Abstract
The nature of the effective interaction responsible for pairing in the hightemperature superconducting cuprates remains unsettled. This question has been studied extensively using the simplified singleband Hubbard model, which does not explicitly consider the orbital degrees of freedom of the relevant CuO_{2} planes. Here, we use a dynamical cluster quantum Monte Carlo approximation to study the orbital structure of the pairing interaction in the threeband Hubbard model, which treats the orbital degrees of freedom explicitly. We find that the interaction predominately acts between neighboring copper orbitals, but with significant additional weight appearing on the surrounding bonding molecular oxygen orbitals. By explicitly comparing these results to those from the simpler singleband Hubbard model, our study provides strong support for the singleband framework for describing superconductivity in the cuprates.
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Introduction
Cuprate superconductivity emerges in their quasitwodimensional (2D) CuO_{2} planes after doping additional carriers into these layers. The undoped parent compounds are chargetransfer insulators due to the large Coulomb repulsion U_{dd} on the Cu 3d orbitals and comparatively smaller charge transfer energy, and, to a good approximation, a spin\(\frac{1}{2}\) hole is located on every Cu 3\({d}_{{x}^{2}{y}^{2}}\) orbital. This situation is welldescribed by a halffilled 2D square lattice Hubbard model or Heisenberg model in the large U_{dd} limit.
Upon doping, the additional holes or electrons primarily occupy the O or Cu orbitals, respectively. The minimal model capturing this asymmetry is the threeband Hubbard model, which explicitly accounts for the Cu \(3{d}_{{x}^{2}{y}^{2}}\), O 2p_{x}, and 2p_{y} orbitals (Fig. 1a)^{1}. Even at finite doping, the low energy sector of the threeband model can be mapped approximately onto an effective singleband Hubbard model^{2}. One expects this in the case of electron doping since the additional carriers go directly onto the Cu sublattice, on which the holes of the undoped materials already reside. The case of hole doping, however, is more subtle. Here, the additional carriers predominantly occupy the O sublattice due to the large U_{dd} on the Cu orbital, and the appropriateness of a singleband model is less clear. In their seminal work, Zhang and Rice^{2} argued that the doped hole effectively forms a spinsinglet state with a Cu hole, the “Zhang–Rice singlet” (ZRS, Fig. 1b), which then plays the same role as a fully occupied or empty site in an effective singleband model, again facilitating a singleband description.
The nature of the singleband 2D Hubbard model’s pairing interaction has been extensively studied^{3,4,5,6,7,8}. Detailed calculations of its momentum and frequency structure using dynamical cluster approximation (DCA) quantum Monte Carlo (QMC)^{3} find that it is welldescribed by a spinfluctuation exchange interaction^{4}. The singleband model, however, cannot provide any information on the orbital structure of the interaction. For example, in the holedoped case, the spins giving rise to the spinfluctuation interaction are located on the Cu sublattice, while the paired holes are moving on the O p_{x/y} sublattice. This situation can produce a different physical picture than if the interaction and the pairs both originate from the same orbital on the same lattice^{9,10,11,12,13}. And indeed, studies have observed twoparticle behavior in a two sublattice system that is not observed in a onelattice system^{14}. Moreover, an analysis of resonant inelastic Xray scattering data has found that a singleband model fails to describe the highenergy magnetic excitations near optimal doping^{15}. For these reasons, numerous numerical studies of the threeband Hubbard model have been carried out to date^{16,17,18,19,20,21,22,23,24,25,26,27,28}; however, the crucial task of studying its effective interaction, and, in particular, determining its orbital structure is currently lacking. Such a study will also provide new insight into the nature of hightemperature superconductivity that is not available from the previous singleband studies. Here, we use a QMCDCA method to explicitly calculate the orbital and spatial structure of the effective interaction in a realistic threeband CuO_{2} model, and compare the results with those obtained from a singleband model.
