Magnetic excitations in quasi-two-dimensional spin-Peierls systems
Post on 11-Apr-2017
PHYSICAL REVIEW B 1 DECEMBER 1997-IIVOLUME 56, NUMBER 22Magnetic excitations in quasi-two-dimensional spin-Peierls systems
Wolfram BrenigInstitut fur Theoretische Physik, Universitat zu Koln Zulpicher Str. 77, 50937 Koln, Germany
~Received 19 May 1997!
A study is presented of a two-dimensional frustrated and dimerized quantum spin system which models theeffect of interchain coupling in a spin-Peierls compound. Employing a bond-boson method to account forquantum disorder in the ground state the elementary excitations are evaluated in terms of gapful triplet modes.Results for the ground-state energy and the spin gap are discussed. The triplet dispersion is found to be inexcellent agreement with inelastic neutron scattering data in the dimerized phase of the spin-Peierls compoundCuGeO3. Moreover, consistent with these neutron scattering experiments, the low-temperature dynamic struc-ture factor exhibits a high-energy continuum split off from the elementary triplet mode.@S0163-1829~97!01642-1#m
The recent discovery of the inorganic spin-Peierls copounds CuGeO3 ~Ref. 1! and a8-NaV2O5 ~Refs. 2 and 3!has greatly stimulated interest in low-dimensional magtism. While many properties of these materials can bescribed in terms of quasi-one-dimensional ~1D! quantumspin systems, clear evidence for a substantial degree oftwo-dimensionality~2D! of their magnetism has been found most noteworthy by inelastic neutron scattering~INS! whichdisplays a sizable transverse dispersion of the magneticcitations in CuGeO3.
46 In the present study I will establisa simple framework to interpret the spin dynamics of a frtrated and dimerized 2D quantum spin model with a partilar focus on the low-temperature phase of CuGeO3.
CuGeO3 is an inorganic spin-Peierls system with a lattidimerization transition at a temperatureTSP.14 K.1,48 Itsstructure comprises of weakly coupled CuO2 chains alongthe c axis, with copper in a spin-1/2 state.9,10 The nearest-neighbor ~NN! exchange coupling between copper spalong the CuO2 chains is strongly reduced by almost othogonal intermediate oxygen states.11 Therefore, next-nearest-neighbor~NNN! exchange in CuGeO3 is relevant.Both, NN and NNN exchanges are antiferromagnetic11,12
~AFM! implying intrachain frustration. In addition, NN awell as NNNinter-chain exchange is present which proceevia the O2 sites.11,13 This exchange is believed to be onorder of magnitude less than the intrachain coupling46,11,13
and comparable toTSP. Therefore the interchain couplinshould be relevant in the dimerized phase. This may be aelement in the INS~Refs. 47! and magnetic Ramanscattering1416 data.
A minimal model of CuGeO3 which includes intra- aswell as interchain interactions is theJ-l-a-m-b model11,17
depicted in Fig. 1. The various line segments labelcoupling strengthsJ, Jl, Jla, Jm, and Jmb between thespins located at the vertices in this figure.J refers to thestrongest or dimer bond the left vertices of which formthe dimer latticelPD. Most important the dimerization inFig. 1 is staggered along theb axis. This is realized both inCuGeO3 ~Ref. 10! as well as ina8-NaV2O5,
18 and turns out560163-1829/97/56~22!/14441~8!/$10.00-
to be relevant for the magnon dispersion. In Fig. 1 an adtional, so-called natural labeling of the intrachain parameters is introduced, i.e.,J1 ,a , and d. This notation is fre-quently used in the context of the 1D dimerized afrustrated spin-chain limit. In CuGeO3 J1 is approximately160 K.19,20 Consensus on the precise magnitude of the inchain frustration ratioa is still lacking. Studies of the magnetic susceptibility, which has been compared only tomodels, have resulted ina'0.24 ~Ref. 19! as well as ina'0.35.20 This would place CuGeO3 in the vicinity of thecritical valueac.0.2411 for the opening of a spin gap soledue to frustration.21 d resembles the lattice dimerizatiowhich is finite for T,TSP only. Values for the zero-temperature dimerizationd(T50) ranging from 0.21 to0.012 have been suggested.11,19,20,22Knowledge on the mag-nitude ofm andmb is limited to umu,umbu!1.11,13
Magnetic excitations in CuGeO3 are clearly distinctamong the uniform~U!, i.e.,T.TSP, and the dimerized~D!,i.e., T,TSP, phase. While the dynamic structure factor ehibits a gapless,c-axis dispersive two-spinon continuumsimilar to that of the 1D Heisenberg chain aboveTSP,
23 well-defined magnonlike excitations with sizablec- and b-axisdispersion have been observed belowTSP.
