Anderson localization: from single particle to many body problems.

(4 lectures)

Igor Aleiner

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

Windsor Summer School, 14-26 August 2012

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-functions extended

Metal Insulator

localized

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

• • • Interactions at finite T lead to finite lim 𝑇→0 𝐿 𝜑 = ∞ 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   energy mismatch (inelastic lifetime) –1 (any mechanism)

Phonon-induced hopping

  energy difference can be matched by a phonon Sir N.F. Mott (1968) Mechanism-dependent prefactor Without Coulomb gap A.L.Efros, B.I.Shklovskii (1975) Optimized phase volume Any bath with a continuous spectrum of delocalized excitations down to w = 0 will give the same exponential

𝜆

𝑒−𝑝ℎ

⟶ 0 ????? (All one-particle states are localized)

“insulator” “metal” Drude Electron phonon Interaction does not enter

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

“insulator” “metal” Drude Electron phonon Interaction does not enter

Q: Can we replace phonons with e-h pairs and obtain phonon-less VRH? A#1: Sure Easy steps: 1) Recall phonon-less AC conductivity: Sir N.F. Mott (1970) 2) Calculate the Nyquist noise (fluctuation dissipation Theorem).

3) Use the electric noise instead of phonons.

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

d Q: Can we replace phonons with e-h pairs and obtain phonon-less VRH? A#1: Sure [Person from the street (2005)] A#2: No way [L. Fleishman. P.W. Anderson (1980)] (for Coulomb interaction in 3D – may be) resonances R g   0 *

Metal-Insulator Transition and many-body Localization: [Basko, Aleiner, Altshuler (2005)] and all one particle state are localized Drude metal insulator (Perfect Ins) Interaction strength

Many-body mobility threshold

[Basko, Aleiner, Altshuler (2005)] All STATES LOCALIZED Many body DoS metal -many-body mobility threshold

“All states are localized “

means

Probability to find an extended state: System volume

Many body localization means any excitation is localized:

Extended Localized

States always thermalized!!!

All STATES EXTENDED All STATES LOCALIZED

States never

Many body DoS

thermalized!!!

Entropy

Is it similar to Anderson transition?

Why no activation?

Many body DoS One-body DoS

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

Fock space localization in quantum dots (AGKL, 1997) No spatial structure ( “0-dimensional” ) ´ - one-particle level spacing;

Fock space localization in quantum dots (AGKL, 1997) 1 -particle excitation 3-particle excitation 5 -particle excitation Cayley tree mapping

Fock space localization in quantum dots (AGKL, 1997) 1 -particle excitation 3-particle excitation 5 -particle excitation 1. Coupling between states: 2. Maximal energy mismatch: 3. Connectivity: - one-particle level spacing;

Metal-Insulator “Transition” in zero dimensions [Altshuler, Gefen, Kamenev,Levitov (1997)] In the paper: Vs. finite T Metal-Insulator Transition in the bulk systems [Basko, Aleiner, Altshuler (2005)] metal insulator Interaction strength

Metal-Insulator “Transition” in zero dimensions [Altshuler, Gefen, Kamenev,Levitov (1997)] - one-particle level spacing; Vs. finite T Metal-Insulator Transition in the bulk systems [Basko, Aleiner, Altshuler (2005)] 1-particle level spacing in localization volume; 1) Localization in Fock space = Localization in the coordinate space.

2) Interaction is local;

Metal-Insulator “Transition” in zero dimensions [Altshuler, Gefen, Kamenev,Levitov (1997)] - one-particle level spacing; Vs. finite T Metal-Insulator Transition in the bulk systems [Basko, Aleiner, Altshuler (2005)] 1-particle level spacing in localization volume; 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

Metal-Insulator “Transition” in zero dimensions [Altshuler, Gefen, Kamenev,Levitov (1997)] - one-particle level spacing; Vs. finite T Metal-Insulator Transition in the bulk systems [Basko, Aleiner, Altshuler (2005)] 1-particle level spacing in localization volume; 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-temperature regime only.

Spatial scales of interest >> 1-particle localization length Otherwise, conventional perturbation theory for disordered metals works. Altshuler, Aronov, Lee (1979); Finkelshtein (1983) – T-dependent SC potential Altshuler, Aronov, Khmelnitskii (1982) – inelastic processes

O

  

tails

a

   1

d

 No spins

j

 

 j 1 l 1 l 2 j 2 Interaction only within the same cell;

Statistics of matrix elements?

Parameters: 

l j

random signs

What to calculate?

