Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson...

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Anderson Localization Looking Forward Boris Altshuler Physics Department, Columbia University Collaborations: Igor Aleiner Also Denis Basko, Gora Shlyapnikov, Vincent Michal, Vladimir Kravtsov, Lecture1 September, 8, 2015

Transcript of Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson...

Page 1: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

Anderson Localization – Looking Forward

Boris AltshulerPhysics Department, Columbia University

Collaborations: Igor Aleiner

Also Denis Basko, Gora Shlyapnikov, Vincent Michal, Vladimir Kravtsov, …

Lecture1

September, 8, 2015

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Outline

1. Introduction

2. Anderson Model; Anderson Metal and Anderson Insulator

3. Phonon-assisted hoping conductivity

4. Localization beyond the real space. Integrability and chaos.

5. Spectral Statistics and Localization

6. Many-Body Localization.

7. Many-Body Localization of the interacting fermions.

8. Many-Body localization of weakly interacting bosons.

9. Many-Body Localization and Ergodicity

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>50 years of Anderson Localization

qp

Page 4: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

…very few believed it [localization] at the time, and even fewer saw its importance; among those who failed to fully understand it at first was certainly its author…

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Part 1.

Introduction and

History

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Diffusion Equation

2 0Dt

Diffusion

constant

,r tCan be density of particles or energy density.It can also be the probability to find a particle at a given point at a given time

The diffusion equation is valid for any random walk provided that there is no memory (markovian process)

Einstein theory of Brownian motion, 1905

Dtr 2

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Diffusion Equation

2 0Dt

Einstein-Sutherland Relation

William Sutherland

(1859-1911)

2 dne D

d

If electrons would be degenerate and form a classical ideal gas

for electric conductivity

T

ntot

Density of states

Page 8: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

Will a fluctuation (wave packet) spread ?

r

t

t

2 2r t r t

2r tD

?

Einstein (1905): Random walk without memory2 0D

t

always diffusion

Diffusion Constant

Anderson (1958): For quantum particles

not always

extended states2

tDr t

localized states2

tconstr

Page 9: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

Basic Quantum Mechanics:

Spectra

-spectrum

ContinuousUnbound states

Extended states

DiscreteBound states

Localized states

2 d

Lr O L

2 r

rr O e

L

d

System size

Number of the spatial dimensions

Potential well

Page 10: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

Localization of single-particle wave-functions.

Continuous limit:

extended

localized

Random potential

Page 11: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

Spin DiffusionFeher, G., Phys. Rev. 114, 1219 (1959); Feher, G. & Gere, E. A., Phys. Rev. 114, 1245 (1959).

MicrowaveDalichaouch, R., Armstrong, J.P., Schultz, S.,Platzman, P.M. & McCall, S.L. “Microwave

localization by 2-dimensional random scattering”. Nature 354, 53-55, (1991).

Chabanov, A.A., Stoytchev, M. & Genack, A.Z. Statistical signatures of photon

localization. Nature 404, 850-853 (2000).

Pradhan, P., Sridar, S, “Correlations due to localization in quantum

eigenfunctions od disordered microwave cavities”, PRL 85, (2000)

Experiment

Localized Extended

f = 3.04 GHz f = 7.33 GHz

Page 12: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

Weaver, R.L. “Anderson localization of ultrasound”.

Wave Motion 12, 129-142 (1990).

H. Hu, A. Strybulevych, J. H. Page, S. E. Skipetrov & B. A. van

Tiggelen “Localization of ultrasound in a three-dimensional elastic

network” Nature Phys. 4, 945 (2008).

Experiment

Localization of Ultrasound

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D. Wiersma, 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).

L.Sapienza, H.Thyrrestrup, S.Stobbe, P. D.Garcia, S.Smolka, P.Lodahl “Cavity Quantum Electrodynamics

with Anderson localized Modes” Science 327, 1352-1355, (2010)

Localization of Light

Page 14: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

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).

