cuprates

PHYSICAL REVIEW B 66, 224512 ~2002!

Magnetic-texture-driven charge pairing in the spin-fermion Hubbard model and superconductivityin the high-Tc cuprates

Eduardo C. Marino1,* and M. B. Silva Neto2,†

1Instituto de Fı´sica, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, Rio de Janeiro, 21941-972, Brazil2Department of Physics, Boston University, Boston, Massachusetts 02215

~Received 28 February 2002; revised manuscript received 15 October 2002; published 31 December 2002!

We propose a model for high-Tc superconductors that considers the spin fluctuations of the localizedmoments of copper and the fluctuations of the doped holes in oxygen as independent degrees of freedom. Weargue that the Coulomb interaction between the doped holes becomes important as the doping concentrationincreases and produces changes in the basic properties of the system. The holes are fractionalized with theircharge being associated with magnetic texture~Skyrmion! excitations of the Copper spin background whereastheir spin is carried by chargeless fermion excitations. Charge pairing is driven by the interaction between themagnetic textures, which in the quantum-disordered phase that precedes the superconducting transition arefully quantum mechanical and gapless. The quantum Skyrmion effective interaction potential is evaluated as afunction of doping and temperature, indicating that Cooper pair formation is determined by the competitionbetween these two types of spin fluctuations~copper and oxygen!. The superconducting transition occurs whenthe effective potential allows for Skyrmion bound states and this is expressed as an algebraic equation relatingTc to other microscopic parameters and doping. Our theoretical predictions for the superconducting phasediagram of La22xSrxCuO4 and YBa2Cu3O61x are in good agreement with experiment in the underdopedregion.

DOI: 10.1103/PhysRevB.66.224512 PACS number~s!: 74.25.Ha, 74.72.2h

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I. INTRODUCTION

The BCS theory of superconductivity was an importabreakthrough towards the understanding of the phenomof frictionless flow of electronic currents in metals. It makclear, for example, that this is the natural ground statenormal metals, like Ni or Al, at sufficiently low temperatures. The formation of Cooper pairs and the consequopening of a gap at the Fermi surface were also wellplained as being originated from the attractive interactbetween the quasiparticles mediated by the lattice vibratiodegrees of freedom, namely, the phonons. For this reathe discovery of superconductivity in the La1.85Ba0.15CuO4

copper oxide1 attracted the attention of the solid-state comunity and still challenges it, since the stoichiometric copound happens to be, rather, an insulator. In addition, itsoon realized that, even in the normal phase, transportoptical properties, such as resistivity (}T), the Hall effect(}1/T), and Raman intensity (}v/T), as well as nuclearmagnetic resonance~NMR! and inelastic neutron scatterin~INS! data, deviated significantly from the usual Fermliquid picture and for these reasons are called normal-sanomalies.2 Moreover, some other results, like the absencecoherence peaks in the spin-lattice relaxation rate3 and thenon-Yosida-like behavior of the spin Knight shift for planCu sites,4 revealed that superconductivity in the cupratesnot likely to arise from the usual BCS singlets-wave pairingmechanism. These high-temperature cuprate superconduactually exhibit a much wider variety of interesting physicphenomena, such as Neél and metal-insulator transitions, incommensurate modulations of antiferromagnetic spin fltuations, charge segregation, pseudogap, and spin-chseparation, and have inspired a large amount of theoreti5

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numerical,6 and experimental7 work for more than 15 yearsIn spite of that, even the nature of the ground state and oelementary excitations have not yet been fully determinand many different pictures have been proposed andcluded along those years.

One of the most fundamental points yet to be understoactually, is the mechanism of charge pairing itself. If, on tone hand, it has been realized that such pairing is not duan attractive interaction mediated by phonons, on the ohand, it is by now well established that antiferromagnespin correlations play an important role in the dynamicsthe system and, most likely, in the pairing mechanism, eafter the destruction of the Neél state. Indeed, different spinfluctuation models have been successfully used to expthe observed spectral weight in angle-resolved photoemsion spectroscopy~ARPES! data of high-Tc materials8 andthe temperature and doping dependences of the uniformceptibility and magnetic correlation length probed by INS9

as well as other anomalies in spin and charge response ftions detected by nuclear quadrupole resonance~NQR! andRaman scattering.10 Moreover, the idea of spin-fluctuatioinduced charge pairing and superconductivity is not newhas been recurrently used.11

The antiferromagnetic nature of the ground state inundoped compounds has opened the possibility for topolcal excitations, or magnetic textures, to arise upon dopFor two-dimensional~2D! quantum antiferromagnets, suctopological excitations can be found, for example, in tform of Skyrmions, spin polarons, or as a spiral phase,pending on the commensurability of the order paramewave vector. In the case of Skyrmions, for example,could naturally expect changes in some properties of theprates as happens to other systems, for example, in manites, where the Skyrmions have been argued to contribut

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E. C. MARINO AND M. B. SILVA NETO PHYSICAL REVIEW B 66, 224512 ~2002!

the colossal magnetoresistance effect,12 or 2D quantum Hallsystems away from the incompressible regime (nÞ1),where the Skyrmions carry the electric charge ofquasiparticles,13 consequently affecting its transport propeties. It is, therefore, natural to expect that, if Skyrmionsintroduced through doping in high-temperatusuperconductors14 and if, as in the case of quantum Haferromagnets, these excitations carry electric charge, tpresence would significantly affect the transport propertiethe cuprates as well. The situation for the case of spiraltortions has been considered in Ref. 15 where its connecwith the spin-glass phase of the high-Tc cuprates has beeclarified. Finally, for the case of spin polarons, a good reviabout its physics and its importance to the physics ofcuprates can be found in Ref. 16.

In this work we propose a theory for high-Tc cuprates thattakes into account the Coulomb interaction betweendoped holes and treats the spin fluctuations of the Cu21 mag-netic ions and of the O22 doped holes as independent dgrees of freedom. Then, as happens in the 2D quantumproblem and as proposed earlier,14,17we associate the chargof the dopants with Skyrmion quantum spin excitationsthe Cu21 background. While in the Neél phase these arfinite energy defects, closely related to their classic counparts, in the quantum-disordered phase they become gapnontrivial, purely quantum mechanical excitations. Asturns out, Skyrmion excitations are dual to spin waves.deed, while in the Neél phase spin waves are gapless withpower-law behavior of its correlation functions, Skyrmiohave a finite excitation energyEsÞ0 and exponentially decaying correlators18 ;e2Esur u. In the quantum-disorderephase, on the other hand, spin waves become gapped,exponentially decaying correlators, while the Skyrmiopurely quantum mechanical, have now zero activation eneEs50 and power-law behavior;1/ur un for its correlationfunctions.18 This duality existing between Skyrmions anspin waves allows one to naturally understand the factSkyrmion excitations should, in principle, be nontrivialthe quantum-disordered phase, even though the corresping classical solutions no longer exist. This scenario enaus to pursue the identification of the doped hole charge wquantum Skyrmion excitations even in the highly dopedgion where the antiferromagnetic order has already beenstroyed. Quantum Skyrmion correlators, evaluated inquantum-disordered phase of our theory, indeed exhibitnontriviality of Skyrmions in that region of doping.

