# Spin waves and three-dimensionality in the high-pressure antiferromagnetic
phase of SrCu2(BO3)2

Ellen Fogh  Laboratory for Quantum Magnetism, Institute of Physics, Ecole
Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
Gaétan Giriat  Laboratory for Quantum Magnetism, Institute of Physics, Ecole
Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
Mohamed E. Zayed  Department of Physics, Carnegie Mellon University in Qatar,
Education City, PO Box 24866, Doha, Qatar  Andrea Piovano  Institut Laue-
Langevin (ILL), 71 avenue des Martyrs, 38042 Grenoble, France  Martin Boehm
Institut Laue-Langevin (ILL), 71 avenue des Martyrs, 38042 Grenoble, France
Paul Steffens  Institut Laue-Langevin (ILL), 71 avenue des Martyrs, 38042
Grenoble, France  Irina Safiulina  Institut Laue-Langevin (ILL), 71 avenue des
Martyrs, 38042 Grenoble, France  Ursula B. Hansen  Institut Laue-Langevin
(ILL), 71 avenue des Martyrs, 38042 Grenoble, France  Stefan Klotz  Sorbonne
Université, UMR CNRS 7590, Institut de Minéralogie, de Physique des Matériaux
et de Cosmochimie (IMPMC), Paris, France  Jian-Rui Soh  Laboratory for Quantum
Magnetism, Institute of Physics, Ecole Polytechnique Fédérale de Lausanne
(EPFL), CH-1015 Lausanne, Switzerland  Ekaterina Pomjakushina  Laboratory for
Multiscale Materials Experiments, Paul Scherrer Institute, CH-5232 Villigen
PSI, Switzerland  Frédéric Mila  Institute of Physics, Ecole Polytechnique
Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland  Bruce Normand
Laboratory for Quantum Magnetism, Institute of Physics, Ecole Polytechnique
Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland  Laboratory for
Theoretical and Computational Physics, Paul Scherrer Institute, CH-5232
Villigen-PSI, Switzerland  Henrik M. Rønnow  Laboratory for Quantum Magnetism,
Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015
Lausanne, Switzerland

###### Abstract

Quantum magnetic materials can provide explicit realizations of paradigm
models in quantum many-body physics. In this context, SrCu2(BO3)2 is a
faithful realization of the Shastry-Sutherland model (SSM) for ideally
frustrated spin dimers, even displaying several of its quantum magnetic phases
as a function of pressure. We perform inelastic neutron scattering (INS)
measurements on SrCu2(BO3)2 at 5.5 GPa and 4.5 K, observing spin waves that
characterize the high-pressure antiferromagnetic phase. The experimental
spectra are well described by linear spin-wave calculations on a SSM with an
inter-layer interaction, which is determined accurately as $J_{c}=0.053(3)$
meV. The presence of $J_{c}$ indicates the need to account for the three-
dimensional nature of SrCu2(BO3)2 in theoretical models, also at lower
pressures. We find that the ratio between in-plane interactions,
$J^{\prime}/J=1.8(2)$, undergoes a dramatic change compared to lower pressures
that we deduce is driven by a sharp drop in the dimer coupling, $J$. Our
results underline the wide horizons opened by high-pressure INS experiments on
quantum magnetic materials.

Frustrated magnetic interactions give rise to fascinating entangled quantum
states, including valence-bond solids, quantum spin liquids, and spin ices
Savary and Balents (2016). Beyond the Heisenberg model on frustrated
geometries such as the triangular and kagome lattices, exotic ground states,
excitations, and quantum phase transitions (QPTs) are found in many systems
where interactions compete in real space or in spin space. Despite multiple
developments in both theoretical and numerical methods, the nature of many
such states, their spectra, and the model phase diagrams remain only partially
understood. Experimental studies of materials providing physical realizations
are therefore an essential factor for our understanding of quantum magnetic
systems.

