Elliptic operators with honeycomb symmetry: Dirac points, Edge States and Applications to Photonic Graphene

J. P. Lee-Thorp M. I. Weinstein Yi Zhu

Mathematical Physics mathscidoc:2204.22003

2018.9
Consider electromagnetic waves in two-dimensional honeycomb structured media. The properties of transverse electric (TE) polarized waves are determined by the spectral properties of the elliptic operator L^A=-\nabla_x·A(x) \nabla_x, where A(x) is Λ_h− periodic (Λ_h denotes the equilateral triangular lattice), and such that with respect to some origin of coordinates, A(x) is PC− invariant (A(x)=\overline{A(-x)}) and 120° rotationally invariant (A(R*x)=R*A(x)R, where R is a 120° rotation in the plane). We first obtain results on the existence, stability and instability of Dirac points, conical intersections between two adjacent Floquet-Bloch dispersion surfaces. We then show that the introduction through small and slow variations of a domain wall across a line-defect gives rise to the bifurcation from Dirac points of highly robust (topologically protected) edge states. These are time-harmonic solutions of Maxwell's equations which are propagating parallel to the line-defect and spatially localized transverse to it. The transverse localization and strong robustness to perturbation of these edge states is rooted in the protected zero mode of a one-dimensional effective Dirac operator with spatially varying mass term. These results imply the existence of uni-directional propagating edge states for two classes of time-reversal invariant media in which C symmetry is broken: magneto-optic media and bi-anisotropic media. Our analysis applies and extends the tools previously developed in the context of honeycomb Schrödinger operators.
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@inproceedings{j.2018elliptic,
  title={Elliptic operators with honeycomb symmetry: Dirac points, Edge States and Applications to Photonic Graphene},
  author={J. P. Lee-Thorp, M. I. Weinstein, and Yi Zhu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220421120223791080089},
  year={2018},
}
J. P. Lee-Thorp, M. I. Weinstein, and Yi Zhu. Elliptic operators with honeycomb symmetry: Dirac points, Edge States and Applications to Photonic Graphene. 2018. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220421120223791080089.
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