Disorder and chaos in quantum systems II. Lecture 3.

1. Disorder and chaos in quantum systems II. Lecture 3. Boris AltshulerPhysics Department, Columbia University

2. Lecture 3. 1.Introduction

3. Previous Lectures: • 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 • Quantum Chaos and Integrability and Localization. • Integrable Poisson; Chaotic Wigner-Dyson • 4. Anderson transition beyond real space • Localization in the space of quantum numbers. • KAM Localized; Chaotic Extended

4. Previous Lectures: • Anderson Localization and Many-Body Spectrum in finite systems. • Q: Why nuclear spectra are statistically the same as • RM spectra – Wigner-Dyson? • A: Delocalization in the Fock space. • Q: What is relation of exact Many Body states and quasiparticles? • A: Quasiparticles are “wave packets” • Anderson Model and Localization on the Cayley tree • Ergodic and Nonergodic extended states • Wigner – Dyson statistics requires ergodicity!

5. Definition: We will call a quantum state ergodic if it occupies the number of sites on the Anderson lattice, which is proportional to the total number of sites : ergodic nonergodic

6. Such a state occupies infinitely many sites of the Anderson model but still negligible fraction of the total number of sites nonergodic states Example of nonergodicity: Anderson ModelCayley tree: transition – branching number ergodicity crossover

7. Resonance is typically far localized Resonance is typically far nonergodic Typically there is a resonance at every step nonergodic Typically each pair of nearest neighbors is at resonance ergodic

8. Lecture 3. 2. Many-Body localization

9. Experiment Cold Atoms J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, P. Lugan1, D.Clément, L.Sanchez-Palencia, P. Bouyer & A. Aspect, “Direct observation of Anderson localization of matter-waves in a controlled Disorder” Nature 453, 891-894 (12 June 2008) 87Rb L. Fallani, C. Fort, M. Inguscio: “Bose-Einstein condensates in disordered potentials” arXiv:0804.2888 Q: What about electrons ? A: Yes,… but electrons interact with each other

10. strength of disorder Strong disorder + moderate interactions ? More or less understand strength of the interaction Wigner crystal Fermi liquid

11. Temperature dependence of the conductivity one-electron picture Chemical potential DoS DoS DoS

12. Temperature dependence of the conductivity one-electron picture Assume that all the states are localized DoS

13.  energy mismatch  Inelastic processestransitions between localized states

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

15. Phonon-assisted hopping   Variable Range Hopping N.F. Mott (1968) is mean localization energy spacing – typical energy separation between two localized states, which strongly overlap Any bath with a continuous spectrum of delocalized excitations down tow = 0will give the same exponential

16. In disordered metals phonons limit the conductivity, but at low temperatures one can evaluate ohmic conductivity without phonons, i.e. without appealing to any bath (Drude formula)! A bath is needed only to stabilize the temperature of electrons. Q1: ? Is the existence of a bath crucial even for ohmic conductivity? ? Can a system of electrons left alone relax to the thermal equilibrium without any bath? Q2:

17. Main postulate of the Gibbs Statistical Mechanics – equipartition (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. Is it always true? No external bath!

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

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

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

21. Q: Can e-h pairs lead to phonon-lessvariable range hoppingin the same way as phonons do? A#1: Sure A#2: No way (L. Fleishman. P.W. Anderson (1980)) A#3: Finite temperatureMetal-Insulator Transition (Basko, Aleiner, BA (2006)) Drude metal insulator s = 0

22. Finite temperatureMetal-Insulator Transition Many body wave functions are localized in functional space Drude Many body localization! metal Interaction strength insulator s = 0 Localization spacing

23. `Main postulate of the Gibbs Statistical Mechanics – equipartition (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. Is it always true? • Many-Body Localization: • It is not localization in a real space! • 2.There is no relaxation in the localized state in the same way as wave packets of localized wave functions do not spread.

24. Finite temperatureMetal-Insulator Transition Includes, 1d case, although is not limited by it. Good (Drude) metal Bad metal

25. There can be no finite temperature phase transitions in one dimension! This is a dogma. • Justification: • Another dogma: every phase transition is connected with the appearance (disappearance) of a long range order • 2.Thermal fluctuations in 1d systems destroy any long range order, lead to exponential decay of all spatial correlation functions and thus make phase transitions impossible

26. There can be no finite temperature phase transitions connected to any long range order in one dimension! Neither metal nor Insulator are characterized by any type of long range order or long range correlations. Nevertheless these two phases are distinct and the transition takes place at finite temperature.

