Journal of the Physical Society of Japan
Vol. 72, No. 12, December, 2003, pp. 3243–3253#2003 The Physical Society of JapanESR Study on the Excited State Energy Spectrum of SrCu2 (BO3 )2—A Central Role of Multiple-Triplet Bound States—
Hiroyuki N OJIRI, Hiroshi K AGEYAMA1 , Yutaka U EDA1 and Mitsuhiro M OTOKAWA2
1

Department of Physics, Okayama University, Tsushimanaka 3-1-1, Okayama 700-8530
Institute for Solid State Physics, University of Tokyo, Kashinoha 5-1-5, Kashiwa, Chiba 277-8581
2
Institute for Materials Research, Tohoku University, Katahira 2-1-1, Sendai 980-8577(Received December 19, 2002)

The excited state energy spectrum of a two-dimensional dimer system SrCu2 (BO3 )2 has been
investigated by high ﬁeld ESR. Besides two non-degenerated one-triplet excitations, various types ofmultiple-triplet bound states are identiﬁed: singlet, triplet, and quintet. The intensity of these bound states
is extremely strong because one-triplet process is strongly suppressed in the orthogonal dimer system.
The bound singlet state is found slightly below the energy gap of the one-triplet state, indicating theproximity to a quantum critical point. A band of the two-triplet bound state splits due to distant neighborinteractions. The bound quintet state is the direct evidence of two-triplet object made up of four S ¼ 1=2
spins. The energy of the lowest quintet becomes lower than that of the one-triplet excitation before theclosing of the lowest spin gap, namely, the quintet touches ﬁrst to the ground singlet state. The present
investigation exhibits an essential role of multiple-triplet bound states in the magnetic excitation
spectrum.
KEYWORDS: SrCu2 (BO3 )2 , spin gap, ESR, magnetic excitation, high magnetic field
DOI: 10.1143/JPSJ.72.3243

1.

Introduction[001]

Coupled chain and dimer systems have been attracted
much attention in this decade. One of the interests is in acomplex and fruitful crossover from a well-known purely
one-dimensional antiferromagnet to a two-dimensional (2D)
antiferromagnet which is the basis of a high-temperature
superconductor.1) A 2D spin gap system is quite rare, incontrast with a rapid growth of new quasi-one dimensional
compounds. One of fundamental diﬃculties is that a quasi
2D system tends to exhibit three dimensional (3D) longrange order. However, the 3D-order is eﬀectively suppressedby introducing frustration. One of the most successful
examples is SrCu2 (BO3 )2 : a 2D dimer system with considerably strong frustration.3)
In this compound, a layer of S ¼ 1=2 Cu2þ ions issandwiched by layers of Sr ions in a tetragonal unit cell.2) A
dimer unit is composed of a neighboring pair of edge-sharedplanar rectangular CuO4 and these dimers connect orthogonally by way of a triangular planar BO3 , providing a unique
2D-network as shown in Fig. 1. This 2D lattice is
topologically equivalent to a 2D square lattice with additional alternating diagonal interactions, for which the direct
product of the singlet pairs is the exact ground state of thesystem, as proven by Shastry and Sutherland.4)
Although the ground state is the unique dimer solid, theexcited state is very rich. A low-lying excited triplet, which
is dominant for spin gap compounds in general, is considerably suppressed due to the extreme localization.5) Higher
order multiple-triplet states are instead stabilized by the
kinetic energy called ‘‘correlated hopping’’. Those interesting features of the excited state have been studied experimentally and theoretically, although a complete understanding has not been obtained so far.6–16)
Another important issue is that the compound is locatedvery close to a quantum critical point. The ratio J2 /J1 isevaluated to be 0.63–0.68, where J1 and J2 are, respectively,[110][1-10]

A

B
[100]
[010]
Fig. 1. 2D-dimer plane viewed along the c-axis with respect to z ¼ 0, or
0.5. Open, shadowed and closed circles, denotes Cu, O and B ions,
respectively. The size of a circle indicates the shift along the c-axis(bigger/smaller, corresponding upward/downward). An arrow on Cushows a tilting of the principle axis of g-tensor. A and B distinguish two
types of dimers (with dashed ellipsoids) which are connected orthogonally each other. Thin square shows a unit cell. J1 (thick dashed line) andJ2 (thin dashed line) are the nearest neighbor and the next-nearestneighbor interaction in the plane, respectively. The inset is the [1–10]axis view of the A-dimer. Ellipsoids schematically represent theanisotropy of g-tensor.

the intradimer and interdimer interactions as shown in Fig.
1(a).17) The value is just below the critical value of J2 =J1 ¼
0:70 between the dimer state and the Néel-ordered state.
More recently, it has been proposed that a resonatingplaquette singlet state exists in between these two states.18)
The present compound is clearly in the dimer phase.
However, a precursor eﬀect of the new ground state may

3243

J. Phys. Soc. Jpn., Vol. 72, No. 12, December, 2003

Experimental Procedure

Submillimeter wave ESR measurements have been performed up to 1.3 THz and in pulsed magnetic ﬁelds up to
40 T. An optical pumped far-infrared laser, backwardtravelling wave tubes and Gunn oscillators have been
employed as radiation sources. We have employed a simple
transmission method with Faraday conﬁguration where thepropagation vector of the incident radiation is aligned
parallel to the external magnetic ﬁeld. The polarization oflight is random in the plane normal to the propagation
vector. An InSb is used as the detector. The detail of the ESR
system is given in the references.15,25)
High-purity bulk single crystals of SrCu2 (BO3 )2 weregrown by the travelling solvent ﬂoating zone (TSFZ) methodfrom a polycrystalline SrCu2 (BO3 )2 and a solvent LiBO2
using FZ-T10000N 10 KW high pressure type (Crystal
System Inc.) with four halogen lamps as heat sources.
Oxygen gas (PO2 ¼ 1 atom, 99.999%) was being ﬂowed
during the growth process.26) By means of Laue X-ray backreﬂection, the grown materials were checked and the crystal
axes were determined. A piece of the single crystal with thedimensions of about 4 mm  4 mm  1 mm was used for the
ESR experiments.
3.

