Pairingmechanism forhigh temperature superconductivity inthecuprates: whatcanwelearn from the two-dimensional t− J model? Huan-Qiang Zhou1 1Centre for Modern Physics and Department of Physics, Chongqing University, Chongqing 400044, The People’s Republic of China Morethantwentyyearshavepassedsincehightemperaturesuperconductivityinthecopperoxides(cuprates) wasdiscoveredbyJ.G.BednorzandK.A.Mu¨llerin1986[1]. Althoughintensetheoreticalandexperimental 0 effortshavebeendevotedtotheinvestigationofthisfascinatingclassofmaterials,thepairingmechanismre- 1 sponsibleforunprecedented hightransitiontemperaturesT remainselusive. Theoretically, thedifficultylies c 0 inthefactthatthisclassofmaterials,asdopedMott-Hubbardinsulators[2],involvestrongelectroniccorrela- 2 tions,whichrendersconventionaltheoreticalapproachesunreliable. Recentprogressinnumericalsimulations n ofstronglycorrelatedelectronsystemsinthecontextoftensornetworkrepresentations[3,4]makesitpossible a togetaccesstoinformationencodedintheground-statewavefunctionsofthetwo-dimensionalt−Jmodel-a J minimalmodel,aswidelybelieved,tounderstandelectronicpropertiesofdopedMott-Hubbardinsulators[5– 9 8]. Inthisregard, anintriguingquestioniswhetheror notthetwo-dimensional t−J modelholdsthekeyto 1 understandinghightemperaturesuperconductivityinthecuprates.Asitturnsout,suchakeyliesinasupercon- ductingstatewithmixedspin-singletd+s−waveandspin-triplet p (p )-wavesymmetriesinthepresenceof x y ] ananti-ferromagneticbackground[9].Here,thed+s-wavecomponentinthespin-singletchannelbreaksU(1) n symmetryinthechargesector,whereasboththeanti-ferromagneticorderandthespin-tripletp (p )-wavecom- o x y ponentbreaksSU(2)symmetryinthespinsector. Therefore,fourgaplessGoldstonemodesoccur. However, c evenifweresorttotheKosterlitz-Thoulesstransition[10],onlythed+s-wavesuperconductingcomponentsur- - r vivesthermalfluctuations. ThisturnsthreegaplessGoldstonemodes,arisingfromSU(2)symmetrybreaking, p intotwo degenerate soft modes, withtwicethe spin-triplet p (p )-wave superconducting energy gap as their u x y characteristicenergyscale: oneisaspin-tripletmodeobservedasaspinresonancemodeininelasticneutron s . scattering,theotherisaspin-singletmodeobservedasaA1gpeakinelectronicRamanscattering.Thescenario at allowsustopredictthatpairingisofd+s-wavesymmetry, withthetwodegenerate softmodesasthelong- m soughtkeyingredientsindeterminingthetransitiontemperatureTc,thusofferingapossiblewaytoresolvethe controversyregardingtheelusivemechanismforhightemperaturesuperconductivityinthecuprates. - d n PACSnumbers:74.20.-z,74.20.Mn,74.20.Rp o c [ Imagine if we would have been able to solve a model etersareindependentofsitesonthelattice. system describing dopedMott-Hubbardinsulators on a two- Now let us switch on thermal fluctuations. Suppose we 1 dimensionalsquarelattice,whoseground-statewavefunction restrict ourselves to a strict two dimensional system. Then, v 8 isasuperconductingstatewithmixedspin-singletd+s−wave eveniftheKosterlitz-Thoulesstransition[10]isinvoked,only 5 andspin-tripletp (p )−wavesymmetriesinthepresenceofan spin-singletd+s-wavesuperconductingcomponentsurvives x y 3 anti-ferromagneticbackground,withtheorderparametersfor thermal fluctuations. However, the non-abelian SU(2) sym- 3 the s-wave, d-wave, and p (p )-wave superconductingcom- metry is not allowed to be broken at any finite tempera- . x y 1 ponents, together with the anti-ferromagnetic order parame- ture [11, 12]. This immediately implies that the Goldstone 0 ter, shown in Fig. 1, in a properdoping range. Note that ∆ modes arising from the spontaneous symmetry breaking of 0 d and ∆ are, respectively, the spin-singlet d-wave and s-wave SU(2) in the spin sector have to be turned into degenerate 1 s : superconducting energy gaps, whereas ∆p is the spin-triplet softmodes,withtwicethespin-tripletpx(py)-wavesupercon- v p (p )-wave superconducting energy gap and N is the anti- ductingenergygapastheircharacteristicenergyscale: oneis i x y X ferromagneticNe´elorderparameter. A fewpeculiarfeatures aspin-tripletmodeassociatedwiththeanti-ferromagneticor- r of this state are: (i) Both the 90 degree (four-fold) rotation der,withthemomentumtransfer(π,π),andtheotherisaspin- a