††thanks: These two authors contributed equally††thanks: These two authors
contributed equally

# Type II t-J model in charge transfer regime in bilayer La3Ni2O7 and trilayer
La4Ni3O10

Hanbit Oh<EMAIL_ADDRESS>Boran Zhou  Ya-Hui Zhang<EMAIL_ADDRESS>William H.
Miller III Department of Physics and Astronomy, Johns Hopkins University,
Baltimore, Maryland, 21218, USA

###### Abstract

Recent observations of an 80 K superconductor in La3Ni2O7 under high pressure
have attracted significant attention. Recent experiments indicate that
La3Ni2O7 may be in the charge transfer regime, challenging the previous models
based purely on the Ni $d_{x^{2}-y^{2}}$ and $d_{z^{2}}$ orbitals. In this
study, we propose a low energy model that incorporates doped holes in the
oxygen $p$ orbitals. Given that the parent nickel state is in the $3d^{8}$
configuration with a spin-one moment, doped hole only screens it down to spin-
half, in contrast to the Zhang-Rice singlet in cuprate. We dub the single hole
state as Zhang-Rice spin-half and build an effective model which includes
three spin-one states ($d^{8}$) and two Zhang-Rice spin-half states
($d^{8}L$). At moderate pressure around $20$ GPa, the dominated oxygen orbital
is an in-plane Wannier orbital with the same lattice symmetry as the
$d_{x^{2}-y^{2}}$ orbital. The resulting model reduces to the bilayer type II
t-J model previously proposed in the Mott-Hubbard regime. Notably, the hopping
between the in-plane $p$ orbitals of the two layers is still suppressed.
Density matrix renormalization group (DMRG) simulation reveals a pairing dome
with the optimal hole doping level at $x=0.4\sim 0.5$, distinct from the hole
doped cuprate where optimal doping occurs around $x=0.19$. Further increasing
pressure initially raises the critical temperature ($T_{c}$) until reaching an
optimal pressure beyond which the $p_{z}$ orbital of oxygen becomes favorable
and superconductivity is diminished. This shift from in-plane $p$ orbital to
$p_{z}$ orbital may elucidate the experimentally observed superconducting dome
with varying pressure. As an extension, we also suggest a trilayer version of
the type II t-J model as the minimal model for pressured La4Ni3O10, which is
distinct from the models in the Mott-Hubbard regime.

Introduction.— The recent discovery of a superconductor with a critical
temperature of approximately 80 K in La3Ni2O7 under high pressureSun  _et al._
(2023); Hou  _et al._ (2023); Zhang  _et al._ (2023a) has sparked considerable
interestLiu  _et al._ (2023a); Yang  _et al._ (2023a); Zhang  _et al._
(2023b); Luo  _et al._ (2023); Zhang  _et al._ (2023c); Yang  _et al._
(2023b); Sakakibara  _et al._ (2023); Gu  _et al._ (2023); Shen  _et al._
(2023); Wú  _et al._ (2023); Christiansson  _et al._ (2023); Liu  _et al._
(2023b); Cao and Yang (2023); Oh and Zhang (2023); Lu  _et al._ (2024a); Qu
_et al._ (2023); Lu  _et al._ (2023); Jiang  _et al._ (2023a); Tian  _et al._
(2023); Zhang  _et al._ (2023d); Qin and Yang (2023); Huang  _et al._ (2023);
Zhang  _et al._ (2023e); Jiang  _et al._ (2023b); Yang  _et al._ (2023c); Qin
and Yang (2023); Zhang  _et al._ (2023); Kitamine  _et al._ (2023); Jiang  _et
al._ (2023); Lange  _et al._ (2023a, b); Schlömer  _et al._ (2023); Jiang  _et
al._ (2024a); Zhan  _et al._ (2024); Lechermann  _et al._ (2024); Wang  _et
al._ (2024); Xue and Wang (2024); Kaneko  _et al._ (2024); Jiang  _et al._
(2024b); Ryee  _et al._ (2023); Liao  _et al._ (2023); Wú  _et al._ (2023).
Additionally, there is emerging evidence of superconductivity in La4Ni3O10
under high pressure, exhibiting critical temperatures ranging from 20-30 K,
which has further fueled research interest in this area Li  _et al._ (2024a);
Zhu  _et al._ (2024); Sakakibara  _et al._ (2024); Zhang  _et al._ (2024a); Lu
_et al._ (2024b); Zhang  _et al._ (2024b); LaBollita  _et al._ (2024); Yang
_et al._ (2024). Identifying a minimal model that captures the essential
physics is a crucial step forward.

According to density functional theory (DFT) Sun  _et al._ (2023), the valence
of the nickel (Ni) atom in the bilayer nickelate La3Ni2O7 is in the $3d^{8-x}$
configuration. The $d_{x^{2}-y^{2}}$ orbital is at density $n_{1}=1-x$ per
site with $x=0.5$, while the $d_{z^{2}}$ orbital has the density $n_{2}\approx
1$ and is near Mott localization. Following this picture, many theoretical
works propose a two-orbital Hubbard or t-J model in terms of the
$d_{x^{2}-y^{2}}$ and the $d_{z^{2}}$ orbital. Especially, Refs.Oh and Zhang
(2023); Lu  _et al._ (2024a) highlight the role of Hund’s coupling $J_{H}$ in
transmitting a large inter-layer spin-spin coupling $J_{\perp}$ from the
$d_{z^{2}}$ to the $d_{x^{2}-y^{2}}$ orbital, providing a plausible
explanation for a high critical temperature ($T_{c}$) even at large doping
$x\approx 0.5$. In Ref.Lu  _et al._ (2024a), the localized $d_{z^{2}}$
orbitals are simply integrated out, so the model is reduced to a one-orbital
bilayer t-J model of $d_{x^{2}-y^{2}}$ with a negligible $t_{\perp}$. But the
integration of $d_{z^{2}}$ orbital is not appropriate in the large $J_{H}$
limit Oh and Zhang (2023); Yang  _et al._ (2023d). Instead, the minimal model
is demonstrated to be a bilayer type-II t-J model Oh and Zhang (2023); Yang
_et al._ (2023d), which includes the spin-one Ni2+ state (3$d^{8}$) and spin-
half Ni3+ state (3$d^{7}$) at each site. The type II t-J model hosts unique
physics such as a dome of pairing gap around $x\approx 0.5$ due to doping
induced Bardeen-Cooper-Schrieffer(BCS) to Bose-Einstein-condensate (BEC)
crossoverYang  _et al._ (2023d). There are also two different Fermi liquids
above $T_{c}$ with a jump of the Fermi surface volume by $1/2$ Brillouin
zoneOh and Zhang (2023); Yang  _et al._ (2023d); Wu  _et al._ (2024). Clearly
the physics is essentially different from the familiar hole doped cuprates.
However, the model is derivedOh and Zhang (2023) assuming that the system is
in the Mott-Hubbard regime of the Zaanen–Sawatzky–Allen classification scheme
Zaanen  _et al._ (1985). Depending on the energy splitting $\Delta$ between
the oxygen $p$ orbital and the Ni $d$ orbital compared to Hubbard U, we have
the Mott-Hubbard regime ($\Delta>U$) or the charge-transfer regime
($\Delta<U$). In the Mott-Hubbard regime, the doped holes enter the 3d
orbitals of the Ni atom and we can ignore the oxygens. However, a recent
experiment suggests that La3Ni2O7 might be within the charge transfer regime
where holes enter the oxygen $p$ orbitals while the nickel atom is pinned to
be in the 3$d^{8}$ configuration with a localized spin-one moment Dong  _et
al._ (2023). This challenges the previous theoretical models. It is thus
critical to derive an effective model for the charge transfer regime, akin to
what has been done in hole doped cuprates Zhang and Rice (1988).

There is a notable distinction between bilayer nickelates and cuprates. The
undoped Ni state is in the spin-one $3d^{8}$ configuration with two electrons
in the two $e_{g}$ orbitals, $(d_{x^{2}-y^{2}},d_{z^{2}})$, strongly coupled
together by a Hund coupling $J_{H}$. Thus doped hole in the oxygen can at best
screen the spin-one moment down to spin-half, contrasting the Zhang-Rice
singlet in cuprate. In this work, we demonstrate that the doped hole enters
the in-plane oxygen orbital and forms a net spin-half moment together with the
Ni spin-one moment at moderate pressure. The hole state state is dubbed as the
Zhang-Rice spin-half. Then by taking the Zhang-Rice spin-half as the primary
state of the single-hole $d^{8}L$ state and keeping the spin-triplet parent
state of the undoped $d^{8}$ state, the type-II t-J model Zhang and Vishwanath
(2020) is shown to be the minimal model also in the charge-transfer regime of
the doped bilayer nickelates. Besides, we reveal that the oxygen $p_{z}$
orbital living between the two layers can only mediate a very small inter-
layer hopping $t_{\perp}$ due to symmetry. From the density matrix
renormalization group (DMRG) simulation, we show that the pairing gradually
decreases with $t_{\perp}$ but remains robust. If we further increase across
an optimal value, the doped hole shifts from the intra-layer $p_{x},p_{y}$
oxygen orbital to the inter-layer $p_{z}$ oxygen and the pairing is
suppressed. We finally point out that the trilayer La4Ni3O10 is also described
by a trilayer version of type II t-J model in the charge transfer regime.

