††thanks: Corresponding author<EMAIL_ADDRESS>
# Neutron Scattering Studies on the High-$T_{c}$ Superconductor La3Ni2O7-δ at
Ambient Pressure

Tao Xie  Mengwu Huo  Xiaosheng Ni  Center for Neutron Science and Technology,
Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices,
School of Physics, Sun Yat-Sen University, Guangzhou, Guangdong 510275, China
Feiran Shen  Spallation Neutron Source Science Center, Dongguan, Guangdong
523803, China  Xing Huang  Center for Neutron Science and Technology,
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  Helen C. Walker  ISIS Neutron and Muon Source, Science and
Technology Facilities Council, Rutherford Appleton Laboratory, Didcot OX11
0QX, United Kingdom  Devashibhai Adroja  ISIS Neutron and Muon Source, Science
and Technology Facilities Council, Rutherford Appleton Laboratory, Didcot OX11
0QX, United Kingdom  Dehong Yu  Australian Nuclear Science and Technology
Organisation, Locked bag 2001, Kirrawee DC, NSW 2232, Australia  Bing Shen
Center for Neutron Science and Technology, Guangdong Provincial Key Laboratory
of Magnetoelectric Physics and Devices, School of Physics, Sun Yat-Sen
University, Guangzhou, Guangdong 510275, China  Lunhua He  Beijing National
Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy
of Sciences, Beijing, 100190, China  Spallation Neutron Source Science Center,
Dongguan, Guangdong 523803, China  Songshan Lake Materials Laboratory,
Dongguan, Guangdong 523808, China  Kun Cao  Meng Wang  Center for Neutron
Science and Technology, Guangdong Provincial Key Laboratory of Magnetoelectric
Physics and Devices, School of Physics, Sun Yat-Sen University, Guangzhou,
Guangdong 510275, China

###### Abstract

After several decades of studies of high-temperature superconductivity, there
is no compelling theory for the mechanism yet; however, the spin fluctuations
have been widely believed to play a crucial role in forming the
superconducting Cooper pairs. The recent discovery of high-temperature
superconductivity near 80 K in the bilayer nickelate La3Ni2O7 under pressure
provides a new platform to elucidate the origins of high-temperature
superconductivity. We perform elastic and inelastic neutron scattering studies
on a polycrystalline sample of La3Ni2O7-δ at ambient pressure. No magnetic
order can be identified down to 10 K. However, we observe a weak flat spin-
fluctuation signal at $\sim$ 45 meV in the inelastic scattering spectra. The
observed spin excitations could be interpreted as a result of strong
interlayer and weak intralayer magnetic couplings for a double-stripe
antiferromagnetic order. The absence of long-range magnetic order in neutron
diffraction measurements may be ascribed to a short-range magnetic correlation
length perpendicular to the stripe direction. Our results provide crucial
information on the spin dynamics and are thus important for understanding the
superconductivity in La3Ni2O7.

Introduction. The mechanism of high-temperature (high-$T_{c}$)
superconductivity is a long-lasting mystery although tremendous progress has
been achieved since the discovery of superconductivity in cuprates and iron-
based compounds. High-$T_{c}$ superconductivity in copper-oxide and iron-based
superconductors always appears after the suppression of the antiferromagnetic
(AFM) order in the parent compounds by chemical doping or pressure and evolves
into a superconducting dome with the highest $T_{c}$ around the critical
doping/pressure where the AFM order disappears Scalapino (2012); Tranquada
_et al._ (2014); Keimer  _et al._ (2015); Dai (2015). As the AFM order is
suppressed by chemical doping, the spin fluctuations persist throughout the
phase diagram and form a neutron spin resonant mode below $T_{c}$ at certain
energies around the AFM wave vectors of the parent compounds. The temperature
dependence of the intensity of the neutron spin resonant mode behaves like a
superconducting order parameter and the resonant energy can be linearly scaled
with the corresponding $T_{c}$ and superconducting gap Eschrig (2006); Sidis
_et al._ (2007); Christianson  _et al._ (2008a); Yu  _et al._ (2009). All
these characteristics suggest that magnetism is crucial to the mechanism of
the high-$T_{c}$ superconductivity, and the AFM spin fluctuations could be the
“glue” for the formation of superconducting Cooper pairs.

