Abstract

We demonstrate that a class of stable Z₂ monopole charge Dirac point (Z₂DP) phases can robustly exist in real materials, which surmounts the understanding: that is, a Z₂DP is unstable and generally considered to be only the critical point of a Z₂ nodal line (Z₂NL) characterized by a Z₂ monopole charge (the second Stiefel-Whitney number w2) 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 Z₂DP 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 Z₂DP 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 Z₂ monopole charge (w2 = 1), which results in higher-order hinge states along the entire Brillouin zone of the kz direction. By breaking some crystal symmetries or tailoring interlayer coupling we are able to access Z₂NL phases or other Z₂DP phases with hinge states of adjustable length. In addition, we present other 3D materials which host Z₂DPs 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 Z₂DP phases, which will provide direct and strong guidance for the realization of the Z₂ monopole semimetal phases in electronic materials, metamaterials and electrical circuits, etc.

Abstract

We demonstrate that a class of stable Z₂ monopole charge Dirac point (Z₂DP) phases can robustly exist in real materials, which surmounts the understanding: that is, a Z₂DP is unstable and generally considered to be only the critical point of a Z₂ nodal line (Z₂NL) characterized by a Z₂ monopole charge (the second Stiefel-Whitney number w2) 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 Z₂DP 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 Z₂DP 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 Z₂ monopole charge (w2 = 1), which results in higher-order hinge states along the entire Brillouin zone of the kz direction. By breaking some crystal symmetries or tailoring interlayer coupling we are able to access Z₂NL phases or other Z₂DP phases with hinge states of adjustable length. In addition, we present other 3D materials which host Z₂DPs 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 Z₂DP phases, which will provide direct and strong guidance for the realization of the Z₂ monopole semimetal phases in electronic materials, metamaterials and electrical circuits, etc.

Abstract

We demonstrate that a class of stable Z₂ monopole charge Dirac point (Z₂DP) phases can robustly exist in real materials, which surmounts the understanding: that is, a Z₂DP is unstable and generally considered to be only the critical point of a Z₂ nodal line (Z₂NL) characterized by a Z₂ monopole charge (the second Stiefel-Whitney number w2) 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 Z₂DP 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 Z₂DP 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 Z₂ monopole charge (w2 = 1), which results in higher-order hinge states along the entire Brillouin zone of the kz direction. By breaking some crystal symmetries or tailoring interlayer coupling we are able to access Z₂NL phases or other Z₂DP phases with hinge states of adjustable length. In addition, we present other 3D materials which host Z₂DPs 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 Z₂DP phases, which will provide direct and strong guidance for the realization of the Z₂ monopole semimetal phases in electronic materials, metamaterials and electrical circuits, etc.

Abstract

We demonstrate that a class of stable Z₂ monopole charge Dirac point (Z₂DP) phases can robustly exist in real materials, which surmounts the understanding: that is, a Z₂DP is unstable and generally considered to be only the critical point of a Z₂ nodal line (Z₂NL) characterized by a Z₂ monopole charge (the second Stiefel-Whitney number w2) 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 Z₂DP 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 Z₂DP 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 Z₂ monopole charge (w2 = 1), which results in higher-order hinge states along the entire Brillouin zone of the kz direction. By breaking some crystal symmetries or tailoring interlayer coupling we are able to access Z₂NL phases or other Z₂DP phases with hinge states of adjustable length. In addition, we present other 3D materials which host Z₂DPs 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 Z₂DP phases, which will provide direct and strong guidance for the realization of the Z₂ monopole semimetal phases in electronic materials, metamaterials and electrical circuits, etc.