Numerical Analysis of the Anderson Localization

Transcription

1 Numerical Analysis of the Anderson Localization Peter Marko² FEI STU Bratislava FzU Praha, November 3. 23

2 . introduction: localized states in quantum mechanics 2. statistics and uctuations 3. metal - insulator transition: nite size scaling method 4. analytical theory vs numerical data

3 Quantum Mechanics Textbook QM: there are two possible quantum states: Bond: Extended: Φ (x) E = E 3 E 2 V(x) E P. W. Anderson (958): The 3rd possible quantum state: localized

4 Example I Disordered sample (24 24 sites). Time evolution of the quantum state Ψ( r, t = ) = δ( r center) Time evolution of four dierent localized states: Localization center is not associated with any potential minima.

5 Example II: D disordered chain.5 W=.5 weak disorder W= W= W= W= N strong disorder Eigenstates of electron clossest to E =.5 for various disorder strength W

6 Example III: Transmission through random chain E =.448 T =. E =.486 T =.84 input T = t 2 E =.5 T =.67 E =.543 T =. E =.587 T = N Transmission R = r Fermi energy Five eigenstates and corresponding transmission T (W = ).

7 Physical model of localization 2 T T 2 2 T R 2 R T 2 2 T R 2 R R 2 R T 2 Classical particle: T2 K = T T 2 + T R 2 R T 2 + T R 2 R R 2 R T Quantum particle: t 2 = t t 2 + t r 2 r t 2 + t r 2 r r 2 r t t = T e iφ t r = Re iφ r Amplitudes t and r are complex - we have more parameters

8 Physical model of localization T K 2 = T T 2 R R 2 T Q 2 = t 2 2 = T T 2 r r 2 2 = T T 2 + R R 2 r r 2 r r 2 = T T 2 + R R 2 2 R R 2 cos φ φ = φ r + φ r 2 + φ L Transmission is determined by phases of reection amplitudes φ r, φ r2 and phase diference φ L given by distance between barriers.

9 Transmission through N barriers Classical particle: Ohm's law Quantum aprticle: Transmission Exponencial localization: lnt α N 2L λ Absence of self-avergaing: var ln T 2 ln T realization of random energies Transmission Transmission p( ln T) W = W = Length of the system N ln T

10 Physical origin of localization wave character of electron Observation of localization of classical wavers: Electromagnetic waves in disordered dielectircs [Gennack] Ultrasund in system of metallic spheres [van Tiggelen] Acoustic sound [van Tiggelen]

11 Anderson model.5 H = W ε n c c n n + t c c. n n n [nn ] P(ε) Gauss Box t =, size L d d dimension W - disorder.5 Symmetry - orthogonal spinless, time invariance - unitary with magnetic eld (Quantum Hall eect) - ε Peierls factor: t x = exp iα, t y = - symplectic spin dependent hopping: t is a matrix chiral, chiral unitary, chiral symplectic...

12 3D model: Phase diagram Disorder W/ W c (E =, W c ) metal insulator E c density of states ρ(e) delocalized states localized states Energy E c Energy E Andersonov prechod: ciritcal disorder W c mobility edge E c = E c (W ). multifractal distribution of electron at the critical point universality along the critical line

13 Critical disorder. For xed Fermi energy: disorder induced metal-insulator transition (due to the increase of the disorder W ) σ (W c -W) s λ (W-W c ) ν critical disorder W c = W c (E F ). - Critical exponent: s = (d 2)ν [Wegner]. Metal W c Insulator disorder W - σ: conductance in the metallic phase (W < W c ). - λ: localization length for insulator (W > W c ).

14 Transmission T vs conductance g A B C D ( ) ( ) C A = S B D ( ) ( ) C A = T D B ( ) t r S = r t ( t r ) t T = rt t r t r Economou - Soukoulis formula (called Landauer formula) g = T = Tr t t (e 2 /h = )

15 Fluctuations of random potential - nonlocal character of electron propagation Disordered lattice contains N = 4 random energies. Changes of sign of a single energy causes changes of the transmission in % 2D samples, disorder W = 2, 4 and 6. Corresponding transmission coecient decreases from T = 5.,.5,.8.

