High-T C superconductivity in La 3 Ni 2 O 7 based on the bilayer two-orbital t-J model

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I. INTRODUCTION
Understanding high transition temperature (T c ) superconductivity remains one of the greatest challenges in the 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 role [2].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 the unconventional superconductivity is found in the ironbased superconductors [4][5][6][7][8], where multiple d−orbitals participate in the pairing process.The discovery of superconductivity in infinite-layer Nd 1−x Sr x NiO 2 thin films has sparked widespread interest due to cuprate-like Ni 1+ (3d 9 ) [9][10][11][12][13][14][15][16].Most recently, a Ruddlesden-Popper nickelate superconductor La 3 Ni 2 O 7 is found with a T c ≈ 80 K [17][18][19] under moderate pressures.On one hand, La 3 Ni 2 O 7 shares similarities with cuprates, as both feature the NiO 2 /CuO 2 plane containing a key d x 2 −y 2 orbital at Fermi level.On the other hand, La 3 Ni 2 O 7 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 that whether the underlying pairing mechanism of La 3 Ni 2 O 7 resembles that of cuprates?How does it differ from the extensively studied cuprates?From a theoretical perspective, a 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 con-struct a superconducting phase diagram of the relevant physical models.
Electronic structure studies [21][22][23][24][25][26][27] and optical experimental probe [28] suggest that in pressurized La 3 Ni 2 O 7 , 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 threepocket Fermi surface structure [21].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 the electron occupancy.La 3 Ni 2 O 7 typically exhibits a nominal valence configuration of d 7.5 [17,29], 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 theory [30] have 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 counterpart [30].Such a significant J ⊥ is likely responsible for the high transition T c in La 3 Ni 2 O 7 [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 La 3 Ni 2 O 7 , using the renormalized mean-field theory (RMFT) [40][41][42].The RMFT approach is a con-arXiv:2308.16564v4[cond-mat.supr-con]15 Aug 2024 crete implementation of Anderson's resonating valence bond (RVB) concept of the unconventional superconductivity [43].Following the Gutzwiller scheme, the renormalization effects from strong electron correlations are accounted on different levels by incorporating the dopingdependent renormalization factors g t , g J [40] in RMFT.The mean-field decomposition of the magnetic exchange couplings J, then enables exploring of 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 phenomena [41].In RMFT, the superconducting order parameter g t ∆ involves two competing energy scales: the phase coherence energy scale associated with g t , and the pairing amplitude represented by ∆, each exhibiting distinct doping dependencies [40].In our study, we observe that the outcome of this competition positions La 3 Ni 2 O 7 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 La 3 Ni 2 O 7 superconductor.Furthermore, we elucidate various possible pairing symmetries across a broad doping range.
The remainder of paper is organized as follows.In Sec.IV, we present the physical model and describe the RMFT method.In Sec.II, 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 Sec.III we discuss the stability of the s ± −pairing.Finally, more details can be found in the Supplementary Information.

II. 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 in the parameter regime that most relevant to the La 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.1. Superconducting order parameters g ν t |∆ ν ℓ | as a function of temperature T .Note that the two purple lines overlap with each other with zero magnitude over the whole temperature range.Since here the doping is fixed as p = 0 (nx = 0.665, nz = 0.835), g ν t is constant according to Eq. (5-6).

A. Tc for pristine compound
We first present the RMFT calculated superconducting transition temperature T c at µ = 0 that corresponds to pristine La 3 Ni 2 O 7 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 = 1.5 [21].In Fig. 1, the superconducting order parameter g ν t |∆ ν | is plotted as a function of T , which clearly demonstrate 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 is somehow coincides with the experiment [17], 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 La 3 Ni 2 O 7 superconductivity in a quantitatively correct way.We note that the s ± −wave pairing has also finite d x 2 −y 2 orbital component, as shown by the dotted lines in Fig. 1, despite that its order parameter 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 in-stability may be ruled out in our model for La 3 Ni 2 O 7 .It is worthy noting that RMFT is originally formulated at zero temperature [40].In Fig. 1 we solve the RMFT equations in the finite temperature regime [47].This extension captures finite temperature effects on the meanfield parameters ∆ and χ, while neglecting thermal effects on the renormalization factors g t , G Jr .We argue that the latter should have insignificant impacts on determining T c , given the hole concentrations in Fig. 1,  (n h x = 1 2. Superconducting order parameter g ν t |∆ ν ℓ | as a function of doping p. 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.

B. Doping evolution
Now we focus on the doping dependence of the superconducting order parameter g ν t |∆ ν ℓ |.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 are 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 ν t |∆ ν s ± | are located at p ≈ −0.04 which is very close to p = 0 for PC.This indicates that, interestingly, the La 3 Ni 2 O 7 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 ν t → 0, reflecting the Mott insulting nature at half-filling [30].
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 , such as ∆ ⊥z d/s ± from decomposing J ⊥ , and ∆

C. Doping phase diagram
To gain further insights into the RMFT result of La 3 Ni 2 O 7 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 occur 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 ν t |∆ ν ℓ | in Fig. 4 , from which one sees that for s ± −wave pairing, the pairing tendencies of ∆ ⊥z s ± (Fig. 4a) with decreasing n z according to Eq. (5)(6).Finally, it is interesting to note that although J ⊥ = 2J || is used in our study, Fig. 4 shows that the the maximal value of d−wave superconducting order parameter are 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 the in-plane counterpart J || , where z = 4.

