Boundary states in the chiral symmetric systems with a spatial symmetry

TL;DR

This paper investigates topological boundary states in systems with combined chiral and spatial symmetries, revealing how perturbations affect symmetry protection and identifying related topological invariants in 1D and 3D models.

Contribution

It introduces a classification of topological states protected by combined chiral and spatial symmetries and connects boundary invariants with bulk topological invariants in 3D systems.

Findings

Perturbations can break one or both chiral symmetries depending on spatial symmetry preservation.

Boundary modes in 3D can be protected by hidden lower-dimensional symmetries.

Topological invariants are derived and linked to bulk properties.

Abstract

We study topological systems with both a chiral and a spatial symmetry which result in an additional spatial chiral symmetry. We distinguish the topologically nontrivial states according to the chiral symmetries protecting them and study several models in 1D and 3D systems. The perturbations breaking the spatial symmetry can break only one of the two chiral symmetries while the perturbations preserving the spatial symmetry always break or preserve both of them. In 3D systems, besides the 3D symmetries, the topologically nontrivial boundary modes may also be protected by the hidden lower dimensional symmetries. We then figure out the corresponding topological invariants and connect them with the 3D invariants.