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

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)