Physica B 294}295 (2001) 14}19

High-"eld ESR on quantum spin systems
H. Nojiri *, T. Asano, Y. Ajiro, H. Kageyama, Y. Ueda, M. Motokawa
Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, JapanDepartment of Physics, Kyushu University, Fukuoka 812-8581, Japan
Institute for Solid State Physics, University of Tokyo, Kashinoha 5-1-5, Kashiwa 277-8581, Japan

Abstract
High-"eld electron spin resonance (ESR) experiments have been performed on quantum spin systems. A distinct"eld-induced crossover between the gap-less spinon regime and the gapped sine-Gordon regime is observed for the
Heisenberg quantum spin chain; CuBenzoate. In a Shastry and Sutherland model compound SrCu (BO ) , a set of
multiple-triplet excitations has been resolved by utilizing high-resolution ESR. The splitting of the two-triplet boundstate into a few discrete energy levels is considered as the evidence of the extreme localization of the triplet excitations inthis compound. The present results indicate the unique and powerful features of ESR in the study of magnetic
excitations.  2001 Elsevier Science B.V. All rights reserved.
Keywords: ESR; Quantum spin systems; Magnetic excitation; Soliton

1. Introduction
Quantum spin systems have been considered asimportant models to study the quantum criticalbehavior in strongly correlated electrons. In a conventional phase transition, critical phenomena aremostly examined as a function of temperature. For
quantum spin systems, however, a magnetic "eldcan be used as the unique handling parameter of thequantum spin #uctuation. The in#uence of magnetic"elds on strong quantum #uctuation is manifested
as drastic anomalies in the magnetic excitation
spectra. These `anomaliesa are, in many cases, nontrivial when a system is located near the quantum
critical point. Hence, studies of the magnetic excita* Corresponding author. Tel.: #81-22-215-2019; fax: #8122-215-2016.
E-mail address: nojiri@imr.tohoku.ac.jp (H. Nojiri).

tions in quantum spin systems in high magnetic"elds have attracted much interest in this decade.
Electron spin resonance (ESR) is considered to
be a unique tool among many di!erent kinds ofexperimental techniques to study magnetic excitations such as inelastic neutron scattering, Raman
scattering and far-infrared spectroscopy, for thefollowing reasons: (1) high sensitivity, (2) high-energy resolution and (3) capability of high-"eldmeasurements by using a pulsed magnetic "eld.
The accessible "eld range, by a non-destructive
pulsed "eld ESR equipment, is more than 40 T andsuch high "elds cannot be used by any means otherthan mentioned above. Experiments in mega-gauss"elds are also possible by using a single-turn coilsystem. In the following, after a brief description of
the experimental setup, several recent examples of
the high-"eld ESR on quantum spin systems arepresented.

0921-4526/01/$ - see front matter  2001 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 1 - 4 5 2 6 ( 0 0 ) 0 0 5 9 8 - 6

H. Nojiri et al. / Physica B 294}295 (2001) 14}19

2. Experimental
The schematic drawing of the experimental setupof the ESR equipment is shown in Fig. 1. For insulating compounds, a simple transmission method is
employed. For metallic materials, several di!erent
techniques are used as given in the reference [1].
Three di!erent types of sources are installed tocover the wide frequency range between 35 GHz
and 7 THz. For the microwave region, conventional Gunn oscillators are employed. Between
250 GHz and 7 THz, an optical pumped farinfrared laser is used. Backward travelling wave
tubes are also used between 100 and 400 GHz. To
cover such wide frequency range, four di!erent
types of detectors: (a) InSb, (b) GaAs, (c) a magnetic-"eld-tuned InSb and (d) GeGa, are installed.
It should be noted that this frequency coverage isequivalent to that available by a neutron triple-axis
spectrometer installed in a reactor. To obtain pulsed "elds up to 40 T, a capacitor bank of 100 kJ
energy is installed. For the generation of magnetic"elds, a conventional wire-wound multi-layer coilis used. The pulse duration time is between 5 and8 ms. The inner diameter of the cryostat can bechanged between 12 and 22 mm depending on theFig. 1. Schematic view of the high-"eld ESR system.

