# Longitudinal modes in quasi-one-dimensional antiferromagnets

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PHYSICAL REVIEW B VOLUME 46, NUMBER 14 1 OCTOBER 1992-II

Longitudinal modes in quasi-one-dimensional antiferromagnets

Ian AfHeckCanadian Institute for Advanced Research and Physics Department, University ofBritish Columbia, Vancouver,

British Columbia, Canada V6T1Zl

Greg F. Wellman'Physics Department, University ofBritish Columbia, Vancouver, British Columbia, Canada V6TIZI

(Received 6 May 1992)

Neutron-scattering data on CsNiC13, a quasi-one-dimensional spin-one antiferromagnet, exhibit ananomalous mode. It was later proposed, based on a Landau-Ginsburg model, that this should be viewedas a longitudinal fluctuation of the sublattice magnetization. This theory is elaborated in more detailhere and compared with experimental data on CsNiC13 and RbNiC13. In particular, we give explicitly arenormalization-group argument for the existence of such modes in Neel-ordered antiferromagnetswhich are nearly disordered by quantum fluctuations, due to quasi-one-dimensionality or other effects.We then discuss the non-Neel case of a stacked triangular lattice such as CsNiC13 where longitudinal andtransverse modes mix. In this case the quantum disorder transition is driven first order by fluctuationsand the longitudinal mode always has a finite width. Effects of a magnetic field on the magnon spectrumare calculated both in conventional spin-wave theory and in the Landau-Ginsburg model and are com-

pared with experimental data on CsNiC13. This model is compared with an alternative Lagrangian-based one that was proposed recently.

I. INTRODUCTION

It was argued by Haldane' that one-dimensionalinteger-spin Heisenberg antiferromagnets have an excita-tion gap above a singlet ground state. The first experi-mental evidence for the Haldane gap was obtained byBuyers et al. in neutron-scattering experiments onCsNiC13. The spin Hamiltonian for this material is high-ly isotropic (i.e., Heisenberg-like) in spin space and ap-parently exhibits a ratio of interchain to intrachain cou-plings of about 2%. This weak interchain coupling pro-duces magnetic order at a temperature of 4.8 K, about 3of the. intrachain coupling. Because the lattice structureis of stacked triangular type, the ordered state has anti-parallel neighboring spins along the chains and neighbor-ing spins at angles of 2n/3 in the planes. (See Fig. 1.)Neutron-scattering experiments at temperatures of about10 K, above the ordering temperature but still quite smallcompared to the exchange energy, indicate the existenceof a gap in the purely one-dimensional case. Experimentsin the ordered phase, below 4.8 K also exhibit anomalousbehavior. Apart from the Goldstone modes predicted by

A

IE

B F

FIG. 1. Orientation of spin vectors on the six inequivalentsublattices (see Fig. 2) for the stacked triangular lattice antifer-romagnet.

spin-wave theory, a portion of another excitation branchwith a finite gap is also observed. This was argued to bea longitudinal mode, i.e., a longitudinal fluctuation of thesublattice magnetization, and a Landau-Ginsburg modelwas constructed to study the problem. In this model thelong-wavelength staggered magnetization field is treatedas a three-vector field, P, of arbitrary magnitude anddirection in spin space. In a magnetically ordered statethis field has a nonzero ground-state expectation value.In a simple Neel state, as would occur for a bipartite lat-tice (in which all spins are parallel or antiparallel), fluc-tuations in the direction of this field give the usual twoGoldstone modes of spin-wave theory. Fluctuations inthe magnitude of the field correspond to the longitudinalmode. The necessity of three modes follows from con-tinuity from the disordered phase where the ground-stateexpectation value vanishes and the magnon is a triplet.The stacked triangular lattice is more complicated. Nowa transverse fluctuation on one site is not orthogonal to alongitudinal one on a neighboring site in the same plane.Consequently, the transverse and longitudinal modes mixin the Landau-Ginsburg model.

The Landau-Ginsburg model predicts that the longitu-dinal mode has a finite decay rate into a pair of Gold-stone modes (even at zero temperature). Consequently, itis possible to view the longitudinal mode as a two-magnon resonance, making contact with the traditionalHolstein-Primakov approach to spin-wave theory. Thisdecay rate depends on the size of the ~P~ coupling in theLandau-Ginsburg model. The width-to-gap ratio van-ishes linearly at weak coupling. If this decay rate is toolarge the longitudinal mode might not be observable. Ingeneral, the observability of the longitudinal mode is anempirical question, but there is one case where we can

46 8934 1992 The American Physical Society

46 LONGITUDINAL MODES IN QUASI-ONE-DIMENSIONAL. . . 8935

predict with confidence that the longitudinal mode isvery long lived. This occurs in the simple Neel case whenthe system is very close to being disordered by quantumfluctuations. This would correspond to the case wherethe sublattice magnetization is very much reduced com-pared to its classical value (s) at T =0 due to quantumfluctuation effects. The strength of these fluctuatione8'ects is determined by the spin Hamiltonian. One wayof enhancing them is by making the system quasi-one-dimensional. As the ratio of interchain to intrachaincouplings is lowered, eventually the order is destroyed,even at T =0. When this ratio is only slightly larger thanthis critica1 value, the longitudinal mode is very longlived. This follows from the fact that this second-order,T =0 quantum phase transition is in the four-dimensional universality class and is consequentlygoverned by the weak-coupling Landau-Ginsburg model(see, for example, Ma ), i.e., the model becomes exact,with a very small coupling constant very close to the crit-ical point. Consequently, at the critical point the gap ofthe longitudinal mode vanishes, as does the width-to-gapratio. Sufficiently close to the critical point, on the or-dered side, the longitudinal mode will then be very lightand highly stable.

However, the magnetic ordering transition in a stackedtriangular antiferromagnet is in a different universalityclass than the simple Neel case. This can be seen fromthe fact that a Neel state is invariant under rotationsabout the unique ordering axis, whereas the triangularstate has no such residual U(1) symmetry. Arenormalization-group analysis in this case indicates thatthe Gaussian fixed point is unstable. This indicates theoccurrence of a fluctuation-induced first-order phasetransition. Since the ~P~ coupling constants do not renor-malize to zero in this case, the longitudinal mode doesnot become perfectly stable.

In general, the question of whether or not the longitu-dinal mode will be sufficiently narrow to be observed is aheuristic one. It is reasonable to expect it to be more ob-servable for systems which are quite close to the quantumdisorder transition.

We emphasize that the renormalization-group argu-ment for the stability of the longitudinal mode dependscrucially on the fact that the transition is in the four-dimensional universality class, since it occurs at T=O.The finite-temperature transition is, of course, in thethree-dimensional universality class and, consequently,exhibits much less trivial critical behavior. There is noreason to expect a stable longitudinal mode in this case.

The outline of the rest of this paper is as follows. InSec. II, we review the Landau-Ginsburg model and thecalculation of the dispersion relation for both Neel andtriangular cases. We also discuss the extent to whichneutron-scattering data on CsNiC13 (Refs. 2 and 6) andRbNiC13 (Ref. 7) agree with this model. While the agree-ment is not completely satisfactory, we argue that theCsNiC13 data clearly call for a nontrivial extensionof spinwave theory. In Sec. III, we give therenormalization-group arguments for the stability of thelongitudinal mode in the Neel case and for the first-ordernature of the transition in the triangular case. In Sec. IV,

II. LANDAU-GINSBURG MODEL

We begin by discussing a single chain, Heisenberg anti-ferromagnet:

H, =2J Q S; S;+) . (2.1)

The continuum limit is defined by introducing' the pairof noncommuting vector fields, P(z} and l(z} representingthe long-wavelength staggered and uniform magnetiza-tion, respectively (z m. easures distances along the chain.We set the lattice spacing equal to one for the time be-ing). Because the integral of l over the entire chain givesthe conserved total magnetization, its commutation rela-tions are fixed to be

[l'(z), lJ(z') ]=i e'J"I"(z)5(zz'),[l'(z), P(z')]=le'J"P"(z)5(z z') .

(2.2}

(We set A'=1.) The commutation relations of the com-ponents of P with themselves are not fixed by any symme-try requirement and depend on the spin magnitude, s.We make the large-s semiclassical approximation thatthey commute. A correct treatment of the large-s limitalso requires that we impose the constraints~P~

~l~ /s =~/~ =1, P 1=0. This defines the non-

linear 0 model upon expanding the Hamiltonian tosecond order in l and dgldz. A perturbative treatmentof the cr model involves expanding P about its ground-state expectation value. This gives two Goldstone modes,the same spectrum as obtained from spin-wave theory (atlong wavelengths). However, this is known to be com-pletely the wrong picture in one dimension. Quantumfluctuations disorder the ground state. Roughly speak-ing, P fluctuates around the unit sphere so that(0~/~0) =0. The spectrum consists of a triplet of mas-sive magnons which correspond to the three componentsof P. [Since the field theory is Lorentz invariant, themagnon dispersion has the relativistic formE(Q n. }=+(vQ) +b, , where Q is the momentum, 6the gap, and v=4Js the spin-wave velocity. Thus, wemay regard b, /v as the rest-mass. ] The Landau-Ginsburg model is designed to give the correct behaviorat a mean-field level. %'e simply relax the constraint onP and replace it by a quadratic plus quartic potential.The full Lagrangian density is given by

2 '2

2v Bt 2 Bz~

2v 4

we calculate the magnetic field dependence of the mag-non dispersion relation, both in ordinary spin-wavetheory and in the Landau-Ginsburg model. It is againclear that spin-wave theory fails to capture, even qualita-tively, trends in the experimental data. It is unclear howgood the agreement with the Landau-Ginsburg model is;a detailed comparison will require the calculation of in-tensities and lifetimes and more experiments. Section Vsummarizes the agreement between experiment andtheory. The Appendix compares the Landau-Ginsburgmodel to another one which was recently proposed.

8936 IAN AFFLECK AND GREG F. WELLMAN

The quartic term is, in general, necessary for stability.The uniform magnetization density is then determinedfrom the commutation relations to be

I =(I/u)y X ay/at . (2.4)

This model becomes essentially exact in the large-n limitof the O(n) o model. We may estimate the normaliza-tion factor for the staggered magnetization asS;=( 1}'sv'g p, where g =&2/s, based on the large n-and large-s limits.

