of 9
ARTICLE
Discovery of quantum phases in the Shastry-
Sutherland compound SrCu
2
(BO
3
)
2
under extreme
conditions of
fi
eld and pressure
Zhenzhong Shi
1,8
, Sachith Dissanayake
1
, Philippe Corboz
2
, William Steinhardt
1
, David Graf
3
,
D. M. Silevitch
4
, Hanna A. Dabkowska
5
, T. F. Rosenbaum
4
, Frédéric Mila
6
& Sara Haravifard
1,7
The 2-dimensional layered oxide material SrCu
2
(BO
3
)
2
, long studied as a realization of the
Shastry-Sutherland spin topology, exhibits a range of intriguing physics as a function of both
hydrostatic pressure and magnetic
fi
eld, with a still debated intermediate plaquette phase
appearing at approximately 20 kbar and a possible decon
fi
ned critical point at higher pres-
sure. Here, we employ a tunnel diode oscillator (TDO) technique to probe the behavior in the
combined extreme conditions of high pressure, high magnetic
fi
eld, and low temperature. We
reveal an extensive phase space consisting of multiple magnetic analogs of the elusive
supersolid phase and a magnetization plateau. In particular, a 10 × 2 supersolid and a 1/5
plateau, identi
fi
ed by in
fi
nite Projected Entangled Pair States (iPEPS) calculations, are found
to rely on the presence of both magnetic and non-magnetic particles in the sea of dimer
singlets. These states are best understood as descendants of the full-plaquette phase, the
leading candidate for the intermediate phase of SrCu
2
(BO
3
)
2
.
https://doi.org/10.1038/s41467-022-30036-w
OPEN
1
Department of Physics, Duke University, Durham, NC 27708, USA.
2
Institute for Theoretical Physics and Delta Institute for Theoretical Physics, University
of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands.
3
National High Magnetic Field Laboratory, Florida State University, Tallahassee, FL
32310, USA.
4
Division of Physics, Math and Astronomy, California Institute of Technology, Pasadena, CA 91125, USA.
5
Brockhouse Institute for Material
Research, McMaster University, Hamilton, ON L8S 4M1, Canada.
6
Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015
Lausanne, Switzerland.
7
Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC 27708, USA.
8
Present address:
Institute for Advanced Study, School of Physical Science and Technology, Soochow University, Suzhou 215006, China.
email:
sara.haravifard@duke.edu
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1
1234567890():,;
W
hile the behavior of individual spins in isolation is well
understood, complex behavior and exotic quantum
states often emerge from networks of such spins,
especially when competing interactions forestall the formation of
simple ordered states, a phenomenon known as magnetic
frustration
1
. A key tool for understanding these states is the
ability to tune parameters such as the relative strength of the
different interactions or the external magnetic
fi
eld. In that
respect, the Shastry
Sutherland (SS) model
2
, a 2-dimensional
(2D) network of orthogonal interacting spin dimers, together
with its experimental realization SrCu
2
(BO
3
)
2
, are prominent
candidates.
The Cu spins 1/2 in SrCu
2
(BO
3
)
2
form weakly coupled 2D
networks of orthogonal dimers topologically equivalent to the SS
model. For the pure Heisenberg model, the exact ground state is a
product of singlet dimers
2
,
3
as long as the inter-dimer coupling
J
0
is not too large as compared to the intra-dimer coupling
J
.
