The underdoped cuprates as fractionalized Fermi liquids: transition to superconductivity
TThe underdoped cuprates as fractionalized Fermi liquids:transition to superconductivity
Eun Gook Moon and Subir Sachdev
Department of Physics, Harvard University, Cambridge MA 02138 (Dated: October 31, 2018)
Abstract
We model the underdoped cuprates using fermions moving in a background with local antifer-romagnetic order. The antiferromagnetic order fluctuates in orientation, but not in magnitude,so that there is no long-range antiferromagnetism, but a ‘topological’ order survives. The normalstate is described as a fractionalized Fermi liquid (FL*), with electron-like quasiparticles coupledto the fractionalized excitations of the fluctuating antiferromagnet. The electronic quasiparticlesreside near pocket Fermi surfaces enclosing total area x (the dopant density), centered away fromthe magnetic Brillouin zone boundary. The violation of the conventional Luttinger theorem islinked to a ‘species doubling’ of these quasiparticles. We describe phenomenological theories of thepairing of these quasiparticles, and show that a large class of mean-field theories generically dis-plays a nodal-anti-nodal ‘dichotomy’: the interplay of local antiferromagnetism and pairing leadsto a small gap near the nodes of the d -wave pairing along the Brillouin zone diagonals, and a largegap in the anti-nodal region. a r X i v : . [ c ond - m a t . s t r- e l ] N ov . INTRODUCTION The nature of the ground state in the underdoped regime of the hole-doped cupratesuperconductors remains a central open issue. Angle resolved photoemision spectroscopy(ARPES) and scanning tunneling microscopy (STM) have been the main tools to exploresuch a regime. In both probes, an unexpected angular dependence of the electron spectralgap function has been revealed: a ‘dichotomy’ between the nodal and anti-nodal regions ofthe Brillouin zone in the superconducting state . Specifically, this dichotomy is realized bydeviations in the angular dependence of the gap from that of a short-range d -wave pairingamplitude ∼ (cos k x − cos k y ).This paper will describe the superconducting instabilities of a recently developed model of the normal state of the underdoped cuprates based upon a theory of fluctuating localantiferromagnetic order . A related normal state model of fluctuating antiferromagnetshas been discussed by Khodas and Tsvelik , who obtained results on the influence of spin-wave fluctuations about the ordered state similar to ours . These results have been foundto agree well with ARPES observations . Another approach using fluctuating antifer-romagnetism to model the underdoped cuprates has been discussed recently by Sedrakyanand Chubukov . We will also connect with the scenario emerging from recent dynamicalmean-field theory (DMFT) studies .The theory of Ref. 6 describes the normal state in the underdoped regime as a fractional-ized Fermi liquid (FFL or FL*), although this identification was not explicitly made in thatpaper. So we begin our discussion by describing the the structure of the FL* phase.The FL* phase is most naturally constructed using a Kondo lattice model describinga band of conduction electrons coupled to lattice of localized spins arising from a half-filled d (or f ) band. The key characteristics of the FL* are ( i ) a ‘small’ Fermi surface whosevolume is determined by the density of conduction electrons alone, and ( ii ) the presenceof gauge and fractionalized neutral spinon excitations of a spin liquid. In the simplestpicture, the FL* can be viewed in terms of two nearly decoupled components, a small Fermisurface of conduction electrons and a spin liquid of the half-filled d band. The FL* shouldbe contrasted from the conventional Fermi liquid, in which there is a ‘large’ Fermi surfacewhose volume counts both the conduction and d electrons: such a heavy Fermi liquid phasehas been observed in many ‘heavy fermion’ rare-earth intermetallics. Recent experiments2n YbRh (Si . Ge . ) have presented evidence for an unconventional phase, which couldpossibly be a FL*.A concept related to the FL* is that of a “orbital-selective Mott transition” (OSMT),as discussed in the review by Vojta . For latter, we begin with a multi-band model, likethe lattice Anderson model of conduction and d electrons, and have a Mott transition toan insulating state on only a subset of the bands (such as the d band in the Andersonmodel). The OSMT has been described so far using dynamical mean field theory (DMFT),which has an over-simplified treatment of the Mott insulator. In finite dimensions, any suchMott insulator must not break lattice symmetries