One-Triplet Excitation

In the present work, we mainly focus on ESR spectra at
1.6 K and below the critical ﬁeld Hc at which the gap closes.
Since this temperature is much lower than the spin gap, all
the signals are assigned to be the excitations from the ground

3.1 Experimental features
Figure 2(a) shows a typical temperature dependence of
ESR spectrum. The intensity of a broad peak at 13.3 T
decreases rapidly below 10 K. It indicates that the peak iscaused by the transitions of the thermally excited triplets asshown by dotted arrows in the inset. At 1.6 K, two weaksignals marked by arrows are observed. Since the intensityof the weak signals increases with decreasing temperature,

1.6 K
3
65
7
8
9
10
15
20
25
Energy2.

state. In this section, we show important features of an onetriplet excitation: a creation of single triplet in a dimer and
we discuss the ESR selection rule considering Dzyaloshinsky–Moriya (DM) interactions and a staggered ﬁeld. To
avoid confusion, we attach subscripts: GS (ground state) andBS (bound state or multiple-triplet excitation), to the terms
such as singlets and triplets. For examples, singletGS ,singletBS and tripletBS represents the singlet ground state,
the excited singlet bound state and the triplet bound state,
respectively.

Transmission (relative)

appear in excited states by the proximity to the quantum
critical point.
An interesting feature also appears in the magnetization
curve with quantized plateaux at one-third, one-quarter andone-eighth of the saturated moment.19,20) These plateaux are
caused by the extreme localization of one-triplet excitation.
Namely, the excited triplets cannot propagate freely in the
2D lattice owing to the orthogonal dimer arrangement andconsequently organize a regular lattice composed of spatially separated triplets when the density of the triplets iscommensurate with the underlying lattice.
As discussed above, it is very interesting to study the
excited state spectrum of the present system, especially inhigh magnetic ﬁelds. Among many methods, ESR has
established its unique status as a probe of magnetic
excitations in high magnetic ﬁelds.21–24) In fact, the onetriplet as well as the multiple-triplet excitations were foundin our previous ESR work.9) However, the richness of thespectrum and the limited frequency range prevented us tocomplete the experimental determination of a full spectrum.
In the present work, a nearly complete excited state energy
spectrum up to the two-triplet continuum and up to 40 T is
obtained.
The format of the paper is as follows. After brieﬂymentioning the experimental procedure, we show details of
experimental results. The energy diagram, selection rule, andthe symmetry of one-triplet and multiple-triplet excitations
are examined. The anomalous behavior of the magnetic
excitation and its relation to the magnetization around thecritical ﬁeld is discussed in terms of the localization of thebound states.

H. NOJIRI et al.

B

0

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8 10 12 14 16
B (T)
(a)

1.6 K

Transmission (relative)

3244

3
4
6
8
15
20
25

0

2

4

6

8 10 12 14 16
B (T)
(b)

Fig. 2. Temperature dependence of ESR spectra at 428.6 GHz (a) forB k c and (b) for B k a. The sharp peak at 15.3 T is the signal of DPPH
used as a ﬁeld maker. The inset shows a schematic energy diagram of
triplet with a spin gap.

J. Phys. Soc. Jpn., Vol. 72, No. 12, December, 2003

ν (GHz)/Absorption (relative)

1200
H//c T=1.6 K
1000
800
600
400
200
0

0

5

10 15 20 25 30 35 40
B (T)
(a)

ν (GHz)/Absorption (relative)

1200
H//a T=1.6 K
1000
800
600
400
200
0

0

5

10 15 20 25 30 35 40
B (T)
(b)

Fig. 3. Summarized magnetic excitation spectra spanning 0–40 T and 0–
1100 GHz at 1.6 K, (a) for B k c, (b) for B k a. To reduce the data
volume, the data points are averaged with the interval of 0.1 T.

these are assigned as the transitions between the groundsinglet and the excited states, which is indicated by the solid
arrows in the inset. While the similar temperature dependence is found for B k a in Fig. 2(b), diﬀerences are found in
the splitting and the intensity of the two peaks. As is wellknown, an ESR transition between the ground singlet and theexcited triplet states due to the mechanism of magnetic
dipolar transition is usually forbidden.27) A non-secular termsuch as DM-interaction, as proposed recently by Cépas etal., may be the leading origin of the break down of theselection rule.16) This point will be discussed later.
In Figs. 3(a) and 3(b), ESR spectra measured at diﬀerent
frequencies are collected together for B k c and B k a,
respectively. In these ﬁgures, each ESR spectrum is plottedin such way that its background coincides with the measured
frequency, so that the resultant plot can be regarded as a kindof the frequency–ﬁeld diagram which furthermore show
intensity variation of excitations. A detailed frequency ﬁeld