SrCu2(BO3)2 is an ideally frustrated system composed of Cu2+ dimers ($S=1/2$
spin pairs) arranged orthogonally both in and between quasi-two-dimensional
(2D) layers, as represented in Figs. 1(a)-(c) Kageyama  _et al._ (1999a). The
in-plane network realizes the Shastry-Sutherland model (SSM) [inset Fig.
1(d)], proposed Shastry and Sutherland (1981) for its exact dimer-singlet
ground state at small $J^{\prime}/J$ and governed by this ratio of competing
interactions Koga and Kawakami (2000); Chung  _et al._ (2001); Läuchli  _et
al._ (2002); Corboz and Mila (2013); Nakano and Sakai (2018); Xi  _et al._
(2023). SrCu2(BO3)2 exhibits the dimer-singlet ground state at ambient
pressure Miyahara and Ueda (1999); Kageyama  _et al._ (1999a, 2000a, 2000b);
Gaulin  _et al._ (2004); Kakurai  _et al._ (2005) and is well described by a
SSM with $J^{\prime}/J=0.63$ and additional weak (3%) Dzyaloshinskii-Moriya
(DM) interactions Cépas  _et al._ (2001); Miyahara and Ueda (2003); Kodama
_et al._ (2005); Miyahara and Mila (2005); Shi  _et al._ (2022). Remarkably,
the phase diagram of SrCu2(BO3)2 under an applied hydrostatic pressure Waki
_et al._ (2007); Zayed  _et al._ (2017); Sakurai  _et al._ (2018), shown in
Fig. 1(d), mirrors that of the SSM with increasing $J^{\prime}/J$ [Fig. 1(e)],
and is studied extensively for its exotic QPTs Guo  _et al._ (2020); Larrea
Jiménez  _et al._ (2021); Guo  _et al._ (2023). Above 4 GPa, where the
tetragonal symmetry changes to monoclinic Loa  _et al._ (2005); Haravifard
_et al._ (2014); Zayed  _et al._ (2014), antiferromagnetic (AFM) order is
found by neutron and X-ray diffraction Haravifard  _et al._ (2014); Zayed  _et
al._ (2017); Zayed (2010) and phase-transition features are reported around 8
K Guo  _et al._ (2020) and above 120 K Haravifard  _et al._ (2014); Zayed  _et
al._ (2014); Guo  _et al._ (2020). However, the high pressures pose a severe
challenge to experiment and rather little is known about either the precise
pressure regimes for the SSM-AFM and monoclinic-AFM phases or the possible
magnetic interactions in either phase.

Here we perform an inelastic neutron scattering (INS) experiment to
characterize the monoclinic-AFM phase of SrCu2(BO3)2. Working at 5.5 GPa and
4.5 K, we observe dispersive spin-wave excitations in 58 mg of sample. Linear
spin-wave theory is appropriate to deduce the parameters of the minimal
magnetic Hamiltonian. The in-plane interactions change dramatically from
$J^{\prime}/J=0.63$ at ambient pressure to 1.8(2) at 5.5 GPa, which we ascribe
to a strong reduction of $J$ arising from its near-critical bond angle. We
also discover that it is necessary to include an inter-layer coupling, which
we fit as $J_{c}=0.053(3)$ meV, to explain the observed 2 meV splitting of
otherwise degenerate branches in the low-energy part of the spectrum. We
discuss the consequences of this 3D nature for the physics of SrCu2(BO3)2.

Figure 1: SrCu2(BO3)2 and the Shastry-Sutherland model (SSM). (a)-(b) Crystal
structure of SrCu2(BO3)2 shown in projection along $(0,0,1)$ (a) and $(1,0,0)$
(b). The dimer interaction, $J$ (solid blue line), inter-dimer interaction,
$J^{\prime}$ (dashed green line), and inter-layer interaction, $J_{c}$ (solid
orange line), are indicated. (c) Illustration of the tetrahedron formed by two
orthogonal dimers stacked along the $(0,0,1)$ direction. (d) Phase diagram of
SrCu2(BO3)2 as a function of temperature and pressure, as deduced from
specific-heat measurements (circles Larrea Jiménez  _et al._ (2021) and dots
Guo  _et al._ (2020)) and neutron diffraction (diamonds) Haravifard  _et al._
(2014); Zayed  _et al._ (2017). The star marks the location in temperature and
pressure of our INS experiments. The inset shows a schematic representation of
the SSM. (e) Phase diagram of the SSM as a function of $J^{\prime}/J$, as
determined by multiple numerical methods Koga and Kawakami (2000); Chung  _et
al._ (2001); Läuchli  _et al._ (2002); Nakano and Sakai (2018); Corboz and
Mila (2013); Xi  _et al._ (2023).

