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Peer Review File
REVIEWER
COMMENTS
Reviewer #1 (Remarks to the Author):
SrCu2(BO3)2 has been known as a realization of the frustrated 2D Shastry
-
Sutherland spin model
and extensively studied over past twenty years mainly along two different directions. One is the
remarkable s
equence of multiple magnetization plateaus, which are induced by applying magnetic
fields to the dimer singlet ground state and understood as crystallization of triplets or bound triplet
pairs. The other direction is the quantum phase transitions induced b
y applying high pressure, that
changes the ratio of intra
-
to interdimer exchange couplings. In this paper, the authors report
surprisingly rich phases of SrCu2(BO3)2 including a new plateau and supersolid phases obtained in
the multiple extreme conditions
combining both high magnetic fields and high pressures. For this
purpose, they used sophisticated experimental and theoretical techniques, namely the tunnel diode
oscillator for precise measurements of magnetization anomalies and iPEPS calculations for
id
entification of the phases. In my opinion, this work has opened a new route to explore even more
exotic quantum states in this remarkable material than known to date. Therefore, I strongly
recommend publication of this work in Nature Communications once th
e authors have considered
the following questions and comments.
1. The authors assert that the sub
-
1/8 anomaly at H1~27.5 T is caused by condensation of S=2 bound
triplet pairs (pages 7 and 8). Are there any experimental evidence for this interpretation?
Since the
sub
-
1/8 anomaly has been seen only for the field in the ab plane and absent for H // c, it seems to
me more reasonable to ascribe its origin to anisotropic interactions such as DM interaction. The
condensation of bound triplets in the SS model sh
ould be an isotropic phenomenon when the fields
are multiplied by the g
-
values.
2. I cannot find any “anomalies at 1.9 GPa and 2.3 GPa around 21.5 T in experiments” (second
paragraph of page 11) either in Fig 1 or Fig. 3.
3. The observation of descendant
relation among the full praquette, 10x2 supersolid, and 1/5
plateau phases is very interesting. Then I would expect the transition directly from 10x2 supersolid
to 1/5 plateau. However, the iPEPS result shows a wide intervening region of 1/3 supersolid ph
ase in
between. Is there any simple explanation for this ?
4. Related to the previous question, since the full praquette phase does not appear in the SS model,
the lowest field phase obtained by iPEPS above 1.9 GPa in Fig. 3 must be the empty praquette ph
ase.
Then It seems to me logically inappropriate to apply the above argument of descendant relation to
interpret the results in Fig. 3.
Reviewer #2 (Remarks to the Author):
The authors present a study of the material SrCu2(BO3)2 using TDO (tunnel Diode
Oscillator)
technique at high pressures, high magnetic fields and low temperatures. The experimental study is
accompanied by a numerical analysis based on iPEPS calculations.
By combining their experimental and numerical results the authors propose a fie
ld
-
pressure phase
diagram for SrCu2(BO3)2 exhibiting many different magnetic phases, some of which were previously
observed in that compound while others are novel.
Experiments at high pressure and furthermore, experiments combining high pressures, high
magnetic fields and low temperatures are notoriously difficult. With TDO measurements, the
authors present a phase diagram that covers a large range, with pressures from 0 to about 2.5 GPa
and magnetic fields from 0 to 45 T. It also contains relatively pre
cise boundaries between the
different proposed phases.
As such, the study is of particular interest for quantum magnetism as it provides novel information
about an important compound and its theoretical counterpart, the Shastry
-
Sutherland model.
In the TDO measurements some of the features are easy to spot while some others require more
analysis and are harder to see. The later depend on consistency checks that the authors have made
and that cannot be easily checked by peer review. However, in my
opinion, the authors have
provided reasonably sufficient justification about the location of the different features mentioned in
the manuscript.
The details of iPEPS calculation are not part of my expertise and I cannot comment on those.
The references g
iven are appropriate.
I would therefore in principle recommend it for publication in Nature communications provided the
issues raised below are answered.
Main concerns and questions
1. The numerical calculations make use of the pressure dependence of J
and J’. However as
mentioned in the text these are not well known. The ab
-
initio paper in Ref 26, itself contains several
values and references. The J(p) and J’(p) chosen by the authors seem not to be any of the previously
published ones. J(p) is not expl
icitly given and J’(p) is taken to vary by 3% from 0 to 1.8 GPa. First, can
the slope of J(p) be given. Second, this raises a question about how robust the numerical analysis is
with respect to the “choice” of the pressure dependence of the exchange parame
ters.
The authors should address this issue and explain or show how changes in the initial values of J and
J’ or in their slope would affect their results. In particular does the “match” between calculations and
experiment in figure 3 strongly depend on th
ose choices?
2. Comparison neutron vs TDO: In Figure 2, the TDO measurements are systematically higher (by
0.25 to 0.5 meV) than the spin gaps reported by neutrons and specific heat measurements in the
dimer phase. In the plaquette phase however there is
no shift for the low energy modes while the
shift remains for the higher energy mode. Could the authors address and discuss this issue?
Is that a due to a difference in dispersion? Is that a systematic of the TDO measurement compared
to other techniques.
Why does the systematically higher TDO values suddenly match the neutron
and specific heat values for the lower modes of the plaquette?
