PHYSICAL REVIEW B 67, 174405 共2003兲

Longitudinal magnon in the tetrahedral spin system Cu2Te2O5Br2 near quantum criticality
C. Gros,1 P. Lemmens,2,4 M. Vojta,3 R. Valentı́,1 K.-Y. Choi,4 H. Kageyama,5 Z. Hiroi,5 N. V. Mushnikov,5 T. Goto,5
M. Johnsson,6 and P. Millet7
1

Fakultät 7, Theoretische Physik, Universität des Saarlands, D-66041 Saarbrücken, Germany
2
Max Planck Institute for Solid State Research, D-70569 Stuttgart, Germany
3
Theoretische Physik III, Universität Augsburg, D-86135 Augsburg, Germany
4
2. Physikalisches Institut, RWTH Aachen, D-52056 Aachen, Germany
5
Institute for Solid State Physics, Univ. of Tokyo, Kashiwa-shi, Chiba 277-8581, Kashiwa, Japan
6
Department of Inorganic Chemistry, Stockholm Univ., S-10691 Stockholm, Sweden
7
Centre d’Elaboration de Matériaux et d’Etudes Structurales, CEMES/CNRS, F-31062 Toulouse, France
共Received 28 January 2003; published 8 May 2003兲
We present a comprehensive study of the coupled tetrahedra compound Cu2 Te2 O5 Br2 by theory and experiments in external magnetic fields. We report the observation of a longitudinal magnon in Raman scattering in
the ordered state close to quantum criticality. We show that the excited tetrahedral-singlet sets the energy scale
for the magnetic ordering temperature T N . This energy is determined experimentally. The ordering temperature
T N has an inverse-log dependence on the coupling parameters near quantum criticality.
DOI: 10.1103/PhysRevB.67.174405

PACS number共s兲: 75.30.Gw, 75.10.Jm, 78.30.⫺j

INTRODUCTION

Quantum fluctuations in antiferromagnetic insulators leadto a reduction of the magnetic moment and to a new mode inwhich the magnitude of the local order parameter oscillates:
the longitudinal magnon 共LM兲 共Ref. 1兲 关see Fig. 1共a兲兴. Thiselementary excitation is absent in classical magnets, where
the excitations perform a precession of the moment around
its equilibrium position and are therefore transversally polarized 关see Fig. 1共b兲兴. The longitudinal mode is difficult toobserve and only recently has a LM been detected by inelastic neutron scattering in quantum spin- 21 共Refs. 2 and 3兲 andspin-1 共Refs. 4 and 5兲 chain compounds.
Quantum spin fluctuations are of special importance inquasi-zero-dimensional systems with weakly coupled spinclusters. These lattices allow for quantum phase transitions
between magnetically ordered states and nonmagnetic phaseswith a spin gap.6 The recently discovered7 spin-tetrahedral
compounds Cu2 Te2 O5 X 2 (X⫽Cl, Br兲 have been shown toorder at transition temperatures T N(Cl) ⫽18.2 K and T N(Br)⫽11.4 K which are strongly suppressed with respect to themagnitude of the intratetrahedral couplings.8 Unconventional
Raman scattering has been found in the magnetic channel,8,9and the occurrence of low-lying singlet excitations has been
proposed.7 Plateaus in the magnetization have been predicted
for a related linear chain of spin tetrahedra.10
The nature of the ordered states in Cu2 Te2 O5 X 2 has not
yet been settled. The ordering temperatures T N(Cl) and T N(Br)
decrease and rise, respectively, with an external magnetic
field.8 This unusual magnetic-field-induced stabilization ofT N(Br) motivated in part the present study. A decrease of T N istypical for an antiferromagnet in the classical limit. We willshow later on that T N may rise near a quantum phase transition. This results then indicates Cu2 Te2 O5 Br2 to be close tocriticality.
Here we report the observation of a LM in Cu2 Te2 O5 Br2
by Raman scattering in a magnetic field, and present the0163-1829/2003/67共17兲/174405共5兲/$20.00

evolution of this mode under the influence of an external
magnetic field. We believe that this study constitutes the firsttime that a longitudinal magnon is detected optically, as
well as the first observation of such a mode in a tetrahedralspin system, i.e., in a system with an even number of
spins per unit cell. Furthermore, high-field magnetizationand other thermodynamic data on pure and substituted
Cu2 Te2 O5 (Brx Cl1⫺x ) 2 are compared via a mean-field analysis which allows one to determine the microscopic parameters for Cu2 Te2 O5 Br2 . We find, interestingly, that the scaleof the ordering temperature T N is set by the 共nonmagnetic兲
excited singlet of the copper tetrahedron and that T N has an
essential singularity at criticality.
MEAN-FIELD APPROACH