46,24These mag-nons are gapful and are split off from a continuum which,zone center, starts at roughly twice the magnon gap.2426,17
FIG. 1. TheJ-l-a-m-b model. Line segments refer to exchangcouplings for spins located at segment vertices. Interchain cplings are shown for a single dimer site only.b5beb andc5cec arethe primitive vectors.14 441 1997 The American Physical Society
14 442 56WOLFRAM BRENIGThe aim of this work is to study the magnetic propertof the J-l-a-m-b model with a focus on theD phase ofCuGeO3. First I will describe a bond-spin representationtheJ-l-a-m-b model which, in turn, is treated by an apprpriate linearization. Next, results for the ground-state enethe spin gap, the magnon dispersion, and the dynamic sture factor are contrasted against other theoretical approaand are compared with experimental findings. Finally, detof an alternative mean-field approach using the bond-srepresentation are provided in the Appendix.
II. BOND-OPERATOR THEORY
In this section the properties of theJ-l-a-m-b model arediscussed by representing thesite-spin algebra in terms obond-spin operators.27 First, the essential features of theoperators are briefly restated. Consider any two spin-1/2eratorsS1 andS2. The eigenstates of the related total spin aa singletus& and three tripletsuta& with a5x,y,z. These canbe created out of a vacuumu0& by applying the bosonicoperatorss and ta
tyu0&5uty&5 i ~ u&1u&)/A2,
where@s,s#51, @s(),ta()#50, and@ ta ,tb
#5dab . The ac-tion of S1 andS2 on this space leads to the representatio
s2 iabgtb tg!, ~2!
for the individual spin operators. Hereabg is the Levi-Civita symbol and a summation over repeated indices isplied hereafter. The upper~lower! subscript on the LHS ofEq. ~2! refers to the upper~lower! sign on the RHS. Thebosonic Hilbert space has to be restricted to the physHilbert space, i.e., either one singlet or one triplet, byconstraint
ss1ta ta51. ~3!
Using Eqs.~2! and ~3! it is simple to check thatS1 and S2satisfy a spin algebra indeed and moreover that
4ta ta . ~4!
In order to transform theJ-l-a-m-b model into the bosonrepresentation a particular distribution l of bonds, i.e.,pairs of spinsSl1 andSl2, has to be selected. Here this seletion will be based on the limit of strong dimerizatio(l,la)(0,0), or equivalently (a ,d)(0,1), and small in-terchain coupling (m,mb)(0,0). In this limit the groundstate is a product of singlets on each dimerbond whileelementary excitations are composed of the corresponlocalized triplets. Therefore it is natural to place the singf
and triplet bosons onto the dimer bonds, i.e.,Sl15Sl andSl25Sl1c with lPD. The transformed Hamiltonian reads
S 2 34 slsl1 14 t la t laD ,H15 (
lmPDa~ l,m!~ t la
tmasm sl1t la
tma smsl1H.c.! ~5!
b~ l,m!~ iabgtma t lb
c~ l,m!~ t la tma
tmbt lb2t la tmb
where each local Hilbert space is subject to the constraint~3!and, if not explicitly stated otherwise, the unit of energy isJhereafter. The interdimer matrix elements can be obtaifrom Fig. 1:
4@l~d ll12dml1!1m~dml32dml21d ll22d ll3!#,
where l1,2,3 are defined in Fig. 1 andt15l(122a),t25m(122b), t35l(112a), and t45m(112b). As an-ticipated, the interdimer matrix elementsa( l,m), b( l,m),and c( l,m) vanish in the strong dimer limit leaving thHamiltonian diagonal insl and t la .
In order to treat the local constraint and the dimer intactions approximations have to be made. To this end I wemploy the Holstein-Primakoff~HP! representation of thebond operators which has been detailed in Refs. 2830this representation the constraint is treated by eliminatingsinglet operator viasl
5sl5(12t la t la)
21/2. Moreover, afterinserting this into the Hamiltonian only terms up to secoorder in the triplet operators are retained. The latter produre is analogous to the linear spin-wave approximationsystems with broken spin-rotational invariance. The lineized ~LHP! Hamiltonian is given by
2(kPB Cka F11ek ekek 11ekGCka , ~7!
