††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work

# Trilayer multi-orbital models of $\mathrm{La_{4}Ni_{3}O_{10}}$

Zhihui Luo  Cui-Qun Chen  Meng Wang  Wéi Wú  Dao-Xin Yao
<EMAIL_ADDRESS>Center for Neutron Science and Technology, Guangdong
Provincial Key Laboratory of Magnetoelectric Physics and Devices, State Key
Laboratory of Optoelectronic Materials and Technologies, School of Physics,
Sun Yat-Sen University, Guangzhou, 510275, China

###### Abstract

Recently, the discovery of superconductivity in Ruddlesden-Popper (RP)
$\mathrm{La_{4}Ni_{3}O_{10}}$ under pressure has further expanded the realm of
nickelate-based superconductor family. In this paper, we performed a first-
principle study of $\mathrm{La_{4}Ni_{3}O_{10}}$ for both $P2_{1}/a$ phase at
ambient pressure and $I4/mmm$ phase at high pressure, with $U$=0, 3.5 eV. Our
results confirmed the characteristic upward shift of Ni-$d_{z^{2}}$ bonding
band under pressure. Moreover, our analysis of electronic spectrum and orbital
occupancy unveil the dynamical mechanism of electronic reconstructions under
pressure, embedded in a critical dual effect. Based on our DFT results, we
further proposed a trilayer two-orbital model by performing Wannier
downfolding on Ni-$e_{g}$ orbitals. Our model reveals four Fermi surface
sheets with $\alpha,\beta,\beta^{\prime},\gamma$ pockets, bearing resemblance
to that of bilayer $\mathrm{La_{3}Ni_{2}O_{7}}$. According to the model, our
calculated spin susceptibility under random phase approximation predicts an
analogous magnetic signal at ${\rm q}=(\frac{\pi}{2},\frac{\pi}{2})$, which is
more associated with nesting within $\beta,\beta^{\prime}$ pocktes. Finally, a
high energy sixteen-orbital model with direct $dp,pp$ hoppings is proposed,
which implies that $\mathrm{La_{4}Ni_{3}O_{10}}$ also lies in charge-transfer
picture within Zaanen-Sawatzky-Allen scheme. Our exposition of electronic
reconstructions and multi-orbital models shed light on theoretical electronic
correlation study and experimental exploration for lower pressure in RP
series.

##  I Introduction

The recently consecutive discoveries of nickelate-based superconductors from
infinite-layer nickelates $R\mathrm{NiO_{2}}$ ($R$=La,Nd,Pr) [1, 2, 3] to
Ruddlesden-Popper (RP) series nickelates ${\rm La}_{n+1}{\rm Ni}_{n}{\rm
O}_{3n+1}$ (n=2,3) [4, 5, 6, 7, 8, 9, 10, 11], have drawn intensive
investigations on the field of high-$T_{c}$ superconductivity [12, 13, 14, 15,
16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34,
35, 36, 37, 38, 39, 40, 41, 42, 43], particularly in the aspect of how
correlations renormalize the low-lying electronic states and potentially drive
the unconventional pairing in these compounds [37, 27, 28, 29, 30, 31, 32, 33,
34, 35, 36, 38, 43, 39, 40, 41]. Also, for $R\mathrm{NiO_{2}}$, the sample-
dependent factors, such as defeats, impurity and domains deserve careful
consideration [44, 2]. For RP series nickelates, pressure is undoubtedly in
the centre of the roadmap, as the observations of zero-resistance always
correlate with a structure transition, and further promote the electronic
reconstructions [4, 8, 10, 11, 12]. The reconstructions exhibit a quite
general trend among this series, which is largely manifested in the upward
shift of Ni-$d_{z^{2}}$ states towards Fermi energy [4, 12, 45]. It is widely
believed that such upward shift should be responsible for the emergence of
superconductivity. However, there still lacks of a comprehension of the
dynamical mechanism of these reconstructions, which is indeed fundamental and
indispensable. Furthermore, inspections of layer dependence is another
important aspect. These are crucial to unveil superconducting mechanism in RP
series.

In bilayer $\mathrm{La_{3}Ni_{2}O_{7}}$, the structure at ambient pressure is
characterized by octahedral-distorted $Amam$ (space group: 63), which transits
to octahedral-regular $Fmmm$ phase (space group: 69) roughly in the range of
10$\sim$15 GPa [9, 5, 6] within orthorhombic lattice. Surprisingly, this range
highly coincides with the full development of superconducting $T_{c}$ from 0
to 80 K [4, 5, 6]. For pressure up to $\sim$19 GPa, a higher symmetric
$I4/mmm$ (space group: 139) phase within tetragonal lattice is emergent [46,
47], where $T_{c}$ exhibits certain amount of decrease [4, 5, 6]. In trilayer
$\mathrm{La_{4}Ni_{3}O_{10}}$, an analogous monoclinic $P2_{1}/a$ (space
group: 14) phase is characterized at ambient pressure, which transits to
$I4/mmm$ phase roughly in the range of 12$\sim$15 GPa. However, the maximal
$T_{c}$ merely reaches 20$\sim$30 K under a much higher pressure of about 43
GPa [8]. These further suggest some critical roles of pressure to be clear in
the series.

In this paper, we perform a comprehensive first-principle study of trilayer
$\mathrm{La_{4}Ni_{3}O_{10}}$ for both $P2_{1}/a$ phase at ambient pressure
(AP) and $I4/mmm$ phase at high pressure (HP). Our resulting electronic
structure and microscopic multi-orbital models strongly suggest an alike
superconducting mechanism as compared with $\mathrm{La_{3}Ni_{2}O_{7}}$.
Moreover, based on our DFT results, we clarify the dynamical mechanism of such
electronic reconstructures, which is of importance for a comprehensive
understanding of superconductivity in RP series.

The paper is organized as follows. In section. II, we firstly analyse the
structure difference of P21/a phase at (AP) and $I4/mmm$ phase at HP, which
can provides readers a general understanding of the pressure effect on the
lattice level. On this basis, in section. III, we present the detailed
settings of our DFT calculations. In section. IV.1, we present our main DFT
results, which include band structure and electronic spectrum for both phases,
with $U$=0, 3.5 eV. The charge transfer processes and valence under pressure
are also particularly analysed in this section. In section. IV.2, we establish
an effective trilayer two-orbital model based on our DFT results. In section.
IV.3, we investigate the spin susceptibility based on the model. In section.
IV.4, we further proposed a high energy sixteen-orbital model. In section. V,
we provide discussions regarding the dynamical mechanism of electronic
reconstructions and layer dependence, followed by the summary in section. VI.

##  II Structure transition under pressure

In trilayer $\mathrm{La_{4}Ni_{3}O_{10}}$, each Ni ion is surrounded by six
oxygen sites forming NiO6 octahedron. The corner-shared stacking of these
octahedrons gives rise to NiO2 trilayers, as well as the important inter-layer
Ni-O-Ni bond along $c$ axis. The main structure difference of AP $P2_{1}/a$
and HP $I4/mmm$ phase are demonstrated in Fig. 1, in which we only draw a
single ${\rm NiO}_{2}$ plane for clarity. The schematic reveals two
differences: (1) For AP phase, there is perceivable octahedral tiltings as
compared with HP phase; (2) Also, both phases are differentiate in the lattice
constants under a $\sqrt{2}\times\sqrt{2}$ lattice, in which AP phase shows
orthorhombic distortion with unequal $a,b$ length, while the HP phase
possesses square structure. It should be noted that, there are also several
other types of structure distortions in RP series, such as octahedral
rotation, bond disproportionation of NiO6 octahedron. But their impact on
electronic structure are insignificant compared with octahedral tilting [47].
In fact, for $\rm La_{4}Ni_{3}O_{10}$, there exists another phase of $Bmab$
(space group: 64) at AP, which survives in high temperature. This phase differ
from $P2_{1}/a$ phase by a relative octahedral rotation, while both band
structures are almost indistinguishable around $E_{\rm F}$ [48].

Figure 1:  NiO2 plane transition under pressure. The green and red balls
represent the oxygen and nickel atoms, respectively. The grey area shapes the
Ni-O octahedra.

##  III METHOD Details

Density functional theory (DFT) calculations were performed by Vienna ab
initio simulation package (VASP) [49, 50], in which the projector augmented
wave (PAW) [51, 52] method within the framework of the local density
approximation (LDA) [53] exchange correlation potential are applied. The
energy cutoff of the plane-wave expansion was set as 600 eV and a
$\Gamma$-centered $20\times 20\times 19$ Monkhorst Pack k-mesh grid was
adopted. In the structural relaxations, we adopted the experimental refined
lattice constants for both phases [45], which is $P2_{1}/a$ phase at ambient
pressure and $I4/mmm$ phase at pressure of 43 GPa. The convergence criterion
of force was set as 0.001 eV/Å and total energy was set as $10^{-7}$eV. In
band structure calculation, we adopted a $\sqrt{2}$$\times$$\sqrt{2}$
primitive cell for $I4/mmm$ phase in order to have a direct comparison with
$P2_{1}/a$ phase, as both contain 6 Ni atoms. To obtain the projected tight-
binding models, we further performed Wannier downfolding as implemented by
WANNIER90 [54] package, in which the good convergences were reached.

