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Physical Review Letters

Empirical relations between static and dynamic exponents for Ising model cluster algorithms

Paul David Coddington, CLIVE F. BAILLIE

Published Feb 17, 1992• 45 citations
Publisher

Abstract

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 ${\mathit{z}}_{\mathrm{i}\mathrm{n}\mathrm{t},}$${\mathit{E}}^{\mathrm{W}}$=\ensuremath{\alpha}/\ensuremath{\nu}. For the Swendsen-Wang algorithm, scaling the autocorrelations by the average maximum cluster size gives either a constant or a logarithm, which implies that ${\mathit{z}}_{\mathrm{i}\mathrm{n}\mathrm{t},}$${\mathit{E}}^{\mathrm{SW}}$=\ensuremath{\beta}/\ensuremath{\nu} for the Ising model.

DOI
10.1103/physrevlett.68.962

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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 {\mathit{z}}_{\mathrm{i}\mathrm{n}\mathrm{t},}$${\mathit{E}}^{\mathrm{W}}=\ensuremath{\alpha}/\ensuremath{\nu}. For the Swendsen-Wang algorithm, scaling the autocorrelations by the average maximum cluster size gives either a constant or a logarithm, which implies that {\mathit{z}}_{\mathrm{i}\mathrm{n}\mathrm{t},}$${\mathit{E}}^{\mathrm{SW}}=\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.
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.
Monte Carlo cluster algorithms are popular for their efficiency in studying the Ising model near its critical temperature. We might expect that this efficiency extends to the bond-diluted Ising model. We show, however, that this is not always the case by comparing how the correlation times τ_{w} and τ_{sw} of the Wolff and Swendsen-Wang cluster algorithms scale as a function of the system size L when applied to the two-dimensional bond-diluted Ising model. We demonstrate that the Wolff algorithm suffers from a much longer correlation time than in the pure Ising model, caused by isolated (groups of) spins which are infrequently visited by the algorithm. With a simple argument we prove that these cause the correlation time τ_{w} to be bounded from below by L^{z_{w}} with a dynamical exponent z_{w}=γ/ν≈1.75 for a bond concentration p<1. Furthermore, we numerically show that this lower bound is actually taken for several values of p in the range 0.5<p<1. Moreover, we show that the Swendsen-Wang algorithm does not suffer from the same problem. Consequently, it has a much shorter correlation time, shorter than in the pure Ising model even. Numerically at p=0.6, we find that its dynamical exponent is z_{sw}=0.09(4).