# The Shastry-Sutherland Compound SrCu2(BO3)2
Studied up to the Saturation Magnetic Field

T. Nomura<EMAIL_ADDRESS>Institute for Solid State Physics,
University of Tokyo, Kashiwa, Chiba 277-8581, Japan  P. Corboz
<EMAIL_ADDRESS>Institute for Theoretical Physics and Delta Institute for
Theoretical Physics, University of Amsterdam, Science Park 904, 1098 XH
Amsterdam, The Netherlands  A. Miyata  Hochfeld-Magnetlabor Dresden (HLD-
EMFL), Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany  S.
Zherlitsyn  Hochfeld-Magnetlabor Dresden (HLD-EMFL), Helmholtz-Zentrum
Dresden-Rossendorf, 01328 Dresden, Germany  Y. Ishii  Institute for Solid
State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan  Y. Kohama
Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba
277-8581, Japan  Y. H. Matsuda  Institute for Solid State Physics, University
of Tokyo, Kashiwa, Chiba 277-8581, Japan  A. Ikeda  Department of Engineering
Science, University of Electro-Communications, Chofu, Tokyo 182-8585, Japan
C. Zhong  Present address: Department of Applied Chemistry, Ritsumeikan
University, Kusatsu, Shiga 525-8577, Japan Graduate School of Engineering,
Kyoto University. Nishikyouku, Kyoto 615-8510, Japan  H. Kageyama  Graduate
School of Engineering, Kyoto University. Nishikyouku, Kyoto 615-8510, Japan
F. Mila  Institute of Theoretical Physics, Ecole Polytechnique Fédérale de
Lausanne (EPFL), 1015 Lausanne, Switzerland

###### Abstract

We report ultrasound velocity and magnetostriction results for the orthogonal-
dimer antiferromagnet SrCu2(BO3)2 up to the saturation magnetic field. In
addition to anomalies that can be associated to plateaus with fractional
magnetization $M/M_{\mathrm{s}}=$ 1/8, 1/4, 1/3, 2/5, and 1/2, we have
detected higher-field sound-velocity and magnetostriction anomalies at 116,
127, and 139 T. Strikingly, the anomalies of the sound velocity of the
$c_{66}$ acoustic mode are very much plateau dependent, with a huge decrease
by $-50$% in the 1/2 plateau state due to the tetragonal-to-orthorhombic
instability triggered by the triplet crystallization into a checkerboard
pattern. This effect, as well as the high-field anomalies, can be well
accounted for by tensor-network calculations (iPEPS) of the Shastry-Sutherland
model. We conclude in particular that the anomaly at 139 T corresponds to the
saturation of magnetization, and we suggest that the other two anomalies at
116 and 127 T are related to supersolid phases whose energies are nearly
degenerate in this field range.

##  I introduction

The discovery of a sequence of magnetization plateaus in SrCu2(BO3)2 (SCBO),
stabilized at fractional magnetization values, is a milestone in quantum spin
physics [1, 2, 3, 4]. In this material, Cu2+ ions with spin $S=1/2$ arrange in
an orthogonal-dimer geometry known as the Shasty-Sutherland lattice [Figs.
1(a, b)] [5, 6]. The effective Hamiltonian including intra- and inter-dimer
interactions ($J$ and $J^{\prime}$, respectively) and the Zeeman term is
defined by

$\mathcal{H}=J\sum\limits_{\langle
i,j\rangle}S_{i}S_{j}+J^{\prime}\sum\limits_{\langle\langle
i,j\rangle\rangle}S_{i}S_{j}-h\sum\limits_{i}S_{i}^{z}.$ (1)

The competing antiferromagnetic (AFM) couplings ($J$ and $J^{\prime}$) result
in a geometrical frustration, giving rise to various phases depending on the
ratio $J^{\prime}/J$ and magnetic field $h$ [7, 8, 9, 10, 11, 12, 13, 14]. The
exchange parameter ratio for SCBO at ambient pressure is estimated to be
$J^{\prime}/J\sim 0.63$ from the magnetization measurement up to 118 T [3].
However, the pantograph-like magnetostriction, which modulates the Cu-O-Cu
angle and, hence, the superexchange interaction, can change the ratio in
applied magnetic fields [15, 16]. At low temperatures, SCBO shows a
fascinating magnetization curve with multiple anomalies, the most prominent
ones are related to the plateau phases at 1/8, 2/15, 1/6, 1/4, 1/3, 2/5, and
1/2 of the saturation magnetization $M_{\mathrm{s}}$ [3, 17, 18, 19]. In the
plateau regions, triplets (or bound states of triplets for the low-field
plateaus) crystallize into magnetic superstructures [20, 21, 22] as a result
of the effective repulsion and the localized nature of triplets inherent to
the frustrated geometry [6, 23, 24, 25, 26, 26]. So far, no experimental
investigation up to the saturation magnetization has been reported.

Figure 1:  (a) Crystal structure of SrCu2(BO3)2 shown in the $ab$ plane. The
blue, red, and green spheres represent Cu, O, and B atoms, respectively. The
black square shows the crystallographic unit cell. (b) Exchange couplings ($J$
and $J^{\prime}$) between Cu2+ ions with $S=1/2$. (c, d) Deformed structure
with the strains (c) $\varepsilon_{xy}$ ($c_{66}$ mode) and (d)
$\varepsilon_{xx}$ ($c_{11}$ mode), resulting in the asymmetric and symmetric
modulations of $J$, respectively. Only the nearest neighbor couplings are
shown in (c) and (d).

