Quantum Hall transition in graphene with correlated bond disorder T. Kawarabayshi (Toho University) Y. Hatsugai (University of Tsukuba) H. Aoki (University of Tokyo) ArXiv: 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 --

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

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

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

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 |

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

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 = 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:

Hall conductance  xy (Chern Number C E ) as a function of E N x =N y =10, 300 samples   =1/50  t =  t = 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

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

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

(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

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