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.

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

Sketches of the Swendsen-Wang algorithm

• 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 sites in different clusters

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

• For each cluster, independently, choose at random (whether with probability 1/2) to flip (invert the value of $S$) or not to flip, every spin on the cluster

References

1. Robert H. Swendsen and Jian-Sheng Wang, Nonuniversal critical dynamics in Monte Carlo simulations, Phys. Rev. Lett. 58, 86 - 88 (1987)