March 2008
EPL, 81 (2008) 67004
doi: 10.1209/0295-5075/81/67004

www.epljournal.org

Field dependence of the quantum ground statein the Shastry-Sutherland system SrCu2(BO3)2
F. Levy1 , I. Sheikin1 , C. Berthier1 , M. Horvatić1 , M. Takigawa2 , H. Kageyama3 , T. Waki2 and Y. Ueda2
1

Grenoble High Magnetic Field Laboratory (GHMFL), CNRS - BP 166, 38042 Grenoble Cedex 09, FranceInstitute for Solid State Physics, University of Tokyo - Kashiwanoha, Kashiwa, 277-8581, Japan3
Department of Chemistry, Graduate School of Science, Kyoto University - Kyoto 606-8502, Japan2

received 30 November 2007; accepted in ﬁnal form 30 January 2008
published online 29 February 200875.10.Jm – Quantized spin models
75.30.Kz – Magnetic phase boundaries (including magnetic transitions, metamagnetism,
etc.)
PACS 75.30.Sg – Magnetocaloric eﬀect, magnetic cooling
PACS
PACS

Abstract – We present magnetic torque measurements on the Shastry-Sutherland quantum spin
system SrCu2 (BO3 )2 in ﬁelds up to 31 T and temperatures down to 50 mK. A new quantum phase
is observed in a 1 T ﬁeld range above the 1/8 plateau, in agreement with recent NMR results. Since
the presence of the DM coupling precludes the existence of a true Bose-Einstein condensationand the formation of a supersolid phase in SrCu2 (BO3 )2 , the exact nature of the new phasein the vicinity of the plateau remains to be explained. Comparison between magnetization andtorque data reveals a huge contribution of the Dzyaloshinskii-Moriya interaction to the torqueresponse. Finally, our measurements demonstrate the existence of a supercooling due to adiabatic
magnetocaloric eﬀects in pulsed ﬁeld experiments.
c EPLA, 2008
Copyright Quantum antiferromagnetic spin systems with singlet
ground states exhibit a variety of magnetic-ﬁeld–induced
quantum phase transitions. Crystalline arrays of S = 1/2
spin dimers, for instance, can present two contrasting
behaviors [1]. When the magnetic ﬁeld exceeds a criticalvalue at which the lowest energy levels cross each other,
the triplet excitations, which can be treated as hard corebosons on a lattice, typically undergo a Bose-Einsteincondensation (BEC) [2–4]. Another possibility, however,is the occurrence of magnetization plateaus at fractional
values of the saturated magnetization. Such plateauscorrespond to the formation of a superlattice of triplets(“magnetic crystal”) and may occur when the kinetic
energy of the triplets is strongly reduced by frustration,
so that the repulsive interactions become dominant. The
best known example for the formation of such plateaus is
SrCu2 (BO3 )2 with its two-dimensional network of orthogonal dimers of S = 1/2 Cu2+ ions [5]. This material shows
an excitation gap ∆0 = 35 K and plateaus at 1/8, 1/4,
and 1/3 of the saturated magnetization [6,7]. A magnetic
superlattice at the 1/8-plateau has actually been observed
in NMR experiments [8]. It has been argued on theoretical basis that some analog of a supersolid phase [9–12],
consisting of the superposition of the magnetic crystal
and a Bose-Einstein condensate of the interstitial triplets,

could occur in the vicinity of the plateau phases. The
presence of such an exotic phase, the magnetic analog of
the highly debated supersolid phase in 4 He, is however
precluded in SrCu2 (BO3 )2 because of the presence ofan intradimer Dzyaloshinskii-Moriya (DM) interaction [13,14] which breaks the U (1) symmetry. However,
NMR measurements have recently revealed the existenceof new magnetic phases above the 1/8 plateau. This
prompted us to re-examine the ﬁeld-temperature (H-T )phase diagram of SrCu2 (BO3 )2 up to 31 T using torquemeasurements. Our experiments indeed conﬁrm the existence of a new phase adjacent to the 1/8 plateau, in which
the magnetization is only slowly increasing. In addition,
we report for the ﬁrst time the ﬁeld dependence of thepure longitudinal magnetization up to 31 T, measured
at the temperature of 60 mK. The results strongly diﬀer
from those obtained by torque measurements, as expectedin the presence of DM interaction within the dimers [15].
In particular, the magnetization jump before the 1/8
plateau is much larger than previously reported [6,16], in
excellent agreement with NMR data.
The torque measurements were performed at the Grenoble High Magnetic Field Laboratory in a 20 MW resistive
magnet equipped with a dilution refrigerator. The sample(∼ 1 × 1 × 0.5 mm3 size) was mounted on a 25 µm thick

67004-p1

F. Levy et al.
1.60

0.7
H up
H down

0.6

1.55

T (K)

0.5

1.50

τ / H (a. u.)

