Statistical Mechanics - Physics at Oregon State University

218 CHAPTER 9. GENERAL METHODS: CRITICAL EXPONENTS.
9.7 Renormalization group theory.
A very powerful way of treating critical phenomena is via renormalization group
theory. The basic idea behind this theory is very simple. Near a critical point
the dominant length scale for any system is the correlation length ξ. Exactly at
the critical temperature this correlation length diverges and somehow a picture
of a given state of a system should be independent of any length scale. In
other words, if we paint spin-up atoms red and spin-down atoms blue, in a
magnetic system the pattern we observe should be independent of any length
scale. If we look at the pattern of spins with atomic resolution we will see the
individual red and blue dots. If we diminish the resolution we will start to see
average colors, obtained by mixing the effects of a few dots. If we decrease
the resolution even more, the average color will depend on the effects of more
dots in a larger region. Imagine a whole series of pictures taken this way, with
decreasing resolution. The pictures with atomic resolution will look somewhat
different, since we see the individual dots. All the other pictures should look
very similar, however, since the divergence of the correlation length has taken
away our measure of length. There is no way to distinguish the pictures by their
pattern and deduce the magnification from the observed pattern. Notice that
in real life this fails since samples are of finite dimensions and at some point
we will see the boundaries of the samples. Theorists do not bother with these
trivial details.
The previous paragraph sketches the concepts of renormalization group theory.
Essentially we apply scaling theory but now start at the microscopic level.
We build the macroscopic theory up from the ground, by averaging over larger
and larger blocks. The next step is to formulate this idea in mathematical terms.
We need to define some kind of transformation which corresponds to decreasing
the magnetization. The easiest way is to define a procedure that tells us how to
average over all variables in a certain volume. The one-dimensional Ising model
will again serve as an example. The atomic sites are labelled with an index i.
Suppose we want to decrease the magnification by an integral factor p. Hence
we define a new index j which also takes all integer values and label the atoms
in group j by i = pj + k , with k = 1, 2, · · · , p. The spin in cell j can be defined
in several ways. For example, we could assign it the average value or the value
of the left-most element of the cell. In the end, our result should be independent
of this detail, and we choose the procedure that is easiest to treat.
This procedure defines a transformation Rp in the set of states that can
be characterized by {· · · , σ−1, σ0, σ1, · · ·}. It is not a one-to-one mapping and
hence the word group is a misnomer, the inverse of Rp does not exist. This
transformation is also applied to the Hamiltonian and the requirement is that
at a critical point the Hamiltonian is invariant under the operations of the
renormalization group.
These ideas are best illustrated by using the one-dimensional Ising ring as
an example, even though this model does not show a phase transition. It is the
easiest one for actual calculations, and it will serve to get the ideas across. The
partition function is given by