Anderson Localization: Multifractality, Symmetries, and Topologies. Alexander D. Mirlin Karlsruhe Institute of Technology

Transcription

1 Anderson Localization: Multifractality, Symmetries, and Topologies Alexander D. Mirlin Karlsruhe Institute of Technology Physics Colloquium, Heidelberg, 11 May 2012

2 Plan Anderson localization: basic properties, field theory Wave function multifractality Symmetries of disordered systems Manifestations of topology in localization theory Influence of electron-electron interaction

3 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 Nobel Prize 1977

4 Anderson Localization: Extended and localized wave functions Schrödinger equation in a random potential [ 2 2m + U(r)]ψ = Eψ U E x Ψ delocalized Ψ localized x x ψ 2 exp{ r r 0 /ξ}

5 Precursor of strong Anderson localization: Weak localization = = = p+q p+q q p p Cooperon loop (interference of timereversed paths) σ WL = e2 2πh σ WL = e2 πh lnl ω l 1 (dq) σ WL σ 0 πν Dq 2 iω ( 1 1 ), 3D L ω (D/ω) 1/2 l L ω, 2D Generally: IR cutoff L ω min{l ω,l φ,l,l H } σ WL = e2 h L ω, quasi-1d e 2 /h (25 kω) 1 conductance quantum

6 Weak localization in experiment: Magnetoresistance Li et al. (Savchenko group), PRL 03 2D electron gas in GaAs heterostructure low field: weak localization Gorbachev et al. (Savchenko group), PRL 07 weak localization in bilayer graphene

7 Anderson Insulators & Metals β = dln(g) / dln(l) g=g/(e 2 /h) d=3 d=2 d=1 Connection with scaling theory of critical phenomena: Thouless 74; Wegner 76 Scaling theory of localization: Abrahams, Anderson, Licciardello, Ramakrishnan 79 scaling variable: dimensionless conductance g = G/(e 2 /h) 0 ln(g) RG for field theory (σ-model) Wegner 79 quasi-1d, 2D : all states are localized d > 2: Anderson metal-insulator transition delocalized critical point localized disorder review: Evers, ADM, Rev. Mod. Phys. 80, 1355 (2008)

8 action: S[Q] = πν 4 Field theory: non-linear σ-model d d r Tr [ D( Q) 2 2iωΛQ], Q 2 (r) = 1 Wegner 79 σ-model manifold: e.g., unitary symmetry class (broken time-reversal symmetry): fermionic replicas: U(2n)/U(n) U(n), n 0 sphere bosonic replicas: U(n, n)/u(n) U(n), n 0 hyperboloid supersymmetry (Efetov 83): U(1, 1 2)/U(1 1) U(1 1) { sphere hyperboloid } dressed by anticommuting variables with electron-electron interaction: Finkelstein 83

9 σ model: Perturbative treatment For comparison, consider ferromagnet model in external magnetic field: [ ] κ H[S] = d d r 2 ( S(r))2 BS(r), S 2 (r) = 1 n-component vector σ-model Target manifold: sphere S n 1 = O(n)/O(n 1) Independent degrees of freedom: transverse part S ; S 1 = (1 S 2 )1/2 H[S ] = 1 d d r [ κ[ S (r)] 2 + BS 2 2 (r) + O(S4 (r))] Ferromagnetic phase: broken symmetry, Goldstone modes spin waves S S q 1 κq 2 + B Similarly S[Q] = πν 4 theory of interacting diffusion modes; Goldstone mode: diffusion propagator d d rstr[d( Q ) 2 iωq 2 + O(Q3 )] Q Q q,ω 1 πν(dq 2 iω)

10 σ-models: What are they good for? reproduce diffuson-cooperon diagrammatics......and go beyond it: metallic samples (g 1): level & wavefunction statistics: random matrix theory + deviations quasi-1d samples: exact solution, crossover from weak to strong localization Anderson transitions: RG treatment, phase diagrams, critical exponents σ-models with non-trivial topologies: Dirac fermions, topological insulators non-trivial saddle-points: nonperturbative effects, asymptotic tails of distributions generalizations: interaction, non-equilibrium (Keldysh)

11 Quasi-1D geometry: Exact solution of the σ-model quasi-1d geometry (many-channel wire) 1D σ-model diffusion on σ-model curved space Localization length Efetov, Larkin 83 t W = Q W, t = x/ξ Exact solution for the statistics of eigenfunctions Fyodorov, ADM Exact g (L/ξ) and var(g)(l/ξ) Zirnbauer, ADM, Müller-Groeling e.g. for orthogonal symmetry class: g n (L) = π [ dλtanh 2 (πλ/2)(λ 2 + 1) 1 p n (1,λ,λ)exp L ] 2 0 2ξ (1 + λ2 ) dλ 1 dλ 2 l(l 2 1)λ 1 tanh(πλ 1 /2)λ 2 tanh(πλ 2 /2) p n (l,λ 1,λ 2 ) l 2N +1 0 σ,σ 1,σ 2 =±1 ( 1 + σl + iσ 1 λ 1 + iσ 2 λ 2 ) 1 exp [ L4ξ ] (l2 + λ 21 + λ22 + 1) p 1 (l,λ 1,λ 2 ) = l 2 + λ λ , p 2 (l,λ 1,λ 2 ) = 1 2 (λ4 1 + λ l4 + 3l 2 (λ λ2 2 ) + 2l2 λ 2 1 λ2 2 2)

