Anderson Localization in Quantum Chaos: Scaling and Universality

A.M. García-García Physics Department, Princeton University, Princeton, New Jersey 08544, USA and The Abdus Salam International Centre for Theoretical Physics, P.O.B. 586, 34100 Trieste, Italy and J. Wang Temasek Laboratories, National University of Singapore, 117542 Singapore and Beijing-Hong Kong-Singapore Joint Center for Nonlinear and Complex Systems, National University of Singapore, 117542 Singapore

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Received: 25 05 2007;

The one-parameter scaling theory is a powerful tool to investigate Anderson localization effects in disordered systems. In this paper we show that this theory can be adapted to the context of quantum chaos provided that the classical phase space is homogeneous, not mixed. The localization problem in this case is defined in momentum, not in real space. We then employ the one-parameter scaling theory to: (a) propose a precise characterization of the type of classical dynamics related to the Wigner-Dyson and Poisson statistics which also predicts in which situations Anderson localization corrections invalidate the relation between classical chaos and random matrix theory encoded in the Bohigas-Giannoni-Schmit conjecture, (b) to identify the universality class associated with the metal-insulator transition in quantum chaos. In low dimensions it is characterized by classical superdiffusion, in higher dimensions it has in general a quantum origin as in the case of disordered systems. We illustrate these two cases by studying 1d kicked rotors with non-analytical potentials and a 3d kicked rotor with a smooth potential.

DOI: 10.12693/APhysPolA.112.635 PACS numbers: 72.15.Rn, 71.30.+h, 05.45.Df, 05.40.-a