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arXiv:cond-mat/0112188 · Physical review. B, Condensed matter

Evolution of spin excitations in a gapped antiferromagnet from the quantum to the high-temperature limit

M. Kenzelmann, Roger Arthur Cowley, W. J. L. Buyers, R. Coldea, M. Enderle, MCMORROW Desmond Francis

Published Nov 8, 2002• 18 citations
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Abstract

We have mapped from the quantum to the classical limit the spin excitation spectrum of the antiferromagnetic spin-1 Heisenberg chain system ${\mathrm{CsNiCl}}_{3}$ in its paramagnetic phase from $T=5$ to $200\mathrm{K}.$ Neutron scattering shows that the excitations are resonant and dispersive up to at least $T=70\mathrm{K}\ensuremath{\simeq}2.7J,$ but broaden considerably with increasing temperature. The dispersion flattens out with increasing temperature as the resonance energy $\ensuremath{\Delta}$ at the antiferromagnetic wave vector increases and the maximum in the dispersion decreases. The correlation length $\ensuremath{\xi}$ between $T=12$ and $50\mathrm{K}$ is in agreement with quantum Monte Carlo calculations for the spin-1 chain. $\ensuremath{\xi}$ is also consistent with the single mode approximation, suggesting that the excitations are short-lived single particle excitations. Below $T=12\mathrm{K}$ where three-dimensional spin correlations are important, $\ensuremath{\xi}$ is shorter than predicted and the experiment is not consistent with the random phase approximation for coupled quantum chains. At $T=200\mathrm{K},$ the structure factor and second energy moment of the excitation spectrum are in excellent agreement with the high-temperature series expansion.

