Multifractality of wave functions: Interplay with interaction, - PowerPoint PPT Presentation

Multifractality of wave functions: Interplay with interaction, classification, and symmetries Alexander D. Mirlin Karlsruhe Institute of Technology & PNPI St. Petersburg I. Burmistrov , Landau Institute I. Gornyi, S. Bera, F. Evers , Karlsruhe I. Gruzberg , Chicago A. Ludwig , Santa Barbara M. Zirnbauer , K¨ oln

Plan • Introduction: Anderson localization and multifractality • Multifractality and interaction • Dephasing and temperature scaling at localization transitions Burmistrov, Bera, Evers, Gornyi, ADM, Annals Phys. 326, 1457 (2011) • Enhancement of superconductivity by Anderson localization Burmistrov, Gornyi, ADM, PRL 108, 017002 (2012) • Classification of composite operators and symmetry properties of scaling dimensions Gruzberg, Ludwig, ADM, Zirnbauer, PRL 107, 086403 (2011); Gruzberg, ADM, Zirnbauer, to be published

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

Anderson transition 2 /h) g=G/(e 1 d=3 β = dln(g) / dln(L) Scaling theory of localization: 0 Abrahams, Anderson, Licciardello, d=2 Ramakrishnan ’79 −1 d=1 Modern approach: RG for field theory ( σ -model) 0 ln(g) quasi-1D, 2D: metallic → localized crossover with increasing L d > 2: metal-insulator transition delocalized localized ��� ��� ��� ��� ��� ��� disorder ��� ��� critical ��� ��� point review: Evers, ADM, Rev. Mod. Phys. 80, 1355 (2008)

Field theory: non-linear σ -model S [ Q ] = πν � d d r Tr [ − D ( ∇ Q ) 2 − 2 iω Λ Q ] , Q 2 (r) = 1 4 Wegner ’79 σ -model manifold: symmetric space e.g. for broken time-reversal invariance: U(2 n ) / U( n ) × U( n ) , n → 0 with Coulomb interaction: Finkelstein’83 supersymmetry (non-interacting systems): Efetov’82

Anderson localization & topology: Integer Quantum Hall Effect IQHE flow diagram von Klitzing ’80 ; Nobel Prize ’85 Khmelnitskii’ 83, Pruisken’ 84 localized localized Field theory (Pruisken): ��� ��� ��� ��� ��� ��� ��� ��� σ -model with topological term critical point − σ xx 8 Tr( ∂ µ Q ) 2 + σ xy � � � d 2 r S = 8 Tr ǫ µν Q∂ µ Q∂ ν Q → n = . . . , − 2 , − 1 , 0 , 1 , 2 , . . . protected edge states QH insulators − → Z topological insulator −

Multifractality at the Anderson transition d d r | ψ (r) | 2q � P q = inverse participation ratio  L 0 insulator   L − τ q � P q � ∼ critical L − d ( q − 1) metal   τ q = d ( q − 1) + ∆ q ≡ D q ( q − 1) multifractality normal anomalous metallic critical d τ q − → Legendre transformation f( α ) − → singularity spectrum f ( α ) α + α − 0 | ψ | 2 large 2 small | ψ | wave function statistics: P (ln | ψ 2 | ) ∼ L − d + f (ln | ψ 2 | / ln L ) α d α 0 L f ( α ) – measure of the set of points where | ψ | 2 ∼ L − α

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 | / ln L } • field theory language: ∆ q – scaling dimensions of operators O ( q ) ∼ ( Q Λ) 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

Dimensionality dependence of multifractality 4 Analytics (2 + ǫ , one-loop) and numerics 3 D q 2 ~ τ q = ( q − 1) d − q ( q − 1) ǫ + O ( ǫ 4 ) 1 f ( α ) = d − ( d + ǫ − α ) 2 / 4 ǫ + O ( ǫ 4 ) 0 0 1 2 3 d = 4 (full) q d = 3 (dashed) 4 d = 2 + ǫ, ǫ = 0 . 2 (dotted) d = 2 + ǫ, ǫ = 0 . 01 (dot-dashed) 3 3 f( α ) 2 2 Inset: d = 3 (dashed) f( α ) ~ 1 ~ vs. d = 2 + ǫ , ǫ = 1 (full) 1 0 0 −1 0 1 2 3 4 5 α Mildenberger, Evers, ADM ’02 −1 0 1 2 3 4 5 6 7 α

Multifractality at the Quantum Hall transition 2.0 Evers, Mildenberger, ADM ’01 1.5 1.0 2.0 ( α ) 0.5 1.5 f L=16 f( α ) L=128 1.0 0.0 L=1024 0.5 −0.5 0.0 0.8 1.2 1.6 2.0 2.4 −1.0 0.5 1.0 1.5 2.0 2.5 α

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

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

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

Dephasing at metal-insulator and quantum Hall transitions Burmistrov, Bera, Evers, Gornyi, ADM, Annals Phys. 326, 1457 (2011) e-e interaction − → dephasing at finite T − → smearing of the transition local. length ξ ∝ | n − n c | − ν , dephasing length L φ ∝ T − 1 /z T δn ∝ T κ , − → transition width κ = 1 /νz T We focus on short-range e-e interaction: • long-range Coulomb interaction negligible because of large dielectric constant • 2D: screening by metallic gate • interacting neutral particles (e.g. cold atoms) Earlier works: Lee, Wang, PRL’96 ; Wang, Fisher, Girvin, Chalker, PRB ’00

