A quantum field theory of phonons and phonon instability in nonprimitive crystals and their liquids

Atom and phonon Green's functions and a gap equation of atoms are derived from the viewpoint of the spontaneous breakdown of symmetry, considering the two concrete Ward-Takahashi relations represented by atom and phonon Green's functions, taking into account all the couplings between the density fluctuations due to the number density the ground state and the first excited state, and the particle-hole pairs. The gap equation plays an important role in constructing phonons and phonon instability. Phonon dispersion relations and phonon instability temperatures are derived for sphalerite structure crystals and their liquids. Phonon dispersion curves of acoustic modes for liquids behave in the same form as those for simple liquids.