Magnetic excitations in quasitwodimensional spinPeierls systems
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PHYSICAL REVIEW B 1 DECEMBER 1997IIVOLUME 56, NUMBER 22Magnetic excitations in quasitwodimensional spinPeierls 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 twodimensional frustrated and dimerized quantum spin system which models theeffect of interchain coupling in a spinPeierls compound. Employing a bondboson method to account forquantum disorder in the ground state the elementary excitations are evaluated in terms of gapful triplet modes.Results for the groundstate 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 spinPeierls compoundCuGeO3. Moreover, consistent with these neutron scattering experiments, the lowtemperature dynamic structure factor exhibits a highenergy continuum split off from the elementary triplet mode.@S01631829~97!016421#m
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eouI. INTRODUCTION
The recent discovery of the inorganic spinPeierls copounds CuGeO3 ~Ref. 1! and a8NaV2O5 ~Refs. 2 and 3!has greatly stimulated interest in lowdimensional magtism. While many properties of these materials can bescribed in terms of quasionedimensional ~1D! quantumspin systems, clear evidence for a substantial degree oftwodimensionality~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 lowtemperature phase of CuGeO3.
CuGeO3 is an inorganic spinPeierls 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 spin1/2 state.9,10 The nearestneighbor ~NN! exchange coupling between copper spalong the CuO2 chains is strongly reduced by almost othogonal intermediate oxygen states.11 Therefore, nextnearestneighbor~NNN! exchange in CuGeO3 is relevant.Both, NN and NNN exchanges are antiferromagnetic11,12
~AFM! implying intrachain frustration. In addition, NN awell as NNNinterchain 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 theJlamb 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 ina8NaV2O5,
18 and turns out5601631829/97/56~22!/14441~8!/$10.00
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to be relevant for the magnon dispersion. In Fig. 1 an adtional, socalled natural labeling of the intrachain parameters is introduced, i.e.,J1 ,a , and d. This notation is frequently used in the context of the 1D dimerized afrustrated spinchain 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 zerotemperature dimerizationd(T50) ranging from 0.21 to0.012 have been suggested.11,19,20,22Knowledge on the magnitude 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,caxis dispersive twospinon continuumsimilar to that of the 1D Heisenberg chain aboveTSP,
23 welldefined magnonlike excitations with sizablec and baxisdispersion have been observed belowTSP.
46,24These magnons are gapful and are split off from a continuum which,zone center, starts at roughly twice the magnon gap.2426,17
FIG. 1. TheJlamb 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
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14 442 56WOLFRAM BRENIGThe aim of this work is to study the magnetic propertof the Jlamb model with a focus on theD phase ofCuGeO3. First I will describe a bondspin representationtheJlamb model which, in turn, is treated by an apprpriate linearization. Next, results for the groundstate 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 meanfield approach using the bondsrepresentation are provided in the Appendix.
II. BONDOPERATOR THEORY
In this section the properties of theJlamb model arediscussed by representing thesitespin algebra in terms obondspin operators.27 First, the essential features of theoperators are briefly restated. Consider any two spin1/2eratorsS1 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
:
su0&5us&5~ u&2u&)/A2,
txu0&5utx&52~ u&2u&)/A2,
~1!
tyu0&5uty&5 i ~ u&1u&)/A2,
tzu0&5utz&5~ u&1u&)/A2,
where@s,s#51, @s(),ta()#50, and@ ta ,tb
#5dab . The action of S1 andS2 on this space leads to the representatio
S12
a5
1
2~6sta6ta
s2 iabgtb tg!, ~2!
for the individual spin operators. Hereabg is the LeviCivita 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
S1aS2
a523
4ss1
1
4ta ta . ~4!
In order to transform theJlamb 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 interchain 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
y,ces
lsin
pe

