We analyze the electronic structure in the three-dimensional (3D) crystal formed by the sp² hybridized orbitals (K₄ crystal), by the tight-binding approach based on the first-principles calculation. We discover that the bulk Dirac-cone dispersions are realized in the K4 crystal. In contrast to the graphene, the energy dispersions of the Dirac cones are isotropic in 3D and the pseudospin S=1 Dirac cones emerge at the Γ and H points of the bcc Brillouin zone, where three bands become degenerate and merge at a single point belonging to the T₂ irreducible representation. In addition, the usual S=1/2 Dirac cones emerge at the P point. By focusing the hoppings between the nearest-neighbor sites, we show an analytic form of the tight-binding Hamiltonian with a 4×4 matrix, and we give an explicit derivation of the S=1 and S=1/2 Dirac-cone dispersions. We also analyze the effect of the spin-orbit coupling to examine how the degeneracies at Dirac points are lifted. At the S=1 Dirac points, the spin-orbit coupling lifts the energy level with sixfold degeneracy into two energy levels with two-dimensional E¯₂ and four-dimensional F¯ representations. Remarkably, all the dispersions near the F¯ point show the linear dependence in the momentum with different velocities. We derive the effective Hamiltonian near the F¯ point and find that the band contact point is described by the S=3/2 Weyl point.

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