16 Fluctuations of random potential Very strong disorder - change of the ln g. No there is no walley in the random potential.

17 Statistics (g) is not self-averaging quantity Metal: Gaussian p(g) and insulator: almost Gaussian P(ln g),7 p(g),6,5,4,3,2, p(ln g) g ln g Critical point W = W c : size - independent distribution - p c (g) -2-3 L= <g>=.2834 L=2 <g>=.2825 L=8 <g>=.289 L=22 <g>= p c (ln g) -4 L= <ln g>=-.929 L=2 <ln g>=-.935 L=8 <ln g>=-.94 L=22 <ln g>= ,5,5 2 2,5 3 g ln g

18 Conductance g = g(l) and wavefunction Ψ( r) (L ) Metal electron is everywhere g = σl d 2 Insulator electron is localized g exp 2L/λ W = W c Ψ is multifractal g = g c = const. Distribution of electron in the metallic, localized and critical regime

19 Scaling theory of localization ln g = β(ln g). ln L Univerzal functon β(ln g): - continuous - monotonic - unknown. Conductance g is an order parameter ln g c β(ln g) - d=3 d=2 d= ln g g g σl d 2 β(ln g) = d 2 g g exp 2L/λ β(ln g) = ln g g = g c β(g c ), s = g c β (g c. g, ln g or percentiles of the distribution p(ln g) were used for the scaling analysis.

20 Finite - size scaling Various length scales: - mean free path l pre pruºný rozptyl, - localization length λ, - potential correlation length l c, In the vicinity of the critical point g becomes a fuction of only one parameter: g = F (L/ξ(W )) ξ(w )... correlation length Since L/ξ we have ξ(w ) W W c ν g g c + αl/ξ(w ) = g c + A(W W c )L /ν + BL y +... (y < ) We nd numerically g = g(w, L) and from Eq. (2) nde critical exponent ν and other parameters we need.

21 2D model with spin (symplectic symmetry).6 W = <g(w)>.4 < g(l) >.4 < g(w,l) >.4.3 L= 7 L= L=4 L= W.35 W = L L/ ξ nite-size scaling nite-size eects ν 2.85 L max =2 L max =4 L min =3 L min =5 L min =7 critical exponent ν 2.8 ± /L max /L min

22 2D model without spin 2D orthogonal system box disorder 2D orthogonal system box disorder W = 2 <g>.. W= W=2 W=3 W=4 W=5 W=6 L <g> L Finite size eects: For small W conductance g(l) increases untill L l (mean fre path) Weak localization: g = g π ln L l

23 Inverse Participation Ratio IPR Electron eigenfunction Φ n ( r) in the critical region: IPR (inverse participation ratio) I q (E n ) = r Φ n (r) 2q { L d(q ) metal I q (E n ) insulator Critical point: wave functions are multifractals: I q (E n ) L d q(q ) d q : universal multifractal dimensions

24 Universality of Anderson transition Scaling of IPR Y (E) = N stat N stat i E E n <δe ln I (E n ) Disorder (E=, W c ) Insulator Metal E c2 E c Ensemble of N stat sampels. The i-th sample has n i Eigenvalues in the energy interval Scaling: E ± δe Y q (E, L) = Y c q d q (q ) ln L+A(E E c )L /ν For Gaussian disorder: W = 2, E c = 6.58, d 5 =.96, ν =.52. <Y 5 (E,L)> Energy E = E c : Y 5 = *ln L E = 6.5 E = L

25 Critical exponent a fractal dimensions.6,4.5 GW c GE c BW c,2 ν d q.4 GW c GE c BW c q, q hould be the same for any critical point.

