Origin of chaos in the spherical nuclear shell model: role of symmetries

To elucidate the mechanism by which chaos is generated in the shell model, we compare three random–matrix ensembles: the Gaussian orthogonal ensemble, French’s two–body embedded ensemble, and the two–body random ensemble (TBRE) of the shell model. Of these, the last two take account of the two–body nature of the residual interaction, and only the last, of the existence of conserved quantum numbers like spin, isospin, and parity. While the number of independent random variables decreases drastically as we follow this sequence, the complexity of the (fixed) matrices which support the random variables, increases even more. In that sense we can say that in the TBRE, chaos is largely due to the existence of (an incomplete set of) symmetries.