Anderson localization from single particle to many body

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;

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 Variable Range Hopping 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

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

Q: Can we replace phonons with e-h pairs and obtain phonon-less VRH? Drude “metal” “insulator” 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: Person from the street (2005) (2005 -2011) 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).

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 d [L. Fleishman. P. W. Anderson (1980)] (for Coulomb interaction in 3 D – may be) is contributed by rare Thus, the matrix element vanishes !!! 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)] metal insulator All STATES EXTENDED All STATES LOCALIZED Many body Do. S -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 Do. S thermalized!!! Entropy

Is it similar to Anderson transition? Why no activation? Many body Do. S One-body Do. S

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

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

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

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

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

Reproduces correct behavior of the tails of one particle wavefunctions No spins

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

Statistics of matrix elements?

Parameters: 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

Probability Distribution metal insulator Look for: Note:

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: Non-ergodic+Drude metal Trouble !!! Insulator ? ? ?

Stability of the insulator Nonlinear integral equation with random coefficients Notice: Linearization: for is a solution

# of interactions # of hops in space Recall: metal insulator h probability distribution for a fixed energy 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: Interaction matrix element Energy mismatch # of possible decay processes 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 1 D 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 1 D; • Remind: “Superconductor”-insulator transition at T=0; • Many-body metal-insulator transiton at finite T;

1. Localization of single-electron wave-functions: extended 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 s(w) for one channel: V. L. Berezinskii, (1973) localized

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 ~h behavior for a given realization probability distribution for a fixed energy

Perturbation theory for the fermionic systems: metal insulator 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: Interaction matrix element Energy mismatch # of possible decay processes 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 1 D See e. g. Altman, Kafri, Polkovnikov, G. Refael, PRL, 100, 170402 (2008); 93, 150402 (2004).

Finite temperature phase transition in 1 D I. A. , Altshuler, Shlyapnikov ar. Xiv: 0910. 434; Nature Physics (2010)

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

High Temperature region I. A. , Altshuler, Shlyapnikov ar. Xiv: 0910. 434; Nature Physics (2010)

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

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

Intermediate Temperature region I. A. , Altshuler, Shlyapnikov ar. Xiv: 0910. 434; Nature Physics (2010)

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

Intermediate temperatures: # of levels with different energies

Intermediate temperatures: Delocalization occurs in all energy strips

Low Temperature region I. A. , Altshuler, Shlyapnikov ar. Xiv: 0910. 434; Nature Physics (2010)

Low temperatures: 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: Occupation

Low temperatures: Start with T=0 Occupation Tunneling between Lifshitz states See Altman, Kafri, Polkovnikov, G. Refael, PRL, 100, 170402 (2008); 93, 150402 (2004).

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

Low temperatures: 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;

Transition line terminate is QPT point I. A. , Altshuler, Shlyapnikov ar. Xiv: 0910. 434; Nature Physics (2010)

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)