Seeking new unconventional high-$T_{c}$ superconductors is the key to
understanding the high-$T_{c}$ mechanism and extending the application of
superconductivity. Nickelates have been proposed to be a promising material
platform to search for high-$T_{c}$ superconductors due to the similarity of
the lattice and electronic structure with those of the cuprates Anisimov  _et
al._ (1999). Among different families of nickelates, $RE$NiO2 ($RE$ = La, Nd,
Pr) were predicted to be the most probable to achieve superconductivity due to
the same electronic configuration of Ni+ (3$d^{9}$) with Cu2+ in the cuprates.
Superconductivity was indeed observed in Nd0.8Sr0.2NiO2 thin films ($T_{c}$
$\sim$ 9-15 K) in 2019 and later confirmed in other thin film samples with
similar compositions Li  _et al._ (2019); Osada  _et al._ (2020); Wang  _et
al._ (2022); Gu and Wen (2022). Additionally, it is interesting to note that
spin-wave-like magnetic fluctuations of Nd1-xSrxNiO2 from resonant inelastic
X-ray scattering (RIXS) measurements give comparable magnetic exchange
couplings to that in copper-oxide superconductors Lu  _et al._ (2021); Coldea
_et al._ (2001).

Recently, superconductivity with $T_{c}$ up to 80 K was reported in single-
crystal samples of the Ruddlesden-Popper phase bilayer nickelate La3Ni2O7
[Fig. 1(a)] with pressure above 14 GPa Sun  _et al._ (2023). Either zero-
resistance or a diamagnetic response corresponding to the superconducting
transition in La3Ni2O7 has been confirmed on single-crystal or polycrystalline
samples in subsequent reports Hou  _et al._ (2023); Zhang  _et al._ (2023a);
Wang  _et al._ (2023); Zhou  _et al._ (2023). X-ray photoemission spectroscopy
measurements indicated the valence state of the Ni ions in La3Ni2O7 is a
mixture of Ni2+ and Ni3+, which may result in a charge density wave (CDW) and
a spin density wave (SDW)Liu  _et al._ (2023a); Taniguchi  _et al._ (1995).
Electrical transport and specific heat measurements on single crystal samples
indeed revealed transition-like anomalies in resistance and heat capacity at
153 and 110 K Liu  _et al._ (2023a). An optical study observed a CDW-like
transition at $\sim$115 K Liu  _et al._ (2023b). A recent $\mu$SR study
provided evidence of the existence of the SDW in a La3Ni2O7-δ powder sample at
$\sim$150 K Chen  _et al._ (2023). However, neutron diffraction measurements
did not observe long-range magnetic order in powder samples Ling  _et al._
(1999). Thus, an elucidation of the nature of the magnetic ground state, the
properties of the spin fluctuations, and the magnitudes of the magnetic
exchange interactions of La3Ni2O7 are crucial to understanding the mechanism
of the high-$T_{c}$ superconductivity.

Although the angle of the bilayer Ni-O-Ni along the $c$ axis evolves from 168∘
to 180∘ under pressure Sun  _et al._ (2023), the spin fluctuations and
magnetic exchange interactions under ambient conditions should be comparable
to those at high pressure. In this paper, we report neutron diffraction and
systematic inelastic neutron scattering (INS) measurements on a
polycrystalline La3Ni2O7-δ sample at ambient pressure. No long-range magnetic
order can be identified at 10 K. The INS data reveal a non-dispersive spin
excitation signal at $\sim 45$ meV, which has been used to estimate the
magnitude of the magnetic exchange couplings. We find that an alternate double
stripe AFM order with only short-range correlations or very small magnetic
moments exhibiting a strong interlayer coupling and weak intralayer couplings
matches the experimental observations. In addition, dispersive and non-
dispersive phonon modes are observed. By employing density functional theory
(DFT) calculations, we have computed the dispersion and the density of states
(DOS) of the phonon modes, which are roughly consistent with the INS spectra.
Our results reveal that the strong interlayer magnetic exchange couplings
dominate the spin fluctuations, which is distinct from the situation for the
cuprates and iron-based superconductors.

