# Superconductivity in nickelate and cuprate superconductors with strong
bilayer coupling

Zhen Fan  The first three authors contributed equally to this work.  Institute
of Physics, Chinese Academy of Sciences, P.O. Box 603, Beijing 100190, China
School of Physical Sciences, University of Chinese Academy of Sciences,
Beijing 100049, China  Jian-Feng Zhang  The first three authors contributed
equally to this work.  Institute of Physics, Chinese Academy of Sciences, P.O.
Box 603, Beijing 100190, China  Bo Zhan  The first three authors contributed
equally to this work.  Institute of Physics, Chinese Academy of Sciences, P.O.
Box 603, Beijing 100190, China  School of Physical Sciences, University of
Chinese Academy of Sciences, Beijing 100049, China  ByteDance Research,
Zhonghang Plaza, No. 43, North 3rd Ring West Road, Haidian District, Beijing
100089, China  Dingshun Lv  ByteDance Research, Zhonghang Plaza, No. 43, North
3rd Ring West Road, Haidian District, Beijing 100089, China  Xing-Yu Jiang
Institute of Physics, Chinese Academy of Sciences, P.O. Box 603, Beijing
100190, China  Bruce Normand<EMAIL_ADDRESS>Laboratory for Theoretical
and Computational Physics, Paul Scherrer Institute, CH-5232 Villigen-PSI,
Switzerland  Institute of Physics, Ecole Polytechnique Fédérale de Lausanne
(EPFL), CH-1015 Lausanne, Switzerland  Tao Xiang<EMAIL_ADDRESS>Institute
of Physics, Chinese Academy of Sciences, P.O. Box 603, Beijing 100190, China
Collaborative Innovation Center of Quantum Matter, Beijing 100190, China
Beijing Academy of Quantum Information Sciences, Beijing, 100190, China.

###### Abstract

The discovery of superconductivity at 80 K under high pressure in La3Ni2O7
presents the groundbreaking confirmation that high-$T_{c}$ superconductivity
is a property of strongly correlated materials beyond cuprates. We use density
functional theory (DFT) calculations of the band structure of La3Ni2O7 under
pressure to verify that the low-energy bands are composed almost exclusively
of Ni 3$d_{x^{2}-y^{2}}$ and O 2$p$ orbitals. We deduce that the Ni
3$d_{z^{2}}$ orbitals are essentially decoupled by the geometry of the high-
pressure structure and by the effect of the Ni Hund coupling being strongly
suppressed, which results from the enhanced interlayer antiferromagnetic
interaction between $d_{z^{2}}$ orbitals and the strong intralayer
hybridization of the $d_{x^{2}-y^{2}}$ orbitals with O 2$p$. By introducing a
tight-binding model for the Fermi surfaces and low-energy dispersions, we
arrive at a bilayer $t$-$t_{\perp}$-$J$ model with strong interlayer hopping,
which we show is a framework unifying La3Ni2O7 with cuprate materials
possessing similar band structures, particularly the compounds La2CaCu2O6,
Pb2Sr2YCu3O8, and EuSr2Cu2NbO8. We use a renormalized mean-field theory to
show that these systems should have ($d$+$is$)-wave superconductivity, with a
dominant $d$-wave component and the high $T_{c}$ driven by the near-optimally
doped $\beta$ band, while the $\alpha$ band adds an $s$-wave component that
should lead to clear experimental signatures.

##  I Introduction

Ever since the seminal discovery of high-temperature superconductivity in
copper oxide materials, scientists have explored the possibility of achieving
similar transition temperatures ($T_{c}$) in other correlated electronic
systems. Particular targets in this search include transition-metal compounds
and especially oxides with crystal structures resembling those of the
cuprates. In this context, the recent identification of superconductivity at
$T_{c}\approx 80$ K in La3Ni2O7 under high pressure [1, 2, 3, 4] constitutes
the long-sought demonstration of cuprate-type $T_{c}$ values in a non-cuprate
system, and as such has attracted intensive research activity within the
condensed-matter community.

La3Ni2O7 behaves as a metal under ambient pressure where, following the
discovery of its superconductivity, the band structure and Fermi surface were
characterized by angle-resolved photoemission spectroscopy (ARPES) [5]. As the
applied pressure is increased, the crystal structure of La3Ni2O7 undergoes a
phase transition from the orthorhombic symmetry $Amam$ to $Fmmm$ at
approximately 14 GPa, where superconductivity emerges [1, 4, 2, 3].
Measurements made at 20.5 GPa showed that, similar to the cuprates, La3Ni2O7
exhibits a resistivity that is linear in temperature above $T_{c}$ [3],
indicating the importance of strong correlations. Measurements of the Hall
coefficient showed further that the transport is dominated by holes [6].
However, the superconducting volume fraction of La3Ni2O7, deduced from a
measurement of the modulated a.c. magnetic susceptibility [6], is around 1%,
implying that the available samples are highly inhomogeneous. Because it is
not possible to probe the band structure under high pressure by ARPES, all the
preliminary insight into the origin of superconductivity in La3Ni2O7 has to
date been provided by density functional theory (DFT) calculations [7, 8, 9,
10, 11, 12, 13, 14, 15].

DFT results representative of the $Fmmm$ structure reveal that three
electronic bands, designated as $\alpha$, $\beta$, and $\gamma$, cross the
Fermi level. The $\alpha$ band forms an electron pocket, while the $\beta$ and
$\gamma$ bands create two distinct hole pockets in the first Brillouin zone.
Considering the two relevant $d$ orbitals of the Ni atom, the $\alpha$ and
$\beta$ bands originate primarily from the Ni 3$d_{x^{2}-y^{2}}$ and O
2$p_{x,y}$ orbitals, whereas the $\gamma$ band arises predominantly from the
Ni 3$d_{z^{2}}$ orbital. Although DFT methods provide only an approximate
treatment of strongly correlated materials, they can be calibrated against
ARPES measurements at ambient pressure, which indicates that the leading
correlation effects in La3Ni2O7 are being captured appropriately. At this
pressure, while the $\alpha$ and $\beta$ pockets exhibit shapes very similar
to those in the high-pressure system, the $\gamma$ band lies below the Fermi
surface.

