Journal of the Physical Society of Japan
Vol. 73, No. 12, December, 2004, pp. 3435–3438#2004 The Physical Society of Japan

Observation of Bose–Einstein Condensation of Triplonsin Quasi 1D Spin-Gap System Pb2 V3 O9
T. W AKI, Y. M ORIMOTO, C. M ICHIOKA, M. K ATO, H. K AGEYAMA, K. Y OSHIMURA,
S. N AKATSUJI1 , O. SAKAI1 , Y. M AENO1;2 , H. M ITAMURA3 and T. G OTO3
1

Department of Chemistry, Graduate School of Science, Kyoto University, Kyoto 606-8502
Department of Physics and Astronomy, Graduate School of Science, Kyoto University, Kyoto 606-8502
2
Kyoto University International Innovation Center, Kyoto 606-8502
3
Institute for Solid State Physics, The University of Tokyo, Kashiwa, Chiba 277-8581(Received June 28, 2004)

We observed a ﬁeld-induced magnetic ordering in the quasi one-dimensional antiferromagnetic
alternating chain system Pb2 V3 O9 for the ﬁrst time. The characteristic temperature dependence of themagnetic susceptibility below TN and the power law behavior of phase boundary between paramagnetic
and ordered phases in the Pb2 V3 O9 system can be comprehended in terms of Bose–Einstein condensationof triplons, as in the case of TlCuCl3 [Phys. Rev. Lett. 84 (2000) 5868]. Our results indicate that BEC of
triplons can be observed in the present strongly-coupled-dimer system, while TlCuCl3 is a weak interdimer coupling system.
KEYWORDS: Bose–Einstein condensation of triplons, field-induced magnetic ordering, alternating chain, spin-gap,
Pb2 V3 O9
DOI: 10.1143/JPSJ.73.3435

1.

Introduction

The Bose–Einstein condensation (BEC) is one of the most
attractive topics in modern physics. The BEC of dilute alkali
atoms has been achieved according to the development of
the laser cooling technique.1,2) For solid state physics, it has
been indicated that ﬁeld-induced triplets (triplons)3) in a
dimer-gap system behave as quasiparticles and can exhibit
the BEC.4) These triplons can be regarded as hard corebosons. When the oﬀ-diagonal mixing between adjacentdimer spins exists, the triplons can move and behave as abose gas in the system. As temperature decreases, thetriplons undergo the BEC at a transition temperature TN . The
triplon BEC is experimentally observed as the antiferromagnetic long range ordering of the transverse spincomponents, perpendicular to an applied magnetic ﬁeld.
Some characteristic behaviors around TN are predicted in the
triplon BEC theory. One of the features is the minimum ofmagnetization at TN . The magnetization is in proportion to
the number of triplons, which are thermally excited andcondensed below TN . The thermally excited triplons decrease as decreasing temperature while the condensed ones
increase below TN . Thus the condensation of triplons gives
the minimum of magnetization at TN . Another feature is
power law behavior of the phase boundary between the BEC
and normal states with respect to temperature in themagnetic ﬁeld phase diagram. The phase boundary can be
expressed by the following power law:
ðg=2Þ½Hc ðTÞ  Hg  / T  ;ð1Þ

where Hg is the critical ﬁeld at T ¼ 0 K, and  the criticalexponent. Although this theory can explain the experimental
data of ﬁeld-induced magnetic ordering in TlCuCl3 qualitatively, some disagreements between the theory andexperimental data are found. One of the problems is thevalue of the critical exponent of the phase boundary. AE-mail: twac@kuchem.kyoto-u.ac.jp

Hartree–Fock approximation gives  ¼ 1:5,4) while thecritical exponent is estimated experimentally to be  ¼ 2:2
for TlCuCl3 .4) The other is a discontinuous jump of themagnetization at TN , predicted by a Hartree–Fock description, which has not been observed experimentally.
The BEC of triplons is expected to be observed in many
isotropic spin-gap systems. Therefore, many spin-gap systems must be investigated from the aspect of the BEC andthe critical exponent of the phase boundary should be
decided experimentally. In this paper we present a possible
perspective of the BEC of triplons in one of quasi onedimensional (1D) systems, Pb2 V3 O9 .
In this compound, there are three kinds of vanadium ionsin the unit cell. Two of them are the nonmagnetic V5þ ions,located in VO4 tetrahedrons, and the rest one is V4þ ion with
S ¼ 1=2 in the VO6 octahedron. Each VO6 octahedron links
in the corner sharing conﬁguration.5) As shown in Fig. 1,
V4þ spins are located in two sites alternately and construct
S ¼ 1=2 alternating spin chain. The Hamiltonian for 1D
alternating Heisenberg chain is given by
N=2
X
H¼
ðJ1 S2i1  S2i þ J2 S2i  S2iþ1 Þ;
ð2Þ
i¼1

where N is the number of spins, J1 the exchange integral of
the nearest neighbor interaction, and J2 that of the next
nearest neighbor interaction. From the numerical results, theground state of this model is of spin-singlet when J2 6¼ 0.6,7)
The temperature dependence of magnetic susceptibility  ofPb2 V3 O9 shows a peak around 20 K, and decreases markedly
toward zero as temperature decreases, indicating spin-singlet
ground state. The behavior can be well analyzed with S ¼
1=2 1D antiferromagnetic alternating Heisenberg chain(1DAFAHC) model equation.6,7)
2.

