1. 1 A new field dependence of Landau levels in a graphene-like structure Petra Dietl, Ecole Polytechnique student Frédéric Piéchon, LPS G. M. PRL 100, 146802 (2008) www.lps.u-psud.fr/users/gilles

2. 2 field dependence of Landau levels Schrödinger electron gasGraphene: Weyl (massless Dirac) electron gas other field dependencies for 2D gas ?

3. 3 new field dependence of Landau levels hybrid 2D electron gas : new dispersion relation

4. 4 tight-binding problem on honeycomb lattice hexagonal Bravais lattice+2 atoms basis Bloch theorem: BA Gap ? Y. Hasegawa et al., 2006

5. 5 Dirac points energy bands graphene: isotropic Dirac gas isotropic Dirac gas

6. 6 Landau levels of graphene Hofstadter-Rammal butterfly honeycomb lattice in a magnetic field low field 1 5 3 -3 -5 1 5 3 -3 -5 Valley degeneracy 2-fold degeneracy of LL

7. 7 anisotropic Dirac gas energy bands moving Dirac points

8. 8 Landau levels of anisotropic Dirac gas low field lifts 2-fold LL degeneracy 3 2 -2 -3 1 Y. Hasegawa, M. Kohmoto, 2006

9. 9 anisotropic Dirac gas moving Dirac points

10. 10 energy bands anisotropic hybrid Schrödinger-Dirac gas Merging of Dirac points

11. 11 “hybrid” gas 2 -2 1

12. 12 hybrid Schrödinger-Dirac gas Density of statesOnsager argument gap opening

13. 13 A simple problem of quantum mechanics Instead of for a linear spectrum

14. 14 Quartic oscillator Harmonic oscillator

15. 15  and Berry’s phase Roth, 1966 Wilkinson

16. 16 Universal features for 2D lattices with two atoms basis model oblique lattice

17. 17 Phase diagram 1 GAP Dirac spectrum graphene GAP “Hybrid” spectrum

18. 18 Summary motion and merging of Dirac points universal features of 2D Bravais lattices with 2 atoms basis tight binding Hamiltonian on honeycomb lattice experimental realization? next nearest neighbor hopping  tilted cones Landau levels with renormalized velocity and Quantum Hall effect (J.N. Fuchs, M.O. Goerbig, F. Piéchon and G.M. ‏ arXiv:0803.0912 Realization of a graphene structure in atomic gases A. Kobayashi, S. Katayama, Y. Suzumura, H. Fukuyama (2006) Massless Dirac fermions in organic conductor a-(BEDT-TTF)2I3 ‏

19. 19 4 atoms basis, 8 hopping integrals Dirac cones in organic conductors

20. 20 I= conic II= conic, but metallic III= gap tilted Dirac cones hybrid model transition line 2atoms basis 4 hopping integrals

21. 21 It is enough to consider… 2 atoms per cell, 4 hopping integrals 4 atoms per cell, 8 hopping integrals

22. 22 I = conic IV= gap

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25. 25 The Chocolate LEDERER PrizeThe NOBEL Prize First recipient : Pascal Lederer FRANCAIS Leder er Chocolate