We assume that the basic spin cluster in this compoundis given by the copper tetrahedron 共see the inset in Fig. 2兲.
We denote by s kl the spin-singlet state of two intratetrahedral sites k and l, and by t ␣kl the respective triplet states,with ␣⫽⫾1,0. We start by considering the eigenstates of theisolated tetrahedra with H t ⫽J 1 关 (S1 ⫹S2 )•(S3 ⫹S4 ) 兴⫹J 2 (S1 •S2 ⫹S3 •S4 ), which consist of two singlets, three
triplets, and one quintuplet.
For J 2 ⬍J 1 the ground state singlet ␺ s1 and the excited
singlet ␺ s2 are

FIG. 1. Schematic representation of 共a兲 a longitudinal magnon
and 共b兲 a transversal magnon.

67 174405-1©2003 The American Physical Society

PHYSICAL REVIEW B 67, 174405 共2003兲

C. GROS et al.

0兩˜␸ 典 ⫽sin ␸ 兩 ␺ s1 典 ⫺cos ␸ 兩 ␺ t1
典,

共4兲with 具 ˜␸ 兩 ␸ 典 ⫽0, and new energies⌬E ␸ , ˜␸ ⫽FIG. 2. Neél temperature 共left scale兲 and spin tetrahedra meanfield moment 共right scale兲 for J 2 /J 1 ⫽0.66. Shown is the analytic
approximation to leading order by Eq. 共7兲 共solid line兲 and the numerical solution of the self-consistency equation 共6兲 共stars兲. The
dashed line 共right scale兲 is the magnetic moment at zerotemperature equation 共8兲. Note the logarithmic singularity in T N for
J c →0.75J 1 . Inset: Cu tetrahedron with exchange couplings J 1 andJ 2 共solid and dashed lines兲.

␺ s1 ⫽

⫺1

冑3

0 0
⫹ ⫺
⫺ ⫹
t 34⫺t 12
t 34⫺t 12
t 34兴 ,
关 t 12

␺ s2 ⫽s 12s 34

0
␺ t1
⫽

⫺
␺ t1
⫽

⫹
␺ t1
⫽

1

冑2
1

冑2
1

冑2

M⫽

(qc)
T N ⯝⌬E s2 log⫺1 关 J (qc)兲兴 .
c / 共 J c ⫺J c

共2兲The quintuplet has the energy ⌬E q ⫽3J 1 . The intertetrahedra
couplings can be described in a mean-field approach by12
H M F ⫽⫺J c M 共 S z1 ⫹S z2 ⫺S z3 ⫺S z4 兲and共3兲with M being the staggered magnetization order parameter.
J c here is the sum over all intertetrahedra couplings.
0,
The mean-field Hamiltonian H M F couples ␺ s1 and ␺ t1
leading to new eigenstates for H⫽H t ⫹H M F ,0兩 ␸ 典 ⫽cos ␸ 兩 ␺ s1 典 ⫹sin ␸ 兩 ␺ t1
典,

2 tan ␸,3 1⫹tan2 ␸

共6兲

The transition temperature T N can be obtained from Eq.
共6兲 by imposing M ⫽0. Assuming that 共i兲 s 2 is the lowest
excited state of a tetrahedron and 共ii兲 at small temperatures
only the leading order in a 1/T expansion is contributing in
Eq. 共6兲, T N can be analytically derived:⫹ 0
0 ⫹t 34⫺t 12
t 34兲 ,
共 t 12

1
M ⫽ 具 S z1 ⫹S z2 ⫺S z3 ⫺S z4 典 ,4冑

RESULTS0 ⫺⫺ 0t 34⫺t 12
t 34兲 ,
共 t 12␣␺ ␣t3 ⫽t 12
s 34 .

e⫺ ␤ E ␸ ⫺e⫺ ␤ E ˜␸
Zwhere ␤ ⫽1/T and Z is the partition function for the coupled
tetrahedra system, i.e., Z⫽e⫺ ␤ E s1 ⫹e⫺ ␤ E s2 ⫹••• . For J c⫽J (qc)c ⫽3J 1 /4 the magnetization M goes to zero and thesystem shows a second-order phase transition at T N .

