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

Zhihui Luo1,†, Biao Lv1,†, Meng Wang1, Wéi Wú1,∗ and Dao-Xin Yao1,∗ 1Center
for Neutron Science and Technology, Guangdong Provincial Key Laboratory of
Magnetoelectric Physics and Devices, State Key Laboratory of Optoelectronic
Materials and Technologies, School of Physics, Sun Yat-Sen University,
Guangzhou, 510275, China  †These authors contributed equally to the work
<EMAIL_ADDRESS>and<EMAIL_ADDRESS>
###### Abstract

The recently discovered high-TC superconductor La3Ni2O7 has sparked renewed
interest in the 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
$s^{\pm}-$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 $T_{c}\sim 80$ K. We establish a
comprehensive superconducting phase diagram in the doping plane. Notably, the
La3Ni2O7 under pressure is found 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 $s^{\pm}-$wave pairing against Hund’s
coupling and other magnetic exchange couplings is discussed.

  * Dec 2023

Keywords: superconductivity, bilayer nickelate, transition temperature,
pairing symmetry, $t-J$ model

##  1 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 essentially lie in the $d_{x^{2}-y^{2}}$ orbital with the
presence of strong Coulomb repulsion between electrons [2]. It is usually
referred as unconventional superconductivity distinguishing from the
traditional Bardeen-Cooper-Schrieffer (BCS) type of superconductivity. Another
prominent example of the unconventional superconductivity is found in the
iron-based superconductors [4, 5, 6, 7, 8], where multiple $d-$orbitals are
often involved in the pairing. Most recently, a new Ruddlesden-Popper
nickelate superconductor La3Ni2O7 is found with a $T_{c}\approx 80$ K [9, 10,
11] under moderate pressures. On one hand, La3Ni2O7 is similar to cuprates, as
they both have the NiO2/CuO2 plane hosting a crucial $d_{x^{2}-y^{2}}$ orbital
at Fermi level. On the other hand, La3Ni2O7 differs from the cuprates as its
apical O-$p_{z}$ and Ni-$d_{z^{2}}$ orbitals come to play a role in low-energy
physics [9, 12]. Given this context, we would like to ask that whether the
underlying pairing mechanism of La3Ni2O7 resemble cuprates? If it belongs to a
novel scenario of superconductivity, how is it different from the extensively
investigated cuprates? From the the theoretical perspective, to address these
questions a first step is to resolve the roles played by the $d_{x^{2}-y^{2}}$
and $d_{z^{2}}$ orbitals in pairing, and map out a superconducting phase
diagram of the relevant prototypical physical models.

Electronic structure studies [13, 14, 15, 16, 17] as well as optical
experimental probe [18] show that in pressurized La3Ni2O7, Ni-$d_{z^{2}}$
orbital is involved in Fermi energy due to the strong inter-layer coupling via
apical oxygen in La3Ni2O7. Via with the in-plane oxygen $p-$orbital,
$d_{z^{2}}$ orbitals can hybridize with $d_{x^{2}-y^{2}}$ orbitals, eventually
giving rise to a three-pocket Fermi surface structure[13]. This kind of Fermi
surface geometry is different from that of the cuprates, which may lead to
pairing symmetry and effective pairing “glue” distinct from that of the
$d-$wave superconductivity in cuprates upon doping a single $d_{x^{2}-y^{2}}$
orbital. Another important issue to be taken into account is the electron
occupancy. La3Ni2O7 has a nominal valence configuration $d^{7.5}$ [9, 19],
indicating 2.5 holes on average for the active $e_{g}$ sub-shell. In previous
studies, the computed electrons densities of the two $e_{g}$ orbitals varies
in different works [20, 21, 22, 23, 24]. However, it in general can be
regarded as a multi-orbital system consisting of the heavily hole-doped
$d_{x^{2}-y^{2}}$ orbital (hole density $\sim 1.5$), and the near half-filled
$d_{z^{2}}$ orbital (hole density $\sim 1.0$) with strong electron
correlations. Regarding the superexchange couplings, studies based on cluster
dynamical mean-field theory [20] pointed out that the $J_{\bot}$ connecting
two apical $d_{z^{2}}$ orbitals may acquire a magnitude being at least $\sim$2
times larger than the intra-layer $d_{x^{2}-y^{2}}$ exchange coupling, with
the latter estimated to be close to the cuprate counterpart [20]. Such large
$J_{\bot}$ is very likely to be responsible for the high transition $T_{c}$ in
La3Ni2O7 [20, 21, 25, 26, 27, 28, 29].

