Transcript Slide 1
Disorder and chaos in quantum systems II. Lecture 2. Boris Altshuler Physics Department, Columbia University Previous Lecture: 1. Anderson Localization as Metal-Insulator Transition Anderson model. Localized and extended states. Mobility edges. 2. Spectral Statistics and Localization. Poisson versus Wigner-Dyson. Anderson transition as a transition between different types of spectra. Thouless conductance P(s) Conductance g s Lecture2. 1. Quantum Chaos, Integrability and Localization P(s) Particular nucleus 166Er P(s) N. Bohr, Nature 137 (1936) 344. 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 ? there exist very “simple” Later it systems with as many as 2 became degrees of freedom (d=2), which demonstrate RMT clear that like spectral statistics Classical (h =0) Dynamical Systems with d degrees of freedom Integrable Systems The variables can be separated and the problem reduces to d onedimensional problems Examples 1. A ball inside rectangular billiard; d=2 • Vertical motion can be separated from the horizontal one • Vertical and horizontal components of the momentum, are both integrals of motion 2. Circular billiard; d=2 • Radial motion can be separated from the angular one • Angular momentum and energy are the integrals of motion d integrals of motion Classical Dynamical Systems with d degrees of freedom Integrable The variables can be separated [ d one-dimensional Systems problems [d integrals of motion Rectangular and circular billiard, Kepler problem, . . . , 1d Hubbard model and other exactly solvable models, . . Chaotic Systems The variables can not be separated [ there is only one integral of motion - energy Examples Sinai billiard B Stadium Kepler problem in magnetic field Classical Chaos h =0 •Nonlinearities •Exponential dependence on the original conditions (Lyapunov exponents) •Ergodicity Quantum description of any System with a finite number of the degrees of freedom is a linear problem – Shrodinger equation Q: What does it mean Quantum Chaos ? h0 Bohigas – Giannoni – Schmit conjecture Chaotic classical analog Wigner- Dyson spectral statistics No quantum numbers except energy Quantum Classical Integrable Chaotic ? ? Poisson WignerDyson Lecture1. 2. Localization beyond real space Kolmogorov – Arnold – Moser (KAM) theory A.N. Kolmogorov, Dokl. Akad. Nauk SSSR, 1954. Proc. 1954 Int. Congress of Mathematics, NorthHolland, 1957 Andrey Kolmogorov Vladimir Arnold Jurgen Moser 0 Integrable classical Hamiltonian Hˆ 0 , d>1: Separation of variables: d sets of action-angle variables I1, 1 21t; ... , I 2 , 2 22t;.. 1D classical motion – action-angle variables Kolmogorov – Arnold – Moser (KAM) theory 0 A.N. Kolmogorov, Dokl. Akad. Nauk SSSR, 1954. Proc. 1954 Int. Congress of Mathematics, NorthHolland, 1957 Integrable classical Hamiltonian Hˆ 0 , d>1: Separation of variables: d sets of action-angle variables I1, 1 21t; ... , I 2 , 2 22t;.. Quasiperiodic motion: set of the frequencies,1 , 2 ,.., d which are in general incommensurate. Actions I i are integrals of motion I i t 0 Andrey Kolmogorov I1 Vladimir Arnold Jurgen Moser 1 I2 2 …=> tori Integrable dynamics: Each classical trajectory is quasiperiodic and confined to a particular torus, which is determined by a set of the integrals of motion space real space phase space: (x,p) energy shell tori Number of dimensions d 2d 2d-1 d For d>1 each torus has measure zero on the energy shell ! Kolmogorov – Arnold – Moser (KAM) theory A.N. Kolmogorov, Integrable classical Hamiltonian Hˆ 0 , d>1: Dokl. Akad. Nauk SSSR, 1954. Proc. 1954 Int. Congress of Mathematics, NorthHolland, 1957 Separation of variables: d sets of action-angle variables I , 2 t;.., I , 2 t;.. 