Ab initio Bogoliubov coupled cluster theory for open-shell nuclei

Background: Ab initio many-body methods have been developed over the past 10 yr to address closed-shell nuclei up to mass $A\approx 130$ on the basis of realistic two- and three-nucleon interactions. A current frontier relates to the extension of those many-body methods to the description of open-shell nuclei. Several routes to address open-shell nuclei are currently under investigation, including ideas that exploit spontaneous symmetry breaking.

Purpose: Singly open-shell nuclei can be efficiently described via the sole breaking of U(1) gauge symmetry associated with particle-number conservation as a way to account for their superfluid character. While this route was recently followed within the framework of self-consistent Green's function theory, the goal of the present work is to formulate a similar extension within the framework of coupled cluster theory.

Methods: We formulate and apply Bogoliubov coupled cluster (BCC) theory, which consists of representing the exact ground-state wave function of the system as the exponential of a quasiparticle excitation cluster operator acting on a Bogoliubov reference state. Equations for the ground-state energy and the cluster amplitudes are derived at the singles and doubles level (BCCSD) both algebraically and diagrammatically. The formalism includes three-nucleon forces at the normal-ordered two-body level. The first BCC code is implemented in $m$ scheme, which will permit the treatment of doubly open-shell nuclei via the further breaking of SU(2) symmetry associated with angular momentum conservation.

Results: Proof-of-principle calculations in an $N_{\text{max}}=6$ spherical harmonic oscillator basis for ${}^{16,18}\mathrm{O}$ and ${}^{18}\mathrm{Ne}$ in the BCCD approximation are in good agreement with standard coupled cluster results with the same chiral two-nucleon interaction, while ${}^{20}\mathrm{O}$ and ${}^{20}\mathrm{Mg}$ display underbinding relative to experiment. The breaking of U(1) symmetry, monitored by computing the variance associated with the particle-number operator, is relatively constant for all five nuclei, in both the Hartree-Fock-Bogoliubov and BCCD approximations.

Conclusions: The newly developed many-body formalism increases the potential span of ab initio calculations based on single-reference coupled cluster techniques tremendously, i.e., potentially to reach several hundred additional midmass nuclei. The new formalism offers a wealth of potential applications and further extensions dedicated to the description of ground and excited states of open-shell nuclei. Short-term goals include the implementation of three-nucleon forces at the normal-ordered two-body level. Midterm extensions include the approximate treatment of triples corrections and the development of the equation-of-motion methodology to treat both excited states and odd nuclei. Long-term extensions include exact restoration of U(1) and SU(2) symmetries.

Figure 1

Normal-ordered contributions to the grand-canonical potential in diagrammatic form. The first line corresponds to the ${\mathrm{\Omega }}^{\left[2\right]}$ terms, the second line to the ${\mathrm{\Omega }}^{\left[4\right]}$ terms, and the final two lines to the ${\mathrm{\Omega }}^{\left[6\right]}$ terms, which are neglected in the present work.

Figure 2

Singly $\left({\mathcal{T}}_1\right)$, doubly $\left({\mathcal{T}}_2\right)$, and triply $\left({\mathcal{T}}_3\right)$ excited quasiparticle cluster operators in diagrammatic form. In the present work, ${\mathcal{T}}_3$ is neglected.

Figure 3

Contributions to $\mathrm{\Delta }{\mathrm{\Omega }}_0$ in diagrammatic form. Excluding ${\mathrm{\Omega }}^{\left[6\right]}$ terms, these three diagrams provide an exact form for the correction to the unperturbed energy, independent of the truncation imposed on $\mathcal{T}$.

Figure 4

Diagrammatic representation of the single-excitation amplitude equations in the BCCSD approximation.

Figure 5

Diagrammatic representation of the double-excitation amplitude equations in the BCCSD approximation.

Figure 6

Explicit labeling of two diagrams for the double-excitation amplitude equations in BCCSD.

Figure 7

Comparison of CCSD and CCD calculations for ${}^{16}\mathrm{O}$ with $N_{\text{max}}=6$ for a variety of HO bases given by $\hslash \omega$. The points denote total energies in the CCSD approximation, with a minimum observed at $-119.211\mathrm{MeV}$ for $\hslash \omega =26\mathrm{MeV}$, while the solid curve provides corresponding results in the more restrictive CCD approximation. The horizontal dotted line represents the extrapolated energy (see text), which is indistinguishable in the CCSD and CCD approximations on the scale of the figure, while the horizontal solid line is the experimental energy.

Figure 8

CCD calculations for ${}^{16}\mathrm{O}$ with $N_{\text{max}}=6$ as a function of $L$. The points denote total energies, with a minimum observed at $-119.110\mathrm{MeV}$ for $\hslash \omega =26\mathrm{MeV}$. Four points, from the calculations with $\hslash \omega =50,53,55,58\mathrm{MeV}$, are fit to Eq. (54) and represented by the solid curve. The horizontal dotted line is the extrapolated energy $E_{\infty }$, while the horizontal solid line is the experimental energy.

Figure 9

BCCD calculations for ${}^{18}\mathrm{O}$ (black circles) and ${}^{18}\mathrm{Ne}$ (red crosses) with $N_{\text{max}}=6$ as a function of $L$. In each nucleus, four points, from the calculations with $\hslash \omega =50,53,55,58\mathrm{MeV}$, are fit to Eq. (54) and represented by the solid curve. The open symbols correspond to results at $\hslash \omega =26\mathrm{MeV}$. The horizontal dotted line is the extrapolated energy $E_{\infty }$. See text for additional details.

Figure 10

BCCD calculations for ${}^{20}\mathrm{O}$ (black circles) and ${}^{20}\mathrm{Mg}$ (red crosses). See caption to Fig. 9 for details.

Figure 11

Diagrammatic representation of the single-excitation amplitude equations in the quasilinear BCCSD approximation.

Figure 12

Diagrammatic representation of the double-excitation amplitude equations in the quasilinear BCCSD approximation.

Figure 13

Diagrammatic representation of the intermediates that enter into the amplitude equations of the quasilinear BCCSD approximation. Note that ${\chi }_{k_1{k}_2}^{11}$ and ${\chi }_{k_1{k}_2}^{11\text{a}}$ have an identical form diagrammatically, but their symmetry factors (i.e., the factor in front of the diagram) are different. To differentiate the two, ${\chi }_{k_1{k}_2}^{11}$ is drawn with a squiggly line to the right, as shown here, while ${\chi }_{k_1{k}_2}^{11\text{a}}$ has a squiggly line to the left, as in Fig. 12.