Statistical Mechanics - Physics at Oregon State University

8.6. BETHE APPROXIMATION. 187
which leads to
In the limit q ≫ 1 we have log( q
e βJ (2 − q) = e −βJ (−q) (8.135)
q
2βJ = log( ) (8.136)
q − 2
kBTc = J
q−2
2
log( q
q−2 )
) ≈ log(1 + 2
q
(8.137)
) ≈ 2
q and hence kBTc ≈ qJ
like in mean field. The mean field results is best when the number of neighbors
is large, which is certainly true in high dimensional systems. As we saw in
Thermodynamics, mean field theory is exact for high dimensions.
The final, important, question is the following. When there is a solution
with a non-zero value of h ′ , does this solution have the lowest energy? Now we
need to address the question of finding f(h ′ ). The energy of the cluster is given
by
〈Ec〉 = −Jq〈σ0σj〉 − h(q + 1)m − h ′ m + f(h ′ ) (8.138)
and the correlation function 〈σ0σj〉 follows from
〈σ0σj〉 = 1
Zc
{σ0,···,σq}
Following the calculation for Sj we find that
σ0σje −βEc(σ0,···,σq) = S0j
S0j = e βh [2 cosh(β(J + h + h ′ ))] q−1 [2 sinh(β(J + h + h ′ ))]−
Zc
(8.139)
e −βh [2 cosh(β(−J + h + h ′ ))] q−1 [2 sinh(β(−J + h + h ′ ))] (8.140)
where the only difference with Sj is the minus sign in front of the second term.
So here we see another advantage of the Bethe approximation, we do have an
estimate for the correlation function.
The value of f(h ′ ) can be determined by considering a situation where all
spins are equal to m and are uncorrelated, in which case we know the energy.
This is too tedious, though. But if we think in terms of Landau theory and
consider h ′ to be a parameter in the energy expression, we can deduce that with
three solutions for h ′ the order has to be minimum-maximum-minimum, and
hence the h ′ = 0 solution, which is always the middle one, has to be a maximum.