Heuristically, we can think of the magnetization as due to mag-
netic particles
T
1
that form when a dimer singlet
S
is replaced by a
triplet polarized along the
fi
eld. These particles have a very small
kinetic energy due to the highly frustrated dimer arrangement,
leading to very
fl
at bands and to Mott insulating phases (i.e.,
magnetization plateaus) at fractional
fi
llings. The
fi
rst magneti-
zation plateaus
4
in SrCu
2
(BO
3
)
2
, at 1/8 and 1/4, were initially
observed in 1999, and the con
fi
rmation that the translational
symmetry is broken in the 1/8 plateau soon followed
5
. At ambient
pressure, additional plateaus have been identi
fi
ed
6
13
to build the
improbable sequence 1/8, 2/15, 1/6, 1/4, 1/3, 2/5, and 1/2. It
required 15 years and the invention of tensor network algorithms
to develop a theory capable of accounting for this remarkable
series
14
. Some of these plateaus (1/4, 1/3, 1/2) can be simply
interpreted as Wigner crystals of
T
1
particles, while the lower
magnetization plateaus are best seen as Wigner crystals of spin-2
bound states that form because of a second-order kinetic term in
J
0
=
J
that leads to a binding between pairs of
T
1
particles on
neighboring parallel dimers
14
. Additionally, the supersolid phases
correspond to adding
T
1
particles to a plateau phase, the hopping
of these extra-particles being due to correlated hopping that takes
advantage of the underlying network of
T
1
particles
15
,
16
. Between
the plateaus, translation invariance is never restored, and it
remains a challenge to establish which of these intermediate
phases are spin-supersolids and which are incommensurate
phases with proliferating domain walls
11
. The excitation spec-
trum is also remarkable, with very
fl
at bands, and it has been
shown that, due to Dzyaloshinskii
Moriya interactions
17
21
,a
small
fi
eld induces topological magnon bands with non-zero
Chern numbers
22
and experimental consequences still to be
explored.
In addition to the rich set of physics revealed by high magnetic
fi
elds, SrCu
2
(BO
3
)
2
is remarkably sensitive to pressure for an
oxide, and two phase transitions have been observed in it
23
29
.
This sensitivity is enabled by the geometry of the Cu-Cu bonds: at
ambient pressure, the intra-dimer Cu
Cu bond is close to 90°, and
applying pressure brings this angle even closer to 90°, reducing
J
and increasing the ratio
J
0
=
J
(ref.
29
,
30
). At ambient pressure, the
ratio
J
0
=
J
0
:
63 for SrCu
2
(BO
3
)
2
12
puts it close to the boundary
of the dimer phase, hence a relatively modest pressure of order 20
kbar is suf
fi
cient to induce a
fi
rst-order transition into another
gapped phase
23
27
, followed at higher pressure by a transition into
another phase still to be fully characterized
27
,
31
. Similarly, the
phase diagram of the SS model has three phases
32
36
: an exact
dimer phase up to
J
0
=
J
0
:
675, an antiferromagnetic (AFM)
phase above
J
0
=
J
¼
0
:
765
ð
15
Þ
(ref.
36
) (in the limit
J
0
=
J
!1
the
SS model is equivalent to the square lattice antiferromagnet), and
an intermediate plaquette phase in between, where strong
J
0
bonds
form around half the empty square plaquettes of the SS lattice.
The transition between the dimer phase and the intermediate
phase is clearly
fi
rst order, and it has been shown very recently
that as a function of temperature it terminates at a critical point
analogous to that of water
31
. By contrast, the nature of the
transition between the intermediate phase and the AFM phase is
still debated, and the interest in this transition has risen recently
after the proposal that it could be a decon
fi
ned quantum critical
point
37
39
. NMR experiments have revealed early on that there
are two types of Cu sites
23
, inconsistent with the intermediate
phase of the SS model, and various experimental results seem to
be rather consistent with a full-plaquette phase where strong
J
0
bonds form around half the square plaquettes that contain a
dimer. The emerging picture for SrCu
2
(BO
3
)
2
is then that of a
system dominated by a tendency to an orthorhombic distortion at
intermediate pressure
23
,
25
,
28
. In the absence of direct probes of
the symmetry of this intermediate phase, its precise nature
remains an open issue, and a very important one because the
nature of the intermediate phase will of course in
fl
uence the
nature of the transition into the AFM phase.
