Symmetry-imposed correlation in nuclear level statistics: The spin distribution

Despite long-term research, the origin of spin cutoff in the angular-momentum (spin) distribution of nuclear level densities remains incompletely elucidated. We demonstrate that this problem can be traced back to Bethe's assumption that nucleons in finite Fermi systems are independent random variables. By constructing a statistical ensemble that enforces rotational invariance through angular-momentum coupling, we obtain an analytical expression for the spin cutoff parameter, which includes a previously unidentified finite-population correction. Our results show that, even in the absence of interactions, nuclear many-body states exhibit non-negligible correlations arising from fermionic antisymmetry and angular-momentum coupling. From this perspective, spin cutoff may be interpreted as a quantitative measure of correlation imposed by symmetry in nuclear level statistics.