We now consider a quasi-one-dimensional system:

on the two sublattices, the potential energy per spin be-comes

V(gu) =(b, /2u 8J's )((}0+(A/4)go .We see that the critical value of J' is

(2.10)

J, '=b, /16vs . (2.11)

Neel state with all nearest-neighbor spins antiparallel.Writing

(2.9)

chains planes

83=J $ S; Si+J' $ S;.SJ .&,)'' (2.5) For larger J' the size of the sublattice magnetization isgiven by

Here the first term is over all nearest-neighbor pairs onthe same chain and the second is over all nearest-neighbor pairs in the same plane, with J'J. (Eachnearest-neighbor pair occurs twice in the above sums. )The Landau-Ginsburg Lagrangian is obtained by intro-ducing a separate field P;(z) for each chain i and thencoupling the staggered magnetization vectors at adjacentpoints on neighboring chains. (A coupling of uniformmagnetizations could also be included but this leads tocorrections of higher order in J'/J and, in any event,does not qualitatively alter our conclusions. ) Thus, thethree-dimensional Lagrangian is given by

T

L, = f dz gX, [P;(z)]2J's g P;(z) P, (z)l (;) '

(2.6)

Here the second sum is over nearest-neighbor chains. Inthe ordered state, (P;(z)) will be constant along eachchain. Dropping the t and z derivative terms, the La-grangian becomes minus the potential energy. We seethat whether or not order occurs is determined by a com-petition between the Haldane gap, 6, and the interchaincoupling J'. The critical value of J' is O(b, /vs). In thedisordered phase, at small J' where (0~/~0) =0, we cal-culate the magnon dispersion relation by simply ignoringthe quartic term. We see that there is a triplet of massivemagnons with a dispersion relation

E,&(g, a, Q~)=(/u Q, +b, +8J'vsf(Qj),where

(2.7}

f(Q, )=,' g e (2.8)

and the sum runs over the vectors 5; to nearest-neighborsites in the planar lattice (assumed to be Bravais). Thisformula is valid for Q, =m.. The shift of Q, by ~ is due tothe fact that P is the staggered magnetization.

Let us now consider the ordered phase which occursfor suf5ciently large J'. We now must distinguish be-tween different lattice types. We first consider the case ofa tetragonal lattice; i.e., a square lattice of chains. (Thefollowing discussion could be trivially generalized to thecase where the transverse lattice is rectangular ratherthan square). The ordered ground state is the simple

Pu=(16J's b, /u)/A. .We expand L to second order in small fluctuations:

4'=(0. 0, do+0.

(2.12)

(2.13)

x and y fluctuations are transverse and z fluctuations arelongitudina1. These do not mix to quadratic order. Wemay then read off the dispersion relations

E,(g, m., Q~)=+u Q, +8J'us[2+f(Qj)],

E (Q, n', Q ) =Qv Q, +8J'vs[2+ f(Q )]+6

(2.14)

(2.15)

where

For the square lattice of chains, of spacing a,

f(Q~) = cos(ag)+ cos(ag ) & 2 . (2.17)Note that E, vanishes at the antiferromagnetic wave vec-tor (~/a, m/a, n ). EL, on the other hand, has a gap brat this wave vector. Note that all three dispersion rela-tions, that of the triplet in the disordered phase, Eq. (2.7),and those of the longitudinal mode and transverse modesin the ordered phase, Eq. (2.15), become identical atJ'=J,' i.e., as we vary J', the spectrum varies continuous-ly, the triplet of the disordered phase splitting up into thetwo transverse modes and one longitudinal mode of theordered phase.

The intensities of these modes take a very simple form.The canonical commutation relation,

[P( xt), P(x', t)]=iu5(x x'},implies that the spin correlation function is

K'(Q, n. ,Qi, E ) = (S'S')(Q, m.,Qi, E)- (y'y')(Q, E)~ 6[EE, (Q)]/E'(Q) . (2.18)

[To obtain the neutron-scattering cross section, S"mustbe multiplied by the Lorentz factor (1Q, ), and a sumover a must be performed, depending on polarization. ]

bL =+2ukgu=+2(16J'us b, )=}/32us(J' J,') .

(2.16)

LONGITUDINAL MODES IN QUASI-ONE-DIMENSIONAL. . . 8937

&-b.= ~0o4.(0'. +Ay ) . (2.20}We see that the decay rate, which goes like the square ofthis coupling constant, is 0(A, ). The Landau-Ginsburgmodel actually reduces to the nonlinear cr model in thelimit where A,~ 00 and b, ~a& with 6 /A, = 1, sincethen the magnitude of P is forced to be exactly one. Inthis limit the mass of the longitudinal mode goes toinfinity, as does its decay rate. How light and narrow thelongitudinal mode is depends on the parameters in themodel. In particular, its mass is controlled by J' J,' andits width by A, . We expect that spin-wave theory will ex-hibit a two-magnon resonance with a finite energy gap.This resonance may then be identified with the longitudi-nal mode. Whether or not the longitudinal mode is stableenough to observe depends on the parameter A, in theLandau-Ginsburg model. %'e will argue, using the renor-ma1ization group, in the next section, that the width-to-gap ratio of the longitudinal mode vanishes in the limitJ'~J,'. This corresponds to the fact that the renormal-ized coupling constant A, vanishes at the critical point.

We now consider the effects of anisotropy. If we con-sider axial anisotropy which breaks the SU(2) symmetrydown to a U(1) subgroup, rotation about the z axis, thenwe must distinguish the easy-plane and easy-axis cases.

Thus, in particular, the intensity of the transverse modes(a =1,2) blows up at the ordering wave vector, whereasthe intensity of the longitudinal mode remains finite. Thetransverse modes would appear in xy polarized experi-ments and the longitudinal mode in z polarized experi-ments. As we shall see, things are considerably morecomplicated in the case of a stacked-triangular lattice.

The transverse modes are simply the standard result oflinear spin-wave theory for Q=m. and J'J. To seethis, note that the standard spin-wave theory spectrum is

EswT(Q) =1/(4Js+8J's) [4Js cosg, +4J'sf(Qi}]2 .(2.19)

Expanding to first order in J'/J and Q,2 inside the squareroot, and using U =4Js, we obtain precisely the first ofEq. (2.15). The longitudinal mode, on the other hand, isnot a standard spin-wave theory result. The reason isthat, in spin-wave theory, the spins are considered to beof fixed length, as in the nonlinear cr model. The longitu-dinal mode occurs in the above treatment simply becausewe have relaxed the constraint on the magnitude of thefield P in passing to the Landau-Ginsburg model. Actu-ally, the two theories are not quite as difFerent as they atfirst appear. To see this note that the longitudinal modeis unstable, even at zero temperature. It can always de-cay into a pair of transverse modes (i.e., Goldstone bo-sons). This is kinematically allowed since the Goldstonemodes are gapless whereas the longitudinal mode has agap. It is allowed by conservation of the z component oftotal spin, since the longitudinal mode has = 0,whereas the two species of transverse modes haveJ'= +1. Decay into a pair of Goldstone modes of oppo-site spin conserves spin. Such a decay vertex occurs inthe theory due to the (A, /4)~P~ term in the Lagrangian.Expanding P as in Eq. (2.13},we obtain a cubic term:

ai = (a/2, +3a /2, 0),az =(a /2, &3a /2, 0),a~ =(0,0,c) .

(2.21)

It is also convenient to define a set of three linearlydependent in-plane lattice vectors, 5;, with i =1,2, 3:

5, :-a, ,52 a2,5~:a, a2=( a, 0,0) . (2.22)

(See Fig. 2.} The reciprocal lattice is also triangular. Theprimitive vectors, b;, i =1,2, 3, defined by the conditions

b'a =2m5"J IJare given by

(2.23)

4m.b, :

4n i/3 1(2.24)

=2K3C

(0,0, 1) .

A B C

AjS

C

FIG. 2. Labeling of sites and lattice vectors in the basalplane.

In the first case, the U(1) symmetry is spontaneously bro-ken so there is a single Goldstone mode. Again the longi-tudinal mode is unstable against decaying into the Gold-stone mode. In the second, easy-axis case, the U(1) sym-metry is not spontaneously broken in the ordered phase,only the Z2 symmetry S'~ S' is broken. Consequent-ly, there are no gapless Goldstone modes. Spin-wavetheory and the Landau-Ginsburg model predict that thetwo branches of would-be Goldstone modes have equalgaps. However, according to the Landau-Ginsburg mod-el, the longitudinal mode still has a vanishing gap at thecritical value of J', and thus becomes lighter than thewould-be Goldstone modes suIciently close to the criti-cal point (on the ordered side). Therefore, it becomeskinematically unable to decay in this region and shouldexist as an infinitely stable excitation.

We now turn to the case of a triangular lattice ofchains. We choose the triangular lattice to lie in the xyplane with links parallel to the x axis, as shown in Fig. 2.The lattice spacing is a. The spacing between spins alongthe chains we take to be c/2, in order to agree with stan-dard conventions for CsNiC13, which has two formulaunits per unit cell. We choose a basis of primitive latticevectors:

8938 IAN AFFLECK AND GREG F. %'ELLMAN

eLA) kA

e&A

LB

e1C

reci procal-lattice point

aramBrilla

q=&/2maaei~maa ~aaaama~+aaa Q

I X

e~e

FIG. 4. Unit vectors in spin space.

Now the critical value of J' isJ,'=b, c/24us, (2.31)

IIIIIII

FIG. 3. The reciprocal lattice, projected onto the basal plane,showing the paramagnetic Brillouin zone, the wave vector cor-responding to (g, g, 1) and the reflection symmetry g~1 g.

(See Fig. 3.) Wave vectors are expanded in reciprocal-lattice vectors:

3

Q=(Qi Qz Qz)= g Q b (2.25)

S; =s( sinQO x;,0, cosQu x;), (2.26)

We also sometimes refer to wave vectors by x, y, and zcomponents, and use Qi to label the projection of thewave vector onto the basal plane. The classical orderedstate now involves three di6'erent directions, making an-gles of 2m. /3 with each other. We choose these to lie inthe xz plane, with one of them lying along the z axis, inagreement with the conventions of previous work. SeeFig. 1. Hence, the classical state at lattice point x, isgiven by

and the sublattice magnetization in the ordered state, forJ'& J,', is

i))u=(24J's/c b, /u)/1, . (2.32)

eL, = ( sinQz x;,0, cosQz x; ),e=(cosQz. x;,0, sinQ2 x;) .

(2.33)

Introducing the transverse xz fluctuation, iI)1;, the trans-verse y fluctuation, Pz;, and the longitudinal (xz) fluctua-tion, (()L, , as well as the unit vector parallel to the y axis,e2, we decompose the field at each site as

To obtain the excitation spectrum, we expand the La-grangian density to quadratic order in small fluctuationsaround the ordered state. For this purpose, it is con-venient to introduce two orthogonal unit vectors, both ly-ing in the zx plane in spin space, at each lattice site, i, eL;,and e. eL; is parallel to (P; ) and e is perpendicular toit. (See Fig. 4.) In terms of the projection of the orderingwave vector onto the basal plane, Qz, we may write

where the ordering wave vector, Qu, is given by

Q =(1 i 1) (2.27)4;=ex;(No+PL, )+ei 4i +eA'2 . (2.34)

in the notation of Eq. (2.25). It is also convenient todefine the projection of the ordering wave vector onto thebasal (xz) plane, Qz..