which increase the size of a unit cell, forotherwise the state reached by the OSMT is indistinguishable from a conventionally orderedstate. Thus the Mott insulator must be realized as a fractionalized spin liquid with collectivegauge excitations; such gauge excitations are not present in the DMFT treatment. With aMott insulating spin liquid, the phase reached by the OSMT becomes a FL*.Returning our discussion to the cuprates, there is strong ARPES evidence for only a singleband of electrons, with a conventional Luttinger volume of 1 + x holes at optimal doping andhigher (here x is the density of holes doped into the half-filled insulator). Consequently, theidea of an OSMT does not seem directly applicable. However, Ferrero et al. argued thatan OSMT could occur in momentum space within the context of a single-band model. Theyseparated the Brillouin zone into the ‘nodal’ and ‘anti-nodal’ regions, and represented thephysics using a 2-site DMFT solution. Then in the underdoped region, the anti-nodal regionunderwent a Mott transition into an insulator, while the nodal regions remained metallic.A similar transition was seen by Sordi et al. in studies with a 4-site cluster . While theseworks offers useful hints on the structure of the intermediate energy physics, ultimately theDMFT method does not allow full characterization of the different low energy quasiparticlesor the nature of any collective gauge excitations.We turn then to the work of Ref. 6, who considered a single band model of a fluctuatingantiferromagnet. Their results amount to a demonstration that a FL* state can be con-structed also in a single band model, and this FL* state will form the basis of the analysis ofthe present paper. The basic idea is that the large Fermi surface is broken apart into pocketsby local antiferromagnetic N´eel order. We allow quantum fluctuations in the orientations ofthe N´eel order so that there is no global, long-range N´eel order. However, spacetime ‘hedge-hog’ defects in the N´eel order are suppressed, so that a spin liquid with bosonic spinons and3 U(1) gauge-boson excitation is realized . Alternatively, the N´eel order could developspiral spin correlations, and suppressing Z vortices in the spiral order realizes a Z spinliquid with bosonic spinons . The Fermi pockets also fractionalize in this process, and weare left with Fermi pockets of spinless fermions; the resulting phase was called the algebraiccharge liquid (ACL). Depending upon the nature of the gauge excitations of the spinliquid, the ACL can have different varieties: the U(1)-ACL and SU(2)-ACL were describedin Refs. , and Z -ACL descends from these by a Higgs transition involving a scalar withU(1) charge 2, as in the insulator .Although these ACLs are potentially stable phases of matter, they are generically suscep-tible to transformation into FL* phases. As was already noted in Ref. 8, there is a strongtendency for the spinless fermions to found bound states with the bosonic spinons, leadingto pocket Fermi surfaces of quasiparticles of spin S = 1 / ± e . Also, as wewill review below, there is a ‘species-doubling’ of these bound states , and this is crucialin issues related to the Luttinger theorem, and to our description of the superconductingstate in the present paper. When the binding of spinless fermions to spinons is carried tocompletion, so that Fermi surfaces of spinless fermions has been completely depleted, we areleft with Fermi pockets of electron/hole-like quasiparticles which enclose a total volume ofprecisely x holes . The resulting phase then has all the key characteristics of the FL* notedabove, and so we identify it here as a FL*. The U(1)-ACL and Z -ACL above lead to theconducting U(1)-FL* and Z -FL* states respectively. Ref. 6 presented a phenomenologicalHamiltonian to describe the band structure of these FL* phases. Thus this is an explicitroute to the appearance of an OSMT in a single-band, doped antiferromagnet: it is the localantiferromagnetic order which differentiates regions of the Brillouin zone, and then drivesa Mott transition into a spin liquid state, leaving behind Fermi pockets of holes/electronswith a total volume of x holes.We should note here that the U(1)-FL* state with a U(1) spin liquid is ultimately unstableto the appearance of valence bond solid (VBS) order at long scales . However the Z -FL* isexpected to describe a stable quantum ground state. The analysis of the fermion spectrumbelow remains the same for the two cases.Phases closely related to the U(1)-FL* and Z -FL* appeared already in the work ofRef. 