Topologically Protected States in One-Dimensional Systems

ISBN-13: 9781470423230 / Angielski

C.F. Fefferman;J.P. Lee-Thorp;M.I. Weinstein

Opis

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.

Spis treści

• 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