Physica B 312-313 (2002) 594–596

Magnetism of a tetrahedral spin-chain

Wolfram Breniga,*, Klaus W. Beckerb, Peter Lemmensc

a Institut f .ur Theoretische Physik, TU Braunschweig, Mendelssohnstr. 3, 38106 Braunschweig, Germanyb Institut f .ur Theoretische Physik, TU Dresden, Dresden, Germanyc II. Physikalisches Institut, RWTH Aachen, Aachen, Germany

Abstract

Magnetic properties of a completely frustrated tetrahedral chain are summarized. Using exact diagonalization, and

bond-operator theory results for the ground-state phase diagram, the one-triplet excitations and the Raman spectrum

are given. The link to novel tellurate materials is clarified. r 2002 Published by Elsevier Science B.V.

Keywords: Quantized spin models; Antiferromagnetics; Quantum critical points

Recently, tellurates of type Cu2Te2O5X2 with X ¼ Cl;Br have been identified as a new class of spin-1

2

tetrahedral-cluster compounds [1]. Bulk thermodynamic

data has been analyzed in the limit of isolated tetrahedra

[1]. Raman spectroscopy, however indicates substantial

inter-tetrahedral coupling [2]. In this brief note we

summarize results on the magnetism of a purely one-

dimensional (1D) chain of tetrahedra which is coupled in

a geometry analogous to that along the c-axis direction

of Cu2Te2O5X2: In this direction the exchange topology

is almost completely frustrated suggesting the spin-

model of Fig. 1. The hamiltonian can be written as a 1D

chain in terms of the total edge-spin operators T1ð2Þl ¼S1ð4Þl þ S3ð2Þl and the dimensionless couplings b ¼ J3=J1and a ¼ J2=J1

H

J1¼

Xl

T1lT2l þ bT2lT1lþ1 þa

2ðT2

1l þ T22lÞ �

3a

2

� �:

ð1Þ

This model displays infinitely many local conservation

laws: ½H ;T2ið¼1;2Þl � ¼ 0; 8l; i ¼ 1; 2: The Hilbert space

decomposes into sectors of fixed distributions of edge-

spin eigenvalues Til ¼ 1 or 0, each corresponding to a

sequences of spin-1 chain-segments with bond-alterna-

tion intermitted by chain-segments of localized singlets.

For aoacðbÞ the ground state is found [3] to be in the

sector of the infinite-length, spin-1 chain with bond-

alternation, i.e. Til ¼ 1 for all i; l; with a dimer phase at

bobc and a Haldane phase for b > bc: For a > acðbÞ theinfinite-length product state of singlets, i.e. Til ¼ 0 for all

i; l is realized. In the latter case the ground state energy isEG ¼ �N3a=2: In the former case we have used exact

diagonalization (ED) on up to 2N ¼ 16 sites, as well as

bond-boson theory to determine the ground state energy

and phase boundaries. The phase diagram is shown in

Fig. 2. Since, by T2lð1lþ1Þ-T1lð2lÞ; Eq. (1) is symmetric

under ðJ1; a; bÞ-ðJ1b; a=b; 1=bÞ; a correspondingly re-

scaled mirror image of Fig. 2 exists. The combination of

both covers the complete parameter space. The critical

value of acðb ¼ 0Þ ¼ 1 for the 1st-order transition from

the dimer to the singlet product state agrees with Ref.

[1], while a2N¼16c ðb ¼ 1Þ ¼ 1:403y agrees with Ref. [4].

For the 2nd-order dimer–Haldane transition we find

bcC35from finite-size extrapolation [3], which is con-

sistent with Ref. [5].

In addition to ED, Fig. 2 displays results of an

analytic bond–boson approach to the spin-1 chain sector

with bond-alternation. Labeling the singlet, triplet and

quintet states of a tetrahedron by bosons s; ta; qa; withthe unit-cell index suppressed, and discarding the high-

energy quintets the edge-spins can be replaced by Ta1=2 ¼

7ffiffiffiffiffiffiffiffi2=3

pðtwas þ swtaÞ � ieabgt

wbtg=2: This transforms Eq. (1)

into an interacting bose gas including a hardcore

constraint sws þ twata þ qwaqa ¼ 1: Condensing the sing-

lets, i.e. sðwÞ ¼ /sS; to either /sS ¼ 1 (Linear Holstein

Primakoff (LHP) approximation) or to a selfconsistently

determined mean field(MFT)-value /sSo1 the model

can be diagonalized on the quadratic level [3]. Contrast-*Corresponding author.

E-mail address: w.brenig@tu-bs.de (W. Brenig).

