# Quantum spin nematic phase in a square-lattice iridate

Hoon Kim  These authors contributed equally to this work.  Jin-Kwang Kim
These authors contributed equally to this work.  Jimin Kim  Hyun-Woo J. Kim
Seunghyeok Ha  Kwangrae Kim  Wonjun Lee  Center for Artificial Low Dimensional
Electronic Systems, Institute for Basic Science (IBS), 77 Cheongam-Ro, Pohang
37673, South Korea  Department of Physics, Pohang University of Science and
Technology, Pohang 37673, South Korea  Jonghwan Kim  Center for Van der Waals
Quantum Solids, Institute for Basic Science (IBS), Pohang, 37673, Korea
Department of Materials Science and Engineering, Pohang University of Science
and Technology, Pohang 37673, Korea  Gil Young Cho  Center for Artificial Low
Dimensional Electronic Systems, Institute for Basic Science (IBS), 77
Cheongam-Ro, Pohang 37673, South Korea  Department of Physics, Pohang
University of Science and Technology, Pohang 37673, South Korea  Hyeokjun Heo
Joonho Jang  Department of Physics and Astronomy, Seoul National University,
Seoul 08826, South Korea  J. Strempfer  G. Fabbris  Y. Choi  D. Haskel  Jungho
Kim  J. W. Kim  Advanced Photon Source, Argonne National Laboratory, Argonne,
Illinois 60439, USA  B. J. Kim<EMAIL_ADDRESS>Center for Artificial Low
Dimensional Electronic Systems, Institute for Basic Science (IBS), 77
Cheongam-Ro, Pohang 37673, South Korea  Department of Physics, Pohang
University of Science and Technology, Pohang 37673, South Korea

Spin nematic (SN) is a magnetic analog of classical liquid crystals, a fourth
state of matter exhibiting characteristics of both liquid and solid Blume and
Hsieh (1969); Andreev and Grishchuk (1984). Particularly intriguing is a
valence-bond SN Mila (2017); Läuchli  _et al._ (2005); Shannon  _et al._
(2006); Sato  _et al._ (2013), in which spins are quantum entangled to form a
multi-polar order without breaking time-reversal symmetry, but its unambiguous
experimental realization remains elusive. Here, we establish a SN phase in the
square-lattice iridate Sr$\bm{}2$IrO$\bm{}4$, which approximately realizes a
pseudospin one-half Heisenberg antiferromagnet (AF) in the strong spin-orbit
coupling limit Kim  _et al._ (2008, 2009); Jackeli and Khaliullin (2009);
Bertinshaw  _et al._ (2019). Upon cooling, the transition into the SN phase at
$\bm{T}_{\rm\bf C}$ $\bm{\approx}$ 263 K is marked by a divergence in the
static spin quadrupole susceptibility extracted from our Raman spectra, and
concomitant emergence of a collective mode associated with the spontaneous
breaking of rotational symmetries. The quadrupolar order persists in the
antiferromagnetic (AF) phase below $\bm{T}_{\rm\bf N}$ $\bm{\approx}$ 230 K,
and becomes directly observable through its interference with the AF order in
resonant x-ray diffraction, which allows us to uniquely determine its spatial
structure. Further, we find using resonant inelastic x-ray scattering a
complete breakdown of coherent magnon excitations at short-wavelength scales,
suggesting a resonating-valence-bond-like quantum entanglement in the AF state
Dalla Piazza  _et al._ (2015); Shao  _et al._ (2017). Taken together, our
results reveal a quantum order underlying the Néel AF that is widely believed
to be intimately connected to the mechanism of high temperature
superconductivity (HTSC) Anderson (1987); Lee  _et al._ (2006); Keimer  _et
al._ (2015).

With its relevance to HTSC in cuprates, spin one-half ($S$ = 1/2) Heisenberg
model on a square lattice has been a subject of extensive research over the
last several decades Keimer  _et al._ (2015); Lee  _et al._ (2006); Scalapino
(2012); Bertinshaw  _et al._ (2019). Although Mermin-Wagner theorem states
that continuous symmetries cannot be spontaneously broken in two dimensions
(2D), it is well established that Néel-type AF orders develop at low
temperatures due to weak inter-layer couplings that form a 3D network in
quasi-2D materials Vaknin  _et al._ (1987). Nevertheless, spins retain much of
their properties in the disordered phase and undergo large quantum zero-point
motions Singh (1989); Coldea  _et al._ (2001). The resulting ground state wave
function is believed to embody a highly non-trivial structure akin to those in
quantum spin liquids Savary and Balents (2016); Zhou  _et al._ (2017). For
instance, the resonant valence bond (RVB), a superposition of states in which
spins pair up to form singlet “valence bonds”, may have substantial overlap
with the ground state Anderson (1987) (Fig. 1). Upon carrier doping, these
singlets evolve into Cooper pairs in a prominent theory of HTSC Lee  _et al._
(2006); Anderson (1987).