##  IV RESULTS

The DFT results are carefully examined, which are in agreement with previous
theoretical calculations [48, 45] as well as the reported ARPES results [55].
On this basis, we adopted a very large k-mesh size as mentioned before in
order to determine the precise Fermi level, which is important given the flat
band feature in this material.

###  IV.1 Electronic structure

Figure 2: Electronic structure of $\mathrm{La_{4}Ni_{3}O_{10}}$. DFT (LDA)
calculated band structures of $\mathrm{La_{4}Ni_{3}O_{10}}$ at $U=0$ eV
(a)-(c) and $U=3.5$ eV (d)-(f) for AP $P2_{1}/a$ phase and HP $I4/mmm$ phase.
The blue and red colors are spectral weight for $d_{3z^{2}}$ and
$d_{x^{2}-y^{2}}$ orbitals, respectively. (c) Projected DOS of
$\mathrm{La_{4}Ni_{3}O_{10}}$ at AP (left panel) and HP (right panel). The
blue, red, orange, green and cyan colors denote the projection of
Ni-$d_{3z^{2}-r^{2}}$, Ni-$d_{x^{2}-y^{2}}$, Ni-$t_{2g}$, O-$p$ and La
orbitals.

In Fig. 2, we present the electronic structures of
$\mathrm{La_{4}Ni_{3}O_{10}}$ for both AP $P2_{1}/a$ and HP $I4/mmm$ phases.
Here the upper and lower panels represent the cases of $U=0$ and 3.5 eV,
respectively. From these band structures we can observe the dominant
Ni-$d_{z^{2}}$ (blue) and $d_{x^{2}-y^{2}}$ (red) orbitals around Fermi
energy, which is quite similar to the bilayer system [4, 12, 56]. For
Ni-$d_{z^{2}}$ band, it clearly exhibits the characteristic 3-branch
structure, which corresponds to bonding, non-bonding and anti-bonding states
with increasing energy, as indicated in Fig. 2(a) [45, 57]. Apart from
Ni-$e_{g}$ sector, there also appears dense bands (grey) away from $E_{\rm
F}$, which can be found in the density of states [Fig. 2(c),(f)] associating
with Ni-$t_{2g}$ sector around -1 eV (orange) and La sector around 3 eV
(cyan).

Next, we discuss the electronic reconstructions under pressure. As expected
that pressure is prone to enhance the itinerancy of electron, which in our
case corresponds to a ratio of $\sim$1.3 to the increase of bandwidth from AP
to HP, as illustrated in Fig. 2(a)-(b) for $U$=0 eV and Fig. 2(d)-(e) for
$U$=3.5 eV. As we further zoom in to the Fermi level, for AP phase (left
panel), we observe a very narrow gap relating the bonding and non-bonding
bands at $\Gamma$ point [48, 45]. For HP phase (middle panel), the bonding
band gets upward shifted as a hole pocket for both $U$=0, 3.5 eV. It is worth
noting that, for $U$=3.5 eV [Fig. 2(e)], such a hole pocket is almost flat
right at $E_{\rm F}$, giving rise to a sharp peak in the density of states
(DOS), further highlighting a crucial link to the superconductivity [58, 59]
in RP series.

Table 1: DFT calculated ($U$=0 eV) orbital occupancies of $\mathrm{La_{4}Ni_{3}O_{10}}$ for both AP $P2_{1}/a$ phase and HP $I4/mmm$ phase. Here $n^{\rm O}_{p_{x}/p_{y}}$ denotes one of the in-plane O-$p_{x}$, $p_{y}$ orbitals that is elongated along its adjacent Ni site. For $n^{\rm O}_{p_{z}}$, the row of “Inner” and “Outer” denote the apical O-$p_{z}$ orbitals that are inside and outside the trilayer, respectively. Note that for AP phase, each value is averaged appropriately over doubled atoms.  | Layer | $n_{x^{2}-y^{2}}^{\rm Ni}$ | $n_{z^{2}}^{\rm Ni}$ | $n^{\rm Ni}_{t_{2g}}$ | $n_{p_{x}/p_{y}}^{\rm O}$ | $n_{p_{z}}^{\rm O}$
---|---|---|---|---|---|---
AP | Inner | 1.199 | 1.289 | 5.874 | 1.695 | 1.656
Outer | 1.173 | 1.241 | 5.942 | 1.698 | 1.861
HP | Inner | 1.066 | 1.022 | 5.996 | 1.675 | 1.641
Outer | 1.077 | 1.100 | 5.993 | 1.696 | 1.843

Given such a critical band upward shift under pressure, we would further
address the corresponding charge transfer process. In Table. 1, we present the
orbital occupanices of Ni-$d$, O-$p$ orbitals for both phases at $U$=0 eV. It
is clear that, both two $e_{g}$ orbitals are over half-filling at AP, while
pressure tends to depress their occupancies, and become almost half-filling at
HP. We noted that the decrease of $n^{\rm Ni}_{x^{2}-y^{2}}$ is hardly
identified in band structure, although it has a relatively smaller amount than
that of $n^{\rm Ni}_{z^{2}}$. But for $t_{2g}$ sector, pressure has an
opposite effect, in which $n^{\rm Ni}_{t_{2g}}$ is increased from about 5.9 to
6, i.e., fully filled at HP. Such trend can also be observed in Fig. 2(c,f),
in which the $t_{2g}$ spectrum is closer to $E_{\rm F}$ for AP phase. Also,
for oxygen occupanies, we find their variations are quite insignificant under
pressure. Finally, the table reveals a slight charge imbalance between inner
and outer layers, and we find that these trends and magnitudes are quite close
to that of DFT+DMFT results at HP [60]. This hightlights that our results at
$U=0$ eV are also valid for the consideration of correlation effect.

Figure 3:  (a) Schematic of trilayer $\mathrm{La_{4}Ni_{3}O_{10}}$ lattice.
The blue and orange shapes denote Ni-$d_{z^{2}}$ and $d_{x^{2}-y^{2}}$
orbitals, respectively. Some of the hoppings of trilayer two-orbtal model are
drawn, whose values can be found in Table. 2, also see text for details. (b)
Electronic configuration $\mathrm{La_{4}Ni_{3}O_{10}}$. The solid arrows
denote nominal configuration $3d^{7.33}$. The dashed arrow indicates the
additional electrons in a realistic material, which corresponds to $3d^{8}$.

In view of the above analysis, we can quantitatively obtain some crucial
charge transfer magnitudes from AP to HP based on Table. 1. Overall, there is
an average $3d^{8.36}\rightarrow 3d^{8.14}$ for Ni ions. Specifically,
2.44$\rightarrow$2.14 for Ni-$e_{g}$ and 5.92$\rightarrow$6.0 for Ni-$t_{2g}$
sectors. However, for oxygen ions, an average $\sim$1.7 valence is persistent.
Here if we assume full ionization of oxygens, we would alternatively obtain an
average $d^{7.35}\rightarrow d^{7.14}$, in which the former value is quite
close to the nominal $d^{7.33}$ configuration. These charge transfers imply a
critical dual effect of pressure on microscopic electronic structure [23]. On
one hand, pressure promotes a charge redistribution within Ni-$3d$ orbitals,
which might further alter the relative renormalization effects of these
orbitals. In particular, the involvement of $t_{2g}$ sector near $E_{\rm F}$
could enhance the charge fluctuation at AP, which is relevant to the observed
charge density wave [61, 8]. On the other hand, the electric neutrality
indicates a pressure driven charge transfer from Ni to La ions. (the transfer
to Ni-4$s$ orbitals is negligible). In fact, from Fig. 2(a,b), we can also see
a notable drop of the lowest La band from about 2 to 0.7 eV at $\Gamma$ point,
which implies a general downward shift of La spectrum. This feature further
indicates an enhanced $d$-$f$ hybridization at HP above ${E_{\rm F}}$,
especially with Ni-$d_{z^{2}}$ orbital, which might be observed in resonant
inelastic x-ray scattering signal [62].

