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arXiv:cond-mat/0409762 · Computer Physics Communications

A study of dynamic finite size scaling behavior of the scaling functions—calculation of dynamic critical index of Wolff algorithm

Semra Gündüç, M. Dilaver, Meral Aydın, Y. Gündüç

Published Jan 13, 2005• 10 citations
arXivHTMLPDF
DOI
10.1016/j.cpc.2004.09.008

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In this work we have calculated the dynamic critical exponent zzz for 2-, 3- and 4-dimensional Ising models using the Wolff's algorithm through dynamic finite size scaling. We have studied time evolution of the average cluster size, the magnetization and higher moments of the magnetization. It is observed that dynamic scaling is independent of the algorithm. In this sense, universality is established for a wide range of algorithms with their own dynamic critical exponents. For scaling, we have used the literature values of critical exponents to observe the dynamic finite size scaling and to obtain the value of zzz. From the simulation data a very good scaling is observed leading to vanishingly small zzz values for all three dimensions.
We calculate the dynamic critical exponent for the Niedermayer algorithm applied to the two-dimensional Ising and XY models, for various values of the free parameter E 0 . For E 0 = − 1 we regain the Metropolis algorithm and for E 0 = 1 we regain the Wolff algorithm. For − 1 < E 0 < 1, we show that the mean size of the clusters of (possibly) turned spins initially grows with the linear size of the lattice, L , but eventually saturates at a given lattice size , which depends on E 0 . For , the Niedermayer algorithm is equivalent to the Metropolis one, i.e., they have the same dynamic exponent. For E 0 > 1, the autocorrelation time is always greater than for E 0 = 1 (Wolff) and, more important, it also grows faster than a power of L . Therefore, we show that the best choice of cluster algorithm is the Wolff one, when comparing against the Niedermayer generalization. We also obtain the dynamic behavior of the Wolff algorithm: although not conclusively, we propose a scaling law for the dependence of the autocorrelation time on L .
We study the dynamic critical behavior of the two-dimensional random-bond three-state Potts model using large-scale Monte Carlo simulations with the Wolff and Swendsen-Wang cluster algorithms. At the disorder-dependent critical temperature, we compute integrated and exponential autocorrelation times and extract the dynamic critical exponent z via finite-size scaling. Critical slowing down is significantly weakened by bond randomness, with the dynamic exponent decreasing from z≈0.54 in the pure system to z≈0.24 at strong disorder. From the finite-size scaling of the Wolff cluster size, we obtain γ/ν≈1.73, where γ and ν are the susceptibility and correlation-length exponents, respectively, consistent with the universality class of the two-dimensional three-state Potts model. These results indicate that bond randomness strongly affects the critical dynamics but leaves the underlying static universality class unchanged. Furthermore, both static and dynamic observables, including the specific heat and the autocorrelation times, show a lack of self-averaging.