2162-5

Advanced Workshop on Anderson Localization, Nonlinearity and Turbulence: a Cross-Fertilization

Boris ALTSHULER

23 August - 3 September, 2010

Columbia University, Dept. of Physics, New York NY

U.S.A.

INTRODUCTORY Anderson Localization - Introduction

Anderson localization - introduction

Boris AltshulerPhysics Department, Columbia University

Advanced Workshop on Anderson Localization, Nonlinearity and Turbulence: a Cross-Fertilization

August 23 – September 03, 2010

OOne--particle Localization

>50 years of Anderson Localization

q.p.

�One quantum particle

�Random potential (e.g., impurities) Elastic scattering

Einstein (1905):Random walk

always diffusion

Dtr �2

diffusion constant

Anderson(1958):For quantum

particles

not always!

constr t���� ��2

It might be that

0�D

as long as the system has no memory

Quantum interference memory

2 0Dt� ��� �

�

�

x

t

t�I V

2 dne Dd

� �

� �

Density of statesConductivity

Einstein Relation (1905)

Diffusion Constant

ALG

VIG

V

�

���

����

�

�

;0

2 dne Dd

� �

� �

Density of statesConductivity

Einstein Relation (1905)

Diffusion Constant

No diffusion – no conductivity

Metal – insulator transition

Localized states – insulatorExtended states - metal

Localization of single-electron wave-functions:

extended

localized

Disorder

IV

Conductanceextended

localizedexp

loc

x y

z

z

L LL

L

�

�

� �� �� �� � �

� �� �� � �� �

���� �

extended

localized

z

Spin DiffusionFeher, G., Phys. Rev. 114, 1219 (1959); Feher, G. & Gere, E. A., Phys. Rev. 114, 1245 (1959).LightWiersma, D.S., Bartolini, P., Lagendijk, A. & Righini R. “Localization of light in a disordered medium”, Nature 390, 671-673 (1997).Scheffold, F., Lenke, R., Tweer, R. & Maret, G. “Localization or classical diffusion of light”, Nature 398,206-270 (1999).Schwartz, T., Bartal, G., Fishman, S. & Segev, M. “Transport and Anderson localization in disordered two dimensional photonic lattices”. Nature 446, 52-55 (2007).C.M. Aegerter, M.Störzer, S.Fiebig, W. Bührer, and G. Maret : JOSA A, 24, #10, A23, (2007)MicrowaveDalichaouch, R., Armstrong, J.P., Schultz, S.,Platzman, P.M. & McCall, S.L. “Microwave localization by 2-dimensional random scattering”. Nature 354, 53, (1991).Chabanov, A.A., Stoytchev, M. & Genack, A.Z. Statistical signatures of photon localization. Nature 404, 850, (2000).Pradhan, P., Sridar, S, “Correlations due to localization in quantum eigenfunctions od disordered microwave cavities”, PRL 85, (2000)

SoundWeaver, R.L. Anderson localization of ultrasound. Wave Motion 12, 129-142 (1990).

Experiment

Localized StateAnderson Insulator

Extended StateAnderson Metal

f = 3.04 GHz f = 7.33 GHz

Billy et al. “Direct observation of Anderson localizationof matter waves in a controlled disorder”. Nature 453,891- 894 (2008).

Localization of cold atoms

87Rb

Roati et al. “Anderson localization of a non-interacting Bose-Einstein condensate“. Nature 453, 895-898 (2008).

Anderson Model

• Lattice - tight binding model

• Onsite energies �i - random

• Hopping matrix elements Iijj iIij

Iij ={-W < �i <Wuniformly distributed

I < Ic I > IcInsulator All eigenstates are localized

Localization length Metal

There appear states extendedall over the whole system

Anderson Transition

I i and j are nearest neighbors

0 otherwise

Q:Why arbitrary weak hopping I is not sufficient for the existence of the diffusion

?Einstein (1905): Marcovian (no memory)

process �� diffusion

j iIij

Quantum mechanics is not marcovian There is memory in quantum propagation!Why?

1�2�

I

���

����

��

2

1ˆ�

�I

IH

Hamiltonian

diagonalize ���

����

��

2

1

00ˆE

EH

! " 221212 IEE #��� ��

���

����

��

2

1ˆ�

�I

IH ! "

III

IEE�����

$#���12

1212221212 ��

������

What about the eigenfunctions ?

1 1 2 2 1 1 2 2, ; , , ; ,E E% � % � & &'

1,212

2,12,1

12

(��

(&

��

���

����

��

#�

���

IO

I

1,22,12,1

12

((&

��

)$

� I

Off-resonanceEigenfunctions are

close to the original on-site wave functions

ResonanceIn both eigenstates the probability is equally

shared between the sites

Anderson insulatorFew isolated resonances

Anderson metalThere are many resonances

and they overlap

Transition: Typically each site is in resonance with some other one

localized and extended never coexist!

