoldstonemodeisstillrobustandgapless.Wendthatthespinwavesaremodiedbytheresonancewhichiscreatedbythesuperconductinggap.Theresultisakinkduetotheinteractionwiththeparticle-holecontinuuminthespinwavedispersion.Wealsocalculatedthespin-wavevelocityinthecoexistencestateasobservedinthetransversesusceptibility.Forthelongitudinalsusceptibility,wefoundaresonancemodeattheincommensuratemomentumclosetoq=(,)duetothesign-changingsuperconductinggap.Theabovestudiesaboutsuperconductivityarebasedonaphenomenologicalmeaneldd)]TJ /F1 11.955 Tf 9.3 0 Td[(wavepairinginteraction.Inordertounderstandthemicroscopictheoryofsuperconductivityinthepresenceofanorderedantiferromagneticstate,weadaptedthebagtheoryproposedbySchrieffer,WenandZhangandgeneralizedittotheelectron-dopedcase.Wealsoincludedthetransversepartofthespinuctuationpairing 78

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vertexwhichwasignoredbySchrieffer.WederivedthefulleffectivepairingHamiltonianintheantiferromagneticstateandseparatedthegapequationsintospinsingletandtripletparts.Tostudytheinstabilityinthesingletchannel,weexpandedthecoherencefactorsandthespinsusceptibilityaroundelectronandholepocketcenterinthelimitofsmallpocketsize.WeusedtheLAHAapproachtoanalyzethesymmetryofthesuperconductinggap.Thesuperconductinggapsofsymmetriesconsistentwiththestaggeredantiferromagneticstatearealsoexpandedforcomparisonwiththepairingpotentials.Westudythechargeandlongitudinalspinpartandthetransversepartoftheuctuationsseparately.Forthechargeandlongitudinalspinpartwendthatwithelectronpockets,theleadingsuperconductinginstabilityhasdx2)]TJ /F9 7.97 Tf 6.59 0 Td[(y2)]TJ /F1 11.955 Tf 9.3 0 Td[(wavesymmetry,whereaswithholepocketstheleadingcontributionhasoddparityp)]TJ /F1 11.955 Tf 9.29 0 Td[(wavesymmetry.Forthetransversepotentials,thesingularitiesintheRPAsusceptibilityareavoidedbecausetheyoccurincombinationwiththeSDWcoherencefactors.Thisresultsinanon-divergentcontributiontothepairingfromthespinwaves.Fortheholecase,thetransverseuctuationsalsosupportap)]TJ /F1 11.955 Tf 9.3 0 Td[(wavesymmetryandfortheelectroncasetheysupportadx2)]TJ /F9 7.97 Tf 6.58 0 Td[(y2)]TJ /F1 11.955 Tf 9.3 0 Td[(wavepairing.Butthepairingstrengthfromtheelectronpocketisweakertoorder(keF)2;thereforethelongitudinaluctuationsdominateinthiscase.Toshowthatourapproximationsforthepotentialswithsmallpocketsarejustied,weplottedthepotentialsalongthepocketwithangledependenceforthefullexpressionandtheapproximatedexpression.Forbothcharge-andlongitudinalspin-uctuationpartandthetransversespin-uctuationpart,theyagreewiththegeneralsymmetryandmagnitudes.Ontheelectrondopedsidethesuperconductivityhasasmoothcrossoverfromthepuresuperconductingstatetothecoexistencestatewhileontheholedopedsidethesuperconductinggaphastochangesymmetrytoanodd-paritysingletp)]TJ /F1 11.955 Tf 9.3 0 Td[(wavetoavoidnodesonthepocket,thereforelessfavorable.Ourndingsregardingtheleadingsuperconductinggapsymmetries,p)]TJ /F1 11.955 Tf 9.3 0 Td[(waveonthehole-dopedanddx2)]TJ /F9 7.97 Tf 6.58 0 Td[(y2)]TJ /F1 11.955 Tf 9.3 0 Td[(waveelectron 79

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dopedsidesuggeststhatthecoexistencestateonthecupratephasediagramcanonlyexistontheelectron-dopedside. 80

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APPENDIXAMEANFIELDQUANTITIESINTHEPUREANTIFERROMAGNETICSTATEThefollowingsectionsarethederivationofWandUself-consistentequationandllinglevel. A.1Antiferromagneticorderparameterequation:derivationStartingwiththedenitionoftheantiferromagneticorderparameterW, W=U 2Xksgn()hcyk+Qcki,(A)wewriteoutthespinandfoldthemomentumindextothereducedBrillouinzone,wehave W=U 2Xk0hcyk+Q"ck"i)-223(hcyk+Q#ck#i+hcyk"ck+Q"i)-222(hcyk#ck+Q#i.(A)UsingtheunitarytransformationinEquation 2 tochangethebasisoftheoperators,weget W=U 2Xk0h()]TJ /F3 11.955 Tf 9.29 0 Td[(vkyk"+ukyk")(ukk"+vkk")i)-223(h(vkyk#)]TJ /F3 11.955 Tf 11.96 0 Td[(ukyk#)(ukk#+vkk#)i+h(ukyk"+vkyk")()]TJ /F3 11.955 Tf 9.3 0 Td[(vkk"+ukk")i)-222(h(ukyk#+vkyk#)(vkk#)]TJ /F3 11.955 Tf 11.95 0 Td[(ukk#)i=UXk0ukvk(hykki+hykki).(A)TheexpectationvaluesofthenumberoperatorscanbereplacedbytheFermifunction, hykki=f(Ek)=1 e)]TJ /F9 7.97 Tf 6.58 0 Td[(Ek=kBT+1=1 2)]TJ /F6 11.955 Tf 13.15 8.09 Td[(1 2tanh(Ek 2kBT)(A)where=,.PluggingintheSDWcoherencefactorsinEquation 2 andreplacingtheexpectationvaluesofthedensityoperatorswiththeFermifunctions,wegettheself-consistentequationforW,andUas W=UXk0W p ("k)]TJ /F10 11.955 Tf 11.96 0 Td[("k+Q)2+4W2tanh(Ek 2kBT))]TJ /F6 11.955 Tf 11.96 0 Td[(tanh(Ek 2kBT).(A) 81

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A.2Theelectronlling:derivationTheelectronllingisdenedby n=1+x=Xk,hcyk,ck,i.(A)WereducedthesumofthemomentuminsidethemagneticBrillouinzoneandchangethebaseoftheoperatorwecanget n=2Xk0u2khyk"k"i+v2khyk#k#i+v2khyk"k"i+u2khyk#k#i.(A)ThenwereplacetheexpectationvaluesofthenumberoperatorwiththeFermifunction.Wecanobtain n=2)]TJ /F12 11.955 Tf 11.95 11.36 Td[(Xk0htanhEk 2kBT+tanhEk 2kBTi.(A) 82

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APPENDIXBDERIVATIONSINTHECOEXISTENCESTATEOFANTIFERROMAGNETISMANDSUPERCONDUCTIVITY B.1Antiferromagneticorderparameterequationinthecoexistencestatewithsuperconductivity:derivationStartingwiththedenitionandtheresultoftheantiferromagneticorderparameterfromAppendix A.1 ,wehave W=UXk0ukvk(hykki+hykki).(B)ThenwechangethebasistotheoperatorsinthecoexistencestatewiththehelpofBCStransformationinEquation 2 ,andobtain W=)]TJ /F3 11.955 Tf 9.3 0 Td[(UXk02ukvk[u2khyk0k0i+v2khklykli)]TJ /F6 11.955 Tf 19.51 0 Td[(u2khyk0k0i)]TJ /F6 11.955 Tf 19.54 0 Td[(v2khklykli].(B)NowwereplacetheexpectationvaluesofthenumberoperatorswiththeFermifunctionforthequasiparticlesinthecoexistencestate,andarriveatthefollowingexpression, W=)]TJ /F3 11.955 Tf 9.3 0 Td[(UXk02ukvk[u2kf(k)+v2k)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(f(k))]TJ /F6 11.955 Tf 12.2 0 Td[(u2kf(k))]TJ /F6 11.955 Tf 12.24 0 Td[(v2k)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(f(k)].(B)ByevaluatingboththeSDWcoherencefactorsandtheBCScoherencefactors,wegetthenalself-consistentequationforWandUinthecoexistencestateas W=UXk0W 2p ")]TJ /F9 7.97 Tf 0 -8.27 Td[(k+W2hEk ktanh 2kBT)]TJ /F3 11.955 Tf 13.24 8.08 Td[(Ek ktanh 2kBTi.(B) B.2Fillinglevelofelectronsinthecoexistencestate:derivationWestartwithEquation A ,thedenitionofelectronllingintheantiferromagneticstatebasis, n=2Xk0u2khyk"k"i+v2khyk#k#i+v2khyk"k"i+u2khyk#k#i.(B) 83

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Forthederivationofthedopinglevelinthecoexistencestateofsuperconductivityandferromagnetism,weusetheBCSBogoliubovtransformation k"=ukk0+vkykl,y)]TJ /F5 7.97 Tf 6.59 0 Td[(k#=)]TJ /F6 11.955 Tf 9.59 0 Td[(vkk0+ukykl(B)tochangethebasisoftheoperatorsfromtheantiferromagneticstatetothecoexistencestate.Weobtain n=2Xk0u2kh(ukyk0+vkkl)(ukk0+vkykl)i+v2kh()]TJ /F6 11.955 Tf 9.58 0 Td[(vkk0+ukykl)()]TJ /F6 11.955 Tf 9.59 0 Td[(vkyk0+ukkl)i+v2kh(ukyk0+vkkl)(ukk0+vkykl)i+u2kh()]TJ /F6 11.955 Tf 9.59 0 Td[(vkk0+ukykl)()]TJ /F6 11.955 Tf 9.58 0 Td[(vkyk0+ukkl)i.(B)Withsomeorganization,wehave n=Xk0u2khyk0k0i+v2khklykli+v2khk0yk0i+u2khyklkli+u2khyk0k0i+v2khklykli+v2khk0yk0i+u2khyklkli.(B)Hereweplugintheexpectationvaluesofthenumberoperatorsofthequasiparticlesinthecoexistencestate,hyklkli=f(k).Weget n=2Xk0u2kf(k)+v2k[1)]TJ /F3 11.955 Tf 11.95 0 Td[(f(k)]+u2kf(k)+v2k[1)]TJ /F3 11.955 Tf 11.95 0 Td[(f(k)]=Xk02)]TJ /F3 11.955 Tf 13.24 8.09 Td[(Ek ktanh(k 2T))]TJ /F3 11.955 Tf 13.24 8.09 Td[(Ek ktanh(k 2T).(B)Whenthesuperconductinggapsgotozero,=k!E=k.Theequationreducestotheresultinthepureantiferromagneticstate,Equation A 84

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B.3Meaneldenergyinthecoexistencestate:derivationTheHamiltonianinthecoexistencestateisinthefollowing.Weseparateitintothreeterms,H=H1+H2+H3 (B)=Xkkcykck+Xk,k0,U 2cykck+Qcyk0+Qck0+Xk,p,q,Vqcyk+qcyp)]TJ /F5 7.97 Tf 6.59 0 Td[(qcpck. (B)Thekineticenergytermafterthesequentialtransformationsis hH1i=Xkkhcykcki=2Xk0("ku2k+v2k"k+Q)(u2kf(k)+v2k(1)]TJ /F3 11.955 Tf 11.96 0 Td[(f(k)))+("kv2k+u2k"k+Q)(u2kf(k)+v2k(1)]TJ /F3 11.955 Tf 11.95 0 Td[(f(k))).(B)ForthesecondpartoftheHamiltonian,weusethemean-eldtreatmenttodecouplethefouroperators, H2=Xk,k0,U 2cykck+Qcyk+Qck0=Xk,k0,U 2hcykck+Qicyk+Qck0+cykck+Qhcyk+Qck0i)-223(hcykck+Qihcyk+Qck0i.(B)WefoldthemomentumintothereducedBrillouinzone.Andaftertheunitarytransforma-tion,weobtain H2=)]TJ /F6 11.955 Tf 9.3 0 Td[(2UXk,k0,0ukvku0kv0k(hykki)-222(hykki)(yk0k0)]TJ /F10 11.955 Tf 11.95 0 Td[(yk0k0)+(ykk)]TJ /F10 11.955 Tf 11.95 0 Td[(ykk)(hyk0k0i)-222(hyk0k0i))]TJ /F6 11.955 Tf 11.95 0 Td[((hykki)-222(hykki)(hyk0k0i)-222(hyk0k0i).(B)Sincehyk0"k0"i=hyk0#k0#i,andW=UPk,0)]TJ /F3 11.955 Tf 10.65 0 Td[(ukvk(hykki)-113(hykki),wecanre-writetheHamiltonianas H2=)]TJ /F3 11.955 Tf 9.3 0 Td[(WhXk0,0)]TJ /F3 11.955 Tf 11.96 0 Td[(u0kv0k(yk0k0)]TJ /F10 11.955 Tf 11.95 0 Td[(yk0k0)+Xk,0)]TJ /F3 11.955 Tf 11.96 0 Td[(ukvk(ykk)]TJ /F10 11.955 Tf 11.96 0 Td[(ykk)i+W2 U=2WXk,0ukvk(ykk)]TJ /F10 11.955 Tf 11.96 0 Td[(ykk)+W2 U.(B) 85

