Fixed points of the RG equations in greater than one dimension next up previous
Next: General linearized RG theory Up: No Title Previous: No Title

Fixed points of the RG equations in greater than one dimension

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., tex2html_wrap_inline556 . 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 tex2html_wrap_inline558 3 blocks, there will be 3 boundary spin pairs mediating the interaction between two blocks:

   figure90
Figure 1:

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:

displaymath98

In a three-dimensional lattice, using tex2html_wrap_inline566 blocks, there would be tex2html_wrap_inline568 spin pairs. Generally, in d dimensions using blocks containing tex2html_wrap_inline572 spins, the RG equation at low T should behave as

displaymath100

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 tex2html_wrap_inline584 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 tex2html_wrap_inline588 must remain a stable fixed point. These two facts suggest that, for d;SPMgt;1, between T=0 and tex2html_wrap_inline588 , 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:

   figure105
Figure 2:

Any perturbation to the left of the unstable fixed point iterates to T=0, tex2html_wrap_inline584 and any perturbation to the right iterates to tex2html_wrap_inline588 and K=0. The unstable fixed point corresponds to a finite, nonzero value of tex2html_wrap_inline604 and a temperature tex2html_wrap_inline606 , 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

eqnarray113

Suppose we start at a point K near tex2html_wrap_inline612 and require n(K) iterations of the RG equations to reach a value tex2html_wrap_inline616 between K=0 and tex2html_wrap_inline604 under the RG flow:

   figure117
Figure 3:

If tex2html_wrap_inline622 is the correlation length at tex2html_wrap_inline624 , which can expect to be a finite number of order 1, then, by the above transformation rule for the correlation length, we have

displaymath125

Now, recall that, as the starting point, K is chosen closer and closer to tex2html_wrap_inline612 , the number of iterations needed to reach tex2html_wrap_inline616 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 tex2html_wrap_inline604 initially, than an infinite number of iterations is needed. This tells us that as K approaches tex2html_wrap_inline612 , 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 tex2html_wrap_inline638 knowing the behavior of the RG equation near the unstable fixed point. Since this is a fixed point, tex2html_wrap_inline612 satisfies, quite generally,

displaymath128

Near the fixed point, we can expand the RG equation, giving

displaymath130

Define an exponent y by

displaymath132

so that

displaymath135

Near the critical point, tex2html_wrap_inline644 diverges according to

displaymath137

Thus,

displaymath147

but

displaymath150

which implies

eqnarray152

which is only possible if

displaymath157

This result illustrates a more general one, namely, that critical exponents are related to derivatives of the RG transformation.


next up previous
Next: General linearized RG theory Up: No Title Previous: No Title

Mark Tuckerman
Tue May 4 23:40:04 EDT 1999