Anderson localization: from theoretical Anderson localization: from theoretical aspects to applicationsaspects to applications

Antonio M. García-Garcíaag3@princeton.edu

http://phy-ag3.princeton.edu Princeton and ICTP

Theoretical aspects

ApplicationsLocalization 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

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

Scaling Scaling

Perturbative locator expansion

Field theory

Computers

50’

70’

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 Localization Lee

But my recollection is that, on the whole, But my recollection is that, on the whole, the attitude was one of humoring me.the attitude was one of humoring me.

Tight binding model

Vij 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!

No control on the approximation.No control on the approximation. It should be a good approx for It should be a good approx for d>>2. d>>2. It predicts correctly localization It predicts correctly localization in 1d and a transition in 3din 1d and a transition in 3d

= 0metal

insulator> 0

metalinsulator

The distribution of the self The distribution of the self energy Senergy Sii (E) is sensitive to (E) is sensitive to localization.localization.

)(Im iESi

Perturbation theory around Perturbation theory around the insulator limit (locator the insulator limit (locator expansion). expansion).

Energy ScalesEnergy Scales1. Mean level spacing:1. Mean level spacing:2. Thouless energy: 2. Thouless energy: ttTT(L) (L) is the travel time to cross a box of size L is the travel time to cross a box of size L

1

TEg Dimensionless Dimensionless

Thouless conductanceThouless conductance

22 ddT LgLLDE Diffusive motion Diffusive motion

without localization without localization correctionscorrections

11

gEgE

T

T

TT thE /

MetalMetal

InsulatorInsulator

Scaling theory of localizationScaling theory of localization Phys. Rev. Lett. 42, 673 (1979), Gang of four. Based on Thouless,Wegner, scaling ideas

Scaling theory of Scaling theory of localizationlocalization

)(lnlog g

Ldgd

0log)(1/)2()(1

/

2

ggegggdgLgg

L

d

The change in the conductance with the system size only depends on the conductance itself)(g

gWeak localizationWeak localization

Predictions of the Predictions of the scaling theory at the scaling theory at the transitiontransition

dttr /22 )(

dd LLDqqD 22 )()(

1. Diffusion becomes anomalous1. Diffusion becomes anomalous

2. Diffusion coefficient become size 2. Diffusion coefficient become size and momentum dependentand momentum dependent

3. g=g3. g=gcc is scale invariant therefore level is scale invariant therefore level statistics are scale invariant as wellstatistics are scale invariant as well

Imry, Slevin

Chalker

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

Predictions of the self Predictions of the self consistent theory at the consistent theory at the

transitiontransition

|||)(| /c

r EEer 42/142

1

ddd

1. Critical 1. Critical exponents:exponents:

2. Transition for d>2

Vollhardt, Wolfle,1982

3. Correct for d ~ 2

Disagreement with numerical simulations!!

Why?

1. Always perturbative around the metallic 1. Always perturbative around the metallic (Vollhardt & Wolfle) or the insulator state (Vollhardt & Wolfle) or the insulator state (Anderson, Abou Chacra, Thouless) .(Anderson, Abou Chacra, Thouless) .

A new basis for localization is neededA new basis for localization is needed

2

2

)()(

d

d

qqDLLD

Why do self Why do self consistent methods consistent methods fail for d = 3?fail for d = 3?

2. Anomalous diffusion at the 2. Anomalous diffusion at the transition (predicted by the scaling transition (predicted by the scaling theory) is not taken into account.theory) is not taken into account.

Proposal:Proposal:

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

2. Right at the transition the quantum dynamics is 2. Right at the transition the quantum dynamics is well described by a process of anomalous diffusion well described by a process of anomalous diffusion with no further localization corrections. with no further localization corrections.

dttr /22 )(

Idea Idea Solve the self consistent equation assuming that Solve the self consistent equation assuming that the diffusion coefficient is renormalized as predicted by the the diffusion coefficient is renormalized as predicted by the scaling theoryscaling theory

AssumptionAssumptions:s:1. All the quantum corrections missing in 1. All the quantum corrections missing in the self consistent treatment are included the self consistent treatment are included by just renormalizing the coefficient of by just renormalizing the coefficient of diffusion following the scaling theory. diffusion following the scaling theory.

