Delocalization of edge states in topological phases (Vol. 50, No. 5-6)

Delocalization of edge states in topological phases
Dispersions in a topological system with positive indirect gap and edge states (left); negative indirect gap and no edge states (right).

Topological properties are a hot topic currently. If the bulk of a system is topologically non-trivial (Chern number), the bulk-boundary correspondence predicts in-gap states in finite samples. These states close the energy gap between bands of different topology so that it can change at boundaries. Conventionally, the in-gap states are localized at these boundaries so that they are edge states. We show, however, that this localization only occurs for positive indirect gap. Generically, without indirect gap the in-gap states become extended by mixing with bulk states despite . This is illustrated for two fundamental lattice models (Haldane and checkerboard model) by adding terms to the Hamiltonians proportional to the identity in momentum space. Thus, the dispersions change while the topology remains unchanged. These terms can close the indirect gap and lead to delocalization of edge states in finite geometries. Thus, discrete topological invariants may exist without localized edge modes. This underlines the vital significance of indirect gaps for the existence of topological edge states and puts the bulk-boundary into perspective.

M. Malki and G. S. Uhrich, Delocalization of edge states in topological phases, EPL 127, 27001 (2019)