2@ t1cos~2kc!12t2cos~kb!cos~kc!#, ~8!
56 14 443MAGNETIC EXCITATIONS IN QUASI-TWO- . . .where D is the number of dimers andk is a momentumvector restricted to a Wigner-Seitz cellB of the reciprocallattice. For comparison with experimental dataB is orientedwith respect to the nondimerized system, instead of the Blouin zone of the dimer lattice, i.e.,k5(kb ,kc) withbkb502p and ckc50p with b,c set to unity here-after. Cka
() is a spinor with Cka 5@ tka
t2ka] andt la 51/AD(ke2 ik ltka
.Equation~7! describes a threefold degenerate set of d
persive triplets. The triplets are renormalized by ground-squantum fluctuations. These are produced by the termtype tka
t2ka and their Hermitian conjugate. The excitatio
spectrumEk follows from a Bogoliubov transformation
kPB,aEkS aka aka1 12D , ~9!
whereaka() are the Bogoliubov quasiparticles which are giv
Cka5Fgk hkhk gkGFka , ~11!hk
2S 11ekEk 21D , hkgk52 12 ekEk ,d.
2S 11ekEk 11D ,whereFka
() is a a spinor withFka 5@aka
a2ka#.To conclude this section, I note that instead of treating
constraint by means of the HP representation one mayapply the so-called bond-operator mean-field theory~MFT!of Ref. 27. The application of this method to theJ-l-a-m-bmodel is detailed in the Appendix. As will become evidentSec. III B this technique seems less well suited in the prescontext.
In the following sections the consequences of the Lrepresentation of the 2DJ-l-a-m-b model will be con-trasted against other known results in the limiting case of1D J1-a-d model as well as INS data observed on CuGeO3.
A. Ground-state energy
From Eq.~9! it is obvious that the ground-state energy plattice site, Eg , is equal to23/8 at the point of completedimerization. This is the proper energy gain for a bare singformation. For arbitraryt1 and t2Eg~ t1 ,t2!529
dkcF @122t2cos~kc!2t1cos~2kc!#21/2 ES 24t2cos~kc!122t2cos~kc!2t1cos~2kc! D G , ~12!dver-ergy
where E is the complete elliptic integral of the second kinFor vanishing interchain coupling this simplifies to
2pA12t1 ES 2t1t121D , ~13!
which can be compared to existing results in various regiof the (a ,d) plane of the frustrated and dimerized spin-1chain. In particular at the isotropic Heisenberg point, i.(a ,d)5(0,0), Eg53/(A2p)29/8'20.4498 which agreesreasonably well with the Bethe ansatz resEg
Bethe51/42 ln(2)'20.4431. In Fig. 2 the ground-state e
ergy along the (a50,d) line, i.e., for a dimerized chain, iscontrasted against results from exact diagonalization.31 De-viations are within the numerical error of the diagonalizatidata ford*0.05. For dimerizations below 0.05 small diffeences are caused by an unphysical extremum in the Lenergy which is visible in the inset. While the agreemenencouraging along the (a50,d) line, only qualitative consis-tency is to be expected along the (a ,d50) line since thisregion is more distant from the strong dimer lim(a ,d);(0,1). This is shown in Fig. 3 which compareEg(a) with exact diagonalization
32 and density-matrix renors
malization group33 ~DMRG! data for the case of a frustratechain. This figure demonstrates that the LHP approach oestimates the frustration induced loss of ground-state enat intermediatea .
FIG. 2. Ground-state energies in the dimerized-chain limit: LHtheory~solid! vs exact diagonalization~Ref. 31! ~solid dots!, errorsare less than marker size. Inset: smalld limit. E!(d)/J152$(11d)Eg@(12d)/(11d),0#2 Eg
14 444 56WOLFRAM BRENIGB. Spin gap
The LHP approximation does not break the sprotational invariance and leaves the system in a quantdisordered ground state. This is consistent with a spin gaDof the triplet dispersion which, for positivet1 andt2, is givenby D5A12t122t2 and is situated atk5(0,0) and (p,p). Ift1 and t2 are such that the triplet modes turn massless,D50, a quantum phase transition towards antiferromnetism~AFM! will occur at T50. In terms ofa andd, theAFM instability line is located ata1d5t2(11d). Beyondthis line the LHP approach breaks down.
