Disorder and chaos in quantum system:

Anderson localization and its generalization

Boris Altshuler (Columbia)Igor Aleiner (Columbia)

(6 lectures)

Lecture # 2• Stability of insulators and Anderson transition• Stability of metals and weak localization

Anderson localization (1957)

extended

localized

Only phase transition possible!!!

Anderson localization (1957)

extended

localized

Strong disorder

Anderson insulator

Weaker disorder

Localized

Localized

Localized

Extended

Extended

d=3

Any disorder, d=1,2

d=3

Anderson Model

• Lattice - tight binding model

• Onsite energies ei - random

• Hopping matrix elements Iij j iIij

-W < ei <W uniformly distributed

Iij =I i and j are nearest neighbors

0 otherwise{ Critical hopping:

Resonant pair

Bethe lattice:

INFINITE RESONANT PATH ALWAYS EXISTS

Resonant pair

Bethe lattice:

INFINITE RESONANT PATH ALWAYS EXISTS

Decoupled resonant pairs

Long hops?

Resonant tunneling requires:

“All states are localized “

means

Probability to find an extended state:

System size

Order parameter for Anderson transition?Idea for one particle localization Anderson, (1958);MIT for Bethe lattice: Abou-Chakra, Anderson, Thouless (1973);Critical behavior: Efetov (1987)

Metal Insulator

Order parameter for Anderson transition?Idea for one particle localization Anderson, (1958);MIT for Bethe lattice: Abou-Chakra, Anderson, Thouless (1973);Critical behavior: Efetov (1987)

InsulatorMetal

Order parameter for Anderson transition?Idea for one particle localization Anderson, (1958);MIT for Bethe lattice: Abou-Chakra, Anderson, Thouless (1973);Critical behavior: Efetov (1987)

InsulatorMetal

Metal Insulator

Idea for one particle localization Anderson, (1958);MIT for Bethe lattice: Abou-Chakra, Anderson, Thouless (1973);Critical behavior: Efetov (1987)

Order parameter for Anderson transition?

h!0metal

insulator

behavior for agiven realization

metal

insulator

~ h

probability distributionfor a fixed energy

Order parameter for Anderson transition?Idea for one particle localization Anderson, (1958);MIT for Bethe lattice: Abou-Chakra, Anderson, Thouless (1973);Critical behavior: Efetov (1987)

Probability Distribution

metal

insulator

Note:

Can not be crossover, thus, transition!!!

On the real lattice, there are multiple pathsconnecting two points:

Amplitude associated with the pathsinterfere with each other:

To complete proof of metal insulator transition one has to show the stability of the metal

Back to Drude formulaFinite impurity density

Drude conductivity

CLASSICAL

Quantum (band structure)

Quantum (single impurity)

Why does classical consideration of multiple scattering events work?

1

2

Classical Interference

Vanish after averaging

Look for interference contributions that survive the averaging

1

2

12

unitarity

Correction toscattering crossection

Phase coherence

Additional impurities do not break coherence!!!

1

2

12

unitarity

Correction toscattering crossection

Sum over all possible returning trajectories

unitarity1

2

12

Return probability forclassical random

work

Quantum corrections (weak localization)(Gorkov, Larkin, Khmelnitskii, 1979)

3D

2D

1D

Finite but singular

2D

1D

Metals are NOT stable in one- and two dimensions

Localization length:

Drude + corrections

Anderson model,

Exact solutions for one-dimensionx U(x)

Nch

Gertsenshtein, Vasil’ev (1959)

Nch =1

Exact solutions for one-dimensionx U(x)

NchEfetov, Larkin (1983)Dorokhov (1983) Nch >>1

Strong localizationWeak localization

Universal conductancefluctuations

Altshuler (1985); Stone; Lee, Stone

(1985)

We learned today:• How to investigate stability of insulators (locator

expansion).• How to investigate stability of metals (quantum

corrections)• For d=3 stability of both phases implies metal

insulator transition; The order parameter for the transition is the distribution function

• For d=1,2 metal is unstable and all states are localized

Next time:

• Inelastic transport in insulators