The complete treatment of the dopant part of the thewe consider in this paper will also depend on whether orthe Coulomb interaction between the doped holes is tainto account. For low enough doping we can neglect tinteraction and integrate out the holes degrees of freedwithin, for example, a nearly free electron picture withparabolic dispersion. In this case it is not difficult to see tour theory will be reduced to the ones already existing inliterature. For higher doping, on the other hand, we shargue that the Coulomb interaction can no longer beglected and that this will give rise to a different kind elemetary excitation, namely, massless Dirac fermions. ThDirac fermions will contribute to the effective interactio

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potential between the quantum Skyrmion topological exctions, which is the quantity we are going to use in orderstudy charge pairing. As will become clear, Cooper pairis, in our formalism, controlled by the competition betwethe spin fluctuations of Cu21 magnetic ions~Schwingerbosons! and those of the O22-doped holes~Dirac fermions!.At low doping the interaction is dominated by the contribtion from the localized spins and is found to be repulsive.high enough doping, however, the fermionic contributiowhich is attractive, takes over and produces an effectivetractive interaction between Skyrmions~charges!. The con-dition for the formation of Skyrmion bound states, and cosequent pairing of charge, will be explicitly obtained in thform of an algebraic equation forTc , the superconductingcritical temperature. Our predictions for theTc line, as afunction of doping, and for both La22xSrxCuO4 andYBa2Cu3O61x compounds are in good agreement with eperiment.

II. CHOICE OF A HAMILTONIAN

It is commonly accepted that the most general startpoint for describing the physics of the cuprates is the thrband Hubbard model19 ~3BHM! on a square lattice. Thesthree bands correspond, respectively, to thedx22y2 band ofcopper, contributing one electron, and thepx andpy bands ofthe oxygen atoms, contributing two electrons each. Theelectrons interact through three different types of on-sCoulomb repulsion:Ud between the electrons of the copporbitals,Up between the electrons of the oxygen orbitals, afinally Upd between the electrons of the copper and oxygorbitals. The Hamiltonian of the 3BHM is then given bH3BHM5H01HI where

H05(i ,s

~ed2m!nisd 1(

j ,s~ep2m!nj s

p

1 (^ i , j &s

t i jpd~dis

† pj s1H.c.!

1 (^ j , j 8&s

t j j 8pp

~pj s† pj 8s1H.c.! ~1!

and

HI5(i

Udni↑d ni↓

d 1 (^ i , j &ss8

Updnj sp nis8

d1(

jUpnj↑

p nj↓p .

~2!

In the above notation,i , j run on the copper and oxygelattices, respectively,dis

† (dis) and pj s8† (pj s8) are the cre-

ation ~annihilation! operators for the electrons in thedx22y2

copper andpx,y oxygen orbitals,ed ,ep are the local energylevels in these orbitals, andm is the chemical potential. Asusual, nis

d and nj s8p are the electron number operators f

copper and oxygen, respectively, andt i jpd and t j , j 8

pp the hop-ping parameters for copper-oxygen and oxygen-oxygen,spectively.

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MAGNETIC-TEXTURE-DRIVEN CHARGE PAIRING IN . . . PHYSICAL REVIEW B66, 224512 ~2002!

The values for the parameters of the 3BHM can betained from quantum chemical, constrained density futional theory methods and multiband cluster calculation20

One typical example isDE[ep2ed52.75, tpd51.5, tpp50.65,Ud58.8, Up56.0, andUpd50.8, all values given inunits of eV. We clearly see that the dominant interaction teis the Coulomb repulsion between the electrons in the coporbitals. In fact,[email protected] , showing that the parent cuprate compounds are actually charge transfer~CT!insulators21 characterized by the so-called charge transgapDE. The above model has been quite successfully uin reproducing the tight-binding band structure observedphotoemission spectroscopy experiments.22 However, theenormous complexity of the model has made it difficult ffurther theoretical developments and the necessity of a spler description of the electronics of the cuprates becaevident.

A. Spin-fermion and t-J models

A simplified version of the above model can be obtainat strong coupling.23 At finite but low doping we can safelyneglectUp and Upd and after integrating out the electrondegrees of freedom of the oxygen we end up with thecalled spin-fermion model24

HSF52tp (^ i , j &,a

~ci ,a† cj ,a1H.c.!

1JK (i ,a,b

SW i•ci ,a† sW abci ,b1Jd(

^ i , j &SW i•SW j , ~3!

whereSW i5di ,a† sW a,bdi ,b represents the localized spins of co

per ions, which interact through the superexchange menism with

Jd54tpd

4

~DE1Upd!2 S 1

Ud1

2

2DE1UpD ,

ci ,a† (ci ,a), a51, . . . ,N52, are now thep-band hole cre-

ation ~annihilation! operators@not to be confused with thep-bandelectroncreation~annihilation! operatorspi ,s

† ,pi ,s],tp is the hopping term for the doped holes, and

JK5tpd2 S 1

DE1

1

Ud2DEDis a strong Kondo-like exchange coupling between the spof Cu21 ions and the spins of O22 holes, obtained in perturbation theory with respect to the small parametertpd /DE.