Minor concern
3. a) In page 7: “H1 is the sub
-
1/8 anomaly that signals the onset of the condensation of triplet
bound
states”. Is that claim based on the current analysis and can some justification be given, or is it
based on a previous work in which case a reference should be given? There is not much further
discussion in the main manuscript, but in the supplementary ma
terial, it appears that it is a new
interpretation. Can some more quantitative justification be provided?
b) Would other such “sub
-
plateau” anomalies be expected for the other plateaus?
4. Doped material. a) Do the indicated percentage correspond to the input composition during
crystal growth or to the final result obtained. b) Were X
-
rays performed to verify the doping levels
and the fact that the Mg impurities did actually go into the C
u positions? Nothing is said about
sample characterization in the methods section.
5. In Supplementary figures 5 and 6. There is no big shift in the dip (and bump) position from x=0 to
x=0.03 doping. At x=0.05 there is a sudden jump. The first dip, for in
stance, stays close to 6.0 GPa all
the way to x=0.03 and then jumps to ~4.8 GPa for x=0.05. The authors should discuss this peculiar
behavior.
6. a) Two orientations were measured: with H parallel to c and H perpendicular to c (i.e in the a
-
b
plane). In t
he second case was the orientation systematically the same (for example H // to a) or did
it change from one run to the next?
b) Should one expect a difference in magnetization or in df/dH when, for instance, H//a compared to
when H is in the a
-
b plane at
45 deg between a and b?
Clarification needed
7. Not clear in text in page 6:
“similar anomalies”. Does this refer to H6 an H7 only or all of H1 to H7?
“this field range”, does it refer to the field for H6 and H7 only?
8. Page 8 “ We first focus on H
1, ...” the structure of the presentation is not very clear. One would
expect then a focus on the other features, but that is not how the text continues.
9. In page 11
-
12, when discussing the discrepancies between calculation and experiment, the
authors in
voke the 3D and DM couplings and then possible structural differences related to open vs
full plaquettes. All of these are acceptable as possible sources of discrepancies. However, it is not
clear at the end of the paragraph why those should be important “
particularly at low fields”.
10. It would be good for the reader, in my opinion, to 1) explain in the text why some of the phases
correspond to a “supersolid” state. For instance, what characteristics of a supersolid do these
particular spin arrangements
display? And 2) justify or explain the naming used such as: “10x2” and
“1/3” supersolids.
11. Supplementary Fig S4. The quality of the data at 0.3K is much better than the data at 0.44K. Is
that uniquely due to the physics or was the 0.44K data (and highe
r T) taken with lower statistics? It
would be good to give this information in the supplementary text.
12. Supplementary Fig 5
-
6. Is the vertical scale the same for all panels?
Reviewer #3 (Remarks to the Author):
The manuscript by Shi et al combines experimental and theoretical efforts to explore the magnetic
phase diagram of SrCu2(BO3)2 under extreme conditions: sub
-
Kelvin temperatures, high pressures
and high field. The anomalies observed in the tunnel diode osci
llator (TDO) experiment are then
used to establish the phase boundaries. By comparing to the iPEPS simulations, the authors claim
the discoveries of a few new phases, in particular the 10x2 supersolid and a 1/5 plateau.
While it is always interesting to f
ind novel magnetic states, for example the 10x2 supersolid and the
1/5 plateau, I am not fully convinced that this work meets the criteria of validity and novelty to
publish under Nature Communications. My major concerns are list below:
1.As the authors a
dmitted themselves (last paragraph in “iPEPS calculation results”), the model they
are using does not even correctly reproduce the “full”
-
plaquette ground state at high pressure and
zero field. Consequently, the theoretical findings at high pressure are si
tting on a wrong base. This
naturally explains their discrepancy in matching the theoretical and experimental data at pressures
larger than 1.9GPa. Consequently, I am not convinced that the 10x2 supersolid phase and the
narrow AF phase are relevant to the
real material.
2.In Fig. 3, There is only one experimental data point near the upper phase boundary of the 10x2
supersolid phase, and there is no matching experimental/numetical data point at the lower phase
boundary. This again questions the validity of
the 10x2 phase.
3.For the 1/5 plateau and 1/3 supersolid, there are only limited experimental data points for the
phase boundaries (three~four for 1/5 plateau and two~three for 1/3 supersolid). The interpretation
of these phase are mostly relying on the m
odel which could not even reproduce the zero phase full
plaquette phase, and there is no direct experimental evidence.
4.Even if the numerically discovered new phases are established, I am not convinced yet that these
results are of high significance to t
he field that qualifies publication in a high profile journal like the
Nature Communication. Besides showing novel magnetic structures, the authors should provide
evidence that such findings have significance either on the conceptual or at the application
level.