15

maximum "eld. A magnetic "eld up to 20 T can begenerated by a split-type pulsed magnet. The split
magnet is used to perform the experiments in Voigt
con"guration.
Two types of cryostats are installed to control
the sample temperature. Below 70 K, a conventional bath-type cryostat is used. The lowest temperature of 0.4 K is obtained by using a He insert.
Above 70 K, a gas-#ow-type cryostat is employed.
The highest temperature available is 450 K.

3. Breather excitation in CuBenzoate
A S" Heisenberg spin chain (S"HSC) is themost well-know model system showing quantum
critical phenomena. A well-known spinon continuum of S"HSC is characterized by a diverging zero energy (gap-less) mode with strong
quantum #uctuations. In high magnetic "elds,
a new soft mode develops at the incommensurate
wave vector and the excitation spectrum is gap-lessup to the saturation of magnetization.
In the last three decades, CuBenzoate
Cu(C H COO) 3H O (CBZ) had been known as  a good model of S"HSC with an exchange coupling J"8.6 K [2]. Recently, the presence of a "eldinduced energy gap was found in the speci"c heat,neutron scattering and susceptibility by Dender et
al. [3]. This unexpected observation is very surprising, because the magnetic excitation of S"HSCshould be governed by the gap-less spinon modeeven in high magnetic "elds. Since no NeH el ordering
was found at least down to 0.1 K, the "eld-induced
gap cannot be attributed to the "eld-induced ordering. Moreover, it is found that the magnitude of the
gap increases with magnetic "eld intensity H as
H. This power}law dependence on H indicatesthat the critical phenomenon of the system is controlled by the application of magnetic "elds.
By using "eld-theoretical approaches, it has been
claimed that the gap is caused by the staggered"elds acting between neighboring spins in a chain
due to the presence of Dzyaloshinsky}Moriya(DM) interactions which alternate their directions
between the neighboring sites [4,5]. The mostimportant point is that CBZ is classi"ed as aquantum sine-Gordon system. As it is well known,

16

H. Nojiri et al. / Physica B 294}295 (2001) 14}19

a sine-Gordon model is the best-known system that
contains non-linear excitations such as solitons,
antisolitons, and their bound states, the so-called`breather excitationsa. The integrability of themodel enables us to calculate many kinds of physical quantities exactly and thus it is very important
to "nd out a good model compound representing
this model.
ESR measurements on this compound below 1 K
were reported more than 20 years ago and a new
ESR peak was found at lowest temperature [6].
This ESR signal was interpreted as the antiferromagnetic resonance, which is not compatible
with the present results as mentioned above. It has
been proposed by Oshikawa and A%eck, that thisnew ESR signal is the manifestation of a breather
excitation [7]. A characteristic feature of this modeis the non-linear "eld dependence of the resonancefrequency  ash"[(g H)#M ],(1)

where g is the g-value and M is the mass of the"rst breather. Since the "rst breather B is thelowest energy excitation in a sine-Gordon model,
the M (energy of B ) is identical to the magnitude
of the "eld-induced gap. In the present ESR experiments,  and H can be precisely measured and thus,
the "eld dependence of M can be determined veryaccurately.
Fig. 2 shows an example of the temperature dependence of ESR spectra for Hc. As the temperature is decreased, the resonance "eld shifts and theline width shows a broadening. Such changes arecommonly observed for a S"HSC and the origin
is considered as the development of the correlation
length of the spinon excitation towards T"0. Below 1.6 K, a new peak appears in low-"eld side
besides the conventional spinon ESR mode (seeRef. [8] for more details). This behavior is similar
to that observed in the previous ESR experiment
performed at much lower frequency. This di!erence
originates in the di!erent magnitude of the gap as
a function of H. To examine the nature of this newmode, the frequency dependence of the resonance"eld is examined at 0.5 K.
Fig. 3 shows the "eld dependence of the gap (M )
as a function of H derived from Eq. (1) in thepresent work. In Eq. (1), g-value, , and H are

Fig. 2. Temperature dependence of ESR spectra observed at
326 GHz for Hc.

