Statistical Mechanics - Physics at Oregon State University

226 CHAPTER 9. GENERAL METHODS: CRITICAL EXPONENTS.
constant. The deviation δ ˜ J ′ corresponds to a different temperature T ′ according
to δ ˜ J = α(T ′ − Tc) and we have
(T ′ − Tc) = λ(T − Tc) (9.182)
On the other hand the correlation length depends on the temperature via
ξ(T ) = c(T − Tc) −ν . By construction the primed system corresponds to a rescaled
system with correlation length ξ
f . The scaling factor f is equal to two in
our example above. Hence we also have
combining these two results gives
(T ′ − Tc) −ν −ν 1
= (T − Tc)
f
(T − Tc) −ν λ −ν −ν 1
= (T − Tc)
f
(9.183)
(9.184)
or λ ν = f. Hence if we calculate the matrix M at the fixed point the
eigenvalues are related to critical exponents. We could also vary h to get similar
results. The one dimensional Ising model is again too simple. The fixed points
are ˜ J ∗ = 0 or ∞ with a value of λ = 0 or 1. If λ approaches zero from
above the value of ν approaches zero. This means that the correlation length is
independent of the temperature. In other words, the correlation length is not
important and can be taken to be zero for all practical purposes, as expected.On
the other hand, for the other fixed point λ approaches one from above, and this
means that ν goes to infinity. The correlation length increases faster than a
power law, which is what we already found from the exact results.
9.8 Problems for chapter 9
Problem 1.
Consider the one-dimensional Ising model in a cluster approximation. A
cluster contains a central atom 0 and the neighboring sites j = ±1.
A. Calculate < σ0σj > in this approach.
B. Assume that there is an additional potential in this model of the form
V = λ
j=±1 σ0σj. Show that we indeed have
∂G
∂λ =< V >λ.
Problem 2.
Suppose the earth magnetic field is due to a linear chain of atoms with
spin one-half polarized along the chain. Take J = 1 eV and N = 10 16 . Use
the results for the one-dimensional Ising model to estimate the time between
spontaneous reversals of the earth magnetic field.