H downH up

d 2 (τ / H) / dH 2 (a. u.)

0.4
0.3

1/ 8

0.2

1.45
0

0.1

1.40

0.0
26

1.35
26
26

27

27

28

28

29
29

30
30

28

29

30

31

H (T)

H (T)

1.30

27

31
31

H (T)

Fig. 1: (Colour on-line) Torque divided by ﬁeld vs. ﬁeld.
The blue continuous line corresponds to a ﬁeld sweep up at
55 ± 10 mK. The red dashed line corresponds to a ﬁeld sweep
down at 90 ± 10 mK. The inset shows the corresponding second
derivatives d2 m⊥ /dH 2 . Six extrema (three transitions) arefound in each sweep and pointed out by arrow for ﬁeld sweep
up, and by squares for ﬁeld sweep down. For a given transition,
the ﬁeld range between the maximum and the minimum ofd2 m⊥ /dH 2 corresponds to the range in which both adjacent
phases coexist.

CuBe cantilever with its c-axis perpendicular to thesurface. In situ rotation allowed us to obtain an angle θ
between the c-axis and the applied magnetic ﬁeld of about
0.4◦ , a conﬁguration which has been used for most of theexperiments. Torque measurements have been performed
at various constant temperatures while sweeping ﬁeld up(at a rate of 100 G/s) and down (at a rate of 200 G/s).
When the sample is placed in a homogeneous ﬁeld, it
is submitted to a torque τ = M × H in which M is themagnetization and H the applied ﬁeld. This torque istransmitted to the cantilever, which can rotate around aﬁxed axis perpendicular to H. So the torque is equal to τ =m⊥ H, where m⊥ is the component of the magnetization
perpendicular to ﬁeld and to the rotation axis of thecantilever. One must be careful that the variation of m⊥as a function of H is not necessarily the same as that of
the longitudinal magnetization mz . However, if the sample
is placed in a ﬁeld gradient, one additionally obtains anaccess to the magnetization parallel to the applied ﬁeld,
provided the torque component becomes negligible with
For this we moved therespect to the force F = −Mµ0 H.
sample by 1 cm above the magnet center. From here on, we
will refer to such measurements as “true” magnetization
measurements.
Figure 1 shows the results obtained at the lowest
temperature. Three anomalies are clearly visible in thesecond derivative of the torque divided by ﬁeld, that isof m⊥ ∝ τ /H. The ﬁrst two correspond to the boundariesof the 1/8 plateau. Above the plateau, there is a second

Fig. 2: (Colour on-line) Temperature-ﬁeld phase diagram.
Position of the extrema of the second derivative d2 m⊥ /dH 2
made at various constant temperature (see ﬁg. 1). Threetransition lines are clearly appearing. For a given transition, thedistance between the two corresponding extrema corresponds
to the ﬁeld range within which the two adjacent phases coexist.
The blue circles correspond to ﬁeld sweep up and the redsquares to ﬁeld sweep down. The NMR points, black triangles,delimit the boundary of the magnetic ordering.

phase which ends up around 29.5 T. In both phases, m⊥
is slightly decreasing with increasing the ﬁeld. Comparing
data acquired in ramping up and down the magnetic ﬁeld,
one observes no hysteresis for the lower boundary of theplateau. This absence of hysteresis as a function of H has
already been observed in NMR experiments [8], in spite of
the fact that the transition is of the ﬁrst order, as indicatedby the coexistence of the two phases. This contrasts withthe two other transitions, which exhibit a rather strong
hysteresis.
The position of the peaks found in the second derivative of m⊥ (see ﬁg. 1) are reported in ﬁg. 2 to establish theﬁeld-temperature phase diagram. The transition temperatures delimiting the phase boundary between the paramagnetic and the ordered magnetic states (black triangles)
are taken from NMR results [17]. The distance between
the two extrema of the second derivative for a given transition should roughly give the range of magnetic ﬁeld
within which the two adjacent phases coexist. This isindeed in excellent agreement with the NMR data: at very
low temperature, the coexistence between the uniform
paramagnetic phase and the triplet superlattice starts
at 26.6 T and the uniform phase disappears at 27 T [8].
The transition from the 1/8 plateau into the adjacent
phase above was found to start at 28.3 T at 0.31 K bothby NMR [17] and torque measurements. Considering thetransition around 29.5 T, one observes that the hysteresis
becomes much stronger below 200 mK. Indeed, a modiﬁcation of the NMR lineshape has been observed at temperatures below 200 mK close to the beginning of the phasefollowing the 1/8 plateau [17]. So, one cannot completely
exclude the possibility that there is a new phase at low
temperature, which would be seen by torque measurements only through the hysteresis of the transition around67004-p2

Field dependence of the quantum ground state in SrCu2 (BO3 )2
60 mK

1/ 8
6

240 mK

1/ 8<SZ> X 10

2

370 mKτ / H (a. u.)