Charge transfer model.— We consider a model on bilayer square lattice (See
Fig. 1), which includes three $p$-orbitals ($2p_{x},2p_{y},2p_{z}$) of oxygen
as well as the two $E_{g}$ $d$-orbitals ($3d_{x^{2}-y^{2}},3d_{z^{2}}$) of
nickel. In the hole picture, we first focus on one-unit cell with two Ni atoms
and nine O atoms, the Hamiltonian is,

$\displaystyle H$ $\displaystyle=$ $\displaystyle
H_{dp}+\sum_{i,l}\left[U_{1}n_{1;i;l;\uparrow}^{d}n^{d}_{1;i;l;\downarrow}+U_{2}n_{2;i;l;\uparrow}^{d}n_{2;i;l;\downarrow}^{d}\right.$
(1)
$\displaystyle+U^{\prime}n^{d}_{1;i;l}n^{d}_{2;i;l}-2J_{H}(\vec{s}^{d}_{1;i;l}\cdot\vec{s}^{d}_{2;i;l}+\frac{1}{4}n^{d}_{1;i;l}n^{d}_{2;i;l}),$
$\displaystyle\left.+\sum_{a}\epsilon_{d,a}n_{a;i,l}^{d}\right]+\sum_{i^{\prime},l,\alpha}\epsilon_{p}n_{\alpha;i^{\prime};l}^{p}+\sum_{i}\epsilon_{p;z}n^{p}_{z;i},$

and

$\displaystyle H_{dp}$ $\displaystyle=$
$\displaystyle\sum_{i,l,a,\sigma}\left[2t_{dp;a}d^{\dagger}_{a;i;l;\sigma}p_{a;i;l;\sigma}+\mathrm{H.c.}\right]$
$\displaystyle+$
$\displaystyle\sum_{i,\sigma}\left[t_{dp;z}(d^{\dagger}_{2;i;t;\sigma}-d^{\dagger}_{2;i;b;\sigma})p_{z;i;\sigma}+\mathrm{H.c.}\right]$

with

$\displaystyle
p_{1;i;l;\sigma}=\frac{1}{2}\left[p_{x;i+\frac{{\widehat{x}}}{2};l;\sigma}-p_{x;i-\frac{{\widehat{x}}}{2};l;\sigma}-p_{y;i+\frac{{\widehat{y}}}{2};l;\sigma}+p_{y;i-\frac{{\widehat{y}}}{2};l;\sigma}\right],$
$\displaystyle
p_{2;i;l;\sigma}=\frac{1}{2}\left[p_{x;i+\frac{{\widehat{x}}}{2};l;\sigma}-p_{x;i-\frac{{\widehat{x}}}{2};l;\sigma}+p_{y;i+\frac{{\widehat{y}}}{2};l;\sigma}-p_{y;i-\frac{{\widehat{y}}}{2};l;\sigma}\right],$

where $a=1,2$ labels $d_{x^{2}-y^{2}}$ and $d_{z^{2}}$ orbital of Ni and
$l=t,b$ labels the top and bottom layer. $\sigma=\uparrow,\downarrow$ labels
the spin. Here $d^{\dagger}_{a;i;l;\sigma}$ creates a hole on the nickel site
$i$ relative to the $3d^{10}$ state. $p^{\dagger}_{a;i;l;\sigma}$ creates a
hole in a superposition of occupying the four oxygen atom sites $i^{\prime}$
around the Ni site $i$. The $p_{x},p_{y}$ orbitals are regrouped as
$p_{1},p_{2}$ orbitals living on the Ni site according to the $D_{4h}$
symmetry. One can check that $p_{1},p_{2}$ have the same lattice symmetry as
the $d_{1},d_{2}$ orbitals. In supplemental material (SM) Sec.I, we provide
the details on the symmetry analysis. $U$($U^{\prime}$) is the intra-(inter-)
orbital onsite repulsion, and $J_{H}>0$ is the Hund coupling. $t_{dp}$ is the
nearest-neighbor hopping between $d-p$ orbitals. There are three different
hopping channels, $(d_{1;l},p_{1;l})$, $(d_{2;l},p_{2;l})$, and
$(d_{2;t}-d_{2;b},p_{z})$ classified by $D_{4h}$ point group, respectively.
The relative size of the onsite energies $\epsilon_{d}$ and $\epsilon_{p}$
decides whether the system is in the charge transfer or Mott Hubbard regime.
We consider the case that the undoped parent Ni state is in $3d^{8}$ with two
holes in the two $e_{g}$ orbitals, forming a $S=1$ localized moment. Then
under further hole doping, additional holes enter the oxygen $p$ orbitals. We
note that the orbital $p_{1}$ and $p_{2},p_{z}$ have different eigenvalues
under the $C_{4}$ rotation and can not hybridize. So the doped hole enters
either $p_{1}$ or $p_{2},p_{z}$ depending on energetics.

Figure 1: The lattice structure and $p,d$ orbitals of the bilayer Nickelates
La3Ni2O7. The green (yellow) sphere denotes Ni (O) atom, respectively. At each
Ni atom, there are two $d$ orbitals, $d_{1},d_{2}$ (Left inset). Around each
Ni atom at site $i$, the $p_{x},p_{y}$ orbitals from the four oxygen atoms
around the Ni site in the same plane form two orbitals $p_{i;1},p_{i;2}$
living on the same site $i$. Meanwhile, there is a $p_{z}$ orbital from the
oxygen at the center of the ${\widehat{z}}$ bond. At moderate pressure, the
doped hole enters the Wannier orbital,
$p_{1;i}=\frac{1}{2}[p_{x;i+{\widehat{x}}/{2}}-p_{x;i-{\widehat{x}}/2}-p_{y;i+{\widehat{y}}/2}+p_{y;i-{\widehat{y}}/2}]$
which has the same symmetry as the $d_{1}$ orbital. The hole in this $p_{i;1}$
orbital couples to the $S=1$ moment of the Ni $3d^{8}$ state and forms a
Zhang-Rice spin-half state.

Zhang-Rice Spin-half.— In the strong coupling limit, $t_{dp}\ll U,U^{\prime}$,
the hole enters the oxygen $p$ orbitals and interact with the localized spin-
one moments from the Ni2+ state through a Kondo coupling,

$\displaystyle H_{K}$ $\displaystyle=$
$\displaystyle\sum_{i,l,a}J_{K;a}\big{[}(p_{a;i;l}^{\dagger}\vec{\sigma}p_{a;i;l})\cdot\vec{s}^{d}_{a;i;l}-\frac{1}{2}n^{p}_{a;i;l}\big{]}$
$\displaystyle+$
$\displaystyle\sum_{i,l}J_{K;z}[(p_{z;i}^{\dagger}\vec{\sigma}p_{z;i})\cdot\vec{s}^{d}_{2;i;l}-\frac{1}{2}n^{p}_{z;i}]$
$\displaystyle-$
$\displaystyle\sum_{i,l}2J_{H}[\vec{s}^{d}_{1;i;l}\cdot\vec{s}^{d}_{2;i;l}+\frac{1}{4}],$

where $\vec{s}^{d}_{a;i;l}$ is the spin 1/2 operator from the $d_{a}$ orbital
of the Ni at site $i$ of layer $l=t,b$. We have

$\displaystyle J_{K;a}$ $\displaystyle=$
$\displaystyle\frac{4t^{2}_{dp;a}}{U_{1}+U^{\prime}+J_{H}-\Delta_{p;a}}+\frac{4t^{2}_{dp;a}}{\Delta_{p;a}+J_{H}-U^{\prime}},$
$\displaystyle J_{K;z}$ $\displaystyle=$
$\displaystyle\frac{t^{2}_{dp;z}}{U_{2}+U^{\prime}+J_{H}-\Delta_{p;2}^{z}}+\frac{t^{2}_{dp;z}}{\Delta_{p;2}^{z}+J_{H}-U^{\prime}}.$
(3)

with $\Delta_{p;a}=\epsilon_{p}-\epsilon_{d;a}$,
$\Delta_{p;a}^{z}=\epsilon_{p;z}-\epsilon_{d;a}$. Here, we assume
$U^{\prime}-J_{H}\ll\Delta_{p,a},\Delta_{p,a}^{z}\ll U_{a}+U^{\prime}$ so the
system is in the charge transfer regime. For illustration, we adopt the
hopping parameters calculated at 29.5GPa by DFT, $t_{dp;1}=1.56$eV,
$t_{dp;2}=-0.75$eV, $t_{dp;z}=1.63$eV Wú  _et al._ (2023). We assume the
similar interaction strength, $U_{a}=10$eV, $U^{\prime}=6$eV, $J_{H}=2$eV and
as in the cuprates Ogata and Fukuyama (2008) and use
$\Delta_{p;a}=\Delta_{p;a}^{z}=9$eV. Within those parameter sets, we found
that $J_{K;z}\simeq J_{K;1}/4$ with $J_{K;1}=3.03$eV and $J_{K;z}=0.83$eV.

Figure 2:  The wave function of Zhang-Rice spin-1/2 states of the bilayer
nickelates. The local $d_{1}$, $d_{2}$ orbitals of Ni atoms surrounded by the
four in-plane $p$ orbitals. The yellow, green, and magenta arrow is for the
spin of the $p_{1}$, $d_{1}$, and $d_{2}$ orbital, respectively. (a) In the
$J_{K;1}\gg J_{H}$ limit, the Zhang-Rice spin-1/2 state is a simple product
state of the Zhang-Rice singlet made by $d_{1},p_{1}$ and the decoupled
$d_{2}$ orbital. (b) In the $J_{K;1}\ll J_{H}$ limit, two $d$ orbital forming
a spin triplet forms a spin $1/2$ along with $p_{1}$ orbitals. Here, for
simplicity, we illustrate only the $\ket{{\uparrow}}$, since
$\ket{{\downarrow}}$ is just a its time-reversal partner. (c) The three
orbital contributions in $S^{z}_{\mathrm{tot}}=\frac{1}{2}$ of
$\ket{\uparrow}$ state. As $J_{K;1}/J_{H}\rightarrow+\infty$, the spin-half
state is dominated by the $d_{2}$ orbital. Generically the state is a
combination of all three orbitals: $p_{1}$, $d_{1}$, and $d_{2}$.

