Quantum transport in disordered systems: Part III. Alexander D. Mirlin

Transcription

1 Quantum transport in disordered systems: Part III Alexander D. Mirlin Research Center Karslruhe & University Karlsruhe & PNPI St. Petersburg mirlin/ Evers, ADM Anderson transitions, arxiv:

2 Magnetotransport resistivity tensor: ρ xx = E x /j x ρ yx = E y /j x Hall resistivity B E x E y j x classically (Drude Boltzmann theory): ρ xx = m e 2 n e τ ρ yx = B n e ec independent of B ρ ρ xy xx B

3 Quantum transport in strong magnetic fields Integer Quantum Hall Effect (IQHE) Fractional Quantum Hall Effect (FQHE)

4 Basics of IQHE 2D Electron in transverse magnetic field Landau levels E n = ω c (n + 1/2) N(E) ω c - cyclotron frequency ν = Φ 0 n B = N e N Φ - filling factor Φ 0 = hc e - flux quantum disorder Landau levels broadened delocalized states localized states E Anderson localization only states in the band center delocalized E F in the range of localized states { plateau in σxy σ xx = 0 Quantization of σ xy?

5 Hall conductivity quantization Laughlin 81, Halperin 82 EH = 1 2πr 1 c dφ dt σxy = j EH = Q Φ0/c Q - charge transported as Φ Φ + Φ0 Gauge invariance states are not changed Localized states are not influenced Delocalized state may transform into another delocalized state = k = 0, 1, 2,... electrons are transported k - number of filled delocalized bands Q = ke = σxy = k e2 h Φ E H j E F E

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

7 Critical behavior: Modeling and numerics Models: a) white-noise disorder projected to the lowest Landau level b) Chalker-Coddington network (models a long-range disorder) Chalker, Coddington 88 Localization length exponent ξ E E c ν ν = 2.35 ± 0.03 Huckestein, Kramer 90,...

8 Multifractal wave functions at the Quantum Hall transition

9 Multifractality at the Quantum Hall critical point QHE: anomalous dimensions f(α) f(α) L=16 L=128 L=1024 q /q(1 q) q α spectrum is parabolic with a high (1%) accuracy: f(α) 2 (α α 0) 2 4(α 0 2), q (α 0 2)q(1 q) with α 0 2 = 0.262±0.003 Evers, Mildenberger, ADM 01 important for identification of the CFT of the Quantum Hall critical point

10 Critical exponents: analytical approaches Numerics summary: ν = 2.35 ± 0.03 (= 7/3?) localization length q (α 0 2)q(1 q) α 0 2 = ± multifractality Analytics -? attempts to identify the Conformal Field Theory Zirnbauer 99 ; Bhaseen, Caux, Kogan, Tsvelik 00 ; Tsvelik 07 models of the Wess-Zumino-Novikov-Witten type term lines of critical points good to accomodate a strange value of α 0 2 ν =? Universality -? σ-model with topological term Pruisken et al present extension of a weak-coupling RG calculation (perturbation theory + instantons) beyond the region of its validity (to σ xx 1) ν 2.8, α accuracy uncontrollable

11 Dynamical scaling: Transition width Transition width δ T κ, ω ζ κ, ζ? Experiment: Wei et al PRL Li et al PRL 05 measured κ depends on the nature of disorder For short-range disorder (presumably) κ 0.42, ν 2.3 Frequency scaling: ζ Engel et al 93 ; Hohls et al 02 What about theory?

