Combining symmetry breaking and restoration with configuration interaction: Extension to $z$-signature symmetry in the case of the Lipkin model

Background: Ab initio many-body methods whose numerical cost scales polynomially with the number of particles have been developed over the past fifteen years to tackle closed-shell mid-mass nuclei. Open-shell nuclei have been further addressed by implementing variants based on the concept of spontaneous symmetry breaking (and restoration). These methods typically access ground states properties and some restricted aspects of spectroscopy.

Purpose: In order to access the spectroscopy of open-shell nuclei more systematically while controlling the numerical cost, we design a novel many-body method that combines the merit of breaking and restoring symmetries with those brought about by low-rank individual excitations.

Methods: The recently proposed truncated configuration-interaction method based on optimized symmetry-broken and -restored states is extended to the $\text{SU}\left(2\right)$ group associated with total angular momentum. Dealing more specifically with the Lipkin Hamiltonian, the present study focuses on the breaking and the restoration of the $z$-signature symmetry associated with a discrete subgroup of $\text{SU}\left(2\right)$. The highly-truncated $N$-body Hilbert subspace within which the Hamiltonian is diagonalized is spanned by a $z$-signature broken and restored Slater determinant vacuum and associated low-rank, e.g., one-particle/one-hole and two-particle/two-hole, excitations. Furthermore, the extent by which the symmetry-unrestricted vacuum breaks $z$-signature symmetry is optimized in the presence of projected low-rank particle-hole excitations. The quality of the method is gauged against exact ground- and excited-state eigenenergies for a large range of values of the two-body interaction strength. Furthermore, results are compared to those obtained from the generator coordinate method, the random-phase approximation, and the self-consistent (second) random-phase approximation.

Results: The proposed method provides an excellent reproduction of the ground-state energy and of low-lying excitation energies of various $z$ signatures and total angular momenta across the full range of internucleon coupling defining the Lipkin Hamiltonian and driving the normal-to-deformed quantum phase transition. In doing so, the successive benefits of (i) breaking the symmetry, (ii) restoring the symmetry, (iii) including low-rank particle-hole excitations, and (iv) optimizing the amount by which the underlying vacuum breaks the symmetry are illustrated. While the generator coordinate method, built on the same deformed vacua, provides results of similar quality, this is not the case for the symmetry-restricted random-phase and self-consistent (second) random-phase approximations in the strongly interacting regime.

Conclusions: The numerical cost of the newly designed variational method is polynomial with respect to the system size. It achieves a good accuracy on the ground-state energy and the low-lying spectroscopy for both weakly and strongly interacting systems. The present study confirms the results obtained previously for the attractive pairing Hamiltonian in connection with the breaking and restoration of $\text{U}\left(1\right)$ global gauge symmetry. These two studies constitute a strong motivation to apply this method to realistic nuclear Hamiltonians in view of providing a complementary, accurate and versatile ab initio description of mid-mass open-shell nuclei.

Figure 1

Potential energy curves (in units of $\varepsilon$) as a function of $\alpha$ for $N=20$ and (a) $\chi =0.5<{\chi }_c$, (b) $\chi =1\equiv {\chi }_c$, (c) $\chi =1.1>{\chi }_c$, and (d) $\chi =3.0\gg {\chi }_c$. The UHF (black solid line) along with the positive (blue dashed line) and negative (red short-dashed line) $z$-signature projected energies are displayed in each case. Arrows point to the global minimum of each curve. A single arrow appears in panel (d) given that the global minimum is the same for the three curves in this case. In all panels the horizontal black dotted-dashed line corresponds to the exact ground-state energy.

Figure 2

Absolute ground-state energy as a function of the coupling strength $\chi \in [0,2]$ for $N=10$ and $J=5$. The three panels compare successive levels of approximations to exact results. (a) RHF and UHF with and without inclusion of associated 2p2h configurations. (b) PHF with and without inclusion of 2p2h configurations all at the deformation of the UHF minimum. (c) TCI results obtained by optimizing the $\alpha$ parameter with and without inclusion of 2p2h configurations. Note that when low-rank excitations are included, the optimization is made in the presence of them. The critical value ${\chi }_c=1$ is indicated by the black dashed vertical line.

Figure 3

Top: Error on the ground-state energy per particle as a function of $\chi$ for $N=16,\cdots ,100$. Results are obtained from the optimized TCI method including 1p1h and 2p2h configurations. Bottom: Maximum error over the interval $\chi \in [0,3]$ on the ground-state energy per particle as a function of the particle number.

Figure 4

Low-lying excitation energies for $N=20$ as a function of $\chi \in [0,3]$. (a) $J=9$ and $\eta =\pm 1$. (b) $J=19/2$ and $\eta =\pm i$. (c) $J=10$ and $\eta =\pm 1$. Results from the TCI method including up to 2p2h excitations (lines) are compared to exact results (symbols). The deformation of the underlying vacuum is optimized in each IRREP $(J,\eta )$ to minimize the energy of the lowest state.

Figure 5

Focus on the second (b) and third excited states (a) shown in panel (c) of Fig. 4. The exact results (filled circles) are systematically compared to the optimized TCI method including up to 2p2h configurations (black solid line), 3p3h configurations (red dotted line), and 4p4h configurations (blue dashed line).

Figure 6

First excited state energy with $J=10$ and (a) $\eta =+1$ or (b) $\eta =-1$ obtained for $N=20$ particles. Exact results (open circles) are compared to those obtained via RPA (blue dashed line) and SCRPA (red short dashed line) [37] as well as via the TCI approach including up to 2p2h (black solid line) and 3p3h (green dashed line) configurations.

Figure 7

Maximum error, over the interval $\chi \in [0,3]$, on the ground-state energy per particle as a function of the particle number. (a) TCI method including up to 2p2h configurations (full circles) and up to 3p3h configurations (empty circles). (b) horizontal GCM method based on $N_{\alpha }=5,7,19$ equidistant discretization points in the interval $[-\pi /4,+\pi /4]$.

Figure 8

Same as panel (b) of Fig. 7 but adding the optimal $\pm |{\alpha }_{\text{RVAP}}|$ mesh points to the equidistant discretization corresponding to $N_{\alpha }=5,7$.

Figure 9

Same as Fig. 4 with the adiabatic GCM using $N_{\alpha }=5$ equidistant discretization points in the interval $[-\pi /4,+\pi /4]$.

Figure 10

Same as Fig. 5 for the adiabatic GCM. Results are displayed for $N_{\alpha }=5$ (black solid line), 7 (red dotted line), and 9 (blue dashed line). Exact energies are shown as open circles.

Figure 11

Same as Fig. 6 for the adiabatic GCM approach with $N_{\alpha }=5,7$ ($N_{\alpha }=7,9$) for $\eta =+1$ ($\eta =-1$).