Title: Composite Weyl nodes stabilized by screw symmetry with and without time reversal,
Abstract: We classify the band degeneracies in 3D crystals with screw symmetry $n_m$
and broken $\mathcal P*\mathcal T$ symmetry, where $\mathcal P$ stands for
spatial inversion and $\mathcal T$ 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 $|\chi|=1$ or
composite nodes with $|\chi|=2$ or $3$, and the possible $\chi$ 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 $\mathcal
T$ symmetry is present and $\mathcal P$ is broken there may exist on some
symmetry lines Weyl nodes pinned to $\mathcal T$-invariant momenta, which in
some cases are unavoidable. In the absence of other symmetries their
classification depends on $n$, $m$, and the type of $\mathcal T$ symmetry. With
spinless $\mathcal T$ such $\mathcal T$-invariant Weyl nodes are always double
nodes, while with spinful $\mathcal T$ they can be single or triple nodes.
$\mathcal T$-invariant triples nodes can occur not only on 6-fold axes but also
on 3-fold 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 $\mathcal T$-broken, $\mathcal
P$-invariant crystal with $6_3$ symmetry, and for trigonal tellurium and
hexagonal NbSi$_2$, which are $\mathcal T$-invariant, $\mathcal P$-broken
crystals with 3-fold and 6-fold screw symmetry respectively.