Technical details: Critical exponents

The critical exponent ν, can be obtained by The critical exponent ν, can be obtained by solving the above equation for with D solving the above equation for with D (ω) = 0.(ω) = 0.

21

21

d

2

|| cEE

Level Statistics:

Starting point:Starting point: Anomalous diffusion Anomalous diffusion predicted by the scaling predicted by the scaling theorytheory

Semiclassically, only “diffusons”

Two levels correlation function

'EEs

1 d

Cayley tree Aizenman, Warzel

dDd

22

Chalker Kravtsov,Lerner

A linear number variance in the 3d case was obtained by Altshuler et al.’88

2/ dcg dc

Shapiro, Abrahams

Comparison with numerical

results

06.078.075.006.084.083.007.003.1106.052.15.1

66

55

44

33

NT

NT

NT

NT

77.066.07.06.048.05.027.033.0

66

55

44

33

NT

NT

NT

NT

21

21

d

1. Critical exponents: Excellent2, Level statistics: OK? (problem with gc)3. Critical disorder: Not better than before

|||)(| /c

r EEer

Problem: Conditions for the absence of localization in 1d

Motivation

Quasiperiodic potentials

Nonquasiperiodic potentials

Work in progress in collaboration with E Cuevas

Your intuition about localization

V(x)

X

Ea

Eb

Ec

For any of the energies above: For any of the energies above: Will the classical motion be Will the classical motion be strongly affected by quantum effects?strongly affected by quantum effects?

0

Random

The effective 1d random potential is correlated

Speckle potentials

tttt

Localization/Delocalization in 1d:

Random uncorrelated potential

Exponential localization for every energy and disorder

Periodic potentialBloch theorem. Absence

of localization. Band theory

In between?

Quasiperiodic potentials

)cos()( xxV Jitomirskaya, Sinai,Harper,Aubry Critical

InsulatorMetal

111

)exp( AkBak

Jitomirskaya, Bourgain

What it is the least smooth potential that can lead to a band of metallic states?

Similar results

k

k kxaxV )2cos()(

Conjecture 2/3)( CnV

No metallic band if V(x) is discontinuous Jitomirskaya, Aubry, Damanik

CnkanVk

k ))(2cos()( 1||

kAak0

Fourier space:

Long range hopping Localization for >0

Levitov FyodorovMirlin

Delocalization in real space

A metallic band can exist for 0)( CnV

Non quasiperiodic potentials

Physics literature

2/1)(

kkS

Neither of them is accurate

1. Izrailev & Krokhin Metallic band if:

)0()()exp()( VxVikxdxkS

cc kkkkS ,0)( )()( kSk

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,TitovDecaying and sparse potentials (Kunz,Simon, Soudrillard): transition but non ergodic

Mathematical literature:

Kotani’s theory of ergodic operators

Non deterministic potentials

No a.c. spectrum

Deterministic potentials More difficult to tell

Discontinuous potentials

No a.c. spectrum

xVxVxB /1)0()()(

No a.c. spectrum

Kotani, Simon, Kirsch, Minami, Damanik.

Damanik,Stolz, Sims

Neighboring values of the potential must be correlated enough in order to avoid destructive interference.

CxV )( > 0 and V(x) and its derivative are bounded.

How to proceed?

A band of metallic states might exist provided

Smoothing uncorrelated random potentials

0)0()(lim)(

VxVxBx

According to the scaling theory in the metallic region motion must be ballistic.

Finite size scaling Finite size scaling analysisanalysis

Thouless, Shklovski, Shapiro 93’

dssPssss nn )(var 22

Spectral correlations are scale invariant at the transition

i

iissP /)( 1

Diffusive Metal

1var)var(0)var(

286.0var)var(

P

WD

ss

s

Clean metal

Insulator

AGG, Cuevas

Savitzsky-Golay

1. Take np values of V(n) around a given V(n0)

2. Replace V(n0) by the

best fit of the np values to a polynomial of M degree

3. Repeat for all n0

Resulting potential is not continuous

0)0()(lim)(

VxVxBx

A band of metallic states does not exist

Fourier filtering

Resulting potential is analytic

1. Fourier transform of the uncorrelated noise.

2. Remove k > kcut

3. Fourier transform back to real space

0)0()(lim)(

VxVxBx

A band of metallic states do exist

Gruntwald Letnikov operator

2/12/1 /1,/1 NNi

0)0()(lim)(

VxVxBx

Resulting potential is C-+1/2

2/1,)( CnVA band of metallic states exists provided

Is this generic?

