Statistical Mechanics - Physics at Oregon State University

224 CHAPTER 9. GENERAL METHODS: CRITICAL EXPONENTS.
Note that we cannot go any further in renormalization than to have one
spin left. If at this point values are converged, fine, but if not, we see finite
size effects. Only in the thermodynamic limit do we get a sharp transition
temperature independent of the initial conditions. For a finite sample we always
find a small range of fixed points.
The application of renormalization theory to a one-dimensional problem is
often straightforward because a one-dimensional space has a very simple topology.
In more dimensions one has to deal with other complications. For example,
we could apply the same set of ideas to a two-dimensional Ising model. The sites
are numbered (i, j) and the easiest geometry is that of a square lattice where
the actual lattice points are given by R = a(iˆx + j ˆy). We could try to sum first
over all sites with i + j odd. The remaining sites with i + j even form again a
square lattice, but with lattice constant a √ 2. Suppose we start with a model
with only nearest-neighbor interactions. It is easy to see that after one step in
this renormalization procedure we have a system with nearest and next-nearest
neighbor interactions! As a result we have to consider the Ising model with
all interactions included and in the end find fixed points which correspond to
nearest neighbor interactions only. Real life is not always as easy as the current
section seemed to suggest. But if we do it correctly, we will find that the twodimensional
Ising model has a non-trivial fixed point even for h = 0. The value
˜Jfp is related to Tc by ˜ Jfp = J
kBTc .
If renormalization theory would only give critical temperatures, its value
would be quite small. The most important aspect of this theory, however, is
that it also yields critical exponents. Suppose that ˜ J ∗ and ˜ h ∗ are the values of
the coupling parameters at a critical point. The critical exponents are related
to the behavior of the renormalization group equations near the critical point.
Assume that we are close to the critical point and that we have ˜ J = ˜ J ∗ + δ ˜ J,
˜h = ˜ h ∗ + δ ˜ h with δ ˜ J and δ ˜ h small. In first approximation we use a linearized
form, valid for small deviations, and we have
and
δ ˜ J ′
∂
=
˜ J ′
∂ ˜
δ
J
˜
∂
J +
˜ J ′
∂˜
δ
h
˜ h (9.172)
δ˜ h ′
∂
=
˜ h ′
∂ ˜
δ
J
˜
∂
J +
˜ h ′
∂˜
δ
h
˜ h (9.173)
where the partial derivatives are calculated at the fixed point.
The deviations in the coupling constants from the fixed point are combined in
a two-vector d. If we have more coupling constants, we have a larger dimensional
space, so it is easy to generalize to include a larger number of coupling constants.
Hence near a critical point we have
d ′ = M d (9.174)