We should view $p_{1},p_{2}$ as Wannier orbitals centered on Ni atom. The
doped hole is favored to enter one of them or the $p_{z}$ orbital by
minimizing the Kondo coupling $J_{K;1},J_{K;2},J_{K;z}$. When the hole
occupies the in-plane $p_{a}$ orbital, it forms a net $S=\frac{1}{2}$ together
with the original spin-one moment from the Ni2+. On the other hand, when the
hole occupies the $p_{z}$ orbital, it couples to the spin-one moments of Ni
from both layers. The ground state energies resulting from the Kondo
interactions per unit cell summed over two layers for each case are

$\displaystyle
E_{G}^{a}=-\big{[}J_{H}+\sqrt{J_{H}^{2}+J_{K;a}J_{H}+J_{K;a}^{2}}+J_{K;a}\big{]},$
$\displaystyle
E_{G}^{z}=-\frac{1}{2}\big{[}2J_{H}+\sqrt{4J_{H}^{2}+8J_{K;z}J_{H}+9{J_{K;z}}^{2}}+3J_{K;z}\big{]}.$

Using the condition $t_{dp;1}\gg t_{dp;2}$, we always have
$E_{G}^{1}(J_{K;1})\gg E_{G}^{2}(J_{K;2})$ and the main competition is between
the $p_{1}$ and $p_{z}$ orbitals. For the estimated parameter of
$J_{K;1}\simeq 4J_{K;z}$, we have $E_{G}^{1}(J_{K;1}\simeq
4J_{K;z})<E_{G}^{z}(J_{K;z})$ and the $p_{1}$ orbital wins. In this regime the
doped hole enters the in-plane oxygen orbital $p_{1}$ and screens the spin-one
Ni2+ down to spin-half. We dub this hole state as Zhang-Rice spin-half state
as analog to the Zhang-Rice singlet in hole doped cuprate. This Zhang-Rice
spin-half consists of the spins from the $p_{1}$ orbital and the original
$d_{1}$ and $d_{2}$ orbital of Ni. The wavefunction depends on the parameter
$r=J_{K;1}/J_{H}$:

$\displaystyle\ket{\sigma}_{l}\sim(1+r+\alpha)\ket{\overline{\sigma},\sigma,\sigma}-(r+\alpha)\ket{\sigma,\overline{\sigma},\sigma}-\ket{\sigma,\sigma,\overline{\sigma}},\\!$
(4)

with $\sigma=\pm\frac{1}{2}$. We used $\alpha(r)=[r^{2}+r+1]^{1/2}$ and here
omitted the normalization factor, for simplicity (See SM Sec.II). Also,
$\overline{\sigma}=-\sigma$, and $\ket{\sigma,\sigma_{1},\sigma_{2}}$ denotes
the spin of $p_{1;l},d_{1;l},d_{2;l}$ orbital respectively. A few remarks are
as follows. In the limit of $J_{H}\ll J_{K;1}$, the state is simply the tensor
product of the Zhang-Rice singlet from only $d_{1},p_{1}$ orbitals and a
decoupled spin-half from the $d_{2}$ orbitals shown in Fig. 2(a). However,
with a general $J_{H}$, the three orbitals are highly entangled and should be
considered together. For instance, in Fig. 2(c), we plot the spin
contributions of the net spin-1/2 from all three orbitals. The bipartite
entropy between the $d_{1},p_{1}$ orbitals and the $d_{2}$ orbital increases
with increasing $J_{H}$, reaching its maximum values in the limit where
$J_{H}\gg J_{K;1}$ (See SM), where we illustrate the state in Fig. 2(b).

The state $|{\sigma}\rangle_{l}$ is derived from solving the Hamiltonian in a
single unit cell, next we should make the states at two different sites $i,j$
orthogonal. We just call this state as $|{\sigma}\rangle_{i;l}$. This can be
simply done by replacing the $p_{1}$ orbital in Eq.(4) with a wannierized
electron operator,
$c_{i;l;\sigma}^{\dagger}=\mathcal{N^{\prime}}\sum_{j}B(i-j)p_{1;j;l;\sigma}$
where $B(i-j)=\frac{1}{N}\sum_{k}e^{ik\cdot(i-j)}\beta(k)$ and
$\beta(k)=[1-\frac{1}{2}(\cos k_{x}+\cos k_{y})]^{-1/2}$ which form orthogonal
and complete sets with
$\\{c_{i;l;\sigma},c_{j;l^{\prime};\sigma^{\prime}}^{\dagger}\\}=\delta_{i,j}\delta_{l,l^{\prime}}\delta_{\sigma,\sigma^{\prime}}\mathcal{N^{\prime}}^{2}$.
Here, we use
$\mathcal{N}^{\prime}(r)=[2(r+\alpha)^{2}+2(r+\alpha)+2]^{1/2}/(r+\alpha+1)$,
monotonic increasing function as increasing $r$, ranging $[1.22,1.41]$. Then,
we can turn to the electron picture in the remaining part of the paper, which
is more intuitive, through using $c_{i;l;\sigma}$. In this context, we have
$n_{i;c}=1-x$, which corresponds to the condition $n_{i;p}=x$ in the hole
representation.

Type-II t-J model.— Now we can see that the low energy model in the charge
transfer regime is still the type II t-J model Zhang and Vishwanath (2020).
The undoped Ni site is in the $d^{8}$ with spin-one, while the single hole
$d^{8}L$ state corresponds to the Zhang-Rice spin-half state above, which is
an analog of the $d^{7}$ states in the Mott-Hubbard regime given that the
$p_{1}$ orbital has the same crystal symmetry as the $d_{1}$ orbital. Then,
the wannierized orbital $c_{i;l;\sigma}$ can be projected in the $2+3=5$
dimensional Hilbert space. We now turn to the electron picture for simplicity.
For a general $r=J_{K;1}/J_{H}$, the electron operator of the $p_{1}$ orbital
is written as,

$\displaystyle c_{i;l;\uparrow}$ $\displaystyle=$
$\displaystyle+\prod_{j<i}{(-1)^{n_{j}}}\left[\ket{\downarrow}_{i;l}\bra{-1}_{i;l}+\frac{1}{\sqrt{2}}\ket{\uparrow}_{i;l}\bra{0}_{i;l}\right],$
$\displaystyle c_{i;l;\downarrow}$ $\displaystyle=$
$\displaystyle-\prod_{j<i}{(-1)^{n_{j}}}\left[\ket{\uparrow}_{i;l}\bra{1}_{i;l}+\frac{1}{\sqrt{2}}\ket{\downarrow}_{i;l}\bra{0}_{i;l}\right],$

with the Jordan-Wigner string, $\prod_{j<i}(-1)^{n_{j}}$. The spin-triplet
states of $d^{8}$ are
$\ket{-1}_{l}=d^{\dagger}_{1;l\downarrow}d^{\dagger}_{2;l;\downarrow}\ket{G}$,
$\ket{0}_{l}=\frac{1}{\sqrt{2}}(d^{\dagger}_{1;l;\uparrow}d^{\dagger}_{2;l;\downarrow}+d^{\dagger}_{1;l;\downarrow}d^{\dagger}_{2;l;\uparrow})\ket{G}$
and $\ket{1}_{l}=d^{\dagger}_{1;l;\uparrow}d^{\dagger}_{2;l;\uparrow}\ket{G}$.
Note that $d_{a;l,\sigma}$ is a hole-operator of $d$ orbital and $\ket{G}$ is
defined as vacuum as a $3d^{10}$ configuration.

Then, the charge-transferred type II t-J model written in the electron picture
is,

$\displaystyle H=T_{K}$ $\displaystyle+$ $\displaystyle
J_{\parallel}^{ss}\sum_{l}\sum_{\langle
ij\rangle}\vec{s}_{i;l}\cdot\vec{s}_{j;l}+J^{dd}_{\parallel}\sum_{l}\sum_{\langle
ij\rangle}\vec{S}_{i;l}\cdot\vec{S}_{j;l}$ $\displaystyle+$ $\displaystyle
J^{sd}_{\parallel}\sum_{l}\sum_{\langle
ij\rangle}(\vec{s}_{i;l}\cdot\vec{S}_{j;l}+\vec{S}_{i;l}\cdot\vec{s}_{j;l})$
$\displaystyle+$ $\displaystyle
J_{\perp}^{ss}\sum_{i}\vec{s}_{i;t}\cdot\vec{s}_{i;b}+J^{dd}_{\perp}\sum_{i}\vec{S}_{i;t}\cdot\vec{S}_{i;b}$
$\displaystyle+$ $\displaystyle
J^{sd}_{\perp}\sum_{i}(\vec{s}_{i;t}\cdot\vec{S}_{i;b}+\vec{S}_{i;t}\cdot\vec{s}_{i;b})+V\sum_{i}n_{i;t}n_{i;b},$

with

$\displaystyle T_{K}$ $\displaystyle=$ $\displaystyle-
t_{\parallel}\sum_{l,\sigma,\langle
i,j\rangle}c_{i;l;\sigma}^{\dagger}c_{j;l;\sigma}-t_{\perp}\sum_{\sigma,i}c_{i;t;\sigma}^{\dagger}c_{i;b;\sigma}+\mathrm{H.c.},$

The spin operators for the spin-1/2 state are
$\vec{s}_{i;l}=\frac{1}{2}\sum_{\sigma\sigma^{\prime}}\ket{\sigma}_{i;l}\vec{\sigma}_{\sigma\sigma^{\prime}}\bra{\sigma^{\prime}}_{i;l}$.
Meanwhile, the spin operators for the spin-one moment are written as
$\vec{S}_{i;l}^{d}=\sum_{\alpha,\beta=-1,0,1}\vec{T}_{\alpha\beta]}\ket{\alpha}_{i;l}\bra{\beta}_{i;l}$.
We have $T^{x}=\frac{1}{\sqrt{2}}\begin{pmatrix}0&1&0\\\ 1&0&1\\\
0&1&0\end{pmatrix}$ and $T^{y}=\frac{1}{\sqrt{2}}\begin{pmatrix}0&-i&0\\\
i&0&-i\\\ 0&i&0\end{pmatrix}$ in the $S^{z}=1,0,-1$ basis. $V$ is the inter-
layer repulsive density interaction. The expressions for $J_{\perp}$,
$J_{\parallel}$ in terms of the microscopic parameters are provided with the
derivation in SM Sec.III. Additionally, we establish the relationships
$J^{ss}_{\perp}=\mathcal{C}(r)J^{sd}_{\perp}=\mathcal{C}(r)^{2}J^{dd}_{\perp}$
and
$J^{ss}_{\parallel}=\mathcal{C}(r)\mathcal{A}J^{sd}_{\parallel}=\mathcal{C}(r)^{2}\mathcal{A}^{2}J^{dd}_{\parallel}$,
where $\mathcal{A}\simeq 0.7705$. Using the parameters mentioned above, we
have $J_{\parallel}^{dd}=0.097$eV, $J_{\perp}^{dd}$=0.076eV, and
$t_{\parallel}$ is in the range of $0.32-0.49$eV. $\mathcal{C}(r)$ ranges from
$4/3$ at $r\rightarrow 0$ limit and $0$ at $r\rightarrow\infty$ limit. Here
$r=J_{K;1}/J_{H}$. The dependence of $t_{\parallel}(r)$ and $\mathcal{C}(r)$
are provided in SM. In the Mott-Hubbard regime, we have
$J^{ss}_{\perp}=2J^{sd}_{\perp}=4J^{dd}_{\perp}=4t_{\perp}^{2}/U$, and
$J^{ss}_{\parallel}=J^{sd}_{\parallel}=0$,
$J^{dd}_{\parallel}=t_{\parallel}^{2}/U$ Yang  _et al._ (2023d). We stress
that the differences in their ratios do not alter the qualitative behavior
significantly.

