One-Dimensional Ising Model in Bethe Approximation

[MathematicalPhysicist]

Homework Statement

The following question and its solution is from Bergersen's and Plischke's:

Calculate the magnetization for the one-dimensional Ising model in
a magnetic field in the Bethe approximation and compare with the exact
result (3.38).

Equation (3.38) is:
$$m = \frac{\sinh (\beta h)}{\sqrt{\sinh^2(\beta h) + e^{-4\beta J}}}$$

Homework Equations

The Attempt at a Solution

They provide the solution in their solution manual which I don't understand how did they come to it.

Let us define: $x=e^{\beta h}; \ y=e^{2\beta J} ; \ z=e^{2\beta h'}$.

We have:
$$(3.2) \ \ Z_C = xyz+xy^{-1}z-1+2x+x^{-1}y^{-1}z+x^{-1}yz^{-1}+2x^{-1}$$
$$(3.3) \ \ \langle \sigma_0 \rangle =\frac{1}{Z_C} [xyz+xy^{-1}z-1+2x-x^{-1}y^{-1}z-x^{-1}yz^{-1}-2x^{-1}]$$
$$\langle \sigma_1 \rangle = \frac{1}{Z_C} [ xyz-xy^{-1}z^{-1}+x^{-1}y^{-1}z-x^{-1}yz^{-1}]$$

I hope I don't have a typo.

After that they are equating between $\sigma_0$ and $\sigma_1$ and solve for $z$,

My problem is how to derive the above three equations, (3.2) and (3.3), I'm lost.[/quote]

 

[MathematicalPhysicist]

Anyone?

 

[Fred Wright]

I don't have a copy of your text but the Bethe approximation considers a cluster of a central spin $\sigma_0$ in an external field h, and two surrounding spins (in the 1-s Ising model), in an effective field h', which mimics the remaining crystalline lattice. The spin Hamiltonian of this cluster is:$$\mathcal H_c=\mathcal J\sigma_0 \left (\sigma_1 + \sigma_2 \right ) - h\sigma_0 -h'\left (\sigma_1 + \sigma_2 \right )$$
and the partition function of this cluster is:$$\mathcal Z_c=\sum_{ \sigma_i } exp\left (-\beta \mathcal H_c \right )$$
The spins in the sum can take the values +1 or -1. Using the notation from your book construct the following table:
$\sigma_0$ $\sigma_1$ $\sigma_2$ $\mathcal Z_c$
$1$ $1$ $1$ $xyz$
$1$ $-1$ $1$ $x$
$1$ $1$ $-1$ $x$
$1$ $-1$ $-1$ $xy^{-1}z^{-1}$
$-1$ $1$ $1$ $x^{-1}y^{-1}z$
$-1$ $-1$ $1$ $x^{-1}$
$-1$ $1$ $-1$ $x^{-1}$
$-1$ $-1$ $-1$ $x^{-1}yz^{-1}$
Having found $\mathcal Z_c$ I assume you can calculate:
$$ < \sigma_0 >=\frac {1} {\beta}\frac {\partial ln \mathcal Z_c} {\partial h}$$
and
$$ < \sigma_1 >=\frac {1} {\beta}\frac {\partial ln \mathcal Z_c} {\partial h'}$$