Classes of projective representations in determining the symmetry of collective spinor excitations and their dispersion in crystals and periodic nanostructures

Abstract

The distribution of electronic elementary excitations in crystals and periodic nanostructures according to irreducible projective representations of the corresponding projective classes of point and spatial symmetry groups and the dependence of projective classes on the structure of nontrivial translations of spatial groups are considered. The main attention is paid to the establishment of two-valued irreducible projective representations and corresponding projective classes taking into account the electron spin, when the wave functions of electronic states are two-valued spinor orbitals. This paper presents methods for constructing factor systems that are projectively equivalent (p-equivalent) factor systems, inherent in a certain projective class of projective representations, and methods for reducing them to the p-equivalent standard form. A new classification of projective classes for hexagonal structures is proposed, and a correct table of symmetric transformations of spinors is constructed. It is shown that the establishment of projective classes of projective representations and their changes for different points of Brillouin zones in crystalline graphite ɣ-C and two-period structure of singlelayer graphene C_L1 makes it possible to provide qualitative symmetric interpretation of electron excitation dispersion in crystalline graphite and single-layer graphene. In particular, it makes it possible to detect spindependent cleavages of electronic states in their Brillouin zones, which are due to the spin-orbit interaction in spinor orbitals.