P O BMBPT (ODE) Projected Bogoliubov Many-Body Perturbation Theory. P P BMBPT (Perturbative) Projected Bogoliubov Many-Body Perturbation Theory. PAV Projection After Variation

Keywords: χEFT Chiral Effective Field Theory, ALM Augmented Lagrangian Method, AMR-T Angular-Momentum-Reduced Tensor, AMR-TN Angular-Momentum-Reduced Tensor Network, BCC Bogoliubov Coupled Cluster, BCI Bogoliubov Configuration Interaction BMBPT Bogoliubov Many-Body Perturbation Theory, CC Coupled Cluster HO Harmonic Oscillator, IMSRG In-Medium Similarity Renormalization Group, IRREP Irreducible Representation, MBPT Many-Body Perturbation Theory, NCSM k-Body approximation, ODE Ordinary Differential Equation PBCC Projected Bogoliubov Coupled Cluster, PBMBPT Projected Bogoliubov Many-Body Perturbation Theory, PES Potential Energy Surface, PHFB Projected Hartree-Fock-Bogoliubov PQNOkB Particle-number-conserving Quasi-Normal-Ordered k-Body approximation, QCD Quantum Chromodynamics SHO Spherical Harmonic Oscillator, SR-T Symmetry-Reduced Tensor, SR-TN Symmetry-Reduced Tensor Network, SRG Similarity Renormalization Group, SU-T Symmetry-Unrestricted Tensor, SU-TN Symmetry-Unrestricted Tensor Network, TN Tensor Network

Here we are, these three years have already passed and I finally got the title of Doctor. It was an incredible opportunity ! The chance to work on theoretical nuclear structure, and to present the results of my work in workshops or international conferences in Darmstadt, Trento, Madrid or even Vancouver. Before switching to french I would like to take the opportunity to thank all the non french natives I had the chance to meet, especially the jury for the time they spent on assessing the work produced during those three years and for coming from Paris, Madrid or East Lansing to attend my defence. A very special thanks to Alexander Tichai whose arrival at Saclay was a turning point of my PhD. It was a real pleasure to work with you (and drink your coffee ;)). Thanks a lot ! D'aussi loin que je me souvienne faire de la recherche a toujours été dans un coin de ma tête. Petit cela ne se présentait que sous la forme d'un goût, d'une aisance pour les mathématiques mais aussi d'une incompréhension envers ceux pour qui ce language ne parlait pas. Puis au cours de mes années au collège on m'a fait comprendre que c'était bien beau de faire le je-sais-tout mais que c'était mieux d'essayer d'expliquer ce qui me paraisait limpide, ce qui s'avérait assez difficile pour moi à l'époque (désolé Line pour mon impatience d'alors). À ce moment l'idée de faire de l'enseignement à commencer à germer.

C'est lors de mon passage au lycée que je me suis découvert un enthousiasme pour la physique. Un mercredi matin en salle de travaux pratiques on me fait faire tomber une goutte d'huile dans un bac rempli d'eau préalablement saupoudrée de talc. Après quelques lignes de calculs que voilà une jolie estimation de la longueur de la queue hydrophobe des lipides, d'approximativement un nanomètre ! Je me souviendrais toujours de ce que j'ai ressenti alors. Que l'on pouvait utiliser les mathématiques pour comprendre le monde dans ses secrets les plus intangibles. Je tiens à remercier Anne Robin, mon professeur de sciences physiques de seconde, pour m'avoir donné ce goût de la physique mais aussi pour m'avoir inculqué la rigueur nécessaire à tout bon scientifique.

C'est alors que je me suis tourné vers les classes préparatoires CPGE (un grand merci à Brigitte Huguet pour m'avoir poussé à faire le bon choix) dans le but de devenir professeur de physique. C'est l'occasion pour moi de remercier Jean-Michel Rey, mon professeur de mathématiques de MPSI3, pour avoir toujours montré que la qualité prévalait sur la quantité ainsi que Jean-Baptiste Paire, mon professeur de physique en PSI*, pour sa bonne humeur communicative ainsi que pour m'avoir orienté vers le magistère de physique fondamentale de l'université Paris-Sud.

Finalement, arrivé à l'université Paris-Sud et entouré d'enseignants-chercheurs l'idée de faire une thèse s'installa peu à peu. Je remercie notamment Christophe Texier et Nicolas Pavloff, dont l'enseignement m'a définitivement convaincu que je voulais faire de la physique théorique, ainsi qu'Elias Khan pour m'avoir montré qu'on avait encore beaucoup de choses intéressantes à découvrir en physique nucléaire.

Arrivé au master NPAC, j'ai eu la chance de faire la rencontre de Jean-Paul Ebran qui enseignait le cours de physique nucléaire théorique. Grâce à toi j'ai pu faire la rencontre de Thomas et c'est ainsi que tout a commencé.

Un grand merci à mes collègues du CEA DAM Ile-de-France pour leur accueil et leur iii

bienveillance. Je garde un très bon souvenir de cette fameuse soirée karaoké (en partie grâce à l'interprétation de Whitney Houston par un certain Guillaume ;)). Merci à tous mes collègues de l'IPNO et du DPhN.

Je tiens à remercier mes directeur, co-directeur et co-encadrant de thèse, Thomas, Jean-Paul et Denis qui m'ont introduit et guidé au sein de la communauté de la structure nucléaire théorique. Denis, je te remercie d'avoir accepté de me prendre en stage et d'avoir continué à me suivre pendant la thèse. Avec toi j'ai pu découvrir la théorie BCS ainsi que le fortran commenté en italien :p. Jean-Paul et Thomas, merci pour m'avoir accordé l'opportunité de faire cette thèse avec vous. Jean-Paul, je te remercie pour les cours que tu m'as donnés ainsi que pour le cours que tu m'as laissé donner (mon premier cours : en anglais, qui dura 3 heures et avec un seul jour de préparation :p) et bien que le sujet de la thèse ait évolué et que je me sois posé au DPhN sache que j'aurais aimé avoir plus d'occasions de travailler et d'échanger avec toi. Enfin Thomas, merci pour m'avoir tant appris, et pas seulement à propos du problème à N corps mais aussi à bien écrire en anglais, à rédiger des notes propres ou à préparer des présentations claires et adaptées. Ton extrême rigueur et exigence m'ont permis de m'améliorer là où je ne pensais jamais pouvoir. Malgré ma façon de travailler particulière, tu as souvent su me canaliser et je pense qu'on a sorti de bons résultats de cette collaboration.

Merci à tous les membres des soirées films, que j'attendais avec impatience les lundis soirs, à tous mes amis danseurs, qu'il me tardait de rejoindre le vendredi soir, ainsi qu'aux membres du club BTK, qui se reconnaîtront... Merci Lucie et Hugo de vous être déplacés pour venir assister à ma soutenance :). Un grand merci à ma famille qui a toujours cru en moi et qui m'a toujours soutenu dans ce que j'entreprenais. Et pour finir, un immense merci à Mélissa qui a été plus que patiente pendant la période de rédaction. Je suis tellement chanceux d'avoir pu faire ta rencontre. Merci pour tout.

Introduction

Nuclear physics is the field of physics that aims at the description of atomic nuclei. Contrary to what is commonly thought about nuclear physics, properties of nuclei are not yet fully understood and the field has recently accomplished major progress both from a theoretical and an experimental point of view.

One of the specificity about nuclear systems is that the internucleonic force that make them bound cannot be directly derived from quantum chromodynamics (QCD). Whereas attempts to connect the nuclear interaction to the underlying QCD through lattice calculations have being made [START_REF] Ishii | Nuclear Force from Lattice QCD[END_REF][2][START_REF] Van Kolck | Few-Nucleon Systems in a Quirky World[END_REF], results of physical relevance are no yet available. Considering that fact, more phenomenological or effective methods have been designed in order to perform realistic calculations. Another difficulty lies in the content of the nuclear interaction itself, e.g. spin-orbit, tensor, quadratic spin-orbit terms, that makes its handling particularly complex. Additionally, there are two sources of non-perturbative character. The first one corresponds to the strong short-range repulsion associated with interpenetration of nucleons, that is a non-perturbative source of the ultraviolet kind. The second one corresponds to the scattering lengths associated with the existence of a weakly bound proton-neutron state, the deuteron, and of a virtual di-neutron state, that is a non-perturbative source of the infrared kind. Furthermore, the treatment of nucleons as point-like particles, while they are in fact composites, requires the use of three-, four-, . . . , up to A-nucleons forces in principle. In practice, the limitation to three-nucleons forces has been shown to provide quite accurate results.

