Cluster algorithms

Cluster 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 shall 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: $\Phi _{ij}/k_{B}T=-K$ or positive $\Phi _{ij}/k_{B}T=+K$, depending on the product: $S_{i}S_{j}$ (See Ising Models for details on the notation)

• For pairs of interacting sites (nearest neighbors) with $\Phi _{ij}/k_{B}T<0$ 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.

The bonding probability $p$ is given by:

$p=1-\exp[-2K]$

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 this point, the whole cluster is flipped (see above)

Invaded Cluster Algorithm

(See Ref 3)

The purpose of this algorithm is to locate critical points (critical temperature). So, in this case the probability of bonding neighboring sites with equal spins is not set a priori.

The algorithm for an Ising system with periodic boundary conditions can be implemented as follows:

Given a certain configuration of the system:

• We consider the possible bonds on the system (pairs of nearest neighbours with favourable interaction)

• Using random numbers we asign a random order to those possible bonds

• The possible bonds are being activated in the order fixed in the previous step (the cluster structure is watched during this process)

• The bond activation stops when one cluster percolates through the entire system (i.e. considering the periodic boundary conditions

the cluster becomes of inifinite size)

• Then, every cluster (as in the Swendsen-Wang algorithm) is flipped with proability 1/2.

• An effective bond probability for the percolation threshold, $p_{per}$ can be computed as:

$p_{per}=M_{act}/M$

with $M_{act}$ being the number of activated bonds when the first cluster percolates, and $$M$ is the number of possible bonds.

The value of $p_{per}$ (in one realisation, or the averaged value over the simulation, See the references for the practical application) can be related with the critical coupling constant, $k_{c}$ as:

$p_{per}\approx 1-\exp \left[-2k_{c}\right]$

Probability-Changing Cluster Algorithm

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

Beyond the Ising and Potts models

The methods described so far, can be used with minor chages in the simulation of Potts models. In addition, extensions have been proposed in the literature (Eeferences will be addes one of these days) to build up very efficient cluster algorithm to simulate more complex lattice systems (XY models, Heisenberg model, etc).

References

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