⫹ ⫺⫺ ⫹t 34⫺t 12
t 34兲 ,
共 t 12␣␣␺ t2⫽s 12t 34,

共5兲

with tan ␸ ⫽⫺⌬E ␸ 冑6/(4J c M ). 兩␸典 is the ground state and兩˜␸ 典 can be identified as a longitudinal magnon excitation.
The physical interpretation of this excitation is as follows.
When J c ⫽0 we have isolated tetrahedra, and 兩 ˜␸ 典 would cor0
respond to the excited intratetrahedral triplet state 兩 ␺ t1
典 . ForJ c ⫽0, 兩 ˜␸ 典 evolves continuously from ␺ t 1 as a function of
the intertetrahedral coupling J c , and becomes soft at thetransition point to the ordered state. The molecular field also
couples ␺ t1 with the quintuplet ␺ q , though we neglect thiscoupling here since we are interested in phases with lowtransition temperatures T N for which the high-energy quintuplet does not contribute significantly.
The calculation of the staggered magnetization M⫽Tr 关 (S z1 ⫹S z2 ⫺S z3 ⫺S z4 )e⫺ ␤ H 兴 /(4Z) 关Eq. 共3兲兴 leads to thefollowing self-consistency equation:

共1兲with eigenenergies E s1 ⫽⫺2J 1 ⫹J 2 /2 and E s2 ⫽⫺3J 2 /2
␣,
⫽E s1 ⫹⌬E s2 , with ⌬E s2 ⫽2J 1 ⫺2J 2 . The three triplets ␺ t1
␣␣
␺ t2 , and ␺ t3 have the 共excitation兲 energies ⌬E t1 ⫽J 1 and⌬E t2 ⫽⌬E t3 ⫽2J 1 ⫺J 2 , with respective eigenstates

J1关 1⫿ 冑1⫹32M 2 J 2c / 共 3J 21 兲兴 ,2

共7兲T N shows an inverse-log singularity close to the quantum. The critical J (qc)
critical point at J c ⫽J (qc)cc ⫽3J 1 /4 is independent of J 2 . In Fig. 2 we plot T N as a function of J c , bothas obtained in the analytic solution Eq. 共7兲 and by numerically solving the self-consistent equation 共6兲. Note that for
the region J c ⬃J (qc)
, Eq. 共7兲 provides a good approximationcfor T N .
The inverse-log dependence of the Néel temperature implies that T N is substantial even near the quantum criticalpoint, as illustrated in Fig. 2, in contrast to the magnitude of
the zero-temperature magnetic moment,
M 共 T⫽0 兲 ⫽

1

冑6

2
冑1⫺ 共 J (qc)c /J c 兲 ,共8兲which has a standard mean-field form13 共compare Fig. 2兲.
For J 2 ⬎J 1 the tetrahedral ground state changes to ␺ s 2 andthe nonmagnetic singlet ␺ s 2 therefore sets the scale for T N .

174405-2

PHYSICAL REVIEW B 67, 174405 共2003兲

LONGITUDINAL MAGNON IN THE TETRAHEDRAL SPIN . . .
STATES IN AN EXTERNAL FIELD

An external longitudinal magnetic field does not induce
additional couplings in between the different eigenstates, but
it leads to shifts in the respective eigenenergies. A transversal
magnetic field B x induces, on the other hand, a coupling in0⫾and ␺ t1
关see Eq. 共2兲兴. The mean-field groundbetween ␺ t1
state, which breaks the rotational invariance, can be written,
in lowest order in B x , as
兩 ␸ ,B x 典 ⫽cos ␣ 兩 ␸ 典 ⫹

