Using both recently developed cluster-algorithm and histogram methods, we have carried out a high-resolution Monte Carlo study of static critical properties of classical ferromagnetic Heisenberg models. Extensive Monte Carlo simulations were performed at several temperatures in the critical region, using an improved cluster-updating scheme, on L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L simple-cubic and body-centered-cubic systems with L\ensuremath{\le}40. Thermodynamic quantities as a function of temperature in the vicinity of the critical point were obtained by an optimized multiple-histogram method, and the critical temperature and static critical exponents were extracted using finite-size scaling. Our best estimates for the inverse critical temperatures are 0.693 035(37) for the simple-cubic system and 0.486 798(12) for the body-centered-cubic system. Estimated static critical exponents for both systems agree with each other within their respective error bars, and the mean estimates \ensuremath{\nu}=0.7048(30) and \ensuremath{\gamma}=1.3873(85) are also in excellent agreement with field-theoretic predictions 0.705(3) and 1.386(4).
The static critical properties of classical, isotropic Heisenberg ferromagnets have been studied numerically. Using an improved cluster algorithm we performed extensive Monte Carlo simulations on L×L×L simple cubic and body-centered cubic systems with L≤40 over a temperature range covering the critical region. Temperature dependences of thermodynamic quantities were determined with an optimized multiple-histogram method, and both critical temperatures and critical exponents were extracted from finite-size scaling of several different thermodynamic quantities. It was found that within their respective error bars the critical exponents for both ferromagnets agree with each other and with field theoretical predictions.
Using extensive Monte Carlo simulations, we study the equilibrium properties of the simple-cubic, classical Heisenberg ferromagnet. We employ very long runs for L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L lattices to obtain high-precision data for the magnetization probability distribution. Using finite-size scaling for L\ensuremath{\le}24 and an optimized multiple-histogram data analysis, we obtain an accurate value of the inverse critical temperature J/
kBTc=0.6929\ifmmode\pm\else\textpm\fi{}0.0001, which is higher than previously accepted estimates. Calculated values of various static exponents are in excellent agreement with renormalization-group and \ensuremath{\epsilon}-expansion predictions.
We have simulated the three-dimensional Heisenberg model on simple cubic lattices, using the single-cluster Monte Carlo update algorithm. The expected pronounced reduction of critical slowing down at the phase transition is verified. This allows simulations on significantly larger lattices than in previous studies and consequently a better control over systematic errors. In one set of simulations we employ the usual finite-size scaling methods to compute the critical exponents \ensuremath{\nu},\ensuremath{\alpha},\ensuremath{\beta},\ensuremath{\gamma},\ensuremath{\eta} from a few measurements in the vicinity of the critical point, making extensive use of histogram reweighting and optimization techniques. In another set of simulations we report measurements of improved estimators for the spatial correlation length and the susceptibility in the high-temperature phase, obtained on lattices with up to
1003 spins. This enables us to compute independent estimates of \ensuremath{\nu} and \ensuremath{\gamma} from power-law fits of their critical divergencies.
Using recently developed histogram techniques and an ultrafast multispin coding simulation algorithm, we have investigated the critical behavior of the d=3 simple-cubic Ising model. We have studied lattice sizes ranging from L=8 to 96 using between 3\ifmmode\times\else\texttimes\fi{}
106 and 12\ifmmode\times\else\texttimes\fi{}
106 Monte Carlo steps (complete lattice updates). By accurately measuring the finite-size behavior of several different thermodynamic quantities, we are able to determine the critical properties with a precision comparable to that obtained with Monte Carlo renormalization-group and sophisticated series-expansion techniques. The best estimate of the inverse critical temperature from our analysis is
Kc=0.221 659 5\ifmmode\pm\else\textpm\fi{}0.000 002 6. The advantages of the histogram technique are discussed, as are the potential problems that can arise at this level of resolution.
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. 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 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.
The critical dynamics of `model A'' of Hohenberg and Halperin has been studied by the Monte Carlo method. Simulations have been carried out in the three-dimensional (3d) simple cubic Ising model for lattices of sizes
L=16 to
L=512. Using Wolff's cluster algorithm, the critical temperature is precisely found as ${\ensuremath{\beta}}_{c}=0.221\phantom{\rule{0.16em}{0ex}}654\phantom{\rule{0.16em}{0ex}}68(5)$. By Fourier transform of the lattice configurations, the critical scattering intensities $I(\stackrel{P\vec}{q}\phantom{\rule{0.16em}{0ex}})$ can be obtained. After circular averaging, the static simulations with
L=256 and
L=512 provide an estimate of the critical exponent $\ensuremath{\gamma}/\ensuremath{\nu}=2\ensuremath{-}\ensuremath{\eta}=1.9640(5)$. The $|\stackrel{P\vec}{q}\phantom{\rule{0.16em}{0ex}}|$-dependent distribution of $I(\stackrel{P\vec}{q}\phantom{\rule{0.16em}{0ex}})$ showed an exponential distribution, corresponding to a Gaussian distribution of the scattering amplitudes for a large
q domain. The time-dependent intensities were then used for the study of the critical dynamics of 3d lattices at the critical point. To simulate results of an x-ray photon correlation spectroscopy experiment, the time-dependent correlation function of the intensities was studied for each $|\stackrel{P\vec}{q}\phantom{\rule{0.16em}{0ex}}|$-value. In the
q region where $I(\stackrel{P\vec}{q}\phantom{\rule{0.16em}{0ex}})$ had an exponential distribution, the time correlations can be fit to a stretched exponential, where the exponent $\ensuremath{\mu}=\ensuremath{\gamma}/\ensuremath{\nu}z\ensuremath{\simeq}0.975$ provides an estimate of the dynamic exponent
z. This corresponds to
z=2.0145, in agreement with the observed variations of the characteristic fluctuation time of the intensity: $\ensuremath{\tau}(q)\ensuremath{\propto}{q}^{\ensuremath{-}z}$, which gives
z=2.015. These results agree with the $\ensuremath{\epsilon}$ expansion of field-theoretical methods (2.017). In this paper, the need to take account of the anomalous time behavior ($\ensuremath{\mu}<1$) in the dynamics is exemplified. This dynamics reflects a nonlinear time behavior of model A, and its large time extension is discussed in detail.