H. NOJIRI et al.

3245

diagram is obtained by tracing diﬀerent types of peaks in theplot. Frequency–ﬁeld diagrams thus obtained are depicted in
Figs. 4(a) and 4(b) for B k c and for B k a, respectively. A
contour map of the intensity in the frequency–ﬁeld plane isshown in Fig. 5. Those ﬁgures are useful to see the overview
of the excited state energy spectra in ﬁeld–frequency plane.
A complicated structure of the spectrum is noticed, thatcannot be expressed well by a simple frequency–ﬁeld plot.
In the following, we show important features of the onetriplet excitation using the set of these ﬁgures.
In Figs. 4(a) and 4(b), it is noticed that two low-lying
triplet excitations: O1 and O2 exist. The values of the zeroﬁeld gaps are close to the spin gap estimated from magnetic
susceptibility and high ﬁeld magnetization;3,19) hence thesemodes are assigned as one-triplet excitation. More precisely,
the zero ﬁeld energy gaps of O1 and O2 are 764  2 GHz and679  2 GHz, respectively for B k c. The splitting between
O1 and O2 is largest in this direction. The spin gap Eg isdetermined to be 722  2 GHz, which is evaluated as theaverage of the zero ﬁeld intercepts of O1 and O2 . The
existence of two modes is generally expected when a unit
cell contains two magnetic ions. The two modes, originallydegenerated, may be split by an anisotropic inter-dimer
interaction and/or other perturbations. In the present compound, the splitting of the two-modes is interpreted fairly
well by including a inter-dimer DM-interaction.16)
When a magnetic ﬁeld is rotated from the c-axis to the aaxis, the splitting shows a monotonic decrease and takes a
non-zero minimum for B k a as we reported in the previous
work (see Fig. 4 of ref. 9). The angular dependence of theupper-going branches of O1 and O2 is more clearly shown in
Fig. 6. The two modes are almost parallel to each other for
B k c, while they become very close each other for B k aaccompanied by the bending around zero ﬁeld. The behavior
indicates that the principle axis of the interaction causing the
splitting points to the c-axis.
It should be stressed here that a small but intrinsic
splitting exists between O1 and O2 for B k a. It should be
noted that the splitting is not caused by the sample
misorientation or anisotropy of g-value. This can be partly
conﬁrmed if one measures the two modes for B k ½110. In
this case, as shown in Fig. 1, the magnetic ﬁeld is parallel to
the A-dimers and is perpendicular to the B-dimers. A larger
splitting would be expected, if it was caused by thediﬀerence of g-value between the A-dimer and B-dimer.
However, it is found that the splitting is nearly identical withthat for B k a (not shown here). The result indicates that thesmall splitting observed for B k a is not caused by the gvalue anisotropy.
The essential diﬀerence between O1 and O2 in B k a isfound in the mixing behaviors as follows. Firstly, it is foundthat the magnitude of the mixing with singletBS states is
diﬀerent between O1 and O2 . In Fig. 4(b), O1 shows a breakaround 860 GHz, while O2 shows another break around
650 GHz. These breaks may be caused by an anti-levelcrossing with an excited singlet state (this point will be
discussed later). Secondly, it is noticed that O1 and O2 roundvery diﬀerently around Hc where a mixing with the groundstate occurs. A mixing is strongly related to the symmetry ofcorresponding modes; hence these ﬁndings indicate that thesymmetry of O1 is diﬀerent from that of O2 .

J. Phys. Soc. Jpn., Vol. 72, No. 12, December, 2003(a) H//c T=1.6 K

1200

0.4

T7
T8
Tw

1050

θ=0 (//a )
θ=15
θ=30
θ=45
θ=60
θ=75
θ=90 (//c)

1000
0.3

950

400
0.1

900

2.3

850
g-value

0.2

600ν (GHz)

800
B

ν(GHz)

1000

T1
T2
T3
T4
T5
T6

H. NOJIRI et al.

M (µ /Cu)

3246

800

200

750
0

5

10 15 20 25 30 35 40
B (T)

700

T=25 K
ν=135 GHz

2.1
2
-90 -60 -30 0 30 60 90
[100]
[001]
[010]
Angle(degree)

0

0

2.2

0

2

4

6
B (T)

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12

0.4

Q1
Q2
Q3

ν(GHz)

1000

Q4
Q5
Qw
B
M

0.3

800
0.2

600

Fig. 6. Angular dependence of O1 and O2 in the ac-plane.  is the angle
between the a-axis and an external ﬁeld. Lines are eye-guides. Anisotropyof g-value is plotted in the inset, which is measured at 25 K with 135 GHz.
The negative angle corresponds to the rotation of an external ﬁeld fromthe a-axis to the b-axis.

B

1200

O1
O2

M (µ /Cu)(b) H//a T=1.6 K

400
0.1

200
0

0

0

5

10 15 20 25 30 35 40
B (T)Fig. 4. Frequency–ﬁeld diagram at 1.6 K (a) for B k c and (b) for B k a.
O, T, Q, B, M are one-triplet, tripletBS , quintetBS , new signal andparamagnetic like signal, respectively. For details, see text. The subscript
W denotes weak signals which are not well classiﬁed. Solid line shows
the paramagnetic resonance line ( ¼ gB H, : frequency, g: g-value).
Dashed lines are eye guides. Magnetization curves taken from ref. 20 are
plotted together.

Fig. 5. Contour plot of ESR absorption intensity for B k a at 1.6 K. The
color scale is arbitrary.

Next we discuss the intensity of the one-triplet excitation.
For B k c, a clear diﬀerence is found in the ﬁeld dependence
of intensity between O1 and O2 as shown in Fig. 3(a). The
intensity is nearly ﬁeld independent for O1 , while it issensitive to the magnetic ﬁeld intensity for O2 . Namely, theintensity of the down-going branch of O2 increases withincreasing ﬁeld and that of the up-going branch decreases
with increasing ﬁeld. In addition to this, a rapid decrease of
intensity is observed for O2 around zero ﬁeld. The behavior
suggests that the origin of the break down of the selection
rule is diﬀerent between O1 and O2 .
In B k a, the behavior is much diﬀerent from that in
B k c. First, the intensity is nearly ﬁeld independent for both
O1 and O2 . Secondly, the intensity is stronger than that in
B k c. The second feature suggests that a non-secular termcausing these transitions should be larger for B k a.
We compare the present result with that of the FIR
spectroscopy.12) It should be reminded that the deﬁnitions ofabsorption intensity are not exactly same each other. In ourcase, the intensity is normalized by the intensity of DPPH
used as maker or by the intensity of paramagnetic signal. In
the FIR experiment, the magnetic intensity is obtained bytaking a diﬀerence between a high temperature spectrum anda low temperature one. In spite of this diﬀerence, two results
are roughly consistent each other. In Fig. 4 of ref. 12, a ﬁeld
is applied along the a-axis. The intensity decreases withincreasing ﬁeld for the c-polarization, while it increases forthe ab-polarization. In case of ESR, un-polarized radiation isused and thus the intensity should be averaged over the twopolarizations. If we average the data of diﬀerent polarizations obtained by FIR spectroscopy, it can happen that theintensity depends on magnetic ﬁeld only weakly for thecancellation, which agrees with the ESR result for B k a.
3.2 Selection rule and DM-interaction
In the following, we discuss possible interactions allowing
the ﬁnite intensity of the one-triplet excitations, which isJ. Phys. Soc. Jpn., Vol. 72, No. 12, December, 2003