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WiththeBCStransformationweobtain H2=2WXk,0ukvku2kyk0k0+v2kklykl+v2kk0yk0+u2kyklkl)]TJ /F6 11.955 Tf 12.19 0 Td[(u2kyk0k0)]TJ /F6 11.955 Tf 12.24 0 Td[(v2kklykl)]TJ /F6 11.955 Tf 12.24 0 Td[(v2kk0yk0)]TJ /F6 11.955 Tf 12.2 0 Td[(u2kyklkl+W2 U.(B)TheexpectationvalueoftheHamiltonianisthen hH2i=4WXk,0ukvkhu2kf(k)+v2k[1)]TJ /F3 11.955 Tf 11.96 0 Td[(f(k)])]TJ /F6 11.955 Tf 12.2 0 Td[(u2kf(k))]TJ /F6 11.955 Tf 12.24 0 Td[(v2k[1)]TJ /F3 11.955 Tf 11.96 0 Td[(f(k])i+W2 U.(B)Thethird(superconducting)partofthetotalHamiltonianis H3=Xk,p,q,Vqcyk+qcyp)]TJ /F5 7.97 Tf 6.58 0 Td[(qcpck.(B)Wehavethesamemean-elddecouplingprocedure, H3=Xk,p,q,Vqhhcyk+qcyp)]TJ /F5 7.97 Tf 6.59 0 Td[(qicpck+cyk+qcyp)]TJ /F5 7.97 Tf 6.59 0 Td[(qhcpcki)-222(hcyk+qcyp)]TJ /F5 7.97 Tf 6.59 0 Td[(qihcpckii.(B) 86

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Withtheunitarytransformation,wehave H3=Xk,p,0Vp)]TJ /F5 7.97 Tf 6.59 0 Td[(k[(u2pu2k+2upvpukvk+v2pv2k)(hyp"y)]TJ /F5 7.97 Tf 6.59 0 Td[(p#i)]TJ /F5 7.97 Tf 6.58 0 Td[(k"k#+yp"y)]TJ /F5 7.97 Tf 6.59 0 Td[(p#hy)]TJ /F5 7.97 Tf 6.59 0 Td[(k"yk#i)-222(hyp"y)]TJ /F5 7.97 Tf 6.58 0 Td[(p#ihy)]TJ /F5 7.97 Tf 6.59 0 Td[(k"yk#i+hyp"y)]TJ /F5 7.97 Tf 6.58 0 Td[(p#iy)]TJ /F5 7.97 Tf 6.58 0 Td[(k"yk#+yp"y)]TJ /F5 7.97 Tf 6.59 0 Td[(p#hy)]TJ /F5 7.97 Tf 6.59 0 Td[(k"yk#i)-222(hyp"y)]TJ /F5 7.97 Tf 6.58 0 Td[(p#ihy)]TJ /F5 7.97 Tf 6.59 0 Td[(k"yk#i)+(u2pv2k)]TJ /F6 11.955 Tf 11.96 0 Td[(2upvpukvk+v2pu2k)(hyp"y)]TJ /F5 7.97 Tf 6.59 0 Td[(p#iy)]TJ /F5 7.97 Tf 6.59 0 Td[(k"yk#+yp"y)]TJ /F5 7.97 Tf 6.59 0 Td[(p#hy)]TJ /F5 7.97 Tf 6.59 0 Td[(k"yk#i)-222(hyp"y)]TJ /F5 7.97 Tf 6.58 0 Td[(p#ihy)]TJ /F5 7.97 Tf 6.58 0 Td[(k"yk#i+hyp"y)]TJ /F5 7.97 Tf 6.58 0 Td[(p#iy)]TJ /F5 7.97 Tf 6.59 0 Td[(k"yk#+yp"y)]TJ /F5 7.97 Tf 6.58 0 Td[(p#hy)]TJ /F5 7.97 Tf 6.59 0 Td[(k"yk#i)-222(hyp"y)]TJ /F5 7.97 Tf 6.59 0 Td[(p#ihy)]TJ /F5 7.97 Tf 6.59 0 Td[(k"yk#i)])]TJ /F3 11.955 Tf 9.29 0 Td[(Vp)]TJ /F5 7.97 Tf 6.58 0 Td[(k+Q[(v2pu2k+2upvpukvk+u2pv2k)(hyp"y)]TJ /F5 7.97 Tf 6.59 0 Td[(p#iy)]TJ /F5 7.97 Tf 6.58 0 Td[(k"yk#+yp"y)]TJ /F5 7.97 Tf 6.59 0 Td[(p#hy)]TJ /F5 7.97 Tf 6.59 0 Td[(k"yk#i)-222(hyp"y)]TJ /F5 7.97 Tf 6.58 0 Td[(p#ihy)]TJ /F5 7.97 Tf 6.59 0 Td[(k"yk#i+hyp"y)]TJ /F5 7.97 Tf 6.58 0 Td[(p#iy)]TJ /F5 7.97 Tf 6.58 0 Td[(k"yk#+yp"y)]TJ /F5 7.97 Tf 6.59 0 Td[(p#hy)]TJ /F5 7.97 Tf 6.59 0 Td[(k"yk#i)-222(hyp"y)]TJ /F5 7.97 Tf 6.58 0 Td[(p#ihy)]TJ /F5 7.97 Tf 6.59 0 Td[(k"yk#i)+(v2pv2k)]TJ /F6 11.955 Tf 11.96 0 Td[(2upvpukvk+u2pu2k)(hyp"y)]TJ /F5 7.97 Tf 6.59 0 Td[(p#iy)]TJ /F5 7.97 Tf 6.59 0 Td[(k"yk#+yp"y)]TJ /F5 7.97 Tf 6.59 0 Td[(p#hy)]TJ /F5 7.97 Tf 6.59 0 Td[(k"yk#i)-222(hyp"y)]TJ /F5 7.97 Tf 6.58 0 Td[(p#ihy)]TJ /F5 7.97 Tf 6.58 0 Td[(k"yk#i+hyp"y)]TJ /F5 7.97 Tf 6.58 0 Td[(p#iy)]TJ /F5 7.97 Tf 6.59 0 Td[(k"yk#+yp"y)]TJ /F5 7.97 Tf 6.58 0 Td[(p#hy)]TJ /F5 7.97 Tf 6.59 0 Td[(k"yk#i)-222(hyp"y)]TJ /F5 7.97 Tf 6.59 0 Td[(p#ihy)]TJ /F5 7.97 Tf 6.59 0 Td[(k"yk#i)].(B)WecanseethatthetermswhicharegeneratedbythefoldingoftheBrillouinzoneoverallhaveadifferentsignfromthetermsinsidethereducedBrillouinzone.Sowecancombinethesetwopartsas H3=Xk,p,0[Vp)]TJ /F5 7.97 Tf 6.59 0 Td[(k(u2pu2k+2upvpukvk+v2pv2k))]TJ /F3 11.955 Tf 11.96 0 Td[(Vp)]TJ /F5 7.97 Tf 6.58 0 Td[(k+Q(v2pu2k+2upvpukvk+u2pv2k)](hypy)]TJ /F5 7.97 Tf 6.59 0 Td[(pi)]TJ /F5 7.97 Tf 6.59 0 Td[(kk+ypy)]TJ /F5 7.97 Tf 6.59 0 Td[(ph)]TJ /F5 7.97 Tf 6.59 0 Td[(kki)-222(hypy)]TJ /F5 7.97 Tf 6.58 0 Td[(pih)]TJ /F5 7.97 Tf 6.59 0 Td[(kki+hypy)]TJ /F5 7.97 Tf 6.59 0 Td[(pi)]TJ /F5 7.97 Tf 6.59 0 Td[(kk+ypy)]TJ /F5 7.97 Tf 6.59 0 Td[(ph)]TJ /F5 7.97 Tf 6.59 0 Td[(kki)-222(hypy)]TJ /F5 7.97 Tf 6.58 0 Td[(pih)]TJ /F5 7.97 Tf 6.58 0 Td[(kki)+[Vp)]TJ /F5 7.97 Tf 6.59 0 Td[(k(u2pv2k)]TJ /F6 11.955 Tf 11.95 0 Td[(2upvpukvk+v2pu2k))]TJ /F3 11.955 Tf 11.96 0 Td[(Vp)]TJ /F5 7.97 Tf 6.58 0 Td[(k+Q(u2pu2k)]TJ /F6 11.955 Tf 11.96 0 Td[(2upvpukvk+v2pv2k)](hypy)]TJ /F5 7.97 Tf 6.59 0 Td[(pi)]TJ /F5 7.97 Tf 6.59 0 Td[(kk+ypy)]TJ /F5 7.97 Tf 6.58 0 Td[(ph)]TJ /F5 7.97 Tf 6.58 0 Td[(kki)-222(hypy)]TJ /F5 7.97 Tf 6.59 0 Td[(pih)]TJ /F5 7.97 Tf 6.59 0 Td[(kki+hypy)]TJ /F5 7.97 Tf 6.59 0 Td[(pi)]TJ /F5 7.97 Tf 6.58 0 Td[(kk+ypy)]TJ /F5 7.97 Tf 6.59 0 Td[(ph)]TJ /F5 7.97 Tf 6.59 0 Td[(kki)-222(hypy)]TJ /F5 7.97 Tf 6.58 0 Td[(pih)]TJ /F5 7.97 Tf 6.59 0 Td[(kki).(B) 87

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ByusingmeaneldtheoryandtheBogoliubovtransformation,wehavetheCooperpairexpectationvalues, hyp"y)]TJ /F5 7.97 Tf 6.59 0 Td[(p#i=hp")]TJ /F5 7.97 Tf 6.59 0 Td[(p#i=p 2ptanh)]TJ /F6 11.955 Tf 7.75 -0.74 Td[(p 2T(B)where=,.ThemeaneldenergyfromthethirdtermoftheHamiltonianproducesthefollowingresult, hH3i=Xk,p0)]TJ /F6 11.955 Tf 11.95 0 Td[(2[Vp)]TJ /F5 7.97 Tf 6.59 0 Td[(k(u2pu2k+2upvpukvk+v2pv2k))]TJ /F3 11.955 Tf 11.96 0 Td[(Vp)]TJ /F5 7.97 Tf 6.59 0 Td[(k+Q(v2pu2k+2upvpukvk+u2pv2k)]p 2ptanh)]TJ /F6 11.955 Tf 7.44 -0.74 Td[(p 2Tk 2ktanh)]TJ /F6 11.955 Tf 7.44 -1.6 Td[(k 2T+p 2ptanh)]TJ /F6 11.955 Tf 7.58 -0.74 Td[(p 2Tk 2ktanh)]TJ /F6 11.955 Tf 7.58 -1.6 Td[(k 2T+[Vp)]TJ /F5 7.97 Tf 6.59 0 Td[(k(u2pv2k)]TJ /F6 11.955 Tf 11.95 0 Td[(2upvpukvk+v2pu2k))]TJ /F3 11.955 Tf 11.96 0 Td[(Vp)]TJ /F5 7.97 Tf 6.59 0 Td[(k+Q(u2pu2k)]TJ /F6 11.955 Tf 11.96 0 Td[(2upvpukvk+v2pv2k)]p 2ptanh)]TJ /F6 11.955 Tf 7.44 -0.73 Td[(p 2Tk 2ktanh)]TJ /F6 11.955 Tf 7.58 -1.59 Td[(k 2T+p 2ptanh)]TJ /F6 11.955 Tf 7.58 -0.73 Td[(p 2Tk 2ktanh)]TJ /F6 11.955 Tf 7.44 -1.59 Td[(k 2T.(B)NowthegapequationinEquation 2 canreplacepartofthisexpression,reducingthenalresultforthethirdtermtothefollowingform, hH3i=)]TJ /F6 11.955 Tf 9.3 0 Td[(2Xk0kk 2ktanh)]TJ /F6 11.955 Tf 7.44 -1.59 Td[(k 2T+kk 2ktanh)]TJ /F6 11.955 Tf 7.58 -1.59 Td[(k 2T.(B)Combiningthekinetictermandtheantiferromagneticterm,weget hH1i+hH2i=2Xk0("ku2k+v2k"k+Q)(u2kf(k)+v2k(1)]TJ /F3 11.955 Tf 11.96 0 Td[(f(k)))+("kv2k+u2k"k+Q)(u2kf(k)+v2k(1)]TJ /F3 11.955 Tf 11.96 0 Td[(f(k)))+2W2 q ")]TJ /F5 7.97 Tf 6.59 0 Td[(2k+W2[u2kf(k)+v2k(1)]TJ /F3 11.955 Tf 11.95 0 Td[(f(k)))]TJ /F6 11.955 Tf 12.2 0 Td[(u2kf(k))]TJ /F6 11.955 Tf 12.24 0 Td[(v2k(1)]TJ /F3 11.955 Tf 11.96 0 Td[(f(k))]+W2 U.(B) 88

