Electron transport in disordered graphene. Alexander D. Mirlin

Transcription

1 Electron transport in disordered graphene Alexander D. Mirlin Research Center Karslruhe & University Karlsruhe & PNPI St. Petersburg P. Ostrovsky, Research Center Karlsruhe & Landau Inst., Chernogolovka I. Gornyi, Research Center Karlsruhe & Ioffe Institute, St.Petersburg A. Schüssler, Research Center Karlsruhe Phys. Rev. B 74, (2006) Phys. Rev. Lett. 98, (2007) Eur. Phys. J. Special Topics 148, 63 (2007) Phys. Rev. B 77, (2008) in preparation Anderson transitions, Evers, ADM, arxiv: , to appear in Rev. Mod. Phys.

2 Plan Localization theory: Symmetries and topologies Experiments: Graphene Disordered Dirac fermions in graphene Conductivity and (de-)localization at the Dirac point Chiral disorder Long-range disorder Anomalous quantum Hall effect From ballistics to diffusion/criticality Related systems: Topological delocalization Summary

3 Anderson Insulators & Metals 1 g=g/(e 2 /h) d=3 Disorder-induced localization Anderson 58 β = dln(g) / dln(l) ln(g) d=2 d=1 Scaling ideas Thouless 74 ; Wegner 76 Scaling theory of localization: Abrahams, Anderson, Licciardello, Ramakrishnan 79 Modern approach: RG for field theory (σ-model) quasi-1d, 2D : all states are localized d > 2: Anderson metal-insulator transition delocalized critical point localized disorder

4 S[Q] = πν 4 Field theory: non-linear σ-model d d r Str [ D( Q) 2 2iωΛQ], Q 2 (r) = 1 Wegner 79 (replicas); Efetov 83 (supersymmetry) σ-model manifold: unitary class: fermionic replicas: U(2n)/U(n) U(n), n 0 bosonic replicas: U(n, n)/u(n) U(n), n 0 supersymmetry: U(1, 1 2)/U(1 1) U(1 1) orthogonal class: fermionic replicas: Sp(4n)/Sp(2n) Sp(2n), n 0 bosonic replicas: O(2n, 2n)/O(2n) O(2n), n 0 supersymmetry: OSp(2, 2 4)/OSp(2 2) OSp(2 2) in general, in supersymmetry: Q { sphere hyperboloid } dressed by anticommuting variables

5 Disordered electronic systems: Symmetry classification Conventional (Wigner-Dyson) classes T spin rot. chiral p-h symbol GOE + + AI GUE +/ A GSE + AII Altland, Zirnbauer 97 Chiral classes T spin rot. chiral p-h symbol ChOE BDI ChUE +/ + AIII ChSE + + CII Bogoliubov-de Gennes classes T spin rot. chiral p-h symbol CI + + C + + DIII + D H = H = ( 0 t t 0 ) ( h h T )

6 Disordered electronic systems: Symmetry classification Ham. RMT T S compact non-compact σ-model σ-model compact class symmetric space symmetric space B F sector M F Wigner-Dyson classes A GUE ± U(N) GL(N, C)/U(N) AIII AIII U(2n)/U(n) U(n) AI GOE + + U(N)/O(N) GL(N, R)/O(N) BDI CII Sp(4n)/Sp(2n) Sp(2n) AII GSE + U(2N)/Sp(2N) U (2N)/Sp(2N) CII BDI O(2n)/O(n) O(n) chiral classes AIII chgue ± U(p + q)/u(p) U(q) U(p, q)/u(p) U(q) A A U(n) BDI chgoe + + SO(p + q)/so(p) SO(q) SO(p, q)/so(p) SO(q) AI AII U(2n)/Sp(2n) CII chgse + Sp(2p + 2q)/Sp(2p) Sp(2q) Sp(2p, 2q)/Sp(2p) Sp(2q) AII AI U(n)/O(n) Bogoliubov - de Gennes classes C + Sp(2N) Sp(2N, C)/Sp(2N) DIII CI Sp(2n)/U(n) CI + + Sp(2N)/U(N) Sp(2N, R)/U(N) D C Sp(2n) BD SO(N) SO(N, C)/SO(N) CI DIII O(2n)/U(n) DIII + SO(2N)/U(N) SO (2N)/U(N) C D O(n)

