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SYMMETRY OF SPIN WAVES AND HALDANE GAP
IN CsNiCl3
Z. Tun, W. Buyers, R. Armstrong, E. Hallman, D. Arovas
To cite this version:
JOURNAL DE PHYSIQUE
Colloque C8, Suppl6ment au no 12, Tome 49, dkcembre 1988
SYMMETRY OF SPIN WAVES AND HALDANE GAP IN
CsNiCL3Z. Tun (I), W. J. L. Buyers (I), R. L. Armstrong (2), E. D. Hallman (3) and D. P. Arovas (4)
(I) Atomic Energy of Canada Limited, Chalk River Nuclear Laboratories, Chalk River, Ontario, KOJ IJO, Canada
(*) University of Toronto, Toronto, Ontario, M5S f A 1 , Canada
(3) Laurentian University, Sudbury, Ontario, P3E 2C6, Canada
(4) University of Chicago, Chicago, Illinois 60637, U.S.A.
Abstract.
-
Neutron scattering experiments and theoretical calculations on CsNiCt3 show that linear theory fails for the spin wave polarization in the 3-D phase. In the 1-D phase the Hafdane mode is observed to be a singlet-to-triplet excitation. We also comment on the wave vector dependence of the dynamic structure factor.Haldane's prediction [I] of a gap in the spectrum of integer spin quantum ~ e i i e n b e r ~ 1-D antiferromagnets remains a subject of theoretical interest. The exper- imental observation of a Haldane gap in a 1-D spin
(S) = 1 antiferromagnet was first reported by Buy- ers et al. [2] for CsNiC13 and has been confirmed by Steiner et al. [3].
Theoretically, the excitation spectrum of the S = 1 Heisenberg chain features a disordered nondegenerate singlet ground state and a gap to a triplet branch, known as the Haldane mode; the gap achieves its min- imum at the magnetic zone center (Q = n). Since the ground state has no broken symmetry, the spectrum should not reflect the n periodicity of antiferromagnets in which N&l order is present. The recent discovery of an isotropic Heisenberg-like S = 1 system with an exactly solvable ground state [4, 51 has strengthened these conclusions.
The symmetry of spin waves was determined by a polarized neutron scattering experiment. A mag- netic field in the vertical (9) direction was applied along [ilO] to obtain a single domain specimen be- low the NBel temperature (4.4 K) with canting entirely in the horizontal (xz) plane. Constant-Q scans per- formed near the magnetic zone centre (0.2, 0.2, 1) are shown in figure 1. At 1.5 K, two magnon branches with frequencies 0.29 THz and 0.33 THz can be seen in the non-spin-flip (NSF) and the spin-flip (SF) channels respectively corresponding to (SYS')
and (SxSx)
+
(SzS") modes. A similar scan with veryhigh counting statistics at (113, 113, 1.03) clearly re- vealed a third branch with a frequency of 0.27 THz in the SF channel. The frequencies of alI three branches at (113 113 1.03) are in good agreement with the linear spin wave calculations [6]. The symmetry of the third branch, however, disagrees with the calculations which predict it t o be a pure (SySY) mode. Steiner et al. [3] have already reported this disagreement. Although it is possible to obtain magnon dispersion curves with the observed symmetries, the very large single site anisotropy required (D = 0.039 THz) is inconsistent
FREQUENCY lTHzl
I
i '0.0
FREQUENCY lTHrl
Fig. 1.
-
Symmetry of spin waves below (a), and above (b)-
with the known anisotropy (D = 0.002 THz), de- stroys the agreement with the measured frequencies, and most importantly does not reproduce the known
spin structure. The linear spin wave theory in its present form cannot predict the correct symmetry of the third rnagnon branch.
When a scan was carried out well above
TN
at (0.2, 0.2, 1) where three-dimensional effects are negligible (2 cos 2?r<+cos 4nC = 0 at<
= 0.19) the scattered in- tensity was found to be the same in the two channels (Fig. lb), in agreement with the observation of Steineret al. [3]. The results of figure lb, where the momen- tum transfer is close to the z direction, thus prove that
(SxSx) = (SY Sy)
.
Steiner's observation of equal inten-sities in the two channels in different Brillouin zones [3] further shows that (SZS") = (SxSx) = (SYSY)
,
i.e. the excitation is unpolarized.From this observation we reach the important con- clusion that the Haldane gap excitations correspond t o a transition between a singlet ground state and a triplet excited state. The singlet follows because we have shown earlier that there is no quasielastic scatter- ing. The triplet excited state is then deduced from the fact that the Haldane gap mode is unpolarized. Note that it is not possible t o obtain a triplet excited state with any conventional sitebased spin wave model.
