Magnetism of a tetrahedral cluster-chain

2 Pages·0.09 MB·English

by  Wolfram Brenig

Magnetism of a tetrahedral cluster-chain
Wolfram Brenig
Institut fu¨r Theoretische Physik, Technische Universita¨t Braunschweig, 38106 Braunschweig, Germany
Klaus W. Becker
Institut fu¨r Theoretische Physik, Technische Universita¨t Dresden, 01062 Dresden, Germany
1
0 Peter Lemmens
0
II. Physikalisches Institut, RWTH Aachen,52056 Aachen, Germany
2
(February 1, 2008)

n
a
J
the ground state is found3 to be in the sector of the
0 Magnetic properties of a completely frustrated tetrahe-
infinite-length, dimerized spin-1 chain, i.e. T = 1 for
3 dral chain are summarized. Usingexact diagonalization, and il
all i,l, with a dimer phase at b < b and a Haldane
c
] bond-operator theory results for the ground-state phase di- phase for b > bc. For a > ac(b) the infinite-length
l agram, the one-triplet excitations and the Raman spectrum product state of singlets, i.e. T = 0 for all i,l is re-
e are given. Thelink to noveltellurate materials is clarified. il
- alized. In the latter case the ground state energy is
r
t PACS numbers: 75.10.Jm, 75.40.-s, 75.40.Mg, 75.50.Ee EG = −N3a/2. In the former case we have used ex-
s
act diagonalization(ED) on up to 2N =16 sites, as well
.
t as bond-boson theory to determine the ground state en-
a
m Recently, tellurates of type Cu Te O X with X=Cl, ergyandphaseboundaries. The phasediagramis shown
2 2 5 2
- Br have been identified as a new class of spin- in fig. 2. Since, by T2l(1l+1) → T1l(2l), (1) is symmet-
d 1/2 tetrahedral-cluster compounds1. Bulk thermody-
1
n namic data has been analyzed in the limit of isolated
o tetrahedra1. Raman spectroscopy, however indicates 0.8 Haldane MFT N=16
c
substantialinter-tetrahedralcoupling2. Inthisbriefnote
[
v1 wdiemseunmsimonaarliz(e1rDe)suclthsaionnothfetemtraaghneedtrisamwohficahpiusrecloyuopnleed- = J/J3100..46 Singlet product
in a geometry analogous to that along the c-axis di- b
0 LHP
rection of Cu Te O X . In this direction the exchange Dimer
6 2 2 5 2 0.2
4 topology is almost completely frustrated suggesting the
1 spin-model of fig. 1. The hamiltonian can be written 0
0 0.5 1 1.5 oo
0 a = J /J
1 1 J 4 J 2 1
1 3 FIG.2. Phases of the tetrahedral chain. Solid lines: exact
0 ... ... diagonalization. Haldane-Dimertransitionatb≈3/5. Stared
/ J
t 2 (circled) dashed lines: bond-boson mean-field/MFT (Hol-
a
stein-Primakoff/LHP)approach. LHPterminatesatb=3/8.
m
3 2
d- l ric under (J1,a,b) → (J1b,a/b,1/b), a correspondingly
n rescaled mirror image of fig. 2 exists. The combination
FIG.1. Thetetrahedralcluster-chain. llabelstheunitcell
o ofbothcoversthecompleteparameterspace. Thecritical
containing spin-1/2 momentsat the vertices 1,...,4.
c value of a (b = 0) = 1 for the 1st-order transition from
c
v: thedimertothesingletproductstateagreeswith1,while
i as a 1D chain in terms of the total edge-spin operators a2N=16(b=1)=1.403...agreeswith4. Forthe2nd-order
X T =S +S and the dimensionless couplings c
1(2)l 1(4)l 3(2)l dimer–Haldane transition we find b ≃ 3/5 from finite–
c
r b=J3/J1 and a=J2/J1 size extrapolation3, which is consistent with5.
a
In addition to ED, fig. 2 displays results of an
H a 3a
J =X[T1lT2l+bT2lT1l+1+ 2(T21l+T22l)− 2 ] (1) analytic bond-boson approach to the dimerized spin-1
1 l chain sector. Labeling the singlet, triplet and quin-
tet states of a tetrahedron by bosons s,t ,q , with
α α
This model displays infinitely many local conservation
the unit-cell index suppressed, and discarding the high-
laws: [H,T2i(=1,2)l] = 0; ∀l,i = 1,2. The Hilbert space energyquintetstheedge-spinscanbereplacedbyTα =
1/2
decomposes into sectors of fixed distributions of edge-
spineigenvaluesT =1or0,eachcorrespondingtoase- ±p2/3(t†αs+s†tα)−iǫαβγt†βtγ/2. This transforms (1)
