General Theorems on Ferromagnetism and Ferromagnetic Spin Waves

Abstract

Ideal ferromagnetism in perfect crystals (and/or in free space), where spin-orbit interactions may be neglected, is investigated at zero temperature under the following conditions: The fermion system considered here should have the inversion symmetry of the space coordinates and the thermodynamic limit. Its ground state is nondegenerate for a fixed eigenvalue of $S_z$. In other respects the ferromagnets considered are quite general and may cover all possible types of ferromagnetism: insulators, metals, and free fermions. Dynamical spin-spin correlation functions are studied. Sum rules for them are developed so as to exclude the contributions from Stoner excitations. Spin waves are considered by means of these sum rules. In the case of complete ferromagnetism (all electron spins being aligned in one direction), it is shown rigorously that no consistent result can be obtained; the excitation energies of magnons cannot be finite in the form of $D{q}^2$, but are vanishing. This suggests that the complete ferromagnetism, if it could exist, must violate one of the above conditions.