Combining symmetry breaking and restoration with configuration interaction: A highly accurate many-body scheme applied to the pairing Hamiltonian

Abstract

Background: Ab initio many-body methods have been developed over the past ten years to address mid-mass nuclei. In their best current level of implementation, their accuracy is of the order of a few per cent error on the ground-state correlation energy. Recently implemented variants of these methods are operating a breakthrough in the description of medium-mass open-shell nuclei at a polynomial computational cost while putting state-of-the-art models of inter-nucleon interactions to the test.Purpose: As progress in the design of inter-nucleon interactions is made, and as questions one wishes to answer are refined in connection with increasingly available experimental data, further efforts must be made to tailor many-body methods that can reach an even higher precision for an even larger number of observable/quantum states/nuclei. It is the objective of the present work to contribute to such a quest by designing and testing a new many-body scheme. Methods:We formulate a truncated configuration interaction method that consists of diagonalizing the Hamiltonian in a highly truncated subspace of the total N -body Hilbert space. The reduced Hilbert space is generated via the particle-number projected BCS state along with projected seniority-zero two and four quasi-particle excitations. Furthermore, the extent by which the underlying BCS state breaks U (1) symmetry is optimized in presence of the projected two and four quasi-particle excitations. This constitutes an extension of the so-called restricted variation after projection method in use within the frame of multi-reference energy density functional calculations. The quality of the newly designed method is tested against exact solutions of the so-called attractive pairing Hamiltonian problem.Results: By construction, the method reproduce exact results for N = 2 and N = 4. For N = (8,16,20), the error on the ground-state correlation energy is less than (0.006, 0.1, 0.15) % across the entire range of inter-nucleon coupling defining the pairing Hamiltonian and driving the normal-to-superfluid quantum phase transition. The presently proposed method offers the advantage to automatically access the low-lying spectroscopy, which it does with high accuracy. Conclusions:The numerical cost of the newly designed variational method is polynomial (N 6 ) in system size. It achieves an unprecedented accuracy on the ground-state correlation energy, effective pairing gap and onebody entropy as well as on the excitation energy of low-lying states of the attractive pairing Hamiltonian. This constitutes a strong enough motivation to envision its application to realistic nuclear Hamiltonians in view of providing a complementary, accurate and versatile ab initio description of mid-mass open-shell nuclei in the future.

Geminal-based configuration interaction

“…Our geminal CI bears close resemblance to some methods which merge correlation and symmetry projection, essentially because AGP is a symmetry-projected meanfield method (AGP is equivalent to number-projected BCS). Although the two methods are distinct, geminal CI is particularly closely related to the projected configuration interaction of Ripoche et al 15 (and also to a spinprojected analog due to Tshuchimochi and Ten-no 16,17 ). A coupled-cluster generalization of our geminal CI would likewise approximate symmetry-projected coupled cluster theory.…”

Variational method for integrability-breaking Richardson-Gaudin models

“…While such models are solvable by U (1)-breaking BCS mean-field theory in the thermodynamic limit, it is important to obtain an accurate description for mediumsize systems as well 46,108,109 . In fact, it has been shown that the Richardson-Gaudin equations are equivalent to the BCS mean-field equations for thermodynamically large systems, and as such the BCS wave function and the Bethe ansatz wave function coincide in this limit 53 .…”

Variational approximations to exact solutions in shell-model valence spaces: Calcium isotopes in the pf shell

“…This is in line with the conclusions drawn in Refs. [56,57] in the context of a truncated configuration-interaction method. There were, however, some discrepancies for some states belonging to many-body Hilbert spaces with large dimensions (i.e.…”