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Reply to the Reviewers
We are grateful to the Reviewers for their insights and for appreciation of the importance of our work. The
constructive suggestions by the Reviewers on content and clarity have prompted us to significantly improve
our
manuscript. We have now revised our manuscript to address the Reviewers’ comments. The changes
are summarized below (comments in blue, replies to comments in black):
Re
viewer #1
SrCu2(BO3)2 has been known as a realization of the frustrated 2D Shastry
-
Suth
erland spin model and
extensively studied over past twenty years mainly along two different directions. One is the remarkable
sequence of multiple magnetization plateaus, which are induced by applying magnetic fields to the dimer
singlet ground state and u
nderstood as crystallization of triplets or bound triplet pairs. The other direction
is the quantum phase transitions induced by applying high pressure, that changes the ratio of intra
-
to
interdimer exchange couplings. In this paper, the authors report su
rprisingly rich phases of SrCu2(BO3)2
including a new plateau and supersolid phases obtained in the multiple extreme conditions combining both
high magnetic fields and high pressures. For this purpose, they used sophisticated experimental and
theoretical t
echniques, namely the tunnel diode oscillator for precise measurements of magnetization
anomalies and iPEPS calculations for identification of the
phases. In my opinion, this work has opened a
new route to explore even more exotic quantum states in this re
markable material than known to date.
Therefore, I strongly recommend publication of this work in Nature Communications once the authors have
considered the following questions and comments.
We thank the Reviewer for the positive assessment and the appreciation of our results, and for the strong
recommendation of publication of our paper in
Nature Communications
. We also appreciate the insightful
questions and comments by the Reviewer. As desc
ribed below, we have carefully considered them and
revised our manuscripts in response to the
R
eviewer’s comments. We hope the Reviewer will find our
response satisfactory and our revised manuscript suitable for publication in
Nature Communications
.
1. The
authors assert that the sub
-
1/8 anomaly at H1~27.5 T is caused by condensation of S=2 bound
triplet pairs (pages 7 and 8). Are there any experimental evidence for this interpretation? Since the sub
-
1/8
anomaly has been seen only for the field in the ab pl
ane and absent for H // c, it seems to me more
reasonable to ascribe its origin to anisotropic interactions such as DM interaction. The condensation of
bound triplets in the SS model should be an isotropic phenomenon when the fields are multiplied by the g
-
values.
We thank the Reviewer for the comment. The sub
-
1/8 anomaly is indeed an interesting observation that we
believe has been mostly omitted in the literature. In fact, this anomaly was first noted in Ref. 5 [Onizuka, K.,
et al.
J. Phys. Soc. Jpn. 69,
1016 (2000)], where it was found that the g
-
factor normalized perpendicular
magnetization and parallel magnetization overlap in the entire field range except for the region near 30 T.
Ref. 5 discussed the possibility of an anisotropic 1/10 plateau but did
not reach a definite conclusion. To
the best of our knowledge, this question has not been addressed in any subsequently published studies.
Therefore, we have carefully considered possible scenarios responsible for the sub
-
1/8 anomaly (in
Supplementary Note
1 of our previous version of the manuscript). First, we agree wit
h
the Reviewer that
anisotropic interactions such as DM interaction have to be important because of the clear anisotropic
behavior of the perpendicular magnetization and parallel magnetizati
on. However, the DM interaction J
DM
,
being much smaller than the exchange interaction J (J
DM
/J = 0.03 ~ 0.04, see Ref. 13 and references
therein), could not directly account for such a large sub
-
1/8 anomaly by itself. It is more likely that the DM
interact
ion alters the stability of some steps in the magnetization process.
In this regard, we think plateaus such as the 1/10 plateau proposed by Ref. 5 are highly unlikely. Some of
us (P.C. and F.M.) have shown in Ref. 13 that a high density of plateaus lies e
nergetically very close below
the 1/8 plateau and are much less stable compared to the 1/8 plateau. Experimental evidence so far for the
higher order plateaus are also inconclusive. Although anisotropic interactions such as intra
-
and inter
-
dimer
DM intera
ctions could stabilize and destabilize certain plateaus, the sub
-
1/8 anomaly is too strong
compared to the 1/8 plateau to be induced by any realistic values of DM interaction strength.
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2
On the other hand, experimental signatures of condensation of S=2 bou
nd states have been elusive and
have not been intentionally searched for, even though it has to occur before the 1/8 plateau where the S=2
bound states crystallize (see Ref. 13). It is
,
therefore
,
likely that the condensation of S=2 bound states
occurs at
a field very near the 1/8 plateau and the two cannot be distinguished in the perpendicular
magnetization. In the parallel magnetization, however, the separation between the condensation of S=2
bound states and the 1/8 plateau could be enhanced by the large
r g
-
factor and by the anisotropic DM
interaction.
In response to the Reviewer’s question, and
also
a similar question by Reviewer 2, we have made the
following changes:
(1) We have added a new
r
eference (Ref. 42), where the strength of the DM
interaction was measured.
The new reference is cited in the revised text mentioned in (2).
(2) We have moved the discussion in Supplementary Note 1 to the main text, and expanded it. The revised
text in pages 7
-
8 now reads (due to the length of the text, w
e refer the Reviewer to the revised manuscript
for the full change):
At even lower fields, we identified H1 and H2 at P=0 (see Fig. 1a and Fig. 1c)...
...This behavior is more pronounced for H||ab than for H||c, likely because the small separation between
the
two field scales is more apparent with the smaller g
-
factor along the a and b axes than along c.
(3) We reworded the first two sentences of the 2nd paragraph in page 8 for a more smooth transition:
At even lower fields, we observe the emergence of a ne
w feature near 7 T ... At 2.2 GPa, this new feature
further splits
.”