Page 15: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

What about charge transport ?

Problem: electrons interact with each other

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2 0Dt

r

t

t

IV

2 dne D

d

Density of statesConductivity

Einstein Relation (1905)

Diffusion Constant

G =I

V

æ

èçö

ø÷V=0

s =GL

A

Extended states - metal

G µ Ld-2;

sL®¥

¾ ®¾¾ const; DL®¥

¾ ®¾¾ const

Localized states - insulator: Gµe-L/x; s = 0; D = 0

At zero temperature

Conductivity

Conductance

Page 17: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

Part 2.

Anderson Model;

Anderson Metal

and

Anderson Insulator

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Anderson

Model

• Lattice - tight binding model

• Onsite energies ei - random

• Hopping matrix elements Iijj i

Iij

Iij ={-W < ei<W

uniformly distributed

I < Ic I > IcInsulator

All eigenstates are localized

Localization length

MetalThere appear states extended

all over the whole system

Anderson Transition

I i and j are nearest neighbors

0 otherwise

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1

ˆ0

0 N

I

H I I

I

e

e

One-dimensional Anderson Model

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

?Einstein (1905): Marcovian (no memory)

process g diffusion

j i

Iij

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

Page 21: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

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

Memory!

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

O

WEAK LOCALIZATION

Constructive interference probability of the return to the origin gets enhanced quantum corrections reduce the diffusion constant.

Tendency towards localization

Phase accumulated when traveling along the loop

The particle can go around the loop in two directions

Memory!pdr

Page 23: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

Localization of single-particle wave-functions.

Continuous limit:

extended

localized

d=1; All states are localized

d=2; All states are localized

d >2; Anderson transition

Random potential

Page 24: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

1e2e

I

2

1ˆe

e

I

IH

Hamiltonian

diagonalize

2

1

0

E

EH

22

1212 IEE ee

Two-site Anderson model

=Two-well potential

Page 25: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

2

1ˆe

e

I

IH diagonalize

2

1

0

E

EH

II

IIEE

12

121222

1212ee

eeeeee

von Neumann & Wigner “noncrossing rule”

Level repulsion

v. Neumann J. & Wigner E. 1929 Phys. Zeit. v.30, p.467

Page 26: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

In general, a multiple spectrum in typical families of quadratic forms is observed only for two or more parameters, while in one-parameter families of general form the spectrum is simple for all values of the parameter. Under a change of parameter in the typical one-parameter family the eigenvalues can approach closely, but when they are sufficiently close, it is as if they begin to repel one another. The eigenvalues again diverge, disappointing the person who hoped, by changing the parameter to achieve a multiple spectrum.

Arnold V.I., Mathematical Methods of Classical Mechanics (Springer-Verlag: New York), Appendix 10, 1989

E

x

H x E x

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2

1ˆe

e

I

IH diagonalize

2

1

0

E

EH

II

IIEE

12

121222

1212ee

eeeeee

von Neumann & Wigner “noncrossing rule”

Level repulsion

v. Neumann J. & Wigner E. 1929 Phys. Zeit. v.30, p.467

What about the eigenfunctions ?

Page 28: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

2

1ˆe

e

I

IH

II

IIEE

12

121222

1212ee

eeeeee

What about the eigenfunctions ?

1 1 2 2 1 1 2 2, ; , , ; ,E E e e

2 1

1,2 1,2 2,1

2 1

I

IO

e e

e e

2 1

1,2 1,2 2,1

Ie e

Off-resonanceEigenfunctions are

close to the original on-site wave functions

ResonanceIn both bonding and anti-bonding

eigenstates the probability is equally shared between the sites

Page 29: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

Anderson insulatorFew isolated resonances

Anderson metalMany resonances and they overlap

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

Page 30: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

Anderson

Model

• Lattice - tight binding model

• Onsite energies ei - random

• Hopping matrix elements Iijj i

Iij

Iij ={-W < ei<W

uniformly distributed

I < Ic I > IcInsulator

All eigenstates are localized

Localization length

MetalThere appear states extended

all over the whole system

Anderson Transition

I i and j are nearest neighbors

0 otherwise

Page 31: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

Condition for Localization:

i j typWe e

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

Page 32: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

Condition for Localization:

