arXiv:2107.02929v1 [cond-mat.str-el] 6 Jul 2021
Phase diagram of the Shastry-Sutherland Compound SrCu
2
(BO
3
)
2
under extreme
combined conditions of field and pressure
Zhenzhong Shi,
1
Sachith Dissanayake,
1
Philippe Corboz,
2
William Steinhardt,
1
David Graf,
3
D.M.
Silevitch,
4
Hanna A. Dabkowska,
5
T.F. Rosenbaum,
4
Fr ́ed ́eric Mila,
6
and Sara Haravifard
1,7,
∗
1
Department of Physics, Duke University, Durham, North Caro
lina 27708, USA
2
Institute for Theoretical Physics and Delta Institute for T
heoretical Physics,
University of Amsterdam, Science Park 904, 1098 XH Amsterda
m, The Netherlands
3
National High Magnetic Field Laboratory, Florida State Uni
versity,Tallahassee, Florida 32310, USA
4
Division of Physics, Math, and Astronomy, California Insti
tute of Technology,Pasadena, California 91125, USA
5
Brockhouse Institute for Material Research, McMaster Univ
ersity, Hamilton, Ontario L8S 4M1, Canada
6
Institute of Physics, Ecole Polytechnique F ́ed ́erale de La
usanne (EPFL), CH-1015 Lausanne, Switzerland
7
Department of Mechanical Engineering and Materials Scienc
e,
Duke University, Durham, North Carolina 27708, USA
(Dated: July 8, 2021)
Motivated by the intriguing properties of the Shastry-Suth
erland compound SrCu
2
(BO
3
)
2
under
pressure, with a still debated intermediate plaquette phas
e appearing at around 20 kbar and a
possible deconfined critical point at higher pressure upon e
ntering the antiferromagnetic phase, we
have investigated its high-field properties in this pressur
e range using tunnel diode oscillator (TDO)
measurements. The two main new phases revealed by these meas
urements are fully consistent with
those identified by infinite Projected Entangled Pair states
(iPEPS) calculations of the Shastry-
Sutherland model, a 1/5 plateau and a 10
×
2 supersolid. Remarkably, these phases are descendants
of the full-plaquette phase, the prominent candidate for th
e intermediate phase of SrCu
2
(BO
3
)
2
.
The emerging picture for SrCu
2
(BO
3
)
2
is shown to be that of a system dominated by a tendency
to an orthorhombic distortion at intermediate pressure, an
important constraint on any realistic
description of the transition into the antiferromagnetic p
hase.
I. INTRODUCTION
While the behavior of individual spins in isolation is
well understood, complex behavior and new quantum
states often emerge from networks of such spins, espe-
cially when competing interactions forestall the forma-
tion 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 field. In that respect, the Shastry-
Sutherland (SS) model, a 2-dimensional (2D) network
of interacting spin dimers, together with its experimen-
tal realization SrCu
2
(BO
3
)
2
, is a prominent candidate.
Starting with the discovery of the first magnetization
plateaus [2] at 1/8 and 1/4 in 1999, and the confir-
mation that the translational symmetry is broken in the
1/8 plateau [3], the interest in this layered frustrated
quantum magnet and the related SS model has never de-
creased, and many new remarkable properties have been
discovered since then. At ambient pressure, additional
plateaus have been identified [4–11] to build the im-
probable sequence 1/8, 2/15, 1/6, 1/4, 1/3, 2/5 and 1/2,
and it took 15 years and the invention of tensor network
algorithms to come up with a theory able to account for
this remarkable series [12]. Between the plateaus, trans-
lation invariance is never restored, and it remains a chal-
lenge to establish which of these intermediate phases are
∗
sara.haravifard@duke.edu
spin-supersolids and which are incommensurate phases
with proliferating domain walls [9]. The excitation spec-
trum is also remarkable, with very flat bands, and it has
been shown that, due to Dzyaloshinskii-Moriya interac-
tions [13–17], a small field induces topological magnon
bands with non-zero Chern numbers [18] and experimen-
tal consequences still to be explored. Finally, this com-
pound is remarkably sensitive to pressure for an oxide,
and two phase transitions have been observed [19–25].
The first one is clearly first order, and it has been shown
very recently that, as a function of temperature, it termi-
nates at a critical point analogous to that of water [26].
Most of these properties are direct consequences
of the peculiar arrangement of the Cu spins 1/2 in
SrCu
2
(BO
3
)
2
, which form weakly coupled 2D networks of
orthogonal dimers topologically equivalent to SS model
introduced in 1981 [27]. For the pure Heisenberg model,
the exact ground state is a product of singlet dimers [27,
28] as long as the inter-dimer coupling
J
′
is not too large
as compared to the intra-dimer coupling
J
. Heuristically,
we can think of the magnetization as due to magnetic
particles
T
1
that form when a dimer singlet
S
is replaced
by a triplet polarized along the field. These particles
have a very small kinetic energy due to the highly frus-
trated dimer arrangement, leading to very flat bands and
to Mott insulating phases (i.e. magnetization plateaus)
at fractional fillings. 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
′
/J
that leads
2
to a binding between pairs of
T
1
particles on neighboring
parallel dimers [12]. 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 [29, 30].