sin ␣

冑2

⫹
⫺
关 兩 ␺ t1
典 ⫹ 兩 ␺ t1
典 ],

共9兲

with tan ␣ ⫽B x sin ␸/(J1⫺⌬E␸). In lowest order in B x , theground-state energy ⌬E ␸ ,B x ⫽E ␸ ,B x ⫺E ␸ ,⌬E ␸ ,B x ⫽⫺B 2x sin2 ␸ / 共 J 1 ⫺⌬E ␸ 兲 ,共10兲

decreases quadratically with B x .
This result has an interesting consequence for the transition temperature. The energy of the excited singlet E s2 is notaffected by B x ; its relative energy to the ground state ⌬E s2
increases consequently with B x 关compare Eq. 共10兲兴. Equation
共7兲 then tells us that the Néel temperature also increases
with B x . 15 An order-of-magnitude estimate of the effectfor Cu2 Te2 O5 Br2 and B x ⫽13 T yields ⌬T N ⬇0.8 K⫽0.56 cm⫺1 , which agrees well with the experimentallyobserved raise of ⬃1 K already reported in Ref. 8.
For the longitudinal magnon, a calculation analogous to
Eq. 共10兲 leads to⌬E ˜␸ ,B x ⫽⫺B 2x cos2 ␸ / 共 J 1 ⫺⌬E ˜␸ 兲 ,共11兲

The resulting change ⌬E ˜␸ ,B x ⫽E ␸˜ ,B x ⫺E ␸˜ in thelongitudinal-magnon energy is positive, as J 1 ⫺⌬E ˜␸ ⬍0, andsubstantially larger than the shift for the ground state ⌬E ␸ ,B x
共and correspondingly for E s2 ) since cos2␸Ⰷsin2␸. As weshall discuss in the next sections, this trend is qualitatively inagreement with the Raman data presented below.

FIG. 3. Mean-field results for the specific heat of spin tetrahedra coupled by J c . The inset shows the specific heat of
Cu2 Te2 O5 (Brx Cl1⫺x ) 2 with x⫽1, 0.66, and 0. The data with x⫽0
and 1 are compiled from Ref. 8.

field results for increasing J c reproduce the continuous shiftof the specific heat anomaly to higher temperatures with decreasing x.
A further support to the interpretation of these systems asthat of coupled tetrahedra with a mean-field J c intertetrahedra coupling and with a T N ordering temperature is obtained
from high-field magnetization measurements which are presented in Fig. 4. We observe a finite slope for all samples at
small fields which increases with decreasing x, and no direct
evidence for a plateau in the magnetization curve is found. A
plateau at M ⫽1/2 is predicted for a one-dimensional chainof spin tetrahedra with parameters placing the system in the
gapped phase.10 The absence of such a plateau in our experimental observations would support the fact that these systems cannot be described as chains of tetrahedra.11 The finite
slope in M (H) at small fields and even for x⫽1 is intrinsic,
and points to the underlying weak Néel state. The corresponding anisotropy is observed in single crystal susceptibil-

SUBSTITUTION EXPERIMENTS

Specific heat and high-field magnetization have beenmeasured on Cu2 Te2 O5 (Brx Cl1⫺x ) 2 powder samples with x⫽1,0.75,0.66, and 0.0.7,8 Substituting Br by Cl leads to acontinuous decrease of the unit cell volume by 7% (x⫽0)
and an increase of the transition temperature from 11.4 to18.2 K. Also, other physical properties change continuouslywith substitution.14 In our coupled-cluster model we expectthat the decrease of the unit cell volume causes an increaseof the coupling constant J c . We have calculated the specificheat
C v ⫽ ␤ 2 具 共 H⫺ 具 H 典 兲 2 典

共12兲

in the mean-field approximation. In Fig. 3 the results areshown for various J c values. In the inset of Fig. 3 the evolution of the experimentally determined specific heat as afunction of substitution x is presented. Note that the mean-

FIG. 4. High-field magnetization of Cu2 Te2 O5 (Brx Cl1⫺x ) 2
powder samples for x⫽1, 0.75, and 0. The data with x⫽0.66 areomitted here for clarity. Dotted lines correspond to high field extrapolations. The kink in M (H,x⫽1) at H SF⫽13.2 T is interpreted
as a spin-flop transition. The inset shows the anisotropic magnetic
susceptibility of Cu2 Te2 O5 Br2 single crystals.

