Swendsen–Wang algorithm: Difference between revisions

Revision as of 17:03, 18 April 2021

Revision as of 18:16, 18 April 2021

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Although not analytically clear from the original paper, the reason why all the values of z obtained with the SW algorithm are much lower than the exact lower bound for single-spin-flip algorithms (<math>z\geq\gamma/\nu</math>) is that the correlation length divergence is strictly related to the formation of percolation clusters, which are flipped together. In this way the relaxation time is significantly reduced. Another way to view this is through the correspondence between the spin statistics and cluster statistics in the [[Random cluster model|Edwards-Sokal representation]].<ref>{{Cite journal|last=Edwards|first=Robert G.|last2=Sokal|first2=Alan D.|date=1988-09-15|title=Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm|url=https://link.aps.org/doi/10.1103/PhysRevD.38.2009|journal=Physical Review D|volume=38|issue=6|pages=2009–2012|doi=10.1103/PhysRevD.38.2009}}</ref>

== Generalizations ==

The algorithm is not efficient in simulating [[Geometrical frustration|frustrated systems]], because the [[Percolation critical exponents|correlation length of the clusters]] is larger than the [[Correlation function (statistical mechanics)|correlation length of the spin model]] in the presence of frustrated interactions.<ref>{{Cite journal|last=Cataudella|first=V.|last2=Franzese|first2=G.|last3=Nicodemi|first3=M.|last4=Scala|first4=A.|last5=Coniglio|first5=A.|date=1994-03-07|title=Critical clusters and efficient dynamics for frustrated spin models|url=https://link.aps.org/doi/10.1103/PhysRevLett.72.1541|journal=Physical Review Letters|volume=72|issue=10|pages=1541–1544|doi=10.1103/PhysRevLett.72.1541}}</ref> Currently, there are two main approaches to addressing this problem, such that the efficiency of cluster algorithms is extended to frustrated systems.

The first approach is to extend the bond-formation rules to more non-local cells, and the second approach is to generate clusters based on more relevant order parameters. In the first case, we have the [[KBD algorithm]] for the [[Domino tiling|fully-frustrated Ising model]], where the decision of opening bonds are made on each plaquette, arranged in a checkerboard pattern on the square lattice.<ref>{{Cite journal|last=Kandel|first=Daniel|last2=Ben-Av|first2=Radel|last3=Domany|first3=Eytan|date=1990-08-20|title=Cluster dynamics for fully frustrated systems|url=https://link.aps.org/doi/10.1103/PhysRevLett.65.941|journal=Physical Review Letters|volume=65|issue=8|pages=941–944|doi=10.1103/PhysRevLett.65.941}}</ref> In the second case, we have [[replica cluster move]] for low-dimensional [[Spin glass|spin glasses]], where the clusters are generated based on spin overlaps, which is believed to be the relevant order parameter.

== See also ==

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* [[Random cluster model]]

* [[Monte Carlo method]]

*[[Wolff algorithm]]

* http://www.hpjava.org/theses/shko/thesis_paper/node69.html

*http://www-fcs.acs.i.kyoto-u.ac.jp/~harada/monte-en.html