Connection topology of step edge state bands at the surface of a three dimensional topological insulator

Topological insulators in the Bi$_2$Se$_3$ family manifest helical Dirac surface states that span the topologically ordered bulk band gap. Recent scanning tunneling microscopy measurements have discovered additional states in the bulk band gap of Bi$_2$Se$_3$ and Bi$_2$Te$_3$, localized at one dimensional step edges. Here numerical simulations of a topological insulator surface are used to explore the phenomenology of edge state formation at the single-quintuple-layer step defects found ubiquitously on these materials. The modeled one dimensional edge states are found to exhibit a stable topological connection to the two dimensional surface state Dirac point.