Advertisement

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

  • J. P. Lee-Thorp
  • M. I. Weinstein
  • Y. Zhu
Article
  • 12 Downloads

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 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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

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.

References

  1. 1.
    Ablowitz, M.J., Curtis, C.W., Zhu, Y.: On tight-binding approximations in optical lattices. Stud. Appl. Math. 129(4), 362–388 (2012)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ablowitz, M.J., Nixon, S.D., Zhu, Y.: Conical diffraction in honeycomb lattices. Phys. Rev. A 79(5), 053830 (2009)ADSCrossRefGoogle Scholar
  3. 3.
    Allaire, G., Palombaro, M., Rauch, J.: Diffractive geometric optics for Bloch wave packets. Arch. Rat. Mech. Anal. 202(2), 373–426 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Allaire, G., Piatnitski, A.: Homogenization of the Schrödinger equation and effective mass theorems. Commun. Math. Phys. 258(1), 1–22. MR2166838 (2006h:35007) 2005ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Ando, Y.: Topological insulator materials. J. Phys. Soc. Jpn. 82(10), 102001 (2013)ADSCrossRefGoogle Scholar
  6. 6.
    Ashcroft, N.W., Mermin, N.D.: Solid State Physics. Harcourt, Orlando, FL (1976). (German)zbMATHGoogle Scholar
  7. 7.
    Avron, J., Simon, B.: Analytic properties of band functions. Ann. Phys. 110, 85–101 (1978)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Berkolaiko, G., Comech, A.: Symmetry and Dirac points in graphene spectrum. J. Spectral Theory 8(3), 1099–1147 (2018)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bernevig, B.A., Hughes, T.L.: Topological Insulators and Topological Superconductors. Princeton University Press, Princeton (2013)CrossRefGoogle Scholar
  10. 10.
    Berry, M.V., Jeffrey, M.R.: Conical Diffraction: Hamilton's Diabolical Point at the Heart of Crystal Optics, Progress in optics, 2007Google Scholar
  11. 11.
    Birman, M.Sh., Suslina, T.A.: Two-dimensional periodic pauli operator. the effective masses at the lower edge of the spectrum. Math. Results Quantum Mech., 13–31 1999Google Scholar
  12. 12.
    Birman, M.Sh., Suslina, T.A.: Threshold effects near the lower edge of the spectrum for periodic differential operators of mathematical physics. Systems, Approximation, Singular Integral Operators, and Related Topics, 71–107 2001Google Scholar
  13. 13.
    Birman, M.Sh., Suslina, T.A.: Periodic differential operators of the second order. threshold properties and homogenization. Algebra i Analyz 15(2), 1–108 2003Google Scholar
  14. 14.
    Birman, MSh, Suslina, T.A.: Homogenization of a multidimensional periodic elliptic operators in a neighborhood of the edge of internal gap. J. Math. Sci. 136(2), 3682–3690 (2006)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Chen, W.J., Jiang, S.J., Chen, X.D., Zhu, B., Zhou, L., Dong, J.W., Chan, C.T.: Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide. Nat. Commun. 5 2014Google Scholar
  16. 16.
    Cheng, X., Jouvaud, C., Ni, X., Mousavi, S.H., Genack, A.Z., Khanikaev, A.B.: Robust reconfigurable electromagnetic pathways within a photonic topological insulator. Nat, Mater (2016)ADSCrossRefGoogle Scholar
  17. 17.
    Makwana, M., Craster, R.V.: Homogenization for hexagonal lattices and honeycomb structures. Q. J. Mech. Appl, Math (2014)zbMATHGoogle Scholar
  18. 18.
    Colin De Verdiere, Y.: Sur les singularites de van hove generiques. Memoires de la S. M. F. serie 2 46, 99–109 1991Google Scholar
  19. 19.
    Delplace, P., Ullmo, D., Montambaux, G.: Zak phase and the existence of edge states in graphene. Phys. Rev. B 84(19), 195452 (2011)ADSCrossRefGoogle Scholar
  20. 20.
    Dimassi, M., Sjoestrand, J.: Spectral Asymptotics in the Semi-classical Limit, London Mathematical Society Lecture Note Series, vol. 268, Cambridge University Press, 1999Google Scholar
