Computational Physics

15.5 Reducing critical slowing down 505

the neighbouring sites of the finer grids, after which we again perform a few MC
steps, and so on.
The algorithm reads, in recursive form:

ROUTINE MultiGridMC(l,ψHl)

Perform a few MC sweeps:ψ→ψ′;

IF (l>0) THEN

Calculate the form of the Hamiltonian

on the coarse grid:Hl− 1 [δφ]=Hl[ψ′+Pl,l− 1 (δφ)];

Setδφequal zero;

MultiGridMC(l−1,δφ,Hl− 1 );

ENDIF;

ψ′′=ψ′+Pl,l− 1 δφ;

Perform a few MC sweeps:ψ′′→ψ′′′;

END MultiGridMC.

The close relation to the multigrid algorithm for solving Poisson’s equation, given
in Appendix A7.2, is obvious.
The MC sweeps consist of a few Metropolis or heat-bath iterations on the fine grid
fieldψ. This step is ergodic as the heat bath and Metropolis update is ergodic. Note
that the coarse grid update in itself is not ergodic because of the restriction imposed
on fine grid changes (equal changes for groups of four spins) – the Metropolis or
heat-bath updates are essential for this property.
We should also check that the algorithm satisfies detailed balance. Again, the
Metropolis or heat-bath sweeps respect detailed balance. The detailed balance
requirement for the coarse grid update is checked recursively. A full MCMG
step satisfies detailed balance if the coarse grid update satisfies detailed balance.
But the coarse grid update satisfies detailed balance if the coarser grid update
satisfies detailed balance. This argument is repeated until we reach the coarsest
level (l = 1). But at this level we perform only a few MC sweeps, which
certainly satisfy detailed balance. Therefore, the full algorithm satisfies detailed
balance.
There is one step which needs to be worked out for each particular field theory:
constructing the coarse HamiltonianHl− 1 from the fine one,Hl. We do not know
a priori whether new interactions, not present in the fine Hamiltonian, will be
generated when constructing the coarse one. This often turns out to be the case. As
an example, consider the scalar interactingφ^4 field theory. The termsφ^2 andφ^4
generate linear and third powers inφwhen going to the coarser grid. Moreover,
the Gaussian coupling(φn−φn+μ)^2 generates a termφn−φn+μ. Therefore, the