Temperature scaling of quantum Hall transition Transition width exponent κ = 1 /νz T = 0 . 42 ± 0 . 01 Wei, Tsui, Paalanen, Pruisken, PRL’88 ; Li et al., PRL’05, PRL’09

Interaction scaling at criticality K 1 = ∆ 2 �� � 2 δ ( E + ω − ǫ α ) δ ( E − ǫ β ) � � � � B αβ ( r 1 , r 2 ) 2 αβ B αβ ( r 1 , r 2 ) = φ α ( r 1 ) φ β ( r 2 ) − φ α ( r 2 ) φ β ( r 1 ) α β α β � µ 2 � | r 1 − r 2 | K 1 ( r 1 , r 2 , E, ω ) = L − 2 d , | r 1 − r 2 | ≪ L ω L ω L ω = L (∆ / | ω | ) 1 /d length scale set by frequency ω

Interaction scaling at quantum Hall critical point Hartree, Fock Hartree – Fock enhanced by multifractality suppressed by multifractality exponent ∆ 2 ≃ − 0 . 52 < 0 exponent µ 2 ≃ 0 . 62 > 0

Interaction-induced dephasing β r r γ 2 3 α γ β r r 3 2 α α δ α r r r r δ 4 4 1 1   4 ImΣ R (0 , 0) ∼ − 1 � � �  U ( r 1 − r 2 ) U ( r 3 − r 4 ) dr j d Ω Ω  2∆ 3 j =1 coth Ω 2 T − tanh Ω � � K 2 ( { r j } , 0 , 0 , ε ′ ∼ T, Ω) × 2 T K 2 ( { r j } , E, ε, ε ′ , Ω) = ∆ 4 � � B ∗ αβ ( r 1 , r 2 ) B δγ ( r 1 , r 2 ) B ∗ γδ ( r 3 , r 4 ) B βα ( r 3 , r 4 ) 8 αβγδ × δ ( E − ǫ α ) δ ( ε ′ + Ω − ǫ β ) δ ( ε ′ − ǫ γ ) δ ( ε + Ω − ǫ δ ) � . � µ 2 � R � α � | r 1 − r 2 | | r 3 − r 4 | K 2 ( { r j } , 0 , 0 , ε ′ ∼ Ω , Ω) = L − 4 d R R L Ω R = ( r 1 + r 2 − r 3 − r 4 ) / 2

Interaction scaling at quantum Hall critical point: Second order black: N=512, red: 768, ρ /R = 1/2 blue: 1024 1/4 1/8 1.0 1.0 1/16 1/32 -2 µ 2 Κ 2 R/N 0.0 0.1 ( ρ /R) 0.5 0.5 1.0 -2 µ 2 Κ 2 ( ρ /R) α =0 α =-0.1 0.2 0.0 0.0 0.0 0.2 0.4 -2 -1 10 10 R/N R/N µ 2 = 0 . 62 ± 0 . 05 in agreement with scaling of first order α = − 0 . 05 ± 0 . 1 (in fact, exactly zero for unintary class; see below) Exponent α drops out of the expression for τ − 1 φ if α > 2 µ 2 − d — fulfilled for QH transition

Scaling at QH transition: Theory and experiment • Theory (short-range interaction): ∝ T p with p = 1 + 2 µ 2 /d τ − 1 − → dephasing rate φ L φ ∝ T − 1 /z T dephasing length z T = d/p z T ν = 1 + 2 µ 2 /d 1 Transition width exponent κ = νd µ 2 ≃ 0 . 62 − → p ≃ 1 . 62 − → z T ≃ 1 . 23 ν ≃ 2 . 35 (Huckestein et al ’92, . . . ) − → κ ≃ 0 . 346 ν ≃ 2 . 59 − → κ ≃ 0 . 314 (Ohtsuki, Slevin ’09) • Experiment (long-range 1 /r Coulomb interaction): κ = 0 . 42 ± 0 . 01 Difference in κ fully consistent with short-range and Coulomb (1 /r ) problems being in different universality classes

Anderson transition: 2 + ǫ dimensions, short-range interaction − dt/d ln L ≡ β ( t ) = ǫt − 2 t 3 − 6 t 5 + O ( t 7 ) (Wegner ’89) t = 1 / 2 πg g – dimensionless conductance � 1 / 2 � 3 / 2 � ǫ − 3 � ǫ + O ( ǫ 5 / 2 ) Metal-insulator transition at t ∗ = 2 2 2 ν = − 1 /β ′ ( t ∗ ) = 1 2 ǫ − 3 Localization length index 4 + O ( ǫ ) Exponents controlling scaling of interaction: √ 2 ǫ − 3 2 ζ (3) ǫ 2 + O ( ǫ 5 / 2 ) α = O ( ǫ 5 / 2 ) µ 2 = Temperature scaling of transition: √ √ 2 ǫ 1 / 2 + 5 ǫ − 4 2 ǫ 3 / 2 + O ( ǫ 2 ) z T = 2 − 2 √ √ 2 ǫ 3 / 2 + ǫ 2 + ǫ 5 / 2 / 2 + O ( ǫ 3 ) κ = ǫ +