Composite Weyl nodes stabilized by screw symmetry with and without time-reversal invariance
AbstractWe classify the band degeneracies in three-dimensional crystals with screw symmetry nm and broken P*T symmetry, where P stands for spatial inversion and T for time reversal. The generic degeneracies along symmetry lines are Weyl nodes: chiral contact points between pairs of bands. They can be single nodes with a chiral charge of magnitude |χ|=1 or composite nodes with |χ|=2 or 3, and the possible χ values only depend on the order n of the axis, not on the pitch m/n of the screw. Double Weyl nodes require n=4 or 6, and triple nodes require n=6. In all cases, the bands split linearly along the axis, and for composite nodes the splitting is quadratic on the orthogonal plane. This is true for triple as well as double nodes, due to the presence in the effective two-band Hamiltonian of a nonchiral quadratic term that masks the chiral cubic dispersion. If T symmetry is present and P is broken, there may exist on some symmetry lines Weyl nodes pinned to T-invariant momenta, which in some cases are unavoidable. In the absence of other symmetries, their classification depends on n,m, and the type of T symmetry. With spinless T such T-invariant Weyl nodes are always double nodes, while with spinful T they can be single or triple nodes. T-invariant triples nodes can occur not only on sixfold axes but also on threefold ones, and their in-plane band splitting is cubic, not quadratic as in the case of generic triple nodes. These rules are illustrated by means of first-principles calculations for hcp cobalt, a T-broken, P-invariant crystal with 63 symmetry, and for trigonal tellurium and hexagonal NbSi2, which are T-invariant, P-broken crystals with threefold and sixfold screw symmetry, respectively.