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

We demonstrate that a class of stable $\mathbb{Z}_2$ monopole charge Dirac point ($\mathbb{Z}_2$DP) phases can robustly exist in real materials, which surmounts the understanding: that is, a $\mathbb{Z}_2$DP is unstable and generally considered to be only the critical point of a $\mathbb{Z}_2$ nodal line ($\mathbb{Z}_2$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. For the first time, we explicitly reveal the higher-order bulk-boundary correspondence in the stable $\mathbb{Z}_2$DP phase. We propose the alternating-twisted multilayer graphene, which can be regarded as 3D twisted bilayer graphene (TBG), as the first example to realize such stable $\mathbb{Z}_2$DP phase and show that the Dirac points in the 3D TBG are essential degenerate at high symmetric 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 $\mathbb{Z}_2$NL phases or other $\mathbb{Z}_2$DP phases with hinge states of adjustable length. In addition, we present other 3D materials which host $\mathbb{Z}_2$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 $\mathbb{Z}_2$DP phases, which will provide direct and strong guidance for the realization of the $\mathbb{Z}_2$ monopole semimetal phases in electronic materials, metamaterials and electrical circuits, etc.

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