Renormalization Group 641Therefore, the correlation lengthξ(K)∼ 1 /ln(tanhK)−→ ∞asT−→0, so that at
T= 0 the correlation length is infinite, which is another indication that anordered
phase exists.
Finally, we examine the behavior of the RG equation at very lowTwhereKis
large. Note that eqn. (16.9.15) can be written as
tanhK′= tanh^3 K= tanhKtanh^2 K= tanhK[
cosh(2K)− 1
cosh(2K) + 1]
. (16.9.25)
The term in brackets is very close to 1 whenKis large. Thus, whenKis large, eqn.
(16.9.25) can be expressed asK′∼K, which is a linearized version of the RG equation.
On an arbitrary spin lattice, interactions between blocks are predominantly mediated
by interactions between spins along the boundaries of the blocks (see Fig. 16.16 for
an illustration of this in two dimensions). In one dimension, this interaction involves
a single spin pair, and thus we expect a block spin transformation in one dimension
to yield a coupling constant of the same order as the original couplingconstant at low
Twhere there is significant alignment between the blocks.
16.10Fixed points of the RG equations in greater than one dimension
Fig. 16.16 Interactions between blocks of a square spin lattice.