Bogoliubov many-body perturbation theory under constraint
Introduction
The long-term objective of the so-called ab initio approach to atomic nuclei is to develop an accurate and universal description of low-energy nuclear systems from first principles. Such a viewpoint stipulates that the atomic nucleus can be appropriately modelled in terms of structureless and strongly interacting neutrons and protons. In this context, the basic interactions between proton and neutron degrees of freedom emerge from the underlying gauge theory of interacting quarks and gluons, i.e., from quantum chromodynamics (QCD). As such, the ab initio endeavour involves two steps:
- (1)
Modelling the elementary inter-nucleon interactions (ideally with an uncertainty estimate);
- (2)
Solving the -body Schrödinger equation (ideally with an uncertainty estimate);
such that the output predictions to be confronted with experimental data are a convolution of these two components. Whenever the uncertainty associated with one of the two above components dominates, the distance to the data can be attributed to it, thus, leading to the necessity to improve on it. Eventually, the ab initio approach offers a systematic path towards a universal theoretical framework to describe nuclear properties ranging from binding energies and charge radii to spectroscopic properties and electroweak transition probabilities.
In the past 15 years, ab initio low-energy nuclear theory has made tremendous progress regarding points (1) and (2) above. First, the ab initio approach has been systematically formulated within the frame of chiral effective field theory (-EFT) [1], [2] in which quark and gluon, as well as heavy hadron, degrees of freedom are integrated out. The long- and mid-range parts of inter-nucleon interactions are mediated by pions, the Goldstone bosons of the spontaneously broken chiral symmetry at low energy, and are complemented with contact interactions accounting for high-energy degrees of freedom that are not explicitly incorporated.1 The low-energy constants (LEC’s) of the EFT Lagrangian are typically fixed by reproducing a selected set of few-body data. As such, -EFT yields (i) a sound connection to QCD, (ii) a clear hierarchy of the importance of two-nucleon (NN) interactions, three-nucleon (3N) interactions, …, (iii) a consistent construction of other, e.g. electroweak, operators, (iv) a mean to estimate uncertainties due to the truncation employed in the systematic construction of the operators and (v) a systematic way to improve on the description if necessary. In addition to their construction within the frame of -EFT, another key development relates to the use of similarity renormalization group (SRG) transformations to ”soften” nuclear Hamiltonians and make them more amenable to many-body calculations [4], [5]. The unitary SRG evolution constitutes a pre-diagonalization of the operator in momentum space, thus suppressing the coupling between high- and low-momentum modes. As a result, many-body applications discussed below based on SRG-evolved operators have shown highly improved model-space convergence, thus facilitating studies of mid-mass nuclei.
As for point (2), the continuous improvement of methods formulated in the 1980s to solve the -nucleon Schrödinger equation, as well as the development of novel ones, have allowed the computation of many more nuclear observables from first principles. As a first step, essentially exact solutions were typically provided by large scale diagonalization methods such as the no-core shell model (NCSM) [6], [7] and by Green’s function Monte Carlo (GFMC) techniques [8], [9]. However, due to the exponentially scaling cost with respect to basis/system size, these approaches are typically limited2 to light nuclei with mass number . In this context, a breakthrough occurred about 15 years ago to access heavier doubly closed-shell nuclei, i.e., nuclei whose neutron and proton numbers are such that the highest occupied single-nucleon shells are fully filled in a simple mean-field description. This breakthrough was made possible thanks to the development and application of non-perturbative methods whose numerical cost scale polynomially with system size. Examples are coupled cluster (CC) [12], [13], [14], [15], [16], in-medium similarity renormalization group (IM-SRG) [17], [18], [19], [20] and self-consistent Green’s function (SCGF) [21], [22], [23], [24], [25] methods. In particular, while SCGF advanced within the field of nuclear physics, CC was successfully transferred back from quantum chemistry where it has been intensively developed over the last four decades to describe molecular properties from first principles. These methods have allowed one to access a variety of observables in a few tens of doubly closed-(sub)shell nuclei with .
