Mesoscopic Physics. Smaller is different

Transcription

1 Mesoscopic Physics Smaller is different 1. Theories of Anderson localization 2. Weak localization: theory and experiment 3. Universality and Random Matrix Theory 4. Metal-Insulator Transitions 5. Mesoscopic physics beyond condensed matter References: 1. Thouless, Phys. Rep. 13, 93 (1974). 2. Lee, Ramkrishnan, Rev. Mod. Phys. 57,387 (1985). 3. Kramer, MacKinnon, Rep. Prog. Phys. 56, 1469 (1993). 4. Boris Altshuler, Boulder Colorado Lectures

2 What is mesoscopic physics? 1. Interference\tunneling effects in a solid. 2. These effects usually occur at intermediate scales and at relatively low temperatures. 3. Disorder plays a role in most materials. Why is mesoscopic physics interesting? 1. Reveals universal features of quantum physics. 2. Continuation of quantum mechanics. 3. Technological applications. Are 4+1 lectures enough? Problems are easy to understand but difficult to solve rigorously.

3 Lecture I: From Anderson to Anderson: perturbative formalism and scaling theory of localization 1. Intuition about quantum dynamics in a disordered potential. Anderson localization 2. Theories of localization: Locator expansions a. Anderson 1957: Absence of diffusion in certain random lattices b. Anderson, Abou-Chacra, Thouless, 1973: A selfconsistent theory of localization 3. Abrahms, Anderson, et al., 1979: Scaling theory of localization

4 Your intuition about localization E a Random V(x) E b 0 E c X Will the classical motion be strongly affected by quantum effects? P. W. Anderson

5 H ˆΨ = α E α Metal ψ α Hˆ = 2 2 m r r V( ) V( ') + V r ( ) 2 r r = V δ( ') 0 r Metal <r 2 > Insulator r 2 ( t) t Insulator r 2 ( t) const t t Abs. Continuous t Pure point spectrum Ψ α ( r ) 1/ V Ψ α ( r) e r / ξ loc k F l >>1 P( t) 0 t P( t) k F l>1 cons t

6 Theories of localization Locator expansions One parameter scaling theory

7 6203 citations! What if I place a particle in a random potential and wait? Tight binding model 1. (Locator) expansion around V=0 2. Probability distribution needed 3. At V=V c perturbation breaks down Metal

8 =0 Metal Increase V until perturbation theory breaks down = Metal

9 However: 1. Problem with small denominators 2. Uncorrelated paths? V > V c V < V c Correctly predicts a metal-insulator transition in 3d and localization in 1d Interactions? Disbelief?, against band theory But my recollection is that, on the whole, the attitude was one of humoring me.

10 250 citations Perturbation theory around the insulator limit (locator expansion). No control on the approximation. It should be a good approx for d>>2. It predicts correctly localization in 1d and a transition in 3d The distribution of the self energy S i (E) is sensitive to localization. metal Γ = ImS i ( E+ iη) > 0 = 0 metal insulator insulator η

11 Solution only provided that the homogenous equation has solutions with λ 1 We need probabilities distributions!! k 1 k 2 Fourier transform of E i, i Neglect E i, s k 1 Guess

12 Accurate for d >> 1

13 Localization in mathematics literature Rigorous proof of localization for strong disorder 1. Absence of diffusion in the Anderson tight binding model for large disorder or low energy Jürg Fröhlich and Thomas Spencer, Comm. Math. Phys. 88, 151 (1983). 2. Localization at Large Disorder and at Extreme Energies: an Elementary Derivation. M. Aizenman, S. Molchanov, Comm. Math. Phys. 157, 245 (1993).

14 Experiments

15 State of the art: d = 1 d =2 d > 2 An insulator for any disorder An insulator for any disorder Localization only for disorder strong enough ρ Still not well understood E Metal Insulator Transition Why?

