The correlation between the vibrational and electron excitation modes in the energy spectra of single-layer graphene and crystalline graphite, as well as the dispersion dependences of those modes, has been studied. The methods of the theory of projective representations of the point and spatial symmetry groups are used for the first time in order to interpret those correlations. The correlations of vibrational and electron excitation spectra and the compatibility conditions for irreducible projective representations in the descriptions of quantum states of graphene and crystalline graphite at various points of their Brillouin zones are determined. For the projective representations of all projective classes belonging to the hexagonal system, standard factor-systems are constructed for the first time. In particular, the factor-systems for electron states are first determined. The results obtained are used to calculate, also for the first time, the correct spinor multiplication tables, i.e. the multiplication tables for elements in double symmetry groups. The developed method is applied to classify all high-symmetry points in the Brillouin zones of single-layer graphene and crystalline graphite with respect to the symmetry type of vibrational excitations.
A Hamiltonian Approach for Obtaining Irreducible Projective Representations and the κ·p Perturbation for Anti-unitary Symmetry Groups
