Journal of the Physical Society of Japan
Vol. 74, No. 12, December, 2005, pp. 3155–3158#2005 The Physical Society of JapanLETTERS

Anomalous Magnetization of Two-Dimensional S ¼ 1=2
Frustrated Square-Lattice Antiferromagnet (CuCl)LaNb2 O7
H. K AGEYAMA, J. YASUDA, T. K ITANO, K. TOTSUKA1 , Y. N ARUMI2 y,
M. H AGIWARA2 , K. K INDO2;3 , Y. BABA, N. OBA, Y. A JIRO and K. YOSHIMURA
Department of Chemistry, Graduate School of Science, Kyoto University, Kyoto 606-8502
1
Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502
2
Research Center for Materials Science at Extreme Conditions, Osaka University, Toyonaka, Osaka 560-8531
3
Division of Physics in Extreme Condition, Institute for Solid State Physics,
University of Tokyo, Kashiwa, Chiba 277-8581(Received July 12, 2005; accepted September 22, 2005)

High-ﬁeld magnetization measurements have been performed up to 56 T for a two-dimensional
S ¼ 1=2 frustrated square-lattice antiferromagnet (CuCl)LaNb2 O7 , a recently discovered spin gap system(=kB ¼ 26:7 K). It is found that the spin gap closes at a surprisingly low ﬁeld Hc1 ¼ 10:3 T comparedwith that expected from the zero-ﬁeld spin gap (=gB ¼ 18:4 T). For H > Hc1 , the magnetization
exhibits a linear increase without any trace of anomalies such as fractional plateaus until it saturates at
Hc2 ¼ 30:1 T. This means that the gapless phase, where the ﬁeld-induced magnetic ordering is expected
to occur at low temperatures, is stable over a wide ﬁeld region. These results suggest strong correlationsof triplet excitations in the layer and the proximity of the spin-liquid phase to the magnetically orderedphase.
KEYWORDS: magnetization, spin liquid, square lattice, J1 –J2 model, ion-exchange reaction, bound state,(CuCl)LaNb2 O7
DOI: 10.1143/JPSJ.74.3155

A soft chemical approach carried out under relatively mild
conditions allows access to metastable phases with novel
physical and chemical features, providing a diverse range ofapplications that include catalysis,1) ionic conduction2) andexotic superconductivity.3) A number of chemical reactions
involving inorganic solids such as dehydration, oxidation,
reduction, and ion exchange are known to proceed topochemically to yield well-deﬁned products that bear adeﬁnite structural relationship with the mother compounds.
In particular, layered perovskites, intergrowths of perovskite
and other structures, have been the target of many lowtemperature topochemical manipulations.4) The A-site ordered compounds A0 ½An1 Bn O3nþ1  (A0 ¼ alkali metals, H,
Ag, Tl, etc.; A ¼ Ca, Sr, Ba, La; B ¼ Ti, Nb, Ta), termed theDion-Jacobson series, have one interlayer cation per formulaunit.5,6) Most of the studies on these compounds havefocused on the A0 -site ion-exchange between monovalent
single ions. It has been realized, however, that insertion of
metal-anion arrays within perovskite hosts to yield, e.g.,(CuCl)LaNb2 O7 and (CuCl)Sr2 Nb3 O10 , is also possible.7–10)
It has been recently demonstrated that the two-dimensional (2D) S ¼ 1=2 frustrated square-lattice system(CuCl)LaNb2 O7 , corresponding to A0 ¼ ½CuCl, A ¼ La,
B ¼ Nb and n ¼ 2, achieves a spin-liquid state with a ﬁnite
gap of =kB ¼ 26:7 K to the lowlying excited triplet state.11)
The lattice is of tetragonal symmetry (P4=mmm) with room and c ¼ 11:728 A.
 7)temperature cell constants a ¼ 3:879 A
As shown in Fig. 1, the structure consists of copper-chlorine
planes that are widely separated from each other by double
perovskite slabs. The magnetism of this compound is due to
S ¼ 1=2 Cu2þ ions, which are arranged in the square lattice.
E-mail: kage@kuchem.kyoto-u.ac.jp
Present address: Division of Physics in Extreme Condition, Institute for
Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581.

y

(a)

(b)
J1

J2

c
b

a

Fig. 1. (a) Schematic representation of structure of (CuCl)LaNb2 O7 . The
closed spheres are coppers, the gray spheres are chlorines, the open
spheres are lanthanums, and the octahedra are NbO6 octahedra.7) (b)Copper–chlorine layer viewed along c-axis. The bold broken and solidlines represent, respectively, the J1 and J2 bonds.

