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arXiv:1810.08804

Ab initio Rayleigh-Schrödinger perturbation calculation including three-body force

B. S. Hu, T. Li, F. R. Xu

Published Oct 20, 2018• 1 citation
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Abstract

We first derive the Rayleigh-Schrödinger many-body perturbation theory up to third order (RSPT3) for Hamiltonians with three-body interaction. The structure of closed-shell nuclei in a wide mass range from 4He to 48Ca has been investigated by the RSPT3 with explicit NN+3N Hamiltonian. The RSPT3 calculations are performed within Hartree-Fock bases. The perturbative contribution of antisymmetrized Goldstone diagrams (diagrammatic expansion for RSPT) with normal-ordered interaction has been analyzed. We demonstrate that the normal-ordered two-body level (NO2B) approximation which neglects the residual three-body term can catch the main effect of full three-body force in RSPT calculation. We also present rigorous benchmarks for RSPT3 with non-perturbative coupled cluster and in-medium similarity renormalization group using the same chiral NNLOsat NO2B interaction. The three methods are in good agreement with each other for binding energies and radii. However, the RSPT3 provides an alternative that is computationally inexpensive but comparable in accuracy to state-of-the-art non-perturbative many-body methods.

DOI
10.48550/arxiv.1810.08804

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We first derive the Rayleigh-Schrödinger many-body perturbation theory up to third order (RSPT3) for Hamiltonians with three-body interaction. The structure of closed-shell nuclei in a wide mass range from 4He to 48Ca has been investigated by the RSPT3 with explicit NN+3N Hamiltonian. The RSPT3 calculations are performed within Hartree-Fock bases. The perturbative contribution of antisymmetrized Goldstone diagrams (diagrammatic expansion for RSPT) with normal-ordered interaction has been analyzed. We demonstrate that the normal-ordered two-body level (NO2B) approximation which neglects the residual three-body term can catch the main effect of full three-body force in RSPT calculation. We also present rigorous benchmarks for RSPT3 with non-perturbative coupled cluster and in-medium similarity renormalization group using the same chiral NNLOsat NO2B interaction. The three methods are in good agreement with each other for binding energies and radii. However, the RSPT3 provides an alternative that is computationally inexpensive but comparable in accuracy to state-of-the-art non-perturbative many-body methods.
We advance the many-body perturbation theory (MBPT) calculations of the ground-state energy and radius of closed-shell nuclei beyond third order. Using automated diagram generation and evaluation up to fifth order, we present ground-state properties of selected closed-shell nuclei up to 78^{78}78Ni with two- and three-nucleon interactions derived from chiral effective field theory. A clear convergence trend is observed for the ground-state energy enabling calculations at improved accuracy. We further investigate in detail the decomposition of the fourth-order contributions. For the ground-state energy, the magnitude of the fourth-order contribution is typically less than half of the third order, and a typical cancellation among different classes of diagrams is observed. Finally, we perform a comprehensive comparison between MBPT and non-perturbative in-medium similarity renormalization group (IMSRG) calculations, with the goal to provide insight into many-body uncertainties associated with the IMSRG(2) truncation.
The solution of the quantum many-body problem for medium-mass nuclei using realistic nuclear interactions poses a superbe challenge for nuclear structure research. Because an exact solution can only be provided for the lightest nuclei, one has to rely on approximate solutions when proceeding to heavier systems. Over the past years, tremendous progress has been made in the development and application of systematically improvable expansion methods and an accurate description of nuclear observables has become viable up to mass number A≈100A \approx 100A≈100. While closed-shell systems are consistently described via a plethora of different many-body methods, the extension to genuine open-shell systems still remains a major challenge and up to now there is no \textit{ab initio} many-body method which applies equally well to systems with even and odd mass numbers. The goal of this thesis is the development and implementation of innovative perturbative approaches with genuine open-shell capabilities. This requires the extension of well-known single-reference approaches to more general vacua. In this work we choose two complementary routes for the usage of generalized reference states. First, we derive a new \emph{ab initio} approach based on multi-configurational reference states that are conveniently derived from a prior no-core shell model calculation. Perturbative corrections are derived via second-order many-body perturbation theory, thus, merging configuration interaction and many-body perturbation theory. The generality of this ansatz enables for a treatment of medium-mass systems with arbitrary mass number, as well as the extension to low-lying excited states such that ground and excited states are treated on an equal footing. In a complementary approach, we use reference states that break a symmetry of the underlying Hamiltonian. In the simplest case this corresponds to the expansion around a particle-number-broken Hartree-Fock-Bogoliubov vacuum which is obtained from a mean-field calculation. Pairing correlations are already absorbed in the reference state. The mild scaling allows for investigations up to tin isotopic chains. All benchmarks were performed using state-of-the-art chiral two- plus three-nucleon interactions thus allowing for a universal description of nuclear observables in the medium-mass regime.