Idea for one particle localization Anderson, (1958); MIT for Cayley tree: Abou-Chakra, Anderson, Thouless (1973); Critical behavior: Efetov (1987) – random quantity No interaction : Metal Insulator

insulator

Probability Distribution

Note: metal Look for:

Iterations:

Cayley tree structure

Nonlinear integral equation with random coefficients after standard simple tricks: Decay due to tunneling Decay due to e-h pair creation + kinetic equation for occupation function

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 ?

So far, we have learned:

Trouble !!!

Insulator ???

Non-ergodic+Drude metal

Stability of the insulator

Nonlinear integral equation with random coefficients Notice: Linearization: for is a solution

# of interactions

Recall:

insulator metal h probability distribution for a fixed energy # of hops in space STABLE unstable

So, we have just learned:

Non-ergodic+Drude metal Metal Insulator

Estimate for the transition temperature for general case 1) Start with T=0; 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:

Energy mismatch Interaction matrix element # of possible decay

p

rocesses of an excitations

allowed by interaction Hamiltonian;

• • • •

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

Lecture #4.

Many body localization and phase diagram of weakly interacting 1D bosons I.A, Altshuler, Shlyapnikov, NATURE PHYSICS 6 (2010) 900-904

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 Exact solution for one channel: M.E. Gertsenshtein, V.B. Vasil’ev, (1959) “Conjecture” for one channel: Sir N.F. Mott and W.D. Twose (1961) Exact solution for w for one channel: V.L. Berezinskii, (1973)

Many-body localization;

Idea for one particle localization Anderson, (1958); MIT for Cayley tree: Abou-Chakra, Anderson, Thouless (1973); Critical behavior: Efetov (1987) – random quantity No interaction: metal insulator h !

0 insulator metal behavior for a given realization ~ h probability distribution for a fixed energy

Perturbation theory for the fermionic systems: insulator metal h probability distribution for a fixed energy + stability of the metallic phase at STABLE unstable

Estimate for the transition temperature for general case 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:

Energy mismatch Interaction matrix element # of possible decay

p

rocesses of an excitations

allowed by interaction Hamiltonian;

Fermionic system: # of electron-hole pairs

Weakly interacting bosons in one dimension

Phase diagram

1

Crossover????

No finite T phase transition in 1D

See e.g.

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

1 

c



c

   1 3 

c



t

1 3 

c

~ 1 1 g I.A., Altshuler, Shlyapnikov arXiv:0910.434; Nature Physics (2010) 1 g

t



T ng

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

1

High Temperature region



c



c

   1 3 

c



t

1 3 

c

~ 1 1 g I.A., Altshuler, Shlyapnikov arXiv:0910.434; Nature Physics (2010) 1 g

t



T ng

t

 g  1 Bose-gas is not degenerate: occupation numbers either 0 or 1 # of bosons to interact with

t

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

1 Intermediate

Temperature region



c



c

   1 3 

c



t

1 3 

c

~ 1 1 g I.A., Altshuler, Shlyapnikov arXiv:0910.434; Nature Physics (2010) 1 g

t



T ng

Intermediate temperatures:

g  1 2 g  1 m 

T

2

T d



ng

,

E

*

T



T d

Bose-gas is degenerate; typical energies ~ | m |<> 1 matrix elements are 

c



t

2 3 g 1 3 g 

t

g  1

Intermediate temperatures:

g  1 2 g  1 # of levels with different energies

Intermediate temperatures:

g  1 2 g  1 Delocalization occurs in all energy strips 

c

   1 3

1 

c

~ 1 Low

Temperature region



c



c

   1 3 

c



t

1 3 1 g I.A., Altshuler, Shlyapnikov arXiv:0910.434; Nature Physics (2010) 1 g

t



T ng

Low temperatures:

t

 g  1 2

Start with T=0

Spectrum is determined by the interaction but only Lifhsitz tale is important; Gross-Pitaevskii mean-field on strongly localized states: Optimal occupation # Random, non-integer:

l

 

l

   

x

Occupation

nl

  g  1 2  1

Low temperatures:

 

l t

 g  1 2

Start with T=0

l

   

Occupation

nl

  g  1 2  1

Tunneling between

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

Low temperatures:

t

 g  1 2

Start with T=0

Everything is determined by the weakest links: T=0 transition: L Interaction relevant: Interaction irrelevant: Insulator “Superfluid”

Low temperatures:

t

 g  1 2

Start with T=0

Insulator: All excitations are localized; many-body Localization transition temperature finite; “Superfluid” Localization length of the low-energy excitations (phonons) diverges As their energy goes to zero; The system is delocalized at any finite Temperature;

1 

c

~ 1 Transition line terminate is QPT point 

c



c

   1 3 

c



t

1 3 1 g I.A., Altshuler, Shlyapnikov arXiv:0910.434; Nature Physics (2010) 1 g

t



T ng

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)