Results
Pairing structure of the threeband model
To study the structure of the pairing interaction, we solved the Bethe–Salpeter equation (BSE) in the particle–particle singlet channel to obtain its leading eigenvalues and eigenvectors (see the online supplementary materials)^{3} (see “Methods”). Figure 1c shows the leading eigenvalue of the BSE for the threeband model as a function of hole concentration n_{h} obtained on a 4 × 4 cluster with β = 1/k_{B}T = 16 eV^{−1}. We find that it always corresponds to a dwave superconducting state^{29} and is larger for hole doping (n_{h} > 1) compared to electron doping (n_{h} < 1). The latter observation suggests a particlehole asymmetry in T_{c} consistent with experiments and prior studies of the single and twoband Hubbard models^{30,31}. (Although λ_{d} is largest at halffilling, we expect that it asymptotically approaches one as the temperature decreases but never actually crosses one due to the opening of a Mott gap. We observe such behavior in explicit calculations on smaller threeband clusters (see Supplementary Fig. 1 in the online supplementary materials).
We now analyze the spatial and orbital structure of the leading eigenvector ϕ_{αβ}(k) (α and β denote orbitals), by Fourier transforming ϕ_{αβ}(k) to real space to obtain \({\phi }_{{{\bf{r}}}_{\beta }}({{\bf{r}}}_{\alpha })\), where r_{β} denotes the position of the orbital taken as the reference site. We employed a 6 × 6 cluster to allow for longranged pairing correlations at T = 0.1 eV. While this relatively high temperature is needed to mitigate the Fermion sign problem, we have found that the leading eigenvector changes very slowly as the system cools (see Supplementary Fig. 2 in the online supplementary materials). For example, we can reach much lower temperatures on 2 × 2 clusters, where we resolve the superconducting T_{c} explicitly (see Supplementary Fig. 1 in the online supplementary materials). In that case, we observe that while the eigenvalue has a strong temperature dependence near T_{c}, its corresponding eigenvector does not vary much with temperature. From here on, we focus on results obtained at optimal (15%) hole or electron doping. We have obtained similar results for different cluster sizes and for finite U_{pp} (see Supplementary Figs. 3 and 4 in the online supplementary materials), indicating that our conclusions are robust across much of the model phase space.
In the singleband Hubbard model, the pairs are largely comprised of carriers on nearestneighbor sites in a dwave state, i.e., with a positive (negative) phase along the x (y)directions. The internal structure of the pairs in the threeband model seems more complicated^{32}. The realspace structure of \({\phi }_{{{\bf{r}}}_{\beta }}({{\bf{r}}}_{\alpha })\) shown in Fig. 2a–f for the hole and electrondoped cases, respectively, display an extended and rich orbital structure. Here, the size and color of the data points indicate the strength and phase of \({\phi }_{{{\bf{r}}}_{\beta }}({{\bf{r}}}_{\alpha })\), respectively, on each site after adopting the central Cu \(3{d}_{{x}^{2}{y}^{2}}\) or O 2p_{x,y} orbital as a reference site at r_{β}. The form factors \({\phi }_{{{\bf{r}}}_{\beta }}({{\bf{r}}}_{\alpha })\) are similar for both electron and hole doping, decaying over a length scale of ~2–3 lattice constants. Moreover, while the dwave pairing between nearest Cu sites dominates, there is also a significant contribution from d–p pairing, with a comparable amplitude for up to the thirdnearest (unitcell) neighbors. The pairing between the individual O 2p_{x,y} orbitals is much weaker in comparison.
Pairing structure in the molecular basis
We now transform the leading eigenvector for the holedoped case from the O p_{x} and p_{y} basis to the bonding L and antibonding \({L}^{\prime}\) basis (Fig. 1d). These combinations, formed from the four O orbitals surrounding a Cu cation, are the relevant states for the ZRS, in which the doped holes are argued to reside in. The bonding L state strongly hybridizes with the central Cu 3\({d}_{{x}^{2}{y}^{2}}\) orbital (Fig. 1b), while the antibonding \({L}^{\prime}\) state does not. The resulting antiferromagnetic exchange interaction between the Cu and L holes is then argued to bind them into the Zhang–Rice spinsinglet state, which provides the basis for the mapping onto a singleband model.