Figure 1: Structure, electronic occupancy, and bulk electrical transport,
magnetic and structural characterization of the La3Ni2O7-δ polycrystalline
sample. (a) Crystal structure of La3Ni2O7-δ in the $Amam$ space group. The
open blue circle represents an oxygen vacancy. (b) Schematic of the
$\sigma$-bonding and anti-bonding states for two Ni2.5+ (3$d^{7.5}$). (c)
Temperature dependence of electrical resistivity and field cooled (FC)
magnetization ($H$ = 1000 Oe). (d) Neutron powder diffraction pattern
collected at 10 K.  Figure 2: INS spectra of La3Ni2O7-δ with different
incident energies at low temperature. (a) INS spectrum collected at PELICAN
with $E_{i}$ = 3.7 meV. (b) and (c) are INS spectra measured at MERLIN with
$E_{i}$ = 22.6 and 160 meV, respectively. (d) INS spectrum up to 350 meV
obtained at MAPS. (e) A constant $|Q|$ cut in the range of [0.6, 1.9] Å-1 for
$E_{i}$ = 3.7 meV. The shadow region indicates the elastic and quasi-elastic
scatterings from the sample. The peak at 1.5 meV is from spurious scattering
which can also be observed in panel (a). (f) Constant energy cuts within $E$ =
[2.5, 3.5] meV at 7 K (the upper panel) and a comparison between 7 and 300 K
(the lower panel) for $E_{i}$ = 22.6 meV. Figure 3: INS spectra after
subtraction of high from low temperature data. (a)-(c) show the resultant data
between 7 and 300 K for $E_{i}$s = 66.3, 160, and 500 meV. The inset of panel
(a) is a constant energy cut with $E$ = [45, 47] meV. (d) Constant $|Q|$ cut
from (b) with $|Q|$ = [1.8, 4] Å-1. The dashed lines mark the centers of the
peaks. (e) The alternate double stripe AFM order used for the SpinW
calculations. The red arrows represent the directions of the ordered spins.
The green solid lines indicate the unit cell of the orthorhombic $Amam$
structure. The in-plane exchange couplings $J_{1}$, $J_{2}$, $J_{4}$, and the
inter-layer exchange coupling $J_{\perp}$ are indicated in the structure. (f)
The calculated spin excitation spectrum from SpinW with $SJ_{1}$ = 5, $SJ_{2}$
= -2, $SJ_{4}$ = 1, and $SJ_{\perp}$ = 36 meV.

Characterizations and experimental details. Figure 1(c) shows basic bulk
characterization of the sample. The temperature dependence of the resistivity
goes up slightly at low temperatures. The magnetization at $H$ = 1000 Oe also
increases towards lower temperatures. Both upward trends at low temperatures
suggest the existence of oxygen vacancies Zhang  _et al._ (1994). Neutron
diffraction experiments were performed on the general purpose powder
diffractometer (GPPD) Chen  _et al._ (2018) at China Spallation Neutron Source
(CSNS) at 10 and 160 K. The data were refined with the Rietveld method using
the GSAS software package Toby (2001). The low-energy INS spectra at 2 and 200
K were collected on the time-of-flight cold neutron spectrometer PELICAN Yu
_et al._ (2013) at the Australian Centre for Neutron Scattering (ACNS) at the
Australian Nuclear Science and Technology Organisation (ANSTO) with an
incident neutron energy $E_{i}$ = 3.7 meV. The high-energy spectra were
obtained from the time-of-flight spectrometers MERLIN Bewley  _et al._ (2006)
and MAPS Ewings  _et al._ (2019) at the ISIS Neutron and Muon Source at UK
with $E_{i}$s = 15, 22.6, 36, 66.3, 150.6, and 500 meV (MAPS) at 7 and 300 K.
We used 5 g powder samples of La3Ni2O7 synthesized by the solid-solid reaction
method Liu  _et al._ (2023a) for the experiments on GPPD and PELICAN. 20 g
powder samples synthesized in the same way were used for the experiments on
MERLIN and MAPS. The INS data from PELICAN were analyzed using the LAMP
software package. The data collected at MERLIN and MAPS were reduced and
analyzed using the software packages MANTIDPLOT Arnold  _et al._ (2014) and
DAVE Azuah  _et al._ (2009). The scattering intensities from aluminum have
been removed by subtracting the data from an empty can for the PELICAN and
MERLIN experiments. The spin excitation simulations were performed using the
SpinW software package Toth and Lake (2015). The phonon dispersions and DOS
were calculated using the finite displacement method as implemented in the
Phonopy Togo and Tanaka (2015) software in combination with the Vienna $ab\
initio$ Simulation Package (VASP) Kresse and Hafner (1993); Kresse and
Furthmüller (1996).