Our INS experiments used a 58 mg disc-shaped single-crystalline sample of
SrCu2(BO3)2, which was grown by the floating-zone method Kageyama  _et al._
(1999b); Jorge  _et al._ (2004) and oriented with ${\bf Q}=(q_{h},q_{k},0)$ in
the horizontal scattering plane. The small sample size is a further challenge
for INS, but recent developments in high-pressure techniques at the Institute
Laue-Langevin (ILL) Komatsu  _et al._ (2014); Klotz  _et al._ (2019); Funnell
_et al._ (2021) made this experiment possible. Using a Paris-Edinburgh press,
the pressure was increased gradually to 5.5 GPa while the lattice parameters
of SrCu2(BO3)2 Loa  _et al._ (2005) were measured for pressure determination.
The sample chamber was pre-cooled with liquid N2 and cooling to 4.5 K was
provided by a closed-cycle refrigerator. We performed two consecutive
experiments at the IN8 and ThALES triple-axis spectrometers with the same
loading. We used fixed momentum transfers $k_{f}=2.66$ Å-1 at IN8 and
$k_{f}=1.50$ Å-1 at ThALES, providing respective instrumental resolutions
(FWHM) of $1.05(1)$ and $0.167(4)$ meV. Scans with constant energy transfer,
$E$, or constant momentum transfer, $\bf Q$, were performed with typical
counting times of 5 min per point. In the Supplemental Material (SM) sup  we
provide detailed descriptions of the pressure loading procedure (Sec. S1A),
instrument configurations (Sec. S1B), pressure determination (Sec. S1C), and
data treatment (Sec. S1D). Supporting diffraction data were collected at IN3,
also at the ILL, and this experiment is described in Sec. S1E.

A selection of constant-$E$ and constant-$\bf Q$ scans is shown in Fig. 2,
with data from both IN8 and ThALES. The width of the elastic line on IN8
allowed for access to energy transfers above approximately 2 meV, whereas at
ThALES 0.5 meV was accessible. Because the atomic positions in the monoclinic
high-pressure phase remain close to those of the tetragonal phase, with in-
plane lattice parameters $a/b\simeq 0.999$ (and monoclinic angle $\beta\simeq
94^{\circ}$) Haravifard  _et al._ (2014), we retain the tetragonal
approximation with $a^{*}$ and $b^{*}=2\pi/a$. Figures 3(a)-(b) collect all
neutron intensities measured respectively along $(0,q_{k},0)$ and
$(q_{h},1,0)$ at constant $E$, and make clear that dispersive modes are
present with origins at $(0,1,0)$ and $(1,1,0)$. Above 5 meV it becomes
difficult to track the mode positions due to vanishing neutron intensity
[Figs. 2(a) and 2(d)]. At 1.5 meV and below, we no longer observe neutron
intensity at $(0,1,0)$ [Figs. 2(b) and 2(c)]. In Figs. 2(a)-(b) we fitted the
data collected along $(q_{h},1,0)$ with Gaussian line shapes centered
respectively at $\pm\delta_{h}$ and $\pm\delta_{h}^{\prime}$ from $(0,1,0)$
and $(1,1,0)$, allowing the peak widths and intensities to vary individually
(Sec. 1D of the SM sup ). Similar fits to the data collected along
$(0,q_{k},0)$ yielded the mode positions along both directions in $\bf Q$. The
dispersion minimum at $(0,1,0)$ is resolved in a constant-$\bf Q$ scan from
ThALES [Fig. 2(c)]. A Gaussian fit yields an energy of 1.90(8) meV and the
magnetic origin of the excitation is demonstrated by its absence at 140 K. In
Fig. 2(d), we show constant-$\bf Q$ scans from IN8 where subtraction of the
incoherent contribution allowed the fitting of one or two Gaussian profiles at
different $\bf Q$. The fitted mode positions are also marked in Figs.
3(a)-(b).

