Disorder and chaos in quantum systems II. Lecture 2. Boris Altshuler Physics Department, Columbia University

Disorder and chaos in Disorder and chaos in quantum systems II.quantum systems II.

Lecture 2.

Boris Altshuler Physics Department, Columbia

University

Previous Lecture:Previous Lecture:1. Anderson Localization as Metal-Insulator Transition

Anderson model. Localized and extended states. Mobility edges.

2. Spectral Statistics and Localization. Poisson versus Wigner-Dyson. Anderson transition as a transition between different types of spectra. Thouless conductance

Conductance g s

P(s)

Lecture2.Lecture2.1. 1. Quantum Quantum Chaos, Chaos, Integrability Integrability and and

LocalizationLocalization

sP(s)

Particular nucleus

166Er

Spectra of several nuclei combined (after spacing)rescaling by the mean level

P(s)

N. Bohr, Nature 137 (1936) 344.

Q: ?Why the random matrix theory (RMT) works so well for nuclear spectra

Original answer:

These are systems with a large number of degrees of freedom, and therefore the “complexity” is high

Later itbecameclear that

there exist very “simple” systems with as many as

2 degrees of freedom (d=2), which

demonstrate RMT - like spectral statistics

Integrable Systems

Classical ( =0) Dynamical Systems withd degrees of freedom

The variables can be separated and the problem reduces to d one-dimensional problems

d integrals of motion

ExamplesExamples1. A ball inside rectangular billiard; d=2• Vertical motion can be

separated from the horizontal one

• Vertical and horizontal components of the

momentum, are both integrals of motion

2. Circular billiard; d=2• Radial motion can be separated from the angular one

• Angular momentum and energy are the integrals of motion

Integrable Systems

Classical Dynamical Systems with d degrees of freedom

Rectangular and circular billiard, Kepler problem, . . . , 1d Hubbard model and other exactly solvable models, . .

The variables can be separated d one-dimensional problems d integrals of motion

Chaotic Systems

The variables can not be separated there is only one integral of motion - energy

ExamplesExamples

Sinai billiard

Kepler problem in magnetic field

B

Stadium

Classical Chaos =0

•Nonlinearities•Exponential dependence on the original conditions (Lyapunov

exponents)

•Ergodicity

Q: What does it mean Quantum Chaos ?

Quantum description of any Quantum description of any System with a finite number of System with a finite number of the degrees of freedom is a the degrees of freedom is a linear problem – Shrodinger linear problem – Shrodinger equation equation

Bohigas – Giannoni – Schmit conjecture

Chaotic classical analog

Wigner- Dyson spectral statistics

0

No quantum numbers except

energy

Wigner-Dyson

?

Classical

Poisson

Quantum

?Chaotic

Integrable

Lecture1.Lecture1.

2. 2. Localization Localization beyond real beyond real spacespace

Andrey Kolmogorov

Vladimir Arnold

JurgenMoser

Kolmogorov – Arnold – Moser (KAM) theoryKolmogorov – Arnold – Moser (KAM) theory

Integrable classical Hamiltonian , d>1:

Separation of variables: d sets of action-angle variables

1 1 1 2 2 2, 2 ; ... , , 2 ;..I t I t

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

0H

0

1D classical motion – action-angle variables

Andrey Kolmogorov

Vladimir Arnold

JurgenMoser

Kolmogorov – Arnold – Moser (KAM) theoryKolmogorov – 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

11I

22I …=>

1 1 1 2 2 2, 2 ; ... , , 2 ;..I t I t

iI0 tI i


0H

0

tori

space Number of dimensions

real space d

phase space: (x,p) 2denergy shell 2d-1

tori d

For d>1 each torus has measure zero on the energy shell !