Q:(1 1 0) (2.28)

L;, = (4J'/c) g fdzP;(z) P.(z) .&i j )

(2.29)

Noting that, for two neighboring chains, P; P = ,'$0,and that each chain has six nearest-neighbor chains, wefind the potential energy per chain, per unit length, in theordered state:

Note that Qz lies on the x axis, at a corner of theparamagnetic Brillouin zone, which is a hexagon, asshown in Fig. 3.

The interchain coupling term in the Lagrangian is nowof the form

We now substitute this decomposition into the Lagrang-ian, Eq. (2.6), and expand to quadratic order in $1, Pz,and PL. It is important to note that we obtain crossterms between the transverse xz mode, Pand the longi-tudinal mode, PL. This does not occur in the case of a bi-partite lattice. It occurs here because longitudinal andtransverse fluctuations on neighboring sites are not or-thogonal to each other as illustrated in Fig. 5. On theother hand, the transverse y mode, Pz, is unmixed (toquadratic order). The terms quadratic in the y mode are

1 Pzi u Pzi

2u dt 2 Bz

(2.35)

V(go)=(b, /2u 12J's/c)$0+(A/4)$0 . (2.30)The terms quadratic in the transverse and longitudinal xzmodes are

LONGITUDINAL MODES IN QUASI-ONE-DIMENSIONAL. . . 8939

a

2v Bt

2 '2 2v ~Pi; a', 1 APL, ; v 54'L.;2 Bz 2u " 2v dt 2 Bz

'2Q2

$2 $2($2 +3/2 )

01 0'Ij '+( )NLi4Lj +(eLl elj)PL elj +(ei' eLj )01 PLj 1&i,j)

(2.36)

Here we have used the fact that eL,- eL. =e1,'e, = ,' for nearest-neighbor sites. The other dot products, e.-eL -, takeon the values +&3/2. In terms of the lattice vectors, 5; [see Eq. (2.22)), the interchain term in X,can be written

3

(4J'sic) g g g [ ,'[P&(x;)Pi(x;+5b)+PL(x;)P&(x;+5b)]i b=1 2

T(~3/2)[p, (x;)QL (x)k5b) pt (x;)pi(x;+5b)] j . (2.37)At this point we Fourier transform. It is convenient to define Fourier modes, $(Q), over the entire paramagnetic

Briilouin zone in the basal plane, shown in Fig. 3. In this way, we only obtain three branches of excitations, corre-sponding to pz and two linear combinations of p, and $L. Of course, the basal plane antiferromagnetic Brillouin zone

as only , the area so there should be 3 times as many excitation branches. This simplifying step is possible because ofthe symmetry of the antiferromagnetic ground state under simultaneous translation by one lattice spacing and rotationof the spina by 2w/3 about the y axis. This symmetry is incorporated into the definition of the components P& and PLvia the rotating coordinate system, e1 and ez. To obtain the experimentally observable neutron-scattering cross section,we must translate the xz branches into the antiferromagnetic zone, giving a total of five branches. Explicitly, we seefrom Eq. (2.33}that

P(Q) = g [niPL (Q+ n Qz)+P, (Q+n Qz) ]/2, $2(Q), g [PL (Q+ n Qz)+nig, (Q+n Q2)]/2 (2.38)n= 1 n= 1

We find that the dispersion relation for the y mode is given by

E2(g, 2m/c, Qi) =Q(vg, ) +(8vJ's/c )[3+2f(Qi)],where now

f(Qi)= cos(2ngi)+ cos(2ng2)+ cos[2m(gi+g2)] .The two xz modes are given by the solution of the eigenvalue equation:

(vg, ) +(8J'svlc)(3 f)i8v 3J'svflcE2

i8~3J'sv j'/c (vg, ) +(8J'su lc )(3 f)+EL,

where

j'(Qi) = sin(2m g, ) + sin(2n gz ) + sin [2m (g, +gz ) ] .Note that this is a slightly different notation than in Ref. (3):

ht 48J'vs/c 2hThe two frequencies are

E~(g, 2n. /c, Q~)=(vg, ) +(8J'vs/c)[3 f(Qi)]+bL /2++(bt /2) +3[8J'suj'(Qi)/c]

(2.39)

(2.40)

(2.41)

(2.42)

(2.43)

(2.44)

The intensities of the modes can be calculated from the eigenvectors obtained from Eq. (2.41). Normalizing theeigenvectors to one )Pi ~ + (Pt ~ =1,we may write

eP~(g, 2n/c, Qi, E)~ 5[EEq(Q)],12

4""(g,2n/c, Qi, E)=4' (g, 2m/c, Q&,E) (2.45}n= 1

(Q+ n Q2)5[E E+ (Q+ n Q2) ] .

The shift of Q by nQ2 (n = 1) results from the rotating coordinate system implicit in the definition of P, and PL, seeEq. (2.38). The constants of proportionality are the same in both of the above equations. Finally, solving the eigenvalueequation, (2.41), we obtain

8940 IAN AFFLECK AND GREG F. WELLMAN

(Sv'3J'svf Ic n[hi /2+ II/(b, l /2) +3[SJ'svf Ic] j )(Sv 3J'sv/cf ) + [bi /2+1/ (bL /2) +3[8J'svflc] ]

(2.46)

We see from its definition, Eq. (2.43), thatbi (48J'vs/c provided that 6 &0. However, b, mustbe regarded as a renormalized parameter and, forsufficiently large J'/J, it may become negative. There-fore, it is interesting to consider the qualitative behaviorof this spectrum as a function of the parameter hz for0&EL & ~, with J' held fixed. Let us first consider thelimit, AL ~~. In this limit, E+ ~ ~ and

E ~(vQ, ) +(8J'vs/c)[3 f(Qi)] .In this limit, E becomes purely transverse and E+purely longitudinal. This limiting formula for E can beseen to be the same as that of conventional spin-wavetheory in the limit of small Q, 2n/c an. d small J'/J.Likewise, the y mode, which is unaffected by EL, has thesame dispersion relation as the corresponding mode inconventional spin-wave theory in this limit. However, aswe reduce hL, the transverse xz fluctuations mix increas-ingly with longitudinal fluctuations so that E deviatesfrom the spin-wave theory result. Likewise, the otherbranch of energy E+ moves down in energy. Finally, atJ'=J,', b L =0 and the xz energies becomeE+(Q, 2'/c, Qi)~(vQ, ) +(SJ'sv/c)

&""(Q,E)=+"(Q,E)=+~"(Q,E)at 41 =0. The neutron-scattering cross section in the or-2=dered phase goes over continuously to that of the disor-dered phase. Note that the way this occurs is that the in-tensity of two of the five branches goes to zero and theother three become degenerate, with zero gap in the limitAL ~0. For a relatively small value of hL we should ex-pect two of the five branches to be very weak and the oth-er three to lie quite close to each other.

The theoretical spectrum is compared with the experi-mental results ' on CsNiC13 in Fig. 6. The theory con-tains three parameters, U, J', and AL . These can be usedto fit the slope BEIBQ,(Qv), the bandwidth of the basalplane dispersion at Q, =2m /c, and the gap of the strongermode of nonzero energy at the ordering wave vector [i.e.,E (b3+ 2Q2)]. This procedure gives 2v /hzc = l.38THz, J'/h& =0.0052 THz, and b L /bhz =0.28 THz. (h~is Planck's constant; we attach the subscript P to distin-guish it from the magnetic field. %e follow the standardconvention in the neutron-scattering literature of quotingfrequencies rather than angular frequencies; hence, they

X[3f(Q, )+&3lf(Q, )]. (2.47)

0.6a=(Tt, T[, 1)

Using the identity

f(Qi)+v'3f (Qi) = 2f (Qi+Q2we see that

Ep(Q, 2m/c, Qi) +(vQ, ) +(8J'sv/c)

(2.48)

X [3+2f(Qi+ eQi) ],(2.49)

where e=sn [f(Qi)]. Comparing with Eq. (2.39) and Eq.(2.7), we see that E,vI(Q), E2(Q), and E+(QkeQ2) allbecome identical at J'=J,'. Furthermore, in this limitthe eigenfunctions take on the simple form$1+ = + Ei P,+ = 1/&2. Thus, in the formula for theneutron-scattering intensity, Eq. (2.45), two of the termsvanish and the other two become identical, giving

N

0.4

UC

UQ)

LL

I

o

I

0.2

01 --( )

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Tl

0, 6Q( 1)

0.4

0.3

0.2

0. 1

0. 1 0.2 0.3 0.4 0.5Tl

0.6 0.7

FIG. 5. A longitudinal Auctuation on sublattice 3 is not or-thogonal to a transverse Auctuation on neighboring sublattice 8.

FIG. 6. Dispersion relation {Ref. 6) in CsNiC13 compared tothe Landau-Ginsburg model: (a) y polarization, (b) xz polariza-tion. Thick, thin, and dotted lines represent strong (I&0.64),medium (0.13&I&0.64), or weak {I&0.13) relative intensity,where the intensity is normalized to one for the y mode at(0,0,1). (See Fig. 7.)

LONGITUDINAL MODES IN QUASI-ONE-DIMENSIONAL. . . 8941

must be multiplied by hp, not R to obtain the correspond-ing energies. } Given the weak intensity of two of thebranches, as shown in Fig. 7, the agreement betweentheory and experiment is fairly satisfactory. The maindiscrepancy is that the second xz branch,E+ (Q+Q2) =E2 [see Fig. 6(b)], is only resolved for smallQ~ where its energy is about 10% lower than the theoreti-cal prediction. Is it important to realize that variouseffects (such as perturbative corrections from the A,Pterm in the Lagrangian) will renormalize the dispersionrelation. It may well be that these effects reduce the energy E2 suSciently that it can only be resolved from E3 atsmall Qj. The prediction of normal spin-wave theory iscompared with experiment in Fig. 8. Here we useJ/hz =,'(2v/c) =0.345 THz and the same value of J' asabove. Both theories fit the y mode quite well; indeedthey make essentially identical predictions. However,conventional spin-wave theory disagrees badly with theobserved xz dispersion. Of course, there are also renor-malizations of spin-wave theory which can be calculatedin a 1/s expansion using the Holstein-Primakov orDyson-Maleev formalism. Is it possible that these mighteventually lead to good agreement between theory andexperiment?