In this paper, we systematically investigate the superconductivity in the
bilayer two orbital $t-J$ model that prototypes the low-energy physics of
pressurized La3Ni2O7, using the renormalized mean-field theory (RMFT) [30, 31,
32]. RMFT approach can be seen as a concrete implementation of Anderson’s
resonating valence bond (RVB) concept of the unconventional
superconductivity[33]. In line with Gutzwiller scheme, the renormalization
effects from strong electron correlations are considered on different levels
by introducing the doping-dependent renormalization factors $g_{t},g_{J}$ [30]
in RMFT. Meanwhile, the mean-field decomposition of the magnetic exchange
couplings $J$ allows searching for the BCS pairing instabilities of the
system. Despite its simple formulations, RMFT has shown to be able to describe
various aspects of the cuprates, such as superfluid density, the dome-shaped
doping-dependence of $T_{c}$, as well as pseudogap [31]. In RMFT, the
superconducting order parameter $g_{t}\Delta$ involves two competing energy
scales, i.e., the phase coherence temperature related to $g_{t}$, and the
pairing amplitude described by $\Delta$, which have fully different doping
dependence[30]. In our study, we find that the outcome of this competition
leaves La3Ni2O7 residing in the optimal doping regime in the superconducting
phase diagram. Our calculation suggests a $T_{c}$ comparable to that of the
experiments, indicating that our considerations are in close relevance to the
La3Ni2O7 superconductor. We also elucidate different possible pairing
symmetries in a broad doping range.

The paper is organized as follows. In Sec. 2, we introduce the model and the
RMFT method. In Sec. 3, we present our results including $T_{c}$ for pristine
compound, doping phase diagram and Fermi surface. The influence of the
strength of superexchanges and Hund’s coupling are also discussed at the end
of the section. In Sec. 4 we discuss the stability of the $s^{\pm}-$pairing.
Section 5 provides a conclusion. Finally, details about the supporting
materials including discussion on the pseudogap physics can be found in the
appendixes.

##  2 Model and Methods

In the strong-coupling limit, the bilayer two orbital Hubbard model [13] can
be mapped into a $t-J$ model as done in Wú, $et\ al.$ ’s work [20],

$\displaystyle\mathcal{H}$ $\displaystyle=\mathcal{H}_{t}+\mathcal{H}_{J}$ (1)
$\displaystyle\mathcal{H}_{t}$
$\displaystyle=\sum_{ijst\sigma}(t_{ij}^{st}-\mu\delta_{ij}\delta_{{}^{st}})c_{is\sigma}^{\dagger}c_{jt\sigma}$
$\displaystyle\mathcal{H}_{J}$
$\displaystyle=J_{\bot}\sum_{i}\boldsymbol{S}_{iz_{1}}\cdot\boldsymbol{S}_{iz_{2}}+J_{//}\sum_{<ij>}^{s=x_{1},x_{2}}\boldsymbol{S}_{is}\cdot\boldsymbol{S}_{js}$
$\displaystyle+J_{xz}\sum_{<i,j>}^{st=x_{1}z_{1},x_{2}z_{2}}\boldsymbol{S}_{is}\cdot\boldsymbol{S}_{jt}+J_{H}\sum_{i}^{s>t}\boldsymbol{S}_{is}\cdot\boldsymbol{S}_{it},$