symmetryandthetranslationsymmetryunderone-siteshifts singletmodeassociatedwiththespin-triplet p (p )-wavesu- x y are spontaneously broken on the square lattice. (ii) Spin- perconductingcomponent,withthemomentumtransfer(0,0). rotation symmetry SU(2) is spontaneously broken, due to Ontheotherhand,thereisnothingtopreventfromthebreak- the simultaneousoccurrenceof boththe p (p )−wave super- ingofthediscretefour-foldrotationsymmetryonthesquare x y conductingcomponentandtheanti-ferromagneticorder. (iii) lattice. Actually,thisbrokensymmetrynotonlymanifestsit- U(1)symmetryinthechargesectorisspontaneouslybroken, selfintheadmixtureofasmalls-wavecomponenttothedom- due to pairing in both spin-singlet and spin-triplet channels. inantd-wavesuperconductingstate(seeFig.1,leftpanel),but Here, we emphasize that the symmetry mixing of the spin- also protectsthe spin-singletsoftmodethat is unidirectional singletandspin-tripletchannelsarisesfromthespin-rotation asitarisesfromthep (p )-wavesuperconductingcomponent. x y symmetrybreaking,simplybecausespinisnotagoodquan- Our argument leads to a scenario that, at any finite tem- tumnumber. (iv)Allsuperconductingcomponentsarehomo- perature, the pairing is of d + s-wave symmetry, with two geneous,inthesensethattheirsuperconductingorderparam- degeneratesoftmodesactingasthe keyingredientsin deter- 2 mining the transition temperature Tc. Actually, two distinct 2∆ 2∆ ewniethrgtyhesccaolensv2en∆t∗ioannadlEsurepsearcreonindvuocltvoerds:,i2n∆a∗marairskeesdfrcoomntrtahset 2∆ 2∆ds ∆∆/sd 2∆ 2∆p* anti-ferromagneticNe´elorderparameterN, whichisrespon- δ sibleforpairing,withitscouplingstrengthdecreasingalmost linearly with doping, whereas E = 2∆ , which is respon- res p sible for condensation. Therefore, E must scale with the δ δ res superconductingtransition temperature T , i.e., E ∼ k T , c res B c withk beingtheBoltzmannconstant[seeFig.1,rightpanel]. FIG. 1: (color online) The doping dependence of the order pa- B Similarly, 2∆∗ scales as 2∆∗ ∼ k T∗, with T∗ being the so- rameters for a model system describing doped Mott-Hubbard in- B calledpseudogaptemperature[13,14]. Inaddition,onemay sulators on a two-dimensional square lattice, whose ground-state expect that E < 2∆ , simply due to the fact that the pre- wave function is a superconducting state with mixed spin-singlet res d d+s−waveandspin-tripletp (p )−wavesymmetriesinthepresence dominant d-wave superconductingcomponentsurvives ther- x y ofananti-ferromagneticbackground, withtheorderparametersfor mal fluctuations. Considering that both the superconducting the s-wave,d-wave,and p (p )-wavesuperconductingcomponents, gap∆d andthetransitiontemperatureTc characterizethesu- togetherwiththeanti-ferroxmaygneticorderparameterN. Leftpanel: perconductivity,theyshouldtrackeachotherintheentiredop- Thesuperconductinggaps∆ and∆ forthespin-singletd-waveand d s ingrange, implying∆d ∼ kBTc. Infact, forthe t− J model, s-wavecomponentsasafunctionofdopingδ,withtheirratio∆s/∆d aolulorwsismuuslattoioenstiinmdaicteateasutnhiavteErsraesl c≈oe1ffi.2c5i∆endt[κ9].≈T5h.3is7inintuthrne sahnodwEnreisn=th2e∆inpsfeot.rRthieghatnptia-nfeerlr:oTmhaegenneetircgyorsdcearleasn2d∆th∗e∼spkBinT-t∗ri∼plNet p (p )−wavesuperconductingorderasafunctionofdopingδ,with scalingrelation: E =κk T . x y res B c Nbeingtheanti-ferromagneticorder,andk theBoltzmannconstant. B Notethatthetwodistinctenergyscalesintheunderdoped Acrossover fromtheBose-Einsteincondensation (BEC)regimeto regimearesplitofffromonesingleenergyscaleinthe(heav- theBardeen-Cooper-Schrieffer (BCS)regimeoccurs, whenthetwo ily) overdoped regime. This naturally results in a crossover energy scales merge into one single energy scale in the (heavily) from the Bose-Einstein condensation (BEC) regime to the overdoped regime. Note that, in general, Eres < 2∆d. Indeed, Bardeen-Cooper-Schrieffer (BCS) regime, as conjectured in Eres ≈1.25∆d,aspredictedfromthetwo-dimensionalt−Jmodel.In addition,E scaleswiththesuperconductingtransitiontemperature Ref.