The orbital structure of the leading eigenvector simplifies considerably after one transforms to the bonding L and antibonding \({L}^{\prime}\) combinations. Figure 3 plots the pairing amplitudes for a hole on Cu paired with another hole on a neighboring Cu (d–d, Fig. 3a) or bonding molecular orbital (d–L, Fig. 3b). Both components exhibit a clear \({d}_{{x}^{2}{y}^{2}}\) symmetry that is dominated by the (nearestneighbor) \(\cos ({k}_{x}a)\cos ({k}_{y}a)\) harmonic; however, both channels also have indications of additional higherorder harmonics [i.e., \(\cos (2{k}_{x}a)\cos (2{k}_{y}a)\) and \(\cos (2{k}_{x}a)\cos ({k}_{y}a)\cos (2{k}_{y}a)\cos ({k}_{x}a)\)]. Interestingly, the contribution from holes occupying neighboring bonding molecular orbitals exhibits similar behavior (L–L, Fig. 3c). The \({L}^{\prime}\)related pairing contributes very little as will be discussed in Fig. 4 and in the supplement (see Supplementary Fig. 5 in the online supplementary materials).
Figure 3a–c establishes that the pairing between the different orbital components of the ZRS all possesses the requisite \({d}_{{x}^{2}{y}^{2}}\) symmetry. This observation indicates that the ZRS picture—a singlet state made up of holes in the d and L orbitals—is valid for describing pairing correlations in the threeband Hubbard model of the cuprates. To show the pair structure for the ZRS, we plot in Fig. 3d the sum over the d–d, d–L, L–d, and L–L components (with a factor of 0.5 applied to all components). One sees that the ZRS pair structure has a vanishing \(\cos (2{k}_{x}a)\cos (2{k}_{y}a)\) component, while higherorder harmonics remain.
To compare this result to the singleband picture, we also computed the realspace structure of the leading particle–particle BSE eigenvector for the singleband Hubbard model. Here, we considered cases with nextnearestneighbor hopping \({t}^{\prime}/t=0\) (panel e), −0.2 (f), −0.3 (g), which are commonly used in the literature, as well as −0.4 (h). The singleband model reproduces the shortrange pairing structure of the threeorbital model (panel d), regardless of the value of \({t}^{\prime}\); however, the longerranged pairing in Fig. 3d is only captured correctly for large \(\left\right.{t}^{\prime}/t\left\right.\). In particular, we observe that with increasing \(\left\right.t^{\prime} /t\left\right.\), the relative amplitude of the thirdnearestneighbor \([\cos (2{k}_{x}a)\cos (2{k}_{y}a)]\) term is suppressed. For \({t}^{\prime}/t=0.4\) (panel h), the singleband pair structure is very similar to that for the ZRS (panel d), with differences appearing at the longest length scales. This value of \({t}^{\prime}\) is close to the value \({t}^{\prime}=0.453t\) that we obtain by downfolding our threeband model parameters onto the singleband model by diagonalizing small Cu_{2}O_{7} clusters^{33,34}. A sizeable negative \({t}^{\prime}\) is also consistent with parametrizations of the bandstructure extracted from angleresolved photoemission spectroscopy^{35}. These results provide remarkable support for the validity of the ZRS construction but also indicate that singleband models may not capture the correct longerranged correlations without a suitable choice of \({t}^{\prime}\). The latter conclusion further underscores the crucial role of \({t}^{\prime}\) for determining the superconducting properties of the singleband model^{8,31,36,37}.
Weights for orbitalresolved pair components
Figure 3 shows that the structure of the leading eigenvector ϕ_{αβ} is closely linked to the orbital structure of the ZRS. Figure 4 examines how this internal structure evolves with doping by plotting the orbitaldependent hole density (panel a) and the orbital composition of the eigenvector ϕ_{αβ} (panel b) on a 4 × 4 cluster (adequate to capture the essential pairing structure) at a lower temperature. Figure 4a shows that the single hole per unit cell in the undoped case has ~65% Cud character, while 35% of the hole is located in the bonding O–L molecular orbital. With electron doping, there is a small decrease of n_{d}/n_{h} indicating that the holes are removed mainly from the Cud orbital. In contrast, with hole doping, there is a significant redistribution of the hole density from the d to the Lorbital, showing that doped holes mainly occupy the O–L molecular orbital. The hole density on the antibonding O\({L}^{\prime}\) orbital is negligible.
