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Anderson localization: from theoretical aspects to applications Antonio M. García-García Princeton and ICTP Theoretical aspects Applications Localization in Quantum Chromodynamics Existence of a band of metallic states in 1d Analytical approach to the 3d Anderson transition Collaborators: Emilio Cuevas, Wang Jiao, James Osborn

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Problem: Get analytical expressions for different quantities characterizing the metal-insulator transition in d 3 such as, level statistics. Locator expansions One parameter scaling theory Selfconsistent condition Quasiclassical approach to the Anderson transition

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Scaling Scaling Perturbative locator expansion Field theory Computers 50’ 70’ 80’ 90’ 00’ Experiments Anderson localization Self consistent conditions Thouless, Wegner, Gang of four, Frolich, Spencer, Molchanov, Aizenman Abou Chakra, Anderson, Thouless, Vollhardt, Woelfle Anderson Efetov, Wegner Aoki, Schreiber, Kramer, Shapiro Aspect, Fallani, Segev Dynamical localization Fishman, Grempel, Prange, Casati Cayley tree and rbm Efetov, Fyodorov,Mirlin, Klein, Zirnbauer,Kravtsov 1d Kotani, Pastur, Sinai, Jitomirskaya, Mott. Weak LocalizationLee

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But my recollection is that, on the whole, the attitude was one of humoring me. Tight binding model V ij nearest neighbors, I random potential What if I place a particle in a random potential and wait? Technique: Looking for inestabilities in a locator expansion Interactions? Disbelief?, against the spirit of band theory Correctly predicts a metal-insulator transition in 3d and localization in 1d Not rigorous! Small denominators 4202 citations!

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

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Energy Scales 1. Mean level spacing: 2. Thouless energy: t T (L) is the travel time to cross a box of size L Dimensionless Thouless conductance Diffusive motion without localization corrections Metal Insulator Scaling theory of localization Phys. Rev. Lett. 42, 673 (1979), Gang of four. Based on Thouless,Wegner, scaling ideas

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Scaling theory of localization The change in the conductance with the system size only depends on the conductance itself g Weak localization

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Predictions of the scaling theory at the transition 1. Diffusion becomes anomalous 2. Diffusion coefficient become size and momentum dependent 3. g=g c is scale invariant therefore level statistics are scale invariant as well Imry, Slevin Chalker

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1.Cooperons (Langer-Neal, maximally crossed, responsible for weak localization) and Diffusons (no localization, semiclassical) can be combined. 3. Accurate in d ~2. Weak localization Self consistent condition (Wolfle-Volhardt) No control on the approximation! Positive correction to the resistivity of a metal at low T

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Predictions of the self consistent theory at the transition 1. Critical exponents: 2. Transition for d>2 Vollhardt, Wolfle, Correct for d ~ 2 Disagreement with numerical simulations!! Why?

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

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Proposal: Analytical results combining the scaling theory and the self consistent condition. and level statistics.

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2. Right at the transition the quantum dynamics is well described by a process of anomalous diffusion with no further localization corrections. Idea Solve the self consistent equation assuming that the diffusion coefficient is renormalized as predicted by the scaling theory Assumptions: 1. All the quantum corrections missing in the self consistent treatment are included by just renormalizing the coefficient of diffusion following the scaling theory.

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

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Level Statistics: Starting point: Anomalous diffusion predicted by the scaling theory Semiclassically, only “diffusons” Two levels correlation function

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Cayley tree Aizenman, Warzel Chalker Kravtsov,Lerner A linear number variance in the 3d case was obtained by Altshuler et al.’88 Shapiro, Abrahams

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Comparison with numerical results 1. Critical exponents: Excellent 2, Level statistics: OK? (problem with g c ) 3. Critical disorder: Not better than before

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Problem: Conditions for the absence of localization in 1d Motivation Quasiperiodic potentials Nonquasiperiodic potentials Work in progress in collaboration with E Cuevas

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Your intuition about localization V(x) X EaEa EbEb EcEc For any of the energies above: Will the classical motion be strongly affected by quantum effects? 0 Random

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The effective 1d random potential is correlated Speckle potentials tt t

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Localization/Delocalization in 1d: Random uncorrelated potential Exponential localization for every energy and disorder Periodic potential Bloch theorem. Absence of localization. Band theory In between?