[15],whichinturnisessentiallyequivalenttothephase res T : E ∼k T fluctuation picture proposed by Emery and Kivelson [16]. c res B c However, there is an importantdifference: the superconduc- tivityweakensintheheavilyunderdopedregime,notonlybe- causeofthelossofphasecoherence,butalsobecauseofthe (heavily)overdopedregimeasaconsequenceoftheevolution decreaseofthesuperconductinggap∆ withunderdoping. of the Fermiarcs in the underdopedregimeto a large Fermi d surface in the (heavily) overdopedregime [30, 31]. We em- Nowafundamentalquestioniswhetherornotsucha sce- phasizethatthepseudogapneartheantinodalregiondoesnot narioisreallyrelevanttothehighT problem. Thisbringsus c characterize a precursor to the superconducting state, in the tothephenomenologyofthehightemperaturecupratesuper- conductors. sensethatthepseudogapsmoothlyevolvesintothesupercon- First, let us focus on the two distinct energy scales 2∆∗ ducting gap at Tc [13, 30, 31]. Instead, it coexists with the superconductinggap of the d + s-wave symmetry in the su- and E . Physically, the two distinct energy scales mea- res perconductingstate. More likely, a precursorpairing occurs sure, respectively, the pairing strength and the coherence of thesuperfluidcondensate. Thisnaturallyleadsto twodiffer- inthenodalregion[32],withitsonsettemperaturelowerthan T∗, but above T , which may be identified with the Nernst entphases: oneischaracterizedbyincoherentpairing,which c regime[33, 34]. As observed, the superconductinggapnear may be identified with the pseudogapphase; the other is as- the nodes scales as k T [29]. This makes a strong case for sociated with the emergenceof a coherentcondensateof su- B c our argument, if one takes into account the smallness of the perconductingpairs,whichmaybeidentifiedwiththesuper- conductingphaseofd+ s-wave symmetry[see Fig.2]. Evi- s-wave superconductinggap. On the other hand, ample evi- dencehasbeenaccumulated,overtheyears,fortheuniversal denceforthetwodistinctenergyscaleswasreportedinangle- scalingrelationE = κ k T ,validforbothsoftmodes,i.e., resolved photoemission spectra [17–20], electronic Raman res B c thespin-tripletresonancemodeininelasticneutronscattering spectra [21–24], scanning tunneling microscopy [25], c-axis experiments[35–41]andthespin-singletmodeobservedasa conductivity[26],Andreevreflection[27],magneticpenetra- A peakinelectronicRamanscattering[21,42–46],respec- tiondepth[28],andotherprobes(forareview,see,Ref.[29]. 1g tively. The experimentally determined κ is around 6, quite Actually,thesestudiesindicatethatthegapneartheantinodal closetoourtheoreticalestimate. Thispresentsapossibleres- region, which is identified as the pseudogap, does not scale olutionto the mysterious A problem[42]: the A modeis with T in the underdopedregime, whereas the gap near the 1g 1g c a charge collective mode as a bound state of (quasiparticle) nodal region may be identified as the superconductingorder parameterinthecuprates[17–23]. Thisidentificationoffersa singletpairsoriginatingfromthefluctuating px(py)-wavesu- perconductingorder. naturalexplanationwhythe twodistinctenergyscalesin the underdopedregimemergeintoonesingleenergyscaleinthe Second, is the pairing symmetry really of d + s-wave na- 3 if the stripe states (for reviews, see, e.g., Ref. [63–66]) were nottouchedupon. Surprisingly,astripe-likestate,i.e.,astate with charge and spin density wave order, coexisting with a spin-tripletp (p )−wavesuperconductingstate,doesoccuras x y T agroundstateinthet−Jmodelfordopingsuptoδ≈0.18[9], N * with J/t = 0.4. Again, all symmetries, including the four- T T PG foldrotationandtranslationlatticesymmetry,SU(2)spinro- tationandU(1)chargesymmetry,arespontaneouslybroken. AF Asimilarlineofreasoningyieldsthat,onlythechargedensity T wave order survives thermal fluctuations, with the concomi- C SC tantoccurrenceofthesoftmodes: theyarisefromtheSU(2) symmetry breaking of the spin density wave order and the δ spin-triplet p (p )−wave superconducting order, with twice x y the spin-triplet p (p )-wave superconducting energy gap as x y their characteristic energyscale. Althoughit remains uncer- FIG.2: (coloronline)Aschematicphasediagramofthehole-doped tain whether or not such a commensurate stripe-like state is high temperature cuprate superconductors plotted as a function of an artifact of our choice of the unit cell for the tensor net- temperatureT andholedopingδ. Here, SC,PG,andAFstandfor