In general, the stability of a band crossing point indicates the presence of a quantized topological number associated with it. In particular, the recent discovery of three-dimensional Dirac semimetals in ${\mathrm{Na}}_3\mathrm{Bi}$ and ${\mathrm{Cd}}_3{\mathrm{As}}_2$ demonstrates that a Dirac point with fourfold degeneracy can be stable as long as certain crystalline symmetries are supplemented in addition to the time-reversal and inversion symmetries. However, the topological charges associated with ${\mathrm{Na}}_3\mathrm{Bi}$ and ${\mathrm{Cd}}_3{\mathrm{As}}_2$ are not clarified yet. In this work, we identify the topological charge of three-dimensional Dirac points. It is found that although the simultaneous presence of the time-reversal and inversion symmetries forces the net chiral charge to vanish, a Dirac point can carry another quantized topological charge when an additional rotation symmetry is considered. Two different classes of Dirac semimetals are identified depending on the nature of the rotation symmetries. First, the conventional symmorphic rotational symmetry which commutes with the inversion gives rise to the class I Dirac semimetals having a pair of Dirac points on the rotation axes. Since the topological charges of each pair of Dirac points have the opposite sign, a pair creation or a pair annihilation is required to change the number of Dirac points in the momentum space. On the other hand, the class II Dirac semimetals possess a single isolated Dirac point at a time-reversal invariant momentum, which is protected by a screw rotation. The nonsymmorphic nature of screw rotations allows the anticommutation relation between the rotation and inversion symmetries, which enables to circumvent the doubling of the number of Dirac points and create a single Dirac point at the Brillouin zone boundary.

• Received 4 June 2015

DOI:https://doi.org/10.1103/PhysRevB.92.165120