In this paper, we present the results of ultrasound and magnetostriction
measurements up to 150 T, reaching to the saturation field. For studying the
magnetism of SCBO at extremely high magnetic fields, the ultrasound and
magnetostriction are powerful techniques due to presence of the spin-strain
interactions. Furthermore, the sensitivity of these techniques is maximal at
the top of the pulsed field, where the sensitivity is strongly reduced for the
magnetization measurements [27]. Besides, the earlier studies have pointed out
that the spin-lattice coupling plays an important role for the high-field
properties of SCBO [17, 15, 16, 28, 29, 30, 31]. In particular, the $c_{66}$
mode (with $\pm\varepsilon_{xy}$ strain) shows extremely large anomalies. In
this study, we discuss the magneto-structural properties of SCBO at ultrahigh
magnetic fields supported by the theoretical calculations based on the
infinite projected entangled pair state (iPEPS) tensor-network algorithm [32,
33, 34]. The elastic properties of novel supersolid phases above 100 T are
discussed in terms of the spin-lattice coupling.

##  II Experimental method

The pulsed magnetic fields up to 150 T were generated with help of the
vertical single-turn-coil system (STC) in the ISSP, University of Tokyo [35].
We used a liquid 4He bath cryostat to keep the sample at 3.2–4.2 K. We note
that the sample temperature can change due to the magnetocaloric effect during
the pulsed field ($\sim 6$ $\mu$s) [17, 30]. High-quality single crystals of
SCBO were grown by a traveling solvent floating zone method [36]. We used the
one ($2\times 1\times 1$ mm3) for the ultrasound and another one ($2\times
1\times 0.3$ mm3) for the magnetostriction measurement. Magnetic fields were
always applied along the $c$ axis in our experiments.

We performed the ultrasound measurements by using the continuous-wave
excitation technique [37]. Ultrasound waves with the frequency of 20–40 MHz
were excited by a LiNbO3 transducer attached to the surface of the crystal.
The transmitted waves were detected by another transducer and recorded by a
digital oscilloscope. The recorded signals were analyzed by using the
numerical lock-in technique, and the phase change was converted to the
relative change of sound velocity $\Delta v/v_{0}$. With this technique, one
can obtain reliable results around the peak of the pulsed field where the
field sweep rate slows down. Therefore, we repeated the measurements with
different peak fields and extracted the reproducible part of the results. We
also performed the ultrasound measurements up to 83 T by using the ultrasound
pulse-echo technique with a dual-pulse magnet in HLD, Dresden [38]. We
measured two in-plane modes, $c_{66}$ ($\bf{k}||[100]$, $\bf{u}||[010]$) and
$c_{11}$ ($\bf{k}||\bf{u}||[100]$), where $\bf{k}$ ($\bf{u}$) is the
propagation (displacement) vector.

We performed the magnetostriction experiments using the Fiber-Bragg grating
(FBG) fixed onto the crystal and the optical filter method [39]. When the
sample length changed, the Bragg wavelength of the reflected light also
changed. By using a band-pass filter with a band edge close to the Bragg
wavelength, the reflection wavelength shift was detected as an amplitude
change. This scheme allows us to measure magnetostriction at a high speed of
100 MHz. We used an amplified spontaneous emission source as an incident
broadband near-infrared light source. In this study, we fixed the FBG using
the low-temperature glue SK-229 to detect the longitudinal magnetostriction
along the $c$ axis, $L_{c}$. In addition, the fiber and sample were put deep
inside a vacuum grease to attenuate the sample vibration caused by the
magneto-structural phase transitions [40, 41]. Because of the large amount of
grease coupled to the FBG, the detected magnetostriction was reduced to $\sim
20$ % of the reported value [17]. Nevertheless, the obtained magnetostriction
was qualitatively reproducible and reflected the magnetic-field-induced phase
transitions.

##  III Experimental results

Figure 2 shows a summary of the ultrasound and magnetostriction results up to
the ultrahigh field of 150 T. For comparison, the magnetization curve at 2.1 K
[3] is shown in Fig. 2(c) with bars representing the regions of magnetization
plateaus. For reproducibility and raw data, see Supplemental Material (SM)
[42].

Figure 2:  The ultrahigh-field data ($H//c$) obtained in SrCu2(BO3)2. (a)
Relative change of the sound velocity for the $c_{66}$ and $c_{11}$ acoustic
modes. The results for the $c_{11}$ mode are multiplied by 10. The results
obtained with the non-destructive magnet at 1.5 K and the STCs at 3.2 K are
shown by the black and colored curves, respectively. The purple, green,
orange, yellow, cyan curves represent the results up to the maximum fields of
$\sim$150, 130, 120, 70, 30 T, respectively. (b) Magnetostriction along the
$c$ axis measured at 4.2 K. (c) Magnetization along the $c$ axis measured at
2.1 K [3]. The plateau field regions suggested by the magnetization
measurement are shown by bars. Anomalies above the 1/2 plateau are denoted by
arrows.

