# RKKY interraction

1. Oct 23, 2012

### 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 $i$?

2. Oct 26, 2012

### M Quack

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

Looks like a bug to me.

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

3. Oct 26, 2012

### LagrangeEuler

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

4. Oct 26, 2012

### 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.

5. Oct 26, 2012

### 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.

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

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