Shell Model Calculations with Realistic Interactions David J. Dean and Steven E. Koonin, California Institute of Technology Recent nuclear structure computations done on the Delta yielded the first simulation results that agree with experiments and are a major step toward solving a problem that has been in existence for 20 years. Many studies of nuclear structure are based on the shell model. This is analogous to the more familiar atomic shell model, but the nuclear case is complicated by a strong two-body residual interaction that moves nucleons among the orbitals and is a major determinant of nuclear properties. The shell model, traditionally formulated as a matrix-diagonalization problem, has been very successful in describing nuclei lighter than calcium, but, unfortunately, the size of the matrix to be diagonalized grows combinatorially with both the number of orbitals and the number of nucleons that occupy them. Thus, moderately sized nuclei such as iron require matrices of the order of 109, for which diagonalization is not currently possible.
Recent advances allow the solution of much larger shell model problems.
The advantage of this method is that the numerical work then grows only as a modest power of the problem size. The Hamiltonian includes pairing and quadrupole terms appropriate for these nuclei. As is seen from the plots, deformation clearly sets in with decreasing temperature. The sign of the field integrand can vary from sample to sample and the net integral results from a delicate cancellation that is difficult to reproduce with a finite number of samples. We have found that all realistic nuclear shell model Hamiltonians suffer from a sign problem.