Computational Physics

506 Computational methods for lattice field theories

N N'

n

n n'

n'

Figure 15.5. Two neighbouring blocks on a fine lattice with coarse lattice sitesN

andN′.

Hamiltonians which we must consider have the form:

H[ψ]=

1

2







∑

〈nn′〉

[

Jn,n′(ψn−ψn′)^2 +

∑

μ

Kn,n′(ψn−ψn′)

]

+

∑

n

[

Lnψn+Mnψ^2 +Tnψn^3 +Gnψn^4

]

}

. (15.100)

Restricting this Hamiltonian to a coarser grid leads to new values for the coupling
constants.
We denote the sites of the new grid byN,N′. Furthermore,

∑

nn′|NN′denotes a
sum over setsn,n′of neighbouring points, which belong to different neighbouring
blocks of four sites belonging to∑ NandN′respectively as inFigure 15.5. Finally,

n|Ndenotes a sum over the fine grid sitesnbelonging to the blockN. With this
notation, the new coupling constants on the coarse grid can be written in terms of
those on the fine grid:

JNN′=

∑

nn′|NN′

Jnn′; KNN′=

∑

nn′|NN′

[Knn′+ 2 Jnn′(ψn−ψn′)];

LN=

∑

n|N

(Ln+ 2 Mnψn+ 3 Tnψn^2 + 4 Gnψn^3 ); (15.101)

MN=

∑

n|N

(Mn+ 3 Tnψn+ 6 Gnψn^2 );

TN=

∑

n|N

(Tn+ 4 Gnψn); GN=

∑

n|N

Gn.

With this transformation, the MCMG method can be implemented straight-
forwardly. It can be shown that critical slowing down is completely eliminated
for Gaussian type actions, so it will work very well for theφ^4 theory close to the
Gaussian fixed point. However, theφ^4 theory has more than one critical point in two
dimensions. One of these points has Ising character: for this point, the coefficient