PHYSICAL REVIEW B 86, 064527 (2012)

Electron-like Fermi surface and in-plane anisotropy due to chain statesin YBa2Cu3O7−δ superconductors

Tanmoy DasTheoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

(Received 12 May 2012; revised manuscript received 21 August 2012; published 29 August 2012)

We present magnetotransport calculations for YBa2Cu3O7−δ (YBCO) materials to show that the electron-likemetallic chain state gives both the negative Hall effect and in-plane anisotropic large Nernst signal. We show thatthe inevitable presence of the metallic 1D CuO chain layer lying between the CuO2 bilayers in YBCO rendersan electron-like Fermi surface in the doping range as wide as p = 0.05 to overdoping. With underdoping, apseudogap opening in the CuO2 state reduces its hole-carrier contribution, and, therefore, the net electron-likequasiparticles dominate the transport properties, and a negative Hall resistance commences. We also show that theobservation of in-plane anisotropy in the Nernst signal—which was taken as a definite evidence of the electronic“nematic” pseudogap phase—is naturally explained by including the “quasiuniaxial” metallic chain state. Finally,we comment on how the chain state can also lead to electron-like quantum oscillations.

DOI: 10.1103/PhysRevB.86.064527 PACS number(s): 71.10.Hf, 71.18.+y, 74.25.F−, 74.72.Kf

I. INTRODUCTION

An understanding of the pairing mechanism in cupratesuperconductors relies on the underlying complex Fermisurface (FS) properties and how they are derived from thenormal-state pseudogap (PG) phase. Important clues to theorigin of the PG state have been put forward by recentHall effect measurements which observed a negative Hallresistance below the PG temperature in YBCO systems.1 Thisresult received further supports from the negative Seebeckcoefficient,2,3 strongly enhanced Nernst effect,4,5 as wellas identification of quantum oscillations6–10 with cyclotronfrequency indicative of electron-like FS. These stimulatingdiscoveries, which, according to the conventional Onsager-Lifshitz paradigm11 implies a closed electron pocket, pointtoward the topological changes in the FS in the PG statewhen compared with its large metallic FS in the overdopedsample.12 Such small electron pockets do not arise naturallyfrom the band-structure calculations considering the CuO2

planes and their total area is inconsistent with the nominaldoping concentration.13

To explain these results, candidate proposals for varioussymmetry-breaking patterns have been offered, leading todrastic FS reconstruction into multiple Fermi pockets ofdifferent natures.14–21 We would, however, expect to seesignatures of these pockets in spectroscopies such as angle-resolved photoemission spectroscopy (ARPES) and scanningtunneling microscopy/spectroscopy (STM/S). Yet, extensiveARPES12,22 and STM23 studies on various cuprate materialsconsistently have so far been able to map out only oneFermi pocket or “Fermi arc”—both would represent thesame FS segment if the intensity on the pocket’s backside is suppressed by the coherence factors—which is againof holelike character.16,17 Furthermore, to explain the largenegative Hall coefficient, one requires to have electron-likequasiparticles dominating over the hole quasiparticles onthe FS, which is prohibited in hole-doped system accordingto the Luttinger theorem. On the basis of this reasoning,the recent experimental evidence of a field-induced chargeordering18 will also find difficulty in explaining the negativeHall coefficient, even if it is assumed to induce an electron

pocket.19 Taken together, the electron-pocket scenarios ofvarious density wave origins14,16,17,19 are inadequate to explainthe negative slope of the magnetotransport data.

In this study, we introduce a fundamentally newperspective—we show that the emergence of the electron-like FS in the PG state can be rationalized by realisticallyconsidering the contributions of the much ignored metallicCuO chain bands in YBCO. The unambiguous existence ofthe chain states on the FS has been well established bymany ARPES studies from an extreme underdoped region(p = 0.05) to overdoping (p = 0.29 and above).12,22 Its contri-bution to c-axis transport24 and quantum oscillation in specificheat25 measurements is also studied earlier, further supportingthe metallic character of chain state. The chain states seem toexhibit neither the PG opening nor any considerable dopingdependence. Nevertheless, it dominates in the electronicspectra in the PG state due to its interplay with the CuO2

states. Our underlying reasoning for this conclusion is thatthe PG opens in the CuO2 bilayers, truncating its full metallicFS into small segments. Irrespective of the specific origin andnature of the small FS, it contains much reduced hole-carrierconcentrations, determining the nominal hole doping of thesample. Therefore, the low-energy electronic state becomeseffectively characterized by electron-like charge carriers ofthe chain state. Above the PG regime both in temperature (T )and doping (p), the large CuO2 state is recovered, restoring thepredominant hole carrier on the FS of YBCO. The resultingtwo-carrier FSs quantify the negative Hall effect1 and theenhanced Nernst signal with prominent in-plane anisotropy4,5

due to the “quasiuniaxial” character of the chain state.We supplement these arguments by an exact analytical

diagonalization of a trilayer quantum lattice model26 andsubsequently confirm their validity by performing numericalcomputation of the Hall coefficient and Nernst effects. In whatfollows we advance two principal ideas, appended by potentialpostulates to reconcile several other experimental observationsof YBCO within a consistent picture. First, we demonstratethat the CuO chain states are predominantly electron-like.Due to its uniaxial dispersive feature, it does not contributeto the Luttinger counting of the doping concentration from

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TANMOY DAS PHYSICAL REVIEW B 86, 064527 (2012)

the area of the FSs. Therefore, by assuming either a residualspin-17 or d-density wave16 for the formation of a hole pocket,we accurately formulate the suppression of the hole-carrierdensity at the FS (with proper counting of hole doping),while retaining its shape and position in accord with ARPESand STM results. The resulting two-carrier FSs quantify thenegative Hall effect.1 Second, the observed in-plane anisotropyin the behavior of the Nernst signal has been taken as anevidence of a discrete rotational symmetry breaking electronic“nematic” liquid crystal phase for the origin of the PG state.We show that even in the absence of any spontaneous rotationalsymmetry breaking electronic state,4,5 the two-carrier FSscan consistently explain the enhanced Nernst signal at thePG state, while the accompanying anisotropy arises naturallyfrom the “quasiuniaxial” CuO chain states. Finally, we presentnumerous justifications that the present formalism can alsoexplain other similar phenomena such as negative Seebeckcoefficients,2,3 nearly doping independent cyclotron mass—proportional to the weakly doping dependent chain states—probed by quantum oscillations,9,10 and in-plane anisotropyobserved in the spin dynamics measured by inelastic neutronscattering in YBCO.27 The results are applicable to doublechain layered YBa2Cu4O8+δ (Y124) sample in which the chainstate is also metallic.13

The rest of the paper is organized as follows: In Sec. II weprovide the tight-binding model for two CuO2 planes and oneCuO chain state, including a pseudogap ordering in the plane.The Hall effect and Nernst effect calculations are given inSec. III. The conclusions are given in Sec. IV. Appendices Aand B give details of the tight-binding model and transportcalculations, respectively.