7 Escaping Anderson localization in 2D Common wisdom : all states are localized in 2D In fact, in 9 out of 10 symmetry classes the system can escape localization! Mechanisms of delocalization & criticality in 2D: broken spin-rotation invariance antilocalization, metallic phase, MIT classes AII, D, DIII topological term π 2 (M) = Z (quantum-hall-type) classes A, C, D : IQHE, SQHE, TQHE topological term π 2 (M) = Z 2 classes AII, CII chiral classes: vanishing β-function, line of fixed points classes AIII, BDI, CII Wess-Zumino term classes AIII, CI, DIII (random Dirac fermions, related to chiral anomaly)

8 Graphene: monoatomic layer of carbon

9 The rise of graphene 2004 breakthrough: Manchester University group (Geim, Novoselov) single-layer graphene on an insulating substrate Novoselov et al. Electric field effect in atomically thin carbon films, Science 306, 666 (2004) Novoselov et al. Two-dimensional atomic crystals, PNAS 102, (2005) 2005: transport measurements; also Columbia Univ. group (Kim, Stormer) Novoselov et al, Two-dimensional gas of masless Dirac fermions in graphene, Nature 438, 197 (2005) Zhang et al, Experimental observation of the quantum Hall effect and Berry s phase in graphene, Nature 438, 201 (2005) : 50 Nature + Science + Nature Physics/Materials/Nanotechnology cond-mat arxiv: # of graphene papers exploding (factor 3 per year) Reviews: Geim and Novoselov. The rise of graphene, Nature Materials 6, 183 (2007) Castro Neto, Guinea, Peres, Novoselov, Geim. The electronic properties of graphene, arxiv: , to appear in Rev. Mod. Phys.

10 Graphene fabrication: Micromechanical cleavage Graphite Slicing Looking

11 Graphene samples

12 Graphene: atomic resolution images

13 Experiments on transport in graphene Novoselov, Geim et al; Zhang, Tan, Stormer, and Kim; Nature 2005 linear dependence of conductivity on electron density ( V g ) minimal conductivity σ 4e 2 /h ( e 2 /h per spin per valley) T independent in the range T = 30 mk 300 K

14 T-independent minimal conductivity in graphene Tan, Zhang, Stormer, Kim 07 T = 30 mk 300 K

15 Graphene in transverse magnetic field Anomalous integer quantum Hall effect: only odd plateaus observed: σ xy = (2n + 1) (2e 2 /h) QHE transition at n = 0 (Dirac point), i.e. at σ xy = 0 visible up to room temperature!

16 Graphene dispersion: 2D massless Dirac fermions 2.46 Å K K m K k 0 K (a) (b) K K Two sublattices: A and B Hamiltonian: H = t k = t [ ] 1 + 2e i( 3/2)k y a cos(k x a/2) ( ) 0 tk t k 0 Spectrum ε 2 k = t k 2 The gap vanishes at 2 points, K, K = (±k 0, 0), where k 0 = 4π/3a. In the vicinity of K, K : massless Dirac-fermion Hamiltonian: H K = v 0 (k x σ x + k y σ y ), H K = v 0 ( k x σ x + k y σ y ) v cm/s effective light velocity, sublattice space isospin

17 Graphene: Disordered Dirac-fermion Hamiltonian Hamiltonian 4 4 matrix operating in: AB space of the two sublattices (σ Pauli matrices), K K space of the valleys (τ Pauli matrices). Four-component wave function: Ψ = {φ AK, φ BK, φ BK, φ AK } T Hamiltonian: H = iv 0 τ z (σ x x + σ y y ) + V (x, y) Disorder: V (x, y) = µ,ν=0,x,y,z σ µ τ ν V µν (x, y)

18 Clean graphene: symmetries Space of valleys K K : Isospin Λ x = σ z τ x, Λ y = σ z τ y, Λ z = σ 0 τ z. Time inversion Chirality T 0 : H = σ x τ x H T σ x τ x C 0 : H = σ z τ 0 Hσ z τ 0 Combinations with Λ x,y,z T x : H = σ y τ 0 H T σ y τ 0 C x : H = σ 0 τ x Hσ 0 τ x T y : H = σ y τ z H T σ y τ z C y : H = σ 0 τ y Hσ 0 τ y T z : H = σ x τ y H T σ x τ y C z : H = σ z τ z Hσ z τ z Spatial isotropy T x,y and C x,y occur simultaneously T and C