Given the model of the nuclear Hamiltonian, the goal is now to achieve the challenge of solving the A-body Schrödinger equation. Atomic nuclei contain up to approximately 300 nucleons such that most of the nuclei cannot be considered as very little nor very big many-body systems. Hence, exact methods available for few-body systems quickly find their theoretical and computational limits, as the number of nucleons increases, while physical effects coming from the finite size of nuclei prevent the use of a statistical approach. Low-energy nuclear theory aims at describing ground-state (mass, radius, deformation, multipolar moments...) and excited-states (single-particle, vibrational, shape and spin isomers, high-spin and super-deformed rotational bands...) properties as well as their various decay modes (strong, electromagnetic and electroweak). Furthermore, a unified description of the nuclear system requires a description of close-and open-shell systems, small-and large-amplitude collective motions, interfacing structure and reaction to access spontaneous and induced fission, fusion and nucleon emission at the drip-line... This challenge is to be achieved over the whole nuclear chart, i.e. for around 3400 nuclei that has already been observed [START_REF] Audi | The NUBASE2016 evaluation of nuclear properties[END_REF] and the thousands still to be discovered. gluons are not explicitly accounted for, such that nucleons are treated as pointlike objects. In this context, collective degrees of freedom, as any other nuclear phenomenon, are meant to emerge from the description of interacting nucleons.

2. Internucleonic interactions are rooted in the underlying QCD to preserve the link with higher-energy physics. The current paradigm in ab initio nuclear physics is to use interactions derived from chiral effective field theory (χEFT) that are fitted in the two-body sector for two-nucleons forces, three-body sector for three-nucleons forces, etc.

3. The solution of the many-body Schrödinger equation is expanded in a systematic way, thus providing control over the truncation of the result and an assessment of the associated uncertainties.

4. Errors on the results, coming from the input Hamiltonian, the analytical truncation and the numerical treatment, are eventually estimated.

These characteristics distinguish ab initio methods from other approaches to the nuclear many-body problem that rely on phenomenological interactions and for which a thorough error assessement is often complicated if not impossible. Ab initio methods providing essentially exact solution of the A-body Schrödinger equation, i.e. Fadeev-Yakubovski [6][START_REF] Glöckle | Alpha-particle binding energies for realistic nucleonnucleon interactions[END_REF][START_REF] Nogga | Triton binding energies for modern NN forces and the π-π exchange three-nucleon force[END_REF], Green's function monte carlo [START_REF] Pudliner | Quantum Monte Carlo calculations of nuclei with A <˜7[END_REF][START_REF] Wiringa | Quantum Monte Carlo calculations for light nuclei[END_REF][START_REF] Wiringa | Quantum Monte Carlo calculations of A = 8 nuclei[END_REF] and the no-core shell model [START_REF] Navrátil | Shell-model calculations for the three-nucleon system[END_REF][START_REF] Navrátil | Large-basis shell-model calculations for p -shell nuclei[END_REF][START_REF] Navrátil | Few-nucleon systems in a translationally invariant harmonic oscillator basis[END_REF][START_REF] Quaglioni | Ab initio no-core shell model and microscopic reactions: Recent achievements[END_REF][START_REF] Navrátil | Recent developments in no-core shell-model calculations[END_REF], have been limited to light nuclei up to A ∼ 12 due to their exponential scaling with A. In the past fifteen years, and with the development of polynomially-scaling methods, a significant extension of ab initio methods with respect to accessible mass numbers has been possible. Ab initio approaches applicable to closed-shell systems typically start from a single-determinantal, e.g. Hartree-Fock (HF), reference state and account for dynamic correlations via the inclusion of particle-hole excitations on top of it. The simplest method in this regards is many-body perturbation theory (MBPT) [START_REF] Goldstone | Derivation of the Brueckner many-body theory[END_REF][START_REF] Hugenholtz | Perturbation theory of large quantum systems[END_REF][START_REF] Shavitt | Many-body methods in chemistry and physics : MBPT and coupled-cluster theory[END_REF]. This method has been abandoned in the 1960s by the nuclear community due to the (believed to be) inherent hard-core character of the nuclear interaction. Only recently with the development of softer Hamiltonians generated through similarity renormalization group (SRG) transformations taming down the ultraviolet source of non-perturbativeness, MBPT has been revisited with great success [START_REF] Tichai | Hartree-Fock many-body perturbation theory for nuclear ground-states[END_REF][START_REF] Hu | Ab initio nuclear many-body perturbation calculations in the Hartree-Fock basis[END_REF][START_REF] Tichai | Open-shell nuclei from No-Core Shell Model with perturbative improvement[END_REF]. Several manybody frameworks resumming all-order perturbative contributions have been developed to describe medium-mass systems, e.g., coupled cluster (CC) theory [START_REF] Kowalski | Coupled Cluster Calculations of Ground and Excited States of Nuclei[END_REF][START_REF] Bartlett | Coupled-cluster theory in quantum chemistry[END_REF][START_REF] Hagen | initio coupled-cluster approach to nuclear structure with modern nucleon-nucleon interactions[END_REF][START_REF] Piecuch | Left-eigenstate completely renormalized equation-of-motion coupled-cluster methods: Review of key concepts, extension to excited states of open-shell systems, and comparison with electronattached and ionized approaches[END_REF][START_REF] Binder | Extension of coupled-cluster theory with a noniterative treatment of connected triply excited clusters to three-body Hamiltonians[END_REF], self-consistent Green function (SCGF) theory [START_REF] Dickhoff | Self-consistent Green's function method for nuclei and nuclear matter[END_REF][START_REF] Cipollone | Chiral three-nucleon forces and the evolution of correlations along the oxygen isotopic chain[END_REF][START_REF] Carbone | Self-consistent Green's functions formalism with three-body interactions[END_REF] or the in-medium similarity renormalization group (IMSRG) approach [START_REF] Tsukiyama | In-Medium Similarity Renormalization Group For Nuclei[END_REF][START_REF] Hergert | In-medium similarity renormalization group with chiral two-plus three-nucleon interactions[END_REF][START_REF] Morris | Magnus expansion and in-medium similarity renormalization group[END_REF][START_REF] Hergert | The In-Medium Similarity Renormalization Group: A novel ab initio method for nuclei[END_REF]. For doubly closed-shell nuclei, all of these non-perturbative methods agree well with quasi-exact NCSM calculations for ground-state energies of nuclei in the A ∼ 20 regime, and are considered to constitute the reference methods for mid-mass nuclei. Such expansion methods have been able, based on realistic chiral Hamiltonians, to extend their reach of the nuclear chart up to A ∼ 130 in the past decade [START_REF] Binder | Ab initio path to heavy nuclei[END_REF], but remained for a long time limited to doubly-closed shell (or neighboring) nuclei. Going away from nuclear shell closures, the single-determinantal description becomes qualitatively wrong due to the degeneracies present in the single-particle spectrum, requiring a proper treatment of static correlations. In order to overcome this drawback, more general reference states are required. A first option to overcome this difficulty is to rely on multi-reference (MR) methods, accounting for the different product states that contribute substantially to the wave function. This idea has been followed to develop MR-IMSRG [START_REF] Hergert | The In-Medium Similarity Renormalization Group: A novel ab initio method for nuclei[END_REF][START_REF] Hergert | Initio Calculations of Even Oxygen Isotopes with Chiral Two-Plus-Three-Nucleon Interactions[END_REF][START_REF] Hergert | In-medium similarity renormalization group for closed and open-shell nuclei[END_REF] in nuclear physics or MR-CC in quantum chemistry [START_REF] Mukherjee | Use of Cluster Expansion Methods in the Open-Shell Correlation Problem[END_REF][START_REF] Jeziorski | Spin-adapted multireference coupled-cluster approach: Linear approximation for two closed-shell-type reference configurations[END_REF][START_REF] Musiał | Multireference coupled-cluster theory: The easy way[END_REF]. More recently this idea was employed the involved tensor networks entering a particular many-body formalism via the development of the AMC program.

