Orbital structure of the effective pairing interaction in the high-temperature superconducting cuprates
Peizhi Mai, Giovanni Balduzzi, Steven Johnston, Thomas A. Maier
OOrbital structure of the effective pairing interaction in the high-temperaturesuperconducting cuprates
Peizhi Mai , Giovanni Balduzzi , Steven Johnston , , Thomas A. Maier , , ∗ Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6164, USA Institute for Theoretical Physics, ETH Z¨urich, 8093 Z¨urich, Switzerland Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996-1200, USA Joint Institute for Advanced Materials at The University of Tennessee, Knoxville, TN 37996, USA and Computational Sciences and Engineering Division,Oak Ridge National Laboratory, Oak Ridge, TN, 37831-6494, USA (Dated: October 1, 2020)The nature of the effective interaction responsible for pairing in the high-temperature supercon-ducting cuprates remains unsettled. This question has been studied extensively using the simplifiedsingle-band Hubbard model, which does not explicitly consider the orbital degrees of freedom ofthe relevant CuO planes. Here, we use a dynamic cluster quantum Monte Carlo approximation tostudy the orbital structure of the pairing interaction in the three-band Hubbard model, which treatsthe orbital degrees of freedom explicitly. We find that the interaction predominately acts betweenneighboring copper orbitals, but with significant additional weight appearing on the surroundingbonding molecular oxygen orbitals. By explicitly comparing these results to those from the simplersingle-band Hubbard model, our study provides strong support for the single-band framework fordescribing superconductivity in the cuprates. Introduction — Cuprate superconductivity emerges intheir quasi-two-dimensional (2D) CuO planes after dop-ing additional carriers into these layers. The undopedparent compounds are charge transfer insulators due tothe large Coulomb repulsion U dd on the Cu 3 d orbitals,and, to a good approximation, a spin- hole is locatedon every Cu 3 d x − y orbital. This situation is well de-scribed by a 2D square lattice Hubbard model or Heisen-berg model in the large U dd limit.Upon doping the additional holes or electrons primar-ily occupy the O or Cu orbitals, respectively. The mini-mal model capturing this asymmetry is the three-bandHubbard model, which explicitly accounts for the Cu3 d x − y , O 2 p x , and 2 p y orbitals (Fig. 1 a ) [1]. Evenat finite doping, the low energy sector of the three-bandmodel can be mapped approximately onto an effectivesingle-band Hubbard model [2]. One expects this in thecase of electron-doping since the additional carriers godirectly onto the Cu sublattice, on which the holes ofthe undoped materials already reside. The case of hole-doping, however, is more subtle. Here, the additionalcarriers predominantly occupy the O sublattice due tothe large U dd on the Cu orbital, and the appropriate-ness of a single-band model is less clear. In their seminalwork, Zhang and Rice [2] argued that the doped holeeffectively forms a spin-singlet state with a Cu hole, the“Zhang-Rice singlet” (ZRS, Fig. 1 b ), which then playsthe same role as a fully occupied or empty site in an ef-fective single-band model, again facilitating a single-banddescription.The nature of the single-band 2D Hubbard model’spairing interaction has been extensively studied [3–8].Detailed calculations of its momentum and frequencystructure using dynamic cluster approximation (DCA)quantum Monte Carlo (QMC) [3] find that it is well described by a spin-fluctuation exchange interaction [4].The single-band model, however, cannot provide any in-formation on the orbital structure of the interaction. Forexample, in the hole doped case, the spins giving riseto the spin-fluctuation interaction are located on the Cusublattice, while the paired holes are moving on the O p x/y sublattice. This situation can produce a differentphysical picture than if the interaction and the pairs bothoriginate from the same orbital on the same lattice [9–13].And indeed, studies have observed two-particle behaviorin a two sublattice system that is not observed in a one-lattice system [14]. Moreover, an analysis of resonantinelastic x-ray scattering studies has found that a single-band model fails to describe the high-energy magneticexcitations near optimal doping [15]. Studying the effec-tive interaction in a three-band model, and, in particu-lar, determining its orbital structure is, therefore, critical.Such a study will also provide new insight into the natureof high-temperature superconductivity that is not avail-able from the previous single-band studies. In this letter,we use a QMC-DCA method to explicitly calculate theorbital and spatial structure of the effective interactionin a realistic three-band CuO model, and compare theresults with those obtained from a single-band model. Model and Methods — The three-band Hubbard modelwe study can be found in Refs. [16, 17]. We adopted aparameter set appropriate for the cuprates [16, 18–20](in units of eV): the nearest neighbor Cu-O and O-Ohopping integrals t pd = 1 . t pp = 0 .
