Parallel and anti-parallel helical surface states for topological semimetals

Abstract

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