Abstract
The Ward-Takahashi relations associated with the spontaneous breakdown of spatially translational invariance are established in terms of Matsubara's finite temperature Green's functions. Atom and phonon Green's functions are constructed in the two band model and in the random phase approximation so that they satisfy the Ward-Takahashi relations including all the coupling of the density fluctuations due to the number density of the ground state, the number density of the first excited and the particle-hole pairs. Phonos are the collective excitations of the particle-hole pairs. The energy spectrum of atoms has gaps on the Brillouin zone boundary. The gaps come from the scattering due to the periodic potential made up by the surrounding atoms and the mixing states.
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