The present document is structured as follows: Chapter 1 presents the formal ingredients needed all throughout the document, in particular to formulate PBMBPT. Chapter 2 introduces the fully non-perturbative projected BCC (PBCC) [START_REF] Duguet | Symmetry broken and restored coupled-cluster. Global gauge symmetry and particle number[END_REF][START_REF] Qiu | Particle-number projected Bogoliubov-coupled-cluster theory: Application to the pairing Hamiltonian[END_REF][START_REF] Qiu | Projected coupled cluster theory[END_REF] formalism with the goal to extract its perturbative version, i.e. PBMBPT. This is done in two variants, based on gauge-rotated amplitudes and unrotated amplitudes. In chapter 3, one version of PBMBPT is revisited such that off-diagonal kernels themselves are expanded in perturbation. Whereas low orders given in chap. 2 have been worked out in Ref. [START_REF] Duguet | Symmetry broken and restored coupled-cluster. Global gauge symmetry and particle number[END_REF], the goal is to design a new version of the code ADG [START_REF] Arthuis | ADG: Automated generation and evaluation of many-body diagrams I. Bogoliubov many-body perturbation theory[END_REF] that can automatically generate all valid off-diagonal BMBPT diagrams to arbitrary order and evaluate their algebraic expression to be implemented for numerical applications. In chapter 4, a framework providing an efficient way to compute matrix elements between two (particle-number-projected) many-body states is designed. The two states may originate from perturbative BMBPT amplitudes, non-perturbative BCC amplitudes or even variational BCI solutions. This framework is at the basis of the application of eigenvector continuation [START_REF] Frame | Eigenvector Continuation with Subspace Learning[END_REF] to BMBPT. Chapter 5 presents an extension of the NO2B approximation, often used in nuclear many-body theory in order to capture the dominant effects of three-nucleon forces while effectively working with two-body operators, that is consistent with symmetries of the Hamiltonian while working with a symmetry broken reference state. This approximation has been employed in Gorkov SCGF [START_REF] Somà | Ab initio self-consistent Gorkov-Green's function calculations of semimagic nuclei: Formalism at second order with a twonucleon interaction[END_REF][START_REF] Somà | Ab initio Gorkov-Green's function calculations of open-shell nuclei[END_REF][START_REF] Somà | Ab initio self-consistent Gorkov-Green's function calculations of semi-magic nuclei: Numerical implementation at second order with a two-nucleon interaction[END_REF] and BMBPT [START_REF] Tichai | Bogoliubov many-body perturbation theory for open-shell nuclei[END_REF][START_REF] Tichai | Pre-processing the nuclear many-body problem[END_REF] in which U(1) symmetry associated with particle-number conservation is spontaneously broken by the (approximate) many-body state. Last but not least, chapter 6 introduces a systematic method, based on the socalled Yutsis graph [START_REF] Yutsis | Mathematical apparatus of the theory of angular momentum[END_REF], used to preform the angular-momentum reduction of many-body equations. The first version of the AMC program that automatically performs this tedious symmetry reduction is presented. Taking the symmetry-unrestricted expression of a generic tensor network as an input, the code provides its angular-momentum-coupled form in an error-safe way in a matter of seconds. Normal-ordered matrix elements, similaritytransformed operator, off-diagonal diagrams and expressions, perturbative amplitudes and angular-momentum-coupled formulae are provided in a set of appendices.

Chapter 1.

Prolegomena 1.1. Master equations

The aim of ab initio nuclear structure theory is to solve the time-independent many-body Schrödinger equation

H|Ψ A n = E A n |Ψ A n . (1.1)
In equation (1.1), H is the Hamiltonian containing elementary inter-nucleon interactions, typically derived within the framework of chiral effective field theory [START_REF] Weinberg | Nuclear forces from chiral lagrangians[END_REF][START_REF] Weinberg | Effective chiral lagrangians for nucleon-pion interactions and nuclear forces[END_REF][START_REF] Epelbaum | Modern theory of nuclear forces[END_REF][START_REF] Meißner | The long and winding road from chiral effective Lagrangians to nuclear structure[END_REF][START_REF] Epelbaum | Nuclear chiral EFT in the precision era[END_REF], while E A n and |Ψ A n denote its A-body eigenenergies and eigenstates, respectively. Presently focusing on many-body methods relying on the breaking and restoration of particle-number symmetry, one must further consider explicitly the particle-number eigenvalue equation

A|Ψ A n = A|Ψ A n , (1.2)
where A is the particle-number operator, the number of nucleons A of the many-body state being its eigenvalue 1 . Combining both Eqs. (1.1) and (1.2) eventually leads to introducing the grand potential eigenvalue equation

Ω|Ψ A n = E A n |Ψ A n , (1.3)
where Ω is the grand potential operator Ω ≡ H -λA , (1.4)

E A n = E A
n -λA its eigenvalues, and λ a Lagrange parameter. The grand potential operator is introduced in order to eventually constrain the particle number in average and to drive the imaginary time evolution in Bogoliubov many-body perturbation theory (BMBPT) [START_REF] Arthuis | Bogoliubov Many-Body Perturbation Theory for Nuclei : Systematic Generation and Evaluation of Diagrams and First ab initio Calculations[END_REF].

In the following, one often makes use of a generic operator O, commuting with Ω and A, whose eigenvalue equation reads in the basis of the Hamiltonian eigenstates as

O|Ψ A n = O A n |Ψ A n .
(1.5)

Operator representation

In this manuscript, many-body operators are represented in second-quantized form. To do so, an arbitrary single-particle basis B 1 ≡ c l , c † l of the one-body Hilbert space H 1 is considered, where fermionic creation and annihilation operators c l and c † l obey standard anti-commutation relations ≡ o [0] + o [2] + o [4] + o [6] + . . . o [2] o 11 o [4] o 22 o [6] o 33

c l 1 , c l 2 = 0 , (1.6a) c † l 1 , c † l 2 = 0 , (1.6b) c l 1 , c † l 2 = δ l 1 l 2 . ( 1
Table 1.1. Contributions to the three-body operator O in normal-ordered form with respect to the particle vacuum |0 and expressed in {c, c † }. The o ij contributions are sorted horizontally according to i -j and vertically according to i + j.

where each term o nn of the particle-number conserving operator O is obviously characterized by an equal number n of particle creation and annihilation operators. The class o [2n] is nothing but the term o nn of n-body character, which is given in the single-particle basis B 1 by

o nn ≡ 1 n!n! l 1 ...l 2n
o nn l 1 ...l n l n+1 ...l 2n c † l 1 . . . c † l n c l 2n . . . c l n+1 .

(1.8)

In Eq. (1.8), matrix elements o nn l 1 ...l n l n+1 ...l 2n are fully antisymmetric with respect to the permutation of the n first, resp. n last, indices o nn l 1 ...l n l n+1 ...l 2n = (σ) o nn σ(l 1 ...l n |l n+1 ...l 2n ) , (

where (σ) refers to the signature of the permutation σ. The notation σ(. . . | . . .) denotes a separation between the n first and the n last indices such that permutations are only considered between members of the same group. If the operator O is hermitian, the n-body matrix elements o nn l 1 ...l n l n+1 ...l 2n satisfy o nn * l 1 ...l n l n+1 ...l 2n = o nn l n+1 ...l 2n l 1 ...l n .

(1.10)

In Eq. (1.8), the operator O is written in normal-ordered form with respect to the particle vacuum |0 , i.e. all creation operators are to the left of all annihilation operators. A graphical representation of the contributions to a three-body operator O is given in Tab. 1.1 where the o ij contributions are sorted horizontally according to i -j and vertically according to i + j. As the operator O commutes with A, there is no contribution with i = j in the present case. This graphical representation will become useful in the discussion of normal ordering with respect to the quasi-particle vacuum |Φ .

In particular, the one-body particle-number operator reads itself as .11) such that its matrix elements are nothing but a 11 l 1 l 2 = δ l 1 l 2 in any basis of H 1 .

A ≡ l c † l c l , ( 1

Global gauge symmetry

In the following, global gauge symmetry associated with particle-number conservation will be considered to be (i) spontaneously broken and (ii) restored. The symmetry group associated with the global rotation of an A-body fermionic system in gauge space is the abelian Lie group U (1) ≡ {R(ϕ), ϕ ∈ [0, 2π]}. The unitary representation of U (1) on Fock space F is given by R(ϕ) = e iAϕ , (1.12) where A is the infinitesimal generator of global-gauge rotations, such that the characters of irreducible representations (IRREPs) read as

Ψ A n |R(ϕ)|Ψ A n = e iAϕ δ AA δ nn , (1.13)
From a group theoretical perspective, the number A appearing on the right-hand side of Eq. (1. [START_REF] Navrátil | Large-basis shell-model calculations for p -shell nuclei[END_REF]) is an integer (A ∈ Z). In practice this number represents the number of nucleons in the system so that it is constrained from a physical perspective to be a natural number (A ∈ N). The orthogonality relation among the IRREPs reads as

R(ϕ)|Ψ

A n = e iAϕ |Ψ A n .

