Abstract
The zero-field Ising Hamiltonian
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- 1.
For large \(n\)
$$\begin{aligned} \ln (n!)=\left( n+{\textstyle \frac{1}{2}}\right) \ln (n)-n+{\textstyle \frac{1}{2}}\ln (2\pi )+\mathrm {O}(1/n). \end{aligned}$$ - 2.
In an accurate treatment, which must include the effect of local fluctuations, the susceptibility divergence is sharper.
- 3.
We are interested is in deriving the part of the phase diagram in the plane
shown in Fig. 4.3a, b. - 4.
Of course, with this layered model on a \(d\)-dimensional hypercubic lattice
, 
, but 
and 
are used in the interests of generality. - 5.
For the simple cubic lattice
, 
. - 6.
Here we encounter the usual problem in statistical mechanics that, whereas the equations take a simpler form in terms of couplings
and 
, physical insight is helped by presenting figures in terms of the fields \(T\) and 
. We shall, therefore, in the discussion make free use of both sets of variables. - 7.
With
, and remembering that in Sect. 6.1
\(:=J/T\) not 
. - 8.
As indicated in Sect. 5.5, for the exponents we simply omit the ‘\(\mathrm {p}\)’.
- 9.
The case shown here is for \(\lambda =1\), \(T_{\mathrm {t}}>0\); the differences that occur when \(\lambda <\frac{1}{3}\), \(T_{\mathrm {t}}<0\) are discussed below.
- 10.
There is also a mirror-image transition line in the
half-plane, but we need consider only the 
line explicitly.
shown in Fig. 
, 
, but 
and 
are used in the interests of generality.
, 
.
and 
, physical insight is helped by presenting figures in terms of the fields 
. We shall, therefore, in the discussion make free use of both sets of variables.
, and remembering that in Sect. 
.
half-plane, but we need consider only the 
line explicitly.Author information
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Lavis, D.A. (2015). Mean-Field Theory. In: Equilibrium Statistical Mechanics of Lattice Models. Theoretical and Mathematical Physics. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9430-5_6
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DOI: https://doi.org/10.1007/978-94-017-9430-5_6
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