We highlight that even in the charge transfer regime, the type-II t-J model
still shows suppressed hopping between the in-plane $p_{1}$ orbitals of the
two layers, with $t_{\perp}\simeq 0$, akin to the behavior observed in the
Mott-Hubbard regime. This can be validated by a simple symmetry argument that
since $p_{1}$ and $p_{z}$ orbitals have different symmetries, there is no
direct coupling channel between them at the lowest order. A finite $t_{\perp}$
can only be generated through virtual hopping to a $p_{z}$ orbital at a
different site and its value should be small. In the following we will show
that a small $t_{\perp}$ has no significant effect and thus as a good
approximation we can set it to be zero.

Figure 3: DMRG simulation results of the type-II t-J model, Eq.(Type II t-J
model in charge transfer regime in bilayer La3Ni2O7 and trilayer La4Ni3O10) in
the two-leg ladder at $t_{\parallel}=1$, $J^{\parallel}_{ss}=0.1$ Here, we
impose $J^{ss}_{\perp}=\frac{4}{3}J^{sd}_{\perp}=\frac{16}{9}J^{dd}_{\perp}$
and
$J^{ss}_{\parallel}=\frac{4}{3}\mathcal{A}J^{sd}_{\parallel}=\frac{16}{9}\mathcal{A}^{2}J^{dd}_{\parallel}$
with $\mathcal{A}=0.7705$. (a) The $t_{\perp}$ dependence of the spin gap,
$\Delta_{s}=E(S_{z}=1)-E(S_{z}=0)$, of the type-II t-J model at $V=0$ and
$x=0.5$ with various $J^{\perp}_{ss}=1,1.5,2,3,3.5$. Here, we use $L_{x}=40$
and $\chi=2400$ for the simulation. The spin gap monotonically decreases as
increasing in the $t_{\perp}$ in the large $J_{\perp}$ limit. (b) The doping
dependence of the spin gap at $t_{\perp}=0.1$, $J_{\perp}^{ss}=1$ with various
$V=0,1,2,2.5$. The superconducting dome is exhibited in the BCS limit, where
we increase $V$, and the substantial spin gap remains even in $x=0.5$. (c) The
pair correlation at $t_{\perp}=0.1$ and $x=0.5$ with various $J_{\perp}^{ss}$
values. It exhibits the power law decaying where the fitted power $\alpha$ is
denoted as the solid line. (d) The scaling of the entanglement entropy and the
correlation length $\xi$ at $x=0.5$ within the infinite DMRG calculation.
Here, we choose $t_{\perp}=0,0.2,0.4$ and $J_{\perp}=2,3$. The extracted
central charge $c$ is carried by the relation $S=\frac{c}{6}\log\xi$.

Numerical simulations.— We perform the density matrix renormalization group
(DMRG) simulations applied to the type-II t-J model described by Eq.(Type II
t-J model in charge transfer regime in bilayer La3Ni2O7 and trilayer
La4Ni3O10) White (1992); Hauschild and Pollmann (2018). We consider the two-
leg ladder configuration ($L_{z}=2,L_{y}=1,L_{x}\rightarrow\infty$), rather
than the full bilayer two-dimensional cubic lattice, due to the well-known
limitation of the DMRG. In our simulations, we set $t_{\parallel}=1$ and
$J_{\parallel}^{ss}=0.1$. In Fig. 3, we present the spin gap varying the
parameters $J_{\perp}^{ss}$, $t_{\perp}$, and the doping ratio. The
convergence of the DMRG results under the system size is also checked an
provided in the SM Sec.IV. The presence of a finite spin gap indicates the
emergence of the Luther-Emily liquid phase Luther and Emery (1974) with power-
law inter-layer pairing correlations. The pairing gap gradually decreases with
$t_{\perp}$ because inter-layer pairing frustrates the $t_{\perp}$ term. But
the pairing remains robust at small $t_{\perp}$. In the bilayer nickelate we
believe $t_{\perp}\ll t$ and hence its effect should be negligible. One of the
remarkable features of the bilayer type-II t-J model is the doping dependence
of the pairing gap. As presented in Fig. 3 (b), the pairing scale shows a dome
centered near $x=0.4-0.5$ in the presence of a repulsion. This is due to the
doping induced BCS to BEC crossoverYang  _et al._ (2023d). Finally, we check
the key characteristics of Luther-Emily liquids, such as power-law pair
correlation functions and a central charge $c=1$, in Figs. 3 (c-d). In the
main figure, we only illustrate the case with $x=0.5$ relevant to the
experiments, but the Luther-Emily liquid phases are manifested in the broad
range of $x$ (See SM Sec.IV).

Discussion.— Our theoretical proposal can provide a potential scenario on the
dome of the $T_{c}$ versus pressure in the experiment Li  _et al._ (2024b).
Increasing pressure should enhance $t_{dp;z}$ and thus the $J_{\perp}$ term.
Initially, $T_{c}$ increasing with the pressure due to the increase of the
$J_{\perp}$. However, when the pressure is larger than an optimal value
$P_{*}$, the hole prefers to stay in the $p_{z}$ orbital and then the pairing
is suppressed (see SM Sec.V). The shift from the in-plane $p_{1}$ orbital to
the $p_{z}$ orbital is likely the origin of the dome with pressure.

Our analysis can also be generalized to the trilayer nickelates, La4Ni3O10 Zhu
_et al._ (2024) (see SM Sec.VI) and a trilayer version of the type II t-J
model is the minimal model. Now the model is very different from the models
proposed assuming Mott-Hubbard regime. For the trilayer case both
$d_{1},d_{2}$ orbitals are assumed to be at fractional filling and the mobile
carriers are argued to be from both orbitals Sakakibara  _et al._ (2024);
Zhang  _et al._ (2024b); Lu  _et al._ (2024b). In contrast, in the charge
transfer picture, both $d_{1},d_{2}$ are still localized which just provide a
spin-one moment at each Ni site. Then the doped hole enters the $p_{1}$
orbital and the final model is still one-orbital like, similar to the bilayer
case. We leave to future to analyze this trilayer model.

Conclusion.— In summary, we provide analytical and numerical study for bilayer
nickelates, La3Ni2O7 within the charge transfer framework. Our primary
discovery is the identification of the Zhang-Rice spin-half state as the
dominant hole state. We emphasize that the type II t-J model again serves as a
minimal model for both the bilayer and the trilayer nickelate. The physics is
distinct from the hole doped cuprates due to the importance of strong inter-
layer spin-spin coupling. This leads to an optimal doping as large as
$40-50\%$ in the bilayer type II t-J model in contrast to $20\%$ in cuprate.

Acknowledgement. H.Oh thanks Hui Yang for valuable help on the density matrix
renormalization group simulation codes. B.Zhou thanks Xinlong Liu for fruitful
discussions on constructing Zhang-Rice spin-half states in the stage of
starting the project. This work was supported by the National Science
Foundation under Grant No. DMR-2237031.

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Supplemental Material for “Type II t-J model in charge transfer regime in
bilayer La3Ni2O7 and trilayer La4Ni3O10”

Hanbit Oh${}^{\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}*}}$, Boran
Zhou, and Ya-Hui Zhang${}^{\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\dagger}}$

_William H. Miller III Department of Physics and Astronomy,
Johns Hopkins University, Baltimore, Maryland, 21218, USA_

###### Contents

  1. I I. Symmetry properties of the Charge transfer Hamiltonian
  2. II II. Zhang-Rice spin-half state with general $J_{K},J_{H}$
  3. III III. Derivation of type-II t-J model from the charge transfer Hamiltonian
    1. III.1 A. Derivation of $J_{\perp}$
    2. III.2 B. Derivation of $J_{\parallel}$
    3. III.3 C. Derivation of $t_{\parallel}$
  4. IV IV. More detailed DMRG simulation results
  5. V V. The pressure dependence of critical temperature
  6. VI VI. Discussion on trilayer La4Ni3O10
    1. VI.1 A. Zhang-Rice spin-half state
    2. VI.2 B. Trilayer layer Type-II t-J model

##  I I. Symmetry properties of the Charge transfer Hamiltonian

The charge transfer Hamiltonian enjoys $D_{4h}=D_{4}\otimes\mathcal{I}$ point
group symmetry defined at Ni $i$ site. While the two $d$ orbitals,
($d_{1},d_{2}$) are already one-dimensional irreducible representations of
$D_{4h}$, $p_{x,y}$ orbitals living at the link of the cubic lattice are not.
To classify them as an symmetry representation, we first identify the symmetry
actions of the $p$ orbitals. For example, under the
$D_{4}=\\{E,2C_{4},C_{2},2C_{2}^{\prime},2C_{2}^{\prime\prime}\\}$, one can
show that

$\displaystyle C_{4}:\quad p_{x;i\pm\frac{x}{2};l}\rightarrow
p_{y;i\pm\frac{y}{2};l},\quad p_{y;i\pm\frac{y}{2};l}\rightarrow-
p_{x;i\mp\frac{x}{2};l},$ $\displaystyle C_{2}:\quad
p_{x;i\pm\frac{x}{2};l}\rightarrow-p_{x;i\mp\frac{x}{2};l},\quad
p_{y;i\pm\frac{y}{2};l}\rightarrow-p_{y;i\mp\frac{y}{2};l},$ $\displaystyle
C_{2}^{\prime}:\quad p_{x;i\pm\frac{x}{2};l}\rightarrow
p_{x;i\pm\frac{x}{2};l},\quad p_{y;i\pm\frac{y}{2};l}\rightarrow-
p_{y;i\mp\frac{y}{2};l},$ $\displaystyle C_{2}^{\prime\prime}:\quad
p_{x;i\pm\frac{x}{2};l}\rightarrow p_{y;i\pm\frac{y}{2};l},\quad
p_{y;i\pm\frac{y}{2};l}\rightarrow p_{x;i\mp\frac{x}{2};l},$

Solving the eigenvalues of the above symmetry actions, we can find that the
following linear combinations,

$\displaystyle p_{1;i;l;\sigma}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[p_{x;i+\frac{{\widehat{x}}}{2};l;\sigma}-p_{x;i-\frac{{\widehat{x}}}{2};l;\sigma}-p_{y;i+\frac{{\widehat{y}}}{2};l;\sigma}+p_{y;i-\frac{{\widehat{y}}}{2};l;\sigma}\right],$
(S1) $\displaystyle p_{2;i;l;\sigma}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[p_{x;i+\frac{{\widehat{x}}}{2};l;\sigma}-p_{x;i-\frac{{\widehat{x}}}{2};l;\sigma}+p_{y;i+\frac{{\widehat{y}}}{2};l;\sigma}-p_{y;i-\frac{{\widehat{y}}}{2};l;\sigma}\right],$
(S2)

become one-dimensional representations with $B_{1g}$, $A_{1g}$ representation.
This can be checked by the character table in Table S1. We tabulate the
characters of the each orbitals and the irreducible representation. Here,
$g(u)$ denotes the even (odd) under the inversion $\mathcal{I}$.