Figure 2: Selection of INS data collected at IN8 and ThALES at 5.5 GPa.
(a)-(b) Constant-$E$ scans along ${\bf Q}=(q_{h},1,0)$. (c)-(d) Constant-$\bf
Q$ scans. Fits to determine the mode positions in $\bf Q$ and $E$ are
illustrated with solid lines and the shaded areas represent the individual
Gaussians. Dashed lines in panel (d) show the incoherent contribution to the
scattering signal measured across the elastic line at $(0.7,1.3,0)$. Figure 3:
Measured and modelled excitation spectrum of SrCu2(BO3)2 in the monoclinic-AFM
phase. (a)-(b) INS intensity collected on IN8 at 5.5 GPa and 4.5 K, shown as a
function of $E$ and $\bf Q$ along the $(0,q_{k},0)$ and $(q_{h},1,0)$
directions. The inset in panel (b) shows lower-energy data (dashed box)
collected at ThALES around $(0,1,0)$. (c)-(d) Comparison with the calculated
spectrum using an instrumental energy resolution of 1 meV. The splittings,
$\Delta$ and $\Delta^{\prime}$, are indicated in panel (c). The mode positions
determined from the INS data are shown with open pink circles (IN8) and open
pink triangles (ThALES). The orientation of the error bars indicates whether a
given data point originates from a constant-$E$ or constant-$\bf Q$ scan. The
grey curves show the dispersion calculated from linear spin-wave theory.

From the near-tetragonal nature of the high-pressure phase, we assume that the
SSM continues to provide a minimal model for the observed spin-wave
dispersion. The physics of the AFM phase is that the stronger $J^{\prime}$
forces the spins to order ferromagnetically (FM) on the dimers and AFM between
dimers, both in and out of plane [Fig. 1(c)]. The 2D AFM phase has two doubly
degenerate spin-wave branches, a Goldstone mode and an optical mode. Although
we have no dispersion measurements in the $c$ direction, the observed in-plane
scattering intensity [Figs. 3(a)-(b)] motivates the introduction of an inter-
layer interaction, $J_{c}$, between orthogonal dimers along the $(0,0,1)$
direction, as shown in Fig. 1(c). This model has the following properties: (i)
the degeneracy of the low-energy mode is lifted, with a splitting
$\Delta=8S\sqrt{J^{\prime}J_{c}}$ above the massless point; (ii) due to the
AFM inter-layer stacking, the intensity of the Goldstone mode vanishes at
${\bf Q}=(0,1,0)$ and $(0,3,0)$, making the spectrum look gapped at these
positions; (iii) there is an intensity asymmetry along $(0,q_{k},0)$ around
$q_{k}=1$ and $3$; (iv) the splitting between the low-energy and optical
branches at the top of the former [e.g. ${\bf Q}=(0,1.5,0)$ and $(0.5,1,0)$]
is $\Delta^{\prime}\approx 2SJ$.

We refine the model parameters based on the experimentally determined mode
positions and intensities with assistance from spinW Toth and Lake (2015), a
software implementation of linear spin-wave theory, to obtain
$J^{\prime}=4.28(14)$ meV, $J=2.36(26)$ meV (i.e. $J^{\prime}/J=1.8(2)$), and
$J_{c}=0.053(3)$ meV. The spectra for an assumed instrumental energy
resolution of 1 meV are shown in Figs. 3(c)-(d), and provide good agreement
with the measurements in Figs. 3(a)-(b). In particular, the fitted spectrum
confirms an AFM layer stacking, because FM stacking would result in Goldstone-
mode intensity at $(0,1,0)$ and $(0,3,0)$, but not at $(1,1,0)$. We note that
our deduced parameter values are not corrected for the quantum renormalization
factor, $Z_{c}$, which to our knowledge has not been evaluated for the AFM
phase of the SSM.