2. I cannot find any “anomalies at 1.9 GPa and 2.3 GPa around 21.5 T in experiments” (second paragraph
of page 11) either in Fig 1 or Fig. 3.
It is indeed a typo, and we are grateful to the Reviewer for pointing it out. The anomalies at 1.9 GP and 2.3
GPa we refer to actually lie just under 20 T, not around 21.5 T. In Fig 3, these are the two data points (black
right triangles) lie near the boun
dary between the 1/3 supersolid and 10x2 supersolid. We have corrected
the error and double checked the entire manuscript for consistency. In page 12, the sentence is now revised
as follows:
Interestingly, the anomalies at 1.9 GPa and 2.3 GPa just under 2
0 T in experiments...
3. The observation of descendant relation among the full praquette, 10x2 supersolid, and 1/5 plateau
phases is very interesting. Then I would expect the transition directly from 10x2 supersolid to 1/5 plateau.
However, the iPEPS result
shows a wide intervening region of 1/3 supersolid phase in between. Is there
any simple explanation for this?
We thank the Reviewer for the comments.
In
deed
,
the 1/5 plateau and the 10x2 supersolid
phases are
closely related, since they are both based on the same
full plaquette phase
(
FPP
)
stripe pattern. However,
the 10x2 supersolid phase is not obtained by just a small deformation of the 1/5 plateau (as it is the case
for the 1/3 supersolid phase
starting from the 1/3 plateau). The 10x2 supersolid phase is obtained starting
from the 1/5 plateau by rotating the spins on each even (odd) stripe by 90 (
-
90) degrees, with an additional
small rotation such that there is a net magnetization in the z
-
direc
tion. Thus, since the deformation is not
small, we do not necessarily expect that the two phases are adjacent.
In response to the Reviewer’s comment, we have
added
a sentence at the end of the section “
Nature of
the 1/5 plateau and 10x2 supersolid
” to cla
rify this point.
Finally, the 10x2 supersolid
phase can be obtained from the 1/5 plateau by an alternating rotation of the
magnetization of successive stripes clockwise or counterclockwise by 90 degrees, and by adding some
magnetic particles between the stripes, see Supplementary Fig. 9. We note that
, because the 10x2
supersolid is not obtained by a just small rotation of the spins of the 1/5 plateau, we a priori do not expect
it to be adjacent to the 1/5 plateau phase
.”
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4. Related to the previous question, since the full praquette phase does not app
ear in the SS model, the
lowest field phase obtained by iPEPS above 1.9 GPa in Fig. 3 must be the empty praquette phase. Then It
seems to me logically inappropriate to apply the above argument of descendant relation to interpret the
results in Fig. 3.
We a
ppreciate the Reviewer
’s
following up on this point
.
The full plaquette phase (FPP) is energetically
very close to the empty plaquette phase (EPP), and it can be stabilized as the ground state in a slightly
deformed Shastry
-
Sutherland model,
please
see Ref
. [24]. Thus, there is a strong competition between the
two states at
zero field (
H
=
0
)
, and a close competition can also be expected at finite
H
. What is now very
interesting is that at sufficiently large
H
we do not find EPP
-
like states, but rather FPP
-
like states (i.e. the
new 1/5 plateau and the 10x2 supersolid phase), already in the standard Shastry
-
Sutherland model without
any deformation. Th
us, the finite field helps to stabilize FPP
-
like states over the EPP
-
like states. In the
deformed model (favoring FPP correlations), these FPP
-
like states naturally remain energetically favored.
Furthermore, they naturally appear as descendants of the FPP,
which rapidly becomes the intermediate
phase upon deforming the model.
We have verified this in additional calculations for the deformed
model
and
added the results in the Supplementary Fig. 12.
Reviewer #2
The authors present a study of the material SrCu
2(BO3)2 using TDO (tunnel Diode Oscillator) technique
at high pressures, high magnetic fields and low temperatures. The experimental study is accompanied by
a numerical analysis based on iPEPS calculations.
By combining their experimental and
numerical results
the authors propose a field
-
pressure phase diagram for SrCu2(BO3)2 exhibiting many different magnetic
phases, some of which were previously observed in that compound while others are novel.
Experiments at high pressure and furthermore,
experiments combining high pressures, high magnetic fields
and low temperatures are notoriously difficult. With TDO measurements, the authors present a phase
diagram that covers a large range, with pressures from 0 to about 2.5 GPa and magnetic fields from
0 to
45 T. It also contains relatively precise boundaries between the different proposed phases.
As such, the
study is of particular interest for quantum magnetism as it provides novel information about an important
compound and its theoretical counterpar
t, the Shastry
-
Sutherland model.
In the TDO measurements some
of the features are easy to spot while some others require more analysis and are harder to see. The later
depend on consistency checks that the authors have made and that cannot be easily checke
d by peer
review. However, in my opinion, the authors have provided reasonably sufficient justification about the
location of the different features mentioned in the manuscript.
The details of iPEPS calculation are not part
of my expertise and I cannot com
ment on those.
The references given are appropriate.
I would therefore in
principle recommend it for publication in Nature communications provided the issues raised below are
answered.
We thank the
R
eviewer for the overall positive assessment of our work
a
nd for recommending its
publication
in
Nature Communications
after the issues raised are addressed
.
