Topologically protected states in one-dimensional continuous systems and Dirac points

We study a class of periodic Schrodinger operators on R that have Dirac points. The introduction of an edge via adiabatic modulation of a periodic potential by a domain wall results in the bifurcation of spatially localized edge states, associated with the topologically protected zero-energymode of an asymptotic one-dimensional Dirac operator. The bound states we construct can be realized as highly robust transverse-magnetic electromagnetic modes for a class of photonic waveguides with a phase defect. Our model captures many aspects of the phenomenon of topologically protected edge states for 2D bulk structures such as the honeycomb structure of graphene.