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The edge states of a two-dimensional quantum spin Hall (QSH) insulator form aone-dimensional helical metal which is responsible for the transport propertyof the QSH insulator. Conceptually, such a one-dimensional helical metal can beattached to any scattering region as the usual metallic leads. We study theanalytical property of the scattering matrix for such a conceptualmultiterminal scattering problem in the presence of time reversal invariance.As a result, several theorems on the connectivity property of helical edgestates in two-dimensional QSH systems as well as surface states ofthree-dimensional topological insulators are obtained. Without addressing realmodel details, these theorems, which are phenomenologically obtained, emphasizethe general connectivity property of topological edge/surface states from themere time reversal symmetry restriction.

Connectivity of edge and surface states in topological insulators