††thanks: Equal contributions††thanks: Equal contributions

# Electronic and magnetic excitations in La3Ni2O7

Xiaoyang Chen  State Key Laboratory of Surface Physics, Department of Physics,
and Advanced Materials Laboratory, Fudan University, Shanghai 200438, China
Jaewon Choi  Diamond Light Source, Harwell Campus, Didcot OX11 0DE, UK
Zhicheng Jiang  National Synchrotron Radiation Laboratory and School of
Nuclear Science and Technology, University of Science and Technology of China,
Hefei, 230026, China  Jiong Mei  Beijing National Laboratory for Condensed
Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing
100190, China  School of Physical Sciences, University of Chinese Academy of
Sciences, Beijing 100190, China  Kun Jiang  Beijing National Laboratory for
Condensed Matter Physics and Institute of Physics, Chinese Academy of
Sciences, Beijing 100190, China  School of Physical Sciences, University of
Chinese Academy of Sciences, Beijing 100190, China  Jie Li  National
Laboratory of Solid State Microstructures and Department of Physics, Nanjing
University, Nanjing 210093, China  Stefano Agrestini  Diamond Light Source,
Harwell Campus, Didcot OX11 0DE, UK  Mirian Garcia-Fernandez  Diamond Light
Source, Harwell Campus, Didcot OX11 0DE, UK  Xing Huang  Guangdong Provincial
Key Laboratory of Magnetoelectric Physics and Devices, School of Physics, Sun
Yat-Sen University, Guangzhou, Guangdong 510275, China  Hualei Sun  School of
Science, Sun Yat-Sen University, Shenzhen, Guangdong 518107, China  Dawei Shen
National Synchrotron Radiation Laboratory and School of Nuclear Science and
Technology, University of Science and Technology of China, Hefei, 230026,
China  Meng Wang  Guangdong Provincial Key Laboratory of Magnetoelectric
Physics and Devices, School of Physics, Sun Yat-Sen University, Guangzhou,
Guangdong 510275, China  Jiangping Hu  Beijing National Laboratory for
Condensed Matter Physics and Institute of Physics, Chinese Academy of
Sciences, Beijing 100190, China  New Cornerstone Science Laboratory, Beijing,
100190, China  Yi Lu<EMAIL_ADDRESS>National Laboratory of Solid State
Microstructures and Department of Physics, Nanjing University, Nanjing 210093,
China  Collaborative Innovation Center of Advanced Microstructures, Nanjing,
210093, China  Ke-Jin Zhou<EMAIL_ADDRESS>Diamond Light Source,
Harwell Campus, Didcot OX11 0DE, UK  Donglai Feng<EMAIL_ADDRESS>National
Synchrotron Radiation Laboratory and School of Nuclear Science and Technology,
University of Science and Technology of China, Hefei, 230026, China  New
Cornerstone Science Laboratory, University of Science and Technology of China,
Hefei, 230026, China  Collaborative Innovation Center of Advanced
Microstructures, Nanjing, 210093, China

The striking discovery of high-temperature superconductivity (HTSC) of 80 K in
a bilayer nickelate La3Ni2O7 under a moderately high pressure of about 14 GPa
ignited a new wave of studying HTSC in nickelates [1, 2, 3, 4, 5, 6, 7, 8].
The properties of the parental phase at ambient pressure may contain key
information on basic interactions therein and bosons that may mediate pairing
giving birth to superconductivity. Moreover, the bilayer structure of La3Ni2O7
may suggest a distinct minimal model in comparison to cuprate superconductors.
Here using X-ray absorption spectroscopy and resonant inelastic X-ray
scattering, we studied La3Ni2O7 at ambient pressure, and found that Ni
3$d_{x^{2}-y^{2}}$, Ni 3$d_{z^{2}}$, and ligand oxygen 2$p$ orbitals dominate
the low-energy physics with a small charge-transfer energy. Remarkably, well-
defined optical-like magnetic excitations were found to soften into a quasi-
static spin-density-wave ordering, evidencing the strong electronic
correlations and rich magnetic properties. Based on a Heisenberg spin model,
we found that the inter-layer effective magnetic superexchange interaction is
much larger than the intra-layer ones, and proposed two viable magnetic
structures. Our results set the foundation for further exploration of La3Ni2O7
superconductor.

