# Longitudinal modes in quasi-one-dimensional antiferromagnets

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<ul><li><p>PHYSICAL REVIEW B VOLUME 46, NUMBER 14 1 OCTOBER 1992-II</p><p>Longitudinal modes in quasi-one-dimensional antiferromagnets</p><p>Ian AfHeckCanadian Institute for Advanced Research and Physics Department, University ofBritish Columbia, Vancouver,</p><p>British Columbia, Canada V6T1Zl</p><p>Greg F. Wellman'Physics Department, University ofBritish Columbia, Vancouver, British Columbia, Canada V6TIZI</p><p>(Received 6 May 1992)</p><p>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-</p><p>pared with experimental data on CsNiC13. This model is compared with an alternative Lagrangian-based one that was proposed recently.</p><p>I. INTRODUCTION</p><p>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</p><p>A</p><p>IE</p><p>B F</p><p>FIG. 1. Orientation of spin vectors on the six inequivalentsublattices (see Fig. 2) for the stacked triangular lattice antifer-romagnet.</p><p>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.</p><p>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</p><p>46 8934 1992 The American Physical Society</p></li><li><p>46 LONGITUDINAL MODES IN QUASI-ONE-DIMENSIONAL. . . 8935</p><p>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.</p><p>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.</p><p>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.</p><p>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.</p><p>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,</p><p>II. LANDAU-GINSBURG MODEL</p><p>We begin by discussing a single chain, Heisenberg anti-ferromagnet:</p><p>H, =2J Q S; S;+) . (2.1)</p><p>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</p><p>[l'(z), lJ(z') ]=i e'J"I"(z)5(zz'),[l'(z), P(z')]=le'J"P"(z)5(z z') .</p><p>(2.2}</p><p>(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~</p><p>~l~ /s =~/~ =1, P 1=0. This defines the non-</p><p>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</p><p>2 '2</p><p>2v Bt 2 Bz~</p><p>2v 4</p><p>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.</p></li><li><p>8936 IAN AFFLECK AND GREG F. WELLMAN</p><p>The quartic term is, in general, necessary for stability.The uniform magnetization density is then determinedfrom the commutation relations to be</p><p>I =(I/u)y X ay/at . (2.4)</p><p>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.</p><p>We now consider a quasi-one-dimensional system:</p><p>on the two sublattices, the potential energy per spin be-comes</p><p>V(gu) =(b, /2u 8J's )((}0+(A/4)go .We see that the critical value of J' is</p><p>(2.10)</p><p>J, '=b, /16vs . (2.11)</p><p>Neel state with all nearest-neighbor spins antiparallel.Writing</p><p>(2.9)</p><p>chains planes</p><p>83=J $ S; Si+J' $ S;.SJ .&,)'' (2.5) For larger J' the size of the sublattice magnetization isgiven by</p><p>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</p><p>T</p><p>L, = f dz gX, [P;(z)]2J's g P;(z) P, (z)l (;) '</p><p>(2.6)</p><p>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</p><p>E,&(g, a, Q~)=(/u Q, +b, +8J'vsf(Qj),where</p><p>(2.7}</p><p>f(Q, )=,' g e (2.8)</p><p>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.</p><p>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</p><p>Pu=(16J's b, /u)/A. .We expand L to second order in small fluctuations:</p><p>4'=(0. 0, do+0. </p><p>(2.12)</p><p>(2.13)</p><p>x and y fluctuations are transverse and z fluctuations arelongitudina1. These do not mix to quadratic order. Wemay then read off the dispersion relations</p><p>E,(g, m., Q~)=+u Q, +8J'us[2+f(Qj)],</p><p>E (Q, n', Q ) =Qv Q, +8J'vs[2+ f(Q )]+6</p><p>(2.14)</p><p>(2.15)</p><p>where</p><p>For the square lattice of chains, of spacing a,</p><p>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.</p><p>The intensities of these modes take a very simple form.The canonical commutation relation,</p><p>[P( xt), P(x', t)]=iu5(x x'},implies that the spin correlation function is</p><p>K'(Q, n. ,Qi, E ) = (S'S')(Q, m.,Qi, E)- (y'y')(Q, E)~ 6[EE, (Q)]/E'(Q) . (2.18)</p><p>[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. ]</p><p>bL =+2ukgu=+2(16J'us b, )=}/32us(J' J,') . </p><p>(2.16)</p></li><li><p>LONGITUDINAL MODES IN QUASI-ONE-DIMENSIONAL. . . 8937</p><p>&-b.= ~0o4.(0'. +Ay ) . (2.20}We see that the decay rate, which goes like the square ofthis coupling constant, is 0(...</p></li></ul>

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