Anderson localization in quantum chaos: scaling and universality

TL;DR

This paper adapts the one parameter scaling theory to quantum chaos, characterizing localization in momentum space and identifying universality classes for metal-insulator transitions, with applications to kicked rotor models.

Contribution

It extends the scaling theory to quantum chaos with homogeneous phase space, linking classical dynamics to localization and universality classes in momentum space.

Findings

Classical chaos relates to localization in momentum space.

Universality classes depend on dimension and classical diffusion.

Quantum effects dominate in higher-dimensional systems.

Abstract

The one parameter scaling theory is a powerful tool to investigate Anderson localization effects in disordered systems. In this paper we show 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 what situations Anderson localization corrections invalidate the relation between classical chaos and random matrix theory encoded in the Bohigas-Schmit-Giannoni 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 1$d$ kicked rotors with non analytical potentials and a 3$d$ kicked rotor with a smooth potential.