II. TRILAYER LATTICE MODEL

Encouraged by the ARPES results12,22 and the first-principle calculations,13 we formulate a unified noninteractingthree-band model, originating from 2D CuO2 bilayers and an1D CuO chain.26,28 To effectively model the reminiscent nodalFS segment of the CuO2 bands in the PG state, we considersome form of density-wave order whose front side would alsoaccommodate the “Fermi arc” property. Constrained by thisassumption, a Hartree-Fock-like order parameter is introducedto the Hamiltonian with a periodic modulation Q = (π,π ),which is commensurate with the CuO2 lattice plane. We canthink of a residual spin-17 or charge-density wave18 (S/CDW)or a d-density wave16 or some other commensurate FSinstability as the microscopic origin of such ordering; however,the macroscopic properties we aim to describe here do notrely on these details.17 On the basis of these experimentalconsiderations, we deduce the general form of the Hamiltonianin the Nambu representation per unit cell,

H =∑kσ,ij

[ξij kc†ikσ cj kσ + Vij kc

†i(k+ Q)σ cj kσ + H.c.], (1)

where c†ikσ (cikσ ) creates (annihilates) an electron with momen-

tum k and spin σ on the i = p,p′,c trilayers. The singe-particledispersions ξij are defined within tight-binding expansionof the Cu dx2−y2 orbitals (hybridized intrinsically with O p

orbitals) hopping on a periodic lattice in the tetragonal crystal

FIG. 1. (Color online) (a) Paramagnetic dispersion obtained fromthe eigenstates of the Hamiltonian in Eq. (1). Dark (blue) lines arethe bonding and antibonding bands arising from the CuO2 planes,while the light (magenta) line depicts the chain state. The shadings ofsame colors represent the holelike and electron-like character of thebands, respectively. (b) Corresponding computed FSs are comparedwith the ARPES data12 (plotted only in one quadrant of the Brillouinzone for visualization). (c) Same as in (a) but computed in thePG state in an underdoped region. The width of each line givesthe associated spectral weight, determined by the spin density wave(SDW) coherence factors. (d) FSs in the PG state host only tiny holepockets centering nodal points.

as given by

ξpp = −2t(φx + φy) − 4t ′φxφy − 2t ′′(φ2x + φ2y) − μp,

ξcc = −2tcφy − μc, (2)

ξpp′ = −2tpp′ (φx − φy)φz/4,ξcp = −2tcpφz/4,

where φαν = cos (αkν) with ν = x,y,z. μp,c are the chemicalpotential for the plane and chain states, respectively, whichencode relative onsite energy differences and other crystaleffects between the two levels. t , t ′, and t ′′ are the first, second,and third nearest-neighbor (NN) hopping matrix elementson the 2D CuO2 plane, tc is the NN hopping on the 1Dchain aligned along y axis, and tpp′(tcp) is the plane-plane(chain-plane) tunneling matrix element along the c axis. Webenchmark the values of the hopping integrals by fittingthe eigenstates of the noninteracting part of the Hamiltonian(setting V = 0) to the experimental FSs [see Fig. 1(a)] andto the first-principles deduced dispersions of Ref. 13. Thechemical potentials are evaluated self-consistently to maintainthe doping concentration, p, of the CuO2 bands.

Next we exemplify how a small hole pocket forms in thePG state. To simplify the analysis, while retaining the salientspectroscopic features, we adopt a k-independent interactionpotential V to be same for both bilayers states, while settingV = 0 for the chain layer, as the latter do not exhibit any PGopening. We set the spin index σ = −σ to construct a SDWmodulation. In this case, V encodes the onsite Hubbard U , anda order parameter S, as Vi = USi , with Si = 〈c†i(k+ Q)σ cikσ 〉(i = p,p′), where the canonical average is taken over the entireBrillouin zone. We use the value of effective U = 1.59 eV,deduced from earlier calculations,17,29 and the order parameter

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S is evaluated self-consistent at each T and doping. SeeAppendix A for details. We emphasize that our choice of aparticular form of density wave does not preclude us fromanalyzing the full problem and, rather, improves the reliabilityof the results with respect to a single parameter U . The choiceof Q = (π,π ) order, however, gives the FS topology whichagrees well with most ARPES data.

In Figs. 1(c) and 1(d), we notice that a direct band gap isopened at EF everywhere in the Brillouin zone, except at thenodal points, giving rise to a hole pocket in the CuO2 states.The width of each line depicts the associated quasiparticleweight, which is in detailed agreement with experiments, giventhat the weak intensity of the shadow bands can be detected byARPES.30,31 It is important to note that while the parametersof the chain states are doping independent, its electronic statesposses a weak doping evolution via the interlayer coupling tothe plane states. It is self-evident from the band curvature ofthe chain states at EF that it continues to host electron-likeFS (although open orbit) at all doping; see Fig. 2 for furtherillustration. The electron-like property of the chain state is alsoevident in the ARPES data of Ref. 12.

III. HALL AND NERNST EFFECTS

Relying on the aforementioned effective normal-stateHamiltonian for YBCO, we now proceed to study thequasiparticle transport properties based on a semiclassicalnumerical calculation. Nernst experiment measures the trans-verse electric field, E, in response to a combination setupof an externally imposed temperature gradient, ∇T , andan orthogonal magnetic field, B.4,5 The net electric currentdensity J produced via this effect is essentially related to Eand ∇T via linear response in-plane Hall conductivity tensorσ and Nernst conductivity tensor α as J = σ E − α∇T . Incase of a perfect cancellation of the net charge current, weobtain a working formula for these two measured quantities:

E = σ−1α∇T = −θ∇T . (3)

Experimentally, the Hall resistance, ρ=σ−1, and the Nernstcoefficient, ν = θ/B, are measured independently via trans-port probes typically by setting a weak magnetic field B = Bz

along the c axis of the lattice.1,4

In theory, σ and α are calculated by solving the standardBoltzmann equations for the low-T DC transport.20,21 Aftersome tedious algebra we derive the formalism for these quan-tities in terms of the quasiparticle states and their associatedweight (see Appendix B),

σ ijμν = βe2

2

∑k,n

Mij

nkvμ

nk�−1nk vν

nksech2

(βEnk

2

), (4)

αijμν = −β2e

2

∑k,n

Mij

nkvμ

nk�−1nk vν

nkEnksech2

(βEnk

2

). (5)

