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arXiv:1412.2696 · Physical Review C

Ab initioBogoliubov coupled cluster theory for open-shell nuclei

A. Signoracci, T. Duguet, G. Hagen, G. R. Jansen

Published Jun 29, 2015• 96 citations
arXivHTMLPDF

Abstract

Background: Ab initio many-body methods have been developed over the past 10 yr to address closed-shell nuclei up to mass $A\ensuremath{\approx}130$ on the basis of realistic two- and three-nucleon interactions. A current frontier relates to the extension of those many-body methods to the description of open-shell nuclei. Several routes to address open-shell nuclei are currently under investigation, including ideas that exploit spontaneous symmetry breaking.Purpose: Singly open-shell nuclei can be efficiently described via the sole breaking of U(1) gauge symmetry associated with particle-number conservation as a way to account for their superfluid character. While this route was recently followed within the framework of self-consistent Green's function theory, the goal of the present work is to formulate a similar extension within the framework of coupled cluster theory.Methods: We formulate and apply Bogoliubov coupled cluster (BCC) theory, which consists of representing the exact ground-state wave function of the system as the exponential of a quasiparticle excitation cluster operator acting on a Bogoliubov reference state. Equations for the ground-state energy and the cluster amplitudes are derived at the singles and doubles level (BCCSD) both algebraically and diagrammatically. The formalism includes three-nucleon forces at the normal-ordered two-body level. The first BCC code is implemented in $m$ scheme, which will permit the treatment of doubly open-shell nuclei via the further breaking of SU(2) symmetry associated with angular momentum conservation.Results: Proof-of-principle calculations in an ${N}_{\text{max}}=6$ spherical harmonic oscillator basis for $^{16,18}\mathrm{O}$ and $^{18}\mathrm{Ne}$ in the BCCD approximation are in good agreement with standard coupled cluster results with the same chiral two-nucleon interaction, while $^{20}\mathrm{O}$ and $^{20}\mathrm{Mg}$ display underbinding relative to experiment. The breaking of U(1) symmetry, monitored by computing the variance associated with the particle-number operator, is relatively constant for all five nuclei, in both the Hartree-Fock-Bogoliubov and BCCD approximations.Conclusions: The newly developed many-body formalism increases the potential span of ab initio calculations based on single-reference coupled cluster techniques tremendously, i.e., potentially to reach several hundred additional midmass nuclei. The new formalism offers a wealth of potential applications and further extensions dedicated to the description of ground and excited states of open-shell nuclei. Short-term goals include the implementation of three-nucleon forces at the normal-ordered two-body level. Midterm extensions include the approximate treatment of triples corrections and the development of the equation-of-motion methodology to treat both excited states and odd nuclei. Long-term extensions include exact restoration of U(1) and SU(2) symmetries.

DOI
10.1103/physrevc.91.064320

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We extend coupled-cluster (CC) theory performed on top of a Slater determinant breaking rotational symmetry to allow for the exact restoration of the angular momentum at any truncation order. The main objective relates to the description of near-degenerate finite quantum systems with an open-shell character. As such, the newly developed many-body formalism offers a wealth of potential applications and further extensions dedicated to the ab initio description of, e.g., doubly open-shell atomic nuclei and molecule dissociation. The formalism, which encompasses both single-reference CC theory and projected Hartree–Fock theory as particular cases, permits the computation of usual sets of connected diagrams while consistently incorporating static correlations through the highly non-perturbative restoration of rotational symmetry. Interestingly, the yrast spectroscopy of the system, i.e. the lowest energy associated with each angular momentum, is accessed within a single calculation. A key difficulty presently overcome relates to the necessity to handle generalized energy and norm kernels for which naturally terminating CC expansions could be eventually obtained. The present work focuses on SU (2) but can be extended to any (locally) compact Lie group and to discrete groups, such as most point groups. In particular, the formalism will be soon generalized to U (1) symmetry associated with particle number conservation. This is relevant to Bogoliubov CC theory that was recently applied to singly open-shell nuclei.
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.
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.
Abstract We have recently extended many-body perturbation theory (MBPT) 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 (Duguet 2015 J. Phys. G: Nucl. Part. Phys. 42 025107 ). Following a similar route, we presently extend Bogoliubov MBPT 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. The long-term goal relates to the ab initio description of near-degenerate finite quantum systems with an open-shell character.
Given elementary inter-nucleon interactions,the resolution of the A-body Schrödinger equation gives access to the properties of the nuclear eigenstates. Many-body perturbation theory (MBPT) based on a Hartree-Fock mean field allows one to treat the dynamical correlations at play in doubly-closed-shell nuclei. Beyond shell closures,breaking U(1) symmetry associated with the conservation of neutron and proton numbers further permits to capture static correlations at the meanfield level and to formulate the well-behaved Bogoliubov many-body perturbation theory (BMBPT).Nevertheless, BMBPT results show contaminations associated with the broken symmetry that is only emergent on finite quantum system such as theatomic nucleus. Thus, the restoration of U(1) symmetry beyond the mean field is necessary for acorrect description and gives rise to the projected BMBPT method (PBMBPT). The goal is to implement PBMBPT in order to perform ab initio calculations of singly-open-shell mid-mass nuclei.The present work provides systematic solutions to the formal and technical problems arising during the implementation of PBMBPT: an automatic and safe method to generate PBMBPT diagrams and associated expressions, a framework to calculate other observables than the energy, an extension of the normal ordering approximation to symmetry breaking methods as well as an automatic and safe tool for spherical reduction of tensor networks.
The reach of ab initio many-body theories is rapidly extending over the nuclear chart. However, dealing fully with three-nucleon, possibly four-nucleon, interactions makes the solving of the A-body Schrodinger equation particularly cumbersome, if not impossible beyond a certain nuclear mass. Consequently, ab initio calculations of mid-mass nuclei are typically performed on the basis of the so-called normal-ordered two-body (NO2B) approximation that captures dominant effects of three-nucleon forces while effectively working with two-nucleon operators. A powerful idea currently employed to extend ab initio calculations to open-shell nuclei consists of expanding the exact solution of the A-body Schrodinger equation while authorizing the approximate solution to break symmetries of the Hamiltonian. In this context, operators are normal ordered with respect to a symmetry-breaking reference state such that proceeding to a naive truncation may lead to symmetry-breaking approximate operators. The purpose of the present work is to design a normal-ordering approximation of operators that is consistent with the symmetries of the Hamiltonian while working in the context of symmetry broken (and potentially restored) methods. Focusing on many-body formalisms in which U(1) global-gauge symmetry associated with particle-number conservation is broken (and potentially restored), a particle-number-conserving normal-orderedk-body (PNOkB) approximation of an arbitrary N-body operator is designed on the basis of Bogoliubov reference states. A numerical test based on particle-number-projected Hartree-Fock-Bogoliubov calculations permits to check the particle-number conserving/violating character of a given approximation to a particle-number conserving operator. The PNOkB approximation of an arbitrary N-body operator is formulated. Based on this systematic approach, it is demonstrated that naive extensions of the normal-ordered two-body (NO2B) approximation employed so far on the basis of symmetry-conserving reference states lead to particle non-conserving operators. Alternatively, the PNOkB procedure is now available to generate particle-number-conserving approximate operators. The formal analysis is validated numerically. Using the presently proposed PNOkB approximation, ab initio calculations based on symmetry-breaking and restored formalisms can be safely performed. The future formulation of an angular-momentum-conserving normal-ordered k-body approximation based on deformed Slater determinant or Bogoliubov reference states is envisioned.