Helical edge modes preserved across transitions from topological insulators with indirect gaps to insulators or semimetals

Helical edge modes are characteristic of topological insulators in two dimensions. This paper demonstrates that helical edge modes remain across transitions to ordinary insulators or to semimetals under certain conditions. Straight and zigzag edges are considered in a tight-binding model on a square lattice. We focus on the case of an indirect gap in bulk topological insulators and obtain the spectrum of the edge modes on both sides of the transitions. For a straight edge, the helical edge mode in topological insulators with strong particle-hole asymmetry has a reentrant region in momentum space. Edge modes show up even in ordinary insulators but are absent in semimetals. In the zigzag edge, the helical edge mode survives in both semimetals and ordinary insulators. However, the edge modes are absent inside the energy gap of ordinary insulators. All results are obtained analytically.