First, we discuss the results of the sound velocity. Figure 2(a) shows the
relative changes of the sound velocity $\Delta v/v_{0}$ for the $c_{11}$ mode
(multiplied by 10) and the $c_{66}$ mode. The results obtained by using the
non-destructive magnet are shown by the black curves. $\Delta v/v_{0}$ is
significantly larger for the $c_{66}$ than $c_{11}$ mode, which is consistent
with the previous study [29]. Clear anomalies are observed at 27, 33, 40, and
74 T, corresponding to the onsets of the 1/8, 1/4, 1/3, and 2/5 plateau
phases, in line with published high-field results [3, 17]. The results above
the 1/2 plateau are obtained by the STC experiments (colored curves).
Therefore, the experimental error ($\pm 10\%$ of $\Delta v/v_{0}$) is larger
than that obtained in a non-destructive pulsed magnet ($\pm 3\%$ of $\Delta
v/v_{0}$). Nevertheless, the relative changes in $\Delta v/v_{0}$ are reliable
and qualitatively well reproduced. The sound velocity of the $c_{66}$ mode
stays constant at $\Delta v/v_{0}=-50\%$ in the 1/2 plateau phase. This is a
surprising result because such a large softening is usually observed at the
phase boundary when a soft mode leads to a lattice distortion [31]. The strong
softening persists up to 116 T, which is slightly higher than the end of the
1/2 plateau (108 T) reported by the magnetization measurement [3]. At 116 T,
$\Delta v/v_{0}$ shows a drastic increase. Another anomaly is observed at 126
T, where the slope of $\Delta v/v_{0}$ changes. One more anomaly is detected
at 140 T, where $\Delta v/v_{0}$ discontinuously increases and saturates at
the level of $+20\%$. In contrast to the $c_{66}$ mode, the $c_{11}$ mode
reveals no clear anomaly above 80 T within our experimental resolution.

Figure 2(b) shows two magnetostriction curves measured along the $c$ axis. The
results exhibit features similar to those of the magnetization; both start to
increase when the spin gap closes ($\sim 25$ T) and stay approximately
constant in the plateau phases. Here, we comment on the reproducibility of our
magnetostriction results. As the previous study shows [17], the
magnetostriction strongly depends on the experimental setting and the strain
between the FBG and the sample. Indeed, we also find that the results vary
slightly depending on the glue and the surrounding grease. Although the
overall magnetostriction depends on the setting, the transition field detected
by this technique is well reproduced. Therefore, in this study, we only focus
on the critical fields and do not discuss the magnitude of the
magnetostriction. Above the 1/2 plateau, we detect four anomalies in $\Delta
L_{c}$ at 108, 116, 128, and 138 T.

Table I summarizes the critical fields obtained by the ultrasound,
magnetostriction, and magnetization experiments. The critical-field notations
($H_{\mathrm{c6}}$, $H_{\mathrm{c7}}$, $H_{\mathrm{c8}}$, and
$H_{\mathrm{c9}}$) are termed after Ref. [3]. The critical-field values agree
well considering the error range of the STC experiments. The only discrepancy
is that the anomaly at 108 T is missing in the ultrasound results. This
anomaly corresponding to the end of the 1/2 plateau is observed in both
magnetization and magnetostriction. Note that the experimental conditions
(sample cooling and sweep rate) are very similar for these three experiments.
Therefore, it suggests that the acoustic modes ($c_{11}$ and $c_{66}$) are not
sensitive to the transition at 108 T. The origin of the observed anomalies is
discussed below.

Table 1: Critical fields with error ranges in the unit of Tesla obtained by ultrasound, magnetostriction, and magnetization experiments. | $H_{\mathrm{c6}}$ | $H_{\mathrm{c7}}$ | $H_{\mathrm{c8}}$ | $H_{\mathrm{c9}}$
---|---|---|---|---
Ultrasound | – | 116(3) | 126(3) | 140(2)
Magnetostriction | 108(2) | 116(2) | 128(3) | 138(3)
Magnetization [3] | 108(1) |  |  |

##  IV theoretical approach

We have used iPEPS to map out the phase diagram at high magnetic fields,
complementing previous results at low [43] and intermediate fields [3, 12]. An
iPEPS is a variational tensor network ansatz to represent 2D ground states in
the thermodynamic limit [32, 33, 34] and can be seen as a higher-dimensional
generalization of matrix product states. The ansatz consists of a unit cell of
tensors which is periodically repeated on the infinite 2D lattice. Here we use
one tensor per dimer and unit cells of different sizes to capture the relevant
magnetic structures. The accuracy of the ansatz is systematically controlled
by the bond dimension $D$ of the tensors. The optimization of the variational
parameters is done based on an imaginary time evolution using a simple update
[44] and cluster update [45] approach, which provides good estimates of ground
state energies while being computationally affordable even for very large unit
cell sizes. The approximate contraction of the 2D tensor network is done by a
variant [46, 47] of the corner-transfer matrix method [48, 49], where the
contraction dimension $\chi$ is kept large enough so that contraction errors
are negligible. For more details on the iPEPS approach we refer to Refs. [50,
51, 52].