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Thetotalmeaneldenergyistherefore EMF=hHi=hH1i+hH2i+hH3i=2Xk0Ek(u2kf(k)+v2k(1)]TJ /F3 11.955 Tf 11.96 0 Td[(f(k)))+Ek(u2kf(k)+v2k(1)]TJ /F3 11.955 Tf 11.95 0 Td[(f(k))))]TJ /F6 11.955 Tf 9.3 0 Td[(2Xk02k 2ktanh)]TJ /F6 11.955 Tf 7.44 -1.59 Td[(k 2T+2k 2ktanh)]TJ /F6 11.955 Tf 7.58 -1.59 Td[(k 2T+2W2 U.(B)Thenalresultforthemeaneldenergyis EMF=Xk0Ek)]TJ /F6 11.955 Tf 11.95 0 Td[(k+Ek)]TJ /F6 11.955 Tf 11.96 0 Td[(k+2k 2ktanh)]TJ /F6 11.955 Tf 7.44 -1.6 Td[(k 2T+2k 2ktanh)]TJ /F6 11.955 Tf 7.58 -1.6 Td[(k 2T+2kf(k)+2kf(k)+W2 U.(B)WecanmakeasimplecheckofthenalresultwiththelimitInthelimitofpureantiferromagneticstate, hHMi=Xk02Ekf(Ek)+2Ekf(Ek)+W2 U.(B)ThelimitinthepureSCstate,W=0,reducestotheexpectationvaluegiveninEq.(3.45)inTinkham[ 89 ]whichistheenergyforthesuperconductingstate, HM=Xk("k)]TJ /F3 11.955 Tf 11.95 0 Td[(Ek+kbk)+XkEk(yk0k0+yk1k1).(B)Intermsofexpectationvalue,itwouldbe hHMi=Xk("k)]TJ /F3 11.955 Tf 11.96 0 Td[(Ek+2k 2Ektanh(Ek 2T))+Xk2Ekf(Ek).(B) 89

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APPENDIXCDERIVATTIONSOFDYNAMICSPINSUSCEPTIBILITYINTHEPUREANTIFERROMAGNETICSTATE C.1Transversedynamicspinsusceptibilityintheantiferromagneticstate:derivationThedenitionofthetransversedynamicspinsusceptibilityis +)]TJ /F5 7.97 Tf -6.59 -7.98 Td[(0(q,q0,i!)=Z0dhTS+q()S)]TJ -1.07 -8.34 Td[()]TJ /F5 7.97 Tf 6.59 0 Td[(q0(0)iei!.(C)Inordertocalculatethetransversepartofthesusceptibility,weneedtousethedeni-tionofthetransversespinoperators: S+q()=Xkcyk+q"()ck#()S)]TJ /F5 7.97 Tf -1.07 -7.89 Td[(q()=Xkcyk+q#()ck"().(C)Nowthesusceptibilityiswrittenintermsofraisingandloweringoperatorsas +)]TJ /F5 7.97 Tf -6.59 -7.98 Td[(0(q,q0,i!)=Z0dXk,k0hTcyk+q"()ck#()cyk0)]TJ /F5 7.97 Tf 6.59 0 Td[(q0#(0)ck0"(0)iei!.(C)WeapplyWicktheoremtodecouplethefour-operator,andget +)]TJ /F5 7.97 Tf -6.58 -7.97 Td[(0(q,q0,i!)=Z0dei!Xk,k0[hTcyk+q"()ck0"(0)ihTck#()cyk0)]TJ /F5 7.97 Tf 6.59 0 Td[(q0#(0)i+hTcyk+q"()cyk0)]TJ /F5 7.97 Tf 6.59 0 Td[(q0#(0)ihTck"()ck0#(0)i].(C)HereweseetwodifferenttermsforthetransversesusceptibilityafterapplyingWicktheorem.OneinvolvestheexpectationvaluesofthenormalstateelectronoperatorsandtheotherinvolvestheexpectationvalueofthesuperconductingCooperpairoperators.Forthecalculationsinthepureantiferromagneticstatewithnosuperconductivity,theexpectationvaluesofCooperpairoperatorswouldbezero.Thereforewecanignorethelatterforthenon-superconductingcalculation.Weonlyhavetoevaluate: +)]TJ /F5 7.97 Tf -6.59 -7.97 Td[(0(q,q0,i!)=Z0dei!Xk,k0hTcyk+q"()ck0"(0)ihTck#()cyk0)]TJ /F5 7.97 Tf 6.59 0 Td[(q0#(0)i.(C) 90

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Transformingkandk0tothereducedBrillouinzone,wehave +)]TJ /F5 7.97 Tf -6.58 -7.97 Td[(0(q,q0,i!)=Z0dei!Xk,k00[hTcyk+q"()ck0"(0)ihTck#()cyk0)]TJ /F5 7.97 Tf 6.59 0 Td[(q0#(0)i+hTcyk+q"()ck0+Q"(0)ihTck#()cyk0)]TJ /F5 7.97 Tf 6.59 0 Td[(q0+Q#(0)i+hTcyk+q+Q"()ck0"(0)ihTck+Q#()cyk0)]TJ /F5 7.97 Tf 6.59 0 Td[(q0#(0)i+hTcyk+q+Q"()ck0+Q"(0)ihTck+Q#()cyk0)]TJ /F5 7.97 Tf 6.58 0 Td[(q0+Q#(0)i.(C)Thetransversesusceptibilityhasadiagonalterm+)]TJ /F5 7.97 Tf -6.59 -7.98 Td[(0(q,q,i!)andtheUmklappterm+)]TJ /F5 7.97 Tf -6.58 -7.97 Td[(0(q,q+Q,i!)duetothebreakingoftranslationalsymmetry.Herewerstcalculatethediagonalpartsofthetransversesusceptibilityandsetq0=q.WeapplytheunitarytransformationofEquation 2 togetthesusceptibilityintheantiferromagneticstate.Whenapplyingthetransformation,wehavetoconsiderbothcase,k+qinsideandoutsideofthereducedBrillouinzone.Duetothesymmetryofthethealgebra,thetwocaseshavethesameSDWcoefcientsforthetransversesusceptibility.Wecanjustassumeonecasethatk+qisinsidethereducedzoneandk+q+Qisoutsideofthereducedzoneandsumoverk0.Thesusceptibilityreducesto: +)]TJ /F5 7.97 Tf -6.58 -7.97 Td[(0(q,q,i!)=Z0dei!Xk0[(u2k+qu2k)]TJ /F6 11.955 Tf 11.95 0 Td[(2uk+qvk+qukvk+v2k+qv2k)hTyk+q"()k+q"(0)ihTk#()yk#(0)i+hTyk+q"()k+q"(0)ihTk#()yk#(0)i+(u2k+qv2k+2uk+qvk+qukvk+v2k+qu2k)hTyk+q"()k+q"(0)ihTk#()yk#(0)i+hTyk+q"()k+q"(0)ihTk#()yk#(0)i].(C)EvaluatingtheFouriertransformoftheMatsubaraGreen'sfunction,weget 91

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Z0dei!mhyk+q()k+q(0)ih0yk0(0)0k0()i=f(Ek+q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(E0k) i!)]TJ /F3 11.955 Tf 11.95 0 Td[(E0k+Ek+q (Ca)Z0dei!mhyk+q()k+q(0)ih0k0(0)0yk0()i=1)]TJ /F3 11.955 Tf 11.96 0 Td[(f(E0k))]TJ /F3 11.955 Tf 11.96 0 Td[(f(Ek+q) i!+E0k+Ek+q (Cb)Z0dei!mhk+q()yk+q(0)ih0yk0(0)0k0()i)=f(E0k)+f(Ek+q))]TJ /F6 11.955 Tf 11.95 0 Td[(1 i!)]TJ /F3 11.955 Tf 11.96 0 Td[(E0k)]TJ /F3 11.955 Tf 11.95 0 Td[(Ek+q (Cc)Z0dei!mhk+q()yk+q(0)ih0k0(0)0yk0()i)=f(E0k))]TJ /F3 11.955 Tf 11.96 0 Td[(f(Ek+q) i!+E0)]TJ /F3 11.955 Tf 11.95 0 Td[(Ek+q (Cd)withf(E)beingtheFermifunctionandEkthequasiparticleenergy.Thenalresultforthetransversedynamicspinsusceptibilityintheantiferromagneticstateis: +)]TJ /F5 7.97 Tf -6.59 -7.97 Td[(0(q,q,i!)=)]TJ /F12 11.955 Tf 13.43 11.35 Td[(Xk,=00(u2k+qu2k)]TJ /F6 11.955 Tf 11.96 0 Td[(2uk+qvk+qukvk+v2k+qv2k)f(Ek+q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(E0k) i!)]TJ /F3 11.955 Tf 11.95 0 Td[(E0k+Ek+q)]TJ /F12 11.955 Tf 13.43 11.35 Td[(Xk,6=00(u2k+qv2k+2uk+qvk+qukvk+v2k+qu2k)f(Ek+q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(E0k) i!)]TJ /F3 11.955 Tf 11.95 0 Td[(E0k+Ek+q(C)where,0arebothand.Ifwepluginthevaluesforthecoherencefactors,wegetthenalresultforthetransverseoff-diagonaldynamicspinsusceptibility +)]TJ /F5 7.97 Tf -6.59 -7.98 Td[(0(q,q,i!)=)]TJ /F6 11.955 Tf 10.5 8.09 Td[(1 2Xk,=000B@1+")]TJ /F5 7.97 Tf 0 -8.28 Td[(k")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q)]TJ /F3 11.955 Tf 11.96 0 Td[(W2 q )]TJ /F10 11.955 Tf 5.48 -9.69 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k2+W2q )]TJ /F10 11.955 Tf 5.48 -9.69 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q2+W21CAf(Ek+q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(Ek) i!)]TJ /F3 11.955 Tf 11.96 0 Td[(Ek+q+Ek)]TJ /F6 11.955 Tf 10.5 8.09 Td[(1 2Xk,6=000B@1)]TJ /F10 11.955 Tf 57.52 9.32 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q)]TJ /F3 11.955 Tf 11.96 0 Td[(W2 q )]TJ /F10 11.955 Tf 5.48 -9.68 Td[(")]TJ /F5 7.97 Tf 0 -8.27 Td[(k2+W2q )]TJ /F10 11.955 Tf 5.48 -9.68 Td[(")]TJ /F5 7.97 Tf 0 -8.27 Td[(k+q2+W21CAf(E0k+q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(Ek) i!)]TJ /F3 11.955 Tf 11.96 0 Td[(E0k+q+Ek. (C) C.2UmklapptermforthetransversedynamicspinsusceptibilityWithoutlossofgenerality,wecanstartwithEquation C .FortheUmklapp(off-diagonal)term,weevaluatethespinsusceptibilitywithq0=q+Qandwehave 92

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+)]TJ /F5 7.97 Tf -6.58 -8.28 Td[(Q(q,i!)=+)]TJ /F5 7.97 Tf -6.59 -7.98 Td[(0(q,q+Q,i!)=Z0dei!Xk,k00[hTcyk+q"()ck0"(0)ihTck#()cyk0)]TJ /F5 7.97 Tf 6.59 0 Td[(q+Q#(0)i+hTcyk+q"()ck0+Q"(0)ihTck#()cyk0)]TJ /F5 7.97 Tf 6.59 0 Td[(q#(0)i+hTcyk+q+Q"()ck0"(0)ihTck+Q#()cyk0)]TJ /F5 7.97 Tf 6.59 0 Td[(q+Q#(0)i+hTcyk+q+Q"()ck0+Q"(0)ihTck+Q#()cyk0)]TJ /F5 7.97 Tf 6.58 0 Td[(q#(0)i.(C)Herethesameasinthecalculationof+)]TJ /F6 11.955 Tf 7.09 -4.34 Td[((q,q0,i!),weuseEquation 2 fortheunitarytransformation,andobtain +)]TJ /F5 7.97 Tf -6.59 -8.28 Td[(Q(q,i!)=Z0dei!Xk,k02RhhT(uk+qyk+q"()+vk+qyk+q"())(uk0k0"(0)+vk0k0"(0))ihT(ukk#()+vkk#())(vk0)]TJ /F5 7.97 Tf 6.59 0 Td[(qyk0)]TJ /F5 7.97 Tf 6.58 0 Td[(q#(0))]TJ /F3 11.955 Tf 11.96 0 Td[(uk0)]TJ /F5 7.97 Tf 6.59 0 Td[(qyk0)]TJ /F5 7.97 Tf 6.58 0 Td[(q#(0))i+hT()]TJ /F3 11.955 Tf 9.3 0 Td[(vk+qyk+q"()+uk+qyk+q"())(uk0k0"(0)+vk0k0"(0))ihT(vkk#())]TJ /F3 11.955 Tf 11.95 0 Td[(ukk#())(vk0)]TJ /F5 7.97 Tf 6.59 0 Td[(qyk0)]TJ /F5 7.97 Tf 6.58 0 Td[(q#(0))]TJ /F3 11.955 Tf 11.96 0 Td[(uk0)]TJ /F5 7.97 Tf 6.59 0 Td[(qyk0)]TJ /F5 7.97 Tf 6.58 0 Td[(q#(0))i+hT(uk+qyk+q"()+vk+qyk+q"())(uk0k0"(0)+vk0k0"(0))ihT(ukk#()+vkk#())(uk0)]TJ /F5 7.97 Tf 6.59 0 Td[(qyk0)]TJ /F5 7.97 Tf 6.59 0 Td[(q#(0)+vk0)]TJ /F5 7.97 Tf 6.59 0 Td[(qyk0)]TJ /F9 7.97 Tf 6.58 0 Td[(q#(0))i+hT(uk+qyk+q"()+vk+qyk+q"())(vk0k0"(0))]TJ /F3 11.955 Tf 11.96 0 Td[(uk0k0"(0))ihT(vkk#())]TJ /F3 11.955 Tf 11.95 0 Td[(ukk#())(vk0)]TJ /F9 7.97 Tf 6.59 0 Td[(qyk0)]TJ /F9 7.97 Tf 6.59 0 Td[(q#(0))]TJ /F3 11.955 Tf 11.95 0 Td[(uk0)]TJ /F9 7.97 Tf 6.59 0 Td[(qyk0)]TJ /F9 7.97 Tf 6.59 0 Td[(q#(0))i.(C) 93

