# Spin fluctuations sufficient to mediate superconductivity in nickelates

Paul Worm  0000-0003-2575-5058 Institute of Solid State Physics, TU Wien, 1040
Vienna, Austria  Qisi Wang  0000-0002-8741-7559 Physik-Institut, Universität
Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland  Department of
Physics, The Chinese University of Hong Kong, Shatin, Hong Kong, China
Motoharu Kitatani  0000-0003-0746-6455 Department of Material Science,
University of Hyogo, Ako, Hyogo 678-1297, Japan  RIKEN Center for Emergent
Matter Sciences (CEMS), Wako, Saitama, 351-0198, Japan  Izabela Biało
0000-0003-3431-6102 Physik-Institut, Universität Zürich, Winterthurerstrasse
190, CH-8057 Zürich, Switzerland  AGH University of Science and Technology,
Faculty of Physics and Applied Computer Science, 30-059 Kraków, Poland  Qiang
Gao  Beijing National Laboratory for Condensed Matter Physics, Institute of
Physics, Chinese Academy of Sciences, Beijing 100190, China  Xiaolin Ren
Beijing National Laboratory for Condensed Matter Physics, Institute of
Physics, Chinese Academy of Sciences, Beijing 100190, China  Jaewon Choi
Diamond Light Source, Harwell Campus, Didcot OX11 0DE, United Kingdom  Diana
Csontosová  Department of Condensed Matter Physics, Faculty of Science,
Masaryk University, Kotlářská 2, 611 37 Brno, Czechia  Ke-Jin Zhou  Diamond
Light Source, Harwell Campus, Didcot OX11 0DE, United Kingdom  Xingjiang Zhou
Beijing National Laboratory for Condensed Matter Physics, Institute of
Physics, Chinese Academy of Sciences, Beijing 100190, China  Zhihai Zhu
Beijing National Laboratory for Condensed Matter Physics, Institute of
Physics, Chinese Academy of Sciences, Beijing 100190, China  Liang Si
0000-0003-4709-6882 Corresponding author<EMAIL_ADDRESS>School of
Physics, Northwest University, Xi’an 710127, China  Institute of Solid State
Physics, TU Wien, 1040 Vienna, Austria  Johan Chang  Physik-Institut,
Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland  Jan
M. Tomczak  0000-0003-1581-8799 Department of Physics, King’s College London,
Strand, London WC2R 2LS, United Kingdom  Institute of Solid State Physics, TU
Wien, 1040 Vienna, Austria  Karsten Held  0000-0001-5984-8549 Corresponding
author<EMAIL_ADDRESS>Institute of Solid State Physics, TU Wien, 1040
Vienna, Austria

###### Abstract

Infinite-layer nickelates show high-temperature superconductivity, and the
experimental phase diagram agrees well with the one simulated within the
dynamical vertex approximation (D$\Gamma$A). Here, we compare the spin-
fluctuation spectrum behind these calculations to resonant inelastic X-ray
scattering experiments. The overall agreement is good. This independent cross-
validation of the strength of spin fluctuations strongly supports the
scenario, advanced by D$\Gamma$A, that spin-fluctuations are the mediator of
the superconductivity observed in nickelates.

##  I Introduction

Contrasting cuprates Bednorz and Müller (1986), to the new nickelate
superconductors Li  _et al._ (2019, 2020); Zeng  _et al._ (2020); Osada  _et
al._ (2020); Zeng  _et al._ (2021); Pan  _et al._ (2021); Osada  _et al._
(2021); Wang  _et al._ (2022a) offers the unique opportunity to understand
high-temperature ($T_{c}$) superconductivity more thoroughly: the two systems
are similar enough to expect a common origin of superconductivity, but at the
same time distinct enough to pose severe restrictions on any theoretical
description. Structurally, both, nickelate and cuprate superconductors,
consist of Ni(Cu)O2 planes that host the superconductivity. These layers are
separated by buffer layers of, e.g., Nd(Ca) atoms in the infinite-layer
compound NdNiO2(CaCuO2). Additionally, both Ni and Cu exhibit a nominal
3$d^{9}$ electronic configuration in the respective parent compound, with a
3$d_{x^{2}-y^{2}}$-derived band that is close to half-filling.

Turning to the differences, a major one is that for cuprates the oxygen 2$p$
bands, that strongly hybridize with the Cu 3$d_{x^{2}-y^{2}}$ band, are below
but close to the Fermi energy. This makes the parent compound a charge-
transfer insulator Zaanen  _et al._ (1985), and the Emery model Emery (1987)
the elemental model for cuprates. For nickelates, on the other hand, these
2$p$ bands are shifted down relative to the 3$d_{x^{2}-y^{2}}$ band which is
fixed to the Fermi energy. As a consequence, the oxygen band is now
sufficiently far away from the Fermi energy. While there is still the
hybridization with the Ni 3$d_{x^{2}-y^{2}}$ band, the oxygen 2$p$ bands do
not host holes if nickelates are doped. Instead, however, the rare earth 5$d$
bands are also shifted down (compared to the Ca bands that are above the Fermi
energy in the cuprate CaCuO2), now even cross the Fermi energy and form two
electron pockets around the $\Gamma$ and $A$ momentum points. This is
evidenced by density functional theory (DFT) calculations Botana and Norman
(2020); Sakakibara  _et al._ (2020); Jiang  _et al._ (2019); Hirayama  _et
al._ (2020); Hu and Wu (2019); Wu  _et al._ (2020); Nomura  _et al._ (2019);
Zhang  _et al._ (2020); Jiang  _et al._ (2020); Werner and Hoshino (2020); Si
_et al._ (2020); Nomura and Arita (2022); Kitatani  _et al._ (2023a) and,
experimentally, by the negative Hall conductivity Li  _et al._ (2019); Zeng
_et al._ (2020) for the infinite-layer compound. In all, this situation
creates a seemingly more complicated multi-band picture already for the
undoped parent compound.

However, one of the pockets, the $\Gamma$ pocket, shifts up and even above the
Fermi energy either when (i) doping into the superconducting regime or (ii)
when replacing Nd by La in DFT+dynamical mean-field theory (DMFT) calculations
Kitatani  _et al._ (2020); Held  _et al._ (2022) (and CaxLa1-xNiO2 shows a
very similar phase diagram as SrxNd1-xNiO2). Thus it appears unlikely that the
$\Gamma$ pocket is the key for superconductivity in nickelates. The $A$
pocket, on the other hand, is more stable but it does not hybridize with the
Ni 3$d_{x^{2}-y^{2}}$ band111This pocket also vanishes when going from
infinite to finite-layer nickelatesWorm  _et al._ (2022).. Hence, in Ref.
Kitatani  _et al._ , 2020 the pockets were justifiably treated as a passive
electron reservoir, largely decoupled from the Ni 3$d_{x^{2}-y^{2}}$ band
222At larger doping, outside the superconducting dome, the Ni 3$d_{z^{2}}$
band crosses the Fermi level in DFT+DMFT Kitatani  _et al._ (2020) and becomes
relevant as well. Because of Hund’s exchange this orbital cannot be treated as
decoupled from the Ni 3$d_{x^{2}-y^{2}}$ band. A similar picture has also been
observed in other DFT+DMFT calculations Karp  _et al._ (2020); Pascut  _et
al._ (2023). In $GW$+DMFT the Ni 3$d_{z^{2}}$ band touches the Fermi level
already at lower dopings at large $k_{z}$ Petocchi  _et al._ (2020), and in
self-interaction corrected (sic) DFT+DMFT the Ni 3$d_{z^{2}}$ band is even
more prominent Lechermann (2020); Kreisel  _et al._ (2022).. A similar picture
has also been advocated in Refs. Karp  _et al._ , 2020, Karp  _et al._ , 2022,
and Pascut  _et al._ , 2023.

While the pockets are important for the (Hall) conductivity, we expect
superconductivity to primarily emerge from the Ni 3$d_{x^{2}-y^{2}}$ band
which is strongly correlated. Indeed, calculations based on this single-band
model, with appropriately calculated doping (to account for the pockets),
Kitatani  _et al._ (2020) using the dynamical vertex approximation
($\textrm{D}\Gamma\textrm{A}$) Toschi  _et al._ (2007); Katanin  _et al._
(2009); Rohringer  _et al._ (2018); Kitatani  _et al._ (2022) were able to
compute the superconducting phase diagram, in good agreement with experiments
Lee  _et al._ (2023),333Note that the different substrate LSAT instead of STO
Lee  _et al._ (2023) mainly allows for defect-free nickelate films, as is also
obvious from the largely reduced resistivity. The change in lattice parameters
is of minor importance, cf. the discussion in Section IV.2., see Fig. 1. In
these $\textrm{D}\Gamma\textrm{A}$ calculations, antiferromagnetic (AFM) spin
fluctuations mediate $d$-wave superconductivity. Despite the agreement of Fig.
1, it is imperative to further test this picture of spin-fluctuation-mediated
superconductivity in nickelates. An important validation of the spin-
fluctuation scenario comes from comparing the spin-wave spectrum predicted by
D$\Gamma$A to that measured in experiment. This is the aim of the present
paper.

