A semiclassical, quantitative approach to the Anderson transition Antonio M. García-García ag3@princeton.edu Princeton University We study analytically.

A semiclassical, quantitative approach A semiclassical, quantitative approach to the Anderson transitionto the Anderson transition

Antonio M. García-Garcíaag3@princeton.edu

Princeton University

We study analytically and numerically the metal-insulator transition in a We study analytically and numerically the metal-insulator transition in a d dimensional, non interacting (short range) disordered system by d dimensional, non interacting (short range) disordered system by combining the self-consistent theory of localization with the one combining the self-consistent theory of localization with the one

parameter scaling theory. parameter scaling theory. 

The upper critical dimension is infinity.The upper critical dimension is infinity.

Level Statistics at the transition: Level Statistics at the transition:

AGG, Emilio Cuevas, AGG, Emilio Cuevas, Phys. Rev. B 75, 174203 (2007), AGG Phys. Rev. B 75, 174203 (2007), AGG arXiv:0709.1292arXiv:0709.1292

2

1

2

1

d

|||)(| /c

r EEer

1-1)()( 2222

if EE

d LL~LnLn=LΣ

EH ˆ1. For a given disorder, E and d, 1. For a given disorder, E and d, how is the quantum dynamics? how is the quantum dynamics? Metal or insulator like?Metal or insulator like?

2. When does a metal insulator 2. When does a metal insulator transition occurs?. How is it transition occurs?. How is it described?described?

)(2

ˆ2

rVm

H

)'()'()( 20 rrVrVrV

locrer

/)( Vr /1)(

ttrt

)(2 consttrt

)(2

MetalMetal

Abs. ContinuousAbs. Continuous

Wigner-Dyson Wigner-Dyson statisticsstatistics

InsulatorInsulator

Pure point spectrum Pure point spectrum

Poisson statisticsPoisson statistics

Main Goals:

TransitionTransitiond

t

ttr/2

)(2

SingularSingular

MultifractalMultifractal

Critical Critical statisticsstatistics

What we What we (physicists)(physicists) know know believebelieve::

d = 1 An insulator for any disorder

d =2 An insulator for any disorder

d > 2 Disorder strong enough: Insulator

Disorder weak enough: Mobility edge

Why?Why?

Metal Insulator Metal Insulator TransitionTransition

E

1.1. Self-consistent theorySelf-consistent theory from the from the insulator side, valid only for d >>2. insulator side, valid only for d >>2. No interference. No interference. Abou-Abou-Chacra, Anderson.Chacra, Anderson.

2.2. Self-consistent theorySelf-consistent theory from the from the metallic side, valid only for d ~ 2. metallic side, valid only for d ~ 2. No tunneling. No tunneling. Vollhardt and Wolfle.Vollhardt and Wolfle.

3.3. One parameter scaling theoryOne parameter scaling theory. . Anderson et al.(1980)(1980) Correct (?) but qualitative Correct (?) but qualitative.

Weak disorder/localization. Perturbation theory. Well understood. Relevant in the transition in d = 2+(Wegner, Hikami, Efetov)

1. Numerical 1. Numerical simulationssimulations

Some of the main results of Some of the main results of the field are already the field are already included in the original included in the original paper by Anderson 1957!!paper by Anderson 1957!!

2. Analytical2. Analytical

Approaches to localizationApproaches to localization::

Currently reliable Currently reliable

Strong Strong disorder/localization.disorder/localization.NONO quantitative theory but:quantitative theory but:

Energy ScalesEnergy Scales

1. 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 quantum without quantum correctionscorrections

1

1

gE

gE

T

T

TT thE /

MetalMetal

InsulatorInsulator

Scaling theory of Scaling theory of localizationlocalization

)(ln

logg

Ld

gd 0log)(1

/)2()(1/

2

ggegg

gdgLggL

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 2. Diffusion coefficient become size and momentum dependentsize and 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

1.Cooperons (Langer-Neal, maximally crossed, responsible for weak localization) and Diffusons (no localization, semiclassical) can be combined.

2. Perturbation theory around the metallic limit.

3. No control on the approximation.

No control on the approximation.No control on the approximation.

Exact for a Cayley tree. It should Exact for a Cayley tree. It should be a good approx for d>>2. be a good approx for d>>2.

= 0metal

insulator

> 0

metal

insulator

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

)(Im iESi

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

Predictions of the self Predictions of the self consistent theoryconsistent theory

|||)(| /c

r EEer 42/1

421

d

dd

1. Critical 1. Critical

exponents:exponents:

2. Critical 2. Critical disorder:disorder:

3. Critical conductance:

d = 4 Upper d = 4 Upper critical critical dimension!dimension!