Main concerns and questions
1. The numerical calculations make use of the pressure dependence of J and J’. However as mentioned in
the text these are not well known. The ab
-
initio paper in Ref 26, itself contains several values and references.
The J(p) and J’(p) chosen by the authors
seem not to be any of the previously published ones. J(p) is not
explicitly given and J’(p) is taken to vary by 3% from 0 to 1.8 GPa. First, can the slope of J(p) be given.
Second, this raises a question about how robust the numerical analysis is with res
pect to the “choice” of
the pressure dependence of the exchange parameters.
The authors should address this issue and explain
or show how changes in the initial values of J and J’ or in their slope would affect their results. In particular
does the “match”
between calculations and experiment in figure 3 strongly depend on those choices?
We thank the
R
eviewer for pointing out the confusion in our discussion. As a starting point at ambient
pressure, we have taken the value of J'/J = 0.63 from Ref. 10 which pr
ovides the best fit to the high
-
field
magnetization plateaus. All previous studies so far have predicted that the change in J' as a function of
pressure is considerably weaker than in J. Here we have chosen J'(p) to decrease linearly by 3% between
zero and
the critical pressure Pc ~ 1.8 GPa (corresponding to J'/J=0.675)
which lies in between the
prediction based on ESR data [22] (~1%) and fits of exact diagonalization results of 20
-
site clusters to the
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4
measured magnetic susceptibility [21] (~5%). With a 3
% change in J'(p)
,
the resulting linear change in J(p)
is ~9.5% between ambient and critical pressure, and the corresponding slopes are ~
-
0.86 K / GPa and ~
-
3.08 K / GPa, respectively. In the revised version we have added this extra information about the
modeling
of the pressure dependence in the iPEPS results section
to clarify it further
. In order to see the dependence
on the choice of J'(p), we have added two additional phase diagrams (
please see
Supplementary Fig. 11),
one with 1% change and the other
one with 5% change, both leading to only small shifts compared to the
case of 3% used for the phase diagram in the main text.
In response to the Reviewer’s comment, we have made the following changes
:
(1) In page 10, we expanded the
discussion,
and the new text now reads
as follows.
Here we model the pressure dependence assuming a linear dependence of J and J’ on pressure and a
small change of 3% in J’ between its value at ambient pressure and its value at the critical pressure p
c
=
1.8 GPa. Th
is choice lies in between the prediction based on ESR data
[
23
]
(~ 1%) and magnetic
susceptibility data
[
22
]
(~ 5%). At ambient pressure we use J = 81.5 K. This value lies in between previously
predicted values
[
11, 44
]
and yields good agreement with the o
nset of the 1/4 and 1/3 plateaus observed
in experiments. The resulting slopes of the linear functions J’(p) and J(p) are
-
0.86 K/GPa and
-
3.08 K/GPa,
respectively
.”
(2) We added a new Figure as Supplementary Fig. 11 to show Phase diagrams obtained for dif
ferent
pressure dependence of J’(p).
2. Comparison neutron vs TDO: In Figure 2, the TDO measurements are systematically higher (by 0.25 to
0.5 meV) than the spin gaps reported by neutrons and specific heat measurements in the dimer phase. In
the plaquette
phase however there is no shift for the low energy modes while the shift remains for the higher
energy mode. Could the authors address and discuss this issue?
Is that a due to a difference in dispersion? Is that a systematic of the TDO measurement compared
to other
techniques. Why does the systematically higher TDO values suddenly match the neutron and specific heat
values for the lower modes of the plaquette?
We thank the Reviewer for pointing out the confusion. We note that the TDO data points are associa
ted
with the right axis (external magnetic field μ
0
H, in the unit of Tesla), while the spin gap reported by neutron
scattering and heat capacity are plotted with the left axis (in the unit of meV). Since the two sets of data
have different units, one could not directly compare them.
Moreover, the sub
-
1/8
anomaly occurs after the spin gap is closed and does not really represent the energy
scale of the spin ga
p
, therefore,
we do not necessarily expect a close resemblance between the pressure
dependence of the sub
-
1/8 anomaly and that of the spin gap. That is
why we
believe
it is more appropriate
to plot our TDO data in terms of the characteristic fields
rather than
the spin gaps, so that we do not mislead
the readers. On the other hand, the low energy mode represents the transition between the plaquette phase
and the AFM phase. The energy scale for such a transition could be detected by the neutron scattering and
heat capacity. It is
,
therefore
,
perhaps not surprising that the pressure dependence of our TDO data follows
a similar trend as that reported by the
other two techniques.
In response to the Reviewer’s comment, we have made the following changes:
(1) In Fig. 2, we exchanged the left and right axes, so that our TDO data is now plotted with the left axis,
and the reported spin gaps are plotted with the r
ight axis,
to further clarify
that our TDO data
is
associated
with the characteristic fields not the spin gaps. The caption of Fig. 2 is revised correspondingly.
(2) We
have
revised the corresponding text in page 9, also
,
in response to the Reviewer’s comm
ent. It now
reads:
Interestingly, when plotted in the same figure, as shown in Fig. 2, the pressure
-
dependence of some of the
characteristic fields [μ
0
H
1
(P) and μ
0
H
0
(P)] and that of the spin gap [
D
(P)] measured by neutron scattering
and heat capacity meas
urements
,
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 provide a broader perspective for the evolution of the spin
gap with pressure in this material.
