Disorder and chaos in quantum system Anderson localization

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

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

Lecture # 1 • Metals and insulators – importance of disorder • Drude theory

Lecture # 1 • Metals and insulators – importance of disorder • Drude theory of metals • First glimpse into Anderson localization

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

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

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

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

Band metals and insulators Metals Gapless spectrum Insulators Gapped spectrum

Band metals and insulators Metals Gapless spectrum Insulators Gapped spectrum

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

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

Gapless spectrum Gapped spectrum But clean systems are in fact perfect conductors: (quasi-momentum is

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 One impurity Probability density Incoming flux Scattering cross-section

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

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

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

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

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 and Diffusion Finite impurity density Diffusion coefficient Einstein relation

Conductivity, Diffusion, Density of States (Do. S) Einstein relation Density of States (Do. S)

Conductivity, Diffusion, Density of States (Do. S) Einstein relation Density of States (Do. S)

Density of States (Do. S) Clean systems

Density of States (Do. S) Clean systems

Density of States (Do. S) Clean systems Metals, gapless Insulators, gapped Phase transition!!!

Density of States (Do. S) Clean systems Metals, gapless Insulators, gapped Phase transition!!!

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

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

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

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

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

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

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 Transition Coexistence of the localized and extended states is not possible!!! extended Rules

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

Temperature dependence of the conductivity Do. S Metal Do. S Insulator Do. S “Perfect”

Temperature dependence of the conductivity Do. S Metal Do. S Insulator Do. S “Perfect” one particle Insulator

To take home: • Conductivity is finite only due to broken translational invariance (disorder)

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

Next time: • Anderson transition in more detail • Non-existence of metals in two-

Next time: • Anderson transition in more detail • Non-existence of metals in two- and onedimensional system (instability with respect to quantum corrections).