49, on-site inter-actions U dd = 8 . U pp = 0, and charge-transfer energy∆ = ε p − ε d = 3 .
24, unless otherwise stated. Since weuse a hole language, half-filling is defined as hole density n h = 1 and n h > <
1) corresponds to hole (electron)-doping. A finite U pp only leads to small quantitativechanges in the results (see Fig. S4 [17]) but worsens a r X i v : . [ c ond - m a t . s up r- c on ] S e p FIG. 1: a The orbital basis of the three-band Hubbard model. b Sketch of the bonding ligand ( L ) molecular orbital sur-rounding a central Cu- d orbital. c Leading BSE eigenvalue λ d vs n h for a 4 × β = 16 eV − . d Sketches ofsome ways a pair can form with a d -wave symmetry. Here, D d and D dL pair a Cu 3 d hole with a hole on the neighboringCu- d and L molecular orbital, respectively, while D LL pairsholes on neighboring O- L orbitals. the sign problem significantly [16]. Therefore, we keep U pp = 0 for this study.We study the single- and three-band Hubbard mod-els using DCA with a continuous time QMC impuritysolver [21–23]. We determine the structure of the pairinginteraction by solving the Bethe-Salpeter equation (BSE)in the particle-particle singlet channel to obtain its lead-ing eigenvalues and (symmetrized) eigenvectors [3, 17]. Atransition to the superconducting state occurs when theleading eigenvalue λ ( T = T c ) = 1, and the magnitude of λ < Results — Figure 1 c shows the leading eigenvalue ofthe BSE for the three-band model as a function of holeconcentration n h obtained on a 4 × β =1 /k B T = 16 eV − . We find that it always corresponds toa d -wave superconducting state [24] and is larger for hole-doping ( n h >
1) compared to electron-doping ( n h < T c consistent with experiments and prior stud-ies of the single and two-band Hubbard models [25, 26].(Although λ d is largest at half-filling, we expect that itasymptotically approaches one as the temperature de-creases but never actually cross one due to the opening ofa Mott gap. We observe such behavior in explicit calcu-lations on smaller three-band clusters, see Fig. S1 [17].)We now analyze the spatial and orbital structure ofthe leading eigenvector φ αβ ( k ) ( α and β denote orbitals),by Fourier transforming φ αβ ( k ) to real space to obtain φ r β ( r α ), where r β denotes the position of the orbitaltaken as the reference site. We employed a 6 × T = 0 . U pp (see Figs. S3 and S4) [17], indicatingthat our conclusions are robust across much of the modelphase space.In the single-band Hubbard model, the pairs arelargely comprised of carriers on nearest neighbor sites ina d -wave state, i.e. with a positive (negative) phase alongthe x - ( y )-directions. The internal structure of the pairsin the three-band model seems more complicated [27].The real-space structure of φ r β ( r α ) shown in Figs.2 a - c and Figs.2 d - f for the hole- and electron-doped cases,respectively, display an extended and rich orbital struc-ture. Here, the size and color indicate the strength andphase of φ r β ( r α ), respectively, on each site after adoptingthe central Cu 3 d x − y or O 2 p x,y orbital as a referencesite at r β . The form factors φ r β ( r α ) are similar for bothelectron and hole doping, decaying over a length scaleof ∼ d -wavepairing between nearest Cu sites dominates, there is alsoa significant contribution from d - p pairing, with a com-parable amplitude for up to the third-nearest neighbors.The pairing between the individual O 2 p x,y orbitals ismuch weaker in comparison.We now transform the leading eigenfunction from theO- p x and p y basis to the bonding L and anti-bonding L combinations (Fig. 1d). These combinations, formedfrom the four O orbitals surrounding a Cu cation, arethe relevant states for the ZRS, in which the doped holesare argued to reside in. The bonding L state stronglyhybridizes with the central Cu 3 d x − y orbital (Fig. 1 b ), a b cd e f % e doped15 % h doped Cu Reference O(x) Reference O(y) Reference