In this paper, our aim is to gain insight into the properties of
SrCu
2
(BO
3
)
2
by studying its high-
fi
eld properties in the relevant
pressure range using tunnel diode oscillator (TDO) technique,
and by an investigation of the high-
fi
eld properties of the SS
model in the corresponding
J
0
=
J
range using tensor network
methods. These regions of the phase diagrams of SrCu
2
(BO
3
)
2
and of the SS model have not been previously explored and here
are demonstrated to host exotic magnetic phases, including a
10 × 2 supersolid and a 1/5 plateau. Furthermore, we show that
the discovered complex magnetic phase diagram of SrCu
2
(BO
3
)
2
,
at high-pressure and high-magnetic
fi
eld, offers insight into the
second phase transition, suggesting it to display a possible
decon
fi
ned quantum critical point with fractional excitations. Our
results set the ground for further studies of SrCu
2
(BO
3
)
2
under
pressure. We, additionally, establish TDO as a viable and effective
technique to be utilized for similar measurements for other
quantum magnets under combined extreme conditions of high
H
,
high
P
, and low
T
.
Results
Experimental results
. The TDO technique has been previously
proven to be a valuable tool
13
,
40
,
41
for probing the behavior of
pure SrCu
2
(BO
3
)
2
in the spin-dimer phase and the ambient-
pressure behavior of doped SrCu
2
x
Mg
x
(BO
3
)
2
. It allows mea-
surement of the change in magnetization at sub-Kelvin tem-
peratures, high pressures, and high magnetic
fi
elds (see
Methods
for details), making it especially well-suited for phy-
sical systems that require all three simultaneously. In Fig.
1
a, we
show TDO magnetic susceptibility measurements for pure
SrCu
2
(BO
3
)
2
in
μ
0
H
up to 45 T (
H
ab
) and
T
=
0.3 K, where
df
/
dH
dM
2
/
d
2
H
(ref.
40
), for a series of pressures spanning the
spin-dimer and the putative 4-spin plaquette phases. The high
sensitivity of the technique allows the identi
fi
cation of weak
magnetization changes that would otherwise be extremely dif
fi
-
cult to detect. For example, at
P
=
0, we identify seven anomalies
in
M
(
H
)at
fi
elds
H
1
~ 27.5 T,
H
2
~ 30.2 T,
H
3
~ 31.8 T,
H
4
~ 33.4 T,
H
5
~ 34.4 T,
H
6
~ 37.6 T, and
H
7
~ 43.6 T, all of
which correspond to jumps or slope changes in magnetization
(see
Methods
for details).
We
fi
rst focus on the two anomalies at the highest
fi
elds at
P
=
0, namely
H
6
and
H
7
(Fig.
1
a), which can be identi
fi
ed
immediately as the onset of the 1/4 and 1/3 magnetization
plateaus, respectively
40
. The natural next step is to follow the two
anomalies to higher pressures. At
P
=
1.1 GPa, two similar
anomalies are also observed, though shifted to lower
fi
elds
( ~ 35 T and ~ 40 T, respectively). In the intermediate plaquette
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phase, at 1.9 GPa and 2.3 GPa, we still can identify two anomalies
in this
fi
eld range, although they are now much weaker and
shifted slightly to even lower
fi
elds. It is tempting to assign these
two anomalies at these high pressures (1.1 GPa, 1.9 GPa, and
2.3 GPa) as extensions of the
H
6
(1/4 plateau) and
H
7
(1/3
plateau) seen at
P
=
0. However, we caution that the fate of the
magnetization plateaus at higher pressure needs to be understood
fi
rst. Indeed, as we show below, the real physical picture is much
more complicated than a simple extension from the ambient
pressure results. In fact, some of these anomalies actually signal
previously unobserved phases that only appear at high pressure
and high
fi
eld, such as a 1/5 plateau phase and a
10 × 2 supersolid phase.
The interpretation of the data obtained at lower
fi
elds also
requires some care. First, at
P
=
0, we identify three anomalies at
H
3
,
H
4
, and
H
5
, between the expected 1/4 and 1/8 plateaus
(Fig.
