15.5 Reducing critical slowing down 495
IF (Frozen(x,y,x,y−1)) THEN
BackTrack(x,y−1);
END IF
END BackTrack.Frozen(x1,y1,x2,y2) is a Boolean function which returns TRUE if the nearest
neighbour bond between(x1,y 1 )and(x2,y 2 )is frozen and FALSE otherwise.
Periodic boundaries should be implemented using a modulo operator or function,
and it is convenient to decide before scanning the cluster whether it is going to be
flipped and, if yes, to do so during the recursive scanning (alongside putting the
Visited flag). On exit, the cluster is scanned and all its sites marked as visited. In a
sweep through all valuesiandj, all clusters will be found in this way; note that the
computer time needed to scan a cluster in the back-track algorithm scales linearly
with the cluster volume (area).
Another algorithm for detecting all the clusters in the system is that of Hoshen
and Kopelman. This algorithm does not use recursion. It scales linearly with the
lattice size and it is more efficient than back-tracking (30–50%) but it is somewhat
more difficult to code. Details can be found in the literature [23].
The time scaling exponentzcan be determined from the simulations. Note that
the time correlation of the magnetisation is useless for this purpose as the clusters
are set to arbitrary spin values after each sweep, so that the magnetisation correlation
time is always of order 1. Therefore, we consider the time correlation function of
the (unsubtracted) susceptibility per site. This is defined as
χ=1
L^2 d〈(
∑
isi) 2 〉
. (15.84)
Its time correlation function is
Cχ(k)=∑N
∑n=^1 χn+kχn
N
n= 1 χn^2(15.85)
where the indicesnandkare ‘time’ indices, measured in MC steps per spin.
The susceptibility can be determined directly from the lattice configuration after
each step using(15.84), but it is possible to obtain a better estimate by realising
that when the system is divided up into clusterscof areaNc, the average value of
χis given by
χ=1
L^2 d〈(
∑
cNcsc