Figure 1: Superconducting phase diagram, $T_{c}$ vs. Sr doping $x$, of
SrxNd1-xNiO2. Following the discovery of nickelate superconductivity (“a
priori”, data taken from Ref. Li  _et al._ , 2019), $T_{c}$ was calculated
(“D$\Gamma$A theory”, from Ref. Kitatani  _et al._ , 2020). The thus predicted
$T_{c}$ vs. doping well agrees with the one measured “a posteriori” (from Ref.
Li  _et al._ , 2020), especially after “defect free” films have been
synthesized (from Ref. Lee  _et al._ , 2023). Pentalayer Nd6Ni5O12 Pan  _et
al._ (2021) which has a 20% doping of the $d_{x^{2}-y^{2}}$-orbital Worm  _et
al._ (2022) also agrees with theory and infinite-layer SrxNd1-xNiO2 at the
same doping of the Ni $d_{x^{2}-y^{2}}$ orbital (upper $x$-axis). The blue
region indicates where only the Ni $d_{x^{2}-y^{2}}$ orbital and the
$A$-pocket cross the Fermi level in multi-orbital DFT+DMFT.

Specifically, signatures of AFM fluctuations have been measured in resonant
inelastic X-ray scattering (RIXS)Lu  _et al._ (2021), nuclear magnetic
resonance (NMR)Cui  _et al._ (2021) and $\mu$SR Fowlie  _et al._ (2022) where
the fluctuation lifetime can exceed that of the muon Fowlie  _et al._ (2022).
Long-range AFM order is, however, absent in infinite-layer nickelates Lin  _et
al._ (2022); Ortiz  _et al._ (2022), a notable difference to cuprates. One
natural explanation is the self-doping of the Ni 3$d_{x^{2}-y^{2}}$ orbital
induced by the $A$\- and $\Gamma$-pockets. At hole-doping levels similar to
that of the nickelate parent compounds, AFM order has also vanished in
cuprates Oda  _et al._ (2004); Keimer  _et al._ (2015). Consequently, electron
doping (or changing a buffer layer to remove the pockets Nomura  _et al._
(2020); Kitatani  _et al._ (2023b)) is presumably needed to stabilize AFM
order in nickelates.

In the present paper, we calculate and analytically continue the magnetic
susceptibility behind the $\textrm{D}\Gamma\textrm{A}$ calculation of Fig. 1.
The non-local scattering amplitude (two-particle vertex) at the heart of this
magnetic susceptibility directly enters the Cooper (particle-particle) channel
as a pairing vertex and thus mediates superconductivity. For details see Ref.
Kitatani  _et al._ , 2022. We also perform RIXS experiments and compare them
to previous RIXS data by Lu et al. Lu  _et al._ (2021). We find theory and
experiment to be consistent. In particular, the strength of the experimental
AFM coupling is similar to the one extracted from
$\textrm{D}\Gamma\textrm{A}$, advocating that it is sufficient to mediate the
$T_{c}$ observed in nickelate superconductors.

The outline of the paper is as follows: In Sec. II.1, we describe the theory
behind the modelling of spin fluctuations and superconductivity by a one-band
Hubbard model, the ab-initio calculated parameters used, and the
$\textrm{D}\Gamma\textrm{A}$ calculations performed (cf. Appendix A).
Similarly, in Sec. II.2 the experimental methods are discussed, specifically
the film growth and RIXS measurements. Sec. II.3 discusses possible
shortcomings and sources of errors when extracting the paramagnon dispersion
in theory and experiment. In Sec. III, we compare the theoretical and
experimental magnetic spectrum. An analysis of the results in terms of a spin-
wave model is presented in Sec. IV.1. Its interaction dependence is elucidated
in Sec. IV.2, and two possible effects of disorder are discussed in in Sec.
IV.3. Finally, Sec. V summarizes our results.

##  II Methods

###  II.1 Theory

####  II.1.1 Modeling

Previous work, based on DFT+DMFTKarp  _et al._ (2020); Kitatani  _et al._
(2020) identified two bands that cross the Fermi-surface in the
superconducting regime of infinite-layer nickelates: one band with Ni
3$d_{x^{2}-y^{2}}$ character and a pocket around the $A$-momentum composed of
Ni 3$d$-$t_{2g}$+Nd-5$d_{xy}$ character (subsequently referred to as the
$A$-pocket; for some dopings and rare-earth cations there is also an
additional $\Gamma$ pocket). However, the Ni 3$d_{x^{2}-y^{2}}$ band and the
$A$-pocket within the same cell do not hybridize by symmetry. Hence, to a
first approximation, they can be regarded as effectively decoupled. In this
picture, superconductivity is expected to emerge from the Ni
3$d_{x^{2}-y^{2}}$ band, which can be described by a one-band Hubbard model:

$\mathcal{H}=\sum_{ij\sigma}t_{ij}\hat{c}_{i\sigma}^{\dagger}\hat{c}^{\phantom{\dagger}}_{j\sigma}+U\sum_{i}\hat{n}_{i\uparrow}\hat{n}_{i\downarrow}.$
(1)

Here, $t_{ij}$ denotes the hopping amplitude from site $j$ to site $i$;
$\hat{c_{i}}^{\dagger}$ ($\hat{c_{j}}$) are fermionic creation (annihilation)
operators and $\sigma$ marks the spin;
$\hat{n}_{i\sigma}=\hat{c}^{\dagger}_{i\sigma}\hat{c}^{\phantom{\dagger}}_{i\sigma}$
are occupation number operators. The Coulomb interaction is, because of
screening, restricted to the on-site interaction $U$. The electrons taken away
by the $A$-pocket are accounted for by properly relating the Sr doping to the
$d_{x^{2}-y^{2}}$ doping. This translation is displayed in the difference
between lower and upper $x$-axis of Fig. 1, which is based on multi-orbital
DFT+DMFT calculationsKitatani  _et al._ (2020).

####  II.1.2 Ab-initio determined parameters

This Hubbard model has been used sucessfully as an effective low-energy model
to calculate the superconducting dome in NdNiO2 Kitatani  _et al._ (2020).
Here, we employ exactly the same ab initio-derived parameters for NdNiO2, see
Table 1, where $t$ is the nearest, $t^{\prime}$ the next-nearest, and
$t^{\prime\prime}$ the next-next nearest neighbor hopping amplitude. The
tight-binding parameters are obtained after full relaxation of the lattice
parameters with VASPKresse and Hafner (1993) using the PBE Perdew  _et al._
(1996) version of the generalized gradient approximation (GGA). In the
presence of a substrate, we fix the in-plane lattice parameters to that of the
substrate. For this crystal structure, the hopping parameters are subsequently
obtained from a DFT calculation using Wien2K Blaha  _et al._ (2019); Schwarz
_et al._ (2002) and Wien2WannierKuneš  _et al._ (2010) to construct maximally
localized Wannier orbitalsMarzari  _et al._ (2012). As one can see from Table
1 the variation of these hopping parameters among different nickelates and
substrates is minimal, cf. the discussion below. Because of this insensitivity
to structural details we restrict ourselves in the following to one
$\textrm{D}\Gamma\textrm{A}$ calculation resembling the hopping parameters of
NdNiO2 (bulk). Specifically, we use the same hopping parameters
($t=0.395\,$eV, $t^{\prime}/t=-0.25$, $t^{\prime\prime}/t=0.12$) as in Ref.
Kitatani  _et al._ , 2020 for calculating the data in Fig. 1 and use a
temperature $T=60\,$K if not stated otherwise.

System |  $t$[eV] | $t^{\prime}/t$ | $t^{\prime\prime}/t$
---|---|---|---
NdNiO2 (bulk) | 0.395 | -0.24 | 0.12
NdNiO2/STO | 0.377 | -0.25 | 0.13
NdNiO2/LSAT | 0.392 | -0.25 | 0.13
LaNiO2 (bulk) | 0.389 | -0.25 | 0.12
LaNiO2/STO | 0.376 | -0.23 | 0.11
LaNiO2/LSAT | 0.390 | -0.22 | 0.11
PrNiO2/STO | 0.378 | -0.25 | 0.13
Table 1: Hopping parameters of the effective Ni 3$d_{x^{2}-y^{2}}$ orbital for
nickelates with different spacer cations, two substrates (STO: SrTiO3; LSAT:
(La,Sr)(Al,Ta)O3) and bulk.