Figure 4b shows that the total weight of the nearestneighbor pairing increases from ~70% in the undoped case to almost 100% with either hole or electron doping. Since the BSE eigenvector reflects the momentum structure of the pairing interaction, this dependence can be understood from an interaction that becomes more peaked in momentum space as n_{h} = 1 is approached. This behavior leads to a more delocalized structure of \({\phi }_{{{\bf{r}}}_{\beta }}({{\bf{r}}}_{\alpha })\) and, therefore, a reduction of the relative weight of the nearestneighbor contribution. The partial contributions to the nearestneighbor pairing weight, D_{d} and D_{dL}, have a doping dependence very similar to the corresponding orbital densities n_{d} and n_{L} in panel a, closely linking the orbital structure of the pairing to the orbital makeup of the ZRS. The weight of the \({L}^{\prime}\) contributions remains negligible over the full doping range (see Supplementary Fig. 5 in the online supplementary materials).
Discussion
We have determined the orbital structure of the effective pairing interaction in a threeband CuO_{2} Hubbard model and shown that it simplifies considerably when viewed in terms of a basis consisting of a central Cud orbital and a bonding L combination of the four surrounding Op orbitals. These states underlie the ZRS singlet construction that enables the reduction of the threeband to an effective singleband model. By explicitly comparing the threeband with singleband results, we showed that the effective interaction is correctly described in the singleband model. In summary, these results strongly support the conclusion that a singleband Hubbard model provides an adequate framework to understand highT_{c} superconductivity in the cuprates.
Methods
Model parameters
The threeband Hubbard model we study can be found in ref. ^{18} (see the online supplementary materials). We adopted a parameter set appropriate for the cuprates^{18,38,39,40} (in units of eV): the nearestneighbor Cu–O and O–O hopping integrals t_{pd} = 1.13, t_{pp} = 0.49, onsite interactions U_{dd} = 8.5, U_{pp} = 0, and chargetransfer energy Δ = ε_{p} − ε_{d} = 3.24, unless otherwise stated. Since we use a hole language, halffilling is defined as hole density n_{h} = 1 and n_{h} > 1( < 1) corresponds to hole (electron)doping. A finite U_{pp} only leads to small quantitative changes in the results (see Supplementary Fig. 4 in the online supplementary materials) but worsens the sign problem significantly^{18}. Therefore, we keep U_{pp} = 0 for this study.
Dynamical cluster approximation
We study the single and threeband Hubbard models using DCA with a continuoustime QMC impurity solver^{41,42,43,44,45}. We determine the structure of the pairing interaction by solving the BSE in the particle–particle singlet channel to obtain its leading eigenvalues and (symmetrized) eigenvectors^{3} (see the online supplementary materials). A transition to the superconducting state occurs when the leading eigenvalue λ(T = T_{c}) = 1, and the magnitude of λ < 1 measures the strength of the normal state pairing correlations. The spatial, frequency, and orbital dependence of the corresponding eigenvector, which is the normal state analog of the superconducting gap, reflects the structure of the pairing interaction^{3,8}.
Data availability
The data that support the findings of this study can be obtained at https://github.com/JohnstonResearchGroup/Mai_etal_3bandPairs_2021.
Code availability
The DCA++ code used for this project can be obtained at https://github.com/CompFUSE/DCA.
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Acknowledgements
The authors would like to thank L. Chioncel, P. Dee, A. Georges, K. Haule, E. Huang, S. Karakuzu, G. Kotliar, H. Terletska, and D.J. Scalapino for useful comments. This work was supported by the Scientific Discovery through Advanced Computing (SciDAC) program funded by the U.S. Department of Energy, Office of Science, Advanced Scientific Computing Research and Basic Energy Sciences, Division of Materials Sciences and Engineering. An award of computer time was provided by the Innovative and Novel Computational Impact on Theory and Experiment (INCITE) program. This research used resources of the Oak Ridge Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DEAC0500OR22725.
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P.M., G.B., and T.A.M. developed the DCA++ code. P.M. carried out the calculations. P.M., S.J., and T.A.M. analyzed the results and wrote the manuscript. S.J. and T.A.M. supervised the project.
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Mai, P., Balduzzi, G., Johnston, S. et al. Orbital structure of the effective pairing interaction in the hightemperature superconducting cuprates. npj Quantum Mater. 6, 26 (2021). https://doi.org/10.1038/s41535021003265
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DOI: https://doi.org/10.1038/s41535021003265
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