Neutron diffraction results. Figure 1(d) presents the neutron powder
diffraction data measured at 10 K. No extra peaks can be seen when comparing
the data between 10 and 160 K (see supplementary information (SI) SI  for
details). Our refinements reveal two structures of La3Ni2O7 with space groups
of $Amam$ (No. 63) and $Fmmm$ (No. 69). The refined compositions at 10 K for
the two phases are 56.1% La3Ni2O6.79 and 41.0% La3Ni2O6.90 for the $Amam$ and
$Fmmm$ structures, respectively SI . A fraction of 2.9% La2NiO4 impurity is
also identified. Based on the refined results, there are more vacancies on the
inner apical oxygen sites for the $Amam$ phase, consistent with a recent
transmission electron microscopy experiment Dong  _et al._ (2023). We thus use
La3Ni2O7-δ to refer to our sample hereafter. It should be noted that La3Ni2O7
with the $Fmmm$ structure has also been reported at ambient pressure Zhang
_et al._ (1994).

Inelastic neutron scattering results. Figures 2(a)-2(d) present the INS
spectra measured with different $E_{i}$s at 2 K on PELICAN and 7 K on MERLIN
and MAPS. Identical measurements at 200 and 300 K are presented in the SI SI .
One cannot see obvious spin fluctuations from the color maps at low $|Q|$. At
higher $|Q|$ and energy ranges weak acoustic phonon modes in Fig. 2(b) and
flat optical phonon modes in Fig. 2(c) can be observed. To inspect the INS
spectra more closely, we performed one-dimensional (1D) cuts along the $|Q|$
and energy axes from both the low-temperature and high-temperature data. In
the low $|Q|$ range of [0.6, 1.9] Å-1 as shown in Fig. 2(e), where the
intensity of phonons should be weak, the difference in intensities around 1
meV may be ascribed to temperature-enhanced spin fluctuations followed by the
Bose factor. In Fig. 2(f), we present 1D cuts along the $|Q|$ axis by
integrating the energies between 2.5 and 3.5 meV from the map in Fig. 2(b).
The cut at 300 K in the lower panel of Fig. 2(f) reveals obvious acoustic
phonons, whose intensities become much weaker at 7 K. Interestingly, obvious
signals above the background at 7 K can be observed below 3 Å-1 as shown in
the upper panel of Fig. 2(f). The intensities observed at low $|Q|$ and low
temperature of 7 K should be magnetic in origin.

Figure 4: Phonon modes and vibrational density of states (DOS) of La3Ni2O7-δ.
(a)-(b) Constant $|Q|$ cuts with the momentum transfer $|Q|$ = [4, 8] Å-1 for
$E_{i}$ = 66.3 meV and $|Q|$ = [11, 12] Å-1 for $E_{i}$ = 160 meV. The inset
of (b) shows a zoomed-in region of the spectrum at 7 K, revealing peaks around
57, 78, and 80 meV. (c) DFT calculations of the dispersion of phonons of
La3Ni2O7. The dispersive modes in the high-energy region dominated by oxygen
are highlighted in olive. (d) The calculated total vibrational DOS and partial
DOS for different elements of La3Ni2O7.

Since weak magnetic excitations at higher energies may be embedded in the INS
spectra, we performed direct data subtractions between 7 and 300 K, as shown
in Fig. 2 and the SI SI . The subtraction results are shown in Fig. 3. Phonon
intensities at 300 K are expected to be much stronger than at 7 K. However, a
flat mode centered at 45 meV appears for the data collected at MERLIN [Figs.
3(a) and 3(b)] and MAPS [Fig. 3(c)]. The intensities decrease as $|Q|$
increases as shown in the inset of Fig. 3(a)], indicating the magnetic origin.
We show a 1D cut in the $|Q|$ range of [1.8, 4] Å-1 along the energy axis to
see the exact energy of the magnetic excitations in Fig. 3(d). The peak of the
magnetic mode centers at 45 meV. The tiny positive intensities around $\sim
70-80$ meV are likely to originate from the mismatch of the line width of the
optical phonon modes between 7 and 300 K. The $|Q|$ dependence of these
intensities does not follow a magnetic form factor. Thus, we only consider the
features around 45 meV to be magnetic excitations.