1
a and Supplementary Fig. 2a). Here, NMR measurements
have found evidence for 2/15 and 1/6 plateaus
11
for
H
c
. After
accounting for the g-factor difference between
H
c
and
H
ab
,we
fi
nd that two of the anomalies (
H
3
and
H
5
) are located at
fi
elds
consistent with the onsets of the 2/15 and 1/6 plateaus
11
(Fig.
1
b).
For
H
4
, it is likely associated with the 1/7 anomaly, the possible
trace of an intermediate plateau
14
stabilized by factors such as
inter-layer coupling.
At even lower
fi
elds, we identi
fi
ed
H
1
and
H
2
at
P
=
0(Fig.
1
a
and Fig.
1
b). Here, at ambient pressure and the
fi
eld expected for
the beginning of the 1/8 plateau
6
, we consistently observe the weak
feature marked as
H
2
(
H
ab
) (Fig.
1
a and Supplementary Fig. 1).
The identi
fi
cation of
H
2
anomaly as the 1/8 plateau is also
supported by comparison with the results for
H
c
after proper
g-factor correction [refs.
6
,
9
,
10
, also see Supplementary Fig. 1].
Below the onset of 1/8 plateau at
H
2
,we
fi
nd a pronounced sub-1/8
anomaly at
H
1
, which seems to only appear for the
H
ab
orientation, and which corresponds to the large jump in
magnetization that was reported in early studies
6
but not studied
in detail. It has been suggested that any anomaly in this
fi
eld range
might be a hallmark of a higher order (e.g., 1/9 or 1/10)
plateau
6
,
9
,
10
. The anisotropic behavior might suggest some role of
the Dzyaloshinskii
Moriya (DM) interactions
14
,
42
, which could
stabilize or destabilize certain plateaus for different
fi
eld orienta-
tions. However, as shown in ref.
14
, the higher-order plateaus are
much less stable compared to the 1/8 plateau, and the experimental
results for higher-order plateaus so far are also not conclusive. It is
thus highly unlikely that the rather weak DM interaction in
SrCu
2
(BO
3
)
2
(
J
DM
/
J
=
0.03 ~ 0.04, refs.
14
,
42
) could enhance the
higher-order plateaus to such a degree. An alternative explanation
is that the pronounced anomaly at
H
1
corresponds to the transition
between the single singlet condensation and the condensation of
the bound states of triplets, which is theoretically expected to occur
before the bound states crystallize at the 1/8 plateau. This behavior
is more pronounced for
H
ab
than for
H
c
, likely because the
small separation between the two
fi
eld scales is more apparent with
the smaller g-factor along the
a
and
b
axes than along
c
.
At even lower
fi
elds, we observe the emergence of an anomaly
near 7 T (Fig.
1
a, right panel), which we refer to as
H
0
, only in the
pressure range where plaquette state appears. At 2.2 GPa, this
anomaly further splits. It is interesting to note that the magnetic
energy scale of this anomaly is comparable to the low-energy
excitation mode observed by inelastic neutron scattering in the
plaquette state
25
, albeit without observing the subsequent splitting
at the higher pressure. Structure factor measurements of this low-
energy mode suggested that the ground state is a full plaquette
featuring diagonal bonds
25
. As shown below, our numerical
results show that the splitting corresponds to a hidden AFM state,
which is possibly connected adiabatically to the AFM ground
state observed by heat capacity measurements above 2.5 GPa.
This is consistent with the expectation that the AFM phase is
favored at higher
T
,
H
, and
P
where entropy is increased
31
.
In Fig.
1
b, we show the characteristic magnetic
fi
elds of all the
anomalies as a function of pressure. Here,
H
2
,
H
6
, and
H
7
correspond to the 1/8, 1/4, and 1/3 plateaus respectively at
ambient pressure;
H
1
is the sub-1/8 anomaly that signals the onset
of the condensation of triplet-bound states;
H
3
,
H
4
, and
H
5
are
attributed to the intermediate magnetization plateaus as discussed
above;
H
0
represents the low-
fi
eld anomalies that appear above
1.7 GPa. As we will show below, these characteristic
fi
elds
constitute a rich phase diagram containing a variety of spin
superstructures.
