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, (2008)

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

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

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

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

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

7 anisotropic Dirac gas energy bands moving Dirac points

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

9 anisotropic Dirac gas moving Dirac points

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

11 “hybrid” gas

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

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

14 Quartic oscillator Harmonic oscillator

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

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

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

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

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

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

22 I = conic IV= gap

23

24

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