1 1 1 2 2 2 Quasiperiodic motion: set of the frequencies, 1, 2 ,.., d which are in general incommensurate Actions 1 I1 Q: Andrey Kolmogorov Vladimir Arnold Jurgen Moser A: Ii are integrals of motion I i I2 2 t 0 …=> Will an arbitrary weak perturbation Vˆ of the integrable Hamiltonian Hˆ 0 destroy the tori and make the motion ergodic (when each point at the energy shell will be reached sooner or later) ? Most of the tori survive KAM weak and smooth enough theorem perturbations Most of the tori survive weak and smooth enough perturbations KAM theorem: I2 I2 Vˆ 0 Each point in the space of the integrals of motion corresponds to a torus and vice versa I1 Finite motion. Localization in the space of the integrals of motion I1 ? Most of the tori survive weak and smooth enough perturbations KAM theorem: I2 I2 Vˆ 0 I1 I2 I1 h0 I1 Energy shell Consider an integrable system. Each state is characterized by a set of quantum numbers. It can be viewed as a point in the space of quantum numbers. The whole set of the states forms a lattice in this space. A perturbation that violates the integrability provides matrix elements of the hopping between different sites (Anderson model !?) Weak enough hopping: Localization - Poisson Strong hopping: transition to Wigner-Dyson Sinai billiard Square billiard Localized momentum space Disordered localized Disordered extended extended Localized real space Glossary Classical Quantum Integrable Integrable H0 H0 I Hˆ 0 E , KAM Localized Ergodic – distributed all over the energy shell Chaotic Extended ? I Extended Level repulsion, anticrossings, Wigner-Dyson spectral statistics states: Localized Poisson spectral statistics states: Invariant (basis independent) definition Chaotic Integrable Square billiard Disordered localized All chaotic systems resemble each other. All integrable systems are integrable in their own way Sinai billiard Disordered extended Consider a finite system of quantum particles, e.g., fermions. Let the oneparticle spectra be chaotic (WignerDyson). Q: What is the statistics of the many-body spectra? ? a.The particles do not interact with each other. Poisson: individual energies are conserving quantum numbers. b. The particles do interact. ???? Lecture 2. 3. Many-Body excitation in finite systems Decay of a quasiparticle with an energy Landau Fermi liquid e Fermi Sea e in Quasiparticle decay rate at T = 0 in a clean Fermi Liquid. I. d=3 e e e Fermi Sea e coupling e ee e constant e F h 2 2 d 3 Reasons: • At small e the energy transfer, , is small and the integration over e and gives the factor e2. ………………………………………………………………… •The momentum transfer, q , is large and thus the scattering probability at given e and does not depend on e , or e Quasiparticle decay rate at T = 0 in a clean Fermi Liquid. II. Low dimensions Small moments transfer, q , become important at low dimensions because the scattering probability is proportional to the squared time of the interaction, e 1/q (qvF. )-2 e eF 2 h ee e e e 2 e F loge F e d 3 d 2 d 1 vF Quasiparticle decay a clean Quasiparticle decayrate rate atatTT= =00inina clean FermiFermi Liquid.Liquid. e e e e h ee e e 2 eF e 2 e F loge F e e d 3 d 2 d 1 Fermi Sea Conclusions: 1. For d=3,2 from e e F it follows that e ee h , i.e., that the qusiparticles are well determined and the Fermi-liquid approach is applicable. 