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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 new modes 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. S5 and S6 and
Supplementary Note 1) ...
Minor concern
3. a) In page 7: “H1 is the sub
-
1/8 anomaly that signals the onset of the condensation of triplet bound states”.
Is that claim based on the current analysis and can some justification be given, or is it based on a pre
vious
work in which case a reference should be given? There is not much further discussion in the main
manuscript, but in the supplementary material, it appears that it is a new interpretation. Can some more
quantitative justification be provided?
We thank
the Reviewer for the question. In fact, a similar question is also raised by the Reviewer 1. We
kindly
ask the Reviewer to
please
refer to our
extended
reply to the first comment by Reviewer 1 and
please
review
our revised manuscript for
the
corresponding
detailed changes.
b) Would other such “sub
-
plateau” anomalies be expected for the other plateaus?
A short answer would be no. We do not expect such “sub
-
plateau” anomaly for other plateaus. As explained
also pre
viously in our reply to the Reviewer 1, the sub
-
1/8 anomaly is most likely associated with the
condensation of the S=2 bound states, which is a prerequisite for all other plateaus associated with
crystallization of the bound states. It is expected theoreti
cally, but its experimental evidence has not been
intentionally searched for, although some experimental signatures have already been reported in Ref. 5
[Onizuka, K., et al. J. Phys. Soc. Jpn. 69, 1016 (2000)]. More detailed explanation of the sub
-
1/8 anom
aly
is now given in the revised text, as mentioned in our reply to the first comment by Reviewer 1.
4. Doped material. a) Do the indicated percentage correspond to the input composition during crystal growth
or to the final result obtained. b) Were X
-
rays performed to verify the doping levels and the fact that the Mg
impurities did actually go into the C
u positions? Nothing is said about sample characterization in the
methods section.
We thank the Reviewer for the question. Our study actually builds upon some of our earlier studies using
the same samples
[
Shi et al. Nat. Commun.
10, 2439 (2019), Ref
40]. The detailed characterization of these
samples has been reported in Ref. 40. To answer the Reviewer’s question, a) yes, the indicated percentage
correspond to the input composition during the crystal growth (nominal doping percentage), and b) we have
conducted susceptibility measurements and extracted the Mg
-
impurity percentage using Curie
-
Weiss law.
The extracted doping level agrees with the nominal value very well (
please s
ee methods and supplementary
Fig. 1 in Ref 40 for details).
5. In Supplementar
y figures 5 and 6. There is no big shift in the dip (and bump) position from x=0 to x=0.03
doping. At x=0.05 there is a sudden jump. The first dip, for instance, stays close to 6.0 GPa all the way to
x=0.03 and then jumps to ~4.8 GPa for x=0.05. The author
s should discuss this peculiar behavior.
We thank the Reviewer for the question. In fact, we realize that this comment is related to the comment #2
and #8 by the Reviewer regarding the discussion of the low energy mode and its doping dependence.
Indeed, w
e are also intrigued by the softening behavior of the low energy mode from ~ 6 T to 4.8 T
for
when
the doping is increased to x=0.05. We would like to note that the data for x=0.02, 0.03 plotted in
Supplementary Fig. 6 were actually taken at a lower pressu
re (2.1 GPa) compared to that (2.4 GPa) for
x=0, 0.05 plotted in Supplementary Fig. 5. Because of technical challenges reaching high pressure using
piston
-
type cells, we do not have 2.4 GPa data for the x=0.02 and x=0.03 samples. We note that the splitting
of the low energy mode is not observed until P is higher than 2.3 GPa. Since our data for x=0.02 and x=0.03
samples are taken at P up to 2.1 GPa, we do not know if such softening of the low energy mode at 2.4 GPa
is a sudden jump or not with doping.
We h
ave made some comments about this behavior in the previous version of our manuscript (the last
sentence above section “
iPEPS calculation results
”, where it says: “
However, the impurity
-
driven
shift in
the low
-
field mode noted above suggests that the dopant
s act to destabilize the plaquette phase and instead
favor the AFM phase
.”
In response to the Reviewer’s comment, we made the following changes:
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6
(1) We further clarify this point with the following added text in page 9:
“...though the doping dependence of t
his softening remains to be explored (see Supplementary Figs. S5
and S6
,
and Supplementary Note 1
).”
(2) We added the following discussion at the end of the Supplementary Note 1:
We note that for x=0.02 and x=0.03, we do not have data up to 2.4 GPa (Supplementary Fig. 6), so the
doping evolution of the softening of the low energy mode requires future exploration
.”
6. a) Two orientations were measured: with H parallel to c and H per
pendicular to c (i.e in the a
-
b plane).
In the second case was the orientation systematically the same (for example H // to a) or did it change from
one run to the next?
b) Should one expect a difference in magnetization or in df/dH when, for instance, H//
a compared to when
H is in the a
-
b plane at 45 deg between a and b?
