Anderson localization: from single particle to many body problems. Igor Aleiner (4 lectures) Windsor Summer School, 14-26 August 2012 ( Columbia University

Anderson localization:from single particle to many

body problems.

Igor Aleiner

(4 lectures)

Windsor Summer School, 14-26 August 2012

( Columbia University in the City of New York, USA )

Lecture # 1-2 Single particle localization

Lecture # 2-3 Many-body localization

I

V

Conductivity:

Conductance:

Insulator

Metal

Transport in solids

Superconductor

I

V

Conductivity:

Conductance:

Insulator

Metal

Transport in solids

Focus ofThe course

Lecture # 1• Metals and insulators – importance of disorder• Drude theory of metals• First glimpse into Anderson localization• Anderson metal-insulator transition (Bethe lattice

argument; order parameter … )

Band metals and insulators

Metals Insulators

Gapless spectrum Gapped spectrum

Metals

Gapless spectrum

Insulators

Gapped spectrum

But clean systems are in fact perfect conductors:

CurrentElectric field

Metals

Gapless spectrum

Insulators

Gapped spectrum

But clean systems are in fact perfect conductors:

(quasi-momentum is conserved, translational invariance)

Finite conductivity by impurity scatteringOne impurity

Incoming flux

Probability density

Scattering cross-section

Finite conductivity by impurity scatteringFinite impurity density

Elastic mean free path

Elastic relaxation time

Finite conductivity by impurity scatteringFinite impurity density

Drude conductivity

CLASSICAL

Quantum (band structure)

Quantum (single impurity)

Conductivity and DiffusionFinite impurity density

Einstein relation

Diffusion coefficient

Conductivity, Diffusion, Density of States (DoS)

Einstein relation

Density of States (DoS)


Clean systems


Clean systems

Metals,gapless

Insulators,gapped

Phase transition!!!

But only disorder makes conductivity finite!!!

Disordered systems

CleanDisorder included

Disordered

Disordered

Spectrum always gapless!!!

No phase transition???Only crossovers???

Lifshitz tail

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

DoS

extended

Anderson Transition

- mobility edges (one particle)

Coexistence of the localized and extended states is not possible!!!

Rules out first order phase transition

Temperature dependence of the conductivity (no interactions)

DoS DoSDoS

Metal Insulator “Perfect” one particleInsulatorNo singularities in any

thermodynamic properties!!!

To take home so far:• Conductivity is finite only due to broken

translational invariance (disorder)• Spectrum (averaged) in disordered system is

gapless• Metal-Insulator transition (Anderson) is

encoded into properties of the wave-functions

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:

One could think that diffusion occurs even for :

Golden rule:

Random walk on the lattice

Pronounce words:Self-consistencyMean-fieldSelf-averagingEffective medium …………..

?

is F A L S E

Probability for the level with given energy on NEIGHBORING sites

Probability for the level with given energy in the

whole system2d attempts

Infinite number of attempts

Perturbative Resonant pair

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

(


InsulatorMetal

(


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?

(

h0metal

insulator

behavior for agiven realization

metal

insulator

~ h

probability distributionfor a fixed energy


Probability Distribution

metal

insulator

Note:

Can not be crossover, thus, transition!!!

But the Anderson’s argument is not complete:

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

Summary of Lecture # 1• Conductivity is finite only due to broken

translational invariance (disorder)• Spectrum (averaged) in disordered system is

gapless (Lifshitz tail)• Metal-Insulator transition (Anderson) is encoded

into properties of the wave-functionsextended

localized

Metal

Insulator

• Distribution function of the local densities of states is the order parameter for Anderson transition

metalinsulator

Resonant pairI < I c

Perturbation theory in (I/W) is convergent!

I > I c

Perturbation theory in (I/W) is divergent!

To establish the metal insulator transition we have to show the convergence of (W/I) expansion!!!

Lecture # 2• Stability of metals and weak localization• Inelastic e-e interactions in metals• Phonon assisted hopping in insulators• Statement of many-body localization and many-

body metal insulator transition

Why does classical consideration of multiple scattering events work?

1

2

Classical Interference

Vanish after averaging

Back to Drude formulaFinite impurity density

Drude conductivity

CLASSICAL

Quantum (band structure)

Quantum (single impurity)

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

Sometimes you may see this…MISLEADING…

DOES NOT EXIST FOR GAUSSIAN DISORDER AT ALL

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

3D

2D

1D

Finite but singular

E. Abrahams, P. W. Anderson, D. C. Licciardello, and T.V. Ramakrishnan, (1979)

Thouless scaling + ansatz:

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)

Other way to analyze the stability of metal

metalinsulator

Metal ???

Explicit calculation yields:

Metal is unstable

To take home so far:

• Interference corrections due to closed loops are singular;

• For d=1,2 they diverges making the metalic phase of non-interacting particles unstable;• Finite size system is described as a good metal,

if , in other words• For , the properties are well described by

Anderson model with replacing lattice constant.

Regularization of the weak localization byinelastic scatterings (dephasing)

e-h pair

Does not interfere with

Regularization of the weak localization byinelastic scatterings (dephasing)

e-h pair

But interferes with

e-h pair

e-h pair e-h pair

Phase difference:

e-h pair e-h pair

Phase difference:

- length of the longest trajectory;

Inelastic rates with energy transfer

Electron-electron interactionAltshuler, Aronov, Khmelnitskii (1982)

Significantly exceeds cleanFermi-liquid result

Almost forward scattering:

diffusive

Ballistic

To take home so far:• Interference corrections due to closed loops are singular;• For d=1,2 they diverges making the metalic phase of non-interacting particles unstable;

• Interactions at finite T lead to finite

• System at finite temperature is described as a good metal, • if ,

in other words

• For , the properties are well described by ??????

Transport in deeply localized regime

Inelastic processes: transitions between localized states

(inelastic lifetime)–1

energymismatch

(any mechanism)

Phonon-induced hopping

energy difference can be matched by a phonon

Any bath with a continuous spectrum of delocalized excitations

down to w = 0 will give the same exponential

Variable Range HoppingSir N.F. Mott (1968)

Without Coulomb gapA.L.Efros, B.I.Shklovskii (1975)

Optimizedphase volume

Mechanism-dependentprefactor

“insulator”

Drude

“metal” Electron phononInteraction does not enter

⟶ 0 ?????

Q: Can we replace phonons with e-h pairs and obtain phonon-less VRH?

“insulator”

Drude

“metal” Electron phononInteraction does not enter

Metal-Insulator Transition and many-body Localization:

insulator

Drude

metal

[Basko, Aleiner, Altshuler (2005)]

Interaction strength(Perfect Ins)

and all one particle state are localized

Documents

Anderson localization: from single particle to many body problems. Igor Aleiner (4 lectures) Windsor Summer School, 14-26 August 2012 ( Columbia University