Zero-pairing limit of Hartree-Fock-Bogoliubov reference states

Abstract

Background:The variational Hartree-Fock-Bogoliubov (HFB) mean-field theory is the starting point of various (ab initio) many-body methods dedicated to superfluid systems. In this context, pairing correlations may be driven towards zero either on purpose via HFB calculations constrained on, e.g., the particle-number variance or simply because internucleon interactions cannot sustain pairing correlations in the first place in, e.g., closed-shell systems. While taking this limit constitutes a text-book problem when the system is of closed-shell character, it is typically, although wrongly, thought to be ill-defined whenever the naive filling of single-particle levels corresponds to an open-shell system. Purpose: The present work demonstrates that the zero-pairing limit of an HFB state is well-defined independently of the average particle number A it is constrained to. Still, the nature of the limit state is shown to depend on the regime, i.e., on whether the nucleus characterizes as a closed-shell or an open-shell system when taking the limit. Finally, the consequences of the zero-pairing limit on Bogoliubov many-body perturbation theory (BMBPT) calculations performed on top of the HFB reference state are illustrated. Methods: The zero-pairing limit of a HFB state constrained to carry an arbitrary (integer) number of particles A on average is worked out analytically and realized numerically using a two-nucleon interaction derived within the frame of chiral effective field theory. Results: The zero-pairing limit of the HFB state is mathematically well-defined, independently of the closedor open-shell character of the system in the limit. Still, the nature of the limit state strongly depends on the underlying shell structure and on the associated naive filling reached in the zero-pairing limit for the particle number A of interest. First, the textbook situation is recovered for closed-shell systems, i.e., the limit state is reached for a finite value of the pairing strength (the well-known BardeenCooperSchrieffer (BCS) collapse) and takes the form of a single Slater determinant displaying (i) zero pairing energy, (ii) nondegenerate elementary excitations, and (iii) zero particle-number variance. Contrarily, a nonstandard situation is obtained for open-shell systems, i.e., the limit state is only reached for a zero value of the pairing interaction (no BCS collapse) and takes the form of a specific finite linear combination of Slater determinants displaying (a) a nonzero pairing energy, (b) degenerate elementary excitations, and (c) a nonzero particle-number variance for which an analytical formula is derived. This nonzero particle-number variance acts as a lower bound that depends in a specific way on the number of valence nucleons and on the degeneracy of the valence shell. All these findings are confirmed and illustrated numerically. Last but not least, BMBPT calculations of closed-shell (open-shell) nuclei are shown to be well-defined (ill-defined) in the zero-pairing limit. Conclusions: While HFB theory has been in...

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

“…At the same time, the contribution of BMBPT( 3) is enlarged. 20 The numerical solution of the PHFB-PT(2) linear system is very stable in the present example such that a small complex shift (γ = 1 MeV) can be used safely. The precision error on PHFB-PT(2) energies is essentially invisible in Fig.…”

“…Second, the PGCM depends on a choice of suitable collective coordinates that must be rich enough to capture all non-perturbative static correlations at play, only leaving weak perturbative corrections to the subsequent PGCM-PT step, while maintaining a low-enough dimensionality to retain its advantage over large-scale diagonalization methods. For example, while adding the triaxial degree of freedom did not impact the dilated PGCM spectrum of 20 Ne at s = 10, 20 MeV −1 , the use of HFB states obtained while adding a cranking constraint breaking time-reversal invariance [27,28] typically compresses the PGCM spectrum as demonstrated in MR-EDF calculations and in recent ab initio studies [26,29]. Obtaining such a compression at the PGCM level is expected to correlate with a further suppression of dynamical correlations on top of the PGCM step.…”

“…As a next step, further static correlations are captured via the restoration of angular momentum and the inclusion of shape fluctuations through the PGCM. As demonstrated in Paper II, the description of 20 Ne strongly benefits from breaking and restoring parity as well as the inclusion of octupole shape fluctuations. Our present calculations are however restricted to axial quadrupole deformation, leaving some room for further improvement in the future.…”

“…C.3 for an illustration. The ground-state TECs of 20 Ne are displayed in Fig. 6 as a function of the axial quadrupole deformation 13 β 2 of the (underlying) HFB vacua.…”

“…The doubly open-shell 20 Ne constitutes a richer and more instructive example. Figure 12 Schrödinger's equation at the PGCM-PT(2) level.…”