Abstract
Weyl points, carrying a Z-type monopole charge C, have bulk-surface correspondence (BSC) associated with helical surface states (HSSs). When |C| >1, multi-HSSs can appear in a parallel manner. However, when a pair of Weyl points carrying C=pm 1 meet, a Dirac point carrying C = 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 ({T})-glide ({G}) symmetry ({tilde{Theta }}=TG), i.e., anti-parallel double/quad-HSSs associated with a new Z_{2}-type monopole charge {Q} 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 C, showing both local and global topology for three kinds of Weyl points, and it leads to parallel multi-HSSs. The other carries the Z_{2}-type monopole charge {Q}, only showing the global topology for {tilde{Theta }}-invariant Dirac points, and it is accompanied by anti-parallel multi-HSSs.
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