Abstract
The recently discovered high-Tc superconductor La3Ni2O7 has sparked renewed interest in unconventional superconductivity. Here we study superconductivity in pressurized La3Ni2O7 based on a bilayer two-orbital t−J model, using the renormalized mean-field theory. Our results reveal a robust s±-wave pairing driven by the inter-layer \({d}_{{z}^{2}}\) magnetic coupling, which exhibits a transition temperature within the same order of magnitude as the experimentally observed Tc ~ 80 K. We establish a comprehensive superconducting phase diagram in the doping plane. Notably, the La3Ni2O7 under pressure is found to be situated roughly in the optimal doping regime of the phase diagram. When the \({d}_{{x}^{2}-{y}^{2}}\) orbital becomes close to half-filling, d-wave and d + is pairing can emerge from the system. We discuss the interplay between Fermi surface topology and different pairing symmetries. The stability of the s±-wave pairing against Hund’s coupling and other magnetic exchange couplings is discussed.
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Introduction
Understanding high transition temperature (Tc) superconductivity remains one of the greatest challenges in condensed matter physics. For cuprate superconductors1,2,3, the fundamental mechanism of the d-wave pairing is believed to primarily lie in the \({d}_{{x}^{2}-{y}^{2}}\) orbital, with strong Coulomb repulsion between electrons playing a crucial role2. This form of superconductivity is usually referred to as unconventional, distinguishing it from the traditional Bardeen–Cooper–Schrieffer (BCS) type of superconductivity. Another prominent example of unconventional superconductivity is found in iron-based superconductors4,5,6,7,8, where multiple d-orbitals participate in the pairing process. The discovery of superconductivity in infinite-layer Nd1−xSrxNiO2 thin films has sparked widespread interest due to cuprate-like Ni1+(3d9)9,10,11,12,13,14,15,16. Most recently, a Ruddlesden–Popper nickelate superconductor La3Ni2O7 is found with a Tc ≈ 80 K17,18,19 under moderate pressures. On the one hand, La3Ni2O7 shares similarities with cuprates, as both feature the NiO2/CuO2 plane containing a key \({d}_{{x}^{2}-{y}^{2}}\) orbital at the Fermi level. On the other hand, La3Ni2O7 differs from cuprates as its apical O-pz and Ni-\({d}_{{z}^{2}}\) orbitals play a significant role in its low-energy physics17,20. Given this context, one would ask whether the underlying pairing mechanism of La3Ni2O7 resembles that of cuprates? How does it differ from the extensively studied cuprates? From a theoretical perspective, the first step in addressing these questions is to resolve the roles played by the \({d}_{{x}^{2}-{y}^{2}}\) and \({d}_{{z}^{2}}\) orbitals in pairing and to construct a superconducting phase diagram of the relevant physical models.
Electronic structure studies21,22,23,24,25,26,27 and optical experimental probe28 suggest that in pressurized La3Ni2O7, Ni-\({d}_{{z}^{2}}\) orbital is involved in Fermi energy due to strong inter-layer coupling via apical oxygen. Through hybridization with the in-plane oxygen p-orbital, \({d}_{{z}^{2}}\) orbitals can mix with \({d}_{{x}^{2}-{y}^{2}}\) orbitals, resulting in a three-pocket Fermi surface structure21. This Fermi surface geometry differs from that of cuprates, which may lead to pairing symmetry and effective pairing “glue” distinct from that of the d-wave superconductivity in cuprates, which arises upon doping a single \({d}_{{x}^{2}-{y}^{2}}\) orbital. Another crucial consideration is electron occupancy. La3Ni2O7 typically exhibits a nominal valence configuration of d7.517,29, indicating an average of 2.5 holes in the active eg sub-shell. While previous studies have shown variations in the computed electron densities of the two eg orbitals30,31,32,33,34, it can be in general viewed as a multi-orbital system comprising a heavily hole-doped \({d}_{{x}^{2}-{y}^{2}}\) orbital (with a hole density ~ 1.5) and a near half-filled \({d}_{{z}^{2}}\) orbital (with a hole density ~ 1.0). Regarding superexchange couplings, investigation based on cluster dynamical mean-field theory30 has pointed out that the exchange coupling between two inter-layer \({d}_{{z}^{2}}\) orbitals J⊥ may be substantially larger, by a factor of at least ~2, than the intra-layer \({d}_{{x}^{2}-{y}^{2}}\) exchange coupling J∣∣, with the latter estimated to be comparable to its cuprate counterpart30. Such a significant J⊥ is likely responsible for the high transition Tc in La3Ni2O730,31,35,36,37,38,39.
