Topologically Protected States in One-Dimensional Systems

Fefferman, C.; Lee-Thorp, J.; Weinstein, M.

doi:10.1090/memo/1173

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Topologically Protected States in One-Dimensional Systems

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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

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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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Topologically Protected States in One-Dimensional Systems

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Topologically Protected States in One-Dimensional Systems