A quantum field theory of phonons in crystals extended to the multi-band model I

A quantum field theory of phonons in crystals developed in a two-band model is extended to a multi-band model. The Ward-Takahashi relations associated with the spontaneous breakdown of spatially translational and rotational symmetry are represented by use of atom and phonon Green's functions. The Ward-Takahashi relations show the existence of Nambu-Goldstone bosons and the gap equation. A dynamical equation for phonons is derived in the random phase approximation. The higher excited states modify phonon dispersion curves in the two-band model with the density distribution functions. At enough low temperatures the dynamical equation reduces to that in the two-band model corresponding to the conventional lattice theory.