Computational Physics

498 Computational methods for lattice field theories

correlation timeτWexpressed in SW time steps:

τW= ̄τW

〈N1C〉

Ld

. (15.89)

The average single cluster size〈N1C〉occurring on the right hand side is the improved
estimator for the (unsubtracted) susceptibility per site:

〈N1C〉=

〈

NSWC

Ld

NSWC

〉

=χ. (15.90)

This formula can be understood by realising that the probability of generating a
SW cluster of sizeNSWCin the single cluster algorithm is equal toNSWC/Ld.To
evaluate the average cluster size we must multiply this probability withNSWCand
take the expectation value of the result.

programming exercise
Implement Wolff’s single cluster algorithm and compare the results with the
SWalgorithm–seealsoRef.[21].
In many statistical spin systems and lattice field theories, the spins are not dis-
crete but they assume continuous values. Wolff’s algorithm was formulated for a
particular class of such models, the O(N)models. These models consist of spins,
which areN-dimensional unit vectors, on a lattice. Neighbouring spinssi,sjinter-
act – the interaction is proportional to the scalar productsi·sj. An example which is
relevant to many experimental systems (superfluid and superconducting materials,
arrays of coupled Josephson junctions ...) is the O( 2 )orXYmodel, in which the
spins are unit vectorssilying in a plane, so that they can be characterised by their
angleθiwith thex-axis, 0≤θi< 2 π.
For simulations, it is important that relevant excitations inO(N)models are
smooth variations of the spin orientation over the lattice (except near isolated
points – see below). This implies that changing the value of a single angleθi
somewhere in the lattice by an amount of order 1 is likely to lead to an improbable
configuration; hence the acceptance rate for such a trial change is on average very
small. The only way of achieving reasonable acceptance rates for changing a single
spin is by considerably restricting the variation in the orientation of the spin allowed
in a trial step. This, however, will reduce the efficiency because many MC steps
are then needed to arrive at statistically independent configurations. A straightfor-
ward generalisation of the SW or single cluster algorithm in which all spins in
some cluster are reversed is bound to fail for the same reason, as this destroys the
smoothness of the variation of the spins at the cluster boundary.
Wolff has proposed a method in which the spins in a cluster are modified to
an extent depending on their orientation [27]. It turns out that his method can be