We thank the Reviewer for the comment. For H parallel to the a
-
b plane, we have oriented the sample using
Laue diffraction before our measurements, and the field is applied parallel to the
a
-
axis (or equivalently, b
-
axis) within ~
5
degrees. The ~
5
degree
s
error comes from experimental difficulty in fixing the sample
position inside the TDO coil and from mounting the TDO coil on sample stage by hand.
Moreover, the Reviewer raised an import
ant question regarding the in
-
plane anisotropy of SCBO. We have
indeed carefully conducted a two
-
axis rotation TDO experiment, where we systematically studied the in
-
plane anisotropy of the system. And our results reveal essentially no change in any of the
characteristic
field for the plateaus, suggesting same g
-
factors within the ab
-
plane (no in
-
plane anisotropy).
In response to the Reviewer’s comment, we have made the following changes in the “
Tunnel diode oscillator
(TDO)
” section in Methods:
The sample
s are oriented using Laue diffraction so that the long edge of the sample is along the a
-
axis (or
equivalently, b
-
axis)... When mounting on the sample stage, the coil is oriented by hand such that the field
is applied parallel to the a
-
axis (or b
-
axis) withi
n ~
5
degrees. The coil with the sample inside and a capacitor
is used to form a LC circuit
.”
Clarification needed
7. Not clear in text in page 6:
“similar anomalies”. Does this refer to H6 an H7 only or all of H1 to H7?
“this
field range”, does it refer to
the field for H6 and H7 only?
We thank the Reviewer for pointing out the confusion. In both places, we refer to H6 and H7 that was
defined at P =0 only. Prompted by the Reviewer’s comment, we have made the following changes for
clarification.
(1) In page 6, we started a new paragraph
where we discussed the evolution of the two anomalies at H6
and H7 with increasing pressure.
(2) We rewrote several sentences that describe the pressure dependence of these two anomalies, and it
now reads:
We first focus on the two anomalies at the highe
st fields at P = 0, namely H6 and H7, which can be
identified 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 fields (~35 T and ~40 T, respectively). 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 even lower fields. It is tempting to assign these two anomalies that we see at
these high pressures (1.1 GPa, 1.9 GPa and 2.3 GPa) as exten
sions of the H6 (1/4 plateau) and H7 (1/3
plateau) seen at P = 0.
However, we caution that the fate of the magnetization plateaus at higher pressure
needs to be understood first
.
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8. Page 8
“We
first focus on H1, ...” the structure of the presentation is no
t very clear. One would expect
then a focus on the other features, but that is not how the text continues.
We thank the Reviewer for prompting us to realize the confusing part in our presentation. Since this
comment is related to the Reviewer’s comment #2,
which also concerns the discussion of Fig. 2, we refer
the Reviewer
to
please see
our response above for detailed changes. These changes indeed significantly
improved our presentation, and we are very grateful for the Reviewer for these comments.
9. In
page 11
-
12, when discussing the discrepancies between calculation and experiment, the authors
invoke the 3D and DM couplings and then possible structural differences related to open vs full plaquettes.
All of these are acceptable as possible sources of dis
crepancies. However, it is not clear at the end of the
paragraph why those should be important “particularly at low fields”.
What we had in mind here is that in particular the transition into the AF phase will be pushed down to lower
fields by including an
inter
-
plane coupling, since it is known that the extent of the plaquette phase gets
reduced with respect to
t
he AF phase with increasing inter
-
plane coupling. Also, in case the ground state
in the material is an FPP ground state (and not an EPP ground sta
te), this may also have an effect on the
critical field into the AF state. At larger fields, which includes large extended phases (in contrast to the
narrow AF phase in between the plaquette phase and 10x2 supersolid phase), there is a weaker
competition (
i.e. larger energy differences) between different phases, and thus we expect a smaller effect
of the additional modifications of the model.
In order to clarify further
, we have extended this sentence in
page 13.
These modifications of the model may also
affect the magnetization process, particularly at low fields,
where the narrow AF phase is energetically closely competing with the plaquette and the 10x2 supersolid
phase. We stress, however, that the 1/5 plateau and the 10x2 supersolid phase remain relev
ant ground
states also in the deformed model (see Supplementary Fig. 12).
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 se
ctions
.”
10. It would be good for the reader, in my opinion, to 1) explain in the text why some of the phases
correspond to a “supersolid” state. For instance, what characteristics of a supersolid do these particular
spin arrangements display? And 2) justi
fy or explain the naming used such as: “10x2” and “1/3” supersolids.
We agree with the
Reviewer
that it would be useful to explain this. We have added additional information
in the iPEPS results section to explain the meaning of the supersolid phase and the naming.
(1)
O
n page 11, the revised text now reads:
At high fields (up to 45 T) the dominant
phases are the 1/4 plateau, the 1/3 plateau, and a 1/3 supersolid
phase [11]. A supersolid phase simultaneously breaks the translational symmetry and the U(1) symmetry
associated with the total Sz conservation. The 1/3 supersolid exhibits the same transla
tional symmetry
breaking pattern as the 1/3 plateau state, but with additional spin components in the transverse direction,
reflecting the broken U(1) symmetry
.”
(2)
O
n page 12, the revised text now reads:
Above the empty plaquette (P) phase in zero and s
mall fields, we find a narrow partially polarized
antiferromagnetic phase (AFM), and a 10x2 supersolid state (obtained in a 10x2 unit cell; hence the name),
followed by the 1/3 supersolid and 1/3 plateau phases
.”