(1.17

)
Particular examples of scalar operators, i.e., operators of rank 0, are the Hamiltonian, the particle-number operator itself and the grand potential. These scalar operators commute with gauge rotation operators, such that the relations

[H, R(ϕ)] = [A, R(ϕ)] = [Ω, R(ϕ)] = 0 , ( 1.18)
hold for any ϕ ∈ [0, 2π]. A key feature for the following is that any integrable 2π-periodic function f (ϕ) can be expanded over the IRREPs of the U (1) group. This constitutes nothing but the Fourier decomposition of the function

f (ϕ) ≡ k∈Z f k e ikϕ , ( 1.19)
which defines the set of expansion coefficient f k . It is possible to define a projection 4 operator P A P A = 1 2π 2π 0 dϕ e -iAϕ R(ϕ) , (1.20)

3 A tensor operator of rank k with respect to the U (1) group is an operator that associates a state of the A-body Hilbert space H A to a state of the (A + k)-body Hilbert space H A+k , i.e., that changes the number of particles by k units. 4 As P A is a projection operator it is also idempotent, such that the relation (P A ) 2 = P A holds.

Bogoliubov algebra

which is able to select the component belonging to a particular IRREP out of a particlenumber wave packet, see Fig. where |Θ A are eigenstates of the particle-number operator A|Θ A = A |Θ A , the application of P A indeed gives

P A |Θ = A P A |Θ A = A 1 2π 2π 0 dϕe i(A -A)ϕ |Θ A = |Θ A (1.22)
where the orthogonality relation (1.14) has been used.

Bogoliubov algebra 1.4.1. Bogoliubov transformation

The Bogoliubov transformation [START_REF] Ring | The nuclear many-body problem[END_REF] is a linear transformation that connects a set of singleparticle creation and annihilation operators {c l , c † l } to a set of quasi-particle creation and annihilation operators {β k , β † k } according to

β k ≡ l U * lk c l + V * lk c † l , (1.23a) β † k ≡ l V lk c l + U lk c † l , (1.23b)
or in matrix form

β β † ≡ W † c c † , ( 1.24)
where the Bogoliubov matrix W is given by

W ≡ U V * V U * . (1.25)
The components U and V of the Bogoliubov matrix W are not arbitrary as quasi-particle operators are enforced to obbey fermionic anticommutation rules

β k 1 , β k 2 = 0 , (1.26a) β † k 1 , β † k 2 = 0 , (1.26b) β k 1 , β † k 2 = δ k 1 k 2 . (1.26c)
This requirement restricts the Bogoliubov matrix W to being unitary

W † W = 1 , (1.27a) WW † = 1 , (1.27b)
which translates into the set of relations5

U † U + V † V = 1 , (1.28a) V T U + U T V = 0 , (1.28b) U U † + V * V T = 1 , (1.28c) V U † + U * V T = 0 .
(1.28d)

Bogoliubov vacuum

The Bogoliubov state 6 |Φ is a vacuum for the complete set of quasi-particles operators, i.e.

β k |Φ = 0 (1.29)
for all k defined through Eq. (1.23). One can explicitly construct this vacuum as the product of all quasi-particle annihilation operators β k acting on the particle vacuum

|Φ ≡ C k β k |0 , ( 1.30)
with C a complex normalization constant. As it is built from quasi-particle operators, the Bogoliubov vacuum is not an eigenstate of A but rather a particle-number wave-packet

A|Φ = A|Φ , (1.31)
i.e. it breaks U (1) symmetry. Alternatively the Bogoliubov vacuum is fully defined by the normal density matrix ρ and the anomalous density matrix 7 κ whose matrix elements in the single-particle basis are defined through

ρ l 1 l 2 ≡ Φ|c † l 2 c l 1 |Φ Φ|Φ , ( 1.32a
)

κ l 1 l 2 ≡ Φ|c l 2 c l 1 |Φ Φ|Φ , ( 1.32b)
or in matrix notation

ρ = V * V T , (1.33a) κ = V * U T = -U V † , (1.33b)
where the Bogoliubov transformation (Eq. (1.23)) has been used. Using fermionic anticommutation relations of single-particle operators (Eq. (1.6)) and the definition of the density matrices (Eq. (1.32)), one can show that the normal density matrix ρ is hermitian and that the anomalous density matrix κ is skew-symmetric, i.e.

ρ † = ρ , (1.34a)

κ T = -κ . (1.34b)
Furthermore using the unitarity of the Bogoliubov transformation (Eq. (1.28)) and the matrix form of ρ and κ (Eq. (1.33)) one can show two relations relating ρ and κ ρ 2 -ρ = κκ * , (1.35a) ρκ = κρ * .

(1.35b)

The complete information encoded in the elementary contractions (Eq. (1.32)) can be compacted by introducing the generalized density matrix

R ≡ ρ κ -κ * 1 -ρ * , (1.36)
which is hermitian and idempotent

R † = R , (1.37a) R 2 = R . (1.37b)
The matrix R is diagonal in the basis of the quasi-particle creation, resp. annihilation, operators with eigenvalue 0, resp. 1,

W † RW = 0 0 0 1 . (1.38)

Gauge-rotated Bogoliubov transformations

The gauge-rotated Bogoliubov vacuum Φ(ϕ)| is defined by the application of the gaugespace rotation operator (Eq. (1.12)) onto the Bogoliubov bra 8 Φ| Φ(ϕ)| ≡ Φ|R(ϕ) , (1.39) and is the vacuum of the gauge-rotated quasi-particle operators defined as

β R (ϕ) β † R (ϕ) ≡ R -1 (ϕ) β β † R(ϕ) , (1.40)
such that each gauge-rotated quasi-particle creation operator annihilates it

Φ(ϕ)|(β † R ) k (ϕ) = 0 , ( 1.41)
for all k.

Single-particle basis to rotated quasi-particle basis

The rotated quasi-particle operators are related to the single-particle operators through a Bogoliubov transformation

β R (ϕ) β † R (ϕ) = W † R (ϕ) c c † , ( 1.42)
where the Bogoliubov matrix W † R (ϕ) is equal to

W † R (ϕ) ≡ U † (ϕ) V † (ϕ) V T (ϕ) U T (ϕ) , ( 1.43)
and U (ϕ) ≡ e -iϕ U , (1.44a)

V (ϕ) ≡ e +iϕ V .

(1.44b)

Proof. Starting from the definition of rotated quasi-particle operators (1.40) and applying Baker-Campbell-Hausdorff's formula

e X Y e -X = k=0 1 k! [X, Y ] (k) = Y + [X, Y ] + 1 2 [X, [X, Y ]] + . . . , (1.45)
for X = -iϕA and Y = β k , one obtains

(β R ) k (ϕ) = e -iϕA β k e iϕA = β k + [-iϕA, β k ] + 1 2 [-iϕA, [-iϕA, β k ]] + . . . . (1.46)
Expanding quasi-particle operators through Eq. (1.23) and using commutation relations of the particle-number operator with single-particle operators

[A, c † l ] = +c † l , (1.47a) [A, c l ] = -c l ,
(1.47b) rotated quasi-particle operators are written as

(β R ) k (ϕ) = l (U * lk c l + V * lk c † l ) + -iϕA, (U * lk c l + V * lk c † l ) + 1 2 -iϕA, -iϕA, (U * lk c l + V * lk c † l ) + . . . = l 1 + iϕ + 1 2 (iϕ) 2 + . . . U * lk c l + 1 -iϕ + 1 2 (iϕ) 2 + . . . V * lk c † l = l e +iϕ U * lk c l + e -iϕ V * lk c † l ≡ l U * lk (ϕ)c l + V * lk (ϕ)c † l , (1.48)
which is indeed the content of the upper row of the Bogoliubov matrix W † R (ϕ) in Eq. (1.43). The lower row can be obtained by the same procedure or invoking the unitarity of the Bogoliubov matrix.

Quasi-particle basis to rotated quasi-particle basis

The rotated quasi-particle operators are related to the unrotated ones through a Bogoliubov transformation

β R (ϕ) β † R (ϕ) ≡ O † R (ϕ) β β † , ( 1.49)
where the Bogoliubov matrix O † R (ϕ) is equal to

O † R (ϕ) ≡ A † (ϕ) B † (ϕ) B T (ϕ) A T (ϕ) , ( 1.50)
with

A(ϕ) ≡ V † V (ϕ) + U † U (ϕ) , (1.51a) B(ϕ) ≡ U T V (ϕ) + V T U (ϕ) .
(1.51b)

Proof. Using the Bogoliubov transformation (1.42) and applying W on both sides of (1.24) one obtains

β R (ϕ) β † R (ϕ) = W † R (ϕ)W β β † . (1.52)
The content of the Bogoliubov matrix O † R (ϕ) is then obtained through the multiplication of W † R (ϕ) and W .

Furthermore it is easy to check that

A(-ϕ) = A † (ϕ) , (1.53a)
B(-ϕ) = B T (ϕ) .

