Anderson Localization Looking Forward

Anderson Localization Looking Forward Boris Altshuler Physics Department, Columbia University Collaborations: Also Igor Aleiner Denis Basko, Gora Shlyapnikov, Vincent Michal, Vladimir Kravtsov, Lecture2 September, 9, 2015

Outline 1. Introduction 2. Anderson Model; Anderson Metal and Anderson Insulator 3. Localization beyond the real space. Integrability and chaos. 4. Spectral Statistics and Localization 5. Many-Body Localization. 6. Many-Body Localization of the interacting fermions. 7. Many-Body localization of weakly interacting bosons. 8. Many-Body Localization and Ergodicity

extended f = 3.04 GHz f = 7.33 GHz Anderson Model W, I localized Localized j I ij i E Ĥ x E x I 2 Finite motion 0 DoS x I 1

Vˆ, Matrix element of the perturbation I 0 I,..., I 1 I d Anderson Model! AL hops are local one can distinguish near and far KAM perturbation is smooth enough

Pradhan & Sridar, PRL, 2000 Sinai billiard Square billiard Disordered localized 0 W Localized momentum space extended Disordered extended Localized real space I 0

Classical Integrable H0 H0 I ; I t 0 Perturbation Glossary Quantum Integrable E Perturbation V; I t 0 Vˆ V, KAM Localized Ergodic (chaotic) Extended? ˆ 0 H,, I

Question: What is the reason to speak about localization if we in? general do not know the space in which the system is localized Need an invariant (basis independent) criterion of the localization

Part 4. Spectral Statistics and Localization

RANDOM MATRIX THEORY N N E s 1 P... s E 1 E 1 E 1 Spectral Rigidity ensemble of Hermitian matrices with random matrix element E - spectrum (set of eigenvalues) - mean level spacing - ensemble averaging - spacing between nearest neighbors - distribution function of nearest neighbors spacing between P s 0 0 Spectral statistics N Level repulsion P s 1 s 1, 2, 4

Wigner-Dyson; GOE Poisson Gaussian Orthogonal Ensemble Unitary 2 Orthogonal 1 Simplectic 4 Poisson completely uncorrelated levels

RANDOM MATRICES N N matrices with random matrix elements. N Dyson Ensembles Matrix elements real complex 2 2 matrices Ensemble orthogonal unitary simplectic 1 2 4 realization T-inv potential broken T-invariance (e.g., by magnetic field) T-inv, but with spinorbital coupling

Hˆ Reason for H H H 11 12 H 12 22 P s 0 0: when s E E H H H 2 2 2 1 22 11 12 small small small Recall the Wigner von Neumann noncrossing rule 1. The assumption is that the matrix elements are statistically independent. Therefore probability of two levels to be degenerate vanishes. 2. If H 12 is real (orthogonal ensemble), then for s to be small two statistically independent variables ((H 22 - H 11 ) and H 12 ) should be small and thus P( s) s 1

Hˆ H H H 11 12 H 12 22 p H H p H 2 2 P E E d H H dh E E H H H 2 1 11 22 12 2 1 22 11 12 11 22 12 Distribution function of the diagonal matrix elements Distribution function of the off-diagonal matrix elements Distribution function of the spacing P s

Hˆ Reason for H H H 11 12 H 12 22 P s 0 0: when s E E H H H 2 2 2 1 22 11 12 small small small 1. The assumption is that the matrix elements are statistically independent. Therefore probability of two levels to be degenerate vanishes. 2. If H 12 is real (orthogonal ensemble), then for s to be small two statistically independent variables ((H 22 - H 11 ) and H 12 ) should be small and thus P( s) s 1 3. Complex H 12 (unitary ensemble) both Re(H 12 ) and Im(H 12 ) are statistically independent three independent 2 random variables should be small P( s) s 2

Anderson Model Lattice - tight binding model Onsite energies i - random j i Hopping matrix elements I ij Hˆ i i aˆ i aˆ i I aˆ i, jn. n. i aˆ j I ij -W < i <W uniformly distributed Is there much in common between Random Matrices and Hamiltonians with random potential? Q: What are the spectral statistics? of a finite size Anderson model

Strong disorder Anderson Transition Weak disorder I < I c Insulator All eigenstates are localized Localization length x The eigenstates, which are localized at different places will not repel each other I > I c Metal There appear states extended all over the whole system Any two extended eigenstates repel each other Poisson spectral statistics Wigner Dyson spectral statistics

Zharekeschev & Kramer. Exact diagonalization of the Anderson model Disorder W

Anderson transition in terms of pure level statistics P(s)

Part 5. Quantum Chaos, Integrability and Localization

ATOMS NUCLEI Wigner: Main goal is to classify the eigenstates in terms of the quantum numbers For the nuclear excitations this program does not work Study spectral statistics of a particular quantum system a given nucleus Spectra: {E } Random Matrices Ensemble Ensemble averaging Atomic Nuclei Spectral averaging (over ) Particular quantum system Nevertheless Statistics of the nuclear spectra are almost exactly the same as the Random Matrix Statistics