Here Enk stands for the quasiparticle band with wave vector kand band index n of the Hamiltonian in Eq. (1). Denoting thecorresponding Bloch eigenstate by ψi

nk, we express the trans-

port matrix element Mij

nk = ψinkψ

j†nk. M essentially projects the

transport tensors from the Bloch quasiparticle representation tothe trilayer lattice indices i,j . v

μ

nk = ∂Enk/h∂kμ is the quasi-particle velocity along the μ = x,y,z axes, and β = 1/kBT ,

FIG. 2. (Color online) [(a)–(d)] 2D color plots of the total“quasi-charge-weight” Cij (k,EF ) (averaged over all layers) plotted atfour representative cases. The light-white-dark (red-white-blue) colorscheme gives the electron-like quasiparticle to no quasiparticle toholelike quasiparticle states at a given k on the FS. (a) “Quasi-charge-weight” calculated at an extreme low-T region for doping p = 0.10,where the Hall coefficient, RH , is negative and Nernst coefficient,ν/T , is small, demonstrating the dominance of the electron-likequasiparticle state on the FS. (b) Chosen at an intermediate T where“quasielectron” and “quasihole” weight compensate each other tolead to zero Hall coefficient and enhanced Nernst signal. At a higherT in (c) SDW disappears and the “quasihole” weight dominates.Here Hall and Nernst coefficients flip sign. The plot in (d) issimilar to that in (a), but at a higher doping, p = 0.14. It shouldbe noted here that, unlike in a strict ambipolar state, the maximumof the Nernst signal shifts from the zero of Hall coefficient, dueto overlapping FS topology and matrix-element effects. (e) Thecomputed T evolution of the Hall coefficient, RH , is comparedwith corresponding experimental data1 at two representative dopingsof YBCO. The excellent agreement below the SDW temperaturecan be clearly marked. (f) Corresponding computed T evolutionof the Nernst coefficient, νa = νxy and νb = νyx , at same T as in(e). The experimental data for YBCO (shown in inset) is availableat a slightly different doping.5 Despite a characteristically goodagreement between theory and experiments, we expect discrepanciesdue to the relaxation time approximation as mentioned in the text.Results of νa and νb reveal a large anisotropy, which is T dependent.The source is this anisotropy is the presence of the “quasiuniaxial”chain state, lying along the y axis of the crystal. The anisotropyis strongest when the Nernst signal attains its maximum, in goodagreement with experimental data taken from Ref. 5.

where kB is Boltzmann constant. The differential operatorthat drives the system into a nonequilibrium state in re-sponse to the applied field is evaluated within relaxation-timeapproximation,32 �nk = [− e

hc(vnk × B) · ∇k + τ−1

k ], wherethe relaxation time is taken to be constant both in k and T 33

and e, h, and c have their usual meanings.The accurate evaluation of the T dependence of the

transport properties requires an experimentally constrainedT -dependent gap parameters, in other words, an accuratedescription of the T evolution of the FS topology. It is wellknown that a mean-field order parameter overestimates thePG transition temperature.17,20 To overcome this difficulty,

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we adopt a phenomenological form of the T dependence ofthe gap parameter V (T )=V

√1 − T/Ts ,20,34 with the onset T

for the density wave order is set to be Ts ≈ 55 K, in closeagreement with Neutron measurement for Nd-doped La-basedcuprate.35 However, at each T , the Fermi level is evaluatedself-consistently to preserve the hole counting.

A. Results

Figure 2 displays the T evolution of the Hall coefficientRH = ρyx/B and Nernst coefficient νyx/T at two dopingsat which the normal state Hall coefficient data are available(the value of B is scaled out in the presented results). Avisual comparison between theory and experiment revealsa systematic agreement between them. To relate the signand magnitude of RH and νyx/T to their correspondingcharge carrier features, we introduce a function called “qua-sicharge weight,” in analogy with the quasiparticle weight,Cij (k,ω = EF ) = ∑

n,k enkψinkψ

j†nkδ(ω − Enk), where enk =

±e at electron- and hole-like kF obtained from the slope ofthe bands at EF , respectively, for nth band. We emphasizethat, in the orthorhombic real-space unit cell, the Hall currentarises due to finite tunneling between two adjacent chain layers(which gives a warped chain FS in the k-space). We plot thetotal the ‘quasi-charge weight’ in Figs. 2(a) to 2(d) for severalrepresentative cases where the Hall coefficient is negative,zero, and positive as a function of T and doping.

The obtained results confirm our aforementioned postulatesthat at low T when the strong PG shrinks the CuO2 statesinto small hole pockets, the dominant chain states producea net electron-like quasiparticle character on the FS. Withincreasing T when the strength of the PG reduces, the areaof the hole pocket gradually increases, while the chain stateremains very much unperturbed (note that with increasinghole-pocket area, the self-consistency allows the associatedcoherence factors to adjust themselves to maintain the samedoping concentrations). At some intermediate temperature,Tm, the “quasihole” and “quasielectron” weights exactly canceleach other to make RH vanish; see Fig. 1(b). At T > Tm, thelarger hole pocket reverses the sign of the Hall resistance, inFig. 1(e). Finally, above the SDW temperature, the metallicstate is restored and the Hall coefficient becomes featureless.The same analysis applies to the higher doping result inthat doping reduces the PG order, rendering a smaller holepocket and, hence, less transport contribution. The results arein excellent agreement with the corresponding experimentaldata,1 plotted with symbols in Figs. 2(e) and 2(f).

The discussion of Nernst effect proceeds similarly, althoughthe fundamental mechanism for the enhanced Nernst signaldiffers somewhat. Experimentally, it is well established thatthe Nernst signal changes sign from negative to positive as wellas it becomes abruptly enhanced below the PG temperature,but much above the superconducting transition temperature,Tc,4,5 see the inset to Fig. 2(f). Despite some proposalsthat there exist vortex-type excitations even above Tc whichcan drive the Nernst signal,36 the widely used ambipolarphenomena suggests that the compensating electron-like andholelike FSs are primarily necessary to obtain an enhanced“normal state” Nernst signal.20,21,37 Within the present theory,such ambipolar state naturally evolves in the vicinity of

the intermediate temperature Tm, where “quasihole” and“quasi-electron weights” become comparable, and a positivemaximal of Nernst signal arises, as shown in Fig. 2(b).Quantum-oscillation measurement has also detected similarcompensating electron- and holelike FSs in YBCO.8 Awayfrom this T region, the positive Nernst signal diminishesas the underlying states depart from predominant ambipolarcharacter, that is, when only one type of “quasi-charge weight”dominates the excitation spectrum. Interestingly, even whenthe metallic state appears above the onset temperature of thedensity wave order, the so-called Sondheimer cancellationof the Nernst signal in a metallic state is still not strictlyapplicable here due to the coexistence of two-carrier chargeson different bands.38 In fact, the finite experimental value ofthe Nernst signal above Ts , which remained an unexplainedpuzzle in the existing theories, is surprisingly well reproducedwithin the present theory. The results capture the salientexperimental features of YBCO data available at a slightlydifferent doping,5,39 although some discrepancies are clearlyvisible. The adopted relaxation time approximation imposeslimitations to quantitatively map out the detailed shape andpeak positions of the Nernst signal. Furthermore, the presentlinear-response theory is mainly applicable to the low magneticfield region; on the other hand, experimentally, it requires asufficiently high field to expose the “normal state” Nernstsignal. Despite these approximations, the Hall and Nernstanomalies are robust features in YBCO, tied to the underlyingelectronic structure that is confirmed by ARPES12,22 andfirst-principles results.13