(1.53b)

Thouless transformation between vacua

Thouless' theorem [START_REF] David | Perturbation theory in statistical mechanics and the theory of superconductivity[END_REF] states that given a Bogoliubov product state Φ 0 | that is a vacuum for the operators {β 0 k }, any Bogoliubov product state Φ 1 | that is not orthogonal to Φ 0 | can be written in the form

Φ 1 | = Φ 1 |Φ 0 Φ 0 |e Z , ( 1.54)
where the one-body Thouless operator

Z ≡ 1 2 k 1 k 2 Z 02 k 1 k 2 β 0 k 2 β 0 k 1 , (1.55)
only contains a pure de-excitation part over |Φ 0 and the skew-symmetric Thouless matrix Z 02 is uniquely determined. Conversely, any wavefunction of the form of Eq. (1.54), where Φ 0 | is a Bogoliubov product state, is also a Bogoliubov product state. Furthermore if the Bogoliubov transformation between the operators {β 0 k } and the operators {β 1 k }, for which Φ 1 | is the vacuum, is given by

β 1 β 1 † = U 10 V * 10 V 10 U * 10 β 0 β 0 † , ( 1.56)
then the Thouless matrix reads as

Z 02 = V 10 U -1 10 .
(1.57)

Particle vacuum to Bogoliubov vacuum

The Thouless theorem can be used in order to express Φ| via a non-unitary Thouless transformation applied to 0| according to Φ| = Φ|0 0|e Z , (1.58) where the one-body Thouless operator

Z ≡ 1 2 l 1 l 2 Z 02 l 1 l 2 c l 2 c l 1 , (1.59)
only contains a pure de-excitation part over |0 . The corresponding Thouless matrix

Z 02 ≡ V U -1 , (1.60)
is expressed in terms of the Bogoliubov transformation connecting single-particle operators to quasi-particle operators (1.23).

Bogoliubov vacuum to rotated Bogoliubov vacuum

As stipulated by Eq. (1.39), the rotated Bogoliubov vacuum is obtained from Φ| via the unitary transformation R(ϕ). One can rather express Φ(ϕ)| via a non-unitary Thouless transformation applied to Φ| according to Φ(ϕ)| = Φ(ϕ)|Φ Φ|e Z(ϕ) , (1.61) where the one-body Thouless operator

Z(ϕ) ≡ 1 2 k 1 k 2 Z 02 k 1 k 2 (ϕ)β k 2 β k 1 , (1.62)
only contains a pure de-excitation part over |Φ . The corresponding Thouless matrix Z 02 (ϕ) ≡ B(ϕ)A -1 (ϕ) , (1.63) is expressed in terms of the Bogoliubov transformation connecting quasi-particle operators to rotated quasi-particle operators (1.50).

Off-diagonal elementary contractions

The elementary contractions of quasiparticle operators that are in use when employing the off-diagonal Wick theorem [START_REF] Balian | Nonunitary bogoliubov transformations and extension of Wick's theorem[END_REF] are given by

R(ϕ) ≡    Φ(ϕ)|β † β |Φ Φ(ϕ)|Φ Φ(ϕ)|β β |Φ Φ(ϕ)|Φ Φ(ϕ)|β † β † |Φ Φ(ϕ)|Φ Φ(ϕ)|β β † |Φ Φ(ϕ)|Φ    ≡ R +-(ϕ) R --(ϕ) R ++ (ϕ) R -+ (ϕ) = 0 0 Z 02 (ϕ) 1
.

(1.64) Most of the above contractions are easily obtained by using the fact that |Φ is the vacuum of the quasiparticle operators, i.e. β k |Φ = 0 for all k. The single non-trivial (anomalous) contraction is obtained on the basis of standard Wick's theorem [START_REF] Wick | The Evaluation of the Collision Matrix[END_REF] as

R ++ k 1 k 2 (ϕ) = Φ(ϕ)|β † k 1 β † k 2 |Φ Φ(ϕ)|Φ = Φ|e Z(ϕ) β † k 1 β † k 2 |Φ = 1 2 kk Z 02 kk (ϕ) Φ|β k β k β † k 1 β † k 2 |Φ = 1 2 Z 02 k 1 k 2 (ϕ) -Z 02 k 2 k 1 (ϕ) = Z 02 k 1 k 2 (ϕ) , (1.65)
and is zero in the diagonal case, i.e. R ++ k 1 k 2 (0) = 0.

Similarity-transformed Bogoliubov transformation

The similarity-transformed quasi-particle operators are defined by the application of the similarity transformation associated with the Thouless operator Z(ϕ) to quasi-particle operators through

β Z (ϕ) β † Z (ϕ)
≡ e Z(ϕ) β β † e -Z(ϕ) , (1.66) and are actually related to unrotated quasi-particle operators via a non-unitary Bogoliubov transformation

β Z (ϕ) β † Z (ϕ) = O † Z (ϕ) β β † , ( 1.67)
where

O † Z (ϕ) ≡ 1 0 Z 02 (ϕ) 1
.

(1.68)

Proof. Starting from the definition of transformed quasi-particle operators (1.66) and applying the Baker-Campbell-Hausdorff formula (1.45) one obtains β † Z (ϕ) = e +Z(ϕ) β † e -Z(ϕ)

= β † + Z(ϕ), β † + 1 2 Z(ϕ), Z(ϕ), β † + . . . . (1.69) Using commutation relations of the Thouless operator with quasi-particle creation operators

Z(ϕ), β † k = 1 2 k 1 k 2 Z 02 k 1 k 2 (ϕ) β k 2 β k 1 , β † k = 1 2 k 1 k 2 Z 02 k 1 k 2 (ϕ) β k 2 δ kk 1 -β k 1 δ kk 2 = (Z 02 (ϕ)β) k , ( 1.70)
transformed quasi-particle creation operators are written as β † Z (ϕ) = β † + Z 02 (ϕ)β , (1.71) which is indeed the content of the lower row of the Bogoliubov matrix O † Z (ϕ) in (1.68). The upper row can be obtained via the same procedure

β Z (ϕ) = e +Z(ϕ) βe -Z(ϕ) = β + [Z(ϕ), β] + 1 2 [Z(ϕ), [Z(ϕ), β]] + . . . = β , (1.72)
where the commutation relation of the Thouless operator with quasi-particle annihilation operators

[Z(ϕ), β k ] = 1 2 k 1 k 2 Z 02 k 1 k 2 (ϕ) β k 2 β k 1 , β k = 0 (1.73)
was used.

Normal ordering

In this section, normal-ordered forms of the operator O, with respect to the Bogoliubov vacuum |Φ , are given both in the single-particle and the quasi-particle bases.

Single-particle basis

The normal ordering of O with respect to the Bogoliubov vacuum |Φ leads to re-expressing the operator under the form

O ≡ N n=0 N i,j=0 i+j=2n 1 i!j! l 1 ...l i+j Λ ij l 1 ...l i+j : c † l 1 . . . c † l i c l i+j . . . c l i+1 : ≡ N n=0 N i,j=0 i+j=2n Λ ij ≡ N n=0
O [2n] (1. [START_REF] Arthuis | Bogoliubov Many-Body Perturbation Theory formalism[END_REF] where : . . . : denotes a normal-ordered product with respect to |Φ . Matrix elements Λ ij l 1 ...l i+j are fully antisymmetric with respect to the permutation of the i first, resp. j last, indices

Λ ij l 1 ...l i l i+1 ...l i+j = (σ) Λ ij σ(l 1 ...l i |l i+1 ...l i+j ) . (1.75) If the operator O is hermitian, matrix elements Λ ij l 1 ...l i l i+1 ...l i+j satisfy Λ ij * l 1 ...l i l i+1 ...l i+j = Λ ji l i+1 ...l i+j l 1 ...l i . (1.76)
The class O [2n] groups all the terms containing a normal-ordered product of 2n singleparticle operators, i.e. terms possibly containing different numbers of single-particle creation and annihilation operators according to

O [2n] ≡ N i,j=0 i+j=2n Λ ij , ( 1.77)
where Λ ij denotes a n-body normal field 9 , resp. anomalous field, if i = j, resp. i = j, containing all the terms with a normal-ordered product of i, resp. j, single-particle creation, resp. annihilation, operators and reads as

Λ ij ≡ 1 i!j! l 1 ...l i+j Λ ij l 1 ...l i l i+1 ...l i+j : c † l 1 . . . c † l i c l i+j . . . c l i+1 : . (1.78)
Applying Wick's theorem [START_REF] Wick | The Evaluation of the Collision Matrix[END_REF] to O, matrix elements of Λ ij receive contributions from all n-body terms o nn with n ≥ max(i, j), i.e.