  21. 21.
    Do, N.T., Kuchment, P.: Quantum graph spectra of a graphyne structure, nanoscale Systems: mathematical modeling. Theory Appl. 2, 107–123 (2013)zbMATHGoogle Scholar
  22. 22.
    Eastham, M.S.P.: The Spectral Theory of Periodic Differential Equations. Scottish Academic Press, London (1973)zbMATHGoogle Scholar
  23. 23.
    Fefferman, C.L., Lee-Thorp, J.P., Weinstein, M.I.: Topologically protected states in one-dimensional continuous systems and dirac points. Proc. Nat. Acad. Sci. 111(24), 8759–8763 (2014)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Fefferman, C.L., Lee-Thorp, J.P., Weinstein, M.I.: Bifurcations of edge states—topologically protected and non-protected—in continuous 2d honeycomb structures. 2D Mater. 3(1), 014008 2016CrossRefGoogle Scholar
  25. 25.
    Fefferman, C.L., Lee-Thorp, J.P., Weinstein, M.I.: Topologically protected states in one-dimensional systems. Memoirs Am. Math. Soc. 247(1173), 2017Google Scholar
  26. 26.
    Fefferman, C.L., Weinstein, M.I.: Honeycomb lattice potentials and Dirac points. J. Am. Math. Soc. 25(4), 1169–1220 (2012)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Fefferman, C.L., Weinstein, M.I.: Wave packets in honeycomb structures and two-dimensional Dirac equations. Commun. Math. Phys. 326(1), 251–286 (2014)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    Fefferman, C.L., Lee-Thorp, J.P., Weinstein, M.I.: Edge states in honeycomb structures. Ann. PDE 2(12), 2016Google Scholar
  29. 29.
    Fefferman, C.L., Lee-Thorp, J.P., Weinstein, M.I.: Honeycomb Schroedinger operators in the strong-binding regime. Commun. Pure Appl. Math. 71(6), 2018CrossRefGoogle Scholar
  30. 30.
    Figotin, A., Kuchment, P.: Band-gap structure of spectra of periodic dielectric and acoustic media. I. scalar model. SIAM J. Appl. Math. 56(6), 68–88 (1996)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Figotin, A., Kuchment, P.: Band-gap structure of spectra of periodic dielectric and acoustic media. II. two-dimentional photonic crystals. SIAM J. Appl. Math. 56(6), 1561–1620 (1996)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Fliss, S., Joly, P.: Solutions of the time-harmonic wave equation in periodic waveguides: asymptotic behaviour and radiation condition. Arch. Rat. Mech. Anal. 219(1), 349–386 (2016)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Fouque, J.-P., Garnier, J., Papanicolaou, G., Solna, K.: Wave Propagation and Time Reversal in Randomly Layered Media, Springer, 2007Google Scholar
  34. 34.
    Geim, A.K., Novoselov, K.S.: The rise of graphene. Nat. Mater. 6(3), 183–191 (2007)ADSCrossRefGoogle Scholar
  35. 35.
    Gesztesy, F., Latushkin, Y., Zumbrun, K.: Derivatives of (modified) fred-holm determinants and stability of standing and traveling waves. J. Math. Pures Appl. 90, 160–200 (2008)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Grushin, V.V.: Multiparameter perturbation theory of Fredholm operators applied to Bloch functions. Math. Notes 86(6), 767–774 (2009)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Haldane, F.D.M.: Raghu, S: Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry. Phys. Rev. Lett. 100(1), 013904 (2008)ADSCrossRefGoogle Scholar
  38. 38.
    Hasan, M.Z., Kane, C.L.: Colloquium: topological insulators. Rev. Mod. Phys. 82(4), 3045 (2010)ADSCrossRefGoogle Scholar
  39. 39.
    Hempel, R., Lienau, K.: Spectral properties of periodic media in the large coupling limit. Commun. PDE 25(7–8), 1445–1470 (2000)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Hempel, R., Post, O.: Spectral gaps for periodic elliptic operators with high contrast: an overview. Prog Anal 577–587 (2003)Google Scholar
  41. 41.
