500 Computational methods for lattice field theories
curves (in two dimensions, and (hyper)surfaces in higher dimensions) on which
the spinssiare more or less perpendicular tou. In other words, if we provide a
directionu, the algorithm will find an appropriate cluster boundary such that the
spin reflection does not require a vast amount of energy. Therefore, the acceptance
rate is still appreciable.
The procedure is ergodic as every unit vectorucan in principle be chosen and
there is a finite probability that the cluster to be swapped consists of a single spin. The
isolated spin-update method is therefore included in the new algorithm. Detailed
balance is satisfied because the new Ising Hamiltonian (15.93) is exactly equivalent
to the originalO(N)Hamiltonian under the restriction that only the reflection steps
described are allowed in the latter.
The implementation of the method for the two-dimensionalXYmodel proceeds
along the same lines as described above for the Ising model. Apart from selecting
a random location from which the cluster will be grown, a unit vectorumust be
chosen, simply by specifying its angle with theX-axis. Each spin is flipped when
added to the cluster. If we try to add a new spinsito the cluster (in routine ‘TryAdd’),
we need the spin value of its neighboursjin the cluster. The freezing probability
Pfis then calculated as
Pf= 1 −exp[− 2 J(si·u)(sj·u)] (15.94)(note that the cluster spinsihas already been flipped, in contrast tosj). The spin
sjis then added to the cluster with this freezing probability. Instead of considering
continuous angles between 0 and 2π, it is possible to consider ann-state clock
model, which is anXYmodel with the restriction that the angles allowed for the
spins assume the values 2jπ/n,j=0,...,n− 1 [29]; see alsoSection 12.6. At
normal accuracies, the discretisation of the angles will not be noticed forngreater
than about 20. The cosines and sines needed in the program will then assumen
different values only and these can be calculated in the beginning of the program
and stored in an array.
programming exercise
Write a Monte Carlo simulation program for theXYmodel, using Wolff’s
cluster algorithm. If the program works correctly, it should be possible to
detect the occurrence of the so-called Kosterlitz-Thouless phase transition
(note that this occurs only in two dimensions). This is a transition which has
been observed experimentally in helium-4 films [30] and Josephson junction
arrays [31]. We shall briefly describe the behaviour of theXYmodel.
Apart from excitations which are smooth throughout the lattice –spin-waves–
theXYmodel exhibits so-calledvortex excitations. A pair of vortices is shown in