# Unraveling higher-order corrections in the spin dynamics of RIXS spectra

Umesh Kumar<EMAIL_ADDRESS>Theoretical Division, Los Alamos National
Laboratory, Los Alamos, New Mexico 87545, USA  Abhishek Nag
<EMAIL_ADDRESS>Diamond Light Source, Harwell Campus, Didcot OX11
0DE, United Kingdom  Jiemin Li  Diamond Light Source, Harwell Campus, Didcot
OX11 0DE, United Kingdom  Beijing National Laboratory for Condensed Matter
Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190,
China  H. C. Robarts  Diamond Light Source, Harwell Campus, Didcot OX11 0DE,
United Kingdom  H. H. Wills Physics Laboratory, University of Bristol, Bristol
BS8 1TL, United Kingdom  A. C. Walters  Diamond Light Source, Harwell Campus,
Didcot OX11 0DE, United Kingdom  Mirian García-Fernández  Diamond Light
Source, Harwell Campus, Didcot OX11 0DE, United Kingdom  R. Saint-Martin
Institut de Chimie Moléculaire et des Matériaux d’Orsay, Université Paris-
Saclay, UMR 8182, 91405 Orsay, France  A. Revcolevschi  Institut de Chimie
Moléculaire et des Matériaux d’Orsay, Université Paris-Saclay, UMR 8182, 91405
Orsay, France  Justine Schlappa  European X-Ray Free-Electron Laser Facility
GmbH, Holzkoppel 4, 22869 Schenefeld, Germany  Thorsten Schmitt  Photon
Science Division, Paul Scherrer Institut, 5232 Villigen PSI, Switzerland
Steve Johnston  Department of Physics and Astronomy, The University of
Tennessee, Knoxville, TN 37996, USA  Ke-Jin Zhou<EMAIL_ADDRESS>Diamond Light Source, Harwell Campus, Didcot OX11 0DE, United Kingdom

###### Abstract

Resonant inelastic x-ray scattering (RIXS) is an evolving tool for
investigating spin dynamics of strongly correlated materials, which
complements inelastic neutron scattering. Both techniques have found that non-
spin-conserving (NSC) excitations in quasi-1D isotropic quantum
antiferromagnets are confined to the two-spinon phase space. Outside this
phase space, only spin-conserving (SC) four-spinon excitations have been
detected using O $K$-edge RIXS. Here, we investigate SrCuO2 and find four-
spinon excitations outside the two-spinon phase space at both O $K$\- and Cu
$L_{3}$-edges. Using the Kramers-Heisenberg formalism, we demonstrate that the
four-spinon excitations arise from both SC and NSC processes at Cu
$L_{3}$-edge. We show that these new excitations only appear in the second-
order terms of the ultra-fast core-hole lifetime expansion and arise from
long-range spin fluctuations. These results thus open a new window to the spin
dynamics of quantum magnets.

The elementary excitations of a system encode information about the
microscopic origin of its behavior. For example, magnetic excitations are
extensively studied in high-$T_{c}$ superconducting cuprates for their
possible role in the pairing mechanism and provide access to critical physical
parameters like their correlation strength, superexchange interactions, or
electronic structure parameters Scalapino (1995); Chubukov  _et al._ (2003);
Orenstein and Millis (2000); Sidis  _et al._ (2004); Scalapino (2012); Lee
_et al._ (2006); Headings  _et al._ (2010). Inelastic neutron scattering (INS)
has traditionally been the method of choice for probing magnetic correlations.
In recent years, however, resonant inelastic x-ray scattering (RIXS) has
emerged as an important complementary technique to INS Ament  _et al._ (2011);
Jia  _et al._ (2016); Braicovich  _et al._ (2010); Minola  _et al._ (2015); Le
Tacon  _et al._ (2011); Dean  _et al._ (2012); Kim  _et al._ (2012); Lu  _et
al._ (2018); Robarts  _et al._ (2021); Nag  _et al._ (2020); Ament  _et al._
(2009); Braicovich  _et al._ (2010); Gel’mukhanov  _et al._ (2021). Its large
scattering cross-section and element specificity are advantageous over INS for
small sample volumes, as well as for complex systems with multiple magnetic
elements Pelliciari  _et al._ (2021); Umetsu  _et al._ (2019).

While the relationship between the INS cross-section and the dynamical spin
structure factor $S({\bf q},\omega)$ is well understood, the same cannot be
said about the RIXS cross-section, which is described by the Kramers-
Heisenberg (KH) formalism Ament  _et al._ (2011). Here, the presence of a
core-hole in the intermediate state allows for a diverse set of scattering
channels including both spin-conserving (SC, $\Delta S_{\mathrm{tot}}=0$) and
non-spin-conserving (NSC, $\Delta S_{\mathrm{tot}}=1$) processes. The RIXS
intensity can only be directly equated with the dynamical structure factors
under certain scattering conditions Jia  _et al._ (2014) or in the limit of
ultra-short core-hole lifetimes Ament  _et al._ (2007), a fact that is
becoming more evident with the recent developments in resolving the
polarization of the scattered x-rays Betto  _et al._ (2021); Minola  _et al._
(2015); Fumagalli  _et al._ (2019). While this aspect complicates the
interpretation of RIXS experiments, it has also led to the observation of
novel excitations such as orbitons Schlappa  _et al._ (2012); Wohlfeld  _et
al._ (2013) and SC four-spinon excitations outside two-spinon phase space
Schlappa  _et al._ (2018); Kumar  _et al._ (2018) in the one-dimensional (1D)
Heisenberg antiferromagnetic (HAFM) spin chain Sr2CuO3.

