# Homework Help: One-Dimensional Ising Model in Bethe Approximation

1. Nov 2, 2017

### MathematicalPhysicist

1. The problem statement, all variables and given/known data
The following question and its solution is from Bergersen's and Plischke's:
Equation (3.38) is:
$$m = \frac{\sinh (\beta h)}{\sqrt{\sinh^2(\beta h) + e^{-4\beta J}}}$$

2. Relevant equations

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

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]

Last edited: Nov 2, 2017
2. Nov 6, 2017

Anyone?

3. Nov 8, 2017

### 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
$$< \sigma_1 >=\frac {1} {\beta}\frac {\partial ln \mathcal Z_c} {\partial h'}$$