In this paper, we systematically investigate superconductivity in the bilayer two orbital t−J model, which serves as a prototype for the low-energy physics of pressurized La3Ni2O7, using the renormalized mean-field theory (RMFT)40,41,42. The RMFT approach is a concrete implementation of Anderson’s resonating valence bond (RVB) concept of unconventional superconductivity43. Following the Gutzwiller scheme, the renormalization effects from strong electron correlations are accounted for on different levels by incorporating the doping-dependent renormalization factors gt, gJ40 in RMFT. The mean-field decomposition of the magnetic exchange couplings J enables exploring the BCS pairing instabilities within the system. Despite its straightforward formulations, RMFT has demonstrated its capability of capturing various aspects of cuprate superconductors, including superfluid density, the dome-shaped doping dependence of Tc, and pseudogap phenomena41. In RMFT, the superconducting order parameter gtΔ involves two competing energy scales: the phase coherence energy scale associated with gt, and the pairing amplitude represented by Δ, each exhibiting distinct doping dependencies40. In our study, we observe that the outcome of this competition positions La3Ni2O7 within the optimal doping regime in the superconducting phase diagram. Our calculation suggests a Tc comparable to experimental observations, indicating the relevance of our considerations to the La3Ni2O7 superconductor. Furthermore, we elucidate various possible pairing symmetries across a broad doping range.
The remainder of the paper is organized as follows. In the section “Methods”, we present the physical model and describe the RMFT method. In the section “Results”, we present our results, including Tc for the pristine compound, the doping phase diagram, and the Fermi surface. Additionally, the impact of the strength of superexchanges and Hund’s coupling is discussed at the end of the section. In the section “Discussion”, we discuss the stability of the s±-pairing. Finally, more details can be found in the Supplementary Information.
Results
Now we present the RMFT result on the superconducting instabilities of the bilayer two-orbital t−J model. In particular, we provide detailed investigations of the parameter regime that is most relevant to the La3Ni2O7 system. The impacts of several key factors, including temperature T, doping levels of \({d}_{{x}^{2}-{y}^{2}}\), and \({d}_{{z}^{2}}\) orbitals: px, pz, as well as the geometry of the Fermi surface are analyzed while eyeing the evolution of the superconducting order parameter \({g}_{t}^{\alpha \beta }| {\Delta }_{\ell }^{\alpha \beta }|\). The definition of \({g}_{t}^{\alpha \beta }| {\Delta }_{\ell }^{\alpha \beta }|\) and RMFT formalism can be found in the “Methods” section. For brevity, we adopt the shorthand notation \({g}_{t}^{\nu }| {\Delta }_{\ell }^{\nu }|\) in the following discussion to represent \({g}_{t}^{\alpha \beta }| {\Delta }_{\ell }^{\alpha \beta }|\), where ℓ = d, s±, denoting pairing symmetries, and ν = ∣∣x, ∣∣z, ⊥z denoting intra-layer \({d}_{{x}^{2}-{y}^{2}}\) [(α, β) = (x1, x1)], intra-layer \({d}_{{z}^{2}}\) [(α, β) = (z1, z1)] and inter-layer \({d}_{{z}^{2}}\) [(α, β) = (z1, z2)] pairing. Without loss of generality, we adopt typical values of J⊥ = 2J∣∣ = 0.18 eV taken from ref. 30 throughout the paper unless otherwise specified.
