Time-reversal degeneracy in the electronic band structure of a magnetic metal

The essential degeneracies that may arise in the electronic band structure of a magnetic metal have been investigated by Falicov and Ruvalds. Because of the existence of a spontaneous magnetic moment the operation of time reversal θ is no longer a symmetry operation of the crystal. However, the crystal still possesses some compound anti-unitary symmetry operations, ignored by Falicov and Ruvalds, which consist of products of θ with various space-group operations. It is shown that for face-centred cubic and body-centred cubic structures the inclusion of these extra anti-unitary operations leads to no extra degeneracy, but that in the hexagonal close-packed structure, for certain magnetization directions and for certain points and lines of symmetry on the top face of the Brillouin zone, these extra anti-unitary symmetry operations lead to additional degeneracies which were not noted by Falicov and Ruvalds.