paper

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

arXiv:1710.03389 · doi:10.1007/s00205-018-1315-4

Abstract

Consider electromagnetic waves in two-dimensional {\it honeycomb structured media}. The properties of transverse electric (TE) polarized waves are determined by the spectral properties of the elliptic operator $\LA=-\nabla_\bx\cdot A(\bx) \nabla_\bx$, where $A(\bx)$ is periodic ( denotes the equilateral triangular lattice), and such that with respect to some origin of coordinates, $A(\bx)$ is invariant ($A(\bx)=\overline{A(-\bx)}$) and rotationally invariant ($A(R^*\bx)=R^*A(\bx)R$, where is a 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 {\it domain wall} across a line-defect gives rise to the bifurcation from Dirac points of highly robust (topologically protected) {\it 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 {\it uni-directional} propagating edge states for two classes of time-reversal invariant media in which 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.

65 pages, 8 figures, To appear in Archive for Rational Mechanics and Analysis

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