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Physical Review Letters

Embedded dynamics for φ4 theory

Richard C. Brower, Pablo Tamayo

Published Mar 6, 1989• 129 citations
Publisher

Abstract

A new Monte Carlo dynamics is proposed for ${\mathrm{\ensuremath{\varphi}}}^{4}$ field theory that reduces critical slowing down. Discrete Ising variables are embedded into the field theory, so that large-scale tunneling events can be induced via a modified Swendsen-Wang algorithm with fluctuating site-bond percolation probabilities. The measured dynamical critical exponents ${\mathrm{z}}_{\mathrm{EL}=0.07\ifmmode\pm\else\textpm\fi{}}$0.07, 0.29\ifmmode\pm\else\textpm\fi{}0.09, and 0.87\ifmmode\pm\else\textpm\fi{}0.20 in one, two, and three dimensions, respectively, are consistent with our conjecture that this embedded dynamics lies in the same universality class as the Swendsen-Wang Ising dynamics.

DOI
10.1103/physrevlett.62.1087

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A new Monte Carlo dynamics is proposed for \ensuremathφ4{\mathrm{\ensuremath{\varphi}}}^{4}\ensuremathφ4 field theory that reduces critical slowing down. Discrete Ising variables are embedded into the field theory, so that large-scale tunneling events can be induced via a modified Swendsen-Wang algorithm with fluctuating site-bond percolation probabilities. The measured dynamical critical exponents zEL=0.07\ifmmode±\else\textpm\fi{\mathrm{z}}_{\mathrm{EL}=0.07\ifmmode\pm\else\textpm\fi{}}zEL=0.07\ifmmode±\else\textpm\fi​0.07, 0.29\ifmmode\pm\else\textpm\fi{}0.09, and 0.87\ifmmode\pm\else\textpm\fi{}0.20 in one, two, and three dimensions, respectively, are consistent with our conjecture that this embedded dynamics lies in the same universality class as the Swendsen-Wang Ising dynamics.
The Fortuin-Kasteleyn mapping between the Ising model and the site-bond correlated percolation model is shown to be only one of an infinite class of exact mappings. These new cluster representations are a result of "renormalized" percolation rules correlated to entire blocks of spins. For example these rules allow for percolation on "virtual" bonds between spins not present in the underlying Hamiltonian. As a consequence we can define new random cluster theories each with its own Monte Carlo cluster dynamics that exactly reproduce the Ising model. By tuning parameters on the critical percolation surface, it is demonstrated numerically that cluster algorithms can be formulated for the 2-d and 3-d Ising model that have smaller autocorrelations than the original Swendsen-Wang algorithm.
In the present work, we present an extensive Monte Carlo simulation study on the dynamical properties of the two-dimensional random-bond Ising model. The correlation time τ of the Swendsen-Wang and Wolff cluster algorithms is calculated at the critical point. The dynamic critical exponent z of both algorithms is also measured by using the numerical data for several lattice sizes up to L=512. It is found for both algorithms that the autocorrelation time decreases considerably and the critical slowing-down effect reduces upon the introduction of bond disorder. Additionally, simulations with the Metropolis algorithm are performed, and the critical slowing-down effect is observed to be more pronounced in the presence of disorder, confirming the previous findings in the literature. Moreover, the existence of the non-self-averaging property of the model is demonstrated by calculating the scaled form of the standard deviation of autocorrelation times. Finally, the critical exponent ratio of the magnetic susceptibility is estimated by using the average cluster size of the Wolff algorithm.
In order to reduce the critical slowing down in Monte Carlo simulations, a numerical study for the Ising model on 3-D random lattice using Swendsen-Wang algorithm is performed. The critical point obtained isβc=0. 075?. 001. Moreover, to analyse the dynamical property of Swendsen-Wang method, the dynamical critical exponent z is estimated. The result is z =0. 74 ±0. 03. This indicates that the Swendsen-Wang method can greatly reduce the critical slowing down for the Ising model on 3-D random lattice.
Nonuniversal dynamics introduced by Swendsen and Wang [Phys. Rev. Lett. 58, 86 (1987)] is studied in a mean-field Ising model. Numerical simulation data indicate that z, the dynamical critical exponent, is 0.98\ifmmode\pm\else\textpm\fi{}0.08. A modified version of this dynamics in which only large clusters are flipped apparently gives the same exponent, although the amplitude is reduced by a factor of 1/2. In addition, a theoretical argument is presented which gives z=1 and provides insight into the dynamics.
We present a new cluster-update Monte Carlo algorithm for a fluctuating interface in the 3D Ising model. Critical slowing down is drastically reduced and the dynamical critical exponent at the roughening temperature of the 3D Ising model is found to be zcl{\mathit{z}}_{\mathrm{cl}}zcl​=0.44\ifmmode\pm\else\textpm\fi{}0.03. For the Metropolis algorithm we obtain zMet{\mathit{z}}_{\mathrm{Met}}zMet​=2.0\ifmmode\pm\else\textpm\fi{}0.1 while the Swendsen-Wang algorithm performs even worse than local updating. A new method which suppresses fluctuations in the two bulk phases is introduced to determine moments of the interfacial thickness.
We study the dynamic critical behavior of Wolff's embedding algorithm combined with the standard Swendsen-Wang algorithm, for the two-dimensional XY\mathrm{XY}XY and O(4) spin models on lattices up to 256\ifmmode\times\else\texttimes\fi{}256. We find that critical slowing down is completely or almost completely eliminated.