The symmetry allowed nearest neighbor hopping between $d,p$ orbitals are given
as

$\displaystyle H_{dp}$ $\displaystyle=$
$\displaystyle\sum_{l,\sigma}\sum_{\langle
a;i,\alpha;i^{\prime}\rangle}\left[t_{dp}^{a;i,\alpha;i^{\prime}}d^{\dagger}_{a;i;l;\sigma}p_{\alpha;i^{\prime};l;\sigma}+\mathrm{H.c.}\right]+\sum_{\sigma}\sum_{i}\left[t_{dp;z}(d^{\dagger}_{2;i;t;\sigma}-d^{\dagger}_{2;i;b;\sigma})p_{z;i;\sigma}+\mathrm{H.c.}\right]$

where $i$($i^{\prime}$) are sites of nickel (oxygen) atoms. The phase factors
should be defined accordingly on the hopping terms,

$\displaystyle t_{dp}^{1;i,\alpha;i^{\prime}}$ $\displaystyle=$ $\displaystyle
t_{dp;1}\sum_{s=\\{\pm
1\\}}(-1)^{s}[\delta_{\alpha,x}\delta_{i^{\prime},i+s\frac{{\widehat{x}}}{2}}-\delta_{\alpha,y}\delta_{i^{\prime},i+s\frac{{\widehat{y}}}{2}}]$
$\displaystyle t_{dp}^{2;i,\alpha;i^{\prime}}$ $\displaystyle=$ $\displaystyle
t_{dp;2}\sum_{s=\\{\pm
1\\}}(-1)^{s}[\delta_{\alpha,x}\delta_{i^{\prime},i+s\frac{{\widehat{x}}}{2}}+\delta_{\alpha,y}\delta_{i^{\prime},i+s\frac{{\widehat{y}}}{2}}].$

Substituting the Eqs.(S1-S2) into $H_{dp}$ simply reduces to,

$\displaystyle H_{dp}$ $\displaystyle=$
$\displaystyle\sum_{i,l,a,\sigma}\left[2t_{dp;a}d^{\dagger}_{a;i;l;\sigma}p_{a;i;l;\sigma}+\mathrm{H.c.}\right]+\sum_{i,\sigma}\left[t_{dp;z}(d^{\dagger}_{2;i;t;\sigma}-d^{\dagger}_{2;i;b;\sigma})p_{z;i;\sigma}+\mathrm{H.c.}\right]$

which is introduced in the main-text.

Orbitals | R | $E$ | $2C_{4}(z)$ | $C_{2}$ | $2C_{2}^{\prime}$ | $2C_{2}^{\prime\prime}$ | $\mathcal{I}$
---|---|---|---|---|---|---|---
$d_{1;l},p_{1;l}$ | $B_{1g}$ | 1 | -1 | 1 | 1 | -1 | 1
$d_{2;l},p_{2;l}$ | $A_{1g}$ | 1 | 1 | 1 | 1 | 1 | 1
$d_{2;t}-d_{2;b},p_{z}$ | $A_{1u}$ | 1 | 1 | 1 | 1 | 1 | -1
Table S1: The irreducible representation (R) and the character table of
$D_{4h}$ point group.

##  II II. Zhang-Rice spin-half state with general $J_{K},J_{H}$

In this section, we provide the energy analysis of possible $d^{8}L$ states
with general values of $J_{K},J_{H}$. In the strong coupling limit
$J_{K},J_{H}\gg 1$, we can consider the local Hamiltonian defined at each
site, neglecting the hopping term,

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{l;a}2J_{K;a}[\vec{s}^{d}_{l;a}\cdot\vec{s}^{p}_{l;a}-\frac{1}{4}n^{p}_{l;a}]+\sum_{l}2J_{K;z}[\vec{s}^{d}_{l;2}\cdot\vec{s}^{p}_{z}-\frac{1}{4}n^{p}_{z}]-\sum_{l}2J_{H}[\vec{s}^{d}_{l;1}\cdot\vec{s}^{d}_{l;2}+\frac{1}{4}],$
(S3)

where $J_{K;a},J_{K;z}$ is the Kondo coupling derived in main text, Eq.(3).
Based on the parameters listed in Ref.Wú  _et al._ (2023), we assume
$t_{dp;1}\simeq t_{dp;z}>t_{dp;2}$ and
$\Delta_{p;1}\simeq\Delta_{p;2}^{z}\simeq\Delta_{p;2}$, leading to the
relation $J_{K;1}\approx 4J_{K;z}>J_{K;2}$. There are three possible states
depending on the $p$ orbital occupancy : $(i)$ $n_{l;1}^{p}=1$, $(ii)$
$n_{l;2}^{p}=1$, and $(iii)$ $n_{l;z}^{p}=1$.

  * •

$n_{l;1}^{p}=1$ or $n_{l;2}^{p}=1$ case: The local Hamiltonian is reduced into
three-site Hamiltonian written in terms of $(p_{a;l},d_{1;l},d_{2;l})$,

$\displaystyle H$ $\displaystyle=$ $\displaystyle
2J_{K;a}[\vec{s}^{d}_{l;a}\cdot\vec{s}^{p}_{l;a}-\frac{1}{4}]-\sum_{l}2J_{H}[\vec{s}^{d}_{l;1}\cdot\vec{s}^{d}_{l;2}+\frac{1}{4}].$

The ground states of the above Hamiltonian is the spin $1/2$ state whose wave
function is given by,

$\displaystyle\ket{{\uparrow}}_{l}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2[(r+\alpha)^{2}+(r+\alpha+1)]}}[(1+r+\alpha)|\downarrow,\uparrow,\uparrow\rangle-(r+\alpha)|\uparrow,\downarrow,\uparrow\rangle-|\uparrow,\uparrow,\downarrow\rangle],$
(S4) $\displaystyle\ket{{\downarrow}}_{l}$ $\displaystyle=$
$\displaystyle\frac{-1}{\sqrt{2[(r+\alpha)^{2}+(r+\alpha+1)]}}[(1+r+\alpha)|\uparrow,\downarrow,\downarrow\rangle-(r+\alpha)|\downarrow,\uparrow,\downarrow\rangle-|\downarrow,\uparrow,\downarrow\rangle].$
(S5)

where $\ket{s_{0},s_{1},s_{2}}$ implies the spin of the $p_{a},d_{1},d_{2}$
orbital respectively. Here, we use $r=J_{K;a}/J_{H}$ and
$\alpha(r)=\sqrt{r^{2}+r+1}$. These states are called as Zhang-Rice spin-half
state, and the schematic illustrations are provided in Fig. S1. The wave
function is obviously entangled by the three ($p_{1},d_{1},d_{2}$) orbitals in
the most range of $r$, except $r=+\infty$. In this specific point $r=+\infty$,
the state becomes the tensor product of the Zhang-Rice singlet only made by
($p_{1},d_{1}$) and decoupled $d_{2}$ orbitals. The subsystem entanglement
entropy is obtained in Fig. S2(b), manifesting its values is finite due to the
entanglement of ($p_{1},d_{1}$) and $d_{2}$. The associated energy is obtained
as,

$\displaystyle E_{G}=-[1+r+\sqrt{r^{2}+r+1}]J_{H},$ (S6)

introducing the dimensionless parameter, $r=J_{K;a}/J_{H}$.

  * •

$n_{z}^{p}=1$ case: The local Hamiltonian is reduced into five-site
Hamiltonian of $(p_{z},d_{1;t},d_{1;b},d_{2;t},d_{2;b})$,

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{l}2J_{K;z}[\vec{s}^{d}_{l;2}\cdot\vec{s}^{p}_{z}-\frac{1}{4}]-\sum_{l}2J_{H}[\vec{s}^{d}_{l;1}\cdot\vec{s}^{d}_{l;2}+\frac{1}{4}].$

The wave function $\ket{\psi}$ of the ground states of the above Hamiltonian
have four-fold degeneracy and are very complicated. However, the important
thing to note is the total spin of the state is
$S_{\mathrm{tot}}=\frac{3}{2}$, i.e.
$S_{tot}^{2}\ket{\psi}=\frac{15}{4}\ket{\psi}$, and
$S_{tot}^{z}\ket{\psi}=\pm\frac{1}{2},\pm\frac{3}{2}\ket{\psi}$, for any
values of positive $J_{K;z}$ and $J_{H}$. The associated ground state energy
is obtained as

$\displaystyle E_{G}^{z}=\frac{1}{2}[-2-3r-\sqrt{4+8r+9r^{2}}]J_{H},$ (S7)

with a dimensionless parameter, $r=J_{K;z}/J_{H}$,

The above-hybridized states are illustrated in Fig. S1. We now compare those
three energies, Eqs.(S6-S7), using the fact $J_{K;1}\approx 4J_{K;z}>J_{K;2}$.
Obviously, $E_{G}^{1}(J_{K;1})<E_{G}^{2}(J_{K;2})$ is established. Next, we
compare the energy between $E_{G}^{1}$ and $E_{G}^{z}$ in Fig. S2 showing that
$E_{G}^{1}(J_{K;1})<E_{G}^{z}(J_{K;z}\simeq 4J_{K;1})$. Hence, we can conclude
that the Zhang-Rice spin-1/2 state, especially with $p_{1}$ hole is the most
stable state.