##  V theoretical results

In Fig. 3, we present the phase diagram as a function of $J^{\prime}/J$
between the 1/2 plateau and full saturation, for values of $J^{\prime}/J$ in
the vicinity of the predicted value $J^{\prime}/J=0.63$ for SCBO from Ref.
[3]. The data has been obtained for a bond dimension $D=8$ and cluster update
optimization, which is sufficiently large so that finite-$D$ errors on the
phase boundaries are small (of the order of the symbol sizes). Representative
spin patterns of the phases are also shown in Fig. 3. Besides the familiar 1/2
plateau phase, we find different types of supersolid phases (SSPs), i.e.
phases which simultaneously break translational symmetry and the U(1) symmetry
associated with the total $S^{z}$ conservation. They all exhibit a diagonal
stripe pattern with a certain period.

Figure 3:  iPEPS phase diagram of the Shastry-Sutherland model at high
magnetic fields up to saturation (obtained for $D=8$ and cluster
optimization), revealing several supersolid phases (SSP) with different
periods in between the 1/2 plateau and saturation. Typical spin patterns for
each phase are shown, where the thickness of the grey bonds scales with the
local bond energy (the thicker the lower the energy).  Figure 4: Upper panel:
Magnetization curve obtained with iPEPS for $J^{\prime}/J=0.63$ ($D=8$ and
cluster update optimization) at high fields up to saturation. Lower panel:
Energy difference in units of $J$ from the lowest energy state. The values on
the horizontal axis in the upper panel have been converted to real units [3]
for the direct comparison with experiments. Phase boundaries are indicated by
vertical dashed lines.

Above the 1/2 plateau the dominant phase is the 1/3 SSP, which has a period 6
and which can also be found at lower fields above the 1/3 plateau (hence the
name 1/3 SSP, see Ref. [3]). Within the 1/3 SSP, there are two distinct
regions which we call type a and type b. The former has 2 different spin
directions, whereas the latter has 6. In between the 1/2 plateau and the 1/3
SSP for $J^{\prime}/J\leq 0.63$, an extremely narrow period-10 SSP appears,
which is energetically very close to their neighboring phases. At higher
magnetic fields we find a transition into a period-14 SSP before reaching
saturation. Close to saturation, the energies of the competing states get very
close. At larger values of $J^{\prime}/J\sim 0.68$ a period-8 SSP is
stabilized before saturation. We did not find evidence of another plateau
above the 1/2 plateau, although a 7/8 plateau gets energetically close to the
supersolid phases, especially at larger values of $J^{\prime}/J$.

In Fig. 4, the magnetization curve for $J^{\prime}/J=0.63$ together with the
energy differences of the competing states is shown, where the vertical dashed
lines indicate the phase boundaries. Converted to real units [3], we find that
the 1/2 plateau terminates at $106$ T, with a jump in magnetization to the
extremely narrow period-10 SSP, followed by the 1/3 SSP. The location of this
transition is compatible with $H_{\mathrm{c6}}$ from the magnetostriction and
magnetization measurements. The transition between the two 1/3 SSPs is found
at $122$ T (there is no anomaly observed in experiments at this value which
could be because the two 1/3 SSPs are very similar states). The transition
into the period-14 SSP occurs around $128$ T which coincides with
$H_{\mathrm{c8}}$. Finally, saturation is reached at $137$ T, in close
agreement with the experimental values of $H_{\mathrm{c9}}$.

##  VI discussion

###  VI.1 Sound velocity of the plateau phases

First, we quantitatively discuss the drastic sound-velocity change of the
$c_{66}$ mode. The sound velocity $v$ is a thermodynamical quantity, related
to the elastic constant $c$, as $c=\rho v^{2}$. Here, $\rho$ is the mass
density. The elastic constant is the second derivative of the free energy with
respect to the strain. In SCBO, strain modulates the Cu-O-Cu angle and
exchange coupling of the dimers, leading to a modulation of the magnetic free
energy. Depending on the elastic modes ($c_{66}$ and $c_{11}$), the exchange
modulation acts on dimers asymmetrically and symmetrically, respectively
[Figs. 1(c, d)]. For the case of the $c_{66}$ mode with $\varepsilon_{xy}$,
the Cu-O-Cu angle of the horizontal (vertical) dimer decreases (increases),
leading to the reduced (enhanced) AFM interaction. This bond alternation
naturally stabilizes the 1/2 plateau phase with the checkerboard pattern; i.e.
vertical dimers with (almost) aligned spins and horizontal dimers with
predominantly singlets (or vice versa). This is in strong contrast to the 1/3
plateau which exhibits polarized spins on both the vertical and horizontal
dimers (separated by two rows with dimers with opposite spins on each dimer
[3]). This odd periodicity leads to a cancellation of contributions when
computing the derivative, leading to a value close to zero. The 1/4 plateau
exhibits an even periodicity (every fourth diagonal row exhibits almost
polarized spins), leading to a finite first derivative, albeit with a smaller
magnitude than the 1/2 plateau, as observed also in the experiment. At
saturation, the state is a product state (all spins aligned) and the
derivatives with opposite signs cancel each other exactly.

