Spin-phonon interactions in a Heisenberg ferromagnet

The effect of an interaction Hamiltonian linear in the phonon field and quadratic in the spin operators on a system of noninteracting spin waves and phonons for an ideal Heisenberg ferromagnet is examined by using the techniques of quantum field theory. Assuming the coupling to be weak, a perturbation expansion is performed, and the renormalized spin wave and phonon energies are calculated to lowest order, together with the lifetimes of the individual excitation modes. The calculations are performed separately for two different approximations for the spin system. The random phase approximation (1) is used to study the behavior of the system near the Curie temperature, and the Holstein-Primakoff approximation (2) is used at low temperatures. The interaction leads to a shift in the Curie temperature, Tc, but the behavior of the magnetization, σ, (calculated in RPA) in the neighborhood of the Curie temperature is virtually unchanged. At low temperatures, in addition to a renormalization of the coefficient of the Bloch T 3 2 term (3), a T4 term is obtained in the magnetization which is entirely due to the interaction. The sound velocity is renormalized by the interaction; the change in the sound velocity vanishes at T = 0 and also for T = Tc in the absence of an external magnetic field. In the neighborhood of the Curie temperature the phonon spectrum is found to have a term linear in σ. Because dσ dT is large (∞ in RPA) at Tc, the phonon contribution to thermodynamic quantities involving ( dω dT )(q, T) like the specific heat, will be peaked at the Curie point. Finally, an expression for the thermal expansion coefficient of the lattice is derived which, in addition to the usual term proportional to the phonon contribution to the specific heat, is found to have a term proportional to the specific heat of the spin system. Therefore, the thermal expansion coefficient will have a discontinuity at Tc, as expected in a phase transition of the second kind. At low temperatures it is porportional to T 3 2 , the spin contribution again being dominant.