We now attempt to estimate the magnetic exchange couplings by assuming a
certain magnetic structure and comparing its magnetic excitation spectrum to
the INS data. Several types of magnetic orders have been tested and a range of
possible combinations of exchange couplings can be found in the SI SI . The
following information was taken to determine the magnetic order and set of
$SJ$s: (1) The signature of SDW revealed by $\mu$SR Chen  _et al._ (2023) and
the absence of magnetic Bragg peaks in neutron diffraction, suggesting that
the magnetic order is short-range or involves only small magnetic moments. (2)
An AFM magnetic structure with alternate stripes of magnetic and non-magnetic
Ni atoms is highly possible, given the short-range character and the mixture
of Ni2+ and Ni3+ in La3Ni2O7 with the same proportion Liu  _et al._ (2023a).
(3) The intensities of the dispersive magnetic excitations at low $|Q|$ and
low energies are weak, while the flat magnetic excitation at 45 meV is
prominent. (4) The interlayer coupling should be much stronger than the
intralayer couplings due to the half-filling of the 3$d_{z^{2}}$ orbital and
the quarter-filling of the 3$d_{x^{2}-y^{2}}$ orbitals Sun  _et al._ (2023);
Yang  _et al._ (2023a).

Figure 3(e) shows a double stripe-type AFM order with magnetic and non-
magnetic Ni atoms. We employ a Heisenberg Hamiltonian by considering the
effective intralayer exchange couplings $SJ_{1}$, $SJ_{2}$, $SJ_{4}$, and the
inter-layer coupling $SJ_{\perp}$. In Fig. 3(f), we present the calculated
magnetic excitation spectrum using $SJ_{1}$ = 5, $SJ_{2}$ = -2, $SJ_{4}$ = 1,
and $SJ_{\perp}$ = 36 meV with the magnetic order shown in Fig. 3(e), which
matches the INS data well. The double stripe-type AFM order in Fig. 3(e) and
the set of magnetic exchange couplings obey the above constraints. We note the
observed INS spectrum cannot give a unique magnetic order. However, the
relative strong interlayer coupling and weak intralayer couplings should be
reliable.

The acoustic phonon branches have been shown in Figs. 2(b) and 2(f). The 1D
constant $|Q|$ cuts focusing on the optical phonons are exhibited in Figs.
4(a) ($E_{i}$ = 66.3 meV) and 4(b) ($E_{i}$ = 160 meV). The most notable
features in Fig. 4(a) are the strong peaks around 11 and 23 meV and weak peaks
around 30, 34, and 40 meV, which all have stronger intensities at 300 K than
that at 7 K, confirming their phononic origin. Similarly, the INS peaks around
58, 68, and 78 meV that correspond to the flat bands in Figs. 2(c) can be
identified [Fig. 4(b)]. The phonon calculations were conducted using the DFT
method. The dispersion of the phonon modes along the high-symmetry directions
is shown in Fig. 4(c). A large number of phonon branches with different levels
of dispersion can be seen. The vibrational density of states is presented in
Fig. 4(d), where the peak features around 10-15, 30, 40, 55, and 70 meV can be
recognized. These peaks are roughly consistent with the experimental INS peaks
in Figs. 4(a) and 4(b). The calculated partial DOS shows that the higher flat
bands are dominated by the vibration of oxygen atoms as expected.

Discussion and Conclusion. Our detailed comparisons of the INS spectra at low
and high temperatures reveal convincing evidence of magnetic excitations in
La3Ni2O7-δ. Although the results are obtained at ambient pressure, the
proposed magnetic exchange couplings are important for understanding the
mechanism of superconductivity and the pairing gap symmetry in the
superconducting state under high pressure Yang  _et al._ (2023b); Zhang  _et
al._ (2023b); Liu  _et al._ (2023c); Qu  _et al._ (2023). The results also
indicate the existence of spin fluctuations is universal among copper-oxide,
iron-based superconductors, and nickelate high $T_{c}$ superconductors.