A key difference between La3Ni2O7 and the cuprate superconductors lies in the
valence-electron numbers of the Ni2+ and Cu2+ ions. Specifically, a Cu ion
($3d^{9}$) possesses only one unpaired valence electron in the
$d_{x^{2}-y^{2}}$ orbital, while the Ni ion (conventionally $3d^{8}$) has
unpaired valence electrons in both the $d_{x^{2}-y^{2}}$ and $d_{z^{2}}$
orbitals. Multiple models have already been proposed that incorporate
different Hubbard- or $t$-$J$-type interactions in a bid to capture the
properties of this multi-orbital system [16, 17, 11, 18, 19, 20, 21]. Most of
these models consider interactions within the $d_{x^{2}-y^{2}}$ and
$d_{z^{2}}$ orbitals, deducing both an intralayer antiferromagnetic (AF)
interaction between the $d_{x^{2}-y^{2}}$ orbitals and an interlayer AF
interaction between the $d_{z^{2}}$ orbitals [8, 22, 23, 24, 25, 26, 27, 28,
29]. In contrast, other models focus exclusively on the latter (interlayer)
interaction [30, 12, 31]. The Hund coupling, $J_{\rm H}$, between the
$d_{x^{2}-y^{2}}$ and $d_{z^{2}}$ orbitals is considered explicitly in certain
two-orbital models [24, 27], and in the limit of strong $J_{\rm H}$ it has
been argued that the interlayer interaction between the $d_{z^{2}}$ orbitals
may induce an effective interlayer AF interaction between $d_{x^{2}-y^{2}}$
spins. This leads to a one-orbital $t$-$J$-$J_{\perp}$ model with bilayer AF
interactions [32, 28, 33, 25, 34, 35].

A key point of contention among these models lies in the interaction terms.
The focus of many studies on the interlayer interaction between $d_{z^{2}}$
orbitals is justified by the fact that the interlayer Ni-O-Ni angle jumps from
$168^{\circ}$ to $180^{\circ}$ at the 14 GPa structural transition, while the
$c$-axis lattice constant is shortened. Both changes should contribute to a
significant enhancement of the interlayer interaction at pressures $P>14$ GPa,
leading to a popular interpretation of the high-pressure superconductivity in
La3Ni2O7 involving enhanced interaction-induced interlayer pairing between
electrons in $d_{z^{2}}$ orbitals [8, 18, 22, 32, 30, 33, 36, 12, 31, 10, 34,
23, 24, 25]. Within this class of models, there are two proposed mechanisms
for the origin of pairing, one involving the metallization of the $\gamma$
band when the high pressure lifts it through the Fermi surface [37, 1, 30].
The other relies on strong hybridization between the $d_{z^{2}}$ and
$d_{x^{2}-y^{2}}$ orbitals, with the latter contributing significantly to the
propagation of paired electrons, whose pairing is induced predominantly by the
interactions between the Ni 3$d_{z^{2}}$ orbitals) [8, 18, 22, 32, 33, 36, 12,
31].

In counterpoint to these $d$-orbital models, another strand of research has
considered the role of the O 2$p$ orbitals [1, 7, 22, 11, 38], whose strong
hybridization with the 3$d_{x^{2}-y^{2}}$ orbitals would suggest the emergence
of Zhang-Rice singlets [1, 22, 7, 29]. Most of these theoretical
investigations nevertheless either address superconductivity in La3Ni2O7
independently or tend to emphasize the differences between La3Ni2O7 and the
cuprates, rather than any common ingredients. Exploration of the pairing
symmetry within these different types of model has also yielded diverse
results, including $s$-wave [12, 8, 18, 28, 33, 39, 17, 16, 40, 23, 34, 10,
31, 17], $d$-wave [11, 29], and more complex cases [32, 19, 24, 21, 28, 41].

In this work, we reconsider the fundamental factors governing the
superconducting properties of La3Ni2O7 and establish a direct connection with
cuprate materials. The crucial insight arises from examining the Fermi surface
of La3Ni2O7 under high pressure, determined by DFT calculations as in Refs.
[7, 8, 9, 10, 11, 12]. We observe striking similarities between the Fermi
surfaces of La3Ni2O7 and those of a number of bilayer cuprates with medium-
high $T_{c}$ values, including La2CaCu2O6 [42, 43, 44], Pb2Sr2YCu3O8 [45, 46],
and EuSr2Cu2NbO8 [47, 48]. As summarized in Ref. [49], the bilayer splitting
is larger in these three systems than in the better-known cuprate
superconductors, presumably because in-plane buckling is larger and the energy
difference between the two $e_{g}$ orbitals is smaller, and this causes the
similarity to La3Ni2O7. In Fig. 1 we show how the Fermi surfaces of the
$\alpha$ and $\beta$ bands in La3Ni2O7 closely mirror those of the three
cuprate superconductors listed. The sole distinction lies in the fact that
La3Ni2O7 possesses an additional $\gamma$ Fermi surface under high pressure.
Given the significant impact of the Fermi-surface shape on the low-energy
physics, this unexpected resemblance prompts a reevaluation of the roles
played by the three bands in La3Ni2O7. As a consequence, we center our
investigation on identifying the common elements shared by these materials,
with the overarching goal of proposing a universal description of
superconductivity applicable both to La3Ni2O7 and to the cuprates.

The subsequent sections of this paper are organized as follows. In Sec. II, we
analyze the low-energy electronic structure of La3Ni2O7, which leads us to put
forward a bilayer $t$-$t_{\perp}$-$J$ model. In Sec. III, we use a
renormalized mean-field theory to calculate the superconducting phase diagram
of this model, focusing on the symmetry of the pairing gap and the effect of
doping. Section IV provides a summary of our results and of their physical
implications for nickelate superconductivity.

