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Journal of the Physical Society of Japan

Dynamical Scaling Analysis and Estimation of Critical Exponent z for Continuous Spin Systems Simulated Using Event-Chain Algorithm

Yuma Osada, Yukiyasu Ozeki

Published Jul 21, 2023• 1 citation
Publisher

Abstract

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 \(z \simeq 2\), it is higher than \(z \simeq 1\) reported in a previous work [Y. Nishikawa et al., Phys. Rev. E 92, 063306 (2015)].

DOI
10.7566/jpsj.92.084004

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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 \simeq 2), it is higher than (z \simeq 1) reported in a previous work [Y. Nishikawa et al., Phys. Rev. E 92, 063306 (2015)].
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.
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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.
Recently the present authors proposed the nonequilibrium-to-equilibrium scaling (NE-ES) scheme for the critical Monte Carlo relaxation process [Nonomura, J. Phys. Soc. Jpn. 83, 113001 (2014)10.7566/JPSJ.83.113001], which scales relaxation data in the whole simulation-time regions regardless of functional forms, namely, both for the stretched-exponential critical relaxation in cluster algorithms and for the power-law critical relaxation in local-update algorithms. In the present study, we generalize this scheme to off-critical relaxation process and scale relaxation data for various temperatures in the whole simulation-time regions. This proposal of the off-critical scaling in cluster algorithms cannot be described by the dynamical finite-size scaling theory based on the power-law critical relaxation. As an example, we investigate the three-dimensional classical Heisenberg model previously analyzed with the NE-ES [Nonomura and Tomita, Phys. Rev. E 93, 012101 (2016)10.1103/PhysRevE.93.012101] in the Swendsen-Wang and Metropolis algorithms.
We study the critical relaxation of the two-dimensional Ising model from a fully ordered configuration by series expansion in time ttt and by Monte Carlo simulation. Both the magnetization (m)(m)(m) and energy series are obtained up to 12th order. An accurate estimate from series analysis for the dynamical critical exponent zzz is difficult but compatible with 2.2. We also use Monte Carlo simulation to determine an effective exponent, ${z}_{\mathrm{eff}}(t)=\ensuremath{-}\frac{1}{8}d\mathrm{ln}t/d\mathrm{ln}m$, directly from a ratio of three-spin correlation to mmm. Extrapolation to $\stackrel{\ensuremath{\rightarrow}}{t}\ensuremath{\infty}$ leads to an estimate $z=2.169\ifmmode\pm\else\textpm\fi{}0.003$.