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arXiv:cond-mat/0010049 · Physical Review Letters

Continuum in the Spin-Excitation Spectrum of a Haldane Chain Observed by Neutron Scattering in CsNiCl3

Igor A. Zaliznyak, S.-H. Lee, S. V. Petrov

Published Jun 15, 2001• 73 citations
arXivHTMLPDF

Abstract

The spin-excitation continuum, expected to dominate the low-energy fluctuation spectrum in the Haldane spin chain around the Brillouin zone center, $q\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$, is directly observed by inelastic magnetic neutron scattering in the $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$ quasi-1D antiferromagnet ${\mathrm{CsNiCl}}_{3}$. We find that the single mode approximation fails, and that a finite energy width appears in the dynamic correlation function $S(q,\ensuremath{\omega})$ for $q\ensuremath{\lesssim}0.5\ensuremath{\pi}$. The width increases with decreasing $q$, while $S(q,\ensuremath{\omega})$ acquires an asymmetric shape qualitatively similar to that predicted for the two-magnon continuum in the nonlinear $\ensuremath{\sigma}$-model.

DOI
10.1103/physrevlett.87.017202

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The spin-excitation continuum, expected to dominate the low-energy fluctuation spectrum in the Haldane spin chain around the Brillouin zone center, q=0q\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0q=0, is directly observed by inelastic magnetic neutron scattering in the S=1S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1S=1 quasi-1D antiferromagnet CsNiCl3{\mathrm{CsNiCl}}_{3}CsNiCl3​. We find that the single mode approximation fails, and that a finite energy width appears in the dynamic correlation function S(q,\ensuremathω)S(q,\ensuremath{\omega})S(q,\ensuremathω) for q\ensuremath≲0.5\ensuremathπq\ensuremath{\lesssim}0.5\ensuremath{\pi}q\ensuremath≲0.5\ensuremathπ. The width increases with decreasing qqq, while S(q,\ensuremathω)S(q,\ensuremath{\omega})S(q,\ensuremathω) acquires an asymmetric shape qualitatively similar to that predicted for the two-magnon continuum in the nonlinear \ensuremathσ\ensuremath{\sigma}\ensuremathσ-model.
We present detailed measurement of the nonhydrodynamic part of the spin excitation spectrum in the model quasi-one-dimensional S=1 antiferromagnet CsNiCl3 by inelastic magnetic neutron scattering. In the better part of the Brillouin zone the effect of interchain coupling is negligible, and the spin dynamics reflects that of a single Haldane chain. We find that at q≲0.6π this quantum spin system ceases to support coherent propagating excitation, which gradually turns into a continuum band of states, whose width grows with decreasing q. This finding is consistently verified under different resolution conditions, obtained with two different high-luminosity experimental setups.
Spin fluctuations in the Haldane ground state should exhibit a quantum spin gap observable with Raman scattering. The Raman spectra of single-crystalline Y2{\mathrm{Y}}_{2}Y2​BaNiO5{\mathrm{BaNiO}}_{5}BaNiO5​, a linear spin-1 Heisenberg antiferromagnet, exhibit broad continuum scattering between \ensuremath{\sim}200 and \ensuremath{\sim}800 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$, in the energy range expected for magnetic excitations (\ensuremath{\sim}J=224 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$). Numerous weak and broad features superimposed on this continuum scattering possibly result from quantization of the magnon continuum by local molecular fields from nearby chains. A strong and relatively narrow mode at \ensuremath{\sim}680 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$, near 3J, unrelated to lattice vibrations, may also be of magnetic origin. All of these features are strongly resonant in the red (${\ensuremath{\lambda}}_{\mathit{L}}$=6328 \AA{}), disappearing for shorter excitation wavelengths. These spectra provide a view of the Haldane spin system excitations complementary to that given by inelastic neutron scattering. Modeling the excitations with Schwinger bosons or spin waves predicts a highly asymmetric scattering feature with a sharp onset near 2\ensuremath{\Delta}(\ensuremath{\sim}0.8J), gradually diminishing toward higher energies. The observed spectra disagree with this theoretical expectation, perhaps due to spin-phonon coupling which may provide an efficient decay channel for the magnon pairs.
We report inelastic time-of-flight and triple-axis neutron scattering measurements of the excitation spectrum of the coupled antiferromagnetic spin-1 Heisenberg chain system CsNiCl3.{\mathrm{CsNiCl}}_{3}.CsNiCl3​. Measurements over a wide range of wave-vector transfers along the chain confirm that above TNCsNiCl3{T}_{N} {\mathrm{CsNiCl}}_{3}TN​CsNiCl3​ is in a quantum-disordered phase with an energy gap in the excitation spectrum. The spin correlations fall off exponentially with increasing distance with a correlation length $\ensuremath{\xi}=4.0(2)$ sites at T=6.2K.T=6.2\mathrm{K}.T=6.2K. This is shorter than the correlation length for an antiferromagnetic spin-1 Heisenberg chain at this temperature, suggesting that the correlations perpendicular to the chain direction and associated with the interchain coupling lower the single-chain correlation length. A multiparticle continuum is observed in the quantum-disordered phase in the region in reciprocal space where antiferromagnetic fluctuations are strongest, extending in energy up to twice the maximum of the dispersion of the well-defined triplet excitations. We show that the continuum satisfies the Hohenberg-Brinkman sum rule. The dependence of the multiparticle continuum on the chain wave vector resembles that of the two-spinon continuum in antiferromagnetic spin-1/2 Heisenberg chains. This suggests the presence of spin-1/2 degrees of freedom in CsNiCl3{\mathrm{CsNiCl}}_{3}CsNiCl3​ for T< 12K,T<~12 \mathrm{K},T< 12K, possibly caused by multiply frustrated interchain interactions.
A continuum of magnetic states has been observed by neutron scattering from the spin-1 chain compound CsNiCl3 in its disordered gapped one-dimensional phase. Results using both triple-axis and time-of-flight spectrometers show that around the antiferromagnetic point Qc = pi, the continuum lies higher in energy than the Haldane gapped excitations. At 6 K the integrated intensity of the continuum is about 12(2)% of the total spectral weight. This result is considerably larger than the 1%-3% weight predicted by the nonlinear sigma model for the three-particle continuum.
Inelastic neutron scattering experiments on the S=1 quasi-one-dimensional bond-alternating antiferromagnet Ni(C9D24N4)(NO2)ClO4 have been performed under magnetic fields below and above a critical field Hc at which the energy gap closes. Normal field dependece of Zeeman splitting of the excited triplet modes below Hc has been observed, but the highest mode is unusually small and smears out with increasing field. This can be explained by an interaction with a low-lying two magnon continuum at q=pi that is present in dimerized chains but absent in uniform ones. Above Hc, we find only one excited mode, in stark contrast with three massive excitations previously observed in the structurally similar Haldane-gap material NDMAP [A. Zheludev et al., Phys. Rev. B 68, 134438 (2003)].
Inelastic neutron scattering experiments on the S = 1 quasi-one-dimensional bond-alternating antiferromagnet Ni(C9D24N4)(NO2)ClO4 have been performed under magnetic fields below and above a critical field Hc at which the energy gap closes. Normal field dependence of Zeeman splitting of the excited triplet modes below Hc has been observed, but the highest mode is unusually small and smears out with increasing field. This can be explained by an interaction with a low-lying two magnon continuum at q(parallel) = pi that is present in dimerized chains but absent in uniform ones. Above Hc, we find only one excited mode, in stark contrast with three massive excitations previously observed in the structurally similar Haldane-gap material NDMAP [A. Zheludev, Phys. Rev. B 68, 134438 (2003)].