Topologically Protected States in One-Dimensional Systems

TL;DR

This paper investigates topologically protected edge states in one-dimensional periodic Schrödinger operators, demonstrating their robustness and connection to Dirac points, with implications for photonic waveguides.

Contribution

It introduces a model linking topologically protected edge states in 1D systems to Dirac points, with potential applications in photonics.

Findings

Edge states bifurcate from Dirac points due to domain walls.

Edge states are topologically protected and robust against perturbations.

The model applies to photonic waveguides with phase defects.

Abstract

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. The states we construct can be realized as highly robust TM- electromagnetic modes for a class of photonic waveguides with a phase-defect.