In fact there is a qualitative feature of the data whichconventional spin-wave theory will not be able to captureincluding arbitrary higher-order corrections. This is thestriking fact that the energy of the xz polarized mode ob-served at Q~=O is at about 2,' times higher energy thanthat of the upper mode at the ordering wave vector.Spin-wave theory predicts that these two frequenciesshould be equal. Furthermore, this is actually a conse-quence of a symmetry and thus should survive allhigher-order corrections. The symmetry argument ismost easily understood by plotting the xz mode in the en-tire paramagnetic Brillouin zone. The observed neutron-scattering intensity is then obtained by translating thisbranch into the reduced zone, Q~Q+Qz. The paramag-netic reciprocal lattice is shown in Fig. 3. The experi-mental results shown in Fig. 6 correspond to movingalong the x axis, g=4ng/a, shown .in Fig. 3. Note thata link of the reciprocal lattice is a perpendicular bisectorof this line, cutting it at g= ,. This implies that thedispersion relation is symmetric about g~1 q. The

3.53

2.52

1 ~ 5CC

1

0.5

0 0 1 0 2 0 3 0 4 0 5 0 6 0 7Tl

FIG. 7. Intensities from the Landau-Ginsburg model forCsNiC13. Curves 1, 2, 3, and 4 refer to xz-polarized modes not-ed in Fig. 6(b).

0.6

Q=(q, q, i)

OICT47

U

0.3

0.2

0.1 0.2 0.3 0.4 0.5 0.6 0.7Tl

FIG. 8. Dispersion relation (Ref. 6) in CsNiC13 with xz polar-ization compared to conventional spin-wave theory.

fact that the excitations can be defined over the entireparamagnetic Brillouin zone is a consequence of the sym-metry of the antiferromagnetic ground state under com-bined translation by one site and rotation by 2m. /3. Thesymmetry g~1 g is a consequence of a symmetry ofthe paramagnetic Brillouin zone. Thus, we expect thissymmetry to survive all higher-order corrections. Upontranslating the spectrum into the antiferromagnetic zone,this symmetry implies that the upper xz branch is sym-metric about g =, and, in particular, implies the equalityof the energies of the upper xz mode at (,', ,', 1) and the xzmode at (0,0,1). Any model which predicts a single xzbranch in the paramagnetic zone will predict that theseenergies are equal. (This includes the alternative modeldiscussed in the Appendix. ) Thus, the observed markeddifference in these energies indicates that there are twodifferent xz branches in the paramagnetic zone, as pre-dicted by the Landau-Ginsburg model. Note that thismodel also obeys the same symmetry. Both xz brancheshave the feature that the upper energy (upon translatinginto the antiferromagnetic zone) at (,', ,', 1) is degeneratewith the energy at (0,0,1).

RbNiC13 is another s = 1 quasi-one-dimensionalHeisenberg antiferromagnet with properties very similarto those of CsNiC13. The only important difference ap-pears to be that the ratio of interchain to intrachain cou-plings is about 80%%uo higher in RbNic13. So far, only un-polarized neutron-scattering experiments have been re-ported on this compound. They exhibit a dispersion re-lation very similar to that of CsNiC13. Again only a sin-gle spin-wave energy is observed near (0,0, 1). Again thisenergy is considerably higher than that of the uppermode at (,', ,', 1). However, in this case it is higher by afactor of approximately 2 rather than 2,'. A fit can be ob-tained to the Landau-Ginsburg model, this time with2v /&pc = 1.94 THz, J'/A p =0.0143 THz, andEL /hp=0. 9 THz, corresponding to a negative value of5, perhaps indicative of large renormalization effectsdue to the interchain coupling. The agreement betweentheory and experiment seems to suffer from the same de-fect as in CsNiC13. See Figs. 9 and 10. Branch number 2of Fig. 9 is again not observed near (,', ,', 1). The situationnear (0,0,1) is more ambiguous since polarized experi-

IAN AFFLECK AND GREG F. %ELLMAN

1.4

1.2

N

0.8OC

0.6

0.4

0.2

even for a cubic lattice. In any event, since the spin isquite large in this case, as is the anisotropy, it seems like-ly that conventional spin-wave theory would provide anat least qualitatively reliable description. So far only asingle peak has been observed near (0,0,1). Polarizedneutron-scattering experiments are needed to determineif this peak again contains contributions from twobranches. The various other branches predicted by aniso-tropic spin-wave theory have not yet been observed.

0 0 05 0 1 0 15 0 2 0 25 0 3 0 35Tt

FIG. 9. Dispersion relation (Ref. 7) in RbNiC13 compared tothe Landau-Ginsburg model. Thick, thin and dotted linesrepresent strong (I& 0.64), medium (0.13 &I & 0.64), or weak(I & 0.13) relative intensity, where the intensity is normalized toone for the y mode at (0,0,1). (See Fig. 10.)

ments have not yet been performed. A single broad peakis observed. Since it has the same shape at two differentequivalent wave vectors, if it results from two branchesthey must by very close together in energy. Since thepredicted splitting between y and xz modes is larger inthis case, due to the increased three-dimensionality, thisis diScult to understand. An alternative possibility isthat the observed peak only results from the y branch.Possibly the higher-energy xz branch is too broad to beobservable in RbNiC13. If this is the case then the spec-trum is not qualitatively difFerent than that predicted byconventional spin-wave theory. The main quantitativedifFerence is that branch 3 is higher in energy than pre-dicted by that theory. This issue could be resolved by po-larized neutron-scattering experiments.

Experiments have also been reported on the s =,'quasi-one-dimensional antiferromagnet, CsMnI3. ' ThisdifFers in several respects from the other two compounds.Apart from having a larger, half-integer spin, it may alsoexhibit more Ising anisotropy; the spins on the B and Csublattices (see Fig. 1) are measured to make an angle of51' with the z axis rather than 59' as in CsNiC13. It is notclear if the Landau-Ginsburg model is ever applicable inthe case of half-integer spin since the zero-temperaturephase transition may be in a different universality class,

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35Tl

FIG. 10. Relative intensities in the Landau-Ginsburg model.Curves 1, 2, 3 and 4 refer to xz-polarized modes noted in Fig. 9.

III. RENORMALIZATION-GROUP ARGUMENTS

In this section we apply renormalization-groupmethods to go beyond the Gaussian approximation to theLandau-Ginsburg model used in Sec. II. We use univer-sality arguments to justify the passage from "hard-spin"to "soft-spin" models. We show that the phase transitionas a function of interchain coupling, J', is second orderfor a bipartite lattice but first order for a stacked triangu-lar lattice. In the former case we show that the longitudi-nal mode becomes stable at the critical point. We verifythe conclusions in this case by comparing with the large-n limit of the 0 (n) nonlinear o model.

It must be emphasized at the outset that we can deduceexact results (subject to generally accepted assumptions)about these problems because of the fact that the phasetransition as a function of J' at zero temperature is in thefour-dimensional universality class. Since this is theupper critical dimension for the phase transition, it isGaussian up to logarithmic corrections. Consequently,conclusions drawn from a weak-coupling analysis of theLandau-Ginsburg model become exact at the criticalpoint.

We begin by considering the bipartite lattice case. Aswe approach the critical point we expect the correlationlength to diverge, both along the chains and also for in-terchain correlations. Thus, we may replace the discretesum over chains in Eq. (2.6) by an integral. Upon rescal-ing lengths in the basal plane appropriately, we obtainthe standard (I) quantum field theory in (3+1) space-time dimensions with Lagrangian density:

X =(1/2v)(BQ/Bt)' (v/2)(V'P)'(&3/2)IyI' (&/4)IyI' . (3.1)

Here V4 represent the gradient in the four-dimensionalspace, and we have absorbed the velocity U into a rescal-ing of the time coordinate. This is precisely the standardclassical Landau-Ginsburg Hamiltonian in four (space)

Here 53 is an effective parameter in this long-wavelengththeory which corresponds roughly to b, 16J'vs/c. Acrucial point is that this model is Lorentz invariant dueto the second-order time derivative. (See the Appendixfor a review of how this second-order term arises. ) Weexpect all breaking of Lorentz invariance to become ir-relevant near the critical point. The Feynman path-integral formulation of the theory is most easily studiedby going to imaginary time (i.e., Euclidean space): t ~i rThe Euclidean space Lagrangian density is

(3.2)

46 LONGITUDINAL MODES IN QUASI-ONE-DIMENSIONAL. . . 8943

dk, /d lnL = (11/8e )A, +(69/64m. )A, + (3.3)The solution, at large length scales and small couplings is

A(L) +A(LO)/[1+(11/86)A(LO) ln(L/Lo)] . (3.4)

The coupling constant flows to zero logarithmically slow-ly at large length scales. Away from the critical point,this decrease of A, ceases at a scale L of order the correla-tion length. (See Fig. 11.) On the ordered side, we mayestimate this scale as the inverse size of the order parame-ter: Lo/L ~ (P). Thus, we obtain the universal predic-tion

A,(L)~8' /11~ ln(P) ~ (3.5)as (P )~0. To lowest nontrivial order in A, , the mass ofthe longitudinal mode, for 53 (0 is given by EL = 2b, 3,and its decay rate is given by

I I =A,bL l32 (3.6)

Thus, the width-to-gap ratio is I'I/b, l =A, /32m. . Therenormalization-group prediction can be obtained fromthis by simply replacing A, by A,(L), the effective coupling

brokensgmmetrg

9~9c

nonlinearcs model

9~9c

FIG. 11. Renormalization-group flow for O(3)~O(2)universality class in four dimensions.

dimensions which would be used to model a finite-temperature classical magnetic transition. The Feynmanpath integral corresponds to the usual Boltzmann sum.Thus, the fact that the classical system in four dimensionsis governed by the Gaussian fixed point applies immedi-ately to the quantum system at T =0 in three dimensions.This result is well known in quantum field theory, ofcourse; it is usually referred to as the triviality of Ptheory.

The renormalization-group flows are shown in Fig. 11.The critical trajectory separating broken and unbrokensymmetry phases flows into 623=A, =O at long length (ortime) scales. This is much difFerent than in lower dimen-sions. In 4e space-time dimensions, the Gaussian(A, =O) fixed point is unstable and the phase transition iscontrolled by a fixed point at I,, of 0(e). In three space-time dimensions, A, is expected to be 0(1) at the criticalpoint. Consequently, the zero-temperature phase transi-tion, as a function of J, is much more trivial than theone that occurs as a function of temperature. In theT =0 case, along the critical trajectory, the effective cou-pling constant is governed by the P function

constant at scale L. (The anomalous dimension factorswhich normally arise cancel between b,L and AL .)Hence,

(3.7)

The longitudinal mode becomes infinitely stable at thecritical point. We note that S' contains a two-magnoncontinuum starting at zero energy as well as the longitu-dinal mode. The strength of this two-magnon contribu-tion vanishes as the critical point is approached. Thismeans that the spin Green's function, expressed as afunction of the invariant momentum p =E v p, hasboth a cut on the positive p axis beginning at p =0 anda pole displaced o8' the axis at p =EL +2ihL1. As weapproach the critical point, the intensity of the cut goesto zero and the pole approaches the origin along a trajec-tory which approaches the real axis. We give an explicitexample of this behavior below, using the large-n limit.

