Chaos, spins and symmetries in nuclear structure

Abstract

A pedagogic introduction to recent developments in "many-body quantum chaos", complicated dynamics at high level density in a finite quantum system with strong interaction between constituents, is given. Topics relating chaotic dynamics to the concept of symmetry are discussed: spectral statistics, extreme sensitivity to perturbations violating the symmetry, correlations between the states of different symmetry, and many examples of applications to nuclear structure. One of manifestations is the strikin~ predominance of ground states with zero spin in a system with arbitrary but rotationally invariant two-body interactions.1 Many-body quantum chaos 1.1 Does quantum chaos exist?The expression "quantum chaos" is controversial. The routine viewpoint [1] is that one' can speak only on "quantum signatures of classical chaos", and quantum chaos does not exist. Let us first try to understand what is the real meaning of such statements.Classical chaos is nowadays a well studied phenomenon that has found its way to the modern textbooks [2]. Moreover, the standard stuff of books in classical mechanics (two-body problem, small amplitude oscillations, rigid body rotation and so on) seems to cover only very exceptional cases of regular (integrable) motion. The majority of realistic problems, starting with the classical three-body problem, include domains of chaotic motion. There are different degrees of chaoticity [3], but for our purpose it is sufficient to characterize typ!cal features of classical chaotic dynamics in very general terms.The symmetry arguments play a key role in all considerations of chaos. Usually the idea of chaotic motion implies that all symmetries of the problem are destroyed. The simplest example can be that of a billiard, a two-dimensional area where a particle moves freely being elastically reflected from the boundaries. In a billiard of good symmetry, as round or rectangular, motion is integrable; it is characterized, apart from energy, by an additional integral of motion, angular momentum or linear momentum of the unfolded trajectory (Fig. 1), respectively. In coordinate space one trajectory can cover uniformly the entire available area; but in momentum space we keep the fixed constant of motion. In a billiard of arbitrary shape, such conservation laws are destroyed. In fact it suffices to match rectilinear and round pieces of the boundary (a stadium billiard, Fig. 2) in order to come to chaotic motion. Now a single trajectory covers an area in phase space.