0921-4526/02/$ - see front matter r 2002 Published by Elsevier Science B.V.

PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 1 4 7 8 - 8

ing the resulting ground state energy against the singlet

product state the stared(circled)-dashed phase bound-

aries of Fig. 2 are obtained. In the dimer-phase region

the agreement with ED is very good, both for LHP and

MFT. In principle, the singlet condensate restricts the

bond–boson approaches to the dimer phase. In fact, the

LHP spin-gap closes at b ¼ 38confining the LHP to

bo38obc: The MFT can be extended into the Haldane

regime, even though the ground state symmetries are

different, yielding a transition line qualitatively still

comparable to ED.

Next, we consider excitations in the dimer phase

which would be a likely candidate for the tellurates

assuming weakly coupled tetrahedra. The excited states

may (i) remain in the spin-1 chain sector with bond-

alternation, or (ii) involve transitions into sectors with

localized edge-singlets. As has been pointed out in Ref.

[1], for a single tetrahedron, the energy of a pair of two

type-(ii) excitations resides in the spin gap of sector (i)

for 12pap1: While analogous, dispersionless singlets are

found in the spin gap of the dimer phase of the lattice

model [3], we only report on one- and two-triplet

excitations of type (i) in this brief note: Fig. 3a displays

the dispersion of the first triplet, comparing ED, LHP,

and MFT. The agreement is very good. Fig. 3b displays

the magnetic Raman spectrum, i.e. total spin-zero, two-

triplet excitations, based on bond-boson theory and a

Loudon–Fleury vertex with c-axis polarization, i.e. for

the electric-field vector of the incoming and outgoing

light parallel to the c-axis. As the Raman susceptibility is

a two-triplet propagator, on the two-particle level, we

can account for quartic triplet-interactions beyond the

LHP approach exactly by summing all ladder-diagrams,

i.e. the T-matrix [3]. Due to the existence of a singlet

bound-state which occurs in the T-matrix, and which

merges with the continuum at zero momentum the

renormalized Raman spectrum deviates strongly from

the bare one. The sharp cut-off at either end of the

continuum is due to the LHP triplet-propagators

displaying a single, coherent and dispersive pole.

Available Raman data on Cu2Te2O5Br2 [2] displays a

sharp mode at 20 cm�1 and a continuum centered at

60 cm�1: One might speculate the sharp mode to

correspond to transitions of the aforementioned type

(ii) and the continuum to correspond to that of Fig. 3b.

However, the measured continuum is definitely more

symmetric than the solid line in Fig. 3b. This leaves the

magnetism of the tellurates an open issue deserving

further studies.

Uncited reference

[6]

Acknowledgements

It is a pleasure to thank R. Valenti, C. Gros, and

F. Mila for stimulating discussions and comments. This

research was supported in part by the Deutsche

Forschungsgemeinschaft under Grant No. BR 1084/1-2.

J1

J2

J3

23

1 4

... ...

lFig. 1. The tetrahedral cluster-chain. l labels the unit cell

containing spin-12moments at the vertices 1;y; 4:

0 0.5 1 1.5a = J

2/J

1

0

0.2

0.4

0.6

0.8

1

b =

J3/J

1

Dimer

Haldane

Singlet product

oo

MFT

LHP

N=16

Fig. 2. Phases of the tetrahedral chain. Solid lines: exact

diagonalization. Haldane–Dimer transition at bE35: Stared

(circled) dashed lines: bond-boson mean-field/MFT (Holstein–

Primakoff/LHP) approach. LHP terminates at b ¼ 38:

0 π 2πk

0

0.5

1

1.5

Ek/J

1

0 0.5 1 1.5 2 2.5ω/J

1

0

5

10

I(ω

) a

.u.

0

N=14, Sz=0 _ _ _ MFT____ LHP

b=0.1 0.2 0.3

2∆ST

_ _ _ bare____ T−matrix

(a)

(b)

Fig. 3. (a) Low lying excitations in the spin-1 chain sector with

bond-alternation. (b) Two-magnon Raman spectrum.

W. Brenig et al. / Physica B 312-313 (2002) 594–596 595

References

[1] M. Johnsson, et al., Chem. Mater. 12 (2000) 2853.

[2] P. Lemmens, et al., in preparation.

[3] W. Brenig, K.W. Becker, in preparation.

[4] A. Honecker, F. Mila, M. Troyer, Eur. Phys. J. B. 15 (2000)

227.

[5] Y. Kato, A. Tanaka, J. Phys. Soc. Japan 63 (1993)

1277.

[6] W. Brenig, Phys. Rev. B 56 (1997) 14441.

W. Brenig et al. / Physica B 312-313 (2002) 594–596596