The copper ion is octahedrally coordinated, bridging
between apical oxygen atoms from the perovskite layerand surrounded by four chlorines in the CuCl plane. The
CuO2 Cl4 octahedra share corners with the NbO6 octahedraof the perovskite slabs, while they share edges with other
CuO2 Cl4 octahedra in the ab plane.
Our dc magnetic susceptibility measurement clearly
indicates a thermally activated behavior at low temperatures.11) The analysis of the data suggests competinginteractions between the ferromagnetic ﬁrst-nearest-neighbor J1 ð< 0Þ and the antiferromagnetic second-nearestneighbor J2 ð> 0Þ. Inelastic neutron scattering (INS) measurements directly probed the one-triplet excitation at
2.3 meV (corresponding to ) as well as the multitriplet
excitation at 5.0 meV. Most likely as a result of the strong
geometrical frustration, we observed a nearly ﬂat dispersionof the lowlying mode and an oscillatory Q-dependence of its

3155

LETTERS

RbLaNb2 O7 þ CuCl2 ! (CuCl)LaNb2 O7 þ RbCl: ð1Þ
RbLaNb2 O7 was prepared by a high-temperature ceramic
method from Rb2 CO3 (99.99%), La2 O3 (99.99%), andNb2 O5 (99.999%). Stoichiometric quantities of La2 O3 andNb2 O5 with a 25% molar excess of Rb2 CO3 were weighed
and ground thoroughly inside an Ar-ﬁlled glove box(UNIlab2000, MBRAUN), annealed overnight at 850 C
followed by an additional thermal treatment at 1050 C for
24 h. The excess of RbCO3 was added to balance that lost
due to volatilization. To prepare (CuCl)LaNb2 O7 , RbLaNb2 O7 was mixed with a two-fold molar excess of ultradry
CuCl2 (99.99%) and then pressed into pellets inside theglove box. Reactions were carried out in sealed, evacuated(< 103 Torr) Pyrex tubes at 320 C for one week. The ﬁnal
product was washed with water to eliminate the excess
copper chloride and alkali-metal byproduct, and dried at120 C. The high-ﬁeld magnetization measurements wereconducted using a pulse magnet installed at the High
Magnetic Field Laboratory, KYOKUGEN, Osaka University. A powder sample of 100 mg was made to ﬁll acylindrical Teﬂon tube with a diameter of 2.4 mm that wasthen closed by two quartz plates at the top and the bottom.
The sample holder attached at one end of a quartz rod wasinserted into the 4 He cryostat combined with the pulse
magnet.
Figure 2 shows the magnetizations M and the diﬀerential
magnetizations dM=dH at 1.3 K (blue) and 4.2 K (green)plotted against magnetic ﬁeld H, where within the experimental accuracy, no hysteresis was observed upon increasing and decreasing the external ﬁeld. Field-induced phasetransitions are observed at Hc1 ¼ 10:3 T and Hc2 ¼ 30:1 T,
as indicated by the arrows in Fig. 2. Here, the critical ﬁelds
are determined as the inﬂection point in the diﬀerential
magnetization curve at 1.3 K. Reﬂecting the spin-singlet
ground state with a ﬁnite spin gap,11) the initial magnetization slope is very small. The slow increase in themagnetization for H < Hc1 is attributable to a small amount

(a)

Hc2

1. 3 K
Hc1
0

4.2 K
0

dM/dH (arb.units)intensity, indicating an extremely localized nature of theexcited triplets. Accordingly, (CuCl)LaNb2 O7 markedly
resembles the Shastry–Sutherland system SrCu2 (BO3 )2 .12–14)
It is known in SrCu2 (BO3 )2 that fractional magnetization
plateaus exist at 1/8, 1/4, and 1/3 of the saturation
magneization,12,15) and anisotropic interactions cause singlet–triplet mixing and eliminate phase transition at the
expected critical ﬁeld Hc for gap closing.16)
Such novel phenomena in the presence of a magnetic ﬁeld
might also be observed in (CuCl)LaNb2 O7 . This letter deals
with a high-ﬁeld magnetization investigation of the (CuCl)LaNb2 O7 powder sample. Among our motivations are (i) toobserve the ﬁeld-induced transition from the disordered tothe ordered phase, (ii) to evidence the point that the spinsinglet dimers are not magnetically isolated, but stronglyinteracting with each other, (iii) to unambiguously evaluate
exchange constants, which was diﬃcult from the dc susceptibility measurement alone, (iv) to reveal the eﬀect ofmultitriplet bound states under a magnetic ﬁeld and (v) tocheck whether or not fractional magnetization plateaus
appear.
The ion-exchange reaction for the synthesis of (CuCl)LaNb2 O7 is expressed as