12 Quasi-1D geometry: Exact solution of the σ-model (cont d) L ξ asymptotics: g (L) = 2ξ L and var(g(l)) = L ξ + 4 ( ) L 2 + O 945 ξ ( ) L 2. ξ L ξ + O ( L ξ ) 3 L ξ asymptotics: g n = 2 3/2 n π 7/2 (ξ/l) 3/2 exp{ L/2ξ} <g>l/ξ var(g) L/ξ L/ξ orthogonal (full), unitary (dashed), symplectic (dot-dashed)

13 From weak to strong localization of electrons in wires GaAs wires Gershenson et al, PRL 97

14 Anderson localization of atomic Bose-Einstein condensate in 1D Billy et al (Aspect group), Nature 2008

15 3D Anderson localization transition in Si:P Stupp et al, PRL 93; Wafenschmidt et al, PRL 97 (von Löhneysen group)

16 3D Anderson localization in atomic kicked rotor kicked rotor H = p2 2 + K cosx[1 + ǫcosω 2tcosω 3 t] δ(t 2πn/ω 1 ) n Anderson localization in momentum space. Three frequencies mimic 3D! Experimental realization: cesium atoms exposed to a pulsed laser beam. Chabé et al, PRL 08

17 Multifractality at the Anderson transition P q = d d r ψ(r) 2q P q inverse participation ratio L 0 insulator L τ q critical L d(q 1) metal τ q = d(q 1) + q D q (q 1) multifractality normal anomalous τ q Legendre transformation singularity spectrum f(α) wave function statistics: P(ln ψ 2 ) L d+f(ln ψ2 /lnl) f(α) d 0 ψ 2 large α α + L f(α) measure of the set of points where ψ 2 L α d α 0 metallic critical ψ 2 small α

18 Multifractality (cont d) Multifractality implies very broad distribution of observables characterizing wave functions. For example, parabolic f(α) implies log-normal distribution P( ψ 2 ) exp{ # ln 2 ψ 2 /lnl} field theory language: q scaling dimensions of operators O (q) (QΛ) q Wegner 80 Infinitely many operators with negative scaling dimensions, i.e. RG relevant (increasing under renormalization) 2-, 3-, 4-,...-point wave function correlations at criticality ψ 2 i (r 1) ψ 2 j (r 2)... also show power-law scaling controlled by multifractality boundary multifractality Subramaniam, Gruzberg, Ludwig, Evers, Mildenberger, ADM, PRL 06

19 Dimensionality dependence of multifractality 4 3 Analytics (2 + ǫ, one-loop) and numerics ~ D q f(α) ~ q ~ f(α) α α τ q = (q 1)d q(q 1)ǫ + O(ǫ 4 ) f(α) = d (d + ǫ α) 2 /4ǫ + O(ǫ 4 ) d = 4 (full) d = 3 (dashed) d = 2 + ǫ, ǫ = 0.2 (dotted) d = 2 + ǫ, ǫ = 0.01 (dot-dashed) Inset: d = 3 (dashed) vs. d = 2 + ǫ, ǫ = 1 (full) Mildenberger, Evers, ADM 02

20 Multifractality at the Quantum Hall transition Evers, Mildenberger, ADM f(α) f(α) L=16 L=128 L= α

21 Symmetry of multifractal spectra ADM, Fyodorov, Mildenberger, Evers 06 LDOS distribution in σ-model + universality exact symmetry of the multifractal spectrum: q = 1 q f(2d α) = f(α) + d α q, 1 q b=4 b=1 b=0.3 b=0.1 f(α) b=0.3 b=0.1 b=1 b= q α probabilities of unusually large and unusually small ψ 2 (r) are related!

22 Symmetries of multifractal spectra (cont d) Relation to invariance of the σ model correlation functions with respect to Weyl group of the σ model target space; generalization to unconventional symmetry classes Gruzberg, Ludwig, ADM, Zirnbauer PRL 11 generalization on full set of composite operators, i.e. also on subleading ones. Important example: Gruzberg, ADM, Zirnbauer, in preparation A 2 = V 2 ψ 1 (r 1 )ψ 2 (r 2 ) ψ 1 (r 2 )ψ 2 (r 1 ) 2 Hartree-Fock matrix element of e-e interaction scaling: A q 2 L (2) q symmetry: (2) q = (2) 2 q

23 Interaction scaling at criticality Hartree, Fock enhanced by multifractality exponent < 0 Hartree Fock suppressed by multifractality exponent (2) > 0 Burmistrov, Bera, Evers, Gornyi, ADM, Annals Phys. 326, 1457 (2011) Temperature scaling at quantum Hall and metal-insulator transitions with short-range interaction

24 Multifractal spectrum of A 2 at quantum Hall transition Numerical data: Bera, Evers, unpublished Confirms the symmetry q 2 q

25 Multifractality: Experiment I Local DOS flucutuations near metal-insulator transition in Ga 1 x Mn x As Richardella,...,Yazdani, Science 10

26 Multifractality: Experiment II Ultrasound speckle in a system of randomly packed Al beads Faez, Strybulevich, Page, Lagendijk, van Tiggelen, PRL 09

27 Multifractality: Experiment III Localization of light in an array of dielectric nano-needles Mascheck et al, Nature Photonics 12

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

29 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)

30 Role of symmetry: 2D systems of Wigner-Dyson classes Orthogonal and Unitary: localization; parametrically different localization length: ξ U ξ O Symplectic: metal-insulator transition Usual realization of Sp class: spin-orbit interaction

31 Role of symmetry: graphene with vacancies at the Dirac point Σ 4e 2 Πh a 0, colored nl 2 No localization at E = 0 (Dirac point): Chiral symmetry! Ostrovsky, Titov, Bera, Gornyi, ADM, PRL 10

32 Symmetry alone is not always sufficient to characterize the system. There may be also a non-trivial topology. It may protect the system from localization.

33 Integer quantum Hall effect von Klitzing 80 ; Nobel Prize 85 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 IQHE flow diagram Khmelnitskii 83, Pruisken 84 localized 11 localized critical point QH insulators n =..., 2, 1,0,1,2,... protected edge states Z topological insulator

34 2D massless Dirac fermions Graphene Geim, Novoselov 04 Nobel Prize 10 Surface of 3D topological insulators BiSb, BiSe, BiTe Hasan group 08 σ-model field theory for disordered 2D Dirac fermions Ostrovsky, Gornyi, ADM 07 Graphene: long-range disorder (no valley mixing) Surface states of 3D TI: no restriction on disorder range

35 2D Dirac fermions: σ-models with topological term Generic disorder (broken TRS) = class A (unitary) S[Q] = 1 8 Str[ σ xx ( Q) 2 + Q x Q y Q ] = σ xx 8 Str( Q)2 + iπn[q] topol. invariant N[Q] π 2 (M) = Z = Quantum Hall critical point σ = 4σ U 4 ( )e2 h Random potential (preserved TRS) = class AII (symplectic) S[Q] = σ xx 16 Str( Q)2 + iπn[q] topological invariant: N[Q] π 2 (M) = Z 2 = {0,1} Topological protection from localization! Ostrovsky, Gornyi, ADM, PRL 98, (2007)

36 Dirac fermions in random potential: numerics Bardarson, Tworzyd lo, Brouwer, Beenakker, PRL 07 Nomura, Koshino, Ryu, PRL 07 absence of localization confirmed log scaling towards the perfect-metal fixed point σ

37 Schematic beta functions for 2D systems of symplectic class AII Conventional spin-orbit systems Dirac fermions (topological protection) surface of 3D top. insulator or graphene without valley mixing

38 Graphene: Experiments Geim-Novoselov and Kim groups Topological terms explain unconventional properties of high-quality graphene samples: absence of localization at Dirac point down to very low temperatures (30 mk), although conductivity e 2 /h per spin per valley anomalous QHE: σ xy = (2n + 1) 2e 2 /h; QHE transition at n = 0 (Dirac point), i.e. at σ xy = 0

39 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 09; Ostrovsky, Gornyi, ADM 10

40 Electron-electron interaction E-e interaction can be incorporated within the same general theoretical framework (σ model); in some cases essentially modifies localization properties

41 MIT in a 2D gas with strong interaction Kravchenko et al 94,... Finkelstein 83; Punnoose, Finkelstein, (number of valleys N 1; in practice, N = 2 sufficient)

42 s s Localization behavior in 2D Symplectic class: Effects of topology and Coulomb interaction t r t t t r t s Ostrovsky, Gornyi, ADM, PRL 10

43 Summary Anderson localization: basic properties, field theory Wave function multifractality Symmetries of disordered systems Manifestations of topology in localization theory Influence of electron-electron interaction Collaboration: F. Evers, A. Mildenberger, I. Gornyi, P. Ostrovsky, I. Protopopov, E. König, S. Bera (Karlsruhe) M. Titov (Edinburgh Nijmegen) M. Zirnbauer (Köln) I. Gruzberg, A. Subramaniam (Chicago) Y. Fyodorov (Notthingham London) A. Ludwig (Santa Barbara) I. Burmistrov (Moscow)