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EPJ Web of Conferences

Recent developments of the projected shell model based on many-body techniques

Yang Sun, Long-Jun Wang, Fangqi Chen, T. Mizusaki, Makito Oi, Peter A. Ring

Published Jan 1, 2015• 0 citations
Publisher

Abstract

Recent developments of the projected shell model (PSM) are summarized. Firstly, by using the Pfaffian algorithm, the multi-quasiparticle configuration space is expanded to include 6-quasiparticle states. The yrast band of 166Hf at very high spins is studied as an example, where the observed third back-bending in the moment of inertia is well reproduced and explained. Secondly, an angular-momentum projected generate coordinate method is developed based on PSM. The evolution of the low-lying states, including the second 0+ state, of the soft Gd, Dy, and Er isotopes to the well-deformed ones is calculated, and compared with experimental data.

DOI
10.1051/epjconf/20159301017

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Recent developments of the projected shell model (PSM) are summarized. Firstly, by using the Pfaffian algorithm, the multi-quasiparticle configuration space is expanded to include 6-quasiparticle states. The yrast band of 166Hf at very high spins is studied as an example, where the observed third back-bending in the moment of inertia is well reproduced and explained. Secondly, an angular-momentum projected generate coordinate method is developed based on PSM. The evolution of the low-lying states, including the second 0+ state, of the soft Gd, Dy, and Er isotopes to the well-deformed ones is calculated, and compared with experimental data.
In a calculation of rotated matrix elements with angular momentum projection, the generalized Wick's theorem may encounter a practical problem of combinatorial complexity when the configurations have more than four quasiparticles (qp's). The problem can be solved by employing the Pfaffian algorithm generally applicable to calculations of matrix elements for Hartree-Fock-Bogoliubov states with any number of qp's. This breakthrough in many-body techniques enables studies of high-spin states in a shell-model framework. As the first application of the Pfaffian algorithm, the configuration space of the projected shell model is expanded to include 6-qp states for both positive and negative parities. Taking 166Hf^{166}\mathrm{Hf}166Hf as an example, we show that 6-qp states become the main configuration of the yrast band beyond spin $I\ensuremath{\approx}34\ensuremath{\hbar}$, which explains the observed third back-bending in moment of inertia. Structures of multi-qp high-KKK isomers in 176Hf^{176}\mathrm{Hf}176Hf are analyzed as another example.
We propose a quantum Monte Carlo diagonalization method (QMCD) for solving the quantum many-body interacting systems. Not only the ground state but also low-lying excited states are obtained with their wavefunctions. Consequently the level structure of low-lying states can be studied with realistic interactions. The developments in the formulation of the QMCD are described with an illustrative and intuitive example. The QMCD is finally characterized as a `importance truncation' scheme to the shell model. After testing this method for Cr, we present first results for energy levels and E2 properties of Ge, indicating its large and -soft deformation. The doubly closed shell probability of Ni is shown to be only 49% in a full pf shell calculation, in contrast to the corresponding probability of Ca which reaches 86%. The prolate deformed excited band is obtained as a result of the QMCD calculation for non-yrast states, in a good agreement to recent experimental data. This band seems to have an SU (3) (-like) structure which was originally suggested by Elliott for sd-shell nuclei.
In spite of missing dynamical correlations, the projected generator coordinate method (PGCM) was recently shown to be a suitable method to tackle the low-lying spectroscopy of complex nuclei. Still, describing absolute binding energies and reaching high accuracy eventually requires the inclusion of dynamical correlations on top of the PGCM. In this context, the present work discusses the first realistic results of a novel multi-reference perturbation theory (PGCM-PT) that can do so within a symmetry-conserving scheme for both ground and low-lying excited states. First, proof-of-principle calculations in a small (emax=4e_{\mathrm{max}}=4emax​=4) model space demonstrate that exact binding energies of closed- (\nucl{O}{16}) and open-shell (\nucl{O}{18}, \nucl{Ne}{20}) nuclei are reproduced within 0.5−1.5%0.5-1.5\%0.5−1.5% at second order, i.e. through PGCM-PT(2). Moreover, profiting from the pre-processing of the Hamiltonian via multi-reference in-medium similarity renormalization group transformations, PGCM-PT(2) can reach converged values within smaller model spaces than with an unevolved Hamiltonian. Doing so, dynamical correlations captured by PGCM-PT(2) are shown to bring essential corrections to low-lying excitation energies that become too dilated at leading order, i.e., at the strict PGCM level. The present work is laying the foundations for a better understanding of the optimal way to grasp static and dynamical correlations in a consistent fashion, with the aim of accurately describing ground and excited states of complex nuclei via ab initio many-body methods.
The shell model Monte Carlo (SMMC) method enables calculations in model spaces that are many orders of magnitude larger than those that can be treated by conventional methods, and is particularly suitable for the calculation of level densities in the presence of correlations. We review recent advances and applications of SMMC for the microscopic calculation of level densities. Recent developments include (i) a method to calculate accurately the ground-state energy of an odd-mass nucleus, circumventing a sign problem that originates in the projection on an odd number of particles, and (ii) a method to calculate directly level densities, which, unlike state densities, do not include the spin degeneracy of the levels. We calculated the level densities of a family of nickel isotopes 59−64Ni and of a heavy deformed rare-earth nucleus 162Dy and found them to be in close agreement with various experimental data sets.
Perturbative and non-perturbative expansion methods already constitute a tool of choice to perform ab initio calculations over a significant part of the nuclear chart. In this context, the categories of accessible nuclei directly reflect the class of unperturbed state employed in the formulation of the expansion. The present work generalizes to the nuclear many-body context the versatile method of Ref. \cite{burton20a} by formulating a perturbative expansion on top of a multi-reference unperturbed state mixing deformed non-orthogonal Bogoliubov vacua, i.e. a state obtained from the projected generator coordinate method (PGCM). Particular attention is paid to the part of the mixing taking care of the symmetry restoration, showing that it can be exactly contracted throughout the expansion, thus reducing significantly the dimensionality of the linear problem to be solved to extract perturbative corrections. While the novel expansion method, coined as PGCM-PT, reduces to the PGCM at lowest order, it reduces to single-reference perturbation theories in appropriate limits. Based on a PGCM unperturbed state capturing (strong) static correlations in a versatile and efficient fashion, PGCM-PT is indistinctly applicable to doubly closed-shell, singly open-shell and doubly open-shell nuclei. The remaining (weak) dynamical correlations are brought consistently through perturbative corrections. This symmetry-conserving multi-reference perturbation theory is state-specific and applies to both ground and excited PGCM unperturbed states, thus correcting each state belonging to the low-lying spectrum of the system under study.
We summarize our recent efforts to develop the Density Matrix Renormalization Group (DMRG) method into a practical truncation strategy for large-scale nuclear shell model calculations. Following an overview of the essential features of the DMRG, we discuss the changes we have implemented for its use in nuclei. In particular, we have found it useful to develop an angular-momentum conserving variant of the method (the JDMRG). We then summarize the principal results we have obtained to date, first reporting test results for 48^{48}48Cr and then more recent test results for 56^{56}56Ni. In both cases we consider nucleons limited to the 2p-1f shell. While both calculations produce a high level of agreement with the exact shell model results, the fraction of the complete space required to achieve this high level of agreement is found to go down rapidly as the size of the full space grows.