Λ ij l 1 ...l i+j ≡ N n=max(i,j) Λ ij(nn) l 1 ...l i+j , (1.79)
such that Λ ij(nn) l 1 ...l i+j accounts for all appropriate contraction patterns and is given by

Λ ij(nn) l 1 ...l i+j ≡ n ρ +2n κ * =n-i n ρ +2n κ =n-j (n ρ ,n κ * ,n κ ) 1 n ρ !n κ * !n κ ! 1 2 n κ * 1 2 n κ Tr[o nn ρκ * κ] (n ρ ,n κ * ,n κ ) l 1 ...l i+j , (1.
(n ρ ,n κ * ,n κ ) l 1 ...l i+j ≡ l i+j+1 ...l 2n o nn l 1 ...l i l i+j+1 ...l j+n l i+1 ...l i+j l j+n+1 ...l 2n ρ l 2n-n ρ +1 l j+n-n ρ +1 . . . ρ l 2n l j+n × κ * l i+j+1 l i+j+2 . . . κ * l j+n-n ρ -1 l j+n-n ρ × κ l j+n+1 l j+n+2 . . . κ l 2n-n ρ -1 l 2n-n ρ . (1.81)
Whenever the Bogoliubov vacuum reduces to a Slater determinant, all anomalous contractions are null, i.e. κ = κ * = 0, such that Λ ij = 0 except for i = j. The proof leading to eqs. (1.80) and (1.81), together with the explicit form of the Λ ij matrix elements associated with a three-body operator, are given in appendix A.

Quasi-particle basis

One can rather choose to express the normal-ordered form of O in the quasi-particle basis

{β k , β † k } such that O ≡ N n=0 2N i,j=0 i+j=2n 1 i!j! l 1 ...l i+j O ij l 1 ...l i+j β † k 1 . . . β † k i β k i+j . . . β k i+1 ≡ N n=0 2N i,j=0 i+j=2n O ij ≡ N n=0 O [2n] , (1.82)
where matrix elements O ij k 1 ...k i+j are fully antisymmetric with respect to the permutation of the i first, resp. j last, indices

O ij k 1 ...k i k i+1 ...k i+j = (σ) O ij σ(k 1 ...k i |k i+1 ...k i+j ) . (1.83) If the operator O is hermitian, matrix elements O ij k 1 ...k i+j satisfy O ij * k 1 ...k i k i+1 ...k i+j = O ji k i+1 ...k i+j k 1 ...k i . (1.84)
The class O [2n] groups all terms O ij of effective n-body character, i.e. containing a normal-ordered product of 2n quasiparticles operators

O [2n] ≡ 2N i,j=0 i+j=2n O ij , (1.85)
where O ij is characterized by its number i (j) of quasiparticle creation (annihilation) operators and reads

O ij ≡ 1 i!j! k 1 ...k i+j O ij k 1 ...k i k i+1 ...k i+j β † k 1 . . . β † k i β k i+j . . . β k i+1 . (1.86)
Because O has been normal-ordered with respect to |Φ , all quasiparticle creation operators (if any) are located to the left of all quasiparticle annihilation operators (if any). As an example, and thus extending to O [6] the results given in Ref. [START_REF] Signoracci | Ab initio Bogoliubov coupled cluster theory for open-shell nuclei[END_REF], the explicit form of the O ij matrix elements associated with a three-body operator O is given in App. A.

Similarity-transformed operator

The gauge-dependent similarity-transformed operator 10 of O is defined through

O Z (ϕ) ≡ e Z(ϕ) Oe -Z(ϕ) .
(1.87)

Taking as an example one term in the normal-ordered expression of O, e.g.,

O ij ≡ 1 i! 1 j! k 1 ...k i+j O ij k 1 ...k i+j β † k 1 . . . β † k i β k i+j . . . β k i+1 , (1.88) 10
The hermitic character of an operator O is lost by the application of a similarity transformation.

1.6. Hartree-Fock-Bogoliubov Theory its transformed partner reads as 11

O (ij) Z (ϕ) ≡ e Z(ϕ) O ij e -Z(ϕ) (1.89) = 1 i! 1 j! k 1 ...k i+j O ij k 1 ...k i+j ( β † Z ) k 1 (ϕ) . . . ( β † Z ) k i (ϕ)(β Z ) k i+j (ϕ) . . . (β Z ) k i+1 (ϕ) ,
where similarity-transformed quasi-particle operators were defined in Eq. (1.66). Exploiting Eqs. (1.67) and (1.68) and normal ordering the resulting terms with respect to |Φ , the similarity-transformed operator in Eq. (1.89) is eventually written as

O (ij) Z (ϕ) ≡ i+j n=j m+n≤i+j m=0 1 m! 1 n! k 1 ...k m+n O mn(ij) k 1 ...k m+n (ϕ)β † k 1 . . . β † k m β k m+n . . . β k m+1 , (1.90)
thus defining a sum of normal-ordered terms. Each term has at least as many quasi-particle annihilation operators (j) as the original operator O ij and possibly up to the total number of original quasiparticle operators (i + j). The number of creation operators ranges from 0 to the original number (i) such that the overall number of quasiparticle operators is bound to remain between j and i + j in each term. One notices that the only structural difference between the original and the transformed normal-ordered operators relates to the fact that matrix elements of the latter depend on the gauge angle. Of course, the original operator is recovered in the unrotated limit, i.e. O Z (0) = O.

Applying the above procedure to the complete operator O provides the normal-ordered form of the transformed operator

O Z (ϕ) ≡ O [0] Z (ϕ) + O [2] Z (ϕ) + O [4] Z (ϕ) + O [6]
Z (ϕ) , (1.91) in which the term O mn Z (ϕ) collects various contributions O mn(ij) Z (ϕ). Each term O mn Z (ϕ) possesses the same operator structure as the corresponding term in Eq. (1.82), except that the original matrix elements are replaced by gauge-dependent ones, e.g.

O 31 k 1 k 2 k 3 k 4 is formally replaced by O 31 k 1 k 2 k 3 k 4 (ϕ)
. The expressions of the matrix elements of each normalordered contribution O mn Z (ϕ) in terms of the matrix elements of the original normal-ordered contributions to an operator O with deg_max = 6 are provided in App. B.

Hartree-Fock-Bogoliubov Theory

The Hartree-Fock-Bogoliubov (HFB) approach is at the basis of ab initio methods invoking the spontaneous breaking of U (1) symmetry. In this approximation, nucleons are described as independent Bogoliubov quasi-particles evolving in a self-consistent mean-field potential. The HFB approach is both an extension of the Hartree-Fock (HF) theory and of the Bardeen-Cooper-Schrieffer (BCS) theory. Breaking particle-number symmetry allows one to access the most general mean-field product-state that consistently includes pairing correlation effects. In this work it serves as a reference state on top of which more elaborate 11 The notation O the variation over the parameters is restricted by the condition that the mean particle number is equal to the actual number of particles in the system

Φ|A|Φ = Tr[ρ] = A . (1.94)
This is achieved using the method of the Lagrange multipliers. The Hamiltonian is thus replaced by the grand potential Ω (Eq. (1.4)) such that the Lagrange multiplier12 λ is fixed by the condition (1.94). The variation thus concerns the expectation value of the grand potential

E[R] ≡ Φ|Ω|Φ Φ|Φ = Ω 00 , ( 1.95)
with the additional constraint that the Bogoliubov transformation defining |Φ must be unitary, or equivalently that the generalized density matrix must be idempotent, i.e.

δ E[R] + Tr[Λ(R 2 -R)] = 0 , (1.96)
where Λ is a matrix of Lagrange multipliers. Introducing the HFB Hamiltonian H defined as

H ≡ ∂E[R] ∂R = h -λ ∆ -∆ * -(h -λ) * , ( 1.97)
where h and ∆ are the so-called Hartree-Fock and Bogoliubov fields13

h -λ ≡ ∂E[R] ∂ρ * = Λ 11 , (1.98) ∆ ≡ ∂E[R] ∂κ * = Λ 20 , (1.99)
the variation with respect to R leads to which ensures the existence of a common eigenbasis of the HFB Hamiltonian and the generalized density matrix in complete analogy to the Hartree-Fock case. As R is diagonal in the basis of the quasi-particle operators (Eq. (1.38)), the diagonalization of the HFB Hamiltonian

H + RΛ + ΛR -Λ = 0 . ( 1
H U V k = h -λ ∆ -∆ * -(h -λ) * U V k = E k U V k , (1.102)
the so-called Hartree-Fock-Bogoliubov equations, uniquely14 determines the U and V matrices of the Bogoliubov transformation (Eq. (1.23)). It can be shown that, employing the quasi-particle basis solution of Eq. (1.102), the grand potential takes the form

Ω = Ω 00 + k E k β † k β k + Ω [n≥4] , (1.103)
i.e. the normal one-body part Ω 11 is diagonal with the quasi-particle energies E k as eigenvalues whereas the anomalous one-body parts Ω 20 and Ω 02 are null. As the normal and pairing fields depend on the elementary contractions and thus on the solution, the problem is intrinsically non-linear and is solved iteratively [START_REF] Ring | The nuclear many-body problem[END_REF].