As in Ref. Kitatani  _et al._ , 2020 the one-site Coulomb repulsion $U$ is
taken from constrained random phase approximation (cRPA) calculations. A
natural first choice would be to simply use $U=U_{\text{cRPA}}(\omega=0)$,
which is about $2.6$ eV for the single-band approximation of LaNiO2Nomura  _et
al._ (2019). However, a slightly enhanced static $U$ is needed to empirically
compensate for the neglected frequency dependence and increase of $U$ above
the effective plasma frequency. Further, previous studies Casula  _et al._
(2012); Han  _et al._ (2018); Honerkamp  _et al._ (2018) showed that cRPA
overscreens the interaction. To take the above into account and in agreement
with common practice, in Ref. Kitatani  _et al._ , 2020 a slightly enhanced
value of $U=8$ $t$ ($3.11$ eV) was considered as the best approximation.
Further, we also use the same scheme to account for the self-doping effect of
the $A$-pocket as in Ref. Kitatani  _et al._ , 2020. That is, the doping of
the one-band Hubbard model is determined from a 5 Ni-$d$ and 5 Nd(La)-$d$
DFT+DMFT calculation for SrxNd(La)1-xNiO2, see Supplemental Material of Ref.
Kitatani  _et al._ , 2020. Both here and in Ref. Kitatani  _et al._ , 2020,
the contribution of the pockets to superconductivity or the magnetic response,
beyond an effective doping, has been neglected. This is justified since, due
to its strong correlations, the $x^{2}-y^{2}$ orbital dominates the magnetic
susceptibility and (presumably) the pairing. Also note that the filling of the
pockets is low.

####  II.1.3 Magnetic susceptibility

We compute the magnetic susceptibility $\chi_{m}$ in
$\textrm{D}\Gamma\textrm{A}$ for the Hubbard model Eq. (1) using the
parameters motivated in the last section. $\textrm{D}\Gamma\textrm{A}$ uses a
DMFT solution as a starting point and introduces non-local correlations via
the Parquet or, in the simplified version used here, the Bethe-Salpeter
equationToschi  _et al._ (2007); Katanin  _et al._ (2009); Valli  _et al._
(2015). We outline in Appendix A, for the sake of completeness, the steps
necessary to obtain $\chi_{m}$; and refer the reader to Ref. Rohringer  _et
al._ , 2018 for a more in-depth discussion of the
$\textrm{D}\Gamma\textrm{A}$, to Ref. Kitatani  _et al._ , 2022 for details on
how to calculate $T_{c}$, and to Ref. Held, 2022 for a first reading.

On a technical note, we solve the Hubbard model with DMFT using continuous-
time quantum Monte-Carlo simulations in the hybridization expansion as
implemented in the w2dynamics package Wallerberger  _et al._ (2019). After
DMFT convergence the two-particle Green’s function of the corresponding
Anderson impurity model (AIM) is obtained and from it the generalized
$\textrm{D}\Gamma\textrm{A}$ susceptibility, as outlined in Appendix A. To
obtain the physical susceptibility we need to sum the generalized
susceptibility over two momenta and frequencies and perform the analytical
continuation described in the next section. To ensure good statistics, we use
order $10^{9}$ measurements in a high statistic run for both one and two-
particle objects.

####  II.1.4 Analytic continuation

In the $\textrm{D}\Gamma\textrm{A}$ calculation, all quantities are defined in
terms of imaginary time, or correspondingly, imaginary frequency (Matsubara
frequency), see Appendix A. When comparing to experiments, however, results on
the real axis are required. To obtain them we use the open-source package
ana$\\_$cont Kaufmann and Held (2023), which employs the maximum entropy
(maxent) method Gubernatis  _et al._ (1991) for bosonic correlation functions
like the physical susceptibility in Eq. (11). Since the physical
susceptibility $\chi^{q={\mathbf{q},\omega},\lambda}_{m}$ depends on momenta
$\mathbf{q}$ and frequency $\omega$, one analytic continuation of the
frequency is performed for each momentum. We fix all hyperparameters of the
maxent routine by employing the "chi2kink" method Kraberger  _et al._ (2017)
as implemented in ana$\\_$cont Kaufmann and Held (2023).

###  II.2 Experiment

####  II.2.1 Nickelate films

The precursor films of Pr1-xSrxNiO3 $(x=0,0.2)$ with $8\,$nm thickness were
grown on (001)-oriented SrTiO3 and (LaAlO3)0.3(Sr2TaAlO6)0.7 (LSAT) substrates
using pulsed laser deposition (PLD). Soft-chemistry reduction using CaH2
powder was then performed to remove the apical oxygens of the precursor films.
After reduction ($300^{\circ}$C, $100$ min), the perovskite phase was
transformed into an infinite-layer phase. The infinite layer Pr1-xSrxNiO2
$(x=0,0.2)$ films were transferred to the PLD chamber to be deposited on their
surfaces with $14$-nm thick SrTiO3 protective top layers. The Pr0.8Sr0.2NiO2
films grown on SrTiO3 substrates are superconducting ones with the onset
temperatures of superconducting transition $\sim$$12$ K.

####  II.2.2 RIXS experiments

Ni $L_{3}$-edge RIXS measurements were carried out at the I21 beamline at the
Diamond Light Source. The energy resolution was set to 39 meV (full-width-at-
half-maximum) at the Ni $L_{3}$ resonance (850.6 eV). Incident x-rays with
$\pi$ polarisation were used to enhance the paramagnon excitations. The
scattering angle was fixed at $154^{\circ}$ to maximize the in-plane momentum
transfer. All RIXS spectra are collected at 16 K and normalized to the weight
of the $dd$ excitations.

###  II.3 Caveats

The $\textrm{D}\Gamma\textrm{A}$ calculation of the magnetic susceptibility
is, as a matter of course, approximate. The DFT (GGA) starting point puts,
e.g., the oxygen orbitals to close to the Fermi level. This is a bit less
relevant for nickelates than for cuprates, and, in particular, when not
including the oxygen orbitals in subsequent DMFT calculations (as done here).
The next step, DMFT, is restricted to local correlations. Here, DMFT is used
for translating the Sr-doping to the doping of the Ni $3d_{x^{2}-y^{2}}$
orbital and for calculating a local vertex of the effective one-band Hubbard
model. From this, we then calculate non-local spin fluctuations in
$\textrm{D}\Gamma\textrm{A}$ through the Bethe-Salpeter ladder, and from
these, in turn, the superconducting pairing glue is obtained. This procedure
neglects how the superconducting fluctuations in the particle-particle channel
feed back to the antiferromagnetic spin fluctuations in the particle-hole
channel, and it also presumes that a local frequency-dependent vertex is a
reasonably good starting point. Generally, this vertex is much more local than
other properties such as the self-energy, even in the superconducting doping
regimeRohringer  _et al._ (2018). The good agreement of
$\textrm{D}\Gamma\textrm{A}$ and diagrammatic quantum Monte-Carlo simulations
has been evidenced in Ref. Schäfer  _et al._ , 2021.

Let us, here, mostly focus on two aspects that we believe are important to
keep in mind when comparing the theoretical spectrum of the magnetic
susceptibility to RIXS experiments: (i) The maxent approach is state-of-the-
art to solve a per se ill-conditioned problem: analytically continuing
imaginary time data to real frequencies. Its error grows with frequency, since
larger real frequencies only enter exponentially suppressed into the imaginary
time (or frequency) data. Further, maxent tends to broaden spectra Kaufmann
and Held (2023). Thus, we may expect the theoretical dispersion to be broader
and that the maxent error possibly even dominates the widths of the
dispersion, especially at higher frequencies.