So far, we have only discussed the Landau-Ginsburgmodel in this section. However, by universality, we ex-pect these arguments to be much more general. TheGaussian fixed point is expected to be the universal stablefixed point governing the symmetry-breaking phase tran-sition SO(3)~SO(2) in four space-time dimensions.Essentially any model which undergoes such a phasetransition is expected to flow to the zero-couplingLandau-Ginsburg fixed point under renormalization.This should be true, for example, of the nonlinear 0 mod-el. Formally, this is obtained from the Landau-Ginsburgmodel by taking b.3~00 with h3/A, held fixed. In thislimit $2=1; longitudinal fluctuations have infinite mass.However, if we assume that the renormalization-groupflows shown in Fig. 11 extend all the way to infinity, thenwe conclude that the nonlinear o model renormalizesfrom A, = ~ all the way to iI =0 at the critical point. Thisflow is noted schematically in Fig. 11. This implies thatthe Landau-Ginsburg model should be better than thenonlinear 0 model for describing the physics close to thecritical point. We expect that the quantum Heisenbergmodel, which is the starting point or microscopic modelfor the study of quantum spin systems, will also be at-tracted to the Gaussian fixed point.

An instructive illustration of the above discussion isprovided by the large-n limits of both Landau-Ginsburgand nonlinear o models. We briefly review this limit, fol-lowing the notation and approach of Ref. 11. The start-ing point is to generalize the three-component vector, P,to n components, and then let n ~ oo. The four-dimensional Langrangian is written exactly as in Eq. (3.1)except that we replace the coupling constant A, by A, /n inorder to have a smooth large-n limit. The nonlinear crmodel is obtained by taking the limit 63~ao with( P ) = nb. 3/1, =n /g held fixed. In this limit longitudi-nal fluctuations of P are frozen, at short wavelengths, soP obeys the constraint P =n/g. The parameter g is thecoupling constant of the nonlinear o model. (The modelis often written in terms of a rescaled field, so that theconstraint becomes P =1 and a factor of n Ig appears infront of the derivative terms in the Lagrangian. ) A con-venient way of dealing with the large-n limit is to intro-

8944 IAN AFFLECK AND GREG F. WELLMAN

+(iy/2)(pi+n b3/A, )]+ [(n 1)/2] tr ln[ B+iy] . (3.9)

We now look for a saddle-point configuration in whichthe fields P, and y are constant and expand the functionalintegral in powers of the fluctuations away from the sad-dle point. It can be easily seen that this gives a series in1/n The .saddle-point configuration is of two possibletypes depending on the values of I, and 5 (or g in the emodel limit}. In the broken-symmetry phase, which weare interested in here, (y) =0 and ($, )%0 at the saddlepoint. The value of (Pi ) is determined by setting to zeroBS/By. This gives the equation, in the large-n limit,

duce an auxiliary field y(x) in terms of which the (Eu-clidean space) Lagrangian is rewritten:

,'V4$ V4$+ny /2k+, (ig/2)(~$~ +nb, 3/A, ) . (3.8)

Upon doing the Gaussian integration over the auxiliaryfield y in the path integral, we obtain the original La-grangian. Note that, the nonlinear 0. model limit, theterm in X quadratic in g vanishes so the effect of the yintegration is to impose the local constraint

nh3/AT. he next step is to integrate over thefields P in the path integral. This can be done exactlysince they now appear only quadratically in X. Becausethere are n P fields, the resulting trace-log terin has a pre-factor n. In order to study the possibility of spontaneoussymmetry breaking in which (P)%0, it is convenient tointegrate only over n 1 of the P fields and leave P, inthe action so that it may obtain an expectation value.The resulting effective action becomes

S(pi,y)= fd x['V4$, V~/, +ny /2A.

d4pS=,' 4

[1/k+&(p') ] X( p)(3.12)

Here ( P ) is given by Eq. (3.10). The progagator (i.e., theFourier transform of the time-ordered Green's function)is given by the inverse of the matrix appearing in S.Thus, in particular, the Pi propagator is given by

i [1/k, +(1/32m )[1+lnA /( p )]]p [1/A, +(1/32m )[1+lnA /( p )]](P)

(3.13)

In the nonlinear o model limit, this takes the simplerform

(i/32m )[1+lnA /( p )]p (1/32m )[1+lnA /( p )](P)

(3.14)

The real part of D gives the 11 components of theneutron-scattering cross section, (I'S'). This illustratesall the general features expected from therenormalization-group arguments. First focus on theweak-coupling limit of the Landau-Ginsburg model, as-suming 1/A, (1/32m )ln(A /(P) ). We see that D(p )has a pole given approximately by p =A, (P) =hi. Thepole is actually displaced slightly off the real axis sinceln( p ) has an imaginary part for positive p~. Thus, atthe pole,

(pi) = nb, 3/A, nf2 2 d k 1(2n) k

(3.10) Imp =A,b,r /32m . (3.15)

We must impose an ultraviolet cutoff A on the momen-tum integral (effectively given by the lattice spacing andthe exchange energy, J, in the quantum spin problem), sothe above integral gives A /16~ . We see that a solutionexists (i.e., the system is in the broken symmetry phase)provided that b,3/A, &A /16m (or 1/g&A /16m inthe o model limit). We now expand the action to quadra-tic order in the fluctuations around the saddle point, writ-ing P, = (P) +Pi. The term quadratic in y involves theintegral

The position of the pole can be written as b,J +2ihz I,where I is the decay rate. Thus, we find

I =Ahi /64m . (3.16)

1/)t, ~ 1/A, ,s.1/A, +(1/32m ) lnA /b, zin Eq. (3.16), with Az determined by

(3.17)

This agrees with Eq. (3.6) up to a factor of 2 which arisesfrom rescaling A, and making the large-n approximation.A better approximation to I is obtained by making thereplacement

~( )p, d4k 1 1(2m. } k (k +p)

(3.11) (P) =hi [1/A. +(1/32m ) ln(A /b, f )] . (3.18)

For momenta much less than the cutoff,

8(p )=(1/32m )[1+lnA /( p )] .

Here we have made the analytic continuation back to realtime and p =E p. Rescaling y by a factor of ~n,the quadratic part of the action, written in momentumspace, in matrix form, becomes

We see that Eq. (3.16) will be approximately correct withthese replacements whenever A,,ff 32m.

Ar ff is just the effective coupling constant determinedfrom the renormalization group (in the large nlimit)-evaluated at the scale hz. We see that, for hz A, theeffective coupling becomes small, even if the bare cou-pling is not. In fact, this remains true even in the non-linear o. model limit, where the bare coupling is infinite.In this case, Eq. (3.17) reduces to

46 LONGITUDINAL MODES IN QUASI-ONE-DIMENSIONAL. . . 8945

I/A, ,s =(I/32m )lnA /hz . (3.19) 4 = (P, +if' )IY2 . (3.26)Thus, we see explicitly, in the large-n limit, that the non-linear o model becomes equivalent to the Landau-Ginsburg model with a small coupling constant, near thecritical point where (P) and b,l vanish. Up to a lnlnterm, we may replace hz by (P) inside the logarithm.

We also see from Eqs. (3.13) and (3.14) that, forI p I h~, the propagator has a cut:

D(p )~ lnA /( p ) .32~'(y)' (3.20)

It is now convenient to change variables to

This arises from the two-Goldstone boson intermediatestate. Thus, the real part of D, which gives the neutron-scattering cross section, has a constant part at small posi-tive p coming from the two-magnon contribution andthen a resonance at p = hz from the longitudinalmode. As (P) ~0, the resonance moves down to p =0,and right at the critical point the propagator reduces toD(p )~i/(p +ie); the real part collapses to a 5 func-tion at p =0. Close to the critical point most of the in-tegrated intensity comes from the resonance, not the cut.

We now turn to the triangular lattice case. It is impor-tant to realize that the order-disorder phase transition isin a different universality class than in the bipartite case.This follows from the fact that there is an unbrokenSO(2) symmetry in the Neel state on a bipartite lattice(rotation of the spins about the unique ordering axis} butnot in the triangular lattice where the ordered state in-volves three different axes making angles of 2n/3 withrespect to each other, as shown in Fig. 1. Now, takingthe continuum limit of the three-dimensional Landau-Ginsburg model, we must introduce three fields, P;,i =1,2, 3 labeling the three inequivalent sublattices in thebasal plane. The quadratic part of the potential energy isof the form

The original spin operators are related to 4 by

S; ~ Re(@e ' '), (3.27)

where Qz is the ordering wave vector projected onto thebasal plane, given in Eq. (2.28). The complete Landau-Ginsburg model may be rewritten in terms of 4 (after el-iminating P, ) as

E'=V44* 7~4+634'4+(A, , /4)(4'4)+(A~/4)(4 4)(4'4') . (3.28)

Here 623 is an effective renormalized gap parameter, asbefore. The two coupling constants, A, , and A,2, are deter-mined by the original single coupling constant, A, . Tolowest order in A, , they have the values, A, , =4k, /3,A,2=2K, /3. They are both positive, resulting in an or-dered state with Re@LIm@. This gives the expected2n/3 structure, from Eq. (3.27). Higher-order correc-tions to the Lagrangian are obtained from integratin~ out

However, these only produce corrections to b, 3, A,and A,z together with terms of higher order in derivativesor in powers of the fields. This follows from the discretesymmetry P& +$2~$3+Pwhich corresponds to4~e ' 4?. This symmetry forbids any other non-derivitive quadratic or quartic terms. [Note that this Z3symmetry is actually enlarged to a U(l} symmetry in theLandau-Ginsburg model. Only by keeping sixth-orderterms in X is the symmetry reduced to the Z3 subgroup. ]Thus, the effective Landau-Ginsburg Lagrangian densitymust have the form of Eq. (3.28) with some effective pa-rameters, 53, A, A,2. The P function has been calculatedfor this model:

d A, , /d lnL = ( I/16' )(7A, (+4k, )A,2+4k,2)+O(A, ),(3.29)

4" =(24' 4z 43}/v'6

4"=(bi+4'2+6)/

(3.23)The resulting renormalization-group flows are shown inFig. 12. Note that only the line A,2=0 flows to the originin coupling constant space. Otherwise all trajectoriesflow to A. , = 00, A,2= + 00. Thus, we see that the Gauss-ian fixed point is not stable, for this phase transition, infour space-time dimensions. The usual interpretation ofthis kind of renormalization-group flow is that the phasetransition is driven first order by fluctuations. This issignified by the negative value of A. 1 upon renormaliza-tion. Including positive ~P~ terms for stability, we find afirst-order phase transition in Landau Theory. There hasbeen some controversy lately over the correspondingphase transition in three dimensions, which would corre-spond to the finite-temperature transition in CsNiC13. Itmay be first or second order. However, in four dimen-sions there seems to be no question; the Gaussian fixed

(3.24)

This diagonalizes the quadratic terms giving

v ,'(5 /U 121's}([y,[ +~/ ~ }+,'(6 /u+24J's)(P, ( (3.25)

We see that, as we increase J', the a and b modes eventu-ally become gapless whereas the c mode gets a larger gap.P, is the ferromagnetic order parameter; its gap vanishesif J' is sufBciently large and negative. Since we are in-terested in the antiferromagnetic case, we may simplydrop the massive mode, P from the low-energy theory.The two remaining modes, P, and Pb, can be combinedinto a complex three-vector field:

(3.22)dA, /d2lnL = (I/16m )(6A, ,A~+3A2)+O(A, ) . (3.30)

8946 IAN AFFLECK AND GREG F. WELLMAN 46

FIG. 12. Renormalization-group fiow for triangular latticeantiferromagnets in four dimensions.

point is unstable so the transition is expected to be firstorder.