26 Properties of IPR IPR depends on the realization of the disorder For a given sample, I 2 possesses values from T 3 up to. k -2 Y = -a E ln L E=6.5 E=6. -4 E= L E = E = 6. E = L = 8 L = L = 32 L = ln I(L) - a E ln L. P( ln I ) E = 7.5 L=24 L=2 L=6 L= ln I For metal (E < E c ) Disorder W/ W c (E =, W c ) insulator.8 metal.6.4 E.2 c Energy and insulator (E > E c )

27 y y There are no localized states in the metallic phase. P( ln I 2 ) P( ln I 2 ) P( ln I 2 ) L = L = L = ln I 2 - d 2 ln L Finite size analysis of the distribution P(ln I 2 ) Typically, I 2 L 3. Probability to nd I 2 > L 2 or I 2 > L 3/2 : L - E = 3. E = L decreases exponentialy when L increases.

28 Green's function analysis Γ q (E, ω) = G(E+ω/2+iɛ, r, r )G(E+ω/2 iɛ, r, r) = 2πρ(E) ia(ω)ω + D(ω)q 2 Γ 5 E = 5 E = 6.58 E = π ρ ( E ) Im.4.8 B(E) L = 6 L = /ω E For q = numerical calculation of ln Im Γ = ln ω + ln B conrms that B = 2πρ(E) and so A.

29 Critical exponent: Theory vs numerical data there is no agreement. 5 d = 3 d = 4 ν Suslov: /(d-2) pfor d<4, /2 for d>4 /ε - expansion Our numerical data numerical.57. mean eld.5 Hikami ɛ-expansion d S 4 4.5

30 Dimension d = 2 + ε Critical dimension d c = 2 Numerical data for d = 3 and for three bi-fractals th d s close to d c 3 ν 2.5 numerical data 2.5 analytical theory / ε

31 Analytical theory: problems disorder is not small - perturbation theory might not work we do not know how to average over randomness: X X X... conductance, diusion coecient, IPR,... transport properties are extremely sesitive to change of the random potential. after averaging, we get another model - maybe nonlinear, but without eects of interference

32 Numerical simulations Any quantity could be calculated and entire statistical ensemble coudl be analyzed. Statistical ensembles with N samles (N in D) Accurate estiamtion of critical exponents, fractal dimensions Correct statistical analysis Consistent results: the same values of critical exponents obtained from scaling analysis of various physical quantities (level spacing, Lyapunov exponents, conductance, IPR,... ) Restricted to relatively small samples this could be avoided by nite size scaling analysis results for larger system size conrm older data for smaller systems. Developed methods applicable also to study of the localization in photonics, acoustics...

33 Weakly disordered systems: numerical data agree with the theory Weak localization and antilocalization in 2D: g = σ = σ ± π ]lnl/l 2D Ando model 2,5 6 2,5 5 W=3.36 W=4.3 W=5.32 <g>, W=3 W=4 E=-3, W=3 <g> 4 3 L 2 L Universal conductance uctiations: var g = g 2 g 2 =.85 (2D), =.34 (3D) and depends on the boundary conditions,8.25 p(g),6,4,2 var g W = W = 2 W = 3 W = g L / mean free path

34 2D models: numerical data conrm the theory Relation between multifractal dimenssions D q and critical conductance σ: 2.9 D(q) q Conductivity (in units of e 2 /h) is Fal'ko - Efetov: D q = 2 Veried for the q 4πσ 2D symplectic model Quantum Hall regime σ = ρhd (ρ is the density of states, D is the diusion coecient)

35 Optimistic Future - strongly disordered systems Numerical data could provide experimental data usefull for the theory. Strongly disordered 3D system - localized regime P(ln g),2,5,,5 3D L=26, W=6 ξ=5.4 <ln g>=-.66 Eq. (83) Γ=.63, γ=γ/2, <ln g>=-.69 3D L=, W=26 ξ=.8 <ln g>= ln g Older paradigma: P(ln g) is Gaussian Numerical data: distribution is not symmetric (symbols) New theory, based on the generalzation of the DMPK Equation

36 Conclusion Perturbation theory and DMPK describe transport in weakly disordered systems: universal conductance uctuations, weak localization, statistics of transmission parameters... There is no analytical theory for critical regime: For d > 2,nuemrical data are in variance with theory Attempt to generalize the DMPK to strongly disordered systems This work was supported by the Slovak Research and Development Agency under the contract No. APVV-8-