490 mK590 mK
650 mK

4

2 26

28
H (T)

30
MZ force
MZ NMR

τ/H
MS NMR

0
5

10

15

20

25

30

H (T)

Fig. 3: (Colour on-line) Torque divided by ﬁeld vs. ﬁeld at
diﬀerent temperatures. The measurements were made for ﬁeld
sweep up. The curves have been arbitrary shifted for clarity.

Fig. 4: (Colour on-line) Comparison of torque and magnetization. Magnetization was measured both by NMR technique, green triangles, and by torque in a ﬁeld gradient,continuous black line. Those measurements are compared withtorque divided by ﬁeld, the continuous red line. The solid red
circles correspond to the transverse staggered magnetization
measured by 11 B NMR.

29.5 T. This should deserve further investigation. We now
consider the temperature dependence of m⊥ shown inﬁg. 3. There are two remarkable features in these data.
First, one can see that the signature of the 1/8 plateau
has fully disappeared at 590 mK, while there is still somereminiscence of the adjacent phase at this temperature.
This again is in excellent agreement with previous NMR
data [8], but contrasts with earlier measurements madein pulsed magnetic ﬁeld performed at 1.4 K [6,16], inwhich the signature of the plateau is still visible. Since thegap between the lowest triplet excitations branch and thesinglet state decreases toward a very small value [14,18] as
the magnetic ﬁeld is approaching the value corresponding
to the entry of the plateau, the “high temperature” observation of the plateau in pulsed magnetic ﬁeld can be interpreted in terms of adiabatic (isentropic) cooling of the spin
system [19,20]. As long as only the lowest triplet branch
and the singlet state are necessary to describe the system,
this is an analog of the cooling of paramagnetic salts byadiabatic demagnetization. The second remarkable feature
in ﬁg. 3 is that between 370 mK and 490 mK the slope ofτ /H changes from negative to positive before the complete
melting of the magnetic superlattice. This corresponds
to the regime in which the triplet superlattice and theparamagnetic phase coexist, which was found by NMR toextend between 360 mK and 520 mK at 27.6 T [21]. Sucha change of sign, related to the coexistence of the superlattice and the uniform phase, can only be explained if
the contribution to τ /H of the superlattice has a negative slope with increasing H, while that of the uniform
paramagnetic phase has a positive one. This is rather
surprising, since within the plateau one would expect
the torque signal to remain constant. In order to elucidate this issue, we have performed a “true magnetization”

measurement in which the bending of the cantilever is now
dominated by the force Fz = −Mz dBz /dz. The results
are shown in ﬁg. 4, together with a torque measurement
recorded at the same temperature. In addition, solid triangles show the amplitude of Sz  as determined by NMR.
The ﬁeld dependence of the longitudinal magnetization
Mz actually strongly diﬀers from the results obtained bytorque. Mz is flat within the 1/8 plateau, as expected,
and within the adjacent phase between 28.4 and  29.5 T
it is nearly ﬂat with only a small increase as approaching the upper boundary. The magnetization data are inexcellent agreement with the NMR results. In particular,
both techniques reveal a large jump of the magnetization
just before the 1/8 plateau, in contrast to all previously
reported data [6,16].
We also remark that recently it has been proposed thatthe 1/8 plateau could be preceded by a 1/9 plateau, andfollowed by a 1/7 plateau [22]. Indeed, the authors ofref. [22] report three transition lines around 27.1, 29 and30.3 T. It turns out that by reducing these three ﬁelds
values by 2% we recover the phase boundaries values
reported in this letter. These latter values agree withthose determined by NMR, a technique which inherently
always provides a precise determination of the magnetic
ﬁeld. Therefore, the values reported in ref. [22] appear toresult from an incorrect ﬁeld scale. Furthermore, looking
carefully at the pulsed ﬁeld measurements of ref. [6],the magnetization values measured for the 1/4 and 1/3
plateaus clearly indicate that the phase extending between
26.7 and 28.3 T can only correspond to the 1/8 plateau. As
far as the adjacent phase is concerned, our measurementsof the longitudinal magnetization demonstrate that it
cannot correspond to a 1/7 plateau, since the increase of
M/Msat is too small.