From our above analysis, we can roughly obtain an electron configuration for
$\mathrm{La_{4}Ni_{3}O_{10}}$ depicted in Fig. 3(b). The schematic demonstrate
a full-filled $t_{2g}$ sector and partial filling of $e_{g}$ one. According to
the nominal valence $3d^{7.33}$, there are 1.33 electrons per Ni per $e_{g}$,
which corresponds to 4 electrons per unit cell per $e_{g}$, as is demonstrated
by the solid red arrows. However, the incomplete ionization of oxygens leads
to an almost half-filling of both Ni-$d_{x^{2}-y^{2}}$ and $d_{z^{2}}$
orbitals, which corresponds to 6 electrons per unit cell per $e_{g}$. The
additional 2 electrons would go to $d_{x^{2}-y^{2}}$ orbital, as demonstrated
by the dashed arrows in the schematic. The schematic also implies a distinct
correlation effect of the two orbitals, as the half-filling of $d_{z^{2}}$
orbital is indeed shared by three Ni site along $c$ axis, while
$d_{x^{2}-y^{2}}$ is much similar to that of cuprates.

###  IV.2 Trilayer two-orbital model

Table 2: Tight-binding parameters of trilayer two-orbital model for $\mathrm{La_{4}Ni_{3}O_{10}}$ under pressure. $t_{[i,j]}^{x/z}$ denotes the hopping term that is connected by $[0,0]$-$[i,j]$ bond within $d_{x^{2}-y^{2}}/d_{z^{2}}$ orbital, while $t^{xz}_{[i,j]}$ denotes the hybridization between them, all in unit of eV. The symmetrically equivalent terms are not shown for clarity.  Layer | $i$ | $j$ | $t_{[i,j]}^{x}$ | $t_{[i,j]}^{z}$ | $t_{[i,j]}^{xz}$
---|---|---|---|---|---
Inner | 0 | 0 | 1.094 | 1.081 | 0
1 | 0 | -0.521 | -0.168 | 0.298
1 | 1 | 0.069 | -0.018 | 0
2 | 0 | -0.076 | -0.018 | 0.040
Outer | 0 | 0 | 0.867 | 0.683 | 0
1 | 0 | -0.511 | -0.143 | 0.274
1 | 1 | 0.065 | -0.015 | 0
2 | 0 | -0.074 | -0.017 | 0.039
Inner-Outer | 0 | 0 | 0.035 | -0.738 | 0
1 | 0 | 0 | 0.033 | -0.057
Outer-outer | 0 | 0 | 0 | -0.078 | 0

To gain more insights to the electronic property that is directly relevant to
the superconductivity, in the following sections, we would focus on the HP
$I4/mmm$ phase. Based on the DFT electronic structure, we further perform
Wannier downfolding on the trilayer Ni-$e_{g}$ orbitals at $U$=0 eV, which
allows us to build an effective trilayer two-orbital model

$\displaystyle\mathcal{H}$
$\displaystyle=\mathcal{H}_{t}+\mathcal{H}_{U},\qquad\mathcal{H}_{t}=\sum_{{\rm
k}\sigma}\Psi_{{\rm k}\sigma}^{\dagger}H({\rm k})\Psi_{{\rm k}\sigma},$ (1)
$\displaystyle\mathcal{H}_{U}$
$\displaystyle=U\sum_{i}^{\alpha=1,2,3}\left(n_{i\uparrow}^{x_{\alpha}}n_{i\downarrow}^{x_{\alpha}}+n_{i\uparrow}^{z_{\alpha}}n_{i\downarrow}^{z_{\alpha}}\right)$
$\displaystyle+U^{\prime}\sum_{i\sigma}^{\alpha=1,2,3}n^{x_{\alpha}}_{i\sigma}n^{z_{\alpha}}_{i\bar{\sigma}}+(U^{\prime}-J_{H})\sum_{i\sigma}^{\alpha=1,2,3}n^{x_{\alpha}}_{i\sigma}n^{z_{\alpha}}_{i\sigma}$
$\displaystyle+J_{H}$
$\displaystyle\sum_{i}^{\alpha=1,2,3}\left(d^{\dagger}_{ix_{\alpha}\uparrow}d^{\dagger}_{ix_{\alpha}\downarrow}d_{iz_{\alpha}\downarrow}d_{iz_{\alpha}\uparrow}-d^{\dagger}_{ix_{\alpha}\uparrow}d_{ix_{\alpha}\downarrow}d^{\dagger}_{iz_{\alpha}\downarrow}d_{iz_{\alpha}\uparrow}+h.c.\right).$

The model $\mathcal{H}$ is composed of the tight-binding $\mathcal{H}_{t}$ and
Coulomb interaction $\mathcal{H}_{U}$. The basis is
$\Psi_{\sigma}=(d_{x_{1}\sigma},d_{x_{2}\sigma},d_{x_{3}\sigma},d_{z_{1}\sigma},d_{z_{2}\sigma},d_{z_{3}\sigma})^{T}$,
in which $d_{x_{\alpha}\sigma}$ denotes annihilation of a $d_{x^{2}-y^{2}}$
electron in the $\alpha^{\rm th}$ layer with spin $\sigma$, $etc$. The
notation of layer is demonstrated in Fig. 3(a). For $\mathcal{H}_{U}$, there
have four terms in full Kanamori form [63] with relation
$U^{\prime}=U-2J_{H}$, and $U,U^{\prime},J_{H}$ are on-site Coulomb repulsion
of the same and different orbitals, and Hund’s coupling, respectively.

We further express the tight-binding kernel as

$\displaystyle H({\rm k})=\left[\begin{array}[]{cc}H^{x}({\rm k})&H^{xz}({\rm
k})\\\ H^{xz}({\rm k})&H^{z}({\rm k})\end{array}\right],$ (4)

in which $H^{x/z}({\rm k})$ is the trilayer tight-binding matrix of
$d_{x^{2}-y^{2}}/d_{z^{2}}$ orbital, and $H^{xz}({\rm k})$ is the
hybridization between them. They are expressed as

$\displaystyle H_{11}^{x/z}$ $\displaystyle=H_{33}^{x/z}=\epsilon^{x/z,{\rm
O}}+2t^{x/z,{\rm O}}_{[1,0]}(\cos k_{x}+\cos k_{y})+4t^{x/z,{\rm
O}}_{[1,1]}\cos k_{x}\cos k_{y}+2t^{x/z,{\rm O}}_{[2,0]}(\cos 2k_{x}+\cos
2k_{y}),$ $\displaystyle H_{22}^{x/z}$ $\displaystyle=\epsilon^{x/z,{\rm
I}}+2t^{x/z,{\rm I}}_{[1,0]}(\cos k_{x}+\cos k_{y})+4t^{x/z,{\rm
I}}_{[1,1]}\cos k_{x}\cos k_{y}+2t^{x/z,{\rm I}}_{[2,0]}(\cos 2k_{x}+\cos
2k_{y}),$ $\displaystyle H^{x/z}_{12}$ $\displaystyle=H^{x/z}_{23}=t^{x/z,{\rm
IO}}_{[0,0]}+2t^{x/z,{\rm IO}}_{[1,0]}(\cos k_{x}+\cos k_{y}),\quad
H^{z}_{13}=t^{z,{\rm OO}}_{[0,0]},$ (5) $\displaystyle H^{xz}_{11}$
$\displaystyle=H^{xz}_{33}=2t^{xz,{\rm O}}_{[1,0]}(\cos k_{x}-\cos
k_{y})+2t^{xz,{\rm O}}_{[2,0]}(\cos 2k_{x}-\cos 2k_{y}),$ $\displaystyle
H^{xz}_{22}$ $\displaystyle=2t^{xz,{\rm I}}_{[1,0]}(\cos k_{x}-\cos
k_{y})+2t^{xz,{\rm I}}_{[2,0]}(\cos 2k_{x}-\cos 2k_{y}),$ $\displaystyle
H^{xz}_{12}$ $\displaystyle=H^{xz}_{23}=2t^{xz,{\rm IO}}_{[1,0]}(\cos
k_{x}-\cos k_{y}).$

Figure 4: (a) Band structure and (b) Fermi surface of trilayer two-orbital
model. The color bar indicates the orbital weight of Ni-$d_{3z^{2}-r^{2}}$ and
$d_{x^{2}-y^{2}}$. The grey lines in (a) are band structure from DFT (LDA,
$U=0$ eV). The diamond shape in (b) indicates the folded Brillouin zone.

We consider the hoppings up to the third-nearest neighbor in order to
accurately describe the low-lying state of our DFT, which yields a total of 25
parameters listed in Table. 2. Some of the major hoppings are demonstrated in
the schematic in Fig. 3(a). Here $t_{[1,0]}^{x,{\rm I}}$ denotes in-plane
nearest-neighbor hopping of $d_{x^{2}-y^{2}}$ orbitals inside inner layer, and
$t^{z,{\rm IO}}_{[0,0]}$ denotes perpendicular hopping of $d_{z^{2}}$ orbital
between inner and outer layers. According to values in Table. 2, we find that
most of the hoppings are enhanced comparably with respect to bilayer
$\mathrm{La_{3}Ni_{2}O_{7}}$, practically, $t^{x}_{[1,0]}$ is enhanced from
-0.483 to -0.521 (inner) or -0.511 (outer), and $t^{z,{\rm IO}}_{[0,0]}$ from
-0.635 to -0.738 [12]. Such enhancement of hopping parameters partially
reflect a relative weaker correlation of trilayer than that of bilayer.

