Spin-Spin and Spin-Phonon Interactions in the Heisenberg Antiferromagnet

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Abstract
The Dyson ideal spin-wave Hamiltonian for the two-sublattice Heisenberg antiferromagent is obtained using a Maleev transformation. A spin-phonon interaction Hamiltonian is derived by expanding the lattice coordinates in small displacements and retaining the linear terms. The static and dynamic properties of the system containing the spin field and the harmonic phonon field are studied using the double-time Green's-function method. This is done in a self-consistent manner, using symmetrized equations of motion, from which a Dyson equation is derived. It is found that if the zeroth-order Hamiltonian describing the Dyson equation contains contributions from the interacting ideal spin waves, the polarization operator is no longer simple. A zeroth-order approximation which contains all the static contributions arising from the interacting ideal spin waves is constructed. Contact is made with the work of previous authors and some aspects of the Callen decoupling procedure are clarified. Using an effective Hamiltonian and a canonical-transformation technique, expressions for the full polarization operator are developed. Finally, expressions for the frequency-dependent susceptibility are obtained and these are used to discuss the line shape for the absorption of energy from an oscillating field at frequencies near and far from resonance.