title: Topologically protected states in one-dimensional continuous systems and Dirac points.
abstract: We study a class of periodic Schrödinger 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-energy mode 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.
authors.affiliation: Department of Mathematics, Princeton University, Princeton, NJ 08544; and cf@math.princeton.edu.
authors.forename: Charles L
authors.initials: CL
authors.lastname: Fefferman
authors.affiliation: Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027.
authors.forename: James P
authors.initials: JP
authors.lastname: Lee-Thorp
authors.affiliation: Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027.
authors.forename: Michael I
authors.initials: MI
authors.lastname: Weinstein
keywords: Floquet–Bloch theory; Hill’s equation; multiple scale analysis; surface states; wave-packets