Figure 1: Fermi surfaces at $k_{z}=0$ calculated by DFT for (a) La3Ni2O7 under
high pressure, and for three cuprates at ambient pressure, La2CaCu2O6 (b),
Pb2Sr2YCu3O8 (c), and EuSr2Cu2NbO8 (d). (e) Low-energy band dispersions
calculated by DFT for La3Ni2O7 under high pressure, shown along the high-
symmetry lines in the two-dimensional (2D) Brillouin zone. (f) Low-energy band
dispersions described by the effective bilayer model of Eq. (1) for the
$\alpha$ and $\beta$ bands. The parameters were determined by fitting to the
DFT results of panel (e) in the energy interval [$-0.4$ eV, 0.4 eV] around the
Fermi level.

##  II Minimal model for La3Ni2O7

To identify the key interactions driving the high-$T_{c}$ superconducting
pairing in La3Ni2O7 under pressure, it is essential to understand two primary
physical effects. The first involves an accurate treatment of the interlayer
coupling and the second centers on the role of the Hund coupling, $J_{\rm H}$.
A noteworthy characteristic of the Fermi surfaces in La3Ni2O7 is the
degeneracy of the $\alpha$ and $\beta$ bands along the two diagonal directions
in the Brillouin zone. As we discuss in Sec. II.1, this degeneracy is the
hallmark of a special form of interlayer tunneling within the bilayer, which
forms a critical piece of the puzzle when constructing a minimal low-energy
model. Hund coupling is inherent to the Ni ion but, as we detail in Sec. II.2,
its effects are suppressed quite considerably at high pressures, where the Ni
$d_{z^{2}}$ orbitals on adjacent layers experience an enhanced interaction. At
the same time, the hybridization of the Ni $d_{x^{2}-y^{2}}$ orbitals with the
O $p$ orbitals, where the holes reside, acts to create both strong intralayer
AF interactions and Zhang-Rice singlets, which further reduce the relevance of
the Hund term. Drawing from these analyses, we deduce that the minimal model
for the low-energy physics of La3Ni2O7 is the single-band bilayer
$t$-$t_{\perp}$-$J$ model with strong interlayer hopping, $t_{\perp}$, and
hence that the high-$T_{c}$ pairing has the same origin as in cuprate
superconductors.

###  II.1 Bilayer coupling in La3Ni2O7

We have performed our own DFT calculations of the electronic band structures
of La3Ni2O7 in its high-pressure $Fmmm$ phase and of the three cuprates
La2CaCu2O6, Pb2Sr2YCu3O8, and EuSr2Cu2NbO8, including correlation effects by
standard methodology. The technical details of these calculations are
presented in App. A and the results shown in Figs. 1(a-e) reproduce well the
results reported by other authors both for La3Ni2O7 at the experimentally
characterized [1] pressure of 29.5 GPa [7, 8, 9, 10, 11, 12, 13, 14, 15] and
for the three cuprates [49]. A distinctive feature of the $\alpha$\- and
$\beta$-bands that recurs in the DFT calculation is their degeneracy along the
two diagonal directions in the Brillouin zone, which is shown at the Fermi
surface in Figs. 1(a,b,d) and for part of the $\Gamma$M direction in Fig.
1(e). This phenomenon is well known in bilayer high-$T_{c}$ cuprate
superconductors [49], having been identified in YBa2Cu3O7-δ, Ba2Sr2CaCu2O8,
HgBa2CaCu2O6, and Tl2Ba2CaCu2O8, as well as in two of the three cuprates whose
Fermi surfaces we show in Fig. 1.

It was recognized very early that the cuprates are charge-transfer insulators,
with the doped holes residing on the O 2$p_{x}$ or 2$p_{y}$ orbitals and
forming Zhang-Rice singlets [50] with the unpaired electron in a Cu
3$d_{x^{2}-y^{2}}$ orbital due to their strong $\sigma$-hybridization. The
wave function of the O hole in this singlet state adopts the $d_{x^{2}-y^{2}}$
symmetry of the Cu orbital and hence the interlayer tunneling of an O hole
depends on its in-plane momentum, $(k_{x},k_{y})$, and vanishes when
$k_{x}=k_{y}$ or $k_{x}=-k_{y}$ [51, 52], leading to the degeneracy of the
bilayer bands along the Brillouin-zone diagonals. It should be emphasized that
direct hopping between Cu 3$d_{x^{2}-y^{2}}$ orbitals along the $z$-axis is
negligible due to the vanishing wave-function overlap in this direction, and
the interlayer tunneling is facilitated by weak admixtures with the Cu
3$d_{z^{2}}$ and 4$s$ orbitals, which are rotation-symmetric around the
$z$-axis.

The $\alpha$ and $\beta$ bands in La3Ni2O7 are a simple consequence of strong
bilayer splitting, which completely alters the curvature of the $\alpha$ Fermi
surface. Although this splitting is stronger than in the most famous examples
of high-$T_{c}$ cuprate superconductors, namely YBa2Cu3O7-δ, Ba2Sr2CaCu2O8,
HgBa2CaCu2O6, and Tl2Ba2CaCu2O8 as listed above, it is completely in line with
the splittings, and hence the Fermi-surface shapes, found in La2CaCu2O6,
Pb2Sr2YCu3O8, and EuSr2Cu2NbO8, as we show in Figs. 1(b-d). Although the
degeneracy is partially broken in Pb2Sr2YCu3O8, the fundamental features of
the Fermi surfaces remain unaltered. We note here that these three cuprates do
not share quite the same high $T_{c}$ values as their better-known analogs; at
hole doping $\delta=0.2$, the superconducting transition temperatures are
approximately 60 K for both La2CaCu2O6 [43] and Pb2Sr2YCu3O8 [46], and 40 K
for EuSr2Cu2NbO8 [48].