16 Anderson localization 50 Perturbative locator expansion Anderson 60 Self consistent condition Abou Chakra, Anderson, Thouless d Weak localization Scaling theory Field theory, RMT Metal-Insulator Computers! Experiments! Kotani, Pastur, Molchanov, Sinai, Jitomirskaya Larkin, Khmelnitskii, Altshuler, Lee Thouless, Wegner, Gang of four, Frolich, Spencer, Molchanov, Aizenman Efetov, Wegner Efetov, Fyodorov, Mirlin, Vollhardt Aoki, Schreiber, Kramer Aspect, Fallani, Segev

17 Scaling ideas (Thouless, 1972) 1. Mean level spacing δ L -d L 2. Thouless energy E T = hd/l 2 δ L = system size; D = diffusion constant δ = mean- level spacing E T = inverse diffusion time ~ sensitivity of the spectrum to change of boundary conditions E E T T >> δ << δ g g >> 1 << 1. Metal Insulator

18 Electric conductivity Ohm s Law for a cubic sample of the size L Conductance Dimensionless conductance Experiments OK

19 L = 2L = 4L = 8L citations E T >> δ g >> 1 Metal E T << δ g <<11 Insulator E T E T E T E T δ δ δ δ Figure from Boris Altshuler Boulder Lectures g g g g d d ( logg) ( logl) =β ( g)

20 Scaling theory of localization β(g) The change in the conductance with the system size only depends on the conductance itself g Figure from Boris Altshuler Boulder Lectures d d log log g L = β ( g) g >> 1 g << 1 g L g e d 2 L/ ζ c > 0? β( g) = ( d 2) c/ g β( g) logg < 0

21 Lecture II: Weak localization: the Russians, the cold war and experiments 1-1 β(g) unstable fixed point g c 1 3D 2D 1D d d log log g L = β g ( g) Is this true? Metal insulator transition in 3D All states are localized for d=1,2

22 NO, Patrick Lee, PRL 42,1492 (1979) Scaling theory is wrong Metal-Insulator transition in d=2! No metallic states in 2d. Scaling theory of localization is right!

23 Interference and weak localization S 2 1 D W 1, W 2 probabilities W 1,2 = A 1,2 2 A = A e ϕ 1,2 1,2 A 1, A 2 probability amplitudes i 1,2 Total probability W 2 ( ) = A + A = W + W + 2Re A A Interference ( ) A A = 2 WW ( ϕ ) 2 Re ϕ cos 1 2 Usually negligible but there are exceptions...

24 Weak localization? A B C Constructive interference between clockwise and counter clockwise enhances return probability to B but suppresss the AC probability. Langer and Neel PRL 16, 984 (1966).? Negative correction to G of a metal at low T

25 O What is the probability P(t) that such a loop is formed at time t? Probability to return to dv around O ( ( ) ) P r t = = dv 0 dv d ( Dt) 2 Figure from B. Altshuler Boulder Lectures dv =D d 1 v dt F ( ) P t = t v dt D d 1 F P ( t ) ( ) d 2 max Dt g τ δ g

26 Decoherence/ dephasing time Thouless time

27 ( ) P t = t v dt d 1 F ( ) d Dt τ D P( t ) 2 δ g g max δ 2 g Dv L 2Dv L F log = F log g D D τ D l d = 2 δ g = 2 log L π 2 β ( g) = π g l Universal

28 Conductivity: The rigorous way Expansion parameter U = Irreducible vertex

29 Conductivity: The rigorous way We must sum geometrical series of maximally crossed diagrams

30 Dolan, Osherhoff, PRL 43, 721 (1979) Resistance of thin metallic stripes increases as T decreases Experimental results also support scaling theory of localization

31 Magnetic field/flux Φ O Figures from B. Altshuler Boulder Lectures O No magnetic field ϕ 1 = ϕ 2 Magnetic field H ϕ 1 ϕ 2 = 2 2π 2π Φ/Φ 0 I = ( ) A A = 2 WW ( ϕ ) 2Re ϕ cos H suppresses weak localization 2. Oscillations in the conductance

32 Negative Magnetoresistance Chentsov (1949) Ref. 3 PRB 21, 5142 (1980)

33 Bohm-Aharonov- effect Theory Altshuler, Aronov, Spivak (1981) Experiment Sharvin & Sharvin (1981) Select exp only those paths whose associated flux is the same Thin metallic cylinders Ref. 3

34 Figure from B. Alshuler Boulder lectures S 2 1 D W 1, W 2 probabilities W 1,2 = A 1,2 2 A = A e ϕ 1,2 1,2 A 1, A 2 probability amplitudes i 1,2 Total probability W 2 ( ) = A + A = W + W + 2Re A A Negative? ( ) A A = 2 W W ( ϕ ) 2 Re ϕ cos 1 2 Weak anti localization?