Currently the most refined estimates for the interaction parameters of
SrCu2(BO3)2 at ambient pressure are $J=81.5$ K (7.0 meV) and $J^{\prime}=51.3$
K (4.4 meV, $J^{\prime}/J=0.63$) Shi  _et al._ (2022). While there is no exact
model for the increase of $J^{\prime}/J$ with pressure, a linear decrease of
both $J$ and $J^{\prime}$, with $\Delta J/P$ falling 3-4 times faster than
$\Delta J^{\prime}/P$ Zayed  _et al._ (2017), reproduces the $(H,P)$ phase
diagram rather well in the dimer and plaquette phases Shi  _et al._ (2022). Ab
initio calculations predict a discontinuity in the parameters on entering the
SSM-AFM phase Badrtdinov  _et al._ (2020), and at higher pressures no
information for $J$ or $J^{\prime}$ was available until now. Although it is
unexpected that $J$ is suppressed so strongly, it is not implausible. Its size
and sign are highly sensitive to the Cu-O-Cu bond angle following the
Goodenough-Kanamori rules Goodenough (1955); Kanamori (1959), especially when
this angle is close to 95∘ Mizuno  _et al._ (1998). Measurements in
SrCu2(BO3)2 [Figs. 1(a)-(b)] indicate a decrease from $98^{\circ}$ at ambient
pressure to $94^{\circ}$ at 3.7 GPa Zayed (2010). The $J^{\prime}$
superexchange path is through a BO3 unit, which remains robust with pressure
and locked in position by a mirror plane, becoming free to rotate out of the
plane only beyond the monoclinic transition. Our measurements therefore
quantify all of these considerations.

Turning to $J_{c}$, we note first that the system is much softer along the
${\hat{c}}$ direction than in the plane, with the $c$ lattice parameter
reduced by around 6% at 5.5 GPa, whereas $a$ and $b$ are reduced by less than
1% Loa  _et al._ (2005); Haravifard  _et al._ (2014). The buckling of the Cu-
BO3 planes shown in Fig. 1(b) leads to two independent $J_{c}$ parameters
Miyahara and Ueda (2003). However, the in-plane INS spectrum is sensitive only
to their combined effect, and thus we fit a single effective $J_{c}$; further
details are presented in Sec. S2A of the SM sup . In this quasi-2D system, the
splitting $\Delta$ is a steep function of $J_{c}$ and in our INS experiment
$\Delta=1.90(8)$ meV is measured to high accuracy, making our estimate of
$J_{c}=0.053(3)$ meV very precise. Its small value reflects the absence of
significant inter-layer superexchange paths, and is likely to be smaller still
at ambient pressure. An indirect estimate of the ambient-pressure $J_{c}$ as a
10% correction to the high-temperature magnetic susceptibility Miyahara and
Ueda (2000, 2003) returned a value of 0.7 meV, an order-of-magnitude
difference from our result that is likely a consequence of strong finite-size
effects inherent to early numerical studies (since remedied by more modern
technology Wietek  _et al._ (2019)).

Although a Heisenberg SSM with an inter-layer coupling captures all the
features of the spin-wave dispersion in the high-pressure phase, SrCu2(BO3)2
is known to have weak DM interactions at ambient pressure Miyahara and Ueda
(2003); Kodama  _et al._ (2005); Shi  _et al._ (2022). These terms also act to
gap the Goldstone mode, but the DM interaction required to create a gap of 1.9
meV in a 2D SSM is comparable to $J$, rendering this an unlikely mechanism to
explain the INS spectra. Although our minimal model has no DM interactions, we
absolutely do not claim that they are no longer present, only that we cannot
resolve them. In Sec. 2B of the SM sup  we explore models including DM
interactions, and show that even 10% DM terms in the lower-symmetry monoclinic
phase of SrCu2(BO3)2 provide only a minor correction to the Heisenberg
interaction parameters.