Figure S1: Possible Zhang-Rice states of $d^{8}L$ electron configurations of
the bilayer nickelates. (a) The spin $1/2$ and (b) spin $3/2$ states are
hybridized by intra-layer $(p_{1},p_{2})$ orbital or inter-layer $p_{z}$
orbital of oxygen atoms. Note that here all of the states are drawn in the
hole description picture.  Figure S2: (a) The $J_{K}/J_{H}=r$ dependence of
$E_{G}^{1}(r)$ and $E_{G}^{z}(r)$. (b)Comparison of the ground states energy
in $(J_{K;1},J_{K;z})$ plane. The blue (green) region indicates the phase
space where $p_{1}$ ($p_{z}$) orbitals energetically favored, so an additonal
holes enter to $p_{1}$ ($p_{z}$) orbitals. Using the estimation $J_{K;1}\sim
4J_{K;z}$ (red dashed line), we conclude that the Zhang-Rice spin-1/2 state
with $p_{1}$ hole is more stable. (c) The bipartite entanglement entropy of
$\ket{\uparrow}$, Eqs.(S4-S5), as a function of $r$. Here, the entanglement
entropy is evaluated by
$S=\mathrm{Tr}_{d_{1}}({\widehat{\rho}}\log{\widehat{\rho}})$, with the
reduced density matrix,
${\widehat{\rho}}=\mathrm{Tr}_{d_{1},p}[\ket{\uparrow}\bra{\uparrow}]$. As
$r\rightarrow+\infty$ limit, Zhang-Rice spin half state is just a tensor
product of the singlet formed by the ($d_{1},p_{1}$) and spin 1/2 of $d_{2}$
orbital, and the entanglement between two subsystems goes to zero. However, in
other value of $r$, the $d_{2}$ orbital does not be decoupled from
$d_{1},p_{1}$ orbitals, forming the Zhang-Rice spin half state together.

##  III III. Derivation of type-II t-J model from the charge transfer
Hamiltonian

In this section, we derive the type-II t-J model starting from the charge
transfer Hamiltonian, Eq. (1). In particular, we show all the spin-exchange
couplings of the type-II t-J model, in terms of the original Hamiltonian. The
spin-exchange interaction of two nickel atoms can be derived by the fourth-
order perturbation in the strong coupling limit. We divide the charge-transfer
Hamiltonian into the kinetic part, $V=H_{dp}$, and remaining part,
$H_{0}=H-V$.

###  III.1 A. Derivation of $J_{\perp}$

First, we show how the $J^{\perp}$ terms are derived, exemplified by
$J^{\perp}_{sd}$, and the generalization into $J^{\perp}_{ss},J^{\perp}_{dd}$
is straightforward. Due to the SU(2) symmetry of the Hamiltonian, it is enough
to evaluate the coefficient of one of the components of
$\frac{1}{2}(s^{-}_{t}S^{+}_{b})=\frac{1}{\sqrt{2}}\ket{s_{t}=-\frac{1}{2},s_{b}=0}\bra{s_{t}=\frac{1}{2},s_{b}=-1}+\cdots$
piece, not all $\vec{s}_{t;i}\cdot\vec{S}_{b;i}$. There are six different
processes (A)-(F) contributing the $s^{-}_{t}S^{+}_{b}$, as illustrated in
Fig.S3 (a),

$\displaystyle\bra{s_{t}=-\frac{1}{2},s_{b}=0}VRVRVRV\ket{s_{t}=\frac{1}{2},s_{b}=-1}$
$\displaystyle=$ $\displaystyle(A)+(B)+(C)+(D)+(E)+(F),$ (S8)

where $R=Q/(E_{0}-H_{0})$ is the projection onto the subspace of all exited
states. Here, $Q=1-P$ is a complementary operator of the projection operator
of the ground state of $H_{0}$ where the doubly occupied states are forbidden.

With some tedious calculations, we find each component is given by

$\displaystyle(A)$ $\displaystyle=$
$\displaystyle\frac{t_{dp,z}}{(\Delta_{p;2}^{z}+\overline{J}_{H})}\bra{-\frac{1}{2}}_{t}\bra{0}_{b}VRVRVp_{z;\uparrow}^{\dagger}[\frac{(1+r+\alpha)}{\mathcal{N}}\ket{s_{d_{1}}=\frac{1}{2},s_{c}=-\frac{1}{2}}_{t}-\frac{1}{\mathcal{N}}\ket{s_{d_{1}}=-\frac{1}{2},s_{c}=\frac{1}{2}}_{t}]\ket{-1}_{b}$
$\displaystyle=$
$\displaystyle\frac{(t_{dp,z})^{2}}{(\Delta_{p;2}^{z}+\overline{J}_{H})(U+2J_{H})}\bra{-\frac{1}{2}}_{t}\bra{0}_{b}VRVd_{2;b;\uparrow}^{\dagger}[\frac{(1+r+\alpha)}{\mathcal{N}}\ket{s_{d_{1}}=\frac{1}{2},s_{c}=-\frac{1}{2}}_{t}-\frac{1}{\mathcal{N}}\ket{s_{d_{1}}=-\frac{1}{2},s_{c}=\frac{1}{2}}]_{t}\ket{-1}_{b}$
$\displaystyle=$
$\displaystyle\frac{-(t_{dp,z})^{3}}{(\Delta_{p;2}^{z}+\overline{J}_{H})^{2}(U+2J_{H})}\bra{-\frac{1}{2}}_{t}\bra{0}_{b}Vp_{z;\downarrow}^{\dagger}[(1+r+\alpha)\ket{s_{d_{1}}=\frac{1}{2},s_{c}=-\frac{1}{2}}_{t}-\ket{s_{d_{1}}=-\frac{1}{2},s_{c}=\frac{1}{2}}_{t}]\frac{\ket{0}_{b}}{\sqrt{2}}$
$\displaystyle=$
$\displaystyle\frac{(t_{dp,z})^{4}}{(\Delta_{p;2}^{z}+\overline{J}_{H})^{2}(U+2J_{H})}\frac{\sqrt{2}(1+r+\alpha)}{\mathcal{N}^{2}}=(D),$

and

$\displaystyle(B)$ $\displaystyle=$
$\displaystyle\frac{(t_{dp,z})^{2}}{2(\Delta_{p;2}^{z}+\overline{J}_{H})^{2}}\bra{-\frac{1}{2}}_{t}\bra{0}_{b}VRVp_{z;\downarrow}^{\dagger}p_{z;\uparrow}^{\dagger}[\frac{(1+r+\alpha)}{\mathcal{N}}\ket{s_{d_{1}}=\frac{1}{2},s_{c}=-\frac{1}{2}}_{t}-\frac{1}{\mathcal{N}}\ket{s_{d_{1}}=-\frac{1}{2},s_{c}=\frac{1}{2}}_{t}]\ket{s_{d_{1}}=-\frac{1}{2}}_{b}$
$\displaystyle=$
$\displaystyle\frac{-(t_{dp,z})^{3}}{2(\Delta_{p;2}^{z}+\overline{J}_{H})^{3}}\bra{-\frac{1}{2}}_{t}\bra{0}_{b}Vp_{z;\downarrow}^{\dagger}[\frac{(1+r+\alpha)}{\mathcal{N}}\ket{s_{d_{1}}=\frac{1}{2},s_{c}=-\frac{1}{2}}_{t}-\frac{1}{\mathcal{N}}\ket{s_{d_{1}}=-\frac{1}{2},s_{c}=\frac{1}{2}}_{t}]\ket{s_{p_{z}}=-\frac{1}{2}}\frac{\ket{0}_{b}}{\sqrt{2}}$
$\displaystyle=$
$\displaystyle\frac{(t_{dp,z})^{4}}{2(\Delta_{p;2}^{z}+\overline{J}_{H})^{3}}\frac{\sqrt{2}(1+r+\alpha)}{\mathcal{N}^{2}}=(C)=(E)=(F),$

with $\alpha=\sqrt{r^{2}+r+1}$ and
$\mathcal{N}=\sqrt{2[(r+\alpha)^{2}+(r+\alpha+1)]}$. By putting everything
together, $J_{sd}^{\perp}$ is obtained as,

$\displaystyle J_{sd}^{\perp}$ $\displaystyle=$
$\displaystyle\mathcal{C}(r)\frac{t_{dp,z}^{4}}{(\Delta_{p;2}^{z}+\overline{J}_{H})^{2}}\left[\frac{1}{U_{2}+2J_{H}}+\frac{1}{\Delta_{p;2}^{z}+\overline{J}_{H}}\right],$
(S9)

with $\mathcal{C}(r)=4(1+r+\alpha)/\mathcal{N}^{2}$, whose $r$ dependence is
illustrated in Fig. S3 (b). Similarly, we can show that

$\displaystyle J_{dd}^{\perp}$ $\displaystyle=$
$\displaystyle\frac{t_{dp,z}^{4}}{(\Delta_{p;2}^{z}+\overline{J}_{H})^{2}}\left[\frac{1}{U_{2}+2J_{H}}+\frac{1}{\Delta_{p;2}^{z}+\overline{J}_{H}}\right],$
(S10) $\displaystyle J_{ss}^{\perp}$ $\displaystyle=$
$\displaystyle\mathcal{C}(r)^{2}\frac{t_{dp,z}^{4}}{(\Delta_{p;2}^{z}+\overline{J}_{H})^{2}}\left[\frac{1}{U_{2}+2J_{H}}+\frac{1}{\Delta_{p;2}^{z}+\overline{J}_{H}}\right],$
(S11)

where we have
$J_{ss}^{\perp}=\mathcal{C}(r)J_{sd}^{\perp}=\mathcal{C}(r)^{2}J_{dd}^{\perp}$.
Here we use $\overline{J}_{H}=J_{H}-U^{\prime}$,
$\Delta_{p;a}=\epsilon_{p}-\epsilon_{d;a}$, and
$\Delta_{p;a}^{z}=\epsilon_{p;z}-\epsilon_{d;a}$.

Figure S3:  (a) All possible processes contributing to the spin exchange
interaction upon the fourth-order perturbation. The green (yellow) sphere
represents an oxygen (nickel) atom. In $(i,a,\sigma)$, $i$ denotes the order
of the process, $a=1,2$ is for labeling $d_{a}$ orbitals and $\sigma$ is for
the spin. These processes specifically contribute to the $s^{-}_{L}s^{+}_{R}$,
$s^{-}_{L}S^{+}_{R}$ or $S^{-}_{L}S^{+}_{R}$ where $L(R)$ denotes the left
(right) nickel atom. (b) The $J_{K}/J_{H}=r$ dependence of $\mathcal{C}(r)$ in
Eqs.(S9-S11). In particular, $\mathcal{C}(r)$ becomes $\frac{4}{3}$ in the
$r\rightarrow 0$ limit.