In Fig. 4, we further show the band structure and Fermi surface of trilayer
two-orbital model, which is in good agreement with our DFT results. The model
is expressed under the primitive cell that contains only three Ni atoms, hence
the flat band will be unfolded from $\Gamma$ to M point, as shown by the
$\gamma$-pocket in Fig. 4(b). The figure also reveals another three pockets
$\alpha,\beta,\beta^{\prime}$ with $\beta,\beta^{\prime}$ very close in each
other. In general, the Fermi surface profile as well as the orbital weights
are quite similar to that of the bilayer $\mathrm{La_{3}Ni_{2}O_{7}}$ [12], as
$\gamma$-point is uniquely characterized by $d_{z^{2}}$ orbital and
$\alpha,\beta,\beta^{\prime}$-pockets by mixing of $d_{z^{2}}$ and
$d_{x^{2}-y^{2}}$ orbitals.

To illustrate the bonding feature in $\mathrm{La_{4}Ni_{3}O_{10}}$, we perform
a rotation of the orbital basis under the trilayer symmetry, in which we
define the new basis as

$\displaystyle\Phi$ $\displaystyle=(c_{-},c_{+},c)^{T},\quad
c_{\pm}=\frac{1}{\sqrt{2}}(d_{1}\pm d_{3}),\quad c=d_{2},$ (6)

with $d$ applied for both $d_{x}$ and $d_{z}$ operators. When along nodal
direction ($|k_{x}|=|k_{y}|$), the two $e_{g}$ orbitals are well decouples as
the off-diagonal $H^{xz}({\rm k})=0$ in Eq. (4). In this case, we can find
that the non-bonding state is decisively characterized by $c_{-}$ state,
namely, a sign-reversed superposition of states from two outer layers, while
the bonding and anti-bonding states are associated with mixing of both
$c_{+}$, $c$ states [64]. For the most concerning bonding band of
Ni-$d_{z^{2}}$ orbital, we can find that it is characterized by the positive
superposition of $c_{+}$ and $c$, which corresponds to a more elongated
Wannier wave function along $c$ axis.

Figure 5: RPA spin susceptibility $\chi_{{}_{S}}^{st,{\rm RPA}}$ of trilayer
two-orbital model. $U=3.5,J=U/10$ are adopted in the calculation. (a) Orbital
sum $\chi_{{}_{S}}^{\rm RPA}=\sum_{st}\chi_{{}_{S}}^{st,{\rm RPA}}$; (b-d)
Orbital resolved $\chi_{{}_{S}}^{st,{\rm RPA}}$. An amplify factor of 2 is
used in (b,c).

###  IV.3 Spin susceptibility

The multi Fermi surface sheets of $\mathrm{La_{4}Ni_{3}O_{10}}$ implies some
possible magnetic instabilities that might be observed in experiments. In Fig.
5, we present the spin susceptibility of trilayer two-orbital model at
$\omega=0$ under random phase approximation. The total orbital-summed
$\chi_{{}_{S}}^{\rm RPA}=\sum_{st}\chi_{{}_{S}}^{st,{\rm RPA}}$ [Fig. 5(a)]
shows a broad magnetic signal over the whole Brillouin zone, just like that in
Ref. [60] as well as in bilayer system [12, 42, 24]. Still, we can observe
notable enhancement at some particular regimes reflecting fine nesting of
Fermi surface, especially at wave vector ${\rm
q}=(\pm\frac{\pi}{2},\pm\frac{\pi}{2})$. From orbital-resolved
$\chi_{{}_{S}}^{st,{\rm RPA}}$ [Fig. 5(b-d)], we find that it has a major
$d_{z^{2}}$ character, also, the contribution from $d_{x^{2}-y^{2}}$ orbital
is nonnegligible. Remarkably, the Fermi surface in Fig. 4(b) shows that such
wave vectors are more associated with nesting within $\beta,\beta^{\prime}$
pockets which possess quasi-1D feature and strong orbital mixing, instead of
$\alpha,\gamma$ pockets. This conjecture is also evidenced in
$\chi_{{}_{S}}^{\rm RPA}$ by the appearance of Moire pattern along these wave
vectors, reflecting the quasi-degeneracy of $\beta,\beta^{\prime}$ pockets.
This highlights that $d_{x^{2}-y^{2}}$ orbital is also important for the
magnetic fluctuation of the ground state in $\mathrm{La_{4}Ni_{3}O_{10}}$,
despite it has a relatively weak correlation than $d_{z^{2}}$ orbital [55].
Recently, the magnetic signal at this wave vector is unambiguously revealed in
a RIXS experiment for bilayer system, indicating a corresponding doubled spin
texture that might exist in the RP series nickelates [65]. Our results for
trilayer $\mathrm{La_{4}Ni_{3}O_{10}}$ further confirm such signal worth of
future RIXS and inelastic neutron scattering experiment.

Figure 6:  (a) Band structure and (b) Fermi surface of sixteen-orbital model.
The color bar indicates the orbital weight of Ni-$d_{3z^{2}-r^{2}}$ and
$d_{x^{2}-y^{2}}$ orbitals. The grey lines in (a) are band structure from DFT
(LDA, $U=0$ eV). The diamond shape in (b) indicates the folded Brillouin zone.
(c) Schematic of sixteen orbitals for sixteen-orbital model.

###  IV.4 Sixteen-orbital model

To gain insights to the charge-transfer property within Zaanen-Sawatzky-Allen
scheme (ZAS) [66, 67], we further perform Wannier downfolding on both
Ni-$e_{g}$ and O-$p$ orbitals. For each of the in-plane oxygen, we only pick
one of $p_{x}/p_{y}$ orbitals that is elongated along its adjacent Ni site, as
normally would do in a CuO2 plane [67]. For apical oxygens, only $p_{z}$
orbitals are selected. These yield a sixteen-orbital basis as demonstrated in
the schematic in Fig. 6(c). In this basis, it is sufficient to consider merely
12 hopping parameters, which include the dominated nearest-neighbor $dp$, $dd$
overlaps as listed in Table. 3. In this table, each hopping term is indicated
by the orbital pairs which are connected by real-space vector [i,j,k], as can
be located in the schematic. Note that the symmetrically equivalent terms are
no shown for clarity. The site-energies of these orbitals are also presented
in the last panel. The resulting band structure and Fermi surface are shown in
Fig. 6(a)-(b), which are all in good agreement with our DFT results. The band
structure covers an energy range of -9$\sim$3 eV, with strong split into two
parts. We find there appear two flat bands in the lower part, which are
originated from two dangling $p_{z}^{\prime}$ orbitals with strong coupling to
apical $d_{z^{2}}$ orbitals. The huge band split indicates a significantly
large energy scale of $dp$ hybridization, from which the effective direct $dd$
couplings emerge, as illustrated in our trilayer two-orbital model. This model
will be important for further study of strong correlation effects and
superexchange couplings.

Table 3: Tight-binding parameters of sixteen-orbital model for $\mathrm{La_{4}Ni_{3}O_{10}}$ under pressure, in unit of eV. The corresponding orbitals are displayed in Fig. 6(b). The symmetrically equivalent terms are no shown for clarity.  Hopping |  $d_{z_{3}}$-$p_{z}$ |  $d_{z_{2}}$-$p_{z}$ |  $d_{x_{2}}$-$p_{x_{2}}$ |  $d_{x_{1}}$-$p_{x_{1}}$
---|---|---|---|---
| [0,0,$\frac{1}{2}$] | [0,0,$\frac{1}{2}$] | [$\frac{1}{2}$,0,0] | [$\frac{1}{2}$,0,0]
$t$ | 1.911 | 1.523 | 1.681 | 1.676
Hopping |  $d_{z_{1}}$-$p_{z}^{\prime}$ |  $d_{z_{2}}$-$p_{x_{2}}$ |  $d_{z_{1}}$-$p_{x_{1}}$ |  $p_{x_{1}}$-$p_{y_{1}}$
| [0,0,$\frac{1}{2}$] | [$\frac{1}{2}$,0,0] | [$\frac{1}{2}$,0,0] | [$\frac{1}{2}$,$\frac{1}{2}$,0]
$t$ | 1.591 | -0.938 | -0.887 | 0.604
Hopping |  $p_{x_{2}}$-$p_{y_{2}}$ |  $p_{z}$-$p_{x_{1}}$ |  $p_{z}$-$p_{x_{2}}$ |  $p_{z}^{\prime}$-$p_{x_{1}}$
| [$\frac{1}{2}$,$\frac{1}{2}$,0] | [$\frac{1}{2}$,0,$\frac{1}{2}$] | [$\frac{1}{2}$,0,$\frac{1}{2}$] | [$\frac{1}{2}$,0,$-\frac{1}{2}$]
$t$ | 0.553 | 0.597 | 0.577 | -0.465
Site- | $\epsilon_{x_{1}}$ | $\epsilon_{z_{1}}$ | $\epsilon_{x_{2}}$ | $\epsilon_{z_{2}}$
energy | -1.431 | -1.354 | -1.075 | -1.027
Site | $\epsilon_{px_{1}}$ | $\epsilon_{px_{2}}$ | $\epsilon_{p_{z}}$ | $\epsilon_{p_{z}^{\prime}}$
energy | -5.109 | -4.757 | -4.494 | -4.149

