504 Computational methods for lattice field theories
The geometric cluster idea has also been used for spin systems formulated on
a lattice [37]. Again, a reflection site is identified at random. Next, for a ran-
domly chosen sitei, the spin is exchanged with that of its reflection partneri′.
Then each neighbourkofiis investigated. If exchanging the spins atkandk′
results in an energy gain(that is, the total energy decreases), the move is
accepted with probability exp(−); if this is not the case,kis left unaltered.
Then the algorithm proceeds with the neighbours ofkjust as in Wolff’s cluster
algorithm. We see that spins are only exchanged in this algorithm, so that the
total spin remains constant: the energy change no longer scales with the cluster
volume.
15.5.4 The multigrid Monte Carlo methodThe multigrid Monte Carlo (MGMC) method [9,38,39] is yet another way of redu-
cing critical slowing down near the critical point. This method is closely related
to the multigrid method for solving partial differential equations described in
Appendix A7.2, and readers not familiar with this method should go through that
section first; see also Problem A7.
Multigrid ideas can be used to devise a new Monte Carlo algorithm which reduces
critical slowing down by moving to coarser and coarser grids and updating these in
an MC procedure with a restricted form of the Hamiltonian.
To be specific, let us start from a grid at levell; a field configuration on this grid
is calledψ. The Hamiltonian on this grid is calledHl[ψ]. The coarse grid is the
grid at levell−1, and configurations on this coarse grid are denoted byφ.Now
consider the prolongation operationPl,l− 1 described inAppendix A7.2, which maps
a configurationφon the coarse grid to a configurationψon the fine grid by copying
the value ofφon the coarse grid to its four nearest neighbours on the fine grid:
Pl,l− 1 :φ→ψ; (15.98a)ψ( 2 i+μ,2j+ν)=φ(i,j), (15.98b)whereμandνare±1. We now consider a restricted HamiltonianHl− 1 [δφ],
which is a function of the coarse grid configurationδφ, depending on the fine
grid configurationψwhich is kept fixed:
Hl− 1 [δφ]=Hl[ψ+Pl,l− 1 (δφ)]. (15.99)We perform a few MC iterations on this restricted Hamiltonian and then we go to
the coarser grid at levell−2. This process is continued until the lattice consists of
a single site, and then we go back by copying the fields on the coarser grid sites to