Three-dimensional higher-spin Dirac and Weyl dispersions in the strongly isotropic $K_4$ crystal

We analyze the electronic structure in the three-dimensional (3D) crystal formed by the $sp^2$ hybridized orbitals ($K_4$ crystal), by the tight-binding approach based on the first-principles calculation. We discover that the bulk Dirac-cone dispersions are realized in the $K_4$ 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 $\Gamma$ and $H$ points of the bcc Brillouin zone, where three bands become degenerate and merge at a single point belonging to the $T_2$ 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 $4\times 4$ 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 $\bar E_2$ and four-dimensional $\bar F$ representations. Remarkably, all the dispersions near the $\bar F$ point show the linear dependence in the momentum with different velocities. We derive the effective Hamiltonian near the $\bar F$ point and find that the band contact point is described by the $S=3/2$ Weyl point.