PRL 101, 037202 (2008)

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PHYSICAL REVIEW LETTERS

NMR Evidence for the Persistence of a Spin Superlattice Beyond the 1=8 Magnetization Plateau
in SrCu2 BO3 2
M. Takigawa,1 S. Matsubara,1,* M. Horvatić,2 C. Berthier,2 H. Kageyama,3 and Y. Ueda1
1

Institute for Solid State Physics, University of Tokyo, Kashiwanoha, Kashiwa, Chiba 277-8581, Japan
2
Grenoble High Magnetic Field Laboratory, CNRS, BP 166 - 38042 Grenoble, France
3
Department of Chemistry, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan(Received 29 October 2007; revised manuscript received 13 April 2008; published 15 July 2008)

We present 11 B NMR studies of the 2D frustrated dimer spin system SrCu2 BO3 2 in the field range
27–31 T covering the upper phase boundary of the 1=8 magnetization plateau, identified at 28.4 T. Ourdata provide a clear evidence that above 28.4 T the spin superlattice of the 1=8 plateau is modified but does
not melt even though the magnetization increases. Although this is precisely what is expected for a
supersolid phase, the microscopic nature of this new phase is much more complex. We discuss the fieldtemperature phase diagram on the basis of our NMR data.
DOI: 10.1103/PhysRevLett.101.037202

PACS numbers: 75.10.Jm, 75.25.+z, 75.30.Kz

0031-9007=08=101(3)=037202(4)(kinetic energy) [19,20], and plateaus at 1=8, 1=4, and
1=3 of the saturated magnetization (see Fig. 1) [10]. A
symmetry-breaking spin superlattice in the 1=8 plateau has
been confirmed by NMR experiments [4,21]. Furthermore,a supersolid phase has been proposed for the ShastrySutherland model above the 1=3 plateau [22]. However,SrCu2 BO3 2 also contains an intradimer DzyaloshinskiMoriya (DM) interaction and a staggered g tensor, which
break the spin-rotation symmetry and prevent true Bose
condensation [23–25]. This explains the gradual increaseof magnetization starting far below the expected criticalfield =gB  23 T and the absence of phase transitions
up to the boundary of the 1=8 plateau (26.5 T). Sinceanisotropic interactions are often present in real materials,

0.3(2/gc )M (µ B /Cu)

A variety of novel quantum phenomena has been discovered by applying a magnetic field to interacting systemsof S  1=2 spin dimers, which have a singlet ground state
and an energy gap  to the triplet excitations at zero field[1–6]. A magnetic field splits the triplets and closes the gap
at the critical field Hc  =gB . Above Hc , a finite density of triplets (magnetization) generally undergoes Bose
condensation, resulting in an antiferromagnetic orderperpendicular to the field [2,7–9]. Another remarkablephenomenon that may occur at higher fields is the magnetization plateau, where the magnetization stays constant at
a fractional value of the saturation over a finite range ofmagnetic field [3,10,11]. In particular, a plateau with noninteger density of triplets per unit cell generally implies
localization of the triplets into a spin superlattice breaking
the translational symmetry of the crystal [12]. Such a stateis stabilized when the kinetic energy of the triplets is
reduced, e.g., by frustration, so that the repulsive interaction becomes dominant.
The nature of phases between the plateaus is particularlyinteresting, since they are the bosonic analog of doped
Mott insulators. A simple scenario for an incommensurate
density of triplets is the melting of the superlattice and theappearance of Bose condensation. A more exotic possibility predicted for certain dimer spin models [13–16] is theformation of a supersolid phase, where the interstitial
triplets undergo Bose condensation in the background of
the commensurate superlattice. This means that the longitudinal spin density modulation coexists with a new transverse staggered order.
To date only a few spin systems are known to show
symmetry-breaking magnetization plateaus. The best example is SrCu2 BO3 2 [3], an orthogonal dimer system ofCu2 ions (S  1=2) with the primary interactions described by the two-dimensional Shastry-Sutherland model[17,18]. This material shows a gap   35 K to the triplet
excitations, which have a very small dispersion width

1
2
5 3
8
7 6 4

0.2

1/4

1/8

3

1

4

2
0.1

0.0
0

10

20

40

30

50

60

H (T)

FIG. 1. The magnetization divided by gc =2 (gc  2:28 [33])is plotted against the pulsed magnetic field along the c axis
(taken from Ref. [10]). The measurement was performed at T 
1:4 K. However, since the 1=8 plateau is not observed above
0.6 K in steady fields [21,29], we believe that the sample wasadiabatically cooled during application of the pulse field. Also
shown are the calculated spin density profiles for the 1=8 and
1=4 plateaus [27]. The solid (open) circles indicate magnetization of Cu ions parallel (antiparallel) to the external field with themagnitude represented by the circle size. The ellipsoid indicatesa triplet unit extending over three dimers.