Next the LHP spin gap att250 is compared to knownresults for the frustrated and dimerized spin-1/2 chain.Fig. 4 D(d) is contrasted against findings of exadiagonalization,31 perturbation theory up to third order il,17,34 and a solution of the MFT equations~A5!~A7!which are discussed in the Appendix. This figure displareasonable agreement between the first three of theseproaches for all values ofd. In addition it shows that theMFT suffers from the inability to close the spin gap at t
FIG. 3. Ground-state energies in the frustrated-chain limit: Ltheory ~solid! vs exact diagonalization~Ref. 32! ~solid dots!, andDMRG ~Ref. 33! ~crosses!, errors are less than marker sizE!(a)/J15 2$Eg@(122a),0#2 Eg
FIG. 4. Spin gaps in the dimerized-chain limit: LHP theo~solid! vs third-order perturbation theory~Refs. 17 and 34!~dashed!, MFT ~dotted!, and exact diagonalization~Ref. 31! ~soliddots!, errors are less than marker size.--
isotropic Heisenberg point.37,38 This caveat renders the MFTunsuitable for the case of a systems with nearly masslessexcitations, e.g., CuGeO3.
As noted in the previous section, analytic approachbased on the strong dimer limit are less reliable if considealong the (a ,d50) line. Nevertheless, a comparisonD(a) as obtained from the LHP theory with various othtechniques is instructive and is shown in Fig. 5. Qualitively, all analytic methods depicted exhibit a tendencythe spin gap to increase asa increases35 however agreemenwith the DMRG data33,36 is absent. In particular, while thirdorder perturbation theory and MFT show no critical frustrtion ratio ac and have a finite gap for all 0,a,0.5 the LHPrepresentation leads to a critical frustrationac50. Quite re-markably there are also substantial differences betweentwo DMRG results fora*0.5.39
C. Triplet dispersion
In this section the relevance of the model with respecthe spin excitations in CuGeO3 will be assessed by comparson of Ek with INS data for theD phase. In particular therole of two-dimensionality will be considered. The essenteffect of a finite hopping amplitudet2 is a mixing of the b-andc-axis dispersion. This mixing is due to the staggeringthe dimerization along theb axis and leads to thet2cos(kb)cos(kc) term in ek . Thus,Ek involves terms of dif-ferent periodicity inkc , i.e., cos(2kc) and cos(kc). Therefore,in contrast to the quasi-1D case, the degeneracy of thelets at the momenta (0,0) and (0,p) is lifted. An identicalreasoning based on the first-order contribution of a thiorder perturbation theory has been given in Ref. 17. Expaing Ek in terms oft1 andt2 I find agreement up to first ordewith the dispersion given in Ref. 17.
In Fig. 6 INS data of the magnon dispersion in CuGeO3are shown for momenta along the edges of the reciprolattice cell fromk5(0,0) to (0,p) as well as from (0,0) to(p/2,0).6 Although data exactly at (0,p) are lacking, itseems very likely from this figure that the triplet excitatioin CuGeO3 arenot degenerate at (0,0) and (0,p). In order to
FIG. 5. Spin gaps in the frustrated-chain limit: LHP theo~solid! vs third-order perturbation theory~Ref. 17! ~dashed!, MFT~dotted!, and two DMRG calculations~Ref. 33! ~crosses!, and~Ref.36! ~solid dots!, errors are less than marker size.
56 14 445MAGNETIC EXCITATIONS IN QUASI-TWO- . . .demonstrate that theJ-l-a-m-b model can account for theobserved dispersion Fig. 6 contains a comparison ofEk withthe INS data. Rather than performing this comparison bleast-squares fit,Ek has been identified with the magnon eergies only atk5(0,0), (0,p/2), and (p/2,0). This fixesJ511.5 meV,t150.859, andt250.054 unambiguously andlimits theb- to c-axis coupling-ratio to roughly 6% which iconsistent with Refs. 46 and 13. While the preceding leaa , d, b, andm undetermined,a can be fixed using an addtional input, i.e.,d50.012. This is within the range of valuesuggested in the literature, i.e.,d50.210.012 ~Refs.11,19,20 and 22! and implies natural parameters ofJ1511.4meV5132 K anda50.059. Allowing for a larger dimerization leads to smallerJ1 and a . In particular a50 withJ15124 K is reached ford50.076. Parameters withslightly smaller ~larger! intra ~inter!-chain exchange havbeen established in Ref. 17, i.e.,J159.8 meV5114 K, a50,and t250.12 however withd50.12.
