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

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: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_{ij}/k_B T= -K } of positive Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_{ij}/k_B T = + K } , depending on the product: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_{i} S_{j} } (See Ising Models for details on the notation)

• For pairs of interacting sites (nearest neighbors) with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_{ij}/k_B T < 0 } create a bond between the two spins with a given probability Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p } (using random numbers)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p } will be chosen to be a function of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S = 1 } or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S = -1 } )

• For each cluster, independently, choose at random with equal probabilities whether to flip (invert the value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

The original paper is shown in Ref 3.

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)