B. In-plane anisotropy

The added benefit of the present theory is that the existenceof the the quasi-1D chain state naturally explains the associatedin-plane anisotropy in the Nernst signal. For the 1D statelying along the (100) direction, the corresponding quasiparticlevelocity is vx � vy , which leads to a similar anisotropy inthe Nernst conductivity α defined in Eq. (5) and, hence, in theNernst coefficient ν/T , plotted in Fig. 2(f). Most importantly,the in-plane anisotropy continues to persist even above wherethe PG phenomena disappears, both in experiment and theory,which adds confidence to our conclusion that the PG phasein YBCO is unlikely to arise from any spontaneous rotationalsymmetry-breaking ordering. Furthermore, the metallic chainstates have been demonstrated by ARPES data12 to be presentfor a doping range as large as p = 0.05 to p = 0.29 (andabove), and all the existing evidence of in-plane anisotropyvia transport4,40 and neutron scattering measurements27 areobtained within this doping range. This clearly indicates thatthe presence of a chain state is responsible for the observedin-plane anisotropy in this system. It is worthwhile mentioningthat most of the evidence for the electronic nematic phase incuprate is obtained in the YBCO family only,4,27,40 with someindication of it is presented in an analysis of the STM data ina Ba-based compound.41

IV. DISCUSSIONS AND CONCLUSIONS

To further support our postulates, we refer to severalother experimental facts where the present mechanism can

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offer a consistent explanation, at least in principle. (1) Animportant finding of the quantum oscillation measurements isthat the cyclotron frequency of the election-like FS—whichis proportional to the quasiparticle density at EF —is verymuch doping independent in the optimal doping region.9,10 Inthis context, a phenomenological model can be recalled whichhas shown that quantum oscillation can arise from open-orbit“Fermi arc,” by taking advantage of the electron-hole mixingof the Cooper pairs.42 This argument finds support froman independent experimental consideration.43 Based on thismechanism, the electron and hole quasiparticles, residing ondifferent parts of the FS in our present model, can also giverise to a quantum oscillation. If so, the doping-independentcyclotron frequency arising from the weakly doping dependentchain state can naturally explain the experimental observationof the cyclotron mass. Another explanation follows in that theopen-orbit chain state becomes “warped FS” and can createclosed FS pockets if the orthorhombic II unit cell is chosen,which, within the conventional framework, will give rise toquantum oscillation. (2) A second important result of recentquantum oscillations studies is that the oscillation disappearsat a critical value of the doping for YBCO, and the cyclotronmass diverges as the critical value is approached from thehigh doping side. Millis et al.15 have argued that the massdivergence is related to a Lifshitz transition where the multipleFS pockets connect to form an open (quasi-1D) FS. Ourargument follows similarly, where we expect that, in extremeunderdoping, the CuO2 states become vanishingly small, andthe residual 1D chain states effectively cause a logarithmicdivergence of the cyclotron mass. A rigorous calculation ofthe quantum oscillation would be of considerable benefit toconfirm our predictions. (3) Finally, we see no fundamentalerror in postulating that the “electron-like” and “quasiuniaxial”chain FS can also explain the negative Seebeck coefficientand thermopower signal,2,3 induced in-plane anisotropy in thespin-susceptibility measured by inelastic neutron scatteringstudy.27,44 Finally, we mention that since the chain stateremains mettalic in double chain layered Y124,13 the abovepostulates continue to hold in this system, which is consistentwith the observation of quantum oscillation in Y124.45

Our realistic framework narrows the range of possible theo-retical models for the PG state in cuprates. It strongly suggeststhat the microscopic route toward understanding the natureof the PG physics may be generic to all cuprates containinga common CuO2 plane. Most of the other accompanyingnormal-state properties then can be explained as added featurescoming from the material specific crystallographic differences,such as apical oxygens surrounding Cu atoms on the CuO2

planes in La- and Bi-based superconductors, chain layersin YBCO, chemically active Cl ions in Ca2−xNaxCuO2Cl2,among others.

Finally, we argue that recent observations of charge densitywave (CDW) do not alter our results. The reasons are asfollows. (1) The CDW has so far been observed in monolayerBi-based cuprate46 and YBCO18,47 and its onset temperatureand doping differ substantially from the PG temperature T ∗.On the other hand, the psuedogap phenomena and similarFS properties are obtained in all hole-doped cuprates. (2) Atheoretical study19 shows that a biaxial CDW-induced nodal FSpocket is electron-like. According to the definition of electron

pocket, it is associated with a band bottom or band foldingbelow EF , implying a gap opening along the nodal directionbelow EF . But ARPES evidence for a gap along this lineis still far from definitive. Based on these facts, we arguethat CDW is a secondary or weak material-specific feature,while the PG has more fundamental and generic origin. (3)As mentioned earlier, irrespective of any origin of electronpockets in the plane state, the hole-carrier concentration isrequired to overcome the electron-carrier concentrations inthis hole-doped cuprates, and, thus, obtaining a large negativeHall effect in this system is unlikely unless an electron-likechain state is included.

ACKNOWLEDGMENTS

The work was supported by the US DOE through the Officeof Science (BES) and the LDRD Program and benefited fromNERSC computing allocation.