Fig. 3. Field dependence of M determined by ESR at T"0.5 K
(closed rectangles). Open circles denote the value of M given inRef. [3]. The horizontal axis is scaled as -power in magnetic
"eld. The solid line is an eye-guide.

experimentally determined precisely and thus M isdetermined without any adjustable parameters.
The "eld dependence of the M agrees well withthat derived from the previous heat capacity
measurement [3]. Moreover, the magnitude of the"eld-induced gap changes as H as proposed bytheory. This agreement indicates that the new ESR
mode is the manifestation of a breather excitation.
It should be noted that the high-energy resolutionof ESR enables us to examine the "eld dependenceof the gap very accurately.

H. Nojiri et al. / Physica B 294}295 (2001) 14}19

Fig. 4. Examples of ESR spectra observed at 0.5 K foe Hc.

Finally, we show an unexpected observation ofhigher-order excitations other than the "rstbreather mode B as depicted in Fig. 4. In thelow-"eld side of B , very small signals marked by
B and B are observed. The energies of these
modes are higher than that of B at the identical"eld intensity. Therefore, it is speculated that thesemodes are related to the higher order breathers.
Although there is no quantitative interpretation ofthese higher-order modes, we believe that the present results stimulate further theoretical investigations. The present results demonstrate the highsensitivity of ESR. Since the spectral weights of
B and B are very small, it is di$cult to obtainthese modes from the thermodynamic quantities
such as heat capacity. Moreover, these modes were
not found, for the limited sensitivity, in the previous
neutron scattering experiments.

4. Multiple-triplet excitation in SrCu2 (BO3 )2
Recently, a two-dimensional (2D) spin-gappedsystem SrCu (BO ) with tetragonal symmetry has
been studied intensively as a realization of the
Shastry and Sutherland model [8]. The Cu> ionsform a unique 2D-network of S" spins as shown
in Fig. 5. Each antiferromagnetic dimer unit ismade up of two neighboring pairs of Cu> ions andthese dimers are orthogonally arranged with re-17

Fig. 5. Schematic spin arrangement in the c-plane. Closed
circles and thick solid line show the Cu> ions and a dimer unit
respectively. Line with arrows denotes a bond for J .
spect to each other. This 2D lattice is topologically equivalent to a 2D square lattice withadditional alternating diagonal interactions, theso-called Shastry and Sutherland model [9].
The orthogonal dimer arrangement of the system
leads to interesting ground state and excited state
properties. First of all, the exact ground state isgiven by the direct product of the singlet dimer
state. As the second point, the propagation of an
excited triplet on a dimer bond to the nearest sites
is considerably suppressed. It has been shown thatthe hopping of a triplet is possible only from thesixth order in the perturbation and thus a triplet
excitation is extremely localized [10]. It should be
noted that this feature originates from the frustration between the intra-dimer interaction (J ) andinter-dimer interaction (J ) and is not caused by
the smallness of the inter-dimer interaction. In fact,
the ratio J /J&0.63 is very close to the criticalvalue of J /J&0.70 between the spin gap state and
the NeH el ordered state [11]. The strong suppressionof the hopping of a triplet is essential for the appearance of quantized plateaux in the magnetization observed in high "elds [12].
The magnetic excitations of SrCu (BO ) havebeen intensively studied by di!erent experimental
techniques [13}16]. In the previous work, we haveinvestigated the spin-gap excitation in the singlet
ground state and found the lowest spin-gap excitation to excite one triplet from the singlet ground18

