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Anderson localization: from single particle to many body problems. Igor Aleiner (4 lectures) Windsor Summer School, August 2012 ( Columbia University in the City of New York, USA )

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Lecture # 1-2 Single particle localization Lecture # 2-3 Many-body localization

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I V Conductivity: Conductance: Insulator Metal Transport in solids Superconductor

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I V Conductivity: Conductance: Insulator Metal Transport in solids Focus of The course

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Lecture # 1 Metals and insulators – importance of disorder Drude theory of metals First glimpse into Anderson localization Anderson metal-insulator transition (Bethe lattice argument; order parameter … )

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Band metals and insulators MetalsInsulators Gapless spectrum Gapped spectrum

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Metals Gapless spectrum Insulators Gapped spectrum But clean systems are in fact perfect conductors: Current Electric field

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Metals Gapless spectrum Insulators Gapped spectrum But clean systems are in fact perfect conductors: (quasi-momentum is conserved, translational invariance)

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Finite conductivity by impurity scattering One impurity Incoming flux Probability density Scattering cross-section

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Finite conductivity by impurity scattering Finite impurity density Elastic mean free path Elastic relaxation time

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Finite conductivity by impurity scattering Finite impurity density Drude conductivity CLASSICAL Quantum (band structure) Quantum (single impurity)

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Conductivity and Diffusion Finite impurity density Einstein relation Diffusion coefficient

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Conductivity, Diffusion, Density of States (DoS) Einstein relation Density of States (DoS)

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Clean systems

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Density of States (DoS) Clean systems Metals, gapless Insulators, gapped Phase transition!!!

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But only disorder makes conductivity finite!!! Disordered systems Clean Disorder included Disordered

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Spectrum always gapless!!! No phase transition??? Only crossovers??? Lifshitz tail

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Anderson localization (1957) extended localized Only phase transition possible!!!

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Anderson localization (1957) extended localized Strong disorder Anderson insulator Weaker disorder Localized Extended d=3 Any disorder, d=1,2 d=3

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DoS extended Anderson Transition - mobility edges (one particle) Coexistence of the localized and extended states is not possible!!! Rules out first order phase transition

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Temperature dependence of the conductivity (no interactions) DoS MetalInsulatorPerfect one particle Insulator No singularities in any thermodynamic properties!!!

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To take home so far: Conductivity is finite only due to broken translational invariance (disorder) Spectrum (averaged) in disordered system is gapless Metal-Insulator transition (Anderson) is encoded into properties of the wave-functions

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Anderson Model Lattice - tight binding model Onsite energies i - random Hopping matrix elements I ij j i I ij -W < i

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One could think that diffusion occurs even for : Golden rule: Random walk on the lattice Pronounce words: Self-consistency Mean-field Self-averaging Effective medium ………….. ?

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is F A L S E Probability for the level with given energy on NEIGHBORING sites Probability for the level with given energy in the whole system 2d attempts Infinite number of attempts

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Perturbative Resonant pair

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Bethe lattice: INFINITE RESONANT PATH ALWAYS EXISTS

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Resonant pair Bethe lattice: INFINITE RESONANT PATH ALWAYS EXISTS Decoupled resonant pairs

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Long hops? Resonant tunneling requires:

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All states are localized means Probability to find an extended state: System size

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Order parameter for Anderson transition? Idea for one particle localization Anderson, (1958); MIT for Bethe lattice: Abou-Chakra, Anderson, Thouless (1973); Critical behavior: Efetov (1987) Metal Insulator

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Order parameter for Anderson transition? Idea for one particle localization Anderson, (1958); MIT for Bethe lattice: Abou-Chakra, Anderson, Thouless (1973); Critical behavior: Efetov (1987) Insulator Metal

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Order parameter for Anderson transition? Idea for one particle localization Anderson, (1958); MIT for Bethe lattice: Abou-Chakra, Anderson, Thouless (1973); Critical behavior: Efetov (1987) Insulator Metal

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Insulator Idea for one particle localization Anderson, (1958); MIT for Bethe lattice: Abou-Chakra, Anderson, Thouless (1973); Critical behavior: Efetov (1987) Order parameter for Anderson transition?