What does this imply about the longitudinal mode'?We should expect that, as the interchain coupling J' isdecreased in the ordered phase, the sublattice magnetiza-tion will decrease smoothly for a while, but eventuallywill make a discontinuous drop to zero at J,'. Corre-spondingly, we expect that the mass of the longitudinalmode will decrease smoothly before dropping abruptly tozero. It also seems plausible that the effective couplingconstants will at first decrease, before the first-order tran-sition point is reached. Thus, we might expect thewidth-to-mass ratio of the longitudinal mode to decreasewith increasing J . Of course, this ratio will never reachzero; before this happens the transition to the disorderedphase will occur. The value of this ratio at the first-ordertransition point, J,', is nonuniversal. Whether or not thelongitudinal mode is observable in a stacked triangularsystem is an empirical question. As we have tried to ar-gue in Sec. II, the experimental evidence in CsNiC13 sug-gests that it is.

IV. NEUTRON-SCATTERING IN A FINITE FIELD

In this section we consider the effect of a magnetic fieldon the magnon dispersion relation in the ordered phasefor the case of a stacked triangular lattice antiferromag-net. ' We first present the result of conventional spin-wave theory. (As far as we know, this simple result hasnot been published before in its entirety. ) We thenpresent the analogous result using the Landau-Ginsburgmodel; i.e., including the longitudinal mode. Some com-parison is then made with finite-field neutron-scatteringexperiments on CsNiC13. Finite-field effects in the disor-dered phase of Haldane gap antiferromagnets have beendiscussed elsewhere. ' In the case under considerationhere, where crystal-field (or exchange) anisotropy can beignored and the Zeeman energy is smaller than the Hal-dane gap, the result is extremely simple. The Haldanetriplet simply undergoes a Zeeman splitting with energies6, 6+gpzh, where h is the magnetic field. As we haveemphasized above, in the Landau-Ginsburg model we gosmoothly from disordered to ordered phases by varyingthe interchain coupling. (However, the transition isdriven to first order by fluctuations in the stacked tri-angular lattice case. ) This remains true when a magnetic

23cos 82

Minimizing E with respect to 8 givesgpzh

s (8J + 18J') (4.1)

We note that, for CsNiC13, J/hz =0.345 THz, so, for themaximum field of 6 T used in the experiment, sin8=0. 06.

To calculate the dispersion relation using conventionalspin-wave theory, we expand around the classical groundstate, calculated above, to quadratic order in magnoncreation and annihilation operators. The calculation canbe considerably simplified by the observation that, as inthe zero-field case, the classical ground state is invariantunder translation by one site together with a rotation by2n/3 in the xz. plane. (This symmetry is not destroyed bycanting. ) For zero canting, the spin operators on the Asublattice (see Fig. 1) are expanded as

S = [&s /2(a +a ),i &s /2( a t a ),s a ta ] . (4.2)

This is the correct representation when the spin points inthe z direction in the classical approximation. Here a(a ) annihilates (creates) a boson. To obtain the correctrepresentation on the six different sublattices, in the pres-ence of canting, we simply rotate S by the rotation ma-trix which produces the correct classical state from theone where the spin points in the z direction. The neededrotation matrices are R the cant (rotation by 0 aboutthe x axis), R2, a rotation by 2n/3 about the y axis,and R3 a rotation by ~ about the y axis. The needed rep-resentation on the six different sublattices, shown in Fig.1, is

S~ =R&S]

S~ =R2S

S~=R2S~,

field is included, with the field lowering J,'. Thus, inweak fields, for J' only slightly bigger than J,' theLandau-Ginsburg model predicts behavior much differentthan does conventional spin-wave theory. This behaviorgoes over smoothly into the simple Zeeman splitting ofthe disordered phase as J'~J,'. The finite-field experi-ments provide evidence for such an effect in CsNiC13.

Following the experimental setup, the spins are or-dered in the xz plane (at h +0), as in Fig. 1 and the mag-netic field, h, is applied along the y axis. We first consid-er the classical problem, regarding each spin as a classicalvector, of length s. This gives the starting point for stan-dard spin-wave theory. (Note that this is not quite thesame as the classical limit of the Landau-Ginsburg modelbecause that model is developed in terms of the staggeredmagnetization. ) Classically, each spin cants in the ydirection by the angle 8, without changing its orientationin the xz plane. The classical energy per spin is

E/N= 2Js cos28 gpzhs sin8

LONGITUDINAL MODES IN QUASI-ONE-DIMENSIONAL. . . 8947

S~ R3Sq,SER3S~,SP=R3S

The Hamiltonian may now'be written in the fullytranslationally invariant form:

A =R1R2R1

1

2

v 3sin82

&3 cosH2

~3 sinH2

2 sin 8cos 82

3 sin8 cos82

v 3cosH2

3 sin8 cos82

cos 8sin 8

2

N~=2 X JSi x. BSi x+s

j=1B=R1R3R1= 0

0

0 0cos28 sin28sin28 cos28

3+J' g Si 'AS) +s, gPahS, ,y, (4.3)X

where the 5 s are the lattice vectors defined in Eq. (2.22)for i =1,2, 3 and 5~=a3, defined in Eq. (2.21}. A and Bare real matrices:

Note that, while B is symmetric, A is not. Thus, theHamiltonian of Eq. (4.3) is not invariant under reflectionsin the basal plane. We now Fourier transform. Since theHamiltonian of Eq. (4.3} has the full lattice translationsymmetry, we may introduce Fourier modes in the fullparamagnetic Brillouin zone. The quadratic terms in Hbecome

N 3cos 8H2 = g 4Js cos28 4Js sin28 cos(cg, /2) +gimii h sinH+ 12J's 1

k=1

+J'sf(Qi)(1 3sin 8) a&ta& [2Js cos 8cos(cg, /2)+ ', J's cos Hf(Qi)](at&at &+a&a &) ' . (4.4)

f(Qi) is defined in Eq. (2.40). We now diagonalize H2 by a Bogliubov transformation, giving the dispersion relationE (Q)= t4Js[1 sin 8 cos(cg, /2)]+ J's[6+f(Qi) 3 sin Hf(Q&)]] [4J, cos 8 cos(cg, /2)+3J'cos Hf(Qi)]

(4.5)

Defining Ec(Q}to be the zero field energy,

Eo(Q)= [4Js+J's[6+f(Qi) ]][4Js cos(cg, /2)+3J'sf (Qi)]

this can be conveniently rewritten as

E(Q)=QEO(Q}+c(Q}s sin 8,where

(4.6)

I

also at the 12 corners of the paramagnetic Brillouin zone.(See Fig. 3.) Consequently, E and E,+ each vanish atone inequivalent wave vector in the antiferromagneticzone. These are the three Goldstone modes correspond-ing to the complete breaking of rotational symmetry. Eand E, vanish in zero field at (ggz,g3)=( ,', ,', 1),the ordering wave vector. [See Figs. 6(a) and 8.] Atnonzero field, E(Q) only vanishes at Q=O since c(Q)vanishes at Q=O but not at the corners of the paramag-

c (Q ) =32J2[ cos (cg, /2) cos( cg, /2 ) ]+8JJ'[5 cos(cg, /2)f(Q )

3f( Qi }6 cos(cg, /2) ]+12J'[f(Qi)' 3f(Qi)] . (4.7)

Ey(Q) =E(Q),

E.~ Q(=)E Q(+ Qo.)(4.8)

(4.9)

Note that, in zero field, Eo(Q} vanishes at Q=O and

Taking into account the rotating reference frame implicitin the definition of the a. 's, we find a single branch ofmagnons with y polarization and two with xz polariza-tion:

0 ' 6

0.5

0.4

0.3

t- 0.2

0.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Tl

FIG. 13. Dispersion relation of a y-polarized mode in a mag-netic field of h =0 and 6 T, according to spin-wave theory andLandau-Ginsburg model.

8948 IAN AFFLECK AND GREG F. %'ELLMAN

I I

y polarization

1t

!

Q=(0,0,1)0.8 tl y polarization0.8 Q=(.39, .39,

zI-OCD

LL

0.6N

0.6C7

CD

UCD

LL

0.4

0.2 0.2l

0 I iI I

4 6 8Magnetic Field (T)

10 124 6

Magnetic Field (T)

10

FIG. 14. Field dependence of - ocies (R f 6

o y-polarized spin-wave frequen-cies e. ) at wave vector (001

an au-Ginsburg model., , ) in CsNiC13, compared to the

FIG. 16. Field dependence of y-polarized spin-wave frequen-

tcies (Ref. 6) at wave vector (0.39 0.39 1)

'

o the Landau-Ginsburg model.in CsNiC13, corn aredp

netic zone. Thus, onl Ey , vanishes at the orderingwave vector. This corresponds to the fas o e act that, classical-

lane costo a ions o the canted spin configuratio

'th

p st zero energy but any rotation involving achange in the y components of th

'y.o e spins costs energy. In

fact, c(Q) is relatively small of O(J'J), o whenever Q, =0.onsequently, the change in the energies of the xz modes

for fields less than 6 T in CsNiCl is ne li ibwave vectors considered in this paper. Only the y modeis significantly effected by a 6-T field and h

a er sma except near the ordering wave vector.e y-mode dispersion relation at fields of 0 and 6 T '

shown in Fi . 13. Wean is

ig. . e do not show the xz dispersion rela-tion at 6 T because to ththe naked eye, it is indistinguish-able from the zero-field result in Fig. 8.