The relationship between the RIXS intensity and more traditional dynamical
structure factors can be derived systematically using the ultra-short core-
hole lifetime (UCL) expansion Jia  _et al._ (2014); Ament  _et al._ (2007);
Forte  _et al._ (2011); Jia  _et al._ (2016). Here, one expands the RIXS
intensity as a series of increasingly complicated multi-particle correlation
functions in powers of the inverse core-hole lifetime $1/\Gamma$. For example,
the transition metal $L$-edge RIXS spectra of a $3d^{9}$ system can be
approximated as $I({\bf q},\omega)\approx|W|^{2}\left[S({\bf
q},\omega)/\Gamma+S^{\prime}({\bf q},\omega)/\Gamma^{2}\right]$ in certain
scattering conditions Jia  _et al._ (2016). Here, $W$ is a matrix element
related to the scattering geometry, $S({\bf q},\omega)$ is the dynamical spin
structure factor, and $S^{\prime}({\bf q},\omega)$ is the leading correction.
Notably, the quantities appearing in the expansion can be computed in the
absence of the core-hole, allowing one to use numerous methods to attack the
problem. For example, the spectra for the SC channel generated via double
spin-flips in 1D cuprate CaCu2O3 Bisogni  _et al._ (2014) were shown to
fractionalize entirely into two-spinons using the Bethe ansatz Klauser  _et
al._ (2011). In the case of spin-ladders, three-triplon bound states were
predicted in the same channel Ferkinghoff  _et al._ (2020) using the
continuous unitary transformation method. However, the extent to which higher-
order corrections to the UCL expansion are required for the NSC excitations at
$L$-edges Jia  _et al._ (2016); Kumar  _et al._ (2019); Forte  _et al._ (2008)
has not been widely studied.

In this letter, we report the spin dynamics of the 1D HAFM SrCuO2 probed by
high-resolution Cu $L_{3}$\- and O $K$-edge RIXS. We observe a two-spinon
continuum, in agreement with INS on this system Zaliznyak  _et al._ (2004).
Unlike INS, however, we also observe considerable spectral weight from four-
spinon excitations located in a region of phase space outside of the two-
spinon continuum. Moreover, the distribution of spectral weight for these
excitations observed at each edge is distinct. The four-spinon excitations
observed at the O $K$-edge arise from the SC channel as found in the related
work on Sr2CuO3 Schlappa  _et al._ (2018). Using detailed theoretical modeling
within the KH formalism for the $t$-$J$ model, we demonstrate that the four-
spinon excitations observed at Cu $L_{3}$-edge spectra arise from both SC and
NSC scattering channels. We further evaluate new dynamical correlation
functions derived from the UCL expansion to show that these four-spinon
excitations can only be accounted for by including second-order terms of the
expansion involving long-range spin excitations. These observations thus open
up possibilities for investigating long-range many-body excitations in quantum
magnets.

Figure 1: (a) A schematic diagram of the RIXS scattering geometry for SrCuO2.
Panels (b) and (c) show the Cu low-energy Cu $L_{3}$\- and O $K$-edge RIXS
spectra, respectively, measured at $q_{\parallel}=0.175$. Panels (d) and (e)
show the corresponding data at $q_{\parallel}=0$. Also shown in (d) is the
RIXS spectrum multiplied by a factor of five and vertically shifted. At
$q_{\parallel}=0$, the energy of the two-spinon excitations and their spectral
weight are zero. For $q_{\parallel}=0.175$, the two-spinons are approximated
by the M$\ddot{\mathrm{u}}$ller Ansatz Karbach  _et al._ (1997). The vertical
dashed line indicates the upper boundary of the four-spinon continuum Caux and
Hagemans (2006). In all cases, additional spectral weight (marked by arrows)
is observed outside the two-spinon phase space but inside the four-spinon
phase space.

SrCuO2 contains edge-shared CuO4 plaquettes arranged in a zig-zag geometry
with negligible inter-chain coupling [Fig. 1(a)] Motoyama  _et al._ (1996);
Kim  _et al._ (1996, 2006); Zaliznyak  _et al._ (2004); Knupfer  _et al._
(1997); Zaliznyak  _et al._ (1999). The Cu sites have $3d^{9}$ valence
configurations in the atomic limit with a hole in the $d_{x^{2}-y^{2}}$
orbital. Due to strong intra-chain coupling, the system effectively behaves as
a 1D spin-$\frac{1}{2}$ HAFM chain above $T_{N}=5$ K Zaliznyak  _et al._
(2004, 1999). The zig-zag arrangement of the plaquettes creates two
inequivalent oxygen sites denoted OA and OB, as shown in Fig. 1(a). Due to the
element specificity of the RIXS process, excitations of the in-chain OB or Cu
sites can be isolated by fixing the incident energy to X-ray absorption
spectra (XAS) peaks from these respective elements. We obtained the resonant
energies of the OB and Cu sites from the XAS collected at O $K$\- and Cu
$L_{3}$-edges, respectively (see Supplementary Materials Fig. S1) Knupfer  _et
al._ (1997); SM . RIXS spectra on SrCuO2 at 13 K were collected at O $K$\- and
Cu $L_{3}$-edges with energy resolutions of $\sim 27$ and $\sim 42$ meV,
respectively, at I21 beamline, Diamond Light Source, United Kingdom I (21).
Throughout, the momentum transfer component along the chain direction
($q_{\parallel}$) is presented in units of 2$\pi/b$.