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) 034503, AMG, J. Osborn

2. Chiral phase transition and Anderson localization in the Instanton Liquid Model for QCD , Nucl.Phys. A770 (2006) 141-161, AMG. J. Osborn

3. Anderson transition in 3d systems with chiral symmetry, Phys. Rev. B 74, 113101 (2006), AMG, E. Cuevas

4. Long range disorder and Anderson transition in systems with chiral symmetry , AMG, K. Takahashi, Nucl.Phys. B700 (2004) 361

5. Chiral Random Matrix Model for Critical Statistics, Nucl.Phys. B586 (2000) 668-685, AMG and J. Verbaarschot

QCD : The Theory of the strong interactionsQCD : The Theory of the strong interactions HighHigh EnergyEnergy g << 1 Perturbativeg << 1 Perturbative 1. Asymptotic freedom Quark+gluons, Well understoodQuark+gluons, Well understood Low Energy Low Energy g ~ 1 Lattice simulationsg ~ 1 Lattice simulations The world around usThe world around us 2. Chiral symmetry breaking2. Chiral symmetry breaking

Massive constituent quark Massive constituent quark 3. Confinement3. Confinement Colorless hadronsColorless hadrons

How to extract analytical information?How to extract analytical information? Instantons , Monopoles, Instantons , Monopoles, VorticesVortices

rrarV /)(

3)240(~ MeV

Deconfinement and chiral restorationDeconfinement 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?

Instantons:Instantons: Non perturbative solutions of the classical Yang Mills Non perturbative solutions of the classical Yang Mills

equation. Tunneling between classical vacuaequation. Tunneling between classical vacua..

1. Dirac operator has a zero mode in the field of an instanton1. Dirac operator has a zero mode in the field of an instanton

2. Spectral properties of the smallest eigenvalues of the Dirac operator are 2. Spectral properties of the smallest eigenvalues of the Dirac operator are controled by instantons controled by instantons

3. 3. Spectral properties related to chiSB. Banks-Casher relation:Spectral properties related to chiSB. Banks-Casher relation:

QCD at T=0, instantons and chiral symmetry breakingQCD at T=0, instantons and chiral symmetry breaking tHooft, Polyakov, Callan, Gross, Shuryak, tHooft, Polyakov, Callan, Gross, Shuryak, Diakonov, Petrov,VanBaalDiakonov, Petrov,VanBaal

300 /10 rrψrDψgA+=D ins

μμ

Vm

imdmDTr

V mm

)(lim)()(1

0

1

3

34 )()ˆ(~)()(

RRuizxiDzxxdT AI

AAIIIA

Instanton liquid Instanton liquid models T = 0models T = 0

Multiinstanton Multiinstanton vacuum?vacuum?

NoNo superpositionsuperposition

00

AI

IA

TT

iD

Variational principles(Dyakonov), Variational principles(Dyakonov), Instanton liquid model (Shuryak). Instanton liquid model (Shuryak).

Non linear equationsNon linear equations

Solution

ILM T > 0 ))(/exp(~ TlRTIA

QCD vacuum as a conductor (T =0)QCD vacuum as a conductor (T =0)Metal:Metal: An electron initially bounded to a single atom An electron initially bounded to a single atom gets delocalized due to the overlapping with nearest gets delocalized due to the overlapping with nearest neighborsneighborsQCD Vacuum:QCD Vacuum: Zero modes initially bounded to an Zero modes initially bounded to an instanton get delocalized due to the overlapping with instanton get delocalized due to the overlapping with the rest of zero modes. (Diakonov and Petrov)the rest of zero modes. (Diakonov and Petrov)

Dis.Sys:Dis.Sys: Exponential decay Exponential decay QCD vacuum:QCD vacuum: Power law Power law

decay decay

DifferencesDifferences

QCD vacuum as a disordered QCD vacuum as a disordered conductorconductor

Instanton positions and color Instanton positions and color orientations varyorientations vary

