High-T_C superconductivity in La₃Ni₂O₇ based on the bilayer two-orbital t-J model

Abstract

The recently discovered high-T_c 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 T_c ~ 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.

Introduction

Understanding high transition temperature (T_c) superconductivity remains one of the greatest challenges in condensed matter physics. For cuprate superconductors^1,2,3, 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 superconductors^4,5,6,7,8, where multiple d-orbitals participate in the pairing process. The discovery of superconductivity in infinite-layer Nd_1−xSr_xNiO₂ 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 T_c ≈ 80 K^17,18,19 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-p_z and Ni-${d}_{{z}^{2}}$ orbitals play a significant role in its low-energy physics^17,20. 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 studies^{21,22,23,24,25,26,27} 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 d^7.517,29, indicating an average of 2.5 holes in the active e_g sub-shell. While previous studies have shown variations in the computed electron densities of the two e_g orbitals^{30,31,32,33,34}, 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 T_c in La₃Ni₂O₇^{30,31,35,36,37,38,39}.

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 g_t, g_J⁴⁰ 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 T_c, and pseudogap phenomena⁴¹. In RMFT, the superconducting order parameter g_tΔ involves two competing energy scales: the phase coherence energy scale associated with g_t, 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 T_c 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 T_c 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: p_x, p_z, 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 T_c 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, n_x = 0.665, n_z = 0.835, and n = n_x + n_z = 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 studies^{36,38,39,44,45,46}. 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 T_c 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 T_c 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 T_c 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 T_c, 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.

Fig. 1: Superconducting order parameters ${g}_{t}^{\nu }| {\Delta }_{\ell }^{\nu }|$ as a function of temperature T.

Note that the two purple lines overlap with each other with zero magnitudes over the whole temperature range. Since here the doping is fixed as p = 0 (n_x = 0.665, n_z = 0.835), ${g}_{t}^{\nu }$ is constant according to Eqs. (5) and (6).

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 n_x = 0.665, n_z = 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 e_g orbitals, namely, p_x/p_z = 2.048 is fixed, such that both e_g orbitals can simultaneously reach half-filling (HF), i.e., n_x = 1, n_z = 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.

Fig. 2: Superconducting order parameter ${g}_{t}^{\nu }| {\Delta }_{\ell }^{\nu }|$ as a function of doping p.

p > 0 represents hole doping the pristine compound case (where n_x = 0.665, n_z = 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 p_x/p_z = 2.048 is maintained. The two e_g orbitals reach half-filling (HF) (n_x = 1.0, n_z = 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 J_r terms in Eq. (4) can be decomposed into different corresponding pairing bonds Δ_δ, 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 p_x−p_z 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 p_x−p_z trajectory along which Fig. 2 is plotted. The black symbols label out sets of (p_x, p_z) 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 ≲ p_z ≲ 0.1 (0.735 ≲ n_z ≲ 0.935), and it is insensitive to the value of p_x. Likewise, the d-wave pairing (purple) prevails in the doping range of −0.335 ≲ p_x ≲ −0.2 (0.465 ≲ n_x ≲ 1.0), and it is, in general, less sensitive to the value of p_z. 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 p_x−p_z 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 (p_x ≈ −0.25), and ${d}_{{z}^{2}}$ becomes half-filling (p_z ~ −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 p_z > 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 p_z shown in Fig. 4d can bee seen vastly as a result of a growing ${g}_{t}^{| | z}$ with decreasing n_z 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.

Fig. 3: Pairing phase diagram constructed by varying doping levels p_x and p_z.

p_α > 0 indicates hole doping and p_α < 0 denotes electron doping relative to the pristine compound case (n_x = 0.665, n_z = 0.835). The dashed white line presents the p_x−p_z trajectory along which Fig. 2 is plotted. The black symbols mark the sets of (p_x, p_z) that are further discussed in Fig. 5. The superexchange couplings applied are J_⊥ = 2J_∣∣ = 0.18.

Fig. 4: Superconducting order parameters ${g}_{t}^{\nu }| {\Delta }_{\ell }^{\nu }|$ with varying doping levels p_x and p_z.