Figures 1(b) and (c) show the RIXS spectra from SrCuO2 for
$q_{\parallel}=0.175$ at the Cu $L_{3}$\- and O $K$-edges, respectively.
Similar to RIXS results on the related compound Sr2CuO3, phonon excitations
and their overtones are observed below $\sim$0.2 eV for both edges Schlappa
_et al._ (2018, 2012). The intense feature close to 0.35 eV energy loss at the
Cu $L_{3}$-edge can be ascribed to a two-spinon excitation and approximated by
the dynamical structure factor computed using the Müller ansatz after,
adopting an exchange coupling $J=0.225$ eV Karbach  _et al._ (1997); Zaliznyak
_et al._ (2004). A weaker feature is also observed at O $K$-edge at a similar
energy. Panels (d) and (e) show the RIXS spectra for $q_{\parallel}=0.0$ at Cu
$L_{3}$\- and O $K$-edges, respectively. For both edges, significant spectral
weight is observed, peaking close to 0.5 eV. Since the weight of the two-
spinon continuum is zero at this momentum point, and the orbital and charge-
transfer excitations lie above 1.5 eV, these features likely arise from four-
spinon excitations Schlappa  _et al._ (2018). Overall, it appears that RIXS
probes similar excitations at both edges but with substantially different
scattering cross-sections, giving rise to distinct spectral weight
distributions. For a better understanding of the RIXS cross-sections, we next
map these excitations in the energy-momentum space.

Figure 2: (a) O $K$-edge and (b) Cu $L_{3}$-edge RIXS intensity maps for
momentum transfer along the chain. The white colour represents the maximum
spectral weight in the colour scale. The two-spinon continuum boundaries are
shown by blue dashed lines. Horizontal dashed green lines mark the higher
energy limit of the phonon excitations. The RIXS spectra shown in Fig. 1(b-e)
correspond to the the vertical black dashed lines. (c) Model intensity maps
for spin excitations at the (c) O $K$\- and (d) Cu $L_{3}$-edge computed using
the Kramers-Heisenberg formalism.

Figures 2(a) and (b) present the momentum resolved RIXS intensity maps at the
O $K$\- and Cu $L_{3}$-edges, respectively. The lower and upper boundaries of
the two-spinon phase space,
$\omega_{\mathrm{2S}}^{\mathrm{l}}(\textbf{q})=\frac{\pi}{2}J|\sin(qb)|$ and
$\omega_{\mathrm{2S}}^{\mathrm{u}}(\textbf{q})=\pi J|\sin(qb/2)|$ Karbach  _et
al._ (1997), are plotted over the maps for $J=0.225$ eV. A highly dispersing
spectral weight is observed within this phase space, consistent with INS
experiments on SrCuO2 Zaliznyak  _et al._ (2004). The two-spinon continuum
composed of fractionalized spin-1/2 entities has been previously observed at
Cu $L_{3}$-edge RIXS for Sr2CuO3, Ca2CuO3 and CaCu2O3 Schlappa  _et al._
(2012, 2018); Fumagalli  _et al._ (2020); Bisogni  _et al._ (2014), where they
arise from a spin-flip scattering process enabled by the strong spin-orbit
coupling in the $2p$ core level. In O $K$-edge RIXS, where a single spin-flip
is not allowed, two-spinon excitations are created by a net spin-zero inter-
site double spin-flip process Forte  _et al._ (2011); Schlappa  _et al._
(2018). Further, due to the longer core-hole lifetime at O $K$-edge, quantum
fluctuations lead to formation of additional domain walls in the intermediate
state Schlappa  _et al._ (2018). Four-spinon excitations are thus formed,
appear as an additional continuum of excitations located outside the two-
spinon phase space [Fig 2(a)] but within the four-spinon phase space extending
up to $\omega_{\mathrm{4S}}^{\mathrm{u}}(\textbf{q})=\pi
J\sqrt{2[1+\cos(qb/2)]}$ ($\sim 1.4$ eV for $J=0.225$ close to
$q_{\parallel}=0$) Caux and Hagemans (2006).

Interestingly, we also observe significant spectral weight outside the two-
spinon phase space in Cu $L_{3}$-edge measurements. Similar features were
reported by Ref. Schlappa  _et al._ (2018), but were not explored further.
Notably, the four-spinon spectral weight distributions are drastically
different between the two edges for the same system. While at the O $K$-edge
they forms a bulbous shape with largest weights close to $q_{\parallel}=0$, at
Cu $L_{3}$-edge they have a flatter distribution with weight collected near
the upper boundary of the two-spinon continuum.

The low-energy spin dynamics of corner-shared cuprates are well described by
$t-J$ Hamiltonian Nocera  _et al._ (2018); Zaliznyak  _et al._ (2004); Walters
_et al._ (2009); Schlappa  _et al._ (2018)

$H=-t\sum_{\langle
i,j\rangle,\sigma}\tilde{c}^{\dagger}_{i,\sigma}\tilde{c}^{\phantom{\dagger}}_{j,\sigma}+J\sum\limits_{i}\big{(}\textbf{S}_{i}\cdot\textbf{S}_{i+1}-\frac{1}{4}n_{i}n_{i+1}\big{)}.$
(1)

Here, $t$ and $J$ are hopping integrals and exchange coupling, respectively,
between the nearest neighbours $\langle i,j\rangle$.
$\tilde{c}^{\dagger}_{i,\sigma}~{}(\tilde{c}^{\phantom{\dagger}}_{i,\sigma})$
is the creation (annihilation) operator for a spin-$\sigma$
($=\uparrow,\downarrow$) hole at site $i$ under the constraint of no double
occupancy;
$n_{i}=\sum_{\sigma}\tilde{c}^{\dagger}_{i,\sigma}\tilde{c}^{\phantom{\dagger}}_{j,\sigma}$
is the number operator; and $\textbf{S}_{i}$ is the spin operator at site $i$.
We use $J=0.225$ eV from the energy-momentum distribution of the two-spinon
continuum in our RIXS experiments and $t=0.3$, typical for cuprates.