ale

engt
and triplet bosons onto the dimer bonds, i.e.,Sl15Sl andSl25Sl1c with lPD. The transformed Hamiltonian reads
H5H01H11H21H3 ,
H05(lPD
S 2 34 slsl1 14 t la t laD ,H15 (
lmPDa~ l,m!~ t la
tmasm sl1t la
tma smsl1H.c.! ~5!
H25 (lmPD
b~ l,m!~ iabgtma t lb
t lgsm1H.c.!,
H35 (lmPD
c~ l,m!~ t la tma
tmbt lb2t la tmb
tmat lb!,
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:
a~ l,m!521
4@ t1dml11t2~dml21dml3!#,
b~ l,m!51
4@l~d ll12dml1!1m~dml32dml21d ll22d ll3!#,
~6!
c~ l,m!521
4@ t3dml11t4~dml21dml3!#,
where l1,2,3 are defined in Fig. 1 andt15l(122a),t25m(122b), t35l(112a), and t45m(112b). As anticipated, 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 HolsteinPrimakoff~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 spinwave approximationsystems with broken spinrotational invariance. The lineized ~LHP! Hamiltonian is given by
HLHP529
4D1
1
2(kPB Cka F11ek ekek 11ekGCka , ~7!
ek521
2@ t1cos~2kc!12t2cos~kb!cos~kc!#, ~8!
ri
istasn
en
thealso
inent
HP
the
er
let
56 14 443MAGNETIC EXCITATIONS IN QUASITWO . . .where D is the number of dimers andk is a momentumvector restricted to a WignerSeitz 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 hereafter. 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 groundsquantum fluctuations. These are produced by the termtype tka
t2ka and their Hermitian conjugate. The excitatio
spectrumEk follows from a Bogoliubov transformation
HLHP529
4D1 (
kPB,aEkS aka aka1 12D , ~9!
Ek5A112ek, ~10!
whereaka() are the Bogoliubov quasiparticles which are giv
by
Cka5Fgk hkhk gkGFka , ~11!hk
251
2S 11ekEk 21D , hkgk52 12 ekEk ,d.
on/2e
uln
onrH
t i
its

l
teof
gk25
1
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 socalled bondoperator meanfield theory~MFT!of Ref. 27. The application of this method to theJlambmodel is detailed in the Appendix. As will become evidentSec. III B this technique seems less well suited in the prescontext.
III. RESULTS
In the following sections the consequences of the Lrepresentation of the 2DJlamb model will be contrasted against other known results in the limiting case of1D J1ad model as well as INS data observed on CuGeO3.
A. Groundstate energy
From Eq.~9! it is obvious that the groundstate 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
81
3
2p2E
0
p
dkcF @122t2cos~kc!2t1cos~2kc!#21/2 ES 24t2cos~kc!122t2cos~kc!2t1cos~2kc! D G , ~12!dverergy
P
where E is the complete elliptic integral of the second kinFor vanishing interchain coupling this simplifies to
Eg~ t1,0!529/813
2pA12t1 ES 2t1t121D , ~13!
which can be compared to existing results in various regiof the (a ,d) plane of the frustrated and dimerized spin1chain. 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 groundstate e
ergy along the (a50,d) line, i.e., for a dimerized chain, iscontrasted against results from exact diagonalization.31 Deviations 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 consistency 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 densitymatrix renors
.,
t
Ps
malization group33 ~DMRG! data for the case of a frustratechain. This figure demonstrates that the LHP approach oestimates the frustration induced loss of groundstate enat intermediatea .
FIG. 2. Groundstate energies in the dimerizedchain limit: LHtheory~solid! vs exact diagonalization~Ref. 31! ~solid dots!, errorsare less than marker size. Inset: smalld limit. E!(d)/J152$(11d)Eg@(12d)/(11d),0#2 Eg
Bethe%.
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esredofertaofta
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HP
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14 444 56WOLFRAM BRENIGB. Spin gap
The LHP approximation does not break the sprotational invariance and leaves the system in a quantdisordered ground state. This is consistent with a spin gaDof the triplet dispersion which, for positivet1 andt2, is givenby D5A12t122t2 and is situated atk5(0,0) and (p,p). Ift1 and t2 are such that the triplet modes turn massless,D50, a quantum phase transition towards antiferromnetism~AFM! will occur at T50. In terms ofa andd, theAFM instability line is located ata1d5t2(11d). Beyondthis line the LHP approach breaks down.
Next the LHP spin gap att250 is compared to knownresults for the frustrated and dimerized spin1/2 chain.Fig. 4 D(d) is contrasted against findings of exadiagonalization,31 perturbation theory up to third order il,17,34 and a solution of the MFT equations~A5!~A7!which are discussed in the Appendix. This figure displareasonable agreement between the first three of theseproaches for all values ofd. In addition it shows that theMFT suffers from the inability to close the spin gap at t
FIG. 3. Groundstate energies in the frustratedchain limit: Ltheory ~solid! vs exact diagonalization~Ref. 32! ~solid dots!, andDMRG ~Ref. 33! ~crosses!, errors are less than marker sizE!(a)/J15 2$Eg@(122a),0#2 Eg
Bethe%.
FIG. 4. Spin gaps in the dimerizedchain limit: LHP theo~solid! vs thirdorder perturbation theory~Refs. 17 and 34!~dashed!, MFT ~dotted!, and exact diagonalization~Ref. 31! ~soliddots!, errors are less than marker size.
.,
n
sap
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 remarkably 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 twodimensionality will be considered. The essenteffect of a finite hopping amplitudet2 is a mixing of the bandcaxis 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 different periodicity inkc , i.e., cos(2kc) and cos(kc). Therefore,in contrast to the quasi1D case, the degeneracy of thelets at the momenta (0,0) and (0,p) is lifted. An identicalreasoning based on the firstorder 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 frustratedchain limit: LHP theo~solid! vs thirdorder 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.
yn
sv
is

