Connectivity of edge and surface states in topological insulators

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.

Figure 1

2D QSH states and helical edge states in a stripe geometry. (Black arrows denote the propagating direction for one state of the edge helical pair). The band insulator (bordered by green dashed lines) is put on the QSH strip (a) on one edge and (b) bridging two edges.

Figure 2

3D TI in a pipe geometry with translationally invariant cross section. The local coordination for the wave vector is drawn for the front and the top faces. If the low-energy Hamiltonian of the front face is characterized by a single Dirac cone, how many Dirac cones are on the top surface?