Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
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8.6. BETHE APPROXIMATION. 181<br />
kBTc = 2J<br />
log(b)<br />
The mean field results for a square l<strong>at</strong>tice is kBT mf<br />
c<br />
(8.107)<br />
= 4J; if b > 1.6 the<br />
Landau estim<strong>at</strong>e of Tc is below the mean-field result. The exact result for a<br />
two-dimensional square Ising l<strong>at</strong>tice is kBTc = 2.27J, or b ≈ 2.4. This is a very<br />
reasonable value for b considering our discussion on the rel<strong>at</strong>ion between the<br />
parameter b and choices of continuing a Bloch wall. The largest value of b we<br />
can have is three, which would give kBTc = 1.8J, and this gives a lower limit<br />
on calcul<strong>at</strong>ions for Tc.<br />
8.6 Bethe approxim<strong>at</strong>ion.<br />
The next question is: how can we improve the mean-field results. In the mean<br />
field approxim<strong>at</strong>ion we do not have any inform<strong>at</strong>ion on the collective behavior<br />
of two neighboring spins. We need th<strong>at</strong> inform<strong>at</strong>ion, and want to find ways to<br />
obtain th<strong>at</strong> inform<strong>at</strong>ion. One possibility is via the density m<strong>at</strong>rices. If we introduce<br />
some correl<strong>at</strong>ion in the approxim<strong>at</strong>e density m<strong>at</strong>rix, we will obtain better<br />
results. M<strong>at</strong>hem<strong>at</strong>ically, this is a rigorous procedure, and can be performed<br />
easily on a computer.<br />
A simple model of introducing correl<strong>at</strong>ions is the Bethe approxim<strong>at</strong>ion,<br />
which does not introduce density m<strong>at</strong>rices explicitly. The basic idea of the<br />
Bethe approxim<strong>at</strong>ion is the following. Consider a cluster of <strong>at</strong>oms. The interactions<br />
between spins in this cluster are taken into account exactly, but the<br />
interactions with the environment are tre<strong>at</strong>ed in a mean field manner.<br />
The basic philosophy of the mean field approxim<strong>at</strong>ion was the following. The<br />
energy <strong>at</strong> a given site depends on the effects of the neighboring sites through<br />
some averaged additional magnetic field:<br />
Emf (σ0) = −(h + h ′ )σ0 + f(h ′ ) (8.108)<br />
where h is the regular magnetic field, and h ′ is the additional field. The term<br />
f(h ′ ) has to be introduced to avoid double counting of bonds. At the central<br />
site all bonds to neighbors are counted, and if we multiply by N th<strong>at</strong> means<br />
th<strong>at</strong> we would count the effect of each bond twice. We determine h ′ by requiring<br />
transl<strong>at</strong>ional symmetry, 〈σ0〉 = m, which leads to m = tanh(β(h + h ′ )). We<br />
need an additional condition to rel<strong>at</strong>e h ′ and m and make the obvious choice<br />
h ′ = qJm.<br />
How can we improve the mean field result? If we want to describe bonds<br />
with neighbors exactly, we need to consider a cluster of <strong>at</strong>oms. All sites in the<br />
cluster are tre<strong>at</strong>ed exactly. There is still an outside of the cluster, and th<strong>at</strong><br />
outside will be represented by some effective field. In this way we do describe<br />
correl<strong>at</strong>ions between spins in the cluster, and obtain more inform<strong>at</strong>ion th<strong>at</strong> we<br />
had before. But since the outside is still represented by an average field, we<br />
expect th<strong>at</strong> some elements of the mean field theory might still survive. We do