A Rayleigh–Schrödinger many-body perturbation theory (MBPT) approach is introduced by making use of a particle-number-breaking Bogoliubov reference state to tackle (near-)degenerate open-shell fermionic systems. By choosing a reference state that solves the Hartree–Fock–Bogoliubov variational problem, the approach reduces to the well-tested Møller–Plesset, i.e., Hartree–Fock based, MBPT when applied to closed-shell systems. Due to its algorithmic simplicity, the newly developed framework provides a computationally simple yet accurate alternative to state-of-the-art non-perturbative many-body approaches. At the price of working in the quasi-particle basis associated with a single-particle basis of sufficient size, the computational scaling of the method is independent of the particle number. This paper presents the first realistic applications of the method ranging from the oxygen to the nickel isotopic chains on the basis of a modern nuclear Hamiltonian derived from chiral effective field theory.
The last few decades in nuclear structure theory have seen a rapid expansion of ab initio theories, aiming at describing the properties of nuclei starting from the inter-nucleonic interaction. Such an expansion relied both on the tremendous growth of computing power and novel formal developments. This work focuses on the development of the recently proposed Bogoliubov Many-Body Perturbation Theory that relies on a particle-number-breaking reference state to tackle singly open-shell nuclei. The formalism is first described in details, and diagrammatic and algebraic contributions are derived up to second order. Its link to standard Many-Body Perturbation Theory is made explicit, as well as its connexion to Bogoliubov Coupled-Cluster theory. An automated extension to higher orders based on graph theory methods is then detailed, and the ADG numerical program generating and evaluating BMBPT diagrams at arbitrary order is introduced. Such a formal development carries implications that are not restricted to the present work, as the developed methods can be applied to other many-body methods. Finally, first numerical results obtained for oxygen, calcium and nickel isotopes are presented. They establish BMBPT as a method of interest for large-scale computations of isotopic or isotonic chains in the mid-mass sector of the nuclear chart.
Rayleigh-Schrödinger time-independent perturbation theory is a key method in quantum mechanics for solving problems where the Hamiltonian of a quantum system cannot be exactly solved. By expanding the solutions in series, this method allows one to obtain approximations for quantum states and eigen-energies, especially for systems where the interactions or perturbing forces are small compared to the principal part of the Hamiltonian. This is crucial for understanding complex phenomena, predicting the behaviours of atomic and molecular systems, and calculating corrections to basic theoretical models, thereby improving the accuracy of quantum predictions. In this work, we will show that using the unitary operators, we arrive at the equations giving the different order of corrections of both energy as well as wave functions of the time-independent perturbation theory. Our approach closely mirrors the famous formalism of Rayleigh-Schrodinger perturbation theory (RSPT). The use of unitary operators simplifies the derivation of higher-order corrections, making the process more accurate and accessible to researchers. This technique harnesses the potential of unitary operators within a perturbation series framework. By combining these two powerful techniques, we offer a comprehensive framework for addressing normalization challenges in quantum mechanical calculations. This methodology not only facilitates a deeper understanding of perturbation theory but also serves as a pedagogical stepping stone, encouraging postgraduate researchers to apply their foundational knowledge and acquired skills in practical problem-solving situations with confidence. This is therefore a second proof or a new method to derive the fundamental equations of Rayleigh Schrodinger perturbation Theory RSPT using the unitary transformations.
We derive coupled-cluster equations for three-body Hamiltonians. The equations for the one- and two-body cluster amplitudes are presented in a factorized form that leads to an efficient numerical implementation. We employ low-momentum two- and three-nucleon interactions and calculate the binding energy of 4He^{4}\mathrm{He}4He. The results show that the main contribution of the three-nucleon interaction stems from its density-dependent zero-, one-, and two-body terms that result from the normal ordering of the Hamiltonian in coupled-cluster theory. The residual three-body terms that remain after normal ordering can be neglected.