Weyl points and Dirac lines protected by multiple screw rotations

In three-dimensional noncentrosymmetric materials two-fold screw rotation symmetry forces electron＇s energy bands to have Weyl points at which two bands touch. This is illustrated for space groups No. 19 （P212121 ） and No. 198 （P213）, which have three orthogonal screw rotation axes. In the case of space groups No. 61 （Pbca） and No. 205 （Pa-3） that have extra inversion symmetry, Weyl points are promoted to four-fold degenerate line nodes in glide-invariant planes. The three-fold rotation symmetry present in the space groups No. 198 and No. 205 allows Weyl and Dirac points, respectively, to appear along its rotation axes in the Brillouin zone and generates four-fold and six-fold degeneracy at the F point and R point, respectively.

In three-dimensional noncentrosymmetric materials two-fold screw rotation symmetry forces electron's energy bands to have Weyl points at which two bands touch.This is illustrated for space groups No.19(P2_12_12_1) and No.198(P2_13),which have three orthogonal screw rotation axes.In the case of space groups No.61(Pbca) and No.205(Pa-3) that have extra inversion symmetry,Weyl points are promoted to fourfold degenerate line nodes in glide-invariant planes.The three-fold rotation symmetry present in the space groups No.198 and No.205 allows Weyl and Dirac points,respectively,to appear along its rotation axes in the Brillouin zone and generates four-fold and six-fold degeneracy at the C point and R point,respectively.