Weak Chaos in a Quantum Kepler Problem

Abstract

Transition from regular to chaotic dynamics in a crystal made of singular scatterers $U(r)=λ|r|^{-σ}$ 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.