)2/ln(2 cc WKW

]2/,2/[1

)( WWW

p

Correctly predicts a transition for d>2

also B. Shapiro and E. Abrahams 1980

Wc3d ~

14Kroha, Wolfle, Kotov, Sadovskii

Anderson, Abou Chacra, Thouless

Vollhardt, Wolfle

Numerical results at Numerical results at the transitionthe transition

1. Scale invariance of the spectral correlations. 1. Scale invariance of the spectral correlations.

2. Intermediate level statistics2. Intermediate level statistics

3. Critical exponents3. Critical exponents

4. Critical disorder4. Critical disorder

5. Anomalous diffusion5. Anomalous diffusion

Finite scale analysis, Shapiro, et al. 93

15.1 43 DD

5.325.16 43 Dc

Dc WW

dttr /22 )(

Disagreement with the selfconsistent theory !

LLnLnL ~)()()(22

2 49.027.0 43 dd

)log()(

)(

2

2

LL

LL

Insulator

Metal

?

Schreiber, Grussbach

Mirlin, Evers, Cuevas, Schreiber, Slevin

Agreement scaling Agreement scaling theorytheory

Agreement scaling Agreement scaling theorytheory

var

dssPssss nn )(var22

SECOND SECOND PARTPART

What we What we did:did:

1. Numerical results for the Anderson transition in d=4,5,6, AGG and E. Cuevas, Phys. Rev. B 75, Phys. Rev. B 75,

174203 (2007),174203 (2007),

Critical exponents, critical disorder, level statistics

2. Analytical results combining the scaling theory and the self consistent condition, AGG, AGG, arXiv:0709.1292arXiv:0709.1292

Critical exponents, critical disorder, level statistics.

Numerical Results: Numerical Results: Anderson model cubic Anderson model cubic

lattice, d=[3,6]lattice, d=[3,6]

dssPssss nn )(var22

2Asse~sP

sesP )(

ii EEs 1

Metal

Insulator

]2/,2/[1

)( WWW

p ii

Critical exponents and Critical Disorder

Cayley tree

Upper critical dimension is infinity

d21

21

43 1 dd

)(dWc/ln(Wc/2)

2

Self consistent theory Self consistent theory

OK butOK but

Level StatisticsLevel Statistics

LdLnLn=LΣ

L if EE

)(~)()(

122

2

1

1

~)(

~)(

s

s

ssP

esP As

)(2 L

1)( dA(d)-1

ln(P(s))

Analytical results

1. Always perturbative around the metallic 1. Always perturbative around the metallic (Wolhardt & Wolfle) or the insulator state (Wolhardt & 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. Anomalous diffusion at the transition 2. Anomalous diffusion at the transition (predicted by the scaling theory) is not taken (predicted by the scaling theory) is not taken into account.into account.

2

2

)(

)(

d

d

qqD

LLD

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

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 well described by a process of anomalous diffusion. with no further localization corrections. diffusion. with no further localization corrections.

dttr /22 )(

Idea! Idea! ((AGG arXiv:0709.1292) Solve the self consistent AGG arXiv:0709.1292) Solve the self consistent equation assuming that the diffusion coefficient is equation assuming that the diffusion coefficient is renormalized as predicted by the scaling theory renormalized as predicted by the scaling 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 solving the above equation for with D (ω) = 0.D (ω) = 0.

2

1

2

1

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

'EE

s

1 d Cayley tree

Aizenman, Warzel

Comparison with numerical results

06.078.075.0

06.084.083.0

07.003.11

06.052.15.1

66

55

44

33

NT

NT

NT

NT

77.066.0

7.06.0

48.05.0

27.033.0

66

55

44

33

NT

NT

NT

NT

2

1

2

1

d

1. Critical exponents: Excellent

2, Level statistics: Good (problem with gc)

3. Critical disorder: Not better than before

CONCLUSIONSCONCLUSIONS

1.We obtain analytical results at the transition by 1.We obtain analytical results at the transition by combining the scaling theory with the self consistent in combining the scaling theory with the self consistent in d>3.d>3.

2. The upper critical dimension is infinity 2. The upper critical dimension is infinity 3. Analytical results on the level statistics agree with 3. Analytical results on the level statistics agree with numerical simulations. numerical simulations.

2

1

2

1

d

1-1)()( 2222

if EE

d LL~LnLn=LΣ

What is next?What is next?

1. Experimental verification.1. Experimental verification.

2. Anderson transition in correlated 2. Anderson transition in correlated potential potential

2

22

0 )(

)(sin)()(

yx

yxVyVxV

Experiments:Experiments:

Our findings may be used to test experimentally the Anderson transition by using ultracold atoms techniques.

One places a dilute sample of ultracold Na/Cs in a periodic step-like standing wave which is pulsed in time to approximate a delta function then the atom momentum distribution is measured.

The classical singularity cannot be reproduced in the lab. However (AGG, W Jiao, PRA 2006) an approximate singularity will still

show typical features of a metal insulator transition.

Spectral signatures of a metal (Wigner-Dyson):Spectral signatures of a metal (Wigner-Dyson):

1. Level Repulsion 1. Level Repulsion

2. Spectral Rigidity2. Spectral Rigidity

Spectral Signatures of an insulator: (Poisson) Spectral Signatures of an insulator: (Poisson)

ii EEAs sse~sP 1

2

1log)()(22

2

if EELL~LnLn=LΣ

)exp()()(2 ssPLL

s

P(s)