Ion Ion InstantonsInstantonsT = 0 TT = 0 TIAIA~ 1/R~ 1/R = = 3<43<4

Diakonov, Petrov, Verbaarschot, Osborn, Shuryak, Zahed,Janik

Shuryak,Verbaarschot, AGG and OsbornShuryak,Verbaarschot, AGG and Osborn

QCD vacuum is a conductorQCD vacuum is a conductor

Electron Electron QuarksQuarks

T>0 TT>0 TIAIA~ e~ e-R/l(T)-R/l(T)

A transition is possibleA transition is possible

nnnQCD iD

0At the same Tc that the Chiral Phase transition

A metal-insulator transition in the Dirac operator A metal-insulator transition in the Dirac operator induces the QCD chiral phase transitioninduces the QCD chiral phase transition

n

n

undergo a metal - insulatormetal - insulator transition

with J. Osborn

Phys.Rev. D75 (2007) 034503

Nucl.Phys. A770 (2006) 141

QCD Dirac operator μμ

QCD gA+=D

Signatures of a metal-insulator transitionSignatures of a metal-insulator transition1. Scale invariance of the spectral correlations.

A finite size scaling analysis is then carried out to determine the transition point.

2.

3. Eigenstates are multifractals.

)1(2 ~)( qDdq

n

qLrdr

Skolovski, Shapiro, Altshuler

1~)(1~)(

sesPsssP

As

Mobility edge Anderson transition

varvar

dssPssss nn )(var 22

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

Metal Metal insulator insulator transitiontransition

ILM with 2+1 massless flavors,

We have observed a metal-insulator transition at T ~ 125 Mev

Spectrum is scale invariant

Instanton liquid model Nf=2, maslessInstanton liquid model Nf=2, masless Localization versus Localization versus chiral transitionchiral transition

Chiral and localizzation transition occurs at the same temperatureChiral and localizzation transition occurs at the same temperature

Problem: To determine the Problem: To determine the importance of Anderson localization importance of Anderson localization effects in deterministic (quantum effects in deterministic (quantum chaos) systems chaos) systems

Scaling theory in quantum chaos

Metal insulator transition in quantum chaos

Quantum chaos studies the Quantum chaos studies the quantum properties of systems quantum properties of systems whose classical motion is whose classical motion is chaotic (or not)chaotic (or not)

Bohigas-Giannoni-Schmit Bohigas-Giannoni-Schmit conjectureconjecture

Classical chaos Wigner-Classical chaos Wigner-DysonDyson

Momentum is not a good quantum number Momentum is not a good quantum number DelocalizationDelocalization

What is quantum chaos?

Energy is the only integral of motionEnergy is the only integral of motion

Gutzwiller-Berry-Tabor Gutzwiller-Berry-Tabor conjectureconjecture

Poisson Poisson statisticsstatistics

(Insulator(Insulator

))

s

P(s)

Integrable classical motion

Integrability

Canonical Canonical momenta are momenta are conservedconserved

System is localized in System is localized in momentum spacemomentum space

Dynamical Dynamical localizationlocalization

n

nTtVpH )()(2

cos)( KV Dynamical localization Dynamical localization in momentum spacein momentum space

2. Harper model2. Harper model3. Arithmetic 3. Arithmetic billiardsbilliards

<p2

>

t

Classical

Quantum

Exceptions to the BGS Exceptions to the BGS conjectureconjecture

1. Kicked systems1. Kicked systems

Fishman, Prange, Casati

Random Deterministic

d = 1,2 d > 2 Strong disorder

d > 2Weak disorder

d > 2Critical disorder

Chaotic motion

Integrable Integrable motionmotion

??????????

Wigner-Dyson

Delocalization

Normal diffusion

Poisson

Localization

Diffusion stops

Critical statistics

Multifractality

Anomalous diffusion

CharacterizationCharacterization

cgg

0g

g

Always?

Bogomolny

Altshuler, Levitov

Casati, Shepelansky

Determine the class of systems in which Determine the class of systems in which Wigner-Dyson statistics applies. Wigner-Dyson statistics applies.

Does this analysis coincide with the BGS Does this analysis coincide with the BGS conjecture?conjecture?

Adapt the one parameter scaling theory in quantum chaos in order to:

Scaling theory and anomalous diffusionScaling theory and anomalous diffusion

2)(

eclas

T

ddLELg clas

clasquanclas 000 quanclas

0)( g

)()( gfg clas

weak weak localization?localization?L Wigner-DysonWigner-Dyson (g) (g)

> 0> 0Poisson Poisson (g) (g) < 0< 0

eddLtq /2 dde e fractal fractal dimension dimension of the of the spectrum.spectrum.