35 Spin-Orbit interactions and anti-localization correction Larkin Hikami Ref. 3 α = 0,1,-1

36 Disorder + Interactions Altshuler Aronov Lee

37 Summary: Weakly localization Boltzmann Picture Both n and A can have any sign σ(h) can be oscillatory n integer > 2 (=2 for e-e collisions) A>0 σ(h) monotonous

38 Ref. 3 Ref. 2

39 g >> 1... Universal properties? Wigner-Dyson statistics??

40 Disordered Systems g >> 1 g = g c Lecture III Quantum Chaos Classically chaotic motion Universality Random Matrix theory Nuclear Physics High energy excitations QCD Infrared spectrum Dirac operator

41 Universality I Field theory approach to disorder systems Denominator problem : Grassmannian variables can help Disorder Is integrated!!

42 Disordered system Effective Field theory Efetov Larkin Wegner Khmelnitskii Density of probability that 2 eigenvalues are separated by ω Q ~ ΦΦ t Q 0 Universal regime s = ω/ Efetov:Supersymmetry in disorder and chaos RANDOM MATRiX THEORY

43 Random Matrix a ij random entries Spectral Properties Level Repulsion P 2 ( ) β As E + E s ~ s e s = 1 δ E i E i Spectral Rigidity Σ 2 ( N ) = n( N) 2 N = n( N) E f E i δ >> 1 2 ~ logn UNIVERSAL spectral correlations

44 Uncorrelated spectrum (Poisson) Σ ( N) = N P( s) = exp( s) 2

45 Não é possível apresentar a imagem. O computador pode não ter memória suficiente para abrir a imagem ou a imagem pode ter sido danificada. Reinicie o computador e, em seguida, abra o ficheiro novamente. Se o x vermelho continuar a aparecer, poderá ter de eliminar a imagem e inseri-la novamente. Universality II Nuclear excitations Bohr 1936 P(s) A statistical approach? Maybe Bohr was right Shell model not feasible P s ( s) = + 1 i ( s [ ] δ ) δ E i E i / ( ) P s ~ s β e As 2

46 Universality III Quantum chaos Bohigas-Giannoni Giannoni-Schmit conjecture Phys. Rev. Lett. 52, 1 (1984) Classical chaos Wigner-Dyson Energy is the only integral of motion Momentum is not a good quantum number Delocalization

47 Gutzwiller-Berry Berry-Tabor conjecture Integrable classical motion Integrability Poisson statistics (Insulator) Canonical momenta are conserved P(s) s System is localized in momentum space

48 Lecture IV Mesoscopic Physics beyond condensed matter QCD vacuum as a disordered medium

49 Inside the Nucleus: What holds the matter together? Quarks Color charge RED, BLUE, GREEN Stable matter: u,d, 3-10MeV Electric Charge 2/3,1/3 Interact by exchanging gluons Hadrons are colorless Strong color forces govern the interaction among quarks. The relativistic quantum field theory to describe quark interactions is quantum chromodynamics (QCD).

50 A two minute course on non perturbative QCD State of the art T = 0 low energy What? Chiral Symmetry breaking and Confinement How? 1. Lattice QCD 2. Instantons. T = T c T > T c Chiral and deconfinement transition Quark- gluon plasma QCD non perturbative! Universality (Wilczek and Pisarski) AdS-CFT N =4 Yang Mills

51 QCD at T=0, instantons and chiral symmetry breaking thooft, Polyakov, Callan, Gross, Shuryak, Diakonov, Petrov,VanBaal Instantons: Non perturbative solutions of the Yang Mills equations D= ( ) ( ) 3 r = 0 ψ r 1 r γ µ ins + ga γ µ µ µ Dψ0 0 / 1. Dirac operator has a zero mode 2. The smallest eigenvalues of the Dirac operator are controlled by instantons ψψ ψψ 1 ρ( λ) = Tr( D+ m) = dλ = V m+ iλ limπ 1 ρ m m Order parameter symmetry breaking 0 ( m) V

52 QCD vacuum as a conductor (T =0) Metal: An electron initially bounded to a single atom gets delocalized due to the overlapping with nearest neighbors QCD Vacuum: Zero modes get delocalized due to the overlapping with the rest of zero modes. (Diakonov and and Petrov) Differences Disorder: Exponential decay QCD vacuum: Power law decay

