Elliptic Operators with Honeycomb Symmetry: Dirac Points, Edge States and Applications to Photonic Graphene

Abstract

Consider electromagnetic waves in two-dimensional honeycomb structured media, whose constitutive laws have the symmetries of a hexagonal tiling of the plane. The properties of transverse electric (TE) polarized waves are determined by the spectral properties of the elliptic operatordenotes the equilateral triangular lattice), and such that with respect to some origin of coordinates,where R is a 120 • rotation in the plane). A summary of our results is as follows: a) For generic honeycomb structured media, the band structure of L A has Dirac points, i.e. conical intersections between two adjacent Floquet-Bloch dispersion surfaces. b) Initial data of wave-packet type, which are spectrally concentrated about a Dirac point, give rise to solutions of the time-dependent Maxwell equations whose waveenvelope, on long time scales, is governed by an effective two-dimensional time-dependent system of massless Dirac equations. c) Dirac points are unstable to arbitrary small perturbations which break either C (complexconjugation) symmetry or P (inversion) symmetry. d) 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. e) 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. arXiv:1710.03389v2 [math-ph] 13 Sep 2018 PreliminariesIn this section, we outline the relevant spectral theory [22,50, 51,76] and introduce terminology and notation for discussing the symmetry properties of the (unperturbed) bulk operator L A . Fourier analysisLet {v 1 , v 2 } be a linearly independent set in R 2 . The lattice generated by {v 1 , v 2 } is the subset of R 2 :A choice of fundamental cell is the parallelogram: Ω = {θ 1 v 1 + θ 2 v 2 : 0 ≤ θ j ≤ 1, j = 1, 2}.

Edge states and the valley Hall effect

“…By results presented in §4, their point spectra are the same. Hence, the point spectrum of / D consists of double eigenvalues; this explains why armchair-type edge state curves come in pairs; see Figure 1, the numerical observations in [22,38] and the photonic results [42]. Remark 1.1.…”

“…Recall that the vertices of the Brillouin zone, B, are high-symmetry quasimomenta generated via 2π/3 rotation of K and K ; see (1.14). 25,38] For a generic choice 2 of honeycomb potential V , the operator −∆ + V has Dirac points at (K, E D ) and (K , E D ).…”

“…Here, 1, τ andτ are the three cube roots of unity (τ = e 2πi/3 ) and L 2 K,ω is the subspace of L 2 K consisting of functions for which Rf = ωf . Using the above observations, it was proved in [24,25] that for generic honeycomb potentials V , the band structure of H 0 = −∆ + V has Dirac points at the vertices of B; see also [1,7,12,28,38]. This means that there exist E D , v F > 0 and b * ≥ 1 such that for K = K, K , the operator H 0 K has a double eigenvalue at energy E D , at which two dispersion surfaces touch conically:…”

“…We consider two types of asymptotic bulk operators, denoted H δ bulk,± = H δ ± and H δ bulk,± = H δ ± ; see [30,36,38,45]: (i) C is preserved and P is broken:…”

“…The displayed edge state curves in Figures 1 and 2 do not change as the size of the computational domain is increased. Figures which display such modes are presented in[38].…”