D. Fermi surfaces
In Fig. 5 we display the paring gap function ∆ ν ℓ 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, Fig. 5b, Fig. 5d) or the one with vanishing superconducting order parameter (Fig. 5c).Fig. 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 point [21].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 antibonding state.Decreasing p x from the PC can drives 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 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 the 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 ∆ ν ℓ instead of the superconducting order parameter g ν t |∆ ν ℓ |, which is why Fig. 5c shows nonvanishing gap for a normal state, see also Supplementary Note 4. Fig. 5d shows the last case with the γ−pocket vanishing from Fermi level.As expected, in this case, only the d−wave pairing with finite 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.See also symbols indicated in Fig. 3 and 4. Note that the gap magnitudes are directly associated with ∆ ν ℓ instead of the superconducting order parameter g ν t |∆ ν ℓ |, 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.
E. Superexchanges J Finally, we investigate the influence of the magnitudes of the superexchanges for the pristine compound.In Fig. 6, we present g ν t |∆ ν ℓ | as a function of J || /J ⊥ .A vertical arrow indicates the value of J || /J ⊥ used in aforementioned 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 La 3 Ni 2 O 7 under pressure, this large value of J || /J ⊥ is however not very realistic [30].Hence, the d−wave pairing instability may can be excluded for 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 ν t |∆ ν ℓ | caused by J xz (green dashed line) and J H (not shown here) are negligible.0.00 0.25 0.50 0.75 1.00 1. 25 6.Superconducting order parameters g ν t |∆ ν ℓ | as a function of J || /J ⊥ for the pristine compound.The solid and dotted lines denote cases of Jxz = 0, dashed lines denote that of Jxz = 0.03 eV, and dash-dotted line denotes the case of JH = −1 eV.Here J ⊥ = 0.18 eV is fixed.The vertical arrow indicates J || /J ⊥ = 0.5 which is used in our above calculations.

III. 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), ∆ ||x is finite although much smaller than ∆ ||z and ∆ ⊥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 formalism [48].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 par-ticular 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 work [31] 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 La 3 Ni 2 O 7 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 La 3 Ni 2 O 7 under pressure.

A. Model
In the strong-coupling limit, the bilayer two orbital Hubbard model [21] can be mapped into a t−J model [30] where H t is the tight-binding Hamiltonian taken from downfolding the DFT band structure [21], which is defined in a basis of Ψ iσ = (c ix1σ , c iz1σ , c ix2σ , c iz2σ ) T , with c iασ representing annihilation of an electron with spin σ on α = x 1 , z

B. RMFT Formalism
To proceed with RMFT [40], we first define the following mean-field parameters where χ αβ ij and ∆ αβ ij are particle-hole and particleparticle pairs relating iα and jβ.Here we assume no magnetic ordering.For each type of exchange couplings J r in Eq. ( 1), including the Hund's coupling J H , the mean-field decomposition of H J introduces condensations of χ and ∆ in the corresponding r−channel H ∆ Jr = − 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 factors [49] G These two quantities essentially reflect the renormalization effects by the electrons 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 = G α t G β t .One sees that, when t αβ ij = t ij δ αβ , the above Hamiltonian reduces to the classical formulas of the single-band t − J model for cuprate superconductors [40], where g t = G 2 t = 2p 1+p , g J = G 2 J = 4 (1+p) 2 , with doping p = 1 − n.In the single-band system, the physical superconducting order parameter is defined as g t |∆| [40].In the multi-orbital system under consideration here, an immediate extension is made by employing the quantity g αβ t |∆ αβ | 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 αβ t is related to the Bose condensation of holons, and ∆ αβ is linked to the spinon pairing in SBMFT.
Fourier transforms to Eq. (4-6), we obtain the meanfield Hamiltonian in momentum space where Φ k = (Ψ T k↑ , Ψ † −k↓ ) T is the corresponding Nambu basis set.This equation can be solved self-consistently, combining Eq. (2-3) to determine the final mean-field parameters , see also Supplementary Note 2 for more.
||x d/s ± from decomposing J || .Note that in our calculation, the pairing components ∆ ||z d/s ± (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 ∆ ||z d/s ± and |∆ ||z d/s ± | 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, ∆||z s ± displays a doping p−dependence similar to that of ∆ ⊥z s ± .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 Section II D.

FIG. 3 .
FIG.3.Pairing phase diagram constructed by varying doping levels px and pz.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.2is 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.

3
Ni 2 O 7 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 , 165) are fairly large.Detailed discussions on this matter can be found in the Supplementary Note 2.
[30]x 2 , z 2 orbital at i site .Here x 1 , x 2 denote the two Ni-3d x 2 −y 2 orbitals situated in the double NiO 2 layer while z 1 , z 2 correspondingly denote the two Ni-3d z 2 orbitals.µisthe chemical potential and ϵ xz is the energy difference between d x 2 −y 2 and d z 2 orbitals.The hopping parameters in H According to the estimated antiferromagnetic correlations in La 3 Ni 2 O 7[30], there can be three major magnetic exchange couplings J ⊥ , J || , J xz , which respectively represents 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.
t used in this work can be found in Supplementary Note 1. H J is the Heisenberg exchange couplings, and the spin operator S iα = 1 2 µν c † iαµ σ µν c iαν .