H. Nojiri et al. / Physica B 294}295 (2001) 14}19

state to the excited state. We have also founda `multiple-tripleta excitation, which is related to
the generation of a bound state of triplets. Usually,
one triple excitation is dominant in a magnetic
excitation spectrum of a spin-gap compound.
Therefore, the observation of such higher order
excitation is very surprising. As it was pointed out
theoretically, a correlated hopping of two-triplets ispossible from the second order in perturbation theory, for one-triplet the sixth order is necessary [17].
Hence, the appearance of a `multiple-tripleta excitation is evidence for the extreme localization of the
one-triplet excitation.
In the present work, we present the results ofhigh-sensitive ESR measurements for the two-triplet excitations. Because of the high resolution of ourequipment, a broad two-triplet band, which is observed in neutron scattering, is resolved into several
independent modes. Moreover, a quintuplet excitation is observed, which is the direct evidence for the
existence of a two-triplet bound state.
Before proceeding to the results, we discuss brie#y the selection rule for ESR. As is well known, thetransition between the ground singlet state and theexcited triplet state is forbidden in principle. However, a non-secular term in a spin Hamiltonianbreaks this selection rule. As a candidate, DMinteraction and a staggered "eld should be considered. As discussed in the previous work, thestaggered "eld mechanism can be ruled out [13].
Recently, it has been proposed that the DM interaction breaks the selection rule for SrCu (BO )[18,19]. The observation of the spin-gap excitations in Faraday con"guration is consistent withthis proposal, however, the angular dependence ofintensity and the zero "eld splitting between theS "1 and S "!1 branches may not be compatX
X
ible with this model. We speculate that the combined e!ect of DM interaction and anisotropic
exchange interaction is important.
Fig. 6 shows the ESR spectrum taken at
1042 GHz. The schematic energy diagram is also
given in the inset. Since the temperature is much
lower than the energy gap of 34.7 K, the signals are
identi"ed as the transitions between the singlet
ground state and the excited states. Since theO mode was reported in the previous work, we
focus on T and Q modes in the following. It is

Fig. 6. ESR spectrum measured at 1.5 K for Ha O, T, andQ denote, one-triplet, two-triplet and quintuplet, respectively. In
the inset, dashed, thick solid, thin solid and dotted lines are O, T,
Q and paramagnetic resonance (not observed at low temperature), respectively.

noticed that the second spin-gap excitations observed as a broad band by neutron scattering experiments are resolved into "ve discrete levels. The
high resolution shows an advantage to use ESR toobserve magnetic excitations. The values of theenergy gaps (in the unit of GHz) are obtained byextrapolating the modes to zero "eld as follows:
1140(T ), 1170(T ), 1190(T ), 1225(T ), 1350(T ),





1390(Q ), 1560(Q ) and 1600(Q ). It should be
noted that linear extrapolation may cause an errorof about 10 GHz for T-modes and about 30 GHz
for Q-mode, for the possible bending of the modes
around zero "eld as observed for the O-modes(neglected in Fig. 6).
In Fig. 6, it is noticed that the relative magnitudes of T-modes to O-modes are not very small.
As mentioned before, the strong suppression of the
one-triplet excitation may cause the shift of thespectral weight from the O-modes to the T-modes.
The geometrical constraint of the triplet hopping
may cause another e!ect; the spectral weightspreading over the continuum concentrate onto
several discrete levels. Next, we discuss the Qmodes. As is well known, a two-triplet bound stateH. Nojiri et al. / Physica B 294}295 (2001) 14}19

19

the interesting non-linear magnetic excitations inthese systems.

Acknowledgements
This work was partly supported by Grant-in-Aid
for Scienti"c Research from the Ministry of Education, Science, Sports and Culture of Japan and by
The Sumitomo Foundation.

References

Fig. 7. Frequency}"eld diagram for Ha.

is an object containing four spins. In the simple
dimer model of four spins with antiferromagentic
couplings, we can expect three di!erent states, suchas singlet (S"0), triplet (S"1), and quintuplet(S"2). It should be noted that the singlet state ofa two-triplet bound state is di!erent from the singlet ground state of the whole system. Singlet statesobserved by means of Raman scattering at "niteenergy may relate to such a level. In a quintuplet,
Zeeman splitting is doubled for S "2 andX
S "!2 branches compared to that of the triplet.
X
Therefore, a Q-mode can be easily identi"ed.
Fig. 7 shows the frequency}"eld diagram obtained at lower frequencies. Besides O-modes,
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its slope is twice that of the O-mode, this mode is
identi"ed as the S "#2 branch of the quintuplet.
X
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