The strong Kondo exchange in Eq.~3! splits the possiblestates of the bonding band of the original 3BHM into tripand singlet states. Since the latter has lower energy,likely that the doped holes will tend to form singlets with thlocalized spins of copper, as proposed originally by Zhaand Rice.25 Using the idea of doped holes forming ZhanRice singlets with the localized spins of copper we can cstruct an even simpler theory for the problem by performperturbation theory in 1/JK for the model in Eq.~3!. Theresult is the so-calledt-J model26

22451

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Ht2J2Cu52td (^ i , j &s

~dis† dj s1H.c.!1Jd(

^ i , j &SW i•SW j , ~4!

where doubly occupied sites should be excluded andtd , thehopping for the electrons of the Cu21 ions, is zero at halffilling ~zero doping!. One of the crucial properties of thmodel is the dual nature of the electronic degrees of freedthat at the same time are expected to form local momentsalso to produce an almost complete Fermi surface inmetallic phase. At zero doping, each site is occupied byactly one Copper spin~see Fig. 1!, and, due to the no-doubleoccupancy constraint, the absence of hopping leads toantiferromagnetic Heisenberg Hamiltonian.27 At finite dop-ing, the formation of Zhang-Rice singlets gives rise to emsites in the system~see Fig. 1!, and the, until now localizedelectrons of Cu21 are then allowed to hop around. Thmodel is perhaps a good starting point for describinglow-doping physics of the cuprates and it has in fact beshown to qualitatively reproduce the band structure of3BHM, with the lower Hubbard band simulating the ZhanRice band.25 For very small doping we indeed find that manother properties of the system are well described by the vous gauge field theory formulations of thet-J model, U~1! orSU~2!, close to half filling.28 For higher dopings, howeverthe situation is less clear. One example is the evolution ofFermi surface with doping, an issue that has not yet bfully clarified. Finally, the above simplified versions of th3BHM cannot account for the effects of the direct Coulominteraction between the doped holes in the oxygen orbitUp , as the dopant concentration becomes significant. In fexcitation spectra exact diagonalization studies have shthat this interaction can only be neglected at sufficiently ldoping.29

B. Generalized spin-fermion Hubbard model

We shall now argue that a different starting point shoube adopted if one attempts to describe the physics ofhigh-Tc materials for higher dopings, close to or beyond tsuperconducting quantum critical point. Our starting powill then be the generalized spin-fermion Hubbard moddescribed by the Hamiltonian30

HSF-H52tp (^ i , j &,a

~ci ,a† cj ,a1H.c.!1Up(

ini ,↑ni ,↓

1JK (i ,a,b

SW i•ci ,a† sW abci ,b1Jd(

^ i , j &SW i•SW j , ~5!

FIG. 1. One-dimensional pictorical representation of the formtion of Zhang-Rice singlets between the localized spins of cop~open squares! and the doped holes~open circles! resulting in themotion of the copper spins as described by thet-J model.

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which also arises from the strong-coupling limit of the threband Hubbard model.23 We stress thatci ,a

† (ci ,a) are the O22

hole creation~annihilation! operators. The most importandifference between Eqs.~3! and~5! is that in Eq.~5! we haveretained the usually ignored on-site Coulomb repulsiontween O22 holes, UpÞ0. The reason is that, as we haalready discussed, realistic estimates from the 3BHM~Ref.20! suggest thatUp /tp;10, which is rather large, and thuwe can perform atp /Up expansion. Second-order perturbtion theory in tp /Up will give rise to a superexchangeJp

52tp2/Up between the Oxygen spins and we end up with

spin-fermiont-J model

HSF-t-J52tp (^ i , j &,a

~ci ,a† cj ,a1H.c.!1Jp(

^ i , j &sW i•sW j

1JK(i

SW i•sW j1Jd(^ i , j &

SW i•SW j , ~6!

wheresW i5ci ,a† sW a,bci ,b .

Let us now see the differences and similarities betwthe above model~6! and the ones discussed previously. Fzero doping the number of holes in the oxygen orbitalszero and we are reduced, as expected, to the Heisenmodel for the copper spins. At very small doping, whenoxygen holes are very dilute, we can neglect the HeisenbexchangeJp between the spins of the holes when compato the Kondo couplingJK . In this case the situation oZhang-Rice singlets is favored and we end up with the ust-J model for the Cu21 spins as discussed in the previosection. For larger dopings, however, where the exchangJpcan no longer be neglected when compared toJK , there willgradually begin to exist a competition between the tendcies of the hole spin to form Zhang-Rice singlets with tCu21 localized moments and to align antiferromagneticaamong themselves~see Fig. 2!. Thus, it is reasonable to expect that many properties of the system, like the nature ofground state and its elementary excitations, will be demined by the competition between the different energy scof the various coupling constants in Eq.~6! as the dopantconcentration is modified. In fact, if we notice that in thlimit of large hole doping we have almost one hole per sitethat is, we are near half filling of the oxygen orbitals—thwe will find that the nearly free electron picture for thdoped holes is no longer applicable. As we shall see, incase a new ground state for the doped holes is obtainamely, thep-flux phase, and a different type of elementaexcitation emerges, the previously mentioned massless Dfermions. In the next section we will then sketch the cotinuum, long-wavelength limit of the above model in orderobtain a field theory for the relevant degrees of freedom

III. CONTINUUM LIMIT

We shall now construct a low-energy, long-wavelengcontinuum field theory for the microscopic theory proposin Eq. ~6!. Our strategy will be as follows. We shall turn othe Kondo couplingJK momentarily and construct the effective field theories for the oxygen~Sec. III A! and copper

22451

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~Sec. III B! parts separately. This will allow us to obtain thdynamics of these objects depending only on their sinteractions. Then we can turn the Kondo interaction back~Sec. III C! to obtain the mutual interaction. Finally we putall together~Sec. III D! and set the theory ready for compuing physical quantities.

A. Field theory for the doped holes

We want to obtain a continuum field theory for the hopart of the Hamiltonian~6!. To this end, we will only con-sider the first two terms of Eq.~6! and setJK5Jd50 mo-mentarily. The result is at-J model for the holes. Then weextend the representation of the spin group from SU~2! to thecase of SU(N) and consider the large-N limit. The mean-field, large-N, solutions of thet-J model are well known andit has been established that, at least at the saddle-pointand near half-filling, ap-flux phase has minimum energamong all the possible ground states.31 Away from half-filling, however, this phase becomes unstable towards otypes of order, and different kinds of elementary excitatioemerge. The important picture we must have in mind is thas discussed previously, from the point of view of our theo~6! the situationclose to half-fillingis that of almost everysite having one doped hole, and this corresponds to thestoichiometric doping for the high-Tc cuprates. For this reason we will be allowed to use the elementary excitationsthe p-flux phase in order to describe the physics of ttheory ~6! at moderately to high hole doping.

The continuum limit of thet-J model with ap-flux phasein the ground state is well known and for that reason we wsimply go through the basic steps of the derivation~for moredetails, see, for example, Ref. 32!. As usual, we write theelectron in terms of a charged spinless bosonm i ~chargon!and a chargeless spin-1/2 fermionf i ,a ~spinon!,

ci ,a† 5m i

†f i ,a† , ~7!

and we decouple the four-particle interactions by introducthe d-wave auxiliary fields

x i j 5^ f i ,a† f j ,a&,

D i j 5^ f i ,↑ f j ,↓2 f i ,↓ f j ,↑&, ~8!

which are nonzero forT,T* , where T* is often calledpseudogap temperature. If we neglect short-distance chfluctuations, namely,m im j

†&.um i u25const, and if we noticethat thed-wave gapD i j vanishes linearly at the nodal poin(6p/2,6p/2), we will find that the lowest-lying excitationsof the p-flux phase are massless, chargeless, spin carrDirac Fermi fields31 whose dynamics is described by thLagrangian32

L5(a,l

i ca,l~g0]t2vFgW •¹W !ca,l[(a,l

i ca,lgm]mca,l .