We solved Eq. (1) using exact diagonalization (ED) on twenty-four site chains
and evaluated the RIXS intensity $I(q,\omega)$ using the KH formalism Ament
_et al._ (2011)

$I(q,\omega)\propto\sum\limits_{f}\left\lvert\sum\limits_{n}\frac{\langle
f\lvert D^{\dagger}_{k_{\mathrm{out}}}\rvert n\rangle\langle n\lvert
D^{\phantom{\dagger}}_{k_{\mathrm{in}}}\rvert
g\rangle}{E_{g}+\omega_{\mathrm{in}}-E_{n}+i\Gamma_{n}}\right\rvert^{2}\delta\left(E_{f}-E_{g}-\omega\right).$
(2)

In the above expression, the incoming (outgoing) photons have energy
$\omega_{\mathrm{in}}$ ($\omega_{\mathrm{out}}$) and momentum
$k_{\mathrm{in}}$ ($k_{\mathrm{out}}$);
$\omega=\omega_{\mathrm{in}}-\omega_{\mathrm{out}}$ and
$q=k_{\mathrm{in}}-k_{\mathrm{out}}$ are the energy and momentum transferred
along the chain direction, respectively; $\lvert g\rangle$, $\lvert n\rangle$,
and $\lvert f\rangle$ are the initial, intermediate, and final states of the
RIXS process with energies $E_{g}$, $E_{n}$, and $E_{f}$, respectively;
$D_{k}=\sum_{i,\sigma}e^{ikR_{i}}D_{i,\sigma}$ is the dipole operator
describing the $2p\rightarrow 3d$
$\left(D_{i,\sigma}=\sum_{\alpha}d^{\phantom{\dagger}}_{i\sigma}p_{i\sigma}^{\dagger}\right)$
and $1s\rightarrow 2p$
$\left(D_{i,\sigma}=[d^{\phantom{\dagger}}_{i+1,\sigma}-d^{\phantom{\dagger}}_{i-1,\sigma}]s_{i\sigma}^{\dagger}\right)$
transition Kumar  _et al._ (2018); Schlappa  _et al._ (2018), where
$p^{\dagger}_{i,\sigma}$ and $s_{i\alpha}^{\dagger}$ are creation operators on
the $2p$ and $1s$ core levels, respectively; and $\Gamma_{n}$ is the core-hole
broadening related to the inverse core-hole lifetime. For our numerical
calculations we use $\Gamma_{n}=0.3$ eV (Cu $L_{3}$-edge) and $0.15$ eV (O
$K$-edge) for all $n$ Jia  _et al._ (2014); Lee  _et al._ (2013). Figs. 2(c)
and (d) show the calculated spectra, which capture the two- and four-spinon
excitations at both edges.

Figure 3: The calculated Cu $L_{3}$-edge RIXS spectra in the (a) NSC and (b)
SC channels, computed using the Kramers-Heisenberg formalism. The white colour
represents the maximum spectral weight in the colour scale. (c) the normalised
intensity profiles of the two spin channels obtained by integrating the region
marked by the red dashed lines in (b) and (c). (d) the integrated intensity
profile for the same region obtained from the Cu $L_{3}$-edge RIXS experiment
on SrCuO2. The continuous line is a weighted sum of the NSC and SC channel
profiles shown in panel (c) and a linear background to match the experimental
profile (see text).

The RIXS spectra at the Cu $L_{3}$-edge has contributions from both the NSC
and SC channels Ament  _et al._ (2009); Jia  _et al._ (2016) while the O
$K$-edge has contributions from only the SC channel. To demonstrate this, Fig.
3 shows the spin-resolved contributions to the Cu $L_{3}$-edge RIXS intensity
from the (a) NSC and (b) SC channels, highlighting the features outside the
two-spinon phase space. Panel (c) shows the computed intensity profiles of the
two channels for momentum transfers along the chain, obtained by integrating
the spectral weight along the energy axis within the region enclosed by the
red dashed lines in panels (a) and (b). The profiles have been normalised to
their maximum values in the probed portion of the Brillouin zone for
comparison. From panels (a) and (c), it is clear that the four-spinon
excitations in the NSC channel have the maximum spectral weight close to the
upper boundary of the two-spinon continuum and a negligible weight at zone
center. Conversely, the four-spinon excitations observed in the SC channel
have maximal spectral weight close to $q_{\parallel}=0$. In panel (d), we show
the experimental Cu $L_{3}$-edge RIXS intensity profile from the same region
(see Fig. 2(b) and Supplementary Materials Fig. S2 for the full map) SM . The
experimental profile can be well described by a sum of the NSC and SC channel
profiles (in a ratio $\sim 1:1.92$) and a linear background. Unlike INS, our
results show that both NSC and SC four-spinons have significant presence
outside the two-spinon phase space when probed using Cu $L_{3}$-edge RIXS.
This result also suggests why the spectral weight distribution in this region
of phase space is substantially different from that at O $K$-edge, where only
SC excitations are allowed.