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Quasiperiodic potentials Jitomirskaya, Sinai,Harper,Aubry Jitomirskaya, Bourgain What it is the least smooth potential that can lead to a band of metallic states? Similar results Conjecture No metallic band if V(x) is discontinuous Jitomirskaya, Aubry, Damanik

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Fourier space : Long range hopping Localization for >0 Levitov Fyodorov Mirlin Delocalization in real space A metallic band can exist for

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Non quasiperiodic potentials Physics literature Neither of them is accurate 1. Izrailev & KrokhinMetallic band if: Born approximation 2. Lyra & Moura 1a. A vanishing Lyapunov exponent does not mean metallic behavior. 1b. Higher order corrections make the Lyapunov exponent > 0 2. Not generic Localization in correlated potentials: Luck, Shomerus, Efetov, Mirlin,Titov Decaying and sparse potentials (Kunz,Simon, Soudrillard): transition but non ergodic

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Mathematical literature: Kotani’s theory of ergodic operators Non deterministic potentials No a.c. spectrum Deterministic potentialsMore difficult to tell Discontinuous potentials No a.c. spectrum Kotani, Simon, Kirsch, Minami, Damanik. Damanik,Stolz, Sims

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Neighboring values of the potential must be correlated enough in order to avoid destructive interference. > 0 and V(x) and its derivative are bounded. How to proceed? A band of metallic states might exist provided Smoothing uncorrelated random potentials According to the scaling theory in the metallic region motion must be ballistic.

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Finite size scaling analysis Thouless, Shklovski, Shapiro 93’ Spectral correlations are scale invariant at the transition Diffusive Metal Clean metal Insulator AGG, Cuevas

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Savitzsky-Golay 1. Take n p values of V(n) around a given V(n 0 ) 2. Replace V(n 0 ) by the best fit of the n p values to a polynomial of M degree 3. Repeat for all n 0 Resulting potential is not continuous A band of metallic states does not exist

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Fourier filtering Resulting potential is analytic 1. Fourier transform of the uncorrelated noise. 2. Remove k > k cut 3. Fourier transform back to real space A band of metallic states do exist

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Gruntwald Letnikov operator Resulting potential is C - +1/2

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A band of metallic states exists provided Is this generic?

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Localization in systems with chiral symmetry and applications to QCD 1. Chiral phase transition in lattice QCD as a metal-insulator transition, Phys.Rev. D75 (2007) , AMG, J. Osborn 2. Chiral phase transition and Anderson localization in the Instanton Liquid Model for QCD, Nucl.Phys. A770 (2006) , AMG. J. Osborn 3. Anderson transition in 3d systems with chiral symmetry, Phys. Rev. B 74, (2006), AMG, E. Cuevas 4. Long range disorder and Anderson transition in systems with chiral symmetry, AMG, K. Takahashi, Nucl.Phys. B700 (2004) Chiral Random Matrix Model for Critical Statistics, Nucl.Phys. B586 (2000) , AMG and J. Verbaarschot

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QCD : The Theory of the strong interactions QCD : The Theory of the strong interactions High Energy g << 1 Perturbative High Energy g << 1 Perturbative 1. Asymptotic freedom Quark+gluons, Well understood Low Energy g ~ 1 Lattice simulations Low Energy g ~ 1 Lattice simulations The world around us The world around us 2. Chiral symmetry breaking 2. Chiral symmetry breaking Massive constituent quark Massive constituent quark 3. Confinement 3. Confinement Colorless hadrons Colorless hadrons How to extract analytical information? Instantons, Monopoles, Vortices

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Deconfinement and chiral restoration Deconfinement: Confining potential vanishes: Chiral Restoration: Matter becomes light: 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). We propose that quantum interference/tunneling plays an important role. How to explain these transitions?

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Instantons: Non perturbative solutions of the classical Yang Mills equation. Tunneling between classical vacua. 1. Dirac operator has a zero mode in the field of an instanton 2. Spectral properties of the smallest eigenvalues of the Dirac operator are controled by instantons 3. Spectral properties related to chiSB. Banks-Casher relation: QCD at T=0, instantons and chiral symmetry breaking tHooft, Polyakov, Callan, Gross, Shuryak, Diakonov, Petrov,VanBaal

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Instanton liquid models T = 0 Instanton liquid models T = 0 Multiinstanton vacuum? No superposition Variational principles(Dyakonov), Instanton liquid model (Shuryak). Variational principles(Dyakonov), Instanton liquid model (Shuryak). Non linear equations Solution ILM T > 0

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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 initially bounded to an instanton get delocalized due to the overlapping with the rest of zero modes. (Diakonov and Petrov) Dis.Sys: Exponential decay QCD vacuum: Power law decay Differences

