Abstract

I summarize some of my recent research on Anderson localization and the metal-insulator transition. In the first part I discuss theoretical aspects: 1. I combine a selfconsistent treatment with the one parameter scaling theory to provide explicit expressions of typical quantities characterizing the metal insulator transition in d > 2 such as the critical exponent which controls the divergence of the localization length. 2. I explore under what circumstances a band of metallic states exists in 1d systems with correlated disorder. I relate the absence of localization with a certain degree of differentiability of the random potential. The second part is devoted to applications: 1. I adapt the one parameter scaling theory to quantum chaos in order to determine in what situations Anderson localization is expected in chaotic but not strictly random potentials. Potential applications to experimental tests of localization in cold atoms physics are briefly discussed. 2. I introduce Anderson localization in the context of Quantum Chromodynamics (QCD). Then I show that at the same time that the chiral phase transition occurs the spectrum of the QCD Dirac operator undergoes a metal-insulator transition. Finally I give a list of challenging problems for the future.