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.
References
Ablowitz, M.J., Curtis, C.W., Zhu, Y.: On tight-binding approximations in optical lattices. Stud. Appl. Math. 129(4), 362–388 (2012)
Ablowitz, M.J., Nixon, S.D., Zhu, Y.: Conical diffraction in honeycomb lattices. Phys. Rev. A 79(5), 053830 (2009)
Allaire, G., Palombaro, M., Rauch, J.: Diffractive geometric optics for Bloch wave packets. Arch. Rat. Mech. Anal. 202(2), 373–426 (2011)
Allaire, G., Piatnitski, A.: Homogenization of the Schrödinger equation and effective mass theorems. Commun. Math. Phys. 258(1), 1–22. MR2166838 (2006h:35007) 2005
Ando, Y.: Topological insulator materials. J. Phys. Soc. Jpn. 82(10), 102001 (2013)
Ashcroft, N.W., Mermin, N.D.: Solid State Physics. Harcourt, Orlando, FL (1976). (German)
Avron, J., Simon, B.: Analytic properties of band functions. Ann. Phys. 110, 85–101 (1978)
Berkolaiko, G., Comech, A.: Symmetry and Dirac points in graphene spectrum. J. Spectral Theory 8(3), 1099–1147 (2018)
Bernevig, B.A., Hughes, T.L.: Topological Insulators and Topological Superconductors. Princeton University Press, Princeton (2013)
Berry, M.V., Jeffrey, M.R.: Conical Diffraction: Hamilton's Diabolical Point at the Heart of Crystal Optics, Progress in optics, 2007
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 1999
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 2001
Birman, M.Sh., Suslina, T.A.: Periodic differential operators of the second order. threshold properties and homogenization. Algebra i Analyz 15(2), 1–108 2003
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)
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 2014
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)
Makwana, M., Craster, R.V.: Homogenization for hexagonal lattices and honeycomb structures. Q. J. Mech. Appl, Math (2014)
Colin De Verdiere, Y.: Sur les singularites de van hove generiques. Memoires de la S. M. F. serie 2 46, 99–109 1991
Delplace, P., Ullmo, D., Montambaux, G.: Zak phase and the existence of edge states in graphene. Phys. Rev. B 84(19), 195452 (2011)
Dimassi, M., Sjoestrand, J.: Spectral Asymptotics in the Semi-classical Limit, London Mathematical Society Lecture Note Series, vol. 268, Cambridge University Press, 1999
Do, N.T., Kuchment, P.: Quantum graph spectra of a graphyne structure, nanoscale Systems: mathematical modeling. Theory Appl. 2, 107–123 (2013)
Eastham, M.S.P.: The Spectral Theory of Periodic Differential Equations. Scottish Academic Press, London (1973)
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)
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 2016
Fefferman, C.L., Lee-Thorp, J.P., Weinstein, M.I.: Topologically protected states in one-dimensional systems. Memoirs Am. Math. Soc. 247(1173), 2017
Fefferman, C.L., Weinstein, M.I.: Honeycomb lattice potentials and Dirac points. J. Am. Math. Soc. 25(4), 1169–1220 (2012)
Fefferman, C.L., Weinstein, M.I.: Wave packets in honeycomb structures and two-dimensional Dirac equations. Commun. Math. Phys. 326(1), 251–286 (2014)
Fefferman, C.L., Lee-Thorp, J.P., Weinstein, M.I.: Edge states in honeycomb structures. Ann. PDE 2(12), 2016
Fefferman, C.L., Lee-Thorp, J.P., Weinstein, M.I.: Honeycomb Schroedinger operators in the strong-binding regime. Commun. Pure Appl. Math. 71(6), 2018
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)
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)
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)
Fouque, J.-P., Garnier, J., Papanicolaou, G., Solna, K.: Wave Propagation and Time Reversal in Randomly Layered Media, Springer, 2007
Geim, A.K., Novoselov, K.S.: The rise of graphene. Nat. Mater. 6(3), 183–191 (2007)
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)
Grushin, V.V.: Multiparameter perturbation theory of Fredholm operators applied to Bloch functions. Math. Notes 86(6), 767–774 (2009)
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)
Hasan, M.Z., Kane, C.L.: Colloquium: topological insulators. Rev. Mod. Phys. 82(4), 3045 (2010)
Hempel, R., Lienau, K.: Spectral properties of periodic media in the large coupling limit. Commun. PDE 25(7–8), 1445–1470 (2000)
Hempel, R., Post, O.: Spectral gaps for periodic elliptic operators with high contrast: an overview. Prog Anal 577–587 (2003)
Hoefer, M.A., Weinstein, M.I.: Defect modes and homogenization of periodic schrödinger operators. SIAM J. Math. Anal. 43(2), 971–996 (2011)
