We study the efficiency and theory behind various Markov chain Monte Carlo update methods (later MCMC) in the classical Heisenberg model in three dimensions. Classical Heisenberg model is a model in statistical physics that describes ferromagnetic phenomena. Classical Heisenberg model is a generalization of the Ising model where the spin is three dimensional unit vector instead of scalar -1 or 1. Both models show a second order phase transition which is the main reason we are interested in these models. The transition in our case describes the loss of magnetization of a ferromagnet as it is heated to its Curie temperature. Monte Carlo simulating the Classical Heisenberg model uses the same MCMC update methods as the Ising model. We introduce the theoretical background of Metropolis, Overrelaxation, and Wolff single cluster updates and study the dynamic critical exponents of Metropolis and Wolff updates. Results of this study are as expected: metropolis suffers from critical slowing down near the phase transition temperature and the autocorrelation time scales to L^5.08 , where L^3 is the size of the lattice. Wolff single cluster update avoids critical slowing down and scales very well with autocorrelation time scaling to L^3.28 . Even though Wolff update scales much better it has its downsides. Parallelization is a very important factor in modern scientific computing and Wolff update is very tricky to parallelize where as Metropolis parallelizes very well. Usually we can avoid this problem by running multiple instances of the simulation at the same time as varying simulation parameters such as the temperature. For large lattices the problems is that getting moderate results requires a lot of time.
We study the efficiency and theory behind various Markov chain Monte Carlo update methods (later MCMC) in the classical Heisenberg model in three dimensions. Classical Heisenberg model is a model in statistical physics that describes ferromagnetic phenomena. Classical Heisenberg model is a generalization of the Ising model where the spin is three dimensional unit vector instead of scalar -1 or 1. Both models show a second order phase transition which is the main reason we are interested in these models. The transition in our case describes the loss of magnetization of a ferromagnet as it is heated to its Curie temperature. Monte Carlo simulating the Classical Heisenberg model uses the same MCMC update methods as the Ising model. We introduce the theoretical background of Metropolis, Overrelaxation, and Wolff single cluster updates and study the dynamic critical exponents of Metropolis and Wolff updates. Results of this study are as expected: metropolis suffers from critical slowing down near the phase transition temperature and the autocorrelation time scales to L^5.08 , where L^3 is the size of the lattice. Wolff single cluster update avoids critical slowing down and scales very well with autocorrelation time scaling to L^3.28 . Even though Wolff update scales much better it has its downsides. Parallelization is a very important factor in modern scientific computing and Wolff update is very tricky to parallelize where as Metropolis parallelizes very well. Usually we can avoid this problem by running multiple instances of the simulation at the same time as varying simulation parameters such as the temperature. For large lattices the problems is that getting moderate results requires a lot of time.
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.
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 zR=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.
Monte Carlo simulations near the phase transition. We compute high precision estimates of the critical coupling Kc, Binder’s parameter U ∗ , and the critical exponents ν, β/ν, η, and α/ν, using extensively histogram reweighting and optimization techniques that allow us to keep control over the statistical errors. Measurements of the autocorrelation time show the expected reduction of critical slowing down at the phase transition as compared to local update algorithms. This allows simulations on significantly larger lattices than in previous studies and consequently a better control over systematic errors in finite-size scaling analyses. 1.
I simulated the two-dimensional Ising model at its exact critical temperature with two Markov chain Monte Carlo methods, the single-spin Metropolis algorithm and the Wolff cluster algorithm, on periodic L×L lattices from L = 8 to L = 64. For each method I measured the integrated autocorrelation time of the energy and of the absolute magnetization, using four independent runs per lattice size. Both methods reproduce the exact finite-lattice energy within statistical error. For Metropolis, the autocorrelation time of |M| grows as L^z with z = 2.11(3) for L ≥ 16 and 2.16(5) for L ≥ 24, in line with the accepted value z ≈ 2.17 for local dynamics. For Wolff, measured in sweep-equivalent units, I get z = 0.29(1) from the energy and 0.15(1) from |M| for L ≥ 24, within the range of published values. In CPU time per statistically independent sample of |M|, Wolff was 5 times faster at L = 8 and 275 times faster at L = 64, and the gap grows roughly as L^2. All code and raw results are available, and the full study runs in under an hour on two CPU cores.
We studied the energy, the magnetization and cumulants of the three-dimesional isotropic Heisenberg model in the phase transition region by using Metropolis algorithm. In order to study the critical temperature we implemented the vectorized Metropolis, hybrid Metropolis and Wolff (the latter one as prescribed by Chen et al (4)). We choseMetropolis:Overrelaxation 1:4 for its fastest independent measurements yield. We implemented Bereau and Swendsen's optimized multi- histogram technique (13) for this study. We obtained through finite size scaling the critical exponents = -0, 0709 +- 0,0099, = 0,3499 +- 0,0076, = 1,3880 +- 0,0060 and = 0,6903 +- 0,0034, the inverse criticaltemperature Kc = 0; 69314 +- 0,00032, and the universal parameter Binder cumulant for an infinite lattice at critical temperature U4B = 0, 62178 +- 0,00049 by means of a strategy involving jackknife to reduce bias. These exponents agree with experimental, theoretical and simulational determinations from literature. We also investigated energy and spin time autocorrelations in unidimensional lattices (an integrable L = 4 system and a non-integrable L = 18 system), by integration with fourth order Runge-Kutta, Predictor-Corrector, Suzuki-Trotter and optimized Forest-Ruth methods. We found that the use of quad precision, an integration step of approximately 10..6, N = 106 sampling, and longspin chains may be sucient to discriminate between an expected and an anomalous spin diffusion. Besides, each algorithm implementation may correspond to a dierent time operator for the non-integrable system.
We simulate the Ising Model with the Monte Carlo method and use the algorithms of Metropolis to update the distribution of spins. We found that, in the specific case of the three-dimensional Ising Model, methods of Metropolis are efficient. Studying the system near the point of phase transition, we observe that the magnetization goes to zero. In our simulations we analyzed the behavior of the magnetization and magnetic susceptibility to verify the phase transition in a paramagnetic to ferromagnetic material. The behavior of the magnetization and of the magnetic susceptibility as a function of the temperature suggest a phase transition around KT/J ≈ 4.5 and was evidenced the problem of finite size of the lattice to work with large lattice.