originally forbidden. We consider an intra-dimer DMinteraction: Dd , the c-axis and the ab-plane components ofinter-dimer DM-interaction: (Dc and Dab , respectively) anda staggered ﬁeld, all of which are allowed to exist by its
crystal structure. We do not consider a novel mechanism:dynamical DM-interaction proposed recently.28,29) Now thepoint to be clariﬁed is what aspect of the experimental result
can be interpreted with those four terms. We would refuse todeny or conﬁrm the existence of the dynamical DMinteraction because the judgement cannot be made withpresent experimental results.
We employ the selection rule established by Sakai, Cépas
and Ziman for DM-interaction.27) Since the present experiment is made in Faraday conﬁguration, one simple rule isapplicable as follows.
DM-rule: Field independent intensity is observed only
when the DM-interaction has a component along theexternal magnetic ﬁeld and the intensity is quadratic to thecomponent.
Among three possible DM-interactions, Dc exists irrespective of the small buckling of the edge-shared CuO4
planes as shown in Fig. 1. Dd and Dab exist only when thisbuckling is taken into account. A staggered ﬁeld becomesalso possible for the buckling as shown in the inset of Fig. 1.
Considering the large anisotropy of g-value in the ac-plane(ga ¼ 2:05 and gc ¼ 2:28) and the small one in the ab-plane(g  0:01, see Fig. 6 inset), a sizable staggered ﬁeld is
expected only for B k c. Those three terms caused by the
buckling may be small, because of the small tilting angle. An
experimental estimate of Dab is made by Kakurai et al.30)
and it is found that the magnitude of Dab is about 30% ofthat of Dc . This value looks overestimated since thebuckling angle is quite small. However, it is important thatthe practical eﬀect is observed with such small buckling.
One should consider even a small Dc because it operates
directly on the two spins in a dimer. From the crystal
symmetry it is evident that Dd lays in the ab-plane. The
direction is perpendicular to a dimer bond. Since two kindsof dimers are orthogonally arranged in the ab-plane, the sumof the squares of a ﬁeld-parallel-component of Dd over the
A-dimers and the B-dimers is constant for any ﬁeld directionin the plane.
Let us examine, if the DM-rule is compatible with theexperimental results. Considering the DM-rule, Dc gives rise
to constant ESR intensity for B k c. This expectation isconsistent with the behavior of O1 in which the intensity isﬁeld-independent. A precise estimation of Dc ¼ 0:18 meV ismade from the dispersion measured by neutron scattering.16)
This value is a standard when we estimate the magnitude of
DM-interaction from the ESR intensities.
Contrary to O1 , the ﬁeld dependence of the intensity of O2
is not compatible with the DM-rule and thus DM-interaction
is not the origin of the transition O2 . As shown in theprevious section, the intensity of O2 shows the characteristicﬁeld dependence. It is noticed that the behavior is similar tothe case of a staggered ﬁeld with Voight conﬁguration.31) In
fact, a staggered ﬁeld is expected for this ﬁeld orientation as
mentioned above. However, it cannot be applicable to thepresent experiment because the Faraday conﬁguration isused. It is clear that neither DM interaction nor a staggeredﬁeld can cause the transition O2 in the lowest order. A

H. NOJIRI et al.

3247

possibility is that a combined higher order term may cause aﬁnite ESR intensity. The fact that the intensity of O2 is much
weaker than that of O1 is consistent with the contribution of
the higher order term. A similar situation is found in
CuGeO3 , in which both optical and acoustic branches areobserved in ESR. This behavior also cannot be interpretedby the lowest order term of DM-interaction.27) As discussed
above, it is rather plausible that a combined higher orderterm of DM interaction and a staggered ﬁeld causes thetransition O2 . A theoretical calculation of the intensity isnecessary to clarify the present proposal.
Next we discuss the case of B k a. Since both Dd and Dab
have the ﬁeld parallel components, the ﬁeld independent
intensity observed in the experiment is compatible with theDM-rule. The fact the intensity is larger in B k a than that in
B k c indicates that the ﬁeld parallel DM-component islarger in B k a. However, the square of the magnitude of
Dab is one-order smaller than that of Dc ; hence Dab causes
the minor contribution to the intensity and Dd should bedominant in B k a. Another piece of the evidence of the
major contribution of Dd is that the intensity of O1 and O2 isalmost constant when we rotate an external magnetic ﬁeld
within the ab-plane (not shown here). It is consistent withthe fact that the sum of the squares of the ﬁeld parallel
components of Dd over A- and B-dimers is constant under
this ﬁeld rotation. Moreover, Dd operates directly on eachdimer, while Dab does not. All these facts indicate that the
Dd is the main cause of the selection rule breaking in B k a.
Finally, we would like to make a comment on an
exchange anisotropy. In our previous work, we did notconsider the inter-dimer DM-interaction and mentioned thatexchange anisotropy may cause the splitting between O1 andO2 and the forbidden transitions. Recent proposal by Cépas
et al. that DM-interaction is the leading term is moreplausible than the exchange anisotropy. Now we consider
that a simple exchange anisotropy term does not cause themixing between the singlet ground and the excited triplet
states. However, a consideration of the problem from moregeneral view point may be interesting.2,32,33) High resolutionof ESR can resolve even a small change of triplet excitation
caused by these terms.
4.

Multiple-Triplet Bound States

As shown in Figs. 3(a) and 3(b), many high energy
excitations: multiple-triplet bound states,15) account for aconsiderable weight in the excited state. In the present
system, a sizable inter-dimer interaction exists. This interaction couples two (or three) triplets on dimers and causes abound sate with total spin S ¼ 2 (or S ¼ 3). A S ¼ 2 boundstate, for example, consists of three sub-states; singlet, triplet
and quintet. Usually such bound state appears as acontinuum. In an orthogonal-dimer system, however, thelocalization of triplets causes the discrete states. In thissection, we describe the behaviors of those bound state
excitations in detail.
4.1 Singlet bound state
A singletBS is an excited singlet state consisting of two
interacting triplets on nearby dimers and the energy is given
by subtracting the binding energy from the twice of Eg . For afrustrated spin gap system, it is expected theoretically that

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J. Phys. Soc. Jpn., Vol. 72, No. 12, December, 2003

H. NOJIRI et al.

ν (GHz)/Absorption (relative)

excited singlets can dominate the states below Eg . In highly
frustrated system, an excited singlet lies in the low energy
range close to the ground state. In another word, themagnitude of the binding energy is a yardstick for the degreeof frustration. In the present case, such a low energy excited
singlet indicates the close distance from a quantum criticalpoint.
A singletBS is observed through the anti-level-crossing
with one-triplet excitation as shown in Figs. 7(a) and 7(b). In

680
660
640
620
600
580
560
0

O2
O1

TS1

O2

1

2

3

4 5
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ν (GHz)/Absorption (relative)

(a)

O2

1000
960

O1

920
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TS2

840

O1

800
0

2

4
6
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10

(b)

1000

ν (GHz)

900
TS2

800

O1
O2

700

TS1

600
500

0

2

4

6

8

10

B (T)
(c)
Fig. 7. (a) ESR spectra around the anti-level-crossing between O2 andTS1 (dotted line). Dashed lines are eye-guides. (b) Same for O1 and TS2(dotted line). (c) Frequency–ﬁeld diagram at 1.6 K, where an appliedmagnetic ﬁeld is tilted 15 from the a-axis in the ac-plane. Dashed lines
are eye-guides.