P(s) Particular nucleus 166 Er N. Bohr, Nature 137 (1936) 344. P(s) s Spectra of several nuclei combined (after spacing) rescaling by the mean level

Q: Why the random matrix? theory (RMT) works so well for nuclear spectra Original answer: These are systems with a large number of degrees of freedom, and therefore the complexity is high Later it became clear that there exist very simple systems with as many as 2 degrees of freedom (d=2), which demonstrate RMT - like spectral statistics

Classical Dynamical Systems with d degrees of freedom Integrable The variables can be separated [ d one-dimensional Systems problems [d integrals of motion Chaotic Systems Examples Rectangular and circular billiard, Kepler problem,..., 1d Hubbard model and other exactly solvable models,.. The variables can not be separated [ there is only one integral of motion - energy B Sinai billiard Stadium Kepler problem in magnetic field

0 Bohigas Giannoni Schmit conjecture Chaotic classical analog Wigner- Dyson spectral statistics No quantum numbers except energy

Classical Integrable? Quantum Poisson Chaotic? Wigner- Dyson

Square billiard Integrable All chaotic systems resemble each other. Sinai billiard Chaotic Disordered localized All integrable systems are integrable in their own way Disordered extended

Sinai billiard Pradhan &Sridar, PRL, 2000 Square billiard Localized momentum space extended Disordered extended Disordered localized Localized real space

Spectral statistics Extended states: Localized states: Level repulsion, avoided crossings, Wigner-Dyson spectral statistics (random matrices) Poisson spectral statistics No level repulsion probability density P(s) nearest neighbor s spacing s Invariant (basis independent) definition

Example: Stadium Localization in the angular momentum space. R a a R - parameter

Localization and diffusion in the angular momentum space a 0 R 0 Integrable circular billiard Angular momentum is the integral of motion Chaotic stadium 0; 1 Diffusion in the angular momentum space 5 D 2

a 0 R 0 Integrable circular billiard Angular momentum is the integral of motion 0; 1 Diffusion in the angular momentum space 5 D 2 Chaotic stadium Localization and diffusion in the angular momentum space Poisson 0.01 g=0.012 Wigner-Dyson 0.1 g=4

Part 6. Many-Body Localization a) Spin systems; Quantum Computer

Example: Random Ising model in the perpendicular field Will not discuss today in detail N N N z z z x ˆ ˆ ˆ ˆ ˆ x ˆ i i ij i j i 0 i i1 i j i1 i1 Hˆ B J I H I i Random Ising model in a parallel field - Pauli matrices, z i i 1,2,..., N; N 1 1 2 Perpendicular field Without perpendicular field all z i commute with the Hamiltonian, i.e. they are integrals of motion

i N N N z z z x ˆ ˆ ˆ ˆ ˆ x ˆ i i ij i j i 0 i i1 i j i1 i1 Hˆ B J I H I - Pauli matrices i 1,2,..., N; N 1 Random Ising model in a parallel field Perpendicular field Without perpendicular field all commute with the Hamiltonian, i.e. they are integrals of motion z i z i determines a site of an N-dimensional hypercube

i H 0 i N N N z z z x ˆ ˆ ˆ ˆ ˆ x ˆ i i ij i j i 0 i i1 i j i1 i1 Hˆ B J I H I - Pauli matrices i 1,2,..., N; N 1 Anderson Model on N-dimensional cube onsite energy Random Ising model in a parallel field z i perp. field x Perpendicular field Without perpendicular field all commute with the Hamiltonian, i.e. they are integrals of motion z i determines a site ˆ ˆ ˆ hoping between nearest neighbors

N N N z z z x ˆ ˆ ˆ ˆ ˆ x ˆ i i ij i j i 0 i i1 i j i1 i1 Hˆ B J I H I Anderson Model on N-dimensional cube Usually: # of dimensions d const Here: # of dimensions d N system linear size L system linear size L 1

6-dimensional cube 9-dimensional cube

Part 6. Many-Body Localization b) Interacting particles

one particle, one level per site, onsite disorder nearest neighbor hoping Hamiltonian: Hˆ Hˆ Vˆ 0 Conventional Anderson Model many (N) particles no } interaction. Individual energies k, k 1,... N are conserved Basis: i, Hˆ Vˆ 0 i I i, jn. n. i i i i labels sites N conservation laws integrable system j i Ĥ E E n k k labels one-particle eigenstates; n - occupation numbers; n

many (N) particles no } interaction. Individual energies k, k 1,... N are conserved N conservation laws integrable system Ĥ E E n k k labels one-particle eigenstates; n - occupation numbers; n Role of the interaction: Transitions between the ideal gas states Basis: Hamiltonian: Hˆ Hˆ ˆ ˆ 0 V, H0 E Interaction: Vˆ Localization in Fock space I,, nn.?

Disorder + Weak Interaction Basis: Hamiltonian: Hˆ Hˆ Vˆ, Hˆ 0 0 E Interaction: Vˆ,, nn.? I Anderson model Q: What is the lattice?

