Waves in Honeycomb Structures
Fefferman, Charles L.1 ; Weinstein, Michael I.2
1 Department of Mathematics Princeton University Princeton, NJ 08540 USA
2 Department of Applied Physics and Applied Mathematics Columbia University New York, NY 10027 USA
Journées équations aux dérivées partielles (2012), article no. 12, 12 p.

We review recent work of the authors on the non-relativistic Schrödinger equation with a honeycomb lattice potential, V. In particular, we summarize results on (i) the existence of Dirac points, conical singularities in dispersion surfaces of H V =-Δ+V and (ii) the two-dimensional Dirac equations, as the large (but finite) time effective system of equations governing the evolution e -iH V t ψ 0 , for data ψ 0 , which is spectrally localized near a Dirac point. We conclude with a formal derivation and discussion of the effective large time evolution for the nonlinear Schrödinger - Gross Pitaevskii equation for small amplitude initial conditions, ψ 0 . The effective dynamics are governed by a nonlinear Dirac system.

DOI : 10.5802/jedp.95
Classification : 00X99
Mots-clés : Periodic structure, Dispersion relation, Dirac point, Dirac equations, Conical point, Graphene, Nonlinear Schrödinger / Gross Pitaevskii equation
Fefferman, Charles L. 1 ; Weinstein, Michael I. 2

1 Department of Mathematics Princeton University Princeton, NJ 08540 USA
2 Department of Applied Physics and Applied Mathematics Columbia University New York, NY 10027 USA 
@incollection{JEDP_2012____A12_0,
     author = {Fefferman, Charles L. and Weinstein, Michael I.},
     title = {Waves in {Honeycomb} {Structures}},
     booktitle = {},
     series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {12},
     pages = {1--12},
     publisher = {Groupement de recherche 2434 du CNRS},
     year = {2012},
     doi = {10.5802/jedp.95},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jedp.95/}
}
TY  - JOUR
AU  - Fefferman, Charles L.
AU  - Weinstein, Michael I.
TI  - Waves in Honeycomb Structures
JO  - Journées équations aux dérivées partielles
PY  - 2012
SP  - 1
EP  - 12
PB  - Groupement de recherche 2434 du CNRS
UR  - http://www.numdam.org/articles/10.5802/jedp.95/
DO  - 10.5802/jedp.95
LA  - en
ID  - JEDP_2012____A12_0
ER  - 
%0 Journal Article
%A Fefferman, Charles L.
%A Weinstein, Michael I.
%T Waves in Honeycomb Structures
%J Journées équations aux dérivées partielles
%D 2012
%P 1-12
%I Groupement de recherche 2434 du CNRS
%U http://www.numdam.org/articles/10.5802/jedp.95/
%R 10.5802/jedp.95
%G en
%F JEDP_2012____A12_0
Fefferman, Charles L.; Weinstein, Michael I. Waves in Honeycomb Structures. Journées équations aux dérivées partielles (2012), article  no. 12, 12 p. doi : 10.5802/jedp.95. http://www.numdam.org/articles/10.5802/jedp.95/

[1] M. Ablowitz, C. Curtis, and Y. Zhu. On tight-binding approximations in optical lattices. Stud. Appl. Math, 129(4):362—388, 2012. | MR

[2] M.J. Ablowitz and Y. Zhu. Nonlinear waves in shallow honeycomb lattices. SIAM J. Appl. Math., 72(240–260), 2012. | MR | Zbl

[3] O. Bahat-Treidel, O. Peleg, and M. Segev. Symmetry breaking in honeycomb photonic lattices. Optics Letters, 33(2251–2253), 2008.

[4] M.V. Berry and M.R. Jeffrey. Conical Diffraction: Hamilton’s diabolical point at the heart of crystal optics. Progress in Optics. 2007.

[5] L. Erdös, B. Schlein, and H.T. Yau. Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems. Invent. Math., 167:515–614, 2007. | MR | Zbl

[6] C.L. Fefferman and M.I. Weinstein. Dynamics of wave packets in honeycomb structures and two-dimensional Dirac equations. submitted, http://arxiv.org/abs/1212.6072.

[7] C.L. Fefferman and M.I. Weinstein. Honeycomb lattice potentials and Dirac points. J. Amer. Math. Soc., 25:1169–1220, 2012. | MR

[8] V. V. Grushin. Multiparameter perturbation theory of Fredholm operators applied to Bloch functions. Mathematical Notes, 86(6):767–774, 2009. | MR | Zbl

[9] F.D.M. Haldane and S. Raghu. Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry. Phys. Rev. Lett., 100:013904, 2008.

[10] R. A. Indik and A. C. Newell. Conical refraction and nonlinearity. Optics Express, 14(22):10614–10620, 2006.

[11] P. Kuchment and O. Post. On the spectra of carbon nano-structures. Comm. Math. Phys., 275:805–826, 2007. | MR | Zbl

[12] J.V. Maloney and A.C. Newell. Nonlinear Optics. Westview Press, 2003. | Zbl

[13] A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, and A.K. Geim. The electronic properties of graphene. Reviews of Modern Physics, 81:109–162, 2009.

[14] K. S. Novoselov. Nobel lecture: Graphene: Materials in the flatland. Reviews of Modern Physics, 837–849, 2011.

[15] O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D.N. Christodoulides. Conical diffraction and gap solitons in honeycomb photonic lattices. Phys. Rev. Lett., 98:103901, 2007.

[16] L. P. Pitaevskii and S. Stringari. Bose Einstein Condensation. Oxford University Press, 2003. | MR | Zbl

[17] M.C. Rechtsman, J.M. Zeuner, Y. Plotnik, Y. Lumer, S. Nolte, M. Segev, and A. Szameit. Photonic Floquet topological insulators. http://arxiv.org/abs/1212.3146.

[18] P.R. Wallace. The band theory of graphite. Phys. Rev., 71:622, 1947. | Zbl

Cité par Sources :