ae
mno
ine
hera
2D
as
dy
m
opthere
o
: aofm
thent
mpinting
56 14 445MAGNETIC EXCITATIONS IN QUASITWO . . .demonstrate that theJlamb model can account for theobserved dispersion Fig. 6 contains a comparison ofEk withthe INS data. Rather than performing this comparison bleastsquares 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 caxis couplingratio 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 ofcaxis ~upper panel! and baxis ~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
es
alf
values ofa obtained suggest that the intrachain frustrationCuGeO3 is significantly smaller than that derived from thpurely 1DJ1ad model,
19,20 i.e., 0.24&a&0.36. The determination 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 wherequantitativeevidence fora'0, based on thirdorder 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
xab~q,t!5^Tt@Sqa~t!Sq
b#&. ~14!
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
Sqa5
1
4~12eiqc!~ tqa
1t2qa!2 i1
4AD~11eiqc!
3 (kPB
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 groundstate 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
~11ek1q1ek2Ek1qEk!~Ek1q1Ek!
Ek1qEk
n~Ek1q!1n~Ek!11
~ ivn!22~Ek1q1Ek!
2G J , ~16!
uumaenon
ednoumthns7
th
ne
le
rtio
csthil
th
theobristica
orsat
gh
me
tiesingthethe
reuicracd asionerion
uree
tent
plet
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 lowenergy 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 recent INS data from theD phase of CuGeO3.
24 In addition tothe magnon these experiments indicate an unexpecthighenergy continuum which is separated from the magby an additional gap. The observed zonecenter continutomagnon gap ratio is approximately 2. Deviations fromlatter value of 2 are possibly due to triplettriplet 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
WT~q!5p
8
12cos~qc!
Eq5
~p,p! p
4D. ~17!
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~dashdotted! vs tripletmode 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
veende
n
e
e
sn
t
n
y
eee
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.
IV. CONCLUSION
In summary I have studied static and dynamic properof a frustrated and dimerized 2D quantum spin model usthe bondoperator method. The groundstate 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 welldefined magnon excitation antemperaturedependent continuum. The magnon dispershows a characteristic lifting of degeneracies which is diffent from purely onedimensional 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 lowtemperatcontinuum 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 incrementalyaxis offset foreach consecutive momentum. Solid diamonds: bounds of tricontinua. Solid square: tripletmode energy. (b5c51, J51, andt1 ,t2 as in Fig. 6.!
ua
fong
u
t b
a

lic
s,o
bens
bin
56 14 447MAGNETIC EXCITATIONS IN QUASITWO . . .ACKNOWLEDGMENTS
I am grateful to P. Fulde and the MaxPlanckInstitut 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. MullerHartmann areacknowledged. This work was supported in part by the Desche Forschungsgemeinschaft through the SFB 341.
APPENDIX: BONDOPERATOR MEANFIELD THEORYMFT
An alternative approach to the Hamiltonian~5! arises byintroduction of a set of local Lagrange multipliersh l to enforce the constraint~3!
H5H2(lPD
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 meanfield~MF!decoupling of all quartic terms leading to an effective qudratic Hamiltonian.27 This Hamiltonian has an overallnegative 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
11
2(kPB Cka F 14 2h1s2ek s2ek
s2ek1
42h1s2ek
GCka ,~A2!, P
v
v

r
t
y

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 (
kPB,aEk
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 ,
14 2h1s
2ek , andEkMFT , respectively. The
Lagrange multiplier and the singlet amplitude have todetermined by solving the saddlepoint equatio^]HMFT /]h&50 and^]HMFT /]s&50. This leads to
05s222
51
3
D (kPB11dek/2
A11dekS ^bkx bkx&1 12D , ~A5!
053
41h2
3
D (kPBek
A11dekS ^bkx bkx&1 12D . ~A6!
These selfconsistency equations can be solved by coming them into a single one for the variabled only:
d5523
D (kPB2
A11dekS ^bkx bkx&1 12D , ~A7!
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