We briefly discuss the reasons for the weakness of the spin fluctuations in
La3Ni2O7-δ. First of all, we should note that La3Ni2O7 is a metallic system
that shows itinerant characteristics. Some of the nickel ions may remain
nonmagnetic or in low spin states. The possible alternate double stripe AFM
order with a short-range correlation perpendicular to the stripe direction
will further suppress the long-range magnetic order and weaken the intensity
of magnetic excitations. Secondly, as shown in Fig. 1(b), the 3$d_{z^{2}}$
electrons can form $\sigma$-bonding and anti-bonding bands within a bilayer
unit and the 3$d_{x^{2}-y^{2}}$ electrons hybridize with the 3$d_{z^{2}}$
electrons within one Ni-O layer, which give rise to nearly half-filled
3$d_{z^{2}}$ orbit and quarter-filled 3$d_{x^{2}-y^{2}}$ orbit. The quarter-
filled character of the 3$d_{x^{2}-y^{2}}$ orbit represents a weak intralayer
exchange coupling which further reduces the intensity of spin fluctuations.
This behavior is different from that in cuprate and iron-based
superconductors, where the dominant intralayer exchange couplings lead to the
formation of in-plane long-range magnetic order/correlation. Thirdly,
polycrystalline samples always give weak signal intensity in INS spectra due
to the averaging effect when compared with the results from single-crystal
samples with a similar mass. In powder samples of iron-based superconductors,
the spin fluctuations are almost invisible Qiu  _et al._ (2008); Christianson
_et al._ (2008b); Mittal  _et al._ (2008), except for the case when there is a
strong neutron spin resonance in superconducting samples Christianson  _et
al._ (2008a).

In conclusion, we report the elastic and inelastic neutron scattering results
of a polycrystalline La3Ni2O7-δ sample at ambient pressure. No long-range
magnetic order can be identified at 10 K. Our INS results reveal magnetic
excitations in La3Ni2O7-δ, indicating that the nickelate superconductors might
be in a similar position to the copper-oxide and iron-based superconductors
when considering the role of magnetism in the mechanism for high-$T_{c}$
superconductivity. Estimating the magnitude of the magnetic exchange couplings
by assuming an alternate double stripe AFM order results in a strong
interlayer coupling and weak intralayer couplings. Our work provides crucial
information to the community for determining a realistic mechanism of
superconductivity in La3Ni2O7.

Acknowledgements. We thank Prof. D. L. Feng for sharing the RIXS data in the
‘Workshop on High-$T_{c}$ Nickelate Superconductors’. Work at SYSU was
supported by the National Key Research and Development Program of China (Grant
Nos. 2023YFA1406500, 2023YFA1406002), the National Natural Science Foundation
of China (Grant nos. 12304187, 12174454, U21301001), the Guangdong Basic and
Applied Basic Research Funds (grant no. 2021B1515120015), Guangzhou Basic and
Applied Basic Research Funds (grant nos. 202201011123, 2024A04J6417),
Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices
(grant no. 2022B1212010008), and the Fundamental Research Funds for the
Central Universities, Sun Yat-sen University (Grant No. 23qnpy57). This work
is based on experiments (Proposal P15658) performed at Australian Centre for
Neutron Scattering (ACNS) at Australian Nuclear Science and Technology
Organisation (ANSTO), Australia. The neutron experiment at ISIS Neutron and
Muon Source in UK was performed under a user program (Proposal RB2300045 and
RB2300079). The neutron diffraction experiment at CSNS in China was performed
under the proposal P1822113000042.

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Supplementary Information: Neutron Scattering Studies on the High-$T_{c}$
Superconductor La3Ni2O7-δ at Ambient Pressure

Figure 5: Neutron powder diffraction results collected on the different
detector banks of the GPPD at 10 K and 160 K. Figure 6: The Rietveld
refinements of the neutron powder diffraction results at 10 K and 160 K.

##  I Neutron Scattering Results

Here we present the neutron scattering raw data that are referred to in the
main text. We first compare the neutron powder diffraction results collected
on GPPD at 10 K and 160 K in Fig. 5. No new peaks or clear intensity gain on
the nuclear peaks from magnetic order can be identified at 10 K. Structural
models are refined against the neutron diffraction data with the Rietveld
method using the GSAS software package. The refined results are shown in Fig.
6, and the refined structure parameters at 10 and 160 K are summarized in
Tables 1, 2, 3, 4.