In principle, the combined use of -EFT Hamiltonians and sophisticated methods to solve the A-body Schrödinger equation provides a universal framework with high predictive power. Most remarkably, benchmark calculations in closed-(sub)shell oxygen isotopes () have demonstrated the consistency among the various many-body techniques and proved that their current level of implementation delivers ground-state observables with an uncertainty better than 2–3% [26]. Following this achievement, many-body calculations also acquired the role of diagnostic tools and have been used to test qualities and deficiencies of input Hamiltonians across the whole range of medium-mass nuclei [27], [28], [29]. At present, a strong effort is devoted to a better understanding of the shortcomings of existing -EFT Hamiltonians with the ambition to improve on the accuracy of ab initio calculations in the future [30], [31], [32].
Many-body theories accessing mid-mass nuclei typically expand the exact ground-state wavefunction around a reference Slater determinant and can meaningfully access doubly closed-shell systems. However, they are not suited to open-shell systems that constitute the large majority of nuclei. This limitation is due to the fact that the ground-state wavefunction of open-shell nuclei is not dominated by a single Slater determinant such that the Hartree–Fock (HF) approximation cannot yield an appropriate reference point for the expansion. Alternatives have been developed to overcome this cutting-edge difficulty. A first option relies on the derivation of effective valence-space Hamiltonians that are subsequently diagonalized to access the spectrum of the target nucleus. While initially developed within a perturbative scheme [33], valence-space interactions have recently been formulated within non-perturbative NCSM [34], CC [27] and IM-SRG [35] frameworks. Still, the dimension of the associated configuration space grows exponentially when moving away from shell closures, which makes it eventually difficult to use such methods beyond . A second flavour of many-body methods applicable to open-shell nuclei are equation-of-motion (EOM) techniques, where one starts from the solution obtained for a closed-shell nucleus and describes neighbouring nuclei via the action of particle-attachment or particle-removal operators. While this has been extensively used in CC theory [16], current implementations are restricted to the attachment/removal of at most two particles [36], which prohibits its use through large degenerate single-particle shells.
Generally speaking, the restriction to a single Slater-determinant reference state is too limiting to design a meaningful expansion method directly in open-shell nuclei due to the degeneracy with respect to elementary particle–hole excitations. The use of more general reference states must be contemplated [37], [38] to lift the degeneracy and tackle, from the outset, strong static correlations associated with it. The first option in this direction relies on reference states mixing a set of appropriately chosen Slater determinants. Those multi-configurational reference states can, for example, be obtained from a prior NCSM calculation in small model spaces or under the form of a particle-number-projected Hartree–Fock–Bogoliubov (PHFB) state. Such reference states have been successfully employed in the multi-reference extension of IM-SRG (MR-IMSRG) [39], [40] or within a perturbative framework yielding multi-configurational perturbation theory (MCPT) [41].
With the objective to maintain a strict polynomial cost with basis/system size and keep the intrinsic simplicity of single-reference expansion methods, another option relies on reference states breaking one or several symmetries, i.e. states that do not carry an eigenvalue of the Casimir operator of a given symmetry of the Hamiltonian as a good quantum number. In semi-magic nuclei, global-gauge symmetry associated with particle-number conservation is allowed to break in order to address Cooper pair’s instability and handle nuclear superfluidity. This leads to expanding the exact solution of the -body Schrödinger equation around a Bogoliubov reference state that reduces to a Slater determinant in closed-shell systems. In doubly open-shell nuclei, rotational symmetry associated with angular-momentum conservation must also be allowed to break, thus, leading to the use of a spatially-deformed reference state. The above considerations have led to the design of non-perturbative Gorkov self-consistent Green’s function (GSCGF) [42], [43] and Bogoliubov coupled cluster (BCC) [44] methods that generalize standard SCGF and CC to open-shell systems. Restricting oneself to a perturbative method, this rational has led to the design of Bogoliubov many-body perturbation theory (BMBPT) [45] that is the focus of the present paper.3
Focusing so far on singly open-shell nuclei, the formal and numerical developments of symmetry-breaking many-body methods have led to unprecedented achievements in the past years. A notable example are the first systematic ab initio calculations along complete chains of oxygen, calcium and nickel isotopes [27], [28], [31] via GSCGF theory. This method has then been applied in the neighbourhood of singly-magic calcium, e.g. in argon [51], potassium [52], [53] and other chains up to chromium [54]. These calculations have contributed to the characterization of the too limited quality of existing chiral EFT Hamiltonians in mid-mass nuclei few years back [27], [28], [52], [55]. Recently, GSCGF was employed to perform the first exploratory calculation of Sn and Xe isotopes [56], extending the current range of applicability of ab initio calculations to . A few years ago, BMBPT was implemented up to third order and shown to provide an accurate description of medium-mass ground-state energies at a significantly lower computational cost than GSCGF, BCC or MR-IMSRG theory [29]. This makes BMBPT an extremely useful candidate to perform large survey calculations, to systematically test next generations of chiral EFT Hamiltonians [57] and to make the future extension to even more challenging doubly open-shell nuclei simpler than in other ab initio frameworks.
