Topological classification of crystalline insulators with space group symmetry

We show that in crystalline insulators, space group symmetry alone gives rise to a topological classification based on the discretization of electric polarization. Using C₃ rotational symmetry as an example, we first prove that the polarization is discretized into three distinct classes, i.e., it can only take three inequivalent values. We then prove that these classes are topologically distinct. Therefore, a Z₃ topological classification exists, with polarization as a topological class index. A concrete tight-binding model is derived to demonstrate the Z₃ topological phase transition. Using first-principles calculations, we identify graphene on a BN substrate as a possible candidate to realize these Z₃ 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 graphenelike systems.