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 (p_x = 0, p_z = 0) corresponds to (n_x = 0.665, n_z = 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 (p_x, p_z) 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 p_x 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 p_x. 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 p_z from the PC case, the γ-pocket vanishes from the Brillouin zone. Consequently, the s^±-wave order parameter vanishes at p_z ~ −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.

Fig. 5: The pairing gap function ${\Delta }_{\ell }^{\nu }$ projected onto the Fermi surface for a few sets of (p_x, p_z).

a Dot: (p_x = 0, p_z = 0), b square: (p_x = −0.28, p_z = 0), d upper triangle: (p_x = −0.2, p_z = −0.15), and c lower triangle: (p_x = 0, p_z = −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 J_xz = 0.03 (dashed line) and J_H = −1 eV (dash-dotted line) are shown in Fig. 6. As one sees that, although both J_xz and J_H 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 J_xz (green dashed line) and J_H (not shown here) are negligible.

Fig. 6: Superconducting order parameters ${g}_{t}^{\nu }| {\Delta }_{\ell }^{\nu }|$ as a function of J_∣∣/J_⊥ for the pristine compound.

The solid and dotted lines denote cases of J_xz = 0, dashed lines denote that of J_xz = 0.03 eV, and the dash-dotted line denotes the case of J_H = −1 eV. Here J_⊥ = 0.18 eV is fixed. The vertical arrow indicates J_∣∣/J_⊥ = 0.5 which is used in our above calculations.

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 p_x, p_z = 0, we note that even when only J_⊥ is considered (with J_∣∣ = 0, J_xz = 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 J_H, 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 J_xz 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 J_H on superconductivity may be underestimated in the RMFT formalism⁴⁸. Additionally, it is important to note that the superconducting T_c 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 T_c 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

$$\begin{array}{l}\;{\mathcal{H}}\,=\,{{\mathcal{H}}}_{t}+{{\mathcal{H}}}_{J}\\ \,{{\mathcal{H}}}_{t}\,=\,\sum _{ij\alpha \beta \sigma }\left({t}_{ij}^{\alpha \beta }-\mu {\delta }_{ij}{\delta }_{\alpha \beta }\right){c}_{i\alpha \sigma }^{\dagger }{c}_{j\beta \sigma }\\\quad\;\; \,+\,{\epsilon }_{xz}\mathop{\sum }\limits_{i\sigma }^{\alpha \beta ={x}_{1}{z}_{1},{x}_{2}{z}_{2}}({n}_{i\alpha \sigma }-{n}_{i\beta \sigma })\\ {{\mathcal{H}}}_{J}\,=\,{J}_{\perp }\mathop{\sum}\limits_{i}{{\boldsymbol{S}}}_{i{z}_{1}}\cdot {{\boldsymbol{S}}}_{i{z}_{2}}+{J}_{| | }\mathop{\sum }\limits_{ < ij > }^{\alpha ={x}_{1},{x}_{2}}{{\boldsymbol{S}}}_{i\alpha }\cdot {{\boldsymbol{S}}}_{j\alpha }\\\quad\;\; \,+\,{J}_{xz}\mathop{\sum }\limits_{ < i,j > }^{\alpha \beta ={x}_{1}{z}_{1},{x}_{2}{z}_{2}}{{\boldsymbol{S}}}_{i\alpha }\cdot {{\boldsymbol{S}}}_{j\beta }+{J}_{{\rm {H}}}\mathop{\sum }\limits_{i}^{\alpha \beta ={x}_{1}{z}_{1},{x}_{2}{z}_{2}}{{\boldsymbol{S}}}_{i\alpha }\cdot {{\boldsymbol{S}}}_{i\beta },\end{array}$$

(1)

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 c_iασ 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 ϵ_xz 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_∣∣, J_xz, 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 ${d}_{{z}^{2}}$. J_H is the intra-atomic Hund’s coupling.