The properties under pressure are the consequence of
another remarkable property of SrCu
2
(BO
3
)
2
. The intra-
dimer Cu-Cu bond is close to ninety degrees, and apply-
ing pressure brings this angle even closer to ninety de-
grees, reducing
J
and increasing the ratio
J
′
/J
[25, 31].
Now, the phase diagram of the SS model has three
phases [32–36]: an exact dimer phase up to
J
′
/J
≃
0
.
675,
an antiferromagnetic phase above
J
′
/J
= 0
.
765(15) [36]
(in the limit
J
′
/J
−→ ∞
the SS model is equivalent to
the square lattice antiferromagnet), and an intermediate
plaquette phase in between where strong
J
′
bonds form
around half the empty square plaquettes of the SS lat-
tice. The transition between the dimer phase and the
intermediate phase is clearly first order, but the nature
of the transition between the intermediate phase and the
AF phase is still debated, and the interest in this tran-
sition has risen recently after the proposal that it could
be a deconfined quantum critical point [37–39].
In SrCu
2
(BO
3
)
2
, the ratio
J
′
/J
≃
0
.
63 [10] puts it
close to the boundary of the dimer phase, and indeed it
turns out that a pressure of about 20 kbar is sufficient
to induce a first-order transition into another gapped
phase [19–23], followed at higher pressure by a transition
into another phase still to be fully characterized [23, 26].
Given the topology of the system, it is natural to ex-
pect the gapped phase to be the intermediate phase of
the SS model, but this does not seem to be the case.
NMR experiments have revealed early on that there are
two types of Cu sites [19], inconsistent with the interme-
diate phase of the SS model, and various experimental
results seem to be rather consistent with a full plaquette
phase where strong
J
′
bonds form around half the square
plaquettes that contain a dimer, likely accompanied by
an orthorhombic lattice distortion [19, 21, 24]. In the
absence of direct probes of the symmetry of this inter-
mediate phase, its precise nature remains an open issue,
and a very important one because the nature of the in-
termediate phase will of course influence the nature of
the transition into the AF phase.
In this paper, our aim is to gain insight into the prop-
erties of SrCu
2
(BO
3
)
2
under pressure by an investiga-
tion of its high-field properties in the relevant pressure
range using Tunnel Diode Oscillator (TDO) measure-
ments, and by an investigation of the high-field properties
of the SS model in the corresponding
J
′
/J
range using
tensor network methods. These regions of the phase di-
agrams of SrCu
2
(BO
3
)
2
and of the SS model have not
been previously explored, and, as we demonstrate, the
new phases that appear in high field shed new light on
the problem and set the ground for further studies of
SrCu
2
(BO
3
)
2
under pressure.
II. EXPERIMENTAL RESULTS
The TDO technique has been previously proven
to be a valuable tool [11, 40, 41] for probing both
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 measurement of the
change in magnetization at sub-Kelvin temperatures,
high pressures, and high magnetic fields (see Ap-
pendix A 2for details). Therefore, the most valuable as-
pect of the technique is revealed when it is used in a
sample environment that combines low temperature, high
pressures and high fields.
In Fig. 1A, we show TDO magnetic susceptibility
measurements for pure SrCu
2
(BO
3
)
2
in
μ
0
H
up to 45 T
(
H
k
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 sensitiv-
ity of the technique allows the identification of weak mag-
netization changes that would otherwise be extremely dif-
ficult to detect. For example, at
P
= 0, we identify seven
anomalies in
M
(
H
) at fields
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 Appendix A 3 for
details).
H
6
and
H
7
can be identified immediately as
the onset of the 1/4 and 1/3 magnetization plateaus [40].
When
P
is increased to 1.1 GPa, similar anomalies are
observed, but shifted to lower fields. In the intermediate
plaquette phase, at 1.9 GPa and 2.3 GPa, we still can
identify two anomalies in this field range, although they
are now much weaker and shifted slightly to lower fields.
It is tempting to assign these two anomalies as exten-
sions of the
H
6
(1/4 plateau) and
H
7
(1/3 plateau) seen
at 0 and 1.1 GPa. However, we caution that the fate of
the magnetization plateaus at higher pressure needs to
be understood first. Indeed, as we will show below, the
real physical picture is much more complicated than a
simple extension from the ambient pressure results. In
fact, some of the anomalies actually signal entirely new
phases that only appear at high pressure and high field,
such as a 1/5 plateau phase and a 10
×
2 supersolid phase.
