Editing Bethe lattice (section)

== In statistical mechanics ==
The Bethe lattice is of interest in statistical mechanics mainly because lattice models on the Bethe lattice are often easier to solve than on other lattices, such as the [[Square lattice|two-dimensional square lattice]]. This is because the lack of cycles removes some of the more complicated interactions. While the Bethe lattice does not as closely approximate the interactions in physical materials as other lattices, it can still provide useful insight.

=== Exact solutions to the Ising model ===
The [[Ising model]] is a mathematical model of [[ferromagnetism]], in which the magnetic properties of a material are represented by a "spin" at each node in the lattice, which is either +1 or -1. The model is also equipped with a constant <math>K</math> representing the strength of the interaction between adjacent nodes, and a constant <math>h</math> representing an external magnetic field.

The Ising model on the Bethe lattice is defined by the partition function

:<math>Z=\sum_{\{\sigma\}}\exp\left(K\sum_{(i,j)}\sigma_i\sigma_j + h\sum_i \sigma_i\right).</math>

==== Magnetization ====
In order to compute the local magnetization, we can break the lattice up into several identical parts by removing a vertex. This gives us a recurrence relation which allows us to compute the magnetization of a Cayley tree with ''n'' shells (the finite analog to the Bethe lattice) as

:<math>M=\frac{e^h-e^{-h}x_n^q}{e^h+e^{-h}x_n^q},</math>

where <math>x_0=1</math> and the values of <math>x_i</math> satisfy the recurrence relation

:<math>x_n=\frac{e^{-K+h}+e^{K-h}x_{n-1}^{q-1}}{e^{K+h}+e^{-K-h}x_{n-1}^{q-1}}</math>

In the <math>K>0</math> case when the system is ferromagnetic, the above sequence converges, so we may take the limit to evaluate the magnetization on the Bethe lattice. We get

:<math>M=\frac{e^{2h}-x^q}{e^{2h}+x_q},</math> where ''x'' is a solution to <math>x=\frac{e^{-K+h}+e^{K-h}x^{q-1}}{e^{K+h}+e^{-K-h}x^{q-1}}</math>.

There are either 1 or 3 solutions to this equation. In the case where there are 3, the sequence <math>x_n</math> will converge to the smallest when <math>h>0</math> and the largest when <math>h<0</math>.

==== Free energy ====
The free energy ''f'' at each site of the lattice in the Ising Model is given by

:<math>\frac{f}{kT}=\frac12[-Kq-q\ln(1-z^2)+\ln(z^2+1-z(x+1/x))+(q-2)\ln(x+1/x-2z)]</math>,

where <math>z=\exp(-2K)</math> and <math>x</math> is as before.<ref>{{cite book | first=Rodney J. | last=Baxter | authorlink=Rodney J. Baxter | title=Exactly solved models in statistical mechanics | publisher=Academic Press | year=1982 | isbn=0-12-083182-1 | zbl= 0538.60093 }}</ref>