1549380323-Statistical Mechanics Theory and Molecular Simulation

Renormalization Group 635

Fig. 16.13Example of the block spin transformation on a 6×6 square lattice. The lattice
on the right shows the four spins that result from applying the transformation to each 3× 3
block.

The proceeding discussion of the RG will be based loosely on the treatment given by
Cardy (1996).
In order to illustrate the RG procedure, let us consider the exampleof a square spin
lattice shown in Fig. 16.13. In the left half of the figure, the lattice is separated into 3× 3
blocks. We now consider defining a new spin lattice from the old by applying a coarse-
graining procedure that replaces each 3×3 spin block with a single spin. Of course,
we need a rule for constructing this new spin lattice, so let us consider the following
simple algorithm: (1) count the number of up and down spins in each block; (2) if the
majority of the spins in the 3×3 block are up, replace the block by a single up spin,
otherwise replace it by a single down spin. For the example on the left inFig. 16.13,
the new lattice obtained by applying this procedure is shown on the right in the figure.
Such a transformation is called ablock spin transformation(Kadanoff, 1966). Near a
critical point, the system will exhibit long-range ordering, hence the coarse-graining
procedure should yield a new spin lattice that is statistically equivalentto the old one;
the spin lattice is then said to possessscale invariance.
Given the new spin lattice generated by the block spin transformation, we now wish
to determine the Hamiltonian of this lattice. Since the new lattice mustbe statistically
equivalent to the original one, the natural route to the transformed Hamiltonian is
through the partition function. Thus, we consider the zero-field (h= 0) partition
function of the original spin lattice using the HamiltonianH 0 in eqn. (16.3.4) for the
Ising model as the starting point:

Q(N,T) =

∑

σ 1

···

∑

σN

e−βH^0 (σ^1 ,...,σN)≡Trσe−βH^0 (σ^1 ,...,σN). (16.9.1)

The transformation functionT(σ′;σ 1 ,...,σ 9 ) that yields the single spinσ′for each 3× 3
block of 9 spin variables can be expressed mathematically as follows: