Helical Edge Modes near Transition to Topological Insulator with Indirect Gap

TL;DR

This paper analytically investigates the persistence of helical edge modes across topological phase transitions, including in ordinary insulators and semimetals, focusing on systems with indirect gaps and different edge geometries.

Contribution

It demonstrates that helical edge modes can survive beyond topological insulators, including in semimetals and ordinary insulators, under specific conditions, with detailed analytical spectra.

Findings

Helical edge modes persist across certain phase transitions.

Edge modes appear even in ordinary insulators with indirect gaps.

Edge modes are absent in semimetals for straight edges but survive in zigzag edges.

Abstract

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 condition. Straight and zigzag edges are considered in a tight-binding model on square lattice. We focus on the case of indirect gap in bulk topological insulators, and obtain the spectrum of edge modes on both sides of transitions. For 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 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.