APPENDIX A: EIGENSTATES OF THE TRILAYERLATTICE MODEL

The unit-cell contains two chain layers and two CuO2

plane layers.44 In order to obtain a minimum low-energymodel, we assume that the interlayer electron tunneling isactive only between the the two nearest-neighbor layers. Indoing so, we are left with a trilayer system which consistsof two CuO2 planes and one CuO chain, while the periodicboundary conditions on infinite lattice is imposed along allthree dimensions. Furthermore, the chain state does not exhibitany gap opening through out the doping range of YBCO. Theplane states undergo FS reconstruction which can be capturedwithin a commensurate spin-density wave with modulationvector Q = (π,π ). The commensurate modulation doublesthe unit cell in real space. Using the standard Nambu notation,we define the trilayer eigenfunction in the magnetic zoneas �

†k = [c†pk↑,c

†p′k↑,c

†ck↑,c

†p(k+ Q)↓,c

†p′(k+ Q)↓,c

†c(k+ Q)↓]. In this

notation, the Hamiltonian presented in the main text in Eq. (1)becomes a 6×6 matrix H = ∑

k �†kHk�k,

Hk =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

ξppk ξpp′k ξcpk Vppk 0 0

ξpp′k ξppk ξcpk 0 Vp′p′k 0

ξcpk ξcpk ξcck 0 0 0

Vppk 0 0 ξpp(k+ Q) ξpp′(k+ Q) ξcp(k+ Q)

0 Vp′p′k 0 ξpp′(k+ Q) ξpp(k+ Q) ξcp(k+ Q)

0 0 0 ξcp(k+ Q) ξcp(k+ Q) ξcc(k+ Q)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

(A1)

In obtaining the noninteracting dispersions ξij k, we assumethat the Cu dx2−y2 orbital contributes to the low-energy scalesof present interest (as commonly used for all cuprates).The Cu dx2−y2 electrons hop to their neighbors via O px,y

orbitals which can be taken into account within an effectivetight-binding formalism.26,48 The hoping terms are depicted inFig. 3, which gives

ξppk = −2t(φx + φy) − 4t ′φxφy − 2t ′′(φ2x + φ2y) − μp,

ξcck = −2tcφy − μc, (A2)

ξpp′k = −2tpp′ (φx − φy)φz/4,ξcp = −2tcpφz/4,

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FIG. 3. (Color online) (a) Schematic phase diagram of cuprate.(b) The doping dependence of the self-consistently evaluated orderparameter S (defined in the text).

where φαμ = cos (αkμ) with μ = x,y,z. μp,c are the chemicalpotential for the plane and chain states, respectively, whichencode relative onsite energy differences and other crystaleffects between the two levels. t , t ′, and t ′′ are the first, second,and third NN hopping on the 2D CuO2 plane, tc is the NNhopping on the 1D chain aligned along y axis, and tpp′(tcp) isthe plane-plane (chain-plane) hopping along the z direction.

The eigenvalues for the upper 3 × 3 interblock matrix isobtained as

ECk = ε+k + E0k, (A3)

EAk = ε+k − E0k, (A4)

EBk = χ−k . (A5)

Here the subscripts A and B denote the antibonding andbonding states created by the bi-CuO2 layers, whereas C

denotes the chain state. ε±k = (χ+

k ± ξcck)/2, χ±k = ξppk ±

ξpp′k, and E0k =√

(ε−k )2 + 2ξ 2

cpk. Of course, the weight of each of thelayers are mixed between each band, which can be determinedby their corresponding eigenvectors: u±

k = 12 [ε−

k ± E0k]/ξcpk.The eigenvalues of the full Hamiltonian can now be expressedas two split bands of the above three bands gapped by SDWorder parameter Vij :16,17

E±Ck = ξ+

Ck ± ξ−Ck, (A6)

E±Ak = ξ+

Ak ±√

(−ξAk)2 + V 2A, (A7)

E±Bk = ξ+

Bk ±√

(ξ−Bk)2 + V 2

B, (A8)

where ξ±ik = (EAk ± EAk)/2 and the self-consistent order

parameters Vi are given below. The eigenvectors for the finaleigenstates given above can be represented in terms of SDW

coherence factors,

α2ik = 1

2

[1 + ξ−

ik/

√(ξ−

ik)2 + V 2i

],

(A9)β2

ik = 1 − α2ik.

Here i = A,B,C. Due to the absence of any gap openingin the chain state, α2

Ck = 1 and β2Ck = 0 at all momentum.

With the definitions of the eigenstates and the coherencefactors, we construct the final unitary matrix Uk that ultimatelydiagonalizes the total Hamiltonian given above in Eq.

Uk =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

u+k u−

k αAk −αBk 0 −u−k βAk βBk

u−k αAk u+

k αAk αBk 0 −u+k βAk −βBk

−αBk αBk 0 0 −βBk 0

0 0 0 0 0 0

u−k βAk u+

k βAk βBk 0 u+k αAk αBk

−βBk βBk 0 0 αBk 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

(A10)

The self-consistent order parameters are defined as VA =USA and VB = USB (and VC = 0), where U is the onsiteHubbard interaction, which is kept the same for both CuO2

planes. SA/B are the two order parameters and p is the dopingconcentration, which is evaluated by solving the following twoself-consistent equations:

p = 2 − 1

N

∑k,i=1,2,σ

〈c†ikσ cikσ 〉

= 2 − 1

N

∑k,i

UiikU†iikf (Eik), (A11)

Si = 1

2(ni↑ − ni↓)

= 1

2N

∑k

[〈c†i(k+ Q)↑cik↑〉 − 〈c†i(k+ Q)↓cik↓〉]

= 1

2N

∑k,j

Uij kU†jikf (Ej k), (A12)

where f is the Fermi function and Ej are the eigenstateslisted in Eqs. (A7) and (A8) where indices i = A,B and j =1–6. ni↑ is the number of up-spin on the i th layer defined as〈c†i(k+ Q)↑cik↑〉, where the thermal average is taken over the

whole Brillouin zone.

1. Tight-binding parameters and self-consistentorder parameters

We find that the tight-binding parameters that give a gooddescription of the FS, in agreement with ARPES and thefirst-principles band structure, are26 (t,t ′,t ′′,t ′′,tc,tpp′ ,tcp) =(0.38,0.0684,0.095,0.25, − 0.01, − 0.0075) eV. We havekept μc = −0.87 eV to be doping independent, while μp iscomputed self-consistently. The band-structure fitting is doneat the overdoped sample of p = 0.29, shown in Fig. 1 in themain text.

