We propose and characterize a new ${\mathbb{Z}}_2$ class of topological semimetals with a vanishing spin-orbit interaction. The proposed topological semimetals are characterized by the presence of bulk one-dimensional (1D) Dirac line nodes (DLNs) and two-dimensional (2D) nearly flat surface states, protected by inversion and time-reversal symmetries. We develop the ${\mathbb{Z}}_2$ invariants dictating the presence of DLNs based on parity eigenvalues at the parity-invariant points in reciprocal space. Moreover, using first-principles calculations, we predict DLNs to occur in ${\mathrm{Cu}}_3\mathrm{N}$ near the Fermi energy by doping nonmagnetic transition metal atoms, such as Zn and Pd, with the 2D surface states emerging in the projected interior of the DLNs. This Letter includes a brief discussion of the effects of spin-orbit interactions and symmetry breaking as well as comments on experimental implications.

We study a class of Dirac semimetals that feature an eightfold-degenerate double Dirac point. We show that 7 of the 230 space groups can host such Dirac points and argue that they all generically display linear dispersion. We introduce an explicit tight-binding model for space groups 130 and 135. Space group 135 can host an intrinsic double Dirac semimetal with no additional states at the Fermi energy. This defines a symmetry-protected topological critical point, and we show that a uniaxial compressive strain applied in different directions leads to topologically distinct insulating phases. In addition, the double Dirac semimetal can accommodate topological line defects that bind helical modes. Connections are made to theories of strongly interacting filling-enforced semimetals, and potential materials realizations are discussed.

We study the band topology and the associated linking structure of topological semimetals with nodal lines carrying $Z_2$ monopole charges, which can be realized in three-dimensional systems invariant under the combination of inversion $P$ and time reversal $T$ when spin-orbit coupling is negligible. In contrast to the well-known $PT$-symmetric nodal lines protected only by the $\pi$ Berry phase, in which a single nodal line can exist, the nodal lines with $Z_2$ monopole charges should always exist in pairs. We show that a pair of nodal lines with $Z_2$ monopole charges is created by a double band inversion process and that the resulting nodal lines are always linked by another nodal line formed between the two topmost occupied bands. It is shown that both the linking structure and the $Z_2$ monopole charge are the manifestation of the nontrivial band topology characterized by the second Stiefel-Whitney class, which can be read off from the Wilson loop spectrum. We show that the second Stiefel-Whitney class can serve as a well-defined topological invariant of a $PT$-invariant two-dimensional insulator in the absence of Berry phase. Based on this, we propose that pair creation and annihilation of nodal lines with $Z_2$ monopole charges can mediate a topological phase transition between a normal insulator and a three-dimensional weak Stiefel-Whitney insulator. Moreover, using first-principles calculations, we predict $ABC$-stacked graphdiyne as a nodal line semimetal (NLSM) with $Z_2$ monopole charges having the linking structure. Finally, we develop a formula for computing the second Stiefel-Whitney class based on parity eigenvalues at inversion-invariant momenta, which is used to prove the quantized bulk magnetoelectric response of NLSMs with $Z_2$ monopole charges under a $T$-breaking perturbation.

Higher-order topological insulators are newly proposed topological phases of matter, whose bulk topology manifests as localized modes at two- or higher-dimensional lower boundaries. In this Letter, we propose the twisted bilayer graphenes with large angles as higher-order topological insulators, hosting topological corner charges. At large commensurate angles, the intervalley scattering opens up the bulk gap and the corner states occur at half filling. Based on both first-principles calculations and analytic analysis, we show the striking results that the emergence of the corner states do not depend on the choice of the specific angles as long as the underlying symmetries are intact. Our results show that the twisted bilayer graphene can serve as a robust candidate material of a two-dimensional higher-order topological insulator.

Exotic topology on the surface: Analyzing the spatial symmetries of three-dimensional (3D) crystal structures has led to the discovery of exotic types of quasiparticles and topologically nontrivial materials. Wieder et al. focus on the symmetry groups of 2D surfaces of 3D materials—the so-called wallpaper groups—and find that some of them allow for an additional topological class. This class hosts a single fourfold-degenerate Dirac fermion on the surface of the material and, on the basis of the authors' calculations, is expected to occur in the compound Sr ₂ Pb ₃ . Science , this issue p. 246