The Kramers degeneracy theorem in a condensed-matter system is proven by the orthogonality of wavefunctions transformed by two time-reversal symmetries: the conventional symmetry Θ of Θ2 = −1 and the supplementary symmetry K of K2 = +1, valid for a vanishing spin–orbit coupling term. The Bloch state may then remain nondegenerate for a finite spin–orbit coupling term, but must become degenerate for a vanishing spin–orbit coupling term at any symmetry point. The theorem is demonstrated for a representative system GaAs on the basis of first-principles calculations. The spin polarization of a wavefunction is attributed to the point-group symmetry of a crystal potential rather than the existence of a spin–orbit coupling term.