Part 6. Many-Body Localization c) Statistical mechanics

Main postulate of the Gibbs StatMechequipartition (microcanonical distribution): In the equilibrium all states with the same energy are realized with the same probability. Without interaction between particles the equilibrium would never be reached each one-particle energy is conserved. Common believe: Even weak interaction should drive the system to the equilibrium. It might be not true for many-body localized states!!!

Localization in Fock space What does it mean? No two-body operator can cause transitions between many-body states that are close in energy. No dissipation due to the external field ideal insulator (glass) The concept of the equilibrium looses its meaning no relaxation to a thermal state. No entropy production Temperature Energy

Time-reversal symmetry = T-invariance Equations of motion are invariant under t t For each classical trajectory there is another trajectory, which is its inversion in time Violation of the T-invariance e.g. by magnetic field H

Statistical mechanics Irreversibility - arrow of time t t x r 2 2 2 r t r t Has nothing to do with the violation of the T-invariance Has everything to do with the delocalization Extended states irreversible dynamics Localized states dynamics is to some extent reversible The same is true for many body systems t

Heat Theorem Nerns, Einstein, Planck, Polany, [Nernst is] not open to reason, because he is not enough of a logician ( der Vernunft nich zuga nglich (zu wenig Logiker)). Einstein, letter to Ehrenfest Is it possible to reach zero temperature? Is it possible to reach thermal equilibrium close to the ground state?

Part 6. Many-Body Localization d) Interacting fermions; phononless transport

Temperature dependence of the conductivity one-electron picture Mobility edge Chemical potential T 0 0 DoS DoS DoS T E T e c F T 0 T there are extended states II c all states are localized II c

Temperature dependence of the conductivity one-electron picture Assume that all the states are localized; e.g. d = 1,2 DoS T 0 T

Inelastic processes transitions between localized states energy mismatch T 0 0 (any mechanism)

Phonon-assisted hopping w w T 0 0 Variable Range Hopping N.F. Mott (1968) Mechanism-dependent prefactor Optimized phase volume Any bath with a continuous spectrum of delocalized excitations down to w= 0 will give the same exponential

Common belief: Given: Anderson Insulator weak e-e interactions? Can hopping conductivity exist without phonons 1. All one-electron states are localized 2. Electrons interact with each other 3. The system is closed (no phonons) 4. Temperature is low but finite Phonon assisted hopping transport Find: DC conductivity (T,w=0) (zero or finite?)

Q: Can e-h pairs lead to phonon-less variable range hopping in the same way as phonons do? A#1: Sure 1. Recall phonon-less AC conductivity: Sir N.F. Mott (1970) 2. Fluctuation Dissipation Theorem: there should be Johnson-Nyquist noise 3. Use this noise as a bath instead of phonons 4. Self-consistency (whatever it means)

Q: Can e-h pairs lead to phonon-less variable range hopping in the same way as phonons do? A#1: Sure A#2: No way (L. Fleishman. P.W. Anderson (1980)) Except maybe Coulomb interaction in 3D is contributed by rare resonances R d R g a matrix element vanishes b 0

No phonons??? No transport T Problem: If the localization length exceeds L, then metal. } At In a metal e e interaction leads to a finite L high enough temperatures conductivity should be finite even without phonons

Q: Can e-h pairs lead to phonon-less variable range hopping in the same way as phonons do? A#1: Sure A#2: No way (L. Fleishman. P.W. Anderson (1980)) A#3: Finite temperature Metal-Insulator Transition (Basko, Aleiner, BA (2006)) insulator 0 metal Drude

Finite temperature Metal-Insulator Transition Many body wave functions are localized Many body in functional space localization! insulator metal Drude Interaction strength 0 d 1 spacing Localization Definitions: Insulator not 0 d dt 0 Metal not 0 d dt 0

Many body Anderson-like Model many particles, several levels per site, onsite disorder local interaction Hamiltonian: H Hˆ Vˆ Vˆ Vˆ 1 0 1 2 Ĥ0 E.., n 1,.., n 1,.., i, j n. n. Vˆ 2,, I U i j Vˆ.., n 1,.., n 1,.., n 1,.., n 1,.. i i i i 1 i n i Basis: labels sites 0,1 Vˆ 2 n i labels levels occupation numbers I U

Stability of the insulating phase: NO spontaneous generation of broadening ( ) 0 is always a solution i linear stability analysis ( x ) 2 2 ( x ) ( x ) 2 After n iterations of the equations of the Self Consistent Born Approximation P n ( ) 3 2 const T ln 1 n first then ( ) < 1 insulator is stable!

Physics of the transition: cascades Conventional wisdom: For phonon assisted hopping one phonon one electron hop It is maybe correct at low temperatures, but the higher the temperature the easier it becomes to create e-h pairs. Therefore with increasing the temperature the typical number of pairs created n c (i.e. the number of hops) increases. Thus phonons create cascades of hops. Typical size of the cascade Localization length