Theory of spin-space groups

The symmetry properties of a magnetically ordered crystal are normally described in terms of the magnetic space group. But the dominant interactions, Heisenberg exchange and anisotropy fields, have more symmetry than this in that the spins may be rotated independently of the lattice. The ‘spin-space groups’ appropriate to this symmetry are defined and described, and methods are given for finding the irreducible representations of their ‘groups of k'. The theory is applicable to discussions of the excitation spectra in these systems, especially spin waves and conduction electrons. Compatibility relations between the spin-space group and the magnetic space group are considered—these allow discussion of the modification of the spectra brought about by the smaller interactions which only have the lower symmetry. The groups for antiferromagnetic rutile structures, spinels and garnets are examined in detail, and applied to spin waves. A group for helical spin structures, as found in rare earth metals, is discussed in relation to the energy band structure.