###  III.2 B. Derivation of $J_{\parallel}$

Next, we show the derivation of $J^{\parallel}$ terms. There are two
significant differences, compared to $J^{\perp}$. First, there are two kinds
of hopping processes with $t_{dp;1}$ and $t_{dp;2}$. Second, the p orbital of
the Zhang-Rice spin-half state can be affected by the hoppings, since the
intermediate-occupied oxygen is also shared by those states. For example, the
hopping process consisting of
$p_{x;i+\frac{{\widehat{x}}}{2};\sigma}^{\dagger}$ eliminates the piece of the
same component in
$\phi_{i;\sigma}=\sum_{j}B(i-j)p_{1;j;\sigma}=c^{\dagger}_{i;\sigma}$, where
$B(i-j)=\frac{1}{N}\sum_{k}e^{ik\cdot(i-j)}\beta(k)$ and
$\beta(k)=[1-\frac{1}{2}(\cos k_{x}+\cos k_{y})]^{-1/2}$. To be specific,
consider the following example,

$\displaystyle
p_{x;i+\frac{{\widehat{x}}}{2};\sigma}^{\dagger}\phi_{i,\sigma^{\prime}}^{\dagger}$
$\displaystyle=$ $\displaystyle
p_{x;i+\frac{{\widehat{x}}}{2};\sigma}^{\dagger}\left[\sum_{j\in\\{i,i+{\widehat{x}}\\}^{c}}B(i-j)p_{1;j,\sigma^{\prime}}^{\dagger}+\sum_{j\in\\{i,i+{\widehat{x}}\\}}B(i-j)\overline{p}_{1;j,\sigma^{\prime}}^{\dagger}\right]=p_{x;i+\frac{{\widehat{x}}}{2};\sigma}^{\dagger}\overline{\phi}_{i,\sigma^{\prime}}^{\dagger},$

where we define the $\overline{\phi}(\overline{p})$ as the $\phi(p)$, but
excluding the $p_{x;i+{\widehat{x}}/2}$ piece,

$\displaystyle\overline{p}_{1;i,\sigma^{\prime}}^{\dagger}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[-p_{x;i-\frac{{\widehat{x}}}{2};l;\sigma}-p_{y;i+\frac{{\widehat{y}}}{2};l;\sigma}+p_{y;i-\frac{{\widehat{y}}}{2};l;\sigma}\right]=p_{1;i,\sigma^{\prime}}^{\dagger}-\frac{1}{2}p_{x;i+\frac{{\widehat{x}}}{2};l;\sigma},$
$\displaystyle\overline{p}_{1;i+{\widehat{x}},\sigma^{\prime}}^{\dagger}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left[p_{x;i+\frac{3{\widehat{x}}}{2};l;\sigma}-p_{y;i+{\widehat{x}}+\frac{{\widehat{y}}}{2};l;\sigma}+p_{y;i+{\widehat{x}}-\frac{{\widehat{y}}}{2};l;\sigma}\right]=p_{1;i,\sigma^{\prime}}^{\dagger}+\frac{1}{2}p_{x;i+\frac{{\widehat{x}}}{2};l;\sigma},$

and

$\displaystyle\overline{\phi}_{i,\sigma^{\prime}}^{\dagger}=\phi_{i,\sigma^{\prime}}^{\dagger}-[\frac{B(0)-B(1)}{2}]p_{x;i+\frac{x}{2};\sigma}^{\dagger}.$

Note that $\bra{\overline{\phi}}\phi\rangle\simeq
1-[\frac{B(0)-B(1)}{2}]^{2}\simeq 0.7705\equiv\mathcal{A}$. Then, the next
step is straightforward to consider all the possible processes depicted in Fig
S3 (a). After finishing all the calculations with $\Delta_{p;1}=\Delta_{p;2}$,
we obtained that

$\displaystyle J_{dd}^{\parallel}$ $\displaystyle=$
$\displaystyle\sum_{a,a^{\prime}}\frac{t_{dp;a}^{2}t_{dp;a^{\prime}}^{2}}{(\Delta_{p}+\overline{J}_{H})^{2}}\left[\frac{1}{U_{a^{\prime}}+2J_{H}}+\frac{1}{\Delta_{p}+\overline{J}_{H}}\right],$
(S12) $\displaystyle J_{sd}^{\parallel}$ $\displaystyle=$
$\displaystyle\mathcal{C}(r)\mathcal{A}\sum_{a,a^{\prime}}\frac{t_{dp;a}^{2}t_{dp;a^{\prime}}^{2}}{(\Delta_{p}+\overline{J}_{H})^{2}}\left[\frac{1}{U_{a^{\prime}}+2J_{H}}+\frac{1}{\Delta_{p}+\overline{J}_{H}}\right],$
(S13) $\displaystyle J_{ss}^{\parallel}$ $\displaystyle=$
$\displaystyle(\mathcal{C}(r)\mathcal{A})^{2}\sum_{a,a^{\prime}}\frac{t_{dp;a}^{2}t_{dp;a^{\prime}}^{2}}{(\Delta_{p}+\overline{J}_{H})^{2}}\left[\frac{1}{U_{a^{\prime}}+2J_{H}}+\frac{1}{\Delta_{p}+\overline{J}_{H}}\right]$
(S14)

Now, we have
$J_{ss}^{\parallel}=\mathcal{C}(r)\mathcal{A}J_{sd}^{\parallel}=\mathcal{C}(r)^{2}\mathcal{A}^{2}J_{dd}^{\parallel}$.
We stress this condition is generally satisfied without imposing
$\Delta_{p;1}=\Delta_{p;2}$.

###  III.3 C. Derivation of $t_{\parallel}$

From the second order perturbation, we can derive the in-plane hopping term
with and without spin dependence
$\sum_{l,\sigma,i,j}-t_{\parallel;i,j}c^{\dagger}_{i;l;\sigma}c_{j;l,\sigma}$.
The main contribution can be classified as the effective O-hopping and spin
exchange between the Ni and O holes, similar as the case in cuprate Zhang and
Rice (1988). We have that:

$\displaystyle t_{\parallel;i,j}=$
$\displaystyle\left(\frac{2t_{dp;1}^{2}}{3(U+U^{\prime}+J_{H}-\Delta_{p})}+\frac{J_{K;1}}{12\sqrt{2}}\right)\delta_{ij,\mathrm{NN}}/\mathcal{N}^{2}(r)-\frac{4J_{K;1}\lambda}{3N}\sum_{\mathbf{k}}\beta^{-1}_{\mathbf{k}}e^{-\mathrm{i}\mathbf{k}\cdot(\mathbf{r_{i}-r_{j}})}/\mathcal{N}^{4}(r),$
(S15)

in which $\delta_{ij,\mathrm{NN}}$ means that only nearest neighbor term is
non-zero. $\lambda$ is defined as
$\sum_{\mathbf{k}}\beta^{-1}_{\mathbf{k}}\simeq 0.96$. Here, we used
$\mathcal{N}(r)=[2(r+\alpha)^{2}+2(r+\alpha)+2]^{1/2}/(r+\alpha+1)$, a
function as increasing $r$, which depends on $r=J_{K;1}/J_{H}$. The function
is monotonic increasing in the range of $[\sqrt{\frac{3}{2}},\sqrt{2}]$. $N$
is the number of the site. In our main text, we only keep the nearest neighbor
term of $t_{\parallel;i,j}$, and denoted by $t_{\parallel}$. The $r$
dependnence is illustrated in Fig.S4.

Figure S4:  The $r$ dependence of $t_{\parallel}(r)$ and $\mathcal{N}(r)$
estimated by the parameters mentioned in the main-text. Note that
$t_{\parallel}$ is plotted in the eV unit, and $\mathcal{N}$ is a
dimensionless quantity.

##  IV IV. More detailed DMRG simulation results

In this section, we provide additional DMRG calculation results, which haven’t
been shown in the main text. Here, we still use $t_{\parallel}=1$ and
$J_{\parallel}^{ss}=0.1$. In Fig. S5, we have shown the $t_{\perp}$ dependence
with various interlayer repulsive interaction strength $V$ to demonstrate,
that the non-monotonic decreasing region diminishes and eventually disappears
increasing $V$. In Fig. S6, we provide some DMRG simulation data at $x=0.2$,
for comparing the data provided in the main text at $x=0.5$ and showing that
the key characteristic behaviors are similar. In Fig. S7, we provide the
convergence check under the system size of the spin-gap results through DMRG
calculations.

Figure S5:  DMRG simulation results of the type-II t-J model with
$L_{z}=2,L_{y}=1$ at $t_{\parallel}=1$, $J_{\parallel}^{ss}=0.1$. (a,b) The
$t_{\perp}$ dependence of the spin gap with various $V=0,0.5,1,1.5,2$ at (a)
$x=0.5$ and (b) $x=0.2$. Here, we choose $t_{\perp}=1$, and use $L_{x}=40$ and
$\chi=2400$ for simulation. The non-monotonically decreasing region is
decreasing as increasing the repulsive interaction where the system goes to
the BCS limit.  Figure S6:  DMRG simulation results of the type-II t-J model,
with $L_{z}=2,L_{y}=1$ at $t_{\parallel}=1$, $J_{\parallel}^{ss}=0.1$. The
doping is fixed at $x=0.2$ to compare the data at $x=0.5$ provided in the
main-text. At $x=0.2$, the results exhibits the evidence of the Luther-Emily
liquid phases. (a) The pair correlation function at $t_{\perp}=0.1$ and
$x=0.2$ shows the powe-law decaying behaviours. (b) The entanglement entropy
and the correlation length at $x=0.2$. The fitted central charge is nearly
$c=1$.  Figure S7:  The system size dependence of DMRG simulation results of
the type-II t-J model with $L_{z}=2,L_{y}=1$ at $t_{\parallel}=1$,
$J_{\parallel}^{ss}=0.1$ (a) $t_{\perp}$ dependence of the spin gap with
various $J_{\perp}$ at $x=0.2$. (b) Doping $x$ dependence of the spin gap with
various $V$ at $J_{\perp}=1$ ($\chi=2400$).

