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

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

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

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

Band metals and insulators Gapped spectrum Gapless spectrum

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

But clean systems are in fact perfect conductors: 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 Incoming flux Probability density Scattering cross-section One impurity

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

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 Insulators, gapped Metals, gapless Phase transition!!!

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

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

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

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

Temperature dependence of the conductivity DoS DoS DoS Metal Insulator “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 one-dimensional system (instability with respect to quantum corrections).