  Dirac semimetal is a class of semi-metallic phase protected by certain types
of crystalline symmetries, and its low-energy effective Hamiltonian is
described by Dirac equations in three dimensions (3D). Despite of various
theoretical studies, theories that describe the topological nature of Dirac
semimetals have not been well established. In this work, we define a
topological invariant for 3D Dirac semimetals by establishing a mapping between
a 3D Dirac semimetal and a topological crystalline insulator in four dimension
(4D). We demonstrate this scheme by constructing a tight-binding model for 4D
topological crystalline insulators that are protected by rotational symmetry. A
new type of topological invariant, "rotational Chern number", is shown to
characterize the topology of this system. As a consequence of the rotational
Chern number, gapless Dirac points are found on the 3D surface of this 4D
system. For a slab with two surfaces, we find that the corresponding low-energy
effective theory of two surface states can be directly mapped to that of a 3D
Dirac semimetal, suggesting that topological nature of 3D Dirac semimetals can
be characterized by rotational Chern number which is defined in 4D. Our scheme
provides a new systematic approach to extract topological nature for
topological semimetal phases.