The calculation of self-consistency is carried out via thefollowing steps. At a given doping p and onsite HubbardU = 1.59 eV, we start our self-consistent cycle with an initial

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ELECTRON-LIKE FERMI SURFACE AND IN-PLANE . . . PHYSICAL REVIEW B 86, 064527 (2012)

guess of chemical potential μp and order parameters SA/B

and calculate doping p and SA/B by solving Eqs. (1) to (12)at T = 0. We iterate the entire loop until the self-consistentvalues of doping p and order parameters SA/B converge. Forthe overdoped sample of p = 0.29, we get SA/B = 0 and μp =−0.54 eV. At an extreme underdoped samples of p = 0.05at which ARPES data are available for fitting in Fig. 1, weget SA(T = 0) ≈ SB(T = 0) = 0.3145 and μp = −0.4. Attwo dopings at which Fig. 2 is obtained in the main text,we find SA(T = 0) ≈ SB(T = 0) = 0.25 and μp = −0.41 atT = 0 to −0.43 eV at T = 55 K for p = 0.10, and SA(T =0) ≈ SB(T = 0) = 0.22 and μp = −0.38 at T = 0 to −0.41eV at T = 55 K for p = 0.14. The results are plotted insupplementary Fig. 2(b) and compared with the schematicphase diagram of YBCO in Fig. 1(a). Note that the temperaturedependence of the order parameters is phenomenologicallyassumed as Si = Si(0)

√1 − T/Ts , where Ts = 55 K is the

SDW transition temperature.20,35

APPENDIX B: DETAILS OF HALL AND NERNSTCOEFFICIENT CALCULATIONS

The Nernst effect experiments measure the transverseelectric field response of a system to a combination setup of anexternally imposed temperature gradient and an orthogonalmagnetic field. The Nernst effect depends sensitively onanisotropies in the band structure. Similarly, the Hall effectis a powerful probe of the Fermi surface of a metal because ofits sensitivity to the sign of charge carriers, which distinguishesbetween electrons and holes. These two probes have recentlygained much attention in the context of cuprates, since theydemonstrate very unusual behavior in the underdoped regionwhen the pseudogap sets in. Apart from the superconductingfluctuation origin of enhanced Nernst effect in cuprates,36

purely quasiparticle picture is extensively proposed in variouscontexts.20,21,37 The quasiparticle Nernst effect has beenstudied on the basis of the linearized Boltzmann equation in therelaxation-time approximation. This is a reliable approach ifthe scattering rate is isotropic, since the neglected “scattering-in” contributions then average out to zero.

We start from the semiclassical Boltzmann transport equa-tion for quasiparticles (charge e) in a weak magnetic field B,driven out of equilibrium by a spatially uniform electric fieldE and temperature gradient ∇T . If we linearize the deviationof the fermion distribution function from its equilibriumstate f (ξk) in both E and ∇T as f ′(ξk) = f + gk, then theBoltzmann equation49 becomes

Vk

(−∂f (ξk)

∂ξk

)− �0

kgk =∑

k′Qk,k′(gk − gk′), (B1)

Vk = vk ·(

−eE − ξk∇T

T

), (B2)

�k = − e

hc(vk × B) · ∇k. (B3)

Here ξk is the quasiparticle state whose velocity is vk =h−1∇kξk. The band index and the eigenvectors are implicitto the quasiparticle states in the above equations. Qk,k′ is thescattering matrix element for the incoming state

∑k′ Qk,k′gk′

and the outgoing state∑

k′ Qk,k′gk. For the present DC

transport property, we assume the elastic scattering withQk,k′ = δ(ξk − ξk′)qk,k′ . Furthermore, we impose the standardrelaxation-time approximation,32 which implies that the sys-tem has enough time to loose its memory, i.e., the scattering-interm is negligible. In that case, the scattering rate is governedentirely by the out-going state as gk

τk= gk

∑k′ Qk,k′ , while

relating its momentum dependence,

1

τk=

∑k′

Qk,k′ = N0〈qk,k′ 〉FS. (B4)

Within this approximation, Eqs. (B1)–(B3) become

gk = �−1k Vk

[−∂f (ξk)

∂ξk

], (B5)

Vk = vk ·(

−eE − ξk∇T

T

), (B6)

�k = −[

e

hc(vk × B) · ∇k + 1

τk

]. (B7)

It is now convenient to introduce the band index n,

gnk = �−1nk Vnk

[−∂f (ξnk)

∂ξnk

], (B8)

From Eq. (B8), the net electrical, J , and thermal currentdensities, Q, that flow on the i th layer (or orbital) can becalculated from the diagonal term of the following matrix:

J ij = −2e∑k,n

Mijn vnkgnk, (B9)

Qij = 2∑k,n

Mijn vnkξnkgnk, (B10)

where M is the matrix element

Mijn (k) = ψi

n(k)ψj†n (k). (B11)

In Eq. (B9), we project the densities from its band basis n [aband index is implicit on both sides of Eq. (B6)] to the layerindices, i,j . The factor 2 is introduced in Eqs. (B9) and (B10)to count spin degeneracy.

1. Linear response theory

The interplay of electrical and thermal effects necessarilyimplies two conductivity tensors σ , α, which relate chargecurrent, J , to electric field, E, and thermal gradient, ∇T ,vectors,

J ij = σ ij E − αij∇T . (B12)

The Nernst response is defined as the electrical field inducedby a thermal gradient in the absence of an electrical currentand is given in linear response by the relation E = −θ∇T . Inabsence of charge current (i.e., when J = 0), Eq. (6) yields

E = (σ ij )−1αij∇T = −θ ij∇T , (B13)

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TANMOY DAS PHYSICAL REVIEW B 86, 064527 (2012)

where Hall resistance, ρ = σ−1, and the Nernst coefficient,ν = θ/B, are measured independently via transport probes bysetting a weak magnetic field B = Bz along the c axis of thelattice.1,5 In Eqs. (B12) and (B13), the layer indices i,j run onboth sides. From Eqs. (B6), (B9), and (B12), we easily deducethat

σ ijμν = 2

∑k,n

Mijn v

μ

nk�−1nk

(∂V ν

nk

∂ E

) [−∂f (ξnk)

∂ξnk

]

= βe2

2

∑k,n

Mij

nkvμ

nk�−1nk vν

nksech2

(βEnk

2

), (B14)

αijμν = 2

∑k,n

Mijn v

μ

nk�−1nk

[∂V ν

nk

∂(−∇T )

] [−∂f (ξnk)

∂ξnk

]

= −β2e

2

∑k,n

Mij

nkvμ

nk�−1nk vν

nkEnksech2

(βEnk

2

),

(B15)

where we have used the identity − ∂f (ξnk)∂ξnk

= β

4 sech2( βEnk2 ). In

the usual manner, the differential operator �−1nk can be arranged

as a perturbative expansion in the magnetic field B20,32 inorder to obtain transport coefficients that do not depend on B.For this purpose we define �nk= ϒnk+�nk [Eq. (B3)], whereϒnk = 1/τnk and �nk is the rest. Then

�−1nk = ϒ−1

nk − ϒ−1nk �nkϒ

−1nk + O(B2). (B16)