RMFT formalism

To proceed with RMFT⁴⁰, we first define the following mean-field parameters

$${\chi }_{ij}^{\alpha \beta }=\frac{1}{2}\left\langle {c}_{i\alpha \uparrow }^{\dagger }{c}_{j\beta \uparrow }+{c}_{i\alpha \downarrow }^{\dagger }{c}_{j\beta \downarrow }\right\rangle =\left\langle {c}_{i\alpha \uparrow }^{\dagger }{c}_{j\beta \uparrow }\right\rangle$$

(2)

$${\Delta }_{ij}^{\alpha \beta }=\frac{1}{2}\left\langle {c}_{i\alpha \uparrow }^{\dagger }{c}_{j\beta \downarrow }^{\dagger }-{c}_{i\alpha \downarrow }^{\dagger }{c}_{j\beta \uparrow }^{\dagger }\right\rangle =\left\langle {c}_{i\alpha \uparrow }^{\dagger }{c}_{j\beta \downarrow }^{\dagger }\right\rangle ,$$

(3)

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 J_r in Eq. (1), including Hund’s coupling J_H, the mean-field decomposition of ${{\mathcal{H}}}_{J}$ introduces condensations of χ and Δ in the corresponding r-channel

$$\begin{array}{l}{H}_{{J}_{r}}^{\chi }\,=\,-\frac{3}{4}{J}_{r}\mathop{\sum }\limits_{ < ij\ > \sigma }({\chi }_{\delta }^{\alpha \beta }{c}_{i\alpha \sigma }^{\dagger }{c}_{j\beta \sigma }+{h.c.})+\frac{3}{2}{J}_{r}N| {\chi }_{\delta }^{\alpha \beta }{| }^{2},\\ {H}_{{J}_{\!r}}^{\Delta }\,=\,-\frac{3}{4}{J}_{r}\mathop{\sum }\limits_{ < ij\ > \sigma }(\sigma {\Delta }_{\delta }^{\alpha \beta }{c}_{i\alpha \sigma }^{\dagger }{c}_{j\beta \bar{\sigma }}^{\dagger }+{h.c.})+\frac{3}{2}{J}_{\!r}N| {\Delta }_{\delta }^{\alpha \beta }{| }^{2}.\end{array}$$

(4)

Assuming translation symmetry, δ = R_jβ−R_iα denotes different bonds in real-space associated with J_r, and N is the total number of sites of the square lattice.

We now introduce two renormalization factors⁴⁹

$$\begin{array}{r}{G}_{t}^{\alpha }=\sqrt{\frac{1-{n}_{\alpha }}{1-{n}_{\alpha }/2}},\qquad {G}_{J}^{\alpha }=\frac{1}{(1-{n}_{\alpha }/2)}.\end{array}$$

(5)

These two quantities essentially reflect the renormalization effects by the electron's repulsions on top of the single-particle Hamiltonian in the Gutzwiller approximation^40,42,50,51, which depend on the orbital occupation number n_α = n_α↑ + n_α↓. This eventually leads to the renormalized mean-field Hamiltonian

$$\begin{array}{rcl}{H}_{t}^{{\rm{MF}}}&=&\mathop{\sum}\limits _{ij\alpha \beta \sigma }{g}_{t}^{\alpha \beta }{t}_{ij}^{\alpha \beta }{c}_{i\alpha \sigma }^{\dagger }{c}_{j\beta \sigma },\\ {H}_{J}^{{\rm{MF}},\chi /\Delta }&=&\sum _{r}{G}_{{J}_{\!r}}^{\alpha }{G}_{{J}_{\!r}}^{\beta }{H}_{{J}_{\!r}}^{\chi /\Delta },\end{array}$$

(6)

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 g_t∣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

$${H}^{{\rm{MF}}}=\sum _{{k}}{\Phi }_{{k}}^{\dagger }\left(\begin{array}{cc}{H}_{t,{k}}^{{\rm{MF}}}+{H}_{J,{k}}^{MF,\chi }&{H}_{J,{k}}^{{\rm{MF}},\Delta} \\ {\left[{H}_{J,{k}}^{{\rm{MF}},\Delta} \right]}^{\dagger }&-{\left[{H}_{t,{k}}^{{\rm{MF}}}+{H}_{J,{k}}^{{\rm{MF}},\chi }\right]}^{* }\end{array}\right){\Phi }_{{k}},$$

(7)

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.