~9!

In the above expression,l51,2 labels the two in-equivalent Fermi points at (p/2,6p/2), ]m5(]t ,vF¹W ), gm

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5(g0,gW)5(isz,sx ,sy), vF52ax is the dopant Fermi velocity~with a being the lattice spacing andx the constant ampli-tude of ux i j u), and

ca,l5S f a,le

f a,lo D ~10!

for odd ~o! and even~e! lattice sites.One comment is in order. Although the dispersion relat

of these Dirac fermions is known to be anisotropic, havdifferent Fermi velocities for different directions in the Brilouin zone, we shall pursue the picture of isotropic Dirfermions. The reason is the same already explained prously; that is, for(ani ,a'x→1 ~near half-filling! the ellip-tical hole pockets around (p/2,6p/2) and symmetry-relatedpoints can be well approximated by circles. At first sight, tseems to be in contradiction to ARPES data,33 which actuallyshow an almost complete Fermi surface at high dopings.emphasize, however, that the correct Fermi surface forholes can only be obtained once thecouplingto the underly-ing antiferromagnetic background is properly taken intocount.

B. Field theory for the localized spins

The long wavelength fluctuations of the localized Cu21

spins, on the other hand, can be described in terms of ea O(N) ~Refs. 34 and 35! or SU(N) ~Ref. 36! spin fluctua-tion field theory of the quantum Heisenberg antiferromag~QHAF! with AF exchangeJ. For the O(N) case, for ex-ample, one can show that starting from the HeisenbHamiltonian and introducing spin coherent states in ordetrace out the states in the partition function, we end up wthe nonlinears model described by the Lagrangian34

LNLsM5rs

0

2 S 1

c02 ~]tn!21~¹in!2D , ~11!

wherers0}J andc0 are the bare spin stiffness and spin-wa

velocity of the spin density fieldn. The above model habeen studied within a two-loop renormalization groapproach34 and also by the large-N expansion35 and it hasbeen shown to correctly reproduce the finite-temperaturehavior of the magnetic correlation lengthj and dynamicstructure factorS(q,w),35 in the so-called renormalized classical regime, where the system exhibits long-range AF orat zero temperature.

For the case of SU(N), on the other hand, we can introduce an overcomplete basis of boson coherent statesSchwinger boson fieldszi

† ,zi , i 51, . . . ,N52, such that

n5zi†sW i j zj . ~12!

The dynamics of thezi† ,zi fields is described by theCPN21

Lagrangian37

L CPN2151

2g0u~]m2 iAm!zi u2, ~13!

22451

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t

rgoh

e-

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the

whereAm52 i zi]mzi is a Hubbard-Stratonovich gauge fielandg05N/2rs

0 is the bare coupling constant. This formultion has been shown to give essentially the same results fjand S(q,w) as the O(N) formalism, and also for the statistaggered and uniform susceptibilitiesxs(q,w) andxu(q,w), respectively.37,38 In what follows we shall consid-ered the SU(N) formulation of the problem since it is ouinterest to work with a gauge field theory.

The phase diagram of the QHAF can be classifiedthree distinct regions in theT3g0 parameter space, depening on the value of the bare coupling with respect to socritical couplinggc . These regimes are~a! g0,gc , renor-malized classical~RC!; ~b! g05gc , quantum critical~QC!;and ~c! g0.gc , quantum disordered~QD!. In each of theseregimes we obtain a different scaling for the thermodynacal quantities as a function of temperature,35 reflecting thedifferences in the nature of the spin fluctuations. Althouthe undoped high-Tc cuprates were shown to belong to thRC regime at low temperatures,34 it has been argued thasome of the properties of the underdoped compounds cbe explained by allowing the system to go over the otphases.35

C. Kondo-like coupling

Now that we have already gone through the continulimit of both the Cu21 spins and O22-doped holes as described by Eq.~3!, we still need to consider the Kondo interaction between these degrees of freedom. To this enwill be convenient to go over the Cu21 spin quantizationreference frame.39 This is equivalent to performing the locacanonical transformationc→Uc, where

U5expFqS z1 2 z2

z2 z1D G , ~14!

with UP SU(2) andq being an arbitrary real constant mesuring the deviation of the Dirac fermion spin orientatiofrom the localized Cu21 spin quantization axis. Now theKondo coupling term in Eq.~3! reduces to a chemical potential term since

U†SW •sW U5sz . ~15!

We should also note that

U†]mU5 iqszAm , ~16!

plus negligible nondiagonal terms,39 and we see that theabove transformation introduces a gauge coupling betwthe Dirac fermions and theCPN21 gauge field with couplingconstant given byq.

D. Effective field theory

We are now ready to collect the results of the previosections, namely, Secs. III A, III B, and III C, and writdown our effective gauge field theory for the spin-fermiHubbard model. The partition function reads

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with

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dtE d2xH (i 51, . . . ,N

1

2g0u~]m2 iAm!zi u2

1 (a51, . . . ,N,l51,2

ca,lgm~ i ]m2qszA m!ca,lJ ,

~18!

whereb51/kBT and we have set, for now,vF51. Observethat the parameterq introduced in the previous section bcomes the coupling constant for the interaction betweenspinons and localized copper spins, which is mediatedAm . In the next section, we will determine how the coupliconstants of the above action depend on doping, whicthereby introduced in the model.

IV. CHARGONS AS QUANTUM SKYRMIONS

The idea of charge-doping-induced topological excitatioemerging in the physics of the high-Tc cuprate superconductors is actually not new.14 In the case of La22xSrxCuO4, forexample, numerical studies by Goodinget al.40 have shownthat the coupling of the hole motion to the transverse fltuations of the Cu21 spins produces a spiraling twisting othe antiferromagnetic order parameter, a state that is tologically equivalent to the Skyrmion excitations of the nolinear s model. The existence of these Skyrmion-like extations at low temperatures was then successfully useexplain many features of the phase diagramLa22xSrxCuO4 in a rather large region of doping,40 namely,from the antiferromagnetic state 0<x<0.02, where the theoretical predictions forj21(x,T50) were found to be ingood agreement with the results of INS by Keimeret al.,9

until the description of the freezing of magnetic momemeasured by muon spin relaxation41 (mSR) in the spin-glassphase 0.02<x<0.05. Although this picture is strictly justified at very low temperatures, where the holes are localinear the Sr impurities, it has been argued40 that the spindistortions produced by the holes as they become mobilfinite temperatures will pretty much resemble those genated by the localized carriers at low temperatures. Forreason we shall, in what follows, consider the picturedoping-induced Skyrmions for a range of doping thatcludes the superconducting part of the phase diagram ofLa22xSrxCuO4 ~LSCO! and YBa2Cu3O61x ~YBCO! andalso at finite temperatures.