Integrable dynamics:Integrable dynamics:Each classical trajectory is quasiperiodic and confined to a particular torus, which is determined by a set of the integrals of motion

Andrey Kolmogorov

Vladimir Arnold

JurgenMoser

Kolmogorov – Arnold – Moser (KAM) theoryKolmogorov – 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

11I

22I …=>

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

iI 0 tI i

Q:Will an arbitrary weak perturbation of the 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

V 0H

KAM theorem


0H

1I

2I

Each point in the space of the integrals of motion corresponds to a torus and vice versa

0ˆ V

1I

2I

Finite motion.Localization in the space of the integrals of motion

?


KAM theorem:

1I

2I

0ˆ V

1I

2I


KAM theorem:

1I

2I 0

Energy shell

Consider an integrable system. Each state is characterized by a set of quantum numbers.

It can be viewed as a point in the space of quantum numbers. The whole set of the states forms a lattice in this space.

A perturbation that violates the integrability provides matrix elements of the hopping between different sites (Anderson model !?)

Weak enough hopping: Localization - Poisson

Strong hopping: transition to Wigner-Dyson

Squarebilliard

Localized momentum

spaceextended

Localized real space

Disordered extended

Disordered localized

Sinaibilliard

Glossary

Classical Quantum

Integrable Integrable

KAM Localized

Ergodic – distributed all over the energy shell Chaotic

Extended ?

IHH

00 IEH

,ˆ0

Extended states:

Level repulsion, anticrossings, Wigner-Dyson spectral statistics

Localized states: Poisson spectral statistics

Invariant (basis independent)

definition

Squarebilliard

Sinaibilliard

Disordered localized

Disordered extended

Integrable ChaoticAll chaotic systems resemble each other.

All integrable systems are integrable in their own way

Q:

Consider a finite system of quantum particles, e.g., fermions. Let the one-particle spectra be chaotic (Wigner-Dyson).

What is the statistics of the many-body spectra? ?a.The particles do not interact with each other.Poisson: individual energies are conserving quantum numbers. b. The particles do interact. ????

Lecture Lecture 2. 2. 3. 3. Many-Body Many-Body excitation in excitation in finite systemsfinite systems

Fermi Sea

Decay of a quasiquasiparticleparticle with an

energy in Landau Fermi liquid

QuasiparticleQuasiparticle decay rate at T = 0 in a clean clean Fermi Liquid.

Fermi Sea

322

constant

coupling

dFee

Reasons:Reasons:• At small the energy transfer, , is small and the integration

over and gives the factor . …………………………………………………………………•The momentum transfer, q , is large and thus the scattering

probability at given and does not depend on , or

I. I. d=3d=3

II. II. Low dimensionsLow dimensions

Small moments transfer, q , become important at low dimensions because the scattering probability is proportional to the squared time of the interaction,

(qvF. )-2

evF

1/q

1

2log

3

2

2

d

d

d

FF

F

ee


Quasiparticle decay rate at T = 0 in a cleanclean Fermi Liquid.

1

2log

32

2

d

d

d

FF

F

ee

Conclusions: Conclusions:

1. For d=3,2 from it follows that , i.e., that the qusiparticles are well determined and the Fermi-liquid approach is applicable.

2. For d=1 is of the order of , i.e., that the Fermi-liquid approach is not valid for 1d systems of interacting fermions.

Luttinger liquids

Fermi Sea


F ee

ee

Fermi Sea

Decay of a quasiparticle with an


Quantum dot – zero-dimensional case ?

Fermi Sea

Decay of a quasiparticle with an


Quantum dot – zero-dimensional case ?

2

1TE

( U.Sivan, Y.Imry & A.Aronov,1994 )Fermi Golden rule:

Mean level spacing

Thouless energy

Decay rate of a quasiparticle with energy

2

1TE

( U.Sivan, Y.Imry & A.Aronov,1994 )Fermi Golden rule:

Mean level spacing

Thouless energy

Decay rate of a quasiparticle with energy in 0d.

Recall: 1

TE g

Zero dimensional system

Thouless conductanc

e

Def:

One particle states are extended all over the system

11 gET

Fermi Sea

zero-dimensional case

one-particle spectrum is discrete

equation

’’can not be satisfied exactly

Recall: in the Anderson model the site-to-site hopping does not conserve the energy

Decay rate of a quasiparticle with energy in 0d.Problem:

’

’+

Offdiagonal matrix element

11,, g

M

Decay rate of a quasiparticle with energy in 0d.