19 Symmetries of various types of disorder in graphene Λ Λ z T 0 T T z C 0 C C z CT 0 CT CT z σ 0 τ 0 α σ {x,y} τ {x,y} β σ x,y τ 0 γ σ 0 τ x,y β z σ z τ z γ z σ z τ x,y β σ 0 τ z γ σ x,y τ z α σ z τ 0 α z Related works: S. Guruswamy, A. LeClair, and A.W.W. Ludwig, Nucl. Phys. B 583, 475 (2000) E. McCann, K. Kechedzhi, V.I. Fal ko, H. Suzuura, T. Ando, and B.L. Altshuler, PRL 97, (2006) I.L. Aleiner and K.B. Efetov, PRL 97, (2006)

20 2D disordered Dirac fermions: ballistic RG (history) random bond Ising model (random mass) V.S. Dotsenko and V.S. Dotsenko 83 quantum Hall effect (random mass, scalar and vector potential) A.W.W. Ludwig, M.P.A. Fisher, R. Shankar, and G. Grinstein 94 disordered d-wave superconductors A. A. Nersesyan, A. M. Tsvelik, and F. Wenger 94, 95 M. Bocquet, D. Serban, and M.R. Zirnbauer 00 A. Altland, B.D. Simons, and M.R. Zirnbauer 02 Dirac fermions + C z chiral disorder (e.g., random-field XY model) S. Guruswamy, A. LeClair, and A.W.W. Ludwig 00 disordered graphene (T 0 invariance + some hierarchy of couplings) I.L. Aleiner and K.B. Efetov 06.

21 2D disordered Dirac fermions: complete one-loop RG dα 0 dlog L = 2α 0(α 0 + β 0 + γ 0 + α + β + γ + α z + β z + γ z ) + 2α α z + β β z + 2γ γ z, dα dlog L = 2(2α 0α z + β 0 β z + 2γ 0 γ z ), dα z dlog L = 2α z(α 0 + β 0 + γ 0 α β γ + α z + β z + γ z ) + 2α 0 α + β 0 β + 2γ 0 γ, dβ 0 dlog L = 2[β 0(α 0 γ 0 + α + α z γ z ) + α β z + α z β + β γ 0 ], dβ dlog L = 4(α 0β z + α z β 0 + β 0 γ 0 + β γ + β z γ z ), dβ z dlog L = 2[ β z(α 0 γ 0 α + α z γ z ) + α 0 β + α β 0 + β γ z ], dγ 0 dlog L = 2γ 0(α 0 β 0 + γ 0 + α β + γ + α z β z + γ z ) + 2α γ z + 2α z γ + β 0 β, dγ dlog L = 4α 0γ z + 4α z γ 0 + β β2 + β2 z, dγ z dlog L = 2γ z(α 0 α + α z β 0 + β β z + γ 0 γ + γ z ) + 2α 0 γ + 2α γ 0 + β β z. dε dlog L = ε(1 + α 0 + β 0 + γ 0 + α + β + γ + α z + β z + γ z ). In most of the cases, this ultraviolet RG describes short-scale behavior but not long-distance one (localization / delocalization / criticality)

22 Conductivity at µ = 0 Drude conductivity (SCBA = self-consistent Born approximation): σ = 8e2 v0 2 d 2 k (1/2τ) 2 π (2π) 2 [(1/2τ) 2 + v0k 2 2 ] = 2e2 2 π 2 = 4 e 2 π h BUT: For generic disorder, the Drude result σ = 4 e 2 /πh at µ = 0 does not make much sense: Anderson localization will drive σ 0. Experiment: σ 4 e 2 /h independent of T Quantum criticality? Can one have non-zero σ in the theory? Yes, if disorder either (i) preserves one of chiral symmetries or (ii) is of long-range character (does not mix the valleys)

23 Realizations of chiral disorder (i) or bond disorder: randomness in hopping elements t ij infinitely strong on-site impurities unitary limit: all bonds adjacent to the impurity are effectively cut (C z -symmetry) (ii) dislocations: random non-abelian gauge field (C 0 -symmetry) (iii) random magnetic field, ripples (both C 0 and C z symmetries) Realizations of long-range disorder (i) (ii) smooth random potential: correlation length lattice spacing charged impurities (iii) ripples: smooth random magnetic field Experimental dependence σ n e and odd-integer QHE = long-range disorder (ii), (iii)

24 Ripples: Chiral long-range disorder Estimated size d and height h: d = 5 nm, h = 0.5 nm (from electron diffraction pattern) Meyer, Geim, Katsnelson, Novoselov, Booth, Roth, Nature 07 d = 10 nm, h = 0.3 nm (from AFM measurements) Tikhonenko, Horsell, Gorbachev, Savchenko, PRL 08

