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arXiv:2110.15737

Multi-reference many-body perturbation theory for nuclei I -- Novel PGCM-PT formalism

Mikaël Frosini, T. Duguet, J.-P. Ebran, V. Somà

Published Oct 29, 2021• 0 citations
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Abstract

Perturbative and non-perturbative expansion methods already constitute a tool of choice to perform ab initio calculations over a significant part of the nuclear chart. In this context, the categories of accessible nuclei directly reflect the class of unperturbed state employed in the formulation of the expansion. The present work generalizes to the nuclear many-body context the versatile method of Ref. \cite{burton20a} by formulating a perturbative expansion on top of a multi-reference unperturbed state mixing deformed non-orthogonal Bogoliubov vacua, i.e. a state obtained from the projected generator coordinate method (PGCM). Particular attention is paid to the part of the mixing taking care of the symmetry restoration, showing that it can be exactly contracted throughout the expansion, thus reducing significantly the dimensionality of the linear problem to be solved to extract perturbative corrections. While the novel expansion method, coined as PGCM-PT, reduces to the PGCM at lowest order, it reduces to single-reference perturbation theories in appropriate limits. Based on a PGCM unperturbed state capturing (strong) static correlations in a versatile and efficient fashion, PGCM-PT is indistinctly applicable to doubly closed-shell, singly open-shell and doubly open-shell nuclei. The remaining (weak) dynamical correlations are brought consistently through perturbative corrections. This symmetry-conserving multi-reference perturbation theory is state-specific and applies to both ground and excited PGCM unperturbed states, thus correcting each state belonging to the low-lying spectrum of the system under study.

DOI
10.48550/arxiv.2110.15737

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Perturbative and non-perturbative expansion methods already constitute a tool of choice to perform ab initio calculations over a significant part of the nuclear chart. In this context, the categories of accessible nuclei directly reflect the class of unperturbed state employed in the formulation of the expansion. The present work generalizes to the nuclear many-body context the versatile method of Ref. \cite{burton20a} by formulating a perturbative expansion on top of a multi-reference unperturbed state mixing deformed non-orthogonal Bogoliubov vacua, i.e. a state obtained from the projected generator coordinate method (PGCM). Particular attention is paid to the part of the mixing taking care of the symmetry restoration, showing that it can be exactly contracted throughout the expansion, thus reducing significantly the dimensionality of the linear problem to be solved to extract perturbative corrections. While the novel expansion method, coined as PGCM-PT, reduces to the PGCM at lowest order, it reduces to single-reference perturbation theories in appropriate limits. Based on a PGCM unperturbed state capturing (strong) static correlations in a versatile and efficient fashion, PGCM-PT is indistinctly applicable to doubly closed-shell, singly open-shell and doubly open-shell nuclei. The remaining (weak) dynamical correlations are brought consistently through perturbative corrections. This symmetry-conserving multi-reference perturbation theory is state-specific and applies to both ground and excited PGCM unperturbed states, thus correcting each state belonging to the low-lying spectrum of the system under study.
In spite of missing dynamical correlations, the projected generator coordinate method (PGCM) was recently shown to be a suitable method to tackle the low-lying spectroscopy of complex nuclei. Still, describing absolute binding energies and reaching high accuracy eventually requires the inclusion of dynamical correlations on top of the PGCM. In this context, the present work discusses the first realistic results of a novel multi-reference perturbation theory (PGCM-PT) that can do so within a symmetry-conserving scheme for both ground and low-lying excited states. First, proof-of-principle calculations in a small (emax=4e_{\mathrm{max}}=4emax​=4) model space demonstrate that exact binding energies of closed- (\nucl{O}{16}) and open-shell (\nucl{O}{18}, \nucl{Ne}{20}) nuclei are reproduced within 0.5−1.5%0.5-1.5\%0.5−1.5% at second order, i.e. through PGCM-PT(2). Moreover, profiting from the pre-processing of the Hamiltonian via multi-reference in-medium similarity renormalization group transformations, PGCM-PT(2) can reach converged values within smaller model spaces than with an unevolved Hamiltonian. Doing so, dynamical correlations captured by PGCM-PT(2) are shown to bring essential corrections to low-lying excitation energies that become too dilated at leading order, i.e., at the strict PGCM level. The present work is laying the foundations for a better understanding of the optimal way to grasp static and dynamical correlations in a consistent fashion, with the aim of accurately describing ground and excited states of complex nuclei via ab initio many-body methods.
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.
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 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.
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 goal of the present paper is twofold. First, a novel expansion many-body method applicable to superfluid open-shell nuclei, the so-called Bogoliubov in-medium similarity renormalization group (BIMSRG) theory, is formulated. This generalization of standard single-reference IMSRG theory for closed-shell systems parallels the recent extensions of coupled cluster, self-consistent Green's function or many-body perturbation theory. Within the realm of IMSRG theories, BIMSRG provides an interesting alternative to the already existing multi-reference IMSRG (MR-IMSRG) method applicable to open-shell nuclei. The algebraic equations for low-order approximations, i.e., BIMSRG(1) and BIMSRG(2), can be derived manually without much difficulty. However, such a methodology becomes already impractical and error prone for the derivation of the BIMSRG(3) equations, which are eventually needed to reach high accuracy. Based on a diagrammatic formulation of BIMSRG theory, the second objective of the present paper is thus to describe the third version (v3.0.0) of the ADG code that automatically (1) generates all valid BIMSRG(n) diagrams and (2) evaluates their algebraic expressions in a matter of seconds. This is achieved in such a way that equations can easily be retrieved for both the flow equation and the Magnus expansion formulations of BIMSRG. Expanding on this work, the first future objective is to numerically implement BIMSRG(2) (eventually BIMSRG(3)) equations and perform ab initio calculations of mid-mass open-shell nuclei.