What is not well known, however, is the fact that such quantum entanglement
between a pair of nearest-neighbor (NN) spins can also manifest as an ordered
spin quadrupole moment when spins are canted Mila (2017); Läuchli  _et al._
(2005); Shannon  _et al._ (2006); Sato  _et al._ (2013) (Fig. 1). Because
quadrupole moment can be non-zero only for $S$ $\geq$ 1, existence of a
quadrupolar order in a $S$ = 1/2 system necessarily implies that spin pairs
are entangled and have $S$ = 1 triplet components. Unlike singlets, however, a
quadrupolar order is, in principle, measurable Barzykin and Gor’kov (1993);
Smerald and Shannon (2013); Savary and Senthil (2015) as it leads to
spontaneous breaking of rotation symmetries Kohama  _et al._ (2019); Orlova
_et al._ (2017). In this Article, we show that a quadrupolar order coexists
with the Néel AF order in Sr2IrO4, a single-layer ($n$=1) member of the
Ruddlesden-Popper series iridates (Fig. 2a), which has received much attention
due to its similarities with superconducting cuprates in their phenomenology
Kim  _et al._ (2008, 2009); Jackeli and Khaliullin (2009); Bertinshaw  _et
al._ (2019). Furthermore, we show that it persists above $T_{\rm N}$ and
realizes a SN phase.

Fig. 1: Spin one-half moments on a square lattice. In a strict 2D, AF order is
not allowed at any finite temperature and spins are disordered but quantum
entangled in a non-trivial way. For example, RVB is a state in which spins
pair up to form singlets and fluctuate among numerous different configurations
covering the square lattice with such dimers. Although most quasi-2D materials
develop AF orders due to weak inter-layer couplings, spins still undergo
strong quantum fluctuations. In a canted AF, quantum entanglement can manifest
as an ordered spin quadrupole moment.

##  I Circular Dichroic
Resonant x-ray Diffraction

We use resonant x-ray diffraction (RXD) to show the coexistence of a
quadruoplar and a dipolar (canted AF) order (Fig. 2a). The dipole-quadrupole
interference contribution to the RXD intensity can be isolated by measuring
circular dichroic (CD) signal of a magnetic Bragg reflection defined as

$I_{\textrm{CD}}=\frac{I_{\textrm{DIFF}}}{I_{\textrm{SUM}}}=\frac{I_{\textrm{LL}}+I_{\textrm{LR}}-I_{\textrm{RR}}-I_{\textrm{RL}}}{I_{\textrm{LL}}+I_{\textrm{LR}}+I_{\textrm{RR}}+I_{\textrm{RL}}}\;,$
(1)

where $I_{\textrm{L(R)L(R)}}$ denotes the intensity of left (right) circular
polarized incident x-ray scattered into left (right) circular outgoing x-ray.
This definition of $I_{\textrm{CD}}$ reflects the fact that the scattered
x-ray polarization is not resolved in our experiment. It is straightforward to
show that the difference spectrum ($I_{\textrm{DIFF}}$) in our particular case
can be expressed as (see Supplementary Note S1)

$\begin{split}I_{\textrm{DIFF}}\propto
F_{1}^{M}e_{1}^{A}F_{2}^{Q}e_{2}^{S}+F_{1}^{Q}e_{1}^{S}F_{2}^{M}e_{2}^{A}\\\
-F_{2}^{M}e_{1}^{A}F_{1}^{Q}e_{2}^{S}-F_{2}^{Q}e_{1}^{S}F_{1}^{M}e_{2}^{A}\;,\end{split}$
(2)

where $F_{1,2}^{Q}$ and $F_{1,2}^{M}$ couple to symmetric ($e_{1}^{S}$) and
antisymmetric ($e_{2}^{A}$) component of the polarization tensor
$e_{\alpha\beta}\equiv\epsilon^{\prime*}_{\alpha}\epsilon_{\beta}$,
respectively, and represent the complex structure factor ($F$ $\equiv$
$F_{1}$+$iF_{2}$) associated with time-reversal-even quadrupoles and time-
reversal-odd dipoles, respectively. Thus, CD arises from the interference
between dipolar and quadrupolar scatterings in the electric-dipole-electric-
dipole ($E1$-$E1$) process of RXD. In Supplementary Note S2, we argue that
this is the unique explanation for the CD in our case among all known
mechanisms.