FIG. 2: The real space components of the leading particle-particle BSE (symmetrized) eigenvector for the three-band model atoptimal doping and β = 10 eV − on a 6 × d (or O p x , p y ) referencesite and all other orbitals as a function of distance. All panels set the Cu- d orbital at the origin, as labelled. while the anti-bonding L state does not. The resultingantiferromagnetic exchange interaction between the Cuand L holes is then argued to bind them into the Zhang-Rice spin-singlet state, which provides the basis for themapping onto a single-band model.The orbital structure of the leading eigenfunction sim-plifies considerably after one transforms to the bond-ing L and anti-bonding L combinations. Fig. 3 plotsthe pairing amplitudes for a hole on Cu paired with an-other hole on a neighboring Cu ( d - d , Fig. 3 a ) or bondingmolecular orbital ( d - L , Fig. 3 b ). Both components ex-hibit a clear d x − y symmetry; however, both channelsalso have indications of a higher momentum harmonic[cos(2 k x a ) − cos(2 k y a )]. Interestingly, the contributionfrom holes occupying neighboring bonding molecular or-bitals exhibits similar behavior ( L - L , Fig. 3 c ). The L -related pairing contributes very little as will be discussedin Fig. 4 and in the supplement (see Fig. S5) [17].Figs. 3 a - c establishes that the pairing between the dif-ferent orbital components of the ZRS all possess the req-uisite d x − y symmetry. This indicates that the ZRSpicture is valid for describing pairing correlations in thethree-band Hubbard model of the cuprates. To confirmthis, we also computed the real-space structure of theleading particle-particle BSE eigenfunction in the single-band Hubbard model. Here, we considered cases withnext-nearest-neighbor hopping t /t = 0 (Fig. 3 d ) and − . e ), which are commonly used in the litera-ture, as well as − .
453 (Fig. 3 f ), which we obtained bydownfolding our three-band model parameters onto thesingle-band model [28, 29]. The single-band model re-produces the short-range pairing structure of the three-orbital model, regardless of the value of t ; however, thesecond and third neighbor pairing is only captured cor-rectly for t /t = − . t . The latter conclusion further underscoresthe crucial role of t for determining the superconductingproperties of the single-band model [8, 30, 31].Figure 3 shows that the structure of the leading eigen-function φ αβ is closely linked to the orbital structure ofthe ZRS. Fig. 4 examines how this internal structureevolves with doping by plotting the orbital-dependenthole density (panel a ) and the orbital composition of theeigenfunction φ αβ (panel b ) on a 4 × a shows that the single hole per unit cellin the undoped case has approximately 65% Cu- d charac-ter, while 35% of the hole is located in the bonding O- L molecular orbital. With electron doping, there is a smalldecrease of n d /n h indicating that the holes are removedmainly from the Cu- d orbital. In contrast, with hole a b cd e f S i ng l e - band T h r ee - band d-d d-L L-Lt’/t=0 t’/t=-0.2 t’/t=-0.453 FIG. 3: A comparison of the orbital structure of the pairs in the three-band and single-band models at 15% hole-doping. Eachpanel plots the real space components of the leading particle-particle BSE (symmetrized) eigenvector. The first row shows d - dd - L and L - L pairing for the three-band model at β = 10 eV − . The second row shows the pair structure for the single-bandmodel ( U = 6 t , β = 5 t − ) at t /t = 0, − . − . doping, there is a significant redistribution of the holedensity from the d - to the L -orbital, showing that dopedholes mainly occupy the O- L molecular orbital. The holedensity on the anti-bonding O- L orbital is negligible.Figure 4 b shows that the total weight of the nearest-neighbor pairing increases from ∼
