Disorder and chaos in quantum system Anderson localization
- Slides: 22
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 of metals • First glimpse into Anderson localization
Transport in solids Metal Superconductor I Insulator Conductance: Conductivity: V
Transport in solids I Metal V Insulator Focus of The course Conductance: Conductivity:
Band metals and insulators Metals Gapless spectrum Insulators Gapped spectrum
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 conserved, translational invariance) Metals Insulators
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 time
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 (Do. S) Einstein relation Density of States (Do. S)
Density of States (Do. S) Clean systems
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
Disordered Spectrum always gapless!!! No phase transition? ? ? Only crossovers? ? ?
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 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” one particle Insulator
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- and onedimensional system (instability with respect to quantum corrections).
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