7. This paper examined ‘quantum disordered’ phases of the Shraiman-Siggia model ,and found states with small Fermi pockets, but no long-range antiferromagnetic order. The4ntiferromagnetic correlations where either collinear or spiral, corresponding to the U(1)and Z cases. However, the topological order in the sector with neutral spinful excitationswas not recognized in this work: these spin excitations were described in terms of a O(3)vector, rather than SU(2) spinor description we shall use here. Indeed, the topological orderis required in such phases, and is closely linked to the deviation from the traditional volumeof the Fermi surfaces. We also note another approach to the description of a FL* state in a single band model,in the work of Ribeiro, Wen, and Ran . They obtain a small Fermi surface of electron-like “dopons” moving in spin-liquid background. However, unlike our approach with gappedbosonic spinons (and associated connections with magnetically ordered phases), their spinonsare fermionic and have gapless Dirac excitation spectra centered at ( ± π/ , ± π/ Z -FL* state with bosonic spinon spin liquid as our modelfor the underdoped cuprates in the present paper. We will investigate its pairing propertiesusing a simple phenomenological model of d -wave pairing. Our strategy will be to use thesimplest possible model with nearest-neighbor pairing with a d -wave structure, constrainedby the requirement that the full square lattice translational symmetry and spin-rotationsymmetry be preserved. Even within this simple context, we will find that our mean-fieldtheories of the FL* state allows us to easily obtain the ‘dichotomy’ in the pairing amplitudeover a very broad range of parameters. We also note that the pocket Fermi surfaces ofthe FL* state will exhibit quantum oscillations in an applied magnetic field with a Zeemansplitting of free spins, and this may be relevant to recent observations .We mention here our previous work on pairing in the parent ACL phase. Thesepapers considered pairing of spinless fermions, while the spin sector was fully gapped: thistherefore led to an exotic superconductor in which the Bogoliubov quasiparticles did notcarry spin. In contrast, our analysis here will be on the pairing instability of the FL* state,where we assume that the fermions have already bound into electron-like quasiparticles, asdiscussed above and in more detail in Ref. 6. The resulting Bogoliubov quasiparticles thenhave the conventional quantum numbers.Our primary results are illustrated in Fig. 1. We also show a comparison to a conven-tional state with co-existing spin density wave (SDW) and d -wave pairing, and to recentexperiments. The left panels illustrate Fermi surface structures in the normal state. Theright panels show the angular dependence of the electron gap in the superconducting states:5 b)(a)(c) (d) (e) (f) FIG. 1: Color online: Our new results for the FL* phase (second row), compared with the Hartree-Fock/BCS theory (top row) and experiments (bottom row). The left panels illustrate Fermi surfacestructures in the normal state. The right panels shows the angular dependence of the electron gapin the superconducting states. (a) Spectral weight of the electron in the normal state with SDWorder at wavevector K = ( π, π ). Here we simply apply a potential which oscillates at ( π, π )to the large Fermi surface in the overdoped region. (b) Minimum electron gap as a function ofazimuthal angle in the Brillouin zone. The full (red) line is the result with a pairing amplitude ∼ (cos k x − cos k y ) co-existing with SDW order, while the dashed (black) line is the normal SDWstate. (c) Spectral weight of the electron in the FL* state, with parameters as in Fig. 3; note thatthe pocket is no longer centered at ( π/ , π/ ; related observations appear in Refs. 12,13. (f) The dichotomy of the spectralgap function from the observations of Ref. 3. See the text for more details. θ , we determine the minimum electron spectral gap along that direction inthe Brillouin zone, and plot the result as a function of θ .The results of the traditional Hartree-Fock/BCS theory on SDW order and d -wave pairingappear in (a) and (b). The SDW order has wavevector K = ( π, π ), and the d -wave pairing isthe conventional (cos k x − cos k y ) form. In the normal state, the Fermi pocket is centered atthe magnetic Brillouin zone boundary, as shown in (a). An important feature of this simpletheory is that the state with co-existing SDW and d -wave pairing has its maximum gap atan intermediate angle, as shown in (b): this reflects the “hot spots” which are points on theFermi surface linked by the SDW ordering wavevector K . No experiment has yet seen sucha maximum at an intermediate angle.One set of our typical results for the FL* theory are shown in (c) and (d). As it was shownin the previous work , the normal state in (c) shows a Fermi pocket which is clearly notcentered the magnetic zone boundary (at ( ± π/ , ± π/ d x − y wave pairing. It is shown that our modelcan reproduce the dichotomy behavior, and we compare our theory with the YRZ modelproposed by Yang, Rice and Zhang , and the related analyses by Wen and Lee . Forcompleteness, it is shown that U (1) gauge fluctuation can mediate the needed d wave pairingin Appendix C. 7 I. EFFECTIVE HAMILTONIAN
