Statistical Mechanics - Physics at Oregon State University

164 CHAPTER 8. MEAN FIELD THEORY: CRITICAL TEMPERATURE.
where the functions heff and H0 are independent of the eigenvalues of the
operators S. We need a criterion to find heff . A simple physical picture of the
state of a spin Si shows all neighboring spins having their average value m, and
hence heff = mqJ as we found in 8.31. The term H0 is determined by requiring
that the energy is correct if all spins have the value m, in other words it removes
the effects of double counting. The internal energy in linearized approximation
is
〈H lin 〉 = −mqJ〈
Si〉 + H0
〈H lin 〉 = −m 2 qJN + H0
i
(8.33)
(8.34)
and this needs to be equal to − 1
2 NqJm2 , which gives H0 as before.
One final remark. The basic form of mean field theory ignores correlations
between fluctuations on different sites. But we also know that fluctuations
are directly related to response functions. Therefore, we expect that response
functions in mean field theory will not be extremely accurate. Especially, we
will see errors in the results for critical exponents.
8.3 Mean Field results.
In order to proceed further we have to calculate the average spin m. We use the
mean-field Hamiltonian to do that and we find
with
m(T, h) = Tr Sie −β(Hmf −hM)
Z mf (T, h, N)
Z mf (T, h, N) = Tr e −β(Hmf −hM)
(8.35)
(8.36)
But this cannot be done directly. The formula for m contains H mf and
hence it depends on m. Therefore, m cannot be chosen arbitrarily, but has to
satisfy the two equations above. This is an implicit equation for m.
The partition function is easy to calculate in this approximation. We use
the basis of eigenstates and we find
e 1
2 βNqJm2
Z mf (T, h, N) =
σ1=±1
· · ·
σ1=±1
σN =±1
i
σi=±1
· · ·
σN =±1
e β(h+mqJ)σi =
i
e β(h+mqJ)σi
N
β(h+mqJ)
e i=1 σi =
=