Experiments

A polycrystalline sample of Pb2 V3 O9 was prepared by theheat treatment of the mixing powders of PbO, V2 O3 andV2 O5 in the molar ratio of 8 : 1 : 5 in a sealed evacuated

3435

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J. Phys. Soc. Jpn., Vol. 73, No. 12, December, 2004

T. WAKI et al.

10
Pb2V3O9

4.071Å

3.305Å

χ (10-3emu/mol)

8

4
2
0
0

[101]

Fig. 1. Schematic V4þ chain in Pb2 V3 O9 . The chain is constructed with
VO6 octahedra linking in the coner sharing conﬁguration along ½101
direction. There are two kinds of V4þ –V4þ distance, forming S ¼ 1=2alternating chain.

silica tube up to 923 K for 2 days. The obtained sample wasfound to be in a single phase by powder X-ray diﬀraction
measurement. The temperature dependence of the magnetic
susceptibility was measured by the SQUID magnetometer
under several magnetic ﬁelds up to 7 T. The high ﬁeld
magnetization was measured at 1.5 K in pulsed magnetic
ﬁeld up to 46 T with a pulse duration of 20 ms. The speciﬁcheat was measured by relaxation method under magnetic
ﬁeld up to 7 T with temperature down to 0.35 K.
3.

Results and Discussion

Figure 2 shows the temperature dependence of  in
Pb2 V3 O9 measured under 1 T. Above 100 K,  follows themodiﬁed Curie–Weiss law, ðTÞ ¼ 0 þ C=ðT  Þ, where0 is a temperature-independent susceptibility, C the Curie
constant and  the Weiss temperature. The obtained
parameters were found to be 0 ¼ 6:7  105 emu/mol,
C ¼ 0:368 emu/(K mol), which corresponding to Peff ¼
1:71 B agreeing with S ¼ 1=2, and  ¼ 12:3 K. As
temperature decreases,  shows maximum around 20 K inagreement with 1D magnetic character. Below the temperature at which  shows the maximum,  decreases markedly.
We ﬁtted the data between 8 and 300 K using the 1DAFAHC
model function with S ¼ 1=2.8,9) Our best-ﬁtted values ofparameters were J1 =kB ¼ 29:0 K and  ¼ 0:65, where  isan alternation parameter deﬁned as  ¼ J2 =J1 . The value of ¼ 0:65 indicates that dimers are strongly couple eachother in this system. These values are similar to thosereported previously as J1 =kB ¼ 29:2 K, J2 =kB ¼ 19:3 K,
 ¼ 0:66.10) If we assume 1DAFAHC model without theother small interactions, the gap can be estimated ascal =kB ¼ 15 K.7) The extrapolation of the ﬁtting line below

J1 = -29.0 K
α = 0.65

6

Data
Alternating chain fitting
40

80

120

T (K)
Fig. 2. Temperature dependence of the magnetic susceptibility of Pb2 V3 O9 .
Open circles represent raw data, the solid line the ﬁtting results using thealternating chain model with J1 ¼ 29:0 K and  ¼ 0:65.

