A 3D Graphene?: Discoveries of materials with exciting electronic properties have propelled condensed matter physics over the past decade. Two of the best-known examples, graphene and topological insulators, have something in common: a linear energy-momentum relationship—the Dirac dispersion—in their two-dimensional (2D) electronic states. Topological insulators also have a more mundane aspect of their electronic structure, characterized by a band gap. Another class of materials, topological Dirac semimetals, has been proposed that has a linear dispersion along all three momentum directions—a bulk Dirac cone; these materials are predicted to have intriguing electronic properties and to be related to other exotic states through quantum phase transitions. Liu et al. (p. 864 , published online 16 January) detected such a state in the compound Na3Bi by using photoemission spectroscopy.

Three-dimensional (3D) Dirac point, where two Weyl points overlap in momentum space, is usually unstable and hard to realize. Here we show, based on the first-principles calculations and effective model analysis, that crystalline A3Bi (A=Na, K, Rb) are Dirac semimetals with bulk 3D Dirac points protected by crystal symmetry. They possess nontrivial Fermi arcs on the surfaces and can be driven into various topologically distinct phases by explicit breaking of symmetries. Giant diamagnetism, linear quantum magnetoresistance, and quantum spin Hall effect will be expected for such compounds.

Three-dimensional topological (crystalline) insulators are materials with an insulating bulk but conducting surface states that are topologically protected by time-reversal (or spatial) symmetries. We extend the notion of three-dimensional topological insulators to systems that host no gapless surface states but exhibit topologically protected gapless hinge states. Their topological character is protected by spatiotemporal symmetries of which we present two cases: (i) Chiral higher-order topological insulators protected by the combination of time-reversal and a fourfold rotation symmetry. Their hinge states are chiral modes, and the bulk topology is Z2 -classified. (ii) Helical higher-order topological insulators protected by time-reversal and mirror symmetries. Their hinge states come in Kramers pairs, and the bulk topology is Z-classified. We provide the topological invariants for both cases. Furthermore, we show that SnTe as well as surface-modified Bi₂TeI, BiSe, and BiTe are helical higher-order topological insulators and propose a realistic experimental setup to detect the hinge states.

In 3D momentum space, a topological phase boundary separating the Chern insulating layers from normal insulating layers may exist, where the gap must be closed, resulting in a ``Chern semimetal'' state with topologically unavoidable band crossings at the Fermi level. This state is a condensed-matter realization of Weyl fermions in (3+1)D, and should exhibit remarkable features, such as magnetic monopoles and Fermi arcs. Here we predict, based on first principles calculations, that such a novel quantum state can be realized in a known ferromagnetic compound HgCr2Se4, with a single pair of Weyl fermions separated in momentum space. The quantum Hall effect without an external magnetic field can be achieved in its quantum-well structure.