The basic setup of the FL* state has been reviewed in some detail in Refs. 6,10, and sowe will be very brief here. The starting point is to transform from the underlyingelectrons c iα to a rotating reference frame determined by a matrix R acting on spinlessfermions ψ p . c iα = R iαp ψ p . (1) R αp is a SU(2) matrix with α = ↑ , ↓ for spin index, p = ± for gauge index, and we parame-terize R i = z i ↑ − z ∗ i ↓ z i ↓ z ∗ i ↑ (2)with | z i | = 1 . In the ACL state, the bosonic z α and the fermionic ψ p are assumed to bethe independent quasiparticle excitations carrying spin and charge respectively. Then weexamined the formation of bound states between these excitations. A key result was thatwas a “doubling” of electron-like quasiparticles, with the availability of two gauge neutralcombinations, F iα ∼ z iα ψ i + , G iα ∼ ε αβ z ∗ iβ ψ i − . (3)This doubling is a reflection of the ‘topological order’ in the underlying U(1) or Z spinliquid; it would not be present e.g. in a SU(2) spin liquid . The F iα and the G iα will bethe key actors in our theory of the FL* phase here. Their effective Hamiltonian is stronglyconstrained by their non-trivial transformations under the space group of the Hamiltonian,which are listed in Table I.From these symmetry transformations, we can write down the following effective8 x R dual π/ I dual x T F α G α G α G α ε αβ F † β G α F α F α F α ε αβ G † β C α C α C α C α ε αβ C † β D α D α D α D α ε αβ D † β TABLE I: Transformations of the lattice fields under square lattice symmetry operations. T x :translation by one lattice spacing along the x direction; R dual π/ : 90 ◦ rotation about a dual latticesite on the plaquette center ( x → y, y → − x ); I dual x : reflection about the dual lattice y axis( x → − x, y → y ); T : time-reversal, defined as a symmetry (similar to parity) of the imaginarytime path integral. Note that such a T operation is not anti-linear. Hamiltonian H tot = H + H int H = − (cid:88) ij t ij ( F † iα F jα + G † iα G jα ) + λ (cid:88) i ( − i x + i y ( F † iα F iα − G † iα G iα ) − (cid:88) i Throughout this paper, we assume that all pairings are d wave, more specifically, d x − y .The assumption of the d wave pairings can be realized by the gauge fluctuation (see Ap-pendix C) or by other channels like conventional spin density wave fluctuations. Then, with11he pairing amplitudes as in Eq. (9), we can write down the mean field Hamiltonian H MFtot = H + H MF ∆ = (cid:88) k C † k , ↑ C − k , ↓ D † k , ↑ D − k , ↓ (cid:15) c ( k ) − ∆ c ( k ) λ − ∆ X ( k ) − ∆ c ( k ) ∗ − (cid:15) c ( k ) − ∆ X ( k ) ∗ − λλ − ∆ X ( k ) (cid:15) d ( k ) − ∆ d ( k ) − ∆ X ( k ) ∗ − λ − ∆ d ( k ) ∗ − (cid:15) d ( k ) C k , ↑ C †− k , ↓ D k , ↑ D †− k , ↓ (10)where ∆ c is the Fourier transform of O c ∆ , ∆ d is the Fourier transform of O d ∆ , and ∆ X is theFourier transform of O cd ∆ + O dc ∆ . For their wavevector dependence we take the forms∆ c ( k )∆ c = ∆ d ( k )∆ d = ∆ X ( k )∆ X = cos k x − cos k y (11)where ∆ c , ∆ d and ∆ X are the respective gap amplitudes.In principle, we could determine these pairing amplitudes from solving a set of BCS-likeself-consistency equations. However, in the absence of detailed knowledge of the pairinginteractions, we will just treat the ∆ c , ∆ d and ∆ X as free parameters. In other words,we are in the deep superconducting phase with adjusted parameters. Then our task is tostudy spectral gap behaviors with given band structures and pairings. More technically, theGreen’s function of the C particle, which determines the electron properties, are studiedfocusing on the pole of the C particles’ Green’s function. The pole basically contains infor-mation about the electron’s dispersion relation, and its minimum determines spectral gapproperties. The latter is defined as the minimum gap along a line from the Brillouin zonecenter at an angle θ : thus the nodal point is at θ = π/ 4, and the anti-nodal point at θ = 0.Although we have three free gap parameters, our results are quite insensitive to theirvalues. For simplicity we will mainly work (in Sections III A