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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Critical dynamics of a dynamical version of the classical Heisenberg model

Dennis C. Rapaport, David Paul Landau

Published May 1, 1996• 11 citations
Publisher

Abstract

We have performed large-scale molecular dynamics simulations of a dynamical variant of the classical Heisenberg model in which the spins are replaced by interacting "linear molecules," each of which has two rotational degrees of freedom. The Hamiltonian consists of rotational kinetic-energy terms as well as nearest-neighbor Heisenberg-like interactions $\ensuremath{-}J{\mathrm{s}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{s}}_{j}$, where ${\mathrm{s}}_{i}$ is a unit vector specifying the orientation of the molecule at lattice site $i$. Systems of size up to ${64}^{3}$ spins on the simple cubic lattice have been studied, and thermostatting is used to maintain strictly constant temperature. We determined the dynamic structure function from the time- and space-displaced correlation functions, and found good agreement with the predictions of the dynamic scaling theory with a dynamic critical exponent of $z=1.5$. Results are compared and contrasted with data from spin-dynamics calculations on classical Heisenberg ferromagnets and antiferromagnets.

DOI
10.1103/physreve.53.4696

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International Journal of Modern Physics C•Jun 1, 1996•1 citation
We have performed large-scale molecular dynamics simulations of a dynamical variant of the classical Heisenberg model in which the spins are replaced by interacting "linear molecules," each of which has two rotational degrees of freedom. The Hamiltonian consists of rotational kinetic-energy terms as well as nearest-neighbor Heisenberg-like interactions \ensuremath−Jsi\ifmmode⋅\else\textperiodcentered\fisj\ensuremath{-}J{\mathrm{s}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{s}}_{j}\ensuremath−Jsi​\ifmmode⋅\else\textperiodcentered\fisj​, where si{\mathrm{s}}_{i}si​ is a unit vector specifying the orientation of the molecule at lattice site iii. Systems of size up to 643{64}^{3}643 spins on the simple cubic lattice have been studied, and thermostatting is used to maintain strictly constant temperature. We determined the dynamic structure function from the time- and space-displaced correlation functions, and found good agreement with the predictions of the dynamic scaling theory with a dynamic critical exponent of z=1.5z=1.5z=1.5. Results are compared and contrasted with data from spin-dynamics calculations on classical Heisenberg ferromagnets and antiferromagnets.
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.
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.
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.
Spin-dynamics simulations have been used to investigate the dynamic behavior of RbMnF_3, treating it as a classical Heisenberg antiferromagnet on a simple cubic lattice. Time-evolutions of spin configurations were determined numerically from coupled equations of motion for individual spins using a new algorithm which is based on 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 value. The dynamic critical exponent was estimated to be z=(1.43\pm 0.03), which is slightly lower than the dynamic scaling prediction, but in good agreement with a recent experimental value. Comparisons are made of both the dispersion curve and the lineshapes obtained from our simulations with very recent experimental results for RbMnF_3 are presented.
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.
The three-dimensional spin-1/2 Heisenberg model on a simple cubic lattice is studied for ferromagnetic and antiferromagnetic interactions using the Decoupled Cell Method for quantum Monte Carlo. Results for the relaxation time τL are determined for both ferromagnetic and antiferromagnetic systems and found to be similar to those found for the classical (s → ∞) Heisenberg model. The scaling of τL with system size is used to extract the dynamical critical exponent z for the two systems. The values of z = 1.98 ± 0.12 for the ferromagnet and z = 1.94 ± 0.09 for the antiferromagnet are in good agreement with theoretical predictions and previous Monte Carlo studies of the classical Heisenberg model.