Multi-reference many-body perturbation theory for nuclei III -- Ab initio calculations at second order in PGCM-PT

Abstract

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, proofof-principle calculations in a small (e max = 4) model space demonstrate that exact binding energies of closed-( 16O) and open-shell ( 18 O, 20 Ne) nuclei are reproduced within 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,

Gorkov algebraic diagrammatic construction formalism at third order

“…( 12) may hold only in a very approximate way. Because both Hamiltonians H and Ω preserve particle number, the requirements (10) and (11) will force |Ψ 0 to be the true ground state |ψ N 0 , with an exact number of particles. The breaking of particle-number symmetry arises naturally, in most cases, whenever approximations have to be made, typically in computing the self-energy.…”

“…Following the above considerations we formulate Gorkov SCGF theory with respect to the ground state of Ω as defined by Eqs. (10) and (11). Since |Ψ 0 breaks particle number symmetry, it is possible to define four Gorkov propagators,…”

“…Following the above discussion one concludes that it is sufficient to derive ADC(3) expressions of the coupling and interaction matrices associated with one particular Gorkov self-energy. While the diagrams contributing to Σ 11 (ω) are presently employed, the other self-energies, Eqs. (29b-29d) were checked to lead to the same results.…”

“…Hence, preserving Eq. (67) will automatically preserve the pairing gaps, the density matrix and all one-body observables, including the average particle number (11) and the point-particle density distributions. Likewise, the Koltun energy sum rule (24) can be expressed in terms of the p = 0, 1 moments…”

“…The possible ways around this issue are either multi-reference approaches or the use of symmetrybreaking reference states. In the first case, all degenerate configurations are diagonalized explicitly, which however adds a costly step to the calculation that scales exponentially with system's size [8], with the notable exception of a recently proposed multi-reference many-body perturbation theory [9][10][11]. The second path relies on using a reference state that explicitly breaks some symmetries of the Hamiltonian, in exchange for lifting the energy degeneracy.…”