T c for pristine compound
We first present the RMFT calculated superconducting transition temperature Tc at μ = 0 that corresponds to pristine La3Ni2O7 under pressure, which is to be dubbed as the pristine compound (PC) case hereafter. For the PC case, we have μ = 0, nx = 0.665, nz = 0.835, and n = nx + nz = 1.521. In Fig. 1, the superconducting order parameter \({g}_{t}^{\nu }| {\Delta }^{\nu }_{\ell}|\) is plotted as a function of T, which clearly demonstrates two dominant branches of the pairing fields at small T: the intra-layer \({d}_{{z}^{2}}\) pairing (dashed line) and the inter-layer \({d}_{{z}^{2}}\) pairing (solid line), forming the s±-wave pairing of the system. This result is in agreement with several other theoretical studies36,38,39,44,45,46. As increasing T, the order parameters decrease in a mean-field manner and eventually drop to zero at around 80 K. The computed value of Tc somehow coincides with the experiment17, highlighting that the various energy scales under our consideration can effectively capture the major physics of the realistic compound. However, we would like to stress that the superconducting Tc from RMFT, in fact dependent on the value of J essentially in a BCS manner. Hence it can be sensitive to the strength of the superexchange couplings. The coincidence between the experimental and RMFT value of Tc should not be taken as the outcome of RMFT capturing La3Ni2O7 superconductivity in a quantitatively correct way. We note that the s±-wave pairing also has a finite \({d}_{{x}^{2}-{y}^{2}}\) orbital component, as shown by the dotted lines in Fig. 1, despite that its order parameters are much smaller than that of the \({d}_{{z}^{2}}\) orbitals. The d-wave order parameters (purple), on the other hand, are fully suppressed, suggesting that the d+ is—wave pairing instability may be ruled out in our model for La3Ni2O7. It is worth noting that RMFT was originally formulated at zero temperature40. In Fig. 1, we solve the RMFT equations in the finite temperature regime47. This extension captures finite temperature effects on the mean-field parameters Δ and χ while neglecting thermal effects on the renormalization factors \({g}_{t},{G}_{{J}_{r}}\). We argue that the latter should have insignificant impacts on determining Tc, given the hole concentrations in Fig. 1 (\({n}_{x}^{{\rm {h}}}=1-{n}_{x}\approx 0.335,{n}_{z}^{{\rm {h}}}=1-{n}_{z}\approx 0.165\)) are fairly large. Detailed discussions on this matter can be found in the Supplementary Note 2.
Doping evolution
Now we focus on the doping dependence of the superconducting order parameter \({g}_{t}^{\nu }| {\Delta }_{\ell }^{\nu }|\). In the following, p > 0 means doping the pristine compound (where nx = 0.665, nz = 0.835) with holes, and p < 0 denotes electron doping. While varying the doping level p of the system, we maintain a fixed ratio between the doping levels of the two eg orbitals, namely, px/pz = 2.048 is fixed, such that both eg orbitals can simultaneously reach half-filling (HF), i.e., nx = 1, nz = 1 at p = − 0.5. From Fig. 2, one learns that the s±-wave pairings (green) are quite robust over a wide range of doping p. The maxima of \({g}_{t}^{\nu }| {\Delta }_{{s}^{\pm }}^{\nu }|\) are located at p ≈ −0.04, which is very close to p = 0 for PC. This indicates that, interestingly, the La3Ni2O7 under pressure corresponds to roughly the optimal doping in our superconducting phase diagram. At extremely large electron dopings (p < −0.25), d-wave pairing can build up, which also exhibits a predominant superconducting dome (purple dotted line). In this doping regime, small s±-wave components of the superconducting order parameter are found to coexist with the d-wave components, indicating the emergence of d+ is—wave pairing. At half-filling in Fig. 2 (p = −0.5), all pairing channels are fully suppressed due to the vanishing renormalization factors \({g}_{t}^{\nu }\to 0\), reflecting the Mott insulting nature at half-filling30.