Note that this field dependence, predicted by spin-wavetheory, is completely different than th t h' ha w ic occurs in

e isordered phase. In this case for th fi ldg e y axis, the y mode, which has &=0, is com-

pletely unaffected by the field a d than e xz modes, whichave 4' =61 and a Zeeman splittin k h.ing g 1MB

ependencee us now consider the experimental field d, ( . ,0.1,1), and (0.39,0.39,1) shown b th

l i Fi 14-1igs. 9. Note that they mode is quite weaklaffected by the field, and its field d de epen ence is quite wellpre icted by conventional spin-wave theory. However

the xz mode showows a stronger field dependence than thei e y conventional spin-mode and is not at all described b c

fre uencies.wave theory which predicts essentiall field-'ia y e -mdependent

quencies. Remarkably, the experimental behmuch better fit b t

n a e avior isy the disordered phase behavior than by

spin-wave theory, despite the fact that thethe ordered hase.

a t e system is inp ase. Since the Landau-Ginsburg model in-

terpolates smoothl bdisordered hase

o y etween spin-wave theory and thp se behavior, we might expect it to give a

e

good description of this behavior.We ne now consider magnetic field dependence in the

Landau-Ginsburg model. The La ran ig gian is obtainedq. . y t e replacement

3 Q2 hxV= g rI);+(A, /4)P, 2U 2U

+(4J's/c)tet 42+42 03+4'3 41 j ~ (4.11)

Choosing the magnetic field to lie along the y axis, we see

Q=(.1, .1, 1) y polarization0.8 0.8

N

C2

CD

DCD

LL

0.6CJ

CD

t3CD

LL0.4 0 4

0

0.2

4 6Magnetic Field (T)

10 4 6Magnetic Field (T)

10

FIG. 15. Field dependence of - olcies (R f.

o y-polarized spin wave frequen-cies e . 6) at wave vector {0.1 0.1 1) ir, , ) in CsNiC13, compared to

e an au-Ginsburg model.

FIG. 17. Field de endenp ence of xz spin-wave frequencies {Ref.6) at wave vector (0,0, 1) in CswiC13, compared to the Landau-

dP/Bt~c)p/c)t+gP hXQ .B (4.10)

We must first recalculate (P ) in the presence of the field.ac su attice of

netic unit cell:ain t e potential energy per antiferrommag-

46 LONGITUDINAL MODES IN QUASI-ONE-DIMENSIONAL. . . 8949

t4zcs

in

Oin

U-

0.8

0.6

0.4 o

0.2

0.8

0.6

0.4

0.2

0 2 4 6 8Magnetic Field (T)

10 12 4 6Magnetic Field p)

10

FIG. 18. Field dependence of xz spin-wave frequencies (Ref.6) at wave vector (0.1,0.1,1) in CsNiC13, compared to theLandau-Ginsburg model.

FIG. 19. Field dependence of xz spin-wave frequencies (Ref.6) at wave vector (0.39,0.39,1) in CsNiC13, compared to theLandau-Ginsburg model.

that the minimizing configuration for the P s is stHl the2n /3 structure in the xz plane, but with a different mag-nitude of P. This might seem to contradict the classicalresult discussed above which involved a canting of thespins in the y direction. However, there is no contradic-tion because this canting is uniform along the chains.Thus, it does not show up in P, the staggered magnetiza-tion, but only in

1= (I/u)fx(ap/Br+gp, ,hxf) =(gp,,/u)(p)'h .(4.12)

$z= [(gp,sh ) +24J'sv lc 6 ]/A, .v (4.14)This sensitive dependence of the sublattice magnetizationon external Beld leads to a strong field dependence ofmagnon energies.

To calculate the dispersion relation, we introduce thesame rotating coordinate system and modes as in Eqs.(2.33) and (2.34) and expand the Lagrangian to quadratic

I

Assuming the 2m/3 structure, the potential energy perchain per unit length becomes

V~ [[6, (gissh ) ]/Zu 12J'sic]iI) +A,P /4 . (4.13)Note that the applied field decreases the elective b andfavors the ordered phase. The sublattice magnetization isgiven by

order. The terms quadratic in the y mode, iI)2, are exactlythe same as previously, Eq. (2.35), except that the valueof (iI)) has changed, now being given by Eq. (4.14). Onthe other hand, the terms quadratic in transverse and lon-gitudinal xz modes, iI), and PL, are the same as previous-ly, Eq. (2.37) (expressed in terms of the new value of((t) )), plus the additional terms

gPa" 4ii~

Pl.i~xz gr Li gr 1 i

I

(4.15)

At this point we Fourier transform with respect to spaceand time. The energy of the y mode is given by

Ei(Q, 2m/c, Qi)= (uQ, ) + (8J'vs/c )[3+2f(Qi) ]+(gpsh )

(4.16)

This is the same result as obtained from conventionalspin-wave theory, Eq. (4.6), for small Q, 2~/c and J'.The agreement with experiment, shown in Figs. 14-16, isquite good. The classical equations of motion mix thetransverse and longitudinal xz modes as before. The clas-sical frequencies are now given by the vanishing deter-minant condition:

Ez+v2Q, +8J'su I(3 f)c2igpshE 8~3iJ'svf Ic-2igijshE+8&3iJ'svf Ic E+v Q, +8J'su(3 f)Ic+AI +2(gpsh ) (4.17)

This gives a quartic equation in E with four real solutionsof both signs. Since f( Qi)= f(Qi), the classicalsolutions at wave vector Q are 1 times the solutionsat wave vector Q. At the quantum-mechanical level, itcan be seen that the magnon energies are given by the ab-solute values of all four classical frequencies, at eachwave vector. The doubling of the number of solutionsoccurs because the modes at Q and Q are mixed. Thesame wave-vector shift occurs, due to the rotating refer-

I

ence frame as at zero field. Thus, there is a single y modeand, in general, eight xz modes given by the absolutevalue of the four solutions of Eq. (4.17) shifted in wavevector by +Q2, defined in Eq. (2.28). The xz frequenciesare quite strongly Beld dependent, due to the field depen-dence of (P), unlike in conventional spin-wave theorywhere the field dependence of xz modes is minute.

The field dependence at wave vector (0,0, 1) of the xzmodes is shown in Fig. 17. There are only four indepen-

8950 IAN AFFLECK AND GREG F. %ELLMAN

dent xz branches at this wave vector. They display be-havior reminiscent of Zeeman splitting, as expected, andas observed experimentally. There is some experimentalevidence that the single zero-field xz peak is actually splitinto two with frequencies of about 0.42 and 0.58 THz.However, due to the low beam intensity in this polarizedinelastic-neutron-scattering experiment, the apparentdouble-peak structure may not be statistically signi6cant.(See Fig. 3a of Ref. 8.) Higher-intensity experiments areneeded to resolve this issue. If this splitting is reallypresent, it is probably a result of crystal-field anisotropy.Such anisotropy can be included in the Landau-Ginsburgmodel. We expect it to split the upper zero-field xz peakinto two components. (It also mixes xz and y modes. )We also show the experimental results and theoreticalpredictions for xz polarization at wave vectors (0.1,0.1,1)and (0.39,0.39,1), in Figs. 18 and 19. The nonzero fielddependence of xz-polarized branches disagrees badly withconventional spin-wave theory which predicts an essen-tially field-independent spectrum given in Fig. 8. It ismore difficult to say how well it agrees with the Landau-Ginsburg model because we have not yet calculated theintensity or width of the branches at finite field and be-cause of the low intensity and resolution of the experi-ments. At (0,0, 1} and (0.1,0.1,1), we expect the lowerbranches to be of very low intensity at small fields. TheZeeman-like behavior of the upper branches is in at leastrough agreement with experiment. Note that, at(0.39,0.39,1), near the ordering wave vector, the lowest,Goldstone, xz branch is essentially field independent,whereas the next lowest branch is split into two by thefield.

V. CONCLUSIONS

We may summarize the agreement between theory andexperiment as follows. In CsNiC13, the y-polarized spin-wave spectrum is in good agreement with both theories,which make essentially the same prediction. The xz-polarized spin-wave spectrum disagrees badly with con-ventional spin-wave theory. In particular, we haveidentified two qualitative features which are missed bythis theory: the large ratio of frequencies of the uppermode at (0,0, 1) and (,', ,', 1) and the strong field depen-dence. It is more dif5cult to estimate the agreement be-tween the Landau-Ginsburg model and experiment forxz-polarized modes. Certain qualitative features are wellexplained: the existence of a mode near (0,0, 1), which isnearly degenerate with the y mode and about 2,' timeshigher than the upper mode at (,', ,', 1); strong fielddependence which is Zeeman-like near (0,0, 1). The mostserious discrepancy is probably the nonobservation, near(,', ,', 1), of xz branch number 2, shown in Fig. 6(b}, pre-dicted at a frequency of 0.28 THz and an intensity ofabout ,' that of branch number 3 at 0.18 THz. However,we can argue with some confidence that it must be theresince it is the continuation of the upper branch which isobserved near (0,0, 1). It seems quite likely that its energyis renormalized downward so that it cannot be resolvedfrom branch 3, near (,', ,', 1). This hypothesis could bechecked by measuring intensities. Another fact to keepin mind is that the upper branches are, in general, of

finite width. These widths have so far not been calculat-ed; in particular, we do not know their wave-vectordependences. Possibly the second branch becomes unob-servably broad near (,', ,', 1).

The number of so far unobserved modes becomeshigher at finite field. Thus, higher-intensity and resolu-tion finite-field experiments may provide a more definitivetest of the Landau-Ginsburg model.

As we have emphasized, the spin-wave spectrum of theLandau-Ginsburg model goes over smoothly from the or-dered to disordered phase (ignoring fiuctuation effectswhich drive the transition first order for a stacked tri-angular lattice). Thus, certain features of the experimen-tal data, taken in the ordered phase, which are reminis-cent of the behavior of the disordered phase, are ex-plained naturally by the Landau-Ginsburg model. Thesefeatures include the fact that the xz modes are nearly de-generate with the y mode near (0,0, 1) and the fact thatthese three modes exhibit a Zeeman-like field depen-dence. These features are approximately reproduced bythe Landau-Ginsburg model with the choice of parame-ters we have made. There is some indication thatCsNiC13 is exhibiting behavior near (0,0,1) which is evenmore like the disordered phase than that of the Landau-Ginsburg model with these parameters. It is quite possi-ble that higher-order corrections to the model mightreproduce this; i.e., these might give an effective Azwhich depends on wave vector and might be smaller near(0,0, 1) than near the ordering wave vector.

A major experimental issue which needs to be resolvedis whether the xz mode at (0,0, 1) is split into two com-ponents even in zero Geld. The experimental situationpresently seems ambiguous. If so, this presumablyrepresents an effect of the Ising anisotropy.

The situation in RbNiC13 remains more ambiguouspending polarized neutron-scattering experiments. Wedo not know whether a portion of branch 2 lies inside thepeak observed at (0,0, 1). It is possible that this peak con-tains only the y mode and that branches 1 and 2 are athigher energy and may be too broad to be observable.