Table 2: First and second derivatives of the energy with respect to the strain $\varepsilon_{xy}$ ($c_{66}$ mode) obtained with iPEPS ($D=6$). The iPEPS estimates of $\Delta c/c_{0}$ in the fourth column have been obtained based on a least-squares fit of the form $\lambda_{1}E^{\prime}+\lambda_{2}E^{\prime\prime}$ to the experimental values of $\Delta c/c_{0}$. $M/M_{\mathrm{s}}$ | $E^{\prime}$ | $E^{\prime\prime}$ | ${\Delta c}/{c_{0}}$ | ${\Delta c}/{c_{0}}$ | ${\Delta v}/{v_{0}}$ | ${\Delta v}/{v_{0}}$
---|---|---|---|---|---|---
|  |  | iPEPS | Exp. | iPEPS | Exp.
1/8 | -0.014 | -0.44 | -0.24 | -0.21(8) | -0.13 | -0.11(4)
1/4 | -0.051 | -0.31 | -0.29 | -0.34(5) | -0.16 | -0.19(3)
1/3 | 0 | -0.18 | -0.08 | -0.06(4) | -0.04 | -0.03(2)
1/2 | -0.23 | -0.034 | -0.74 | -0.73(11) | -0.49 | -0.48(7)

Table 2 summarizes the first and second derivatives of the free energy
($E^{\prime}$ and $E^{\prime\prime}$) for the $c_{66}$ mode. Here, we have
evaluated the energy shift with respect to $\varepsilon_{xy}$, which reduces
(enhances) the AFM exchange coupling of the horizontal (vertical) dimer. As
discussed, $E^{\prime}$ is much larger in the 1/2 plateau than in other
plateau phases. This is a common feature with the experimental $\Delta
v/v_{0}$, where the sound velocity significantly decreases in the 1/2 plateau
but does not in the 1/3 plateau. In fact, the experimental $\Delta v/v_{0}$
(and $\Delta c/c_{0}$) of the major plateau phases at 1/4, 1/3 and 1/2 is
nearly proportional to $E^{\prime}$ (see the second, fifth, and seventh
columns of Table 2), suggesting that $E^{\prime}$ gives a major contribution
to the elastic constant as compared to $E^{\prime\prime}$ although the elastic
constant is the second derivative of the free energy. In fact, $E^{\prime}$
contributes to the elastic constant via the anharmonic potential of the
lattice (see SM for details [42]). This anharmonicity is specifically
important for the 1/2 plateau phase because the triplet crystallization into
the checkerboard pattern leads to the cooperative exchange striction from
tetragonal to orthorhombic. Because of this exchange striction, Cu2+ ions move
from the equilibrium positions at zero field to strained positions where the
magnetic superstructure is better stabilized. At the strained position, the
effect of the anharmonic potential becomes relevant, which is a key to connect
$E^{\prime}$ and the elastic constant. Including the first- and second-
derivatives contributions as
$\lambda_{1}E^{\prime}+\lambda_{2}E^{\prime\prime}$, the experimental $\Delta
v/v_{0}$ is reasonably explained (see the sixth and seventh columns of Table
2). The coefficients $\lambda_{1}$ and $\lambda_{2}$ are optimized by a least-
squares fit.

###  VI.2 High-field supersolid phases

The iPEPS calculation predicts four SSPs between the 1/2 plateau and the
saturation. Even if $J^{\prime}/J$ changes under high fields because of the
exchange striction, the phase boundary does not change greatly. In our
experiments, the period-10 SSP and the transition from the 1/3 plateau type b
to type a would not be detected because of the limited resolution and
precision. Thus, we focus on the major phase boundaries, the end of the 1/2
plateau, the end of the 1/3 SSP, and the saturation field. The latter two are
detected in both ultrasound and magnetostriction experiments as
$H_{\mathrm{c8}}$ and $H_{\mathrm{c9}}$. The quantitative agreement between
the experiment and theory is excellent considering the experimental challenges
at ultrahigh magnetic fields. The slight difference in the saturation field
might be due to the magnetic-field dependence of $J$.

The assignments of the lower-field boundaries $H_{\mathrm{c6}}$ and
$H_{\mathrm{c7}}$ require careful discussions. The transition at
$H_{\mathrm{c6}}$ is the end of the 1/2 plateau, where magnetization starts to
increase discontinuously. This anomaly is observed in the magnetostriction and
magnetization experiments but is not detected in the ultrasound experiment.
Another transition at $H_{\mathrm{c7}}=116$ T is observed both in ultrasound
and magnetostriction experiments but is not predicted by the iPEPS
calculation. Experimentally, the $\Delta v/v_{0}$ of the $c_{66}$ mode shows a
drastic increase at $H_{\mathrm{c7}}$ [Fig. 2(a)].