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Afterorganizing,wehavethetransverseUmklappspinsusceptibilityreducingto +)]TJ /F5 7.97 Tf -6.59 -8.28 Td[(Q(q,i!)=Z0dei!Xk,[(ukvk)]TJ /F3 11.955 Tf 11.96 0 Td[(uk+qvk+q)[hTyk+q"()k+q"(0)ihTk#()yk#(0)i)-222(hTyk+q"()k+q"(0)ihTk#()yk#(0)i])]TJ /F6 11.955 Tf 11.96 0 Td[((uk+qvk+q+ukvk)[hTyk+q"()k+q#(0)ihTk"()yk#(0)i)-222(hTyk+q"()k+q#(0)ihTk"()yk#(0)i].(C)WiththehelpoftheGreen'sfunctionFouriertransformationsinEquation D andpluggingtheexpressionsforthecoherencefactors,wegetthenalresultfortheUmklapptermofthetransverespinsusceptibility, +)]TJ /F5 7.97 Tf -6.58 -7.97 Td[(0(q,q+Q,i!)=+)]TJ /F5 7.97 Tf -6.59 -8.27 Td[(Q(q,i!)=W 2Xk00@1 q )]TJ /F10 11.955 Tf 5.48 -9.68 Td[(")]TJ /F5 7.97 Tf 0 -8.27 Td[(k+q2+W2)]TJ /F6 11.955 Tf 46.73 8.09 Td[(1 q )]TJ /F10 11.955 Tf 5.48 -9.68 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k2+W21A f(Ek+q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(Ek) i!)]TJ /F3 11.955 Tf 11.96 0 Td[(Ek+q+Ek)]TJ /F3 11.955 Tf 13.15 9.32 Td[(f(Ek+q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(Ek) i!)]TJ /F3 11.955 Tf 11.95 0 Td[(Ek+q+Ek!)]TJ /F12 11.955 Tf 11.29 24.03 Td[(0@1 q )]TJ /F10 11.955 Tf 5.48 -9.68 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q2+W2+1 q )]TJ /F10 11.955 Tf 5.48 -9.69 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k2+W21A f(Ek+q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(Ek) i!)]TJ /F3 11.955 Tf 11.96 0 Td[(Ek+q+Ek)]TJ /F3 11.955 Tf 13.15 9.1 Td[(f(Ek+q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(Ek) i!)]TJ /F3 11.955 Tf 11.95 0 Td[(Ek+q+Ek!. (C) C.3ThelongitudinaldynamicspinsusceptibilityThedenitionofthelongitudinalspinsusceptibilityis zz0(q,q0,!)=Zdti 2NhTSzq(t)Sz)]TJ /F5 7.97 Tf 6.59 0 Td[(q(0)iei!t (C) withthespinoperator Szq()=Xkcyk+q()ck().(C)Intermsoftheraisingandloweringoperators,thesusceptibilityiswrittenas zz0(q,q0,!)=Z0dei!Xkk000hT(cyk+q()ck()cyk0)]TJ /F5 7.97 Tf 6.59 0 Td[(q00(0)ck00(0)i.(C) 94

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Writingoutthespinindicesandsettingq0=qforthenon-Umklappspinsusceptibility,weget: zz0(q,q,!)=Z0dei!Xkk0hT(cyk+q"()ck"()cyk0)]TJ /F5 7.97 Tf 6.59 0 Td[(q"(0)ck0"(0))]TJ /F3 11.955 Tf 11.95 0 Td[(cyk+q"()ck"()cyk0)]TJ /F5 7.97 Tf 6.59 0 Td[(q#(0)ck0#(0))]TJ /F3 11.955 Tf 11.96 0 Td[(cyk+q#()ck#()cyk0)]TJ /F5 7.97 Tf 6.59 0 Td[(q"(0)ck0"(0)+cyk+q#()ck#()cyk0)]TJ /F5 7.97 Tf 6.59 0 Td[(q#(0)ck0#(0))i.(C)ByusingWicktheoremtoseparatethefouroperatorsandignoringthetermswhichhaveexpectationvalueszero,wehave zz0(q,q,!)=Z0dei!Xkk0hTcyk+q"()ck0"(0)ihTck"()cyk0)]TJ /F5 7.97 Tf 6.59 0 Td[(q"(0)i+hTcyk+q#()ck0#(0)ihTck#()cyk0)]TJ /F5 7.97 Tf 6.59 0 Td[(q#(0)i.(C)ReducingthesumoverktothereducedBrillouinzoneandapplyingtheunitarytransfor-mation,weget zz0(q,q,!)=Z0dei!Xk0(u2ku2k+q+2ukvkuk+qvk+q+v2kv2k+q)(hT(yk()k(0)ihTk+q()yk+q(0)i+hTyk()k(0)ihTk+q()yk+q(0)i)+(u2kv2k+q)]TJ /F6 11.955 Tf 11.95 0 Td[(2ukvkuk+qvk+q+v2ku2k+q)(hTyk()k(0)ihTk+q()yk+q(0)i+hTyk()k(0)ihTk+q()yk+q(0)i).(C)Nowpluggingtheexpectationvaluesfortheoperators,weobtainthenalresultas 95

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zz0(q,q,!)=)]TJ /F6 11.955 Tf 10.49 8.09 Td[(1 2Xk,=000B@1+")]TJ /F5 7.97 Tf 0 -8.28 Td[(k")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q+W2 q )]TJ /F10 11.955 Tf 5.48 -9.68 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k2+W2q )]TJ /F10 11.955 Tf 5.48 -9.68 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q2+W21CAf(Ek+q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(Ek) i!)]TJ /F3 11.955 Tf 11.95 0 Td[(Ek+q+Ek)]TJ /F6 11.955 Tf 10.49 8.09 Td[(1 2Xk,6=000B@1)]TJ /F10 11.955 Tf 57.52 9.32 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q+W2 q )]TJ /F10 11.955 Tf 5.48 -9.69 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k2+W2q )]TJ /F10 11.955 Tf 5.48 -9.69 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q2+W21CAf(Ek+q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(E0k) i!)]TJ /F3 11.955 Tf 11.95 0 Td[(Ek+q+E0k. (C) C.4ThelongitudinalUmklappsusceptibilityTheUmklapptermforthelongitudinaldynamicspinsusceptibilityiszeroduetothefactthatthereisnosymmetrybreakingalongthex)]TJ /F3 11.955 Tf 11.97 0 Td[(yplanewhenthesystemgoesintoantiferromagneticstate.Thisfactcanalsobecalculatedalgebraically.Startingwiththedenitionofthelongitudinaldynamicspinsusceptibilitywithq0=q+QinEquation C ,wehave zz0(q,q+Q,i!)=Z0dei!Xkk000hT(cyk+q()ck()cyk0)]TJ /F5 7.97 Tf 6.59 0 Td[(q+Q0(0)ck00(0)i.(C)UsingtheWicktheorem,weget zz0(q,q+Q,i!)=Z0dei!Xkk0hTcyk+q()ck0(0)ihTck()cyk0)]TJ /F5 7.97 Tf 6.59 0 Td[(q+Q0(0)i.(C)Transferringkandk0intothereducedBrillouinzone,wehave zz0(q,q+Q,i!)=Z0dei!Xk0hTcyk+q()ck0(0)ihTck()cyk0)]TJ /F5 7.97 Tf 6.59 0 Td[(q+Q(0)i+hTcyk+q+Q()ck0(0)ihTck+Q()cyk0)]TJ /F5 7.97 Tf 6.59 0 Td[(q+Q(0)i+hTcyk+q()ck0+Q(0)ihTck()cyk0)]TJ /F5 7.97 Tf 6.59 0 Td[(q(0)i+hTcyk+q+Q()ck0+Q(0)ihTck+Q()cyk0)]TJ /F5 7.97 Tf 6.58 0 Td[(q(0)i.(C) 96

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Usingtheunitarytransformation,wehave zz0(q,q+Q,!)=Z0dei!Xk0hhT(uk+qyk+q()+vk+qyk+q())(uk+qk+q(0)+vk+qk+q(0))ihT(ukyk()+vkyk())(sgn())()]TJ /F3 11.955 Tf 9.3 0 Td[(vkk(0)+ukk(0))i+hT(sgn())()]TJ /F3 11.955 Tf 9.29 0 Td[(vk+qyk+q()+uk+qyk+q())(uk+qk+q(0)+vk+qk+q(0))i(sgn())hT()]TJ /F3 11.955 Tf 9.29 0 Td[(vkyk()+ukyk())(sgn())()]TJ /F3 11.955 Tf 9.3 0 Td[(vkk(0)+ukk(0))i+hT(uk+qyk+q()+vk+qyk+q())(sgn())()]TJ /F3 11.955 Tf 9.3 0 Td[(vk+qk+q(0)+uk+qk+q(0))ihT(ukyk()+vkyk())(ukk(0)+vkk(0))i+hT()]TJ /F3 11.955 Tf 9.3 0 Td[(vk+qyk+q()+uk+qyk+q())()]TJ /F3 11.955 Tf 9.3 0 Td[(vk+qk+q(0)+uk+qk+q(0))ihT(sgn())()]TJ /F3 11.955 Tf 9.29 0 Td[(vkyk()+ukyk())(ukk(0)+vkk(0))i.(C)Aftersomeorganization,weget: zz0(q,q+Q,i!)=Z0dei!Xk0n)]TJ /F3 11.955 Tf 11.96 0 Td[(sgn()(uk+qvk+q+ukvk)hhT(yk+q()k+q(0)ihTk()yk(0)i)-223(hTyk+q()k+q(0)ihTk()yk(0)ii)]TJ /F3 11.955 Tf 9.3 0 Td[(sgn()(uk+qvk+q)]TJ /F3 11.955 Tf 11.95 0 Td[(ukvk)hhTyk+q()k+q(0)ihTk()yk(0)i)-222(hTyk+q()k+q(0)ihTk()yk(0)iio.(C)TheGreen'sfunctionvaluedoesnotdependonspins.Butthecoherencefactorsresultindifferentsignsforoppositespins.ThereforetheevaluationoftheUmklapptermofthelongitudinaldynamicspinsusceptibilityiszero. 97

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C.5AnalyticprooffortheformationoftheGoldstonemodeTheGoldstonemodeonlyappearsinthetransversepartofthespinsusceptibility,andoccurswhentherealpartofthesusceptibilitysatisesthecondition1)]TJ /F3 11.955 Tf 12.85 0 Td[(U+)]TJ /F5 7.97 Tf -6.59 -7.97 Td[(0atkequalstotheorderingmomentumQ=(,)and!=0.ThisiswhentheRPAsusceptibilityhasapoleinthedenominator.Hereweknowthattheself-consistentequationforUandW,Equation 2 isalwayssatised.Withq=Qand!=0,thebaretransversePaulisusceptibilityreducesto +)]TJ /F5 7.97 Tf -6.58 -7.98 Td[(0(Q,0)=2Xk0f(Ek))]TJ /F3 11.955 Tf 11.95 0 Td[(f(Ek) Ek)]TJ /F3 11.955 Tf 11.95 0 Td[(Ek.(C)HerewehaveusedtherelationEk+Q=Ek.FortheUmklapptransversesusceptibilityequation(14),theperiodiccondition(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+Q)2=(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k)2makesthecoherencefactoralwaystobezero,thereforetheUmklapptermiszeroandweonlyhavetoconsiderthediagonalpartofthetransversesusceptibilitymatrixinEquation 3 .ComparingEquation C withtheself-consistentequationforUandW,Equation 2 ,Wehavesimply +)]TJ /F5 7.97 Tf -6.59 -7.97 Td[(0(Q,0)=1 U.(C)ThisguaranteesthattheGoldstonemodealwayshappensatq=Q. 98

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APPENDIXDDERIVATIONSOFDYNAMICSPINSUSCEPTIBILITYINTHECOEXISTENCESTATEOFANTIFERROMAGNETICANDSUPERCONDUCTIVITY D.1DerivationsoftransversedynamicspinsusceptibilityinthecoexistencestateofantiferromagnetismandsuperconductivityWecanstartthederivationfromtheintermediatestepsofthesusceptibilityinthepureantiferromagneticstate.Forthenon-Umklapp(diagonal)termofthetransversedynamicspinsusceptibility,westartwithEquation C whichis +)]TJ /F5 7.97 Tf -6.58 -7.98 Td[(0(q,q0,i!)=Z0dei!Xk,k0[hTcyk+q"()ck0"(0)ihTck#()cyk0)]TJ /F5 7.97 Tf 6.59 0 Td[(q0#(0)i+hTcyk+q"()cyk0)]TJ /F5 7.97 Tf 6.59 0 Td[(q0#(0)ihTck"()ck0#(0)i].(D)Notethatinthesuperconductingstate,wehavetoincludetheCooperpairingoperatorswhichwereomittedinthecaseofnon-superconductingstate.Wedenethat +)]TJ /F5 7.97 Tf -6.58 -7.98 Td[(0(q,q0,i!)=+)]TJ /F9 7.97 Tf -6.59 -7.9 Td[(nor(q,q0,i!)++)]TJ /F9 7.97 Tf -6.59 -8.28 Td[(SC(q,q0,i!)(D)with +)]TJ /F9 7.97 Tf -6.59 -7.89 Td[(nor(q,q0,i!)=Z0dei!Xk,k0hTcyk+q"()ck0"(0)ihTck#()cyk0)]TJ /F5 7.97 Tf 6.58 0 Td[(q0#(0)i(D)and +)]TJ /F9 7.97 Tf -6.59 -8.28 Td[(SC(q,q0,i!)=Z0dei!nXk,k0hTcyk+q"()cyk0)]TJ /F5 7.97 Tf 6.59 0 Td[(q0#(0)ihTck"()ck0#(0)i.(D)Forthenormalstatepartofthetransversespinsusceptibility+)]TJ /F9 7.97 Tf -6.59 -7.29 Td[(nor(q,q,!),wealreadyhavetheresultfromthepureantiferromagneticstatecalculationwhichis 99

