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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.Törnross,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 excitations, 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 correspond to transitions of the aforementioned type (ii) and 2