174405-3

PHYSICAL REVIEW B 67, 174405 共2003兲

C. GROS et al.

mode and as an excitation to the second singlet ␺ s2 in Eq.
共1兲, respectively.
EXCITED SINGLET

We interpret the additional higher energy mode at ␯ sing
⫽23.2 cm⫺1 presented in Fig. 5 as a transition to the second
singlet ␺ s2 ⫽s 12s 34 . Since this system is noncentrosymmetric, there is a nonzero Dzyaloshinski-Moriya 共DM兲 共Ref. 16兲
interaction. Assuming a DM contribution to the Raman operator, i.e., H R(DM ) ⬃Di j •(Si ⫻Sj) we find a nonzero Raman
matrix element

具 ␺ s2 兩 H R(DM ) 兩 ␸ ,B x 典 ⬃sin ␣ ⬃B x M .
FIG. 5. Raman spectra of Cu2 Te2 O5 Br2 in a magnetic field. The
insets show a兲 spectra with higher resolution and b兲 the shift of the␯ sing ⫽23.2 cm⫺1 mode 共upper open symbols兲 and the ␯ long
⫽16.3 cm⫺1 mode 共lower full and open symbols兲 as a function of
the magnetic field. The dashed line shows the field dependence ofthe transition temperature 共Ref. 8兲, and the full line is a fit to thedata proportional to the square of the magnetic field. The open 共full兲symbols show data with high resolution 共normal resolution兲.

ity shown in the inset of Fig. 4. The transition is evident as akink in ␹ (T) with a magnetic field perpendicular to thec-axis.
RAMAN SPECTROSCOPY

We have performed Raman scattering experiments on
c-axis oriented single crystals with diameters ⬇0.2 mm
and length l⬇1 mm. The used scattering geometry in 共cc兲light scattering polarizations corresponds to A symmetry.8
The magnetic field has been applied perpendicular to thelight scattering polarization. In Fig. 5 we present Raman data
for Cu2 Te2 O5 Br2 in magnetic fields up to 6 T. We observe ashift of the low-energy magnetic mode at ␯ long⫽16.3 cm⫺1
⫽23.4 K 共for B⫽0) to higher energies as a function of B
and the appearance of an additional, magnetic field-induced
mode at ␯ sing⫽23.2 cm⫺1 ⫽33.2 K (B⫽0). In the inset Fig.
5共a兲 the low energy spectrum is displayed, with a smaller
optical slit width and a higher spectral resolution. The intensity of the higher energy mode increases with field. It is notobservable for B⫽0.
In Fig. 5共b兲 the energies of the respective modes areshown as a function of the magnetic field. While the higher
energy mode does not show an appreciable magnetic fielddependence 共upper open symbols兲, the lower energy modeshows a nonlinear dependence on the magnetic field 共lower
full and open symbols兲. The full line in Fig. 5共b兲 is a fit to thedata proportional to the square of the magnetic field. The
dashed curve describing the higher energy mode is proportional to the weaker, positive field dependence of the transition temperature as determined by specific heat and magnetic
susceptibility measurements.8 In the following we shall argue that these two modes at ␯ long⫽16.3 cm⫺1 and ␯ sing
⫽23.2 cm⫺1 can be identified as a longitudinal magnon

The 23.2-cm⫺1 -mode would therefore be observable only inthe ordered phase and in an external magnetic field, consistent with experiment. Having identified this mode with thetransition to ␺ s2 we have then that⌬E s2 ⫽2J 1 ⫺2J 2 ⫽ ␯ sing⬃33 K.