H. KAGEYAMA et al.

dM/dH (arb.units)

J. Phys. Soc. Jpn., Vol. 74, No. 12, December, 2005

1.0

(b)
0.8

M / Ms

3156

0.6

1.3 K
4.2 K
1.3 K (corrected)
impurity

0.4
0.2(CuCl)LaNb2O7

0.0
0

10

20

30

40

50

H (T)
Fig. 2. (a) Diﬀerential magnetization curves dM=dH measured at T ¼ 1:3
and 4.2 K. Arrows point to the transition ﬁelds. (b) Magnetization curves
M measured at T ¼ 1:3 K (blue) and T ¼ 4:2 K (green). The broken line
represents the impurity term, given by the Brillouin function for 2.0% of
noninteracting S ¼ 1=2 ions and T ¼ 1:3 K. After subtracting the impurity term from the 1.3 K data, the (corrected) intrinsic magnetization (red)is obtained.

of impurities or lattice defects as also seen in the dc
magnetic susceptibility for T < 5 K.11) Assuming 2.0% of
noninteracting magnetic ions of S ¼ 1=2, the intrinsic
magnetization curve for (CuCl)LaNb2 O7 is obtained (seethe red line in Fig. 2). A slight increase in the magnetization
observed above around 4 T might be spurious due toinappropriate estimation of the impurity/defect term. However, we cannot totally exclude the possibility that such abehavior is an intrinsic property of (CuCl)LaNb2 O7 . The
magnetization curve has several remarkable features. First ofall, the value of the lower critical ﬁeld Hc1 is not what is
expected from . Since the onset of magnetization istriggered by the softening (or the Zeeman splitting) of the
gapped triplet magnon, the lower critical ﬁeld Hc1 should
coincide with =gB . However, the obtained value of Hc1(¼ 10:3 T) is by far much smaller than that expected fromthe known zero-ﬁeld gap (=gB ¼ 18:4 T), where weassume g ¼ 2:17 estimated from ESR experiments.17)
Above Hc1 , the magnetization grows progressively andalmost linearly with increasing magnetic ﬁeld until it reaches
the saturation magnetization Ms at around Hc2 . No indicationof fractional plateaus is seen, in contrast to that observed in
SrCu2 (BO3 )2 12,15) and [Ni2 (medpt)2 (-ox)(-N3 )]ClO4 0.5H2 O.18) The ground state in this ﬁeld region is gapless.
Accordingly, long-range magnetic ordering, described by the
Bose–Einstein condensation (BEC) of magnetic excitations,is expected to take place at low temperatures.19) Here, thetransverse spin components perpendicular to the appliedﬁeld are responsible for the ﬁeld-induced magnetic ordering.
The second important point to be stressed is that the gaplessphase is stable over a wide ﬁeld range, i.e., Hc2  Hc1 ¼
19:8 T. This fact suggests the existence of sizable andrepulsive interactions among magnetic excitations, because