Following the reasoning described above, we have pposed in previous works17 a model for describing the dopinprocess in quantum Heisenberg antiferromagnets thatcessfully reproduced the magnetization curves and thepart of theT3x phase diagrams of both LSCO and YBCOOne of the important consequences of that model wasobservation that each hole added to the CuO2 planes createsa Skyrmion topological defect on the Cu21 spin background.

22451

ey

is

s

-

o-

-tof

t

d

atr-isf-th

-

c-F

e

The dopant charge, in particular, was found to be attachethe Skyrmion number and consequently the whole dynamof holes becomes totally determined by the quantum Skmion correlation functions. Despite the fact that the moproposed in Ref. 17 is restricted to the antiferromagnetic pof the phase diagram, we shall nevertheless pursue theture in which Skyrmions are in general the charge carriersthe doped holes, for the reasons already explained. Thisworking hypothesis to be verified later on by comparisonour results with the experiments. However, it will allow ustreat the bosonic variablem i introduced in Eq.~7! as a quan-tum Skyrmion operator. The large-distance behavior ofquantum correlators of this operator can then be evaluawithin the continuum model given by Eq.~18!. In particular,we shall exploit this idea in the quantum-disordered phd>dAF , where the Skyrmions are purely quantum mechacal and gapless. The picture above is supported by thethat the quantum Skyrmion correlation functions possesnontrivial behavior even in this disordered phase, where csical Skyrmions no longer exist. Indeed, as we shall seelow and as was already mentioned, these functions exhibexponentially large-distance decay in the ordered Neél phasethat is modified to a power-law decay in the disordered oThis fact clearly shows that Skyrmions are dual to the Cu21

spin-wave excitation and thus are nontrivial excitationsthe disordered phase, even though their activation energzero, exactly in the same way that magnons are nontrigapless excitations in the ordered Neél phase.

The full treatment of quantum Skyrmions for a theorythe kind described by Eq.~18! has been carried out in Re18. The Skyrmion creation operator is given by

m~xW ,t !5expH 2 i2pExW

`

dj ie i j P j~jW ,t !J [eiw(xW ,t), ~19!

where P i(jW ,t) is the momentum canonically conjugateA i . The Skyrmion number density, which will be herebidentified with the hole charge, is given by

rc~xW ,t !51

2pe i j ] iAj~xW ,t !. ~20!

It is easy to show that

@w~xW ,t !,r~yW ,t !#5 id~xW2yW !, ~21!

which is precisely the commutation relation betweencharge density and the phase of the chargon operator induced in Ref. 31. This shows once and for all that our idtification of the chargon operator with the quantum Skyrmioperator and, consequently, the hole charge with the Smion number is consistent with the electron fractionalizatdescribed in Ref. 31.

Following the method of Ref. 18 we can evaluate tquantum Skyrmion correlation function in the renormalizclassical regime of our model given by Eq.~18!, whereg0,gc (gc58p/L). We obtain the expression

^m~x!m†~y!&5e22prsux2yu

ux2yuq2/2

, ~22!

2-6

se

-

tonlert

d

s

an

e

n

nend

er

s-derc-ichoming

a

esrge

ifer-how-nsro-

m-aruntct,ldsthere-

e of

r-el

we

r,

e


wherers is the spin stiffness, given byrs51/g021/gc , andq is the spinon coupling. Conversely, for the theory propoto describe the low-doping AF phase of the cuprates,17 thecorresponding Skyrmion correlator was found to be

^m~x!m†~y!&5e22prs(d)ux2yu

ux2yua(d), ~23!

where the expressions forrs(d) and a(d) have been carefully determined in Ref. 17. The exponenta(d) is given by

a~d!5F 64

p21161

aEM

4p2G ~nd!2, ~24!

with n51 for YBCO and n54 for LSCO.42 aEM is theelectromagnetic fine structure constant corresponding tocontribution of the electromagnetic coupling to the Skyrmicorrelation function. We immediately see that it is negligibwhen compared with the first term. We therefore hencefodrop it. Thers(d) function obtained in Ref. 17 is given by

rs~d!5rs~0!F12g\c

paDrs~0!d2G ~25!

for YBCO and

rs~d!5rs~0!F1216g\c

paDrs~0!d22

8A2\c

ars~0!dG1/2

~26!

for LSCO.43 In the above expressions,g532p(9p2

216)/(p2116)2.10.9, a53.8 Å is the lattice spacing, anaD5a/A252.68 Å.

In order to obtain thed dependence of the spin stiffnesrs and of the spinon couplingq in our model~18!, we nowmatch the two correlation functions in Eqs.~22! and ~23!~ordered phase!. This guarantees that the physics of the qutum Skyrmions of the present model~18! coincides with thatobtained from the model in Ref. 17 in the AF phase. Wthereby obtainrs5rs(d), with rs(d) given by Eqs.~25! and~26!, respectively, for YBCO and LSCO and

q~d!5F 128

p2116G 1/2

~nd!. ~27!

The sublattice magnetization in the ordered phase is give

M ~d!

M ~0!5Ars~d!

rs~0!, ~28!

and consequentlydAF can be obtained fromr(dAF)50.Both the magnetization curves as a function of doping athe quantum critical doping for the destruction of AF ordare in good agreement with experiment for YBCO aLSCO.17 Furthermore, from Eq.~28!, we can infer the criti-cal exponentb for the magnetization. This is read asb51/4 for LSCO~Ref. 44! and b51/2 for YBCO. We shalluse this later on when comparing our results to the expmental data.

