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Physical review. B, Condensed matter

Critical dynamics of the body-centered-cubic classical Heisenberg antiferromagnet

Alex Bunker, Kun Chen, David Paul Landau

Published Oct 1, 1996• 31 citations
Publisher

Abstract

We have used spin dynamics techniques to perform large-scale simulations of the dynamic behavior of the L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L body-centered-cubic classical Heisenberg antiferromagnet with L\ensuremath{\le}48 at a range of temperatures above and below as well as at the critical point ${\mathit{T}}_{\mathit{c}}$. The temporal evolutions of the spin configurations were determined numerically from coupled equations of motion for individual spins by a fourth-order predictor-corrector method, with initial spin configurations generated by Monte Carlo simulations. The neutron scattering function S(q,\ensuremath{\omega}) was calculated from the space- and time-displaced spin-spin correlation function. We used a previously developed dynamic finite-size scaling theory to extract the dynamic critical exponent z from S(q,\ensuremath{\omega}) at ${\mathit{T}}_{\mathit{c}}$. Our results are in agreement with the theoretical prediction of z=1.5 and with experimental results; however, we find that the asymptotic regime was only entered at L\ensuremath{\approxeq}30. In the analysis of the form of the transverse and longitudinal components of S(q,\ensuremath{\omega}) we found that a central diffusion peak appears below ${\mathit{T}}_{\mathit{c}}$ predominantly in the longitudinal component and remains present through and above ${\mathit{T}}_{\mathit{c}}$. The transverse component of the spin-wave peak is Lorentzian below ${\mathit{T}}_{\mathit{c}}$ but for T\ensuremath{\geqslant}${\mathit{T}}_{\mathit{c}}$ is described best by a more complex functional form. Below ${\mathit{T}}_{\mathit{c}}$ we see evidence of multiple spin-wave peaks in the longitudinal component. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.54.9259