and III B) with the case witha single gap parameter ∆ c (cid:54) = 0, and others are set to zero ∆ d = ∆ X = 0. We will brieflyconsider the case with multiple gap parameters in Section III C, and find no significantchanges from single gap case. 12 k x k y 45 60 75 9000.10.20.30.4 45 60 75 9000.10.20.30.4 (a) (b)(c) (d) FIG. 2: Color online: Spectral gap functions and the Fermi surfaces with the Case I, ( t = 0 . t , t = − . t , ˜ t = − . t , ˜ t = 0, ˜ t = 0, ˜ t = − . t , µ = − . t , λ = 0 . t . (a) The spectralweight of the electron Green function with the relaxation time τ t = 200. (b) Fermi surfaces of (cid:15) c ( k ) (dashed inner (red)) , (cid:15) d ( k ) (dashed outer (blue)), and the eigenmodes (thick (black)) of H .The dotted line is the magnetic zone boundary. (c) The spectral gap function with and without∆ c . The dotted (black) line is for the normal case. The thick (red) line is for superconducting statewith ∆ c = 0 . t . (d) The spectral gap function with and without ∆ d,X . The dotted (black) lineis for the normal case. The thick (green) line is for the superconducting state with ∆ X = 0 . t .The dashed (blue) line is the superconducting state with ∆ d = 0 . t . A. Single Gap : case I We consider the case with t = 0 . t , t = − . t , ˜ t = − . t , ˜ t = 0, ˜ t = 0,˜ t = − . t , µ = − . t , λ = 0 . t in Fig. 2. In (a), the calculated spectral weight of the Cparticle is illustrated following the previous paper. The shape is obviously pocket-like, butits spectral weight depends on position on the Fermi surface. In (b), we illustrate the bareenergy Fermi surfaces and their eigenmode Fermi surface. Note that the two bare energybands ( (cid:15) c,d ( k )) are different from the usual SDW formations with Brillouin zone folding. Inthe latter, there is only one electron band, and SDW onset divides the Brillouin zone two13ieces ( (cid:15) ( k ) , (cid:15) ( k + K )). But in our case, the two bands have different energy spectrumsof the electron-like particle ( C ) and the emergent particle ( D ). And λ determines mixingenergy scale between the C and D particles.In (c), the spectral gap function with and without a given pairing, ∆ c is illustrated. Nearthe node, it is obvious that the pairing gap contributes to the spectral gap in a d wave pairingway as expected. However, between the node and anti-node, there is a huge peak. The peakposition is nothing but the mixing point between C, D particles. Therefore, the peak existswhether there is a pairing or not. Near the anti-node, the spectral gap is bigger than thenear-node’s but much smaller than the mixing point peak. It indicates there is tendencyto make electron pockets near the anti-node. For example, if we decrease the magnitudeof λ , which basically represent the mixing energy scale, then the gap near the anti-nodebecomes smaller, and eventually the electron-like pockets appears near the anti-node withthe pre-existing hole type pockets. (See the Appendix) Note that this situation is formallythe same as the pairing with the SDW fluctuation mediating pairing case (see Fig. 1).The “hot spot” between the node and the anti-node has the largest gap magnitude, whichcorresponds to our mixing point. Such a spectral gap behavior is not the experimentallyobserved one. Therefore, we cannot have the needed dichotomy near the anti-node in thiscase; the anti-nodal gap is always smaller than the one of the maximum mixing point.Following the similar reasoning, the experimentally observed dichotomy does not appearin the conventional SDW theory unless additional consideration beyond mean-field theoryis included. In (d), we illustrate other pairing cases (∆ d,X ). As we can see, the role ofthe pairings are similar to the conventional one (∆ c ), and qualitatively they are the same.Therefore, it is not possible to achieve the observed dichotomy by considering the exoticpairings. They cannot push the maximum peak of the normal state to the anti-nodal region.The message of this calculation is simple. With the band structure we considered here, theobserved dichotomy in the spectral gap function cannot be obtained, even though the normalstate can explain experimentally observed Fermi surface structures. Moreover, it also impliesthat it is difficult to explain the observed dichotomy with the Hartree-Fock/BCS mean-fieldtheory of the Fermi liquid.However, we now show how our FL* theory gets a route to explain the dichotomy below.14 k x k y 45 60 75 9000.10.20.30.40.5 45 60 75 9000.10.20.30.40.5 (a) (b)(c) (d) FIG. 3: Color online: Spectral gap functions and the Fermi surfaces for the Case II ( t = 0 . t , t = − . t , ˜ t = − . t , ˜ t = 0, ˜ t = 0, ˜ t = − . t , µ = − . t , λ = 0 . t ). Note that theonly change from Fig. 2 is in the values of µ and λ . (a) The spectral weight of the electron Greenfunction with the relaxation time τ t = 200. (b) Fermi surfaces of ε c (dashed inner (red)) , ε d (dashed outer (blue)), and the eigenmodes (thick (black)) of H . The dotted line is the magneticzone boundary. (c) The spectral gap function with and without ∆ c . The dotted (black) line is forthe normal case with ∆ c = 0. The thick (red) line is the superconducting state with ∆ c = 0 . t .