Symmetry analysis of electronic states for crystals with spiral magnetic order. I. General properties

This paper is devoted to the systematic study of questions concerning the use of spin-space groups (SSG) in calculation and qualitative analysis of the electronic states for crystals with spiral magnetic order. The types of operators that may enter into the symmetry group of a spiral are investigated. Introduction of the wavevector of an electronic state on the basis of the generalized Bloch theorem is discussed, and it is shown that the possibility of the choice of the wavevector is not unique. The condition imposed on the symmetry operations belonging to the group of a given Bloch vector and the formula describing the symmetry of eigenvalues in reciprocal space are obtained and appear to be substantially different from counterparts used in traditional cases of collinear and non-magnetic crystals. It is shown that, contrary to the traditional cases, there are spiral structures whose spectral symmetry is described by non-symmorphic space groups. The irreducible domain of reciprocal space is found for a number of concrete spiral structures. Methods of construction of the SSG double-valued irreducible representations and also of their basis functions are suggested. Special attention is devoted to the possibility of using the corresponding tables of ordinary space groups. In particular, it is shown that for spirals with hexagonal close-packed crystal structure the traditional tables may be used after minor corrections. The peculiarities of allowance for the operation of time inversion are discussed.