Topological quantum criticality of Dirac fermions: From graphene to topological insulators. Alexander D. Mirlin

Transcription

1 Topological quantum criticality of Dirac fermions: From graphene to topological insulators Alexander D. Mirlin Karlsruhe Institute of Technology & PNPI St. Petersburg P. Ostrovsky, Karlsruhe Institute of Technology & Landau I., Chernogolovka I. Gornyi, Karlsruhe Institute of Technology & Ioffe I., St.Petersburg

2 Plan Localization theory: Symmetries and topologies Graphene and 2D Dirac fermions Conductivity and topological delocalization at the Dirac point Anomalous quantum Hall effect Topological insulators (TIs): General classification 2D and 3D Z 2 TIs in time-reversal-invariant systems with spin-orbit interaction Interaction-induced self-organized quantum criticality at the surface of 3D TI Quantum spin Hall transition between 2D TI and normal insulator

3 50 years of Anderson localization Philip W. Anderson 1958 Absence of diffusion in certain random lattices sufficiently strong disorder quantum localization eigenstates exponentially localized, no diffusion Anderson insulator The Nobel Prize in Physics 1977

4 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

5 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

6 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 )

7 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)

8 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)

9 Magnetotransport in 2D: Integer Quantum Hall Effect resistivity tensor: ρ xx = E x /j x ρ yx = E y /j x Hall resistivity B E x E y j x Klaus von Klitzing Nobel Prize 1985

10 IQH transition IQHE flow diagram Khmelnitskii 83, Pruisken 84 localized 11 localized critical point Field theory (Pruisken): σ-model with topological term { S = d 2 r σ xx 8 Tr( µq) 2 + σ } xy 8 Trǫ µνq µ Q ν Q

11 Graphene: monoatomic layer of carbon

12 Graphene fabrication: Micromechanical cleavage Graphite Slicing Looking

13 Graphene samples

14 Graphene: atomic resolution images

15 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

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

17 To compare: Disordered semiconductor systems: From metal to insulator with lowering T quasi-1d geometry (long wires) 2D geometry Gershenson et al 97 Hsu, Valles 95

18 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!

19 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

20 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)

21 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

22 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)

23 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)

24 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)

25 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

26 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 (log L) 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 4σU odd C z -chirality Gade-Wegner phase C 0 -chirality Wess-Zumino-Witten term Λ z -symmetry decoupled valleys θ = π topological term

27 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 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 Random scalar potential vector potential mass Symmetries: 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 )

28 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 DOS: ρ(ε) ε (1 α )/(1+α ) Conductivity: σ = 4e2 πh

29 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

30 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

31 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!

32 Long-range potential disorder: numerics Bardarson, Tworzyd lo, Brouwer, Beenakker, PRL 07 Nomura, Koshino, Ryu, PRL 07 absence of localization confirmed logarithnic scaling towards the perfect-metal fixed point σ

33 Anomalous (odd-integer) 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 Σ 2e 2 h Ν generic (valley-mixing) disorder = conventional IQHE weakly valley-mixing disorder = even plateaus narrow, emerge at low T

34 Anomalous quantum Hall effect Experimental observation of anomalous (odd-integer) QHE long-range disorder (almost decoupled valleys) absence of localization in B = 0 (topological delocalization)

35 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

36 Topological Insulators Topological Insulators = Bulk insulators with topologically protected delocalized states on their boundary Well-known example: Quantum Hall Effect QH insulators n =..., 2, 1, 0, 1, 2,... edge states Z topological insulator Novel class: Z 2 TIs: n = 0 or n = 1 First realization: 2D Quantum Spin Hall Effect insulators

37 Periodic table of Topological Insulators Symmetry classes Topological insulators p H p R p S p π 0 (R p ) d=1 d=2 d=3 d=4 0 AI BDI CII Z Z 1 BDI BD AII Z 2 Z BD DIII DIII Z 2 Z 2 Z DIII AII BD 0 Z 2 Z 2 Z 0 4 AII CII BDI Z 0 Z 2 Z 2 Z 5 CII C AI 0 Z 0 Z 2 Z 2 6 C CI CI 0 0 Z 0 Z 2 7 CI AI C Z 0 0 A AIII AIII Z 0 Z 0 Z 1 AIII A A 0 Z 0 Z 0 H p symmetry class of Hamiltonians R p sym. class of classifying space (of Hamiltonians with eigenvalues ±1) S p symmetry class of compact sector of σ-model manifold Kitaev 09; Schnyder, Ryu, Furusaki, Ludwig 08-09