The agreement displayed in Fig. 6 is satisfying and deonstrates the main point of this section, i.e., that diagotriplet hopping can account for the observed asymmetrythe INS data in theD phase of CuGeO3. Moreover, the
FIG. 6. INS data ofc-axis ~upper panel! and b-axis ~lowerpanel! dispersion of the triplet mode in theD phase of CuGeO3 atT51.8 K ~Ref. 6! ~solid diamonds! vs LHP theory ~solid! forJ511.5 meV,t150.859, andt250.054 (b5c51).a
values ofa obtained suggest that the intrachain frustrationCuGeO3 is significantly smaller than that derived from thpurely 1DJ1-a-d model,
19,20 i.e., 0.24&a&0.36. The deter-mination of model parameters however is in need of furtstudy to improve on the quantitative reliability of the frustrtion dependence of presently available approaches for thecase. This is in contrast to Ref. 17 wherequantitativeevi-dence fora'0, based on third-order perturbation theory, hbeen suggested.
D. Dynamic structure factor
Magnetic excitations are observed by measuring thenamic structure factorS(q,v) which is related by analyticcontinuation and the fluctuation dissipation theoreS(q,v)5 Im@x(q,v)#/(12e2v/T) to the dynamic spin susceptibility
Hereq is the momentum,t the imaginary time, andTt refersto time ordering. Since the physical spin is a compositeerator of the bond bosons the information obtained fromdynamic susceptibility is not restricted to the triplet dispesion even at the LHP level. This will be clarified in thremainder of this section.
Within the LHP representation the spin operator in mmentum space is
abgtk1qb tkg . ~15!
Two qualitatively different excitations appear on the RHSsharp triplet mode due to the first term and a continuumtwo triplet states due to the second. The momentudependent form factors@17exp(iqc)# lead to a vanishingweight of the triplet mode~continuum! at qc50 (qc5p).SinceHLHP does not conserve the bare triplet number,continuum is divided in between two excitations of differenature, i.e., virtual excitations at energiesEk1q1Ek and realexcitations at energiesEk1q2Ek . While the former are dueto ground-state quantum fluctuations and occur at all teperatures, the latter result from excitations across the sgap and are present only at finite temperatures. Evaluathe dynamic susceptibility by standard methods I obtainxab~q,vn!5dab1
4H @cos~qc!21# 1~ ivn!22Eq2 1@cos~qc!11# 12D (kPB F11ek1q1ek1Ek1qEkEk1qEk n~Ek1q!2n~Ek!ivn1Ek1q2Ek1
2G J , ~16!
14 446 56WOLFRAM BRENIGwherevn52npT is a Bose Matsubara frequency.As anticipated the dynamical susceptibility~16! exhibits
finite spectral intensity atEk , Ek1q2Ek , and Ek1q1Ek .This is summarized in Fig. 7 which displays the two continwith respect to the triplet mode along a particular momentspace direction. Parameters identical to those of Fig. 6 hbeen chosen. The triplet mode is situated in a gap betwthe low-energy continuum due to thermal excitations athat at high energies due to quantum fluctuations. At the zcenter the latter is gaped by 2D. This is consistent with re-cent INS data from theD phase of CuGeO3.
24 In addition tothe magnon these experiments indicate an unexpecthigh-energy continuum which is separated from the magby an additional gap. The observed zone-center continuto-magnon gap ratio is approximately 2. Deviations fromlatter value of 2 are possibly due to triplet-triplet interactiowhich are beyond the LHP approach. Additionally, Fig.shows the weight of the triplet mode, as well as that ofcombined continua atT50.05J. At T!J the continuumweight is almost completely due to quantum fluctuatiowhile, independent of temperature, the triplet weight is givby
The leftmost expression refers to the maximum of the tripweight. The corresponding momentum, i.e., (p,p), indicatesthe instability towards AFM asD0.