Figure 4: The generalized dynamical structure factors
$S^{k}_{\mathrm{(N)SC}}(q,\omega)$ appearing in higher-order terms of the UCL
expansion. (a) and (b) show the intensity maps of the first-order ($k=1$)
terms in the NSC and SC channels, respectively. (c) and (d) show the intensity
maps of the second-order ($k=2$) terms in the NSC and SC channels,
respectively. The white colour represents the maximum spectral weight in the
colour scale.

While the KH formalism captures the spectral weight distribution of both the
two- and four-spinon excitations, it is desirable to identify the correlation
functions leading to these excitations using the UCL expansion Ament  _et al._
(2007); Forte  _et al._ (2008); Jia  _et al._ (2016). For a half-filled
antiferromagnetic $d^{9}$ cuprate, the zeroth-order term in the expansion of
the SC and NSC channels are elastic scattering and the conventional dynamical
spin structure factor $S(q,\omega)$, respectively SM ; Jia  _et al._ (2016);
Kumar  _et al._ (2019). Corrections to these terms are provided by the higher-
order terms in the UCL expansion. The next-order generalized dynamical
structure factors in the NSC and SC channels are shown in Fig. 4. The spectra
at $k^{th}$ order are

$S^{k}_{\mathrm{(N)SC}}(q,\omega)=\sum_{f}\left|\langle
f|O^{\mathrm{(N)SC}}_{q,k}|g\rangle\right|^{2}\delta\left(E_{f}-E_{g}-\omega\right),$

where $k$ indexes the $k^{\text{th}}$ order in the UCL approximation,
$O^{\mathrm{(N)SC}}_{q,k}=\frac{1}{\sqrt{N}}\sum_{i}e^{iqR_{i}}O_{i,k}^{\mathrm{(N)SC}}$,
and
$O_{i,k}^{\mathrm{NSC}}=S_{i}^{z}\left(\textbf{S}_{i}\cdot(\textbf{S}_{i+1}+\textbf{S}_{i-1})\right)^{k}$
and
$O_{i,k}^{\mathrm{SC}}=\left(\textbf{S}_{i}\cdot(\textbf{S}_{i+1}+\textbf{S}_{i-1})\right)^{k}$
are the effective operators in each channel. The results in Figs. 4(a) and (b)
demonstrate that the spectral weight remains confined within the two-spinon
phase space Forte  _et al._ (2011); Klauser  _et al._ (2011) for the first-
order corrections to both the NSC and SC channels. Spectral weight outside the
two-spinon phase space in either channel appears only with the second-order
corrections [Fig. 4(c) and (d)]. The four-spinons seen in Cu $L_{3}$-edge RIXS
experiment are, therefore, beyond the scope of first order nearest-neighbour
correlation functions and represent higher order and longer-range spin
correlations (see SM, SM ).

In conclusion, we have demonstrated that both O $K$\- and Cu $L_{3}$-edge RIXS
can probe four-spinon excitations outside the two-spinon phase space in the
low-dimensional quantum magnet SrCuO2. We further showed that these four-
spinon excitations involve long-range spin-flips due to the lifetime of the
intermediate state’s core-hole. Higher-order corrections in the UCL expansion
in the SC channel have been explored previously in the context of two-magnon
excitations in Cu $K$-edge RIXS studies of two-dimensional cuprates Forte  _et
al._ (2008); Hill  _et al._ (2008). But the relevance of these corrections for
the Cu $L_{3}$-edge RIXS has not been widely explored Jia  _et al._ (2016).
Unexpectedly, we find that corrections up to the second-order in the UCL
expansion are needed for both the SC and NSC channels to achieve even a
qualitative description of the four-spinon excitations at Cu $L_{3}$-edge. Our
work illustrates that the Cu $L_{3}$-edge RIXS is more sensitive to long-range
quantum spin fluctuations in low-dimensional quantum magnets than previously
thought. In particular, the NSC channel at this edge probes completely new
long-range correlation functions, which one can exploit for exploring higher-
order many-body spin dynamics.

## Acknowledgements

U. K. acknowledges support from U.S. DOE NNSA under Contract No.
89233218CNA000001 through the LDRD Program. S. J. acknowledges support from
the National Science Foundation under Grant No. DMR-1842056. T. S. was
supported by the Swiss National Science Foundation (SNSF) through the SNSF
research grants 200021$\\_$178867 and 200021L$\\_$141325\. We acknowledge
Diamond Light Source for providing the beam time on I21 beam line under
Proposal No. NR21184. We acknowledge T. Rice for the technical support
throughout the beam time. We also thank G. B. G. Stenning and D. W. Nye for
help on the Laue instrument in the Materials Characterisation Laboratory at
the ISIS Neutron and Muon Source.

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Supplemental Materials: Unraveling higher-order corrections in the spin
dynamics of RIXS spectra

##  S-I Experimental details

Figure S1: Determination of the resonant X-ray energies corresponding to OB
and Cu sites for SrCuO2 shown in Fig. 1(a) of main text. (a) O $K$-edge X-ray
absorption (XAS) with $\pi$ polarisation along ($90^{\circ}$) and normal to
the spin chain direction ($10^{\circ}$), resonating with OB and OA sites,
respectively. (b) Cu $L_{3}$-edge XAS with $\pi$ polarisation along
($90^{\circ}$) the spin chain direction, resonating with Cu site. Figure S2:
(a) Full Cu-$L_{3}$-edge RIXS intensity map for SrCuO2. The two-spinon
continuum boundaries are shown by blue dashed lines. In Fig. 3(d) of main
text, the integrated intensity profile for the region marked by the red dashed
lines is shown. (b) Cu $L_{3}$-edge RIXS line spectra close to
$q_{\parallel}$=0 highlighting the flat four-spinon spectral weights.