DOI
10.1103/physrevb.66.174412

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We have mapped from the quantum to the classical limit the spin excitation spectrum of the antiferromagnetic spin-1 Heisenberg chain system CsNiCl3{\mathrm{CsNiCl}}_{3}CsNiCl3​ in its paramagnetic phase from T=5T=5T=5 to 200K.200\mathrm{K}.200K. Neutron scattering shows that the excitations are resonant and dispersive up to at least T=70K\ensuremath≃2.7J,T=70\mathrm{K}\ensuremath{\simeq}2.7J,T=70K\ensuremath≃2.7J, but broaden considerably with increasing temperature. The dispersion flattens out with increasing temperature as the resonance energy \ensuremathΔ\ensuremath{\Delta}\ensuremathΔ at the antiferromagnetic wave vector increases and the maximum in the dispersion decreases. The correlation length \ensuremathξ\ensuremath{\xi}\ensuremathξ between T=12T=12T=12 and 50K50\mathrm{K}50K is in agreement with quantum Monte Carlo calculations for the spin-1 chain. \ensuremathξ\ensuremath{\xi}\ensuremathξ is also consistent with the single mode approximation, suggesting that the excitations are short-lived single particle excitations. Below T=12KT=12\mathrm{K}T=12K where three-dimensional spin correlations are important, \ensuremathξ\ensuremath{\xi}\ensuremathξ is shorter than predicted and the experiment is not consistent with the random phase approximation for coupled quantum chains. At T=200K,T=200\mathrm{K},T=200K, the structure factor and second energy moment of the excitation spectrum are in excellent agreement with the high-temperature series expansion.
Exact diagonalization calculations of finite antiferromagnetic spin-1 Heisenberg chains at finite temperatures are presented and compared to a recent inelastic neutron scattering experiment for temperatures T up to 7.57.57.5 times the intrachain exchange constant J. The calculations show that the excitations at the antiferromagnetic point q=1q=1q=1 and at q=0.5q=0.5q=0.5 remain resonant up to at least T=2J,T=2J,T=2J, confirming the recent experimental observation of resonant high-temperature domain wall excitations. The predicted first and second moments are in good agreement with experiment, except at temperatures where three-dimensional spin correlations are most important. The ratio of the structure factors at q=1q=1q=1 and 0.5 is well predicted for the paramagnetic infinite-temperature limit. For T< 2J,T<~2J,T< 2J, however, we found that the experimentally observed intensity is considerably less than predicted. This suggests that domain wall excitations on different chains interact up to temperatures of the order of the spin band width.
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.
Well-defined spin-wave excitations were observed by using pulsed neutrons from CsVCl3{\mathrm{l}}_{3}l3​, an S=3/2S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3/2S=3/2 Heisenberg antiferromagnetic chain. The observed spin-wave dispersion relation shows an enhancement of the excitation energy by a factor of 1.26, which must be compared with the quantum renormalization factor of $\ensuremath{\pi}/2$ for an S=1/2S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2S=1/2 system. The linewidth of the spectrum at 40 K was broad and independent of qqq. The qqq-independent linewidth is the first experimental observation, which was predicted by rigorous theories for a classical chain at sufficiently low temperature.
Neutron-scattering measurements on the one-dimensional gapped S=1S=1S=1 antiferromagnet, CsNiCl3,{\mathrm{CsNiCl}}_{3},CsNiCl3​, have shown that excitation corresponding to the Haldane mass gap $\ensuremath{\Delta}$ at low temperatures persists as a resonant feature to high temperatures. We find that the strong upward renormalization of the gap excitation, by a factor of three between 5 and 70 K, is more than enough to overcome its decreasing lifetime. We find that the gap lifetime is substantially shorter than that predicted by the scaling theory of Damle and Sachdev based on a classical dispersion for the Haldane excitations, but when we include their expected relativistic dispersion, the theory, in its low-temperature range of validity, gives a good account of experiment. The upward gap renormalization agrees with the nonlinear sigma model at low temperatures and even up to T of order $2\ensuremath{\Delta}$ provided an upper momentum cutoff is included.
Many phenomena in magnetism can be described using classical models, yet magnetism is a quantum mechanical effect. In addition to thermal fluctuations, quantum fluctuations arise due to the quantum uncertainty of the ground state. These fluctuations are enhanced by low dimensionality and low spin. Spin chains show interesting excitations associated with quasiparticles. Model materials for spin chains can be realized in compounds where the interchain interactions are weak enough. This thesis presents neutron scattering studies of two model materials for spin chains with different spin symmetry. The spin2 Heisenberg antiferromagnetic chain CuSO4*5D2O has a main exchange of J ∼ 3 K, which permits to study its excitation spectrum up to temperatures of ten times the exchange. The spin2 Heisenberg magnet has been studied widely both theoretically and experimentally. The excitation spectrum of the spin2 Heisenberg antiferromagnet consists of quasiparticles called spinons, and calculations for it exist for T = 0 and as a function of temperature. A quantitative comparison between experimental data and theoretical models across a range of temperatures is presented. The excitation spectrum of CuSO4*5D2O is excellently described by the excitation spectrum of a spin2 Heisenberg antiferromagnetic chain at finite temperatures. Energy-temperature scaling is observed at temperatures up to 2J . For these temperatures the Luttinger liquid description of the spin2 Heisenberg antiferromagnetic chain is valid. A spinon continuum is confirmed at all measured temperatures. Therefore, even at ten times the main exchange (T ≈ 10 J), CuSO4*5D2O shows the behavior of a quantum magnet, and not of a classical system at infinite temperatures, for which one would expect single spin flips. The Ising chain is one of the most studied models in magnetism, one model compound for it is RbCoCl3. Optical spectroscopy measurements suggest a lower gap than for similar compounds, which makes it better suited for high resolution neutron spectroscopy. Magnetic correlations along the chains arise in RbCoCl3 below 80 K. A gapped continuum is observed, which arises from isolated antiferromagnetic Ising chains. Below the first magnetic ordering temperature TN1 ≈ 28 K bound states are observed, which arise due to correlations which form between the chains. Strong diffuse scattering leads to the conclusion that disorder is still present. At the second magnetic ordering transition, at TN2 ≈ 14 K, the bound states split as the correlations between the chains change. Fits indicate that the correlations change continuously as a function of temperature. Above TN2 the excitation spectrum of RbCoCl3 can be described by a spin1 2 Ising chain with an XY-component which is treated as a perturbation. Below TN2 the main features are reproduced correctly, but deviations from the details indicate that additional terms in the Hamiltonian may be necessary. This thesis was carried out at the Institut Laue-Langevin, the Laboratory for Neutron Scattering and Imaging at the Paul Scherrer Institute and ETH Zurich.
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