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?


1 Answer 1


You are correct. If this is not enough, these operators correspond to different degrees of freedom and you can safely assume that they are commuting.

  • $\begingroup$ Can i formalize it ? $\endgroup$
    – Giovanni
    Dec 12, 2019 at 13:26
  • $\begingroup$ Yes, but it is trivial. Ladder operators act on occupation numbers that are a different degree of freedom with respect to spin. So, simply $[{\hat a}_{i,\sigma},{\hat S}^\alpha_j]=0$. $\endgroup$
    – Jon
    Dec 12, 2019 at 13:29

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.