p > 0 represents hole doping the pristine compound case (where nx = 0.665, nz = 0.835), and p < 0 for electron doping. Black dot at p = 0 indicates the pristine compound (PC). We vary p while a fixed ratio of px/pz = 2.048 is maintained. The two eg orbitals reach half-filling (HF) (nx = 1.0, nz = 1.0) at p = − 0.5, as indicated by the diamond symbol.
It is worth noting that, as a general prescription of the mean-field approaches, different Jr terms in Eq. (4) can be decomposed into different corresponding pairing bonds Δδ, such as \({\Delta }_{d/{s}^{\pm }}^{\perp z}\) from decomposing J⊥, and \({\Delta }_{d/{s}^{\pm }}^{| | x}\) from decomposing J∣∣. Note that in our calculation, the pairing components \({\Delta }_{d/{s}^{\pm }}^{| | z}\) (dashed lines in Fig. 2) that represent the intra-layer pairing of \({d}_{{z}^{2}}\) orbital, do not have a corresponding J term in the Hamiltonian. Their values are not determined by the competition between \({\Delta }_{d/{s}^{\pm }}^{| | z}\) and \(| {\Delta }_{d/{s}^{\pm }}^{| | z}{| }^{2}\) terms in minimizing the free-energy. Instead, it should be interpreted as the pairing instability induced by the pre-existing inter-layer \({d}_{{z}^{2}}\) pairing. Indeed, as shown in Fig. 2, \({\Delta }_{{s}^{\pm }}^{| | z}\) displays a doping p-dependence similar to that of \({\Delta }_{{s}^{\pm }}^{\perp z}\). Finally, we notice that a small tip appears at hole doping p ~ 0.4, which can be attributed to the van Hove singularity associated with the β-sheet of the Fermi surface, see also the section “Fermi surfaces”.
Doping phase diagram
To gain further insights into the RMFT result of the La3Ni2O7 system, we obtain a phase diagram in the px−pz doping plane where now the two dopings are independent variables. As shown in Fig. 3, RMFT reveals that d, s±, and d+ is pairing symmetries, as well as the normal state occurs in different doping regimes. Here a dashed white line indicates the px−pz trajectory along which Fig. 2 is plotted. The black symbols label out sets of (px, pz) parameters on which the result will be further discussed in Fig. 5. The major feature of Fig. 3 is that the d-wave and s±-wave pairings span roughly a vertical and a horizontal stripe, respectively, in the phase diagram. In other words, s±-wave pairing (green) dominates the regime where −0.1 ≲ pz ≲ 0.1 (0.735 ≲ nz ≲ 0.935), and it is insensitive to the value of px. Likewise, the d-wave pairing (purple) prevails in the doping range of −0.335 ≲ px ≲ −0.2 (0.465 ≲ nx ≲ 1.0), and it is, in general, less sensitive to the value of pz. As a result, the d+ is—wave pairing (orange) naturally emerges at the place where the two stripes overlap. In order to have a better understanding of this phase diagram, we show the magnitudes of the four major pairing bond \({g}_{t}^{\nu }| {\Delta }_{\ell }^{\nu }|\) in Fig. 4, from which one sees that for s±-wave pairing, the pairing tendencies of \({\Delta }_{{s}^{\pm }}^{\perp z}\) (Fig. 4a) and \({\Delta }_{{s}^{\pm }}^{| | z}\) (Fig. 4b) show a similar pattern in the px−pz plane, in consistence with Fig. 2. For the d − wave pairing, the situation is however different. For intra-layer pairing, \({g}_{t}^{| | x}| {\Delta }_{d}^{| | x}|\) is enhanced when \({d}_{{x}^{2}-{y}^{2}}\) is heavily electron doped (px ≈ −0.25), and \({d}_{{z}^{2}}\) becomes half-filling (pz ~ −0.2). This is because the large electron doping drives the γ-pocket of \({d}_{{z}^{2}}\) orbital centered at M-point descends into the Fermi sea. Such that the system becomes effectively a single band system of the active \({d}_{{x}^{2}-{y}^{2}}\) orbital. Hence the dominant d-wave pairing of the single-band t−J model is recovered for \({d}_{{x}^{2}-{y}^{2}}\) orbital in this limit, mimicking the physics of cuprates. On the other hand, the \({d}_{{z}^{2}}\) component of the d-wave pairing \({g}_{t}^{| | z}| {\Delta }_{d}^{| | z}|\) is enhanced when