共13兲

Considering Eq. 共13兲 and the magnetic susceptibility7,8 forT⬎T N we find a good fit with J 1 ⬃47 K and J 2 ⬃31 K
which yields J 2 /J 1 ⬃0.66. The experimental transition
temperature8 of T N(Br) ⫽11.4 K for Cu2 Te2 O5 Br2 implies via
self-consistency condition 共6兲, that J c ⬃0.85J 1 .
Moreover, recalling Eqs. 共7兲 and 共10兲, for B x ⫽6 T theenergy ⌬E s2 of the excited singlet shifts by about
0.12 cm⫺1 . This increase is too small to be resolved by Raman, although the data presented in inset 共b兲 of Fig. 5 seemto indicate a small increase.
LONGITUDINAL MAGNON

We observe that the mean-field Hamiltonian 关Eq. 共3兲兴leads to a Q⫽0 ordering for J c ⬎0, and that the soft longitudinal magnon 兩 ˜␸ 典 should be directly observable in Raman
scattering. For J c ⬍0 ordering with Q⫽␲ would occur andadditional backfolding to the zone center via residual lattice
distortions would be necessary. The matrix element
具 ˜␸ 兩 H R 兩 ␸ 典 of the Raman operator H R ⬃Si •S j (i, j⫽1, . . . ,4)is ⬃cos ␸ sin ␸. It vanishes in the decoupled-tetrahedra limit
J c ⫽0, ␸⫽0, and the transition should be observable only inthe ordered phase.
Summarizing, the Raman mode at ␯ long ⫽16.3 cm⫺1
⫽23.4 K in Fig. 5 共i兲 has been shown8 to become soft at theordering temperature, and 共ii兲 it is observable only in thecondensed phase and its energy increases quadratically关compare Eq. 共11兲兴 with the field. We therefore interpret it asa longitudinal magnon.
The energy of this mode is strongly suppressed below its
mean-field energy E ˜␸ ⫺E ␸ ⫽54 K by dispersion. We can estimate the magnitude of this suppression by comparison withthe results of a bond-operator theory for a coupled dimer
system17,18 共alternatively one may use a generalized random
phase approximation approach19兲. The effective dimer states␣). The gap of the longitudinal magnon has theare ( ␺ s1 , ␺ t1
17
form

174405-4

PHYSICAL REVIEW B 67, 174405 共2003兲

LONGITUDINAL MAGNON IN THE TETRAHEDRAL SPIN . . .
2⌬ long⫽⌬ max冑1⫺ 共 J (qc)c /J c 兲 .

共14兲

2
For Cu2 Te2 O5 Br2 we have J c ⫽0.85J 1 关 (J (qc)c /J c ) ⫽0.78兴
and ⌬ long⬇0.47⌬ max . The energy scale ⌬ max occuring in Eq.
共14兲 is set by the longitudinal magnon gap in the classicalNéel ordered state, i.e., in the limit of strong interdimer 共tetrahedra兲 couplings, where it has the value ⌬ max→J c . WithJ c ⬇0.85J 1 and J 1 ⬇47 K we then find for Cu2 Te2 O5 Br2 that⌬ long⬇19 K, which is qualitatively in agreement with the(expt)⫽ ␯ long⫽23.4 K; see the inset, Fig.
experimental value ⌬ long
5共b兲. Due to the renormalization of ⌬ long by the fluctuations
near the quantum-critical point 关see Eq. 共14兲兴, we cannot
easily quantitatively estimate its dependence on an external
magnetic field, as presented in inset 共b兲 of Fig. 5.

COMPARISON WITH Cu2Te2O5Cl2

The magnitude of the intratetrahedral parameters havebeen estimated7 from susceptibility measurements to be similar both for Cu2 Te2 O5 Br2 and the isostructural
Cu2 Te2 O5 Cl2 . The substantially enhanced Néel temperature8of T N(Cl) ⫽18.2 K indicates a larger interdimer coupling for
the Cl compound. This notion is consistent with the Raman
results for doped compounds, which indicate a hardening of
the excitations with the Cl content, as predicted by Eq.
共14兲.14 We did not attempt a quantitative analysis of the coupling parameters for Cu2 Te2 O5 Cl2 since we were not able to

