Z2 Dirac points with topologically protected multihelicoid surface states

Abstract

In some Dirac systems with time-reversal ($\mathcal{T}$) and glide ($\mathcal{G}$) symmetries, multihelicoid surface states (MHSSs) appear, as discussed in various systems such as electronic and photonic ones. However, the topological nature and the conditions for the appearance of the MHSSs have not been understood. Here we show that MHSSs result from bulk-surface correspondence for the ${Z}_{2}$ monopole charge $\mathcal{Q}$, which cannot be defined as a local quantity associated with the Dirac point, unlike the $Z$ monopole charge characterizing Weyl points. The previously known formula of $\mathcal{Q}$ turns out to be non-gauge-invariant and thus cannot characterize the MHSSs. This shortcoming of the definition of $\mathcal{Q}$ is amended by redefining $\mathcal{Q}$ as a global topological invariant in $k$-space. Surprisingly, the newly defined $\mathcal{Q}$, characterizing $\mathcal{GT}$ invariant gapless systems, is equal to the $\mathcal{G}$-protected ${Z}_{2}$ topological invariant $\ensuremath{\nu}$, which is nontrivial only in $\mathcal{T}$-breaking gapped systems. This global definition of $\mathcal{Q}$ automatically guarantees the appearance of MHSSs even when the Dirac point splits into Weyl points or a nodal ring by lowering the symmetry, as long as the $\mathcal{GT}$ symmetry is preserved. $\mathcal{Q}$ can be simplified to symmetry-based indicators when two vertical $\mathcal{G}\mathrm{s}$ are preserved, and filling-enforced topological crystalline insulators are diagnosed in several cases when a $\mathcal{T}$-breaking perturbation is induced. Material candidate ${\mathrm{Li}}_{2}{\mathrm{B}}_{4}{\mathrm{O}}_{7}$ together with a list of space groups preserving MHSSs are also proposed.