8 K deviates downward from the experimental data, because
the used function can be available above T ¼ 0:25J1 =kB .8,9)
The fact that  approaches zero with decreasing temperature indicates the spin-singlet ground state of this compound. In addition to this fact, the direct proof of the spinsinglet ground state was obtained by the magnetization
measurement at 1.52 K using pulsed high-magnetic ﬁelds asshown in Fig. 3(a). With increasing magnetic ﬁeld, themagnetization is ﬁrst suppressed, and then, the discontinuous
jump of dM=dH is observed at 5.2 T. This behavior
obviously suggests that the ground state of this compound
is of a gapped spin-singlet. If we assume that 1.52 K isenough low for the estimation of the spin-gap, we canrecognize the jump of dM=dH as the lowest excitation
between singlet and triplet states. In this case, the gap energy
was estimated to be =kB  7 K from Hc1 ¼ 5:2 T withusing the most simpliﬁed relation of the spin-gap in themagnetic ﬁeld, ðHÞ ¼   gB H. This value is diﬀerent
from cal =kB ¼ 15 K estimated using the ﬁtting parametersof the alternating model.7) This may be ascribed to theeﬀects of the interchain interaction J3D . On the other hand,
the magnetic ﬁeld 5.2 T can be regarded as the critical point
between the paramagnetic and antiferromagnetic states at
1.52 K. We put this point on the H–T phase diagram shownin Fig. 6. The magnetization increases gradually over thewhole external ﬁeld range and saturates to about 1 B =f.u.
above 40 T, which is consistent with the S ¼ 1=2 saturation
moment. Furthermore, there is anomaly in dM=dH at 36.8 T.
This ﬁeld corresponds to the critical ﬁeld Hc2 at 0 K.6) In
Fig. 3(b), we compare the experimental magnetization curve
and that calculated by Bonner et al.6) Hc2 of 1DAFAHC
model can be written as Hc2 ¼ jJ1 þ J2 j=gB .6) The value of
Hc2 was calculated to be 35.6 T with the values of J1 and J2
estimated above. The result well agrees with the experiment,
while calculated Hc1 ð0Þ (¼ cal =gB ¼ 11:1 T) is diﬀerent

J. Phys. Soc. Jpn., Vol. 73, No. 12, December, 2004

T. WAKI et al.

(a)

(a)
0.5

0.12
T = 1.52K

0.10
0.08

5.2 T

0.5
Pb2V3O9
0T
7T

0.4

2.52K

TN

0.3
0.2

0.3

2.32K
2.11K
1.90K

0.2
1.75 K
1.56 K

0.1

0.06

0.1

0.04

0.0
0

1.35 K
1.15 K
0.95 K

0.4
0.2

2

10

20
H (T)

30

40

0.00

(b)
1.2

4

6

8

6

7

B (T)

5.0

0.8

4.5

Pb2V3O9

1.0

0.8
0.7
0.2

0.9

α=0.0

0.4
0.6

10

20

30

χ (10-3emu/mol)

0.6

0.0
0

5

Hc2

T = 1.52K

0.2

4

Fig. 4. (a) Temperature dependence of speciﬁc heat of Pb2 V3 O9
measured under magnetic ﬁeld of 0 T (solid circle) and 7 T (open circle).
Increase of speciﬁc heat at low temperatures measured under 7 T is due tothe nuclear spins under the magnetic ﬁeld. (b) Field dependence ofspeciﬁc heat at several temperatures down to 0.35 K. There are cusp-likeanomalies at the ﬁelds which we determined as Hc represented by allows.

Pb2V3O9

1.0

0.4

0.0

10

T (K)

0.02

0.0
0

M (µB/f. u.)

Pb2V3O9
0.4
C (J/mol K)

0.6

0.14
C/T (J/mol K2)

M (µB/f.u.)

0.8

36.8 TPb2V3O9

dM/dH

1.0

(b)

0.6

1.2

3437

3.5
3.0

40

H (T)
Fig. 3. (a) Magnetization curve of Pb2 V3 O9 at 1.52 K. The left verticalaxis indicates the magnetization, and the right one dM=dH. The criticalﬁeld Hc1 ¼ 5:2 T at 1.52 K, corresponding to the spin-gap energy, isdetermined from the anomaly of the dM=dH curve. The magnetization
curve saturates to 1 B /f.u. at about 40 T. (b) The calculated magnetization curves for several fraction value  in the alternating model
together with the observed one.6) Hc1 is the ﬁeld where the curveintersects the horizontal axis of the graph and Hc2 is the ﬁeld where thecurve saturates. The shape of the magnetization curve agrees qualitatively
with the calculated curves. The diﬀerence in detail may come from theeﬀects of J3D left out of account in 1DAFAHC model.

from the experimental result.
Figure 4(a) is the temperature dependence of the speciﬁcheat C divided by T under 0 and 7 T. There are no tendenciesof transition down to 0.35 K in C=T measured under H ¼
0 T, consistent with the fact that the ground state of thiscompound is of spin-singlet. On the other hand, C=T under
7 T shows an obvious peak at 2.5 K, indicating an antiferromagnetic transition at the temperature, below which an
antiferromagnetic long range ordering is realized in the spin
triplet state. We determined the peak temperature of C=T as
the Néel temperature TN . There is a steep increase of C=T
below 0.5 K measured under 7 T due to the contribution of