11. Supplementary Fig S4. The quality of th
e data at 0.3K is much better than the data at 0.44K. Is that
uniquely due to the physics or was the 0.44K data (and higher T) taken with lower statistics? It would be
good to give this information in the supplementary text.
We thank the Reviewer for the a
cute observation which prompted us to re
-
examine the data shown in
Supplementary Fig. 4. This figure contains three subpanels, labeled as (a), (b), and (c), and the Reviewer’s
question regards the data at 1.9 GPa shown in panel (b). Here, each of the df/dH
vs H trace between 0.44
K and 0.95 K is actually an average of two measurements (one is with field up
-
sweep and the other is with
field down
-
sweep). For 0.3 K, we repeated the measurements for six times (three up
-
sweeps and three
down
-
sweeps) and calculat
ed the average. By repeating each measurement multiple times and follow
ing
the temperature dependence, we are able to confirm the reproducibility of the identified anomalies.
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8
Prompted by the Reviewer’s comment, we
noticed
that
when plotting the results, the data for 0.3 K trace
was scaled
by an additional factor of 2
compared to
the rest of the temperatures
. This
has
le
d
to the
impression that the
quality of the
data at 0.3 K is
better
than that at 0.44 K
and higher
temperatures
. We
have corrected
this
in the revised version of the Supplementary Fig. 4. In the corrected figure, one can see
that the temperature evolution of the df/dH is gradual without sudden changes.
We note, however, this does
not affect any of our conclusions.
We are grateful to the Reviewer for
pointing this out
. We have also
carefully checked all other figures and made sure they are all correct.
12. Supplementary Fig 5
-
6. Is the vertical scale the same for all panels?
The vertical scale in Supplementary Fig. 5 and 6 are
in
arbitrary units
, and thus different
. This is because
in TDO measurements
,
we measure the
change
in frequency of the TDO circuit
. Additionally, the actual
frequency of the TDO circuit can be differen
t for each setup, however, the relative change of frequency with
regards to magnetic field (thus the characteristic field at which we identify the anomalies) should be
reproducible
this has indeed been the case for the presented results
.
Reviewer #3
The
manuscript by Shi et al combines experimental and theoretical efforts to explore the magnetic phase
diagram of SrCu2(BO3)2 under extreme conditions: sub
-
Kelvin temperatures, high pressures and high field.
The anomalies observed in the tunnel diode oscilla
tor (TDO) experiment are then used to establish the
phase boundaries. By comparing to the iPEPS simulations, the authors claim the discoveries of a few new
phases, in particular the 10x2 supersolid and a 1/5 plateau.
While it is always interesting to find
novel magnetic states, for example the 10x2 supersolid and the 1/5
plateau, I am not fully convinced that this work meets the criteria of validity and novelty to publish under
Nature Communications. My major concerns are list below:
1.As the authors admitt
ed themselves (last paragraph in “iPEPS calculation results”), the model they are
using does not even correctly reproduce the “full”
-
plaquette ground state at high pressure and zero field.
Consequently, the theoretical findings at high pressure are sitting
on a wrong base. This naturally explains
their discrepancy in matching the theoretical and experimental data at pressures larger than 1.9GPa.
Consequently, I am not convinced that the 10x2 supersolid phase and the narrow AF phase are relevant to
the real
material.
We are grateful to the Reviewer for careful reading of our
manuscript,
and w
e
fully
understand
the
importance to address the criticism raised.
We would like to begin with stressing on the fact
that the Shastry
-
Sutherland model (SSM) has been an
excellent starting point to understand several physical phenomena in SCBO over the past 20 years. At
ambient pressure it reproduces the sequence of magnetization plateaus observed in experiments [10,12],
and also thermod
ynamic quantities are in excellent agreement [43]. At finite pressure, it reproduces the
transition into a gapped plaquette phase, followed by a transition into an antiferromagnetic phase, and it
correctly reproduces a critical point at finite temperature
[26], in agreement with experiments. Thus, there
is clear evidence from many previous studies, that the model is relevant for SCBO, and for this reason, we
have used this model also as a starting point in the present study.
Now, there exist indications fro
m experiments that the intermediate plaquette phase is not the empty
plaquette
phase (EPP)
, but a full plaquette phase
(FPP)
whether this is really the case is still an open
question. However, in any case, these two phases lie energetically very close. I
n Ref. [24] it was shown
that already a small deformation of the standard SSM stabilizes the FPP phase over the EPP phase.
S
ince
the deformed model lies very close to the original model in parameter space, we a priori do not expect
substantial changes of t
he phase diagram at high magnetic fields upon slightly modifying the model.
Another concern raised by the Reviewer
may be that if there is a strong competition between EPP and FPP
phases for h=0, the same may be true at finite h, which would imply a stro
ng sensitivity of the results upon
deforming the model. However, this is not the case. In fact, the new phases we find (i.e. the new 1/5 plateau
and the 10x2 supersolid) are FPP
-
like states, and not EPP
-
like states. These FPP
-
like states will naturally
re
main the ground state also of the deformed model (which favors FPP correlations). To demonstrate this
numerically we have performed additional simulations for the deformed model, for two different types of