Dynamic critical index of the Swendsen–Wang algorithm by dynamic finite-size scaling

doi:10.1016/j.cpc.2006.07.007

Computer Physics Communications

Volume 175, Issue 8, 15 October 2006, Pages 553-558

https://doi.org/10.1016/j.cpc.2006.07.007 Get rights and content

Abstract

In this work we have considered the dynamic scaling relation of the magnetization in order to study the dynamic scaling behavior of 2- and 3-dimensional Ising models. We have used the literature values of the magnetic critical exponents to observe the dynamic finite-size scaling behavior of the time evolution of the magnetization during early stages of the Monte Carlo simulation. In this way we have calculated the dynamic critical exponent Z for 2- and 3-dimensional Ising Models by using the Swendsen–Wang cluster algorithm. We have also presented that this method opens the possibility of calculating z and $x_{0}$ separately. Our results show good agreement with the literature values. Measurements done on lattices with different sizes seem to give very good scaling.

Introduction

Fortuin and Kasteleyn's solution of the Potts model by using percolating clusters [1] has been an inspiration for the Swendsen–Wang cluster algorithm [2]. This algorithm uses the Hamiltonian of the Potts model in order to identify the clusters of the spins with the same orientations. In defining a cluster, starting from a seed spin, a new spin is added to the already growing cluster with the probability $P = 1 - e^{- β}$ , where β is the inverse temperature. After obtaining all possible clusters on the lattice, clusters are flipped with equal probability. Immediately after the work by Swendsen and Wang, Wolff proposed an new algorithm [3], which is basically a modification to the Swendsen–Wang algorithm. Despite the fact that the Wolff algorithm is an alternative method of updating clusters, decorrelation times have shown to be very different between Wolff and Swendsen–Wang algorithms. Following these two cluster update algorithms many alternative cluster update algorithms are introduced with decorrelation times always higher than that of the Wolff algorithm. For 2-dimensional Ising model Heerman and Burkitt [4] suggested that the autocorrelation data are consistent with a logarithmic divergence, but it is very difficult to distinguish between the logarithm and a small power [5].

With the introduction of cluster algorithms, a great improvement in the simulations of the magnetic spin systems has been possible since it has been shown that the dynamic critical exponents of these algorithms are much less than that of local algorithms such as Metropolis and Heat Bath. The efficiencies (dynamic behavior and the dynamic critical exponents) have been discussed by many authors by using various spin systems at thermal equilibrium [2], [3], [4], [5], [6], [7], [8].

Recently we have studied the dynamic behavior of 2-, 3-, and 4-dimensional Ising models by using the Wolff cluster algorithm [9]. We based our work on the dynamic scaling which exists in the early stages of the quenching process in the system [10]. The efficiency of the Wolff algorithm is directly related to the size of the updated clusters, hence the efficiency increases during the quenching process as the number of iterations increases. In our calculations, we have observed that our results are consistent with vanishing dynamic critical exponent.

In this work we aimed to discuss the dynamic critical exponent of the Swendsen–Wang algorithm by using dynamic finite-size scaling. This will give us an opportunity to compare the efficiencies of cluster algorithms.

Section snippets

The method

In this work we have employed 2-, 3-, and 4-dimensional Ising models which are described by the Hamiltonian $- β H = K \sum_{〈 i j 〉} S_{i} S_{j} .$

Here $β = 1 / k T$ and $K = J / k T$ , where k is the Boltzmann constant, T is the temperature and J is the magnetic interaction between the spins. In the Ising model the spin variables take the values $S_{i} = \pm 1$ .

To observe the critical behavior of the systems exhibiting second-order phase transitions we use dynamic scaling which exists in the early stages of the quenching process in the

Results and discussion

We have studied 2-, 3-, and 4-dimensional Ising Models evolving in time by using the Swendsen–Wang algorithm. Following our previous work [9], we have prepared lattices with vanishing magnetization and total random initial configurations are quenched at the corresponding infinite lattice critical temperature. We have used the lattices $L = 256$ , 384, 512, 640 and $L = 32$ , 48, 64, 80, $L = 16$ , 20, 24 for 2-, 3-, and 4-dimensional Ising models, respectively. Twenty bins of two thousand runs have been

Conclusion

In this work we have considered the dynamic scaling behavior of moments of magnetization, $(〈 S^{n} 〉)$ , Binder's cumulant ( $B_{2} (t)$ ) and the renormalization function ( $F (t)$ ) for 2-, 3-, and 4-dimensional Ising models using the Swendsen–Wang algorithm. The values of dynamic critical exponent $(z)$ obtained using this algorithm vary between 0.25 and 1.00, depending on the dimension, as shown in Table 1. One can see from the results of dynamic scaling that scaling is very good, the errors are very small, and

Acknowledgements

We gratefully acknowledge Hacettepe University Research Fund (Project no: 01 01 602 019) and Hewlett–Packard's Philanthropy Programme.

References (24)

U. Wolff
Phys. Lett. B
(1989)
S. Gündüç et al.
Comput. Phys. Comm.
(2005)
P.M.C. de Oliveira
Physica A
(1994)
P. Tamayo
Physica A
(1993)
G. Ossola et al.
Nucl. Phys. B
(2004)
P.W. Kasteleyn et al.
J. Phys. Soc. Japan
(1969)
R.H. Swendsen et al.
Phys. Rev. Lett.
(1987)
U. Wolff
Phys. Rev. Lett.
(1989)
D.W. Heerman et al.
Physica A
(1990)
C.F. Baillie et al.
Phys. Rev. B
(1991)

N. Ito et al.

Int. J. Modern Phys. C

(1990)

P. Tamayo et al.

J. Stat. Phys.

(1990)

Cited by (1)

Static and dynamic critical behavior of thin magnetic Ising films
2015, Physica A: Statistical Mechanics and its Applications
This work presents a study of the effect of film thickness on the static and dynamic critical behavior of thin magnetic Ising films. Monte Carlo simulations using the Wolff algorithm were performed to determine the static and dynamic critical exponents of the films. A dimensionality crossover from 2D to 3D (due to the finiteness of the films) in the static and dynamic critical behavior was observed as the film thickness increases. In addition, a slight increase in the effective dimension $d_{e f f}$ and a considerable increase in the critical temperature $T_{c} (\infty)$ were found. Small values for the dynamic critical exponents were obtained, indicating that the Wolff algorithm is a very efficient method for these magnetic systems.

View full text

Dynamic critical index of the Swendsen–Wang algorithm by dynamic finite-size scaling

Abstract

Introduction

Section snippets

The method

Results and discussion

Conclusion

Acknowledgements

Phys. Lett. B

Comput. Phys. Comm.

Physica A

Physica A

Nucl. Phys. B

J. Phys. Soc. Japan

Phys. Rev. Lett.

Phys. Rev. Lett.

Physica A

Phys. Rev. B

Int. J. Modern Phys. C

J. Stat. Phys.