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Equation C +)]TJ /F9 7.97 Tf -6.58 -7.89 Td[(nor(q,q,i!)=Z0dei!Xk0(u2k+qu2k)]TJ /F6 11.955 Tf 11.95 0 Td[(2uk+qvk+qukvk+v2k+qv2k)hhTyk+q"()k+q"(0)ihTk#()yk#(0)i+hTyk+q"()k+q"(0)ihTk#()yk#(0)ii+(u2k+qv2k+2uk+qvk+qukvk+v2k+qu2k)hhTyk+q"()k+q"(0)ihTk#()yk#(0)i+hTyk+q"()k+q"(0)ihTk#()yk#(0)ii.(D)Inthesuperconductingstate,wehavetocontinuewiththeBCStransformationinEquation 2 .Thequasiparticleoperatorsinthepureantiferromagneticstate,kandkwillbereplacedbyk0,klandk0,kl.ButtheSDWcoherencefactorsremainunchanged.WecanjustevaluatetheMatsubaraFouriertransformationofGreen'sfunc-tion,R0dei!hTyk+q"()k+q"(0)ihTk#()yk#(0)i.Laterwereplacetheintegrationoftheexpectationvaluewithitsgeneralexpression.Togetthegeneralexpressionoftheexpectationvalue,weperformtheBCStransformationandget Z0dei!hTyk+q"()k+q"(0)ihTk#()yk#(0)i=Z0dei!mhu2k+qv2khyk+q0()k+q0(0)ihyk0()k0(0)i+u2k+qu2khyk+q0()k+q0(0)ihkl()ykl(0)i+v2k+qv2khk+ql()yk+ql(0)ihyk0()k0(0)i+v2k+qu2khk+ql()yk+ql(0)ihkl()ykl(0)ii(D)wherehi=hi. 100

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NextweusethethefollowingrelationsfortheMatsubaraintegrationoftheGreen'sfunction, Z0dei!hyk+q0()k+q0(0)ih0yk0(0)0k0()i=f(k+q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(0k) i!)]TJ /F6 11.955 Tf 11.95 0 Td[(0k+k+q (Da)Z0dei!hyk+q0()k+q0(0)ih0kl(0)0ykl()i=1)]TJ /F3 11.955 Tf 11.96 0 Td[(f(0k))]TJ /F3 11.955 Tf 11.96 0 Td[(f(k+q) i!+0k+k+q (Db)Z0dei!hk+ql()yk+ql(0)ih0yk0(0)0k0()i)=f(0k)+f(k+q))]TJ /F6 11.955 Tf 11.95 0 Td[(1 i!)]TJ /F6 11.955 Tf 11.96 0 Td[(0k)]TJ /F6 11.955 Tf 11.95 0 Td[(k+q (Dc)Z0dei!hk+ql()yk+ql(0)ih0kl(0)0ykl()i)=f(0k))]TJ /F3 11.955 Tf 11.96 0 Td[(f(k+q) i!+0k)]TJ /F6 11.955 Tf 11.96 0 Td[(k+q (Dd)whereand0areand.Andweget Z0dei!hTyk+q"()k+q"(0)ihTk#()yk#(0)i=u2k+qv2kf(k)+f(k+q))]TJ /F6 11.955 Tf 11.95 0 Td[(1 i!+k+k+q+u2k+qu2kf(k))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k+q) i!)]TJ /F6 11.955 Tf 11.95 0 Td[(k+k+q+v2k+qv2kf(k+q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(k) i!+k)]TJ /F6 11.955 Tf 11.96 0 Td[(k+q+v2k+qu2k1)]TJ /F3 11.955 Tf 11.96 0 Td[(f(k))]TJ /F3 11.955 Tf 11.96 0 Td[(f(k+q) i!)]TJ /F6 11.955 Tf 11.95 0 Td[(k)]TJ /F6 11.955 Tf 11.96 0 Td[(k+q.(D)Forthesamecalculationfor,wejustreplaceallsintheaboveequationwith.Followingthesamecalculationswecanalsogettheinterbandtermswhichare Z0dei!hTyk+q"()k+q"(0)ihTk#()yk#(0)i=u2k+qv2kf(k)+f(k+q))]TJ /F6 11.955 Tf 11.95 0 Td[(1 i!+k+k+q+u2k+qu2kf(k))]TJ /F3 11.955 Tf 11.96 0 Td[(f(k+q) i!)]TJ /F6 11.955 Tf 11.95 0 Td[(k+k+q+v2k+qv2kf(k+q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k) i!+k)]TJ /F6 11.955 Tf 11.96 0 Td[(k+q+v2k+qu2k1)]TJ /F3 11.955 Tf 11.95 0 Td[(f(k))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k+q) i!)]TJ /F6 11.955 Tf 11.95 0 Td[(k)]TJ /F6 11.955 Tf 11.95 0 Td[(k+q.(D) 101

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Theotherinterbandtermcanbeobtainedbyinterchangeandintheaboveequation.Thenwegettheexpressionforthenormalpartofthetransversespinsusceptibility: +)]TJ /F9 7.97 Tf -6.59 -7.9 Td[(nor(q,q,i!)=Xk0(u2k+qu2k)]TJ /F6 11.955 Tf 11.95 0 Td[(2uk+qvk+qukvk+v2k+qv2k)hu2k+qv2kf(k)+f(k+q))]TJ /F6 11.955 Tf 11.96 0 Td[(1 i!+k+k+q+u2k+qu2kf(k))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k+q) i!)]TJ /F6 11.955 Tf 11.96 0 Td[(k+k+q+v2k+qv2kf(k+q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k) i!+k)]TJ /F6 11.955 Tf 11.95 0 Td[(k+q+v2k+qu2k1)]TJ /F3 11.955 Tf 11.96 0 Td[(f(k))]TJ /F3 11.955 Tf 11.96 0 Td[(f(k+q) i!)]TJ /F6 11.955 Tf 11.96 0 Td[(k)]TJ /F6 11.955 Tf 11.96 0 Td[(k+qi+Xk6=00(u2k+qv2k+2uk+qvk+qukvk+v2k+qu2k)hu2k+qv02kf(0k)+f(k+q))]TJ /F6 11.955 Tf 11.95 0 Td[(1 i!+0k+k+q+u2k+qu02kf(0k))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k+q) i!)]TJ /F6 11.955 Tf 11.96 0 Td[(0k+k+q+v2k+qv02kf(k+q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(0k) i!+0k)]TJ /F6 11.955 Tf 11.95 0 Td[(k+q+v2k+qu02k1)]TJ /F3 11.955 Tf 11.96 0 Td[(f(0k))]TJ /F3 11.955 Tf 11.96 0 Td[(f(k+q) i!)]TJ /F6 11.955 Tf 11.95 0 Td[(0k)]TJ /F6 11.955 Tf 11.95 0 Td[(k+qi.(D)NextweevaluatetheotherpartofthetransvasesusceptibilitythathasCooperpairoperators.AfterapplyingtheunitarytransformationtoEquation D ,weget +)]TJ /F9 7.97 Tf -6.59 -8.28 Td[(SC(q,q,i!)=1 VZ0dei!Xk,0[(u2k+qu2k)]TJ /F6 11.955 Tf 11.95 0 Td[(2uk+qvk+qukvk+v2k+qv2k)hTyk+q"()y)]TJ /F5 7.97 Tf 6.58 0 Td[(k)]TJ /F5 7.97 Tf 6.58 0 Td[(q#(0)ihTk"())]TJ /F5 7.97 Tf 6.59 -0.01 Td[(k#(0)i+hTyk+q"()y)]TJ /F5 7.97 Tf 6.59 0 Td[(k)]TJ /F5 7.97 Tf 6.59 0 Td[(q#(0)ihTk"())]TJ /F5 7.97 Tf 6.59 -0.01 Td[(k#(0)i+(u2k+qv2k+2uk+qvk+qukvk+v2k+qu2k)hTyk+q"()y)]TJ /F5 7.97 Tf 6.58 0 Td[(k)]TJ /F5 7.97 Tf 6.58 0 Td[(q#(0)ihTk"())]TJ /F5 7.97 Tf 6.59 0 Td[(k#(0)i+hTyk+q"()y)]TJ /F5 7.97 Tf 6.59 0 Td[(k)]TJ /F5 7.97 Tf 6.58 0 Td[(q#(0)ihTk"())]TJ /F5 7.97 Tf 6.59 0 Td[(k#(0)i].(D)WecanseethatthistermhasthesameSDWcoherencefactorsas+)]TJ /F9 7.97 Tf -6.59 -7.29 Td[(nor(q,q,i!).Thesameasforthenormalpartofthetransversesusceptibility,weperformBCSBogoliubovtransformationtogetnewquasiparticleoperatorsinthecoexistencestate.Foreach 102

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four-operatortermwehave, Z0dei!hyk+q()y)]TJ /F5 7.97 Tf 6.58 0 Td[(k)]TJ /F5 7.97 Tf 6.59 0 Td[(q(0)ih)]TJ /F5 7.97 Tf 6.58 0 Td[(k(0)k()i=Z0dei!uk+qvk+qukvk(hyk+q0()k+q0(0)ihyk0(0)k0()i)-222(hyk+q0()k+q0(0)ihkl(0)ykl()i)-221(hk+ql()yk+ql(0)ihyk0(0)k0()i)+hk+ql()yk+ql(0)ihkl(0)ykl()i).(D)WeevaluatetheMatsubaraintegrationofthequasiparticleoperatorsofthecoexistencestate.EachindividualtermsfollowstherelationsinEquations D .Weget Z0dei!hyk+q()y)]TJ /F5 7.97 Tf 6.59 0 Td[(k)]TJ /F5 7.97 Tf 6.58 0 Td[(q(0)ih)]TJ /F5 7.97 Tf 6.59 0 Td[(k(0)k()i=uk+qvk+qukvkhf(k+q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k) i!)]TJ /F6 11.955 Tf 11.95 0 Td[(k+k+q)]TJ /F6 11.955 Tf 13.15 9.1 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(f(k))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k+q) i!+k+k+q)]TJ /F3 11.955 Tf 13.15 9.1 Td[(f(k)+f(k+q))]TJ /F6 11.955 Tf 11.96 0 Td[(1 i!)]TJ /F6 11.955 Tf 11.96 0 Td[(k)]TJ /F6 11.955 Tf 11.95 0 Td[(k+q+f(k))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k+q) i!+k)]TJ /F6 11.955 Tf 11.95 0 Td[(k+qi.(D)Wecangettheintrabandvaluesbyreplacingallthesintheaboveequationwiths.Fortheinterbandvalues,wehave Z0dei!hyk+q()y)]TJ /F5 7.97 Tf 6.59 0 Td[(k)]TJ /F5 7.97 Tf 6.59 0 Td[(q(0)ih)]TJ /F5 7.97 Tf 6.59 0 Td[(k(0)k()i=uk+qvk+qukvkhf(k+q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k) i!)]TJ /F6 11.955 Tf 11.95 0 Td[(k+k+q)]TJ /F6 11.955 Tf 13.16 9.1 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(f(k))]TJ /F3 11.955 Tf 11.96 0 Td[(f(k+q) i!+k+k+q)]TJ /F3 11.955 Tf 13.15 9.09 Td[(f(k)+f(k+q))]TJ /F6 11.955 Tf 11.95 0 Td[(1 i!)]TJ /F6 11.955 Tf 11.95 0 Td[(k)]TJ /F6 11.955 Tf 11.95 0 Td[(k+q+f(k))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k+q) i!+k)]TJ /F6 11.955 Tf 11.95 0 Td[(k+qi.(D) 103