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We have used spin dynamics techniques to perform large-scale simulations of the dynamic behavior of the L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L body-centered-cubic classical Heisenberg antiferromagnet with L\ensuremath{\le}48 at a range of temperatures above and below as well as at the critical point Tc{\mathit{T}}_{\mathit{c}}Tc​. The temporal evolutions of the spin configurations were determined numerically from coupled equations of motion for individual spins by a fourth-order predictor-corrector method, with initial spin configurations generated by Monte Carlo simulations. The neutron scattering function S(q,\ensuremath{\omega}) was calculated from the space- and time-displaced spin-spin correlation function. We used a previously developed dynamic finite-size scaling theory to extract the dynamic critical exponent z from S(q,\ensuremath{\omega}) at Tc{\mathit{T}}_{\mathit{c}}Tc​. Our results are in agreement with the theoretical prediction of z=1.5 and with experimental results; however, we find that the asymptotic regime was only entered at L\ensuremath{\approxeq}30. In the analysis of the form of the transverse and longitudinal components of S(q,\ensuremath{\omega}) we found that a central diffusion peak appears below Tc{\mathit{T}}_{\mathit{c}}Tc​ predominantly in the longitudinal component and remains present through and above Tc{\mathit{T}}_{\mathit{c}}Tc​. The transverse component of the spin-wave peak is Lorentzian below Tc{\mathit{T}}_{\mathit{c}}Tc​ but for T\ensuremath{\geqslant}Tc{\mathit{T}}_{\mathit{c}}Tc​ is described best by a more complex functional form. Below Tc{\mathit{T}}_{\mathit{c}}Tc​ we see evidence of multiple spin-wave peaks in the longitudinal component. \textcopyright{} 1996 The American Physical Society.
Using spin-dynamics techniques we have performed large-scale computer simulations of the dynamic behavior of the L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L body-centered-cubic classical Heisenberg ferromagnet with L\ensuremath{\le}40 in the vicinity of the critical point Tc{\mathit{T}}_{\mathit{c}}Tc​. The temporal evolutions of spin configurations were determined numerically from coupled equations of motion for individual spins by a fourth-order predictor-corrector method with initial spin configurations generated by Monte Carlo methods. The space- and time-displaced spin-spin correlation functions and their space-time Fourier transforms were calculated to determine the neutron-scattering functions. We developed a dynamic finite-size scaling theory for the neutron-scattering function at Tc{\mathit{T}}_{\mathit{c}}Tc​ and used it to extract the dynamic critical exponent z. Within our resolution limit, the value of z was estimated to be 2.478(28), in excellent agreement with the dynamic scaling prediction, and dynamic scaling was found to be valid for momentum transfer q up to 0.4\ensuremath{\pi} in the (1,0,0) direction.
Using spin-dynamics techniques we have performed large-scale computer simulations of the dynamic behavior of the classical three component XY model (i.e., the anisotropic limit of an easy-plane Heisenberg ferromagnet), on square lattices of size up to 1922{192}^{2}1922, for several temperatures below, at, and above TKT{\mathit{T}}_{\mathrm{KT}}TKT​. The temporal evolution of spin configurations was determined numerically from coupled equations of motion for individual spins by a fourth-order predictor-corrector method, with initial spin configurations generated by a hybrid Monte Carlo algorithm. The neutron-scattering function S(q,\ensuremath{\omega}) was calculated from the resultant space-time displaced spin-spin correlation function. Pronounced spin-wave peaks were found both in the in-plane and the out-of-plane scattering function over a wide range of temperatures. The in-plane scattering function Sxx{\mathit{S}}^{\mathit{xx}}Sxx also has a large number of clear but weak additional peaks, which we interpret to come from two-spin-wave scattering. In addition, we observed a small central peak in Sxx{\mathit{S}}^{\mathit{xx}}Sxx, even at temperatures well below the phase transition. We used dynamic finite-size scaling theory to extract the dynamic critical exponent z. We find z=1.00(4) for all T\ensuremath{\le}TKT{\mathit{T}}_{\mathrm{KT}}TKT​, in excellent agreement with theoretical predictions, although the shape of S(q,\ensuremath{\omega}) is not well described by current theory. \textcopyright{} 1996 The American Physical Society.
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.
Spin dynamics simulations have been used to study dynamic critical behavior of classical Heisenberg magnets. The temporal evolutions of the spin configurations were determined numerically from coupled equations of motion by a fourth-order predictor corrector method, with initial spin configurations generated by Monte-Carlo simulations. The neutron scattering function, S (q, ω), was calculated from the space and time displaced spin-spin correlation function and the dynamic critical exponent z was extracted using dynamic finite size scaling theory. For both ferromagnetic and antiferromagnetic models we find good agreement with theoretical predictions and experimental results.
Monte Carlo and spin dynamics simulations have been used to study the dynamic critical behavior of RbMnF3,{\mathrm{RbMnF}}_{3},RbMnF3​, treated as a classical Heisenberg antiferromagnet on a simple cubic lattice. In an attempt to understand the difference in the value of the dynamic critical exponent z between experiment and theory, we have used larger lattice sizes than in our previous simulations to better probe the asymptotic critical region in momentum. We estimate $z=1.49\ifmmode\pm\else\textpm\fi{}0.03,$ in good agreement with the renormalization-group theory and dynamic scaling predictions. In addition, the central peak in the dynamic structure factor at Tc,{T}_{c},Tc​, seen in experiments and previous simulations, but absent in the renormalization-group and mode-coupling theories, is shown to be solely in the longitudinal component.
Using Monte Carlo and spin-dynamics methods, we have investigated the dynamic behavior of the classical, antiferromagnetic XY model on a triangular lattice with linear sizes L< 300.L<~300.L< 300. The temporal evolutions of spin configurations were obtained by solving numerically the coupled equations of motion for each spin using fourth-order Suzuki-Trotter decompositions of exponential operators. From space- and time-displaced spin-spin correlation functions and their space-time Fourier transforms we obtained the dynamic structure factor S(q,w)S(\mathbf{q},w)S(q,w) for momentum q\mathbf{q}q and frequency $\ensuremath{\omega}.$ Below TKT{T}_{\mathrm{KT}}TKT​ (Kosterlitz-Thouless transition), both the in-plane (Sxx){(S}^{\mathrm{xx}})(Sxx) and out-of-plane (Szz){(S}^{\mathrm{zz}})(Szz) components of $S(\mathbf{q},\ensuremath{\omega})$ exhibit very strong and sharp spin-wave peaks. Well above TKT,{T}_{\mathrm{KT}},TKT​, Sxx{S}^{\mathrm{xx}}Sxx and Szz{S}^{\mathrm{zz}}Szz apparently display a central peak, and spin-wave signatures are still seen in Szz.{S}^{\mathrm{zz}}.Szz. In addition, we also observed an almost dispersionless domain-wall peak at high $\ensuremath{\omega}$ below Tc{T}_{c}Tc​ (Ising transition), where long-range order appears in the staggered chirality. Above Tc,{T}_{c},Tc​, the domain-wall peak disappears for all q. The line shape of these peaks is captured reasonably well by a Lorentzian form. Using a dynamic finite-size scaling theory, we determined the dynamic critical exponent z=1.002(3).z=1.002(3).z=1.002(3). We found that our results demonstrate the consistency of the dynamic finite-size scaling theory for the characteristic frequency ${\ensuremath{\omega}}_{m}$ and the dynamic structure factor $S(\mathbf{q},\ensuremath{\omega})$ itself.