12 Dynamical scaling: Theory vs experiment Transition width δ(t ) T κ is found from L φ (T ) ξ(δ) Dephasing length L φ T 1/z T κ = 1/z T ν analogously, for frequency scaling: δ(ω) ω ζ, L ω ω 1/z, ζ = 1/zν short-range interaction: interaction irrelevant in the RG sense, fixed point unchanged ν 2.35, z = 2 unchanged ζ 0.21 z T 1.2 (numerics) κ 0.36 Coulomb interaction: interaction relevant, fixed point changed, indices not known Experiment z, z T 1 ( Coulomb interaction relevant) but ν = 2.3 (as for non-interacting system) persistent puzzle

13 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 )

14 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)

15 Disordered wires: DMPK approach transfer matrix Dorokhov 82 ; Mello, Pereyra, Kumar 88 ( ) ( ) R out L in = M R in L out current conservation: R out 2 R in 2 = L in 2 L out 2 M G = U(N, N) Cartan decomposition M = ( u 0 0 u ) ( cosh ˆx sinh ˆx sinh ˆx cosh ˆx ) ( v 0 0 v ) ˆx = diag(x 1,..., x N ) radial coordinates ; left/right matrices ( angular coordinates ) elements of K = U(N) U(N) Brownian motion on G/K, described by DMPK equation dp dl = 1 2lγ N i=1 J(x) J 1 (x) P. x i x i J(x) Jacobian of transformation to the radial coordinates

16 Disordered wires: DMPK approach (cont d) WD and BdG classes: 0 < x 1 < x 2 <... < x N, J(x) = N sinh(x i ± x j ) m o N sinh 2x k m l N sinh x l m s i<j ± k l Chiral classes: x 1 < x 2 <... < x N, J(x) = N i<j sinh(x i x j ) m o γ = { mo (N 1) + m l + 1, WD and BdG ; 1 2 [2 + m o(n 1)], chiral. m 0, m l, m s multiplicities of ordinary, long, and short roots. Conductance G = s N T n, T n = n=1 1 cosh 2 x n, s degeneracy

17 Disordered wires: Transfer matrix spaces Ham. transfer matrix tr.matr. m o m l m s class symmetric space class Wigner-Dyson classes A U(p + q)/u(p) U(q) AIII p q AI Sp(2N, R)/U(N) CI AII SO (2N)/U(N) N even N odd DIII-e DIII-o 4 1 chiral classes AIII GL(N, C)/U(N) A BDI GL(N, R)/O(N) AI CII U (2N)/Sp(2N) AII Bogoliubov - de Gennes classes C Sp(2p, 2q)/Sp(2p) Sp(2q) CII p q CI Sp(2N, C)/Sp(2N) C BD O(p, q)/o(p) O(q) BDI 1 0 p q DIII SO(N, C)/SO(N) N even N odd D B

18 Disordered wires: Conventional localization standard localization, Wigner-Dyson classes (m l = 1, m o = β, m s = 0). long-wire limit: 1 T 1 T 2... dp(x k ) dl = 1 2 P 2γl x 2 k 1 ξ k P x k, ξ 1 k = [1 + β(k 1)]/γl advection-diffusion equation, diffusion constant 1/2γl, drift velocity 1/ξ 1 solution: Gaussian form with x k = L/ξ k, var(x k ) = L/γl log of conductance g s/ cosh 2 x 1 has Gaussian distribution with ln g = 2L/γl ; var(ln g) = 4L/γl. On the side of atypically large g the distribution is cut at g 1; average conductance is determined by this cutoff: ln g = L/2γl. Typical and average localization length: ξ typ = γl ; ξ av = 4γl. BdG classes C, CI similar behavior

19 Delocalization in disordered wires: Chiral classes Chiral classes (AIII, BDI, CII), BdG classes (BD, DIII): m l = 0 no repulsion between x i and x i odd number of channels N x (N+1)/2 = 0 ln g = ( 8L πγl ) 1/2 ( ; var(ln g) = 4 8 ) L π γl ; g = (2γl/πL) 1/2 ; var(g) = (8γl/9πL) 1/2. stretched-exponential decay of typical conductance very strong fluctuations very slow L 1/2 decay of average conductance (slower than Ohm s law!) Slightly away from criticality (E 0): Localization length ξ typ ln E ; ξ av ln E 2 DOS ρ(e) 1/ E ln 3 E Dyson 53;...