The use of a perturbation theory relies on the hope that the associated Taylor series converges or possesses asymptotic properties that justify the use of the first few orders as a meaningful estimate of the full series. This question has been addressed in Refs. [58] and [59] for standard MBPT appropriate to doubly closed-shell nuclei. Despite softening the interaction via an SRG transformation [60] to tame down its ultra-violet source of non-perturbative character [61], [62], [63], MBPT was shown [58] to typically diverge in small model spaces for . The algebraic Padé resummation method was successfully invoked to recover physical quantities from the diverging series. Ref. [59] further revealed that using the variationally optimized HF Slater determinant as a reference state (i.e. using the Møller–Plesset scheme) in combination with an SRG-softened interaction provides a convergent MBPT series. These results are consistent with what was found earlier on in quantum chemistry [64].
The objective of the present paper is to extend this study to (singly) open-shell nuclei studied through BMBPT. Strong static correlations of infrared origin are ”regularized” via the breaking of symmetry such that a meaningful expansion can at least be defined on top of the reference state. While low orders have indeed been shown to provide sound results [29], the behaviour of the associated Taylor series remains to be characterized more thoroughly by pushing BMBPT to high orders.
Furthermore, it happens that the breaking of, e.g. , symmetry has profound consequences on the characteristics of the series produced through the perturbative expansion. While the reference state and the perturbatively corrected states are not eigenstates of the Casimir operator of the symmetry group, one wishes to impose that the targeted eigenvalue is at least reproduced on average.4 Breaking symmetry thus implies that states at play are not eigenstates of the particle-number operator but must carry at least the physical number of particle on average.5 In this context, the difficulty relates to the fact that the average particle number typically changes at each perturbative order. In Ref. [29], low orders were addressed in such a way that the shift of the average particle number occurring at each order was accounted for by an unsubstantiated a posteriori correction. A more robust formalism was suggested by the authors in which the average particle number is actually adjusted to the correct value at each perturbative order. This idea leads to a new type of unexplored perturbative sequence characterized by the following two features:
- (1)
Even though the exact solution obtained as the limit of the sequence must lie within the Hilbert space associated with -particle systems, the sequence itself is not restricted to that Hilbert space and spans the entire Fock space.
- (2)
The expansion involves in fact two coupled sequences associated with the energy and the average particle number such that the latter is constrained to match the targeted physical value at each working order. The coupling between the two sequences and the need to deliver the physical particle number on average at each order makes the approach intrinsically iterative and at variance with standard MBPTs.
Eventually, one is led to considering a new type of expansion coined as many-body perturbation theory under constraint.6 The presently introduced constrained version of BMBPT is denoted as BMBPT while the unconstrained form is indicated as BMBPT. The third variant of BMBPT employed in Ref. [29] makes use of an a posteriori correction and is denoted as BMBPT.
In this context, the objective of the present study is to investigate the following, yet unexplored, aspects of BMBPT (or rather BMBPT):
- (1)
What is the high-order behaviour of the perturbative expansion under constraint?
- (2)
In absence of convergence, does this behaviour authorize the use of standard or novel resummation methods delivering the correct result?
