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Topologically Protected States in One-Dimensional Systems
 
C. L. Fefferman Princeton University, New Jersey
J. P. Lee-Thorp Columbia University, New York, NY
M. I. Weinstein Columbia University, New York, NY
Topologically Protected States in One-Dimensional Systems
eBook ISBN:  978-1-4704-3707-7
Product Code:  MEMO/247/1173.E
List Price: $75.00
MAA Member Price: $67.50
AMS Member Price: $45.00
Topologically Protected States in One-Dimensional Systems
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Topologically Protected States in One-Dimensional Systems
C. L. Fefferman Princeton University, New Jersey
J. P. Lee-Thorp Columbia University, New York, NY
M. I. Weinstein Columbia University, New York, NY
eBook ISBN:  978-1-4704-3707-7
Product Code:  MEMO/247/1173.E
List Price: $75.00
MAA Member Price: $67.50
AMS Member Price: $45.00
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2472017; 118 pp

    The authors study a class of periodic Schrödinger operators, which in distinguished cases can be proved to have linear band-crossings or “Dirac points”. They 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. The authors' 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 the authors construct can be realized as highly robust TM-electromagnetic modes for a class of photonic waveguides with a phase-defect.

  • 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
  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2472017; 118 pp

The authors study a class of periodic Schrödinger operators, which in distinguished cases can be proved to have linear band-crossings or “Dirac points”. They 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. The authors' 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 the authors construct can be realized as highly robust TM-electromagnetic modes for a class of photonic waveguides with a phase-defect.

  • 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
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.