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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)
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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Table of Contents


  • 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


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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