Colloquium: Random matrices and chaos in nuclear spectra

Chaos occurs in quantum systems if the statistical properties of the eigenvalue spectrum coincide with predictions of random-matrix theory. Chaos is a typical feature of atomic nuclei and other self-bound Fermi systems. How can the existence of chaos be reconciled with the known dynamical features of spherical nuclei? Such nuclei are described by the shell model (a mean-field theory) plus a residual interaction. The question is answered using a statistical approach (the two-body random ensemble): The matrix elements of the residual interaction are taken to be random variables. Chaos is shown to be a generic feature of the ensemble and some of its properties are displayed, emphasizing those which differ from standard random-matrix theory. In particular, the existence of correlations among spectra carrying different quantum numbers is demonstrated. These are subject to experimental verification.