Commutation rules between itinerant and localized electron operators in s-f Model

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?

• Yes, but it is trivial. Ladder operators act on occupation numbers that are a different degree of freedom with respect to spin. So, simply $[{\hat a}_{i,\sigma},{\hat S}^\alpha_j]=0$.