According to the table, the site-energy differences within NiO2 plane are
calculated as $\epsilon_{x_{2}}-\epsilon_{px_{2}}=$3.42 eV and
$\epsilon_{x_{1}}-\epsilon_{px_{1}}=$3.68 eV, respectively for inner and outer
layers. Along the apical direction, the energy difference are
$\epsilon_{z_{2}}-\epsilon_{p_{z}}$=3.47 eV or
$\epsilon_{z_{1}}-\epsilon_{p_{z}}$=3.14 eV. Compared with bilayer, these
magnitudes are quite similar, and both suggest a relative smaller energy
difference between $d_{z^{2}}$ and apical $p_{z}$ orbitals than that of
cuprate counterpart. This implies that $\mathrm{La_{4}Ni_{3}O_{10}}$ should
also lies in charge-transfer picture [23], and would favor a strong inter-
layer superexchange coupling [68, 69]. However, for trilayer, the microscopic
superexchange couplings is supposed to be more complicated as there involve at
least three $d_{z^{2}}$ and $p_{z}$ orbitals. For example, the superexchange
process of $d_{z^{2}}-p_{z}-d_{z^{2}}$ between two adjacent layers, which is
believed to be dominant in bilayer system [69], would be depressed in trilayer
due to interference from other $d_{z^{2}}$ and $p_{z}$ along $c$ axis, and we
believe this is the main reason for a much lower transition temperature in
$\mathrm{La_{4}Ni_{3}O_{10}}$. Also, such speculation is in line with the
fundamental concept that correlation is depressed in higher dimensions.

##  V Discussion

The microscopic charge transfer processes as well as the associating dual
effect reflect two impulses from pressure. For the first one, the charge
transfer from Ni-$e_{g}$ to $t_{2g}$ under pressure can be directly attributed
to the octahedral tilting as illustrated in Section. II. Meanwhile, a lattice
that possesses higher symmetry is generally preferred by a uniform pressure,
which corresponds to $I4/mmm$ phase in the case of RP series. This can be
simply understood from the definition of enthalpy $H=U+PV$, which shows that
pressure would push the lattice to a higher energy state, where the symmetry
unbreaking state is preferred. For the second one, the partial charge transfer
from Ni-$d$ to La ions essentially reflect an urge from lattice collapse,
which is strongly reminiscent of lanthanide contraction effect [70]. In
lanthanide series, the removal of correlated $4f$ electron under pressure (or
lowering temperature) causes a decrease of screening from nucleus charge, so
that the outer valence electrons are sucked in closer to the nucleus, further
causing a decrease in ion radius [70]. Meanwhile, the partial filling of these
correlated orbitals give rise to valence fluctuation phenomenon. In RP series
compounds, an analogous dynamical mechanism should be applied, which suggests
a likewise enhanced valence fluctuation under pressure. In this sense, the
superconductivity can be interpreted as a unexpected outcome of valence
fluctuation within Ni-$e_{g}$ orbitals.

Based on above analysis, we further elucidate the dynamical mechanism of
electronic reconstructions under pressure. As gradually applying pressure onto
the lattice, both two effects would happen simultaneously, jointly leading to
the upward shift of Ni-$e_{g}$ spectrum. Note that their respective
contributions should be different, and, according to our results, the second
effect has the dominated role. However, the first effect might also be
indispensable, as there is conjecture that the straightening of apical Ni-O-Ni
bond angle can significantly enhance the perpendicular superexchange
$J_{\bot}$, which is crucial for the occurrence of superconductivity. Once the
pressure is high enough for such bond angle to reach formal 180∘, there
corresponds to a structure transition. For even higher pressure, there appears
new situation. In this stage, the oxygens within NiO2 plane start to move
along $c$ axis, leading to a separation of the basal oxygen and Ni planes, but
maintains invariance of $I4/mmm$ structure [47]. This structure variation is
related to a further lattice collapse along $a/b$ axis, while the collapse
along $c$ axis has already depleted in early stage. In this stage, it can be
expected that there appears a charge transfer from Ni-$d_{z^{2}}$ to
$d_{x^{2}-y^{2}}$ orbital, further pushing Ni-$d_{z^{2}}$ spectrum towards
higher energy. Overall, we believe our exposition of the dynamical mechanism
of electronic reconstructions is particularly helpful for theoretical
understanding of electronic properties RP series, which can inspire various
further theoretical investigations on various aspects, such as valence
fluctuation, superexchanges, correlation renormalization effect [43, 71],
orbital selective property [72, 40, 65, 73, 12, 72], strange metal behavior
[74, 8], and the interplay with spin/charge density wave fluctuations [61].
Moreover, it can sheds light on exploration of new superconductor with a lower
pressure via chemical substitute.

Finally, regarding the layer dependence, our investigations on electronic
properties of $\mathrm{La_{4}Ni_{3}O_{10}}$ have strongly recommended a
similar superconducting mechanism among RP series, in which the increase of
layer seems trivially plays an negative effect. If so, it would be
unsurprising that superconductivity is fully depressed in quadlayer
$\mathrm{La_{5}Ni_{4}O_{13}}$ even which possesses an alike Fermi surface
profile [48]. Nevertheless, such an experimental observation is still of
significance. All these features highlight that RP series is a perfect
platform for theoretically understanding of the unconventional superconducting
mechanism. Also, the role of the even-odd effect, which governs the excitation
gap in local spin-$\frac{N}{2}$ chain, remains to be clear in the series [57,
75, 76].

##  VI Summary

In summary, we performed a comprehensive DFT calculations on RP series
$\mathrm{La_{4}Ni_{3}O_{10}}$ for both AP $P2_{1}/a$ and HP $I4/mmm$ phases,
with $U$= 0, 3.5 eV. Our results clearly reveal the characteristic upward
shift of Ni-$d_{z^{2}}$ bonding band under pressure, suggesting a crucial link
to the superconductivity in RP series. Our analysis of electronic spectrum as
well as orbital occupancies indicate a charge transfer of
$3d^{3.35}\rightarrow 3d^{3.14}$ under pressure, specifically,
2.44$\rightarrow$2.14 for Ni-$e_{g}$ and 5.92$\rightarrow$6.0 for Ni-$t_{2g}$
sectors. These trends indicate to a critical dual effect of pressure, in which
pressure on one hand promotes a charge redistribution within Ni-$e_{g}$ and
$t_{2g}$ sectors. On the other hand, pressure also drives a partial charge
transfer from Ni to La ions. On this basis, in the discussion, we fully unveil
the dynamical mechanism of electronic reconstructions under pressure, which
sheds light on theoretical understanding of electronic correlations,
superconductivity, as well as to experimental exploration of new
superconductor with lower pressure.

Based on our DFT results, we perposed a trilayer two-orbital model by
performing Wannier downfolding on Ni-$e_{g}$ orbitals. This model reveals
$\alpha,\beta,\beta^{\prime},\gamma$, which is very close to that of
$\mathrm{La_{4}Ni_{3}O_{10}}$ and suggests a similar superconducting pairing
mechanism in RP series. Also, our calculated RPA spin susceptibility predicts
an alike magnetic signal at ${\rm q}=(\frac{\pi}{2},\frac{\pi}{2})$ as oberved
in $\mathrm{La_{3}Ni_{2}O_{7}}$. Moreover, we find such wave vector is more
associated with nesting with $\beta,\beta^{\prime}$ pockets.