Introduction

Unlike cuprate superconductors, often characterized by a single Zhang-Rice
singlet band consisting of Cu 3$d_{x^{2}-y^{2}}$ and O 2$p$ orbitals, multiple
$d$ orbitals and Ni-O bilayer structure play critical roles in La3Ni2O7 [1, 2,
9, 10, 11, 4, 12]. In particular, the molecular bonding between the two Ni
3$d_{z^{2}}$ orbitals through the apical O $p_{z}$ orbital, together with Ni
3$d_{x^{2}-y^{2}}$ orbital, is widely established by theory and deemed as an
essential ingredient for the low-energy electronic structure of La3Ni2O7 [13,
2, 12, 4, 14, 7, 15]. However, the exact orbital occupancy and orbital
character of La3Ni2O7 remains elusive. If La3Ni2O7 is viewed in close
proximity to cuprates, $i.e.$, at the limit of a strong electronic correlation
and a small charge-transfer energy, significant amount of electron holes would
occupy the oxygen ligands, giving rise to Zhang-Rice-like physics [16]. On the
other hand, supposing it was a sibling of infinite-layer nickelate
superconductors, where the charge-transfer energy is rather comparable to the
Coulomb repulsion, the participation of the oxygen ligands in the low-energy
electronic structure would be much reduced [17, 18, 19].

Figure 1:  XAS and the incident energy dependent RIXS maps in La3Ni2O7. a,
Schematic top view of the NiO2 plane in La3Ni2O7. The solid black square
represents the pseudo-tetragonal unit cell with a lattice constant $a^{T}$
$\sim$ 3.833 Å, while the dashed black square represents the real orthorhombic
in-plane unit cell when considering the tilting of Ni-O octahedra. b, In-plane
Brillouin zone (BZ) for the pseudo-tetragonal unit cell. c, Sketch of the RIXS
experimental geometry. Details of the setup are described in Method. d,e,
$\sigma$ polarised XAS spectra of La3Ni2O7 (red filled circles) taken at the O
$K$-edge (d) and Ni $L_{3}$-edge (e), respectively. The latter is displayed
after subtracting the background of La $M_{4}$-edge. The calculated Ni
$L_{3}$-XAS (red curve) is also displayed. XAS spectra measured on NiO (Ni2+)
and NdNiO3 (Ni3+) (black-filled circles) are shown as references. f,g, RIXS
intensity maps measured as a function of incident photon energy with
$\sigma$\- (f) and $\pi$-polarized photons. (g), respectively. The
corresponding XAS spectrum is superimposed as a solid white curve on each map.
Both XAS and RIXS spectra were collected at 20 K at a grazing-in incident
angle of 20∘. h, Integral of RIXS spectra in (f) and (g) over the incident
energy range [851.8 eV, 853.4 eV]. The grey solid bars display the multiplet
calculations for the Ni $L_{3}$ RIXS.

The bilayer structure and the multi-orbital nature of La3Ni2O7 have profound
impact on its magnetism as well, which plays a pivotal role in theories on
this novel superconductor, resembling the cuprate case [20, 1, 4, 21]. Some
suggest the importance of the interlayer antiferromagnetic coupling $J_{z}$
between $d_{z^{2}}$ orbitals [1, 4]; some others advocate that the strong
interlayer coupling would cause the bilayer splitting of band structure, while
in-plane magnetic exchange interactions play a dominant role in
superconductivity [11, 13]. The intricate magnetic exchange may as well lead
to other exotic phases with broken symmetries that have complex interplay with
superconductivity, as seen in the cuprate superconductors [22, 23]. In the as-
grown La3Ni2O7 crystal at ambient pressure, resistivity measurements have
found a kink-like transition at around 153 K, which responds to the external
out-of-plane magnetic field, implying a possible spin-density-wave (SDW)
therein [24]. A recent $\mu$SR experiment on polycrystalline La3Ni2O7
suggested that a static long-range magnetic order emerges below 148 K,
consistent with an SDW internal field distribution [25]. Moreover, traces of a
possible density wave have been discovered in a latest NMR report [26].
However, the exact nature of this density-wave state is still unclear.

Given the currently limited knowledge on the essential electronic and magnetic
energy scales, such as the charge-transfer gap and exchange interactions,
experimental verification is indispensable. In this work, we employ X-ray
absorption spectroscopy (XAS) and resonant inelastic X-ray scattering (RIXS)
at both Ni $L_{3}$-edge and O $K$-edge of La3Ni2O7 single crystal at ambient
pressure. These spectroscopic and scattering techniques are sensitive to low-
energy electronic and magnetic structures together with elementary
excitations, and thus they are ideally suited for tackling the core issues in
La3Ni2O7.

Electronic structure of La3Ni2O7

As-grown La3Ni2O7 crystallizes in an orthorhombic structure with the space
group of $Amam$ [1]. We define the reciprocal space index $(H,K,L)$ based on
the pseudo-tetragonal unit cell (Figs. 1a and 1b, Method). Figure 1c shows the
experimental geometry, in which the incident X-ray is linearly polarised,
while the scattered X-ray is typically non-polarised but otherwise polarised
if stated explicitly (see Method).