Fig. 7(a), as the frequency increases toward 646 GHz, O2
deviates from O1 and shifts to the low-ﬁeld side with aconsiderable broadening. Above this frequency, O2 approaches to O1 from the high-ﬁeld side. For O1 , a conventional successive Zeeman shift is found. The bending of O1
around zero ﬁeld is caused by the anisotropic inter-dimer
DM-interaction as discussed in the previous section. The
break of O2 shown in Figs. 7(a) and 3(b) is a typicalbehavior of an anti-level-crossing. Although the singletBScannot be observed directly by ESR, the anti-level-crossing
exhibits clearly the existence of the singlet state TS1 at
646 GHz. A similar anti-level-crossing is found at 860 GHz
between the singlet TS2 and O1 as shown in Fig. 7(b). Whenthe frequency is very close to TS2 , O1 disappears and only asingle peak of O2 is found. The magnitude of the anti-levelcrossing at TS2 is larger than that at TS1 . It indicates that the
coupling between the singletBS and the one-triplet state islarger in TS2 .
The present result also exhibits that the symmetry of the
one-triplet excitations O1 and O2 are diﬀerent each other as
mentioned in the previous section. In Fig. 7(c), a frequency–
ﬁeld diagram is plotted, where a magnetic ﬁeld is in the acplane but not parallel to the a- nor the c-axes. An anti-levelcrossing is found in both O1 and O2 at both TS1 and TS2(four breaks in total). This behavior shows that two-modes
O1 and O2 are mixed each other when an external ﬁeld istilted from the principle-axes (a or c) of the crystal. It
implies that the anti-level-crossing is closely related to theanisotropic part of exchange coupling such as DM-interaction.
Let us compare the present results with the Raman
scattering experiments, where several singlet bound states
are found.8) A clear peak was observed at 30 cm1(900 GHz) in B1 polarization (see Fig. 1 of ref. 8). In B2
polarization, a peak was also found at 30 cm1 with ashoulder at 24 cm1 (720 GHz). The shoulder appeared alsoin B1 polarization and a small low energy shift was found at
6 T. Consequently, the shoulder was assigned as one-triplet
excitation being weakly allowed for some defects. Theenergy of TS2 is very close to 30 cm1 and thus TS2 canattribute to the 30 cm1 Raman peak. For TS1 , no corresponding Raman peak exists; hence TS1 located at 646 GHz(21.5 cm1 ) is assigned as the lowest singlet bound state. It
is notable that the energy is lower than the spin gap of the
one-triplet excitation. This fact indicates that the system israther close to the quantum critical point where the energy ofthe bound state should fall to zero.
Let us compare the present results with the theoreticalinvestigations on various bound states.10,14,35) As discussed
by Totsuka, Miyahara and Ueda, the representations E andB2 are infrared (also ESR) active and those A2 and B1 are
infrared inactive in D2d symmetry. The one-triplet belongs to
the E representation. In ref. 35, the lowest singletBS is foundto have the A2  B1 symmetry (additional small splitting isfound in ref. 14). The second lowest singletBS coming fromthe third neighbor pair in ref. 35 has an infrared active A1 
B2 symmetry. Let us assign tentatively the lowest energy
mode and the second lowest energy mode to TS1 and to TS2 ,respectively. It is natural to assume that the one-triplet shows
the stronger mixing with the infrared active mode. The
strong/weak anti-level-crossings of TS2 /TS1 are consistentJ. Phys. Soc. Jpn., Vol. 72, No. 12, December, 2003

H. NOJIRI et al.

with the fact that the lower singletBS (TS1 ) is infrared
inactive and that the upper one (TS2 ) is infrared active.
However, since only two modes are found in the present
experiments, the deﬁnite assignment is diﬃcult. A theoretical investigation by considering the mixing eﬀect is needed
for further understanding. Such a calculation is useful to
examine the wave function of the bound states.
4.2 Triplet bound state
In Fig. 8, an example of ESR spectrum is shown for B k a.
Besides one-triplet excitations O1 and O2 , eight peaks arefound. These additional peaks are easily classiﬁed into
tripletsBS (T1 –T5 ) and quintetsBS (Q1 –Q3 ) by Zeeman shifts
in the frequency–ﬁeld diagram. In the inset of Fig. 8,
temperature dependence of intensity is plotted for diﬀerent
peaks. Below 7 K, the intensity increases with decreasing
temperature with similar temperature dependence for all
three peaks. The behavior conﬁrms that the additional peaks
are the excitations from the ground state.
A most signiﬁcant feature in Fig. 8 is the sizable intensityof the bound state excitations. It is the evidence of the large
suppression of the one-triplet process in the present system.
It also supports the idea that a bound state propagating withcorrelated motion has considerable spectral weight.35) In
general, ESR intensity is not a spectral weight itself because
it is mediated by transition matrix elements. However, theintensity can be compared directly between the one-triplet
state and the tripletBS . The argument is based on the fact thata total spin number of the system changes by one in bothtypes of excitations. From this basic view point, we noticethat the magnitude of matrix element may not be so diﬀerent(at least in the same order) between these two processes. In
fact, the intensity ratio of the sum of O1 and O2 to T1 iscomparable with that of the ﬁrst excited peak to the second
one in neutron scattering (see Fig. 4 of ref. 16).16) This fact
supports our presumption that relative ESR intensity of theTransmission (arb. units)