##  S-II Ultra-short core-hole lifetime expansion of Kramers-Heisenberg
Formalism at Cu $L_{3}$-edge

The Kramers-Heisenberg (KH) expression for the RIXS intensity can be re-
expressed as a series of “simpler” correlation functions using the ultra-short
core-hole lifetime (UCL) approximation Forte  _et al._ (2008); Jia  _et al._
(2016). The UCL approximation at the Cu $L_{3}$-edge should be a good
approximation to the RIXS intensity, as long as $J/\Gamma<1$.

The RIXS intensity in the KH formalism is given by

$I(q,\omega)\propto\sum\limits_{f}\left\lvert\langle f\lvert
D^{\dagger}_{k_{\mathrm{out}}}\mathcal{O}D^{\phantom{\dagger}}_{k_{\mathrm{in}}}\rvert
g\rangle\right\rvert^{2}\delta\left(E_{f}-E_{g}-\omega\right),$ (S1)

where

$\mathcal{O}=\frac{1}{\omega_{\text{in}}-\mathcal{H}+i\Gamma}=\sum_{|N\rangle}\frac{|N\rangle\langle
N|}{\omega_{\text{in}}-\mathcal{E}_{N}+i\Gamma}.$ (S2)

Here, $|N\rangle$ are the eigenstates of intermediate Hamiltonian,
$\mathcal{H}=\tilde{H}+H_{\lambda}$, where
$\tilde{H}=H_{0}+U_{c}\sum_{i}n_{i}n_{i}^{c}$,
$H_{0}=J\sum_{i}\textbf{S}_{i}\cdot\textbf{S}_{i+1}$ and
$H_{\lambda}=\epsilon_{c}^{\alpha}\sum_{i\alpha}n_{i\alpha}^{c}$, $n_{i}$ and
$n_{i}^{c}$ are occupation in the valence and core-level respectively, and
$U_{c}$ is the Coulomb interaction between the two levels. In
$n_{i\alpha}^{c}$, $\alpha\in\\{L_{2},L_{3}\\}$ and characterizes the spin-
orbit coupling in the core-level with energy $\epsilon_{c}^{\alpha}$. We work
in the hole language throughout this derivation. Also, we have ignored the
hopping term in $H_{0}$ since we are interested only in the half-filled case
and it should not contribute significantly to the intermediate state in the
limit of large core-hole potential as the charges are immobile.

Due to the large $U_{c}$ on the core-hole site, hopping to the $d$-orbital is
largely prohibited in the intermediate state. Assuming that hopping to this
site is completely suppressed, we can decompose the intermediate states as
$|N\rangle=|n\rangle|n_{c}\rangle,~{}\mathcal{E}_{n}=E_{n}+E_{c}$. Here,
$|N\rangle~{}(\mathcal{E}_{N})$, $|n\rangle~{}(E_{n})$ and
$|n_{c}\rangle~{}(E_{c})$ are the eigenstates (energies) of the $\mathcal{H}$,
$\tilde{H}$ and $H_{\lambda}$ respectively. One can rewrite the operator
$\mathcal{O}$ in its spectral representation as

$\frac{1}{\omega_{\text{in}}-\mathcal{H}+i\Gamma}=\sum_{|n,n_{c}\rangle}\frac{|n\rangle\langle
n|n_{c}\rangle\langle n_{c}|}{\omega_{\text{in}}-E_{n}-E_{c}+i\Gamma}.$ (S3)

$E_{c}$ can be tuned to either the $L_{2}$ ($E_{L_{2}}$) or $L_{3}$ resonances
($E_{L_{3}}$), which are well separated in energy due to large spin-orbit
coupling in the core $2p$-level of the Cu atom. In our discussion, we discuss
the simplification in the context of $L_{3}$ edge. Though it can be easily
extended to $L_{2}$-edge by replacing $L_{2}$ with $L_{3}$ and considering the
relevant approximations. For $L_{3}$-edge, we have

$\displaystyle\mathcal{O}=\sum_{|n\rangle}\frac{|n\rangle\langle
n|L_{3}\rangle\langle L_{3}|}{\omega_{\text{in}}-E_{n}-E_{L_{3}}+i\Gamma}.$
(S4)

It is important to note that $|L_{3}\rangle$ can either flip or conserve the
spin in core-hole and hence lead to non-spin-conserving (NSC) $\Delta
S_{\mathrm{tot}}=1$ and spin-conserving (SC) $\Delta S_{\mathrm{tot}}=0$
channels, respectively. We define $\omega_{\text{in}}-E_{L_{3}}=\Delta$ such
that

$\displaystyle\sum_{|n\rangle}\frac{|n\rangle\langle n|L_{3}\rangle\langle
L_{3}|}{\omega_{\text{in}}-E_{n}-E_{L_{3}}+i\Gamma}$
$\displaystyle=|L_{3}\rangle\langle L_{3}|\frac{1}{\Delta-\tilde{H}+i\Gamma}=$
(S5) $\displaystyle=|L_{3}\rangle\langle
L_{3}|\frac{1}{\Delta+i\Gamma}\sum_{l=0}^{\infty}\bigg{(}\frac{\tilde{H}}{\Delta+i\Gamma}\bigg{)}^{l}.$

