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# Topologically Protected States in One-Dimensional Systems

### About this Title

**C. L. Fefferman**, *Department of Mathematics, Princeton University, Princeton, New Jersey*, **J. P. Lee-Thorp**, *Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York* and **M. I. Weinstein**, *Department of Applied Physics and Applied Mathematics and Department of Mathematics, Columbia University, New York, New York*

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

MSC: Primary 35J10, 35B32; Secondary 35P, 35Q41, 37G40, 34B30

### Table of Contents

**Chapters**

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