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