Our calculations verify that, as noted above, the $\alpha$ and $\beta$ bands
are formed primarily from the electronic contributions of the Ni
3$d_{x^{2}-y^{2}}$ and in-plane O 2$p_{x}$ and 2$p_{y}$ orbitals. This
observation implies that the nodal degeneracy observed in La3Ni2O7 is
unrelated to the Ni 3$d_{z^{2}}$ and apical O 2$p_{z}$ orbitals. Indeed, the
fact that the energy dispersions of the $\alpha$ and $\beta$ bands in
La3Ni2O7, and the shapes of their Fermi surfaces, are so directly comparable
to those in La2CaCu2O6, Pb2Sr2YCu3O8, and EuSr2Cu2NbO8, strongly suggests a
common origin for the high-$T_{c}$ pairing in these materials, encoded within
the $\alpha$ and $\beta$ bands alone.

Here we comment that, although the $d_{z^{2}}$ orbitals make a negligible
contribution to the $\alpha$ and $\beta$ bands around the Fermi level, they do
have a finite involvement arising from hybridization with the in-plane O 2$p$
orbitals. This contribution has a notable effect on the high-energy states in
both bands, breaking the degeneracy between the $\alpha$ and $\beta$ bands
along the two zone-diagonal directions, as shown in Fig. 1(e). If this
additional hybridization is neglected in a minimal low-energy model, the two
bands remain degenerate along the entirety of the zone diagonals, as shown in
Fig. 1(f).

###  II.2 Effective bilayer $t$-$t_{\perp}$-$J$ model

The above discussion implies that the electronic states around the Fermi level
in the $\alpha$ and $\beta$ bands can be described effectively by a bilayer
model of Ni 3$d_{x^{2}-y^{2}}$ electrons coupled through an interlayer
tunneling matrix element $t_{\perp\mathbf{k}}$. This model is expressed by the
Hamiltonian

$H_{0}=\sum_{\mathbf{k},\sigma}\left(\begin{array}[]{cc}c^{\dagger}_{1,\mathbf{k}}&c^{\dagger}_{2,\mathbf{k}}\end{array}\right)\left(\begin{array}[]{cc}\varepsilon_{\mathbf{k}}&t_{\perp\mathbf{k}}\\\
t_{\perp\mathbf{k}}&\varepsilon_{\mathbf{k}}\end{array}\right)\left(\begin{array}[]{c}c_{1,\mathbf{k}}\\\
c_{2,\mathbf{k}}\end{array}\right),$ (1)

where $c_{n,\mathbf{k}}$ is the annihilation operator of a $d_{x^{2}-y^{2}}$
electron in layer $n=1$, 2 and

$\displaystyle\varepsilon_{\mathbf{k}}$ $\displaystyle=$
$\displaystyle-2t\left(\cos k_{x}-\cos k_{y}\right)-4t^{\prime}\cos k_{x}\cos
k_{y}$ (2) $\displaystyle-2t^{\prime\prime}(\cos 2k_{x}-\cos 2k_{y})-\mu$

is the dispersion relation of electrons in each layer, with respective hopping
integrals $t$, $t^{\prime}$, and $t^{\prime\prime}$ between two nearest-,
next-, and next-next-neighbor sites; $\mu$ is the chemical potential.

In Eq. (1), $t_{\perp\mathbf{k}}$ is the interlayer hopping term,

$\displaystyle t_{\perp\mathbf{k}}=4t_{\perp}\\!\left[\cos k_{x}-\cos
k_{y}+\eta(\cos 2k_{x}-\cos 2k_{y})\right]^{2}\\!,\;\;$ (3)

whose form arises from the hopping of an orbital possessing $d_{x^{2}-y^{2}}$
symmetry and hence contains only ($\cos k_{x}-\cos k_{y}$) and its higher
harmonics. Complementing the ($\cos k_{x}-\cos k_{y}$) term used
conventionally to describe the $c$-axis hopping, we introduce the $\eta$ term
to enhance the accuracy of the fit to the DFT bands.

To obtain a tight-binding fit to the $\alpha$ and $\beta$ bands obtained from
DFT, shown for high-pressure La3Ni2O7 in Fig. 1(e), we focus on the energy
range from $-0.4$ eV to 0.4 eV. By fitting the eigenenergies obtained from the
Hamiltonian of Eq. (1), we determine the parameter values for a pressure
equivalent to 30 GPa as $t=0.39$ eV, $t^{\prime}/t=-0.18$,
$t^{\prime\prime}/t=0.05$, $t_{\perp}/t=-0.14$, $\eta=0.15$, and $\mu=-0.73$
eV. These parameters provide a very accurate fit to the Fermi surface of
La3Ni2O7 [Fig. 1(a)], as shown in App. A. We draw attention to the fact that
$t$ has quantitatively the same magnitude as in the well-known bilayer
high-$T_{c}$ cuprates; $t^{\prime}$ and $t^{\prime\prime}$ naturally have the
same relation to $t$, required to generate the familiar hole-like Fermi
surface, while $t_{\perp}$ is four times larger than the value used to model
YBa2Cu3O7-δ and Ba2Sr2CaCu2O8 in early literature [53]. We comment again that
these effective values are optimized to reproduce the low-energy electronic
structure of the $\alpha$ and $\beta$ bands only, and deviate significantly
from their band structures beyond the energy range of the fit [Fig. 1(f)]; as
noted above, the primary reason for this deviation is the increasing Ni
3$d_{z^{2}}$ orbital content that sets in at energies of order 2 eV.

