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Nuclear Physics B - Proceedings Supplements

Dynamic exponents for potts model cluster algorithms

Paul David Coddington, CLIVE F. BAILLIE

Published Jan 1, 1992• 1 citation
Publisher

Abstract

We have studied the Swendsen-Wang and Wolff cluster update algorithms for the Ising model in 2, 3 and 4 dimensions. The data indicate simple relations between the specific heat and the Wolff autocorrelations, and between the magnetization and the Swendsen-Wang autocorrelations. This implies that the dynamic critical exponents are related to the static exponents of the Ising model. We also investigate the possibility of similar relationships for the Q-state Potts model.

DOI
10.1016/0920-5632(92)90355-v

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We have studied the Swendsen-Wang and Wolff cluster update algorithms for the Ising model in 2, 3 and 4 dimensions. The data indicate simple relations between the specific heat and the Wolff autocorrelations, and between the magnetization and the Swendsen-Wang autocorrelations. This implies that the dynamic critical exponents are related to the static exponents of the Ising model. We also investigate the possibility of similar relationships for the Q-state Potts model.
We have measured the autocorrelations for the Swendsen-Wang and the Wolff cluster update algorithms for the Ising model in two, three, and four dimensions. The data for the Wolff algorithm suggest that the autocorrelations are linearly related to the specific heat, in which case the dynamic critical exponent is zint,{\mathit{z}}_{\mathrm{i}\mathrm{n}\mathrm{t},}zint,​EW{\mathit{E}}^{\mathrm{W}}EW=\ensuremath{\alpha}/\ensuremath{\nu}. For the Swendsen-Wang algorithm, scaling the autocorrelations by the average maximum cluster size gives either a constant or a logarithm, which implies that zint,{\mathit{z}}_{\mathrm{i}\mathrm{n}\mathrm{t},}zint,​ESW{\mathit{E}}^{\mathrm{SW}}ESW=\ensuremath{\beta}/\ensuremath{\nu} for the Ising model.
We have measured the dynamical critical exponent z for the Swendsen-Wang and the Wolff cluster update algorithms, as well as a number of variants of these algorithms, for the q=2 and q=3 Potts models in two dimensions. We find that although the autocorrelation times differ considerably between algorithms, the critical exponents are the same. For q=2, we find that although the data are better fitted by a logarithmic increase in the autocorrelation time with lattice size, they are also consistent with a power law with exponent z\ensuremath{\approxeq}0.25, especially if there are non-negligible corrections to scaling.
We formulate a single-cluster Monte Carlo algorithm for the simulation of the random-cluster model. This algorithm is a generalization of the Wolff single-cluster method for the q-state Potts model to noninteger values q>1. Its results for static quantities are in a satisfactory agreement with those of the existing Swendsen-Wang-Chayes-Machta (SWCM) algorithm, which involves a full-cluster decomposition of random-cluster configurations. We explore the critical dynamics of this algorithm for several two-dimensional Potts and random-cluster models. For integer q, the single-cluster algorithm can be reduced to the Wolff algorithm, for which case we find that the autocorrelation functions decay almost purely exponentially, with dynamic exponents z(exp)=0.07 (1), 0.521 (7), and 1.007 (9) for q=2, 3, and 4, respectively. For noninteger q, the dynamical behavior of the single-cluster algorithm appears to be very dissimilar to that of the SWCM algorithm. For large critical systems, the autocorrelation function displays a range of power-law behavior as a function of time. The dynamic exponents are relatively large. We provide an explanation for this peculiar dynamic behavior.
Simulations of the two-dimensional Ising and three-state Potts models at their critical points are performed using the invaded cluster (IC) algorithm. It is argued that observables measured on a sublattice of size l should exhibit a crossover to Swendsen-Wang (SW) behavior for l sufficiently less than the lattice size L, and a scaling form is proposed to describe the crossover phenomenon. It is found that the energy autocorrelation time ${\ensuremath{\tau}}_{\ensuremath{\varepsilon}}(l,L)$ for an $l\ifmmode\times\else\texttimes\fi{}l$ sublattice attains a maximum in the crossover region, and a dynamic exponent zIC{z}^{\mathrm{IC}}zIC for the IC algorithm is defined according to ${\ensuremath{\tau}}_{\ensuremath{\varepsilon},\mathrm{max}}\ensuremath{\sim}{L}^{{z}^{\mathrm{IC}}}.$ Simulation results for the three-state model yield ${z}^{\mathrm{IC}}=0.346\ifmmode\pm\else\textpm\fi{}0.002,$ which is smaller than values of the dynamic exponent found for the SW and Wolff algorithms and also less than the Li-Sokal bound. The results are less conclusive for the Ising model, but it appears that zIC<0.21{z}^{\mathrm{IC}}<0.21zIC<0.21 and possibly that ${\ensuremath{\tau}}_{\ensuremath{\varepsilon},\mathrm{max}}\ensuremath{\sim}\mathrm{ln}L$ so that zIC=0{z}^{\mathrm{IC}}=0zIC=0---similar to previous results for the SW and Wolff algorithms.
Using a nonequilibrium relaxation method, we calculate the dynamic critical exponent z of the two-dimensional Ising model for the Swendsen–Wang and Wolff algorithms. We examine dynamic relaxation processes following a quench from a disordered or an ordered initial state to the critical temperature T c , and measure the exponential relaxation time of the system energy. For the Swendsen–Wang algorithm with an ordered or a disordered initial state, and for the Wolff algorithm with an ordered initial state, the exponential relaxation time fits well to a logarithmic size dependence up to a lattice size L = 8192. For the Wolff algorithm with a disordered initial state, we obtain an effective dynamic exponent z exp = 1.19(2) up to L = 2048. For comparison, we also compute the effective dynamic exponents through the integrated correlation times. In addition, an exact result of the Swendsen–Wang dynamic spectrum of a one-dimensional Ising chain is derived.
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.