RKKY Interaction: Defining $\hat{H}$

[LagrangeEuler]

Hamiltonian of interaction between fixed ion and conducting electron
$\hat{H}=-J\sum_{i=1}^2\{\hat{s}_i^z\hat{S}_i^z+\frac{1}{2}(\hat{s}_i^+\hat{S}_i^{-}+\hat{s}_i^-\hat{S}_i^{+})\}$
Why both electron and ion are defined with subscript $i$?

 

[M Quack]

...and why does i run from 1 to 2?

Looks like a bug to me.

One would think that you need separate indices i and j to label the atoms and the (conduction) electrons, and that these would run over the number of particles in question.

 

[LagrangeEuler]

There is two fixed ions. One ion polarised electron and that electron interract with secound ion. So between ions is indirect interraction.

 

[M Quack]

OK, that makes sense. However, I would still expect the electron index to be independent of the ion. Remember that RKKY occurs in the conduction electrons of metals.

 

[LagrangeEuler]

Well I think that if electron is near ion you could approximate that it is on the same lattice site ion lattice site. I asked question about calculating matrix element in this problem.
Here is that question.
https://www.physicsforums.com/showthread.php?t=646675

I would be grateful if you could look and maybe answer this question.