70% in the undopedcase to almost 100% with either hole or electron dop-ing. Since the BSE eigenvector reflects the momentumstructure of the pairing interaction, this dependence canbe understood from an interaction that becomes morepeaked in momentum space as n h = 1 is approached.This behavior leads to a more delocalized structure of φ r β ( r α ) and, therefore, a reduction of the relative weightof the nearest-neighbor contribution. The partial contri-butions to the nearest-neighbor pairing weight, D d and D dL , have a doping dependence very similar to the cor-responding orbital densities n d and n L in panel a , closelylinking the orbital structure of the pairing to the orbitalmakeup of the ZRS. The weight of the L contributionsremains negligible over the full doping range [17]. Conclusions — We have determined the orbital struc-ture of the effective pairing interaction in a three-bandCuO Hubbard model and shown that it simplifies con-siderably when viewed in terms of a basis consisting ofa central Cu- d orbital and a bonding L combination of the four surrounding O- p orbitals. These states underliethe ZRS singlet construction that enables the reductionof the three-band to an effective single-band model. Byexplicitly comparing the three-band with single-band re-sults, we show that the effective interaction is correctlydescribed in the single-band model. In summary, theseresults strongly support the conclusion that a single-bandHubbard model provides an adequate framework to un-derstand high- T c superconductivity in the cuprates.Acknowledgements — The authors would like to thankL. Chioncel, P. Dee, A. Georges, K. Haule, E. Huang, S.Karakuzu, G. Kotliar, H. Terletska, and D. J. Scalapinofor useful comments. This work was supported by theScientific Discovery through Advanced Computing (Sci-DAC) program funded by the U.S. Department of En-ergy, Office of Science, Advanced Scientific ComputingResearch and Basic Energy Sciences, Division of Ma-terials Sciences and Engineering. This research usedresources of the Oak Ridge Leadership Computing Fa-cility, which is a DOE Office of Science User Facilitysupported under Contract DE-AC05-00OR22725. TheDCA++ code used for this project can be obtained athttps://github.com/CompFUSE/DCA. ab n d / n h n L / n h n L ' / n h n h O r b i t a l - den s i t y D tot D d / D tot D dL / D tot D LL / D tot n h W e i gh t o f NN pa i r i ng FIG. 4: a Ratios of the orbital hole densities to the totaldensity n h . b Weights of different orbital compositions ofthe nearest-neighbor pairs, as defined in Fig.1, and their totalweight. All results were obtained on a 4 × β =16 eV − .[1] V. J. Emery, Phys. Rev. Lett. , 2794 (1987).[2] F. C. Zhang and T. M. Rice, Phys. Rev. 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Lett , 237001 (2005).[26] A. Macridin, M. Jarrell, T. A. Maier and G. A. Sawatzky,Phys. Rev. B , 134527 (2005).[27] A. Moreo and E. Dagotto, Phys. Rev. B , 214502(2019).[28] S. Johnston, F. Vernay, B. Moritz, Z.-X. Shen, N. Na-gaosa, J. Zaanen and T. P. Devereaux, Phys. Rev. B ,064513 (2010).[29] H. Eskes and G. A. Sawatzky, Phys. Rev. B , 9656(1991).[30] Y.-F. Jiang, J. Zaanen, T. P. Devereaux, and H.-C. Jiang,arXiv:1907.11728 (2019).[31] M. Qin et al. , arXiv:1910.08931 (2019).[32] E. Gull, P. Werner, O. Parcollet and M. Troyer, Eur.Phys. Lett. , 57003 (2008).[33] E. Gull, P. Staar, S. Fuchs, P. Nukala, M. S. Summers,T. Pruschke, T. Schulthess and T. Maier, Phys. Rev. B, , 075122 (2011). rbital structure of the effective pairing interaction in the high-temperaturesuperconducting cuprates – Supplementary Material Peizhi Mai , Giovanni Balduzzi , Steven Johnston , , Thomas A. Maier , , ∗ Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6164, USA Institute for Theoretical Physics, ETH Z¨urich, 8093 Z¨urich, Switzerland Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996-1200, USA Joint Institute for Advanced Materials at The University of Tennessee, Knoxville, TN 37996, USA and Computational Sciences and Engineering Division,Oak Ridge National Laboratory, Oak Ridge, TN, 37831-6494, USA (Dated: October 1, 2020)
THE SINGLE- AND THREE-BAND HUBBARD MODELS