The confirmation of a small $J_{c}$ is essential information for assessing the
suitability of a 2D SSM to describe SrCu2(BO3)2. The fact that the dimer ($J$)
bonds are forced into a FM configuration in the AFM phase makes the four
$J_{c}$ bonds in an inter-layer tetrahedron fully unfrustrated [Fig. 1(c)],
and hence they contribute an energy of order 1 K per site to the stability of
a 3D AFM phase. By contrast, in the dimer-singlet phase these bonds are simply
ineffective. Concerning the plaquette phase, it was shown rather early that
its very existence depends on having only a weak $J_{c}$ Koga (2000), and the
propensity of $J_{c}$ to destabilize the plaquette phase has been used
recently to extract a value $J_{c}\approx 0.2$ meV Vlaar and Corboz (2023).
Our definitive $J_{c}$ can be used to refine the estimates of intra-layer
energetics and their pressure evolution on which that estimate was based.
Another delicate issue is whether the plaquette singlets form around the
squares that contain the dimer bond (“full plaquette”) Zayed  _et al._ (2017)
or the squares that do not (“empty plaquette”) Corboz and Mila (2013). The
minimal $J_{c}$ we consider is thought not to affect plaquette formation Vlaar
and Corboz (2023), and rather too little is known about the 3D plaquette order
to make a statement on whether more complex inter-layer coupling could do so.
Finally, because the plaquette-AFM transition of the 2D SSM has been proposed
as a candidate deconfined quantum critical point (DQCP), and SrCu2(BO3)2 as a
candidate material for its observation Lee  _et al._ (2019); Yang  _et al._
(2022); Cui  _et al._ (2023), the effects of inter-layer coupling should
definitely be included as part of evaluating such a scenario.

Finally, we remark that our results do not provide an immediate understanding
of the reported phase transitions at 8 K Guo  _et al._ (2020) and 120 K
Haravifard  _et al._ (2014); Zayed  _et al._ (2014); Guo  _et al._ (2020), the
parameters of our quasi-2D model being more likely to favor a single ordering
transition at an intermediate temperature (of order $J^{\prime}/2$ Yasuda  _et
al._ (2005)). We performed detailed diffraction studies of the $(0,1,0)$
magnetic Bragg peak, as described in Sec. S1E of the SM sup , finding an
unconventional thermal evolution with a characteristic temperature of 88 K.
While one may speculate about the onset of 2D and of true 3D order, these
contradictions underline the need for more systematic combined studies of the
structure and magnetism at and above 3 GPa.

In conclusion, we have performed INS on a single crystal of 58 mg to measure
spin waves in the monoclinic-AFM phase of SrCu2(BO3)2 at a pressure of 5.5
GPa. The observed spectra are well described by a Shastry-Sutherland model
with an inter-layer coupling. Consequently, when modelling the physics of the
SrCu2(BO3)2 phase diagram, it may be necessary to consider the three-
dimensionality. Our work highlights the importance of the technical capability
to perform INS under extreme conditions of high pressure and low temperature
for the discovery and investigation of quantum magnetic states.

Acknowledgements – We are very grateful to C. Payre and J. Maurice for their
technical assistance with the high-pressure equipment at the ILL. We
acknowledge the financial support of the European Research Council through the
Synergy network HERO (Grant No. 810451) and of the Swiss National Science
Foundation through Project Grant No. 188648. We thank the ILL for the
allocation of neutron beam time for this study on the IN8, ThALES, and IN3
spectrometers. Collected data are available as Refs. Fogh  _et al._ (2021a,
b). This publication was made possible in part by the generous support of the
Qatar Foundation through the Seed Research Funding Program of Carnegie Mellon
University in Qatar. The statements made herein are solely the responsibility
of the authors.

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