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, 14-26 August 2012

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

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

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

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 … )

Band metals and insulators Gapped spectrum Gapless spectrum

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

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

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

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

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

Conductivity and Diffusion Finite impurity density Diffusion coefficient Einstein relation

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

Density of States (DoS) Clean systems

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

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

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

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

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

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

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

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

{ 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

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 ………….. ?

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

Resonant pair Perturbative

INFINITE RESONANT PATH ALWAYS EXISTS Resonant pair Bethe lattice: INFINITE RESONANT PATH ALWAYS EXISTS

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

Long hops? Resonant tunneling requires:

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

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

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

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

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

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

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

But the Anderson’s argument is not complete:

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

Amplitude associated with the paths interfere with each other:

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

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

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

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

Perturbation theory in (I/W) is divergent!

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

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

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

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

Look for interference contributions that survive the averaging Phase coherence 2 Correction to scattering crossection 1 2 1 unitarity

Additional impurities do not break coherence!!! 2 Correction to scattering crossection 1 2 1 unitarity

Sum over all possible returning trajectories 1 2 unitarity Return probability for classical random work

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

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:

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

Exact solutions for one-dimension U(x) Nch Gertsenshtein, Vasil’ev (1959) Nch =1

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

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

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.

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

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

Phase difference: e-h pair e-h pair

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

Inelastic rates with energy transfer

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

Almost forward scattering: Ballistic diffusive

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 ??????

Transport in deeply localized regime

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

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

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

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

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