25 Absence of localization of Dirac fermions in graphene with chiral or long-range disorder Disorder Symmetries Class Conductivity QHE Vacancies C z, T 0 BDI 4e 2 /πh normal Vacancies + RMF C z AIII 4e 2 /πh normal σ z τ x,y disorder C z, T z CII 4e 2 /πh normal Dislocations C 0, T 0 CI 4e 2 /πh chiral Dislocations + RMF C 0 AIII 4e 2 /πh chiral Ripples, RMF C 0, Λ z 2 AIII 4e 2 /πh odd-chiral Charged impurities Λ z, T 2 AII or 4σSp odd random Dirac mass: σ z τ 0,z Λ z, CT 2 D 4e 2 /πh odd Charged imp. + RMF/ripples Λ z 2 A 4σ U odd C z -chirality Gade-Wegner phase C 0 -chirality Wess-Zumino-Witten term Λ z -symmetry decoupled valleys θ = π topological term

26 C 0 -chirality: H = σ z Hσ z (ripples, dislocations) Disorder structures: σ x,y τ z, σ x,y τ x,y, σ x,y τ 0 Couplings: α (breaks T 0 ), β (preserves T 0 ), γ (preserves T 0 ) Symmetry classes: C 0 + no T 0 = AIII (ChUE); C 0 + T 0 = CI (BdG) Generic C 0 -disorder Non-Abelian vector potential problem Nersesyan, Tsvelik, and Wenger 94, 95; Fendley and Konik 00; Ludwig 00 RG equations (cf. Altland, Simons, and Zirnbauer 02): α log L = 0, β log L = 4β γ, γ log L = β2. DoS: ρ(ε) ε 1/7 (generic C 0 ); ρ(ε) ε (1 α )/(1+α ) (α only)

27 Conductivity at µ = 0: C 0 -chiral disorder Current operator j = ev 0 τ z σ relation between G R and G A & σ z j x = ij y, σ z j y = ij x, transform the conductivity at µ = 0 to RR + AA form: σ xx = 1 π [ ] d 2 (r r ) Tr j α G R (0; r, r )j α G R (0; r, r) σ RR. α=x,y Gauge invariance: p p + ea constant vector potential σ RR = 2 π 2 A 2 Tr ln GR σ RR = 0 (?!) But: contribution with no impurity lines anomaly: UV divergence shift of p is not legitimate (cf. Schwinger model 62).

28 Universal conductivity at µ = 0 for C 0 -chiral disorder Calculate explicitly (δ infinitesimal ImΣ) σ = 8e2 v 2 0 π d 2 k δ 2 (2π) 2 (δ 2 + v0 2k2 ) 2 = 2e2 π 2 = 4 e 2 π h for C 0 -chiral disorder σ(µ = 0) does not depend on disorder strength! Alternative derivation: use Ward identity ie(r r )G R (0; r, r ) = [G R jg R ](0; r, r ) and integrate by parts only surface contribution remains: σ = ev 0 4π 3 dk n Tr [ j G R (k) ] = e2 π 3 dkn k k 2 = 4 π e 2 h Related works: Ludwig, Fisher, Shankar, Grinstein 94; Tsvelik 95

29 C z chirality: H = σ z τ z Hσ z τ z (vacancies) Disorder structures: σ x,y τ 0, σ 0 τ x,y, σ x,y τ z, σ z τ x,y Couplings: γ (preserves T 0,z ), β z (breaks T z ), β 0 (breaks T 0 ), α (breaks T 0,z ) Symmetry classes: C z + no T 0,z = AIII (ChUE); C z + T 0 = BDI (ChOE); C z + T z = CII (ChSE) Related works: Gade, Wegner 91; Hikami, Shirai, Wegner 93; Guruswamy, LeClair, Ludwig 00; Motrunich, Damle, Huse 02; Ryu, Mudry, Furusaki, Ludwig 07 RG equations: α log L = 2β 0β z, β 0 log L = β z log L = 2α (β 0 + β z ), γ log L = β2 0 + β2 z DoS: ρ(ε) ε 1 exp( # log ε 2/3 ) Conductivity: σ = 4e2 πh [ (β 0 β z ) 2 + ]