where $\mathcal{H}_{t}$ is the tight-binding Hamiltonian taken from
downfolding the DFT band structure [13], which is defined in a basis of
$\Psi_{\sigma}=(c_{x_{1}\sigma},c_{z_{1}\sigma},c_{x_{2}\sigma},c_{z_{2}\sigma})^{T}$,
with $c_{s\sigma}$ representing annihilation of an electron on
$s=x_{1},z_{1},x_{2},z_{2}$ orbital with spin $\sigma$. $\mu$ is the chemical
potential. The hopping parameters in $\mathcal{H}_{t}$ can also be found in A.
$\mathcal{H}_{J}$ is the Heisenberg exchange couplings, and the spin operator
$\boldsymbol{S}_{is}=\frac{1}{2}\sum_{\alpha\beta}c^{\dagger}_{is\alpha}\boldsymbol{\sigma}_{\alpha\beta}c_{is\beta}$.
According to the estimated antiferromagnetic correlations in
$\mathrm{La_{3}Ni_{2}O_{7}}$ [20], there can be three major magnetic exchange
couplings $J_{\bot},J_{//},J_{xz}$, 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
$d_{z^{2}}$.

To proceed with RMFT [30], we first define the following mean-field
parameters,

$\displaystyle\chi_{ij}^{st}=\frac{1}{2}\langle
c_{is\uparrow}^{\dagger}c_{jt\uparrow}+c_{is\downarrow}^{\dagger}c_{jt\downarrow}\rangle=\langle
c_{is\uparrow}^{\dagger}c_{jt\uparrow}\rangle$ (2)
$\displaystyle\Delta_{ij}^{st}=\frac{1}{2}\langle
c_{is\uparrow}^{\dagger}c_{jt\downarrow}^{\dagger}-c_{is\downarrow}^{\dagger}c_{jt\uparrow}^{\dagger}\rangle=\langle
c_{is\uparrow}^{\dagger}c_{jt\downarrow}^{\dagger}\rangle,$ (3)

where $\chi_{ij}^{st}$ and $\Delta_{ij}^{st}$ are particle-hole and particle-
particle pairs relating $is$ and $jt$. Here we assume no magnetic ordering.
For each type of $J_{r}$, the mean-field decomposition of $\mathcal{H}_{J}$
generates condensations of $\chi$ and $\Delta$ in the corresponding
$r-$channel,

$\displaystyle H_{J_{r}}^{\chi}$
$\displaystyle=-\frac{3}{4}J_{r}\sum_{<ij>\sigma}(\chi^{st}_{\delta}c^{\dagger}_{i\sigma}c_{j\sigma}+h.c.)+\frac{3}{2}J_{r}N|\chi^{st}_{\delta}|^{2}$
$\displaystyle H_{J_{r}}^{\Delta}$
$\displaystyle=-\frac{3}{4}J_{r}\sum_{<ij>\sigma}(\sigma\Delta^{st}_{\delta}c^{\dagger}_{i\sigma}c^{\dagger}_{j\bar{\sigma}}+h.c.)+\frac{3}{2}J_{r}N|\Delta_{\delta}^{st}|^{2}$
(4)

Assuming transnational symmetry, $\delta=R_{j}-R_{i}$ denotes different bonds
in real-space associated with $J_{r}$, and $N$ is the total size of the
lattice.

We now introduce two renormalization factors [34],

$\displaystyle G_{t}^{s}=\sqrt{\frac{1-n_{s}}{1-n_{s}/2}},\qquad
G_{J}^{s}=\frac{1}{(1-n_{s}/2)}.$ (5)

These two quantities essentially reflects the renormalization effects by the
electrons repulsions on top of the single-particle Hamiltonian in the
Gutzwiller approximation [30, 35, 32, 36], with $n_{s}=\sum_{\sigma}\langle
c_{s\sigma}^{\dagger}c_{s\sigma}\rangle$ representing density of orbital $s$.
This eventually leads to the renormalized mean-field Hamiltonian,

$\displaystyle H^{MF}_{t}$
$\displaystyle=\sum_{ijst\sigma}G_{t}^{s}G_{t}^{t}t_{ij}^{st}c_{is\sigma}^{\dagger}c_{jt\sigma},$
(6) $\displaystyle H_{J}^{MF}$
$\displaystyle=\sum_{r}G_{J_{r}}^{s}G_{J_{r}}^{t}(H_{J_{r}}^{\chi}+H_{J_{r}}^{\Delta}).$

