Topologically Protected States in One-Dimensional Systems

Understand

We study a class of periodic Schr\"odinger operators, which in distinguished cases can be proved to have linear band-crossings or "Dirac points".

• We then show that the introduction of an "edge", via adiabatic modulation of these periodic potentials by a domain wall, results in the bifurcation of spatially localized "edge states".
• These bound states are associated with the topologically protected zero-energy mode of an asymptotic one-dimensional Dirac operator.
• Our model captures many aspects of the phenomenon of topologically protected edge states for two-dimensional bulk structures such as the honeycomb structure of graphene.