Symmetry Properties of Wave Functions in Magnetic Crystals

The symmetry properties of wave functions in magnetic crystals are discussed in terms of the irreducible representations of magnetic space groups. The specific effects of the magnetic ordering on the crystal eigenstates are found to be of three types: (1) There is a lifting of some eigenfunction degeneracies because the crystal symmetry is reduced in the magnetic state. (2) New Brillouin zone surfaces are introduced if there is a reduction in translational symmetry. (3) The symmetry of the energy band in $K$ space may be reduced. The rutile structure is considered as a specific example, and the space groups of Mn${\mathrm{F}}_2$ and Mn${\mathrm{O}}_2$ in their magnetic and nonmagnetic states are obtained. A magnetic structure of Mn${\mathrm{O}}_2$ where the ${\mathrm{Mn}}^{2+}$ spins point toward the nearest-neighbor oxygens is assumed. The space groups considered are $\frac{P{4}_2}{\mathrm{mnm}} \left({D_{4h}}^{14}\right)$, $\mathrm{Pnnm} \left({D_{2h}}^{12}\right)$, $I\overline{4}2d \left({D_{2d}}^{12}\right)$, $\frac{P{4}_2}{\mathrm{mnm}} 1^{\prime }$, $\frac{P{4_2}^{\prime }}{\mathrm{mn}{m}^{\prime }}$, and $I_c\overline{4}2d$. The theory is applied to spin-wave states, and it is found that the structure of the spin-wave energy bands throughout the Brillouin zone may be obtained.