The interpretation of the data obtained at lower fields
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 (see Fig. 1 and Supplementary Fig.
S2A [42]). Here, NMR measurements have found evi-
dence for 2/15 and 1/6 plateaus [9] for
H
k
c
. After
accounting for the g-factor difference between
H
k
c
and
H
k
ab
, we find that two of the anomalies (
H
3
and
H
5
)
are located at fields consistent with the onsets of the 2/15
and 1/6 plateaus [9]. For
H
4
, it is likely associated with
the 1/7 anomaly, the possible trace of an intermediate
plateau [12] stabilized by factors such as inter-layer cou-
pling. Further details of our results at ambient pressure
are discussed in Appendix B.
Unlike at high field, the low-field behavior in the pla-
quette state can be immediately seen to be qualitatively
3
0.0
0.5
1.0
1.5
2.0
2.5
0
10
20
30
40
4
6
8
10
0
10
20
30
40
H
5
H
6
H
3
H
0
H
1
H
7
H
4
H
(T)
P
(GPa)
H
2
0.3 K
SrCu
2
(BO
3
)
2
B
P
(GPa) =
0
H
(T)
2.4
2.0
1.5
2.3
2.1
1.7
1.3
1.0
0.5
x
= 0
H
4
H
3
H
5
1.9
H
1
2.4
2.3
1.1
A
H
7
H
6
H
2
d
f
/d
H
P
(GPa) =
0
0
H
(T)
2.3
1.7
1.3
1.0
0.5
Figure 1
.
P
-dependence of the magnetization plateaus and
emergence of low-field 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 sam-
ples using a 18 T superconducting magnet, a 35 T resistive
magnet, and a 45 T hybrid magnet (
H
k
ab
for all measure-
ments). Red arrows denote
H
1
,
H
2
,
H
3
,
H
4
,
H
5
,
H
6
, and
H
7
at
P
= 0. Here
H
1
,
H
2
,
H
6
, and
H
7
corresponding to
the sub-1/8 anomaly and the 1/8, 1/4, 1/3 plateaus, respec-
tively.
H
3
,
H
4
, and
H
5
are likely intermediate 2/15, 1/7, and
1/6 plateaus. The 1/8 plateau is identified as the shoulder
that appears at a slightly higher field than the large sub-1/8
anomaly (see Supplementary Fig. S1 [42]; for identification
of
the other features such as the 1/5 plateau, see Appendix A 3
and Supplementary Fig. S2 [42]).
A
(right panel): Magnified
view of the low-field behavior, showing the emergence of the
low-field anomaly, which splits above
P
∼
2.2 GPa as in-
dicated 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. Traces in
a
and
b
are shifted vertically for clarity.
B
:
H
−
P
phase
diagram showing all the anomalies.
H
1
to
H
7
indicate the
sub-1/8 anomaly and the magnetization plateaus at ambient
pressure, and
H
0
indicates the low-field anomaly. The red
open symbols indicate the splitting of the low-field anomaly
at higher
P
.
different from the spin-dimer phase, with the emergence
of a new feature near 7 T (Fig. 1A, right panel), which
we refer to as
H
0
. This feature further splits at 2.2 GPa.
It is interesting to note that the magnetic energy scale
of this anomaly is comparable to the low-energy excita-
tion mode observed by inelastic neutron scattering in the
plaquette state [21], albeit without observing the subse-
quent 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 [21]. 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 [26].
In Fig. 1B, we show the characteristic magnetic fields
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-field anomalies that appear
above 1.7 GPa. As we will show below, these character-
istic fields constitute a rich phase diagram containing a
variety of spin superstructures.
Finally we have also investigated the effect of Mg
dopants in the system (see Appendix C) and found that
the results can be consistently explained along the lines
of the impurity-induced spin structures we established
for ambient pressure [40].
We note that our low-field 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 [20], but it does not completely close before enter-
ing the plaquette state [21]. Inelastic neutron scatter-
ing measurements within the plaquette phase found the
emergence of a low-energy mode along with a slight hard-
ening of the triplon gap [21]. The pressure dependence of
the former was tracked via heat capacity and was found
to decrease with increasing pressure [23]. 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 field. Therefore, we focus on
the pressure dependence of the characteristic fields iden-
tified by our TDO measurements.
We first focus on
H
1
, i.e. the condensation field of
the spin-2 bound states (see Appendix B), and its pres-
sure dependence. Interestingly, as shown in Fig. 2, there
is a similarity between the pressure-dependence of some
of the characteristic fields (
μ
0
H
1
and
μ
0
H
0
) and that of
the spin gap measured by neutron scattering and heat
capacity measurements. However, some notable differ-
ences of the two types of pressure dependence are also