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TheresultfortheSCpartofthetransversesusceptibilityis +)]TJ /F9 7.97 Tf -6.58 -8.28 Td[(SC(q,q,i!)=Xk,0(u2k+qu2k)]TJ /F6 11.955 Tf 11.95 0 Td[(2uk+qvk+qukvk+v2k+qv2k)uk+qvk+qukvkhf(k+q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(k) i!)]TJ /F6 11.955 Tf 11.96 0 Td[(k+k+q)]TJ /F6 11.955 Tf 13.15 9.32 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(f(k))]TJ /F3 11.955 Tf 11.96 0 Td[(f(k+q) i!+k+k+q)]TJ /F3 11.955 Tf 13.15 9.32 Td[(f(k)+f(k+q))]TJ /F6 11.955 Tf 11.96 0 Td[(1 i!)]TJ /F6 11.955 Tf 11.96 0 Td[(k)]TJ /F6 11.955 Tf 11.96 0 Td[(k+q+f(k))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k+q) i!+k)]TJ /F6 11.955 Tf 11.95 0 Td[(k+qi+Xk,6=00[(u2k+qv2k+2uk+qvk+qukvk+v2k+qu2k)uk+qvk+qukvkhf(k+q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(k) i!)]TJ /F6 11.955 Tf 11.96 0 Td[(k+k+q)]TJ /F6 11.955 Tf 13.15 9.1 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(f(k))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k+q) i!+k+k+q)]TJ /F3 11.955 Tf 13.15 9.1 Td[(f(k)+f(k+q))]TJ /F6 11.955 Tf 11.96 0 Td[(1 i!)]TJ /F6 11.955 Tf 11.96 0 Td[(k)]TJ /F6 11.955 Tf 11.96 0 Td[(k+q+f(k))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k+q) i!+k)]TJ /F6 11.955 Tf 11.95 0 Td[(k+qi.(D)Weclearlyseethatthecoherencefactorukvkisproportionaltothesuperconductinggap.Inthenon-superconductinglimit,thistermwoulddisappear.Andwewouldgettheresultthesameasinthepureantiferromagneticstate.Combiningthetwoterms,wegetthenalresultforthenon-Umklapptransversespinsusceptibilityas +)]TJ /F5 7.97 Tf -6.59 -7.97 Td[(0(q,q,!)=Xk,01 40B@1+")]TJ /F5 7.97 Tf 0 -8.28 Td[(k")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q)]TJ /F3 11.955 Tf 11.96 0 Td[(W2 q )]TJ /F10 11.955 Tf 5.48 -9.68 Td[(")]TJ /F5 7.97 Tf 0 -8.27 Td[(k2+W2q )]TJ /F10 11.955 Tf 5.48 -9.68 Td[(")]TJ /F5 7.97 Tf 0 -8.27 Td[(k+q2+W21CA1+EkEk+q+kk+q kk+qf(k+q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k) !+i)]TJ /F6 11.955 Tf 11.95 0 Td[(k+q+k+1 21)]TJ /F3 11.955 Tf 13.15 9.32 Td[(EkEk+q+kk+q kk+qf(k+q)+f(k))]TJ /F6 11.955 Tf 11.95 0 Td[(1 !+i+k+q+k+1)]TJ /F3 11.955 Tf 11.95 0 Td[(f(k+q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(k) !+i)]TJ /F6 11.955 Tf 11.96 0 Td[(k+q)]TJ /F6 11.955 Tf 11.96 0 Td[(k+Xk,6=001 40B@1)]TJ /F10 11.955 Tf 57.51 9.32 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q)]TJ /F3 11.955 Tf 11.95 0 Td[(W2 q )]TJ /F10 11.955 Tf 5.48 -9.69 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k2+W2q )]TJ /F10 11.955 Tf 5.48 -9.69 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q2+W21CA("1+EkE0k+q+k0k+q k0k+q#f(0k+q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k) !+i+0k+q)]TJ /F6 11.955 Tf 11.96 0 Td[(k+1 2"1)]TJ /F3 11.955 Tf 13.15 9.32 Td[(EkE0k+q+k0k+q k0k+q# f(0k+q)+f(k))]TJ /F6 11.955 Tf 11.96 0 Td[(1 !+i+0k+q+k+1)]TJ /F3 11.955 Tf 11.95 0 Td[(f(0k+q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k) !+i)]TJ /F6 11.955 Tf 11.96 0 Td[(0k+q)]TJ /F6 11.955 Tf 11.96 0 Td[(k!).(D) 104

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Forthepurposeofcoding,wefurthersimplifytheexpressionbyshiftingkandchange!!)]TJ /F10 11.955 Tf 25.15 0 Td[(!insometerms.Wegetthesusceptibilitytobethesumofthefollowingsixterms: +)]TJ /F5 7.97 Tf -6.59 -7.97 Td[(1(q,q,!)=)]TJ /F12 11.955 Tf 11.29 11.35 Td[(Xk1 41+")]TJ /F5 7.97 Tf 0 -8.28 Td[(k")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q)]TJ /F3 11.955 Tf 11.96 0 Td[(W2 q ")]TJ /F5 7.97 Tf 6.59 0 Td[(2k+W2q ")]TJ /F5 7.97 Tf 6.59 0 Td[(2k+q+W21+Ek+qEk+k+qk kk+qf(k+q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(k) i!)]TJ /F6 11.955 Tf 11.96 0 Td[(k+k+q,(D) +)]TJ /F5 7.97 Tf -6.59 -7.97 Td[(2(q,q,!)=)]TJ /F12 11.955 Tf 11.29 11.36 Td[(Xk1 81+")]TJ /F5 7.97 Tf 0 -8.28 Td[(k")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q)]TJ /F3 11.955 Tf 11.96 0 Td[(W2 q ")]TJ /F5 7.97 Tf 6.59 0 Td[(2k+W2q ")]TJ /F5 7.97 Tf 6.59 0 Td[(2k+q+W21)]TJ /F3 11.955 Tf 13.15 9.1 Td[(Ek+qEk+k+qk kk+qf(k+q)+f(k))]TJ /F6 11.955 Tf 11.96 0 Td[(1 i!+k+k+q+1)]TJ /F3 11.955 Tf 11.96 0 Td[(f(k+q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k) i!)]TJ /F6 11.955 Tf 11.96 0 Td[(k)]TJ /F6 11.955 Tf 11.95 0 Td[(k+q,(D) +)]TJ /F5 7.97 Tf -6.59 -7.97 Td[(3(q,q,!)=)]TJ /F12 11.955 Tf 11.29 11.36 Td[(Xk1 41)]TJ /F10 11.955 Tf 41.18 9.32 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q)]TJ /F3 11.955 Tf 11.96 0 Td[(W2 q ")]TJ /F5 7.97 Tf 6.59 0 Td[(2k+W2q ")]TJ /F5 7.97 Tf 6.59 0 Td[(2k+q+W21+Ek+qEk+k+qk kk+qf(k+q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(k) i!)]TJ /F6 11.955 Tf 11.96 0 Td[(k+k+q+f(k))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k+q) i!)]TJ /F6 11.955 Tf 11.96 0 Td[(k+q+k,(D) +)]TJ /F5 7.97 Tf -6.58 -7.97 Td[(4(q,q,!)=)]TJ /F12 11.955 Tf 11.96 11.35 Td[(Xk1 41)]TJ /F10 11.955 Tf 43.84 9.32 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q)]TJ /F3 11.955 Tf 11.96 0 Td[(W2 q ")]TJ /F5 7.97 Tf 6.59 0 Td[(2k+W2q ")]TJ /F5 7.97 Tf 6.59 0 Td[(2k+q+W21)]TJ /F3 11.955 Tf 13.15 9.1 Td[(Ek+qEk+k+qk kk+qf(k+q)+f(k))]TJ /F6 11.955 Tf 11.96 0 Td[(1 i!+k+k+q+1)]TJ /F3 11.955 Tf 11.96 0 Td[(f(k+q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k) i!)]TJ /F6 11.955 Tf 11.96 0 Td[(k)]TJ /F6 11.955 Tf 11.95 0 Td[(k+q,(D) +)]TJ /F5 7.97 Tf -6.59 -7.97 Td[(5(q,q,!)=)]TJ /F12 11.955 Tf 11.29 11.36 Td[(Xk1 81+")]TJ /F5 7.97 Tf 0 -8.28 Td[(k")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q)]TJ /F3 11.955 Tf 11.96 0 Td[(W2 q ")]TJ /F5 7.97 Tf 6.59 0 Td[(2k+W2q ")]TJ /F5 7.97 Tf 6.59 0 Td[(2k+q+W21)]TJ /F3 11.955 Tf 13.15 9.32 Td[(Ek+qEk+k+qk kk+qf(k+q)+f(k))]TJ /F6 11.955 Tf 11.96 0 Td[(1 i!+k+k+q+1)]TJ /F3 11.955 Tf 11.96 0 Td[(f(k+q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k) i!)]TJ /F6 11.955 Tf 11.96 0 Td[(k)]TJ /F6 11.955 Tf 11.96 0 Td[(k+q,(D) 105

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+)]TJ /F5 7.97 Tf -6.59 -7.97 Td[(6(q,q,!)=)]TJ /F12 11.955 Tf 11.29 11.36 Td[(Xk1 41+")]TJ /F5 7.97 Tf 0 -8.28 Td[(k")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q)]TJ /F3 11.955 Tf 11.96 0 Td[(W2 q ")]TJ /F5 7.97 Tf 6.59 0 Td[(2k+W2q ")]TJ /F5 7.97 Tf 6.59 0 Td[(2k+q+W21+Ek+qEk+k+qk kk+qf(k+q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(k) i!)]TJ /F6 11.955 Tf 11.96 0 Td[(k+k+q.(D) D.2TheUmklapptermforthetransversedynamicspinsusceptibilityFortheUmklapp(off-diagonal)transversesusceptibility,weevaluatethespinsusceptibilityatq0=q+Q.TheUmklapptransversesusceptibilityinthecoexistencestatecanalsobeseparatedintonormaltermandsuperconductingtermas +)]TJ /F5 7.97 Tf -6.59 -8.28 Td[(Q(q,i!)=+)]TJ /F5 7.97 Tf -6.59 -7.98 Td[(0(q,q+Q,i!)=+)]TJ /F5 7.97 Tf -6.59 -8.28 Td[(Qnor(q,i!)++)]TJ /F5 7.97 Tf -6.59 -8.28 Td[(QSC(q,i!)(D)with +)]TJ /F5 7.97 Tf -6.59 -8.28 Td[(Qnor(q,i!)=Z0dei!Xk,k0hTcyk+q"()ck0"(0)ihTck#()cyk0)]TJ /F5 7.97 Tf 6.59 0 Td[(q+Q#(0)i(D)and +)]TJ /F5 7.97 Tf -6.59 -8.28 Td[(QSC(q,i!)=Z0dei!nXk,k0hTcyk+q"()cyk0)]TJ /F5 7.97 Tf 6.58 0 Td[(q+Q#(0)ihTck"()ck0#(0)i.(D)Toevaluatethevalueof+)]TJ /F5 7.97 Tf -6.59 -8.28 Td[(Qnor(q,i!),withoutlossofgenerality,wecanstartwithresultinthepureantiferromagneticstate.FromEquation C .,wehave +)]TJ /F5 7.97 Tf -6.59 -8.28 Td[(Qnor(q,i!)=Z0dei!Xk0[(ukvk)]TJ /F3 11.955 Tf 11.95 0 Td[(uk+qvk+q)[hTyk+q"()k+q"(0)ihTk#()yk#(0)i)-222(hTyk+q"()k+q"(0)ihTk#()yk#(0)i])]TJ /F6 11.955 Tf 11.96 0 Td[((uk+qvk+q+ukvk)[hTyk+q"()k+q#(0)ihTk"()yk#(0)i)-223(hTyk+q"()k+q#(0)ihTk"()yk#(0)i].(D) 106

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WereplacetheFouriertransformofthefour-operatorswithEquations D D .Wehave +)]TJ /F5 7.97 Tf -6.59 -8.28 Td[(Qnor(q,i!)=Xk,0(ukvk)]TJ /F3 11.955 Tf 11.95 0 Td[(uk+qvk+q)hu2k+qv2kf(k)+f(k+q))]TJ /F6 11.955 Tf 11.96 0 Td[(1 i!+k+k+q+u2k+qu2kf(k))]TJ /F3 11.955 Tf 11.96 0 Td[(f(k+q) i!)]TJ /F6 11.955 Tf 11.96 0 Td[(k+k+q+v2k+qv2kf(k+q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k) i!+k)]TJ /F6 11.955 Tf 11.96 0 Td[(k+q+v2k+qu2k1)]TJ /F3 11.955 Tf 11.95 0 Td[(f(k))]TJ /F3 11.955 Tf 11.96 0 Td[(f(k+q) i!)]TJ /F6 11.955 Tf 11.95 0 Td[(k)]TJ /F6 11.955 Tf 11.95 0 Td[(k+qi)]TJ /F12 11.955 Tf 16.08 11.36 Td[(Xk,6=00(uk+qvk+q+ukvk)hu2k+qv02kf(0k)+f(k+q))]TJ /F6 11.955 Tf 11.96 0 Td[(1 i!+0k+k+q+u2k+qu02kf(0k))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k+q) i!)]TJ /F6 11.955 Tf 11.95 0 Td[(0k+k+q+v2k+qv02kf(k+q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(0k) i!+0k)]TJ /F6 11.955 Tf 11.96 0 Td[(k+q+v2k+qu02k1)]TJ /F3 11.955 Tf 11.96 0 Td[(f(0k))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k+q) i!)]TJ /F6 11.955 Tf 11.96 0 Td[(0k)]TJ /F6 11.955 Tf 11.96 0 Td[(k+qi.(D)Weapplytheunitarytransformation,Equation 2 tothesuperconductingpartoftheUmklapptransversesusceptibilityEquation C andget +)]TJ /F5 7.97 Tf -6.59 -8.28 Td[(QSC(q,i!)=1 VZ0dei!Xk,0(uk+qvk+q+ukvk)[hTyk+q"()y)]TJ /F5 7.97 Tf 6.58 0 Td[(k)]TJ /F5 7.97 Tf 6.59 0 Td[(q#(0)ihTk"())]TJ /F5 7.97 Tf 6.58 0 Td[(k#(0)i)-222(hTyk+q"()y)]TJ /F5 7.97 Tf 6.59 0 Td[(k)]TJ /F5 7.97 Tf 6.59 0 Td[(q#(0)ihTk"())]TJ /F5 7.97 Tf 6.58 0 Td[(k#(0)i]+(uk+qvk+q)]TJ /F3 11.955 Tf 11.95 0 Td[(ukvk)hTyk+q"()y)]TJ /F5 7.97 Tf 6.58 0 Td[(k)]TJ /F5 7.97 Tf 6.59 0 Td[(q#(0)ihTk"())]TJ /F5 7.97 Tf 6.59 0 Td[(k#(0)i)-223(hTyk+q"()y)]TJ /F5 7.97 Tf 6.58 0 Td[(k)]TJ /F5 7.97 Tf 6.59 0 Td[(q#(0)ihTk"())]TJ /F5 7.97 Tf 6.58 0 Td[(k#(0)i].(D) 107