From the experimental point of view, the transition from the 1/2 plateau to
the 1/3 SSP, where the magnetic unit cell drastically reconstructs, should be
detected by the ultrasound technique. The ultrasound technique is generally
sensitive to this kind of symmetry change at phase transitions [31]. Indeed,
the calculated $E^{\prime}$ is significantly larger in the 1/2 plateau because
of the checkerboard pattern of triplets with the even period, while it is
almost cancelled with the odd period of the 1/3 SSP. The ultrasound results
without anomaly at $H_{\mathrm{c6}}$ suggest that another SSP with an even
period might appear as an intermediate phase between the 1/2 plateau and the
1/3 SSP. One possible scenario to explain the experimental results is that a
1/2 SSP appears at $H_{\mathrm{c6}}$ just above the 1/2 plateau. In this case,
the translational symmetry is the same for the 1/2 phases, and the calculated
$\lambda_{1}E^{\prime}+\lambda_{2}E^{\prime\prime}$ takes similar values.
Thus, the ultrasound technique might be less sensitive to detect the
transition from the 1/2 plateau to a 1/2 SSP. In contrast, the transition from
a 1/2 SSP to the 1/3 SSP would be clearly detected because the magnetic unit
cell drastically changes. The anomaly at $H_{\mathrm{c7}}$ might corresponds
to this phase transition.

From the theoretical point of view, one can identify a metastable 1/2 SSP, but
it is higher in energy than the 1/3 SSP for $J^{\prime}/J\sim 0.63$.
Therefore, the simple Shastry-Sutherland model does not explain the proposed
scenario. To stabilize the 1/2 SSP, some additional terms need to be included
in Eq. (1). One magnetic interaction neglected in the iPEPS is the interlayer
coupling [53, 54]. Since the interlayer coupling is AFM, it is energetically
not favorable to have two dimers of aligned spins on top of each other. Thanks
to the checkerboard structure of the 1/2 plateau and 1/2 SSP, it is possible
to always have a polarized dimer on top of a singlet dimer in 3D, leading to
lower interlayer energy compared to that of the 1/3 SSP. The same holds true
for the 1/2 SSP with the checkerboard-like structure, indicating that the 1/2
SSP could be stabilized with this term. Since the method used in the present
work, iPEPS, is a purely two-dimensional approach, the inclusion of the
interlayer coupling is left for future work. Another important magnetic
interaction is the Dzyaloshinskii-Moriya (DM) term, which stabilizes canted
spin orientations like in SSPs. However, this term would stabilize both the
1/2 SSP and the 1/3 SSP similarly. Thus, the DM term would not change the
relative stability of these phases. Finally, the last term which might be
relevant for the 1/2 plateau and SSP is the spin-lattice coupling term [55].
In these phases, the checkerboard pattern of triplets leads to a tetragonal-
orthorhombic instability of the lattice. The orthorhombic distortion
alternatively modulates the intra-dimer coupling $J$, stabilizing the magnetic
energy at the expense of the elastic energy. This energy gain might be
sufficiently large to stabilize the 1/2 SSP compared to the 1/3 SSP. Further
work is needed to fully understand how to improve the agreement between
experiment and theory in this field range.

##  VII conclusion

In this study, we have discussed the ultrahigh-field properties of SCBO,
reaching magnetic saturation for the first time. Above 100 T, our ultrasound
and magnetistriction experiments have detected four anomalies at
$H_{\mathrm{c6}}=108$, $H_{\mathrm{c7}}=116$, $H_{\mathrm{c8}}=127$, and
$H_{\mathrm{c9}}=139$ T. Quite remarkably, the ultrasound results reveal a
very significant softening of the $c_{66}$ acoustic mode in the 1/2 plateau
phase. Most of these results can be understood as consequences of the peculiar
geometry of the layers of SCBO, as shown by a careful comparison with the
results of iPEPS simulations of the Shastry-Sutherland model. In particular,
the anomalous softening of the $c_{66}$ acoustic mode in the 1/2 plateau phase
as compared to the other plateaus can be attributed to the tetragonal-
orthorhombic instability triggered by the checkerboard pattern of triplets in
this plateau. Besides, the iPEPS results clearly identify $H_{\mathrm{c9}}$ as
the saturation field. Furthermore, iPEPS simulations of the Shastry-Sutherland
finds a sequence of competing SSPs between the 1/2 plateau and saturation,
suggesting that the anomalies in this field range are due to transitions
between such SSPs, with in particular a transition between a 1/3 SSP and a
period-14 SSP at a field in good agreement with $H_{\mathrm{c8}}$. At lower
field, the absence of ultrasound anomaly at the end of the 1/2 plateau at
$H_{\mathrm{c6}}=108$ T suggest that the 1/3 SSP only sets in at
$H_{\mathrm{c7}}=116$ T, with possibly a 1/2 SSP in between. This assignment
needs to be finalized however since the 1/2 SSP is only metastable for the
Shastry-Sutherland model. Further work is needed to check if additional
interactions such as interlayer or spin-lattice coupling can stabilize this
phase. This goes beyond the scope of the present paper however.

## acknowledgments

We acknowledge the support of the HLD at HZDR, member of the European Magnetic
Field Laboratory (EMFL) and the Deutsche Forschungsgemeinschaft (DFG) under
SFB 1143. This work was partly supported by JSPS KAKENHI, Grant-in-Aid for
Scientific Research (Nos. JP20K14403, JP22H00104) and JSPS Bilateral Joint
Research Projects (JPJSBP120193507), by the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement Nos. 677061 and 101001604), and by the Swiss National Science
Foundation (grant No. 182179). The crystal structure figures have beeen
created by using the visualization software VESTA [56].

