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Physical Review Letters

Low-Temperature Spin-Wave Dynamics in Three-Component Classical Heisenberg Chains

P. Heller, Martin Blume

Published Oct 10, 1977• 38 citations
Publisher

Abstract

A detailed quantitative study of the low-temperature spin-wave dynamics has been made for the classical Heisenberg-coupled chain using computer simulation. Results for the relaxation rates are presented and compared with dynamic scaling and other predictions.
DOI
10.1103/physrevlett.39.962

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Combining high-temperature expansions with the recursion method and quantum Monte Carlo simulations with the maximum entropy method, we study the dynamics of the spin-1/2 Heisenberg chain at temperatures above and below the coupling J. By comparing the two sets of calculations, their relative strengths are assessed. At high temperatures, we find that there is a low-frequency peak in the momentum-integrated dynamic structure factor, due to diffusive long-wavelength modes. This peak is rapidly suppressed as the temperature is lowered below J. Calculation of the complete dynamic structure factor S(k,\ensuremath{\omega}) shows how the spectral features associated with the two-spinon continuum develop at low temperatures. We extract the nuclear spin-lattice relaxation rate 1/T1{\mathrm{T}}_{1}T1​ from the \ensuremath{\omega}\ensuremath{\rightarrow}0 limit, and compare with recent experimental results for Sr2{\mathrm{Sr}}_{2}Sr2​CuO3{\mathrm{CuO}}_{3}CuO3​ and CuGeO3{\mathrm{CuGeO}}_{3}CuGeO3​. We also discuss the scaling behavior of the dynamic susceptibility, and of the static structure factor S(k) and the static susceptibility \ensuremath{\chi}(k). We confirm the asymptotic low-temperature forms S(\ensuremath{\pi})\ensuremath{\sim}[ln (T)]3/2{]}^{3\mathrm{/}2}]3/2 and \ensuremath{\chi}(\ensuremath{\pi})\ensuremath{\sim}${\mathrm{T}}^{\mathrm{\ensuremath{-}}1}$[ln (T)]1/2{]}^{1\mathrm{/}2}]1/2, expected from previous theoretical studies.
Computer studies of the approach to equilibrium in the classical 3-component Heisenberg chain, starting from a configuration with a large staggered magnetization (i. e., far from equilibrium) are reported. The equal time transverse and reduced longitudinal correlations g⊥(ℓ, t) and g∥(ℓ, t), respectively, were monitored for times long enough to show the approach to equilibrium. Results for g⊥ confirm predictions by Nelson and Fisher of wave-like behavior with a propagation speed equal to twice the zero-temperature spin wave velocity. The equilibration of g∥ was found to be slower and diffusive in nature. The fitted diffusivity appears to be comparable to the value of the spin diffusivity for the corresponding equilibrium problem.
We present rigorous perturbation results for the dynamics of the classical nearest-neighbor Heisenberg chain, using the temperature as a small parameter. The lifetime and the frequency shift of the spin waves are determined. For the antiferromagnet we find that there is a region of wave vectors close to the zone boundary where the dynamical scaling fails and we argue that this is a general property of the Heisenberg chain close to T=0T=0T=0.
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.
Spin-dynamics techniques have been used to perform large-scale simulations of the dynamic behavior of the classical Heisenberg antiferromagnet in simple cubic lattices with linear sizes L< 60.L<~60.L< 60. This system is widely recognized as an appropriate model for the magnetic properties of RbMnF3.{\mathrm{RbMnF}}_{3}.RbMnF3​. Time evolutions of spin configurations were determined numerically from coupled equations of motion for individual spins using an algorithm implemented by Krech et al., which is based on fourth-order Suzuki-Trotter decompositions of exponential operators. The dynamic structure factor was calculated from the space- and time-displaced spin-spin correlation function. The crossover from hydrodynamic to critical behavior of the dispersion curve and spin-wave half-width was studied as the temperature was increased towards the critical temperature. The dynamic critical exponent was estimated to be $z=(1.43\ifmmode\pm\else\textpm\fi{}0.03),$ which is slightly lower than the dynamic scaling prediction, but in good agreement with a recent experimental value. Direct, quantitative comparisons of both the dispersion curve and the line shapes obtained from our simulations with very recent experimental results for RbMnF3{\mathrm{RbMnF}}_{3}RbMnF3​ are presented.