Finally, we have also investigated the effect of Mg dopants in
the system (Supplementary Note 1 and Supplementary Fig. 10)
and found that the results can be consistently explained by the
impurity-induced spin structures that we established for ambient
pressure
40
.
Fig. 1
P
-dependence of the magnetization plateaus and emergence of
low-
fi
eld anomalies in SrCu
2
(BO
3
)
2
.a
(left panel):
df
/
dH
vs.
H
for
P
up to
2.4 GPa at 0.3 K. The data consist of results from multiple runs on different
samples using a 18 T superconducting magnet, a 35 T resistive magnet, and
a 45 T hybrid magnet (
H
ab
for all measurements). Red arrows denote
H
1
,
H
2
,
H
3
,
H
4
,
H
5
,
H
6
, and
H
7
at
P
=
0.
H
1
,
H
2
,
H
6
, and
H
7
correspond to the
sub-1/8 anomaly and the 1/8, 1/4, 1/3 plateaus, respectively.
H
3
,
H
4
, and
H
5
are likely intermediate 2/15, 1/7, and 1/6 plateaus. The 1/8 plateau is
identi
fi
ed as the shoulder that appears at a slightly higher
H
than the large
sub-1/8 anomaly (see Supplementary Fig. 1; for identi
fi
cation of the other
features such as the high-
P
1/5 plateau, see
Methods
and Supplementary
Fig. 2).
a
(right panel) Magni
fi
ed view of the low-
fi
eld behavior, showing the
emergence of the low-
fi
eld anomaly, which splits above
P
~ 2.2 GPa as
indicated by the two red arrows. The 2.3 GPa and 2.4 GPa traces are from
measurements on two different samples using a resistive magnet and
superconducting magnet respectively. Traces in
a
and
b
are shifted
vertically for clarity.
b
H
P
phase diagram showing all anomalies (red solid
symbols).
H
1
to
H
7
indicate the sub-1/8 anomaly and the magnetization
plateaus at ambient pressure;
H
0
indicates the low-
fi
eld anomaly. The blue
and light blue symbols represent the 1/8, 1/4, 1/3 plateaus extracted from
previous ambient-
P
measurements in ref.
6
and ref.
12
, respectively. The red
open symbols indicate the splitting of the low-
fi
eld anomaly at higher
P
.
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3
We note that our low-
fi
eld results are consistent with previous
results in the entire pressure range. At ambient pressure,
SrCu
2
(BO
3
)
2
has a 3 meV gap separating the spin singlet ground
state and the triplet excited state
43
. Applying pressure within the
dimer phase suppresses this gap
24
, but it does not completely
close before entering the plaquette state
25
. Inelastic neutron
scattering measurements within the plaquette phase found the
emergence of a low-energy mode along with a slight hardening of
the triplon gap
25
. The pressure dependence of the former was
tracked via heat capacity and was found to decrease with
increasing pressure
27
. The spin gap can also be suppressed by the
Zeeman mechanism, where the lowest excited state is brought
down in energy by the application of the magnetic
fi
eld.
Interestingly, when plotted in the same
fi
gure, as shown in Fig.
2
,
the pressure dependence of some of the characteristic
fi
elds
[
μ
0
H
1
(
P
) and
μ
0
H
0
(
P
)] and that of the spin gap [
Δ
(
P
)] measured
by neutron scattering and heat capacity measurements show
similar behaviors. On the other hand, some notable differences of
the two types of pressure dependence are also observed at
P
2.3 GPa. Here,
μ
0
H
0
splits, signaling the emergence of the
AFM state. Our observations thus provide a broader perspective
for the evolution of the spin gap with pressure in this material.