il into an interacting bose gas including a hardcore con-
quences of dimerized spin-1 chain-segments intermitted
straint s†s+t†t +q†q = 1. Condensing the singlets,
by chain-segments of localized singlets. For a < a (b) α α α α
c
1
i.e. s(†) = hsi, to either hsi = 1 (Linear Holstein Pri- the continuum to correspond to that of fig. 3b). How-
makoff(LHP)approximation)ortoaselfconsistentlyde- ever,themeasuredcontinuumisdefinitelymoresymmet-
termined mean field(MFT)–value hsi< 1 the model can ric than the solid line in fig. 3b). This leaves the mag-
bediagonalizedonthequadraticlevel3,6. Contrastingthe netism of the tellurates an open issue deserving further
resultinggroundstateenergyagainstthesingletproduct studies.
state the stared(circled)-dashed phase boundaries of fig. Acknowledgements: This work has been presented
2 are obtained. In the dimer-phase region the agree- at the ’Japanese-German bilateral Seminar’, September
ment with ED is very good, both for LHP and MFT. In 2000, in Sapporo, Japan.
principle,thesingletcondensaterestrictsthebond-boson It is a pleasure to thank R. Valenti, C. Gros,
approaches to the dimer phase. In fact, the LHP spin- and F. Mila for stimulating discussions and comments.
gapclosesatb=3/8confining the LHP to b<3/8<b . This research was supported in part by the Deutsche
c
TheMFTcanbeextendedintotheHaldaneregime,even Forschungsgemeinschaftunder Grant No. BR 1084/1-1.
though the ground-state symmetries are different, yield-
ingatransitionlinequalitativelystillcomparabletoED.
0 π k 2π
1.5
1
/Jk 1 a) 1M.Johnsson,K.W.T¨ornross,F.Mila,andP.Millet,Chem.
E 0.5 S N=14, Sz=0 ____ LHP _ _ _ MFT Mater. 12, 2853 (2000)
b=0.1 0.2 0.3 2P.Lemmens, et al., to be published
100 3W.Brenig, et al., to be published
ω) a.u. 468 __ ___ __ Tba−rmeatr2ix∆ b) 54(YA2..0K0H0ao)tn.oe,ckAe.r,TFan.aMkail,aJ,.MP.hTysr.oySeorc,.EJupr..6P3h,y1s2.7J7.(B1.99135), 227
ST
I( 2 6S.SachdevandR.N.Bhatt,Phys.Rev.B41,9323(1990);
00 A.V.ChubukovandTh.Jolicoeur, Phys.Rev.B44,12050
0 0.5 1 ω/J 1.5 2 2.5 (1991); W. Brenig, Phys.Rev.B 56, 14441 (1997).
1 7O.P. Sushkov and V.N. Kotov Phys. Rev. Lett. 81, 1941
FIG. 3. (a) Low lying excitations in the dimerized spin-1
(1998); C. Jurecka and W. Brenig Phys. Rev. B 61, 14307
chain sector. (b) Two-magnon Raman spectrum.
(2000).
Nextweconsiderexcitationsinthe dimerphasewhich
would be a likely candidate for the tellurates assuming
weakly coupled tetrahedra. The excited states may (i)
remain in the dimerized spin-1 chain sector, or (ii) in-
volvetransitionsinto sectorswithlocalized edge-singlets.
As has been pointed out in1, for a single tetrahedron,
the energy of a pair of two type-(ii) excitations resides
in the spin-gap of sector (i) for 1/2 ≤ a ≤ 1. While
analogous, dispersionless singlets are found in the spin
gap of the dimer phase of the lattice model3, we only
report on one- and two-triplet excitations of type (i) in
this brief note: Figure 3a) displays the dispersion of the
first triplet, comparing ED, LHP, and MFT. The agree-
ment is very good. Figure 3b) displays the magnetic
Raman spectrum, i.e. total spin-zero, two-triplet excita-
tions, based on bond-boson theory and a Loudon-Fleury
vertex with c-axis polarization. For the solid line triplet
interactions havebeen accountedfor exactly onthe two-
particle level beyond the LHP approach by a T-matrix
calculation3,7. Due to the existence of a singlet bound-
state which merges with the continuum at zero momen-
tumtherenormalizedRamanspectrumdeviatesstrongly
from the bare one.
Available Raman data on Cu Te O Br 2 displays a
2 2 5 2
sharp mode at 20cm−1 and a continuum centered at
60cm−1. One might speculate the sharp mode to corre-
spond to transitions of the aforementioned type (ii) and
2