Incommensurate standard map

TL;DR

This paper introduces the incommensurate standard map, extending the classical and quantum dynamics to systems with multiple incommensurate frequencies, revealing complex behaviors like boundedness, unbounded diffusion, and localization transitions.

Contribution

It presents the first study of the incommensurate standard map with multiple harmonics, analyzing both classical and quantum behaviors, including localization and metal-insulator transitions.

Findings

Classical dynamics bounded by KAM surfaces at low kick amplitudes.

Quantum system exhibits a metal-insulator transition in space.

Quantum localization in momentum akin to 2D Anderson localization.

Abstract

We introduce and study the extension of the Chirikov standard map when the kick potential has two and three incommensurate spatial harmonics. This system is called the incommensurate standard map. At small kick amplitudes the dynamics is bounded by the isolating Kolmogorov-Arnold-Moser surfaces while above a certain kick strength it becomes unbounded and diffusive. The quantum evolution at small quantum kick amplitudes is somewhat similar to the case of Aubru-Andr\'e model studied in mathematics and experiments with cold atoms in a static incommensurate potential. We show that for the quantum map there is also a metal-insulator transition in space while in momentum we have localization similar to the case of 2D Anderson localization. In the case of three incommensurate frequencies of space potential the quantum evolution is characterized by the Anderson transition similar to 3D case of disordered potential. We discuss possible physical systems with such map description including dynamics of comets and dark matter in planetary systems.