In the last lecture, we noted that interactions between block of spins in a spin lattice are mediated by boundary spins. In one dimension, where there is only a single pair between blocks, the block spin transformation yields a coupling constant that is approximately equal to the old coupling constant at very low temperature, i.e., . Let us now explore the implications of this fact in higher dimensions.
Consider a two-dimensional spin lattice as we did in the previous lecture. Now interactions between blocks can be mediated by more than a single pair of spin interactions. For the case of 3 3 blocks, there will be 3 boundary spin pairs mediating the interaction between two blocks:
Since 3 boundary spin pair interactions mediate the block-block interaction, the result of a block spin transformation should yield, at low T, a coupling constant K' roughly three times as large as the original coupling constant, K:
In a three-dimensional lattice, using blocks, there would be spin pairs. Generally, in d dimensions using blocks containing spins, the RG equation at low T should behave as
The number b is called the length scaling factor. The above RG equation implies that for d;SPMgt;1, K';SPMgt;K for low T. Thus, iteration of the RG equation at low temperature should now flow toward and the fixed point at T=0 is now a stable fixed point. However, we know that at high temperature, the system must be in a paramagnetic state, so the fixed point at must remain a stable fixed point. These two facts suggest that, for d;SPMgt;1, between T=0 and , there must be another fixed point, which will be an unstable fixed point. To the extent that an RG flow in more than one dimension can be considered a one-dimensional flow, the flow diagram would look like:
Any perturbation to the left of the unstable fixed point iterates to T=0, and any perturbation to the right iterates to and K=0. The unstable fixed point corresponds to a finite, nonzero value of and a temperature , and corresponds to a critical point.
To see that this is so, consider the correlation length evolution of the correlation length under the RG flow. Recall that for a length scaling factor b, the correlation length transform as
Suppose we start at a point K near and require n(K) iterations of the RG equations to reach a value between K=0 and under the RG flow:
If is the correlation length at , which can expect to be a finite number of order 1, then, by the above transformation rule for the correlation length, we have
Now, recall that, as the starting point, K is chosen closer and closer to , the number of iterations needed to reach gets larger and larger. (Recall that near an unstable fixed point, the initial change in the coupling constant is small as the iteration proceeds). Of course, if initially, than an infinite number of iterations is needed. This tells us that as K approaches , the correlation length becomes infinite, which is what is expected for an ordered phase. Thus, the new unstable fixed point must correspond to a critical point.
In fact, we can calculate the exponent knowing the behavior of the RG equation near the unstable fixed point. Since this is a fixed point, satisfies, quite generally,
Near the fixed point, we can expand the RG equation, giving
Define an exponent y by
Near the critical point, diverges according to
which is only possible if
This result illustrates a more general one, namely, that critical exponents are related to derivatives of the RG transformation.