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

TL;DR

This paper investigates the spectral properties of elliptic operators with honeycomb symmetry, demonstrating the existence of Dirac points and topologically protected edge states in photonic media, with implications for unidirectional wave propagation.

Contribution

It extends analysis of honeycomb Schrödinger operators to electromagnetic waves, revealing Dirac points and robust edge states in photonic graphene-like structures.

Findings

Existence and stability of Dirac points in honeycomb structured media.

Formation of topologically protected edge states at line defects.

Potential for unidirectional wave propagation in certain media.

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 $\Lambda_h-$ periodic ($\Lambda_h$ denotes the equilateral triangular lattice), and such that with respect to some origin of coordinates, $A(\bx)$ is $\mathcal{P}\mathcal{C}-$ invariant ($A(\bx)=\overline{A(-\bx)}$) and $120^\circ$ rotationally invariant ($A(R^*\bx)=R^*A(\bx)R$, where $R$ is a $120^\circ$ 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 $\mathcal{C}$ symmetry is broken: magneto-optic media and bi-anisotropic media. Our analysis applies and extends the tools previously developed in the context of honeycomb Schr\"odinger operators.