work representation of quantum states, an important lesson thesuperconducting, pseudogap, andanti-ferromagneticphases,re- wehavelearnedfromoursimulationisthat, thet− J model spectively. T ,T∗,andT represent,respectively,thesuperconduct- c N exhibitsastrongtendencytowardsastripestateintheunder- ingtransitiontemperature,thepseudogaptemperature,andtheanti- doped regime, consistent with the density matrix renormal- ferromagnetic Ne´el temperature. In our scenario, the PG phase is characterizedbyincoherentpreformedpairs,withapseudogapopen- izationgroup[67]foramorerealisticstripepatternatdoping ing near the anti-nodal region, leaving the remnant gapless Fermi 1/8 and J/t = 0.35. From this we conclude that (i) static arcsnear thenodes inthemomentum space. Inaddition, thefour- charge density wave order is compatible with the supercon- foldrotationsymmetryandthetranslationsymmetryarebrokenon ductivity,soitspossibleroleisdeservedtobeexploredinthe thesquarelattice. Superconducivityoccurs,whenlongrangephase formationof the pseudogap; (ii) static spin density wave or- coherencedevelopsatTc. derisdetrimentaltothed+s-wavesuperconductivity,because nod+ s-wave superconductingcomponentcoexistswith the charge and spin density order in the ground state; (iii) fluc- ture? Many theoretical proposals, mainly based on the res- tuatingspindensitywaveorder,togetherwiththefluctuating onating valence bond scenario [2], predicted a pure d-wave spin-triplet p (p )−wavesuperconductingorder,equallywell x y superconductivityinthecuprates[6,47–49]. Althoughavari- account for the Bose-Einstein condensation in our scenario. etyofexperimentshavedemonstratedapredominantd-wave Therefore,the fluctuatingstripe orderis intrinsic to many, if gap [50–52], strong evidence points to an admixture of an notall,familiesofthehightemperaturecupratesuperconduc- s-wave component to the dominant d-wave superconductiv- tors. ity in muon spin rotation studies [53, 54], electronic Raman Fourth,doesourpredictionaboutthespontaneousbreaking measurements[55], angle-resolvedelectron tunnelingexper- of the four-fold rotation symmetry and the translation sym- iments [56], and neutron crystal-field spectroscopy experi- metryunderone-siteshifts in thepseudogapphase represent ments [57]. In addition, a universal scaling relation of the aphysicalreality? Inourscenario,thePGphaseischaracter- superfluiddensity,ρ (0),atabsolutezero,withtheproductof s ized by incoherentpreformedpairs, which occur in the anti- the dc conductivity σ (T ), measured at T , and the transi- dc c c nodal regime in the momentum space, leaving the remnant tion temperature T , indicates that a pure d-wave supercon- c gaplessFermiarcsinthenodalregime. Inaddition,thefour- ductivityisrealizedinthecuprates[58]. Thisscalingrelation foldrotationsymmetryandthetranslationsymmetryarebro- may be regardedas a modified form of the Uemura relation ken on the square lattice. In the superconductingphase, the between the superfluid density ρ (0) and the transition tem- s four-fold rotation symmetry breaking manifests itself in the peratureT [59,60],whichworksreasonablywellintheun- c admixture of an s-wave component to the dominant d-wave derdoped regime. However, a significant deviation from the state. Inthepseudophase,thebrokensymmetryprotectsthe scalingrelationwassubsequentlyobserved[61],withasalient spin-singletsoft mode that is unidirectionalas it arises from featurethatthedeviationincreaseswithdoping. Thisfeature the p (p )-wave superconductingfluctuations, as well as the x y stronglysuggeststhatthediscrepancyshouldbeaccountedfor fluctuatingspindensityorderandpossiblystatic chargeden- by removing an extra contribution from the s-wave compo- sity order. One may expect that measurable physical effects nentinthecontextofthed+s-wavepairingsymmetry. This arise from the coupling of electrons with the unidirectional issue has been addressed recently [62], thus supporting our spin-singletsoftmode. Indeed,alargein-planeanisotropyof scenario. theNernsteffectinYBa Cu O wasreportedthatsetsinex- 2 3 y Third, any theory regarding the underlying mechanism of actlyatthepseudogaptemperatureT∗[68]. 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