TN

4.0

2.5
2.0
0

7.0T
6.8T
6.6T
6.4T
6.2T
6.0T
5.8T
5.6T
5.4T
5.2T
5.0T
1

2

3

4

5

6

T (K)
Fig. 5. Temperature dependence of  measured under several ﬁelds.
Above 5.6 T, a minimum of M=H was observed similar to the case of
TlCuCl3 .4)

the nuclear spins under external ﬁelds. Namely, we found aﬁeld-induced magnetic phase transition in Pb2 V3 O9 . We alsomeasured the magnetic ﬁeld dependence of the speciﬁc heatat several temperatures down to 0.35 K. The cusp-likeanomaly indicating long range ordering was also observed
and we determine the ﬁeld where C takes maximum as Hc[Fig. 4(b)].
Figure 5 shows the temperature dependence of  measured under several external ﬁelds down to 1.8 K. We
observed a minimum of  in external magnetic ﬁelds above
5.6 T. The temperature at which  shows the minimum
measured at 7 T almost coincides with the TN determined by
the speciﬁc heat measurement under 7 T. Therefore, we canconsider that the anomalous behavior of  is attributed to theﬁeld induced antiferromagnetic transition. As shown by3438

J. Phys. Soc. Jpn., Vol. 73, No. 12, December, 2004

Pb2V3O9

12

Condensed State10

χ minimumSpecific HeatHigh Field

H (T)

14

8

T. WAKI et al.

6
4
2
0
0

φ = 1.9 (best)
φ = 1.5
φ = 3.0

Gapped State
1

2

3

4

5

T (K)
Fig. 6. H–T phase diagram of Pb2 V3 O9 . The ﬁtting curve of the boundary
using eq. (1) with the value  ¼ 1:9 and Hc ð0Þ ¼ 4:1 T is presented bysolid line together with estimated curves with  ¼ 1:5 (dotted line) and ¼ 3:0 (dashed two dots) for references.

arrows in the ﬁgure, TN increases gradually as the ﬁeld
increases. The characteristic convex temperature dependence of  below TN was observed similar to the case of
TlCuCl3 . One cannot expect convex behavior of  without
the consideration of the condensation of triplons. For
example, the value of  is known to be invariant below TN
on the basis of the mean ﬁeld theory.11) Hence, the minimumof  can be comprehended in terms of the BEC state of
triplons.4) Therefore, the ﬁeld induced magnetic ordering
state in this system is thought to be just the BEC state of
triplons.
In Fig. 6, we show the phase diagram, in which the phaseboundary between paramagnetic and the BEC regions wasestimated by the speciﬁc heat and magnetization measurements. From the least-square-ﬁtting by eq. (1), we obtained
 ¼ 1:9 and Hg ¼ 4:1 T in the case of Pb2 V3 O9 . The powerlaw behavior of the phase boundary is another feature oftheory of triplon BEC: the critical exponent is predicted tobe  ¼ 1:5 by using Hartree–Fock approximation withsimple quadratic dispersion.4) On the other hand,  ¼ 2:2
has been reported experimentally for TlCuCl3 .4) The criticalexponent  ¼ 1:9 we obtained for Pb2 V3 O9 also deviatesfrom the theoretical prediction. Recently, some theories havebeen proposed in concerned with the critical exponent.
Misguich and Oshikawa reobtained  ¼ 1:5 with selfconsistent Hatree–Fock–Popov calculations.12) The regionof  ¼ 1:5 is limited to very low temperature region. Taking
into account of a realistic dispersion of triplons of TlCuCl3 ,they also reproduce the experimental phase boundary. The
actual dispersion of triplons of Pb2 V3 O9 remains to bestudied. On the other hand, Crisan et al. have obtained thevalue of  ¼ 2, which is in a better agreement with  ¼ 2:2
on TlCuCl3 system as well as the present case of  ¼ 1:9,
including ﬂuctuation eﬀect by the renormalization groupmethod.13) We need further experimental and theoreticalstudies of Pb2 V3 O9 and in other isotropic spin-gap systems
for understanding this phenomenon.
In summary, we observed a ﬁeld-induced magnetic
ordering as well as the gapped spin-singlet ground state in
the 1D antiferromagnetic spin-chain compound Pb2 V3 O9
with S ¼ 1=2. Furthermore, our results suggest that the BECof triplons is observable not only in weak coupled dimer
systems but also in strongly coupled dimer system. From the
H–T phase diagram, critical exponent seems to depend on
the dimensionality (or dispersion) of the system as suggested
from recent works. Further intensive experimental andtheoretical works are needed for fully understanding of thisphenomenon.
Acknowledgement
We thank N. Tsujii for suggesting high ﬁeld magnetization measurements and Y. Itoh for helpful discussion on
this experiment. This study was supported by a Grant-in-Aid
on priority area ‘‘Novel Quantum Phenomena in Transition
Metal Oxides’’, from the Ministry of Education, Culture,
Sports, Science and Technology (12046241), and also
supported by Grants-in-Aid Scientiﬁc Research of the Japan
Society for the Promotion of Science (12440195 and
12874038).

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