pz > 0.2, as shown in Fig. 4d. This can be understood considering the fact that since intra-layer exchange of \({d}_{{z}^{2}}\) orbital J∣∣z = 0 in our study, the d-wave instability driven by J∣∣ of the \({d}_{{x}^{2}-{y}^{2}}\) orbital is less sensitive to the details of \({d}_{{z}^{2}}\) orbital. Hence, the superconducting order parameter \({g}_{t}^{| | z}| {\Delta }_{d}^{| | z}|\) increasing with pz shown in Fig. 4d can bee seen vastly as a result of a growing \({g}_{t}^{| | z}\) with decreasing nz according to Eqs. (5) and (6). Finally, it is interesting to note that although J⊥ = 2J∣∣ is used in our study, Fig. 4 shows that the maximal value of d-wave superconducting order parameter is roughly two times larger than that of the s±-wave pairing. This is expected since the vertical exchange coupling J⊥ has a smaller coordination number, z = 1, compared to its in-plane counterpart, J∣∣, where z = 4.
pα > 0 indicates hole doping and pα < 0 denotes electron doping relative to the pristine compound case (nx = 0.665, nz = 0.835). The dashed white line presents the px−pz trajectory along which Fig. 2 is plotted. The black symbols mark the sets of (px, pz) that are further discussed in Fig. 5. The superexchange couplings applied are J⊥ = 2J∣∣ = 0.18.
Upper panel: s±-wave pairing for a inter-layer and b intra-layer \({d}_{{z}^{2}}\) orbitals. Lower panel: d-wave pairing for intra-layer c \({d}_{{x}^{2}-{y}^{2}}\) and d \({d}_{{z}^{2}}\) orbitals. Again, here (px = 0, pz = 0) corresponds to (nx = 0.665, nz = 0.835). Symbols denote typical dopings to be analyzed in Fig. 5.
Fermi surfaces
In Fig. 5 we display the paring gap function \({\Delta }_{\ell }^{\nu }\) projected onto the Fermi surfaces for four typical sets of dopings (px, pz) with each characterizing one type of pairing symmetries (Fig. 5a, b, d) or the one with vanishing superconducting order parameter (Fig. 5c). Figure 5a shows the FS of the PC case with s± pairing, which has three sheets of FS with one around Γ point and two around M point21. Also, the sign structure is consistent with that in refs. 36,39,45. The key feature here is that the sign of the pairing gap is the same within the α, γ bands, but it reverses between α/γ and β pockets. This is because the former two components come from the bonding state while the latter is from the anti-bonding state. Decreasing px from the PC can drive the \({d}_{{x}^{2}-{y}^{2}}\) orbital closer to half-filling. As shown in Fig. 5b, the α, β-sheet of FS as a whole is also driven closer to the folded Brillouin zone (FBZ) edge (dashed lines), which is accompanied by the pairing symmetry evolving from s± to d + is- wave. In this case, we see that the α, β pockets exhibit the d-wave sign structure while the γ pocket maintains the s-wave symmetry, and its profile is less affected by the changing of px. The occurrence of the d-wave pairing at this doping level unambiguously signals the importance of intra-orbital physics in \({d}_{{x}^{2}-{y}^{2}}\) orbital as it approaches half-filling. Fig. 5c displays that as lowing pz from the PC case, the γ-pocket vanishes from the Brillouin zone. Consequently, the s±-wave order parameter vanishes at pz ~ −0.15. In this case, similar to PC, no finite d-wave order parameter is observed. Note that the gap magnitude in Fig. 5 is associated with \({\Delta }_{\ell }^{\nu }\) instead of the superconducting order parameter \({g}_{t}^{\nu }| {\Delta }_{\ell }^{\nu }|\), which is why Fig. 5c shows nonvanishing gap for a normal state, see also Supplementary Note 4. Figure 5d shows the last case with the γ-pocket vanishing from the Fermi level. As expected, in this case, only the d-wave pairing with finite \({g}_{t}^{| | x}| {\Delta }_{d}^{| | x}|\) is found. As discussed above, here the physics of the system can be essentially captured by the single-band t−J model with the presence of only \({d}_{{x}^{2}-{y}^{2}}\) orbital.
