Connectivity of edge and surface states in topological insulators

TL;DR

This paper investigates the fundamental connectivity properties of edge and surface states in topological insulators using analytical scattering matrix methods, emphasizing the role of time reversal symmetry.

Contribution

It provides general theorems on the connectivity of topological edge and surface states based solely on symmetry considerations, without relying on specific models.

Findings

Theorems on the connectivity of helical edge states in 2D QSH insulators.

Theorems on the connectivity of surface states in 3D topological insulators.

Connectivity properties are dictated by time reversal symmetry alone.

Abstract

The edge states of a two-dimensional quantum spin Hall (QSH) insulator form a one-dimensional helical metal which is responsible for the transport property of the QSH insulator. Conceptually, such a one-dimensional helical metal can be attached to any scattering region as the usual metallic leads. We study the analytical property of the scattering matrix for such a conceptual multiterminal scattering problem in the presence of time reversal invariance. As a result, several theorems on the connectivity property of helical edge states in two-dimensional QSH systems as well as surface states of three-dimensional topological insulators are obtained. Without addressing real model details, these theorems, which are phenomenologically obtained, emphasize the general connectivity property of topological edge/surface states from the mere time reversal symmetry restriction.