The Hamiltonian of the three-band Hubbard model is H = ( ε d − µ ) X i,σ n di,σ + ( ε p − µ ) X j,σ n p α j,σ + X h i,j i ,σ t ij ( d † iσ p αjσ + p † αjσ d iσ )+ X h j,j i ,σ t jj ( p † αjσ p α j σ + p † α j σ p αjσ ) + U dd X i n di ↑ n di ↓ + U pp X j n p α j ↑ n pj ↓ . (S1)Here, d † i,σ ( d i,σ ) creates (annihilates) a spin σ (= ↑ , ↓ ) hole in the copper d x − y orbital at site i ; p † αjσ ( p αjσ ) creates(annihilates) a spin σ hole in the oxygen p α ( α = x, y ) orbital at site j ; for nearest neighbor, j = i ± ˆ x/ y/ n diσ = d † iσ d iσ and n p α jσ = p † αjσ p αjσ are the number operators; (cid:15) d and (cid:15) p are the onsite energies of the Cu and O orbitals,respectively; µ is the chemical potential; t i,j is the nearest neighbor Cu-O hopping integral; t j,j is the nearest neighborO-O hopping integral; and U dd and U pp are the on-site Hubbard repulsions on the Cu and O orbitals, respectively.The hopping integrals are parameterized [5] as t ij = P ij t pd and t jj = Q jj t pp , where P ij = 1 for j = i + ˆ y/ j = i − ˆ x/ P ij = − j = i − ˆ y/ j = i + ˆ x/ Q jj = 1 for j = j − ˆ x/ y/ j = j + ˆ x/ − ˆ y/ Q jj = − j = j + ˆ x/
2+ ˆ y/ j = j − ˆ x/ − ˆ y/
2. Throughout, we adopted (in units of eV): t pd = 1 . t pp = 0 . U dd = 8 . U pp = 0, and ∆ = ε p − ε d = 3 .
24 [5–8], unless otherwise stated. Since we use a hole language, half-filling isdefined as hole density n h = 1 and n h > n h < U pp only leads to small quantitative changes in the pair structure (see Fig. S4), but worsens the sign problemsignificantly [5]. Therefore, we keep U pp = 0 for this study except for the results presented in Fig. S4.The downfolded single-band Hubbard model is H = − µ X iσ n iσ − X h i,j i σ t ij c † iσ c jσ + U X i (cid:18) n i ↑ − (cid:19) (cid:18) n i ↓ − (cid:19) , (S2)where c † iσ ( c iσ ) creates (annihilates) an electron with spin σ at site i , t i,j = t and t for nearest- and next-nearest-neighbor hoping, respectively, and zero otherwise. U is the on-site Hubbard repulsion, and µ is the chemical potential,which is adjusted to fix the electron filling. Throughout, we set t = 1, U = 6 t , and vary t as indicated in the text. SYMMETRIZED EIGENVECTORS OF THE BETHE-SALPETER EQUATION
To determine the structure of the effective pairing interaction, we solve the Bethe-Salpeter equation in the particle-particle singlet channel − TN c X K,α ,α Γ c,ppα,β,α ,α ( K, K ) ¯ χ α ,α ,α ,α ( K ) φ R,να α ( K ) = λ ν φ R,ναβ ( K ) . (S3)Here, K = ( K , iω n ), and ¯ χ α ,α ,α ,α ( K , ω n ) = ( N c /N ) P k [ G α α ( K + k , ω n ) G α α ( − K − k , − ω n )] is the coarse-gained bare particle-particle propagator. The irreducible particle-particle vertex Γ c,pp is extracted from the two-particle cluster Green’s function G ,cα ,α ,α ,α ( K, K ) with zero center of mass momentum and frequency by inverting a r X i v : . [ c ond - m a t . s up r- c on ] S e p the cluster Bethe-Salpeter equation G ,cα ,α ,α ,α ( K, K ) = ¯ G α ,α ( K ) ¯ G α α ( − K ) δ K,K + TN c X K ,α ...α ¯ G α ,α ( K ) ¯ G α ,α ( − K )Γ c,ppα ,α ,α ,α ( K, K ) G ,cα ,α ,α ,α ( K , K ) . (S4)To remove the ambiguity between left and right eigenvectors of the eigenvalue equation (S3), we symmetrize thepairing kernel entering Eq. (S3). Using matrix notation in ( K, α, β ), we first diagonalize the bare particle-particlepropagator, ¯ χ D = U − ¯ χU , where χ D is a diagonal matrix, to introduce the symmetrized BSE − TN c U p χ D U − Γ c,pp U p χ D U − φ ν = λ ν φ ν . (S5)We use the eigenvectors of the symmetrized BSE, φ ναβ ( K ), for the analysis presented in the main text. They arerelated to the right eigenvectors of the BSE in Eq. (S3) by φ ν = U p χ D U − φ R,ν . (S6) THE BASIS TRANSFORMATION TO THE MOLECULAR L , L ORBITALS
The construction of the Zhang-Rice singlet relies on a transformation from the oxygen p x , p y orbital basis to bondingand anti-bonding molecular orbitals, denoted here as L and L , respectively. The two basis are related by a unitarytransformation [1–3] defined in k -space L k σ = iγ k h sin (cid:0) k x a (cid:1) p x k σ − sin (cid:16) k y a (cid:17) p y k σ i , (S7)and L k σ = − iγ k h sin (cid:16) k y a (cid:17) p x k σ + sin (cid:0) k x a (cid:1) p y k σ i , (S8)where γ k = sin ( k x a/
2) + sin ( k y a/ p α k σ = N − / c P j p αjσ exp( − i k · R j ), and we have set the lattice constant a = 1. In this basis, only the L state hybridizes with the Cu- d orbital, while the L state only hybridizes with the L state. The Fourier transform of the L and L orbitals to real-space is defined as L iσ = N − / P k L k σ exp( − i k · R i ), L i σ = N − / P k L k σ exp( − i k · R i ) where i = i + ˆ x/ y/ SUPERCONDUCTING TRANSITION TEMPERATURE IN THE THREE-BAND HUBBARD MODEL