2. For d=1 e ee is of the order of h , i.e., that the Fermi-liquid approach is not valid for 1d systems of interacting fermions. Luttinger liquids Decay of a quasiparticle with an energy e in Landau Fermi liquid Quantum dot – zero-dimensional case ? e e e1 e1 Fermi Sea Decay of a quasiparticle with an energy e in Landau Fermi liquid Quantum dot – zero-dimensional case ? e e e1 e1 Fermi Sea Decay rate of a quasiparticle with energy ( U.Sivan, Y.Imry & A.Aronov,1994 ) Fermi Golden rule: e e 1 ET Mean level spacing 2 Thouless energy e Decay rate of a quasiparticle with energy e in 0d. ( U.Sivan, Y.Imry & A.Aronov,1994 ) Fermi Golden rule: e e 1 ET Mean level spacing Def: Zero dimensional system 2 Thouless energy Recall: ET 1 g Thouless conductance ET e 1 g 1 One particle states are extended all over the system Decay rate of a quasiparticle with energy e in 0d. Problem: e zero-dimensional case e e1 e1 Fermi Sea one-particle spectrum is discrete equation e1e2 e’1 e’2 can not be satisfied exactly Recall: in the Anderson model the site-to-site hopping does not conserve the energy Decay rate of a quasiparticle with energy e in 0d. e e e’+ e’ Offdiagonal matrix element M , e , e 1 g 1 Chaos in Nuclei – Delocalization? 1 2 3 4 e e’ e1’ .... 5 6 generations Can be mapped (approximately) to the problem of localization on Cayley tree 1 e1 Fermi Sea Delocalization in Fock space 2 3 4 5 Conventional Anderson Model •one particle, •one level per site, •onsite disorder •nearest neighbor hoping Basis: i , i labels sites Hamiltonian: H Hˆ 0 e i i i i Hˆ 0 Vˆ Vˆ I i i , j n.n. j e 0d system; no interactions eb many (N ) particles no interaction: Individual energies e and thus occupation numbers n are conserved e conservation laws “integrable system” N e integrable system Hˆ E n E n e 0d system with interactions e Basis: eb e e n occupation 0,1 numbers Hamiltonian: Hˆ 0 E n labels levels H Hˆ 0 Vˆ Vˆ , I .., n 1,.., n b 1,.., n 1,.., n 1,.. Conventional Anderson Model Basis: i i Hˆ e i i i i i , j n .n . I i Basis: labels sites Many body Andersonlike Model j labels levels , n n 0,1 occupation numbers Hˆ E I , “nearest .., n 1,.., n b 1,.., n 1,.., n 1,.. neighbors”: Isolated quantum dot – 0d system of fermions Exact many-body states: Exact means that the imaginary Ground state, excited states part of the energy is zero! Quasiparticle excitations: Finite decay rate Q: What is the connection ? S gate source QD drain D current No e-e interactions – resonance tunneling S gate source QD drain D current g Mean level spacing 1 No e-e interactions – resonance tunneling VSD S gate source QD current g drain D The interaction leads to No e-e interactions – additional peaks – resonance tunneling many body excitations VSD S S D Resonance tunneling Peaks D Inelastic cotunneling Additional peak S gate source QD current g drain S The interaction leads to additional peaks – many body excitations VSD S gate source QD drain D current g Landau quasiparticle with the width SIA NE loc Ergodic - WD VSD Landau quasiparticle with the width SIA NE loc Localized - finite # of the satelites Extended - infinite # of the satelites Ergodic - WD extended VSD (for finite e the number of the satelites is always finite) Ergodic – nonergodic crossover! Anderson Model on a Cayley tree Anderson Model on a Cayley tree I, W K – branching number W 1 Ic K ln K W I K Resonance at every generation W W I K ln K K Sparse resonances will call a quantum state Definition: We ergodic if it occupies the number of sites N on the Anderson lattice, which is proportional to the total number of sites N : N N 0 N nonergodic N N N const 0 ergodic Localized states are const obviously not ergodic: N N Q: A: Is each of the extended state ergodic In 3D probably yes For d>4 most likely no ? nonergodic states Such a state occupies infinitely many sites of the Anderson model but still negligible fraction of the total number of sites Example of nonergodicity: Anderson Model Cayley tree: transition K – branching number W Ic K ln K ergodicity n ln N I erg ~ W crossover I W K ln K Resonance is typically far n const localized W K I W K ln K Typically there is no resonance at the next step n ~ ln N nonergodic n ~ ln N nonergodic W I W K Typically there is a resonance at every step I W Typically each pair of nearest neighbors is at resonance n~N ergodic