Anderson localization: from single particle to many body problems

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

Summary of Lectures # 1,2• 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

• Interference corrections due to closed loops are singular; For d=1,2 they diverges making the metalic phase of non-interacting particles unstable;• Finite T alone does not lift localization;

metalinsulator

• 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 ??????

Lecture # 3• Inelastic transport in deep insulating regime

• Statement of many-body localization and many-body metal insulator transition

• Definition of the many-body localized state and the many-body mobility threshold

• Many-body localization for fermions (stability of many-body insulator and metal)

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 ????? (All one-particle states are localized)

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

“insulator”

Drude

“metal” Electron phononInteraction does not enter


A#1: Sure

3) Use the electric noise instead of phonons.

1) Recall phonon-less AC conductivity:Sir N.F. Mott (1970)

2) Calculate the Nyquist noise (fluctuation dissipation Theorem).

4) Do self-consistency (whatever it means).

Easy steps: Person from the street (2005)(2005-2011)


A#1: Sure [Person from the street (2005)]

A#2: No way (for Coulomb interaction in 3D – may be)

[L. Fleishman. P.W. Anderson (1980)]

is contributed by rare resonances

d

g

R

Thus, the matrix element vanishes !!!

0 *

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

Many-body mobility threshold

insulator

metal

[Basko, Aleiner, Altshuler (2005)]

Many body DoS

All STATES LOCALIZED

All STATES EXTENDED -many-bodymobility threshold

“All states are localized “

meansProbability to find an extended state:

System volume

Many body localization means any excitation is localized:

Extended

Localized

Entropy

Many body DoS

All STATES LOCALIZED

All STATES EXTENDED

States always thermalized!!!

States never thermalized!!!

Many body DoS

Is it similar to Anderson transition?

One-body DoS

Why no activation?

0 Physics: Many-body excitations turn out to be localized in the Fock space

Fock space localization in quantum dots (AGKL, 1997)

- one-particle level spacing;

e

´ ´

´

ee

e

e

´

´ ´

No spatial structure ( “0-dimensional” )


1-particle excitation



Cayley tree mapping






1. Coupling between states:

2. Maximal energy mismatch:

3. Connectivity:

insulator

metal

Vs. finite T Metal-Insulator Transition in the bulk systems[Basko, Aleiner, Altshuler (2005)]

Interaction strength

Metal-Insulator “Transition” in zero dimensions[Altshuler, Gefen, Kamenev,Levitov (1997)]

In the paper:




1-particle level spacing in localization volume;

1) Localization in Fock space = Localization in the coordinate space.

2) Interaction is local;





1,2) Locality:

3) Interaction matrix elements strongly depend on the energy transfer, w:

Matrix elements:

????

In the metallic regime:

Matrix elements:

????

In the metallic regime:

2

Matrix elements:

????

2





1,2) Locality:

3) Interaction matrix elements strongly depend on the energy transfer, w:

Effective Hamiltonian for MIT.We would like to describe the low-temperatureregime only.

Spatial scales of interest >>

1-particle localization length

Otherwise, conventional perturbation theory fordisordered metals works. Altshuler, Aronov, Lee (1979); Finkelshtein (1983) – T-dependent SC potential Altshuler, Aronov, Khmelnitskii (1982) – inelastic processes

, jj

O

d

1

No spins

a

Reproduces correct behavior of the tails of one particle wavefunctions

j1

j2

l1

l2

Interaction only within the same cell;

Statistics of matrix elements?

Parameters:jl

randomsigns

What to calculate?

– random quantity

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

No interaction:

Metal Insulator

Probability Distribution

metal

insulator

Note:

Look for:

Iterations:

Cayley tree structure

after standard simple tricks:

+ kinetic equation for occupation function

Nonlinear integral equation with random coefficients

Decay due to tunneling

Decay due to e-h pair creation

Stability of metallic phase

Assume is Gaussian:

>>( )2

Probability Distributions“Non-ergodic” metal

Drude metal

Kinetic Coefficients in Metallic Phase

Kinetic Coefficients in Metallic Phase Wiedemann-Frantz law ?

Non-ergodic+Drude metal

So far, we have learned:

Trouble !!!

Insulator

???

Nonlinear integral equation with random coefficients

Stability of the insulator

Notice: for is a solution

Linearization:

metal

Recall:

insulator

h

probability distributionfor a fixed energy

# of interactions # of hops in space

unstable

STABLE

So, we have just learned:

Insulator

Metal

Non-ergodic+Drude metal

Estimate for the transition temperature for general case

Energy mismatch

Interaction matrix element

# of possible decay processes of an excitationsallowed by interaction Hamiltonian;

2) Identify elementary (one particle) excitations and prove that they are localized.