For d.dAF , on the other hand, wherers50, we shallassume that the expression forq(d) still holds. This is quite

22451

d

he

h

-

by

dr

i-

reasonable sinceq was introduced by a local canonical tranformation, and at least locally there is short-range AF orbeyonddAF . Moreover, this is also consistent with the piture proposed and verified numerically in Ref. 40 and whis valid for the whole range of the phase diagram going frthe antiferromagnetic until deep into the superconductphases. We therefore have

^m~x!m†~y!&51

ux2yuq2(d)/2, ~29!

for d.dAF (rs50). Notice that this implies that0umu0&50 in this phase, whereu0& represents the ground state. Asconsequence the Skyrmion statesum&5mu0& are orthogonalto the ground state and therefore nontrivial. This providsupport to our picture that quantum Skyrmions are the chacarriers of doped holes even in the regime where the antromagnetic order has been already destroyed. Observe,ever, that it is only in the presence of the Dirac fermio~spinons!, manifested through the existence of a nonzecoupling q, that the Skyrmion correlators will have a nontrivial, power-law, behavior in the disordered phase. This iplies that, within our picture, considering simply a nonlines model in the quantum-disordered phase could not accofor a correct description of the dynamics of the holes. In faif we hadq50 in the quantum-disordered phase, we wouthen have 0umu0&Þ0, indicating that the Skyrmion statewould be equivalent to the vacuum. We conclude thatdescription of Skyrmions as the hole charge carriers, thefore, only makes sense with the simultaneous presencspinons~Dirac fermions!.

V. COOPER PAIR FORMATION—EFFECTIVESKYRMION INTERACTION

Let us now investigate the conditions for Cooper pair fomation. In order to do this, we must derive out of our modthe effective dynamics of Skyrmions. For this purpose,shall first introduce the Skyrmion current

J m51

2pemab]aAb ~30!

in the partition function~17! by means of the identity

15E DJmdS Jm21

2pemab]aAbD . ~31!

Integrating overzi† ,zi and ca ,ca , we obtain, at leading or-

der, the effective action for theAm field, namely,

Se f f@Am#5N

2E dtdt8E d2xd2yAm~x,t!

3Pmn~x2y,t2t8!An~y,t8!, ~32!

where Pmn(x2y,t2t8), the vacuum polarization tensohas a Fourier transform given byPmn(p,i em)5PB

mn(p,i em)1PFmn(p,i em). These are, respectively, th

contributions coming from the complex scalar fieldszi

2-7

onti

cti-

reetio

ttitum

d

h

te

of

dert

itergencepa-

ely,

the

allt

ini-

er-

eldf

thehe-

heertole

ty,

e-the


~Schwinger bosons! and fermionsca,l ~spinons! to theCPN21 gauge field self-energy tensor; see the graphsFig. 3.

In order to obtain the effective current-current interactibetween Skyrmions, we use an exponential representafor the d function in Eq. ~31! and perform the quadratiintegrations overAm and the corresponding Lagrange mulplier field. The result is

Z5E DJmexpH 22p2E dt dt8E d2xd2yJm~x,t!

3Smn~x2y,t2t8!Jn~y,t8!J , ~33!

where Smn(p,i em)5Pmn(p,i em)/(p21em2 ). The real-time

effective interaction Skyrmion energy can then be inferfrom Eq. ~33!. We want to investigate the threshold for thformation of bound states and therefore the relevant situacorresponds to the static case (em50). Precisely the sameprocedure is adopted, for instance, in order to derive, ouquantum electrodynamics, the effective interaction potenbetween the proton and electron that includes the quancorrections leading to the Lamb shift, in a hydrogen atoThen, for static Skyrmions we obtain

HI52p2E d2xE d2yrc~x!S00~x2y;0!rc~y!, ~34!

where rc(x)5J0(x,0) is the dopant charge density anS00(x2y;0) has a Fourier transform given byS00(p)5PB(p)1PF(p) with

PB~p!52D

2p1

1

2pE0

1

dxAupu2x~12x!1m2

3cothSAupu2x~12x!1m2

2kBT D ~35!

and

PF~p!5q2

p E0

1

dxAupu2x~12x!tanhSAupu2x~12x!

2kBT D .

~36!

In the above expressions,m is the inverse correlation lengtof the quantum-disordered phase of theCPN21 model (g0.gc), whereD58p(1/gc21/g0) andrs50. At orderN, itis given by37

m~T!5D12kBTe2D/kBT. ~37!

For two charges at positionsx1 and x2, we haverc(x)5d (2)(x2x1)1d (2)(x2x2). After discarding self-interactions, we obtain (r5x12x2)

V~r !5E d2pS00~p!eip•r1Vl~r !, ~38!

where we have also introduced the centrifugal barrier potial between the two charges that form the Cooper pair,

22451

in

on

d

n

ofalm.

n-

Vl~r !5l ~ l 11!\2

2M* r2, ~39!

with l specifying the relative orbital angular momentumthe pair andM* the effective mass of the charges.45

A. Zero-temperature limit

For studying the zero-temperature limit, we must consiS00 at T50 in Eq. ~38!. Before doing this, however, leus remember the well-known fact that in high-Tc cuprates,the Cooper pairs form at relatively short-distances. In spof the fact that our theory describes the physics at a lawavelength scale, we may still analyze the short-distaregime, provided this remains larger than several latticerameters. In this limit, which corresponds to largeupu, wehave, atT50,

V~r !→E d2pF 1

8upu2

2q2

8upuGeip•r1Vl~r !, ~40!

where the two terms between brackets come, respectivfrom the bosonic and fermionic contributions toS00. Theabove expression clearly shows a competition betweenspin fluctuations of the Cu21 spins~first term! and those ofthe O22-doped spins~second term!, the former opposing andthe latter favoring the occurrence of pairing. For smenough dopingq2,1/2, the potential is always repulsive ashort-distances and there is no charge pairing. Forq2.1/2,on the other hand, the potential has a short distance mmum and consequently charge~Skyrmion! pairing occurs.We conclude that the critical doping for the onset of supconductivity is determined by the condition

q2~dSC!51/2. ~41!

Observe that without considering the antiferromagnetic Cu21

background, the interaction potential~40! would always havebound states for anyqÞ0, at zero temperature, and wwould find dSC50. This is what happens in the mean-fiephase diagram of Kotliar and Liu.46 We see that the effect othe Cu21 background is to shift the value ofdSC to its cor-rect position in the phase diagram, showing once moreimportance of the antiferromagnetic fluctuations to the pnomena of superconductivity.

Finally, although the above result was obtained within tlarge-N limit, we have carefully checked that higher-ord(1/N) corrections to the graphs of Fig. 1, correspondingfluctuations of theAm and constraint fields about the saddpoint, do not contribute to the short-distance potential~40!and therefore do not change the result above.

