Exact solutions of the nuclear shell-model secular problem: Discrete Non-Orthogonal Shell Model within a Variation After Projection approach

TL;DR

This paper demonstrates that non-orthogonal Slater determinants, used within a variational after projection approach, can exactly solve the nuclear shell-model secular problem, capturing pairing correlations and outperforming traditional methods in certain cases.

Contribution

It introduces a discrete non-orthogonal shell model framework that provides exact solutions and improved accuracy for low-lying nuclear states using a variational approach with non-orthogonal wavefunctions.

Findings

Exact solutions for low-lying states in shell-model spaces.

Full capture of pairing correlations via Slater determinants.

Convergence to lower energies than traditional diagonalization methods.

Abstract

We investigate the capacity of non-orthogonal many-body expansions in the resolution of the nuclear shell-model secular problem. Exact shell-model solutions are obtained within the variational principle using non-orthogonal Slater determinants as the variational ansatz. These results numerically prove the realization of the Broeckhove-Deumens theorem on the existence of a discrete set of non-orthogonal wavefunctions that exactly span the full shell-model space for low-lying states of interest. With the angular-momentum variation after projection, pairing correlations are shown to be fully captured by Slater determinants as exemplified in the backbending phenomenon occurred in $^{48}$Cr. The resulting discrete non-orthogonal shell model developed in such variation after projection method is further examined in the case of $^{78}$Ni, an exotic doubly magic nucleus at the edge of currently feasible diagonalization limits. Its ground state binding energy is shown to converge to a lower value than the largest large-scale shell-model diagonalization ever done by the conventional tridiagonal Lanczos method, revealing an outstanding performance of non-orthogonal Slater determinantal wavefunctions to describe the eigensolutions of shell-model Hamiltonians.