3) Consider a one particle excitation at finite T and the possible paths of its decays:

1) Start with T=0;

Summary of Lecture # 3:• Existence of the many-body mobility threshold

is established.• The many body metal-insulator transition is not

a thermodynamic phase transition.• It is associated with the vanishing of the

Langevine forces rather the divergences in energy landscape (like in classical glass)

• Only phase transition possible in one dimension (for local Hamiltonians)

Detailed paper: Basko, I.A.,Altshuler, Annals of Physics 321 (2006) 1126-1205Shorter version: …., cond-mat/0602510; chapter in “Problems of CMP”

I.A, Altshuler, Shlyapnikov, NATURE PHYSICS 6 (2010) 900-904

Lecture #4.

Many body localization and phase diagram of weakly interacting 1D bosons

Outline:

• Remind: Many body localization and estimate for the transition temperature;

• Remind: Single particle localization in 1D;• Remind: “Superconductor”-insulator transition at

T=0; • Many-body metal-insulator transiton at finite T;

1. Localization of single-electron wave-functions:

extended

localized

d=1; All states are localized

M.E. Gertsenshtein, V.B. Vasil’ev, (1959)Exact solution for one channel:

“Conjecture” for one channel:Sir N.F. Mott and W.D. Twose (1961)

Exact solution for ( )s w for one channel:V.L. Berezinskii, (1973)

Many-body localization;

– random quantity

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

No interaction:

h!0metal

insulator

behavior for agiven realization

metal

insulator

~ h


metal

insulator

h


unstableSTABLE

Perturbation theory for the fermionic systems:

+ stability of the metallic phase at

Estimate for the transition temperature for general case

Energy mismatch

Interaction matrix element

# of possible decay processes of an excitationsallowed by interaction Hamiltonian;

1) Identify elementary (one particle) excitations and prove that they are localized.

2) Consider a one particle excitation at finite T and the possible paths of its decays:

Fermionic system:

# of electron-hole pairs

Weakly interacting bosons in one dimension

Phase diagram

1 Crossover????No finite T phase transitionin 1D

See e.g.Altman, Kafri, Polkovnikov, G.Refael, PRL, 100, 170402 (2008); 93,150402 (2004).

1 1 1

1 32c t

1 3c t c t

t T ng

~ 1c

Finite temperature phase transition in 1D

I.A., Altshuler, Shlyapnikov arXiv:0910.434; Nature Physics (2010)

I.M. Lifshitz (1965);Halperin, Lax (1966);Langer, Zittartz (1966)

1 1 1

1 32c t

1 3c t c t

t T ng

~ 1c


High Temperature region

Bose-gas is not degenerate: occupation numbers either 0 or 1

1t

# of bosons to interact with

Bose-gas is not degenerate: occupation numbers either 0 or 1

1t

1 1 1

1 32c t

1 3c t c t

t T ng

~ 1c


Intermediate Temperature region

Intermediate temperatures:

dTT

Bose-gas is degenerate; typical energies ~ |m|<<T occupation numbers >>1 matrix elements are enhanced

*2 ,EngTT d

13132 tttc

1 2 1t

# of levels with different energies

1 2 1t Intermediate temperatures:

1 2 1t Intermediate temperatures:

1 32c t

Delocalization occurs in all energy strips

1 1 1

1 32c t

1 3c t c t

t T ng

~ 1c


Low Temperature region

Low temperatures: 1 2t

x

l l

Occupation

1 2 1nl

Start with T=0

Spectrum is determined by the interaction but only Lifhsitz tale isimportant;

Gross-Pitaevskii mean-field on strongly localized states:

Optimal occupation # Random, non-integer:

Low temperatures: 1 2t

l l

Occupation

1 2 1nl

Start with T=0

Tunneling betweenLifshitz states

See Altman, Kafri, Polkovnikov, G.Refael, PRL, 100, 170402 (2008); 93,150402 (2004).

Low temperatures: 1 2t Start with T=0

Everything is determined by the weakest links:

T=0 transition:

L

Interaction relevant:

Interaction irrelevant:

Insulator

“Superfluid”

Low temperatures: 1 2t Start with T=0

Insulator:

“Superfluid”

All excitations are localized; many-bodyLocalization transition temperature finite;

Localization length of the low-energy excitations (phonons) divergesAs their energy goes to zero;

The system is delocalized at any finite Temperature;

1 1 1

1 32c t

1 3c t c t

t T ng

~ 1c


Transition line terminate is QPT point

Disordered interacting bosons in two dimensions(conjecture)

Summary of Lectures # 3,4:• Existence of the many-body mobility threshold

is established.• The many body metal-insulator transition is not

a thermodynamic phase transition.• It is associated with the vanishing of the

Langevine forces rather the divergences in energy landscape (like in classical glass)

• Only finite T phase transition possible in one dimension (for local Hamiltonians)

Documents

Anderson localization: from single particle to many body problems