##  V V. The pressure dependence of critical temperature

In this section, we analyze the binding energy for predicting the critical
temperature under changing the pressure. Applying pressure effectively
increases the hopping between $p_{z}$ and inplane $d$ orbitals. Hence,
according to our energy estimations, we expect there is a crossover pressure,
satisfying $E_{G}^{1}=E_{G}^{z}$, as illustrated in Fig.S8 (a). At $P<P_{*}$,
the additional holes are occupied in $p_{1}$ orbital, while at $P>P_{*}$, the
$p_{z}$ orbital becomes dominant.

Then, we analyze the binding energy the estimate the critical temperature. For
each regime of $P<P_{*}$ and $P>P_{*}$, the binding energy is differently
estimated, since the additional hole is occupied by $p_{1}$ and $p_{z}$
respectively,

$\displaystyle P<P*:\quad
E_{B}^{1}=E[n_{p_{1}}=1]-(E[n_{p_{1}}=2]+E[n_{p_{1}}=0])/2,$ $\displaystyle
P>P*:\quad E_{B}^{z}=E[n_{p_{z}}=1]-(E[n_{p_{z}}=2]+E[n_{p_{z}}=0])/2.$

The pressure, (or equivalently $t_{dp;z}$) dependence of the binding energy
per unit cell, is illustrated in Fig. S8 (b). In the regime where $p_{1}$
dominates, the binding energy exhibits a positive trend, steadily increasing
under pressure. This arises from the stabilization of the two-hole state,
occupied by $p_{1}$ at each layer, which enhanced inter-layer spin coupling by
larger $t_{dp;z}$. Conversely, in the regime where $p_{z}$ dominates, the
binding energy demonstrates a negative trend, decreasing under pressure. This
stems from the fact that the two hole states occupied by one $p_{z}$ orbital
blocks the hopping between two layers of nickel, potentially leading the
inter-layer spin coupling as zero.

Figure S8: (a) $t_{dp;z}$ dependence of $E_{G}^{1}-E_{G}^{z}$ per unit cell.
There is a transition that hole prefers to stay from $p_{1}$ to $p_{z}$ at
$t_{dp;z}^{*}=2.50$eV. Here, we used the parameter introduced in the main-
text, $t_{dp;1}=1.56$eV, $t_{dp;2}=-0.75$eV, $t_{dp;z}=1.63$eV, $U_{a}=10$eV,
$U^{\prime}=6$eV, $J_{H}=2$eV, and $\Delta_{a}=9$ eV. (b) $t_{dp;z}$
dependence of the binding energy $E_{B}$ per unit cell. In the $p_{1}$
dominant regime, the binding energy $E_{B,1}$ is increasing, implying the rise
of $T_{c}$ under the pressure. Meanwhile in the $p_{z}$ dominant regime, the
binding energy $E_{B,z}$ tends to decrease under the pressure.

##  VI VI. Discussion on trilayer La4Ni3O10

In this section, we expand the theoretical framework of trilayer nickelates
La4Ni3O10 in the charge-transfer regime. According to the DFT data, the
valence of the Ni atom of La4Ni3O10 is in the $3d^{7.33}$ configuration
(Ni+2.67) with the $d_{x^{2}-y^{2}}$ orbital is close to one-third occupied
$n_{1}=1/3$ per site, while the $d_{z^{2}}$ orbital is nearly half-filled
$n_{2}\approx 1$ Sakakibara  _et al._ (2024); Zhang  _et al._ (2024b); Lu  _et
al._ (2024b).

Figure S9: The lattice structure and $p,d$ orbitals of the bilayer Nickelates
La4Ni3O10. The green (yellow) sphere denotes Ni (O) atom, respectively. At
each Ni atom, there are two $d$ orbitals, $d_{1},d_{2}$ at top/middle/bottom
layers. At each O atom at each layer, either $p_{x}$,$p_{y}$ orbitals reside
and can hybridize with the Ni atom at each layer. Meanwhile, the O atoms
living in the middle of the three layers have $p_{z}$ and should be shared by
two Ni atoms at adjacent layers. In our effective charge-transfer model, we
focus on the 17 atoms consisting of 3 Ni and 14 O atoms.

###  VI.1 A. Zhang-Rice spin-half state

We consider the trilayer cubic lattice with nickel $d_{l;1},d_{l;2}$ orbitals
and oxygen $p_{l;1},p_{l;2}$ orbitals defined at each layer $l=t,m,b$. The
oxygen $p_{l^{\prime};z}$ orbitals are living between each layer with
$l^{\prime}=t,b$ (See Fig. S9). Now, in each unit cell, we have to consider
the total 17 sites of atoms consisting of three Ni atoms and fourteen O atoms.
Starting from the local trilayer version of charge-transfer Hamiltonian
generalized from Eq.(1) in the main text, we again obtain the Kondo
Hamiltonian,

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{l=t,m,b}\sum_{a}2J_{K;a}[\vec{s}^{d}_{l;a}\cdot\vec{s}^{p}_{l;a}-\frac{1}{4}n^{p}_{l;a}]+\sum_{l=t,b}2J_{K;z}[(\vec{s}^{d}_{l;2}+\vec{s}^{d}_{m;2})\cdot\vec{s}^{p}_{z}-\frac{1}{2}n^{p}_{l;z}]-\sum_{l=t,m,b}2J_{H}[\vec{s}^{d}_{l;1}\cdot\vec{s}^{d}_{l;2}+\frac{1}{4}],$
(S16)

where $J_{K;a},J_{K;z}$ is expressed in Eq.(3) in the main-text. The only
difference compared to the bilayer model is that $p_{z}$ orbital now has a
layer index. There are still three possible one-hole states depending on the
$p$ orbital occupancy : $(i)$ $n_{l;1}^{p}=1$, $(ii)$ $n_{l;2}^{p}=1$, and
$(iii)$ $n_{l;z}^{p}=1$. For the first two cases ($n_{l;1}^{p}=1$ or
$n_{l;2}^{p}=1$) with $l=t,m,b$, the local Hamiltonian is

$\displaystyle H$ $\displaystyle=$ $\displaystyle
2J_{K;a}[\vec{s}^{d}_{l;a}\cdot\vec{s}^{p}_{l;a}-\frac{1}{4}]-\sum_{l^{\prime}=t,m,b}2J_{H}[\vec{s}^{d}_{l^{\prime};1}\cdot\vec{s}^{d}_{l^{\prime};2}+\frac{1}{4}],$

whose ground state energy is obtained as,

$\displaystyle E_{G}=-[2+r+\sqrt{r^{2}+r+1}]J_{H},$ (S17)

with a dimensionless parameter, $r=J_{K;a}/J_{H}$. For the last case
($n_{l;z}^{p}=1$) with $l=t,b$, the local Hamiltonian is,

$\displaystyle H$ $\displaystyle=$ $\displaystyle
2J_{K;z}[(\vec{s}^{d}_{l;2}+\vec{s}^{d}_{m;2})\cdot\vec{s}^{p}_{l;z}-\frac{1}{2}]-\sum_{l^{\prime}=t,m,b}2J_{H}[\vec{s}^{d}_{l^{\prime};1}\cdot\vec{s}^{d}_{l^{\prime};2}+\frac{1}{4}],$

whose ground state energy is obtained as,

$\displaystyle E_{G}^{z}=\frac{1}{2}[-4-3r-\sqrt{4+8r+9r^{2}}]J_{H},$ (S18)

with a dimensionless parameter, $r=J_{K;z}/J_{H}$. The above results are just
a constant shift by $-J_{H}$ from the Eqs.(S6-S7), which leads to the same
conclusion. Based on the parameters listed in Ref.Luo  _et al._ (2024), we
assume $t_{dp;1}\simeq t_{dp;z}>t_{dp;2}$ and
$\Delta_{p;1}\simeq\Delta_{p;2}^{z}\simeq\Delta_{p;2}$, leading to the
relation $J_{K;1}\approx 4J_{K;z}>J_{K;2}$. Using the fact $J_{K;1}\approx
4J_{K;z}>J_{K;2}$, we again have $E_{G}^{1}(J_{K;1})<E_{G}^{2}(J_{K;2})$ and
$E_{G}^{1}(J_{K;1})<E_{G}^{z}(J_{K;z}\simeq 4J_{K;1})$. Hence, we can conclude
that the Zhang-Rice spin-1/2 state, especially with $p_{1}$ hole is the most
stable state even for the trilayer nickelates.

###  VI.2 B. Trilayer layer Type-II t-J model

Then by taking the Zhang-Rice spin-half as the primary state of the $d^{8}L$
state and keeping a spin-triplet doublon state of the $d^{8}$ state, the
minimal model of the hole-doped trilayer bilayer nickelates is the type-II t-J
model Zhang and Vishwanath (2020); Oh and Zhang (2023). The trilayer type-II
t-J model Hamiltonian is given by

$\displaystyle H=H_{K}$ $\displaystyle+$
$\displaystyle\sum_{l=t,m,b}\sum_{\langle
i,j\rangle}\left[J_{ss}^{\parallel}\vec{s}_{i;l}\cdot\vec{s}_{j;l}+J_{sd}^{\parallel}(\vec{s}_{i;l}\cdot\vec{S}_{j;l}+\cdot\vec{S}_{i;l}\cdot\vec{s}_{j;l})+J_{dd}^{\parallel}\vec{S}_{i;l}\cdot\vec{S}_{j;l}\right]$
$\displaystyle+$
$\displaystyle\sum_{l=t,b}\sum_{i}\left[J_{ss}^{\perp}\vec{s}_{i;l}\cdot\vec{s}_{i;m}+J_{sd}^{\perp}(\vec{s}_{i;l}\cdot\vec{S}_{i;m}+\vec{S}_{i;l}\cdot\vec{s}_{i;m})+J_{dd}^{\perp}\vec{S}_{i;l}\cdot\vec{S}_{i;m}\right]+V\sum_{l=t,b}\sum_{i}n_{i;l}n_{i;m},$

with

$\displaystyle H_{K}$ $\displaystyle=$ $\displaystyle-
t_{\parallel}\sum_{l=t,m,b}\sum_{\sigma,\langle
i,j\rangle}c_{i;l;\sigma}^{\dagger}c_{j;l;\sigma}-t_{\perp}\sum_{l=t,b}\sum_{\sigma,i}c_{i;l;\sigma}^{\dagger}c_{i;m;\sigma}+\mathrm{H.c.},$

where the coefficients $J_{\parallel},J_{\perp}$ are listed in Eqs.(S9-S14).