Turning from the $t$ term to the $J$ term, a notable complication in
constructing a minimal model for the interactions in La3Ni2O7 is the Hund
coupling between the Ni $d_{z^{2}}$ and $d_{x^{2}-y^{2}}$ orbitals. However,
as we summarized at the beginning of this section, two effects conspire to
quench the effect of the Hund coupling in the superconducting phase of the
bilayer nickelate. The first effect is the pressure-induced structural
transition occurring at 14 GPa in La3Ni2O7, which changes the $c$-axis Ni–O–Ni
bond angle from 168∘ to precisely 180∘ while also shortening the bond lengths,
and thus enhancing significantly the superexchange interaction between the two
Ni $d_{z^{2}}$ orbitals in adjacent layers of a bilayer. As detailed in Ref.
[41], the strengthening of this AF interaction causes a concomitant
suppression of the effect of the Hund coupling of the $d_{z^{2}}$ to the
$d_{x^{2}-y^{2}}$ orbitals.

In a similar vein, the second effect is the strong hybridization of the Ni
$d_{x^{2}-y^{2}}$ orbitals with the in-plane O $p$ orbitals. In the absence of
doped holes on the O ions, this gives rise to a robust intralayer AF coupling
of Ni 3$d_{x^{2}-y^{2}}$ spins, and in the presence of a doped hole it causes
the formation of a Zhang-Rice singlet, which quenches the Ni
3$d_{x^{2}-y^{2}}$ spin. Both situations constitute the dominant correlations
of the $d_{x^{2}-y^{2}}$ orbitals, contributing again to making their Hund
coupling to the $d_{z^{2}}$ orbitals ineffective. Although this hybridization
is further enhanced in the high-pressure $Fmmm$ phase, where the NiO2 planes
become significantly flatter, it is generically strong in the $\sigma$-bonding
geometry. Given the nontrivial valence state of a Ni atom in La3Ni2O7, which
diverges from the straightforward $+2$ configuration, we make the case that
the two $e_{g}$ orbital types are effectively decoupled into separate
subsystems, visible in the quite different energies of their respective bands
[Fig. 1(e-f)], and this allows them to be treated independently for the
purpose of revealing the low-energy physics.

This discussion leads to the conclusion that the $d_{z^{2}}$ orbitals are not
relevant to the superconducting properties of La3Ni2O7, which remain governed
by the Zhang-Rice physics of the $d_{x^{2}-y^{2}}$ orbitals familiar from the
cuprate superconductors. The minimal model for the interaction in this
situation is a regular in-plane superexchange term, leading to the Hamiltonian

$H=H_{0}+J\sum_{n\langle
ij\rangle}\left(\mathbf{S}_{ni}\cdot\mathbf{S}_{nj}-{\textstyle{\frac{1}{4}}}n_{ni}n_{nj}\right),$
(4)

where $\mathbf{S}_{ni}$ denotes the spin operator of the electron in the
$d_{x^{2}-y^{2}}$ orbital at site $i$ in layer $n$ of the bilayer. This
Hamiltonian is defined within the subspace obeying the constraint of no double
occupancy at each lattice site. Thus we arrive at a standard one-orbital
bilayer $t$-$t_{\perp}$-$J$ model, albeit with a sufficiently strong
interlayer hopping, $t_{\perp}$, that the $\alpha$ band is converted from
hole-like to electron-like.

The model we have deduced differs from all of the previous proposals made for
La3Ni2O7 in two aspects. First, the pairing interaction is not a consequence
of the AF interlayer interaction between $d_{z^{2}}$ spins but instead is
governed by the AF intralayer interaction between $d_{x^{2}-y^{2}}$ spins.
Second, the interlayer hopping term for the $d_{x^{2}-y^{2}}$ holes is strong
and anisotropic in ${\bf k}$, and drives the emergence of the
characteristically different $\alpha$ and $\beta$ bands. This significant
bilayer-splitting effect has been overlooked in most studies to date.

##  III Superconducting phase diagram

An analytical treatment of the $t$-$J$ model is made extremely difficult by
the hard local constraint of no double site occupancy. In Ref. [54], Zhang and
coworkers introduced a renormalized mean-field theory that aimed to capture
the fundamental consequences of the constrained (three-state) site basis at
the lowest order as a renormalization of both the hopping and the interaction
terms. In a one-orbital model on a square lattice with only a dilute
concentration, $\delta$, of doped holes, the renormalized coefficients are

$\displaystyle t_{\mathrm{eff}}$ $\displaystyle=$
$\displaystyle\frac{2\delta}{1+\delta}t,$ (5) $\displaystyle J_{\mathrm{eff}}$
$\displaystyle=$ $\displaystyle\frac{4}{(1+\delta)^{2}}J,$ (6)

(and similarly for the other hopping integrals). These expressions contain a
strong reduction of the effective bandwidth of the mobile holes, arising from
the difficulty of propagating in an AF background, and a strong enhancement of
the spin correlations arising from the increased weight of the contributing
configuration in the constrained manifold. To gain insight into the physics of
the bilayer $t$-$J$ model for La3Ni2O7 under pressure and its analogous
cuprates, we apply this renormalized mean-field theory by replacing the
coupling constants in Eq. (4) with their renormalized counterparts, thereby
releasing the constraint of no double occupancy at each lattice site.

