Topological Classification of Crystalline Insulators with Point Group Symmetry

We show that in crystalline insulators point group symmetry alone gives rise to a topological classification based on the quantization of electric polarization. Using $C_3$ rotational symmetry as an example, we first prove that the polarization is quantized and can only take three inequivalent values. Therefore, a $Z_3$ topological classification exists. A concrete tight-binding model is derived to demonstrate the $Z_3$ topological phase transition. Using first-principles calculations, we identify graphene on BN substrate as a possible candidate to realize the $Z_3$ topological states. To complete our analysis we extend the classification of band structures to all 17 two-dimensional space groups. This work will contribute to a complete theory of symmetry conserved topological phases and also elucidate topological properties of graphene like systems.