Effective indirect multi-site spin-spin interactions in the s-d(f) model

Using the diagram technique for Matsubara Green's functions it is shown that the dynamics of the localized spin subsystem in the s–d(f) model can be described in terms of an effective spin model with multi-site spin–spin interactions. An exact representation of the action for the effective purely spin model is derived as an infinite series in powers of s–d(f) exchange interaction J. The indirect interactions of the 2nd, 3rd and 4th order are discussed.