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Mathematics and Physics of Anderson localization: 50 Years After

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14th July 2008 to 19th December 2008
Yan Fyodorov University of Nottingham, Queen Mary, University of London
Ilya Goldsheid Queen Mary, University of London
Tom Spencer [Institute for Advanced Study], Institute for Advanced Study, Princeton
Martin Zirnbauer [Institut für Theoretische Physik, Köln], Universität zu Köln


Programme Theme

In his seminal paper Absence of diffusion in certain random lattices (1958) Philip W. Anderson discovered one of the most striking quantum interference phenomena: particle localization due to disorder. Cited in 1977 for the Nobel prize in physics, that paper was fundamental for many subsequent developments in condensed matter theory. In particular, in the last 25 years the phenomenon of localization proved to be crucial for the understanding of the Quantum Hall effect, mesoscopic fluctuations in small conductors as well as some aspects of quantum chaotic behaviour.

Random Schrödinger operators are an area of very active research in mathematical physics and mathematics. Here the main effort is to clarify the nature of the underlying spectrum. In particular, it has been proved that in dimension one all states are localized, and in any dimension the random Schrödinger operator has dense point spectrum for large enough disorder. Some open mathematical problems of major importance include the long-time evolutions of a quantum particle in a weakly disordered medium and existence of absolutely continuous spectrum in three dimensions. The expected transition from localized (point spectrum) to extended eigenstates (absolutely continuous spectrum) will also be addressed.

The goal of the program is to bring together the world leaders in spectral theory of random Schrödinger operators and theoretical physicists successfully working on the problem of Anderson localization. Among the topics that will be addressed during the program are: The nature of critical phenomena associated with localization-delocalization transitions; The existence and statistical properties of extended states for D > 2 and the behaviour in the critical dimension D = 2; rigorous version of supersymmetric methods and of the nonlinear σ-model techniques; the localization-delocalization phenomena associated with the Integer Quantum Hall effect; rigorous mathematical understanding of the relation between magnetic Schrödinger operators and network models, and the connection with quantum percolation; localization in the presence of a random magnetic field; behaviour of products of random matrices and associated Lyapunov exponents; localization and delocalization in disordered systems characterised by non-selfadjoint operators; dynamical localization in Quantum Chaotic systems; localization in systems with aperiodic potential, as well as in models with correlated or long-ranged disorder; localization in systems with nonlinearities, and localization-delocalization phenomena in disordered systems of interacting quantum particles.

Final Scientific Report: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons