Regarding three-dimensional (3D) topological insulators and semimetals as a stack of constituent two-dimensional (2D) topological (or sometimes nontopological) systems is a useful viewpoint. Here, we perform a comparative study of the paradigmatic 3D topological phases: Weyl semimetal (WSM), strong and weak topological insulators (STI/WTI), and Chern insulator (CI). By calculating the $\mathbb{Z}$ and ${\mathbb{Z}}_2$ indices for the thin films of such 3D topological phases, we follow dimensional evolution of topological properties from 2D to 3D. It is shown that the counterparts of STI and WTI in the time-reversal symmetry broken CI system are, respectively, WSM and CI phases. The number ${\mathcal{N}}_D$ of helical Dirac cones emergent on the surface of a topological insulator is shown to be identical to the number ${\mathcal{N}}_W$ of the pairs of Weyl cones in the corresponding WSM phase: ${\mathcal{N}}_D={\mathcal{N}}_W$. To test the robustness of this scenario against disorder, we have studied the transport property of disordered WSM thin films, taking into account both the bulk and surface contributions.

• Received 10 June 2016
• Revised 12 September 2016

DOI:https://doi.org/10.1103/PhysRevB.94.235414