Wave function multifractality at Anderson localization transitions. Alexander D. Mirlin

Transcription

1 Wave function multifractality at Anderson localization transitions Alexander D. Mirlin Forschungszentrum Karlsruhe & Universität Karlsruhe mirlin/ F. Evers, A. Mildenberger (Karlsruhe) Y. Fyodorov (Nottingham, UK) I. Gruzberg, A. Subramaniam (Chicago), A. Ludwig (UC Santa Barbara)

2 Mesoscopic Physics Key ingredients: wave-like electron propagation, quantum interference disorder Coulomb interaction between electrons switched off in this talk Non-interacting mesoscopics: electron in a random potential: H = ˆp2 2m +U(r), U(r)U(r ) = W(r r ) e.g. white noise = 1 2πντ δ(r r ) disorder ensemble statistical treatment: mesoscopic fluctuations Classical analogue: Electromagnetic/acoustic waves in random media

3 Quantum interference in disordered mesoscopic conductors weak-localization correction to resistivity G i A i 2 = A i 2 + i j A i A j generically A i A j but: time-reversed paths: A i = A i mesoscopic conductance fluctuations (δg) 2 ( i j A i A j) 2 i j A i 2 A j 2 Calculus: Green s functions (retarded and advanced), diagrammatics 1 Key objects: diffusons & cooperons Π D (q, ω) Dq 2 iω Weak localization: G/G d d q/dq 2 = Π D (r, r) return probability Conductance fluctuations: (δg) 2 /G 2 d d q/(dq 2 ) 2 = drdr Π 2 D (r, r )

4 Wave function statistics Fyodorov, ADM 92;... Review: ADM, Phys. Rep. 2 distribution function P( ψ 2 (r) ), correlation functions ψ 2 (r 1 )ψ 2 (r 2 ) etc. perturbative (diagrammatic) approach not sufficient Field-theoretical method: σ-model Wegner 79, Efetov 82 (SUSY) S[Q] d d r Str[ D( Q(r)) 2 2iωΛQ(r)] Q 2 (r) = 1 Q {sphere hyperboloid} dressed by Grassmannian variables σ-model contains all the diffuson-cooperon diagrammatics + much more (strong localization; Anderson transition & RG; non-perturbative effects) zero mode (Q = const) RMT distribution P( ψ 2 (r) ) ψ(r) uncorrelated Gaussian random variables diffusive modes [Π D (r 1, r 2 )] deviations from RMT, long-range spatial correlations parameter g = D/L2 g 1: metal Thouless energy level spacing = G e 2 /h g 1: strong localization quasi-1d: g = ξ/l, ξ localization length dimensionless conductance

5 Experiment: Wave function statistics in disordered microwave billiards Kudrolli, Kidambi, Sridhar, PRL 95

6 Wave function statistics: numerical verification numerics: Uski, Mehlig, Römer, Schreiber 1 quasi-1d, metallic regime, distribution of t = V ψ 2 (r) body : 1/g corrections to RMT tail exp(... t) Physics of the slowly decaying tail : anomalously localized states

7 Anomalously localized states: imaging numerics: Uski, Mehlig, Schreiber 2 spatial structure of an anomalously localized state (ALS): ψ(r) 2 δ(v ψ() 2 t) with t atypically large ADM 97 ALS determine asymptotic behavior of distributions of various quantities (wave function amplitude, local and global density of states, relaxation time,... ) Altshuler, Kravtsov, Lerner, Fyodorov, ADM, Muzykantskii, Khmelnitskii, Falko, Efetov...

8 Wave function statistics: numerical verification. 2D. numerics: Uski, Mehlig, Römer, Schreiber 1 body of the distribution: (1/g) ln(l/l) corrections tail exp(... ln 2 t) Falko, Efetov 95 precursors of Anderson criticality

9 Anderson transition β = dln(g) / dln(l) 1 1 g=g/(e 2 /h) d=3 d=2 d=1 Scaling theory of localization: Abrahams, Anderson, Licciardello, Ramakrishnan 79 Modern approach: RG for field theory (σ-model) ln(g) quasi-1d: metallic localized crossover with decreasing g = ξ/l d > 2: Anderson metal-insulator transition (sometimes also in d = 2) delocalized localized critical point disorder Continuous phase transition with highly unconventional properties!