LETTERS

otherwise the magnetization would abruptly jump to saturateat Hc1 (¼ Hc2 ). The existence of sizable interactions and theextremely localized nature of the triplet excitations observedin the INS experiments11) appear to contradict each other.
However, these facts remind us of what was observed in
SrCu2 (BO3 )2 , where the lowest dimer-triplet excitation has atiny bandwidth13) despite a fairly large coupling between
dimers. Our observation may indicate the existence of fairly
strong geometrically frustrated interactions among S ¼ 1=2
spins which, for some reasons, suppress the particle propagation. We may expect that a system with a small kinetic
energy (or tiny magnon bandwidth) and large repulsive
interactions has several intermediate plateaus in its magnetization curve, since magnetization plateaus can be attributed
to a kind of crystalline states. However, this is not the casefor the present system. This point will be discussed later.
To estimate the values of exchange interactions, we takethe S ¼ 1=2 J1 –J2 model as the simplest theoretical model,
which is suggested by the crystal structure. Knowledge ofthe Weiss constant s ¼ J1 þ J2 and the saturation ﬁeld Hc2
enables us to determine the values of J1 and J2 . The
saturation ﬁeld of the J1 –J2 model is obtained by calculating
the energy of one-magnon excitation in the fully polarizedstate, provided that we have no multimagnon bound states.
In the cases of relevance,20) it is given either by Hc2 ¼
2ðJ1 þ 2J2 Þ=ðgB Þ [collinear anitiferromagnetic (CAF)
phase21) J2 > jJ1 j=2] or by Hc2 ¼ 4J1 =ðgB Þ [Néel antiferromagnetic (NAF) phase J2 < J1 =2]. The observed values ofs and Hc2 yield the following sets: J1 ¼ 2:74 K, J2 ¼
12:34 K for CAF and J1 ¼ 10:97 K, J2 ¼ 1:37 K for NAF.
Unfortunately, these sets fail to explain the existence of aﬁnite spin gap since the J1 –J2 model with J1 and J2 given by
the above parameters is believed to be in an ordered phase.21)(We also checked that the high-temperature part of thesusceptibility data yielded a similar result.) From this, we
may conclude that a simple analysis based on a pure J1 –J2model does not work.
To highlight the peculiarity of the observed magnetization
curve, the experimental data at 1.3 K is compared with themagnetization curve of a simple phenomenological model.
Let us consider a spin cluster which has a singlet and a triplet
separated by a ﬁnite spin gap , where the triplet is regarded
as an excited magnetic particle. If we consider a 2-spincluster and set  ¼ J (= coupling between two S ¼ 1=2
spins), this is simply the isolated dimer model. The M–Hcurve of the isolated dimer model is given by22)
Md ðHÞ ¼

NgB sinhðgB H=kB TÞ; ð2Þ
1 þ expð=kB TÞ þ 2 coshðgB H=kB TÞ

where N denotes the number of S ¼ 1=2 ions. The calculated
magnetization for =kB ¼ 26:7 K and T ¼ 1:3 K, shown by
the dotted line in Fig. 3, exhibits a considerable deviationfrom the experimental one.
In order to reproduce the obtained magnetization curve
having a gentle slope in the gapless region, a fairly large
kinetic energy (or the interdimer coupling J 0 in the dimer
model) should be taken into account. Keeping only two
states jS; Sz i ¼ j0; 0i and j1; 1i of each cluster excitation, thesystem reduces to that of pseudospin-1/2 and we can apply
the theory of Tachiki and Yamada23) to obtain the magnetization curve. The lower and upper critical ﬁelds at T ¼ 0

H. KAGEYAMA et al.

3157

1.0
0.8

M / MsJ. Phys. Soc. Jpn., Vol. 74, No. 12, December, 2005

0.6
0.4
experiment (1.3 K)
isolated dimer (1.3 K)
Tachiki & Yamada (0 K)

0.2
0.0
0

10

20

30

40

50

H (T)
Fig. 3. The solid line is the corrected experimental magnetization at T ¼
1:3 K. The dotted and broken lines are, respectively, the theoreticalmagnetizations based on the isolated dimer model with =kB ¼ 26:7 K,
T ¼ 1:3 K, and g ¼ 2:17, and on the Tachiki–Yamada model at 0 K.23)

