492 Computational methods for lattice field theories
is a special case, but we treat them in this chapter because the ideas behind the
methods developed for field theories and statistical mechanics are very similar.
As the local character of the standard algorithms seems to be responsible for
the critical slowing down present in the standard methods, the idea common to the
methods to be discussed is to update the stochastic variablesglobally, that is, all in
one step. How this is done can vary strongly from one method to the other, but the
underlying principle is the same for all of them.
15.5.1 The Swendsen–Wang methodWe start with the cluster method of Swendsen and Wang (SW)[20], and explain
their method for the Ising model inddimensions, discussed already inSection 7.2
and 10.3.1. The SW method is a Monte Carlo method in which the links, rather
than the sites, of the Ising lattice are scanned in lexicographic order. For each link
there are two possibilities:
- The two spins connected by this link are opposite. In that case the interaction
between these spins is deleted. - The two spins connected by the link are equal. In that case we either delete the
bond or ‘freeze’ it, which means that the interaction is made infinitely strong.
Deletion occurs with probabilitypd=e−^2 βJand freezing with probability
pf= 1 −pd.
This process continues until we have visited every link. In the end we are left with
a model in which all bonds are either deleted or ‘frozen’, that is, their interaction
strength is either 0 or∞. This means that the lattice is split up in a set of disjoint
clusters and within each cluster the spins are all equal. This model is simulated
trivially by assigning at random a new spin value+or−to each cluster. Then the
original Ising bonds are restored and the process starts again, and so on.
Of course we must show that the method does indeed satisfy the detailed balance
condition. Before doing so, we note that the method does indeed lead to a reduction
of the dynamic critical exponentzof the two-dimensional Ising model to the value
0.35 presented by Swendsen and Wang,^7 which is obviously an important improve-
ment with respect to the valuez=2.125 for the standard MC algorithm. The reason
the method works is that flipping blocks involves flipping many spins in one step.
In fact, the Ising (or more generally, the Potts model) can be mapped on a cluster
model, where the distribution of clusters is the same as for the SW clusters [22].
The average linear cluster size is proportional to the correlation length, and this will
diverge at the phase transition. Therefore, the closer we are to the critical point, the
larger the clusters are and the efficiency will increase accordingly.
(^7) From a careful analysis, Wolff has found exponentsz=0.2 andz=0.27 for the 2D Ising model,
depending on the physical quantity considered [21].