53 QCD vacuum as a disordered conductor Diakonov, Petrov, Verbaarschot, Osborn, Shuryak, Zahed,Janik Instanton positions and color orientations vary Ion Instantons T = 0 T IA ~ 1/R α, α = 3<4 Conductor. RMT applies? Electron Quarks T>0 T IA ~ e -R/l(T) A transition is possible Shuryak,Verbaarschot

54 Universality E < E c (L) QCD Dirac operator Chiral random matrix model Shuryak, Verbaarshot 1993

55 Deconfinement and chiral restoration Deconfinement: Confining potential vanishes: Chiral Restoration: Matter becomes light: How to explain these transitions? 1. Effective, simple, model of QCD close to the phase transition (Wilczek,Pisarski,Yaffe): Universality. 2. Classical QCD solutions (t'hooft): Instantons (chiral), Monopoles and vortices (confinement).

56 QCD Dirac D QCD = + ga µ µ µ operator Phys.Rev. D75 (2007) Nucl.Phys. A770 (2006) 141 with J. Osborn γ µ D QCD µ ψ n = i λ n ψ n λ n ψ n At the same T c that the Chiral Phase transition ψψ 0 undergo a metal - insulator transition A metal-insulator transition in the Dirac operator induces the QCD chiral phase transition

57 (L,W) Characterization of a metal/insulator Hψ = n E n ψ n 1. Eigenvector statistics: 2. Eigenvalue statistics: Random Matrix P D ( s) 2 ~ IPR d 4 d d D 2 = L ψ n ( r ) d r ~ L ( s [ λ ] ) P ( s ) 1 λ / d ~ se As 2 Wigner Dyson statistics 0.6 No correlations D2 ~ 0 P( s) = e s = δ i + i Poisson statistics var W = i L=10 (square) 12 (star) 16 (circle) 20 (diamond) 2 n n var = s s s s P( s) ds Insulator Poisson statistics Disordered metal WD statistics

58 Metal insulator transition ILM, close to the origin, 2+1 flavors, N = 200

59 Spectrum is scale invariant ILM with 2+1 massless flavors, We have observed a metal-insulator transition at T ~ 125 Mev

60 Localization versus chiral transition Instanton liquid model Nf=2, masless Chiral and localizzation transition occurs at the same temperature

61 g = g c Lecture V Metal Insulator transitions New Window of universality

62 Insulator Critical Metal Typical Multifractal eigenstate Kramer et al. 1999

63 (L,W) Signatures of a metal-insulator transition 1. Scale invariance of the spectral correlations. Skolovski, Shapiro, Altshuler,90 s P( s) P( s) ~ ~ s e β As s << 1 s >> 1 var L=10 (square) 12 (star) 16 (circle) 20 (diamond) W 3. Eigenstates are multifractals ψ ( r) n 2q d d r~ L D ( q 1 ) q 2 = 2 n n var = s s s s P( s) ds 4. Difussion is anomalous

64 Self consistent approach to the transition (Wolfle-Volhardt, Imry, Shapiro) 1.Cooperons (Langer-Neal, maximally crossed, responsible for weak localization) and Diffusons (no localization, semiclassical) can be combined. 3. Accurate in d ~2. No control on the approximation!

65 Predictions of the self consistent theory at the transition 1. Critical exponents: Vollhardt, Wolfle,1982 r/ ξ ψ ( r ) pe ξ E E c ν ν ν 1 = d 2 d < = 1 / 2 d > Transition for d>2 Disagreement with numerical simulations!! 3. Correct for d ~ 2 Why?

66 Why do self consistent methods fail for d = 3? 1. Always perturbative around the metallic (Vollhardt & Wolfle) or the insulator state (Anderson, Abou Chacra, Thouless). A new basis for localization is needed 2. Anomalous diffusion at the transition (predicted by the scaling theory) is not taken into account. D D ( ( L ) q ) L q 2 d d 2

67 Analytical results combining the scaling theory and the self consistent condition. Critical exponents, critical disorder, level statistics.

68 Technical details: Critical exponents 2 The critical exponent ν, can be obtained by solving the ν above equation for ξ E E with D (ω) = 0. c ν = d 1 2

69 Comparison with numerical results = d ν ν ξ ξ ψ ) ( / c r E E r pe ± = = ± = = ± = = + = = N T N T N T N T ν ν ν ν ν ν ν ν