20 Disordered wires with perfectly conducting channels m s 0 repulsion from zero zero eigenvalue(s) x i = 0 perfectly transmitting channels Two types: classes A, C, BD: arbitrary number ( p q ) of perfectly transmitting modes. Realization: Quantum Hall edge states (IQHE, SQHE, TQHE) classes AII, DIII : single perfectly transmitting mode σ-model language: π 1 (M F ) = Z 2 topological term with θ = π allowed. Realizations: models with spin-orbit interaction and odd number of channels: nanotubes (quasi-1d graphene-based structures) with decoupled valleys edge states in quantum spin Hall effect Zirnbauer 92; ADM, Müller-Groeling, Zirnbauer 94; Ando, Suzuura 02; Takane 04; Kane and Mele 05; Bernevig, Hughes, Zhang, Science 06

21 Mechanisms of Anderson criticality in 2D Common wisdom : all states are localized in 2D In fact, in 9 out of 10 symmetry classes the system can escape localization! variety of critical points Mechanisms of delocalization & criticality in 2D: broken spin-rotation invariance classes AII, D, DIII antilocalization, metallic phase, MIT 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 (random Dirac fermions, related to chiral anomaly) classes AIII, CI, DIII

22 Spin quantum Hall effect disordered d-wave superconductor (class C): charge not conserved but spin conserved time-reversal invariance broken: d x 2 y 2 + id xy order parameter strong magnetic field Haldane-Rezayi d-wave paired state of composite fermions at ν = 1/2 SQH plateau transition: spin Hall conductivity quantized ( ) j Z x = σs xy dbz (y) dy Model: SU(2) modification of the Chalker-Coddington network Kagalovsky, Horovitz, Avishai, Chalker 99 ; Senthil, Marston, Fisher 99

23 Spin quantum Hall effect (cont d) Similar to IQH transition but: DoS critical ρ(e) E µ mapping to percolation: analytical evaluation of DOS exponent µ = 1/7 localization length exponent ν = 4/3 lowest multifractal exponents: 2 = 1/4, 3 = 3/4 numerics: analytics confirmed multifractality spectrum: q, f(α) not parabolic 0.14 ρ(e) L 1/ ρ(e) L 1/ E/δ E/δ q / q(1 q) f(α) q α Gruzberg, Ludwig, Read 99; Beamond, Cardy, Chalker 02; Evers, Mildenberger, ADM 03

24 Thermal quantum Hall effect disordered p-wave superconductor (class D): neither charge nor spin conserved; only energy conservation TQH plateau transition: thermal Hall conductivity quantized very rich phase diagram: insulator, quantized Hall, and metallic phases both QH and Anderson (metal-insulator) transitions multicritical point 10-1 disorder metal σ xy =0 σ xy =1 Ising p ρ(e) ρ(e) p=0.02, L=64 p=0.02, L=128 p=0.04, L=64 p=0.04, L=128 p=0.08, L=64 p=0.08, L=128 p=0.13, L=64 p=0.13, L= E E Senthil, Fisher 00; Bocquet et al 00; Read, Green 00, Read, Ludwig 01; Chalker et al 02; Mildenberger, Evers, Narayanan, Chalker, ADM 07

25 Electron transport in disordered graphene Ostrovsky, Gornyi, ADM, Phys. Rev. B 74, (2006) Phys. Rev. Lett. 98, (2007) Eur. Phys. J. Special Topics 148, 63 (2007) arxiv:

26 Graphene: monoatomic layer of carbon

27 How graphene is made? Drawing Slicing Looking

28 Graphene samples

29 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

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

31 Graphene in transverse magnetic field Anomalous integer quantum Hall effect: σ xy = (n+ 1 2 ) (4e2 /h) 4 = 2 (spin) 2 (number of Dirac points) Observed up to room temperature!

32 nts 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 the spectrum is of massless Dirac-fermion type: 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

33 Clean graphene: basis and notations 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 is chosen as Ψ = {φ AK, φ BK, φ BK, φ AK } T. Hamiltonian in this basis has the form: Green function: H = v 0 τ 3 (σ 1 k x + σ 2 k y ) G R(A) 0 (ε, k) = ε + v 0τ 3 (σ 1 k x + σ 2 k y ) (ε ± i0) 2 v 2 0 k2.

