Topological charge and bulk-surface correspondence for quad-helicoid surface states in topological semimetals with two glide-time-reversal symmetries

Quad-helicoid surface states (QHSSs) are unique surface states with two pairs of helicoid surface states in topological semimetals such as Dirac semimetals. So far, topologically protected QHSSs are shown to appear in spinless systems with two $\mathcal{GT}$ symmetries and $\mathcal{T}$ symmetry ($\mathcal{G}$: glide, $\mathcal{T}$: time-reversal). In this paper, we show that topologically protected QHSSs also appear in spinful/spinless systems with only two $\mathcal{GT}$ symmetries by defining new topological charges and establishing the bulk-surface correspondence. We first define a local $Z_2\times Z_2$ monopole charge for gapless nodes at $\mathcal{GT}$-invariant high-symmetry points and a global $Z_2$ charge reflecting the global topological feature of $\mathcal{GT}$-symmetric topological semimetals. Next, we show that the latter $Z_2$ classification corresponds to the presence or absence of QHSSs on the surface with two $\mathcal{GT}$ symmetries. In addition, we provide simplified formulas of the $Z_2$ charge under additional symmetries, and clarify some symmetry conditions where QHSSs are filling-enforced.