Computational Physics

15.5 Reducing critical slowing down 503

0

0.2

0.4

0.6

0.8

1

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

kBT/J

Γ/

J

L=8

L=12

L=20

L=30

L=40

Figure 15.4. The helicity modulus in units of the coupling constantJof theXY

model vs. the inverse coupling constant in units ofkBT. The intersection of the

helicity modulus curves with the straight line gives the value from which the

helicity modulus starts dropping to zero.

If we switch on a magnetic field, the spin-flip symmetry is broken and the cluster
algorithm can no longer be used. Another problem is that it is not immediately
clear whether and how cluster ideas may be generalised to systems that are not
formulated on a lattice. A step towards a solution was made by Dress and Krauth
[35], who usedgeometricsymmetries to formulate a cluster algorithm for particles
moving in a continuum. Usually, a reflection of the particles with respect to a point
chosen randomly in the system is used. The interaction between these particles
is considered to be a hard-core interaction, but long(er)-ranged interaction may
also be present. The problem with the algorithm is that the decision to displace
a particle is made based on the hard-core part. Other interactions are included
in the acceptance criterion, and this leads to many rejections. This problem was
solved by Liu and Luijten who take all interactions into account [36]. They start by
identifying a random reflection point and then choose an initial particle at random.
This and other particles having nonnegligible interaction with the first particles are
then candidates to be reflected. This is done one by one, taking all interactions into
account, and each time a reflection of a particle would result in a decreaseof the
energy, the particle is reflected with probability exp(−||). If the energy increases,
the particle is not reflected. This algorithm promises to be valuable for the analysis
of dense liquids.