For the 6 × × T c fromthe temperature where the leading eigenvalue of the Bethe-Salpeter equation crosses 1, i.e. λ d ( T = T c ) = 1. Thistemperature can be reached on a 2 × T c ( n h ) can be determined as a function of hole density n h in that case. Fig. S1 shows the DCA results for T c ( n h ) obtained in a 2 × c for the leading eigenvalue λ d ( n h )of the particle-particle Bethe-Salpeter equation, as well as the asymmetry found in experiments, the T c versus n h phase diagram exhibits a higher maximum T c on the hole doped side than on the electron-doped side. Moreover,these results are similar to previous DCA 2 × T c vanishes is reduced compared to those earlier calculations. This difference may originatein the difference in model parameters, in particular the neglect of the direct oxygen-oxygen hopping t pp in the earliertwo-band model calculations. DEPENDENCE OF THE LEADING EIGENFUNCTION ON TEMPERATURE, CLUSTER SIZE ANDOXYGEN COULOMB REPULSION
While the leading eigenvalue λ d ( T ) shows a very strong increase with decreasing temperature, the temperaturedependence of the corresponding eigenfunction is found to be rather weak. Fig. S2 shows how the orbital and spatial n h T c ( e V ) FIG. S1:
Superconducting transition temperature as a function of filling for a N Cu = × . Thetransition temperature T c is estimated by finding the temperature at which the leading BSE eigenvalue goes to 1. The modelparameters are (in units of eV) ∆ = 3 . t pd = 1 . t pp = 0 . U pp = 0, and U dd = 8 .
5. Our calculations find twosuperconducting domes on the hole-doped and electron-doped sides of the phase diagram, respectively, with a maximum T c that is higher for the hole-doped case, consistent with experiments. structure of the leading eigenfunction φ r β ( r α ) of the (symmetrized) Bethe-Salpeter equation changes with decreasingtemperature between β = 1 /T = 10 eV − (top panels a - c from Fig. 2 in the main text) and β = 12 eV − (bottompanels d - f ). We only observe small quantitative changes between these two temperatures.The cluster size dependence of the leading eigenfunction is studied in Fig. S3, which shows the results of an N Cu = 4 × N Cu = 6 × × × U pp on the oxygen orbitals.An additional U pp = 4 . U pp . Only a very slightsuppression of the components that involve the O- p orbitals is observed. This justifies the neglect of U pp in most ofour calculations, and provides evidence that our main conclusions reached from those calculations are general and notaffected by U pp . ROLE OF THE ANTI-BONDING MOLECULAR L ORBITAL
Finally, we show the components of the leading eigenfunction that involve the antibonding L orbital in the bottomrow of Fig. S5, compared to the bonding L components that were already shown in Fig. 3 of the main text. Fromthe results for the orbital hole densities in Fig. 4 a of the main text, it is clear that the L molecular orbital remainsalmost completely unoccupied over the entire doping range considered, despite the finite hybridization between the L and L states. As a consequence, and as seen from the bottom row of Fig. S5, the L state is not involved in thepairing. This provides strong support for the Zhang-Rice picture, which only considers the d - and L -states in themapping to an effective single-band model. [1] F. C. Zhang and T. M. Rice, Phys. Rev. B, , 3759(R) (1988).[2] M. B. Z¨olfl, T. A. Maier, T. Pruschke and J. Keller, Eur. Phys. J. B , 47 (2000).[3] A. Avella, F. Mancini, F. P. Mancini and E. Plekhanov, Eur. Phys. J. B , 265 (2013).[4] A. Macridin, M. Jarrell, T. Maier and G. A. Sawatzky, Phys. Rev. B , 134527 (2005). a b cd e f 𝛽 = e V - 𝛽 = e V - Cu Reference O(x) Reference O(y) Reference
FIG. S2:
Temperature dependence of the orbital structure of the pairs in the three-band model for the CuO plane . Each panel plots the real space components of the leading (symmetrized) eigenvector of the Bethe-Salpeter equation.The top and bottom rows show results obtained on a N Cu = 6 × n h = 1 .