B. Determination of dSC—effects of disorder

From the expression ofq in terms of d, Eq. ~27!, andfrom the critical condition for the onset of superconductiviEq. ~41!, we immediately conclude thatdSC

YBCO50.318 anddSC

LSCO50.079, respectively, which are in fairly good agrement with experiment. In particular, we should note thatgeneral expressions fordSC

YBCO and dSCLSCO satisfy dSC

YBCO

2-8

f.

e

of

ethe.l-.

g

.

nt

he

a

ne

o

deres,n

es.

ed

ally

-ro-

n

--ion


54dSCLSCO, a result that comes from the different values on

for the two compounds17 and that is verified experimentallyIndeed, the experimental values are approximatelydSC

YBCO

50.22 anddSCLSCO50.055. This can be seen, provided w

take into account the relation existing betweend and thestoichiometric doping parameterx, namely,d5x for LSCOandd5x20.20 for YBCO. This almost universal naturethe superconducting critical dopingdSC is one of the nicepredictions of the model.17

Let us now show that the introduction of disorder furthimproves the above results. Disorder may be modeled inordered Neél phase of a doped antiferromagnet by considing a continuous random distribution of spin stiffnesses47

For this purpose we shall introduce in the original modeGaussian3ur2rsul21 distribution, with exponentially suppressed magnetic dilution, as has been considered in RefThe effect of this is to produce a correction given bya(d)→a8(d)5a(d)1l ~Ref. 47! in Eq. ~24!. Choosingl5 1

8

for both compounds, the expression of the spinon couplinq(d) becomes

q~d!5A 128

p2116~nd!21

1

4~42!

instead of the one given by Eq.~27!. The critical doping forthe superconducting transition atT50, corresponding to Eq~41! becomes, therefore,

dSC51

nAp2116

512. ~43!

This implies xSCYBCO50.425 andxSC

LSCO50.056 or dSCYBCO

50.225 anddSCLSCO50.056, which are in good agreeme

with the experiment. We emphasize that a single choicethe parameterl was enough to produce good results for ttwo materials. The general relationdSC

YBCO54dSCLSCO is pre-

served.

C. Finite temperature

At finite temperature, we can no longer evaluatePB andPF exactly and we, therefore, resort to temperature expsions. ForPB we shall expand inkBT/m, since it is clearthat m(T).kBT,;T. PB will then be simply given by itszero-temperature limit, wherem5D. For PF , on the otherhand, such a low-T expansion is not necessarily valid evefor upu@kBT. We will then have to split the integral over thFeynman parameterx in Eq. ~36! into three parts. For 0<x<xc and 12xc<x<1, xc5(kBT/upu)2, we will use a high-T expansion, while forxc<x<12xc we use the low-T ex-pression. Performing these expansions, we obtain

S00~p!5122q2

8upu2

m

pupu21

m2

2upu31

16q2T324m3

3pupu4.

~44!

Inserting Eq. ~44! in Eq. ~38!, we get the finite-temperature Skyrmion interacting potential as a functiondoping,

22451

re

r-

a

47.

of

n-

f

V~r !5~122q2!

16p

1

ur u1

m

2p2lnS ur u

L D2m2

4pur u

2~m324q2T3!

6p2ur u2lnS ur u

L D1Vl~r !, ~45!

whereL is the linear size of the system. We can now consithe threshold conditions for the formation of bound statnamely,V8(r0)50 andV9(r0)50. From these, we obtain aalgebraic equation for the superconducting temperatureTcgiven by

~kBTc!352

p~122q2!a3

512q21

Da2

32q22

3pD2a

128q21

D3

4q2

23p2l ~ l 11!a4

q2M* vF2

. ~46!

In the above expression,a5\vF /r 0 , D5mc2, r 05ur0u isthe minimum of the potential~which also measures the sizof the Cooper pair!, and we have restored all physical unitFinally, we remark that inserting the value ofdSC at T50 inEq. ~46! we can get a relation that fixesM* vF

2 with respectto r 0.

VI. COMPARISON WITH THE EXPERIMENT

In order to make contact with experimental data we nethe doping dependence ofD(d). SinceD(d) is the inversecorrelation length at zero temperature, we can generwrite

D~d!5D0F S d

dAFD 2

21Gn

, ~47!

where the exponentn can be inferred from the scaling relations of the thermodynamical quantities near the zetemperature quantum critical point separating the Neél andquantum-disordered phases. In the case of theCPN21 non-linears model at largeN, we have the hyperscaling relatio

b51

2~D221h!n, ~48!

which relates the critical exponentsb andn, associated, re-spectively, with the order parameter~sublattice magnetization! and spin correlation lengthj to the anomalous dimension h of the order parameter. Also, in the above expressD is the spacetime dimensionD5d11, which impliesD53 for our problem in two space dimensions. Sinceh50 atleading order in the 1/N expansion, it follows that, for ourpurposes, we can use the relation

b5n

2. ~49!

Thus, knowledge of the critical exponentb will allow us todetermine the functional form ofD(d) since, at low tempera-tures, we havej215D(d). For YBCO we have obtainedb

2-9

g

-

s

e

in

y.e

othhyha

m

-

-fod

heal-

itionuc-llestosi-

is

istionledds

i-

ghad

est


51/2, which impliesn51, as seen from the above scalinrelations. For this reason we shall write, for YBCO,

D~d!5D0F S d

dAFD 2

21G . ~50!

For LSCO, on the other hand, we haveb51/4, which im-plies n51/2. For this reason we shall use in this case

D~d!5D0F S d

dAFD 2

21G1/2

. ~51!

The T50 AF quantum critical pointdAF has been determined experimentally to bedAF50.22 for YBCO anddAF50.02 for LSCO. Finally, we should remark that, in the caof LSCO, for d@dAF , we havej(x5d,0);1/x, which isthe correct behavior obtained experimentally by Keimet al.9 and numerically by Goodinget al.40

A. Case of La2ÀxSrxCuO4

Let us now plot the curve~46! for La22xSrxCuO4. Weshall use r 0538 Å and l 52 (d-wave pairing!, \vF50.1 eV Å, andD050.87 meV. The result is presentedFig. 2. We should remark that Eq.~46! has no real solutionsfor l .2 and that the minimum of the potential forl 51 isabove the minimum of the interesting casel 52, showingthat thed-wave pairing really gives the minimum energMoreover, we have also noticed that the agreement withperiment is extremely sensible to the choiceD0. A variationof 1% in D0 produces a variation of 10% inTc .

The dashed part of the curve~46! in Fig. 4 is presumablyinside the region whereT.T* , whereT* is the pseudogaptemperature. Hence, we should move to a new saddle pwhere the elementary excitations differ from the ones inp-flux phase and our model is no longer valid. That is wour theoretical curve does not fit the experiments in tregion.

