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

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 percent 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 internucleon interactions to the test.

Purpose: As progress in the design of internucleon 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 or nuclei. The objective of the present work is 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-quasiparticle excitations. Furthermore, the extent by which the underlying BCS state breaks $\text{U}\left(1\right)$ symmetry is optimized in the presence of the projected two- and four-quasiparticle excitations. This constitutes an extension of the so-called restricted variation after projection method in use within the frame of multireference 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 reproduces exact results for $N=2$ and $N=4$. For $N=(8,16,20)$, the error in the ground-state correlation energy is less than (0.006%, 0.1%, 0.15%) across the entire range of internucleon coupling defining the pairing Hamiltonian and driving the normal-to-superfluid quantum phase transition. The presently proposed method offers the advantage of automatic access to 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. This method achieves unprecedented accuracy for the ground-state correlation energy, effective pairing gap, and one-body entropy as well as for the excitation energy of low-lying states of the attractive pairing Hamiltonian. This constitutes a sufficiently strong 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.

Figure 1

Error ${(\mathrm{\Delta }E/E)}_c$ in the ground-state correlation energy as a function of $g$ for $N=\mathrm{\Omega }=20$. Results are shown for (i) second-order particle-number unprojected MBPT energy based on a projective formula (gray dotted line), (ii) second-order particle-number unprojected MBPT energy based on a Hermitian expectation value formula (blue dashed-dotted line), (iii) second-order particle-number projected ${\mathrm{MBPT}}_N$ energy based on a Hermitian expectation value formula (green solid line), and (iv) for VAP-BCS [58] (brown dashed line). The critical value $g_c$ is indicated by the black dashed vertical line.

Figure 2

${(\mathrm{\Delta }E/E)}_c$ as a function of $g$ for $N=\mathrm{\Omega }=16$. Results are shown for the truncated CI (red solid line), ${\mathrm{PoST}}_{\alpha }$ [59] (dark blue dashed line), and ${\mathrm{PoST}}_x$ [59] (yellow dashed-dotted line) calculations.

Figure 3

Eigenvalues $n_N^{\zeta }\left(g\right)$ (relative to the largest of them) of the overlap matrix [cf. Eq. (16)] ordered in increasing values for $N=16$ and $g/\mathrm{\Delta }e=0.8$. A logarithmic scale is used for the vertical axis. Results are shown for $|{\mathrm{\Psi }}_N\left(g\right)\rangle$ made of projected 0qp, 2qp, and 4qp configurations (red solid line), made of projected 0qp and 4qp configurations (green filled circles) or made of projected 0qp and 2qp configurations (purple dashed line).

Figure 4

${(\mathrm{\Delta }E/E)}_c$ from truncated CI calculations as a function of $g$ for $N=\mathrm{\Omega }=16$. Upper panel shows results for the full set of projected 0qp, 2qp, and 4qp configurations (red solid line) as well as using 0qp and 4qp (green filled circles) or 0qp and 2qp (purple dashed line) configurations only. Lower panel shows results for 0qp and 4qp configurations (green filled circles) as well as for 4qp configurations only (pink dashed-dotted line).

Figure 5

${(\mathrm{\Delta }E/E)}_c$ as a function of $g$ for $N=\mathrm{\Omega }=16$. Results are shown for the the VAP-BCS (brown dashed line), the standard PAV-BCS (dark red solid line), and for a PAV-BCS calculation based on an optimized order parameter (dark green dashed-dotted line). The latter corresponds to the RVAP method.

Figure 6

Total binding energy from truncated CI calculations based on projected 0qp, 2qp, and 4qp configurations as a function of $g_{\mathrm{aux}}$. The calculations are performed for $N=\mathrm{\Omega }=16$ and the specific value of $g$ indicated by the arrow. (a) $g=0.15<g_c$. (b) $g=0.4>g_c$. (c) $g=0.8\gg g_c$.

Figure 7

Same as Fig. 2 but with the optimal order parameter $g_{\mathrm{opt}}$ defining the basis at each value of the coupling $g$ for the truncated CI (light blue solid line). Results of ${\mathrm{PoST}}_{\alpha }$ (dark blue dashed line) and PoSTx [59] (yellow dashed-dotted line) are shown for comparison.

Figure 8

Same as Fig. 7 for (a) $N=\mathrm{\Omega }=8$ and (b) $N=\mathrm{\Omega }=20$.

Figure 9

Ground-state effective pairing gap [Eq. (20)] as a function of $g$ for $N=16$. Top panel shows exact results (black solid line) against BCS (purple dashed line), PAV-BCS (red dot-dashed line), and ${\mathrm{MBPT}}_N$ (green filled squares). Lower panel shows exact results against truncated CI based on nonoptimized (red cross) or optimized (blue circles) projected 0qp, 2qp, and 4qp configurations.

Figure 10

Same as Fig. 9 for the one-body entropy.

Figure 11

Low-lying excitation spectrum as a function of $g$ obtained with $N=\mathrm{\Omega }=16$ and $g_{\mathrm{aux}}=g$. Exact results are compared with truncated CI calculations based on projected 0qp, 2qp, and 4qp configurations. Upper panel shows absolute energies of the 10 lowest excited states for exact (black solid lines) and truncated CI (symbols) results. Lower panel shows relative errors on excitation energies for the five lowest excited states.

Figure 12

Same as Fig. 11 for $g_{\mathrm{aux}}=g_{\mathrm{opt}}$.