The Anderson transition in quantum chaos

TL;DR

This paper explores how classical singularities influence quantum chaos, revealing a universal link between classical anomalous diffusion and quantum localization resembling an Anderson transition.

Contribution

It demonstrates that classical singularities lead to universal quantum localization properties, establishing a new universality class in quantum chaos.

Findings

Classical singularities cause anomalous diffusion in phase space.

Quantum eigenstates exhibit power-law localization with a universal exponent.

Quantum properties resemble an Anderson transition with multifractal eigenstates.

Abstract

We investigate the effect of classical singularities in the quantum properties of non-random Hamiltonians. We present explicit results for the case of a kicked rotator with a non-analytical potential though extensions to higher dimensionality or conservative systems are straightforward. It is shown that classical singularities produce anomalous diffusion in the classical phase space. Quantum mechanically, the eigenstates of the evolution operator are power-law localized with an exponent universally given by the type of classical singularity. For logarithmic singularities, the classical motion presents $1/f$ noise and the quantum properties resemble those of an Anderson transition, namely, multifractal eigenstates and critical statistics. Neither the classical nor the quantum properties depend on the details of the potential but only on the type of singularity. It is thus possible to define new universality class in quantum chaos by the relation between classical singularities (anomalous diffusion) and quantum power-law localization.