Monte Carlo evaluation of path integrals for the nuclear shell model

We present in detail a formulation of the shell model as a path integral and Monte Carlo techniques for its evaluation. The formulation, which linearizes the two-body interaction by an auxiliary field, is quite general, both in the form of the effective `one-body' Hamiltonian and in the choice of ensemble. In particular, we derive formulas for the use of general (beyond monopole) pairing operators, as well as a novel extraction of the canonical (fixed-particle number) ensemble via an activity expansion. We discuss the advantages and disadvantages of the various formulations and ensembles and give several illustrative examples. We also discuss and illustrate calculation of the imaginary-time response function and the extraction, by maximum entropy methods, of the corresponding strength function. Finally, we discuss the "sign-problem" generic to fermion Monte Carlo calculations, and prove that a wide class of interactions are free of this limitation.Comment: 38 pages, RevTeX v3.0, figures available upon request; Caltech Preprint #MAP-15

Spin-isospin SU(4) symmetry in sd- and fp-shell nuclei

For all even-A nuclei in the sd shell we evaluate the overlap between several low-lying states, obtained by diagonalizing the realistic Wildenthal interaction, and the eigenstates of the SU(4) Casimir operator. We find that the J=0+ ground states of even-even nuclei near the middle of the shell have rather small overlaps (less than 0.5) with the lowest SU(4) Young tableaux, while the J=0+ and 1+ lowest states of odd-odd nuclei have noticeably larger overlaps (0.6–0.7). We also find that the expansion in the SU(4) Young tableaux converges quite rapidly, and that the two or three lowest tableaux usually account for more than 90% of the nuclear wave function. We then extend the calculation to the fp shell and evaluate the overlaps between the J=0+ ground states obtained by diagonalizing a realistic interaction and the SU(4) eigenstates for even-even nuclei with maximum isospin. For the fp shell the overlaps are even smaller than in the sd shell. Since we observe such sizable SU(4) symmetry breaking effects in the relatively simple sd and pf nuclei, we are rather pessimistic about the prospects of using conclusions based on SU(4) in heavier, more complex nuclei

Oblique-Basis Calculations for 44^{44}Ti

The spectrum and wave functions of $^{44}$Ti are studied in oblique-basis calculations using spherical and SU(3) shell-model states. Although the results for $^{44}$Ti are not as good as those previously reported for $^{24}$Mg, due primarily to the strong spin-orbit interaction that generates significant splitting of the single-particle energies that breaks the SU(3) symmetry, a more careful quantitative analysis shows that the oblique-basis concept is still effective. In particular, a model space that includes a few SU(3) irreducible representations, namely, the leading irrep (12,0) and next to the leading irrep (10,1) including its spin S=0 and 1 states, plus spherical shell-model configurations (SSMC) that have at least two valence nucleons confined to the $f_{7/2}$ orbit -- the SM(2) states, provide results that are compatible with SSMC with at least one valence nucleon confined to the $f_{7/2}$ orbit -- the SM(3) states.Comment: 3 pages, no figures, contribution to Computational and Group Theoretical Methods in Nuclear Physics, Playa del Carmen, Mexico, February 18-21, 200

Giant-dipole Resonance and the Deformation of Hot, Rotating Nuclei

The development of nuclear shapes under the extreme conditions of high spin and/or temperature is examined. Scaling properties are used to demonstrate universal properties of both thermal expectation values of nuclear shapes as well as the minima of the free energy, which can be used to understand the Jacobi transition. A universal correlation between the width of the giant dipole resonance and quadrupole deformation is found, providing a novel probe to measure the nuclear deformation in hot nuclei.Comment: 6 pages including 6 figures. To appear in Phys. Rev. Lett. Revtex