Fig. 2: Dipole-quadrupole interference in CD-RXD. a, A schematic for the Néel
and SN orders overlaid on the crystal structure of Sr2IrO4 having the
tetragonal $I4_{1}/acd$ space group. The magnetic and quadrupole moments
mutually constrain their orientations below $T_{N}$ (left), whereas ($Q_{zx}$,
$Q_{yz}$) transforms as an $E_{\rm g}$ doublet above $T_{N}$ (right). b, c,
Resonance profile of the circular dichroic signal at (0 0 21) magnetic
reflection, measured at $\Psi$ = 30∘ (b) and $\Psi$ = -56∘ (c). $I_{\rm RC}$
(red) and $I_{\rm LC}$ (blue) denote the diffraction intensity for right- and
left-circularly polarized incident x-rays, respectively. d, e, $I_{\rm
SUM}\,\equiv\,I_{\rm RC}+I_{\rm LC}$ (d) and $I_{\rm DIFF}\,\equiv\,I_{\rm
RC}-I_{\rm LC}$ (e) at the resonance for the full range of $\Psi$ measured. S1
and S2 are obtained from two independent measurements. f, Comparison of
$I_{\rm CD}\,\equiv\,I_{\rm DIFF}/I_{\rm SUM}$ to the simulations for
different irreducible representations. The best fit to the data using
$\Gamma_{1}$ profile is shown in d-f, and its structure is shown in a.

Figures 2b and 2c show the representative resonance profile of the (0 0 21)
reflection, arising from the net FM moment due to canting of the spins Kim
_et al._ (2009), at two different azimuth angles ($\Psi$), defined as the
angle between the crystallographic $a$-axis and the vertical scattering plane.
$I_{\textrm{DIFF}}$ is non-zero at and close to the resonance, and positive
(negative) at $\Psi$ = 30∘ ($\Psi$ = -56∘). Figures 2d and 2e show
$I_{\textrm{SUM}}$ and $I_{\textrm{DIFF}}$ at the resonance for the full range
of $\Psi$ measured. The $\Psi$-dependence of $I_{\textrm{CD}}$ (Fig. 2f)
allows us to determine the spatial structure and symmetry of the quadrupolar
order. To systematically find all symmetry-allowed $\mathbf{q}$ = 0 bond-
centered quadrupole structures (see Supplementary Note S3), we first note that
the reflection appearing at (0 0 21) implies that both the dipolar and the
quadrupolar orders break the body-center translation symmetry. Such structures
are represented by one of the four irreducible representations (IRs)
$\Gamma_{1}$, $\Gamma_{2}$, $\Gamma_{3}$, and $\Gamma_{4}$, which are all 2D.
For example, it is known that the magnetic structure shown in Fig. 2a belongs
to $\Gamma_{1}$ IR Ye  _et al._ (2013).

While there are many different possible quadrupole structures (see
Supplementary Note S3), there are only a few that have non-zero structure
factors, which involve $yz$/$zx$ quadrupoles for $\Gamma_{1}$ and
$\Gamma_{2}$, $xy$ for $\Gamma_{3}$, and $x^{2}-y^{2}$ / $3z^{2}-r^{2}$ for
$\Gamma_{4}$. In Fig. 2f, the $\Psi$-dependence of $I_{\textrm{CD}}$ is
simulated for each of these structures and compared with the data. We find
that only the $\Gamma_{1}$ structure, depicted in Fig. 2a, is consistent with
the experimental data, whose key feature is the sign change at $\Psi$ = 0∘.
The total scattering amplitude is given by

$\displaystyle F=F^{M}+rF^{Q}$ (3)
$\displaystyle=i\left(\begin{matrix}0&0&-cM_{y}\\\ 0&0&c^{*}M_{x}\\\
cM_{y}&-c^{*}M_{x}&0\end{matrix}\right)+r\left(\begin{matrix}0&0&cQ_{zx}\\\
0&0&c^{*}Q_{yz}\\\ cQ_{zx}&c^{*}Q_{yz}&0\end{matrix}\right),$

where $\mathbf{M}$ = ($M_{x}$, $M_{y}$) and $\mathbf{Q}$ = ($Q_{yz}$,
$Q_{zx}$) are 2D real basis vectors (whose transformation matrices are listed
in Supplementary Note S3) for $\Gamma_{1}$ representing the net FM component
of the magnetic order and the quadrupolar order, respectively, normalized to
one when they have the maximal values; and $c$ = $4(1-i)$ is a constant
arising from the structure factor. $r$ is a dimensionless factor accounting
for the fact that quadrupoles only become visible by high-order RXD processes
(see Supplementary Note S4 for possible microscopic mechanisms).

For the magnetic order, it is known that the easy axis is along $<$100$>$
Porras  _et al._ (2019), and the fact that $I_{\textrm{SUM}}$ has a minimum at
$\Psi$ = 0∘ means that the magnetic domain being measured has the FM (AF)
component along the $b$-($a$-)axis, and hence $M_{x}$ = 0. As we shall see
below, fixing the direction of $\mathbf{M}$ along $y$ constrains $Q_{yz}$ to
be zero. Thus, it follows that

$I_{\textrm{CD}}=\frac{(rQ_{zx}/M_{y})\;\sin{\alpha}\sin{2\Psi}}{1-\cos{2\alpha}\cos{\Psi}^{2}+(rQ_{zx}/M_{y})^{2}\sin{\Psi}^{2}},$
(4)

where $\alpha$ is incidence angle to the sample surface ($\sim 27^{\circ}$).
Thus, the data can be fit with $r{Q_{zx}}/M_{y}$ as the only fitting
parameter, and the best fit is obtained when $r{Q_{zx}}/M_{y}$ $\approx$ 0.04
(Fig. 2f). With a quantitative estimate of the $r$ factor, which is beyond the
scope of this work, one can in principle compare the relative magnitudes of
$Q_{zx}$ and $M_{y}$.

##  II Two-site S = 1/2 model

Table 1: Quadrupole and dipole moments from the two-site ${\bm{S}}$=1/2 model calculation. Type | Order |  Expression
---|---|---
AF | $\langle N_{x}\rangle$ |  ${\sin 2\theta}{\cos\phi_{c}}$
FM | $\langle M_{y}\rangle$ |  ${\sin 2\theta}{\sin\phi_{c}}$
Quadrupole | $\langle Q_{zx}\rangle$ | $\frac{1}{2}\left({\cos 2\theta}\,{\sin 2\phi}\,{\sin\phi_{c}}\right)$
Singlet | $\langle\text{-}\mathbf{S}^{1}\cdot\mathbf{S}^{2}\text{-}\frac{1}{4}\rangle$ | $\frac{1}{2}\left({\cos 2\theta}\,{\cos 2\phi}\,{\cos^{2}\phi_{c}}-\sin^{2}\phi_{c}\right)$

When a quadrupolar order coexists with a dipolar order, its allowed structure
is constrained by the orientation of the dipolar order. For example, for a
single $S$ = 1 spin, it is easy to show that if its dipole moment is along
$z$, then $yz$ and $zx$ quadrupole moments are zero. For the present case, we
proceed with a simple two-site $S$ = 1/2 model (see Supplementary Note S5). An
arbitrary wavefunction for a pair of NN spins can be expressed in the singlet-
triplet basis ($s$, $T_{x}$, $T_{y}$, $T_{z}$) in terms of two parameters
$\theta$ and $\phi$ as $\mathbf{u}$ \+ $i\mathbf{v}$, where

$\begin{split}\mathbf{u}&\,=\,\cos\theta(\cos{\phi}\cos{\phi_{c}},-\sin{\phi},\,0,\,\cos{\phi}\sin{\phi_{c}})\;,\\\
\mathbf{v}&\,=\,\sin\theta\,(\sin{\phi}\cos{\phi_{c}},\quad\cos{\phi},\,0,\,\sin{\phi}\sin{\phi_{c}})\;,\end{split}$
(5)

under the constraints that the AF(FM) component of the ordered moment is along
$a$($b$) axis, and the canting angle $\phi_{c}$ $\approx$ 12∘; i.e., $\langle
N_{y}\rangle$ = $\langle N_{z}\rangle$ = $\langle M_{x}\rangle$ = $\langle
M_{z}\rangle$ = 0 and $\langle M_{y}\rangle/\langle N_{x}\rangle$ =
$\tan\phi_{c}$, (where $N_{\alpha}$ $\equiv$ $S^{1}_{\alpha}-S^{2}_{\alpha}$
and $M_{\alpha}$ $\equiv$ $S^{1}_{\alpha}+S^{2}_{\alpha}$).

Fig. 3: Phase transition into the spin-nematic phase. a, Raman spectra in the
$E_{\rm g}$ symmetry channel measured on cooling. b, The real part of the
uniform static susceptibility (black) obtained by integrating the $E_{\rm g}$
Raman conductivity $\chi^{\prime\prime}/\omega$. The inverse susceptibility
(red) exhibits a Curie-Weiss behaviour with $T_{C}\,\simeq$ 263 K. c, e,
$A_{\rm 1g}$ phase mode emerges at $T_{C}$ (c), and the $B_{\rm 2g}$ magnon
below $T_{\rm N}$ (e). d, f, Temperature dependence of the intensities (d) and
energies (f) of $A_{\rm 1g}$ and $B_{\rm 2g}$ modes extracted by fitting the
spectra to Fano profiles
$I(\omega)\,=\,\frac{I_{0}}{\Gamma(1-q^{2})}(1-\frac{(q+\epsilon)^{2}}{1+\epsilon^{2}})$,
where $\epsilon\,=\,(\omega-\omega_{0})/\Gamma$, $\Gamma$ is the linewidth,
and $q$ is the asymmetry parameter.

Table 1 compares the magnitudes of magnetic and nonmagnetic orders calculated
as functions of $\theta$ and $\phi$. First, we find that $\langle
Q_{xy}\rangle$ = $\langle Q_{yz}\rangle$ = 0 for all values of $\theta$ and
$\phi$, but $\langle Q_{zx}\rangle$ can be nonzero only when $\phi_{c}$ is
nonzero. This suggests that Dzyloshinskii-Moriya (DM) interaction plays an
important role in stabilizing the quadrupolar order, since the spin canting
arises from the same origin Jackeli and Khaliullin (2009). Second, it is clear
that $\theta$ parameterizes the competition between the magnetic and
nonmagnetic sectors. For example, at $\theta$ = $\frac{\pi}{4}$ the magnetic
moment fully saturates and $\langle Q_{zx}\rangle$ = 0. Third, in the non-
magnetic sector $\phi$ controls the magnitude of $\langle Q_{zx}\rangle$,
which anti-correlates with that of the dimer correlation
$\langle-\mathbf{S}^{1}\cdot\mathbf{S}^{2}-\frac{1}{4}\rangle$, which takes
the values $\frac{1}{2}$, 0, $-\frac{1}{2}$ for the singlet, Néel AF state,
and triplets, respectively. In other words, just as the net FM moment has its
origin in the canting of AF spins, $\langle Q_{zx}\rangle$ arises from the
underlying singlet correlation by “canting” in angle $\phi$. This raises an
intriguing possibility that a SN may arise from a smooth deformation of a RVB
state that has non-zero singlet correlation for every NN pair spins. For
instance, recent studies find that rotating one of the spins by 180∘ in every
dimer in a singlet RVB leads to a triplet RVB, which inherits some of the
generic properties of the RVBs, such as quasiparticles with fractional quantum
numbers König  _et al._ (2022); Shindou and Momoi (2009).

##  III Raman Spectroscopy

Next, we use Raman spectroscopy, which has been suggested as a sensitive probe
for SN Mila (2017), to show that the quadrupolar order persists above $T_{\rm
N}$, armed with the information that the order should appear in the YZ or ZX
polarization channel, which correspond to $E_{g}$ symmetry channel of the
tetragonal $D_{4h}$ point group. This requires the laser beam to be incident
on the side surface of a thin plate-like crystal, and thus has not been
measured in previous Raman studies on Sr2IrO4 Cetin  _et al._ (2012); Gim  _et
al._ (2016); Gretarsson  _et al._ (2016). Figure 3a shows the Raman spectra
measured in ZX scattering geometry. Upon cooling down from $T$ $=$ 325 K, we
observe a broad quasi-elastic feature developing with its tail extending up to
$\lesssim$ 10 meV, which is indicative of slow fluctuations of spins (Fig.
3a). Integrating the Raman conductivity $\chi^{\prime\prime}/\omega$ over a
sufficiently large energy window (0.85 meV$\sim$ 23.4 meV), we obtain in Fig.
3b the real part of the static susceptibility, which follows the Curie-Weiss
temperature dependence with $T_{\rm C}$ $\approx$ 263 K.

Concomitantly, an $A_{1g}$ mode emerges (Fig. 3c and Extended Data Fig. 1),
which is barely visible above our instrumental low-energy cutoff $\approx$
0.85 meV at $T_{\rm C}$, but upon further cooling becomes well resolved as the
peak moves to higher energies. We interpret this peak as the phase mode
associated with the quadrupolar order, analogous to spin-wave modes in the AF
phase. The order parameter has $E_{g}$ symmetry, which can be continuously
rotated about the $c$-axis. This allows the phase mode to have the energy
(1$\sim$2 meV) an order of magnitude smaller than the temperature scale of
$T_{\rm C}$ ($\sim$20 meV), in accordance with the Goldstone theorem. Together
with the divergence in the static susceptibility, this collective excitation
constitutes direct evidence of a thermodynamic phase transition at $T_{C}$
$\sim$ 263 K, well above the AF transition temperature $T_{\rm N}$ $\sim$ 230
K.

The $A_{1g}$ mode persists as the sample is cooled down through $T_{\rm N}$
(Figs. 3c and 3d), at which temperature the $B_{2g}$ single-magnon mode
emerges (Fig. 3e). The $A_{1g}$ mode becomes gapped close to $T_{\rm N}$ as
the AF order further reinforces the breaking of the rotational symmetry (Fig.
3f). The intensity of the $A_{1g}$ mode continues to grow in the AF phase
(Fig. 3d), implying coexistence of dipolar and quadrupolar orders. As the AF
order sets in, however, the stacking pattern of the quadrupoles changes, which
is governed by weak inter-layer interactions of $\lesssim$10 $\mu$eV energy
scales, four orders of magnitudes smaller than those for intra-layer
interactions Porras  _et al._ (2019) (Fig. 2a). We note that the $A_{1g}$ mode
is almost insensitive to the onset of the dipolar order, indicating that spins
entanglement remains intact below $T_{\rm N}$.

##  IV Polarization-resolved
Resonant inelastic x-ray scattering

Fig. 4: Complete breakdown of magnon at short wavelength scales. a, A
schematic of the RIXS geometry. X-rays with $\pi$ polarization are incident to
the sample with the angle $\alpha\approx 75^{\circ}$, and scattered x-rays are
collected without polarization analysis. A single crystal of Sr2IrO4 is placed
between two permanent magnets that apply a magnetic field of 0.3 T along the
$b$-axis ($a$-axis) at the sample position aligning the magnetic domains,
i.e., $\mathbf{M}$ $\parallel$ $\hat{b}$ ($\mathbf{M}$ $\parallel$ $\hat{a}$).
The direction of the applied field can be changed by rotating the outer disk
on which the magnets are mounted while the sample stays fixed. The angle
between the incident and outgoing x-rays is fixed close to 90∘ to suppress
elastic Thomson scattering. b-d, RIXS spectra for $(\pi,\pi)$ (b),
$(\pi/2,\pi/2)$ (c), and $(\pi,0)$ (d). The transverse mode (T) and the
longitudinal mode (L) are displayed along with the sum of the two (T+L). Solid
lines are guides to the eye. Each components are extracted from the raw
spectra measured in different magnetic field directions with a fixed
scattering geometry (see Extended Data Table 1 and Supplementary Note S6).

If the quadrupolar order comprises a substantial part of the ground state
(small $\theta$ limit in Eq. 5), excitation spectra must exhibit a qualitative
departure from the classical spin-wave theory (SWT), which largely reproduces
the experimental spectra of cuprate square-lattice AFs Aeppli  _et al._
(1989); Coldea  _et al._ (2001); Braicovich  _et al._ (2014); Peng  _et al._
(2017). Figure 4 shows the RIXS spectra for the spin components transverse (T)
and longitudinal (L) to the ordered moment direction, which we resolve by
measuring the spectra for two different magnetic moment directions aligned
using a small permanent magnet (see Extended Data Table 1 and Supplementary
Note S6). At the ordering wavevector ($\pi$, $\pi$) (Fig. 4b), the spectral
weight is mostly in the T channel, consistent with the SWT description, which
expects a divergent intensity for a gapless Goldstone mode. The T magnon mode
is also clearly visible at ($\pi/2$, $\pi/2$), although a significant weight
is transferred to the L channel (Fig. 4c). By contrast, at ($\pi$, 0) the
spectrum is completely isotropic and shows no sharp feature that can be
identified as a magnon mode (Fig. 4d).

In the cuprates, the ‘($\pi$,0) anomaly’ is ubiquitously observed across many
different materials Christensen  _et al._ (2007); Headings  _et al._ (2010);
Dalla Piazza  _et al._ (2015); Martinelli  _et al._ (2022); Tsyrulin  _et al._
(2009, 2010), and from its isotropic nature the continuum has been interpreted
as deconfined fractional quasiparticles (spinons) Dalla Piazza  _et al._
(2015); Shao  _et al._ (2017). We note, however, that the T mode still
constitutes the dominant part of the total intensity in most cases. To the
best of our knowledge, the complete loss of the coherent magnon intensity is
unprecedented, and renders alternative multi-magnon scenarios Powalski  _et
al._ (2018) unlikely in our case. Rather, attributing the isotropic continuum
to deconfined spinons in turn suggests that the quadrupolar order may be a
byproduct of a RVB-like state.

##  V Discussion

Theoretically, SNs are predicted to arise in certain spin models with
competing interactions Blume and Hsieh (1969); Chen and Levy (1971); Chandra
and Coleman (1991); Läuchli  _et al._ (2006); Zhitomirsky and Tsunetsugu
(2010). For example, in a square-lattice, four-spin exchange ($J_{c}$)
competing with ferromagnetic (FM) NN Heisenberg exchange ($J$) can lead to a
SN Läuchli  _et al._ (2005); Shannon  _et al._ (2006). In this regard, 5$d$
transition-metal oxides, classified as ‘weak’ Mott insulators stabilized by
strong spin-orbit coupling, may be a promising ground to search for a SN,
because the large spatial extent of the 5$d$ orbitals can bring about strong
competing interactions with NN Heisenberg interactions Witczak-Krempa  _et
al._ (2014); Bertinshaw  _et al._ (2019). Indeed, presence of interaction
terms beyond NN in Sr2IrO4 is clear from its steep spin-wave dispersion along
the magnetic zone-boundary Kim  _et al._ (2012), which can be equally well
fitted with a model including $J_{c}$ or further-neighbor couplings ($J_{2}$
and $J_{3}$). Further, our study suggests that the large canting angle of the
spins resulting in a sizable net FM moment in each layer may be favourable for
stabilizing the SN phase even in an Néel AF.

The fact that $L$ edges of 5$d$ transition-metal elements are in the hard
x-ray regime with access to a wide region in the momentum space is also
advantageous for direct detection of spin quadrupoles, as x-rays become
sensitive to quadrupoles under resonance conditions Savary and Senthil (2015);
Smerald and Shannon (2013); McMorrow  _et al._ (2001). Most other experimental
probes are insensitive to spin quadrupoles, and evidence for a SN has so far
been indirectly provided from thermodynamic Mourigal  _et al._ (2012); Kohama
_et al._ (2019) and nuclear magnetic resonance measurements Orlova  _et al._
(2017). In Sr2IrO4, evidence for a symmetry breaking order above $T_{\rm N}$
has been found in studies using second harmonic generation Seyler  _et al._
(2020), inelastic neutron scattering Jeong  _et al._ (2017), and magnetic
torque measurements Murayama  _et al._ (2021). These studies suggest loop
currents as the possible order. While our result is inconsistent with such a
time-reversal symmetry breaking order (see Extended Data Fig. 2), it is
possible that these experiments are probing yet another order of different
nature.

Our results vividly demonstrate that mulitple orders are intertwinned even in
the Mott insulating phase. The discovery of the SN phase is not only
significant on its own, but also pave a new pathway to investigate the
entanglement structure of quantum spins and the possible connection between SN
and RVB through resonant x-ray scattering processes Savary and Senthil (2015).
The quantum entanglement between NN spins manifests as a spin quadrupolar
order, and possibly accounts for the complete loss of coherent magnons at
momentum ($\pi$, 0). This in turn suggests that the quadrupolar order may be
exploited to follow the singlets formed by NN spins. Along this direction, if
quadrupolar orders can be detected through charge-quadrupole interference,
their temperature and doping evolution can be followed in the absence of a
magnetic order, with the ultimate goal of elucidating the relationship between
the role of magnetic correlations in myriad exotic phases displayed by doped
Mott insulators.

## References

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## Methods

Crystal Growth Single crystals of Sr2IrO4 were grown by the standard flux
growth method. Powders of IrO2, SrCO3, and SrCl ${}_{2}\cdot$ 6H2O were mixed
and placed in an iridium crucible covered with a lid. The mixture was melted
and soaked at $T$ = 1300∘C, slowly cooled down to 900∘C at 8∘C$/$h, and then
furnace cooled to room temperature. We note that our crystals grown in an
iridium crucible have the lattice structure of space group $I4_{1}/acd$ (ref.
54) and shows no sign of glide symmetry breaking distortions previously
reported Torchinsky  _et al._ (2015).

Resonant x-ray diffraction (RXD) RXD experiments were carried out at the 1C
beamline of Pohang Accelerator Laboratory and 4-ID-D beamline of the Advanced
Photon Source. Incident x-ray was tuned at the Ir $L_{3}$ edge (11.217 keV).
The focused beam having spatial resolution better than $\sim$ 100 $\mu$m was
used. The sample was mounted on the cold finger of a closed-cycle cryostat and
temperature was kept at 10 K throughout the experiment. The left and right
circular polarized x-ray was generated using a diamond phase retarder. The
helicity was switched at every data point to measure the flipping ratio of
circular dichroic signal.

Resonant inelastic x-ray sacttering RIXS spectra were measured at the 27-ID-B
beamline of the Advanced Photon Source. Incident x-ray was tuned to the Ir
$L_{3}$ edge (11.215 keV). Using a diamond (1 1 1) high-heat-load
monochromator in combination with a Si (8 4 4) channel-cut monochromator
reduced the energy bandpass down to 14.8 meV. The beam was then focused by a
set of Kirkpatrick-Baez mirrors, producing a spot size of $40\times 10$
(H$\times$V) µm2 FWHM at the sample position. Scattered photons were analyzed
by a Si (8 4 4) diced spherical analyzer with a radius of 2 m and with a mask
of 2 cm diameter. The overall energy resolution was about 30 meV. A horizontal
scattering geometry was used with the incident $\pi$-polarization and the
outgoing polarization was not resolved. All RIXS spectra were taken around (3
0 28.5) in a normal incident scattering geometry and a small magnetic field
($\sim 0.3$ T) was applied along the either $a$\- or $b$-axis to align
magnetic moments. The scattering angle (2$\theta$) was kept around 90∘ to
suppress elastic Thomson scatterings.

Raman spectroscopy Raman spectroscopy was performed with a home-built setup
equipped with a 633 nm He-Ne laser and a liquid nitrogen cooled CCD (Princeton
instruments). The elastic signal is removed by grating-based notch filters
(Optigrate, BragGrate${}^{\textrm{TM}}$ Notch filters). The spectra were
acquired on as-grown surfaces of Sr2IrO4 crystals mounted in a closed cycle
optical cryostat (Montana instruments). The laser power and beam spot size are
0.8 mW and 2 $\mu$m, respectively, which resulted in an almost temperature
independent laser heating of 25 K as determined from the Stokes to Anti-Stokes
intensity ratio. All Raman spectra are Bose-corrected.

Magneto-optical Kerr measurement We performed magneto-optical Kerr
measurements on Sr2IrO4 using an oblique-incidence zero-area Sagnac
interferometer operating at 1550 nm wavelength to measure the in-plane
magnetization Zhu  _et al._ (2021). The relative Kerr angle $\Delta\theta_{K}$
is obtained by subtracting temperature-independent backgrounds, coming from
the instrumentation offset, measured at $T$ = 300 K, and converted to
magnetization (in $\mu_{B}$/ion) by a quantitative comparison of our 0.35 T
Kerr data with the corresponding magnetization data in the Ref. 28. Throughout
the measurements, the incident optical power was maintained below 1 mW so that
the effect of optical heating was smaller than 1 K.

## Acknowledgments

We thank Nic Shannon, Giniyat Khaliullin and Yong Baek Kim for helpful
discussions. This project is supported by IBS-R014-A2 and and National
Research Foundation (NRF) of Korea through the SRC (No. 2018R1A5A6075964). The
use of the Advanced Photon Source at the Argonne National Laboratory was
supported by the U. S. DOE under Contract No. DE-AC02-06CH11357. G.Y.C. is
supported by the NRF of Korea (Grant No. 2020R1C1C1006048) funded by the
Korean Government (MSIT) as well as the Institute of Basic Science under
project code IBS-R014-D1. G.Y.C. is also supported by the Air Force Office of
Scientific Research under Award No. FA2386-22-1-4061 and G.Y.C. also
acknowledges Samsung Science and Technology Foundation under Project Number
SSTF-BA2002-05. H.H. and J.J. are supported by the National Research
Foundation of Korea (Grant No. 2020R1A5A1016518), and the Creative-Pioneering
Researchers Program through Seoul National University.

## Authors contributions

B.J.K. conceived and managed the project. H.K., K.K. and Jo.K. performed Raman
experiments. H.K., S.H., J.S., G.F., Y.C., D.H. and J.W.K. performed RXD
experiments. H.K., J.-K.K., H.-W.J.K. and Ju.K. performed RIXS experiments and
analyzed the data. Ji.K. grew single crystals. H.H. and J.J. performed Kerr
measurements. H.K. and B.J.K. performed representation analysis. W.L. and
G.Y.C. assisted interpretation of the data. H.K., J.-K.K. and B.J.K. wrote the
manuscript with inputs from all authors.

## Competing interests

The authors declare no competing interests.

## Data and materials availability

All data is available in the manuscript or the supplementary information.

Extended Data Fig. 1: Temperature evolution of the low-energy Raman modes. The
Raman spectra of the $A_{\rm 1g}$ and $B_{\rm 2g}$ modes shown in Figs. 2c and
2e are displayed with vertical offset for clarity. The $A_{\rm 1g}$ (red) and
$B_{\rm 2g}$ (black) spectra were measured on the same crystal under the same
experimental conditions including laser power and acquisition time, and the
spectra are plotted in the same arbitrary unit. Extended Data Fig. 2: Magneto-
optical Kerr measurement. a, Relative Kerr angle in the 0.35 T in-plane
magnetic field (red line) and ambient magnetic field near 0 T (black line).
The relative Kerr angle (left axis) is converted to magnetization (right axis)
using a conversion factor of 7.7$\times 10^{-4}$ ($\mu_{B}$/Ir ion)/(µrad). b,
Magnified plot of the 0 T data in a. Dashed line indicates the standard
deviation for the $T$ = 230$\,\sim\,$300 K range. The Kerr signal at $B$ = 0 T
in the range of 230 K $<\,T\,<\,$300 K shows that no net magnetization is
present within our experimental resolution, 3.4$\times 10^{-5}\,\mu_{B}$/ion
(dashed line), thus confirming the preservation of time-reversal symmetry
above $T_{\rm N}\,\sim\,$230 K. Extended Data Table 1:  Sensitive mode for
each combination of specific scattering geometry and field direction in RIXS
measurements on Sr2IrO4. Here, L stands for the longitudinal mode, T for the
in-plane transverse mode, and T′ for the out-of-plane transverse mode. Please
note that ‘Mode’ represents the most sensitive mode in a specific scattering
geometry and a field direction.
Incident angle | Field direction | Moment direction | Polarization | Sensitive element | Mode
---|---|---|---|---|---
Normal | [100] | [010] |  $e_{\pi}$$\times$$e_{\pi^{\prime}}$ | [010] | L
$e_{\pi}$$\times$$e_{\sigma^{\prime}}$ | [001] | T′
[010] | [100] |  $e_{\pi}$$\times$$e_{\pi^{\prime}}$ | [010] | T
$e_{\pi}$$\times$$e_{\sigma^{\prime}}$ | [001] | T′
Grazing | [100] | [010] |  $e_{\pi}$$\times$$e_{\pi^{\prime}}$ | [010] | L
$e_{\pi}$$\times$$e_{\sigma^{\prime}}$ | [100] | T
[010] | [100] |  $e_{\pi}$$\times$$e_{\pi^{\prime}}$ | [010] | T
$e_{\pi}$$\times$$e_{\sigma^{\prime}}$ | [100] | L