(d) The spectral gap function with and without ∆ d,X . The dotted (black) line is for the normalcase. The thick (green) line has ∆ X = 0 . t . The dashed (blue) line has ∆ d = 0 . t . B. Single Gap : case II In Fig. 3 we illustrate the case with t = 0 . t , t = − . t , ˜ t = − . t , ˜ t = 0, ˜ t = 0,˜ t = − . t , µ = − . t , λ = 0 . t . These parameters are as in Section III A, except thatthe values of µ and λ have changed. As we discuss below, this changes the structure of thedispersion of the ‘bare’ C and D particles in a manner which leaves the normal state Fermisurface invariant, but dramatically modifies the spectral gap in the superconducting state.As we can see in (a), the calculated spectral weight of the C particle is qualitativelythe same as the Fig. 2’s. The shape is obviously pocket-like, and its spectral weight alsodepends on position of the Fermi surface similarly. Therefore, in the normal state, there isno way to distinguish the two cases because the low energy theory are all determined by15he Fermi pocket structures. However, in (b), the bare energy Fermi surfaces of (cid:15) c ( k ) and (cid:15) d ( k ) are clearly different from the previous one’s. Even though the bare Fermi surfaces lookunfamiliar, they are irrelevant for the observed Fermi surface which is determined by theeigenmodes of H (black line), and which is qualitatively the same as the Case I.We illustrate our spectral gap behavior with and without the pairing, ∆ c , in (c), whichwas already shown in the introductory section. Without the given pairing, the normal statehas the finite gapless region where the pockets exist, and there is a stable spectral gap in theanti-node. It is easy to check the anti-nodal gap depends on the mixing term, λ , betweenthe C and D particles. With the pairing, the Fermi pockets are gapped and only the noderemains gapless. The spectral gap function has expected d wave type gap near the node, andthe observed dichotomy is clearly shown. Therefore, the origin of the two gaps are manifest;the nodal gap is obviously from the C particle pairing and the anti-nodal gap is originatedfrom the mixing term, which is inherited from the spin-fermion interaction term. In (d),we illustrate other exotic pairings (∆ d,X ). As we can see, role of the pairings are similar tothe conventional pairing (∆ c ), and qualitatively they are the same. So, there is no way todistinguish what pairings are dominant only by studying spectral gaps.Now let us compare our results to the ones of the YRZ model . In the YRZ model,based on a specific spin liquid model, the pseudo-gap behavior is pre-assumed by puttingan explicit d x − y gap function in the spectrum, which means the characteristic of the anti-nodal gap is another input parameter. With the two d wave gaps (pairing and pseudo-gap),the experimental results were fitted.In our FL* theory, the anti-nodal gap behavior is determined by the interplay between λ and the bare spectrum (cid:15) c,d ( k ) Indeed, the pseudo-gap corresponding term, λ , is s wavetype in terms of YRZ terminology. The λ term represents local antiferromagnetism, andthis ‘competing’ order which plays a significant role in the anti-nodal gap. The parameter λ is just input for making the Fermi pockets in the normal state with other dispersionparameters. As mentioned before, it explains the distinct origins of the nodal and anti-nodal gaps. Also, although our theory contains other pairings, ∆ d,X , we did not need thatfreedom to obtain consistency with experimental observations.Of course, non-local terms of λ could be considered. And it is easy to show that the d x − y like terms are not allowed because of the rotational symmetry breaking. Putting thenon-local λ term is secondary effect, and we do not consider it here.16 Θ H deg L D m i n (cid:144) t 45 60 75 9000.10.20.30.40.5 Θ H deg L D m i n (cid:144) t FIG. 4: Multiple gaps. The left panel is the same as the Fig. 2 with two superconducting gaps∆ d = 0 . t and ∆ c = 0 . t . And the right panel is the same as the Fig. 3 with ∆ d = 0 . t and∆ c = 0 . t . In both, the dashed (green) line is with the two