Statistical Mechanics - Physics at Oregon State University

200 CHAPTER 9. GENERAL METHODS: CRITICAL EXPONENTS.
approximation, the answers in the Bethe approximation and in mean-field theory
will be different. If, on the other hand, the correlation length ξ is much larger
than the size of the cluster, the size of the cluster is not important and the
problem is essentially solved in a mean-field approach. Since ξ diverges near
Tc, the Bethe approximation is equivalent to a mean field theory in this limit.
Therefore, the Bethe approximation will give the same critical exponents as
mean-field theory. In order to make a real improvement, we need to take into
account correlations of infinite range.
We can improve the analysis, however. We will use the susceptibility as
an example, but the analysis woks for all critical exponents. Near the critical
temperature we have χ ≈ A±|T −Tc| γ , plus higher order terms. This means that
near the critical temperature we have log(χ) =≈ log(A±) + γ log |T − Tc|. The
second term will dominate if we are close enough to the critical temperature.
Hence we have
log(χ)
log |T − Tc|
→ γ (9.57)
near the critical temperature. Hence if we plot the ratio on the left as a function
of temperature, we should be able to extract the value of γ.
Very simply stated, so what can be wrong. First of all, we assume that we
know the value of the critical temperature. That might be true or not. What
happens if we are wrong? If we do not use the exact value of the critical temperature
a plot of the ratio of the logarithms will show two peaks. The ratio
will go to zero when the denominator goes to infinity, or when the temperature
reaches the erroneous value of the critical temperature. The ratio will go to
infinity when the enumerator diverges, at the true value of the critical temperature.
For example, see figure 9.1 where we have used χ = 1
|t−1| (1 + 0.1|t − 1|)
and moved Tc by a small amount to Tc = 1.01. We can use this behavior to find
the critical temperature. If we have sufficient data near the phase transition we
try different values of Tc and see when the peak and zero merge.
But what is near the critical temperature? One could guess that the correction
in χ is a factor of the form 1 + B±|T − Tc|. This gives an extra term in
the denominator equal to the logarithm of this expression, and near the critical
temperature we now find
log(χ)
log |T − Tc|
|T − Tc|
→ γ + B±
log |T − Tc|
(9.58)
The second term goes to zero, indeed. But how fast? We do not know. See figure
9.2 for a typical case. We used the same form as before, but now with the correct
value of the critical temperature. In addition, the previous argument assumes
that the behavior near the critical temperature is given by powers only. There
are theoretical results where near the critical temperature we have logarithmic
factors, χ ∝ |T − Tc| γ log |T − Tc|. Because the logarithm goes to infinity slower
than any inverse power, such a factor does not change the value of the critical
exponent, but it certainly changes the behavior near the phase transition. We