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Supplemental Material of ”The Shastry-Sutherland Compound SrCu2(BO3)2 Studied
up to the Saturation Magnetic Field”

This Supplemental Material will be separated in the published manuscript.

##  VIII Experimental details

In this study, we used two types of pulsed magnets, the single-turn coil (STC)
in the ISSP, University of Tokyo and the dual-coil pulsed magnet at the HLD in
Dresden. Figure S1 shows the typical waveforms of these magnets. The STC is a
semi-destructive technique where the field duration is limited to several
$\mu$s. Because of this limitation, special experimental efforts are needed to
measure the properties of materials in STC experiments. In this section, we
discuss the experimental technique and present some further results.

Fig. S 1:  Magnetic-field waveforms by the single-turn coil (blue, inset) and
the dual-pulse magnet (black).

For studying the ultrasonic properties up to 150 T, we used the continuous-
wave (CW) excitation technique [37]. The experimental setting [Fig. S2(a)] is
very similar with the conventional pulse-echo technique, where the sample is
sandwiched by two LiNbO3 transducers. In the CW technique, the transducer is
excited continuously without pulse modulation. Therefore, the ultrasound
signal is also continuously detected even in the $\mu$s timescale. However,
the detected signal includes cross-talk and many reflections, which affects
the phase and amplitude of the acoustic waves. As discussed in Ref. [37], this
effect becomes negligibly small if the sample length and shape are optimized.
The relative change of the sound velocity $\Delta v/v_{0}$ is proportional to
the relative change of the phase $\Delta\Phi/\Phi_{0}$ as

$\Delta v/v_{0}=-\Delta\Phi/\Phi_{0}=-\Delta\Phi/(2\pi f\tau_{0}),$ (2)

where $f$ and $\tau_{0}$ are the ultrasound frequency and the time delay of
the ultrasound for a single transmitted signal. In this study, we first tested
the installation up to 40 T in the STC and confirmed that the obtained data
are consistent with the pulse-echo results. Here, we used the sample length of
1.1 mm, with which the time delay of the ultrasound is reasonably small.

Fig. S 2:  Schematic settings of the STC (a) ultrasound and (b)
magnetostriction experiments. (c) FBG setting view from two sides.

Figure S3 shows the raw ultrasound data up to 125 T obtained at 3.2 K. The
magnetic field profile is shifted by taking into account the propagation time
of the ultrasound [from blue to black in Fig. S3(a)]. The detected signal is
shown in Fig. S3(b). By applying the numerical lock-in, the phase shift
($\Delta\Phi$) and amplitude of the signal are extracted as shown in Fig.
S3(c) and Fig. S3(d), respectively. Anomalies related to the phase transitions
are observed both in the field-up and field-down sweeps. Note, that the
detected phase has always $\pm 2\pi$ ambiguity. It means that if the phase
change is very large and fast, the correct phase shift might be overlooked.
This might be specifically the case at the 2/5-1/2 plateau region ($\sim 80$
T, highlighted by light green), where the sound velocity changes by $-40\%$
per 5 T. Such a drastic sound velocity change is very challenging to follow in
the STC experiment, because the sound velocity changes significantly within
the sound propagation time even for 1.1 mm sample. Therefore, we rely on the
results obtained in the non-destructive pulsed magnet by the conventional
pulse-echo technique, and shifted accordingly the STC results in this field
range [Fig. S3(c)].

Fig. S 3:  Raw data of the ultrasound experiment for the $c_{66}$ mode at 3.2
K up to 125 T. (a) Magnetic-field profile. The black curve is shifted from the
original profile (blue) by taking into account the ultrasound-propagation
time. (b) Detected ultrasound signal. (c) Phase change and (d) amplitude of
the detected signal. The exact phase change is challenging to extract in the
regions highlighted by light green because of the rapid change of the sound
velocity in a very narrow magnetic field range.

For studying the magnetostriction, we used the fiber-Bragg grating (FBG) [39].
The schematic setting is shown in Fig. S2(b). The FBG is fixed to the sample,
and the sample deformation is monitored by the wavelength of the reflected
light. By using the optical filter, the wavelength shift is detected as the
amplitude change of the reflected light (for detail, see Ref. [39]). With this
technique, the sample length change can be optically detected, which is a
great advantage in the STC experiments. The biggest challenge performing the
FBG measurement with STC is the sample vibration [40, 41]. When a magneto-
structural phase transition occurs, a slight lattice deformation triggers a
vibration of a crystal. When the sample shape is regular with parallel
surfaces, this vibration results in the resonant oscillation of the crystal.
Such oscillations lead to a periodic noise with the resonant frequency
determined by the sample dimensions and the sound velocity. To attenuate such
oscillations, we used (i) the sample with irregular shape and (ii) grease
covering the whole sample [Fig. S2(c)]. To weaken the resonant oscillation,
the quality factor of the resonance has to be reduced. In this study, we used
the crystal with one polished surface, but without any parallel surfaces. For
such crystal, the phases of the reflected acoustic waves are random, and the
resonant oscillations are strongly damped. Additionally, with vacuum grease,
the sample is mechanically coupled with the sample stage. In this case, the
quality factor of the resonance is strongly reduced by the mass-loading
effect, and the resonant frequency becomes much lower than the STC
characteristic frequency. Figure S4 shows the results of the magnetostriction
experiment with and without vacuum grease. The resonant oscillations are
strongly damped by the grease, although the size of the anomaly is also
reduced. Therefore, in this study, we do not discuss the magnitude of the
magnetostriction and only focus on the critical fields. The anomalies at
$H_{\mathrm{c6}}$ and $H_{\mathrm{c7}}$ are well reproduced for the field-up
and field-down sweeps.

