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H 1 Dispersive waves in novel 2d media; Honeycomb structures, edge states and the strong binding regime

Auteurs : Weinstein, Michael (Auteur de la Conférence)
CIRM (Editeur )

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    Résumé : We discuss the 2D Schrödinger equation for periodic potentials with the symmetry of a hexagonal tiling of the plane. We first review joint work with CL Fefferman on the existence of Dirac points, conical singularities in the band structure, and the resulting effective 2D Dirac dynamics of wave-packets. We then focus on periodic potentials which are superpositions of localized potential wells, centered on the vertices of a regular honeycomb structure, corresponding to the single electron model of graphene and its artificial analogues. We prove that for sufficiently deep potentials (strong binding) the lowest two Floquet-Bloch dispersion surfaces, when appropriately rescaled, are uniformly close to those of the celebrated two-band tight-binding model, introduced by PR Wallace (1947) in his pioneering study of graphite. We then discuss corollaries, in the strong binding regime, on (a) spectral gaps for honeycomb potentials with PT symmetry-breaking perturbations, and (b) topologically protected edge states for honeycomb structures with "rational edges. This is joint work with CL Fefferman and JP Lee-Thorp. Extensions to Maxwell equations (with Y Zhu and JP Lee-Thorp) will also be discussed.

    Codes MSC :
    35B32 - Bifurcation (PDE)
    35J10 - Schrödinger operator
    37G40 - Symmetries, equivariant bifurcation theory
    35Q41 - Time-dependent Schrödinger equations, Dirac equations

      Informations sur la Vidéo

      Langue : Anglais
      Date de publication : 22/06/17
      Date de captation : 13/06/17
      Collection : Partial Differential Equations
      Sous collection : Research talks
      Format : MP4
      Domaine : PDE
      Durée : 00:54:42
      Audience : Chercheurs ; Doctorants , Post - Doctorants
      Download : https://videos.cirm-math.fr/2017-06-13_Weinstein.mp4

    Informations sur la rencontre

    Nom de la rencontre : French-American conference on nonlinear dispersive PDEs / Conférence franco-américaine sur les EDP dispersives non linéaires
    Organisateurs de la rencontre : Carles, Rémi ; Holmer, Justin ; Roudenko, Svetlana
    Dates : 12/06/17 - 16/06/17
    Année de la rencontre : 2017
    URL Congrès : http://conferences.cirm-math.fr/1510.html

    Citation Data

    DOI : 10.24350/CIRM.V.19183103
    Cite this video as: Weinstein, Michael (2017). Dispersive waves in novel 2d media; Honeycomb structures, edge states and the strong binding regime. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19183103
    URI : http://dx.doi.org/10.24350/CIRM.V.19183103

    Voir aussi


    1. Fefferman, C.L., Lee-Thorp, J.P., & Weinstein, M.I. (2017). Honeycomb Schroedinger operators in the strong binding regime. - https://arxiv.org/abs/1610.04930

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