where one sees that, when $t_{ij}^{st}=t_{ij}\delta_{s,t}$, the above
Hamiltonian reduces to the classical formulas for the single-band $t-J$ model
of cuprate superconductors [30], which shows $g_{t}=G_{t}^{2}=\frac{2p}{1+p},\
g_{J}=G_{J}^{2}=\frac{4}{(1+p)^{2}}$, with doping $p=1-n$. Now the physical
superconducting order parameter can be defined as
$g_{t}^{s}|\Delta^{\alpha}_{s}|$ [30] for $s-$orbital component and
$\alpha-$pairing symmetry. It is worth noting that at zero temperature $T=0$,
the approximate correspondence between the RMFT and $U(1)$ slave boson mean-
field theory (SBMFT) [37, 2] self-consistent equations can be established if
one assumes $g_{t}^{s}$ is related to the Bose condensation of holons, and
$\Delta_{ij}^{st}$ is linked to the spinon pairing in SBMFT.

Applying Fourier transforms to Eq. (2-6), we obtain the mean-field Hamiltonian
in momentum space,

$\displaystyle H^{MF}=\sum_{\rm k}$ $\displaystyle\Phi^{\dagger}_{\rm
k}\left(\begin{array}[]{cc}H_{t,\rm k}^{MF}+H_{J,\rm k}^{MF,\chi}&H_{J,\rm
k}^{MF,\Delta}\\\ {[}H_{J,\rm k}^{MF,\Delta}{]}^{\dagger}&-{[}H_{t,\rm
k}^{MF}+H_{J,\rm k}^{MF,\chi}{]}^{*}\end{array}\right)\Phi_{\rm k}$ (9)

where $\Phi_{\rm k}=(\Psi^{T}_{\rm{k}\uparrow},\Psi^{\dagger}_{-\rm
k\downarrow})^{T}$ is the corresponding Nambu basis set. This equation can be
solved self-consistently, combining Eq. (2-3) to determine the final mean-
field parameters.

##  3 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
$\mathrm{La_{3}Ni_{2}O_{7}}$ system. The impacts of several key factors
including temperature $T$, doping levels of $d_{x^{2}-y^{2}}$, and $d_{z^{2}}$
orbitals: $p_{x},p_{z}$, as well as the geometry of Fermi surface are analyzed
while eyeing on the evolution of the superconducting order parameter
$g_{t}^{s}|\Delta^{\alpha}_{s}|$. To be clear, in our convention,
$\alpha=d,s^{\pm}$ are pairing symmetries, while $s=//x,//z,\bot z$ denote
orbitals that Cooper pairs reside on, representing respectively intra-layer
$d_{x^{2}-y^{2}}$, intra-layer $d_{z^{2}}$ and inter-layer $d_{z^{2}}$
pairing. Without loss of generality, we adopt typical values of
$J_{\bot}=2J_{//}=0.18$ eV taken from Ref. [20] throughout the paper unless
otherwise specified.

###  3.1 $T_{c}$ for pristine compound

We first present the RMFT calculated superconducting transition temperature
$T_{c}$ at $\mu=0$ 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 $\mu=0,n_{x}\approx 0.665,n_{z}\approx 0.835$, and $n=n_{x}+n_{z}=1.5$
[13]. In Fig. 1, the superconducting order parameter $g_{t}|\Delta^{\alpha}|$
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 $s^{\pm}-$wave pairing of the system. This result is in agreement
with several other theoretical studies [29, 26, 38, 28, 39, 40]. 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 [9], 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
quantitative correct way. We note that the $s^{\pm}-$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
instability can be ruled out in our model for La3Ni2O7. We note that RMFT is
originally formulated at zero temperature [30]. In Fig. 1 we have directly
extended it to the finite temperature study and have neglected the project
entropy term [41], as we have verified that the latter has a negligible
contribution to the total free energy at low temperatures in our study.

Figure 1: Superconducting order parameters $g_{t}|\Delta^{\alpha}|$ 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 doping
is fixed as $p=0$, $g_{t}$ is constant according to Eq. 5.  Figure 2:
Superconducting order parameter $g_{t}|\Delta^{\alpha}|$ as a function of
doping $p$, with $p>0$ for hole doping and $p<0$ for electron doping. The
round shape at $p=0$ indicates the pristine compound (PC). We asymptotically
vary $p$ with a fixed ratio of $p_{x}/p_{z}=2.048$ so as to reach the half-
filling (HF) of both $e_{g}$ orbitals, as indicated by the diamond shape at
$p=-0.5$.