26

27

28

29

30

31

H (T)

67004-p3

F. Levy et al.
6

At 60 mKθθ'

τ / H (a. u.)

DJ

y

Ms
x

6.0

4

5.5

2

J’

17

19

21

23

26

28

25

27

30

29

31

H (T)

Cu2+dimer

Fig. 5: (Colour on-line) Schematic structure of the Cu2+
dimers. The ﬁgure sketches the orthogonal network of Cu2+
dimers. Black circles represent the Cu2+ ions. Intradimer interactions are represented by dotted black lines, and interdimer
interaction J  by dashed red lines. Thick green arrows indicate
the direction of D-vectors for the intradimer DzyaloshinskiiMoriya interaction D · Si × Sj , where the bond “direction”
i → j is shown by black dotted arrows. The solid blue arrows
represent the staggered magnetization Ms induced by a ﬁeld
applied along the c-axis (z-axis), within the ﬁeld range below
the ﬁrst magnetization plateau. Interdimer DM interactions,which are less eﬀective to generate a transverse magnetization,
are not shown here.

SrCu2 (BO3 )2 crystallizes in a tetragonal structure withalternative layers of Sr and Cu(BO3 ) planes along thec-axis. At low temperature, due to the buckling of theBO3 -Cu-O-Cu-BO3 bonding, the ab-plane is no longer amirror plane [23], allowing the existence of an in-plane
intradimer DM interaction as shown on ﬁg. 5, as well asa staggered g-tensor. Interdimer DM interactions are alsopresent [24], but their role is mainly to partially restore
some kinetic energy to the triplets. The strong diﬀerence
between the longitudinal magnetization Mz (H) and thetorque signal as shown in ﬁg. 4 is indeed the signature of
the intradimer DM interaction and the staggered g tensor.
The presence of the intradimer DM interaction has beenshown to generate a transverse staggered magnetization,observed by 11 B NMR and computed by exact diagonalization [14]. While this transverse staggered magnetization
has no eﬀect on the torque, it has been shown recently
that there is an additional uniform transverse component
generated by the DM interaction, which is smaller by anorder of D/J [15]. In the low ﬁeld limit and for an isolated
dimer, this component for each dimer has the symmetryof D × D × H. Within the low-ﬁeld approximation, thetorque per spin dimer can be expressed as
τ = (χab − χc ) sin θ cos θH 2 − gµB D2 /4J 3 sin θ cos θH 2 ,where θ is the angle between the c-axis and the appliedmagnetic ﬁeld, and χ is the part of the susceptibility

Fig. 6: (Colour on-line) Dzyaloshinski-Moriya contribution
within the 1/8 plateau and its adjacent phase. Torque data
have been recorded at θ = +0.4◦ and at −|θ |. The latter data
have been rescaled to give the same signal at 17 and 26 T.
Two signals strongly diﬀer within the plateau and its adjacent
phase, which is attributed to a symmetry breaking.

which only depends on the symmetric part of theg-tensor.
The ﬁrst term is the standard contribution which
is proportional to the longitudinal magnetization forsmall values of θ, while the second one results fromthe DM interaction. Both terms have the same angulardependence, and vanish for θ = 0 (H  c). In ﬁg. 4 the ﬁeld
dependence of τ /H is compared to that of the staggeredmagnetization determined from NMR measurements [14].
In the low ﬁeld limit, in which the torque signal is
only (as long as Mz = 0) or mainly due to the DM
interaction, their variation is quite similar, which is inagreement with the theory predicting that both quantities
vary linearly with H. When Mz = 0 the torque becomes
the sum of two contributions and a direct comparison
is no longer possible. Exact diagonalization calculations
are required to determine the full ﬁeld dependence of
the “uniform” transverse magnetization in the ShastrySutherland geometry.
We now consider the τ /H variation within the plateau
and its adjacent phase and try to understand the originof the negative slope observed within the 1/8 plateau.
Figure 6 shows torque measurements recorded at θ andat −|θ |. Since, as expected, the corresponding raw data
have diﬀerent sign, those corresponding to −|θ | havebeen renormalized in order to give the same values at
17 and 26 T. One can see that the ﬁeld variation of bothsignals are identical in the uniform phase, as expected
if they only diﬀer by a factor sin θ cos θ. However, theystrongly diﬀer within the 1/8 plateau and its adjacent
phase, in which an extra contribution is observed. Thisis expected, at least for the 1/8 plateau, for the following
reason. Within the uniform phase, the DM interaction
is conserved by the three symmetry operations of thecrystallographic structure: the mirror plane zx, the mirror
plane yz and a C2 rotation at the intersection of thesemirror planes. However, within the 1/8 plateau, thestructure determined for the magnetic superlattice [8] has