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metal insulator behavior for a given realization metal insulator probability distribution for a fixed energy Order parameter for Anderson transition? Idea for one particle localization Anderson, (1958); MIT for Bethe lattice: Abou-Chakra, Anderson, Thouless (1973); Critical behavior: Efetov (1987)

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Probability Distribution metal insulator Note: Can not be crossover, thus, transition!!!

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But the Andersons argument is not complete:

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On the real lattice, there are multiple paths connecting two points:

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Amplitude associated with the paths interfere with each other:

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To complete proof of metal insulator transition one has to show the stability of the metal

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Summary of Lecture # 1 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 localized Metal Insulator

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

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Resonant pair Perturbation theory in (I/W) is convergent!

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Perturbation theory in (I/W) is divergent!

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To establish the metal insulator transition we have to show the convergence of (W/I) expansion!!!

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Lecture # 2 Stability of metals and weak localization Inelastic e-e interactions in metals Phonon assisted hopping in insulators Statement of many-body localization and many- body metal insulator transition

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Why does classical consideration of multiple scattering events work? 1 2 ClassicalInterference Vanish after averaging

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Back to Drude formula Finite impurity density Drude conductivity CLASSICAL Quantum (band structure) Quantum (single impurity)

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Look for interference contributions that survive the averaging unitarity Correction to scattering crossection Phase coherence

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Additional impurities do not break coherence!!! unitarity Correction to scattering crossection

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Sum over all possible returning trajectories unitarity Return probability for classical random work

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Sometimes you may see this… MISLEADING… DOES NOT EXIST FOR GAUSSIAN DISORDER AT ALL

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Quantum corrections (weak localization) (Gorkov, Larkin, Khmelnitskii, 1979) 3D 2D 1D Finite but singular E. Abrahams, P. W. Anderson, D. C. Licciardello, and T.V. Ramakrishnan, (1979) Thouless scaling + ansatz:

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2D 1D Metals are NOT stable in one- and two dimensions Localization length: Drude + corrections Anderson model,

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Exact solutions for one-dimension x U(x) N ch Gertsenshtein, Vasilev (1959) N ch =1

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Exact solutions for one-dimension x U(x) N ch Efetov, Larkin (1983) Dorokhov (1983) N ch >>1 Strong localization Weak localization Universal conductance fluctuations Altshuler (1985); Stone; Lee, Stone (1985)

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Other way to analyze the stability of metal metal insulator Metal ??? Explicit calculation yields: Metal is unstable

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To take home so far: Interference corrections due to closed loops are singular; For d=1,2 they diverges making the metalic phase of non-interacting particles unstable; Finite size system is described as a good metal, if, in other words For, the properties are well described by Anderson model with replacing lattice constant.

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Regularization of the weak localization by inelastic scatterings (dephasing) e-h pair Does not interfere with

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Regularization of the weak localization by inelastic scatterings (dephasing) e-h pair But interferes with e-h pair

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Phase difference:

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e-h pair Phase difference: - length of the longest trajectory;

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Inelastic rates with energy transfer

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Electron-electron interaction Altshuler, Aronov, Khmelnitskii (1982) Significantly exceeds clean Fermi-liquid result

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Almost forward scattering: diffusive Ballistic

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To take home so far: Interference corrections due to closed loops are singular; For d=1,2 they diverges making the metalic phase of non-interacting particles unstable; Interactions at finite T lead to finite System at finite temperature is described as a good metal, if, in other words For, the properties are well described by ??????

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Transport in deeply localized regime

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Inelastic processes: transitions between localized states (inelastic lifetime) –1 energy mismatch (any mechanism)

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

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insulator Drude metal Electron phonon Interaction does not enter

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Q: Can we replace phonons with e-h pairs and obtain phonon-less VRH? insulator Drude metal Electron phonon Interaction does not enter

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

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