DoS DoS

all states arelocalized

I < IcI > Ic

Anderson Transition

- mobility edges (one particle)

extended

Condition for Localization:

i j typW� �� � �

energy mismatch# of n.neighborsI<

energy mismatch

2d�# of nearest neighbors

A bit more precise:

Logarithm is due to the resonances, which are not nearest neighbors

Condition for Localization:

Is it correct?Q:A1:Is exact on the Cayley tree

,lncWI KK K

�is the branching number

2K �

Anderson Model on a Cayley tree

Condition for Localization:

Is it correct?Q:A1:Is exact on the Cayley tree

,lncWI KK K

�is the branching number

2K �

����Is a good approximation at high dimensions.Is qualitatively correct for 3d *

2. Add an infinitesimal Im part i++ to E�

1 2

4 1)2) 0N+

���limits

insulator

metal

1. take descrete spectrum E� of H0

3. Evaluate Im, �

Anderson’s recipe:

4. take limit but only after ��N5. “What we really need to know is the

probability distribution of Im,, not its average…” !

0�+

Probability Distribution of --=Im ,,

metal

insulator

Look for:

V

+ is an infinitesimal width (Impart of the self-energy due to a coupling with a bath) of one-electron eigenstates

Condition for Localization:

Is it correct?Q:A1:Is exact on the Cayley tree

,lncWI KK K

�is the branching number

2K �

A2:For low dimensions – NO. for All states are localized. Reason – loop trajectories

cI � � 1,2d �

����Is a good approximation at high dimensions.Is qualitatively correct for 3d *

Gertsenshtein & Vasil'ev, 1959

1D Localization

Exactly solved: all states are localized

Mott & Twose, 1961Conjectured:. . .

Condition for Localization:

Is it correct?Q:A2:For low dimensions – NO. for

All states are localized. Reason – loop trajectories cI � � 1,2d �

OPhase accumulated when traveling along the loop

The particle can go around the loop in two directions

Memory!

.� rdp ��( 21( (�

Due to the localization effects diffusion description fails at large scales.Quantum interference Memory

Einstein: there is no diffusion at too shortscales – there is memory, i.e., the process is not marcovian.

Large scales are important diffusion constant depends on the system size

O

21( (�

Conductance2dG L� ���

for a cubic sample of the size L

! "! "

2

2d

g L

e DhG Lh L

�����

�

Einstein relation for the conductivity �

2 dne Dd

� �

� �

! "2

1 dhD Lg LL

�Thouless energy

mean level spacing�

DimensionlessThouless conductance

Scaling theory of Localization(Abrahams, Anderson, Licciardello and Ramakrishnan

1979)

L = 2L = 4L = 8L ....

ET�/L-2 01/�/L-d/

without quantum corrections

ET ET ET ET

01 /01 /01 /01

g g g g

d log g! "d log L! "�1 g! "

g = Gh/e2g = ET / 02Dimensionless Thouless

conductance

d log g! "d log L! "�1 g! "

1 – function is

Universal, i.e., material independentButIt depends on the global symmetries, e.g., it is different with and without T-invariance (in orthogonal and unitary ensembles)Limits:

! " ! "2 11 2dg g L g d Og

1� � ��� � � � # � �

� �

! "1 log 0Lg g e g g 1� � $

0 2?? 2

0 2

ddd

� ��

1 - function ! "gLdgd 1�

loglog

1(g)

g

3D

2D

1D-1

1

1$cg

unstablefixed point

Metal – insulator transition in 3DAll states are localized for d=1,2

RG approach

Effective Field Theory of Localization –Nonlinear �� - model

1 - function ! "gLdgd 1�

loglog

1(g)

g

3D

2D

1D-1

1

1$cg

unstablefixed point

! " 2

2

22

log 2

d

dcl

c ddg L L

Lc dl

� �

3�

� �� � �� �� �

! " 2

? ?

d

d

cg d

cg

1 � � #

� )

!! "! "

12

td F

dv dtP tDt4

� 5�

5.� ! "maxg P tg0

$

2 2log logF Fg v L v Lg D D D l0

4� �

�� �$

1Fv 6 ��

g D � �

2 log Lgl

06

��} 2( )gg

16

� � Universal !!!

O

For d=1,2 all states are localized.

Phase accumulated when traveling along the loop

The particle can go around the loop in two directions

Memory!

.� rdp ��( 21( (�

Weak Localization:The localization length can be large

Inelastic processes lead to dephasing, which is characterized by the dephasing length

If , then only small corrections to a conventional metallic behavior

�

L(L(� ��

R.A. Chentsov “On the variation of electrical resistivity of tellurium in magnetic field at low temperatures”, Zh. Exp. Theor. Fiz. v.18, 375-385, (1948).