To gain insights to the charge-transfer property within ZAS scheme, we further
propose a high energy sixteen-orbital model based on Wannier downfolding on
both Ni-$e_{g}$ and O-$p$ orbitals. The model also well reproduce the low-
lying band structure, with merely 12 nearest-neighbor $dd,dp$ hoppings. The
site-energies of the model show that the energy differences between Ni-$d$ and
O-$p$ orbitals are close to that in bilayer, which are 3.42 and 3.68 eV,
respectively for inner and outer layers between $d_{x^{2}-y^{2}}-p_{x}/p_{y}$,
and 3.47, 3.14 eV for apical $d_{z^{2}}-p_{z}$ orbitals. These implies that
$\mathrm{La_{4}Ni_{3}O_{10}}$ also lies in the charge-transfer picture within
ZAS scheme. In short, the sixteen-orbital model as well as the trilayer two-
orbital model are important for future study of strong correlation effects and
unconventional pairing symmetry.

###### Acknowledgements.

We are grateful to Qiang-Hua Wang, Guang-Ming Zhang, and Xun-Wu Hu for
fruitful discussions. Work at Sun Yat-Sen University was supported by the
National Key Research and Development Program of China (Grants No.
2022YFA1402802, 2018YFA0306001, 2023YFA1406500), the National Natural Science
Foundation of China (Grants No. 92165204, No. 12174454, No. 11974432, No.
12274472), the Guangdong Basic and Applied Basic Research Foundation (Grants
No. 2022A1515011618, No. 2021B1515120015), Guangzhou Basic and Applied Basic
Research Funds (grant nos. 2024A04J6417), Guangdong Provincial Key Laboratory
of Magnetoelectric Physics and Devices (Grant No. 2022B1212010008), and
Shenzhen International Quantum Academy (Grant No. SIQA202102).

## References

  * Li  _et al._ [2019] D. Li, K. Lee, B. Y. Wang, M. Osada, S. Crossley, H. R. Lee, Y. Cui, Y. Hikita, and H. Y. Hwang, Superconductivity in an infinite-layer nickelate, Nature 572, 624 (2019).
  * Zeng  _et al._ [2022] S. Zeng, C. Li, L. E. Chow, Y. Cao, Z. Zhang, C. S. Tang, X. Yin, Z. S. Lim, J. Hu, P. Yang, and A. Ariando, Superconductivity in infinite-layer nickelate La1-xCaxNiO2 thin films, Science Advances 8, eabl9927 (2022).
  * Osada  _et al._ [2020] M. Osada, B. Y. Wang, B. H. Goodge, K. Lee, H. Yoon, K. Sakuma, D. Li, M. Miura, L. F. Kourkoutis, and H. Y. Hwang, A superconducting praseodymium nickelate with infinite layer structure, Nano Letters 20, 5735 (2020).
  * Sun  _et al._ [2023] H. Sun, M. Huo, X. Hu, J. Li, Z. Liu, Y. Han, L. Tang, Z. Mao, P. Yang, B. Wang, J. Cheng, D.-X. Yao, G.-M. Zhang, and M. Wang, Signatures of superconductivity near 80 K in a nickelate under high pressure, Nature 621, 493 (2023).
  * Wang  _et al._ [2023a] G. Wang, N. Wang, J. Hou, L. Ma, L. Shi, Z. Ren, Y. Gu, X. Shen, H. Ma, P. Yang, Z. Liu, H. Guo, J. Sun, G. Zhang, J. Yan, B. Wang, Y. Uwatoko, and J. Cheng, Pressure-induced superconductivity in polycrystalline La3Ni2O7 (2023a), arXiv:2309.17378 [cond-mat.supr-con] .
  * Hou  _et al._ [2023a] J. Hou, P. T. Yang, Z. Y. Liu, J. Y. Li, P. F. Shan, L. Ma, G. Wang, N. N. Wang, H. Z. Guo, J. P. Sun, Y. Uwatoko, M. Wang, G. M. Zhang, B. S. Wang, and J. G. Cheng, Emergence of high-temperature superconducting phase in the pressurized La3Ni2O7 crystals (2023a), arXiv:2307.09865 [cond-mat.supr-con] .
  * Hou  _et al._ [2023b] J. Hou, P.-T. Yang, Z.-Y. Liu, J.-Y. Li, P.-F. Shan, L. Ma, G. Wang, N.-N. Wang, H.-Z. Guo, J.-P. Sun, Y. Uwatoko, M. Wang, G.-M. Zhang, B.-S. Wang, and J.-G. Cheng, Emergence of high-temperature superconducting phase in pressurized La3Ni2O7 crystals, Chinese Physics Letters 40, 117302 (2023b).
  * Zhu  _et al._ [2024] Y. Zhu, E. Zhang, B. Pan, X. Chen, D. Peng, L. Chen, H. Ren, F. Liu, N. Li, Z. Xing, J. Han, J. Wang, D. Jia, H. Wo, Y. Gu, Y. Gu, L. Ji, W. Wang, H. Gou, Y. Shen, T. Ying, X. Chen, W. Yang, C. Zheng, Q. Zeng, J. gang Guo, and J. Zhao, Superconductivity in trilayer nickelate La4Ni3O10 single crystals (2024), arXiv:2311.07353 [cond-mat.supr-con] .
  * Zhou  _et al._ [2023] Y. Zhou, J. Guo, S. Cai, H. Sun, P. Wang, J. Zhao, J. Han, X. Chen, Q. Wu, Y. Ding, M. Wang, T. Xiang, H. kwang Mao, and L. Sun, Evidence of filamentary superconductivity in pressurized La3Ni2O7 single crystals (2023), arXiv:2311.12361 [cond-mat.supr-con] .
  * Zhang  _et al._ [2023a] M. Zhang, C. Pei, X. Du, Y. Cao, Q. Wang, J. Wu, Y. Li, Y. Zhao, C. Li, W. Cao, S. Zhu, Q. Zhang, N. Yu, P. Cheng, J. Zhao, Y. Chen, H. Guo, L. Yang, and Y. Qi, Superconductivity in trilayer nickelate La4Ni3O10 under pressure (2023a), arXiv:2311.07423 [cond-mat.supr-con] .
  * Li  _et al._ [2024] Q. Li, Y.-J. Zhang, Z.-N. Xiang, Y. Zhang, X. Zhu, and H.-H. Wen, Signature of superconductivity in pressurized La4Ni3O10, Chinese Physics Letters 41, 017401 (2024).
  * Luo  _et al._ [2023a] Z. Luo, X. Hu, M. Wang, W. Wú, and D.-X. Yao, Bilayer two-orbital model of $\mathrm{La_{3}Ni_{2}O_{7}}$ under pressure, Physical Review Letters 131, 126001 (2023a).
  * Luo  _et al._ [2023b] Z. Luo, B. Lv, M. Wang, W. Wú, and D.-X. Yao, High-TC superconductivity in $\mathrm{La_{3}Ni_{2}O_{7}}$ based on the bilayer two-orbital t-J model (2023b), arXiv:2308.16564 [cond-mat.supr-con] .
  * Tam  _et al._ [2022] C. C. Tam, J. Choi, X. Ding, S. Agrestini, A. Nag, M. Wu, B. Huang, H. Luo, P. Gao, M. Garcia-Fernandez, L. Qiao, and K. Zhou, Charge density waves in infinite-layer NdNiO2 nickelates, Nature Materials 21, 1116 (2022).
  * Liu  _et al._ [2020] Z. Liu, Z. Ren, W. Zhu, Z. Wang, and J. Yang, Electronic and magnetic structure of infinite-layer NdNiO2: trace of antiferromagnetic metal, npj Quantum Materials 5, 31 (2020).
  * Liu  _et al._ [2022] Z. Liu, H. Sun, M. Huo, X. Ma, Y. Ji, E. Yi, L. Li, H. Liu, J. Yu, Z. Zhang, Z. Chen, F. Liang, H. Dong, H. Guo, D. Zhong, B. Shen, S. Li, and M. Wang, Evidence for charge and spin density waves in single crystals of La3Ni2O7 and La3Ni2O6, Science China Physics, Mechanics & Astronomy 66, 217411 (2022).
  * Hu and Wu [2019] L.-H. Hu and C. Wu, Two-band model for magnetism and superconductivity in nickelates, Phys. Rev. Res. 1, 032046 (2019).
  * Zhang  _et al._ [2024] Y. Zhang, L.-F. Lin, A. Moreo, T. A. Maier, and E. Dagotto, Electronic structure, magnetic correlations, and superconducting pairing in the reduced ruddlesden-popper bilayer ${\mathrm{La}}_{3}{\mathrm{Ni}}_{2}{\mathrm{O}}_{6}$ under pressure: Different role of ${d}_{3{z}^{2}-{r}^{2}}$ orbital compared with ${\mathrm{La}}_{3}{\mathrm{Ni}}_{2}{\mathrm{O}}_{7}$, Phys. Rev. B 109, 045151 (2024).
  * Shen  _et al._ [2023] Y. Shen, M. Qin, and G.-M. Zhang, Effective bi-layer model hamiltonian and density-matrix renormalization
group study for the high-$T_{\rm c}$ superconductivity
in La3Ni2O7 under high pressure, Chin. Phys. Lett. 40, 127401 (2023).