037202-1© 2008 The American Physical Society

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PHYSICAL REVIEW LETTERS

understanding their influence on the potential supersolid
phase is an important issue. For SrCu2 BO3 2 , steady fields
above the 1=3 plateau are yet unavailable, and the magnetization shows a direct jump from 1=4 to 1=3 plateaus(Fig. 1). Therefore, the vicinity of the 1=8 plateau provides
a unique opportunity to look for such novel phenomena.
In this Letter, we report results of NMR experiments onboron sites in SrCu2 BO3 2 in the field range 27–31 T. We
identified the upper boundary of the 1=8 plateau phase at
28.4 T from a clear splitting of NMR lines. The range of theinternal field distribution, however, is nearly unchanged
across the boundary. This provides direct evidence thatspatial order of highly polarized triplet dimers persists
above the 1=8 plateau even though the magnetization
changes continuously, pointing to an analog of the supersolid phase. In this study we used the same single crystal as
was employed in our previous high field NMR experiments[4,21,23]. The NMR measurements were performed using
a dilution refrigerator installed in a 20 MW resistive magnet at the Grenoble High Magnetic Field Laboratory. The caxis of the crystal was aligned along the field direction
in situ within 1 by observing the angular variation of the
NMR spectra at 18 T.
Figure 2 shows the 11 B NMR spectra at T  0:19 K
obtained at three different field values, Hext  27:5, 28.7,
and 29.9 T. The upper blue lines represent the distribution of the frequency shift divided by the nuclear gyromagnetic ratio, f=N , where f  f  N Hext , N 
13:66 MHz=T, and f is the resonance frequency. The
spectrum taken at 27.5 T is representative for the entire
1=8 plateau as its shape does not change in the whole field

27.5 T

28.7 T

29.9 T

-0.3

-0.2

-0.1
0.0
δf /γN or Hint (T)

0.1

0.2

FIG. 2 (color online). The upper blue lines show the 11 B NMR
spectra at 27.5 T (inside the 1=8 plateau) and at 28.7 and 29.9 T(outside the plateau) obtained at T  0:19 K. The lower redlines show the distribution of the internal field Hint (see the text).
The arrows in the lowest panel show an example of the quadrupole split three peaks deconvoluted into a single peak. The global
similarity of all three spectra confirms the presence of spin
superlattice above the 1=8 plateau.

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range 26.8–28.2 T [4] (data not shown). The distinct
spectral shapes at 28.7 and 29.9 T indicate that these fields
are outside the plateau. Note that the 11 B NMR spectrum isdetermined by the distribution of the internal magnetic
field Hint at nuclear sites, produced by the field-induced
spin density, as well as by the quadrupole splitting Q for
the spin 3=2 nuclei
fi =N  Hint  iQ =N ;i  1; 0; 1:(1)

The spectrum thus represents convolution of the distribution function of Hint with the sum of three  functions
separated by Q =N . One can recognize the threefold
periodic structure for all the spectra shown in Fig. 2 (anexample is indicated by the arrows in the lowest panel)with Q  1:25 MHz. This value is the same as obtained in
low fields [26], indicating that magnetic field does not
induce strong lattice distortion. The distribution of Hint
can then be obtained by deconvoluting the spectra using
the inverse Fourier transform (the lower red lines in Fig. 2).
The distribution of Hint at 27.5 T agrees with the previous results obtained in a more primitive way [21]. It was
shown in Ref. [21] that the many sharp peaks distributed
over a wide range of Hint are well reproduced by the spin
density distribution of the Shastry-Sutherland model calculated by exact diagonalization [4,27], which predicted asuperlattice of highly polarized triplets surrounded byoscillating spin density shown in Fig. 1. In particular, theleftmost peak with the largest absolute value of the internal
field, Hint  0:2 T, is assigned to the B site nearest to theCu site bearing the largest magnetization (site 1 in Fig. 1)[21]. Such distribution of Hint disappears completely when
the field is decreased below 26.5 T, which is the lower
boundary of the 1=8 plateau [4]. In particular, there is noother ordered phase corresponding to a smaller fraction ofmagnetization such as 1=9, in contradiction to Ref. [28].
The situation is very different when the field is increased
above the plateau. The overall width of the distribution ofHint in Fig. 2 remains nearly unchanged. This is the direct
evidence that highly polarized triplets are still present in
the high field phase even though the magnetization deviates from a commensurate fraction. However, unlike in the
1=8 plateau, the spectral shape changes in the high field
phase, indicating some evolution of the spin structure.
We now examine how the spectrum changes in moredetail. Figure 3 shows the low-frequency part of the spectrum corresponding to Hint  0:13 T with changing fieldor temperature: (a) With increasing field at T  0:31 K,
the two most negative peaks get split above Hext  28:4 T,
marking the upper boundary of the 1=8 plateau. The presence of the split and unsplit peaks at 28.3 T indicatescoexistence of the two phases, pointing to a first-ordertransition. The transition field remains the same at T 
0:12 K (data not shown). (b) With decreasing temperature
at Hext  28:7 T, i.e., slightly above the plateau boundary,
one of the split peaks disappears suddenly between 0.19
and 0.16 K, suggesting another phase boundary. (c) With