(ii) On the experimental side, RIXS measurements do not probe magnetic
excitations exclusively, but rather elementary excitations in general Ament
_et al._ (2011). To extract the paramagnon dispersion, one fits several
functions to the RIXS raw data. For example, the authors of Ref. Lu  _et al._
(2021) used a Gaussian for the elastic peak, a damped harmonic oscillator
(DHO) for the magnon, an anti-symmetrized Lorentzian for phonons and the tail
of an anti-symmetrized Lorentzian for the high-energy background, see
supplementary information of Ref. Lu  _et al._ , 2021. Similarly, in our
fitting function, a DHO convoluted with the energy resolution function is used
to model the single magnon. The elastic peak is described with a Gaussian. The
smoothly varying background is mimicked by a polynomial function. An
additional Gaussian is included to describe the phonon mode at $\sim$$70$ meV
when it becomes visible at a large in-plane momentum ($>0.15$ r.l.u.). For
some dopings and momenta, we have a clear peak structure for the magnon and
the error involved in this fitting is mild. In other situations, e.g., close
to the $X$ momentum, there is only a minor hump or shoulder, and the magnon
energy is much more sensitive to the fitting procedure.

##  III Magnetic response in nickelate superconductors

With the good agreement of the theoreticalKitatani  _et al._ (2022) and
experimental phase diagramsLee  _et al._ (2023); Li  _et al._ (2020) in Fig.
1, we here aim at analyzing whether the underlying magnetic fluctuations that
mediate $d$-wave superconductivity in theory also agree with experiment. The
magnetic spectrum and paramagnon dispersion for the two parent compounds
NdNiO2 and PrNiO2 is shown in Fig. 2 (b) and (c), respectively. Here, Fig. 2
(a) displays the imaginary part of the magnetic susceptibility
$\chi_{\text{m}}^{\prime\prime}(\omega,q)$ as computed in
$\textrm{D}\Gamma\textrm{A}$ using a filling $n=0.95$ of the Ni
$3d_{x^{2}-y^{2}}$ orbital, originating from the self-doping due to the rare-
earth pockets in NdNiO2. For PrNiO2 this self-doping is minimally smaller
(3%). For the hopping parameters and Coulomb interaction, see Sec. II.1.2. The
dispersion is shown along the high-symmetry path in the Brillouin zone (BZ)
from $\Gamma$ to $X$ to $M$/2 to $\Gamma$ that is shown in the inset of Fig.
2(b). The data of Fig. 2(c) is our own measurement, that of Fig. 2(b) was
extracted from the RIXS measurements of Ref. Lu  _et al._ , 2021444As for the
dispersion, we plot the fitted damped harmonic oscillator, which is used as a
model to describe the (para)magnon. The fit was performed by us as described
by the authors of Ref. Lu  _et al._ , 2021 in their supplemental material..
Given that we did not adjust any parameters555Except for the magnitude, which
has been adjusted in both experiments and in theory to a similar scale., the
agreement between theory and the magnon dispersion extracted from RIXS is
quite good. This indicates that experimental spin fluctuations are similar to
those leading to $d$-wave superconductivity in the
$\textrm{D}\Gamma\textrm{A}$ calculations.

Figure 2: Nickelate magnetic susceptibility (colormap; arbitrary units) and
paramagnon dispersion (symbols). (a) $\textrm{D}\Gamma\textrm{A}$ for NdNiO2
at $T=60\,$K and $n=0.95$ because of self-doping. (b) RIXS measurements for
NdNiO2 on STO at $T=15$ K from Ref. Lu  _et al._ , 2021. (c) Our RIXS
measurements for PrNiO2 on STO at $T={16}$ K. Red, orange and blue symbols
mark the peak maximum. Inset in (b): Chosen $k$-path through the Brillouin
zone, where $\Gamma=(0,0,0)$, $X$ $=(\pi,0,0)$ and $M$/2 $=(\pi/2,\pi/2,0)$.

Looking more into the details, we see that the overall paramagnon bandwidth is
systematically a bit larger in $\textrm{D}\Gamma\textrm{A}$. For example, the
peak of $\chi_{\text{m}}(q=X,\omega)$ is at $\omega_{\rm peak}\sim 260$ meV in
$\textrm{D}\Gamma\textrm{A}$, while the measured one is close to $\sim 190$
meV. While the tendency of the maximum entropy method to broaden spectra might
also slightly affect the position of the maximum and the spectrum around the
$X$ point is more blurred in theory and experiment, overall the difference is
beyond the maxent error. In agreement with the overall width, the slope of the
linear dispersion around $\Gamma$ deviates somewhat and hence we conclude: the
overall width of the paramagnon dispersion in theory is noticeable larger than
in experiment. This difference corresponds to a larger effective spin coupling
$J$ in $\textrm{D}\Gamma\textrm{A}$, as we discuss in more detail in Sec.
IV.1.

Furthermore, the “dip” observed in the dispersion around the $M$/2 momentum,
which corresponds to a next-next nearest neighbor exchange $J^{\prime\prime}$
in a spin-wave picture, is more pronounced in theory than in experiment. In
Sec. IV.2, we show that using a larger $U=9\,t$ (instead of $U=8\,t$) results
in a better agreement of the spin wave dispersion and also of the phase
diagram of SrxNd1-xNiO2 on STO, which has considerably lower $T_{c}$’s than
SrxNd1-xNiO2 on LSAT. Please note that the origin for this experimental
difference is not the minute change in lattice constant, but that growing
SrxNd1-xNiO2 on LSAT results in cleaner films without stacking faults Lee  _et
al._ (2023). These “defect-free” films have a much lower resistivity and
higher $T_{c}$’s, and agree better with our best estimate $U=8\,t$.

Let us now turn toward the doped compounds. $\textrm{D}\Gamma\textrm{A}$
results for Sr0.2Nd0.8NiO2 with $x=0.125$ (effective filling $n=0.875$) and
$x=0.225$ ($n=0.80$) are displayed in Fig. 3(a) and (b), respectively, and
compared to the experimental dispersion (orange pentagons). Consistent with
experiment, we observe a shift towards lower energies around the $M$/2
momentum. Furthermore, the amplitude of $\chi_{\text{m}}$ decreases as
$q\to$$X$, which was also observed in Ref. Lu  _et al._ , 2021666See
supplementary material of Ref. Lu  _et al._ , 2021 Fig. S6(b)..

Figure 3: Paramagnon dispersion of SrxNd1-xNiO2 for (a) $x=0.125$ ($n=0.875$)
and (b) $x\simeq 0.225$ ($n=0.8$) in $\textrm{D}\Gamma\textrm{A}$. The red
dots mark the maximum of the $\textrm{D}\Gamma\textrm{A}$ dispersion at each
$k$-point, while the orange pentagons are the corresponding RIXS peak
locations from Ref. Lu  _et al._ , 2021.

Similar to the parent compound (Fig. 2), the bandwidth in
$\textrm{D}\Gamma\textrm{A}$ is larger also at finite doping. Particularly the
peak position at the $X$ momentum shows a substantial deviation compared to
the one extracted from RIXS. This may be partially (but not fully) attributed
to the bias introduced both on the theoretical and experimental sides. On the
one hand, we expect a worse performance of the numerical analytic continuation
for large frequencies. A spectrum already relatively flat at the $X$ momentum
might be additionally broadened because of the maximum entropy analytic
continuation. On the other hand, the intensity of the paramagnon peak is also
reduced in RIXS Lu  _et al._ (2021), making the experimental fitting procedure
more difficult. Notwithstanding, there is a larger theoretical dispersion (or
$J$) than in experiment. Qualitatively similar, $\textrm{D}\Gamma\textrm{A}$
and RIXS show that the minimum at the $M$/2 momentum of the parent compounds
turns into a flat dispersion or even a local maximum with doping.

##  IV Discussion

###  IV.1 Effective spin-wave picture

In the limit of a large Hubbard interaction $U$, the Hubbard model (Eq. 1)
reduces to an effective Heisenberg Hamiltonian. We refer the reader to Ref.
Delannoy  _et al._ , 2009 and references therein for an extensive discussion
for the one-orbital Hubbard model. While this mapping provides a direct
relation between $t$, $U$ and the effective spin couplings $J$, the
temperature does not enter, nor does the doping. Indeed, strictly speaking,
the mapping onto the spin model is possible only for an insulator (at half-
filling). Yet, the parent compound Nd(Pr,La)NiO2 is –in contrast to cuprates–
neither half-filled nor Mott-insulating because of the finite pockets.
Furthermore, the mapping onto a spin model becomes rather tedious in the
presence of hoppings $t^{\prime}$ and $t^{\prime\prime}$ beyond nearest-
neighbors Delannoy  _et al._ (2009). Nevertheless, the spin model and the
spin-wave dispersion provide a somewhat intuitive picture for understanding
the characteristics of spin fluctuations also in the present case of
nickelates.