15 at an inverse temperature β = 10 eV − and β = 12 eV − respectively. The remaining model parameters are (in units of eV) t pd = 1 . t pp = 0 . . U pp = 0, and U dd = 8 .
5. The left column describes the pairing between a Cu d reference site and all other orbitalsas a function of distance. The middle column describes pairings with respective to a p x orbital reference. The right columndescribes pairings with respective to a p y orbital orbital reference. All panels set the Cu 3 d orbital at the origin, as labeled bythe open ring. Only slight changes are observed in the pair structure between these two temperatures.[5] Y. F. Kung et al ., Phys. Rev. B , 155166 (2016).[6] M. T. Czy˙zyk and G. A. Sawatzky, Phys. Rev. B , 14211 (1994).[7] S. Johnston, F. Vernay, and T. P. Devereaux, Eur. phys. Lett. , 37007 (2009).[8] Y. Ohta, T. Tohyama, and S. Maekawa, Phys. Rev. B , 2968 (1991). a b cd e f % e doped15 % h doped Cu Reference O(x) Reference O(y) Reference
FIG. S3:
The orbital structure of the pairs in the three-orbital model for the CuO plane. Each panel plots the realspace components of the leading (symmetrized) eigenvector of the Bethe-Salpeter equation. The top and bottom rows showresults for hole- ( n h = 1 .
15) and electron-doping ( n h = 0 . N Cu = 4 × β = 16 eV − . The remaining model parameters are (in units of eV) t pd = 1 . t pp = 0 .
45, ∆ = 3 . U pp = 0, and U dd = 8 .
5. The left column describes the pairing between a Cu d reference site and all other orbitals as a function of distance.The middle column describes pairings with respect to a p x orbital reference. The right column describes pairings with respectto a p y orbital orbital reference. All panels set the Cu 3 d orbital at the origin, as labeled by the open ring. Compared withFig. 2 in the main text, the 4 × × a b cd e f % e doped15 % h doped Cu Reference O(x) Reference O(y) Reference
FIG. S4:
Effect of finite U pp on the orbital structure of the pairs in the three-band model with finite oxygenCoulomb repulsion U pp . Each panel plots the real space components of the leading (symmetrized) eigenvector of the Bethe-Salpeter equation. The first and second rows show results for hole- ( n h = 1 .
15) and electron-doping ( n h = 0 . N Cu = 4 × β = 10 eV − and finite U pp = 4 .
1. The remaining modelparameters are (in units of eV) t pd = 1 . t pp = 0 .
45, ∆ = 3 .
24 and U dd = 8 .
5. Compared to Fig. S3, the effect of a finite U pp is very weak, with only a very slight suppression of the components that involve the O- p orbitals. a b cd e f d-d d-L L-Ld-L’ L-L’ L’-L’ FIG. S5:
The L -related components of the Cooper pairs in the three-band model for the CuO plane. d - L , L - L , L - L pairing components are presented in panel d , e , f , respectively, as compared with Panel a - c from Fig.3 in the maintext. All results were obtained on a N Cu = 6 × n h = 1 .
15 and at an inverse temperature β = 10 eV − .The remaining model parameters are (in units of eV) t pd = 1 . t pp = 0 . U pp = 0, and U dd = 8 .
5. The same scale is usedfor the size of the points in the top and bottom rows. The pairing with the L0