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APPENDIX band limit band theory Bloch functions coefficients collective excitations collective mode complete set consider continuous limit continuum correlation function corresponding creation operator Curie point degenerate kernel approximation density matrix dispersion relation eigenfunction eigenstate electron and hole electron-hole elementary excitations equation 25 equation 38 equation 43 equation of motion exchange interaction expectation value Fermi distribution Fourier components free electron gas free electron limit frequency gives Hamiltonian Heisenberg model integral equation Izuyama Kubo formalism last term lattice low temperatures magnetization curve matrix elements model of ferromagnetism momentum transfer normal coordinates normal modes number operator obtain outer electrons particle and hole Phys random phase approximation reduced magnetization renormalized result Sawada scattering modes second term selection rules set of excitations single spin spin reversals spin-down bands spin-flip operator spin-up and spin-down Stoner model Substituting equation theory of ferromagnetism tion total momentum Wannier functions Wannier representation wave zero