gaps. And the plain and dotted linesare the same as the previous plots. C. Multiple gaps So far, we have only considered the cases with one pairing gap. Of course, multiple gapsare possible and we illustrate possible two cases in Fig. 4, which contain ∆ c,d with the twonormal band structures. Here, we choose the same phase in both pairings. The spectralgap behaviors are not self-destructive, which means the magnitude of spectral gap with twopairings is bigger than the one with the single pairings. One comment is that even multiplegaps do not change qualitative behavior of the spectral gap functions, which means that theCase I could not have the observed dichotomy even with the multiple gaps.In Fig. 5, two pairings with the opposite sign are illustrated. Clearly, we can see theself-destructive pattern with the same gap magnitudes. Even a node appears beyond thenodal point. Therefore, it is clear that the relative phase between two pairings plays animportant role to determine the gap spectrum. IV. CONCLUSIONS This paper has presented a simple phenomenological model for pairing in the underdopedcuprates, starting from the FL* normal state described in Ref. 6. This is an exotic normalstate in which the Cu spins are assumed to form a spin liquid, and the dopants then occupystates with electron-like quantum numbers. A key feature of this procedure , is that thereis a ‘doubling’ of the electron-like species available for the dopants to occupy: this appears17 Θ H deg L D m i n (cid:144) t FIG. 5: Multiple gaps with the relative phase difference. Details are the same as Fig. 3. The redline is for two superconducting gaps with the same sign gaps, ∆ d = 0 . t and ∆ c = 0 . t . Andthe green line is for the opposite sign gaps, ∆ d = − . t and ∆ c = 0 . t . to be a generic property of such doped FL* states.Our previous work showed how this model could easily capture the Fermi surface struc-ture of the underdoped normal state. In particular, a mixing between the doubled fermion F and G species from the analog of the ‘Shraiman-Siggia’ term led to Fermi pockets whichwere centered away from the antiferromagnetic Brillouin zone boundary.Here we considered the paired electron theory, assuming a generic d -wave gap pairing ofthe cos k x − cos k y variety. Despite this simple gap structure, we found two distinct types ofelectron spectral gaps in this case, illustrated in Figs. 2 and 3. The distinction arose mainlyfrom the strength of a parameter, λ , determining the strength of the local antiferromagneticorder.For weaker local antiferromagnetic order, and with a normal state Fermi surface as inFig. 2a, the angular dependence of the gap had a strong maximum near the intermediate“hot spot” on the underlying Fermi surface. A similar structure is seen in the traditionalHartree-Fock/BCS theory of SDW and d -wave pairing on a normal Fermi liquid, and thisstructure is incompatible with existing experiments.For stronger local antiferromagnetic order, we were able to maintain the normal stateFermi surface as in Fig. 3a, but then found a gap function which had the form shown inFigs. 3c,d, which displays the ‘dichotomy’ of recent observations. Thus in this theory, it isthe fluctuating local antiferromagnetism which controls the dichotomy.Finally, we compare our theory with model proposed by Yang, Rice, and Zhang , andclosely related results of Wen and Lee . Their phenomenological form of the normal state18lectron Green’s function has qualitative similarities to ours , but there are key differencesin detail:( i ) The ‘back end’ of the YRZ hole pocket is constrained to be at ( π/ , π/ ii ) The electron spectral weight vanishes in the YRZ theory at ( π/ , π/ iii ) Our theory allows for a state with both electron and hole and pockets, while only holepockets are present in the YRZ theory.These differences can be traced to the distinct origins of the ‘pseudogap’ in the two theories.Our pseudogap has connections to local antiferromagnetism which fluctuates in orientationwhile suppressing topological defects. Pairing correlations also play an important role inthe pseudogap, but these are neglected in our present mean-field description: these wereexamined in our previous fluctuation analyses of the ACL . The YRZ pseudogap is dueto a d -wave ‘spinon pairing gap’ in a resonating valence bond spin liquid. All approacheshave a similar transition to superconductivity, with a d -wave pairing gap appearing overthe normal state spectrum, and a nodal-anti-nodal dichotomy: thus any differences in thesuperconducting state can be traced to those in the normal state.The differences between our normal state theory with bosonic spinons, and other workbased upon fermionic spinons become more pronounced when we consider a transi-tion from the normal state to a state with long-range antiferromagnetic order. In our theory,such a transition is naturally realized by condensation of bosonic spinons, with universalcharacteristics discussed earlier . Such a natural connection to the antiferromagneticallyordered state is not present in the YRZ theory. Acknowledgments We thank P. Johnson, M. Randeria, T. M. Rice, and T. Senthil for useful discussions.This research was supported by the National Science Foundation under grant DMR-0757145and by a MURI grant from AFOSR. 