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We demonstrate that a class of stable $\mathbb{Z}_2$ monopole charge Diracpoint ($\mathbb{Z}_2$DP) phases can robustly exist in real materials, whichsurmounts the understanding: that is, a $\mathbb{Z}_2$DP is unstable andgenerally considered to be only the critical point of a $\mathbb{Z}_2$ nodalline ($\mathbb{Z}_2$NL) characterized by a $\mathbb{Z}_2$ monopole charge (thesecond Stiefel-Whitney number $w_2$) with space-time inversion symmetry but nospin-orbital coupling. For the first time, we explicitly reveal thehigher-order bulk-boundary correspondence in the stable $\mathbb{Z}_2$DP phase.We propose the alternating-twisted multilayer graphene, which can be regardedas 3D twisted bilayer graphene (TBG), as the first example to realize suchstable $\mathbb{Z}_2$DP phase and show that the Dirac points in the 3D TBG areessential degenerate at high symmetric points protected by crystal symmetriesand carry a nontrivial $\mathbb{Z}_2$ monopole charge ($w_2=1$), which resultsin higher-order hinge states along the entire Brillouin zone of the $k_z$direction. By breaking some crystal symmetries or tailoring interlayer couplingwe are able to access $\mathbb{Z}_2$NL phases or other $\mathbb{Z}_2$DP phaseswith hinge states of adjustable length. In addition, we present other 3Dmaterials which host $\mathbb{Z}_2$DPs in the electronic band structures andphonon spectra. We construct a minimal eight-band tight-binding lattice modelthat captures these nontrivial topological characters and furthermore tabulateall possible space groups to allow the existence of the stable $\mathbb{Z}_2$DPphases, which will provide direct and strong guidance for the realization ofthe $\mathbb{Z}_2$ monopole semimetal phases in electronic materials,metamaterials and electrical circuits, etc.

Stable Higher-Order Topological Dirac Semimetals with $\mathbb{Z}_2$ Monopole Charge in Alternating-twisted Multilayer Graphenes and beyond