Figures 1d and 1e illustrate XAS spectra of La3Ni2O7 taken near the O $K$-edge
and Ni $L_{3}$-edge, respectively. A sizable pre-edge peak at $\sim$ 528.5 eV
is observed near the O $K$-edge, originating from oxygen 1$s$ electron
excitations into the unoccupied oxygen 2$p$ ligand hole state near the Fermi
level, as observed for the Zhang-Rice singlet state in cuprate superconductors
[27]. The Ni $L_{3}$-XAS data show a sharp resonant peak around 852.4 eV,
followed by a broader satellite peak at a higher energy. As the Ni valence
2.5+ of La3Ni2O7 falls in between the archetypal nickelates NiO and NdNiO3,
the XAS spectra of La3Ni2O7 can be qualitatively understood in relation to
these two. NiO resides in the charge-transfer regime of the Zaanen-Sawatzky-
Allen scheme, whose large charge-transfer energy $\Delta$ ($\approx$ 5 eV)
suppresses the charge fluctuations between the Ni 3$d$ and ligand oxygen 2$p$
orbitals despite their large orbital hopping integral [28]. Consequently, its
ground state is well described by
$\alpha|3d^{8}\rangle+\beta|3d^{9}\underline{L}\rangle$
($\alpha^{2}+\beta^{2}\lesssim 1$ and $\underline{L}$ denotes a ligand hole)
with a dominant 3$d^{8}$ character ($\alpha^{2}\approx$ 0.8) [29, 30, 31, 32].
On the other hand, the perovskite N d NiO3 with a nominal 3$d^{7}$
configuration is widely acknowledged as a negative charge-transfer system,
where electrons from ligand oxygen spontaneously transfer onto Ni cations,
resulting in a ground state with a leading 3$d^{8}\underline{L}$ contribution
[33]. Such a substantial ligand hole concentration is underscored by the
pronounced pre-edge hole peak in the O $K$-edge XAS of NdNiO3, similar to that
of La3Ni2O7 (Fig. 1d). This is distinct from NiO, where the pre-peak is
absent, and the unoccupied ligand states are at an elevated energy across the
charge-transfer gap. For the Ni $L_{3}$-XAS, the prominent resonant peak of
La3Ni2O7 is also observed for NiO and NdNiO3 at a similar energy (Fig. 1e),
which was previously identified as the Ni 2$p$ $\rightarrow$ 3$d^{8}$ or
3$d^{8}$ \+ 3$d^{8}\underline{L}$ $e_{g}$ transitions, respectively [34, 33].
A broader satellite peak at a higher energy is likewise seen for NdNiO3,
originating mainly from a part of its ground state wavefunction that contains
additional ligand holes [35, 33, 36]. The above comparison between the
spectral features of La3Ni2O7 and NiO/NdNiO3 suggests a ground state with
primarily 3$d^{8}$ occupancy on the Ni cation, accompanied by non-negligible
ligand holes for the former.

Figures 1f and 1g show the incident-energy dependent RIXS measurements of
La3Ni2O7 across the Ni $L_{3}$-edge. A clear low-energy excitation ($\sim$ 70
meV) is observed near the elastic peak which will be discussed in the next
section. The sharp XAS resonance at $\sim$ 852.4 eV decays mainly to a final
state of a localized excitation at around 1 eV, known as the
$t_{2g}\rightarrow e_{g}$ $dd$ orbital excitation similar to NiO and NdNiO3
[37, 38, 35]. The band-like fluorescence excitation, decaying from the broad
satellite XAS peak, stems from the delocalized Ni-O hybridized continuum
states [35, 36]. The intensity distribution of the fluorescence is more
confined with $\pi$ polarization that couples stronger to the 3$d_{z^{2}}$
orbital, reflecting the more restricted electron kinetics perpendicular to the
bilayer structure. It is noteworthy that, distinct from NdNiO3, two extra $dd$
excitations show up in La3Ni2O7 (at around 0.4 eV and 1.6 eV) and possess
stronger intensities under $\pi$ polarization, suggestive of an enhancement
along the 3$d_{z^{2}}$ orbital direction.

