Cluster algorithms

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Cluster algorithms in Monte Carlo Simulation.

These algorithms are mainly used in the simulation of Ising-like models. The essential feature is the use of collective motions of particles (spins) in a single Monte Carlo step.

An interesting property of some of these application is the fact that the percolation analysis of the clusters can be used to study phase transitions.

Swendsen-Wang algorithm

As an introductory example we will discuss the Swendsen-Wang technique (Ref 1) in the simulation of Ising Models.

Recipe

In one Monte Carlo step of the algorithm the following recipe is used:

• Consider every pair interacting sites (spins)

In the current configuration the pair interaction can be either negative: $u_{ij}/k_{B}T=-K$ of positive $u_{ij}/k_{B}T=+K$, depending on the product: $S_{i}S_{j}$ (See Ising Models for details on the notation)

• For pairs of interacting sites (nearest neighbors) with $u_{ij}/k_{B}T<0$ create a bond between the two spins with a given probability $p$ (using random numbers)

$p$ will be chosen to be a function of $K$

• The bonds generated in the previous step are used to build up clusters of sites (spins).

• Build up the partition of the system in the corresponding clusters of spins.

In each cluster all the spins will have the same state (either $S=1$ or $S=-1$)

• For each cluster, independently, choose at random with equal probabilities whether to flip (invert the value of $S$) or not to flip the whole

set of spins belonging to the cluster.

THIS RECIPE HAS TO BE COMPLETED, BE PATIENT

Wolff algorithm

See Ref 2 for details.

The procedure to create a given bond is the same as in the Swendsen-Wang algorithm. However in Wolff's method the whole set of interacting pairs is not tested to generate (possible) bonds. In stead, a single cluster is built.

• The initial cluster contains one site (selected at random)

• Possible bonds between the initial site and other sites of the system are tested:

The bonded sites are included in the cluster

• Then recursively, one checks the existence of bonds between the new members of the cluster and sites of the

system to add, if bonds are generated, new sites to the growing cluster, until no more bonds are generated.

• At his point, the whole cluster is flipped (see above)

Invaded Cluster Algorithm

See Ref 3.

Probability-Changing Cluster Algorithm

This method was proposed by Tomita and Okabe (See Ref 4)

References

1. Robert H. Swendsen and Jian-Sheng Wang, Nonuniversal critical dynamics in Monte Carlo simulations, Phys. Rev. Lett. 58, 86 - 88 (1987)
2. Ulli Wolff, Collective Monte Carlo Updating for Spin Systems , Phys. Rev. Lett. 62, 361 - 364 (1989)
3. J. Machta, Y. S. Choi, A. Lucke, T. Schweizer, and L. V. Chayes, Invaded Cluster Algorithm for Equilibrium Critical Points , Phys. Rev. Lett. 75, 2792 - 2795 (1995)
4. Yusuke Tomita and Yutaka Okabe, Probability-Changing Cluster Algorithm for Potts Models, Phys. Rev. Lett. 86, 572 - 575 (2001)