Particle-number projected Bogoliubov-coupled-cluster theory: Application to the pairing Hamiltonian

Background: While coupled-cluster theory accurately models weakly correlated quantum systems, it often fails in the presence of strong correlations where the standard mean-field picture is qualitatively incorrect. In many cases, these failures can be largely ameliorated by permitting the mean-field reference to break physical symmetries. Symmetry-broken coupled-cluster, e.g., Bogoliubov-coupled-cluster, theory can indeed provide reasonably accurate energetic predictions, but the broken symmetry can compromise the quality of the resulting wave function and predictions of observables other than the energy.

Purpose: Merging symmetry projection and coupled-cluster theory is therefore an appealing way to describe strongly correlated systems. One indeed expects to inherit and further improve the energetic accuracy of broken-symmetry coupled cluster while retaining proper symmetries.

Methods: Independently, two different but related formalisms have been recently proposed to achieve this goal. The two formalisms are contrasted in this manuscript, with results tested on the Richardson pairing Hamiltonian. While the present paper focuses on the breaking and restoration of $U\left(1\right)$ global-gauge symmetry associated with particle-number conservation, the symmetry-projected coupled-cluster formalism is applicable to other symmetries such as rotational (i.e., spin) symmetry.

Results: Both formalisms are based on the disentangled cluster representation of the symmetry-rotated coupled-cluster wave function. However, they differ in the way that the disentangled clusters are solved. One approach sets up angle-dependent coupled-cluster equations, while the other involves first-order ordinary differential equations. The latter approach yields energies and occupation probabilities significantly better than those of number-projected Bardeen-Cooper-Schrieffer (BCS) and BCS coupled cluster and, when the disentangled clusters are truncated at low excitation levels, has a computational cost not too much larger than that of BCS coupled cluster.

Conclusions: The high quality of results presented in this manuscript indicates that symmetry-projected coupled cluster is a promising method that can accurately describe both weakly and strongly correlated finite many-fermion systems.

Figure 1

Energies for 100 levels at half filling. Left panel: Fraction of the correlation energy recovered. Right panel: Energy errors in units of $\mathrm{\Delta }\epsilon$.

Figure 2

Numerical derivative of the energy with respect to $G$ for 12 levels at half filling. Contrarily to BCS-CCSD, the calculation coined as BCS(P)-CCSD relies on the BCS reference state extracted from the VAP PBCS calculation and not on the plain variational BCS state.

Figure 3

Fraction of the correlation energy recovered at half filling as a function of the number of levels. Two different values of $G/G_c$ are employed, knowing that the precise value of $G_c$ actually depends on the number of levels.

Figure 4

Fraction of correlation energy recovered for 100 levels as a function of the filling fraction. The interaction strength is $G=1.5G_c$, where here $G_c$ is the critical $G$ at half filling.

Figure 5

Deviations from exact occupation numbers for 100 levels at half filling. The interaction strength is $G/\mathrm{\Delta }\epsilon =1\left(G/G_c=5.5\right)$.

Figure 6

Dispersion of particle number for 100 levels at half filling as a function of $G/G_c$. Results are displayed for BCS, BCS-CCSD, PBCS, and PBCS-CCSD.

Figure 7

Energy variation with respect to the chemical potential of the underlying BCS reference state. Results are in units of $\mathrm{\Delta }\epsilon$ for 100 levels at half filling. The interaction strength is $G/\mathrm{\Delta }\epsilon =1\left(G/G_c=5.5\right)$.

Figure 8

Connected part $h\left(\phi \right)$ of the off-diagonal Hamiltonian kernel as a function of the gauge angle. Left panel: norm of $h\left(\phi \right)$ in units of $\mathrm{\Delta }\epsilon$. Right panel: phase of $h\left(\phi \right)$ in radians. The calculation is performed for 100 levels at half filling using $G=1.5G_c=0.27336$. Kernels are displayed at the PBCS and PBCS-CCSD levels. At the PBCS-CCSD level, gauge-dependent cluster amplitudes are either obtained from the gauge-dependent BCC equations (“proj”) introduced in Sec. 2e1 or from the ODEs (“ode”) introduced in Sec. 2e2.

Figure 9

Same as Fig. 8 for the norm kernel $\mathcal{N}\left(\phi \right)$.