# Magnetic excitations in quasi one-dimensional antiferromagnets

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<ul><li><p>Physica B 180 & 181 (1992) 153-157 North-Holland PHMCA !d </p><p>Magnetic excitations in quasi one-dimensional antiferromagnets </p><p>K. Kakurai Institute for Solid State Physics, University of Tokyo, Roppongi 7-22-1, Minato-ku, 106 Tokyo, Japan </p><p>Neutron scattering experiments have been making considerable contributions in understanding the quantum many-body problem of the one-dimensional antiferromagnet. Advanced techniques like the neutron polarization analysis and high energy chopper spectroscopy combined with the conventional triple axis spectroscopy are able to shed more light on this old, but still topical, problem. To illustrate this point the neutron scattering experiments on quasi one-dimensional antiferromagnets of ABX, type are described </p><p>1. Introduction </p><p>Static and dynamic properties of quasi one-dimen- sional (1-D) spin systems have been attracting the interest of theoretical and experimental physicists for a long time [l]. Though started as a theoretically tract- able study case for the real 3-D system, it soon became clear that the dimensionality of a system is very important for its critical behaviour. All the 1-D systems do not show long range order (LRO) at finite temperatures. These systems remove their entropy by gradually developing a strong short-range correlation when the temperature (T) is reduced. It is therefore possible to study critical properties (with strong fluc- tuations) over a wide temperature range. The exten- sive studies in the 70s and 80s had the aim to clarify the thermodynamical role of large amplitude fluctua- tions (topological excitations, nonlinear excitations) besides the small amplitude linear excitations (spin waves). In these studies neutron scattering played an essential role to observe thermally excited solitons directly [2, 31. Most of the thermodynamics of these topological excitations could be interpreted classically. In some other cases like, e.g., spin-energy coupling [4] the neutron scattering results could only be explained by taking into account the quantum nature of the spin system [5]. </p><p>In the Heisenberg antiferromagnetic (AF) chain one knew the importance of the quantum fluctuations in the extreme case of S = $ For this case Bethe [6] could obtain the ground state eigenfunctions and dem- onstrate that the classical N&e1 state is not the true ground state. Des Cloizeaux and Pearson (dC-P) [7] subsequently calculated the spectrum of the lowest excited eigenstates. Their energies turn out to be 7r/2 times the energy values of the classical spin wave (SW) dispersion by Anderson [8]. Again inelastic neutron scattering experiments performed on CuCI, 2N(C,D,) (CPC) [9] clearly demonstrated sharp SW excitations with the dispersion predicted by dC-P. A more recent theoretical study revealed that the dC-P </p><p>dispersion curve is the lower bound of a triplet spin wave double continuum [lo]. The asymmetry of the CPC SW peak [ 1 l] and the more recent spin excitation spectrum in KCuF, [12] indeed indicate the existence of such a continuum. </p><p>The discussion on the quantum aspect in the 1-D AF has recently become very vivid, when Haldane predicted spin-value-dependent dynamical properties (Haldane conjecture) [13]. According to his prediction there is a clear separation in the low temperature properties between half-integer and integer spin Heisenberg AF chain systems: Gapless excitations and power law decay of the spin correlation for the for- mer, and an energy gap from the non-magnetic ground state to the excited triplet state and an exponential decay of the spin correlation for the latter. This conjecture, when put forward by Haldane in 1982, aroused a lot of controversies among the theoretical and experimental physicists. Finite chain calculations with finite size scaling technique [14], Monte Carlo calculations [15], field theoretical treatment [16] and the study of an exact solvable Hamiltonian with the ground state properties as predicted for the S = 1 AF Heisenberg system [17] on the theoretical side, and studies of NENP, CsNiCl, using magnetization, sus- ceptibility measurements [18] and neutron scattering [19-211 on the experimental side have been per- formed. After roughly a decade a consensus has emerged that the Haldane conjecture is accepted as the Haldane effect. </p><p>In this paper I describe the neutron scattering ex- periments on quasi 1-D AF of ABX, type as an example how neutron scattering can contribute to shed more light on the quantum aspect of magnetic correla- tions and excitations in the quasi 1-D antifer- romagnets. </p><p>2. Systems </p><p>Hexagonal ABX, type compounds are well known to represent quasi 1-D magnetic systems. It consists of </p><p>0921-4526/92/$0.5.00 0 1992 - Elsevier Science Publishers B.V. All rights reserved </p></li><li><p>154 K. Kakurui I Mugneric excmtions in yum one-dimmsionul untiferromugne/s </p><p>Parameters for the investigated systems. </p><p>T\ IKI Ref. .I WI J IKI 3 [mcV] CaNiCl, T,, = 4.84 [I </p><p>. I I\ = 4.40 CsNiBr, T,, = 