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

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At the same T c that the Chiral Phase transition A metal-insulator transition in the Dirac operator induces the QCD chiral phase transition metal - insulator undergo a metal - insulator transition with J. Osborn Phys.Rev. D75 (2007) Nucl.Phys. A770 (2006) 141 QCD Dirac operator

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Signatures of a metal-insulator transition 1. Scale invariance of the spectral correlations. A finite size scaling analysis is then carried out to determine the transition point Eigenstates are multifractals. Skolovski, Shapiro, Altshuler Mobility edge Anderson transition var

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

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

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

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Problem: To determine the importance of Anderson localization effects in deterministic (quantum chaos) systems Scaling theory in quantum chaos Metal insulator transition in quantum chaos

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Quantum chaos studies the quantum properties of systems whose classical motion is chaotic (or not) Bohigas-Giannoni-Schmit conjecture Classical chaos Wigner-Dyson Momentum is not a good quantum number Delocalization What is quantum chaos? Energy is the only integral of motion

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

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Dynamical localization in momentum space 2. Harper model 3. Arithmetic billiards t Classical Quantum Exceptions to the BGS conjecture 1. Kicked systems Fishman, Prange, Casati

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RandomDeterministic d = 1,2 d > 2 Strong disorder d > 2 Weak disorder d > 2 Critical disorder Chaotic motion Integrable motion ?????????? Wigner-Dyson Delocalization Normal diffusion Poisson Localization Diffusion stops Critical statistics Multifractality Anomalous diffusion Characterization Always? Bogomolny Altshuler, Levitov Casati, Shepelansky

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Determine the class of systems in which Wigner-Dyson statistics applies. Does this analysis coincide with the BGS conjecture? Adapt the one parameter scaling theory in quantum chaos in order to:

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Scaling theory and anomalous diffusion weak localization? Wigner-Dyson (g) > 0 Poisson (g) < 0 d e fractal dimension of the spectrum. Two routes to the Anderson transition 1. Semiclassical origin 2. Induced by quantum effects Compute g Universality

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Wigner-Dyson statistics in non-random systems 1. Estimate the typical time needed to reach the “boundary” (in real or momentum space) of the system. In billiards: ballistic travel time. In kicked rotors: time needed to explore a fixed basis. 2. Use the Heisenberg relation to estimate thedimensionless conductance g(L). Wigner-Dyson statistics applies if and

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1D =1, d e =1/2, Harper model, interval exchange maps (Bogomolny) =2, d e =1, Kicked rotor with classical singularities (AGG, WangJiao) 2D =1, d e =1, Coulomb billiard (Altshuler, Levitov). =2, d e =1, Kicked rotor with classical singularities (AGG, WangJiao) 2D =1, d e =1, Coulomb billiard (Altshuler, Levitov). 3D =2/3, d e =1, 3D Kicked rotor at critical coupling. Anderson transition in quantum chaos Conditions: 1. Classical phase space must be homogeneous. 2. Quantum power-law localization. 3. Examples:

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3D kicked rotator Finite size scaling analysis shows there is a transition at k c ~ 2.3 At k = k c ~ 2.3 diffusion is anomalous At k = k c ~ 2.3 diffusion is anomalous

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1D kicked rotor with singularities 1D kicked rotor with singularities Classical Motion Quantum Evolution Anomalous Diffusion Quantum anomalous diffusion No dynamical localization for <0 Normal diffusion

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AGG, Wang Jjiao, PRL > 0 Localization Poisson 2. < 0 Delocalization Wigner-Dyson 3. = 0 MIT Critical statistics Anderson transition for log and step singularities Results are stable under perturbations and sensitive to the removal of the singularity Possible to test experimentally

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Analytical approach: From the kicked rotor to the 1D Anderson model with long-range hopping Analytical approach: From the kicked rotor to the 1D Anderson model with long-range hopping Explicit analytical results are possible, Fyodorov and Mirlin Insulator for 0 Fishman,Grempel, Prange 1d Anderson model T m pseudo random Always localization

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Conclusions: 1. Anderson localization depends on the degree of differentiability of the potential. 2. Critical exponents and level statistics are acessible to analytical techniques 3. The adaptation of the scaling theory to quantum chaos provides a powerful tool to predict localization effects in non random systems 4. Anderson localization plays a role in the chiral phase transition of QCD Thanks!

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NEXT 1. Find a way to compute analytically the critical disorder and others quantities that characterize the Anderson transition. 2. Adapt localization theories to the peculiarities of cold atoms. 3. Mathematicians: Prove delocalization

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