###  3.2 Doping evolution

Now we focus on the doping dependence of the superconducting order parameter
$g_{t}|\Delta^{\alpha}|$. In the following, $p<0$ means doping the pristine
compound with electrons, and $p>0$ denotes hole doping. In varying the doping
level $p$ of the system, a fixed ratio of the doping levels of the two $e_{g}$
orbitals is kept, namely, $p_{x}/p_{z}=2.048$, such that both $e_{g}$ orbitals
are half-filled, i.e., $n_{x}=1,n_{z}=1$ at $p=-0.5$. From Fig. 2, one learns
that the $s^{\pm}-$wave pairings (green) are quite robust over a wide range of
doping $p$. The maxima of $g_{t}|\Delta^{s\pm}|$ are located at
$p\approx-0.04$ 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 $s^{\pm}-$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
$g_{t}\rightarrow 0$, reflecting the Mott insulting nature at half-filling
[20].

It is worth noting that, as a general prescription of the mean-field
approaches, different $J_{r}$ terms in Eq. 2 can be decomposed into different
corresponding pairing bonds $\Delta_{\delta}$, such as $J_{\bot}$ (for
$\Delta_{\bot z}^{d/s\pm}$) and $J_{//}$ (for $\Delta_{//x}^{d/s\pm}$).
Nevertheless, in our calculation, the pairing component
$\Delta_{//z}^{d/s\pm}$ (dashed line in Fig. 2) that represents the intra-
layer pairing of $d_{z^{2}}$ orbital, does not have a corresponding $J$ term
in the Hamiltonian. Hence its value is not determined by the competition
between the $\Delta_{//z}^{d/s\pm}$ and $|\Delta_{//z}^{d/s\pm}|^{2}$ terms in
minimizing the free-energy. Instead, it should be interpreted as the pairing
instability driven by the pre-existing inter-layer $d_{z^{2}}$ pairing.
Indeed, as shown in Fig. 2, $\Delta_{//z}^{s\pm}$ and $\Delta_{\perp
z}^{s\pm}$ displays similar behavior as a function of doping $p$. Finally, we
notice that a small tip appears at hole doping $p\sim 0.4$, which can be
attributed to the van Hove singularity (VHS) associated with the $\beta-$sheet
of the Fermi surface, see also Section 3.4.

###  3.3 Doping phase diagram

Figure 3: Pairing phase diagram with varying doping $p_{x}$ and $p_{z}$. The
dashed white line indicates the $p_{x}-p_{z}$ trajectory along which Fig. 2 is
plotted. The black symbols label sets of ($p_{x},p_{z}$) that are further
discussed in Fig. 5. $J_{\bot}=2J_{//}=0.18$ are applied.

To gain further insights into the RMFT result of 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 $d$, $s^{\pm}$,
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 $s^{\pm}-$wave
pairings span roughly a vertical and a horizontal stripe respectively in the
phase diagram. In other words, $s^{\pm}-$wave pairing (green) dominates the
regime where $-0.1\lesssim p_{z}\lesssim 0.1$ ($0.735\lesssim n_{z}\lesssim
0.935$), and it is insensitive to the value of $p_{x}$. Likewise, the $d-$wave
pairing (purple) prevails in the doping range of $-0.34\lesssim p_{x}\lesssim
0.2$ ($0.465\lesssim n_{x}\lesssim 1.005$), 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 $g_{t}|\Delta^{\alpha}|$ in Fig. 4 ,
from which one sees that for $s^{\pm}-$wave pairing, the pairing tendencies of
$\Delta^{s\pm}_{\bot z}$ (Fig. 4a) and $\Delta_{//z}^{s\pm}$ (Fig. 4b) show
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, $g^{x}_{t}\Delta_{//x}^{d}$ is enhanced when $d_{x^{2}-y^{2}}$ is
heavily electron doped ($p_{x}\approx-0.25$), and $d_{z^{2}}$ becomes half-
filling ($p_{z}\sim-0.2$). This is because the large electron doping drives
the $\gamma-$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, thus mimicking cuprates. On the other hand, the $d_{z^{2}}$
component of the $d-$wave pairing $g^{z}_{t}\Delta_{//z}^{d}$ is enhanced when
$p_{z}>0.2$, as shown in Fig. 4d. This can be understood considering the fact
that since $J_{//z}=0$ 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 order parameter $g^{z}_{t}\Delta_{//z}^{d}$
increasing with $p_{z}$ shown in Fig. 4d can bee seen vastly as a result of a
growing $g^{z}_{t}$ with decreasing $n_{z}$ according to Eq. 5. Finally, it is
interesting to note that although $J_{\bot}=2J_{//}$ 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 $s^{\pm}-$wave pairing. This is
expected since the vertical exchange coupling $J_{\bot}$ has a smaller
coordination number $z=1$, compared to its the in-plane counterpart $J_{//}$,
where $z=4$.

