Coupling between Impurity Spins in a Magnetic Host

The presence of an impurity spin in a Heisenberg ferromagnet can give rise to localized modes whose energies may lie above or within the spin-wave band, depending upon the magnitude of the impurity spin and its coupling to the host. We have calculated the eigenvalues of the $p$-like modes for a one-dimensional chain and a simple cubic lattice when two such impurities are present. In the case where the single-impurity mode lies outside the spin-wave band, we find two modes symmetrically located with respect to this original mode. This mode splitting defines an effective impurity-impurity interaction. We find that this interaction can be quite large when the modes lie close to the spin-wave band. When the single-impurity mode lies within the band, it is characterized by a redistribution of the density of states. We have also investigated this redistribution for the case of two impurities. It is found that these results are considerably different from those obtained by a spin-wave scattering approach. This difference illustrates the importance of the contribution of bound states to indirect coupling.