14.25 [23] </p><p>I ,= Il.75 RbNiCl~ T;; = 1 I .365 -c 0.004 1211 </p><p>7,, = I I. IO) + 0.002 CSVCI i T, = 13.3 WI </p><p>linear chains of face sharing octahedra running along the c direction. In the chains the magnetic B transi- tion metal ions are interacting strongly via the superexchange over three X halogen ions. These chains are separated by the large nonmagnetic A ions. The ratio of the interchain to the intrachain interaction ranges from 10~~ to 10 -. The Hamiltonian can be written as </p><p>H = -J c S,S, - J c S,S, - D c (St) , </p><p>where J and J are the interchain and intrachain exchange, respectively, and D is the single site aniso- tropy. i and j sum over the nearest neighbour spins in the chain and k and 1 over the nearest neighbour chains. The systems we have studied are CsNiCI,, CsNiBr, and RbNiCl,, all S = 1, and CsVCl, a S = 3 system. In all substances intra- and interchain ex- change are antiferromagnetic. in the first three com- pounds D is positive (easy axis anisotropy) and in CsVCl, D is negative (easy plane anisotropy). In all cases the spin system orders three-dimensionally be- cause of the small but finite J. In the substances with the easy axis anisotropy there is an ordering of the spin component in the easy axis direction at T,, , while the complete order of all the components is reached at T,, [22-241. T,, and TN2 are listed in the first column of table 1. The ordered spin structure is in all cases AF in the chain direction. and nearly 120 structure per- pendicular to the chain direction. The plane of rota- tion is perpendicular to the c-plane in the easy axis case [25]. and in the c-plane in the easy plane case </p><p>Phi. </p><p>2.1. S=l </p><p>In the course of the detailed studies of CsNiCI, it was discovered that the SW dispersion even in the 3-D ordered state (T < T,,) could not be explained consis- tently with classical SW calculation when the polariza- tion of spin fluctuations experimentally observed by polarized neutron scattering [27] is taken into account. This discrepancy of the experimental observation and the result of the classical SW theory was then regarded as the consequence of the Haldane quantum gap of </p><p>~ 16.6 ~0.36 0.90 -t 0.07 </p><p>-21 .x ~O.XX I .x 0.55 ? </p><p>-23.x -0.76 I .I?5 0.30 -c </p><p>-lhSi-4 -0.046 </p><p>the S = 1 AF chains. Indeed, Affleck could explain the experimental results of the ordered S = I AF chains using a phenomenological model of the Landau- </p><p>0.7 </p><p>0.6 </p><p>0.5 </p><p>0.4 </p><p>0.3 </p><p>0.2 </p><p>0.1 </p><p>0.0 0.0 0.1 0.2 0.3 h 0.4 0.5 0.6 0.7 </p><p>6 RbNiCl3 Q=(h.h,l) </p><p>0.0 0.1 0.2 h </p><p>0.3 04 0.5 </p><p>6 </p><p>CsNiBr3 Q=(h,h,l) </p><p>0.0 0.1 0.2 h </p><p>0.3 04 0.5 </p><p>The dispersion relation perpendicular to the chain direction for (a) CsNiCI,. (h) RhNiCI, and (c) CsNiBr, at T</p></li><li><p>K. Kakurai I Magnetic excitations in quasi one-dimensional antiferromagnets 155 </p><p>Ginsburg Hamiltonian with parameters chosen to re- produce the expected dispersion relation for the single S = 1 AF chain with the gap energy A (in the limit of D = 0) [28]. In fig. 1 the dispersion curves perpen- dicular to the chain direction for studied S = 1 cases are shown. One can see that the dispersion relation in all three cases look very alike and one can therefore assume the same polarization of the SW branches as discussed in detail in CsNiCl, (271. Indicated in fig. 1 are the fits with the phenomenological approach by Affleck. It is clear that the overall agreement here is much better than in the comparison with the classical SW theory (see ref. [27]). The fit parameters deduced from these fits are listed in table 1 (D = 0 case!!). J is deduced from the chain direction zone boundary ener- gy. The gap energies agree quite well with the most recent numerical estimation of the Haldane gap [15] (A = 0.41jJ]) and hence support the phenomenological approach by Affleck. But one should note that in all three cases the upper two xz modes have not been observed. And from the high held measurement in CsNiCI, [29] one knows that the single mode observed at (0 0 1) is three-fold degenerate corresponding to the one y and two xz modes. This is the triplet with the mass gap expected theoretically in the limit of true one-dimensionality [30]. </p><p>Somehow the theoretical approach seems to over- estimate the effect of J on the short wavelength fluctuations in the ordered state. J determined by the classical interpretation of the ESR data [31] is smaller by an order of magnitude. It is interesting to note that the classical interpretation of the SW dispersion around the ordering wave vector ( f , 4, 1) with the correct mode assignment measured by neutron scatter- ing gives roughly the same small J. This may be taken as an indication for the overestimation of the correla- tions or the neglect of quantum fluctuations in the classical interpretation because it is based on the mean field of the classically ordered state. </p><p>2.2. s = 5 </p><p>In the earlier studies of SW in TMMC, a S = $ AF chain system, it was shown that there is a good agreement between experimental results and the clas- sical theory [32]. This was attributed to the fact that the spin value 4 is sufficiently large to be regarded as a .S = r limit. Hence the ground state can be well ap- proximated by the AF Neel state and quantum fluctua- tions can be neglected. In the previous section we have demonstrated that for S = 1 the spin dispersion are indeed influenced by the quantum fluctuations. The question connected to the Haldane conjecture is what is the relevant quantum effect in the half-integer spin value system next to 1 towards the classical limit. We therefore studied the SW dispersion in CsVCl,, a S = $ Heisenberg system with small planar anisotropy. </p><p>In the earlier studies it was already indicated that the classical spin wave theory does not fully explain the experimental observations. There is an inconsistency in J determined by the bulk susceptibility measure- ment [33] and by the inelastic neutron SW velocity measurement [34]. Because of the rather large value of J, the whole dispersion relation along the chain was not accessible using the conventional triple axis tech- nique. The dispersion relation perpendicular to the chain direction has been studied in detail by Kadowaki et al. [35] and it was found that the intensity of the low lying mode near (f , t, l), the 3-D ordering point, could not be interpreted in the frame of the classical SW theory. </p><p>To complete the picture of the magnetic excitations in CsVCl, we performed an inelastic neutron scatter- ing experiment using a chopper spectrometer to follow the dispersion along the chain direction. I am not going to describe in detail the advantageous use of the chopper spectrometer for high energy excitations in 1-D systems because it will be discussed elsewhere [36], but just mention the effective use of the small angle detectors when the chain direction is oriented parallel to the incident beam. In fig. 2 the background corrected spectrum for the incident energy E, = 200 meV obtained by MAR1 at ISIS is displayed. The SW excitation up to 85 meV energy transfer can be clearly seen. The inset of fig. 3 displays all the SW peak positions along the chain direction observed with different incident energies. The fit with the classical dispersion 4S]J] ]sin 41 is shown by the solid line and yields an intrachain exchange J = - 165 K. This J is to be compared with J = - 111 K determined by the bulk susceptibility measurement [32]. A consistent interpre- tation applying quantum theory for both the bulk </p><p>-5 _ </p><p>I 0 50 100 150 </p><p>Energy [meV] </p><p>Fig. 2. Background corrected spectrum of CsVCl, for the incident energy of 200 meV. The solid line is the resolution convoluted spectrum expected from the classical SW picture. Inset: The dispersion along the chain direction. The solid line is a fit with the classical SW dispersion. </p></li><li><p>156 K. Kakurai I Magnetic excitations in quasi one-dimensional antiferromagnets </p><p>8, I 1 I Acknowledgements </p><p>.C cl.1 0.2 0.3 0.4 </p><p>h </p><p>Fig. 3. The dispersion perpendicular to the chain direction in CsVCl,. Open points are from ref. [34] and full points are from our measurement. The solid line is a model dispersion for the case of .I = - 150 K, J = -0.042 K and D = 0.52 K from ref. [34]. </p><p>susceptibility and SW dispersion measurement as in the case of S = $ is highly desirable. Furthermore, the observed line widths cannot be explained by the 6 function-like SW line shape. (See the solid line in fig. 2 representing the resolution convoluted classical SW dispersion.) Figure 3 depicts the dispersion perpen- dicular to the chain direction. Note that the size of the small energy gap observed at (0 0 1) in the ordered state can be understood classically by taking into account the interchain exchange J and the planar anisotropy D as shown by the solid line. </p><p>One should however note that the upper energy mode at (0 0 1) predicted by the classical SW theory is an out-of-plane mode and therefore should not be visible at (0 0 1). This may point to the importance of the longitudinal fluctuations which may occur upon making a renormalization group transformation as in the case of CsNiCl, [28]. These results demonstrate the deficiency of the classical standard SW theory applied for S = 2 quasi 1-D AF. A more detailed investigation of the spin excitations using polarized neutrons is desirable. </p><p>3. Summary </p><p>Advanced techniques in neutron scattering like the polarization analysis and high energy transfer chopper spectroscopy combined with the conventional triple axis spectroscopy can considerably contribute to clarify the role of quantum fluctuations in the quasi 1-D AF systems. In the S = 1 case the phenomeno- logical approach including the quantum effect is suc- cessful to describe the experimental findings. In the S = ; case the experimental results indicate the de- ficiency of the classical standard SW theory pointing to the importance of the quantum corrections. </p><p>I would like to thank M. Arai. M. Enderle, Y. Endoh, K. Hirakawa, K. ho, S. Itoh, H. Kadowaki, J.K. Kjems, K. Nakajima, R. Pynn, M. Steiner, and K. Tanaka without whose help this contribution cou...</p></li></ul>