Figure 4: Magnitudes of different superconducting order parameters
$g_{t}|\Delta^{\alpha}|$ with varying doping $p_{x}$ and $p_{z}$. Symbols
denote typical dopings to be analyzed in Fig. 5.

###  3.4 Fermi surfaces

In Fig. 5 we display Fermi surfaces for four typical dopings with each
characterizing one type of pairing symmetries (a-c) or the one with vanishing
superconducting order parameter (d). Fig. 5a shows the FS of the PC case with
$s^{\pm}$ pairing, which has three sheets of FS with one around $\Gamma$ point
and two around $M$ point [13]. 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
$\alpha,\beta-$sheet of FS as a whole is also driven closer to the magnetic
Brillouin zone (MBZ) edge (green dashed lines), which is accompanied by the
evolving of the pairing symmetry from $s^{\pm}$ to $d+is-$wave. 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. On the other hand, the $\gamma-$pocket that hosts the
$s^{\pm}-$wave, is less affected by the changing of $p_{x}$, as shown in Fig.
5b. Fig. 5c displays that as lowing $p_{z}$ from the PC case, the
$\gamma-$pocket vanishes from the Brillouin zone. Consequently, the
$s^{\pm}-$wave order parameter vanishes at $p_{z}\sim-0.15$. In this case,
similar to PC, no finite $d-$wave order parameter is observed. Fig. 5d shows
the last case with the $\gamma-$pocket vanishing from Fermi level. As
expected, in this case, the $d-$wave pairing with only finite
$g^{x}_{t}|\Delta_{//x}^{d}|$ 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.

Figure 5: Fermi surfaces for a few sets of ($p_{x},p_{z}$) labeled by
different symbols, as also indicated in Fig. 3 and 4. The green and purple
dashed lines indicate the nodes of $s^{\pm}$ and $d-$wave, respectively. The
green line can also denote the antiferromagnetic magnetic Brillouin zone.

###  3.5 Superexchanges $J$

Finally, we investigate the influence of the magnitudes of the superexchanges
for the pristine compound. In Fig. 6, we present $g_{t}|\Delta^{\alpha}|$ as a
function of $J_{//}/J_{\bot}$. The dashed vertical line indicates the values
used for aforementioned calculations, where $s^{\pm}-$wave is found for PC. As
decreasing/increasing $J_{//}/J_{\bot}$, $s^{\pm}-$wave order parameters
(green lines) decrease/increase very slightly with $J_{//}$. When
$J_{//}/J_{\bot}$ $\sim 1.1$, $d-$wave (purple solid line) starts to build up
at $d_{x^{2}-y^{2}}$ orbital. For La3Ni2O7 under pressure, this large value of
$J_{//}/J_{\bot}$ is however not very realistic [20]. Hence, the $d-$wave
pairing instability should be excluded for the realistic pristine compound in
our study. To check the stability of the pairings, the results for
$J_{xz}=0.03$ (dashed line) and $J_{H}=-1$ (dotted line) are also shown in
Fig. 6. As one can see 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 $s^{\pm}-$pairing, the changes of the order
parameter $g_{t}|\Delta^{\alpha}|$ introduced by $J_{xz}$ (green dashed line)
and $J_{H}$ (not shown here) is negligible.