In Fig. 7, the corresponding high-temperature INS spectra to those shown in
Fig. 2 of the main text are presented. These results were subtracted from the
corresponding low-temperature data in Fig. 2 to give the difference shown in
Fig. 3 of the main text. In Fig. 8, the raw data used for the 1D INS spectra
in Fig. 4(a) are presented.

Table 1: Refined crystallographic parameters of the $Amam$ phase (No. 63) (56.1%) at 10 K for our La3Ni2O7-δ sample. Lattice |  $Amam$ (No. 63) |  $a$ = 5.3956(4)Å |  $b$ = 5.4491(4)Å |  $c$ = 20.598(3)Å
---|---|---|---|---
atom | $x$ | $y$ | $z$ | occupancy
La1 | 0.2500 | 0.2453 | 0.3031 | 1
La2 | 0.2500 | 0.2362 | 0.5000 | 1
Ni | 0.2500 | 0.2509 | 0.1059 | 1
O1 | 0.5000 | 0.0000 | 0.0861 | 1
O2 | 0.2500 | 0.2126 | 0.1996 | 1
O3 | 0.0000 | 0.5000 | 0.1055 | 1
O4 | 0.2500 | 0.3014 | 0.0000 | 0.79(6)
Table 2: Refined crystallographic parameters of the $Fmmm$ phase (No. 69) (41.0%) at 10 K for our La3Ni2O7-δ sample. Lattice |  $Fmmm$ (No. 69) |  $a$ = 5.3816(2)Å |  $b$ = 5.4538(3)Å |  $c$ = 20.470(2)Å
---|---|---|---|---
atom | $x$ | $y$ | $z$ | occupancy
La1 | 0.0000 | 0.0000 | 0.5000 | 1
La2 | 0.0000 | 0.0000 | 0.3178 | 1
Ni | 0.0000 | 0.0000 | 0.0942 | 1
O1 | 0.0000 | 0.0000 | 0.0000 | 0.98(8)
O2 | 0.0000 | 0.0000 | 0.2064 | 1
O3 | -0.2500 | 0.2500 | 0.0934 | 0.98(2)
Table 3: Refined crystallographic parameters of the $Amam$ phase (No. 63) (55.8%) at 160 K for our La3Ni2O7-δ sample. Lattice |  $Amam$ (No. 63) |  $a$ = 5.4015(4)Å |  $b$ = 5.4491(4)Å |  $c$ = 20.616(3)Å
---|---|---|---|---
atom | $x$ | $y$ | $z$ | occupancy
La1 | 0.2500 | 0.2453 | 0.3031 | 1
La2 | 0.2500 | 0.2362 | 0.5000 | 1
Ni | 0.2500 | 0.2509 | 0.1059 | 1
O1 | 0.5000 | 0.0000 | 0.0861 | 1
O2 | 0.2500 | 0.2126 | 0.1996 | 1
O3 | 0.0000 | 0.5000 | 0.1055 | 1
O4 | 0.2500 | 0.3014 | 0.0000 | 0.81(6)
Table 4: Refined crystallographic parameters of the $Fmmm$ phase (No. 69) (41.3%) at 160 K for our La3Ni2O7-δ sample. Lattice |  $Fmmm$ (No. 69) |  $a$ = 5.3880(2)Å |  $b$ = 5.4531(3)Å |  $c$ = 20.489(2)Å
---|---|---|---|---
atom | $x$ | $y$ | $z$ | occupancy
La1 | 0.0000 | 0.0000 | 0.5000 | 1
La2 | 0.0000 | 0.0000 | 0.3178 | 1
Ni | 0.0000 | 0.0000 | 0.0942 | 1
O1 | 0.0000 | 0.0000 | 0.0000 | 0.98(8)
O2 | 0.0000 | 0.0000 | 0.2064 | 1
O3 | -0.2500 | 0.2500 | 0.0934 | 0.99(2)
Figure 7: The corresponding high-temperature INS spectra of La3Ni2O7-δ with
different incident energies to those shown in Fig. 2 of the main text. Panels
(a)-(d) are the INS spectra for $E_{i}$ = 3.7, 22.6, 160, and 350 meV. Figure
8: The raw data used for the 1D cuts in Fig. 4(a) of the main text with
$E_{i}$ = 66.3 meV.