19 a) (b)(c) (d) k y k x 45 60 75 9000.10.2 45 60 75 9000.10.2 FIG. 6: Color online: Spectral gap functions and the Fermi surfaces for the Case II ( t = 0 . t , t = − . t , ˜ t = − . t , ˜ t = 0, ˜ t = 0, ˜ t = − . t , µ = − . t , λ = 0 . t ). Note that the onlychange from Fig. 3 is in the value of λ . (a) The spectral weight of the electron Green function withthe relaxation time τ t = 200. (b) Fermi surfaces of ε c (dashed inner (red)) , ε d (dashed outer(blue)), and the eigenmodes (thick (black)) of H . The dotted line is the magnetic zone boundary.(c) The spectral gap function with and without ∆ c . The dotted (black) line is for the normal casewith ∆ c = 0. The thick (red) line is the superconducting state with ∆ c = 0 . t . (d) The spectralgap function with and without ∆ d,X . The dotted (black) line is for the normal case. The thick(green) line has ∆ X = 0 . t . The dashed (blue) line has ∆ d = 0 . t . Appendix A: Electron pockets We consider the case with t = 0 . t , t = − . t , ˜ t = − . t , ˜ t = 0, ˜ t = 0,˜ t = − . t , µ = − . t , λ = 0 . t in Fig. 6. These parameters are as in Section III A,except that the value of λ has lowered. In other words, the ‘bare’ spectrums are the same,but electron pockets near the anti-node appear due to the low mixing term.As we can see in (a), the calculated spectral weight of the C particle shows the hole andelectron pockets with different spectral weights. We illustrate our spectral gap behavior withand without the pairing in (c) and (d). Without pairings, the normal state has the finitegapless region where the pockets exist, and there is an intermediate region peak similar to20 Θ H deg L D m i n (cid:144) t FIG. 7: Color online : Spectral gap behaviors varying with λ . The thick(red), dotted(black) anddashed(green) lines are for λ/t = 0 . , . , . 25 with the same pairing magnitude, ∆ X = 0 . t . the SDW case. With pairings, the Fermi pockets are gapped and only the node remainsgapless. The spectral gap function shows similar behavior as in our case I. In Fig. 7, spectralgap function varying the the mixing term is illustrated to see the evolution of the dip nearthe anti-node.We note that electron pockets can also appear in the YRZ formulation, but have verydifferent shapes . Appendix B: Invariant pairings There are four combinations of invariant pairing terms of the F and G : O A ∆ ( i, j ) = ε αβ ( F i,α F j,β + G i,α G j,β ) O B ∆ ( i, j ) = ε αβ ( F i,α G j,β + G i,α F j,β ) O a ∆ ( i, j ) = ε αβ ( − j x + j y ( F i,α F j,β − G i,α G j,β ) O b ∆ ( i, j ) = ε αβ ( − j x + j y ( G i,α F j,β − F i,α G j,β ) (B1)In Table II, we illustrate the transformation of various pairing terms. The four pairingshavean interesting exchange symmetry. Obviously O A,B ∆ have even under the exchange oper-ation. If we consider nearest neighbor sites,( i, j ), it is easy to show that O b ∆ is even and O a ∆ is odd under the exchange. Therefore, for the d x − y symmetry, the O a does not contribute21 x R dual π/ I dual x T ε αβ F α F β ε αβ G α G β ε αβ G α G β ε αβ G α G β - ε αβ F † β F † α ε αβ G α G β ε αβ F α F β ε αβ F α F β ε αβ F α F β - ε αβ G † β G † α ε αβ F α G β ε αβ G α F β ε αβ G α F β ε αβ G α F β - ε αβ G † β F † α ε αβ G α F β ε αβ F α G β ε αβ F α G β ε αβ F α G β - ε αβ F † β G † α TABLE II: Transformations of the pairing terms. We suppress the lattice index( i, j ) before and aftertransformations. Note that the Time Reversal column ( T ) contains ( − ) term and the conjugatepartner also have the ( − ) sign. to pairings.The conversion between the two representations are as follows: O c ∆ ( i, j ) = ε αβ C i,α C j,β = 12 ( O A ∆ + O B ∆ )( i, j ) O d ∆ ( i, j ) = ε αβ D i,α D j,β = ( − ∆ x +∆ y O A ∆ − O B ∆ )( i, j ) O cd ∆ ( i, j ) = ε αβ C i,α D j,β = 12 ( O a ∆ + O b ∆ )( i, j ) O dc ∆ ( i, j ) = ε αβ D i,α C j,β = ( − ∆ x +∆ y O a ∆ − O b ∆ )( i, j ) , (B2)where ∆ x + ∆ y is coordinates’ difference between two particles, for example, zero for the s wave and one for the d wave. Appendix C: Pairing Instability