Self-Consistent Approximations for Itinerant Ferromagnetism Below the Phase-Transition Point

The self-consistent and "conserving" (in particular the total spin is conserved) approximation scheme, developed in a previous paper for itinerant ferromagnetism, is extended to the region below the phase-transition point. The former general condition for the transformation properties of any approximate self-energy part with regard to infinitesimal transformations in spin space is applied to the case where the transformation is a finite rotation in spin space; these rotations correspond to rotations of the direction of the spontaneous magnetization which is fixed here by an infinitesimal auxiliary field H. This transformation property of the self-energy part ensures that the equations determining the corresponding Green's and correlation functions are covariant in form with respect to rotations of H. Two examples are considered : the "Hartree-Fock" and the "particle-hole T-matrix" approximations for the contact interaction or Hubbard model. In the first example the resulting susceptibilities are shown to be causal response functions, provided the Stoner equation is fulfilled. In the second example the self-energy part and the T matrix are discussed for the two cases where H is parallel and perpendicular to the axis of spin quantization. In fact, these expressions can be transformed into each other by means of the corresponding rotation matrix in spin space.