a Dot: (px = 0, pz = 0), b square: (px = −0.28, pz = 0), d upper triangle: (px = −0.2, pz = −0.15), and c lower triangle: (px = 0, pz = −0.15). See also the symbols indicated in Figs. 3 and 4. Note that the gap magnitudes are directly associated with \({\Delta }_{\ell }^{\nu }\) instead of the superconducting order parameter \({g}_{t}^{\nu }| {\Delta }_{\ell }^{\nu }|\), which is why (c) shows nonvanishing gap for a normal state. The dashed lines indicate the folded Brillouin zone (FBZ) of the two inplane Ni.
Superexchanges J
Finally, we investigate the influence of the magnitudes of the superexchanges for the pristine compound. In Fig. 6, we present \({g}_{t}^{\nu }| {\Delta }_{\ell }^{\nu }|\) as a function of J∣∣/J⊥. A vertical arrow indicates the value of J∣∣/J⊥ used in the afore-mentioned calculations, where s±-wave is found for PC. As decreasing/increasing J∣∣/J⊥, s±-wave order parameters (green lines) decrease/increase very slightly with J∣∣/J⊥. When J∣∣/J⊥ ~ 1.1, d-wave (purple solid line) starts to build up at \({d}_{{x}^{2}-{y}^{2}}\) orbital. For La3Ni2O7 under pressure, this large value of J∣∣/J⊥ is, however, not very realistic30. Hence, the d-wave pairing instability may be excluded from the realistic materials in our study. To check the stability of the superconductivities, results for Jxz = 0.03 (dashed line) and JH = −1 eV (dash-dotted line) are shown in Fig. 6. As one sees that, although both Jxz and JH act as pair-breaking factors, they do not significantly modify the result we obtained above. In particular, for the s±-pairing, the changes of the order parameter \({g}_{t}^{\nu }| {\Delta }_{\ell }^{\nu }|\) caused by Jxz (green dashed line) and JH (not shown here) are negligible.
