Topologically protected states in one-dimensional systems
Author(s)
Bibliographic Information
Topologically protected states in one-dimensional systems
（Memoirs of the American Mathematical Society, no. 1173）
American Mathematical Society, c2017
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Note
"Volume 247, number 1173 (sixfth of 7 numbers), May 2017"
Includes bibliographical references (p. 117-118)
Description and Table of Contents
Description
The authors study a class of periodic Schrodinger 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
Introduction and outline
Floquet-Bloch and Fourier analysis
Dirac points of 1D periodic structures
Domain wall modulated periodic Hamiltonian and formal derivation of topologically protected bound states
Main Theorem--Bifurcation of topologically protected states
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
References
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