T1 T2 T3 Q//a

Q2 Q T5

1

3

T4

Intensity (arb. units)//c140
120

T1

100

T2

80

O1 + O 2

60

O1O2

20
0

0

2

2

4
6
T (K)

4

6

8

triplet states (in both the one-triplet and the bound state)roughly corresponds to the diﬀerence in the spectral weight.
Another important ﬁnding is that the bound state spectrum
splits into many sharp discrete peaks instead of a conventional two-triplet continuum. It suggests that many diﬀerent
types of bound state is formed for diﬀerent interactions
acting between the distant triplets on dimers. The fact thatthese diﬀerent states do not merge into a single bandsuggests that the dispersion of the bound states is very weak.
Let us evaluate the zero ﬁeld energy gap of each boundstate. We simply evaluate a gap by extrapolating each modewith a linear equation. This procedure may cause nonnegligible error, if the level of a bound state has a large
bending around zero ﬁeld as is observed in O1 and O2 . To
check this point, we compare a zero ﬁeld gap obtained for
B k a with that for B k c and found that the diﬀerence is at
most 10 GHz. It is much smaller than that of O1 and O2(40 GHz) and is comparable with experimental errors.
Hence, the above mentioned procedure is justiﬁed. Theenergy gaps of the bound states thus evaluated are listed inTable I. We speculate that the anisotropy of the zero ﬁeld
splittings of the bound states are reduced eﬀectively by the
hopping eﬀect or by the spread of the wave function overdistant dimers. Another ﬁnding is that the anisotropy of theintensity of the tripletsBS except T1 is similar to that of the
one-triplet excitations. It shows that the origin of the break
down of the selection rule may be the same for both cases.
Next we discuss ESR spectra around the one-triplet gap asshown in Figs. 9(a) and 9(b). The spectra change drasticallyboth in the shifts and in the intensity in the tiny range of
frequency. This behavior may be caused by the increasing
mixing between the tripletBS and the one-triplet state. Most
spectacular feature is found at 736.1 GHz for both B k a andB k c. A broad and intense peak consists of many diﬀerent
sub-peaks is found in high ﬁeld side. Since the anomaly isfound at the identical frequency irrespective of ﬁeld
direction, it may be caused by the mixing with a horizontal
level lying at this particular energy. Considering the nomixing of the tripletBS with TS2 [no break is found in Figs.
4(a) and 4(b)] and the no-mixing of the one-triplet stateswith the new horizontal level, it may not be a singlet boundstate. The Sz ¼ 0 branch of the one-triplet at zero wave
vector point is located at 722 GHz and it cannot be thecandidate of the new singlet state. It is noticed that an Sz ¼ 0
branch of the one-triplet at the (1.5, 0, 0) point is foundexactly at this energy by neutron scattering (see Fig. 3 ofref. 30). This Sz ¼ 0 branch may be responsible for the

Table I. Zero ﬁeld energy gap of the bound states. The notation is
identical to that in Figs. 4(a) and 4(b).

40

0

3249

10

8
B (T)

10

12

14

Fig. 8. Anisotropy of ESR spectrum at 1024 GHz and at 1.6 K. Dottedarrows indicate the splitting of O1 and O2 . Lines connect corresponding
peaks. Temperature dependence of intensity is shown in the inset. See text
for the notation of peaks.

TripletGap (GHz)

QuintetGap (GHz)

T1

1140  5

Q11390  15

T21170  5

Q2

1560  15

T31190  5

Q31600  15

T41225  5

Q41652  15

T51350  5Q51729  15

T61220  5

T71264  5

T81303  5

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H. NOJIRI et al.

have more bound states in calculation, the interactions
between distant dimers should be taken into account (aninter-layer coupling may also contribute). Finally wemention the instability of a tripletBS due to the transition
to another ground state, proposed in theoretical work.10) In
spite of detailed experiments, no such anomalous behavior isfound in the present work.

ν (GHz)/Absorption (relative)820(a) H//c T=1.6 K
787.8 GHz

B

780
761.6

740

B

736.1
716.2

700

692.9
670.9

660

0

5

10
15
B (T)

20

25

ν (GHz)/Absorption (relative)820(b) H//a T=1.6 K
787.8 GHz

B

780
B

761.6

740

736.1
716.2

700

692.9
670.9

660

0

5

10
15
B (T)

20

25

Fig. 9. ESR spectra around the one-triplet gap at 1.6 K (a) for B k c and(b) for B k a. Dashed and solid lines are guides of T1 and T4 ,respectively. Vertical bars show O1 and O2 . B is a peak appearing
around Hc .

anomaly at 736.1 GHz. Such mixing between the excitation
at zero-wave vector points [(0, 0, 0) and equivalent points]
and that at -wave vector point such as (1.5, 0, 0) is possible
by some non-secular terms such as a staggered ﬁeld. Weaker
but similar anomalies are found in Fig. 5 as horizontal highintensity lines, which may be caused by the similar origin.
Let us compare the present results with theoretical
calculations. As shown in Table I, all eight bound satesare below the bottom of the two-triplet continuum (2Eg ¼
1444 GHz) and thus these states are stable as a two-triplet
bound state. In ref. 35, only ﬁve modes (reducing to threewith degeneracy) are found below the bottom. All the mode
are infrared active and thus we can attribute at least the ﬁve
from the all eight modes to the theoretical ones. In the exact
diagonalization result which is more reliable around aquantum critical point, two peaks are found side by sidearound 5 meV (1208 GHz). The energy is close to those of
T1 and T2 . In other theoretical works, the number of states isalso less than that found in the present experiment.10,14) To