Fermi Sea

generations

1 2 3 4 5 6

. . . .

Delocalization in Fock space

’

’

Can be mapped (approximately) to the problem of localization on Cayley tree

Chaos in Nuclei – Delocalization?

1 2 3 4 5

Conventional Anderson Model

Basis: ,i i

i

i iiH 0ˆ

..,

ˆnnji

jiIV

Hamiltonian: 0ˆ ˆH H V

•one particle,•one level per site, •onsite disorder•nearest neighbor hoping

labels sites

many (N ) particles no interaction:Individual energies and thus occupation numbers are conserved

N conservation laws“integrable system”

EH

nE

n

n

0d system; no interactions

integrable system

0H E

BasisBasis: 0,1n

HamiltonianHamiltonian:0

ˆ ˆH H V

n occupatio

n numbers

labels levels

,

V I

.., 1,.., 1,.., 1,.., 1,..n n n n

0d system with interactions

Conventional Conventional Anderson Anderson

ModelModel

Many body Anderson-Many body Anderson-like Modellike Model

Basis:Basis: ilabels sites

, . .

ˆi

i

i j n n

H i i

I i j

,

H E

I

.., 1,.., 1,.., 1,.., 1,..n n n n

BasisBasis: ,0,1n

n occupatio

n numbers

labels levels

i

“nearest neighbors”:

Isolated quantum dot – 0d system of fermions

Exact many-body states:Ground state, excited states

Exact means that the imaginary part of the energy is zero!

Quasiparticle excitations:Finite decay rate

Q:What is the connection?

source drain

gate

QDQD

current

S

D

No e-e interactions – resonance tunneling

source drain

gate

QDQD

current

S

D

VSD

gNo e-e interactions – resonance tunneling

Mean level spacing

source drain

gate

QDQD

current

S

D

VSD

gNo e-e interactions – resonance tunneling

The interaction leads to additional peaks – many body excitations

D

S S

D

Resonance tunnelingPeaks

Inelastic cotunnelingAdditional peak

source drain

gate

QDQD

current

S

S

VSD

g The interaction leads to additional peaks – many body excitations

source drain

gate

QDQD

current

S

D

VSD

g

loc

Ergodic - WDNE

Landau quasiparticle with

the widthSIA

VSDloc

Ergodic - WDNE

Landau quasiparticle with

the widthSIA

Localized - finite # of the satelites

Extended - infinite # of the satelites

(for finite the number of the satelites is always finite)

Ergodic – nonergodic crossover!

extended

Anderson Model on a Cayley tree

Anderson Model on a Cayley tree

I, W K – branching number

1

lnc

WI

K K

WI

K Resonance at

every generation

ln

W WI

K K K Sparse

resonances

Definition: We will call a quantum state ergodic if it occupies the number of sites on the Anderson lattice, which is proportional to the total number of sites :

N

N

0 NN

N

nonergodic

0 constN

NN

ergodic

Localized states are obviously not ergodic: constN

N

Is each of the extended state ergodicQ:Q: ??A:A: In 3D probably yes

For d>4 most likely no

Example of nonergodicity: Anderson ModelAnderson Model Cayley treeCayley tree:

nonergodic states

Such a state occupies infinitely many sites of the Anderson

model but still negligible fraction of the total number of sites

Nlnn

– branching number

KK

WIc ln

K

ergodicity

WIerg ~

transition

crossover

KWIW Typically there is a resonance at every step

WI Typically each pair of nearest neighbors is at resonance

Nn ln~

Nn ~

nonergodic

ergodic

lnW K I W K K Typically there is no resonance at the next step

Nn ln~ nonergodic

lnI W K KResonance is typically far n const localized

Documents

Disorder and chaos in quantum systems II. Lecture 2. Boris Altshuler Physics Department, Columbia University