Self-consistent treatment of a crystal-like model

A self-consistent consideration of a crystal-like model is presented to show how mutually interacting molecules arrange themselves in lattice structure. It is shown that certain bosons with gapless energies are dynamically created and that their Bose-Einstein condensations manifest themselves as lattice patterns. These bosons correspond to the well-known quanta of lattice oscillations. The translational invariance of the system is preserved by the self-consistent regulation of the condensation of these gapless bosons despite the emergence of translationally noninvariant crystals. When the lattice length vanishes the system becomes homogeneous and behaves like a superfluid system. The corresponding computations of the present article do not require the two-body potential to be a δ-function.