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

Topological invariants for three dimensional Dirac semimetals and four dimensional topological rotational insulators