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**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, August 2012

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**Lecture # 1-2 Single particle localization**

Lecture # Many-body localization

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

Superconductor I Metal V Insulator Conductance: Conductivity:

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

Metal V Insulator Focus of The course Conductance: Conductivity:

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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**

Gapped spectrum Gapless spectrum

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**Current Metals Insulators**

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

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**But clean systems are in fact perfect conductors:**

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

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**Finite conductivity by impurity scattering**

Incoming flux Probability density Scattering cross-section One impurity

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**Finite conductivity by impurity scattering**

Finite impurity density Elastic relaxation time Elastic mean free path

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**Finite conductivity by impurity scattering**

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

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**Conductivity and Diffusion**

Finite impurity density Diffusion coefficient Einstein relation

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**Conductivity, Diffusion, Density of States (DoS)**

Einstein relation Density of States (DoS)

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**Density of States (DoS)**

Clean systems

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**Density of States (DoS)**

Clean systems Insulators, gapped Metals, gapless Phase transition!!!

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**But only disorder makes conductivity finite!!!**

Disordered systems Clean Disordered Disorder included

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**Lifshitz tail No phase transition??? Only crossovers???**

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

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**Anderson localization (1957)**

extended localized Only phase transition possible!!!

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**Anderson localization (1957)**

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

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**Anderson Transition extended - mobility edges (one particle)**

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

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**Temperature dependence of the conductivity (no interactions)**

DoS DoS DoS Metal Insulator “Perfect” 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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**{ I i and j are nearest Iij = 0 otherwise**

Anderson Model Lattice - tight binding model Onsite energies ei - random Hopping matrix elements Iij j i Iij Iij = I i and j are nearest neighbors 0 otherwise { Critical hopping: -W < ei <W uniformly distributed

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**One could think that diffusion occurs even for :**

Random walk on the lattice Golden rule: Pronounce words: Self-consistency Mean-field Self-averaging Effective medium ………….. ?

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**Infinite number of attempts**

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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Resonant pair Perturbative

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**INFINITE RESONANT PATH ALWAYS EXISTS**

Resonant pair Bethe lattice: INFINITE RESONANT PATH ALWAYS EXISTS

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**INFINITE RESONANT PATH ALWAYS EXISTS**

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

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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) 𝜈 𝑖 (𝜀)= 𝛼 𝜓 𝛼 𝑖 2 𝛿(𝜀− 𝜉 𝛼 ) 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) 𝜈 𝑖 (𝜀)= 𝛼 𝜓 𝛼 𝑖 2 𝛿(𝜀− 𝜉 𝛼 ) 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) 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) 𝜈 𝑖 (𝜀)= 𝛼 𝜓 𝛼 𝑖 2 𝛿(𝜀− 𝜉 𝛼 ) 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) metal insulator insulator h→0 metal ~ h behavior for a given realization probability distribution for a fixed energy

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**Probability Distribution**

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

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**But the Anderson’s 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 insulator metal

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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 Vanish after averaging 2 Classical Interference

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**Back to Drude formula CLASSICAL Quantum (single impurity)**

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

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**Look for interference contributions that survive the averaging**

Phase coherence 2 Correction to scattering crossection 1 2 1 unitarity

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**Additional impurities do not break coherence!!!**

2 Correction to scattering crossection 1 2 1 unitarity

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**Sum over all possible returning trajectories**

1 2 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) Finite but singular 3D 2D 1D 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**

U(x) Nch Gertsenshtein, Vasil’ev (1959) Nch =1

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**Exact solutions for one-dimension**

U(x) Nch Efetov, Larkin (1983) Dorokhov (1983) Nch >>1 Universal conductance fluctuations Altshuler (1985); Stone; Lee, Stone (1985) Weak localization Strong localization

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**Other way to analyze the stability of metal**

insulator Explicit calculation yields: Metal ??? 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)**

Does not interfere with e-h pair

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**Regularization of the weak localization by inelastic scatterings (dephasing)**

But interferes with e-h pair e-h pair

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Phase difference: e-h pair e-h pair

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**Phase difference: e-h pair e-h pair**

- length of the longest trajectory; e-h pair e-h pair

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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:**

Ballistic diffusive

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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**

energy mismatch (inelastic lifetime)–1 (any mechanism)

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**Phonon-induced hopping**

Variable Range Hopping Sir N.F. Mott (1968) energy difference can be matched by a phonon 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

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**𝜆 𝑒−𝑝ℎ ⟶ 0 ????? “metal” Drude “insulator” Electron phonon**

𝜆 𝑒−𝑝ℎ ⟶ 0 ????? Drude “metal” Electron phonon Interaction does not enter “insulator”

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**Q: Can we replace phonons with e-h pairs and obtain phonon-less VRH?**

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

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**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

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