34 Clean graphene: symmetries Space of valleys K K : Isospin Λ x = σ 3 τ 1, Λ y = σ 3 τ 2, Λ z = σ 0 τ 3. Time inversion Chirality T 0 : H = σ 1 τ 1 H T σ 1 τ 1 C 0 : H = σ 3 τ 0 Hσ 3 τ 0 Combinations with Λ x,y,z T x : H = σ 2 τ 0 H T σ 2 τ 0 C x : H = σ 0 τ 1 Hσ 0 τ 1 T y : H = σ 2 τ 3 H T σ 2 τ 3 C y : H = σ 0 τ 2 Hσ 0 τ 2 T z : H = σ 1 τ 2 H T σ 1 τ 2 C z : H = σ 3 τ 3 Hσ 3 τ 3 Spatial isotropy T x,y and C x,y occur simultaneously T and C

35 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 {1,2} τ {1,2} β σ 1,2 τ 0 γ σ 0 τ 1,2 β z σ 3 τ 3 γ z σ 3 τ 1,2 β σ 0 τ 3 γ σ 1,2 τ 3 α σ 3 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)

36 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 + v0 2k2 ] = 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? Yes, if disorder either (i) preserves one of chiral symmetries or (ii) is of long-range character (does not mix the valleys)

37 Types of chiral disorder (i) bond disorder: randomness in hopping elements t ij (C z -symmetry) (ii) infinitely strong on-site impurities unitary limit: all bonds adjacent to the impurity are effectively cut = bond disorder (C z -symmetry) (iii) dislocations: random non-abelian gauge field (C 0 -symmetry) (iv) random magnetic field, ripples (both C 0 and C z symmetries)

38 Critical states of disordered graphene 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 3 τ 1,2 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 Λ z, C 0 2 AIII 4e 2 /πh odd-chiral Charged impurities Λ z, T 2 AII 4σSp random Dirac mass: σ 3 τ 0,3 Λ z, CT 2 D 4e 2 /πh odd Charged imp. + (RMF, ripples) Λ z 2 A 4σU odd

39 Conductivity at µ = 0: C 0 -chiral disorder Current operator j = ev 0 τ 3 σ relation between G R and G A & σ 3 j x = ij y, σ 3 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).

40 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 ) = 2e2 2 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

41 Long-range disorder Smooth random potential does not scatter between valleys Reduced Hamiltonian: H = v 0 σk + σ µ V µ (r) Ludwig, Fisher, Shankar, Grinstein 94; Ostrovsky, Gornyi, ADM Disorder couplings: α 0 = V 2 0 2πv 2 0, α = V 2 x + V 2 2πv 2 0 y, 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)

42 Long-range disorder: unitary symmetry Generic long-range disorder breaks all symmetries = class A (GUE) Effective infrared theory is Pruisken s unitary sigma model with 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 ln Σ d ln L 0 Σ U Θ Π Θ 0 ln Σ

43 Long-range disorder: symplectic symmetry Diagonal disorder α 0 [charged impurities] preserves T -inversion symmetry = class AII (GSE) Partition function real = Im S = 0 or π = Im S topological invariant Symplectic sigma-model action: S[Q] = σ xx 16 Str( Q)2 + iθn[q] with topological term N[Q] Z 2 θ = 0 or π Explicit calculation = θ = π! = Novel critical point? Conductivity: σ = 4σ Sp Alternative scenario: β function everywhere positive, σ Numerics needed!

44 Long-range potential disorder: numerics Bardarson, Tworzyd lo, Brouwer, Beenakker arxiv: Nomura, Koshino, Ryu arxiv: absence of localization confirmed scaling towards a perfect-metal fixed point σ open question: can interaction induce an intermediate fixed point?

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

46 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 Estimate for Coulomb scatterers: T mix 100 mk ; even plateau width δn even 0.05

47 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 Σ 2e 2 h Decoupled valleys (ripples) Weakly mixed valleys (dislocations) Σ 2e 2 h Ν Ν