In the bilayer system with strong $t_{\perp}$, we observe a large difference
between the electron occupancies, $n_{\alpha}$ and $n_{\beta}$, of the
$\alpha$ and $\beta$ bands [Figs. 1(a-d)]. When $t_{\perp}$ is rather weak,
which is the high-$T_{c}$ cuprate paradigm realized in YBa2Cu3O7-δ,
Ba2Sr2CaCu2O8, and similar materials, then the hole dopings
$\delta_{\alpha}=1-n_{\alpha}$ and $\delta_{\beta}=1-n_{\beta}$ can both
remain in the range around optimal doping. With strong $t_{\perp}$, however,
one of the bands becomes strongly overdoped and the other strongly underdoped
($\delta_{\beta}$ may even become negative), as shown in Figs. 1(b-d),
presumably accounting for the lower $T_{c}$ values in the cuprates La2CaCu2O6,
Pb2Sr2YCu3O8, and EuSr2Cu2NbO8. In high-pressure La3Ni2O7 [Fig. 1(a)], the
lower total electron count results in $\delta_{\beta}=0.15$ being close to
optimal doping, whereas $\delta_{\alpha}\approx 0.75$ is extremely large.
Because the renormalized mean-field theory [54] is formulated in real space,
the relevant doping parameter is
$\delta={\textstyle\frac{1}{2}}(\delta_{\alpha}+\delta_{\beta})$, which due to
the large value of $\delta_{\alpha}$ lies in the range 30-50%. While one may
question the applicability of the renormalized mean-field framework of Eqs.
(5-6) at such large $\delta$ values, we do not enter into these considerations
for our present qualitative purposes; we note only that both renormalization
factors become rather small (of order 2) and then the theory approaches, but
presumably remains superior to, the unrenormalized and unconstrained limit.

Figure 2: Phase diagram of the superconducting ground state of the bilayer
$t$-$J$ model deduced from the renormalized mean-field approach as a function
of the $\beta$-band hole doping, $\delta_{\beta}$, and the interaction ratio,
$J/t$. The solid black line is the phase boundary between states of $d$\- and
($d$+$is$)-wave pairing. The dashed white line represents the hole doping of
the $\beta$ band in La3Ni2O7 under the representative pressure of our DFT
calculations [Fig. 1(a)]. The color contours represent the magnitude of the
superconducting gap in units of $t$.

To apply the renormalized mean-field theory, we decouple the interaction term
of Eq. (4) into the pairing channel using the mean-field approximation

$\displaystyle 2\mathbf{S}_{ni}\cdot\mathbf{S}_{nj}$ $\displaystyle-$
$\displaystyle{\textstyle\frac{1}{2}}n_{ni}n_{nj}$ (7) $\displaystyle\approx$
$\displaystyle-\left(\Delta_{nij}\hat{\Delta}_{nij}^{\dagger}+h.c.\right)+\left|\Delta_{nij}\right|^{2},$

where
$\hat{\Delta}_{nij}=c_{n,i,\uparrow}c_{n,j,\downarrow}-c_{n,i,\downarrow}c_{n,j,\uparrow}$
and $\Delta_{nij}$ is the pairing order parameter in real space. In momentum
space, the mean-field Hamiltonian becomes

$H_{\rm
MF}=\sum_{\mathbf{k}}\Psi_{\mathbf{k}}^{\dagger}\begin{pmatrix}\xi_{\mathbf{k}}&\Delta_{1,\mathbf{k}}&-\xi_{\perp\mathbf{k}}&0\\\
\Delta_{1,\mathbf{k}}^{\star}&-\xi_{\mathbf{k}}&0&\xi_{\perp\mathbf{k}}\\\
-\xi_{\perp\mathbf{k}}&0&\xi_{\mathbf{k}}&\Delta_{2,\mathbf{k}}\\\
0&\xi_{\perp\mathbf{k}}&\Delta_{2,\mathbf{k}}^{\star}&-\xi_{\mathbf{k}}\end{pmatrix}\Psi_{\mathbf{k}},$
(8)

where
$\Psi_{\mathbf{k}}^{\dagger}=\big{(}c_{1,\mathbf{k},\uparrow}^{\dagger},c_{1,-\mathbf{k},\downarrow},c_{2,\mathbf{k},\uparrow}^{\dagger},c_{2,-\mathbf{k},\downarrow}\big{)}$,
$\xi_{\mathbf{k}}=t_{\delta}\varepsilon_{\mathbf{k}}$, and
$\xi_{\perp\mathbf{k}}=t_{\delta}t_{\perp\mathbf{k}}$ with
$t_{\delta}=2\delta/(1+\delta)$. $\Delta_{n,\mathbf{k}}$ is the gap function
in layer $n$ and may have different components with different pairing
symmetries; anticipating that this cuprate-type problem will yield singlet
superconductivity, the most general $d$+$is$-pairing state would have

$\Delta_{n,\mathbf{k}}=\Delta_{nd}\beta_{\mathbf{k}}+i\Delta_{ns}\gamma_{\mathbf{k}},$
(9)

where $\beta_{\mathbf{k}}=2(\cos k_{x}-\cos k_{y})$ and
$\gamma_{\mathbf{k}}=2(\cos k_{x}+\cos k_{y})$. The order parameters
$\Delta_{nd}$ and $\Delta_{ns}$ are then determined from the gap equations

$\displaystyle\Delta_{nd}$ $\displaystyle=-$
$\displaystyle\frac{J_{\mathrm{eff}}}{2N}\sum_{\mathbf{k}}\beta_{\mathbf{k}}\langle
d_{n,-\mathbf{k},\downarrow}d_{n,\mathbf{k},\uparrow}\rangle,$ (10)
$\displaystyle\Delta_{ns}$ $\displaystyle=$
$\displaystyle\frac{iJ_{\mathrm{eff}}}{2N}\sum_{\mathbf{k}}\gamma_{\mathbf{k}}\langle
d_{n,-\mathbf{k},\downarrow}d_{n,\mathbf{k},\uparrow}\rangle,$ (11)

where $N$ is the number of sites in the finite lattice on which we solve the
mean-field tight-binding problem. We comment here that we did not enforce
$\Delta_{1,\mathbf{k}}=\Delta_{2,\mathbf{k}}$ in order to leave open the
possibility of a solution with ($d$+$is$) symmetry in one layer and
($d$$-$$is$) in the other, which would be a state preserving time-reversal
symmetry, or indeed a solution with $s_{+-}$ symmetry ($s$-symmetric with
opposite signs in the two layers). However, in all of the results we present
below, the ground state was found to obey
$\Delta_{1,\mathbf{k}}=\Delta_{2,\mathbf{k}}$.