30 Long-range disorder Intervalley scattering suppressed neglect it reduced Hamiltonian: H = iv 0 (σ x x + σ y y ) + σ µ V µ (x, y) Disorder couplings: α 0 = V 2 Symmetries: 0 2πv 2 0 Ludwig, Fisher, Shankar, Grinstein 94 y, α = V 2 x + V 2 2πv 2 0, α z = V 2 z 2πv 2 0 α 0 disorder T-invariance H = σ y H T σ y AII (GSE) α disorder C-invariance H = σ z Hσ z AIII (ChUE) α z disorder CT-invariance H = σ x H T σ x D (BdG) generic long-range disorder A (GUE) Ultraviolet RG: α 0 ln L = 2(α 0 + α z )(α + α 0 ), α ln L = 4α 0α z, α z ln L = 2(α 0 + α z )(α α z )

31 Class D (random mass): Long-range disorder (cont d) Disorder is marginally irrelevant = diffusion never occurs DoS: ρ(ε) = ε πv 2 02α z log ε Conductivity: σ = 4e2 πh Class AIII (random vector potential): C 0 /C z chiral disorder; considered above DOS: ρ(ε) ε (1 α )/(1+α ) Conductivity: σ = 4e2 πh

32 Long-range disorder: unitary symmetry (ripples + charged imp.) Generic long-range disorder (no symmetries) = class A (GUE) Effective infrared theory is Pruisken s unitary σ-model with topological θ-term: S[Q] = 1 4 Str [ σ xx 2 ( Q)2 + ( σ xy + 1 ) ] Q x Q y Q 2 σ xx 8 Str( Q)2 +iπn[q] Compact (FF) sector of the model: Q FF M F = U(2n) U(n) U(n) Topological term takes values N[Q] π 2 (M F ) = Z Vacuum angle θ = π in the absence of magnetic field due to anomaly = Quantum Hall critical point σ = 4σ U 4 ( )e2 h d lnσ d ln L 0 Σ U Θ Π Θ 0 lnσ

33 Long-range disorder: symplectic symmetry (charged imp.) Random scalar potential α 0 preserves T-inversion symmetry = class AII (GSE) Partition function is real = Im S = 0 or π Compact sector: O(4n) Q FF M F = O(2n) O(2n) = π 2 (M F ) = { Z Z, n = 1; Z 2, n 2 At n = 1 M F = S 2 S 2 /Z 2 [Cooperons] [diffusons] = Im S = θ c N c [Q] + θ d N d [Q] T-invariance θ c = θ d = 0 or π Z 2 subgroup Explicit calculation Anomaly θ c,d = π At n 2 we use M F n=1 M F n 2 = Im S = πn[q] possibility of Z 2 topological term: Fendley 01

34 Long-range potential disorder (cont d): symplectic σ-model with Z 2 topological term Symplectic sigma-model with θ = π term: Topological delocalization : S[Q] = σ xx 16 Str( Q)2 + iπn[q] as for Pruisken σ-model of QHE at criticality, instantons suppress localization = possible scenarios: β function everywhere positive, intermediate attractive fixed point, conductivity σ σ = 4σ Sp Numerics needed!

35 Long-range potential disorder: numerics Bardarson, Tworzyd lo, Brouwer, Beenakker, PRL 07 Nomura, Koshino, Ryu, PRL 07 absence of localization confirmed scaling towards the perfect-metal fixed point σ From ballistics to diffusion: single-parameter scaling??? open question: can e-e interaction induce an intermediate fixed point?

36 Odd quantum Hall effect Decoupled valleys + magnetic field = unitary sigma model with anomalous topological term: S[Q] = 1 4 Str [ σ xx 2 ( Q)2 + ( σ xy + 1 ) ] Q x Q y Q 2 = odd-integer QHE 3 2 Σ 2e 2 h Ν generic (valley-mixing) disorder = conventional IQHE weakly valley-mixing disorder = even plateaus narrow, emerge at low T

37 Quantum Hall effect: Weak valley mixing S[Q K, Q K ] = S[Q K ] + S[Q K ] + ρ τ mix Str Q K Q K 3 2 Σ xx 2e 2 h 2g U g U Σ xy 2e 2 h Ρ ħω c 0 ħω c Ε 2 0 2k 1 2k 2k 1 Σ xy 2e 2 h n Even plateau width (τ/τ mix ) 0.45, visible at T < T mix /τ mix Estimate for Coulomb scatterers: T mix 100 mk ; even plateau width δn even 0.05 Cf. splitting of delocalized states in ordinary QHE by spin-orbit / spin-flip scattering, Khmelnitskii 92; Lee, Chalker 94