Discussion
In our study, the RMFT equations are solved in such a way that the pairing fields on different bonds are varied independently, namely, no specific pairing symmetry is presumed in the self-consistent process. The symmetries of the electron pairing naturally emerge as a result of energy minimization in our calculations, preventing the potential overlooking of pairing symmetries. Regarding the dominant s± pairing for the pristine compound at px, pz = 0, we note that even when only J⊥ is considered (with J∣∣ = 0, Jxz = 0), Δ∣∣x is finite although much smaller than Δ∣∣z and Δ⊥z. Given the smaller effective mass of the \({d}_{{x}^{2}-{y}^{2}}\) orbital compared to the \({d}_{{z}^{2}}\) orbital, it may still contribute significantly to the superfluid density in the superconducting state of the system. The dual effects of the Hund’s coupling JH, namely, the alignment of the on-site spins of the \({d}_{{x}^{2}-{y}^{2}}\) and \({d}_{{z}^{2}}\) orbitals, and the enhancement of J∣∣ and Jxz couplings, show no significant impact on the dominant s± pairing in our RMFT study, as indicated in Fig. 6. However, we acknowledge that the implications of JH on superconductivity may be underestimated in the RMFT formalism48. Additionally, it is important to note that the superconducting Tc obtained by RMFT is generally overestimated because both temporal and spatial fluctuations are neglected. This is particularly true when considering that the pairing fields originate from the local inter-layer \({d}_{{z}^{2}}\) magnetic couplings, where phase fluctuations can play a more significant role in suppressing Tc comparing to its single-band t−J model counterpart in cuprate superconductors. Finally, a recent theory work31 proposes that within the framework of the two-component pairing theory in a composite system, the phase fluctuations can be suppressed by the hybridization effects between \({d}_{{x}^{2}-{y}^{2}}\) and \({d}_{{z}^{2}}\) orbitals. Verifying this conjecture is, however, beyond the scope of this work.
Employing the renormalized mean-field theory, we have established a comprehensive superconducting phase diagram for the bilayer two-orbital t−J model. A robust s±-wave pairing is found to exist in the parameter regime relevant to La3Ni2O7 under pressure, which, in general, corresponds to the optimal doping of the superconducting phase diagram. We have carefully investigated the dependence of the pairing instabilities on doping levels, exchange couplings, and the effects of Hund’s coupling. Our study provides significant insights into the theoretical understanding of the superconductivity of La3Ni2O7 under pressure.
Methods
Model
In the strong-coupling limit, the bilayer two orbital Hubbard model21 can be mapped into a t−J model30
where \({{\mathcal{H}}}_{t}\) is the tight-binding Hamiltonian taken from downfolding the DFT band structure21, which is defined in a basis of \({\Psi }_{i\sigma }={({c}_{i{x}_{1}\sigma },{c}_{i{z}_{1}\sigma },{c}_{i{x}_{2}\sigma },{c}_{i{z}_{2}\sigma })}^{\rm {{T}}}\), with ciασ representing annihilation of an electron with spin σ on α = x1, z1, x2, z2 orbital at i site. Here x1, x2 denote the two Ni-\(3{d}_{{x}^{2}-{y}^{2}}\) orbitals situated in the double NiO2 layer while z1, z2 correspondingly denote the two Ni-\(3{d}_{{z}^{2}}\) orbitals. μ is the chemical potential and ϵxz is the energy difference between \({d}_{{x}^{2}-{y}^{2}}\) and \({d}_{{z}^{2}}\) orbitals. The hopping parameters in \({{\mathcal{H}}}_{t}\) used in this work can be found in Supplementary Note 1. \({{\mathcal{H}}}_{J}\) is the Heisenberg exchange couplings, and the spin operator \({{\boldsymbol{S}}}_{i\alpha }=\frac{1}{2}{\sum }_{\mu \nu }{c}_{i\alpha \mu }^{\dagger }{{\boldsymbol{\sigma }}}_{\mu \nu }{c}_{i\alpha \nu }\). According to the estimated antiferromagnetic correlations in La3Ni2O730, there can be three major magnetic exchange couplings J⊥, J∣∣, Jxz, which, respectively, represent nearest-neighbor inter-layer exchange of \({d}_{{z}^{2}}\) orbital, intra-layer exchange of \({d}_{{x}^{2}-{y}^{2}}\) and intra-layer exchange between \({d}_{{x}^{2}-{y}^{2}}\) and \({d}_{{z}^{2}}\). JH is the intra-atomic Hund’s coupling.
