1. Quantum Hall transition in graphene with correlated bond disorder T. Kawarabayshi (Toho University) Y. Hatsugai (University of Tsukuba) H. Aoki (University of Tokyo) ArXiv:0904.1927 Outlines 1. Landau levels in graphene 2. Roles of disorder and Numerical model 3. Density of states 4. Hall conductivity 5. Summary -- Unconventional Hall transition at n=0 Landau level --

2. K K’ 2D Honeycomb Lattice Dirac cones (K and K’) at E=0 (Fermi energy) Landau levels around K and K’ n=0 n=1,2,3,…-3,-2,-1 n=0 (E=0) Landau level Characteristic band structure and Landau levels of graphene

3. n=0 Landau level Essential to anomalous quantum Hall effect Zheng & Ando (2002) (per spin) Novoselov et al. Nature 2005 Robustness the index theorem for Dirac fermions Criticality: Dirac fermions + random potential (long-range) Nomura et al. (2008) mixing of two valleys (K and K’) Koshino, Ando (2007) Schweitzer, Markos(2008) Ostrovsky et al. (2008) Sensitivity

4. Roles of Disorder Key concepts (A) Chiral symmetry (UHU -1 = -H) (B) Mixing of two valleys (K and K’) Random bonds Random magnetic fields Random potential Short-range disorder Long-range disorder Yes No Yes No 2D Honeycomb Lattice Model + Spatially Correlated Disorder How these properties show up in the Landau level structure ? t  a Systematic study on the correlation dependence Chiral symmetric, n=0 (E=0) To control (B) the mixing between K and K’ ~1.42Å

5. Ripples A.H. Castro Neto et al. Rev. Mod. Phys. (2009) An intrinsic disorder in graphene Disorder in transfer integrals A model with disordered transfer integrals Chiral symmetry Correlation length  Gaussian with  Region with large t(r) Region with large t(r) Region with Small t(r) Region with Small t(r) A typical landscape (  /a=5) e1e1 e2e2 L x =N x |e 1 | L y =N y |2e 2 -e 1 |

6.   =1/50  t = 0.115 n=0 n=1 n=2 n=-2 n=-1 Anomaly for n=0 Landau level Correlation length Density of states: correlation dependence  t = 0.000625  /a >1 The Green function Method Schweitzer, Kramer, MacKinnon (1984) N x =5000, N y =100

7. Hall conductivity  xy in terms of Chern number C E Thouless, Hohmoto, Nightingale, den Nijs (1982) Aoki, Ando (1986), Niu, Thouless, Wu (1985) Sum over many filled Landau bands Contributions mostly cancel out E ~ 0, weak fields EFEF C1C1 CMCM Hatsugai (2004,2005) Accurate numerical method for C l N x =N y =10 300 samples  /a=1.5  /a=0 E/t n=1 n=0 n=-1 Unconventional n=0 Hall transition for  /a >1 Fukui, Hatsugai, Suzuki (2005) T.K., Y. Hatsugai, H.Aoki, ArXiv:0904.1927

8. Hall conductance  xy (Chern Number C E ) as a function of E N x =N y =10, 300 samples   =1/50  t = 0.115  t = 0.000625 N x =5000, N y =100 E/t  /a=0  /a=1.5 Transition at E=0 is sensitive to the range of bond disorder n=0 n=1 n=-1

9. n=0 n=1 n=-1 n=0 N x =N y =5 N x =N y =10 Size-independent Additional potential disorder[-w/2, w/2] w=0 w=0.4  /a=1.5 Breakdown of chiral symmetry insensitivesensitive E/t

10. Other disorder with chiral symmetry Disordered magnetic fields  r   /a   =1/41    = 0.237 < 1  t = 0.000625 N x =5000, N y =82    = 2.37 > 1 small disorder large disorder Anomaly at the n=0 level

11. (Meyer, Geim et al, Nature 2007) Summary The n=0 Landau level : anomalously robust against long-range bond disorder No broadening for  /a > 1 Chiral symmetry,Absence of scattering between K and K’ Scale of ripples in graphene : 10 ~ 15 nm >> a No broadening of n=0 level by bond disorder by ripples Other disorder (ex. potential disorder) should be responsible for the broadening of n=0 level Possibility to observe this anomaly at n=0 in clean graphene without potential disorder from substrates Classical Hall transition ? Ostrovsky et al. (2008) Consistent with the index theorem

12. Other disorder without chiral symmery Potential disorder V(r) No anomaly   =1/41  t = 0.000625 N x =5000, N y =82  S  t = 0.288