Recently, some $Z_2$ monopole charges were defined for Dirac semimetals with $\mathcal{GT}$ symmetry ($\mathcal{G}$: glide, $\mathcal{T}$: time-reversal) in previous works, and the charges are believed to lead to double helicoid surface states. However, no proof of the bulk-surface correspondence is given there. In this paper, we define a generalized $Z_2$ charge, which enables us to find various properties of $\mathcal{GT}$-protected Dirac semimetals. Then, by using the $Z_2$ charge, we give a proof of the bulk-surface correspondence. We also show that the new $Z_2$ charge is equal to the $Z_2$ invariant for $\mathcal{G}$-protected topological crystalline insulators, and to the second Stiefel-Whitney number for $\mathcal{PT}$-protected nodal line semimetals, under some symmetries.

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• Surface states
• Symmetry protected topological states
• Dirac semimetal
• Topological materials