For these reasons, we employ here the same approach as in experiment also for
the $\textrm{D}\Gamma\textrm{A}$ data: that is, we fit the spin-wave
dispersion of the Heisenberg model to our $\textrm{D}\Gamma\textrm{A}$ results
in order to extract information about effective spin-couplings $J$ and
$J^{\prime}$. Including only the nearest neighbor ($J$) and next-nearest
neighbor spin-exchange $J^{\prime}$, the effective classical spin-wave
dispersion for a spin-$1/2$ system is given by Coldea  _et al._ (2001);
Delannoy  _et al._ (2009); Ivashko  _et al._ (2019)

$\omega_{\textbf{k}}=Z_{C}\sqrt{\textrm{A}_{\textbf{k}}^{2}-\textrm{B}_{\textbf{k}}^{2}},$
(2)

where $Z_{C}$ is the spin-wave renormalization factor that accounts for the
effects of quantum fluctuations and

$\begin{split}\textrm{A}_{\textbf{k}}&=2J+2J^{\prime}[\cos(k_{x})\cos(k_{y})-1],\\\
\textrm{B}_{\textbf{k}}&=J[\cos(k_{x})+\cos(k_{y})].\end{split}$ (3)

Figure 4: Paramagnon dispersion obtained in $\textrm{D}\Gamma\textrm{A}$ for
LaNiO2 as a function of temperature. (a) $\textrm{D}\Gamma\textrm{A}$
dispersion (dots) and spin-wave fit (lines) using Eq. (2). (b) Magnetic
susceptibility $\chi_{m}(\omega_{\textrm{peak}})$ at the peak frequency
$\omega_{\textrm{peak}}$ of panel (a). Red: $361$ K; blue: $60$ K.

To better compare with the values obtained in experiment, we fix $Z_{C}=1.18$
as in Ref. Lu  _et al._ , 2021. Fig. 4 shows the paramagnon dispersion and the
corresponding value of the magnetic susceptibility
$\chi_{m}(\omega_{\textrm{peak}})$.

To a first approximation (if $J^{\prime}\ll J$), the width of the spin wave
dispersion is $2J$ in Eq. (2), with
$\omega_{{\mathrm{p}eak}\,{\mathbf{k}}}=\omega_{{\mathrm{p}eak}\,M/2}=2J$. A
finite $J^{\prime}$ adds a skewness to this as
$\omega_{{\mathrm{p}eak}\,X}=2J-4J^{\prime}$, whereas
$\omega_{{\mathrm{p}eak}\,M/2}=2J-2J^{\prime}$. The fact that the maximum of
the dispersion is at ${\mathbf{k}}=X$ thus implies a ferromagnetic (negative)
$J^{\prime}$. This is qualitatively similar to cuprates which have, however, a
considerably larger $J$ (e.g., $J=112\,$meV and $J^{\prime}=-15\,$meV for
La2CuO4Coldea  _et al._ (2001)). It is, on the other hand, different from
other nickelates such as La2NiO4Biało  _et al._ (2023) that show an
antiferromagnetic (positive) $J^{\prime}$ and opposite skewness. This is
because the latter require a multi-band description with Ni 3$d_{x^{2}-y^{2}}$
and Ni 3$d_{z^{2}}$ orbital, resulting in a larger effective $U$Biało  _et
al._ (2023).

The $\textrm{D}\Gamma\textrm{A}$ skewness is well described by the spin-wave
fit, but there is a pronounced minimum at the $M$/2-point (for the parent
compound) which is not well captured by the ($J$-$J^{\prime}$) spin-wave fit.
We presume higher-order couplings, which are difficult to fit to the numerical
$\textrm{D}\Gamma\textrm{A}$ data, are needed to account for such a minimum.
Specifically, a next-next neighbor exchange $J^{\prime\prime}$ adds a term

$\textrm{A}_{\textbf{k}}\rightarrow\textrm{A}_{\textbf{k}}-2J^{\prime\prime}(1-[\cos(2k_{x})+\cos(2k_{y})]/2)$
(4)

in Eq. (2) Coldea  _et al._ (2001). For positive (antiferromagnetic)
$J^{\prime\prime}$ this results in the observed local minimum at
$M/2=(\pi/2,\pi/2)$. The change of this minimum to a maximum as observed with
doping, then implies a change of sign of $J^{\prime\prime}$.

Similar to the cuprate superconductor La2CuO4 Coldea  _et al._ (2001) we
observe that the dispersion along the antiferromagnetic zone boundary becomes
more pronounced as temperature is lowered. This mode-hardening is mimicked by
a ferromagnetic next-nearest spin-coupling whose strength increases from
$J^{\prime}=-13$ meV at $361$ K to $J^{\prime}=-23$ meV at $60$ K. For $J$ the
trend is opposite and its value gets reduced from $76$ meV at $361$ K to $62$
meV at $60$ K. Experimentally, $J=63.6\pm 3.3$ meV and $J^{\prime}=-10.3\pm
2.3$ meV at $20$ K were reported in Ref. Lu  _et al._ , 2021 for NdNiO2 .
Similarly, we obtain $J=70.0\pm 5.5$ meV and $J^{\prime}=-8.0\pm 3.8$ meV at
$16$ K from RIXS for PrNiO2, see Table 2.

$U$ [t] | $8$ | $8$ | $9$ | $10$ | RIXS PrNiO2 | RIXS NdNiO2 Lu  _et al._ (2021)
---|---|---|---|---|---|---
$T$ [K] | $361$ | $60$ | $60$ | $60$ | 16 | $20$
$J$ [meV] | $76$ | $62$ | $64$ | $44$ | $70.0\pm 5.5$ | $63.6\pm 3.3$
$J^{\prime}$ [meV] | $-13$ | $-23$ | $-12$ | $-12$ | $-8.0\pm 3.8$ | $-10.3\pm 2.3$
Table 2: Effective spin-coupling $J$ and $J^{\prime}$ for NdNiO2 and PrNiO2
obtained with $\textrm{D}\Gamma\textrm{A}$ and measured in RIXS Lu  _et al._
(2021) given in units of meV. We list results for fits to
$\textrm{D}\Gamma\textrm{A}$ dispersions at different interaction values
$U=\\{8,9,10\\}$ in units of the hopping $t=0.389$ eV and two different
temperatures $T=\\{300\,\text{K},60\,\text{K}\\}$ for $U=8$ $t$.

###  IV.2 Interaction dependence

Table 2 suggests that the $\textrm{D}\Gamma\textrm{A}$-fitted $J$ value agrees
better with experiment if a larger $U=9\,t$ value is considered. Indeed,
$U=9\,t$ has been considered in Ref. Kitatani  _et al._ , 2020 to be the
maximal $U$ still consistent with the cRPA, while $U=8\,t$ is the best
estimate.

Fig. 5 shows the magnetic susceptibility calculated in the same way as in Fig.
2 and Fig. 3, but now for an interaction value $U=9\,t$. The colormaps in
(a,b,c) show $\chi_{\text{m}}^{q,\omega}$ along the same high-symmetry path as
shown in the inset of Fig. 2(b). The red dots mark the maximum at each
momentum, while the blue diamonds correspond to the peak maxima reported in
RIXS Lu  _et al._ (2021). Finally, Fig. 5(d) compares the peak location of the
paramagnon dispersion for several interaction values.

Figure 5: Paramagnon dispersion of SrxNd1-xNiO2 using a larger interaction
$U=9\,t\simeq 3.5$ eV. (a) Parent compound $x=0.0$ ($n=0.95$) at $T\simeq
60K$; (b) $x=0.125$ ($n=0.875$); (c) $x=0.225$ ($n=0.8$). The red dots mark
the maximum of the dispersion at each $k$-point, while the blue dots are the
corresponding peak locations measured in RIXS (taken from Ref. Lu  _et al._ ,
2021). (d) Paramagnon dispersion of the parent compound for different values
of the interaction $U$.

The overall width of the dispersion is reduced for larger $U$, as expected
from the spin-wave picture discussed in the previous subsection. Subsequently,
the agreement with experimental measurements is improved compared to the
results of $U=8$ $t$ in Fig. 2 and Fig. 3. This seemingly suggests that $U=9$
$t$ is more appropriate (but cf. our discussion below). Let us, also, compare
the superconducting transition temperature and discuss it along with the
paramagnon dispersion. Fig. 6 shows the $\textrm{D}\Gamma\textrm{A}$ phase
diagram for the two interaction values $U=8\,t$ (light red) and $U=9\,t$ (dark
red) together with two experimentally measured domes (blue curves). The
theoretical values are taken from Ref. Kitatani  _et al._ , 2020 and the
experimental ones are from Ref. Lee  _et al._ , 2023 [NdNiO2 on LSAT] and Ref.
Li  _et al._ , 2020 [NdNiO2 on STO].