Is it correct?Q:

A1:For low dimensions – NO. for All states are localized. Reason – loop trajectories

cI 1,2d

A2:Works better for larger dimensions 2d

A3:Is exact on the Bethe lattice

Page 33: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

Condition for Localization:

Is it correct?Q:

A1:For low dimensions – NO. for All states are localized. Reason – loop trajectories

cI 1,2d

A2:Works better for larger dimensions 2d

A3:Is exact on the Bethe lattice

Rule of the thumb

At the localization transition a typical site is in resonance with another one

At the localization transition the hoping matrix element is of the order of the typical energy mismatch divided by the number of nearest neighbors

Page 34: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

Anderson’s recipe:

Consider an open system (Anderson Model).Particle escape “broadening” of each eigenstate

; ImE E i

States localized inside the system – small Extended states – large

- scape rate (inverse dwell time)

0 0I

Page 35: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

12

4 1)

2) 0

N

limits

insulator

metal

1. take descrete spectrum E of H0

2. Add an infinitesimal Im part i to E

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

Page 36: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

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

Page 37: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

localized and extended never coexist!

DoS DoS

all states are

localized

I < IcI > Ic

Anderson Transition

- mobility edges

extended

Page 38: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

Chemicalpotential

Temperature dependence of the conductivity

one-electron picture

DoS DoSDoS

00 T T

E Fc

eT

e

TT 0

cI I

there are extended states

Mobility edge

cI I

all states are localized

Page 39: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

Part 3.

Localization beyond real space

Integrability and chaos

Page 40: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

Localization in the angular momentum space

Page 41: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

Quantum and Classical Dynamical Systems

Conventional Boltzmann-Gibbs Statistical PhysicsEquipartition Postulate

Ergodicity: time average = space (ensemble) average

Chaos

Hamiltonian

Integrable Systemsdegrees of freedomintegrals of motion

Ergodicity is violated

Invariant tori dimension

Hamiltonian

Energy shell, dimension

? ,i iH p q

Large number of the degrees of freedom

0 ,i iH p q0H H V

dd

d

2 1d

0KAM regionArnold diffusion

Non-ergodic

Ergodicity Equipartition Fermi, Pasta, Ulam system

(connected nonlinear oscillators)

Solar system

. . .

Classical Dynamics

Quantum Dynamics ???

1d

Page 42: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

Andrey

Kolmogorov

Vladimir

ArnoldJurgen

Moser

Kolmogorov – Arnold – Moser (KAM) theory

Integrable classical Hamiltonian

d >1:

Separation of variables: d sets of action-angle variables

Quasiperiodic motion: set of the frequencies, , which are in general incommensurate. Actions are

integrals of motion

1 2, ,.., d

1

1I2

2I …=>

;..2,;..,2, 222111 tItI

iI

0 tI i

A.N. Kolmogorov, Dokl. Akad. Nauk SSSR, 1954. Proc. 1954 Int. Congress of Mathematics, North-Holland, 1957

0H

tori

Page 43: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

Kolmogorov – Arnold – Moser (KAM) theory

1

1I2

2I …=>

Q:Will an arbitrary weak perturbation of an integrable Hamiltonian destroy the tori and make the motion ergodic (when each point at the energy shell will be reached sooner or later) ?

A: Most of the tori survive weak and smooth enough perturbations

V0H

KAM theorem

A.N. Kolmogorov, Dokl. Akad. Nauk SSSR, 1954. Proc. 1954 Int. Congress of Mathematics, North-Holland, 1957

Given the set of the integrals of motion all trajectories belong to a torus

I

Page 44: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

Classical, d>>1 degrees of freedom

I1, I2, . . ., Id integrals of motion

Quantum, d>>1 degrees of freedomIntegrals of motion are quantized – quantum numbers

Form a “lattice”.