depend on the sign of J 0 . One obtains Hc1 ¼ ðJ þ Þ=gB
and Hc2 ¼ J=gB when J 0 < 0, while Hc1 ¼ ðJ  Þ=gB
and Hc2 ¼ ðJ þ 2Þ=gB is obtained when J 0 > 0, where denotes zJ 0 =2 with z being the number of interacting
neighbors. Supposing z ¼ 4 for simplicity, comparison with
our experimental data led to two sets of coupling constants:
J=gB ¼ 30:1 T and J 0 =gB ¼ 9:9 T for J 0 < 0, or
J=gB ¼ 16:9 T and J 0 =gB ¼ 3:3 T for J 0 > 0. Despite
the apparent agreement between the Tachiki–Yamada
theory and our experiment, the theory fails to explain thezero-ﬁeld gap of 2.3 meV observed by INS.11) This is not
surprising since such simple models of gapped magnons
always predict  ¼ gB Hc1 .
Therefore, the discrepancy in the present compoundsuggests the necessity to go beyond this simple assumption
and consider a contribution from other states seriously. One
possibility is that the branch at 2.3 meV observed in INS is
not the lowest one, which might have been missed in INS forsome reasons (e.g., by selection rules). In this case, however,
the experimental magnetic susceptibility cannot be reproduced. Another possibility, which is more plausible, is that
there exists a bound state with a higher spin (e.g., S ¼ 2
quintet); the bound quintet mode, given suﬃcient binding
energy, is able to touch the spin-singlet ground state fasterthan the one-triplet mode does. A relevant behavior can beseen in the ﬁeld-frequency diagram of the ESR spectra of
SrCu2 (BO3 )2 ;24) a simple linear extrapolation of the boundquintet mode to zero energy gives a critical ﬁeld smaller
than that of the one-triplet mode. In this case, however, aDzyaloshinsky–Moriya interaction and a staggered g tensor
cause singlet–triplet mixing and make the spin gap remain
open until the system enters the 1/8 plateau phase.16) For thisreason, (CuCl)LaNb2 O7 possibly provides the ﬁrst experimental example of magnetic ordering driven by a boundstate. It is noted that INS detects in principle the transition
involving S  S0 ¼ 1 and 0. Thus, the bound quintet statefrom the singlet ground state is not observable by INS. In
order to check the validity of this scenario, other spectroscopic measurements such as ESR and Raman scattering arehighly desired. We should address the fact that the quintet
level (2 =kB ¼ 29:9 K), derived from the critical ﬁeld Hc1 ,is slightly above the triplet level so that the ﬁtting of the3158

J. Phys. Soc. Jpn., Vol. 74, No. 12, December, 2005

LETTERS

magnetic susceptibility may not be very successful. However, such an analysis involves a diﬃculty because of thenonnegligible contribution from impurities/defects andaveraged anisotropic factors due to the use of a polycrystalline sample.11) Therefore, for the unambiguous analysis of
the magnetic susceptibility it is important to obtain single
crystals of high purity.
As an alternative scenario, it is considered that the spin–
liquid phase in (CuCl)LaNb2 O7 may be located in thevicinity of the ordered phase. The spin–liquid state would
then be easily and markedly changed by external parameterssuch as magnetic ﬁeld. Here, the geometrical spin frustration
may have a crucial role to play. As suggested in ref. 11, theﬂat dispersion of the triplet excitation in the absence of amagnetic ﬁeld should not be as a result of the spatial
isolation of spin clusters, but of the strong spin frustration
brought about by the geometry. Thus, the triplet excitations
must be potentially correlated strongly to each other. In
other words, without the geometrical spin frustration, thetriplet excitations in (CuCl)LaNb2 O7 would have a dispersion whose bandwidth is comparable to the magnitude of
exchange interactions.
Finally, let us make a brief comment on the absence ofintermediate plateaus in the magnetization, which is distinct
from the case of SrCu2 (BO3 )2 , where 1/8, 1/4, and 1/3
plateaus exist. From the arguments discussed above,(CuCl)LaNb2 O7 is expected to have strong interactions
and a small kinetic energy, which usually favor formation ofplateaus. We consider that correlated hopping processes
proposed25) to explain the unusual magnetic properties of
SrCu2 (BO3 )2 may give some hints to understand this point.
To conclude, we measured the high-ﬁeld magnetization of
the 2D S ¼ 1=2 frustrated square-lattice antiferromagnet(CuCl)LaNb2 O7 . A ﬁeld-induced magnetic phase transition
from the gapfull to the gapless phases has been observed.
The lower critical ﬁeld is signiﬁcantly smaller than that
expected from the zero-ﬁeld spin gap and the Zeeman
splitting of the triplet state. To explain this anomaly, thecrucial role of the bound states and the quantum criticality
are suggested. Since we are in possession of the series ofðCuXÞAn1 Bn O3nþ1 (n ¼ 2, 3; X ¼ Cl, Br),7–10) the study ofthis family will not only help to understand the magnetism of
the title compound but also oﬀer a wide variety of new
phenomena occurring in the S ¼ 1=2 square lattice.
This work has been supported by Grants-in-Aid for Young
Scientists (A) (No. 17684018) and Science Research on
Priority Area (No. 16076210) from the Ministry of Education, Culture, Sports, Science and Technology of Japan, andby the 21st Century COE Program, Kyoto University
Alliance for Chemistry.