Two routes to the Anderson transitionTwo routes to the Anderson transition

1. Semiclassical origin 1. Semiclassical origin 2. Induced by quantum effects2. Induced by quantum effects

Compute g

Universality

Wigner-Dyson statistics in non-random Wigner-Dyson statistics in non-random systemssystems

02)(

eclas

T

ddLELg clas

tq 2

1. Estimate the typical time needed to 1. Estimate the typical time needed to reach the “boundary” (in real or reach the “boundary” (in real or momentum space) of the system.momentum space) of the system.

In billiards: ballistic travel time.In billiards: ballistic travel time.In kicked rotors: time needed to explore a fixed In kicked rotors: time needed to explore a fixed basis.basis.2. Use the Heisenberg relation to estimate 2. Use the Heisenberg relation to estimate

thedimensionless conductance g(L) .thedimensionless conductance g(L) .

Wigner-Dyson statistics applies Wigner-Dyson statistics applies ifif

and

0quan

tk

ddLELg

eclas

T clas 202)(

1D 1D =1, d=1, dee=1/2, Harper model, interval exchange maps =1/2, Harper model, interval exchange maps (Bogomolny)(Bogomolny)

=2, d=2, dee=1, Kicked rotor with classical singularities =1, Kicked rotor with classical singularities (AGG, WangJiao)(AGG, WangJiao)

2D 2D =1, d=1, dee=1, Coulomb billiard =1, Coulomb billiard (Altshuler, Levitov).(Altshuler, Levitov).

3D 3D =2/3, d=2/3, dee=1, 3D Kicked rotor at critical coupling.=1, 3D Kicked rotor at critical coupling.

Anderson transition in quantum chaosAnderson transition in quantum chaos Conditions:Conditions:

1. 1. Classical phase space must be homogeneous. Classical phase space must be homogeneous. 2. 2. Quantum power-law localization. Quantum power-law localization. 3. 3.

Examples:Examples:

3D kicked rotator3D kicked rotator

Finite size scaling analysis shows Finite size scaling analysis shows there is a transition at kthere is a transition at kc c ~ 2.3~ 2.3

)cos()cos()cos(),,( 321321 kV

3/22 ~)( ttpquan

At k = kAt k = kc c ~ 2.3 diffusion ~ 2.3 diffusion is anomalous is anomalous

1D kicked rotor with singularities 1D kicked rotor with singularities

||)( V

)4

exp()/)(exp()4

exp(ˆ2

2

2

2

TiVTU

n

nTtVpH )()(2

cos)( KV Classical Motion

Quantum Evolution

Anomalous Diffusion

11||/1),( tkktkP

'' ||/1),( tkktkP Quantum anomalous diffusion

No dynamical localization for <0

Normal diffusion

||log)( V

11||)(

tkLELg clas

T clas

AGG, Wang Jjiao, PRL 2005

1. 1. > 0 Localization Poisson > 0 Localization Poisson 2. 2. < 0 Delocalization Wigner-Dyson < 0 Delocalization Wigner-Dyson 3. 3. = 0 MIT Critical statistics = 0 MIT Critical statistics Anderson transition Anderson transition for for

log and step singularitieslog and step singularities

Results are Results are stablestable under perturbations and under perturbations and sensitive to the removal of the singularitysensitive to the removal of the singularity

Possible to test experimentally

Analytical approach: From the kicked rotor to the 1D Analytical approach: From the kicked rotor to the 1D Anderson model with long-range hopping Anderson model with long-range hopping

),()()(),(21),( 2

2

tntVttt

in

1

1 r

Wr

Explicit analytical results are possible, Fyodorov and Mirlin

Insulator for 0

0r

mrmrmm EuuWuT

cos)( KV

||)( V

Fishman,Grempel, PrangeFishman,Grempel, Prange

1d Anderson 1d Anderson modelmodel

Tm pseudo random

1,1, rrrW Always localization

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! ag3@princeton.eduhttp://phy-ag3.princeton.edu

ag3@princeton.eduhttp://phy-ag3.princeton.edu

NEXT1. Find a way to compute analytically the critical disorder and others quantities that characterize the Anderson transition.

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3. Mathematicians: Prove delocalization