To gain a quantitative understanding of XAS and RIXS measurements, we built a
double-cluster model capturing the bilayer structure of La3Ni2O7 and then
carried out multiplet calculations for Ni $L_{3}$\- XAS and RIXS spectra (see
details in Section 2 of Supplementary Information). Systematic optimizations
of the calculated spectra suggest that the charge-transfer energy $\Delta$
falls between 0 and 2 eV, pointing out the rather small-charge-transfer nature
of La3Ni2O7 [39]. This result is reasonable since $\Delta$ is $\sim$ 5 eV and
$\sim$ 0 for NiO and NdNiO3, respectively [40, 36]. With the estimated range
of $\Delta$, the ground state wavefunction of La3Ni2O7 can be deduced to
approximately
$\alpha|3d^{8}\rangle+\beta|3d^{8}\underline{L}\rangle+\gamma|3d^{7}\rangle$
with leading $\alpha^{2}$ and $\beta^{2}$. The calculated XAS for $\Delta=$
0.5 eV is shown in Fig. 1e, which corresponds to a ground state with
$(\alpha^{2},\beta^{2},\gamma^{2})$ $\approx$ (0.4, 0.3, 0.2). The
corresponding RIXS calculation agrees well with the experiment, showing $dd$
excitations identified at comparable energies (Fig. 1h). Notably, we found
that both the XAS line shape and the lower $dd$ excitation ($\sim$ 0.4 eV) in
RIXS show marked difference upon tuning the inter-layer hopping strength
mediated by the 3$d_{z^{2}}$ -$O_{AP}$ 2$p_{z}$ \- 3$d_{z^{2}}$ orbital
overlap in the calculation ($O_{AP}$ stands for the apical oxygen),
underlining the importance of the inter-layer coupling for the electronic
structure (Section 2 of Supplementary Information). This result is consistent
with previous experimental report [1], and lends support to several recent
theoretical works emphasizing on the importance of the bilayer structure [41,
12, 42, 43, 44, 4, 13, 2, 12, 4, 14].

Figure 2:  Energy-momentum dependent magnon in La3Ni2O7. a, RIXS intensity
maps along high-symmetry directions as indicated in the insets. Data were
collected at 20 K using 852.4 eV, $\sigma$-polarised X-ray at the Ni
$L_{3}$-edge. The red circles depict the peak positions of magnetic
excitations here and throughout all panels of this figure. b, RIXS spectra at
representative projected in-plane momentum transfers. The weaker excitations
at $\sim$120 meV may result from the multi-magnons. c, $L$ scan of RIXS
spectra at at $q_{\parallel}$ = (0.035, 0.035). d, Polarimetric RIXS data at
$q$ = (0.035, 0.035, 2.727). The spectra are decomposed into
$\pi-\pi^{\prime}$, $\pi-\sigma^{\prime}$, $\sigma-\sigma^{\prime}$ and
$\sigma-\pi^{\prime}$ components.

Magnetic excitations

Figure 2 summarises the detailed energy-momentum dependence of low-energy
excitations in La3Ni2O7 taken at the incident energy of 852.4 eV corresponding
to the resonance of Ni $L_{3}$-XAS. Figures 2a and 2b show strongly dispersive
excitations along directions illustrated in insets. The excitations reach
maximal energy of about 70 meV at (0, 0) and (0.5, 0) while soften to zero
energy (within the experimental energy resolution) at (0.25, 0.25), suggesting
the presence of a quasi-static ordering. Similar excitations also appear when
excited by $\pi$ incident X-rays polarisation (Fig. S6). Along the out-of-
plane direction, this mode does not exhibit sizable dispersion as a function
of $L$, indicating its quasi-two-dimensional nature (Fig. 2c).

As both magnon and phonon excitations could appear in RIXS spectra,
particularly within 100 meV that is closely relevant to both, the polarimetric
RIXS was employed to analyze the outgoing X-rays linear polarisation for
unraveling the origin of these excitations (see Methods). Clearly, as shown in
Fig. 2d, the inelastic excitation is present under the $\pi-\pi^{\prime}$,
$\pi-\sigma^{\prime}$, and $\sigma-\pi^{\prime}$ channels, while gets much
reduced under the $\sigma-\sigma^{\prime}$ channel. Such behaviour is in
excellent agreement with the assumption of a magnetic origin of the scattering
[45]. Our multiplet RIXS calculation of magnetic excitations in the double-
cluster model confirmed the outgoing polarisation dependence (Fig. S5).
Concerning phonons, in principle, their spectra weight should be present in
the $\sigma-\sigma^{\prime}$ channel. However, the corresponding polarimetric
RIXS spectrum shows negligible spectral weight hence a minute contribution to
the Ni $L_{3}$-RIXS (Fig. 2d). We therefore conclude that the low-energy
excitations observed at the Ni $L_{3}$-edge are dominated by magnons. This
immediately infers a quasi-static SDW ordering at (0.25, 0.25). Remarkably, an
SDW ordering was reported to exist near (0.25, 0.25) in the half-doped
nickelate La3/2Sr1/2NiO4 which has the same hole-doping level, $i.e.$, a
nominal Ni2.5+ valence state, as that of La3Ni2O7 [46]. In both cases, the
magnon softens to almost zero energy near the SDW ordering wavevector, while
their dispersions approaching $\Gamma$ point deviate drastically: there is an
acoustic-like magnon in La3/2Sr1/2NiO4, whereas it is absent here in Fig. 2a.