Finally, we note that while the introduction of Mg doping does
not qualitatively change the behavior of
μ
0
H
1
(
P
)and
μ
0
H
0
(
P
), the
anomalies presaging the AFM state in
μ
0
H
0
(
P
) are shifted to lower
energy compared to that in pure SrCu
2
(BO
3
)
2
, though the doping
dependence of this softening remains to be explored (see
Supplementary Fig. 5 for
x
=
0, 0.05 data collected at 2.4 GPa,
Supplementary Fig. 6 for
x
=
0.02, 0.03 data collected at 2.1 GPa,
and Supplementary Note 1). In the spin-dimer phase, adding
impurities has been found not to move the onset
fi
elds of the 1/n
plateaus, although increased impurity concentration does soften the
spin superstructures and enhances the probability of forming
impurity pairs and impurity-induced triplet states
40
. This suggests
that the superstructures of the triplet-bound states have excitation
energies independent of impurity doping. As shown in Fig.
2
,this
similarity between the pure and doped cases extends into the
plaquette phase, indicating that the triplon excitation is likewise
insensitive to impurity doping. However, the impurity-driven shift
in the low-
fi
eld mode
μ
0
H
0
(
P
) noted above suggests that the
dopants act to destabilize the plaquette phase and instead favor the
AFM phase, which is perhaps not surprising as we will show below
that the AFM phase is also favored at higher
T
,
H
,and
P
.
iPEPS calculation results
. We have performed iPEPS simulations
(Methods) of the SS model in a magnetic
fi
eld, given by the
Hamiltonian:
H
¼
J
h
i
;
j
i
S
i

S
j
þ
J
0
hh
i
;
j
ii
S
i

S
j

h
i
S
z
i
;
ð
1
Þ
where
J
and
J
0
are the intra-dimer and inter-dimer couplings,
respectively, and the strength of the external magnetic
fi
eld is
controlled by
h
. At ambient pressure a ratio
J
0
=
J
¼
0
:
63 was
determined from a
fi
t to high magnetic
fi
eld data
12
.
Applying pressure leads to an effective increase of the ratio
J
0
=
J
; however, the precise pressure dependence of
J
0
and
J
is not
known. Here we model the pressure dependence by linear
functions for
J
(
p
) and
J
0
ð
p
Þ
, with a change of 5% in
J
0
between its
value at ambient pressure and its value at the critical pressure
p
c
=
1.8 GPa [corresponding to
J
0
=
J
¼
0
:
675 (ref.
36
)]. A change
of 5% is also predicted from ESR data
26
, and is close to the
estimate (4%) obtained in a recent ab-initio study
29
(in contrast, a
substantially larger value (17%) was found from
fi
ts to magnetic
susceptibility data
25
). At ambient pressure, we use
J
=
81.5 K
which lies in between previously predicted values
12
,
44
and yields
good agreement with the onset of the 1/4 and 1/3 plateaus
observed in experiments. The resulting slopes of the linear
functions
J
0
ð
p
Þ
and
J
(
p
) are
1.43 K/GPa and
5.13 K/GPa,
respectively. In Supplementary Figs. 11 and 12 and Supplemen-
tary Table 1, we present alternative phase diagrams using
different parameter sets, in order to illustrate the dependence of
the phase boundaries on the various parameters. The boundaries
change very little as long as the variation of
J
0
is only a few
percent. The change by 17% deduced from
fi
tting the
susceptibility
25
would by contrast lead to a phase diagram whose
boundaries depart signi
fi
cantly from our experimental data and
thus can be discarded.
We focus in the following on some of the most prominent
features in the phase diagram as a function of
J
0
=
J
(or pressure)
and magnetic
fi
eld. In particular, we concentrate on the phase
boundaries of the magnetization plateaus and the supersolid
phases at high magnetic
fi
elds, and the competing low-energy
states at high pressure. The results, summarized in Fig.
