Dynamic critical exponents and sample independence times for the classical Heisenberg model.

Abstract

The critical dynamics of the scalar order parameter autocorrelation function, the total vector spin autocorrelation function, and the energy autocorrelation function for the classical O(3) three-dimensional Heisenberg ferromagnet are studied both algebraically and with heat bath Monte Carlo simulations. It is shown from a group decomposition of the canonical Markov transitions used in a Monte Carlo study of this model ferromagnet that the vector spin autocorrelation relaxation time currently used to estimate both the dynamical critical exponent and the statistical sampling efficiency of the scalar order parameter (and other quantities of interest) contains an uncontrolled dependence upon irrelevant symmetric diffusion processes and is therefore unsuitable for this purpose. The central limit theorem is quantified as the fundamental basis of estimates of error and, separately, estimates of statistical sampling efficiency in importance sampling Monte Carlo simulations. A simple but interesting estimate of ${\mathit{z}}_{\mathit{R}}$=d-2\ensuremath{\beta}/\ensuremath{\nu} for the dynamical critical exponent of the ergodicity-restoring thermal-rotational motion of the total vector spin is presented. Various relaxation times are numerically measured and their dynamic critical exponents extracted via finite-size scaling theory to illustrate these points and compare with theoretical predictions. \textcopyright{} 1996 The American Physical Society.