Boris Altshuler. Columbia University, New York, USA

Anderson Localization and Beyond (15h)

1. Introduction.

History of the problem. Einstein relation. Diffusion and “memory”. Experimental evidences of the localization.

2. Anderson Model.

Models of Disorder. Anderson model. Von Neumann & Wigner “no crossing rule”. Resonances and qualitative physical picture. Localization in high dimensions–Anderson’s arguments.

3. Localization at low dimensions. Scaling theory.

History of the localization in one and two dimensions. Thouless energy and dimensionless Thouless conductance. Intuitive scaling picture. β–function and its asymptotic behavior.

4. Weak Localization.

Physics of the 2D localization. Quantum corrections to the conductivity. AC conductivity. Dephasing rate. Temperature dependence of the conductivity.

5. Anomalous Magnetoresistance.

History of the problem. Localization in the presence of magnetic field. Aharonov–Bohm effect.

6. Theory of the dephasing.

Sources of the dephasing. Magnetic impurities. Inelastic collisions.

7. Mesoscopic (sample to sample) fluctuations.

8. Interaction between the electrons in the weak localization regime.

9. Spectral Statistics and Localization.

Random Matrix theory. “Repulsion” of the energy levels. Wigner-Dyson and Poisson statistics. Dyson ensembles. Spectral statistics as signatures of the localization.

10. Localization beyond the real space.

Examples. Localization and quantum chaos. Quantum localization and KAM–theorem for the classical dynamical systems.

11. Many-Body Localization.

General remarks and examples. Generalized Ising model and localization on the hypercube. Irreversibility.

12. Many-Body Localization of the interacting fermions.

Extended Anderson model. Idea of the calculation. Hoping conductivity and cascades. Ergodic and non-ergodic metals.

13. Many-Body localization of weakly interacting bosons.

Superfluid-Glass transition in the presence of disorder. Normal fluid at finite temperatures.