Spin-Peierls Instabilities vs Antiferromagnetic Transition and Phonon Dynamics

We discuss a competition between a spin-Peterls state and the antiferromagnetic state for a Heisenberg antiferromagnetic linear chain system including a phonon Hamiltonian. A Hartree-Fock approximation is applied to a Hamiltonian redescribed in terms of spin-less fermions. We get Dyson's equation of phonons which is consistent with this Hartree-Fock approximation. We point out that Pytte's work is not still complete even in the framework of the Hartree-Fock theory since a term was left out there which gave a quantitative difference between the Heisenberg model and the X Y model. We find the softening of phonons in the antiferromagnetic state which has overcome the spin-Peierls state.