Resonance condition and UCL approximation — When the incident energy is tuned
to resonate with the $L_{3}$-edge such that $\Delta\rightarrow 0$, the
propagator in the above expression can be written as

$\begin{split}\frac{1}{i\Gamma-\tilde{H}}=\frac{1}{i\Gamma}\sum_{l=0}^{\infty}\bigg{(}\frac{\tilde{H}}{i\Gamma}\bigg{)}^{l}.\end{split}$
(S6)

We now consider the application of the dipole operator at the Cu $L_{3}$ edge
$D_{k_{\text{in}}}=\sum_{i,\sigma}e^{ikR_{i}}D_{i,\sigma}$, where
$D_{i,\sigma}=\sum_{\alpha}d^{\phantom{\dagger}}_{i\sigma}p_{i\alpha}^{\dagger}$
annihilates a hole in the $d$-orbital at site $i$. In this case,
$H_{i}D_{i,\sigma}|g\rangle=0$, where
$H_{i}=J~{}\textbf{S}_{i}\cdot(\textbf{S}_{i+1}+\textbf{S}_{i-1})$. Since the
core-hole potential term $H_{c}$ inhibits hopping to site $i$ and
superexchange coupling to the spin at the core-hole site, we can write the
intermediate state Hamiltonian as $\tilde{H}=H_{0}-H_{i}$. One can further see
that $H_{0}-H_{i}$ commutes with $D_{i,\sigma}$ and we have
$\tilde{H}|g\rangle=(H_{0}+H_{c})|g\rangle=0|g\rangle$. (Here, we have set
ground state energy, $H_{0}|g\rangle=0|g\rangle$.) Using these properties, one
can write

$\displaystyle\sum_{l=0}^{\infty}\frac{\tilde{H}^{l}}{(i\Gamma)^{l+1}}D_{i,\sigma}|g\rangle$
$\displaystyle=\sum_{l=0}^{\infty}D_{i,\sigma}\frac{(H_{0}-H_{i})^{l}}{(i\Gamma)^{l+1}}|g\rangle$
(S7)
$\displaystyle=D_{i,\sigma}\frac{1}{i\Gamma}\bigg{(}1-\frac{H_{i}}{i\Gamma}+\frac{H_{i}^{2}}{(i\Gamma)^{2}}-\frac{H_{0}H_{i}}{(i\Gamma)^{2}}+\dots\bigg{)}|g\rangle$

Now, one can see that RIXS cross-section can be written in terms of
correlation functions involving powers of $H_{i}$.

Using the relation,
$d_{i\sigma}^{\dagger}d^{\phantom{\dagger}}_{i\sigma}p^{\phantom{\dagger}}_{i\alpha}|L_{3}\rangle\langle
L_{3}|p_{i\alpha}^{\dagger}=n_{i\sigma}$ and
$d_{i\bar{\sigma}}^{\dagger}d^{\phantom{\dagger}}_{i\sigma}p^{\phantom{\dagger}}_{i\alpha}|L_{3}\rangle\langle
L_{3}|p_{i\alpha}^{\dagger}=S_{i}^{z}$ Ament  _et al._ (2009); Kourtis  _et
al._ (2012), the scattering amplitude with the core-hole term in the expansion
can be written as

$\displaystyle\langle
f|e^{i\textbf{q}\cdot\textbf{R}_{i}}D_{i,\sigma^{\prime}}^{\dagger}|L_{3}\rangle\langle$
$\displaystyle
L_{3}|D_{i,\sigma}\sum_{l=0}^{\infty}e^{i\textbf{q}\cdot\textbf{R}_{i}}\frac{(H_{0}-H_{i})^{l}}{(i\Gamma)^{l+1}}|g\rangle=\begin{cases}\langle
f|\sum_{i}e^{i\textbf{q}\cdot\textbf{R}_{i}}n_{i,\sigma}\sum_{l=0}^{\infty}\frac{(H_{0}-H_{i})^{l}}{(i\Gamma)^{l+1}}|g\rangle&\text{if
$\sigma^{\prime}=\sigma$}\\\ \langle
f|\sum_{i}e^{i\textbf{q}\cdot\textbf{R}_{i}}S_{i}^{z}\sum_{l=0}^{\infty}\frac{(H_{0}-H_{i})^{l}}{(i\Gamma)^{l+1}}|g\rangle&\text{if
$\sigma^{\prime}\neq\sigma$}.\end{cases}$ (S8)

Neglecting the cross terms, the SC and NSC channels with leading correction
terms corresponding to $H_{i}$ and $H_{i}^{2}$ are given by

$\displaystyle\mathcal{I}_{SC}^{UCL}(\textbf{q},\omega)=$
$\displaystyle\Bigg{(}\frac{1}{\Gamma^{2}}\sum_{f}\Big{|}\langle
f|\sum_{i}e^{i\textbf{q}\cdot\textbf{R}_{i}}n_{i,\sigma}|g\rangle\Big{|}^{2}+\frac{J^{2}}{\Gamma^{4}}\sum_{f}\Big{|}\langle
f|\sum_{i}e^{i\textbf{q}\cdot\textbf{R}_{i}}\textbf{S}_{i}\cdot(\textbf{S}_{i+1}+\textbf{S}_{i-1})|g\rangle\Big{|}^{2}$
(S9) $\displaystyle+\frac{J^{4}}{\Gamma^{6}}\sum_{f}\Big{|}\langle
f|\sum_{i}e^{i\textbf{q}\cdot\textbf{R}_{i}}\big{(}\textbf{S}_{i}\cdot(\textbf{S}_{i+1}+\textbf{S}_{i-1})\big{)}^{2}|g\rangle\Big{|}^{2}+\dots\Bigg{)}\delta(\Omega+E_{g}-E_{f})$

