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Model

Hamiltonian and Partition Function

We have N spins $\sigma_i$ , i=1, 2,... on a lattice with coordinate number z. Each spin can be either up or down:

$\begin{displaymath} \sigma_i = \begin{cases} 1,& \text{``up''}\ -1,& \text{``down''} \end{cases}\end{displaymath}$

$\begin{figure} \includegraphics{ising}\end{figure}$

1.

Each spin interacts with external field H:

$\begin{displaymath} E_1 = -H\sum_i \sigma_i \end{displaymath}$

2.

Each pair of neighbors interacts:

This can be written as $-J\sigma_1\sigma_2$ . The total interaction:

$\begin{displaymath} E_0 = -\frac{J}{2}\sum_{(i,j)}\sigma_i\sigma_j \end{displaymath}$

Here (i,j) means sum over all pairs of nearest neighbors

Result:

$\begin{displaymath} E = -H\sum_i \sigma_i -\frac{J}{2}\sum_{(i,j)}\sigma_i\sigma_j\end{displaymath}$

(1)

Microscopic state of the system--combination of all spins:

$\begin{displaymath} \{\sigma\} = \{\sigma_1,\sigma_2,\dots\}\end{displaymath}$

Partition function:

$\begin{displaymath} Z_N = \sum_{\{\sigma\}} \exp\bigl(-\beta E(\{\sigma\})\bigr)\end{displaymath}$

We don't have momenta here--this is actually a configuration integral rather than partition function! Free energy:

$\begin{displaymath} A = -kT\ln Z_N\end{displaymath}$

Average magnetization (H is conjugated to $N\sigma$ !)

$\begin{displaymath} M(H,T) = \left\langle \sigma\right\rangle = -\frac{1}{N}\frac{\partial A(H,T) }{\partial H}\end{displaymath}$

Susceptibility:

$\begin{displaymath} \chi = \frac{\partial M(H,T) }{\partial H} = -\frac{1}{N}\frac{\partial^2 A(H,T) }{\partial H^2} \ge 0\end{displaymath}$

Energy:

$\begin{displaymath} E = A - T \frac{\partial A(H,T) }{\partial T} = -T^2 \frac{\partial A(H,T)/T }{\partial T} \end{displaymath}$

Specific heat:

$\begin{displaymath} C = \frac{\partial E(H,T) }{\partial T}\end{displaymath}$