7

Magnetoresistance

NNo magnetic field ///(2/�/(8

With magnetic field H/////(2�/(8�/8986/7:7;

O O

Length Scales

Magnetoresistance measurements allow to study inelasticcollisions of electrons with phonons and other electrons

Magnetic length LH = (hc/eH)1/2

Dephasing length L( = (D 4()1/2

! " HdLg H fL(

0� �

� � �� �� �

Universalfunctions

Negative Magnetoresistance

Weak Localization

Aharonov-Bohm effectTheory B.A., Aronov & Spivak (1981)

Experiment Sharvin & Sharvin (1981)

Chentsov (1949)

Chemicalpotential

Temperature dependence of the conductivity one-electron picture

DoS DoSDoS

! " 00 ��T� ! " TE Fc

eT�

��

�� ! " TT <� 0�

Assume that all the states

are localized;e.g. d = 1,2 DoS

! " TT <� 0�

Temperature dependence of the conductivity one-electron picture

Inelastic processestransitions between localized states

=

1 energymismatch

(any mechanism)00 �>� �T

Phonon-assisted hopping

Any bath with a continuous spectrum of delocalized excitations down to ?/= 0 will give the same exponential

=

1

Variable Range HoppingN.F. Mott (1968)

Optimizedphase volume

Mechanism-dependentprefactor

1= ��? ���?�

! " 00 ��T�

SSpectral statistics and LLocalization

E== - spectrum (set of eigenvalues)- mean level spacing, determines the density of states- ensemble averaging

- spacing between nearest neighbors

- distribution function of nearest neighbors spacing between

Spectral Rigidity

Level repulsion

! "

! " 4211

00

,,��

��

11ssP

sP

! "sP1

1

0== EEs �

� #

==0 EE �� #11

......

RANDOM MATRIX THEORY

N @/N N �/�ensemble of Hermitian matrices with random matrix element

Spectral statistics

Orthogonal 11�2

Poisson – completely uncorrelated levels

Wigner-Dyson; GOEPoisson

GaussianOrthogonalEnsemble

Unitary1�8

Simplectic1�A

P(s)

( )P s s1�

Anderson Model

• Lattice - tight binding model

• Onsite energies �i - random

• Hopping matrix elements Iijj iIij

-W < �i <Wuniformly distributed

Q: What are the spectral statistics of a finite size Anderson model ?

Is there much in common between Random Matrices and Hamiltonians with random potential ?

I < Ic I > IcInsulator

All eigenstates are localizedLocalization length

MetalThere appear states extended

all over the whole system

Anderson Transition

The eigenstates, which are localized at different places

will not repel each other

Any two extended eigenstates repel each other

Poisson spectral statistics Wigner – Dyson spectral statistics

Strong disorder Weak disorder

I0Localized states

InsulatorExtended states

MetalPoisson spectral

statisticsWigner-Dyson

spectral statistics

Anderson Localization andSpectral Statistics

Ic

Extended states:

Level repulsion, anticrossings, Wigner-Dyson spectral statistics

Localized states: Poisson spectral statistics

Invariant (basis independent) definition

In general: Localization in the space of quantum numbers.KAM tori localized states.

GlossaryClassical Quantum

Integrable Integrable

KAM LocalizedErgodic – distributed all over the energy shellChaotic

Extended ?

! "IHH�

00 �IEH�

��B ����

� ,ˆ0

MMany--BBody LLocalization

BA, Gefen, Kamenev & Levitov, 1997Basko, Aleiner & BA, 2005. . .

01 1 1

ˆ ˆˆ ˆ ˆ ˆ ˆN N N

z z z x xi i ij i j i i

i i j i iH B J I H I� � � � �

� 3 � �

� # # � #B B B B

1,2,..., ; 1i

i N N�� ��

�

Perpendicular fieldRandom Ising model

in a parallel field

- Pauli matrices,

Example: Random Ising model in the perpendicular field

12

zi� � )

Will not discuss today in detail

Without perpendicular field all commute with the Hamiltonian, i.e. they are integrals of motion

zi�

01 1 1

ˆ ˆˆ ˆ ˆ ˆ ˆN N N

z z z x xi i ij i j i i

i i j i iH B J I H I� � � � �

� 3 � �

� # # � #B B B B

C D! "0 iH �onsite energy

ˆ ˆ ˆx� � �# �� #hoping between nearest neighbors

Anderson Model on N-dimensional cube

1,2,..., ; 1i

i N N�� ��

�

Perpendicular fieldRandom Ising model

in a parallel field

- Pauli matrices

C Dzi� determines a site

Without perpendicular field all commute with the Hamiltonian, i.e. they are integrals of motion

zi�

Anderson Model on N-dimensional cubeUsually:# of dimensions

system linear size

d const�

L��

Here:# of dimensions

system linear size

d N� ��

1L �

01 1 1

ˆ ˆˆ ˆ ˆ ˆ ˆN N N

z z z x xi i ij i j i i

i i j i iH B J I H I� � � � �

� 3 � �

� # # � #B B B B

6-dimensional cube 9-dimensional cube

Anderson Model on N-dimensional cube

01 1 1

ˆ ˆˆ ˆ ˆ ˆ ˆN N N

z z z x xi i ij i j i i

i i j i iH B J I H I� � � � �

� 3 � �

� # # � #B B B B

Localization: •No relaxation•No equipartition•No temperature•No thermodynamics

Glass ??