Solving the gap equations self-consistently at a given $\delta$ yields the
pairing symmetry of the superconducting ground state. In Fig. 2, we show the
phase diagram as a function of hole doping and of the ratio of real
interactions, $J/t$. While $J/t$ is not well characterized for nickelate
systems, we assume from the cuprate analogy we find throughout our study that
it will not differ strongly from the familiar cuprate values, and hence we
consider a relatively wide range $0.15\leq J/t\leq 0.35$ centered on the
canonical cuprate value of 1/4. For the hole doping, we subtract out the very
large value of $\delta_{\alpha}$, on the assumption that this extremely
overdoped band plays little or no role in the superconductivity, and show our
results as a function of $\delta_{\beta}$.

Figure 3: Magnitude of the superconducting energy gap, $|\Delta|/t$, shown as
a function of $\delta_{\beta}$ for the renormalized mean-field Hamiltonian
with $J/t=0.3$. Solid circles represent a ($d$+$is$)-wave ground state and
open circles a pure $d$-wave one. The inset shows the ratio between the gap
amplitudes of the $s$\- and $d$-wave components, $\Delta_{s}$ and
$\Delta_{d}$, of the ($d$+$is$)-wave state as a function of $\delta_{\beta}$.

In this broad parameter range, we find that the superconducting ground state
may exhibit either $d$\- or ($d$+$is$)-wave pairing symmetry (Fig. 2). At low
net hole concentrations, down to half-filling of the $\beta$ band, pure
$d$-wave pairing is favored at all but the highest values of $J/t$. However,
increasing the hole content favors an additional $s$-wave component, such that
a ($d$+$is$)-wave state appears over much of the phase diagram. Figure 3 shows
how the gap amplitude varies with $\delta_{\beta}$ for $J/t=0.3$, where the
$s$-wave component sets in above $\delta_{\beta}=0.05$ and, as the inset makes
clear, grows systematically with the doping to become comparable with the
$d$-wave component at high $\delta_{\beta}$. Regarding the overall gap
amplitude, $|\Delta|$ is very robust at lower dopings and higher $J/t$,
approaching the cuprate values of 20-40 meV.

All of these observations are consistent with the scenario that the near-
optimally doped $\beta$ band strongly favors $d$-wave pairing symmetry, as in
the highest-$T_{c}$ cuprates, while the effect of the $\alpha$ band in
favoring an additional $s$-wave component becomes increasingly important as
the net doping is increased. This ($d$+$is$)-wave pairing state, if
substantiated by calculations more rigorous than our present ones, then stands
as a key feature distinguishing nickelate superconductivity from the $d$-wave
cuprate superconductivity realized in the highest-$T_{c}$ materials. The
pairing symmetry realized in the cuprates La2CaCu2O6, Pb2Sr2YCu3O8, and
EuSr2Cu2NbO8 may also deviate from pure $d$-wave because both the $\alpha$ and
$\beta$ band-fillings deviate quite significantly from optimal. The
($d$+$is$)-wave state has two characteristic properties that can be measured
in experiment [55, 56, 57]. One is the fact that the ($d$+$is$)-wave gap has
no node, in contrast to a $d$-wave gap, such that the “V-shaped” single-
particle tunneling spectrum of a $d$-wave superconductor [58] is replaced by a
“U-shaped” spectrum for ($d$+$is$) symmetry. The second is that the
($d$+$is$)-wave state breaks time-reversal symmetry, which can be detected by
$\mu$SR measurements in zero field [59].

Considering the doping in more detail, the Ni 3$d_{x^{2}-y^{2}}$ and O
2$p_{x,y}$ character of the $\alpha$ and $\beta$ bands means that $\delta$ is
to a good approximation the number of $d_{x^{2}-y^{2}}$ holes (Zhang-Rice
singlets) in a single NiO2 plane. The $\gamma$ band consists predominantly of
Ni 3$d_{z^{2}}$ and apical O 2$p_{z}$ orbitals, and thus has no role in this
calculation when it does not cut the Fermi surface. However, the fact that the
$\gamma$ band emerges above the Fermi level under pressure, forming a hole
pocket around the M point, is important because it extracts holes from the
$\alpha$ and $\beta$ bands, allowing the value of $\delta$ to be altered by
the applied pressure. The Fermi-surface volume of the $\beta$ band of La3Ni2O7
at ambient pressure yields an approximate hole doping $\delta_{\beta}=0.26$,
which corresponds to the heavily overdoped regime in cuprates, where
superconductivity is almost completely suppressed. The emergence of the
$\gamma$ band above the Fermi level at the pressure represented in Fig. 1(a)
reduces $\delta_{\beta}$ to 0.15, bringing the $\beta$ band into the optimal
doping regime that we conclude drives the high-$T_{c}$ pairing in this
material.

This straightforward reasoning also accounts for the absence of
superconductivity in La3Ni2O6, which has $\alpha$ and $\beta$ Fermi surfaces
comparable to those of La3Ni2O7 at ambient pressure [9, 49], meaning with the
$\beta$ band well in the overdoped regime. We draw attention here to the
implications of possible O nonstoichiometry in La3Ni2O7, in that the presence
of O vacancies could cause a rapid suppression of $T_{c}$ and thus be a source
of sample inhomogeneity. Similar to La3Ni2O6, the cuprate superconductors with
comparable Fermi surfaces [Figs. 1(b-d)] do not possess a $\gamma$ band, and
thus neither of the $\alpha$ or $\beta$ bands is close to optimal doping,
presumably accounting for the lower $T_{c}$ values of superconductivity in
these systems.