3
, are
obtained for
D
=
8 which already provides an accurate estimate of
the phase boundaries (e.g., the relative error on the phase
boundaries of the 1/4 and 1/3 plateaus is <2% compared to the
results extrapolated to the in
fi
nite
D
limit
12
).
At high
fi
elds (up to 45 T) the dominant phases are the 1/4
plateau, the 1/3 plateau, and a 1/3 supersolid phase
12
.A
supersolid phase simultaneously breaks the translational symme-
try and the U(1) symmetry associated with the total
S
z
conservation. The 1/3 supersolid exhibits the same translational
symmetry breaking pattern as the 1/3 plateau state, but with
additional spin components in the transverse direction, re
fl
ecting
the broken U(1) symmetry. The 1/4 plateau has a
fi
nite extent up
to
J
0
=
J
¼
0
:
675
ð
5
Þ
after which the intermediate
fi
eld region is
dominated by the 1/3 supersolid phase. The 1/3 plateau remains
stable over the entire range of
J
0
=
J
considered here. Below the 1/4
plateau at ambient pressure there is a sequence of small
magnetization plateaus (crystals of triplet-bound states)
14
,
denoted as
intermediate plateaus
in Fig.
3
, which are stable
up to
J
0
=
J
¼
0
:
675
ð
5
Þ
. We also add a characteristic line indicating
a lower bound for the onset of the 1/8 plateau. This line is
obtained by intersecting the energy of the 1/8 plateau with that of
the 1/9 plateau, a plateau which is however probably unstable
towards a condensate of spin-2 bound states (see above).
At intermediate
fi
elds and ~
J
0
=
J
¼
0
:
68, we
fi
nd a 1/5 plateau
that has not been observed previously (we call it high-
P
1/5
plateau); and it is different from the 1/5 plateau made of localized
triplet-bound states appearing at smaller
J
0
=
J
(or lower pressure,
Fig. 2
H
P
phase diagram of the sub-1/8 anomaly and the LE mode.
The
H
P
phase diagram from our TDO measurements is compared with the
Δ
P
phase diagram established by neutron scattering
25
and heat
capacity
27
. (Left axis)
μ
0
H
vs.
P
; Red and green stars are characteristic
fi
elds from the TDO results for the
x
=
0 and
x
=
0.05 samples. (Right axis)
Δ
vs.
P
; Orange squares and blue circles are the spin gap values reported by
studies of neutron scattering
25
and heat capacity
27
, respectively. Similar
pressure dependence are observed for
μ
0
Hvs
.
P
and
Δ
vs
.
P
.
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ref.
14
). The spin structure of this plateau exhibits a stripe pattern
parallel to one set of dimers, as shown in Fig.
3
, where along each
stripe a strong dimer triplet alternates with a pair of weaker dimer
triplets. We will discuss the physical origin of this rather unusual
structure in the next section.
At ambient pressure as well as at 1.1 GPa, we
fi
nd a very good
agreement between the phase boundaries of the 1/4 plateau and
the critical
fi
elds of the anomalies found in experiments. At
1.9 GPa and 2.3 GPa, the anomalies at 40 T are close to the iPEPS
phase boundary of the 1/3 plateau, and the anomalies at 34 T are
in good agreement with the upper edge of the high-
P
1/5 plateau.
At 1.7 GPa and 1.9 GPa, we also identify two anomalies consistent
with the lower edge of the high-
P
1/5 plateau. All these features
are thus well captured by the standard SS model and by the
simple model for the pressure dependence of
J
and
J
0
used here.
Finally, we turn our focus to the low-
fi
eld region at high
pressure. Above the empty plaquette (P) phase in zero and small
fi
elds, we
fi
nd a narrow partially polarized AFM phase, and a
10 × 2 supersolid state (obtained in a 10 × 2 unit cell; hence the
name), followed by the 1/3 supersolid and 1/3 plateau phases. The
corresponding spin patterns are displayed in Fig.
3
, and examples
of magnetization curves are shown in the Supplementary Figs. 7
and 8. Note that the pattern of the 10 × 2 supersolid phase is
different from that of the stripe phase reported in ref.