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AfterreplacingtheFouriertransformofthefour-operatorbyusingEquations D and D ,weget +)]TJ /F5 7.97 Tf -6.59 -8.28 Td[(QSC(q,i!)=Xk,0(uk+qvk+q+ukvk)uk+qvk+qukvkhf(k+q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k) i!)]TJ /F6 11.955 Tf 11.95 0 Td[(k+k+q)]TJ /F6 11.955 Tf 13.16 9.32 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(f(k))]TJ /F3 11.955 Tf 11.96 0 Td[(f(k+q) i!+k+k+q)]TJ /F3 11.955 Tf 13.15 9.32 Td[(f(k)+f(k+q))]TJ /F6 11.955 Tf 11.95 0 Td[(1 i!)]TJ /F6 11.955 Tf 11.95 0 Td[(k)]TJ /F6 11.955 Tf 11.96 0 Td[(k+q+f(k))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k+q) i!+k)]TJ /F6 11.955 Tf 11.95 0 Td[(k+qi+Xk,6=00(uk+qvk+q)]TJ /F3 11.955 Tf 11.95 0 Td[(ukvk)uk+qvk+qu0kv0khf(k+q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(0k) i!)]TJ /F6 11.955 Tf 11.96 0 Td[(0k+k+q)]TJ /F6 11.955 Tf 13.15 9.33 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(f(0k))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k+q) i!+0k+k+q)]TJ /F3 11.955 Tf 13.15 9.32 Td[(f(0k)+f(k+q))]TJ /F6 11.955 Tf 11.96 0 Td[(1 i!)]TJ /F6 11.955 Tf 11.95 0 Td[(0k)]TJ /F6 11.955 Tf 11.95 0 Td[(k+q+f(0k))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k+q) i!+0k)]TJ /F6 11.955 Tf 11.95 0 Td[(k+qi.(D)TheUmklapptransversespinsusceptibilityisthesumofthefollowingterms: +)]TJ /F5 7.97 Tf 6.59 0 Td[((a)Q=)]TJ /F12 11.955 Tf 11.95 11.36 Td[(Xk1 4hW p ("k+q)]TJ /F10 11.955 Tf 11.95 0 Td[("k+q+Q)2+4W2)]TJ /F3 11.955 Tf 64.39 8.09 Td[(W p ("k)]TJ /F10 11.955 Tf 11.95 0 Td[("k+Q)2+4W2ih(Ek+q k+q)]TJ /F3 11.955 Tf 13.24 8.09 Td[(Ek k)f(k)+f(k+q))]TJ /F6 11.955 Tf 11.95 0 Td[(1 i!+k+k+q+(Ek k)]TJ /F3 11.955 Tf 13.94 9.1 Td[(Ek+q k+q)1)]TJ /F3 11.955 Tf 11.96 0 Td[(f(k))]TJ /F3 11.955 Tf 11.96 0 Td[(f(k+q) i!)]TJ /F6 11.955 Tf 11.96 0 Td[(k)]TJ /F6 11.955 Tf 11.95 0 Td[(k+q)]TJ /F6 11.955 Tf 11.95 0 Td[(()]TJ /F3 11.955 Tf 10.59 8.08 Td[(Ek k+Ek+q k+q)f(k)+f(k+q))]TJ /F6 11.955 Tf 11.96 0 Td[(1 i!+k+k+q)]TJ /F6 11.955 Tf 11.96 0 Td[((Ek k)]TJ /F3 11.955 Tf 13.95 9.32 Td[(Ek+q k+q)1)]TJ /F3 11.955 Tf 11.95 0 Td[(f(k))]TJ /F3 11.955 Tf 11.96 0 Td[(f(k+q) i!)]TJ /F6 11.955 Tf 11.96 0 Td[(k)]TJ /F6 11.955 Tf 11.95 0 Td[(k+qi,(D) 108

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+)]TJ /F5 7.97 Tf 6.58 0 Td[((b)Q=)]TJ /F12 11.955 Tf 11.96 11.36 Td[(Xk1 4hW p ("k+q)]TJ /F10 11.955 Tf 11.96 0 Td[("k+q+Q)2+4W2+W p ("k)]TJ /F10 11.955 Tf 11.95 0 Td[("k+Q)2+4W2ih2()]TJ /F3 11.955 Tf 10.58 8.09 Td[(Ek k+Ek+q k+q)f(k)+f(k+q))]TJ /F6 11.955 Tf 11.96 0 Td[(1 i!+k+k+q+2(Ek k+Ek+q k+q)f(k))]TJ /F3 11.955 Tf 11.96 0 Td[(f(k+q) i!)]TJ /F6 11.955 Tf 11.95 0 Td[(k+k+q+2()]TJ /F3 11.955 Tf 10.58 8.08 Td[(Ek k)]TJ /F3 11.955 Tf 13.95 9.09 Td[(Ek+q k+q)f(k+q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(k) i!+k)]TJ /F6 11.955 Tf 11.95 0 Td[(k+q+2(Ek k)]TJ /F3 11.955 Tf 13.95 9.09 Td[(Ek+q k+q)1)]TJ /F3 11.955 Tf 11.96 0 Td[(f(k))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k+q) i!)]TJ /F6 11.955 Tf 11.96 0 Td[(k)]TJ /F6 11.955 Tf 11.96 0 Td[(k+qi,(D) +)]TJ /F5 7.97 Tf 6.59 0 Td[((c)Q=XkhW p ("k+q)]TJ /F10 11.955 Tf 11.95 0 Td[("k+q+Q)2+4W2+W p ("k)]TJ /F10 11.955 Tf 11.95 0 Td[("k+Q)2+4W2ihk+qk 4k+qkf(k+q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k) i!)]TJ /F6 11.955 Tf 11.96 0 Td[(k+k+q+1)]TJ /F3 11.955 Tf 11.96 0 Td[(f(k))]TJ /F3 11.955 Tf 11.96 0 Td[(f(k+q) i!+k+k+q+f(k)+f(k+q))]TJ /F6 11.955 Tf 11.95 0 Td[(1 i!)]TJ /F6 11.955 Tf 11.95 0 Td[(k)]TJ /F6 11.955 Tf 11.96 0 Td[(k+q+f(k))]TJ /F3 11.955 Tf 11.96 0 Td[(f(k+q) i!+k)]TJ /F6 11.955 Tf 11.96 0 Td[(k+q)]TJ /F6 11.955 Tf 14.33 9.32 Td[(k+qk 4k+qkf(k+q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(k) i!)]TJ /F6 11.955 Tf 11.96 0 Td[(k+k+q+1)]TJ /F3 11.955 Tf 11.96 0 Td[(f(k))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k+q) i!+k+k+q+f(k)+f(k+q))]TJ /F6 11.955 Tf 11.95 0 Td[(1 i!)]TJ /F6 11.955 Tf 11.95 0 Td[(k)]TJ /F6 11.955 Tf 11.95 0 Td[(k+q+f(k))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k+q) i!+k)]TJ /F6 11.955 Tf 11.95 0 Td[(k+qi,(D) +)]TJ /F5 7.97 Tf 6.59 0 Td[((d)Q=XkhW p ("k+q)]TJ /F10 11.955 Tf 11.96 0 Td[("k+q+Q)2+4W2)]TJ /F3 11.955 Tf 64.4 8.09 Td[(W p ("k)]TJ /F10 11.955 Tf 11.96 0 Td[("k+Q)2+4W2ihk+qk 4k+qkf(k+q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(k) i!)]TJ /F6 11.955 Tf 11.96 0 Td[(k+k+q+1)]TJ /F3 11.955 Tf 11.96 0 Td[(f(k))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k+q) i!+k+k+q+f(k)+f(k+q))]TJ /F6 11.955 Tf 11.96 0 Td[(1 i!)]TJ /F6 11.955 Tf 11.96 0 Td[(k)]TJ /F6 11.955 Tf 11.96 0 Td[(k+q+f(k))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k+q) i!+k)]TJ /F6 11.955 Tf 11.95 0 Td[(k+q)]TJ /F6 11.955 Tf 14.33 9.32 Td[(k+qk 4k+qkf(k+q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k) i!)]TJ /F6 11.955 Tf 11.95 0 Td[(k+k+q+1)]TJ /F3 11.955 Tf 11.95 0 Td[(f(k))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k+q) i!+k+k+q+f(k)+f(k+q))]TJ /F6 11.955 Tf 11.96 0 Td[(1 i!)]TJ /F6 11.955 Tf 11.96 0 Td[(k)]TJ /F6 11.955 Tf 11.95 0 Td[(k+q+f(k))]TJ /F3 11.955 Tf 11.96 0 Td[(f(k+q) i!+k)]TJ /F6 11.955 Tf 11.96 0 Td[(k+qi.(D) 109

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Afterorganizing,wehavetheUmklapptransversespinsusceptibilityreducingto +)]TJ /F5 7.97 Tf -6.58 -7.97 Td[(0(q,q+Q,!)=W 4Xk,00@1 q )]TJ /F10 11.955 Tf 5.48 -9.69 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q2+W2)]TJ /F6 11.955 Tf 46.73 8.08 Td[(1 q )]TJ /F10 11.955 Tf 5.48 -9.68 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k2+W21AEk+q k+q+Ek kf(k+q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k) !+i)]TJ /F6 11.955 Tf 11.95 0 Td[(k+q+kEk+q k+q)]TJ /F3 11.955 Tf 13.24 8.09 Td[(Ek k1)]TJ /F3 11.955 Tf 11.95 0 Td[(f(k+q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k) !+i)]TJ /F6 11.955 Tf 11.96 0 Td[(k+q)]TJ /F6 11.955 Tf 11.96 0 Td[(k+f(k+q)+f(k))]TJ /F6 11.955 Tf 11.95 0 Td[(1 !+i+k+q+k+W 4Xk,6=000@1 q )]TJ /F10 11.955 Tf 5.48 -9.69 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q2+W2+1 q )]TJ /F10 11.955 Tf 5.48 -9.68 Td[(")]TJ /F5 7.97 Tf 0 -8.27 Td[(k2+W21A( Ek+q k+q+E0k 0k!f(k+q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(0k) !+i)]TJ /F6 11.955 Tf 11.95 0 Td[(k+q+0k Ek+q k+q)]TJ /F3 11.955 Tf 13.24 8.09 Td[(E0k 0k! 1)]TJ /F3 11.955 Tf 11.95 0 Td[(f(k+q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(0k) !+i)]TJ /F6 11.955 Tf 11.96 0 Td[(k+q)]TJ /F6 11.955 Tf 11.96 0 Td[(0k+f(k+q)+f(0k))]TJ /F6 11.955 Tf 11.95 0 Td[(1 !+i+k+q+0k!).(D) D.3ThelongitudinaldynamicspinsusceptibilityThelongitudinaldynamicspinsusceptibility,intermsoftheraisingandloweringoperators,iswrittenas zz0(q,q,!)=Z0dei!Xkk0hT(cyk+q"()ck"()cyk0)]TJ /F5 7.97 Tf 6.59 0 Td[(q"(0)ck0"(0))]TJ /F3 11.955 Tf 11.95 0 Td[(cyk+q"()ck"()cyk0)]TJ /F5 7.97 Tf 6.59 0 Td[(q#(0)ck0#(0))]TJ /F3 11.955 Tf 11.96 0 Td[(cyk+q#()ck#()cyk0)]TJ /F5 7.97 Tf 6.59 0 Td[(q"(0)ck0"(0)+cyk+q#()ck#()cyk0)]TJ /F5 7.97 Tf 6.59 0 Td[(q#(0)ck0#(0))i.(D)ByusingWicktheoremtoseparatethefouroperatorsandignoringthetermswhichhaveexpectationvaluesbeingzero,wehave zz0(q,q,!)=zznor(q,q,!)+zzSC(q,q,!)(D)with zznor(q,q,!)=Z0dei!Xkk0,hTcyk+q()ck0(0)ihTck()cyk0)]TJ /F5 7.97 Tf 6.58 0 Td[(q(0)i(D) 110

