[tex]H = -J\sum_i S_iS_{i+1}[/tex]

Since each spin can only take either +1 or -1, we can write the transition matrix as

[tex]

T =

\left(

\begin{matrix}

e^{K} & e^{-K} \\

e^{-K} & e^{K}

\end{matrix}

\right)

[/tex]

where [tex]K=\beta J[/tex]

Now I try to learn 2D case, I read some book on it but seems quite complicated, so I started with the simplest case (no external field, only nearest interaction, rectangular lattics with only 2 rows and N columns and perodic boundary condition). The hamiltonian is

[tex]H = -J\sum_{i=1}^{2}\sum_{j=1}^N S_{ij}S_{i, j+1}[/tex]

right?

What I am really consfuing is how to find the transition matrix? Now each site has four nearest neighbor (of course, to avoid double counting, we only need to count two one at a time, let's say we count the one next to and below the current site) and each spin can take 2 values, so what's the dimension of the transition matrix? and what does [tex]T_{ij}[/tex] means?