RKKY interraction

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 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.

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.
http://www.physicsforums.com/showthread.php?t=646675

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

LagrangeEuler	RKKY interraction 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 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.

M Quack	RKKY interraction 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.