Computational Physics

15.5 Reducing critical slowing down 499

u u

s

s'

s

s'

Figure 15.2. Spin flips in the Wolff algorithm for theO( 3 )model.

formulated as an embedding of an Ising model into anO(N)model [28]. First a
random unit vectoruis chosen. Every spinsiis then split into two components: the
component alonguand that perpendicular tou:

s‖i=(si·u)u (15.91a)

s⊥i =si−s‖i. (15.91b)

We keeps⊥i and|s‖i|fixed. The only freedom left for theO(N)spins is to flip their
parallel component:
si=s⊥i +i|s‖i|u, i=±1. (15.92)

A flip in the signi corresponds to a reflection with respect to the hyperplane
perpendicular tou(seeFigure 15.2). The interaction of the model with the restriction
on the fluctuations that only flips of the parallel components are allowed, can now
be described entirely in terms of thei:

H[i]=

∑

〈ij〉

Jijij (15.93a)

Jij=J|s‖i||s‖j|. (15.93b)

This Ising Hamiltonian is now simulated using the single cluster or the SW
algorithm. After choosing the unit vectoru, we calculate theifor the actual ori-
entations of the spins and then we allow for reflections of thesi(that is, for spin
flips in theisystem).
This method is more efficient than the standard single spin-update method
because large clusters of spins are flipped at the same time. But why is the accept-
ance rate for such a cluster update not exceedingly small? The point is that the
amount by which a spin changes, depends on its orientation (see Figure 15.2): for
a spin more or less perpendicular tou, the change in orientation is small. This
translates itself into the couplingJijbeing small for spinssi,sjnearly perpendicular
tou. For spins parallel tou, the coupling constantJijis large and these spins will
almost certainly be frozen to the same cluster. The cluster boundaries will be the