##  IV Summary

We have introduced a bilayer $t$-$t_{\perp}$-$J$ framework to explain the
high-temperature superconductivity found in La3Ni2O7 at high pressure. Our
model is based on two key observations. The first is that the $\alpha$ and
$\beta$ bands that dominate the Fermi-surface topology are composed almost
exclusively of Ni 3$d_{x^{2}-y^{2}}$ and in-plane O 2$p_{x}$ and 2$p_{y}$
orbitals at low energies. The second is that the exact 180∘ structure of the
bilayer leads to an effective quenching of the Hund coupling between the Ni
3$d_{x^{2}-y^{2}}$ and 3$d_{z^{2}}$ orbitals, which arises from the strong
interactions within their spatially orthogonal subsystems (respectively the
planar NiO2 $d_{x^{2}-y^{2}}$-$p_{x,y}$ network and the $c$-axis
$d_{z^{2}}$-$p_{z}$-$d_{z^{2}}$ units), and hence to a near-complete
decoupling. An essential consequence of the $d_{x^{2}-y^{2}}$ symmetry of the
single active low-energy orbital is the $\mathbf{k}$-dependence of the
interlayer tunneling, and the resulting degeneracy of the $\alpha$ and $\beta$
bands along the nodal directions in the Brillouin zone is an important
fingerprint of this physics. Other orbitals (such as Ni 4$s$) and processes in
the system mean that the prefactor of the interlayer hopping term is
nevertheless substantial, which is the origin of the qualitative change in
Fermi-surface topology from the best-known bilayer cuprates. However, we
observe that the Fermi surfaces of the $\alpha$ and $\beta$ bands in the high-
pressure phase of La3Ni2O7 are very similar to several cuprate materials with
strong bilayer coupling, specifically La2CaCu2O6, Pb2Sr2YCu3O8, and
EuSr2Cu2NbO8, where the $t$-$J$ model is expected to describe the high-
temperature superconductivity.

By solving the superconducting gap equations of the bilayer
$t$-$t_{\perp}$-$J$ model within the renormalized mean-field approximation
over a wide range of electron fillings and interaction ratios, we find that
the pairing order parameter may have either $d$\- or ($d$+$is$)-wave symmetry.
We assume that the interaction ratio, $J/t$, is similar to that in cuprates,
and find an increase of the gap amplitude with stronger $J$. By decomposing
the hole doping into the contributions from the $\alpha$ and $\beta$ bands, we
deduce that the high-$T_{c}$ superconductivity and $d$-wave pairing are driven
by the near-optimal doping of the $\beta$ band. An essential contribution to
optimizing this doping is made by the $\gamma$ band, which removes holes into
the Ni 3$d_{z^{2}}$ orbitals and provides the mechanism by which the pressure
controls the doping level. The pressure (doping) then induces a transition
from $d$-wave to ($d$+$is$)-wave symmetry, which at the mean-field level of
our calculations appears to be favored as the pairing symmetry in La3Ni2O7.
This ($d$+$is$)-wave state then stands as the leading difference between
nickelate superconductivity and the purely $d$-wave superconductivity realized
in the highest-$T_{c}$ cuprates (YBa2Cu3O7-δ, Ba2Sr2CaCu2O8, HgBa2CaCu2O6, and
Tl2Ba2CaCu2O8). The ($d$+$is$)-wave state is nodeless and breaks time-reversal
symmetry allowing its verification by a number of experimental methods that
can be applied under pressure.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China
(under Grant No. 11888101) and the Postdoctoral Science Foundation of China
(under Grant No. 2022M723355).

##  Appendix A DFT Calculations

In order to calculate the band structure of La3Ni2O7 under pressure, and to
compare it with the cuprates La2CaCu2O6, Pb2Sr2YCu3O8, and EuSr2Cu2NbO8 as
shown in Fig. 1, we performed first-principles DFT calculations for all four
materials. For our calculations we employed the projector augmented wave (PAW)
method [60] implemented in the VASP package [61, 62]. A generalized gradient
approximation (GGA) of the Perdew-Burke-Ernzerhof (PBE) type [63] was chosen
for the exchange-correlation functional. The kinetic energy cutoff of the
plane-wave basis was set to 500 eV and a 12$\times$12$\times$12
$\mathbf{k}$-space mesh was used for sampling the Brillouin zone. A Gaussian
broadening with a width of 0.05 eV was implemented for the determination of
the Fermi surface (illustrated in Fig. 4). We used the GGA+U formalism of Ref.
[64] to incorporate the effect of electron correlations, setting the effective
Hubbard $U$ to 3 eV for Ni in La3Ni2O7 (similar to the value of 3.5 eV used in
Ref. [5]) and to 6 eV for all three of the cuprates [65, 66].

Figure 4: Comparison of the Fermi surface obtained from the tight-binding
Hamiltonian ($H_{0}$) with that calculated for La3Ni2O7 at an effective
pressure of 30 GPa by DFT. The solid lines show the DFT Fermi surface and the
open circles show Fermi-surface points obtained from $H_{0}$ with the
parameters given in Sec. II.2.

Our calculations yielded the electronic band structures of the type shown in
Fig. 1(e) for La3Ni2O7 at a pressure of approximately 30 GPa and for each of
the three cuprates at ambient pressure. The lattice constants were fully
optimized in every case, with the internal atomic positions relaxed until the
forces on all atoms were smaller than 0.01 eV/Å. To generate the band
dispersions of the type depicted in Fig. 1(f), we diagonalized the
Hamiltonian, $H_{0}$ [Eq. (1)], with parameters determined by fitting the DFT
band structures of the $\alpha$ and $\beta$ bands in the energy interval
[$-0.4$ eV, 0.4 eV] around the Fermi level. The values of these fitting
parameters are given in the main text for La3Ni2O7 under pressure. Figure 4
illustrates how this procedure reproduces accurate Fermi-surface contours for
the DFT calculations. We note that our results reproduce accurately the giant
bilayer splitting in all four materials, consistent with recent results for
La3Ni2O7 [7, 8, 9, 10, 11, 12] and earlier results for the three bilayer
cuprates [49].

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