45
.
Interestingly, the anomalies at 1.9 GPa and 2.3 GPa just under 20
T in experiments lie close to the phase boundary between the
10 × 2 and 1/3 supersolid phase. As we shall see below, the
10 × 2 supersolid phase can be seen as a descendant of the
unusual 1/5 plateau.
The narrow AFM region, which vanishes around
J
0
=
J

0
:
686
and which becomes broader with increasing
J
0
=
J
, is qualitatively
compatible with the splitting of the two anomalies observed at
low
fi
elds in experiments. However, quantitatively we
fi
nd that
these phase boundaries occur at higher
fi
elds than in experiments.
We believe the main reason for this discrepancy is the lack of
inter-layer coupling in our model, which is of order of 0.09
J
(ref.
18
) and which is expected to enhance the stability of the AFM
phase compared to the plaquette state
18
leading to a shift of the
phase boundary to smaller critical
fi
elds. Additionally,
Dzyaloshinskii
Moriya (DM) interactions
17
21
of order of a few
percent of
J
may affect the location of the phase boundaries. We
note
fi
nally that the empty plaquette state is different from the
full-plaquette state implied by experiments, which can be
obtained using a deformed Shastry
Sutherland model
28
,
46
that
includes two types of intra- and inter-dimer interactions. These
modi
fi
cations of the model may also affect the magnetization
process, particularly at low
fi
elds, where the narrow AF phase is
energetically closely competing with the plaquette and the 10 × 2
phase. We stress, however, that the high-
P
1/5 plateau and the
10 × 2 supersolid phase remain relevant ground states also in the
deformed model (Supplementary Fig. 13). In fact, they tend to be
further stabilized by the deformation, a logical tendency since
they correspond to descendants of the full-plaquette state, as we
discuss in the following sections.
Nature of the high-
P
1/5 plateau and 10 × 2 supersolid
.As
stated in the Introduction, the high-
fi
eld plateaus can be thought
of as Wigner crystals of
T
1
magnetic particles, while the lower
plateaus are better interpreted as Wigner crystals of spin-2 bound
states of such
T
1
particles
14
. The resulting structures are very
simple to visualize. The high-
fi
eld plateaus build diagonal stripes
(in a geometry where dimers are horizontal or vertical), while the
Wigner crystals of spin-2 bound states consists in putting the
bound states as far as possible from each other.
The two phases discovered in the present paper, the high-
P
1/5
plateau and the 10 × 2 supersolid, are completely different and
cannot be understood in these terms. Since the 10 × 2 supersolid
can be understood as its descendant, let us
fi
rst concentrate on
the high-
P
1/5 plateau. Its main properties are, (i) The stripes are
vertical, and not diagonal; (ii) The state is not a simple Wigner
crystal of
T
1
particles, but half the magnetic particles are
delocalized over two dimers; (iii) This gain of kinetic energy
cannot be achieved with the mechanism that explains the other
Fig. 3
H
P
phase diagram (theory vs. experiment).
Black symbols and lines correspond to anomalies found in experiments, the colored symbols and
lines are based on iPEPS results (
D
=
8). The corresponding coupling ratios
J
0
=
J
are shown on the top axis. The colored phase regions are determined by
the iPEPS data points. Dotted lines are a guide to the eye. The experimental data and the iPEPS data agree well, except for two places: near the 1/8 platea
u
and the Plaquette-AFM-10 × 2-supersolid transitions. The iPEPS calculation does not capture the sub-1/8 anomaly which is very pronounced for
H
ab
but
invisible for
H
c
(ref.
6
) because of the isotropic nature of the standard SS model. On the right-hand side, typical spin patterns of the phases at high
pressure are drawn. The size of the spins scale with the magnitude of the local magnetic moment, where black (red) arrows point along (opposite to) the
external magnetic
fi
eld. The thickness of the gray bonds scales with the local bond energy (the thicker the lower the energy).
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