In the case of SC-channel, the leading term corresponds to elastic scattering
at half-filling. The first order correction is the spin-exchange dynamical
factor $S^{\text{exch}}(\textbf{q},\omega)$ Klauser  _et al._ (2011), which is
important for indirect RIXS. Similarly for the spin flip channel,

$\displaystyle\mathcal{I}_{NSC}^{UCL}(\textbf{q},\omega)=$
$\displaystyle\Bigg{(}\frac{1}{\Gamma^{2}}\sum_{f}\Big{|}\langle
f|\sum_{i}e^{i\textbf{q}\cdot\textbf{R}_{i}}S_{i}^{z}|g\rangle\Big{|}^{2}+\frac{J^{2}}{\Gamma^{4}}\sum_{f}\Big{|}\langle
f|\sum_{i}e^{i\textbf{q}\cdot\textbf{R}_{i}}S_{i}^{z}\textbf{S}_{i}\cdot(\textbf{S}_{i+1}+\textbf{S}_{i-1})|g\rangle\Big{|}^{2}$
(S10) $\displaystyle+\frac{J^{4}}{\Gamma^{6}}\sum_{f}\Big{|}\langle
f|\sum_{i}e^{i\textbf{q}\cdot\textbf{R}_{i}}S_{i}^{z}\big{(}\textbf{S}_{i}\cdot(\textbf{S}_{i+1}+\textbf{S}_{i-1})\big{)}^{2}|g\rangle\Big{|}^{2}+\dots\Bigg{)}\delta(\Omega+E_{g}-E_{f}).$

The response function for first and second order correction is presented in
the Fig. 4 of the main text.

Simplification of Second Order — The second order correction can further be
simplified

$\displaystyle H_{i}^{2}/J^{2}=$
$\displaystyle\big{(}\textbf{S}_{i}\cdot(\textbf{S}_{i-1}+\textbf{S}_{i+1})\big{)}\big{(}\textbf{S}_{i}\cdot(\textbf{S}_{i-1}+\textbf{S}_{i+1})\big{)}$
(S11) $\displaystyle=$
$\displaystyle(\textbf{S}_{i}\cdot\textbf{S}_{i-1})^{2}+(\textbf{S}_{i}\cdot\textbf{S}_{i+1})^{2}+(\textbf{S}_{i}\cdot\textbf{S}_{i-1})(\textbf{S}_{i}\cdot\textbf{S}_{i+1})+(\textbf{S}_{i}\cdot\textbf{S}_{i+1})(\textbf{S}_{i}\cdot\textbf{S}_{i-1}).$

Notice that only the last two terms can contribute to the longer-range
correlation functions. Using the vector triple product and identity reported,
one can rephrase the above operators as
$(\textbf{S}_{i}\cdot\textbf{S}_{i+1})(\textbf{S}_{i}\cdot\textbf{S}_{i-1})=(\textbf{S}_{i-1}\cdot\textbf{S}_{i+1})\textbf{S}_{i}^{2}-\textbf{S}_{i+1}\times(\textbf{S}_{i}\times\textbf{S}_{i-1})\cdot\textbf{S}_{i}$
Notice that in the first term $\langle\textbf{S}_{i}^{2}\rangle=\frac{3}{4}$
as we have a spin-1/2 system. It is also consistent with the Eq.(37) of Ref.
Forte  _et al._ (2008), where long-range correlation functions for the SC
channel were reported. Longer-range correlation functions has also be proposed
and explored in Ref. Igarashi and Nagao (2012) in the context of 1D systems.
From the first term in the vector triple product, it is clear that the second-
order correction has a term with the long-range correlation with the form

$\displaystyle H_{i}^{2}\propto J^{2}\textbf{S}_{i-1}\cdot\textbf{S}_{i+1}.$
(S12)

Figure S3:  $S_{\mathrm{(N)SC}}^{l=2}(q,\omega)$ response. (a) and (b) show
the result for non-spin conserving and spin conserving channel result,
respectively, for the next-nearest neighbour spin-flip operators given by Eq.
(S13) that arise in the second order term of the UCL approximation.

We plot the next nearest-neighbor response given by this term in Fig. S3,
where the correlation operator is given by

$S_{\mathrm{(N)SC}}(q,\omega)=\sum_{f}\left|\langle
f|O^{\mathrm{(N)SC}}_{q}|g\rangle\right|^{2}\delta\left(E_{f}-E_{g}-\omega\right),$
(S13)

where
$O^{\mathrm{(N)SC}}_{q}=\frac{1}{\sqrt{N}}\sum_{i}e^{iqR_{i}}O_{i}^{\mathrm{(N)SC}}$,
and
$O_{i}^{\mathrm{NSC}}=S_{i}^{z}\left(\textbf{S}_{i-1}\cdot\textbf{S}_{i+1}\right)$
and $O_{i}^{\mathrm{SC}}=\left(\textbf{S}_{i-1}\cdot\textbf{S}_{i+1}\right)$
are the effective operators in each channel. The above correlation function
generates excitations in the new phase space, as shown in the Fig. S3.