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Topologically Protected States in One-Dimensional Systems

C.F. Fefferman (author) J.P. Lee-Thorp (author)
M.I. Weinstein (author)

ISBN: 9781470423230

Publication Date: May 2017

Format: Paperback

Examines a class of periodic Schrodinger operators, which in distinguished cases can be proved to have linear band-crossings or "Dirac points". The authors 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".
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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.
Pages 118
Dimensions 254 x 178
Date Published 30 May 2017
Publisher American Mathematical Society
Series Memoirs of the American Mathematical Society
Series Part Volume: 247 Number: 1173
Subject/s Calculus & mathematical analysis   Mathematical physics  
  • 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
C. F. Fefferman, Princeton University, New Jersey.

J. P. Lee-Thorp, Columbia University, New York, NY.

M. I. Weinstein, Columbia University, New York, NY.

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