Joannopoulos, J.D., Johnson, S.G., Winn, J.N., Meade, R.D.: Photonic Crystals: Molding the Flow of Light. Princeton University Press, Princeton (2011)
Keller, R.T., Marzuola, J., Osting, B., Weinstein, M.I.: Spectral Band Degeneracies of \(\frac{\pi }{2}-\) rotationally Invariant Periodic Schrödinger Operators (2018, submitted)
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)
Kuchment, P.: The mathematics of photonic crystals, in ``Mathematical Modeling in Optical Science''. Front. Appl. Math. 22 2001
Kuchment, P., Levendrskii, S.: On the structure of spectra of periodic elliptic operators. Trans. Am. Math. Soc. 354, 537–569 (2001)
Kuchment, P., Pinchover, Y.: Integral representations and liouville theorems for solutions of periodic elliptic equations. J. Funct. Anal. 181, 402–446 (2001)
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)
Kuchment, P., Post, O.: On the spectra of carbon nano-structures. Commun. Math. Phys. 275, 805–826 (2007)
Kuchment, P.A.: Floquet Theory for Partial Differential Equations, vol. 60. Birkhäuser, Basel (2012)
Kuchment, P.A.: An Overview of Periodic Elliptic Operators, . Bull. Amer. Math. Soc. 53, 343–414 (2016)
Lee, M.: Dirac cones for point scatterers on a honeycomb lattice. SIAM J. Math. Anal. 48(2), 1459–1488 (2016)
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)
Lipton, R., Viator, R.: Bloch waves in crystals and periodic high contrast media. ESAIM: Math. Model. Num. Anal. 51(3), 889–918 2017
Lipton, R., Viator, R.: Creating Band Gaps in Periodic Media. Multiscale Model. Simul. 15, 1612–1650 (2017)
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, 2012
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)
Mackay, T.G.: Lakhtakia, A: Electromagnetic Anisotropy and Bian-isotropy: A Field Guide. World Scientific, Singapore (2010)
Mousavi, S.H., Khanikaev, A.B., Wang, Z.: Topologically protected elastic waves in phononic metamaterials. Nat. Commun. 6 2015
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)
Newton, R.G.: Relation between the three-dimensional Fredholm determinant and the Jost functions. J. Math. Phys. 13(2), 880–883 (1972)
De Nittis, G., Lein, M.: Effective light dynamics in perturbed photonic crystals. Commun. Math. Phys. 332(1), 221–260 (2014)
De Nittis, G., Lein, M.: On the role of symmetries in the theory of photonic crystals. Ann. Phys. 350, 568–587 (2014)
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)
De Nittis, G., Lein, M.: Symmetry Classification of Topological Photonic Crystals, (arXiv:1710.08104)
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)
Ortmann, F., Roche, S., Valenzuela, S.O.: Topological Insulators: Fundamentals and Perspectives. Wiley, Weinheim (2015)
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)
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)
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)
Poo, Y., Lee-Thorp, J.P., Tan, Y., Wu, R., Weinstein, M.I., Yu, Z.: Global Phase-Modulated Defect States, in preparation
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)
Raghu, S., Haldane, F.D.M.: Analogs of quantum-hall-effect edge states in photonic crystals. Phys. Rev. A 78(3), 033834 (2008)
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)
Reed, M., Simon, B.: Analysis of Operators, vol. iv of Methods of Modern Mathematical Physics, Academic Press, New York 1978
Simon, B.: Trace Ideals and Their Applications, Second Edition, Mathematical Surveys and Monographs, vol. 120, AMS, 2005
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)
Suslina, T.A.: On averaging of a periodic Maxwell system. Funct. Anal. Appl. 38(234–237), 2004
Wallace, P.R.: The band theory of graphite. Phys. Rev. 71(9), 622 (1947)
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)
Wu, L.-H., Hu, X.: Scheme for achieving a topological photonic crystal by using dielectric material. Phys. Rev. Lett. 114(22), 223901 (2015)
Yang, Z., Gao, F., Shi, X., Lin, X., Gao, Z., Chong, Y., Zhang, B.: Topological acoustics. Phys. Rev. Lett. 114(11), 114301 (2015)
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by S. Serfaty
Rights and permissions
About this article
Cite this article
Lee-Thorp, J.P., Weinstein, M.I. & Zhu, Y. Elliptic Operators with Honeycomb Symmetry: Dirac Points, Edge States and Applications to Photonic Graphene. Arch Rational Mech Anal 232, 1–63 (2019). https://doi.org/10.1007/s00205-018-1315-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-018-1315-4