We demonstrate that a class of stable ${\mathbb{Z}}_2$ monopole charge Dirac point (${\mathbb{Z}}_2\mathrm{DP}$) phases can robustly exist in real materials, which is counterintuitive: that is, a ${\mathbb{Z}}_2\mathrm{DP}$ is unstable and generally considered to be only the critical point of a ${\mathbb{Z}}_2$ nodal line (${\mathbb{Z}}_2\mathrm{NL}$) characterized by a ${\mathbb{Z}}_2$ monopole charge (the second Stiefel-Whitney number $w_2$) with space-time inversion symmetry but no spin-orbital coupling. We explicitly reveal the higher-order bulk-boundary correspondence in the stable ${\mathbb{Z}}_2\mathrm{DP}$ phase. We propose the alternating-twist multilayer graphene, which can be regarded as 3D twisted bilayer graphene (TBG), as the first example to realize such stable ${\mathbb{Z}}_2\mathrm{DP}$ phase and show that the Dirac points in the 3D TBG are essentially degenerate at high-symmetry points protected by crystal symmetries and carry a nontrivial ${\mathbb{Z}}_2$ monopole charge ($w_2=1$), which results in higher-order hinge states along the entire Brillouin zone of the $k_z$ direction. By breaking some crystal symmetries or tailoring interlayer coupling we are able to access ${\mathbb{Z}}_2\mathrm{NL}$ phases or other ${\mathbb{Z}}_2\mathrm{DP}$ phases with hinge states of adjustable length. In addition, we present other 3D materials which host ${\mathbb{Z}}_2\mathrm{DPs}$ in the electronic band structures and phonon spectra. We construct a minimal eight-band tight-binding lattice model that captures these nontrivial topological characters and furthermore tabulate all possible space groups to allow the existence of the stable ${\mathbb{Z}}_2\mathrm{DP}$ phases, which will provide direct and strong guidance for the realization of the ${\mathbb{Z}}_2$ monopole semimetal phases in (among others) electronic materials, metamaterials, and electrical circuits.

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• Electronic structure
• Topological phases of matter
• Dirac semimetal
• Twisted bilayer graphene
• Density functional theory
• Tight-binding model
• Wannier function methods
• k dot p method