statistical mechanics - Hamiltonian of a simple graph

I have a spin system:

As shown in the picture, there are two spins S1 and S2, and a pair of interactions between them. One is a ferromagnetic interaction and the other is anti ferromagnetic interaction. I am trying to calculate the Hamiltonian of this system.

The Hamiltonian of the system is:

$$ H = -J_F S1_z S2_z +J_{AF} S1_z S2_z $$

$S1_z$ is the spin matrix for Z direction for spin 1 and $S2_z$ is the spin matrix for Z direction for spin 2. If we allow two random values for $J_F$ and $J_{AF}$, -0.5 and 0.5 respectively the Hamiltonian of the system is as follows.

$$ H = 0.5 S1_z S2_z + 0.5 S1_z S2_z $$ $$ = S1_z S2_z $$ $$ = \begin{pmatrix} 1 & 0 \\ 0 & -1 \\ \end{pmatrix} \times \begin{pmatrix} 1 & 0 \\ 0 & -1 \\ \end{pmatrix} $$ $$ = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix} $$

Am I able to calculate the Hamiltonian correctly?

Answer

The Hamiltonian of this system lives in a 4-dimensional Hilbert space since you have two spin $1/2$. Therefore, you should represent the spin matrix in this four dimensional space like this:

$S_1^z=\begin{pmatrix} -0.5 & 0 &0 &0 \\ 0&-0.5 &0 &0 \\ 0 &0 &0.5 &0 \\ 0 &0 &0 &0.5 \end{pmatrix}$ , $S_2^z=\begin{pmatrix} -0.5 & 0 &0 &0 \\ 0&0.5 &0 &0 \\ 0 &0 &-0.5 &0 \\ 0 &0 &0 &0.5 \end{pmatrix}$

The order of the four states along the rows and columns is $|DD\rangle,|DU\rangle, |UD\rangle, |UU\rangle$ where $U$ stands for spin up and $D$ stands for spin down.

In this case $S_1^z.S_2^z=\begin{pmatrix} 0.25 & 0 &0 &0 \\ 0&-0.25 &0 &0 \\ 0 &0 &-0.25 &0 \\ 0 &0 &0 &0.25 \end{pmatrix}$