By fitting the magnon spectra to a damped harmonic oscillator (DHO) function
$\chi^{{}^{\prime\prime}}(q,\omega)$, we extracted the peak energy and width
of the magnon (Section 4 of Supplementary Information) [47]. Three possible
spin configurations consistent with the spin order at $Q=(0.25,0.25)$ can be
constructed: the diagonal spin-charge stripe order as in half-doped
La3/2Sr1/2NiO4 where Ni2+ spin and nominal Ni3+ charge stripes intertwined
(Stripe-1, Fig. 3a) [46]; the SDW order could also be realised without the
charge inhomogeneity, $i.e.$, a double-spin stripe order (Stripe-2, Fig. 3b);
by exchanging the charge stripe positions, a third spin configuration could be
achieved as a double spin-charge stripe order (Stripe-3 in Fig. S10c. For all
these SDW orders, owing to the strong bilayer bonding, spins are
antiferromagnetically aligned in the top and bottom NiO2 layers. To obtain the
magnetic superexchange interaction parameters, we constructed an effective
$J_{1}$-$J_{2}$-$J_{z}$ Heisenberg model:
$H=\sum_{i}J_{z}\vec{S}_{i}^{t}\cdot\vec{S}_{i}^{b}+\sum_{\langle
ij\rangle\alpha}J_{1}\vec{S}_{i}^{\alpha}\cdot\vec{S}_{j}^{\alpha}+\sum_{\langle\langle
ij\rangle\rangle\alpha}J_{2}\vec{S}_{i}^{\alpha}\cdot\vec{S}_{j}^{\alpha}$,
where $\alpha$ is the layer index for the bottom (b) or top (t) layer. $J_{z}$
is the inter-layer exchange coupling along the $c$-axis. $J_{1}$ and $J_{2}$
are the nearest-neighbor and next-nearest-neighbor exchange couplings,
respectively, in a single Ni-O layer. The magnon dispersion within the linear
spin wave theory was solved using the torque equation formalism [48] (Section
6 of Supplementary Information). We found that the magnon dispersion as well
as its spectral weight distribution based on both Stripe-1 and Stripe-2 spin
configurations agree with our RIXS result (Fig. 3). Owing to the scattering
matrix effect, the simulated acoustic magnon spectra are significantly weaker
than the optical magnon, consistent with the experimental findings. In
general, the inter-layer effective superexchange interaction is an order of
magnitude larger than that of the intra-layer. The finding of a dominant
magnetic interaction along the molecular bonding direction is in good
accordance with previous theoretical calculation [2]. Interestingly, the
intra-layer effective superexchange interaction between next-nearest-neighbour
spins on Ni2+ sites shows comparable strength to that in the half-doped
La3/2Sr1/2NiO4 [46]. Based on the above results and the currently limited
information, we can conjecture the true spin configuration of La3Ni2O7 is
either Stripe-1 or Stripes-2, with the double spin stripe order just slightly
lower in energy than the spin-charge stripe according to our calculations (see
details in Section 5 of Supplementary Information).

Figure 3:  The spin configuration and the magnon dispersion of La3Ni2O7 a, The
spin configurations for the spin-charge stripe order (Stripe-1). To simplify
the sketch only nickel cations are shown. The blue, red and black circles
represent spin up Ni2+, spin down Ni2+, and the nominal Ni3+ sites,
respectively. The solid lines illustrate the in-plane pseudo-tetragonal unit
cells and the grey cubics represent the Ni-O octahedra. The fitted values of
$J_{1}S$, $J_{2}S$, and $J_{z}S$ based on this spin configuration are noted
(see details in Section 6 of Supplementary Information). b, The spin
configuration for the double spin stripe (Stripe-2), and the fitted value of
$J_{1}S$, $J_{2}S$, and $J_{z}S$. c, The experimental magnon dispersion
$\epsilon_{q}$ (red filled circles) and damping factor $\gamma_{q}$ (black
open circles) versus projected in-plane momentum transfer $q_{\parallel}$
along high-symmetry directions at 20 K. See fitting details in Section 4 of
Supplementary Information. Error bars of $\epsilon_{q}$ were estimated by
combining the uncertainty of the elastic peak position, linear background, and
the standard deviation of the fits. Error bars of $\gamma_{q}$ were estimated
by combining the standard deviation of the fits. The horizontal dashed line
marks the total energy resolution (36 meV). Fittings for an effective
$J_{1}$-$J_{2}$-$J_{z}$ Heisenberg model based on the Stripe-1 order are
overlaid. The blue curves represent the dispersion of two magnon modes, where
the thickness of the lines and the depth of their color represent the mode
intensity. The detailed parameters are listed in Section 6 of Supplementary
Information.