- (3)
If so, do low orders provide a fair approximation of the resummed series?
- (4)
Is the a posteriori correction employed in Ref. [29] justified such that the iterative and costly character of BMBPT can be entirely bypassed in actual applications via the use of BMBPT?
To address these various points, and contrary to its original derivation based on a time-dependent formalism [45], [66], BMBPT is presently introduced on the basis of a more traditional time-independent approach. In this context, BMBPT is easily formulated via a recursive scheme from which corrections up to high, e.g., 20th or 30th, order can be efficiently computed. Still, doing so in numerical applications requires to limit oneself to a small, i.e. schematic, portion of Fock space such that a severe truncation on the set of eigenstates of the unperturbed Hamiltonian must be considered. It is the price to pay to be able to investigate the series up to high orders and one hopes that the truncation effects do not invalidate the general conclusions reached in this way.
The document is organized as follows. In Section 2, basic equations of the many-body problem are stated and notations are introduced. Section 3 introduces the basic ingredients of the Bogoliubov framework. In Section 4, the BMBPT formalism is developed and compared to standard unconstrained MBPT, i.e. BMBPT. After introducing the Taylor series associated with strict perturbation theory, the well-known Padé resummation scheme and the recently proposed eigenvector continuation (EC) technique [67], [68] are introduced. In Section 5, results from calculations performed within a small model-space are displayed and analysed. The specificities of BMBPT, BMBPT and BMBPT, as well as of the resummation methods built on them, are probed and validated against exact diagonalization. Lastly, conclusions and perspectives are provided in Section 6.
Section snippets
Eigenvalue equations
Ab initio nuclear structure calculations aim at solving the time-independent many-body Schrödinger equation where is the Hamiltonian defined from elementary inter-nucleon interactions, while and denote its A-body eigenstates and eigenenergies, respectively. As testified by the carried quantum number7 A, the Hamiltonian commutes
Bogoliubov framework
The novelty of single-reference BMBPT is to employ a particle-number breaking Bogoliubov reference state as a way to handle open-shell nuclei in a controlled fashion. The present section introduces the basics of Bogoliubov algebra, Bogoliubov vacua and the associated Wick’s theorem.
Bogoliubov many-body perturbation theory
Bogoliubov many-body perturbation theory expands exact eigenstates of around a Bogoliubov reference state breaking particle-number symmetry. It presents the tremendous advantage that static, i.e. pairing, correlations at play in (singly) open-shell nuclei are already largely accounted for by the reference state. In doing so, the degeneracy of Slater determinants with respect to particle–hole excitations is lifted from the outset,11
Results
In this section, results obtained from BMBPT calculations and from resummation methods built on it are presented and systematically benchmarked against the corresponding BCI results.
Conclusions and outlook
Convergence properties of the so-called Bogoliubov many-body perturbation theory (BMBPT), suited to the description of open-shell atomic nuclei have been investigated at length.
The capacity of BMBPT to capture strong ”static” correlations originates in the allowed breaking of global gauge symmetry associated with the conservation of particle number. As a result, BMBPT was formulated as a perturbative expansion under the constraint that the particle number is correct in average.
CRediT authorship contribution statement
P. Demol: Formal analysis, Investigation, Methodology, Software, Visualization, Writing - original draft, Discussing the results and reviewing the manuscript. M. Frosini: Formal analysis, Investigation, Methodology, Software, Discussing the results and reviewing the manuscript. A. Tichai: Conceptualization, Methodology, Software, Supervision, Discussing the results and reviewing the manuscript. V. Somà: Conceptualization, Supervision, Discussing the results and reviewing the manuscript. T.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
We thank Robert Roth for providing us with nuclear matrix elements. This publication is based on work supported in part by Research Foundation Flanders (FWO, Belgium), by GOA/2015/010 (BOF KU Leuven), by the framework of the Espace de Structure et de réactions Nucléaires Théorique (ESNT) at CEA, and the Deutsche Forschungsgemeinschaft, Germany through contract SFB 1245. Calculations were performed by using HPC resources from GENCI-TGCC, France (Contracts No. A005057392 and A007057392).
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