Another type of theoretical prediction that we havemade involves the dependence on interchain coupling.We have predicted a first-order transition for a stackedtriangular lattice but second order for a tetragonal lat-tice. If a material could be found with an interchain tointrachain coupling ratio very close to the critical value,then it might be possible to study the transition by apply-ing pressure to the sample. One could search for such amaterial by looking for ordered systems with very smallantiferromagnetic moments. We note that, in the tri-angular case where the transition is first order, the mo-ment would decrease, upon decreasing the ratio, to somelimiting nonzero value, before dropping discontinuouslyto zero. Since this limiting value is not known, it isdifficult to estimate how close CsNiC13 is to the phasetransition.

ACKNO%'LKDGMKNTS

We would like to thank Dan Arovas, Bill Buyers,Matthew Fisher, K. Kakurai, Lon Rosen, Michael

46 LONGITUDINAL MODES IN QUASI-ONE-DIMENSIONAL. . . 8951

Steiner, and Zin Tun for useful questions, discussions,and suggestions.

APPENDIX: FORM OF THE LAGRANGIAN S/d

we obtain the Euler-Lagrange equation

sBS(t)/Bt= S(t)XBH/BS(t) . (A2)For a lattice of coupled quantum spins with a Hamiltoni-an of the general form

H=s Q JJS; SJ, (A3)E,J

this gives the first-order classical equation

BS, /Bt = s g J,"S,XSJ

(A4)

Note that this is the classical torque equation. The con-tinuum limit is obtained, for an antiferromagnet, by keep-ing only long-wavelength fluctuations of the uniform andstaggered magnetization density, 1 and sP, respectively;i.e., we approximate

sS; =1(x;) +( I )'sP(x; ), (A5)where we obtain a plus or minus sign for the two sublat-tices, and we have again set the lattice spacing to one.For a stacked triangular lattice, i labels distance along

An alternative Lagrangian was proposed recently forquasi-one-dimensional antiferromagnets which also con-tains longitudinal fiuctuations but gives a rather differentspin-wave spectrum. The purpose of this appendix is tocompare the two approaches and justify the form of theLagrangian used here.

The path-integral formulation of a single quantum spinis written in terms of a unit vector S(t) with an actionS=sA[S(t}]fdt H[sS(t}], where A is the orientedarea swept out on the unit sphere by the closed path S(t),and H (sS) is the Hamiltonian. ' (See Fig. 20.} There is a4m ambiguity in the choice of area, A, but the weight, e'in the path integral is single valued since s must be an in-teger or half-integer. The equations of motion are de-rived by varying the action with respect to aninfinitesimal change in the path 8(t}+S(t)+58(t). Using(see Fig. 21)

5A =fdt58 [SXBS/Bt], (Al)

FIG. 21. The change in the area resulting from anin6nitesimal deformation of the path.

the chain. Note that fdx l(x} is the total conserved spinand sP is the Neel-order parameter. In the large-s limit,we expect 1 and P to both be of 0 (1). Assuming that 1and P vary slowly, the unit-vector constraint on S; be-comes

(A6)

We substitute this form, Eq. (A5), into the action, assum-ing that both fields 1 and P vary slowly over one latticespacing. Noting that A [S(t)] is odd under S(t)~S(t),we see that the leading term cancels between neighboringsites. There is a correction, of O(1/s}, which couples 1and P:

5A =(1/s) fdt 1 [QXBQ/Bt] . (A7)

=v fdx [(g/2)l +(I /g2)(dg /dx) ]

In general, it is also important to keep the other correc-tion which is a triple product of P, Bp/Bt, and Bp/Bx.This gives the topological term in the nonlinear cr modelin (1+1) dimensions. However, for integer s, this termhas no effect. The one-dimensional Heisenberg Hamil-tonian is also rewritten in terms of 1 and P:

+2Js S; Sj~s fdx[ 1+,'(dP/dx) +21 /s ]

+const, (A8)

with v =4Js and g =2/s. Including the time-derivativeterm, we obtain the Lagrangian in the form

L =f dx [I (Q XBp/Bt }(.vg /2 }1'(v/2g)(BQ/Bx )'] . (A9)

FIG. 20. Classical path traced out on the unit sphere by thetime evolution of the spin variable.

Note that, written in this form, the Lagrangian containsonly a first time derivative. If we replace P by ( P ) in thetime-derivative term, then it becomes identical to the oneused in Ref. 7 in the long-wavelength approximation. Toobtain the second-order time-derivative term of Eq. (2.3)

8952 IAN AFFLECK AND GREG F. WELLMAN 46

we simply eliminate I, using the Euler-Lagrange equation Indeed, a Lagrangian density of the form

ugl =yxay/at . (A10) X=(1/2v)(QXBQIBt) (v/2)(BQ/Bx) + (A13)This is equivalent to integrating out I in the path integralwhere it appears quadratically. This gives a new term inI, of the form

Lk;;,=( I/2vg)(QX BQIBt ) (Al 1)

Finally, using the large-s result~ P ~ = 1, we may replace

(yxay/at)2 (ay/at)'. (A12)

Thus, we see that the 5rst- and second-order time-derivative forms of the Lagrangian are actuallyequivalent. The difference between the results of thepresent approach and those of Ref. 8 lies in the passagefrom hard to soft spins. The approach that we have tak-en consists of beginning with the Lagrangian in second-order form, Eq. (2.3), and then relaxing the constraint ofthe field P and introducing a phenomenological potentialenergy with quadratic and quartic terms. No derivation isgiven in Ref. 8 of the effective Lagrangian used there.Furthermore, it was not claimed to describe the disor-dered phase or the vicinity of the transition. It does seemclear that an essential feature of that approach is that theconstraint is relaxed before the field l (i.e., near zerowave-vector component of the spin operator) is eliminat-ed. It may help to clarify the difference between the twoapproaches to attempt to derive a somewhat generalizedLagrangian, which shares some essential features withthat of Ref. 8, but is also, in principle, applicable in thedisordered phase. Thus, we begin with the Lagrangian infirst order form,-Eq. (A9), and then follow the same stepsas above, i.e., we relax the constraints, P = 1, l /=0 andintroduce a potential energy. We may still eliminate lafter removing the constraint. However, the kinetic termis now ( I/2vg)(p X Bp/dt ) rather than ( I/2vg)(BQ/Bt ) .These two forms are no longer equivalent with the con-straint removed. If we replace P by (P) in the kineticterm, then we obtain, up to a multiplicative constant, theterm that is effectively used in Ref. 8. This alternativeform of the Lagrangian contains, to quadratic order, notime derivatives of the longitudinal component of P.Consequently, the Euler-Lagrange equation 51./5gt =0becomes a constraint equation. For a standard Neelstate, it simply imposes the constraint PL =0, but in thetriangular lattice case where the Lagrangian containscross terms between longitudinal and transverse com-ponents of P, the constraint determines the longitudinalcomponent of P to be proportional to the transverse part.Thus, the number of excitations is not increased relativeto ordinary spin-wave theory; there is no extra branch.However, the mixing in of the longitudinal componentwith the transverse ones can substantially modify thedispersion relation and intensities.

Which of these approaches is corrects %e present ar-guments here in favor of the approach used in this paper.First of all, as discussed in Sec. III, the passage fromhard-spin to soft-spin modes can be accomplished usingthe large-n limit of the O(n) nonlinear cJ model. Usingthis approach, the kinetic energy has the (BQIBt )z form.

is not Lorentz invariant. The spin-wave velocity is nolonger v but is rather given by v /(P). Thus, as we de-crease the interchain coupling, the velocity increases andwould actually diverge at the critical point in the Neelcase. The disordered phase would not contain harmonicmagnon excitations. If we begin with a Lorentz-invarianthard-spin long-wavelength theory, such as the nonlinearu model, then we should expect that whatever renormal-ization processes produce an effective soft-spin modelshould preserve the Lorentz invariance, and hence notchange the spin-wave velocity. Including small breakingof Lorentz invariance, some renormalization of the spin-wave velocity would occur but there is no reason to ex-pect it to diverge at the critical point. A possible solutionto this problem might be to also modify the spatialderivative term, taking a Lagrangian density of the form

X =(I/2v)(y X ay/at ) (v /2)(p X ay/ax p+(A14)

This is now Lorentz invariant and the spin-wave velocityno longer depends on ( P ) . However, the disorderedphase would again not contain harmonic magnons. Fur-thermore, there is no reason why the (BP/Bx) term,present in the hard-spin Lagrangian, should be excludedfrom the soft-spin Lagrangian.

From a more general perspective, (Bf/i3x ) and(BQIBt) are a couple of perfectly good terms whichrespect all the symmetries of the problem and there is noreason to exclude them from the Lagrangian. The sameis true of (QXBQ/Bt) and (QXBQ/Bx) . In general, weshould include all four terms (together with quartic inter-chain couplings) in the elfective Lagrangian. The quarticterms were omitted in our treatment in the usual spirit ofLandau-Ginsburg theory. If we are suf6ciently close tothe critical point so that (P) is small, then they areunimportant. Further from the critical point they couldbe included and would modify the spin-wave spectrum.(Indeed by including them with several additional free pa-rameters, we could presumably get a better 6t to the ex-perimental data. ) However, they do not change the quali-tative picture presented here. In particular, as long asthe (BPIBt) term is present in the Lagrangian, the extrabranch will be present.

Apart from these theoretical arguments, there is an ex-perimental reason to prefer the Lagrangian used here. Asdiscussed above, the main qualitative feature which dis-tinguishes the present approach from the alternative ofRef. 8 and from conventional spin-wave theory is thepresence of an additional excitation branch in theparamagnetic zone. %e argued in Sec. II that the experi-mental evidence for this branch in CsNiC13 is very com-pelling. The presence of an xz polarized mode at (0,0, 1)with an energy 2,' times higher than that of the uppermode at (,', ,', 1) implies the existence of a second xz-polarized branch in the paramagnetic zone since with asingle branch these two modes would have to be degen-erate. In both conventional spin-wave theory and in the

46 LONGITUDINAL MODES IN QUASI-ONE-DIMENSIONAL. . . 8953

model of Ref. 8, this degeneracy cannot be lifted byhigher-order corrections due to the symmetry argumentspelled out in Sec. II. We note that in Ref. 8 the theory isnot compared to the polarized data of Ref. 6. The agree-ment then looks very good since the extra xz branch ismasked by the y-polarized branch near (0,0,1). As theabove discussion indicates, it is crucial to compare thetheory with the polarized neutron-scattering experiments

of Ref. 6. The present theory gives an extra xz-polarizedbranch, albeit with an energy which is about 10% higherthan experiment near (0,0, 1). The alternative of Ref. 8does not contain this experimental feature at all.

In conclusion, both theoretical and experimental argu-ments favor the Lagrangian used here with a (t)pltlt)term, or perhaps better still, a combination of both typesof terms.

'Present address: Physics Department, Princeton University,Princeton, NJ 08544.

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