Spin-density-wave order

We then took an explicit examination on the SDW order. Polarimetric RIXS was
used to confirm the magnetic origin of low-energy excitations, likewise, it
was applied to characterize this SDW order in La3Ni2O7. Similar to the
behaviour of magnons, the momentum-dependent quasi-elastic SDW scattering peak
shows the same trend, $i.e.$, sizable scattering intensities under
$\pi-\pi^{\prime}$, $\pi-\sigma^{\prime}$, and $\sigma-\pi^{\prime}$ except
for $\sigma-\sigma^{\prime}$ (Figs. 4a and 4b), confirming the magnetic origin
of such SDW order. Further insight into the nature of the SDW was gained
through the energy dependence of the SDW scattering at its order wavevector
across the Ni $L_{3}$-edge (Fig. 4c). Unlike the XAS spectra where La $M_{4}$
shows a greater absorption intensity than that of Ni $L_{3}$, the SDW
scattering predominantly results from the Ni 3$d$ \- O 2$p$ hybridised states.
Furthermore, the SDW scattering peak exhibits a colossal polarisation
dependence, namely, its intensity probed under $\pi$ polarisation is about 30
times higher than that with $\sigma$ polarisation. Figure 4d gives an example
taken with 852.4 eV photons, which may indicate its strong association with Ni
3$d_{z^{2}}$ orbital. The half-width at half-maximum $\Gamma$ =
0.0022$\pm$0.0002 r.l.u. of the scattering peak corresponds to a relatively
short in-plane correlation length ($\xi_{H}$ = 1/$\Gamma$) of $\sim$27.7 nm. A
much broader peak is observed as a function of $L$ along the direction of
(0.25, 0.25, $L$) establishing the quasi-two-dimensional nature of such SDW
order (Fig. 4e).

The temperature dependence of the SDW order illustrates a substantial
reduction in both the intensity and the correlation length when the
temperature is raised above $\sim$150 K (Figs. 4f-4h). While the SDW
wavevector does not exhibit a discernible temperature dependence (Fig. 4i).
The discovery of the SDW with a characteristic temperature of around 150 K
agrees well with previous transport, NMR and $\mu$SR measurements on La3Ni2O7
[24, 26, 25].

Figure 4:  SDW order at (0.25, 0.25) of La3Ni2O7.  a,b Polarimetric RIXS data.
The spectra are decomposed into $\pi\pi^{\prime}$, $\pi\sigma^{\prime}$,
$\sigma\sigma^{\prime}$ and $\sigma\pi^{\prime}$ components. c, SDW peaks
intensities as a function of incident photon energy and polarization. The
inset shows the XAS spectra at the La $M_{4}$-edge and the Ni $L_{3}$-edge. d,
SDW peak intensity integrated over an energy window of 36.5 meV (the total
energy resolution) as a function of projected momentum transfer
($q_{\parallel}$) along the ($H$, $H$) direction. The fitted peak center
values are $\sim$ 0.25 r.l.u. and the corresponding half width at the half
maximum (HWHM) is 0.0022 r.l.u. e, $L$ scan of RIXS spectra at $q_{\parallel}$
= (0.25,0.25). f, SDW peaks and their Lorentzian fits along the ($H$, $H$)
direction at various temperatures. g-i, Temperature dependence of the SDW peak
area (g), the correlation length (h) and the SDW wave vector position (i).

Discussion and Conclusion

Our RIXS and XAS measurements revealed the dispersive magnon and SDW order
below 150 K in La3Ni2O7. Detailed analysis suggests that Ni
3$d_{x^{2}-y^{2}}$, Ni 3$d_{z^{2}}$, and O 2$p$ orbitals dominate the low-
energy physics with charge-transfer energy less than 2 eV, and the inter-layer
effective magnetic superexchange interaction is much larger than the intra-
layer ones. These give critical information for constructing the minimal model
for La3Ni2O7 superconductor.

