Particle-number projected Bogoliubov coupled cluster theory. Application to the pairing Hamiltonian
Résumé
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(1) 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.