Today: 5 July 2008 ٢

Anderson localization M. Reza Rahimi Tabar IPM 5 July 2008 ١

Today: 5 July 2008 ٢

Short History of Anderson Localization ٣

Publication 1) F. Shahbazi, etal. Phys. Rev. Lett. 94, 165505 (2005) 2) A. Esmailpour, etal. Phys. Rev. B 74, 024206 (2006) 3) A. Bahraminasab, etal. Phys. Rev. B75, 064301 (2007) 4) A. Esmailpour, etal. J. Stat. Mech. P09014 (2007) 5) N. Abedpour, etal. Phys. Rev. B 76, 195407 (2007) 6) R. Seperhrinia, etal. Phys. Rev. B 77, 014203 (2008) 7) S. Mehdi Vaez Allaei, etal. J. Stat. Mech. P03016 (2008) 8) R. Sepehrinia, etal. Phys. Rev. B. 77, 104202 (2008) 9) A. Sheikhan, etal. Phys. Rev. B. (2008) (sub) 10) R. Sepehrinia, etal. Phys. Rev. B. (2008) 11) A. Bahraminasab, etal. Phys. Rev. B (2008) 12) A. Esmailpour, etal. Phys. Rev. B (2008)(sub) ٤

Localized and extended states in 1D chain 0.15 0.1 Extended state 0.4 Localized state 0.05 0.2 n ψ 0 n ψ 0-0.05-0.2-0.1-0.15 100 200 300 400 500 Site index (n) -0.4 100 200 300 400 500 Site index (n) ٥

A non-interacting electron moving in random potential Quantum interference of scattering waves Anderson localization of electrons extended localized localized localized E c E critical extended ٦

Scaling theory (1979) Conductance changes when system size is changed. Metal: Insulator: All wave functions are localized below two dimensions! A metal-insulator transition at g=g c is continuous (d>2). But the localization phenomena is Multifractal! ٧

Multifractality: scaling behavior of moments of (critical) wave functions Critical wave function at a metal-insulator transition point multifractal exponents In a metal fractal dimension Continuous set of independent and universal critical exponents : anomalous scaling dimensions singularity spectrum ٨

Anderson Localization (1957 1957) What? Quantum diffusion in a random potential stops due to interference effects. <r 2 > D clas t D quan t Goal of localization theory D quan t a a =? D quan = f(d,dis)? Delocalization t Localization Ψ α ( r ) 1/ V Ψ α ( r) e r / ξ loc ٩

Self-consistent theory of Anderson localization ١٠

Self-consistent theory of Anderson localization The presence of loops increases return probability as compared to normal diffusion Diffusion slows down Diffusion constant should be renormalized ١١

Generalization to open media Loops are less probable near the boundaries Slowing down of diffusion is spatially heterogeneous Diffusion constant becomes position-dependent ١٢

Classical localization ψ N L = exp( ) ψ 1 ξ ζ ( E) = Lim N 1 N ψ N ln ψ 1 ١٣

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Multiple Scattering of Waves Incident wave Detector t Random medium ١٥

Multiple Scattering of Waves Incident wave Detector t λ l Random medium L ١٦

From Single Scattering to Anderson Localization Wavelength λ Mean free paths l, l * Localization length ξ Size L of the medium Ballistic propagation (no scatteri ng) Single scattering Multiple scattering (diffusion) Strong (Anderson) localization 0 Strength of disorder ١٧

Experimental Evidence Exponential scaling of average transmission with L Diffusive regime: Localized regime: L ١٨

What if We Look at the Dynamics? L ١٩

Outline 1) Non-perturbative method in 1D a) random mass oscillators b) Disordered Schrödinger equation c) Magnon modes of spin glass 2) Perturbative method a) Mapping the quenched random differential equation to a Martin-Siggia-Rose action (elastic wave ) 3) Numerical method a) Red-Shift of spectral density ( light localization) 4) Acoustic wave in random Dimer media ٢٠

Part I a) Random Mass (1D) ٢١

Random Mass ٢٢

Random Mass ٢٣

Random Mass ٢٤

Random Mass (gerenelized PDF ) Dos: ٢٥

Random Mass ٢٦

Random Mass ٢٧

Lyapunov Exponents ٢٨

Lyapunov Exponents ٢٩

b) Disordered Schrodinger Equation (1D) ٣٠

c) HARMONIC MAGNON MODES HEISENBERG--MATTIS SPIN GLASSES ٣١

HEISENBERG--MATTIS SPIN GLASSES ٣٢

HEISENBERG--MATTIS SPIN GLASSES ٣٣

Part (II-a) Elastic wave localization Martin-Siggia-Rose action ٣٤

II-a) The Model (Scalar Field) ٣٥

Propagation of Wave Component with Frequency ω ٣٦

The Martin-Siggia-Rose Sgg Action Two coupling constants: g 0 =D 0 /λ 02, g ρ = Dρ/λ 2 ρ RG analysis to one-loop order in the limit, ω 2 /λ 0 0, to determine the two beta functions. ٣٧

Diagrammatic Representation and One-Loop Corrections ٣٨

The Beta Functions ٣٩

Phase Space and Fixed Points Three sets of fixed points for 0 <ρ < d/2: Trivial FP (Gaussian) at g 0 *=g ρ *=0 (stable) Non-trivial FPs, one at g 0 * = d/8, g ρ * = 0, and the other at Stable in one eigendirection, but unstable in the other eigendirection. ٤٠

Phase Space (Continued) Thus, a system with uncorrelated disorder is unstable against disorder with long-range correlations towards anewfp. Thus, with increasing disorder, extended localized ٤١

Phase Space (Continued) ٤٢

Phase Space (Continued) Two sets of fixed points for ρ > d/2: Gaussian FP, stable on the g 0 axis, but not on the g ρ axis Non-trivial FP at g 0 * = d/8, g ρ * = 0, unstable in all directions. Thus, power-law disorder relevant, but no new FP. ٤٣

Phase Space (Continued) ٤٤

Frequency-Dependence of Localization Length ٤٥

Wave Front ٤٦

Roughness of Wave Front: Self-Affine Fronts Computing the correlation function C(r) () =< [d(x) [()-d(x + r)] 2 > d(x) = distance from the source along the propagation direction C(r) ~ r 2α α = H = ρ-1 S. M. Vaez Allaei and M. Sahimi, PRL 96, 075507 (2006). ٤٧

The Shape of Wave Front and its Evolution H=0.3 H=0.75 ٤٨

Part III Numerical Method Red-Shift of Spectral Density (Light Localization) ٤٩

Transfer Matrix (random mass) Oseledec theorem: To ensure the stability of the numerical method, after multiplying the transfer matrices M times, we checked the length of the resulting vector, normalized it again, and ٥٠ then continued with the new vector

Transfer Matrix (random mass) The Lyapunov exponent is then expressed in terms of vector lengths obtained after N normalization of v. ٥١

Dynamics of Pulse in Random Media 3D Light localization ٥٢

Red-Shift of Spectral Density ٥٣

Localized Behavior ٥٤

Localization length ٥٥

Black-Body radiation ٥٦

Red-shift vs Distance from the source.. For small disorder strength ٥٧

Red-shift vs strength of disorder ٥٨

Measured Temperature at distance Z (in weak disorder limit) ٥٩

Acoustic waves in the random dimer media ٦٠

Random Dimer ٦١

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Critical exponents ٦٣

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Thanks ٦٦