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 σ or λ. We map the problem to a localization problem, and find that in all space dimensions the transition occurs at σ = 1, i.e., Coulomb potential has marginal singularity. We study the critical line σ = 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 Htheorem. The RG results for σ = 1 are used to derive scaling laws for transport and to calculate critical exponents.
 Publication:

Physics Reports
 Pub Date:
 September 1997
 DOI:
 10.1016/S03701573(97)000380
 arXiv:
 arXiv:condmat/9704122
 Bibcode:
 1997PhR...288..487A
 Keywords:

 Condensed Matter
 EPrint:
 28 pages, ReVTeX, 4 EPS figures, to appear in the I. M. Lifshitz memorial volume of Physics Reports