Projection on particle number and angular momentum: Example of triaxial Bogoliubov quasiparticle states

Abstract

Background: Many quantal many-body methods that aim at the description of self-bound nuclear or mesoscopic electronic systems make use of auxiliary wave functions that break one or several of the symmetries of the Hamiltonian in order to include correlations associated with the geometrical arrangement of the system's constituents. Such reference states have been used already for a long time within self-consistent methods that are either based on effective valence-space Hamiltonians or energy density functionals, and they are presently also gaining popularity in the design of novel ab-initio methods. A fully quantal treatment of a self-bound many-body system, however, requires the restoration of the broken symmetries through the projection of the many-body wave functions of interest onto good quantum numbers.Purpose: The goal of this work is three-fold. First, we want to give a general presentation of the formalism of the projection method starting from the underlying principles of group representation theory. Second, we want to investigate formal and practical aspects of the numerical implementation of particle-number and angularmomentum projection of Bogoliubov quasiparticle vacua, in particular with regard of obtaining accurate results at minimal computational cost. Third, we want to analyze the numerical, computational and physical consequences of intrinsic symmetries of the symmetry-breaking states when projecting them.Methods: Using the algebra of group representation theory, we introduce the projection method for the general symmetry group of a given Hamiltonian. For realistic examples built with either a pseudo-potential-based energy density functional or a valence-space shell-model interaction, we then study the convergence and accuracy of the quadrature rules for the multi-dimensional integrals that have to be evaluated numerically and analyze the consequences of conserved subgroups of the broken symmetry groups. Results:The main results of this work are also threefold. First, we give a concise, but general, presentation of the projection method that applies to the most important potentially broken symmetries whose restoration is relevant for nuclear spectroscopy. Second, we demonstrate how to achieve high accuracy of the discretizations used to evaluate the multi-dimensional integrals appearing in the calculation of particle-number and angular-momentum projected matrix elements while limiting the order of the employed quadrature rules. Third, for the example of a point-group symmetry that is often imposed on calculations that describe collective phenomena emerging in triaxially deformed nuclei, we provide the group-theoretical derivation of relations between the intermediate matrix elements that are integrated, which permits for a further significant reduction of the computational cost of the method. These simplifications are valid whatever the number parity of the quasiparticle states and therefore can be used in the description of even-even, odd-mass, and odd-odd nuclei. Conclusions:The quantum-number...

Angular Momentum of Fission Fragments from Microscopic Theory

“…The scission configurations have axially-symmetric densities that are dumbbell-shaped and can readily be divided into heavy (z < z N ) and light (z > z N ) FFs, where z N locates the minimum of the density. By adapting standard symmetry restoration techniques [41][42][43] to the case of FFs, angular momentum distributions of the heavy (F = H) and the light fragment (F = L) for each configuration |Φ(q) can be calculated as…”