Disorder and chaos in quantum system: Anderson localization and its generalization (6 lectures) Boris Altshuler (Columbia) Igor Aleiner (Columbia)

Lecture # 2 Stability of insulators and Anderson transition Stability of metals and weak localization

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

{ 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) 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) 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) 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!!!

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

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

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

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

Quantum corrections (weak localization) (Gorkov, Larkin, Khmelnitskii, 1979) Finite but singular 3D 2D 1D

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

We learned today: How to investigate stability of insulators (locator expansion). How to investigate stability of metals (quantum corrections) For d=3 stability of both phases implies metal insulator transition; The order parameter for the transition is the distribution function For d=1,2 metal is unstable and all states are localized

Next time: Inelastic transport in insulators