Weak Chaos in a Quantum Kepler Problem

TL;DR

This paper investigates the transition from regular to chaotic dynamics in a quantum system with singular scatterers, identifying a critical Coulomb-like potential at sigma=1 and analyzing its universality and scaling laws.

Contribution

It introduces a renormalization group approach to study the critical transition at sigma=1 in a quantum Kepler problem, revealing universality classes and critical exponents.

Findings

Transition occurs at sigma=1, the Coulomb potential's marginal singularity.

RG flow leads to a universal stationary distribution of coupling parameters.

Scaling laws for transport and critical exponents are derived at the transition.

Abstract

Transition from regular to chaotic dynamics in a crystal made of singular scatterers $U(r)=\lambda |r|^{-\sigma}$ can be reached by varying either sigma or lambda. We map the problem to a localization problem, and find that in all space dimensions the transition occurs at sigma=1, i.e., Coulomb potential has marginal singularity. We study the critical line sigma=1 by means of a renormalization group technique, and describe universality classes of this new transition. An RG equation is written in the basis of states localized in momentum space. The RG flow evolves the distribution of coupling parameters to a universal stationary distribution. Analytic properties of the RG equation are similar to that of Boltzmann kinetic equation: the RG dynamics has integrals of motion and obeys an H-theorem. The RG results for sigma=1 are used to derive scaling laws for transport and to calculate critical exponents.