Topologically Protected States in One-Dimensional Systems.

Cover -- Title page -- Chapter 1. Introduction and Outline -- 1.1. Motivating example-Dimer model with a phase defect -- 1.2. Outline of main results -- 1.3. Outline -- 1.4. Notation -- 1.5. Acknowledgements -- Chapter 2. Floquet-Bloch and Fourier Analysis -- 2.1. Floquet-Bloch theory-1D -- 2.2. Poisson summation in ²_{ } -- Chapter 3. Dirac Points of 1D Periodic Structures -- 3.1. The family of Hamiltonians, ( ), and its Dirac points for =1/2 -- 3.2. (1/2)=-\Dₓ²+ ( -- 1/2) has an additional translation symmetry -- 3.3. The action of -\Dₓ²+ _{\ee}( ) on ²_{ _{⋆}= } -- 3.4. Spectral properties of ^{(\eps=0)}=-\Dₓ² in ²_{ } -- 3.5. Sufficient conditions for occurrence of a 1D Dirac point -- 3.6. Expansion of Floquet-Bloch eigenfunctions near a Dirac point -- 3.7. Genericity of Dirac points at = _{⋆} -- Chapter 4. Domain Wall Modulated Periodic Hamiltonian and Formal Derivation of Topologically Protected Bound States -- 4.1. Formal multiple scale construction of "edge states" -- 4.2. Spectrum of the 1D Dirac operator and its topologically protected zero energy eigenstate -- Chapter 5. Main Theorem-Bifurcation of Topologically Protected States -- Chapter 6. Proof of the Main Theorem -- 6.1. Rough strategy -- 6.2. Detailed strategy: Decomposition into near and far energy components -- 6.3. Analysis of far energy components -- 6.4. Lyapunov-Schmidt reduction to a Dirac system for the near energy components -- 6.5. Analysis of the band-limited Dirac system -- 6.6. Proof of Proposition 6.10 -- 6.7. Final reduction to an equation for = ( ) and its solution -- Appendix A. A Variant of Poisson Summation -- Appendix B. 1D Dirac points and Floquet-Bloch Eigenfunctions -- B.1. Conditions ensuring a 1D Dirac point -- proof of Theorem 3.6 -- B.2. Expansion of Floquet-Bloch modes near a 1D Dirac -- proof of Proposition 3.7.

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 -- References -- Back Cover.

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.

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