Conical Points in the Dispersion Relation and Related Phase Transitions

The existence of conical points in the dispersion relation in some hexagonal system is group-theoretically discussed. Invariants, including the Lifshitz invariant and the domain wall invariant, are obtained for the representation responsible for the appearance of the conical point. The phase transitions into the incommensurate and commensurate phases induced by the instability of such representation are discussed.