Sites of the “lattice” –eigenstate of the integrable system

I2

I1

0

I2

I1

0

Classical Dynamics: from KAM to Chaos

Quantum Dynamics: Many – Body Localization

Space of the integrals of motion

0H H V

Energy Shell

Page 45: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

1I

2I 0

Ly

Lx

Two integrals of motion

1 1;x yI p I p

Rectangular billiard

;yx

x y

x y

nnp p

L L

Energy shell:

2 2 2x yp p mE

Page 46: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

Classical, d>>1 degrees of freedom

I1, I2, . . ., Id integrals of motion

Quantum, d>>1 degrees of freedomIntegrals of motion are quantized – quantum numbers

Form a “lattice”.

Sites of the “lattice” –eigenstate of the integrable system

Perturbation – coupling of the different sites of the “lattice” - bonds

I2

I1

0

I2

I1

KAM

Most of the tori survive weak and

smooth perturbation

Energy Shell

0

Classical Dynamics: from KAM to Chaos

Quantum Dynamics: Many – Body Localization

Space of the integrals of motion

0H H V

Page 47: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

Classical, d>>1 degrees of freedom

I1, I2, . . ., Id integrals of motion

Quantum, d>>1 degrees of freedom

Integrals of motion are quantized –quantum numbers

Form a “lattice”.

Sites of the “lattice” –eigenstate of the integrable system

Perturbation – coupling of the different sites of the “lattice” -bonds

Wave functions localized in the space of quantum numbers

I2

I1

Finite motion

0

I2

I1

KAM

Most of the tori survive weak and

smooth perturbation

Energy Shell

0

Classical Dynamics: from KAM to Chaos

Quantum Dynamics: Many – Body Localization

Space of the integrals of motion

0H H V

Page 48: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

Classical, d>>1 degrees of freedom

I1, I2, . . ., Id integrals of motion

Quantum, d>>1 degrees of freedomIntegrals of motion are quantized – quantum numbers

Form a “lattice”.

Sites of the “lattice” –eigenstate of the integrable system

Perturbation – coupling of the different sites of the “lattice” - bonds

Quantum random walk

Localization?

I2

I1

Finite motion

0

I2

I1

KAM

Most of the tori survive weak and

smooth perturbation

Energy Shell

0

Classical Dynamics: from KAM to Chaos

Quantum Dynamics: Many – Body Localization

Space of the integrals of motion

0H H V

Many – Body Localization is an analog ofthe Anderson Localization in a finite-dimensional space of a quantum particlesubject to a random potential

Page 49: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

1I

2I

0ˆ V

1I

2I

Most of the tori survive weak and smooth enough perturbations

KAM theorem:

1I

2I 0

Energy shell2 2

2

x yp pE

m

0The integrals of motion are quantized

dIII ,...,1

0

I

0c

Page 50: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

,V

dIII ,...,1

Matrix element of the perturbation

0

I

AL hops are local – one can distinguish “near” and “far”

KAM perturbation is smooth enough

Anderson Model !

Page 51: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

Square

billiard

Localized momentum space extended

Pradhan

& Sridar,

PRL, 2000

Localized real space

Disordered

extended

Disordered

localized

Sinai

billiard

Page 52: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

Glossary

Classical Quantum

Integrable Integrable

Perturbation Perturbation

KAM Localized

Ergodic (chaotic) Extended ?

0 0 ; 0H H I I t IEH

,ˆ0

; 0V I t ,

,

V V

Page 53: Anderson Localization Looking Forward1).pdf · Billy et al. “Direct observation of Anderson localization of matter waves in a controlled disorder”. Nature 453, 891- 894 (2008).

Question:

What is the reason to speak about localization if we in general do not know the space in which the system is localized ?

Need an invariant (basis independent) criterion of the localization