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arXiv:1508.05661 · Physical Review E

Event-chain algorithm for the Heisenberg model: Evidence for z≃1 dynamic scaling

Yoshihiko Nishikawa, Manon Michel, Werner Krauth, Koji Hukushima

Published Dec 14, 2015• 42 citations
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Abstract

We apply the event-chain Monte Carlo algorithm to the three-dimensional ferromagnetic Heisenberg model. The algorithm is rejection-free and also realizes an irreversible Markov chain that satisfies global balance. The autocorrelation functions of the magnetic susceptibility and the energy indicate a dynamical critical exponent z≈1 at the critical temperature, while that of the magnetization does not measure the performance of the algorithm. We show that the event-chain Monte Carlo algorithm substantially reduces the dynamical critical exponent from the conventional value of z≃2.

DOI
10.1103/physreve.92.063306

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We apply the event-chain Monte Carlo algorithm to the three-dimensional ferromagnetic Heisenberg model. The algorithm is rejection-free and also realizes an irreversible Markov chain that satisfies global balance. The autocorrelation functions of the magnetic susceptibility and the energy indicate a dynamical critical exponent z≈1 at the critical temperature, while that of the magnetization does not measure the performance of the algorithm. We show that the event-chain Monte Carlo algorithm substantially reduces the dynamical critical exponent from the conventional value of z≃2.
The event-chain Monte Carlo (ECMC) algorithm is rejection-free and irreversible for Markov chain Monte Carlo simulations. Previous studies have revealed that the ECMC dynamics reduces the dynamical exponent z. In this study, we performed dynamical scaling analysis on systems simulated by the ECMC algorithm and estimated the exponent z for several systems. We show three things as follows. 1. For each model, an appropriate dynamical order parameter is proposed given that it is rotation-invariant, and its applicability to the ECMC dynamics is verified for nonequilibrium relaxation analysis. 2. These values of z are smaller than those obtained using Metropolis dynamics, which confirms that the ECMC dynamics is faster. 3. Although z in the three-dimensional ferromagnetic Heisenberg model is much less than z≃2z \simeq 2z≃2, it is higher than z≃1z \simeq 1z≃1 reported in a previous work [Y. Nishikawa et al., Phys. Rev. E 92, 063306 (2015)].
We analyze the convergence of the irreversible event-chain Monte Carlo algorithm for continuous spin models in the presence of topological excitations. In the two-dimensional XY model, we show that the local nature of the Markov-chain dynamics leads to slow decay of vortex-antivortex correlations while spin waves decorrelate very quickly. Using a Fréchet description of the maximum vortex-antivortex distance, we quantify the contributions of topological excitations to the equilibrium correlations, and show that they vary from a dynamical critical exponent at the critical temperature to in the limit of zero temperature. We confirm the event-chain algorithm's fast relaxation (corresponding to z = 0) of spin waves in the harmonic approximation to the XY model. Mixing times (describing the approach towards equilibrium from the least favorable initial state) however remain much larger than equilibrium correlation times at low temperatures. We also describe the respective influence of topological monopole-antimonopole excitations and of spin waves on the event-chain dynamics in the three-dimensional Heisenberg model.
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.
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{\mathit{z}}_{\mathit{R}}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.
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).
Using a Monte Carlo procedure with a self-consistent-field boundary condition, the magnetization of a simple cubic classical Heisenberg ferromagnet with nearest-neighbor interactions only is calculated. For reduced temperatures and reduced fields in the range $0.02<~|\frac{T}{{T}_{c}}\ensuremath{-}1|$, HJ< 0.5\frac{H}{J}<~0.5JH​< 0.5 the data of this computer experiment can be described in terms of the "effective" critical exponents $0.31<~\ensuremath{\beta}<~0.35$, $1.33<~\ensuremath{\gamma}<~1.39$, and $4.9<~\ensuremath{\delta}<~5.3$. Although the present data are not very close to the critical point they do obey the homogeneity requirement and determine the scaling function rather precisely. This function agrees very well with the scaling function for the face-centered-cubic classical Heisenberg magnet derived recently by Milo\ifmmode \check{s}\else \v{s}\fi{}evi\ifmmode \acute{c}\else \'{c}\fi{} and Stanley using high-temperature---series-expansion techniques. This agreement supports their hypothesis that neither critical exponents nor the scaling function depend on the lattice structure in the Heisenberg model.