  * Yao and Carlson [2007] D. X. Yao and E. W. Carlson, Spin-wave dispersion in half-doped ${\mathrm{La}}_{3/2}{\mathrm{Sr}}_{1/2}{\mathrm{NiO}}_{4}$, Phys. Rev. B 75, 012414 (2007).
  * Ryee  _et al._ [2020] S. Ryee, H. Yoon, T. J. Kim, M. Y. Jeong, and M. J. Han, Induced magnetic two-dimensionality by hole doping in the superconducting infinite-layer nickelate ${\mathrm{Nd}}_{1-x}{\mathrm{Sr}}_{x}{\mathrm{NiO}}_{2}$, Phys. Rev. B 101, 064513 (2020).
  * Qin and Yang [2023] Q. Qin and Y.-f. Yang, High-${T}_{c}$ superconductivity by mobilizing local spin singlets and possible route to higher ${T}_{c}$ in pressurized ${\mathrm{La}}_{3}{\mathrm{Ni}}_{2}{\mathrm{O}}_{7}$, Phys. Rev. B 108, L140504 (2023).
  * Wú  _et al._ [2023] W. Wú, Z. Luo, D.-X. Yao, and M. Wang, Charge transfer and zhang-rice singlet bands in the nickelate superconductor $\mathrm{La_{3}Ni_{2}O_{7}}$ under pressure (2023), arXiv:2307.05662 [cond-mat.str-el] .
  * Botzel  _et al._ [2024] S. Botzel, F. Lechermann, J. Gondolf, and I. M. Eremin, Theory of magnetic excitations in multilayer nickelate superconductor La3Ni2O7 (2024), arXiv:2401.16151 [cond-mat.supr-con] .
  * Nomura  _et al._ [2019] Y. Nomura, M. Hirayama, T. Tadano, Y. Yoshimoto, K. Nakamura, and R. Arita, Formation of a two-dimensional single-component correlated electron system and band engineering in the nickelate superconductor ${\mathrm{NdNiO}}_{2}$, Phys. Rev. B 100, 205138 (2019).
  * Botana and Norman [2020] A. S. Botana and M. R. Norman, Similarities and differences between ${\mathrm{LaNiO}}_{2}$ and ${\mathrm{CaCuO}}_{2}$ and implications for superconductivity, Phys. Rev. X 10, 011024 (2020).
  * Jiang  _et al._ [2020] M. Jiang, M. Berciu, and G. A. Sawatzky, Critical nature of the ni spin state in doped ${\mathrm{NdNiO}}_{2}$, Phys. Rev. Lett. 124, 207004 (2020).
  * Zhang  _et al._ [2020a] G.-M. Zhang, Y.-f. Yang, and F.-C. Zhang, Self-doped mott insulator for parent compounds of nickelate superconductors, Phys. Rev. B 101, 020501 (2020a).
  * Zhang and Vishwanath [2020] Y.-H. Zhang and A. Vishwanath, Type-II $t{\rm-}{J}$ model in superconducting nickelate ${\mathrm{Nd}}_{1-x}{\mathrm{Sr}}_{x}{\mathrm{NiO}}_{2}$, Phys. Rev. Res. 2, 023112 (2020).
  * Werner and Hoshino [2020] P. Werner and S. Hoshino, Nickelate superconductors: Multiorbital nature and spin freezing, Phys. Rev. B 101, 041104 (2020).
  * Bernardini  _et al._ [2020] F. Bernardini, V. Olevano, and A. Cano, Magnetic penetration depth and ${T}_{c}$ in superconducting nickelates, Phys. Rev. Res. 2, 013219 (2020).
  * Hepting  _et al._ [2020] M. Hepting, D. Li, C. J. Jia, H. Lu, E. Paris, Y. Tseng, X. Feng, M. Osada, E. Been, Y. Hikita, Y. D. Chuang, Z. Hussain, K. J. Zhou, A. Nag, M. Garcia-Fernandez, M. Rossi, H. Y. Huang, D. J. Huang, Z. X. Shen, T. Schmitt, H. Y. Hwang, B. Moritz, J. Zaanen, T. P. Devereaux, and W. S. Lee, Electronic structure of the parent compound of superconducting infinite-layer nickelates, Nature Materials 19, 381 (2020).
  * Gu  _et al._ [2020] Y. Gu, S. Zhu, X. Wang, J. Hu, and H. Chen, A substantial hybridization between correlated Ni-d orbital and itinerant electrons in infinite-layer nickelates, Communications Physics 3, 84 (2020).
  * Chang  _et al._ [2020] J. Chang, J. Zhao, and Y. Ding, Hund-heisenberg model in superconducting infinite-layer nickelates, The European Physical Journal B 93, 220 (2020).
  * Kitatani  _et al._ [2020] M. Kitatani, L. Si, O. Janson, R. Arita, Z. Zhong, and K. Held, Nickelate superconductors-a renaissance of the one-band hubbard model, npj Quantum Materials 5, 59 (2020).
  * Krieger  _et al._ [2022] G. Krieger, L. Martinelli, S. Zeng, L. E. Chow, K. Kummer, R. Arpaia, M. Moretti Sala, N. B. Brookes, A. Ariando, N. Viart, M. Salluzzo, G. Ghiringhelli, and D. Preziosi, Charge and spin order dichotomy in ${\mathrm{NdNiO}}_{2}$ driven by the capping layer, Phys. Rev. Lett. 129, 027002 (2022).
  * Wu  _et al._ [2020] X. Wu, D. Di Sante, T. Schwemmer, W. Hanke, H. Y. Hwang, S. Raghu, and R. Thomale, Robust ${d}_{{x}^{2}-{y}^{2}}$-wave superconductivity of infinite-layer nickelates, Phys. Rev. B 101, 060504 (2020).
  * Luo  _et al._ [2023c] Z. Luo, D.-X. Yao, and W. Wú, Superconductivity in the two-orbital hubbard model of infinite-layer nickelates (2023c), arXiv:2310.12250 [cond-mat.supr-con] .
  * Fan  _et al._ [2023] Z. Fan, J.-F. Zhang, B. Zhan, D. Lv, X.-Y. Jiang, B. Normand, and T. Xiang, Superconductivity in nickelate and cuprate superconductors with strong bilayer coupling (2023), arXiv:2312.17064 [cond-mat.supr-con] .
  * Shilenko and Leonov [2023] D. A. Shilenko and I. V. Leonov, Correlated electronic structure, orbital-selective behavior, and magnetic correlations in double-layer ${\mathrm{La}}_{3}{\mathrm{Ni}}_{2}{\mathrm{O}}_{7}$ under pressure, Phys. Rev. B 108, 125105 (2023).
  * Yang  _et al._ [2023a] Q.-G. Yang, D. Wang, and Q.-H. Wang, Possible ${s}_{\pm{}}$-wave superconductivity in ${\mathrm{La}}_{3}{\mathrm{Ni}}_{2}{\mathrm{O}}_{7}$, Phys. Rev. B 108, L140505 (2023a).
  * Gu  _et al._ [2023] Y. Gu, C. Le, Z. Yang, X. Wu, and J. Hu, Effective model and pairing tendency in bilayer ni-based superconductor La3Ni2O7 (2023), arXiv:2306.07275 [cond-mat.supr-con] .
  * Yang  _et al._ [2023b] J. Yang, H. Sun, X. Hu, Y. Xie, T. Miao, H. Luo, H. Chen, B. Liang, W. Zhu, G. Qu, C.-Q. Chen, M. Huo, Y. Huang, S. Zhang, F. Zhang, F. Yang, Z. Wang, Q. Peng, H. Mao, G. Liu, Z. Xu, T. Qian, D.-X. Yao, M. Wang, L. Zhao, and X. J. Zhou, Orbital-dependent electron correlation in double-layer nickelate La3Ni2O7 (2023b), arXiv:2309.01148 [cond-mat.supr-con] .
  * Li  _et al._ [2020] Q. Li, C. He, J. Si, X. Zhu, Y. Zhang, and H.-H. Wen, Absence of superconductivity in bulk Nd1-xSrxNiO2, Communications Materials 1, 16 (2020).
  * Li  _et al._ [2023] J. Li, C.-Q. Chen, C. Huang, Y. Han, M. Huo, X. Huang, P. Ma, Z. Qiu, J. Chen, X. Hu, L. Chen, T. Xie, B. Shen, H. Sun, D.-X. Yao, and M. Wang, Structural transition, electric transport, and electronic structures in the compressed trilayer nickelate La4Ni3O10 (2023), arXiv:2311.16763v3 [cond-mat.supr-con] .
  * Wang  _et al._ [2023b] L. Wang, Y. Li, S. Xie, F. Liu, H. Sun, C. Huang, Y. Gao, T. Nakagawa, B. Fu, B. Dong, Z. Cao, R. Yu, S. I. Kawaguchi, H. Kadobayashi, M. Wang, C. Jin, H. kwang Mao, and H. Liu, Structure responsible for the superconducting state in La3Ni2O7 at high pressure and low temperature conditions (2023b), arXiv:2311.09186 [cond-mat.supr-con] .