Figure 6: Superconducting phase diagram $T_{c}$ as a function of Sr doping for
SrxNd1-xNiO2, comparing $\textrm{D}\Gamma\textrm{A}$ with $U=8\,t$($U=9\,t$)
from Ref. Kitatani  _et al._ , 2020 to experiment on a LSAT(STO) substrate
from Ref. Lee  _et al._ , 2023 (Ref. Li  _et al._ , 2020). The increase of
$T_{c}$ in $\textrm{D}\Gamma\textrm{A}$ when using LSAT instead of STO in-
plane lattice parameters is much weaker; the difference between the two
substrates hence most likely reflects the improvement thanks to cleaner,
defect-free films for the LSAT substrate.

The measurements for the same nickelate NdNiO2 on different substrates, one on
STO Li  _et al._ (2020) (light blue) and one on LSAT Lee  _et al._ (2023)
(dark blue), show about a factor of two difference in the superconducting
$T_{c}$. The higher $T_{c}$ for the LSAT substrate is, by the authors of Ref.
Lee  _et al._ , 2023, attributed to the difference in film cleanliness, with
fewer lattice defects observed for the LSAT substrate. This conclusion is
supported by scanning tunneling microscopy that show fewer Ruddlesden-Popper
stacking faults. Further support comes from a strongly reduced resistivity for
the samples grown on LSAT. A dependence of $T_{c}$ on the residual
resistivity, which was taken as a proxy measure for disorder and lattice
defects, was also reported in Ref. Hsu  _et al._ , 2022 and cleaner films show
a larger $T_{c}$ in a manner “not too different from cuprate superconductors”.
Indeed, cuprates show a decrease in $T_{c}$ for an increasing in-plane
Fukuzumi  _et al._ (1996), out-of-plane Hobou  _et al._ (2009); Eisaki  _et
al._ (2004); Fujita  _et al._ (2005) resistivity and when magnetic or non-
magnetic impurities are introduced Bonn  _et al._ (1994).

On the theoretical side, we observe a similar difference in transition
temperature between calculations using $U=8$ $t$ and $U=9$ $t$, respectively.
However, the respective Wannier tight-binding parameters for LaNiO2777It is
common practice to use La instead of Nd to avoid the (wrong) appearance of
$f$-bands around the Fermi energy. For treating Nd directly the Nd $f$-shell
electrons are usually considered as core electrons. with in-plane lattice
constants fixed to those of STO and LSAT are quite similar, see Sec. II.1.2.
That is, we find that the nearest-neighbor hopping $t$ increases by about
$\sim 4\%$ from STO to LSAT, while the ratios of $t^{\prime}/t$ and
$t^{\prime\prime}/t$ remain essentially the same. An increase of $t$ is not
surprising as the smaller in-plane lattice constant of LSAT increases the
orbital overlap. On the other hand, the Hubbard interaction
($U_{\text{cRPA}}=2.6$ eV) essentially does not change when performing
constrained random phase approximation calculations for LaNiO2 with the $a-b$
lattice parameters fixed to either that corresponding to LSAT or that of STO,
respectively.

Considering these changes of the effective single-band Hamiltonian, we expect
samples grown on LSAT to have an intrinsically larger $T_{c}$, since $t$ sets
the energy scale and a smaller $U/t$ is also beneficial Kitatani  _et al._
(2020). That being said, the expected difference in $T_{c}$, as a result of
the slightly different intrinsic models, is closer to $10-15\%$888About $\sim
4\,\%$ because of a change in $t$ and another $5-10\,\%$ because of the change
in $U/t$., but almost certainly not a factor of two 999Larger changes of the
hopping parameters are possible by applying pressure, see Ref. Di Cataldo  _et
al._ , 2023.. For this reason we conclude that changes in our effective
single-band Hubbard model do not explain differences in the measured $T_{c}$’s
for different substrates. The difference has to lie somewhere else, and the
reduced number of defects when growing nickelates on LSAT is the most likely
explanation for the enhanced $T_{c}$ and reduced resistivity, as also
originally suggested in Ref. Lee  _et al._ , 2023.

Following this argument, the appropriate Hubbard interaction for the effective
single-band description of infinite-layer nickelates should be close to our
best estimate $U=8\,t$ with an intrinsic $T_{c}^{\text{max}}\simeq 30$ K,
comparable to that measured on LSAT101010Effects beyond the single-band model
will always lead to some discrepancy. For this reason, we refrain from fine-
tuning parameters.. Consequently, we would expect the $T_{c}$ of samples grown
on STO to be similar (within 4%) once sample of comparable quality are
synthesized. What remains to be understood is how defects and lattice disorder
influence the paramagnon dispersion and $T_{c}$.

###  IV.3 Effect of disorder and stacking faults on the paramagnon dispersions

To fully address the influence of impurities and lattice defects on the
paramagnon dispersion, large-scale calculations for supercells that include
these defects would be required. Such calculations are not feasible at the
moment, at least not for $\textrm{D}\Gamma\textrm{A}$ calculations or similar
many-body methods that include non-local fluctuations. For this reason, we
will restrict ourselves here instead to qualitative considerations.

One possibility is that defects reduce the effective antiferromagnetic
coupling strength $J$ and, with reduced antiferromagnetic fluctuations, also
$T_{c}$. While estimating the absolute influence of such local defects in RIXS
is very difficult, if not impossible, samples that show a different $T_{c}$
can be compared. Such a study would include measurements of several samples of
the same “species”, e.g., Sr0.2Nd0.8NiO2 on STO, which show a sample-to-sample
variation in $T_{c}$. Along the same lines, comparing the paramagnon
dispersions for samples grown on different substrates (e.g., STO and LSAT)
would yield valuable information about the connection between the paramagnon
dispersion and $T_{c}$. A study similar to the latter has already been
performed for the related PrNiO2 compound Gao  _et al._ (2022). The
measurements suggest that the paramagnon dispersion and $J$ are similar for
samples grown on LSAT and STO. However, those measurements were done on the
non-superconducting parent compound. Hence, it would be interesting to check
if the reported results remain unchanged if samples with different $T_{c}$’s
are measured directly. Let us, in this context, mention that for cuprates it
was possible to correlate the increase of $T_{c}$ with the increase of $J$ for
different cupratesWang  _et al._ (2022b). Along this line of thinking, the
better agreement of $T_{c}$ and the RIXS spectrum of NdNiO2 on STO for the
larger $U=9\,t$ might just mimic the suppression induced by disorder which is
not included in our calculations.

Let us also point out another way how lattice defects might influence $T_{c}$:
decreasing the magnetic correlation length $\xi$. Particularly, if stacking
faults and similar defects introduce artificial “grain boundaries”111111See
for examples Fig. 1 in Ref. Lee  _et al._ , 2023., $\xi$ might be restricted
to stay below the typical grain-boundary distance, without directly changing
the effective antiferromagnetic coupling strength $J$. Though conceptually
somewhat different, the $\lambda$-correction in $\textrm{D}\Gamma\textrm{A}$
Katanin  _et al._ (2009) has a similar effect in the sense that $\lambda$
causes a decrease in the magnetic correlation length. Such a reduced
correlation length (added mass), however, essentially does not change the
paramagnon dispersion, see Fig. 7 and the discussion in the next paragraphs.
Furthermore, the intensity primarily changes around the M $=(\pi,\pi)$
momentum, where the susceptibility and $\xi$ are the largest.

Such an effective paramagnon-mass enhancement or reduced $\xi$ is difficult to
extract from RIXS, which cannot access the $M$ point in nickelates. Yet, it is
precisely the strength of the susceptibility around the $M$ momentum, which,
from a spin-fluctuation or $\textrm{D}\Gamma\textrm{A}$ perspective, is most
important for $T_{c}$. The $M$ point and a prospective difference in
correlation length $\xi$ for different substrates might be accessible in
neutron scattering. If measurements of the magnetic correlation length $\xi$
for superconducting samples grown on different substrates show a suppressed
$\xi$ at the M point for samples grown on STO compared to those grown on LSAT,
this would support this second disorder scenario.

