1. QUANTUM CHAOS IN GRAPHENE Spiros Evangelou is it the same as for any other 2D lattice? 1

2.  DISORDER: diffusive to localized quantum interference of electron waves in a random medium TOPOLOGY: integrable to chaotic  TOPOLOGY: integrable to chaotic 2 quantum interference of classically chaotic systems |ψ|

3. Anderson localization Anderson localization (averages over disorder W) 3 random matrix theory! quantum chaos quantum chaos (averages over energy E) energy level-statistics

4. localized to diffusive P(S) level-spacing distribution 4 at the transition? integrable to chaotic Poisson Wigner to

5. graphene 5 a sheet of carbon atoms on a hexagonal lattice

6. 6  linear small-k dispersion near Dirac point  two bands touch at the Dirac point E=0  electrons with large velocity and zero mass Dirac cones near E=0 6 fundamental physics & device applications DOS E

7. armchair and zigzag edges …edge states in graphene 7 chirality nanoribbons flakes:

8. 8 destructive interference for zigzag edges A atoms B atoms edge states

9. in the presence of disorder (ripples, rings, defects,…) 9 what is the level-statistics of the edge states close to DP? diagonal disorder (breaks chiral symmetry) off-diagonal disorder (preserves chiral symmetry)

10. 3D localization Poisson Wigner intermediate statistics?

11. disordered nanotubes 11 energy level-statisticsparticipation ratios energy spacing Amanatidis & Evangelou PRB 2009 L W

12. 12 participation ratio: distribution of PR the E=0 state

13. 13 From PR(E=0) vs L fractal dimension Kleftogiannis and Evangelou (to be published)

14. level-statistics 14 from semi-Poisson to Poisson

15. 15 zero disorder: ballistic motion (Poisson stat) is graphene the same as any 2D lattice? graphene lies between a metal and an insulator! weak disorder: fractal states & weak chaos (semi-Poisson statistics) strong disorder: localization & integrability (Poisson statistics) Amanatidis et al (to be published)