Constrained Hartree-Fock-Bogoliubov theory 1.7.1. Constrained Hartree-Fock-Bogoliubov equations

Let us consider an observable O. In order to constrain the expectation value of this observable in the HFB solution one can add another Lagrange multiplier Λ O into the energy functional. The new functional to be minimized is

E [R] ≡ E[R] -λ O O[R] , (1.104)
where

O[R] is the expectation value of O in the Bogoliubov vacuum O[R] ≡ Φ|O|Φ Φ|Φ . (1.105)
The constrained Hartree-Fock and Bogoliubov fields are given by

h -λ = ∂E [R] ∂ρ * = h -λ -λ O ∂O ∂ρ * = Λ 11 -λ O Λ 11 O , (1.106) ∆ = ∂E [R] ∂κ * = ∆ -λ O ∂O ∂κ * = Λ 20 -λ O Λ
O = l 1 l 2 o 11 l 1 l 2 c † l 1 c l 2 , (1.108)
the normal and pairing contributions are given by

(Λ 11 O ) l 1 l 2 = o 11 l 1 l 2 , (1.109a) (Λ 20 O ) l 1 l 2 = 0 . (1.109b)
In practice, constrained calculations are performed with the augmented Lagrangian method (ALM) [START_REF] Staszczak | Augmented Lagrangian method for constrained nuclear density functional theory[END_REF]. In this method the linear constraint, driven by a Lagrange multiplier λ, is accompanied by a quadratic constraint, driven by a constant c > 0 called the penalty parameter.

Constraint on particle-number variance

In the following, one is interested in the constraint on the particle-number variance defined as

Var(A) ≡ Φ|(A -Φ|A|Φ ) 2 |Φ = Φ|A 2 |Φ -Φ|A|Φ 2 . (1.110)
This is particularly interesting in order to adjust the extent by which the U (1) symmetry is broken by the Bogoliubov reference state and thus to taylor the many-body expansion performed on top of it later on. In order to evaluate the particle-number variance, one needs to consider the square of the particle-number operator

A 2 = α c † α c α β c † β c β = α c † α c α + αβ c † α c † β c β c α = l 1 l 2 (a 2 ) l 1 l 2 c † l 1 c l 2 + 1 4 l 1 l 2 l 3 l 4 (a 2 ) l 1 l 2 l 3 l 4 c † l 1 c † l 2 c l 4 c l 3 , (1.111)
with fully antisymmetric matrix elements

(a 2 ) l 1 l 2 ≡ δ l 1 l 2 (1.112a) (a 2 ) l 1 l 2 l 3 l 4 ≡ 2(δ l 1 l 3 δ l 2 l 4 -δ l 1 l 4 δ l 2 l 3 ) . (1.112b)
The vacuum expectation value of this operator reads as

Φ|A 2 |Φ = αβ κ * αβ κ αβ -ρ βα ρ αβ + ρ αα ρ ββ + α ρ αα = -Tr[κκ * ] -Tr[ρ 2 ] + Tr[ρ] 2 + Tr[ρ] .
(1.113)

The average particle-number variance is thus given by allows one to vary the proportion of particle-number variance associated with normal and anomalous densities. Adding (2α -1) times equation (1.115), where α is a real number, to the particle-number variance (Eq. (1.114)) allows one to define

Var(A) = (-Tr[κκ * ] -Tr[ρ 2 ] + Tr[ρ] 2 + Tr[ρ]) -(Tr[ρ]) 2 = -Tr[κκ * ] -Tr[ρ 2 ] + Tr[ρ] , ( 1
Var(A) α ≡ -2αTr[κκ * ] + 2(1 -α)(Tr[ρ] -Tr[ρ 2 ]) , (1.116)
which is equal to the particle-number variance for all values of α Var(A) α = Var(A) .

(1.117)

The derivatives of the particle-number variance with respect to density matrices gives

∂Var(A) α ∂ρ * l 1 l 2 = 2(1 -α)(δ l 1 l 2 -2ρ l 1 l 2 ) , (1.118a) ∂Var(A) α ∂κ * l 1 l 2 = 4ακ l 1 l 2 , (1.118b)
which lead to constrained Hartree-Fock and Bogoliubov fields of the form

h l 1 l 2 = h l 1 l 2 -2λ α var (1 -α)(δ l 1 l 2 -2ρ l 1 l 2 ) , (1.119) ∆ l 1 l 2 = ∆ l 1 l 2 -4λ α var α κ l 1 l 2 .
(1.120)

The generalized Hartree-Fock-Bogoliubov Hamiltonian is thus given by

H (α) = h -λ -2λ α var (1 -α)(1 -2ρ) ∆ -4λ α var ακ -(∆ -4λ α var ακ) * -(h -λ -2λ α var (1 -α)(1 -2ρ)) * = H (α = 0) + 2αλ α var 1 -2ρ -2κ 2κ * -(1 -2ρ * ) . (1.121) Using Eq. (1.38), one can show that 1 -2ρ -2κ 2κ * -(1 -2ρ * ) = W 1 0 0 -1 W † (1.122)
such that the solution of HFB equations is indeed independent of α whereas quasi-particle energies are modified according to

E k (α) = E k (α = 0) + 2αλ α var .
(1.123)

Introduction

In this chapter, the fully non-perturbative projected Bogoliubov coupled cluster (PBCC) [START_REF] Duguet | Symmetry broken and restored coupled-cluster. Global gauge symmetry and particle number[END_REF][START_REF] Qiu | Particle-number projected Bogoliubov-coupled-cluster theory: Application to the pairing Hamiltonian[END_REF][START_REF] Qiu | Projected coupled cluster theory[END_REF] formalism is introduced with the goal to extract its perturbative version, i.e., the projected Bogoliubov many-body perturbation theory (PBMBPT). This is done in two variants, based on (i) gauge-rotated amplitudes and (ii) unrotated amplitudes. Whereas the presentation of PBCC is reproduced from Ref. [START_REF] Qiu | Particle-number projected Bogoliubov-coupled-cluster theory: Application to the pairing Hamiltonian[END_REF], the extraction of the two variants of PBMBPT constitutes an original work. The numerical implementation associated with this work is close to completion but results will not appear in this document.

Particle-number-projected Bogoliubov coupled cluster theory

In this section, the PBCC [START_REF] Duguet | Symmetry broken and restored coupled-cluster. Global gauge symmetry and particle number[END_REF][START_REF] Qiu | Particle-number projected Bogoliubov-coupled-cluster theory: Application to the pairing Hamiltonian[END_REF][START_REF] Qiu | Projected coupled cluster theory[END_REF] formalism is introduced. This many-body expansion method is an extension of Bogoliubov coupled cluster (BCC) theory [START_REF] Signoracci | Ab initio Bogoliubov coupled cluster theory for open-shell nuclei[END_REF][START_REF] Henderson | Quasiparticle coupled cluster theory for pairing interactions[END_REF]. While both formalisms can model strongly correlated quantum systems by allowing the mean-field reference state to break U (1) symmetry, PBCC inherits and further improves the energetic accuracy of BCC theory by retaining U (1) symmetry.

Wave function ansatz

Before introducing the symmetry restoration, it is necessary to first outline the symmetrybroken version of the formalism. In this section, the many-body method of reference is the BCC theory [START_REF] Signoracci | Ab initio Bogoliubov coupled cluster theory for open-shell nuclei[END_REF][START_REF] Henderson | Quasiparticle coupled cluster theory for pairing interactions[END_REF]. BCC theory builds on a Bogoliubov reference state |Φ (Eq. (1.30)) whose Bogoliubov transformation W (Eq. (1.25)) is typically obtained by solving HFB equations1 (Eq. (1.102)). The main benefit of this choice of reference state, inherently related to the inclusion of static pairing correlations at the mean-field level, relates to the lifting of its degeneracy with respect to elementary excitations. This translates into the apparition of a gap in the spectrum of quasi-particle excitations that authorizes the design of a well-defined many-body expansion in open-shell systems. Many-body correlations are then built on top of |Φ via the usual exponential ansatz of coupled cluster theory

|Ψ BCC ≡ e U |Φ = n=1 e U n |Φ , (2.1)
where U n is a n-tuple (i.e. 2n quasiparticle) connected excitation operator with respect to |Φ