C8 - 1432 JOURNAL DE PHYSIQUE
As shown by Affleck et al. 141 the extended Heisen- berg model
possesses a simple, exact valence bond solid (VBS) ground state when the coefficient X = 1/3. Al- though the Heisenberg interaction of the ~ i chains ~ + in CsNiCL3 does not involve the second term it is nat- ural t o base a theory on this exactly solvable model. The excitation spectrum of the VBS model has been investigated numerically by Haldane [7]. There is an isolated triplet excitation which at Q = a achieves its minimum energy, but remains sharp throughout the region a
<
Q5
3a/2; a t Q=
3a/2, it merges into a twemagnon continuum and becomes damped. As Q approaches 2a, the nuclear zone centre, the bottom of the two-magnon continuum f d s t o a value 2A, twice the Haldane gap energy. The exact first and second moments of the dynamic structure factor have been evaluated in reference [5]:10
dw w S (Q, w) =
-
2 7 (5 f 3 cos aq) (2)where Q = a q is the interspin phase and S (Q) = C
/
S(4,
w) dw. It should be noted that WQ andI?$
J
are not periodic in Q -+ Q
+
a, reflecting the unbro- ken symmetry of the ground state. Both WQ and are thus expected to keep increasing beyond the mag- netic zone boundary and tend to a maximum at the nuclear zone center (q = 2). It should be emphasized that in the region where the Haldane mode is sharp, the second momentr$
is nonzero due to contributions from higher lying excited states separated by a second gap from the elementary Haldane magnon, and is notreflecting any intrinsic width t o that mode. For Q
>
3a/2, the second moment may be interpreted as a true width of a triplet resonance.
To test these predictions we have extended our neu- tron scattering measurements into the nuclear zone. Constant-Q scans were taken at (1/2, 1/2, q) posi- tions, t o avoid phonons, and the observed frequencies are shown in figure 2. The excitations were verified as magnetic by comparison with scans at higher Q. The broken line of figure 2 shows the first moment of the VBS spectrum, equation (2) fitted to 1.46 THz at q = 1.5. While the widths do increase monotonically with q, it is clear that the observed peak energies do
not continue to rise in the nuclear zone, but 'instead fall. This is hard to reconcile with the theoretical pic- ture presented above.
The solid curve shown in figure 2 is the dispersion expected for a broken symmetry Heisenberg antifer- romagnet (HAFM) with a fictitious anisotropy intro-
- HAFM WITH A FICTITIOUS GAP
B OBSERVED WIDTH
--- FIRST-MOMENT OF VBS SPECTFUM 0 - d
1 0 1.2 1.4 1.6 1.8 2.0
REDUCED WAVE-VECTOR COMPONENT 7 ( ~ a , ~ )
Fig. 2. - Measurement of the spin w.ave frequency and width beyond the magnetic wne bouiidary in the one- dimensional phase.
duced t o simulate the 3-D renormalized Haldane gap of 0.22 THz at (1/2, 1/2, 1) and a maxiinum of 1.46 THz at (1/2, 1/2, 1.5). Although this model is not applica- ble, because of the small real anisotropy and the known triplet character of the Haldane mocle, it is nonethe- less interesting to note that the predicted frequencies are in substantial agreement with experiment. To rec- oncile this result with the VBS model would require a surprisingly large spectral weight in the continuum at frequencies above WQ. Our results cannot determine whether the frequency tends to A or 2A as Q --+ 2a.
We are left with two unresolved issues. One is the measured peak widths (more than insbrumental resolu- tion) in the region where theoreticall:? the elementary magnon should be sharp. The other ki the unexpected
downturn in frequencies. It is possible that additional (e.g. 3-D) interactions present in CsNiCL3 are respon- sible for such deviations.
[I] Haldane, F. D. M., Phys. Rev. Lett. 50 (1983) 1153.
[2] Buyers, W. J. L., Morra, R. M., Armstrong, R.
L., Hogan, M. J., Gerlach, P. and Hirakawa, K.,
Phys. Rev. Lett. 56 (1986) 371.
[3] Steiner, M., Kakurai, K., Kjemls, J. K., Petit- grand, D. and Pynn, R., J. Appl. Phys. 6 1 (1987) 3953.
[4] B e c k , I., Kennedy, T., Lieb, E . H. and Tasaki, H., Phys. Rev. Lett. 59 (1987) 799.
[5] Arovas, D. P., Auerbach, A. and Haldane, F. D.
M., Phys. Rev. Lett. 60 (1988) 531.
[6] Morra, R. M., Buyers, W. J. L., Armostrong, R.
L. and Hirakawa, K., Phys. Rev. B 38 (1988) 543.
