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.
C. F. Fefferman, Princeton University, New Jersey.J. P. Lee-Thorp, Columbia University, New York, NY.M. I. Weinstein, Columbia University, New York, NY.
Introduction and outlineFloquet-Bloch and Fourier analysisDirac points of 1D periodic structuresDomain wall modulated periodic Hamiltonian and formal derivation of topologically protected bound statesMain Theorem--Bifurcation of topologically protected statesProof of the Main TheoremAppendix A. A variant of Poisson summationAppendix B. 1D Dirac points and Floquet-Bloch eigenfunctionsAppendix C. Dirac points for small amplitude potentialsAppendix D. Genericity of Dirac points - 1D and 2D casesAppendix E. Degeneracy lifting at Quasi-momentum zeroAppendix F. Gap opening due to breaking of inversion symmetryAppendix G. Bounds on leading order terms in multiple scale expansionAppendix H. Derivation of key bounds and limiting relations in the Lyapunov-Schmidt reductionReferences