To investigate the influence of a suppressed correlation length onto the
paramagnon dispersion, we compare $\chi_{m}$ between DMFT and
$\textrm{D}\Gamma\textrm{A}$ in Fig. 7 along a high-symmetry path through the
BZ now including the $M$ momentum [see inset in panel (d) for the Brillouin
zone]. We choose the overdoped compound ($x=0.225$) since DMFT shows no
antiferromagnetic order for this doping. Hence, we can directly compare
$\chi_{\text{m}}$ between DMFT (no $\lambda$ correction) and
$\textrm{D}\Gamma\textrm{A}$ (with $\lambda$ correction), cf. Ref. Rohringer
_et al._ , 2018. The DMFT with a large correlation length $\xi$ is shown in
panel (c), while $\textrm{D}\Gamma\textrm{A}$ with a shorter $\xi$ is
displayed the latter in panel (a).

Figure 7: Paramagnon dispersion of SrxNd1-xNiO2 for $x\sim 0.225$ ($n=0.8$)
within the effective single-orbital Hubbard model scenario. (a) Colormap of
$\chi_{\text{m}}^{q,\omega}$ calculated by (a) $\textrm{D}\Gamma\textrm{A}$
and (c) DMFT. (b) Comparison of the maximum of the dispersion corresponding to
the lines on top of the colormaps in (a) and (c), respectively. (d) Peak value
of the magnetic susceptibility; inset: $k$-path in the BZ, where
$\Gamma=(0,0,0)$, $X$ $=(\pi,0,0)$ and $M$ $=(\pi,\pi,0)$. The main difference
between DMFT and $\textrm{D}\Gamma\textrm{A}$ is the reduced correlation
length. Parameters: $t=0.389\,$eV, $t^{\prime}/t=-0.25$,
$t^{\prime\prime}/t=0.12$, $U=8\,t$, $T=60\,$K.

We draw the location of the peak in $\chi_{\text{m}}$ at each momentum in Fig.
7(b). It remains essentially unchanged in the presence of a
$\textrm{D}\Gamma\textrm{A}$ $\lambda$-correction, i.e., with a reduced $\xi$.
The magnitude of the respective susceptibility at the peak is, however,
drastically different around the M momentum, as can be seen in Fig. 7(d). A
suppression of $\chi_{m}$ is strongest when the DMFT susceptibility is large,
which is no surprise since $\chi_{m}\rightarrow 1/(1/\chi_{m}+\lambda)$ in
$\textrm{D}\Gamma\textrm{A}$. Hence, a reduction of the correlation length for
antiferromagnetic fluctuations is expected to be virtually invisible in RIXS
measurements, which do not access the M point. At the same time, the reduction
of $T_{c}$ can be dramatic.

Let us further note that we observe incommensurate antiferromagnetic
fluctuations, evidenced by the shift of the maximum amplitude slightly away
from $M$ in Fig. 7(d). This is typical for overdoped cuprates Lipscombe  _et
al._ (2009); Fujita  _et al._ (2012),121212Let us note that the prototypical
cuprate L2CuO4 is a 214 layered system compared to the 112 layer structure of
nickelates.. For nickelates, measurements that could distinguish commensurate
from incommensurate spin fluctuations have, to the best of our knowledge, not
been performed yet.

##  V Conclusion

The pairing symmetry obtained in $\textrm{D}\Gamma\textrm{A}$ for nickelates
is $d$-wave, reminiscent of that in cuprates Keimer  _et al._ (2015). On the
experimental side, the pairing symmetry of nickelates remains an open question
as results are still inconclusiveGu  _et al._ (2020); Harvey  _et al._ (2022);
Chow  _et al._ (2022). Also the mechanism for superconductivity remains highly
controversial, not only for nickelates but also for cuprates. Many different
mechanisms have been proposed Lee  _et al._ (2006); Scalapino (2012); Fradkin
_et al._ (2015); Keimer  _et al._ (2015). In $\textrm{D}\Gamma\textrm{A}$ spin
fluctuations mediate superconductivity131313 Cf. Refs. Kitatani  _et al._ ,
2019 and Dong  _et al._ , 2022 for an in-depth analysis of the pairing vertex
in the Hubbard model and its fluctuation diagnostic Schäfer and Toschi
(2021)..

Further, in the case of nickelates, even the minimal model is hotly debated.
Among others, the relevance of multi-orbital physics Kreisel  _et al._ (2022);
Werner and Hoshino (2020); Petocchi  _et al._ (2020), Kondo physics Yang and
Zhang (2022), or even phonons Alvarez  _et al._ (2022) have been suggested.
The single-band Hubbard model with an appropriately calculated doping of the
Ni 3$d_{x^{2}-y^{2}}$ orbital, that was used in the present paper, is arguably
the simplest possible model for spin fluctuations and superconductivity in
nickelates. It correctly reproduces the doping range of superconductivity, the
absolute value of $T_{c}$, and also the skewness of the phase diagram. Here,
the skewness is a consequence of the largely decoupled ligand pockets which
accommodate part of the holes from the Sr-doping in a non-linear fashion. This
skewness of the superconducting dome is a notable difference to cuprates.

Given the good agreement of the superconducting phase diagram and
antiferromagnetic fluctuations at its origin, a critical check whether or not
spin fluctuations with similar characteristics are observed in experiment is
imperative. To achieve this validation, we compared the paramagnon dispersion
calculated in $\textrm{D}\Gamma\textrm{A}$ with the one extracted from RIXS,
both from our measurements and those from Ref. Lu  _et al._ , 2021. We find
the spin spectrum is overall similar, especially when considering biases
expected both from theory and experiment. This means that the experimental
spin fluctuations are consistent with the observed $T_{c}$ in nickelates
within the spin-fluctuation scenario of superconductivity. Then, the weaker
spin fluctuations (or $J$) in nickelates, compared to cuprates, might also
explain their lower $T_{c}$.

The agreement is however not perfect. The total width of the paramagnon
dispersion (or $J$) is somewhat smaller in experiment than in theory. Using an
enhanced Coulomb interaction $U=9\,t$ in $\textrm{D}\Gamma\textrm{A}$ much
better matches the RIXS spectrum and also the superconducting phase diagram of
SrxPr1-xNiO2 and SrxNd1-xNiO2 on STO. However, this might be by chance. The
larger $T_{c}$’s are observed for SrxNd1-xNiO2 on a LSAT substrate. The
difference of LSAT and STO in-plane lattice constants and thus also the
$\textrm{D}\Gamma\textrm{A}$ hopping parameters are way too small to explain
the by a factor-of-two higher $T_{c}$.

Having much less defects (Ruddlesden-Popper stacking faultsLee  _et al._
(2023)) is instead expected to be the origin of the higher $T_{c}$ and better
conductivity of SrxNd1-xNiO2 on LSAT. Increasing $U$ might only be an
imperfect way to mimic this disorder reduction of antiferromagnetic spin
fluctuations and $T_{c}$. As discussed in Sec. IV.3 local disorder can reduce
$J$. From this perspective, SrxNd1-xNiO2 on LSAT with less disorder should
show a larger $J$ and dispersion in RIXS. However, if the disorder is rather
cutting-off the spin correlation length –and Ruddlesden-Popper stacking faults
might hint in this direction–, we show that the effect on the RIXS dispersion
can be negligible. In this scenario, disorder only affects the peak height at
$M=(\pi,\pi)$, which is not accessible in RIXS but in neutron scattering
experiments.

In all, we believe that our joint theoretical and experimental investigation
strengthens the case for spin-fluctuation mediated superconductivity in
nickelates.

Acknowledgments We thank Simone Di Cataldo, Oleg Janson, and Jan Kuneš for
helpful discussions. We further acknowledge funding through the Austrian
Science Funds (FWF) projects I 5398, I 6142, P 36213, SFB Q-M&S (FWF project
ID F86), the Swiss National Science Foundation under 200021$\\_$188564, Grant-
in-Aids for Scientific Research (JSPS KAKENHI) Grants No. JP21K13887 and No.
JP23H03817, the Research Grants Council of Hong Kong (ECS No. 24306223), and
Research Unit QUAST by the Deutsche Foschungsgemeinschaft (DFG; project ID
FOR5249) and FWF (project ID I 5868). L.S. is thankful for the starting funds
from Northwest University. I.B. acknowledges support from the Swiss Government
Excellence Scholarship under the project number ESKAS-Nr: 2022.0001.
Calculations have been done in part on the Vienna Scientific Cluster (VSC).
For the purpose of open access, the authors have applied a CC BY public
copyright licence to any Author Accepted Manuscript version arising from this
submission.