4.3 Quintet bound state
In the following, we show the ﬁrst observation of aquintetBS in this compound. The nature of the magnetization
appearing around the Hc is also discussed.
As shown in Fig. 8, the intensity of a quintetBS is weaker
by one order of magnitude than that of tripletBS . It is becauseof the higher order excitation process of the total spinchange: S ¼ 2. Five quintetBS modes are found easily forthe typical double Zeeman shift. The zero energy gap ofquintetBS has been extrapolated from the experimental data.
Since the number of data points is limited, the error of the
gap is about 15 GHz as listed in Table I. The anisotropy of
the gap is not found within the experimental error.
It is notable that only Q1 is below the two-triplet
continuum threshold starting at 2Eg ¼ 1444 GHz, while theother four modes are located above the threshold. Since theenergy of Q1 is below the threshold, it is assigned as aquintetBS made up of two-triplets (four spins). Let us discuss
the nature of other high energy quintetsBS . In the isolated
dimer limit of four spins, the tripletBS and quintetBS are
separated by the eﬀective antiferromagnetic interaction
acting among the four spins. In this limit, the higher energyof Q2 –Q5 can attribute to this interaction and those peaks are
considered as two-triplet bound state. This interpretation has
a diﬃculty because a multi-particle bound state is unstable
when the energy is above the corresponding incoherent
continuum in general. It is totally unclear if the quintet
sector of the bound state can separate clearly from thecontinuum spreading above 2Eg . A more natural idea is toassign Q2 –Q5 to three-triplet bound state and in fact such
three triplet state is found in theoretical calculations, whichis shown in later. The existence of three-triplet bound stateshows the signiﬁcant contribution of higher order excitations.
The existence of Q1 below the threshold causes thepeculiar feature that the quintetBS touches the ground state
before the closing of one-triplet gap. In spite that a ﬁnite
mixing between the ground and excited states modiﬁes thissimple situation to the more complex one, a characteristic
contribution of a quintet is found experimentally. The
enlarged spectra of Figs. 4(a) and 4(b) around the Hc are
shown in Figs. 10(a) and 10(b). It is noticed that ESR modes
bend around the Hc and the behavior is much diﬀerent
between B k c and B k a. This anisotropy of ESR spectra
should correspond to the anisotropy of the magnetization
curves. It is because the magnetization around the Hc iscaused by the mixing between the ground and the excitedstates. In another word, the magnetization should bedominated by the wave function component of the modes
enhanced by the mixing. Keeping this point in mind, let us
examine the magnetization and the frequency–ﬁeld diagram
carefully.
For B k c, the magnetization becomes ﬁnite above 15 T

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H. NOJIRI et al.

(a) H//c T=1.6 K

200
150
100
50
15

20
B (T)

25

(b) H//a T=1.6 K
200
150
100
50
15

20

B (T)

25

30

Fig. 10. The low energy part of ESR spectra around Hc (a) for B k c and(b) for B k a. The dashed lines are eye-guides. The vertical bar denotes
Q1 .

and at the same time O2 starts to deviate from a straight line.
This fact shows that the one-triplet component dominates themagnetization. The intensity of Q1 is enhanced around 20 T
presumably for the weak anti-level-crossing with O1 .
Around 21 T, Q1 disappears and this fact indicates that thequintet component in the magnetization is small. After theanti-level-crossing, O1 bends considerably and loses theintensity gradually. It is noticed that O2 shows a turn aroundlike behavior above 20 T, which indicates the strong mixing
with the ground state. On the other hand O1 goes down again
above 26 T and disappears suddenly at the jump of magnetization to the 1=8-plateau as shown in Fig. 4(a). The originof the ﬂat part of O1 around 25 T might be a anti-levelcrossing with a new singletBS , however, a further investigation is necessary for this point.
The strong mixing of O2 with the ground state indicatesthe dominant contribution of the one-triplet component to
the magnetization. It is also noticed that Q3 is enhanced witha large rounding above 20 T, which shows the mixing of
three-triplet state with the ground state. The contribution of
diﬀerent components to the magnetization indicates that themagnetization is not uniform and homogeneous. It is much
diﬀerent from a usual spin gap system in which a magnetization consists of homogeneously distributed triplets. This
characteristic behavior may be caused by the localization of
excited states in the present system. Recently, a homogeneous internal ﬁeld is found by NMR below the 1=8-plateau,however the diﬀerence is presumably due to the low
observation frequency in NMR, in which the distribution

3251

of an internal ﬁeld is averaged.38)
Let us discuss the case of B k a shown in Fig. 10(b).
While the behavior of O1 is similar to that in B k c, both Q1and O2 continuously go down toward the ground state. The
enhancement of Q1 and the disappearance of O2 indicatesthat the quintetBS component dominates the magnetization.
Other higher energy quintetBS states also exhibit enhancements and bendings, which indicate the strong mixing of thequintetsBS with the ground state. An observation of a small
break of O2 in B k a [see Fig. 4(b)] and at 22 T may becaused by the anti-level-crossing with unknown singletBS at
100 GHz, which is also found in B k c.
As shown above, the magnetization below the 1=8-plateau
contains diﬀerent types of states and does not consist of
homogeneously distributed triplets. A dominant contributionof a quintetBS is found for B k a. It is an interesting point, if
a quintetBS coupled by an eﬀective repulsive interaction
shows a condensation at Hc as proposed by Momoi andTotuska.13) Experimentally the domination of the quintetBS
is found clearly in the present work. However, it may be aquenched and localized state rather than a condensation. In
the case of a condensation, a magnetization should notcontains a several diﬀerent states as observed in the present
case. The localization of excited states and the resulting
heavy eﬀective mass of a quintetBS may prevent thecondensation of quintetBS .
Finally we compare the present results with the theoretical
calculation of the quintetBS . In ref. 35, the lowest mode has
the A2  B1 symmetry, which is infrared inactive. Since aquintetBS shows very weak intensity, an infrared inactivemode should not be excluded from the candidate. The factthat the number of two-triplet quintetBS is only a few isconsistent with the present result. However, a large numberof a three-triplet quintetBS found in the experiment calls forfurther theoretical investigations.
5.