67004-p4

Field dependence of the quantum ground state in SrCu2 (BO3 )2
only one symmetry left, which is a C2 rotation aroundthe middle of the most polarized dimer. We thus expectthat the angular dependence of the torque signal becomes
diﬀerent, and starts to depend also on the angle between
the projection of H on the ab-plane and the crystallographic axes. Recently, the investigation of frustrated
ladders with DM interactions in a magnetic ﬁeld [25]has been extended to the situation where the ﬁeld isneither parallel nor perpendicular to the D-vector [26].
It has been shown that the torque induced by the DM
interaction develops peaks upon entering and leaving the
1/2 magnetization plateau. While the torque produced by
the misalignment of the ﬁeld with a principal axis of theg-tensor increases monotonously with the ﬁeld, the torqueinduced by the DM interaction is non-monotonous inside
the plateau, in qualitative agreement with the present
observation. Whether this anomalous contribution disappears or not above 29.5 T, where we know from NMR
that the “magnetic crystal” persists [17], is not clear at
the moment, and would require new measurements.
What is the nature of the phase adjacent to the 1/8
plateau? Recent NMR experiments have shown that themagnetic superlattice, analogous to a magnetic crystal,does not melt when additional triplets are introduced.
One can then immediately suspect that this new phaseis the analog of a supersolid phase, in which the additional triplets would undergo a Bose-Einstein condensation. However, the presence of the intradimer DM interaction and the resulting staggered magnetization break theU (1) symmetry around the applied magnetic ﬁeld and thus
remove the continuous symmetry of a supersolid phase.
So, some more sophisticated theoretical description of theexact nature of this phase has to be provided in the future.
In conclusion, we have determined the phase diagramof the Shastry-Sutherland quantum antiferromagnet
SrCu2 (BO3 )2 in the (H-T )-plane up to 31 T, using bothmagnetic torque and “pure longitudinal” magnetization
measurements. We show that the torque measurementsallow the detection of the phase transitions between
successive quantum ground states, but cannot give accessto the true variation of the longitudinal magnetization
Mz . This is due to the existence of an intradimer
Dzyaloshinskii-Moriya interaction, which generates anadditional uniform transverse magnetization providing
a strong contribution to the torque. In the low-ﬁeld
limit this contribution to τ /H scales linearly with thetransverse staggered magnetization measured by NMR.
The phase boundaries of the 1/8 magnetization plateau
are found to be in agreement with NMR data: at 60 mK,
the coexistence between the uniform paramagnetic phaseand the 1/8 plateau extends from 26.6 to 27 T, and theplateau ends through a ﬁrst-order transition starting at
28.3 T. The temperature corresponding to the complete
melting of the spin superlattice in the 1/8 plateau is at
most 570 mK. This demonstrates that the observation of
the 1/8 plateau at 1.4 K in pulsed ﬁeld measurements isdue to isentropic adiabatic cooling of the spin system. The
most important ﬁnding of this study is the evidence of anew phase adjacent to the 1/8 plateau and extending up
to 29.3 T. The magnetization within this phase is nearlyﬁeld independent and only increases when approaching its
upper boundary. However, its value does not correspondto any simple rational value of the saturation magnetization. Recent NMR measurements show that the “magnetic
crystal” does not melt in that phase. However, since the
Dzyaloshinskii-Moriya interaction removes the continuous
rotation symmetry in SrCu2 (BO3 )2 , this phase has to be
more complex than a simple analog of a supersolid, which
would correspond to the coexistence of the 1/8 plateau“magnetic crystal” and a Bose-Einstein condensate of theinterstitial triplets. This clearly shows that our understanding of the physics of interacting hard-core bosons on
a lattice has to be improved, and that further theoreticaland experimental investigation are necessary to clarify
the evolution of the quantum ground states between the
1/8 and the 1/4 plateaus in this model compound.
∗∗∗We acknowledge F. Mila for enlightening discussions.
This work was supported by the the French ANR grant06-BLAN-0111 and Grant-in-Aids for Scientiﬁc Research(Nos. 16076204 and 19052004) from the MEXT Japan.
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