# Topologically Protected States in One-Dimensional Systems

### About this Title

**C. L. Fefferman**, **J. P. Lee-Thorp** and **M. I. Weinstein**

Publication: Memoirs of the American Mathematical Society

Publication Year:
2017; Volume 247, Number 1173

ISBNs: 978-1-4704-2323-0 (print); 978-1-4704-3707-7 (online)

DOI: https://doi.org/10.1090/memo/1173

Published electronically: February 1, 2017

Keywords:Schrödinger equation, Dirac equation, Floquet-Bloch theory, topological
protection, edge states, Hill’s equation, domain wall

### Table of Contents

**Chapters**

- Chapter 1. Introduction and Outline
- Chapter 2. Floquet-Bloch and Fourier Analysis
- Chapter 3. Dirac Points of 1D Periodic Structures
- Chapter 4. Domain Wall Modulated Periodic Hamiltonian and Formal Derivation of Topologically Protected Bound States
- Chapter 5. Main Theorem—Bifurcation of Topologically Protected States
- Chapter 6. Proof of the Main Theorem
- Appendix A. A Variant of Poisson Summation
- Appendix B. 1D Dirac points and Floquet-Bloch Eigenfunctions
- Appendix C. Dirac Points for Small Amplitude Potentials
- Appendix D. Genericity of Dirac Points - 1D and 2D cases
- Appendix E. Degeneracy Lifting at Quasi-momentum Zero
- Appendix F. Gap Opening Due to Breaking of Inversion Symmetry
- Appendix G. Bounds on Leading Order Terms in Multiple Scale Expansion
- Appendix H. Derivation of Key Bounds and Limiting Relations in the Lyapunov-Schmidt Reduction

### Abstract

We study a class of periodic Schrödinger operators, which in distinguished cases can be proved to have linear band-crossings or âĂĲDirac pointsâĂİ. We then show that the introduction of an âĂĲedgeâĂİ, via adiabatic modulation of these periodic potentials by a domain wall, results in the bifurcation of spatially localized âĂĲedge statesâĂİ. 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.

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