10 P q = d d r ψ(r) 2q Multifractality at the Anderson transition inverse participation ratio L insulator P q L τ q critical L d(q 1) metal τ q = d(q 1) + q D q (q 1) multifractality normal anomalous d metallic critical τ q Legendre transformation singularity spectrum f(α) f(α) ψ 2 large α α + ψ 2 small P(ln ψ 2 ) L d+f(ln ψ2 / ln L) wave function statistics L f(α) measure of the set of points where ψ 2 L α Multifractality is characteristic for a variety of complex systems: turbulence, strange attractors, diffusion-limited aggregation,... d α α Peculiarity: Statistical ensemble f(α) may become negative

11 Multifractality and the field theory q scaling dimensions of operators O (q) (QΛ) q d = 2 + ɛ: q = q(q 1)ɛ + O(ɛ 4 ) Wegner 8 Infinitely many operators with negative scaling dimensions 1 = Q = Λ naive order parameter uncritical Transition described by an order parameter function F (Q) Zirnbauer 86, Efetov 87 distribution of local Green functions and wave function amplitudes ADM, Fyodorov 91

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

13 Power-law random banded matrix model (PRBM) Anderson transition: dimensionality dependence: d = 2 + ɛ: weak disorder/coupling d 1: strong disorder/coupling Evolution from weak to strong coupling? PRBM ADM, Fyodorov, Dittes, Quezada, Seligman 96 N N random matrix H = H H ij 2 = i j 2 /b 2 1D model with 1/r long range hopping < b < parameter Critical for any b family of critical theories! b 1 analogous to d = 2 + ɛ b 1 analogous to d 1 (?) Analytics: b 1: σ-model RG b 1: real space RG Numerics: efficient in a broad range of b Evers, ADM 1

14 Power-law random banded matrix model evolution of P(ln P 2 ) with N for b = Distribution (ln P 2 ) 1 1 scale invariance fractal dimension D 2.75 D ln P 2 fractal dimension D 2 (b) b 1.5 f(α).5 Multifractality spectrum f(α) for b = 4., 1.,.25 and.1.5 b=4. b=1 b=.25 b=.1 parabola b=4. parabola b=1. f (α*2b)/2b Lines: analytics for b 1 and b 1 Symbols: numerics α Evers, ADM 1

15 Non-linear σ-model Relations between multifractal exponents distribution of local Green function G R (r, r)/π ρ = u i ρ P (u, ρ) = 1 2π ρ P ( (u 2 + ρ 2 + 1)/2 ρ ) symmetry of LDOS distribution: P ρ ( ρ) = ρ 3 P ρ ( ρ 1 ) Recently: ADM, Fyodorov 94 (β = 2) more complete derivation (β = 1, 2, 4) via relation to a scattering problem: system with a channel attached at a point r S-matrix S = (1 ik)/(1 + ik) K = 1 2 V G R (r, r)v = u i ρ distribution P (S) invariant with respect to the phase θ of S = re iθ. Fyodorov, Savin, Sommers 4-5

16 Relations between multifractal exponents (cont d) LDOS distribution in σ-model + universality ADM, Fyodorov, Mildenberger, Evers, cond-mat 6 exact symmetry of the multifractal spectrum: q = 1 q f(2d α) = f(α) + d α Consequence: support of the singularity spectrum f(α) is bounded by the interval [, 2d] q, 1 q 1 2 b=4 b=1 b=.3 b=.1 f(α) b=.3 b=.1 b=1 b=4 f(α) q α α

17 Relations between multifractal exponents (cont d): Anderson transition in symplectic class in 2D Mildenberger, Evers, cond-mat/6856 Symmetry of the spectrum: q = 1 q Non-parabolicity of the spectrum δ(q) q q(1 q) const Conformal invariance 2D quasi-1d strip Λ c = 1/πδ δ.172, Λ c πδ Λ c =.999 ±.3 Related work: Obuse, Subramaniam, Furusaki, Gruzberg, Ludwig, in preparation

