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To be more specific, consider the eigenfunction of some space-group element (R,t) fk =c fk (R is a point-group operation and t a translation vector). The action of time-reversal symmetry θ and inversion symmetry (I,0) generates new eigenfunctions with eigenvalues c and e2ikt c, respectively, as can be checked by noting that θ and (R,t) commute while (R,t) (I,0) = (E,2t) (I,0) (R,t), where E is the point-group identity. It is then evident that symmetries involving nonprimitive translations generate different phase factors for the different pseudospin triplet components at different points in the zone.
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To be more specific, consider the eigenfunction of some space-group element (R,t) fk =c fk (R is a point-group operation and t a translation vector). The action of time-reversal symmetry θ and inversion symmetry (I,0) generates new eigenfunctions with eigenvalues c and e2ikt c, respectively, as can be checked by noting that θ and (R,t) commute while (R,t) (I,0) = (E,2t) (I,0) (R,t), where E is the point-group identity. It is then evident that symmetries involving nonprimitive translations generate different phase factors for the different pseudospin triplet components at different points in the zone.
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Notice that the characters are those of double groups, and therefore for a 2π rotation, χ [(2z 2z, 0)] =-1.
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Notice that the characters are those of double groups, and therefore for a 2π rotation, χ [(2z 2z, 0)] =-1.
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