Figure 6: Superconducting order parameters $g_{t}|\Delta^{\alpha}|$ as a
function of $J_{//}/J_{\bot}$ for pristine compound. The solid and dotted
lines deonte $J_{xz}=0$, dashed lines denote $J_{xz}=0.03$, and dash-dotted
line denotes $J_{H}=-1$. $J_{\bot}=0.18$ eV is fixed, hence the vertical
dotted grey line at $J_{//}/J_{\bot}=0.5$ indicates the default values for
previous calculations.

##  4 Discussion

In our study, the RMFT equations are solved in such a way that the pairing
fields on different bonds can be varied independently, namely, no specific
pairing symmetry is presumed in the self-consistent process. The symmetries of
the electron pairing emerge naturally as a result of the minimized energy in
our calculations. This protocol prevents potential overlooking of pairing
symmetries. Concerning the dominant $s^{\pm}$ pairing for the pristine
compound at $p_{x},p_{z}$ = 0, we note that even when only $J_{\perp}$
($J_{//}=0,J_{xz}=0$) is considered, $\Delta_{//x}$ is finite despite being
much smaller than $\Delta_{//z}$ and $\Delta_{\perp z}$. Since the effective
mass of $d_{x^{2}-y^{2}}$ orbital is small comparing to the $d_{z^{2}}$
orbital, it may still contribute a significant portion of the superfluid
density in the superconducting state of the system. The two-folded 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, as well as the enhanced $J_{//}$
and $J_{xz}$ couplings, exhibit no crucial impact on the dominant $s^{\pm}$
pairing in our RMFT study according to Fig. 6. We note that the
superconducting $T_{c}$ obtained by RMFT should be seen in general
overestimated, since where both the temporal and spacial fluctuations are
neglected. This is particular true when considering here the pairing fields
originate from the local inter-layer $d_{z^{2}}$ magnetic couplings, where
phase fluctuations can play a more important role in suppressing $T_{c}$
comparing to its single-band $t-J$ model counterpart in cuprate
superconductors. Finally we note that a recent theory work [21] proposes that
the in the framework of the two-component pairing theory in a composite
system, the phase fluctuations could 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.

##  5 Conclusions

Employing the renormalized mean-field theory, we have established a
comprehensive superconducting phase diagram for the bilayer two-orbital $t-J$
model. A robust $s^{\pm}-$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. The dependence of the
pairing instabilities on doping levels, exchange couplings, as well as the
Hund’s coupling effects are carefully investigated. Our study will have
significant impact on the theoretical understanding of the superconductivity
La3Ni2O7 under pressure.

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 No. 2022YFA1402802, 2018YFA0306001), the
National Natural Science Foundation of China (Grants No. 92165204,
No.12174454, No. 11974432, No.12274472), the Guangdong Basic and Applied Basic
Research Foundation (Grants No. 2022A1515011618, No. 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).

##  Appendix A Model Details

The tight-binding Hamiltonian $\mathcal{H}_{t}$ in Eq. (1) is taken from our
previous works [13, 20]. Here we would like to rewrite it again for reader’s
convenience.

$\displaystyle\mathcal{H}_{t}$ $\displaystyle=\sum_{\rm k\sigma}\Psi_{\rm
k\sigma}^{\dagger}H(\rm k)\Psi_{\rm k\sigma},$ (10)

with the matrix

$\displaystyle H(\rm{k})_{1,1}=H(\rm{k})_{3,3}$ $\displaystyle=2t_{1}^{x}(\cos
k_{x}+\cos k_{y})$ $\displaystyle\quad+4t_{2}^{x}\cos k_{x}\cos
k_{y}+\epsilon^{x}$ $\displaystyle H(\rm{k})_{2,2}=H(\rm{k})_{4,4}$
$\displaystyle=2t_{1}^{z}(\cos k_{x}+\cos k_{y})$
$\displaystyle\quad+4t_{2}^{z}\cos k_{x}\cos k_{y}+\epsilon^{z}$
$\displaystyle H(\rm{k})_{1,2}=H(\rm{k})_{3,4}$
$\displaystyle=2t^{xz}_{3}(\cos k_{x}-\cos k_{y})$ $\displaystyle
H(\rm{k})_{1,4}=H(\rm{k})_{2,3}$ $\displaystyle=2t^{xz}_{4}(\cos k_{x}-\cos
k_{y})$ $\displaystyle H(\rm{k})_{1,3}$ $\displaystyle=t^{x}_{\bot}$
$\displaystyle H(\rm{k})_{2,4}$ $\displaystyle=t^{z}_{\bot}.$ (11)

