Topological Classification of Crystalline Insulators with Point Group Symmetry

TL;DR

This paper demonstrates that point group symmetry in crystalline insulators leads to a Z3 topological classification based on electric polarization quantization, with potential realization in graphene on BN substrate.

Contribution

It introduces a topological classification based solely on point group symmetry, extending to all 17 two-dimensional space groups, and identifies a candidate material for Z3 topological states.

Findings

Polarization is quantized into three values under C3 symmetry.

A tight-binding model shows a Z3 topological phase transition.

Graphene on BN substrate may realize Z3 topological states.

Abstract

We show that in crystalline insulators point group symmetry alone gives rise to a topological classification based on the quantization of electric polarization. Using C3 rotational symmetry as an example, we first prove that the polarization is quantized and can only take three inequivalent values. Therefore, a Z3 topological classification exists. A concrete tight-binding model is derived to demonstrate the Z3 topological phase transition. Using first-principles calculations, we identify graphene on BN substrate as a possible candidate to realize the 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 graphene like systems.