Fig. S 4:  Raw magnetostriction data ($\Delta L_{c}$) at 4.2 K up to 122 T.
Reflected light intensity (left) and magnetic field (right, blue curve) are
plotted as a function of time. The results with and without grease are
compared. The results (black curve) are averaged with the time window of 40
ns.

Figure S5 summarizes the maximum-field dependence of the results. Figure S5(a)
shows the phase of the ultrasound signal of the $c_{66}$ mode. As discussed,
the rapid change of the sound velocity is challenging to capture by this
technique, which is the case in the region highlighted by light green and
blue. Nevertheless, the overall features of the sound velocity are well
reproduced; the softening in the 1/4 plateau, the hardening in the 1/3
plateau, the softening towards the 1/2 plateau, the constant velocity up to
$H_{\mathrm{c7}}=116$ T, and the slope change around $H_{\mathrm{c8}}=127$ T.
Figure S5(b) shows the magnetostriction measured for a single FBG setting,
which survives even after the STC experiments. The slight difference might be
due to the covering conditions with vacuum grease, which melts each time after
the experiment. Again, the overall features are well reproduced; the rapid
decrease at 80 T, the increase at $H_{\mathrm{c6}}=108$ T, the maximum at
$H_{\mathrm{c7}}=116$ T, and the kink at $H_{\mathrm{c8}}=127$ T. The anomaly
at $H_{\mathrm{c9}}=139$ T, which corresponds to the saturation of
magnetization, is also reproduced for the ultrasound and magnetostriction
measurements.

Fig. S 5:  (a) Phase shift of the acoustic wave for the $c_{66}$ mode at 3.2 K
and (b) magnetostriction along the $c$ axis at 4.2 K measured to various
maximum fields in the STC. The curves are deliberately shifted along the
vertical axis for clarity. It is very challenging to reconstruct correctly the
phase change in the green and blue area because very large and fast changes in
the sound velocity in this magnetic field region.

##  IX Magnetic contributions to the $\mathbf{c_{66}}$ mode

In this section, we discuss the relation between the first and the second
derivatives of the magnetic energy with respect to the displacement
$\varepsilon$ associated with the mode probed by $c_{66}$. Without the
magnetic contribution, and up to a constant, the structural energy is minimal
for $\varepsilon=0$, and its expansion in $\varepsilon$ reads:

$E_{struc}=b\varepsilon^{2}+c\varepsilon^{3}$

where $b$ describes the harmonic potential and $c$ the cubic anharmonic
contribution.

As shown by iPEPS calculations, the magnetic contribution to the energy has
both linear and quadratic contributions:

$E_{mag}=a\varepsilon+b^{\prime}\varepsilon^{2}$

with $a=dE_{mag}/d\varepsilon$ and
$b^{\prime}=(1/2)d^{2}E_{mag}/d\varepsilon^{2}$.

Because of the linear term of $E_{mag}$, the minimum of the total energy
$E_{tot}=E_{struc}+E_{mag}$ occurs at a non-zero value $\varepsilon_{0}$ which
is solution of the equation

$\frac{dE_{tot}}{d\varepsilon}=a+2(b+b^{\prime})\,\varepsilon+3\,c\,\varepsilon^{2}=0.$

If $\varepsilon_{0}$ is very small, which will be the case if $a,c\ll b$, then
it is given by

$\varepsilon_{0}\simeq-\frac{a}{2(b+b^{\prime})}.$

Now let us expand the total energy around the new minimum by setting
$\varepsilon=\varepsilon_{0}+\delta$. Up to the second order in $\delta$, we
get:

$\displaystyle E_{tot}(\delta)$ $\displaystyle=$ $\displaystyle
a\varepsilon_{0}+(b+b^{\prime})\varepsilon_{0}^{2}+c\varepsilon_{0}^{3}$
$\displaystyle+$
$\displaystyle(a+2(b+b^{\prime})\varepsilon_{0}+3c\varepsilon_{0}^{2})\,\delta$
$\displaystyle+$ $\displaystyle(b+b^{\prime}+3c\varepsilon_{0})\,\delta^{2}.$

The coefficient of the linear term vanishes because of the condition satisfied
by $\varepsilon_{0}$, and, in the limit of small $\varepsilon_{0}$, the
coefficient of the quadratic term is given by

$b+b^{\prime}-\frac{3c}{2(b+b^{\prime})}a.$

As anticipated, it contains a contribution proportional to $b^{\prime}$, hence
to $d^{2}E_{mag}/d\varepsilon^{2}$, and a contribution proportional to $a$,
hence to $dE_{mag}/d\varepsilon$. The contribution of the first derivative
comes from the anharmonicity - its coefficient is proportional to $c$.