The basis is defined as
$\Psi_{\sigma}=(c_{x_{1}\sigma},c_{z_{1}\sigma},c_{x_{2}\sigma},c_{z_{2}\sigma})^{T}$.
The hopping parameters take the following values [13]: $t_{1}^{x}$=-0.483,
$t_{2}^{x}$=0.069, $t_{1}^{z}$=-0.110, $t_{2}^{z}$=-0.017, $t_{3}^{xz}$=0.239,
$t_{4}^{xz}$=-0.034, $t_{\bot}^{x}$=0.005, $t_{\bot}^{z}$=-0.635, and the site
energies: $\epsilon^{x}$=0.776, $\epsilon^{z}$=0.409, in unit of eV.

##  Appendix B BENCHMARK

To certify our calculations, we present a benchmark of our RMFT results for
one-band $t-J$ model. In Fig. 7, mean-field order parameters are shown as a
function of $p$. The lines are from our calculations and markers are from Ref.
[30], with $J/t=0.2$, $T=0$. As can be seen they are in good agreement, as
both capture the most striking domed-shape doping dependence of $T_{c}$
featured by $g_{t}\Delta^{d}$.

Figure 7: RMFT results of one-band $t-J$ model, which show mean-field order
parameters as a function of $p$. The lines are from our calculations and
markers are from Zhang’s results [30] . $J/t=0.2$, $T=0$ are applied.

##  Appendix C Doping phase diagram of $\Delta$

Figure 8: Intensities of different pairing gap parameters $|\Delta^{\alpha}|$
with vary doping $p_{x},p_{z}$. The renormalization factor $g_{t}$ here is
removed compared with Fig. 4. A factor of 0.25 is incorporated in (c) for
clarity. Also, a much smaller intensity range is used in the last panel.
$J_{\bot}=2J_{//}=0.18$ are applied.

Figure. 8 shows the magnitudes of six major pairing gap parameters
$|\Delta^{\alpha}|$ with varying $p_{x},p_{z}$. Compared with Fig. 4, the
renormalization factor $g_{t}$ is removed from the plot. As can be seen the
intensity distributions is roughly consistent with the former. The major
difference appears in the lower boundary in Fig. 8a and left boundary in Fig.
8c, in which the magnitudes are notably enhanced. This is expected since
$g_{t}\rightarrow 0$ when approaching half-filling. Physically it reflects
suppression of charge motion by strong correlation effect, so as to suppress
formation of pair bonds. We also note that, the stripe-like shape of intensity
distributions in Fig. 8(a-b) seem overall move downward compared with Fig.
4(a-b), which means, in another word, the optimal doping regime is more
deviated from PC if $g_{t}$ is ignored. In Fig. 8(e-f) we also show the
$s^{\pm}-$wave pairing for $d_{x^{2}-y^{2}}$ orbital, which have a much
smaller magnitude compared with the $d_{z^{2}}$ counterparts.

##  Appendix D Pseudogap

Figure. 9 shows the superconducting order parameter $g_{t}^{z}|\Delta_{\bot
z}^{s\pm}|$, and the magnitude of the pairing gap $|\Delta_{\bot z}^{s\pm}|$
of $s^{\pm}-$wave pairing as a function of $p$. Here the quantity
$|\Delta_{\bot z}^{s\pm}|$ (black) is much larger than the order parameter
(green). Since the latter is related to the superconducting transition
$T_{c}$, the former could represent the corresponding pseudogap above $T_{c}$.
This figure suggests a non-monotonic dependence of pseudogap on doping $p$,
unlike that of the one-band $t-J$ model shown in Fig. 7.

Figure 9: Pairing gap parameter $|\Delta_{\bot z}^{s\pm}|$ (black) and
superconducting order parameter $g_{t}^{z}|\Delta_{\bot z}^{s\pm}|$ (green) as
a function of $p$.

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