##  II Linear Spin-Wave Theory Simulation

Since only weak and non-dispersive magnetic fluctuations can be observed in
our data, we cannot do quantitative fitting to the data but perform
qualitative estimations based on the results. Our strategy to simulate the
spin excitations is to assume a magnetic structure and suitable exchange
couplings to produce a similar excitation spectrum in the experiment. We note
the linear spin-wave theory (LSWT) is suitable for long-range ordered magnetic
systems with localized moments. Our sample shows no long-range magnetic order.
The assumed magnetic structure is only used for LSWT simulation in SpinW
software package.

Several magnetic structures with different combinations of exchange couplings
are compared below. We first exclude the well-known Néel-type and A-type
antiferromagnetic orders [Fig. 9], since the calculated powder-averaged INS
spectra give very different features from our experimental results.

We considered two types of stripe AFM orders with the stripes along the
$a$-axis of the crystal. The first type of stripe AFM order is alternate
single stripes of magnetic and non-magnetic Ni atoms [Fig. 9(c)], named single
stripe-type AFM, and is excluded due to the deviation from the experimental
results. The second stripe AFM order is the alternate double stripes of
magnetic and non-magnetic Ni atoms [Fig. 10(a)], named double stripe-type AFM,
and we find a range of exchange parameters that can reproduce the experimental
spectra.

The weak and non-dispersive magnetic fluctuation can only be observed in the
subtracted data at high energy, which might indicate the spin excitations are
rather weak at low-energy regions. This feature reminds us that one cannot
only consider the nearest neighbor (NN) $J_{1}$ as shown in Fig. 10(b), since
it can generate strong intensities in the low-energy region. The $J_{2}$ and
$J_{4}$ are necessary to push the spectral weight from the low-energy region
to the high-energy region as shown in Figs. 10(c)-(e). Thus, we focused on the
combination of $J_{1}$, $J_{2}$, $J_{4}$, and $J_{\perp}$ to estimate the
range of the couplings. Since La3Ni2O7 is an itinerant system, which means
that there may be no local moments, we directly use the effective $SJ_{1}$,
$SJ_{2}$, $SJ_{3}$, $SJ_{4}$, and $SJ_{\perp}$ below. We fix the exchange
couplings that are further than $J_{2}$ and only discuss the relation between
$SJ_{1}$ and $SJ_{\perp}$. Figure 11 presents the LSWT simulations of the
spectra with a wide range of the parameters of $SJ_{1}$ and $SJ_{\perp}$ (from
dominant $SJ_{1}$ to dominant $SJ_{\perp}$). One can see the dominant
$SJ_{\perp}$ cases in Figs. 11(d)-(i) can describe the experimental results
better than the dominant $SJ_{1}$ cases. Interestingly, the combinations of
$SJ_{1}$ and $SJ_{\perp}$ that can well reproduce the experimental spectra
follow a linear relation of $SJ_{\perp}$ = $-$5 $\times$ $SJ_{\perp}$ \+ 61 as
that shown in Fig. 12. At last, we estimate the parameters of $SJ_{1}$ and
$SJ_{\perp}$ that can reproduce the spectra similar to the experimental
results and present them in Fig. 12 between the two dashed lines.

Figure 9: Different magnetic structures and the corresponding LSWT simulations
with different effective exchange parameters for La3Ni2O7. (a) The Néel-type
magnetic order. (b) The A-type AFM order. (c) The single stripe-type AFM order
with only half of the magnetic Ni ions. Figure 10: LSWT simulations with
different effective exchange parameters for the double stripe-type AFM order
with only half of the magnetic Ni ions in La3Ni2O7.  Figure 11: LSWT
simulations with considering different combinations of the in-plane $J_{1}$,
$J_{2}$, $J_{4}$, and inter-layer $J_{\perp}$ for the double stripe-type AFM
order in La3Ni2O7. Panels (b)-(i) give a wide range of the parameters of
$SJ_{1}$ and $SJ_{\perp}$ (from dominant $SJ_{1}$ to dominant $SJ_{\perp}$
with fixed $SJ_{2}$ = -2 and $SJ_{4}$ = 1 meV).  Figure 12: A linear relation
of $SJ_{1}$ and $SJ_{\perp}$ (with fixed $SJ_{2}$ = -2 and $SJ_{4}$ = 1 meV)
for the double stripe-type AFM order that can reproduce the experimental
spectra. The dashed lines mark a reasonable range of parameters.