##  Appendix A Magnetic susceptibility in $\textrm{D}\Gamma\textrm{A}$

As a first step to calculate the magnetic susceptibility $\chi_{m}$, we solve
the Hubbard model Eq. (1) within DMFT and subsequently sample the local two-
particle Green’s function ($G^{(2)}$) for the corresponding Anderson impurity
model (AIM):

$G^{2}(\nu_{1},\nu_{2},\nu_{3},\nu_{4})=\int_{0}^{\beta}\int_{0}^{\beta}\int_{0}^{\beta}\int_{0}^{\beta}d\tau_{1}d\tau_{2}d\tau_{3}d\tau_{4}G^{2}(\tau_{1},\tau_{2},\tau_{3},\tau_{4})e^{{i\mkern
1.0mu}\nu_{1}\tau_{1}}e^{-{i\mkern 1.0mu}\nu_{2}\tau_{2}}e^{{i\mkern
1.0mu}\nu_{3}\tau_{3}}e^{-{i\mkern 1.0mu}\nu_{4}\tau_{4}},$ (5)

where the two-particle Green’s function in terms of imaginary time ($\tau$) is
defined as:

$G^{2}_{\begin{subarray}{c}1234\end{subarray}}(\tau_{1},\tau_{2},\tau_{3},\tau_{4})\equiv\langle{\mathcal{T}[\hat{c}_{1}^{\phantom{\dagger}}(\tau_{1})\hat{c}_{2}^{\dagger}(\tau_{2})\hat{c}_{3}^{\phantom{\dagger}}(\tau_{3})\hat{c}_{4}^{\dagger}(\tau_{4})]}\rangle.$
(6)

Since our Hamiltonian Eq. (1) does not explicitly depend on time, one
frequency can be removed by using energy conservation. If not stated otherwise
we will use the particle-hole (ph) notation:

$\displaystyle\underline{\text{ph-notation:}}$
$\displaystyle\underline{\text{pp-notation:}}$ $\displaystyle\nu_{1}=\nu$
$\displaystyle\nu_{1}=\nu$ $\displaystyle\nu_{2}=\nu-\omega$
$\displaystyle\nu_{2}=\omega-\nu^{\prime}$
$\displaystyle\nu_{3}=\nu^{\prime}-\omega$ $\displaystyle\nu_{3}=\omega-\nu$
$\displaystyle\nu_{4}=\nu^{\prime}$ $\displaystyle\nu_{4}=\nu^{\prime}$

The two-particle Green’s function in Eq. (5) can be expressed in terms of
disconnected (free) and connected (vertex) parts:

$\displaystyle G^{(2),\omega\nu\nu^{\prime}}_{\sigma\sigma^{\prime}}$
$\displaystyle=$ $\displaystyle\delta_{\omega
0}G^{\nu}_{\sigma}G^{\nu^{\prime}}_{\sigma^{\prime}}-\delta_{\nu\nu^{\prime}}\delta_{\sigma\sigma^{\prime}}G^{\nu}_{\sigma}G^{\nu-\omega}_{\sigma^{\prime}}$
(7)
$\displaystyle+\frac{1}{\beta}G^{\nu}_{\sigma}G^{\nu-\omega}_{\sigma}F^{\omega\nu\nu^{\prime}}_{\sigma\sigma^{\prime}}G^{\nu^{\prime}-\omega}_{\sigma^{\prime}}G^{\nu^{\prime}}_{\sigma^{\prime}},$

where $F$ is the so-called vertex function which encodes all scattering events
on the two-particle level and
$G(\nu_{1},\nu_{2})=-\int_{0}^{\beta}\int_{0}^{\beta}d\tau_{1}d\tau_{2}\;\langle{\mathcal{T}c_{1}(\tau_{1})c^{\dagger}(\tau_{2})}\rangle
e^{\nu_{1}\tau_{1}}e^{-\nu_{2}\tau_{2}}$ is the one-particle Green’s function.
All diagrams contained in the vertex function $F$ can be classified
unambiguously by the parquet decomposition

$F^{\omega\nu\nu^{\prime}}_{\sigma\sigma^{\prime}}=\Lambda^{\omega\nu\nu^{\prime}}+\Phi^{\omega\nu\nu^{\prime}}_{\text{pp},\sigma\sigma^{\prime}}+\Phi^{\omega\nu\nu^{\prime}}_{\overline{\text{ph}},\sigma\sigma^{\prime}}+\Phi^{\omega\nu\nu^{\prime}}_{\text{ph},\sigma\sigma^{\prime}}\,.$
(8)

based on their two-particle reducibility, i.e., whether or not a diagram
decomposes if one “cuts” two one-particle Green’s function lines. For a
diagrammatic depiction and the definition of $\Lambda$ and $\Phi_{r}$ see Fig.
8.

Figure 8: Parquet decomposition of the full vertex function $F$ into its
components based on the two-particle reducibility. $\Lambda$ is the fully
irreducible vertex, and $\phi_{r}$ with
$r=\\{\textrm{ph},\overline{\textrm{ph}},\textrm{pp}\\}$ is reducible part of
$F$ in the particle-hole (ph), particle-hole transversal
($\overline{\textrm{ph}}$) and particle-particle (pp) channel.

We now also define the irreducible vertices $\Gamma_{r}=F-\Phi_{r}$, where
$r=\\{\textrm{ph},\overline{\textrm{ph}},\textrm{pp}\\}$ is either one of the
three scattering channels. Furthermore, we assume SU(2) symmetry, i.e.,
restrict ourselves to the paramagnetic phase. This spin symmetry leads to only
two independent spin combinations
$F_{m}=F_{\uparrow\uparrow}-F_{\uparrow\downarrow}$ (magnetic) and
$F_{d}=F_{\uparrow\uparrow}+F_{\uparrow\downarrow}$ (density) Rohringer  _et
al._ (2018). The vertex $F$ can also be expressed directly in terms of any
irreducible vertex via the respective Bethe-Salpeter equation (BSE):

$F^{\omega\nu\nu^{\prime}}_{d/m}=\Gamma^{\omega\nu\nu^{\prime}}_{d/m,\text{ph}}+\frac{1}{\beta}\sum_{\nu_{1}}\Gamma^{\omega\nu\nu^{\prime}}_{d/m,\text{ph}}G^{\nu_{1}}G^{\nu_{1}-\omega}F^{\omega\nu\nu^{\prime}}_{d/m}.$
(9)

Here, we have written the BSE only for $\Gamma_{\textrm{ph}}$, and we use the
inverse of Eq. (9) to extract the local $\Gamma_{\textrm{ph}}$. To obtain a
$q$-dependent susceptibility we use a BSE-like equation for the generalized,
now momentum-$\mathbf{k}$ dependent susceptibility:

$\chi_{d/m}^{qkk^{\prime}}=\chi_{0}^{qkk^{\prime}}-\chi_{0}^{qkk}\frac{1}{\beta^{2}}\sum_{k_{1}}\Gamma^{qkk_{1}}_{\text{d/m}}\chi_{d/m}^{qk_{1}k^{\prime}}.$
(10)

Here $k=({\mathbf{k}},\nu)$ and $q=({\mathbf{q}},\omega)$ are fermionic and
bosonic four-point vectors in generalization of local, frequency-only-
dependent quantities of the AIM. Further, we approximate
$\Gamma^{qkk_{1}}_{\text{d/m}}=\Gamma^{\omega\nu\nu^{\prime}}_{\text{d/m}}$,
i.e., the respective irreducible DMFT vertex. The physical susceptibility is
finally obtained by summing over $kk^{\prime}$, i.e.
$\chi_{\text{phys},d/m}=\frac{1}{\beta^{2}}\sum_{kk^{\prime}}\chi_{d/m}^{qkk^{\prime}}$.
However, a susceptibility constructed this way contains divergences stemming
from the mean-field phase transitions in DMFT, which largely overestimates
critical temperatures Rohringer  _et al._ (2011). This is even more true in
two dimensions where the Mermin-Wagner theorem holdsMermin and Wagner (1966).
We remedy this artifact by employing a regularization parameter
$\lambda$Katanin  _et al._ (2009); Rohringer  _et al._ (2018) (instead of
doing fully fledged parquetValli  _et al._ (2015) or self-consistent
$\textrm{D}\Gamma\textrm{A}$ Kaufmann  _et al._ (2021)). Here, $\lambda$ is
fixed by enforcing the sum-rule

$\frac{1}{2\beta}\sum_{q}\big{(}\chi^{q,\lambda_{m}}_{m}+\chi^{q,\lambda_{d}}_{d}\big{)}=\frac{n}{2}\cdot\big{(}1-\frac{n}{2}\big{)},$
(11)

and
$\chi^{q,\lambda}_{d/m}=\big{[}\big{(}\chi^{q}_{d/m}\big{)}^{-1}+\lambda_{d/m}\big{]}^{-1}$
is the $\lambda$-corrected susceptibility.

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