Strong and weak second-order topological insulators with hexagonal symmetry and $\mathbb{Z}_3$ index

TL;DR

This paper introduces second-order topological insulators with hexagonal symmetry, characterized by a $ ext{Z}_3$ index, demonstrating the emergence of hinge and corner states in 3D and 2D systems, respectively.

Contribution

The work constructs 3D and 2D SOTIs with hexagonal symmetry and a $ ext{Z}_3$ index, revealing the origin of strong and weak phases and their topological hinge and corner states.

Findings

Identification of $ ext{Z}_3$ topological index protected by $IT$ and $C_6I$ symmetries.

Demonstration of hinge states in 3D SOTIs with hexagonal prisms.

Observation of corner states in 2D hexagonal building blocks.

Abstract

We propose second-order topological insulators (SOTIs) whose lattice structure has the hexagonal symmetry $C_{6}$ in three and two dimensions. We start with a three-dimensional weak topological insulator constructed on the stacked triangular lattice, which has only side topological surface states. We then introduce an additional mass term which gaps out the side surface states but preserves the hinge states. The resultant system is a three-dimensional SOTI. The bulk topological quantum number is shown to be the $\mathbb{Z}_{3}$ index protected by the inversion time-reversal symmetry $IT$ and the rotoinversion symmetry $C_{6}I$. We obtain three phases; trivial, strong and weak SOTI phases. We argue the origin of these two types of SOTIs. A hexagonal prism is a typical structure respecting these symmetries, where six topological hinge states emerge at the side. The building block is a hexagon in two dimensions, where topological corner states emerge at the six corners in the SOTI phase. Strong and weak SOTIs are obtained when the interlayer hopping interaction is strong and weak, respectively. They are characterized by the emergence of hinge states attached to or detached from the bulk bands.