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(b)

T=0.31 K

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PHYSICAL REVIEW LETTERS

PRL 101, 037202 (2008)1.0

H=28.7 T

0.8
0.44 K

28.2 T

0.19 K

Uniform
T (K)27.9 T

28.3 T
0.16 K

28.4 T
28.7 T(c)

0.06 K
H=28.7 T(d)0.59 K

0.38 K
-0.14

I1

1/4

I2

1/8
26

28

32

30

34

36

H (T)

0.60 K

0.44 K-0.24 -0.22 -0.20 -0.18 -0.16
Hint (T)0.0

H=29.9 T

0.63 K

0.50 K

0.4
0.2

intensity / 2

0.65 K

0.56 K

0.6-0.24 -0.22 -0.20 -0.18 -0.16 -0.14
Hint (T)FIG. 3. Variation of the low-frequency part of the 11 B NMR
spectra across the phase boundaries. See the text for details.

increasing temperature at Hext  28:7 T, the intensity rapidly decreases and the peaks completely disappear at Tc 
0:60 K (zero intensity data are not shown), marking thetransition to the paramagnetic uniform phase. Indeed, at
higher temperatures only three quadrupole split NMR lines
are observed corresponding to a small uniform internal
field of 0:03 T. (d) With increasing field, Tc becomes
higher: Tc  0:66 K at Hext  29:9 T and Tc  0:71 K at
Hext  30:9 T (data not shown). We notice that while theintensity of the peaks is reduced near Tc , their positions do
not change significantly. This means that the amplitude of
the spin density modulation is only slightly reduced but thevolume fraction of the ordered phase vanishes towards Tc ,indicating again a first-order transition.
The phase boundaries thus determined are plotted inFig. 4, together with the previous NMR data on the transition between the 1=8 plateau and the uniform phase [21].
Our H-T phase diagram is in good agreement with therecent magnetization and torque measurements in steadymagnetic fields [29], which revealed an abrupt change
without hysteresis upon entering into the 1=8 plateau anda jump with hysteresis upon leaving the 1=8 plateau. In
addition, another first-order transition with hysteresis near
29.5 T was observed as shown in Fig. 4, which was only
weakly detected by NMR (data not shown). [Implicitly,
this is clear from the difference between the spectra ofFigs. 3(c) and 3(d)]. Thus there are at least two distinct
intermediate phases (I1 and I2 in Fig. 4) between the 1=8
and the 1=4 plateaus. Our Tc data extrapolate smoothly tothe peak temperatures of the specific heat at higher fields[30] also shown in Fig. 4. From our NMR results it is now
clear that the primary order parameter is the superlattice
modulation of the longitudinal magnetization hSz i. In theintermediate phases, Tc increases linearly with field. Suchbehavior is distinct from the domelike variation in a typical

FIG. 4. The phase diagram constructed from the present (solid
circles) and the previous (solid triangles [21]) NMR results andthe specific heat data (open triangles [30]). The bars indicate thewidth of the transition. The transitions near 29.5 T detected by
the torque measurements [29] are shown by open circles (increasing field) and open squares (decreasing field). The two data
points at the same temperature represent the width of thetransition. The lines and the shades are guides to the eye.