Apart from the extraordinary bilayer structure and the associated predominant
magnetic exchange interaction, the electronic structure of La3Ni2O7 fits in
general into the family of Ruddlesden-Popper (RP) nickelates. The formation of
the Zhang-Rice-like hole band, the small charge-transfer energy, and the well-
defined dispersive magnon allude to its nature of the strong electronic
correlations [49]. The above are typical characteristics of the strongly
correlated cuprates where charge- and spin-density modulation can take place.
Moreover, the occurrence of SDW order at (0.25, 0.25) is reminiscent of that
in the half-doped single-layer La3/2Sr1/2NiO4, where a spin-charge stripe
order exists, and implies the tendency to a charge-density-wave instability in
La3Ni2O7 as illustrated in the scenario of Stripe-1 (Fig.3a) [46, 39]. Indeed
in layered half-doped RP nickelates, manganites, and cobaltates, the spin-
charge intertwined order is prevailing [46, 50, 51]. On the other hand, the
double spin stripe order accommodating homogeneous charge-density (Stripe-2,
Fig. 3b) may be possible too as the 3$d_{x^{2}-y^{2}}$ orbitals are more
itinerant in-plane than the 3$d_{z^{2}}$ orbitals. This is similar to the
collinear double spin stripe state in the correlated FeTe [52]. Verifying the
existence of the charge-density-wave order could unequivocally clarify the
spin configuration of La3Ni2O7.

Finally, we would like to extrapolate our findings to superconducting La3Ni2O7
under a moderately high pressure: here, a structural phase transition
accompanied by a few percent shrinkage of the lattice constants, and the Ni-O-
Ni bonding angles between adjacent NiO6 octahedra straighten to 180∘ [1].
Consequently, the electronic hopping is likely to increase, potentially
suppressing density waves that compete with the superconductivity [53, 54].
Furthermore, the magnetic superexchange $J_{z}$ may get significantly enlarged
due to the increased hopping along Ni-OAP-Ni, highlighting the vital role of
the inter-layer AFM interaction in the HTSC of such a bilayer nickelate.

###### Acknowledgements.

This work was supported by National Natural Science Foundation of China
(Nos.11888101, U2032208, 12274207, 12174428, and 12174454), the New
Cornerstone Science Foundation, and the National Key R&D Program of China
(Nos. 2023YFA1406304, 2022YFA1403000, 2023YFA1406500, and 2023YFA1406002). We
acknowledge Diamond Light Source for providing beamtime at I21 Beamline under
Proposal MM35805 and the science commissioning beamtime for using the
polarimeter at I21. Work at SYSU was as well supported by the Guangdong Basic
and Applied Basic Research Funds (No. 2021B1515120015), Guangzhou Basic and
Applied Basic Research Funds (Nos. 202201011123, 2024A04J6417), and Guangdong
Provincial Key Laboratory of Magnetoelectric Physics and Devices (No.
2022B1212010008).

## References

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Methods
Sample fabrication: La3Ni2O7 sample was fabricated by the high oxygen pressure
floating zone technique and the details are described in [24]. The sample
quality was checked by X-ray diffraction (XRD) and Laue diffraction (see
details in Fig. S1). Samples were cleaved to get a flat, clean surface before
RIXS measurements.

XAS and RIXS measurements: XAS and RIXS measurements were performed at
Beamline I21 at Diamond Light Source [55]. In this work, we describe the
structural properties of La3Ni2O7 referencing to a pseudo-tetragonal unit cell
with cell parameters $a^{T}$ = $b^{T}$ $\sim$ 3.833 Å and $c$ = 20.45 Å.
Reciprocal lattice units (r.l.u.) are defined (where
$2\pi/a^{T}=2\pi/b^{T}=2\pi/c=1$) with
$\mathbf{Q}=H\mathbf{a}^{T*}+K\textbf{b}^{T*}+L\textbf{c}^{*}$. The
crystallographic $a^{T}$–$c$ ($b^{T}$–$c$) plane of La3Ni2O7 single crystal
was aligned within the horizontal scattering plane (Fig. 1c). The polar
angular offsets ($\theta$ and $\chi$) of the crystal were aligned by the (002)
diffraction peak, and the azimuthal offset ($\phi$) by SDW order peak, such
that the $c^{*}$ axis lays in the scattering plane. The spectrometer arm was
at a fixed position of $\Omega=154^{\circ}$ except for $L$ scans where
variable $\Omega$ was employed.

XAS spectra were collected with a grazing incidence angle of
$\theta_{0}=20^{\circ}$ to probe both in-plane and out-of-plane unoccupied
states. All XAS measurements were done at a temperature of 20 K with the exit
slit opening to 30 $\mu$m. Total electron yield XAS spectra were collected
using the draincurrent and normalised to the incoming beam intensity. Both
linear vertical ($\sigma$) and horizontal ($\pi$) polarisations were used.

Energy-dependent RIXS measurements were performed at the grazing incidence
angle of $\theta_{0}=20^{\circ}$ and the temperature of 20 K. The exit slit
was open to 30 $\mu$m corresponding to an average energy resolution of 40 meV
(FWHM). The incident energy range went from 851 to 855 eV in steps of 0.2 eV
to fully capture the resonance behaviour across the Ni-$L_{3}$ absorption
peaks.

