Computational Physics

496 Computational methods for lattice field theories

wherescis the spin value of clusterc. We can write this as

χ=

1

L^2 d

〈

∑

c

Ncsc

∑

c′

Nc′sc′

〉

, (15.87)

and by summing oversc=±1 for all the clusters we obtain the average of this value
for all possible cluster-spin configurations. Then only the termsc=c′survive and
we are left with

χ=

1

L^2 d

〈

∑

c

Nc^2

〉

. (15.88)

This is the so-called ‘improved estimator’ for the unsubtracted susceptibility
[24, 25]. This estimator gives better results because the average over all possible
cluster-spin configurations is built into it.
The correlation time can be determined from the values ofχat the subsequent
MC steps in the usual way (seeSection 7.4). For a detailed analysis of the dynamic
exponentforvariouscases,seeRef.[21].

programming exercise

Code the SW algorithm for the two-dimensional Ising model. Determine the

time relaxation exponent and compare this with the value found for the single-

spin flip algorithm.

Wolff has carried out the cluster algorithm in the microcanonical ensemble[24],
using a microcanical MC method proposed by Creutz [26]. He fixed the num-
ber of unequal bonds to half the number of total bonds and found considerable
improvement in the efficiency.

15.5.2 Wolff’s single cluster algorithm

Wolff has proposed a different cluster method for eliminating critical slowing down
for Potts spin systems, and an extension of this method and the SW method to a
special class of continuous spin models [27]. We start with Wolff’s modification of
the SW method for the Ising model. In Wolff’s method, at each step asinglecluster
is generated, as opposed to the SW model in which the entire lattice is partitioned
into clusters. The single cluster is constructed according to the same rules as the
SW clusters. We start with a randomly chosen spin and consider its neighbours.
Only equal neighbours can be linked to the cluster by freezing the bonds between
them – this happens with probability 1−e−^2 βJ. The cluster is extended in this way
until no more spins are added to it. Then all the spins in the cluster are flipped.