The diagonal entries in Eq. (B15) (in the μ,ν basis, notin the i,j basis) are obtained from the zeroth order inB in Eq. (B16), while the lowest-order contribution tothe off-diagonal coefficients arises from the linear orderin B in the expansion of Eq. (B16). To this purpose, theexpressions Eq. (B15) can be simplified in the form of the

expressions

σ ijμμ = βe2

2

∑k,n

Mij

nk

(v

μ

nk

)2τ0sech2

(βEnk

2

), (B17)

σij

μ �=ν = βe3B

2hc

∑k,n

Mij

nkvμ

nkτ20

(vν

nk∂vν

nk

∂kμ − vμ

nk

∂vνnk

∂kν

)

× sech2

(βEnk

2

), (B18)

αijμμ = −β2e

2

∑k,n

Mij

nk

(v

μ

nk

)2τ0Enksech2

(βEnk

2

), (B19)

αij

μ �=ν = −β2e2B

2hc

∑k,n

Mij

nkvμ

nkτ20

(vν

nk∂vν

nk

∂kμ − vμ

nk

∂vνnk

∂kν

)

×Enksech2

(βEnk

2

). (B20)

In the simplest model, we assume a momentum- andtemperature-independent value of relaxation time τk = τ0. Atemperature-dependent value of τ0 can improve the results,especially for the Nernst signal; however, with this simplestform, we are able to capture the salient features of both the Hallcoefficient and the Nernst coefficient. The results in the maintext are normalized to the constant values of B and τ0. Theabove expressions match exactly with the one proposed earlierfor various forms of the density wave order,20,21 except thematrix-element terms. It is important to note that, if no othercrystal symmetry is broken, then the Mott relation νxy = −νyx

for the Nernst coefficient can be recovered from the Boltzmanntheory.50 This implies that the choice of sign convention forthe Nernst effect is arbitrary. To faciliate direct comparison,we have taken the same sign for |νa| and |νb|.

1D. LeBoeuf, N. Doiron-Leyraud, J. Levallois, R. Daou, J.-B.Bonnemaison, N. E. Hussey, L. Balicas, B. J. Ramshaw, RuixingLiang, D. A. Bonn, W. N. Hardy, S. Adachi, C. Proust, andL. Taillefer, Nature 450, 533 (2007).

2J. Chang, R. Daou, C. Proust, D. LeBoeuf, N. Doiron-Leyraud,F. Laliberte, B. Pingault, B. J. Ramshaw, R. Liang, D. A. Bonn,W. N. Hardy, H. Takagi, A. B. Antunes, I. Sheikin, K. Behnia, andL. Taillefer, Phys. Rev. Lett. 104, 057005 (2010).

3F. Lalibert, J. Chang, N. Doiron-Leyraud, E. Hassinger, R. Daou,M. Rondeau, B. J. Ramshaw, R. Liang, D. A. Bonn, W. Hardy,S. Pyon, T. Takayama, H. Takagi, I. Sheikin, L. Malone, C. Proust,K. Behnia, and L. Taillefer, Nat. Commun. 2, 432 (2011).

4R. Daou, J. Chang, D. LeBoeuf, O. Cyr-Choiniere, F. Laliberte,N. Doiron-Leyraud, B. J. Ramshaw, R. Liang, D. A. Bonn, W. N.Hardy, and L. Taillefer, Nature 463, 519 (2010).

5J. Chang, J. Chang, N. Doiron-Leyraud, F. Lalibert, R. Daou,D. LeBoeuf, B. J. Ramshaw, R. Liang, D. A. Bonn, W. N. Hardy,C. Proust, I. Sheikin, K. Behnia, and L. Taillefer, Phys. Rev. B 84,014507 (2011).

6N. Doiron-Leyraud, C. Proust, D. LeBoeuf, J. Levallois, J. B.Bonnemaison, R. Liang, D. A. Bonn, W. N. Hardy, and L. Taillefer,Nature 447, 565 (2007).

7A. F. Bangura, J. D. Fletcher, A. Carrington, J. Levallois,M. Nardone, B. Vignolle, P. J. Heard, N. Doiron-Leyraud,D. LeBoeuf, L. Taillefer, S. Adachi, C. Proust, and N. E. Hussey,Phys. Rev. Lett. 100, 047004 (2008).

8S. E. Sebastian, N. Harrison, P. A. Goddard, M. M. Altarawneh,C. H. Mielke, Ruixing Liang, D. A. Bonn, W. N. Hardy, O. K.Andersen, and G. G. Lonzarich, Phys. Rev. B 81, 214524 (2010).

9S. E. Sebastian, N. Harrison, M. M. Altarawneh, C. H. Mielke,Ruixing Liang, D. A. Bonn, W. N. Hardy, and G. G. Lonzarich,Proc. Natl. Acad. Sci. USA 107, 6175 (2010).

10J. Singleton, C. de la Cruz, R. D. McDonald, S. Li, M. Altarawneh,P. Goddard, I. Franke, D. Rickel, C. H. Mielke, X. Yao, and P. Dai,Phys. Rev. Lett. 104, 086403 (2010).

11L. Onsager, Phil. Mag. 43, 1006 (1952).12D. Fournier, G. Levy, Y. Pennec, J. L. McChesney, A. Bostwick,

E. Rotenberg, R. Liang, W. N. Hardy, D. A. Bonn, I. S. Elfimov,and A. Damascelli, Nat. Phys. 6, 905 (2010).

13A. Carrington and E. A. Yelland, Phys. Rev. B 76, 140508(R)(2007).

14A. J. Millis and M. R. Norman, Phys. Rev. B 76, 220503 (2007).15M. R. Norman, J. Lin, and A. J. Millis, Phys. Rev. B 81, 180513(R)

(2010).

064527-8

ELECTRON-LIKE FERMI SURFACE AND IN-PLANE . . . PHYSICAL REVIEW B 86, 064527 (2012)

16S. Chakravarty and H. Y. Kee, Proc. Nat. Acad. Sci. USA 105, 8835(2008).

17T. Das, R. S. Markiewicz, and A. Bansil, Phys. Rev. B 77, 134516(2008); T. Das, R. S. Markiewicz, A. Bansil, and A. V. Balatsky,ibid. 85, 224535 (2012).

18T. Wu, H. Mayaffre, S. Kramer, M. Horvatic, C. Berthier, W. N.Hardy, R. Liang, D. A. Bonn, and M.-H. Julien, Nature 477, 191(2011).

19N. Harrison and S. E. Sebastian, Phys. Rev. Lett. 106, 226402(2011).

20A. Hackl and S. Sachdev, Phys. Rev. B 79, 235124 (2009);A. Hackl, M. Vojta, and S. Sachdev, ibid. 81, 045102 (2010).

