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Register a new userParticle-number projected Bogoliubov coupled cluster theory. Application to the pairing Hamiltonian
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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. Merging symmetry projection and coupled cluster theory is therefore an appealing way to describe strongly correlated systems. 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. Both formalisms are based on the disentangled cluster representation of the symmetry-rotated coupled cluster wavefunction. 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 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. 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.
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We describe the second version (v2.0.0) of the code ADG that automatically (1) generates all valid off-diagonal Bogoliubov many-body perturbation theory diagrams at play in particle-number projected Bogoliubov many-body perturbation theory (PNP-BMBPT) and (2) evaluates their algebraic expression to be implemented for numerical applications. This is achieved at any perturbative order $p$ for a Hamiltonian containing both two-body (four-legs) and three-body (six-legs) interactions (vertices). All valid off-diagonal BMBPT diagrams of order $p$ are systematically generated from the set of diagonal, i.e., unprojected, BMBPT diagrams. The production of the latter were described at length in https://doi.org/10.1016/j.cpc.2018.11.023 dealing with the first version of ADG. The automated evaluation of off-diagonal BMBPT diagrams relies both on the application of algebraic Feynmans rules and on the identification of a powerful diagrammatic rule providing the result of the remaining $p$-tuple time integral. The new diagrammatic rule generalizes the one already identified in https://doi.org/10.1016/j.cpc.2018.11.023 to evaluate diagonal BMBPT diagrams independently of their perturbative order and topology. The code ADG is written in Python3 and uses the graph manipulation package NetworkX. The code is kept flexible enough to be further expanded throughout the years to tackle the diagrammatics at play in various many-body formalisms that already exist or are yet to be formulated.
Recently, the zero-pairing limit of Hartree-Fock-Bogoliubov (HFB) mean-field theory was studied in detail in arXiv:2006.02871. It was shown that such a limit is always well-defined for any particle number A, but the resulting many-body description differs qualitatively depending on whether the system is of closed-(sub)shell or open-(sub)shell nature. Here, we extend the discussion to the more general framework of Finite-Temperature HFB (FTHFB) which deals with statistical density operators, instead of pure many-body states. We scrutinize in detail the zero-temperature and zero-pairing limits of such a description, and in particular the combination of both limits. For closed-shell systems, we find that the FTHFB formulism reduces to the (zero-temperature) Hartree-Fock formulism, i.e. we recover the textbook solution. For open-shell systems, however, the resulting description depends on the order in which both limits are taken: if the zero-temperature limit is performed first, the FTHFB density operator demotes to a pure state which is a linear combination of a finite number of Slater determinants, i.e. the case of arXiv:2006.02871. If the zero-pairing limit is performed first, the FTHFB density operator remains a mixture of a finite number of Slater determinants with non-zero entropy, even as the temperature vanishes. These analytical findings are illustrated numerically for a series of Oxygen isotopes.
We solve the Hartree-Fock-Bogoliubov (HFB) equations for a spherical mean field and a pairing potential with the inverse Hamiltonian method, which we have developed for the solution of the Dirac equation. This method is based on the variational principle for the inverse Hamiltonian, and is applicable to Hamiltonians that are bound neither from above nor below. We demonstrate that the method works well not only for the Dirac but also for the HFB equations.
The variational Hartree-Fock-Bogoliubov (HFB) mean-field theory is the starting point of various (ab initio) many-body methods dedicated to superfluid systems. While taking the zero-pairing limit of HFB equations constitutes a text-book problem when the system is of closed-(sub)shell character, it is typically, although wrongly, thought to be ill-defined whenever the naive filling of single-particle levels corresponds to an open-shell system. The present work demonstrates that the zero-pairing limit of an HFB state is mathematically well-defined, independently of the closed- or open-shell character of the system in the limit. Still, the nature of the limit state strongly depends on the underlying shell structure and on the associated naive filling reached in the zero-pairing limit for the particle number A of interest. All the analytical findings are confirmed and illustrated numerically. While HFB theory has been intensively scrutinized formally and numerically over the last decades, it still uncovers unknown and somewhat unexpected features. From this general perspective, the present analysis demonstrates that HFB theory does not reduce to Hartree-Fock theory even when the pairing field is driven to zero in the HFB Hamiltonian matrix.
We have recently extended many-body perturbation theory and coupled-cluster theory performed on top of a Slater determinant breaking rotational symmetry to allow for the restoration of the angular momentum at any truncation order [T. Duguet, J. Phys. G: Nucl. Part. Phys. 42 (2015) 025107]. Following a similar route, we presently extend Bogoliubov many-body perturbation theory and Bogoliubov coupled cluster theory performed on top of a Bogoliubov reference state breaking global gauge symmetry to allow for the restoration of the particle number at any truncation order. Eventually, formalisms can be merged to handle $SU(2)$ and $U(1)$ symmetries at the same time. Several further extensions of the newly proposed many-body formalisms can be foreseen in the mid-term future. The long-term goal relates to the ab initio description of near-degenerate finite quantum systems with an open-shell character.
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