RMFT formalism
To proceed with RMFT40, we first define the following mean-field parameters
where \({\chi }_{ij}^{\alpha \beta }\) and \({\Delta }_{ij}^{\alpha \beta }\) are particle–hole and particle–particle pairs relating iα and jβ. Here we assume no magnetic ordering. For each type of exchange coupling Jr in Eq. (1), including Hund’s coupling JH, the mean-field decomposition of \({{\mathcal{H}}}_{J}\) introduces condensations of χ and Δ in the corresponding r-channel
Assuming translation symmetry, δ = Rjβ−Riα denotes different bonds in real-space associated with Jr, and N is the total number of sites of the square lattice.
We now introduce two renormalization factors49
These two quantities essentially reflect the renormalization effects by the electron's repulsions on top of the single-particle Hamiltonian in the Gutzwiller approximation40,42,50,51, which depend on the orbital occupation number nα = nα↑ + nα↓. This eventually leads to the renormalized mean-field Hamiltonian
with \({g}_{t}^{\alpha \beta }={G}_{t}^{\alpha }{G}_{t}^{\beta }\). One sees that, when \({t}_{ij}^{\alpha \beta }={t}_{ij}{\delta }_{\alpha \beta }\), the above Hamiltonian reduces to the classical formulas of the single-band t−J model for cuprate superconductors40, where \({g}_{t}={G}_{t}^{2}=\frac{2p}{1+p},\,{g}_{J}={G}_{J}^{2}=\frac{4}{{(1+p)}^{2}}\), with doping p = 1−n. In the single-band system, the physical superconducting order parameter is defined as gt∣Δ∣40. In the multi-orbital system under consideration here, an immediate extension is made by employing the quantity \({g}_{t}^{\alpha \beta }| {\Delta }^{\alpha \beta }|\) to represent the αβ-orbital component of the superconducting order parameter. It is worth noting that at zero temperature T = 0, approximate correspondence between the RMFT and U(1) slave boson mean-field theory (SBMFT)2,52 self-consistent equations can be established if one assumes that \({g}_{t}^{\alpha \beta }\) is related to the Bose condensation of holons, and \({\Delta}^{\alpha\beta}\) is linked to the spinon pairing in SBMFT.
Fourier transforms to Eqs. (4)–(6), we obtain the mean-field Hamiltonian in momentum space
where \({\Phi }_{{k}}={({\Psi }_{{k}\uparrow }^{{\rm {T}}},{\Psi }_{-{k}\downarrow }^{\dagger })}^{{\rm {T}}}\) is the corresponding Nambu basis set. This equation can be solved self-consistently, combining Eqs. (2) and (3) to determine the final mean-field parameters, see also Supplementary Note 2 for more.
Data availability
The data is available at https://github.com/ZhihuiLuo/RMFT_Ni327.
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Acknowledgements
The authors thank the helpful discussions with Xunwu Hu, Zhong-Yi Xie, and Guang-Ming Zhang. This project was supported by the National Key Research and Development Program of China (Grants Nos. 2022YFA1402802, 2018YFA0306001), the National Natural Science Foundation of China (Grants Nos. 92165204, 12174454, 11974432, 12274472), the Guangdong Basic and Applied Basic Research Foundation (Grants Nos. 2022A1515011618, 2021B1515120015), Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices (Grant No. 2022B1212010008), Shenzhen International Quantum Academy (Grant No. SIQA202102), and Leading Talent Program of Guangdong Special Projects (201626003).
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D.-X.Y. and W.W. conceived and designed the project. Z.L. wrote the code. Z.L. and B.L. performed the theoretical calculations and corresponding analysis under the supervision of D.-X.Y. and W.W. All authors contributed to the interpretation of the results and wrote the paper.
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Luo, Z., Lv, B., Wang, M. et al. High-TC superconductivity in La3Ni2O7 based on the bilayer two-orbital t-J model. npj Quantum Mater. 9, 61 (2024). https://doi.org/10.1038/s41535-024-00668-w
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DOI: https://doi.org/10.1038/s41535-024-00668-w