field-induced Bose condensed phase [6]. The transition atTc occurs in a narrow temperature window (  20 mK) in
contrast to the broad width (more than 100 mK) observed
for the transition between the 1=8 plateau and the uniform
phase [21].
What is the nature of the intermediate phases ? Our main
conclusion is that in both I1 and I2 phases triplet dimers
with large static magnetization are still present; i.e., a spin
superlattice persists. The similarity of the internal field
distribution for different fields in Fig. 2 suggests that the
spin structure is locally similar to the one in the 1=8
plateau. In particular, the triple-dimer structure of thetriplet unit consisting of the central dimer with parallel
moments and the two neighboring orthogonal dimers withstaggered moments (the ellipsoid in Fig. 1) should bepreserved.
The clear line splitting at the transition from the 1=8
plateau to the I1 phase [Fig. 3(a)] suggests additional
symmetry breaking, as expect for a supersolid phase.
Indeed, the kinetic energy of triplets at high fields isenhanced due to the correlated hopping process in the
Shastry-Sutherland model [22], and this should promote
the formation of a supersolid phase [16]. However, as
mentioned earlier, the DM interaction and the staggered
g tensor in SrCu2 BO3 2 break the continuous spin-rotation
symmetry in the ab plane, which is a prerequisite for a truesupersolid transition. Nevertheless, it is possible that thetransition causes breaking of some remaining discrete
symmetries. The anisotropic interactions and the resultant
transverse staggered magnetization [23] obey all the discrete symmetries of the crystal including the C2 rotation
around the c axis. We note that the spin superstructure of
the 1=8 plateau shown in Fig. 1 is also invariant under theC2 rotation at the center of the triplet dimer, giving thesame value of Hint for the two B nuclei nearest to the Cu atsite 1. However, if a new long-range order of the transverse
magnetization breaking the C2 symmetry occurs in the I1037202-3

PRL 101, 037202 (2008)

PHYSICAL REVIEW LETTERS

phase, the NMR line from these B sites should split asobserved in Fig. 3(a) [31]. This would be analogous to asupersolid phase since the diagonal long-range order (superlattice of hSz i) coexists with some kind of off-diagonal
long-range order (transverse staggered magnetization).
In the I2 phase, the spectrum gets broadened and somewhat featureless, suggesting increased disorder or an incommensurate structure. We note that the stripe structure
predicted for the 1=4 plateau can be obtained by doping
triple-dimer triplet units into the vacant sites in the rhomboid cell of the 1=8 plateau (Fig. 1). This will maintainsimilar local spin structure but introduce certain disorder if
the doped triplets localize randomly. The same process ispossible by starting from the square cell of the 1=8 plateau,
which is obtained by shifting one of the two stripes of therhomboid cell by half of its period. Since both supercells
are energetically nearly degenerate [27], they may alternate in the I1 phase, doubling the unit cell. This could be
another explanation for the line splitting at the upperboundary of the 1=8 plateau and may lead to an incommensurate structure at higher doping.
Finally, we mention possible relevance of the theory
describing effects of the DM interaction on the frustrated
spin ladders exhibiting the half-magnetization plateau [32].
It was found that the DM interaction, if it competes with
the interdimer exchange, stabilizes the superlattice modulation of hSz i far beyond the original plateau region in thephase diagram, similar to what occurs in SrCu2 BO3 2 . In
this model, however, the DM interaction also transforms
the phase transitions separating the plateau and the neighboring phases into a crossover, in contrast to our observation in SrCu2 BO3 2 . This is because the DM interaction
mixes the singlet and the triplet states; hence, hSz i nolonger represents the conserved boson density. To clarify
the effects of the DM interaction in the more complicated
Shastry-Sutherland model would be a future theoreticalchallenge.
In conclusion, our NMR results in SrCu2 BO3 2 demonstrate that a superlattice of highly magnetized triplet
dimers persists above the 1=8 magnetization plateau,
where the magnetization is no longer constant. Although
the presence of the DM interaction prevents the formationof a genuine supersolid phase, the NMR line splitting at theupper boundary of the 1=8 plateau suggests a new orderedphase of different symmetry. The magnetic phase diagram
between the 1=8 and the 1=4 plateau is much more complex than expected and requires further experimental andtheoretical work.
We are grateful for stimulating discussions with F. Mila,
S. Miyahara, O. Tchernyshyov, and C. D. Batista on theoretical aspects and with F. Lévy and I. Sheikin on themagnetization and torque data. The work was supported
by Grant-in-Aids for Scientific Research on Priority Areas
No. 16076204 and No. 19052004 from the MEXT Japan

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18 JULY 2008and by the French ANR Grant No. 06-BLAN-0111.

*Present address: Hitachi Central Research Laboratory,
Kokubunji, Tokyo 185-8601, Japan.
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