    Hoefer, M.A., Weinstein, M.I.: Defect modes and homogenization of periodic schrödinger operators. SIAM J. Math. Anal. 43(2), 971–996 (2011)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Joannopoulos, J.D., Johnson, S.G., Winn, J.N., Meade, R.D.: Photonic Crystals: Molding the Flow of Light. Princeton University Press, Princeton (2011)zbMATHGoogle Scholar
  43. 43.
    Keller, R.T., Marzuola, J., Osting, B., Weinstein, M.I.: Spectral Band Degeneracies of \(\frac{\pi }{2}-\) rotationally Invariant Periodic Schrödinger Operators (2018, submitted)Google Scholar
  44. 44.
    Khanikaev, A.B., Mousavi, S.H., Tse, W.-K., Kargarian, M., MacDonald, A.H., Shvets, G.: Photonic topological insulators. Nat. Mater. 12(3), 233–239 (2013)ADSCrossRefGoogle Scholar
  45. 45.
    Kuchment, P.: The mathematics of photonic crystals, in ``Mathematical Modeling in Optical Science''. Front. Appl. Math. 22 2001Google Scholar
  46. 46.
    Kuchment, P., Levendrskii, S.: On the structure of spectra of periodic elliptic operators. Trans. Am. Math. Soc. 354, 537–569 (2001)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Kuchment, P., Pinchover, Y.: Integral representations and liouville theorems for solutions of periodic elliptic equations. J. Funct. Anal. 181, 402–446 (2001)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Kuchment, P., Pinchover, Y.: Liouville theorems and spectral edge behavior on abelian coverings of compact manifolds. Trans. Am. Math. Soc. 359(12), 5777–5815 (2007)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Kuchment, P., Post, O.: On the spectra of carbon nano-structures. Commun. Math. Phys. 275, 805–826 (2007)ADSMathSciNetCrossRefGoogle Scholar
  50. 50.
    Kuchment, P.A.: Floquet Theory for Partial Differential Equations, vol. 60. Birkhäuser, Basel (2012)zbMATHGoogle Scholar
  51. 51.
    Kuchment, P.A.: An Overview of Periodic Elliptic Operators, . Bull. Amer. Math. Soc. 53, 343–414 (2016)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Lee, M.: Dirac cones for point scatterers on a honeycomb lattice. SIAM J. Math. Anal. 48(2), 1459–1488 (2016)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Lee-Thorp, J.P., Vukićević, I., Xu, X., Yang, J., Fefferman, C.L., Wong, C.W., Weinstein, M.I.: Photonic realization of topologically protected bound states in domain-wall waveguide arrays. Phys. Rev. A 93, 033822 (2016)ADSCrossRefGoogle Scholar
  54. 54.
    Lipton, R., Viator, R.: Bloch waves in crystals and periodic high contrast media. ESAIM: Math. Model. Num. Anal. 51(3), 889–918 2017MathSciNetCrossRefGoogle Scholar
  55. 55.
    Lipton, R., Viator, R.: Creating Band Gaps in Periodic Media. Multiscale Model. Simul. 15, 1612–1650 (2017)MathSciNetCrossRefGoogle Scholar
  56. 56.
    Logg, Anders, Mardal, Kent-Andre, Wells, Garth: Automated Solution of Differential Equations by the Finite Element Method: The Fenics Book, Vol. 84, Springer Science & Business Media, 2012Google Scholar
  57. 57.
    Ma, T., Khanikaev, A.B., Mousavi, S.H., Shvets, G.: Guiding electromagnetic waves around sharp corners: Topologically protected photonic transport in metawaveguides. Phys. Rev. Lett. 114(12), 127401 (2015)ADSCrossRefGoogle Scholar
  58. 58.
    Mackay, T.G.: Lakhtakia, A: Electromagnetic Anisotropy and Bian-isotropy: A Field Guide. World Scientific, Singapore (2010)Google Scholar
  59. 59.
    Mousavi, S.H., Khanikaev, A.B., Wang, Z.: Topologically protected elastic waves in phononic metamaterials. Nat. Commun. 6 2015Google Scholar
  60. 60.
    Neto, A.H.C., Guinea, F., Peres, N.M.R., Novoselov, K.S., Geim, A.K.: The electronic properties of graphene. Rev. Mod. Phys. 81(1), 109 (2009)ADSCrossRefGoogle Scholar
  61. 61.
    Newton, R.G.: Relation between the three-dimensional Fredholm determinant and the Jost functions. J. Math. Phys. 13(2), 880–883 (1972)ADSMathSciNetCrossRefGoogle Scholar
  62. 62.
    De Nittis, G., Lein, M.: Effective light dynamics in perturbed photonic crystals. Commun. Math. Phys. 332(1), 221–260 (2014)ADSMathSciNetCrossRefGoogle Scholar
  63. 63.
