Anderson localization: fifty years old and still growing Antonio M. García-García Princeton University Experiments with cold atoms Theories of localization Metal Insulator transition Quantum Chaos & (dynamical) localization MIT in quantum chaos Localization beyond condensed matter What is localization? What is new and why is still a hot topic? What is next? What do we know about localization?

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

Anderson Localization (1957) a = ? D quan = f(d,dis)? t D clas t D quan t D quan t a Quantum diffusion in a random potential stops due to interference effects. LocalizationDelocalization What? Goal of localization theory

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

Scaling theory Scaling theory Anderson paper and earlier theories of localization Weak localization Computers! Dynamical localization 50’ 60’ 70’ 80’ 90’ 00’ Experiments! Quantum chaos Mesoscopic physics Cold atoms Multifractality Perturbation theory and band theory History Renormalization group Theory of phase transitions

Theories of localization Locator expansions Weak localization expansions One parameter scaling theory

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! 4002 citations!

No control on the approximation. It should be a good approx for d>>2. It should be a good approx for d>>2. Predicts correctly localization in 1d and a MIT 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).

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 quantum corrections Metal Insulator Scaling theory of localization Phys. Rev. Lett. 42, 673 (1979), Abraham, P. W. Anderson, Licciardello,, Ramakrishnan.

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

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

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 Experimental verification 80’

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

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.

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

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.

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

Comparison with numerical results 1. Critical exponents: Excellent 2. Level statistics: Good (problem with g c )

Localization and metal insulator transition in deterministic (quantum chaos) pseudo random systems Universality in quantum chaos Dynamical localization (experiments) Scaling theory in quantum chaos Metal insulator transition in quantum chaos

1. Quantum classical transition. 2. Nano-Meso physics. Quantum engineering. 3. Systems with interactions for which the exact Schrödinger equation cannot be solved. Quantum Chaos Disordered systems Random versus chaotic Impact of classical chaos in quantum mechanics Quantum mechanics in a random potential 1. Scaling theory. 2. Ensemble average. 3. Anderson localization. ? 1. Semiclassical techniques. 2. BGS conjecture. 3. Localization (?) Relevantfor

s P(s) Disordered metal Wigner-Dyson statistics Insulator Poisson statistics Efetov 1. Eigenvector statistics: 2. Eigenvalue statistics: Characterization of a metal/insulator Random Matrix Uncorrelated Spectrum Wigner Dyson statistics Poisson statistics

Signatures of a metal-insulator transition Skolovski, Shapiro, Altshuler var I have created a monster! Scale invariance of the spectral correlations. Finite size scaling analysis. 3. Eigenstates are multifractals = Critical statistics

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

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

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

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?

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:

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

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

Determine the universality class in quantum chaos related to the metal-insulator transition.

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:

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

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

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

V(x)=  log|x| ANDERSON TRANSITON IN QUANTUM CHAOS 1. Scale Invariant Spectrum 2. Level repulsion 3. P(s)~exp(-As) s >> 1 4. Multifractal wavefunctions

3D kicked rotator Finite size scaling analysis shows there is a transition a MIT at k c ~ 3.3 In 3D, for  =2/3

Localization beyond condensed matter physics

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

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

QCD vacuum as a disordered conductor Instanton positions and color orientations vary Instanton positions and color orientations vary Impurities Instantons T = 0 long range hopping 1/R    = 3<4 Diakonov, Petrov, Verbaarschot, Osborn, Shuryak, Zahed,Janik AGG and Osborn, PRL, 94 (2005) QCD vacuum is a conductor for any density of instantons Electron Quarks

QCD at finite T: Phase transitions QCD at finite T: Phase transitions Quark- Gluon Plasma perturbation theory only for T>>T c J. Phys. G30 (2004) S1259 At which temperature does the transition occur ? What is the nature of transition ? Péter Petreczky

Deconfinement and chiral restoration Deconfinement: Confining potential vanishes: Chiral Restoration: Matter becomes light: 1. Effective model of QCD close to the phase transition (Wilczek,Pisarski,Yaffe): Universality, epsilon expansion.... too simple? 2. Classical QCD solutions (t'Hooft): Instantons (chiral), Monopoles and vortices (confinement). Instanton do not dissapear at the transiton (Shuryak,Schafer). Anderson localization We propose that quantum interference and tunneling, namely, Anderson localization plays an important role. How to explain these transitions?

Main contribution 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

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

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

Experimental studies of localization

Experiments: Difficult!!!! Localization is quite fragile phenomenon: broken by inellastic scaterring, even at low temperatures, unquenched disorder…. No control over interactions or details of the disordered potential. People see things ….but is it really localization? No control on absorption by the medium. Interactions and disorder are controlled with great precision!!!! Electronic systems: Light: Cold atoms: Weak localization in 2D. OK

The effective random potential is correlated Speckle potentials

The effective random potential is correlated Π 0 (t) population with zero velocity Π 0 (t) = constant Π 0 (t) ~t -1/2 Insulator arXiv: Metal

Test of quantum mechanics?

Why is the localization problem still interesting? 1. Universal quantum phenomenon. Studies beyond condensed matter. 2. No accurate experimental verification yet!!! Electrons (problem with interactions), light (problem with absorbtion) 3. No conclusive theory for the metal-insulator transition. Why is it interesting now ? Cold atoms in speckle (and kicked) potentials promise a very, very precise verification of Anderson localization and quantum mechanics itself Conclusions: