Transcript Slide 1
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 I Superconductor V Insulator 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 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 (DoS) Einstein relation Density of States (DoS) Density of States (DoS) Clean systems Density of States (DoS) 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 Weaker disorder d=3 Localized Extended Localized Anderson insulator Anderson Transition Coexistence of the localized and extended states is not possible!!! extended Rules out first order phase transition DoS - mobility edges (one particle) Temperature dependence of the conductivity DoS Metal DoS Insulator DoS “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).