27. Conventional Anderson Model • one particle, • one level per site, • onsite disorder • nearest neighbor hoping labels sites Basis: Hamiltonian:

28. Many body Anderson-like Model

29. Many body Anderson-like Model • many particles, • several levels per site, spacing • onsite disorder • Local interaction Basis: labels levels labels sites Hamiltonian: occupation numbers I U

30. Conventional Anderson Model Many body Anderson-like Model Basis: Basis: labels levels labels sites labels sites occupation numbers N sites M one-particle levels per site Two types of “nearest neighbors”:

31. Anderson’s recipe: 1. take discrete spectrumEmof H0 insulator 2. Add an infinitesimalImpart isto Em 3. EvaluateImSm 1 2 4 limits 4. take limitbut only after metal ! 5. “What we really need to know is the probability distributionof ImS, not its average…”

32. Probability Distribution of G=Im S his aninfinitesimal width(Impart of the self-energy due to a coupling with a bath) of one-electron eigenstates metal insulator Look for: V

33. Stability of the insulating phase: NO spontaneous generation of broadening is always a solution linear stability analysis After n iterations of the equations of the Self Consistent Born Approximation first (…) < 1 – insulator is stable ! then

34. Stability of the metallic phase: Finite broadening is self-consistent as long as (levels well resolved) quantum kinetic equation for transitions between localized states (model-dependent)

35. s > 0 insulator metal s = 0 localization spacing Conductivity s Many body localization! interaction strength Drude metal Badmetal temperature T ? Q: Does “localization length” have any meaning for the Many-Body Localization

36. 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 nc (i.e. the number of hops) increases. Thus phonons create cascades of hops. Size of the cascade nc “localization length”

37. 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 nc (i.e. the number of hops) increases. Thus phonons create cascades of hops. At some temperature This is the critical temperature . Above one phonon creates infinitely many pairs, i.e., the charge transport is sustainable without phonons.

38. Many-body mobility edge mobility edge transition!

39. Metallic States Large E (high T): extended states good metal ergodic states bad metal nonergodic states mobility edge transition! Such a state occupies infinitely many sites of the Anderson model but still negligible fraction of the total number of sites

40. Large E (high T): extended states good metal ergodic states crossover bad metal nonergodic states ? mobility edge transition! No relaxation to microcanonical distribution – no equipartition

41. Large E (high T): extended states good metal ergodic states bad metal nonergodic states mobility edge transition! ? Why no activation

42. Temperature is just a measure of the total energy of the system No activation: good metal bad metal mobility edge transition!

43. Lecture 3. 3. Experiment

44. What about experiment? 1. Problem: there are no solids without phonons With phonons 2. Cold gases look like ideal systems for studying this phenomenon.

45. F. Ladieu, M. Sanquer, and J. P. Bouchaud, Phys. Rev.B 53, 973 (1996) G. Sambandamurthy, L. Engel, A. Johansson, E. Peled & D. Shahar, Phys. Rev. Lett. 94, 017003 (2005). M. Ovadia, B. Sacepe, and D. Shahar, PRL (2009). V. M. Vinokur, T. I. Baturina, M. V. Fistul, A. Y.Mironov, M. R. Baklanov, & C. Strunk, Nature 452, 613 (2008) S. Lee, A. Fursina, J.T. Mayo, C. T. Yavuz, V. L. Colvin, R. G. S. Sofin, I. V. Shvetz and D. Natelson, Nature Materials v 7 (2008) YSi } InO Superconductor –Insulator transition TiN FeO4 magnetite

46. } M. Ovadia, B. Sacepe, and D. Shahar PRL, 2009 Kravtsov, Lerner, Aleiner & BA: Switches Bistability Electrons are overheated: Low resistance => high Joule heat => high el. temperature High resistance => low Joule heat => low el. temperature

47. Electron temperature versus bath temperature Electron temperature LR unstable HR Phonon temperature cr Tph Arrhenius gap T0~1K, which is measured independently is the only “free parameter” Experimental bistability diagram (Ovadia, Sasepe, Shahar, 2008)

48. Common wisdom: no heating in the insulating state no heating for phonon-assisted hopping Heating appears only together with cascades Kravtsov, Lerner, Aleiner & BA: Switches Bistability Electrons are overheated: Low resistance => high Joule heat => high el. temperature High resistance => low Joule heat => low el. temperature

49. Low temperature anomalies 1. Low T deviation from the Ahrenius law “Hyperactivated resistance in TiN films on the insulating side of the disorder-driven superconductor-insulator transition” T. I. Baturina, A.Yu. Mironov, V.M. Vinokur, M.R. Baklanov, and C. Strunk, 2009 Also: • D. Shahar and Z. Ovadyahu, Phys. Rev. B (1992). • V. F. Gantmakher, M.V. Golubkov, J.G. S. Lok, A.K. Geim,. JETP (1996)]. • G. Sambandamurthy, L.W. Engel, A. Johansson, and D.Shahar, Phys. Rev. Lett. (2004).

50. Low temperature anomalies 2. Voltage dependence of the conductance in the High Resistance phase Theory : G(VHL)/G(V 0) < e Experiment: this ratio can exceed 30 Many-Body Localization ?