  * Geisler  _et al._ [2023] B. Geisler, J. J. Hamlin, G. R. Stewart, R. G. Hennig, and P. J. Hirschfeld, Structural transitions, octahedral rotations, and electronic properties of $a_{3}$ni2o7 rare-earth nickelates under high pressure (2023), arXiv:2309.15078 [cond-mat.supr-con] .
  * Jung  _et al._ [2022] M.-C. Jung, J. Kapeghian, C. Hanson, B. Pamuk, and A. S. Botana, Electronic structure of higher-order ruddlesden-popper nickelates, Phys. Rev. B 105, 085150 (2022).
  * Kresse and Furthmuller [1996a] G. Kresse and J. Furthmuller, Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set, Computational Materials Science 6, 15 (1996a).
  * Kresse and Furthmuller [1996b] G. Kresse and J. Furthmuller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Physical Review B 54, 11169 (1996b).
  * Blochl [1994] P. E. Blochl, Projector augmented-wave method, Physical Review B 50, 17953 (1994).
  * Kresse and Joubert [1999] G. Kresse and D. Joubert, From ultrasoft pseudopotentials to the projector augmented-wave method, Physical Review B 59, 1758 (1999).
  * Kohn and Sham [1965] W. Kohn and L. J. Sham, Self-consistent equations including exchange and correlation effects, Physical Review 140, A1133 (1965).
  * Mostofi  _et al._ [2008] A. A. Mostofi, J. R. Yates, Y.-S. Lee, I. Souza, D. Vanderbilt, and N. Marzari, wannier90: A tool for obtaining maximally-localised wannier functions, Computer Physics Communications 178, 685 (2008).
  * Li  _et al._ [2017] H. Li, X. Zhou, T. Nummy, J. Zhang, V. Pardo, W. E. Pickett, J. F. Mitchell, and D. S. Dessau, Fermiology and electron dynamics of trilayer nickelate La4Ni3O10, Nature Communications 8, 704 (2017).
  * Christiansson  _et al._ [2023] V. Christiansson, F. Petocchi, and P. Werner, Correlated electronic structure of ${\mathrm{La}}_{3}{\text{Ni}}_{2}{\mathrm{O}}_{7}$ under pressure, Phys. Rev. Lett. 131, 206501 (2023).
  * Sakakibara  _et al._ [2023] H. Sakakibara, M. Ochi, H. Nagata, Y. Ueki, H. Sakurai, R. Matsumoto, K. Terashima, K. Hirose, H. Ohta, M. Kato, Y. Takano, and K. Kuroki, Theoretical analysis on the possibility of superconductivity in a trilayer ruddlesden-popper nickelate La4Ni3O10 under pressure and its experimental examination: comparison with La3Ni2O7 (2023), arXiv:2309.09462 [cond-mat.supr-con] .
  * Kamihara  _et al._ [2006] Y. Kamihara, H. Hiramatsu, M. Hirano, R. Kawamura, H. Yanagi, T. Kamiya, and H. Hosono, Iron-based layered superconductor: LaOFeP, Journal of the American Chemical Society 128, 10012 (2006).
  * Lebègue [2007] S. Lebègue, Electronic structure and properties of the fermi surface of the superconductor LaOFeP, Phys. Rev. B 75, 035110 (2007).
  * Leonov [2024] I. V. Leonov, Electronic structure and magnetic correlations in trilayer nickelate superconductor La4Ni3O10 under pressure (2024), arXiv:2401.07350 [cond-mat.str-el] .
  * Zhang  _et al._ [2020b] J. Zhang, D. Phelan, A. S. Botana, Y. S. Chen, H. Zheng, M. Krogstad, S. G. Wang, Y. Qiu, J. A. Rodriguez-Rivera, R. Osborn, S. Rosenkranz, M. R. Norman, and J. F. Mitchell, Intertwined density waves in a metallic nickelate, Nat Commun 11, 6003 (2020b).
  * Norman  _et al._ [2023] M. R. Norman, A. S. Botana, J. Karp, A. Hampel, H. LaBollita, A. J. Millis, G. Fabbris, Y. Shen, and M. P. M. Dean, Orbital polarization, charge transfer, and fluorescence in reduced-valence nickelates, Phys. Rev. B 107, 165124 (2023).
  * Georges  _et al._ [2013] A. Georges, L. d. Medici, and J. Mravlje, Strong correlations from hund’s coupling, Annual Review of Condensed Matter Physics 4, 137 (2013).
  * Sun  _et al._ [2021] W. Sun, Y. Li, X. Cai, J. Yang, W. Guo, Z. Gu, Y. Zhu, and Y. Nie, Electronic and transport properties in ruddlesden-popper neodymium nickelates ${\mathrm{Nd}}_{n+1}{\mathrm{Ni}}_{n}{\mathrm{O}}_{3n+1}$ ($n=1-5$), Phys. Rev. B 104, 184518 (2021).
  * Chen  _et al._ [2024] X. Chen, J. Choi, Z. Jiang, J. Mei, K. Jiang, J. Li, S. Agrestini, M. Garcia-Fernandez, X. Huang, H. Sun, D. Shen, M. Wang, J. Hu, Y. Lu, K.-J. Zhou, and D. Feng, Electronic and magnetic excitations in La3Ni2O7 (2024), arXiv:2401.12657 [cond-mat.supr-con] .
  * Zaanen  _et al._ [1985] J. Zaanen, G. A. Sawatzky, and J. W. Allen, Band gaps and electronic structure of transition-metal compounds, Phys. Rev. Lett. 55, 418 (1985).
  * Kowalski  _et al._ [2021] N. Kowalski, S. S. Dash, P. Semon, D. Senechal, and A. M. Tremblay, Oxygen hole content, charge-transfer gap, covalency, and cuprate superconductivity, Proc Natl Acad Sci U S A 118, 10.1073/pnas.2106476118 (2021).
  * Been  _et al._ [2021] E. Been, W.-S. Lee, H. Y. Hwang, Y. Cui, J. Zaanen, T. Devereaux, B. Moritz, and C. Jia, Electronic structure trends across the rare-earth series in superconducting infinite-layer nickelates, Phys. Rev. X 11, 011050 (2021).
  * Lu  _et al._ [2023] C. Lu, Z. Pan, F. Yang, and C. Wu, Interplay of two $e_{g}$ orbitals in superconducting La3Ni2O7 under pressure (2023), arXiv:2310.02915 [cond-mat.supr-con] .
  * Lawrence  _et al._ [1981] J. M. Lawrence, P. S. Riseborough, and R. D. Parks, Valence fluctuation phenomena, Reports on Progress in Physics 44, 1 (1981).
  * Tian  _et al._ [2023] Y.-H. Tian, Y. Chen, J.-M. Wang, R.-Q. He, and Z.-Y. Lu, Correlation effects and concomitant two-orbital $s_{\pm}$-wave superconductivity in La3Ni2O7 under high pressure (2023), arXiv:2308.09698 [cond-mat.supr-con] .
  * Zhang  _et al._ [2023b] Y. Zhang, L.-F. Lin, A. Moreo, and E. Dagotto, Electronic structure, dimer physics, orbital-selective behavior, and magnetic tendencies in the bilayer nickelate superconductor ${\mathrm{La}}_{3}{\mathrm{Ni}}_{2}{\mathrm{O}}_{7}$ under pressure, Phys. Rev. B 108, L180510 (2023b).
  * Yang  _et al._ [2023c] Y.-f. Yang, G.-M. Zhang, and F.-C. Zhang, Interlayer valence bonds and two-component theory for high-${T}_{c}$ superconductivity of ${\mathrm{La}}_{3}{\mathrm{Ni}}_{2}{\mathrm{O}}_{7}$ under pressure, Phys. Rev. B 108, L201108 (2023c).
  * str [2023] High-temperature superconductivity with zero-resistance and strange metal behaviour in La3Ni2O7 (2023), arXiv:2307.14819 [cond-mat.str-el] .
  * Dagotto and Rice [1996] E. Dagotto and T. M. Rice, Surprises on the way from one- to two-dimensional quantum magnets: The ladder materials, Science 271, 618 (1996).
  * Cheng  _et al._ [2022] J.-Q. Cheng, J. Li, Z. Xiong, H.-Q. Wu, A. W. Sandvik, and D.-X. Yao, Fractional and composite excitations of antiferromagnetic quantum spin trimer chains, npj Quantum Materials 7, 3 (2022).