    De Nittis, G., Lein, M.: On the role of symmetries in the theory of photonic crystals. Ann. Phys. 350, 568–587 (2014)ADSMathSciNetCrossRefGoogle Scholar
  64. 64.
    De Nittis, G., Lein, M.: Derivation of ray optics equations in photonic crystals via a semi-classical limit. Ann. Henri. Poincare 18(5), 1789–1831 (2017)ADSMathSciNetCrossRefGoogle Scholar
  65. 65.
    De Nittis, G., Lein, M.: Symmetry Classification of Topological Photonic Crystals, (arXiv:1710.08104)
  66. 66.
    Novoselov, K.S., Geim, A.K., Morozov, S.V., Jiang, D., Katsnelson, M.I., Grigorieva, I.V., Dubonos, S.V., Firsov, A.A.: Two-dimensional gas of massless dirac fermions in graphene. Nature 438(7065), 197–200 (2005)ADSCrossRefGoogle Scholar
  67. 67.
    Ortmann, F., Roche, S., Valenzuela, S.O.: Topological Insulators: Fundamentals and Perspectives. Wiley, Weinheim (2015)CrossRefGoogle Scholar
  68. 68.
    Peleg, O., Bartal, G., Freedman, B., Manela, O., Segev, M., Christodoulides, D.N.: Conical diffraction and gap solitons in honeycomb photonic lattices. Phys. Rev. Lett. 98, 103901 (2007)ADSCrossRefGoogle Scholar
  69. 69.
    Plotnik, Y., Rechtsman, M.C., Song, D., Heinrich, M., Zeuner, J.M., Nolte, S., Lumer, Y., Malkova, N., Xu, J., Szameit, A., Chen, Z., Segev, M.: Observation of unconventional edge states in 'photonic graphene'. Nat. Mater. 13(1), 57–62 (2014)ADSCrossRefGoogle Scholar
  70. 70.
    Poo, Y., Lee-Thorp, J.P., Tan, Y., Wu, R., Weinstein, M.I., Yu, Z.: Observation of highly robust phase-defect induced photonic states, Lasers and electro-optics (cleo). Conference on 2016, 1–2 (2016)CrossRefGoogle Scholar
  71. 71.
    Poo, Y., Lee-Thorp, J.P., Tan, Y., Wu, R., Weinstein, M.I., Yu, Z.: Global Phase-Modulated Defect States, in preparationGoogle Scholar
  72. 72.
    Poo, Y., Wu, R., Lin, Z., Yang, Y., Chan, C.T.: Experimental realization of self-guiding unidirectional electromagnetic edge states. Phys. Rev. Lett. 106(9), 093903 (2011)ADSCrossRefGoogle Scholar
  73. 73.
    Raghu, S., Haldane, F.D.M.: Analogs of quantum-hall-effect edge states in photonic crystals. Phys. Rev. A 78(3), 033834 (2008)ADSCrossRefGoogle Scholar
  74. 74.
    Rechtsman, M.C., Plotnik, Y., Zeuner, J.M., Song, D., Chen, Z., Szameit, A., Segev, M.: Topological creation and destruction of edge states in photonic graphene. Phys. Rev. Lett. 111(10), 103901 (2013)ADSCrossRefGoogle Scholar
  75. 75.
    Reed, M., Simon, B.: Analysis of Operators, vol. iv of Methods of Modern Mathematical Physics, Academic Press, New York 1978Google Scholar
  76. 76.
    Simon, B.: Trace Ideals and Their Applications, Second Edition, Mathematical Surveys and Monographs, vol. 120, AMS, 2005Google Scholar
  77. 77.
    Singha, A., Gibertini, M., Karmakar, B., Yuan, S., Polini, M., Vignale, G., Katsnelson, M.I., Pinczuk, A., Pfeiffer, L.N., West, K.W., Pellegrini, V.: Two-dimensional mott-hubbard electrons in an artificial honeycomb lattice. Science 332(6034), 1176–1179 (2011)ADSCrossRefGoogle Scholar
  78. 78.
    Suslina, T.A.: On averaging of a periodic Maxwell system. Funct. Anal. Appl. 38(234–237), 2004Google Scholar
  79. 79.
    Wallace, P.R.: The band theory of graphite. Phys. Rev. 71(9), 622 (1947)ADSCrossRefGoogle Scholar
  80. 80.
    Wang, Z., Chong, Y.D., Joannopoulos, J.D.: Solja\(\check{{\rm c}}\)ić, M.: Reflection-free oneway edge modes in a gyromagnetic photonic crystal. Phys. Rev. Lett. 100(1), 013905 (2008)ADSCrossRefGoogle Scholar
  81. 81.
    Wu, L.-H., Hu, X.: Scheme for achieving a topological photonic crystal by using dielectric material. Phys. Rev. Lett. 114(22), 223901 (2015)ADSCrossRefGoogle Scholar
  82. 82.
    Yang, Z., Gao, F., Shi, X., Lin, X., Gao, Z., Chong, Y., Zhang, B.: Topological acoustics. Phys. Rev. Lett. 114(11), 114301 (2015)ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

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

Personalised recommendations