The geometric tensor for classical states

TL;DR

This paper introduces a classical geometric tensor using Liouville eigenfunctions, linking it to adiabatic gauge potential and Hannay curvature, and explores its role in integrability and chaos transition.

Contribution

It defines a classical geometric tensor and relates it to known classical and quantum concepts, providing new insights into integrability and chaos.

Findings

Imaginary part of the tensor relates to Hannay curvature

Singularities in the tensor link to chaos onset

Transition from integrability to chaos analyzed through formalism

Abstract

We use the Liouville eigenfunctions to define a classical version of the geometric tensor and study its relationship with the classical adiabatic gauge potential (AGP). We focus on integrable systems and show that the imaginary part of the geometric tensor is related to the Hannay curvature. The singularities of the geometric tensor and the AGP allows us to link the transition from Arnold-Liouville integrability to chaos with some of the mathematical formalism of quantum phase transitions.