38 Chiral quantum Hall effect C 0 -chiral disorder random vector potential Atiyah-Singer theorem: In magnetic field, zeroth Landau level remains degenerate! (Aharonov and Casher 79) Within zeroth Landau level Hall effect is classical Decoupled valleys (ripples) 3 2 Weakly mixed valleys (dislocations) 3 2 Σ 2e 2 h Σ xy 2e 2 h Ν 3 Ρ ħω c 0 ħω c Ε n

39 Ballistic transport W L large aspect ratio W L ballistic regime L l perfect metallic leads (highly doped graphene) Transfer matrix transmission eigenvalues T n of t t conductance G = 4e2 h Tr(t+ t) Fano factor F = 1 Tr(t+ t) 2 Tr(t + t) [ clean limit T py (x) = 1 + p2 ( ) ] 1 y p 2 y sinh2 p 2 ǫ2 y ǫ2 x Tworzyd lo et al 06; Titov 07 perturbation theory: leading-order disorder-induced correction ballistic Renormalization Group

40 Random potential: Two-loop Renormalization Group 2D action for Dirac fermions in random potential: [ ] S[ψ] = d 2 x ψσ ψ + iǫ ψψ + πα 0 ( ψψ) 2 Energy Disorder 1-loop 2-loop dǫ d log Λ = ǫ( α 0 + α 2 0 /2) dα 0 d log Λ = 2α α3 0

41 Solution to RG equations and phase diagram α 0 (Λ) = 1 2 log(l 0 /Λ) ǫ(λ) = ǫ 2α0 log(l 0 /Λ) l 0 /a = α 0 e 1/2α 0 mean free path; γ 0 = e 1/2α 0 inverse mean free time log L L l 0 UB D B RG stops when Λ L = ultra-ballistic (UB) ǫ(λ)λ 1 = ballistic (B) α 0 (Λ) 1 = diffusive (D) 0 Ε Γ 0 ΕL Crossover between regimes: UB D: L l 0 zero-energy mean free path UB B: L λ F (ǫ) = 2α 0 log(ǫ/γ 0 )/ǫ Fermi wave length B D: L l(ǫ) = [2α 0 log(ǫ/γ 0 )] 3/2 /ǫ finite-energy mean free path

42 Zero energy Single parameter scaling Gaussian white-noise random potential Transmission distribution is universal! P(T) = W σ 2πLT with σ = 1 T { 1 + 2α 0 (L), ultra-ballistics G πh/4e 2, diffusion (diffusive limit: Dorokhov 83) d log σ d log L Unified scaling d log σ d log L = { (σ 1) 2 /σ, ballistic 1/σ, diffusive 1 σ[4e 2 /πh]

43 Related realizations of Z 2 topological metals and insulators Long-range potential disorder Z 2 topological metal of symplectic class Related systems with Z 2 topology: quasi-1d symplectic-class wires with odd number of channels: one mode remains delocalized Zirnbauer 92; ADM, Müller-Groeling, Zirnbauer 94; Takane 04 Realizations: carbon nanotubes with long-range disorder Ando, Suzuura 02 quantum spin Hall effect: 2D symplectic class Z 2 topological insulator edge state topologically protected Theory: Kane, Mele 05 Recent experimental observation: HgTe/CdTe quantum wells Koenig,..., Molenkamp, Science 07 3D symplectic-class topological insulator surface state Z 2 topological metal the same as in graphene with decoupled valleys! Experimental observation: BiSb crystal, Hsieh,..., Hasan, Nature 08 Theory of 3D topological insulators: Schnyder, Ryu, Furusaki, Ludwig 08

44 Summary Graphene: 2D massless Dirac fermions with disorder Type of disorder symmetry and topology of field theory crucially affect localization Absence of localization for chiral and long-range disorder chiral disorder: conductivity σ = 4 e 2 /πh long-range disorder: topological delocalization random potential + ripples (random magnetic field) quantum Hall critical point, σ = 4 σ U random potential Z 2 topological metal (symplectic class) Anomalous quantum Hall effects: decoupled valleys odd-integer QHE valley mixing even plateaus appear chiral disorder (ripples) no localization, classical Hall effect at the lowest Landau level From ballistics to diffusion: phase diagram ; white-noise potential, ǫ = 0 single-parameter scaling of transm. distr. Long-range random potential: relation to other realizations of Z 2 topological metals and insulators