Disorder and chaos in quantum system Anderson localization

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

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

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

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

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

Resonant pair Bethe lattice: INFINITE RESONANT PATH ALWAYS EXISTS

Resonant pair Bethe lattice: Decoupled resonant pairs

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 h!0 insulator metal ~h behavior for a given realization probability distribution for a fixed energy

Probability Distribution metal Note: 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 Finite impurity density CLASSICAL Quantum (single impurity) Drude conductivity Quantum (band structure)

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

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

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

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

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

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

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

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