H. KAGEYAMA et al.

1) K. Domen, J. Yoshihama, T. Sekine, A. Tanaka and T. Onishi: Catal.
Lett. 4 (1990) 1970.
2) S. H. Byeon and K. Park: J. Solid State Chem. 121 (1996) 430.
3) K. Takada, H. Sakurai, E. Takayama-Muromachi, F. Izumi, R. A.
Dilanian and T. Sasaki: Nature 422 (2003) 53.
4) R. E. Schaak and T. E. Mallouk: Chem. Mater. 14 (2002) 1455.
5) M. Dion, M. Ganne and M. Tournoux: Mater. Res. Bull. 16 (1981)
1429.
6) A. J. Jacobson, J. W. Johnson and J. T. Lewandowski: Inorg. Chem. 24(1985) 3727.
7) T. A. Kodenkandath, J. Lalena, W. L. Zhou, E. Carpenter, C.
Sangregorio, A. Falster, W. Simmons, C. O’Connor and J. B. Wiley:
J. Am. Chem. Soc. 121 (1999) 10743.
8) L. Viciu, G. Caruntu, N. Royant, J. Koenig, W. L. Zhou, T. A.
Kodenkandath and J. B. Wiley: Inorg. Chem. 41 (2002) 3385.
9) H. Kageyama, L. Viciu, G. Caruntu, Y. Ueda and J. B. Wiley: J. Phys.:
Condens. Matter 16 (2004) S590.
10) H. Kageyama, T. Kitano, R. Nakanishi, J. Yasuda, N. Oba, Y. Baba,
M. Nishi, Y. Ueda, Y. Ajiro and K. Yoshimura: to be published inProg. Theor. Phys. Suppl.
11) H. Kageyama, T. Kitano, N. Oba, M. Nishi, S. Nagai, K. Hirota,
L. Viciu, J. B. Wiley, J. Yasuda, Y. Baba, Y. Ajiro and K. Yoshimura:
J. Phys. Soc. Jpn. 74 (2005) 1702.
12) H. Kageyama, K. Yoshimura, R. Stern, N. V. Mushnikov, K. Onizuka,
M. Kato, K. Kosuge, C. P. Slichter, T. Goto and Y. Ueda: Phys. Rev.
Lett. 82 (1999) 3168.
13) H. Kageyama, M. Nishi, N. Aso, K. Onizuka, T. Yosihama, K. Nukui,
K. Kodama, K. Kakurai and Y. Ueda: Phys. Rev. Lett. 84 (2000) 5876.
14) S. Miyahara and K. Ueda: Phys. Rev. Lett. 82 (1999) 3701.
15) K. Onizuka, H. Kageyama, Y. Narumi, K. Kindo, Y. Ueda and
T. Goto: J. Phys. Soc. Jpn. 69 (2000) 1016.
16) K. Kodama, S. Miyahara, M. Takigawa, M. Horvatic, C. Berthier,F. Mila, H. Kageyama and Y. Ueda: J. Phys.: Condens. Matter 17(2005) L61.
17) K. Nakajima and J. Yamauchi: private communication.
18) Y. Narumi, M. Hagiwara, R. Sato, K. Kindo, H. Nakano and
M. Takahashi: Physica B 246 – 247 (1998) 509.
19) T. Nikuni, M. Oshikawa, A. Oosawa and H. Tanaka: Phys. Rev. Lett.
84 (2000) 5868.
20) The saturation ﬁeld is diﬀerent for the three phases of the classical J1 –J2 model; among them, we can safely exclude the ferromagnetic phasebecause it would contradict the magnetization data. Note that for thequantum case (S ¼ 1=2) a spin gap is expected only in narrow regionsof the above phases (see, e.g., ref. 21 and references cited therein for
more details). More information on the magnetization curve of the J1 –J2 model may be found in S. Gluzman: Phys. Rev. B 50 (1994) 6264;
A. Honecker: Can. J. Phys. 79 (2001) 1557, and references citedtherein.
21) N. Shannon, B. Schmidt, K. Penc and P. Thalmeier: Eur. Phys. J. B 38
(2004) 599.
22) Y. Sasago, M. Hase, K. Uchinokura, M. Tokunaga and N. Miura:
Phys. Rev. B 52 (1995) 3533.
23) M. Tachiki and T. Yamada: J. Phys. Soc. Jpn. 28 (1970) 1413.
24) H. Nojiri, H. Kageyama, Y. Ueda and M. Motokawa: J. Phys. Soc. Jpn.
72 (2003) 3243.
25) K. Totsuka, S. Miyahara and K. Ueda: Phys. Rev. Lett. 86 (2001) 520.