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)

References

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