Parallel and anti-parallel helical surface states for topological semimetals

Weyl points, carrying a Z-type monopole charge [Formula: see text] , have bulk-surface correspondence (BSC) associated with helical surface states (HSSs). When |[Formula: see text] | [Formula: see text] , multi-HSSs can appear in a parallel manner. However, when a pair of Weyl points carrying [Formula: see text] [Formula: see text] meet, a Dirac point carrying [Formula: see text] = 0 can be obtained and the BSC vanishes. Nonetheless, a recent study in Zhang et al. (Phys Rev Res 4:033170, 2022) shows that a new BSC can survive for Dirac points when the system has time-reversal ([Formula: see text] )-glide ([Formula: see text] ) symmetry ([Formula: see text] =TG), i.e., anti-parallel double/quad-HSSs associated with a new [Formula: see text] -type monopole charge [Formula: see text] appear. In this paper, we systematically review and discuss both the parallel and anti-parallel multi-HSSs for Weyl and Dirac points, carrying two different kinds of monopole charges. Two material examples are offered to understand the whole configuration of multi-HSSs. One carries the Z-type monopole charge [Formula: see text] , showing both local and global topology for three kinds of Weyl points, and it leads to parallel multi-HSSs. The other carries the [Formula: see text] -type monopole charge [Formula: see text] , only showing the global topology for [Formula: see text] -invariant Dirac points, and it is accompanied by anti-parallel multi-HSSs.