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

  • J. P. Lee-Thorp
  • M. I. WeinsteinEmail author
  • Y. Zhu


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 polarized waves are determined by the spectral properties of the elliptic operator \({\mathcal{L}^{A}=-\nabla_{\bf x}\cdot A({\bf x}) \nabla_{\bf x}}\), where A(x) is \({{\Lambda}_h}\)-periodic (\({{\Lambda}_h}\) denotes the equilateral triangular lattice), and such that with respect to some origin of coordinates, A(x) is \({\mathcal{P}\mathcal{C}}\)-invariant (\({A({\bf x})=\overline{A(-{\bf x})}}\)) and \({120^\circ}\) rotationally invariant (\({A(R^*{\bf x})=R^*A({\bf x})R}\), where R is a \({120^\circ}\) rotation in the plane). A summary of our results is as follows: (a) For generic honeycomb structured media, the band structure of \({\mathcal{L}^{A}}\) 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 wave-envelope, 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 \({\mathcal{C}}\) (complex-conjugation) symmetry or \({\mathcal{P}}\) (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 unidirectional 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.


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The authors wish to thank C. L. Fefferman, L. Lu, M. Rechtsman, D. Ketcheson, V. Quenneville-Bélair and N. Yu for stimulating discussions. This research was supported in part by NSF grants: DMS-1412560, DMS-1620418, DGE-1069420 and Simons Foundation Math + X Investigator grant #376319 (MIW); and the NSF grant DMR-1420073 (JPL-T). YZ acknowledges the hospitality of the Department of Applied Physics and Applied Mathematics during academic visits to Columbia University, supported by Tsinghua University Initiative Scientific Research Program # 20151080424 and NSFCgrants #11471185 and #11871299.


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Authors and Affiliations

  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  2. 2.Department of Applied Physics and Applied Mathematics and Department of MathematicsColumbia UniversityNew YorkUSA
  3. 3.Zhou Pei-Yuan Center for Applied MathematicsTsinghua UniversityBeijingChina

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