I. Affleck and G.F. Wellman, Phys. Rev. B 46, 8934 共1992兲; H.J.
Schulz, Phys. Rev. Lett. 77, 2790 共1996兲; F.H.L. Essler, A.M.
Tsvelik, and G. Delfino, Phys. Rev. B 56, 11 001 共1997兲.
2
B. Lake, D.A. Tennant, and S.E. Nagler, Phys. Rev. Lett. 85, 832
共2000兲.
3
A. Zheludev, M. Kenzelmann, S. Raymond, T. Masuda, K. Uchinokura, and S.-H. Lee, Phys. Rev. B 65, 014402 共2002兲.
4
S. Raymond et al. Phys. Rev. Lett. 82, 2382 共1999兲.
5
M. Enderle, Z. Tun, W.J.L. Buyers, and M. Steiner, Phys. Rev. B
59, 4235 共1999兲.
6
P. Lemmens, C. Gros, and G. Güntherodt, Phys. Rep. 375, 1
共2003兲.
7
M. Johnsson, K.W. Törnroos, F. Mila, and P. Millet, Chem. Mater.
12, 2853 共2000兲.
8P. Lemmens, K.-Y. Choi, E.E. Kaul, C. Geibel, K. Becker, W.
Brenig, R. Valenti, C. Gros, M. Johnsson, P. Millet, and F. Mila,
Phys. Rev. Lett. 87, 227201 共2001兲.
9
W. Brenig and K.W. Becker, Phys. Rev. B 64, 214413 共2001兲.
10
K. Totsuka and H-J. Mikeska, Phys. Rev. B 66, 054435 共2002兲.
11
R. Valentı́, T. Saha-Dasgupta, C. Gros, and H. Rosner1

observe the second singlet as in Cu2 Te2 O5 Br2 . Indeed, ageneralized tight-binding analysis of band-structure
calculations11 indicates that the ratio of the intradimer couplings J 2 /J 1 in Cu2 Te2 O5 Cl2 is smaller than in
Cu2 Te2 O5 Br2 . These findings also suggest that for
Cu2 Te2 O5 Cl2 the excited singlet with energy E s2 ⫽2J 1⫺2J 2 should probably be located in the energy range of themagnetic continuum and thus not be observable separately.
CONCLUSIONS

We have presented a comprehensive set of theoretical andexperimental data indicating that the isostructural spintetrahedral compounds Cu2 Te2 O5 (Brx Cl1⫺x ) 2 constitute a
series of systems with a systematic variation of the microscopic parameters with respect to a quantum critical transition. We have pointed out the importance of the low-lying
singlet for the magnetic state, and reported the observation ofa low-energy longitudinal magnon.
ACKNOWLEDGMENT

We acknowledge the support of the German Science
Foundation 共DFG SP1073, SFB484兲, INTAS 01-278, andimportant discussions with Wolfram Brenig, Frederic Mila,
Jens Jensen, and T. Saha-Dasgupta.

cond-mat/0301119 共unpublished兲.
Other possible ordering patterns, like 具 S z1 ⫹S z2 ⫹S z3 ⫹S z4 典 , lead to
first-order phase transitions.
13
For J c /J 1 →⬁ the coupling to the quintuplet ␺ q neglected in Eq.
共8兲 would contribute and the zero-T moment would take themean-field value of 1/2.
14
P. Lemmens, K.-Y. Choi, A. Ionescu, J. Pommer, G. Güntherodt,
R. Valentı́, C. Gros, W. Brenig, M. Johnsson, P. Millet, and F.
Mila, J. Phys. Chem. Solids 63Õ6-8, 1115 共2002兲.
15
Note that there is a smaller decreasing contribution in B x to theNéel temperature coming from the log-term in Eq. 共7兲 due to an
enhancement of the effective J 1 .
16
I.J. Dzyaloshinskii, J. Phys. Chem. Solids 4, 241 共1958兲; T.
Moriya, Phys. Rev. 120, 91 共1960兲.
17
T. Sommer, M. Vojta, and K.W. Becker, Eur. Phys. J. B 23, 329
共2001兲.
18
B. Normand and T.M. Rice, Phys. Rev. B 56, 8760 共1997兲.
19
J. Jensen and A.R. Mackintosh, Rare Earth Magnetism: Structures and Excitations 共Oxford University Press, Oxford, 1991兲.
12174405-5