21C. Zhang, S. Tewari, and S. Chakravarty, Phys. Rev. B 81, 104517(2010).

22V. B. Zabolotnyy, A. A. Kordyuk, D. Evtushinsky, V. N.Strocov, L. Patthey, T. Schmitt, D. Haug, C. T. Lin, V. Hinkov,B. Keimer, B. Buchner, and S. V. Borisenko, Phys. Rev. B 85,064507 (2012).

23Y. Kohsaka, Y. Kohsaka, C. Taylor, P. Wahl, A. Schmidt, JhinhwanLee, K. Fujita, J. W. Alldredge, Jinho Lee, K. McElroy, H. Eisaki,S. Uchida, D.-H. Lee, and J. C. Davis, Nature 454, 1072 (2008).

24N. E. Hussey, M. Kibune, H. Nakagawa, N. Miura, Y. Iye, H. Takagi,S. Adachi, and K. Tanabe, Phys. Rev. Lett. 80, 2909 (1998); N. E.Hussey, H. Takagi, Y. Iye, S. Tajima, A. I. Rykov, and K. Yoshida,Phys. Rev. B 61, 6475(R) (2000).

25S. C. Riggs, O. Vafek, J. B. Kemper, J. B. Betts, A. Migliori,W. N. Hardy, Ruixing Liang, D. A. Bonn, and G. S. Boebinger,Nat. Phys. 7, 332 (2011).

26W. A. Atkinson, Phys. Rev. B 59, 3377 (1999).27V. Hinkov, D. Haug, B. Fauque, P. Bourges, Y. Sidis,

A. Ivanov, C. Bernhard, C. T. Lin, and B. Keimer, Science 319,597 (2008).

28With doubling the tetragonal unit cell along the x axis, we recoverthe orthorhombic II phase of this system, and the computedelectronic states in both cases match with each other and also withthe first-principles calculations.13 Note that when the periodicity isimposed on the chain state via considering orthorhombic unit cell,the open-orbit chain FS transforms into closed pockets.

29T. Dahm, V. Hinkov, S. V. Borisenko, A. A. Kordyuk, V. B.Zabolotnyy, J. Fink, B. Bchner, D. J. Scalapino, W. Hanke, andB. Keimer, Nat. Phys. 5, 217 (2008).

30J. Chang, Y. Sassa, S. Guerrero, M. Mansson, M. Shi, S. Pailhes,A. Bendounan, R. Mottl, T. Claesson, O. Tjernberg, L. Patthey,M. Ido, N. Momono, M. Oda, C. Mudry, and J. Mesot, New J.Phys. 10, 103016 (2008).

31R. H. He, X. J. Zhou, M. Hashimoto, T. Yoshida, K. Tanaka, S.-K.Mo, T. Sasagawa, N. Mannella, W. Meevasana, H. Yao, M. Fujita,T. Adachi, S. Komiya, S. Uchida, Y. Ando, F. Zhou, Z. X. Zhao,A. Fujimori, Y. Koike, K. Yamada, Z. Hussain, and Z.-X. Shen,New J. Phys. 13, 013031 (2011).

32J. M. Ziman, Electrons and Phonons (Oxford University Press,Oxford, UK, 1960).

33Since the conductivity tensors are proportional to the relaxationtime, this particular choice of featureless τ has no major influenceon the overall shape of the coefficients.

34The values of RH at T = 0 and Ts are constrained by the hole-dopingconcentration and density wave transition temperature, respectively,while its T evolution between them can be well reproducedby assuming a competing order origin of pseudogap with thegap function V (T ) = V (0)(1 − T/Ts)r , with the critical exponentr = 0.5 as obtained within BCS theory. This T evolution also hasbeen used in earlier theoretical studies20 to fit experimental the data[D. D. Prokof’ev, M. P. Volkov, and Yu. A. Boikov, Phys. SolidState 45, 1223 (2003)].

35N. Ichikawa, S. Uchida, J. M. Tranquada, T. Niemoller, P. M.Gehring, S.-H. Lee, and J. R. Schneider, Phys. Rev. Lett. 85, 1738(2000).

36K. Behnia, J. Phys.: Condens. Matter 21, 113101 (2009).37V. Oganesyan and I. Ussishkin, Phys. Rev. B 70, 054503 (2004).38E. H. Sondheimer, Proc. R. Soc. London A 193, 484 (1948).39We have chosen the low-field Nernst effect data from Ref. 5 for

comparison, since the other experimental data near Hc2 relied onthe acquisition of contributions from vortex-induced fluctuatingpairs, which is not included in our theory.

40Y. Ando, K. Segawa, S. Komiya, and A. N. Lavrov, Phys. Rev. Lett.88, 137005 (2002).

41M. J. Lawler, K. Fujita, J. Lee, A. R. Schmidt, Y. Kohsaka, C. KooKim, H. Eisaki, S. Uchida, J. C. Davis, J. P. Sethna, and E.-A. Kim,Nature 466, 347 (2010).

42T. Pereg-Barnea, H. Weber, G. Refael, and M. Franz, Nat. Phys. 6,44 (2009).

43J. M. Tranquada, D. N. Basov, A. D. LaForge, and A. A. Schafgans,Phys. Rev. B 81, 060506(R) (2010).

44T. Das, Phys. Rev. B 85, 144510 (2012).45P. M. C. Rourke, A. F. Bangura, C. Proust, J. Levallois, N. Doiron-

Leyraud, D. LeBoeuf, L. Taillefer, S. Adachi, M. L. Sutherland,and N. E. Hussey, Phys. Rev. B 82, 020514(R) (2010).

46W. D. Wise, K. Chatterjee, M. C. Boyer, Takeshi Kondo,T. Takeuchi, H. Ikuta, Zhijun Xu, Jinsheng Wen, G. D. Gu, YayuWang, and E. W. Hudson, Nat. Phys. 5, 213 (2009).

47G. Ghiringhelli, M. Le Tacon, M. Minola, S. Blanco-Canosa,C. Mazzoli, N. B. Brookes, G. M. De Luca, A. Frano, D. G.Hawthorn, F. He, T. Loew, M. Moretti Sala, D. C. Peets,M. Salluzzo, E. Schierle, R. Sutarto, G. A. Sawatzky, E. Weschke,B. Keimer, and L. Braicovich, Science 337, 821 (2012).

48R. S. Markiewicz, S. Sahrakorpi, M. Lindroos, Hsin Lin, andA. Bansil, Phys. Rev. B 72, 054519 (2005).

49D. I. Pikulin, C. Y. Hou, and C. W. J. Beenakker, Phys.Rev. B 84,035133 (2011).

50A. Hackl and M. Vojta, Phys. Rev. B 80, 220514(R) (2009).

064527-9