B. Case of YBa2Cu3O6¿x

We can now plot the curve~46! for YBCO. We shall againuse r 0538 Å and l 52, but now\vF51.15 eV Å, andD058.0 meV. The result is presented in Fig. 5. The saanalysis concerning the solutions of Eq.~46! for different l asdiscussed for La22xSrxCuO4 applies here. Also the discussion about the sensibility of the fitting with the choice ofD0does apply in this case.

For YBa2Cu3O61x , however, the following considerations must be made. It is known from experiments thatYBCO, the charge transfer to the CuO2 planes is associate

FIG. 2. One-dimensional pictorical representation of the hidoping limit of the spin-fermion Hubbard model. Note that insteof forming singlets with the localized moments of copper~opensquares!, now the spins of oxygen~open circles! tend to align anti-ferromagnetically among themselves.

22451

e

r

x-

inte

t

e

r

with structural transitions of the interplane atoms.48 At verysmall doping, for example, almost all the holes go to tout-of-plane O-Cu-O chains and for this reason we seemost no change in the Neél temperature for 0<x<0.2. Forx>0.2, however, the system undergoes a structural transfrom tetragonal to orthorhombic which causes the introdtion of holes into the CuO2 planes. For this reason we shaused5x20.2 in this region. The charge transfer continufrom x50.2 until x'0.52, where the holes once again gothe out-of-plane O-Cu-O chains. No further structural trantions are observed untilx'0.7, and we have simulated theffect by shifting our continuous curve fromx50.52 to x50.7. Forx'0.7, however, a second structural transitionobserved experimentally, namely, a change in the orientaof the out-of-plane O-Cu-O chains, a transition often calorthorhombic type I to type II. This charge transfer proceeuntil approximatelyx'0.8, beyond which no further transtions are observed until saturation atx51.0. We again simu-late this effect by shifting our continuous curve fromx50.8 tox51.0.

-

FIG. 3. Bosonic~Schwinger bosons! and fermionic~Dirac fer-mions! contributions to the gauge field self-energy tensor~vacuumpolarization!.

FIG. 4. Superconducting phase diagramTc3x for LSCO. Thesolid line is our theoretical prediction~46! with r 0538 Å, l 52,\vF50.1 eV Å, andD050.87 meV. The dashed part of the curvis where we presumably haveT.T* and another ground state mube considered. Experimental data from Ref. 9.

2-10

n

ethun

t

sin

htusce

nganrgt

tupopofesoistrelts

n

s.ap-

offor

on,has02/nd

uc,

tt

e-

is,iftsons

f.


In view of the ideas discussed above, it is worth mentioing that our model predicts that a higherTc for YBCO~dashed line in Fig. 2! would be obtained, provided thmechanism which prevents the introduction of holes inCuO2 planes in the absence of the structural transitions cobe avoided, or else if structural transitions, and consequecharge transfer, could be induced through some sorchemical substitution.

VII. CONCLUSION

We have presented a continuum field theory that servean efficient instrument for performing computations withthe spin-fermion Hubbard model~5!. We apply it in the de-scription of the mechanism of charge pairing in the higtemperature cuprate superconductors. An important feaof the theory is the fractionalization of the dopant holeTheir charge is associated with Skyrmion topological defeon the Cu21 spin background whereas their spin is describby chargeless spin-1/2 fields~spinons!. Spin fluctuations ofthe dopant holes, associated with the O22 ions, as well as theCu21 spin fluctuations are then considered on equal footiCooper pairing is driven by the interaction among the qutum Skyrmions, which become yoked to the dopant cha~chargon!. In the quantum-disordered phase that precedessuperconducting transition, these are gapless fully quanmechanical excitations, which in our picture are the comnents of the Cooper pairs. Whenever the inter-Skyrmiontential becomes attractive charge pairing occurs. The eftive quantum Skyrmion interacting potential is evaluated afunction of temperature and doping and thereby the threshcondition for the formation of Cooper pairs is obtained. This clearly determined by the competition between the conbutions of the two types of spin degrees of freedom, namthose of the Cu21 background and those of the dopanthemselves~spinons!. Our prediction for theTc curve is ingood agreement with the experiment both for LSCO a

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189 ~1986!.2Y. Iye, in Physical Properties of High Temperature Supercond

tivity, edited by D. M. Ginsberg~World Scientific, Singapore1992!, Vol. 3, p. 285.

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Franz and Z. Tesanovic, Phys. Rev. B63, 064516~2001!.9B. Keimeret al., Phys. Rev. B46, 14 034~1992!.

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11T. Moriya, K. Ueda, and Y. Takahashi, J. Phys. Soc. Jpn.49, 2905~1990!; P. Monthoux, A. Balatsky, and D. Pines, Phys. Rev. Le

22451

-

eldtlyof

as

-re.tsd

.-ehem--

c-ald

i-y,

d

YBCO in the underdoped regime where our model applieThe results obtained here encourage us to pursue the

plication of the model in the description of other featureshigh-Tc cuprates such as resistivity in the normal phase,instance. We are presently investigating this point.49

ACKNOWLEDGMENTS

The authors are grateful to A. H. Castro Neto, C. Chamand E. Novais for discussions and comments. This workbeen supported in part by FAPERJ and PRONEX - 66.201998-9. E.C.M. was partially supported by CNPq aM.B.S.N. by FAPERJ and CNPq.

-

.

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FIG. 5. Superconducting phase diagramTc3x for YBCO. Thedashed line corresponds to our theoretical prediction~46! with theconcentration of holes varying continuously with doping, thatd5x20.20. The solid line is obtained after considering the shdescribed in Sec. VI B associated with the structural transitiknown to occur in this material. We have usedr 0538 Å, l 52,\vF51.15 eV Å, andD058.0 meV. Experimental data from Re22.

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m

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22451

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43The different dependence on the doping parameter obtained intwo expressions above, respectively for the two compoundsascribed to the peculiar form of the Fermi surface in each caThe square root occurring in the case of LSCO, in particular,been shown to follow from the presence of stripes in this copound~Ref. 17!.

44We remark that our resultb51/4 is in good agreement with thempirical valueb50.236 obtained by Borsaet al., Phys. Rev. B52, 7334~1995!.

45Use of a two-dimensional centrifugal barrier would only redefiM* .

46G. Kotliar and J. Liu, Phys. Rev. B38, 5142~1988!.47E. C. Marino, Phys. Rev. B65, 054418~2002!.48P. Burletet al., in Dynamics of Magnetic Fluctuations in High-Tc

Superconductors, Vol. 246 of NATO Advanced Study InstituteSeries B: Physics, edited by G. Reiter, P. Horsch, and G. CPsaltakis~Plenum, New York, 1991!.

49E. C. Marino and M. B. Silva Neto~unpublished!.

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