Figure 8 depicts the spectral intensity Im@xxx(q,vn2 iv1n)] of the continua as a function of frequency fotwo temperatures and for various momenta along a direcin reciprocal space identical to that of Fig. 7. Thek sums ofEq. ~16! have been performed numerically on a 6003300lattice. In order to obtain sufficient smoothing the frequenhas been shifted off the real axis byn50.05. The parametercorrespond to those of Fig. 6. As a guide to the eyeposition of the triplet mode is labeled by solid squares whsolid diamonds in this figure label the exact locations of
FIG. 7. Energy bounds,VC(q), for both triplet continua~dash-dotted! vs triplet-mode energy,E(q) ~solid!, as well as combinedspectral weight of continua,WC(q) ~dotted!, at T50.05 vs spectralweight of triplet mode, WT(q) ~dashed!, for momenta alongqb5qc . (b5c51, J51 andt1 ,t2 as in Fig. 6.!a
spectral bounds of the continua. Although smeared due toimaginary broadening, van Hove singularities are clearlyservable at the spectral bounds as well as other characteenergies within the continua. Evidently the quantum fluctutions exhibit largest weight at intermediate wave vectwhile at higher temperatures additional weight appearssmaller momentum due to thermal excitations. Even thouthe intensity in Fig. 8 decreases both, asq approaches (0,0)and (p,p), only in the latter case this is due to the forfactor in Eq.~15! while in the former case this is a consquence of the conservation of the total spin.
In summary I have studied static and dynamic properof a frustrated and dimerized 2D quantum spin model usthe bond-operator method. The ground-state energy andspin gap have been gauged against known results from1D limiting cases of this model. Effects of dimerization afound to be described almost quantitatively while the inflence of frustration is captured qualitatively. The dynamstructure factor has been analyzed and displays two chateristic features, i.e., a well-defined magnon excitation antemperature-dependent continuum. The magnon dispershows a characteristic lifting of degeneracies which is diffent from purely one-dimensional models. The dispersagrees very well with INS data on CuGeO3 for
@J1 ,d,a ,m(122b)#5@132 K, 0.012, 0.059, 0.054# in termsof the coupling parameters of Fig. 1. The low-temperatcontinuum of the dynamic structure factor exhibits a zoncenter gap twice that of the magnon. This is also consiswith INS experiments on CuGeO3.
FIG. 8. Triplet continua for various momenta alongqb5qc andfor two temperaturesT50.05~a! andT50.15~b!. The bottommostcurve corresponds toq50 with 0.1 incrementaly-axis offset foreach consecutive momentum. Solid diamonds: bounds of tricontinua. Solid square: triplet-mode energy. (b5c51, J51, andt1 ,t2 as in Fig. 6.!
56 14 447MAGNETIC EXCITATIONS IN QUASI-TWO- . . .ACKNOWLEDGMENTS
I am grateful to P. Fulde and the Max-Planck-Institut frPhysik komplexer Systeme for their kind hospitality. It ispleasure to thank G. Uhrig for helpful comments andcommunicating his results prior to publication. Stimulatidiscussions with B. Buchner and E. Muller-Hartmann areacknowledged. This work was supported in part by the Desche Forschungsgemeinschaft through the SFB 341.
APPENDIX: BOND-OPERATOR MEAN-FIELD THEORYMFT
An alternative approach to the Hamiltonian~5! arises byintroduction of a set of local Lagrange multipliersh l to en-force the constraint~3!
h l~slsl1t la
t la21!. ~A1!
To treat this Hamiltonian one replaces the local constraina global one, i.e.,h l5h, and introduces a mean-field~MF!decoupling of all quartic terms leading to an effective qudratic Hamiltonian.27 This Hamiltonian has an overallnega-tive prefactor to thesl
sl term which implies Bose condensation of the singlets. Thereforesl
5sl5^sl&5s is assumed.Moreover it can be shown that contributions from the tripand quartic triplet termsH2 and H3 to the MFT can beneglected.40,41 The MF Hamiltonian reads
HMFT5DS 2 38 2 34 s22hs21 52 h D
2(kPB Cka F 14 2h1s2ek s2ek
GCka ,~A2!, P
with notations equivalent to Eq.~7!. Analogous to the latterequation~A2! represents three dispersive triplet excitationhowever, with a modified dispersion relation. After a Bogliubov transformation one gets42
HMFT5DS 2 38 2 34 s22hs21 52 h D1 (
MFTS bka bka1 12D , ~A3!Ek
MFT5S 14 2h DA11dek, ~A4!whered52s2/(1/42h). Thebka quasiparticles follow fromexpressions identical to Eq.~11! with ek , 11ek , and Ekreplaced bys2ek ,
2ek , andEkMFT , respectively. The
Lagrange multiplier and the singlet amplitude have todetermined by solving the saddle-point equatio^]HMFT /]h&50 and^]HMFT /]s&50. This leads to
A11dekS ^bkx bkx&1 12D , ~A5!
A11dekS ^bkx bkx&1 12D . ~A6!
These self-consistency equations can be solved by coming them into a single one for the variabled only:
A11dekS ^bkx bkx&1 12D , ~A7!
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