Momentum-dependent RIXS measurements were performed at the resonant energy of
852.4 eV at a temperature of 20 K with the exit slit opening to 20 $\mu$m
corresponding to an average energy resolution of 36 meV (FWHM). RIXS spectra
were collected using both $\sigma$ and $\pi$ polarisations. The grazing out
geometry ($\theta>\Omega/2$) was applied for the acquisition of RIXS spectra
shown in the main text.

Polarimetric RIXS apparatus employs a graded multilayer designed for the Ni
$L_{3}$-edge with a grazing incidence angle of 20∘ lying perpendicular to the
scattering plane. Measurements were performed at Q = (0.035, 0.035, $L$) and
around (0.25, 0.25, $L$) to analyse the outgoing X-rays linear polarisation of
the magnon and SDW ordering, respectively. The total energy resolution of the
polarimetric RIXS is $\sim$ 55 meV (FWHM). Since the multilayer does not work
at the exact Brewster’s angle, the outgoing polarised RIXS (the indirect RIXS)
from the reflection of the multilayer will be a mixture of linearly polarised
spectra. The direct and indirect RIXS spectral intensities are then given by
the following formula:

$I_{direct}=I_{\sigma^{\prime}}+I_{\pi^{\prime}}$ (1)
$I_{indirect}=R_{\sigma^{\prime}}I_{\sigma^{\prime}}+R_{\pi^{\prime}}I_{\pi^{\prime}}$
(2)

where $I_{direct}$ and $I_{indirect}$ stands for the outgoing nonpolarised and
mixed polarised RIXS spectral intensity, respectively. From the above formula,
the outgoing $\sigma^{\prime}$ and $\pi^{\prime}$ polarised RIXS spectra can
be deduced:

$I_{\pi^{\prime}}=\frac{I_{indirect}-R_{\sigma^{\prime}}I_{direct}}{R_{\pi^{\prime}}-R_{\sigma^{\prime}}}$
(3)
$I_{\sigma^{\prime}}=\frac{I_{indirect}-R_{\pi^{\prime}}I_{direct}}{R_{\sigma^{\prime}}-R_{\pi^{\prime}}}$
(4)

In the above, R${}_{\sigma^{\prime}}$ (R${}_{\pi^{\prime}}$) refers to the
multilayer reflectivity of the outgoing $\sigma^{\prime}$ ($\pi^{\prime}$)
polarised X-ray photon. At the Ni $L_{3}$-edge, R${}_{\sigma^{\prime}}$ and
R${}_{\pi^{\prime}}$ is 14.1% and 9.1%, respectively, based on the calibration
of the multilayer.

Theoretical calculations: The Ni $L_{3}$-edge XAS and RIXS calculations shown
in Figure 1 were performed employing a fully correlated Ni2O11 cluster model,
accounting for the two corner-sharing NiO6 octahedra within the pseudo-
tetragonal unit cell. The noninteracting part of the Hamiltonian integrates
material-specific on-site energies and hybridizations involving Ni 3$d$ and O
2$p$ orbitals, along with spin-orbit coupling within the Ni core 2$p$ and 3$d$
shells. Full Coulomb interactions within the Ni 3$d$ shell and between the Ni
2$p$ and 3$d$ shells are included, with parametrization by Slater integrals
scaled at 0.8 based on atomic Hartree-Fock values [56]. Comprehensive details
regarding model construction and relevant parameters are described in Section
2 of Supplementary Information. The model was solved using the exact
diagonalization method as implemented in Quanty [57].

The DFT calculations employ the Vienna ab-initio simulation package (VASP)
code [58] with the projector augmented wave (PAW) method [59]. The Perdew-
Burke-Ernzerhof (PBE) exchange-correlation functional [60] is used. The energy
cutoff energy for expanding the wave functions into a plane-wave basis is set
to be 500 eV. The $\Gamma$-centered k-mesh is used in KPOINTS files which are
generated by VASPKIT [61] with the KPT-resolved value equal to 0.02 for
different unit cells. The SDW orders are calculated using the simplified
rotation invariant approach based on the DFT+U method introduced by Dudarev
$et$ $al.$ [62]. Then, the effective Heisenberg interactions for the SDW
orders are constructed. The magnon dispersion within the linear spin wave
theory are calculated using the torque equation formalism[48, 17]. The RIXS
intensity for the magnon mode in the $\sigma$-$\pi$ polarization channel is
calculated following the reference [63]. More details can be found in Sections
5 and 6 of Supplementary Information.

Competing interests
Authors declare that they have no competing interests.

Data availability
All data are available in the main text and Supplementary Information.