High Field Phase

In this section, we discuss ESR signals observed above Hcand their concerns with the magnetization plateaux. In thisregion of magnetic ﬁeld, the most of the intensity concentrates into the signals B and M as shown in Figs. 3(a) and3(b). These two signals alternate at the magnetization jump
to the 1=8-plateau as shown in Fig. 11. The position of M is
identical with that of the corresponding paramagnetic
resonance and the extrapolation of M crosses the origin of
the frequency–ﬁeld diagram for both B k c and B k a asshown in Figs. 4(a) and 4(b). On the other hand, B shows theasymptotic behavior to the paramagnetic line. When a spin
gap is closed, a 3D ordered-state or a disordered ground state
consists of gap-less triplets appears. In the former, a ﬁeld
variation of a spin wave mode is observed as an antiferromagnetic resonance (AFMR). In the latter, the Zeeman
splitting of a gapless triplet causes ESR at the identicalresonance ﬁeld with that of paramagnetic resonance. The
strong correlation among the spins does not cause a shift ofresonance ﬁeld because the spin operator commutes with theisotropic exchange coupling.39)
The behavior of M shows that the signal is the ESR of adisordered state. It is noticed that M continues at 1=4-plateau
and a gapped low-energy excitation is absent. Such low
energy discrete excitation is observed at 1=3-plateau, whichν (GHz)/Absorption (relative)

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J. Phys. Soc. Jpn., Vol. 72, No. 12, December, 2003

1100
1000
900

H. NOJIRI et al.

ν
Sz=0ν 1 ν2
Sz=1 ν3
H
SGS=0

800
700
600
15

20

25
B (T)

30

35

Fig. 11. The ESR spectra around the 1=8-plateau for B k c at 1.6 K. The
thick and thin dashed lines indicate the paramagnetic line and the ﬁeld of
the magnetization jump to the 1=8-plateau, respectively. The dotted line isan eye-guide for the signal B. The inset shows the schematic energy levelof an isolated S ¼ 1=2 antiferromagnetic dimer. SGS ¼ 0 is the singlet
ground state. For others, see text.

corresponds to the creation of a new triplet into a superlattice structure at a plateau.36) Momoi and Totuska proposedthat a ‘‘supersolid’’ state appears between the plateaux, in
which a spin density wave and superﬂuid of spins coexist.13)
The latter component gives rise to a gap-less ESR. The
present result indicates that a superﬂuid component (gap-lessmode) is also signiﬁcant at 1=4-plateau and that the plateau
is not completely static at 1.6 K. Recently, the static superlattice structure is found by NMR in the 1=8-plateau at 30
mK.38) For 1=4-plateau, a static super structure may also
appear if we measure in much lower temperature below
1.6 K. This point remains for further experimental investigations.
Although the behavior of M is rather conventional, B
shows unusual features: (1) location above the paramagnetic
line, (2) non-linear ﬁeld dependence, and (3) disappearance
at the jump to the 1=8-plateau. Considering these features,
we propose two possible origins of B: (a) AFMR, (b)transition Sz ¼ 1 ! Sz ¼ 0 in the lowest triplets. First we
discuss the possibility of AFMR. In AFMR, a deviation fromthe paramagnetic line is caused by anisotropy and thus thedeviation should change for diﬀerent ﬁeld orientation.
However, the deviation of B is nearly equal between B k c
and B k a. The most fundamental diﬃculty of the interpretation by AFMR is that no evidence of 3D-ordering is foundbelow the 1=8-plateau. Actually, in NMR measurement, no
static internal ﬁeld is found in this ﬁeld range, which shows
non-existence of 3D-order.38)
Next we discuss the possibility of the case (b). In the most
simple picture of a isolated dimer system, the Sz ¼ 1 level isbelow the singlet ground state above Hc as shown in the inset
of Fig. 11. The transition Sz ¼ 1 ! Sz ¼ 0 shown by thearrow (2 ) represents ESR of a disordered magnetic state andthe resonance ﬁeld is identical with that of a paramagnetic
resonance. In the present system, the level scheme is
modiﬁed for the anti-level-crossing as shown by the dottedlines. The transition Sz ¼ 1 ! Sz ¼ 0 also changes to that ofthe dashed arrow (1 ). The signal B may be assigned as thistransition 1 . Qualitatively, the features (1) and (2) mentioned above are consistent with this interpretation. The
strong intensity of B is also consistent with this picture
because it is the allowed transition from the ground state.
However, a quantitative agreement is diﬃcult with thesimple model shown in the inset. In this case, the relation
1 ¼ 2 þ 3 should hold at each ﬁeld, where 3 is thetransition SGS ¼ 0 ! Sz ¼ 1 (note that arrows in the inset
are shifted to prevent the overlapping). It is because of theanti-symmetry of the anti-level-crossing around the Hc .
However, this relation is not found among O1 , O2 and B as
can be seen clearly in Figs. 4(a) and 4(b). For example while
the deviation of B is rather independent of the ﬁeld direction,
O1 and O2 are anisotropic. As shown above, the model (b)cannot explain perfectly the behavior of the signal B. We
may consider that the nature of the state below the 1=8plateau seems to be unconventional, which requires further
microscopic investigations by using diﬀerent types ofprobes.
6.

Summary

To be summarized, the following important features arefound.
(1) Diﬀerence of the symmetry between the nearly
degenerated one-triplet excitations O1 and O2 arefound in the ﬁeld dependences and in the anti-levelcrossing.
(2) Two-singlet bound states are found by the anti-levelcrossing with the one-triplet excitation. The one ofthose is located below the spin gap of the one-triplet,
which indicates the proximity to the quantum criticalpoint.
(3) Several triplet bound states are found. The zero ﬁeld
energy gaps are all below the two-particle continuum
threshold.
(4) The gap of the lowest quintetBS is below the twoparticle continuum threshold; hence it crosses with the
one-triplet excitation before the closing of the onetriplet spin gap. Some of the quintetBS modes are
assigned as a three-triplet bound state.
(5) The ﬁnite magnetization appearing below the 1=8plateau contains various types of magnetic components
caused by the anisotropic mixing.
(6) Two models are proposed for the unusual ESR mode B
observed below the 1=8-plateau.
(7) The domination of gap-less ESR signal M indicatesthat the 1=4-plateau is rather soft compared to the 1=3plateau.
(8) Qualitative interpretation is made for the selection
rules in the one-triplet excitation by considering intradimer DM-interaction, inter-dimer DM-interaction anda staggered ﬁeld.
The present work exhibits various novel features of higher
order multiple-particle excitations in a quantum spin gap
system.
Acknowledgements
The authors would like to thank T. Sakai, T, Ziman, O.
Cépas, K. Totsuka and Y. Fukumoto for valuable discussions. This work was partly supported by a Grant-in-Aid for

J. Phys. Soc. Jpn., Vol. 72, No. 12, December, 2003

Scientiﬁc Research on Priority Areas from the Ministry of
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