  A three-dimensional (3D) Dirac semimetal is the 3D analog of graphene whose
bulk band shows a linear dispersion relation in the 3D momentum space. Since
each Dirac point with four-fold degeneracy carries a zero Chern number, a Dirac
semimetal can be stable only in the presence of certain crystalline symmetries.
In this work, we propose a general framework to classify stable 3D Dirac
semimetals. Based on symmetry analysis, we show that various types of stable 3D
Dirac semimetals exist in systems having the time-reversal, inversion, and
uniaxial rotational symmetries. There are two distinct classes of stable 3D
Dirac semimetals. In the first class, a pair of 3D Dirac points locate on the
rotation axis, away from its center. Moreover, the 3D Dirac semimetals in this
class have nontrivial topological properties characterized by 2D topological
invariants, such as the $Z_{2}$ invariant or the mirror Chern number. These 2D
topological indices give rise to stable 2D surface Dirac cones, which can be
deformed to Fermi arcs when the surface states couple to the bulk states on the
Fermi level. On the other hand, the second class of Dirac SM phases possess a
3D Dirac point at a time-reversal invariant momentum (TRIM) on the rotation
axis and do not have surface states in general.