38 Classification of Topological insulators Two ways to detect existence of TIs of class p in d dimensions: (i) by inspecting the topology of classifying spaces R p : { { TI of type Z Z π 0 (R p d ) = TI of type Z 2 Z 2 (ii) by analyzing homotopy groups of the σ-model manifolds: { TI of type Z π d (S p ) = Z Wess-Zumino term TI of type Z 2 π d 1 (S p ) = Z 2 θ = π topological term WZ and θ = π terms make boundary excitations non-localizable TI in d topological protection from localization in d 1 Bott periodicity: π d (R p ) = π 0 (R p+d ), periodicity 8

39 2D Z 2 TIs: Quantum Spin Hall Effect Kane, Mele 05; Sheng, Sheng, Ting, Haldane 05; Bernevig, Zhang 06 Symmetry class AII (symplectic): time-reversal invariance T 2 = 1 Simple model: two copies of QHE, magnetic field B for spin and B for spin σ xy ( ) = e 2 /h σ xy ( ) = e 2 /h spin Hall conductivity σ xy ( ) σ xy ( ) = 2e 2 /h generic spin-orbit interaction spin not conserved anymore but Kramers degeneracy holds one popagating edge mode in each direction backscattering forbidden: topological protection! Earlier results on symplectic-class wires with odd number of channels: one mode remains delocalized Zirnbauer 92; ADM, Müller-Groeling, Zirnbauer 94; Takane 04 realization: carbon nanotubes with long-range disorder Ando, Suzuura 02

40 Quantum Spin Hall Effect in graphene with SO interaction Kane, Mele 05

41 QSHE in CdTe/HgTe/CdTe quantum wells: Theory Bernevig, Hughes, Zhang 06 H eff (k) = h(k) = ( h(k) 0 0 h (k) ( m(k) Ak Ak + m(k) k ± = k x ± ik y ) m(k) = M + B(k 2 x + k2 y ) HgTe: inverted band structure M < 0 for d > d c TI )

42 QSHE in CdTe/HgTe/CdTe quantum wells: Experiment Molenkamp group 07 I normal insulator, d = 5.5 nm II, III, IV inverted quantum well structure, topological insulator d = 7.3 nm

43 3D Topological Insulators Fu, Kane, Mele 07 Tight-binding model on a diamond lattice with spin-orbit interaction

44 3D Topological Insulator: Bi 1 x Sb x Hasan group 08 Other realizations: BiTe, BiSe

45 2D Dirac surface states of a 3D TI: Phenomenology

46 2D Dirac surface states of a 3D TI (cont d) Surface of 3D Z 2 TI: single 2D massless Dirac mode (more generally: odd number) single-valley graphene! With disorder: Topological protection from localization, RG flow towards supermetal What is the effect of Coulomb interaction? Topological protection from localization persists But interaction may destroy the supermetal phase!

47

48 Interaction-induced quantum criticality in 3D TI Interaction tendency to localization at g 1 Topology protection from strong localization (no flow towards g 1) novel quantum critical point should emerge at g 1 analogous to QHE, but here induced by interaction

49 β functions for symplectic class: Interaction and Topology no interaction with interaction usual spin-orbit β(g) 0 insulator g 1.4 supermetal g β(g) 0 insulator g Dirac fermions β(g) 0 supermetal g β(g) 0 critical state g

50 2D TIs: QSHE phase diagram In the presence of disorder, TI and normal insulator phases are separated by the supermetal phase transitions TI supermetal and supermetal NI are in the coventional symplectic symmetry class Onoda, Avishai, Nagaosa 07; Obuse et al 07 Effect of Coulomb interaction on phase diagram?

51 2D TIs: QSHE phase diagram (cont d) no interaction with interaction critical supermetal 0disorder QSH insulator inverted 0 band gap normal insulator normal 0disorder QSH insulator inverted 0 band gap normal insulator normal Coulomb interaction kills the supermetal phase, thus restoring a direct transition between two insulator phases quantum critical point of Quantum Spin Hall transition

52 Interaction-induced quantum critical points of Z 2 TIs We thus have two novel 2D quantum critical point: on surface of 3D TI 2D QSH transition They share many common properties: symplectic symmetry Z 2 topological protection interaction-induced criticality conductivity of order unity (probably universal) This suggests that these two critical points may be equivalent

53 Plan Localization theory: Symmetries and topologies Graphene and 2D Dirac fermions Conductivity and topological delocalization at the Dirac point Anomalous quantum Hall effect Topological insulators (TIs): General classification 2D and 3D Z 2 TIs in time-reversal-invariant systems with spin-orbit interaction Interaction-induced self-organized quantum criticality at the surface of 3D TI Quantum spin Hall transition between 2D TI and normal insulator