Statistical properties of many-particle spectra V. Fluctuations and symmetries

The manner in which energy-level and strength fluctuations change as a good symmetry is gradually broken is studied for a wide variety of chaotic systems describable in terms of random-matrix models. The local parameter Λ which governs the transition is identified as the mean-square symmetry-breaking matrix element in units of the local average spacing; its basic significance is clarified by exhibiting it as the “time” variable in a hierarchic set of diffusion equations. While many transitions, including Poisson → GOE, are considered, the transition GOE → GUE, from Gaussian orthogonal to Gaussian unitary ensembles, is studied in particular detail because of its connection with the breaking of time-reversal invariance (TRI), especially in complex nuclei for which the relevance of the GOE has been experimentally confirmed (GOE → GUE enters also into the study of chaotic magnetic systems). It is shown that the ensemble theory is directly applicable to real systems so that the transition equations derived for various spectral and strength measures can be applied to fluctuation data to give multiparticle bounds on the TRI-breaking part of the Hamiltonian. Analyses of spectral and strength fluctuations in the nuclear neutron-resonance and proton-resonance regions give bounds on the symmetry-breaking matrix elements as approximately one-tenth of the local average spacing. The analysis however must not stop with determining a bound on Λ; although the mere existence of a nonzero Λ could be of interest, indicating the occurrence of a symmetry breaking, the significance, if any, of an upper bound will usually not be obvious. It is therefore essential to determine, from the local multibody parameter Λ, the global (two-body) quantity α, the relative norm of the time-reversal noninvariant (TRNI) part of the nucleon-nucleon interaction. We do that in the following paper.