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

Abstract

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.

PRB Letter, in press