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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)
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

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Table of Contents


  • 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


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.

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