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and zzSC(q,q,!)=Z0dei!Xkk0,hTcyk+q()cyk0)]TJ /F5 7.97 Tf 6.59 0 Td[(q(0)ihTck()ck0(0)i.(D)ReducingthesumoverktothereducedBrillouinzoneandapplyingtheunitarytransfor-mationtozznor(q,q,!),weget zznor(q,q,!)=Z0dei!Xk0(u2ku2k+q+2ukvkuk+qvk+q+v2kv2k+q)(hT(yk()k(0)ihTk+q()yk+q(0)i+hTyk()k(0)ihTk+q()yk+q(0)i)+(u2kv2k+q)]TJ /F6 11.955 Tf 11.95 0 Td[(2ukvkuk+qvk+q+v2ku2k+q)(hTyk()k(0)ihTk+q()yk+q(0)i+hTyk()k(0)ihTk+q()yk+q(0)i).(D)Pluggingtheexpectationvaluesforthefour-operators,weobtaintheexpressionas zznor(q,q,!)=Xk01 2(1+")]TJ /F5 7.97 Tf 0 -8.28 Td[(k")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q+W2 q ")]TJ /F5 7.97 Tf 6.58 0 Td[(2k+W2q ")]TJ /F5 7.97 Tf 6.58 0 Td[(2k+q+W2)hu2k+qv2kf(k)+f(k+q))]TJ /F6 11.955 Tf 11.95 0 Td[(1 i!+k+k+q+u2k+qu2kf(k))]TJ /F3 11.955 Tf 11.96 0 Td[(f(k+q) i!)]TJ /F6 11.955 Tf 11.95 0 Td[(k+k+q+v2k+qv2kf(k+q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k) i!+k)]TJ /F6 11.955 Tf 11.96 0 Td[(k+q+v2k+qu2k1)]TJ /F3 11.955 Tf 11.95 0 Td[(f(k))]TJ /F3 11.955 Tf 11.96 0 Td[(f(k+q) i!)]TJ /F6 11.955 Tf 11.96 0 Td[(k)]TJ /F6 11.955 Tf 11.95 0 Td[(k+qi+Xk6=001 2(1)]TJ /F10 11.955 Tf 43.84 9.32 Td[(")]TJ /F5 7.97 Tf 0 -8.27 Td[(k")]TJ /F5 7.97 Tf 0 -8.27 Td[(k+q+W2 q ")]TJ /F5 7.97 Tf 6.59 0 Td[(2k+W2q ")]TJ /F5 7.97 Tf 6.59 0 Td[(2k+q+W2)hu2k+qv02kf(0k)+f(k+q))]TJ /F6 11.955 Tf 11.96 0 Td[(1 i!+0k+k+q+u2k+qu02kf(0k))]TJ /F3 11.955 Tf 11.96 0 Td[(f(k+q) i!)]TJ /F6 11.955 Tf 11.96 0 Td[(0k+k+q+v2k+qv02kf(k+q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(0k) i!+0k)]TJ /F6 11.955 Tf 11.96 0 Td[(k+q+v2k+qu02k1)]TJ /F3 11.955 Tf 11.96 0 Td[(f(0k))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k+q) i!)]TJ /F6 11.955 Tf 11.96 0 Td[(0k)]TJ /F6 11.955 Tf 11.96 0 Td[(k+qi.(D) 111

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FortheSCpartofthelongitudinalspinsusceptibilityfromEquation D ,aftertheunitarytransformation,wehave zzSC(q,q,!)=20 2Z0dei!nXk2R(u2ku2k+q+2ukvkuk+qvk+q+v2kv2k+q)(hT(yk()y)]TJ /F5 7.97 Tf 6.59 0 Td[(k(0)ihTk+q())]TJ /F5 7.97 Tf 6.59 0 Td[(k)]TJ /F5 7.97 Tf 6.58 0 Td[(q(0)i+hTyk()y)]TJ /F5 7.97 Tf 6.59 0 Td[(k(0)ihTk+q())]TJ /F5 7.97 Tf 6.59 0 Td[(k)]TJ /F5 7.97 Tf 6.59 0 Td[(q(0)i)+(u2kv2k+q)]TJ /F6 11.955 Tf 11.96 0 Td[(2ukvkuk+qvk+q+v2ku2k+q)(hTyk()y)]TJ /F5 7.97 Tf 6.59 0 Td[(k(0)ihTk+q())]TJ /F5 7.97 Tf 6.59 0 Td[(k)]TJ /F5 7.97 Tf 6.59 0 Td[(q(0)i+hTyk()yk(0)ihTk+q())]TJ /F5 7.97 Tf 6.59 0 Td[(k)]TJ /F5 7.97 Tf 6.59 0 Td[(q(0)i).(D)Pluggingtheexpectationvaluesforthefour-operators,weobtaintheexpressionas zzSC(q,q,!)=)]TJ /F12 11.955 Tf 9.3 11.35 Td[(Xk01 2(1+")]TJ /F5 7.97 Tf 0 -8.28 Td[(k")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q+W2 q ")]TJ /F5 7.97 Tf 6.59 0 Td[(2k+W2q ")]TJ /F5 7.97 Tf 6.59 0 Td[(2k+q+W2)h(uk+qvk+qukvk)(f(k+q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k) i!)]TJ /F6 11.955 Tf 11.95 0 Td[(k+k+q)]TJ /F6 11.955 Tf 13.15 9.32 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(f(k))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k+q) i!+k+k+q)]TJ /F3 11.955 Tf 13.15 9.32 Td[(f(k)+f(k+q))]TJ /F6 11.955 Tf 11.96 0 Td[(1 i!)]TJ /F6 11.955 Tf 11.96 0 Td[(k)]TJ /F6 11.955 Tf 11.96 0 Td[(k+q+f(k))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k+q) i!+k)]TJ /F6 11.955 Tf 11.95 0 Td[(k+q)i+Xk6=001 2(1)]TJ /F10 11.955 Tf 43.85 9.32 Td[(")]TJ /F5 7.97 Tf 0 -8.27 Td[(k")]TJ /F5 7.97 Tf 0 -8.27 Td[(k+q+W2 q ")]TJ /F5 7.97 Tf 6.59 0 Td[(2k+W2q ")]TJ /F5 7.97 Tf 6.59 0 Td[(2k+q+W2)(uk+qvk+qu0kv0k)hf(k+q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(0k) i!)]TJ /F6 11.955 Tf 11.96 0 Td[(0k+k+q)]TJ /F6 11.955 Tf 13.15 9.32 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(f(0k))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k+q) i!+0k+k+q)]TJ /F3 11.955 Tf 13.15 9.32 Td[(f(0k)+f(k+q))]TJ /F6 11.955 Tf 11.95 0 Td[(1 i!)]TJ /F6 11.955 Tf 11.96 0 Td[(0k)]TJ /F6 11.955 Tf 11.95 0 Td[(k+q+f(0k))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k+q) i!+0k)]TJ /F6 11.955 Tf 11.95 0 Td[(k+qi.(D) 112

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TheSCtermhasthesameSDWcoefcients.Thenalexpressionforthetotallongitudi-nalspinsusceptibility,zz0(q,q,!)is zz0(q,q,!)=Xk,01 40B@1+")]TJ /F5 7.97 Tf 0 -8.27 Td[(k")]TJ /F5 7.97 Tf 0 -8.27 Td[(k+q+W2 q )]TJ /F10 11.955 Tf 5.48 -9.68 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k2+W2q )]TJ /F10 11.955 Tf 5.48 -9.68 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q2+W21CA1+EkEk+q+kk+q kk+qf(k+q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k) !+i)]TJ /F6 11.955 Tf 11.95 0 Td[(k+q+k+1 21)]TJ /F3 11.955 Tf 13.15 9.32 Td[(EkEk+q+kk+q kk+qf(k+q)+f(k))]TJ /F6 11.955 Tf 11.95 0 Td[(1 !+i+k+q+k+1)]TJ /F3 11.955 Tf 11.95 0 Td[(f(k+q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(k) !+i)]TJ /F6 11.955 Tf 11.96 0 Td[(k+q)]TJ /F6 11.955 Tf 11.96 0 Td[(k+Xk,6=001 40B@1)]TJ /F10 11.955 Tf 57.51 9.32 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q+W2 q )]TJ /F10 11.955 Tf 5.48 -9.69 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k2+W2q )]TJ /F10 11.955 Tf 5.48 -9.69 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q2+W21CA("1+EkE0k+q+k0k+q k0k+q#f(0k+q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k) !+i)]TJ /F6 11.955 Tf 11.96 0 Td[(0k+q+k+1 2"1)]TJ /F3 11.955 Tf 13.15 9.32 Td[(EkE0k+q+k0k+q k0k+q# f(0k+q)+f(k))]TJ /F6 11.955 Tf 11.96 0 Td[(1 !+i+0k+q+k+1)]TJ /F3 11.955 Tf 11.95 0 Td[(f(0k+q))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k) !+i)]TJ /F6 11.955 Tf 11.96 0 Td[(0k+q)]TJ /F6 11.955 Tf 11.96 0 Td[(k!).(D)Forthepurposeofcoding,wesimplifytheexpressionbyshiftingkandchange!!)]TJ /F10 11.955 Tf 25.53 0 Td[(!insometermsThetotallongitudinalspinsusceptibility,zz0(q,q,!)isalsothesumofthefollowingsixterms: zz(1)0(q,q,!)=)]TJ /F12 11.955 Tf 11.96 11.36 Td[(Xk2R1 4(1+")]TJ /F5 7.97 Tf 0 -8.28 Td[(k")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q+W2 q ")]TJ /F5 7.97 Tf 6.59 0 Td[(2k+W2q ")]TJ /F5 7.97 Tf 6.59 0 Td[(2k+q+W2)1+Ek+qEk+k+qk k+qkf(k+q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(k) i!)]TJ /F6 11.955 Tf 11.96 0 Td[(k+k+q,(D) zz(2)0(q,q,!)=)]TJ /F12 11.955 Tf 11.29 11.36 Td[(Xk2R1 8(1+")]TJ /F5 7.97 Tf 0 -8.28 Td[(k")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q+W2 q ")]TJ /F5 7.97 Tf 6.59 0 Td[(2k+W2q ")]TJ /F5 7.97 Tf 6.59 0 Td[(2k+q+W2)1)]TJ /F3 11.955 Tf 13.15 9.1 Td[(Ek+qEk+k+qk k+qkhf(k)+f(k+q))]TJ /F6 11.955 Tf 11.96 0 Td[(1 i!+k+k+q+1)]TJ /F3 11.955 Tf 11.96 0 Td[(f(k))]TJ /F3 11.955 Tf 11.96 0 Td[(f(k+q) i!)]TJ /F6 11.955 Tf 11.96 0 Td[(k)]TJ /F6 11.955 Tf 11.95 0 Td[(k+qi,(D) 113

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zz(3)0(q,q,!)=)]TJ /F12 11.955 Tf 11.29 11.35 Td[(Xk2R1 4(1)]TJ /F10 11.955 Tf 43.85 9.32 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q+W2 q ")]TJ /F5 7.97 Tf 6.58 0 Td[(2k+W2q ")]TJ /F5 7.97 Tf 6.58 0 Td[(2k+q+W2)1+Ek+qEk+k+qk k+qkhf(k+q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(k) i!)]TJ /F6 11.955 Tf 11.96 0 Td[(k+k+q+f(k))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k+q) i!)]TJ /F6 11.955 Tf 11.96 0 Td[(k+q+ki,(D) zz(4)0(q,q,!)=)]TJ /F12 11.955 Tf 11.29 11.36 Td[(Xk2R1 4(1)]TJ /F10 11.955 Tf 43.85 9.32 Td[(")]TJ /F5 7.97 Tf 0 -8.28 Td[(k")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q+W2 q ")]TJ /F5 7.97 Tf 6.58 0 Td[(2k+W2q ")]TJ /F5 7.97 Tf 6.58 0 Td[(2k+q+W2)1)]TJ /F3 11.955 Tf 13.15 9.1 Td[(Ek+qEk+k+qk k+qkh1)]TJ /F3 11.955 Tf 11.96 0 Td[(f(k))]TJ /F3 11.955 Tf 11.95 0 Td[(f(k+q) i!)]TJ /F6 11.955 Tf 11.96 0 Td[(k)]TJ /F6 11.955 Tf 11.96 0 Td[(k+q+f(k)+f(k+q))]TJ /F6 11.955 Tf 11.96 0 Td[(1 i!+k+k+qi,(D) zz(5)0(q,q,!)=)]TJ /F12 11.955 Tf 11.29 11.36 Td[(Xk2R1 8(1+")]TJ /F5 7.97 Tf 0 -8.28 Td[(k")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q+W2 q ")]TJ /F5 7.97 Tf 6.58 0 Td[(2k+W2q ")]TJ /F5 7.97 Tf 6.58 0 Td[(2k+q+W2)1)]TJ /F3 11.955 Tf 13.15 9.32 Td[(Ek+qEk+k+qk k+qkhf(k)+f(k+q))]TJ /F6 11.955 Tf 11.96 0 Td[(1 i!+k+k+q+1)]TJ /F3 11.955 Tf 11.96 0 Td[(f(k))]TJ /F3 11.955 Tf 11.96 0 Td[(f(k+q) i!)]TJ /F6 11.955 Tf 11.96 0 Td[(k)]TJ /F6 11.955 Tf 11.95 0 Td[(k+qi(D)and zz(6)0(q,q,!)=)]TJ /F12 11.955 Tf 11.29 11.35 Td[(Xk2R1 4(1+")]TJ /F5 7.97 Tf 0 -8.28 Td[(k")]TJ /F5 7.97 Tf 0 -8.28 Td[(k+q+W2 q ")]TJ /F5 7.97 Tf 6.59 0 Td[(2k+W2q ")]TJ /F5 7.97 Tf 6.59 0 Td[(2k+q+W2)1+Ek+qEk+k+qk k+qkf(k+q))]TJ /F3 11.955 Tf 11.96 0 Td[(f(k) i!)]TJ /F6 11.955 Tf 11.96 0 Td[(k+k+q.(D) 114

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BIOGRAPHICALSKETCH WenyaRowewasborninTaiwanin1980.ShestudiedinFu-JenCatholicUniversityandreceivedherbachelordegreesinPhysicsandMathematicsin2003.ShereceivedherMasterdegreefromCheng-KungUniversityin2005.In2007sheenteredtheUniversityofFlorida.ShereceivedherPhDdegreeinMay2014. 124