The s-f Model is a model who could describe the $\textbf{magnetic 4 $\textit{f}$ systems}$, i.e systems where we could identify localized electrons in $4\,\textit{f}$ orbitals and conductions electrons
The interaction between conduction electrons and $4\,\textit{f}$ electrons,could be described as an intra-atomic exchange interaction between the spins $\hat{{\sigma}}$ of the conduction electrons and the spin $\hat{S}$ of $4\,\textit{f}$ electrons:
\begin{equation} \hat{H}_{sf} = -g \sum_{i}\,\hat{\sigma}_{i}\,\cdot\,\hat{S}_{i} \end{equation}
Here the index $i$ indentify the lattice site in $\textbf{R}_{i}$, $g$ is the exchange coupling constant. For semplicity it is assumed to be identical for all pairs of conduction and $4\,\textit{f}$ electrons
Using the following relations for the spin operators $\hat{\sigma}_{i}$ and $\hat{S}_{i}$
\begin{align} \hat{S}_{j}^{x} &= \frac{1}{2} \left(\hat{S}_{j}^{+} + \hat{S}_{j}^{-}\right) \nonumber \\ \hat{S}_{j}^{y} &= \frac{1}{2} \left(\hat{S}_{j}^{+} - \hat{S}_{j}^{-}\right) \nonumber \\ \hat{S}_{j}^{z} &= \hat{S}_{j}^{z} \end{align}
and the second quantization rapresentation for the spin operators of conduction electrons
\begin{align} \hat{ \sigma}_{i}^{+} &= \hbar\,\hat{a}^{\dagger}_{i,\uparrow}\hat{a}_{i,\downarrow} \nonumber \\ \hat{ \sigma}_{i}^{-} &= \hbar\,\hat{a}^{\dagger}_{i,\downarrow}\hat{a}_{i,\uparrow} \nonumber \\ \hat{ \sigma}_{i}^{z} &= \frac{\hbar}{2}\, \left( \hat{a}^{\dagger}_{i,\uparrow}\hat{a}_{i,\uparrow} - \hat{a}^{\dagger}_{i,\downarrow}\hat{a}_{i,\downarrow} \right) \end{align}
The interaction term $\hat{H}_{sf}$ could be written in the following way
\begin{equation} \hat{H}_{sf} = - \frac{1}{2} \, g \, \hbar \sum_{i} \left[ \hat{a}^{\dagger}_{i,\uparrow}\hat{a}_{i,\downarrow} \hat{S}_{i}^{-} + \hat{a}^{\dagger}_{i,\downarrow}\hat{a}_{i,\uparrow} \hat{S}_{i}^{+} + \left( \hat{a}^{\dagger}_{i,\uparrow}\hat{a}_{i,\uparrow} - \hat{a}^{\dagger}_{i,\downarrow}\hat{a}_{i,\downarrow} \right) \, \hat{S}_{i}^{z}\right] \end{equation}
Since i have to compute commutators like $\left[\hat{a}_{i,\sigma}, \hat{H}\right]$, where $H$ is the s-f Hamiltonian,i have to know the commutation relations between the annihliation and creation operators of conduction electrons $\hat{a}^{\dagger}_{i,\sigma},\,\hat{a}_{i,\sigma}$ and the spin operator of localized electrons $\hat{S}_{i}^{z}, \hat{S}_{i}^{+}, \hat{S}_{i}^{-}$
I am thinking , since these are two different kind of electrons the commutator between them is 0 , right?
