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arXiv:cond-mat/0112219 · Physical review. B, Condensed matter

Temperature dependence of single particle excitations in a S=1 chain: Exact diagonalization calculations compared to neutron scattering experiments

M. Kenzelmann, Paolo Maria Santini

Published Nov 27, 2002• 6 citations
arXivHTMLPDF

Abstract

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 $7.5$ times the intrachain exchange constant J. The calculations show that the excitations at the antiferromagnetic point $q=1$ and at $q=0.5$ remain resonant up to at least $T=2J,$ 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 $q=1$ and 0.5 is well predicted for the paramagnetic infinite-temperature limit. For $T<~2J,$ 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.

DOI
10.1103/physrevb.66.184429

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Physical Review Letters•Mar 20, 1995•30 citations
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 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 $T=70\mathrm{K}\ensuremath{\simeq}2.7J,$ 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=12T=12T=12 and 50K50\mathrm{K}50K 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=12KT=12\mathrm{K}T=12K 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=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.
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.
The research presented here focuses on the theoretical investigation of spinons in quasi one-dimensional quantum spin chains from the perspective of neutron scattering, with emphasis placed on the calculation of dynamic spin structure factors for both unpolarised and polarised incident neutrons. Two anisotropic versions of the Heisenberg spin chain are considered for such an examination: the spin-1/21/21/2 antiferromagnetic XXZXXZXXZ model and the spin-1/21/21/2 ferromagnetic XYZXYZXYZ model.The XYZXYZXYZ model supports spinon scattering (modelled at finite temperature) and is related to the so-called Villain mode. The introduction of a perturbation in the form of an external transverse magnetic field or Dzyaloshinskii-Moriya interaction (DMI) leads to the emergence of incommensurability in the system; such a result is a signature of fractional excitations. The presence of these interactions also give rise to a chiral response associated with the spinons in the polarised term of the scattering cross section. The inelastic scattering cross section is computed for both unpolarised and polarised neutrons.In relation to the antiferromagnetic XXZXXZXXZ model, emphasis is on spinon pair creation (modelled at T=0rmKT = 0 {rm K}T=0rmK). Motivated by the presence of a chiral response in the case of the XYZXYZXYZ model mentioned above, a new technique is developed in order to incorporate DMI and the transverse magnetic field so that one can determine if there is a chiral property associated with the two-spinon system. This technique is numerical in nature and is based on Green function methods. Using the results produced by the Green function approach, the inelastic cross section in the presence of an external transverse magnetic field is computed for the first time. The result is compared with experimental neutron scattering data; good agreement is demonstrated between theory and experiments on rmCsCoBr3{rm CsCoBr}_3rmCsCoBr3​.Finally, the Green function approach is generalised to account for the interactions between spin chains in magnetic compounds described by a staggered field. Various dynamic structure factors are computed for this scenario with results compared to experiment; good agreement between the theory and experiments on rmRbCoCl3{rm RbCoCl}_3rmRbCoCl3​ is shown.
Quasi-one-dimensional magnetic materials provide important test beds for rigorous models of interacting many-body systems. Nevertheless, there have been few studies of finite temperature effects on spin correlations in low-spin systems, though these are most likely to exhibit quantum effects. Copper benzoate, Cu(C6D5COO)2⋅3D2O, is a good realization of the S=1/2 Heisenberg antiferromagnetic chain (HAFC). We have performed ac susceptibility and inelastic neutron scattering experiments on this material which establish the Hamiltonian and an excitation spectrum which is consistent with the spinon continuum ansatz of Müller et al. We present inelastic neutron scattering measurements of the temperature evolution of the dynamic spin–spin correlation function Szz(π,ω). Comparison will be made to the classical prediction and to a finite temperature field theory of Schulz as applied to this S=1/2 HAFC.
We study dynamical properties at finite temperature (T)(T)(T) of Heisenberg spin chains with random antiferromagnetic exchange couplings, which realize the random singlet phase in the low-energy limit, using three complementary numerical methods: exact diagonalization, matrix-product-state algorithms, and stochastic analytic continuation of quantum Monte Carlo results in imaginary time. Specifically, we investigate the dynamic spin structure factor $S(q,\ensuremath{\omega})$ and its $\ensuremath{\omega}\ensuremath{\rightarrow}0$ limit, which are closely related to inelastic neutron scattering and nuclear magnetic resonance (NMR) experiments (through the spin-lattice relaxation rate 1/T1)1/{T}_{1})1/T1​). Our study reveals a continuous narrow band of low-energy excitations in $S(q,\ensuremath{\omega})$, extending throughout the qqq space, instead of being restricted to $q\ensuremath{\approx}0$ and $q\ensuremath{\approx}\ensuremath{\pi}$ as found in the uniform system. Close to $q=\ensuremath{\pi}$, the scaling properties of these excitations are well captured by the random-singlet theory, but disagreements also exist with some aspects of the predicted qqq dependence further away from $q=\ensuremath{\pi}$. Furthermore we also find spin diffusion effects close to q=0q=0q=0 that are not contained within the random-singlet theory but give non-negligible contributions to the mean 1/T11/{T}_{1}1/T1​. To compare with NMR experiments, we consider the distribution of the local relaxation rates 1/T11/{T}_{1}1/T1​. We show that the local 1/T11/{T}_{1}1/T1​ values are broadly distributed, approximately according to a stretched exponential. The mean 1/T11/{T}_{1}1/T1​ first decreases with TTT, but below a crossover temperature it starts to increase and likely diverges in the limit of a small nuclear resonance frequency ${\ensuremath{\omega}}_{0}$. Although a similar divergent behavior has been predicted and experimentally observed for the static uniform susceptibility, this divergent behavior of the mean 1/T11/{T}_{1}1/T1​ has never been experimentally observed. Indeed, we show that the divergence of the mean 1/T11/{T}_{1}1/T1​ is due to rare events in the disordered chains and is concealed in experiments, where the typical 1/T11/{T}_{1}1/T1​ value is accessed.
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.