1549380323-Statistical Mechanics Theory and Molecular Simulation

642 Critical phenomena

Fig. 16.16 shows a two-dimensional spin lattice and the interactions between two
blocks, which are mediated by the boundary spins. In more than onedimension, these
interactions are mediated by more than a single spin pair. For the case of the 3×3 blocks
shown in the figure, there are three boundary spin pairs mediating the interaction
between blocks. Consequently, the result of a block spin transformation should yield,
at lowT, a coupling constantK′roughly three times as large as the original coupling
constantK, i.e.,K′ ∼ 3 K. In a three-dimensional lattice, using 3× 3 ×3 blocks,
interactions between blocks would be mediated by 3^2 = 9 spin pairs. Generally, ind
dimensions using blocks ofbdspins, the RG equation at lowTshould behave as

K′∼bd−^1 K. (16.10.1)

The numberbis called thelength scaling factor. Eqn. (16.10.1) implies that ford >1,
K′> Kat lowT. Thus, iteration of the RG equation at low temperature yields an RG
flow towardsK=∞, and the fixed point atT= 0 becomes stable (in one dimension,
this fixed point was unstable). However, we know that at high temperature, the system
must be in a disordered state, and hence the fixed point atT=∞must remain a stable
fixed point, as it was in one dimension. These two observations suggest that ford >1,
there must be a third fixed point with coupling constant ̃xbetweenT= 0 andT=∞.
Moreover, an iteration initiated withx 0 = ̃x+ǫ(ǫ >0) must iterate tox= 1 where
T= 0, and an iteration initiated fromx 0 = ̃x−ǫmust iterate tox= 0 whereT=∞.
Hence, this fixed point is unstable and is, therefore, a critical pointwithK ̃ =Kc.
To the extent that an RG flow in more than one dimension can be represented as a
one-dimensional process, the flow diagram would appear as in Fig. 16.17. Since this

..

Stable Stable

x = 1 x = 0

K =^8 K = 0

.

K = Kc

.

K = K 0

Unstable

Fig. 16.17Renormalization group flow in more than one dimension. The figure shows the
iteration to each stable fixed point starting from the unstable fixed point and an arbitrary
pointK=K 0.

unstable fixed point corresponds to a finite, nonzero temperatureTc, it is a physical
critical point.
This claim is further supported by the evolution of the correlation length under
the RG flow. Recall that for a length scaling factorb, the correlation length transforms
asξ(K′) =ξ(K)/borξ(K) =bξ(K′). Suppose that we start at a pointKnearKc
and thatn(K) iterations of the RG equation are required to reach a valueK 0 between
K= 0 andK=Kc. Ifξ 0 is the correlation length atK=K 0 , which should be a finite
number of order 1, then by eqn. (16.9.22) we find that