18 Relations between multifractal exponents (cont d) Wigner delay time: t W = θ(e)/ E Relation between wave function (P y ) and delay time (P W ) distribution in the σ-model: P W ( t W ) = t 3 W P y( t 1 W ) t W = t W /2π Ossipov, Fyodorov 5 + universality exact relation between multifractal exponents for closed (wave functions, τ q ) and open (delay times, γ q ) systems γ q = τ 1+q ADM, Fyodorov, Mildenberger, Evers, cond-mat 6

19 Magnetotransport: Resistivity tensor: Integer Quantum Hall Transition ρ xx = E x /j x ρ yx = E y /j x Hall resistivity B E x E y j x Quantum Hall plateau transition: localized σ xy = n e2 h critical point localized σ xy = (n + 1) e2 h Delocalization in 2D?! Topological term in σ-model (Pruisken) Closely related to existence of edge states

20 Multifractal wave functions at the Quantum Hall transition

21 Multifractality at the Quantum Hall critical point QHE: anomalous dimensions f(α).5. f(α) L=16 L=128 L=124 q /q(1 q) q α spectrum is parabolic with a very high (1%) accuracy: f(α) = 2 (α α ) 2 4(α 2), q = (α 2)q(1 q) with α 2 =.262±.3 Evers, Mildenberger, ADM 1 important for identification of the CFT of the Quantum Hall critical point

22 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 = ( t t ) ( h h T )

23 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 effective magnetic field) SQH plateau transition: spin Hall conductivity quantized Kagalovsky, Horovitz, Avishai, Chalker 99; Senthil, Marston, Fisher 99 Similar to IQH transition but: DoS critical mapping to percolation: ρ(e) E 1/7 Gruzberg, Ludwig, Read 99; Beamond, Cardy, Chalker 2

24 Multifractality at the spin quantum Hall transition Evers, Mildenberger, ADM 3 mapping to percolation: analytical evaluation of lowest multifractal exp s: 2 = 1/4, 3 = 3/4 numerics: q, f(α) not parabolic.14 ρ(e) L 1/ ρ(e) L 1/ E/δ E/δ q / q(1 q) f(α) q α

25 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 Multifractality spectrum: freezing? f(α = ) = τ q 1 = Critical state has a localized core with multifractal tails disorder metal σ xy = σ xy =1 Ising.5 1 p f(α) α Mildenberger, Evers, Chalker, ADM, in preparation/progress

26 Surface multifractality Subramaniam, Gruzberg, Ludwig, Evers, Mildenberger, ADM, cond-mat/51241 r near the boundary L d 1 ψ(r) 2q L τ s q τ s q = d(q 1) + qµ s q τ s q, s q surface multifractal exponents SQHE analytical evaluation for q = 1, 2, 3 via mapping to percolation µ = 1/12 s 2 = 1/3 (bulk: -1/4) s 3 = 1 (bulk: -3/4) <ψ 2 >L 2+µ P q L s q q /q(1 q) µ=1/12 q=2: 2 = 1/3 q=3: 3 = 1 1 L 1 bulk surface analytical: 1/6 analytical: 1/ q f s (α), f b (α) 2 1 bulk surface α

27 Surface multifractality in 2D or (2 + ɛ)d bulk : τ b q = 2(q 1) + γq(1 q) γ = (βπg) 1 1 surface : τ s q = 2(q 1) γq(1 q) τ q 2(q 1) 2 2 q q + bulk surface q 1 2 f(α) α + s α + b surface bulk α b α s α 2.5 f b (α) = 2 (α 2 γ) 2 /4γ f s (α) = 1 (α 2 2γ) 2 /8γ

28 Summary and Outlook Anderson localization transition wave functions multifractal at criticality dimensionality dependence: from weak (2 + ɛ) to strong (d = 3, 4,...) MF PRBM: 1d model with 1/r random hopping: family of critical theories from weak (b 1) to strong (b 1) multifractality 2d + magnetic field QH transitions: IQHE, SQHE, TQHE exact relations for MF spectra: symmetry q 1 q; closed open surface multifractality Open problems / Current activity: Multifractality at the Anderson transition in large d: qualitatively similar to PRBM with b 1? Anderson and quantum Hall transitions in novel universality classes Criticality in 2D: strong coupling fixed points. CFT? Effect of interaction on multifractality Experimental observation of wave function multifractality?