U n ≡ 1 (2n)! k 1 ...k 2n U 2n0 k 1 ...k 2n β † k 1 . . . β † k 2n , (2.2)
such that [U p , U q ] = 0. The full BCC wave function does represent an exact eigenfunction of both H and A. In practical applications, however, the expansion of the connected excitation operator U = n U n must be truncated such that the BCC wave function is no longer a particle-number eigenfunction. The idea behind the PBCC theory [START_REF] Duguet | Symmetry broken and restored coupled-cluster. Global gauge symmetry and particle number[END_REF][START_REF] Duguet | Symmetry broken and restored coupled-cluster theory: I. Rotational symmetry and angular momentum[END_REF][START_REF] Qiu | Particle-number projected Bogoliubov-coupled-cluster theory: Application to the pairing Hamiltonian[END_REF][START_REF] Qiu | Projected coupled cluster theory[END_REF] is to obtain an eigenfunction of the particle-number operator independently of the truncation order by applying a particle-number projection operator (Eq. (1.20)) to the BCC wave function2

|Ψ PBCC ≡ P A |Ψ BCC . (2.3)
In a beyond mean-field theory that consistently restores a broken symmetry, it is in fact beneficial to start from a mean-field calculation that is informed of the symmetry restoration. Such a goal can be achieved using a variation after projection (VAP) scheme in which the Bogoliubov transformation W defining the reference state minimizes the projected mean-field energy (Eq. (2.30a)) rather than the straight mean-field one (Eq. (1.102)). Numerical methods implementing this scheme are expensive and one can rather employ a restricted variation after projection (RVAP) scheme [START_REF] Ripoche | Combining symmetry breaking and restoration with configuration interaction: A highly accurate many-body scheme applied to the pairing Hamiltonian[END_REF][START_REF] Ripoche | Combining symmetry breaking and restoration with configuration interaction: Extension to z -signature symmetry in the case of the Lipkin model[END_REF][START_REF] Tomás | Restricted variation after projection and the Lipkin-Nogami methods[END_REF] in which the Bogoliubov transformation W minimizes the projected mean-field energy built on top of the potential energy surface (PES) generated via the use of the particle-number variance in constrained HFB, see Sec. 1.7.2.

Ground-state observables and off-diagonal kernels

Inserting the PBCC wave-function ansatz into Schrödinger's equation (Eq. (1.1)), commuting H, resp. A, with the particle-number projection operator and left-multiplying by Φ| leads to writing the ground-state energy3 , resp. particle number, under the form

E A 0 = Φ|P A He U |Φ Φ|P A e U |Φ , (2.4a) A = Φ|P A Ae U |Φ Φ|P A e U |Φ .
(2.4b)

Expanding the particle-number projector P A according to Eq. (1.20) and introducting off-diagonal, i.e., left-rotated, unexcited norm, Hamiltonian and particle-number kernels

N (ϕ) ≡ Φ(ϕ)|e U |Φ , (2.5a) H(ϕ) ≡ Φ(ϕ)|He U |Φ , (2.5b) A(ϕ) ≡ Φ(ϕ)|Ae U |Φ , (2.5c)
the PBCC ground-state energy and particle number are rewritten as

E A 0 = 2π 0 dϕ e -iAϕ H(ϕ) 2π 0 dϕ e -iAϕ N (ϕ) , ( 2.6a)
A = 2π 0 dϕ e -iAϕ A(ϕ) 2π 0 dϕ e -iAϕ N (ϕ) . ( 2.6b)
Left-rotated kernels (Eq. (2.5)) are better expressed in terms of similarity-transformed operators e U Z (ϕ) ≡ e Z(ϕ) e U e -Z(ϕ) ,

(2.7a)

H Z (ϕ) ≡ e Z(ϕ) He -Z(ϕ) , (2.7b) A Z (ϕ) ≡ e Z(ϕ) Ae -Z(ϕ) , ( 2.7c)
see Sec. 1.5.3 for details. With Eq. (2.7) at hand, Eq. (2.5) can indeed be rewritten as

N (ϕ) = Φ(ϕ)|Φ Φ|e U Z (ϕ) |Φ , (2.8a) H(ϕ) = Φ(ϕ)|Φ Φ|H Z (ϕ)e U Z (ϕ) |Φ , (2.8b) A(ϕ) = Φ(ϕ)|Φ Φ|A Z (ϕ)e U Z (ϕ) |Φ , (2.8c)
where use was made of the identity e Z(ϕ) |Φ = |Φ as Z(ϕ) is a pure de-excitation operator over |Φ . The introduction of similarity-transformed operators (Eq. (2.7)) translates the evaluation of left-rotated kernels (Eq. (2.5)) via the off-diagonal Wick's theorem [START_REF] Balian | Nonunitary bogoliubov transformations and extension of Wick's theorem[END_REF] into their evaluations (Eq. (2.8)) via the standard, i.e. diagonal, Wick's theorem [START_REF] Wick | The Evaluation of the Collision Matrix[END_REF]. In order to make the best use of diagonal Wick's theorem, similarity-transformed operators need to be re-expressed in terms of untransformed quasi-particle operators and re-normal ordered with respect to |Φ , see App. B for details.

At this point, one can notice that U Z (ϕ) is not a pure excitation operator even though U is. This constitutes a difficulty in view of obtaining a terminating expansion of the exponential and thus a closed form of the kernels. In order to achieve that goal, the angle-dependent state e U Z (ϕ) |Φ can be written in coupled cluster form

e U Z (ϕ) |Φ = e W (ϕ) |Φ , ( 2.9)
where the angle-dependent operator W (ϕ) ≡ n=0 W n (ϕ) is made out of n-fold excitation operators and includes a normalization constant W 0 (ϕ). The gauge-rotated cluster operators W n (ϕ) are obtained by expanding e U Z (ϕ) , normal ordering each term with respect to |Φ and retaining the non-zero terms of e U Z (ϕ) |Φ , i.e., the excitations over |Φ

W n (ϕ) ≡ e U Z (ϕ) 2n0 c = ∞ p=0 1 p! (U Z (ϕ)) p 2n0 c ≡ 1 2n! k 1 ...k 2n W 2n0 k 1 ...k 2n (ϕ)β † k 1 . . . β † k 2n , ( 2.10)
where c denotes the normal-ordered part of n-tuple excited contributions to e U Z (ϕ) obtained by contracting strings of U Z (ϕ) operators. The excitation operators W n (ϕ) are denoted as disentangled clusters because the process by which they are obtained is equivalent to disentangling the excitations, de-excitations and quasiparticle-number conserving operators. In this process, W n (ϕ) receives contributions from all U k and from all excitation ranks in the expansion of e U Z (ϕ) such that even if U is truncated at some low order of excitation, W (ϕ) will generally be non-zero for all excitation orders. In practice, this requires that both symmetry-broken U and disentangled W (ϕ) cluster operators are truncated. While Eq. (2.10) provides a formal definition of the gauge-rotated cluster operators W n (ϕ), it is not used in practice for their evaluations because the sum is too slowly converging.

The process by which gauge-dependent cluster operators W n (ϕ) are actually obtained is detailed in Sec. 2.2.3. With these disentangled clusters at hand, the left-rotated kernels are rewritten as

N (ϕ) = Φ(ϕ)|Φ e W 0 (ϕ) , ( 2.11a)
H(ϕ) = Φ|H Z (ϕ)e T (ϕ) |Φ N (ϕ) , (2.11b) A(ϕ) = Φ|A Z (ϕ)e T (ϕ) |Φ N (ϕ) , (2.11c)
where the angle-dependent operator T (ϕ) ≡ W (ϕ) -W 0 (ϕ) has no zero-body part. The factorized form of off-diagonal kernels in Eq. (2.11), which represents a fundamental property, leads to introducing the correlated part of the unexcited norm kernel N (ϕ) as well as the connected part of the Hamiltonian H(ϕ) and particle-number A(ϕ) kernels via

n(ϕ) ≡ N (ϕ) Φ(ϕ)|Φ = e W 0 (ϕ) , (2.12a) h(ϕ) ≡ H(ϕ) N (ϕ) = Φ|H Z (ϕ)e T (ϕ) |Φ c , (2.12b) a(ϕ) ≡ A(ϕ) N (ϕ) = Φ|A Z (ϕ)e T (ϕ) |Φ c , (2.12c)
where the index c stipulates the connected character of the kernel and thus the terminating behaviour of the exponential. In the case of a two-body Hamiltonian, the exponential in the expression of h(ϕ) (Eq. (2.12b)) can be expanded in powers of the gauge-angle cluster amplitudes and terminates at the double-excitation level according to

h(ϕ) = Φ|H Z (ϕ)(1 + W 1 (ϕ) + 1 2 W 2 1 (ϕ) + W 2 (ϕ))|Φ c , (2.13a) a(ϕ) = Φ|A Z (ϕ)(1 + W 1 (ϕ))|Φ c , (2.13b)
whereas in a(ϕ) (Eq. (2.12c)), the exponential terminates at the single-excitation level given that A is a one-body operator. At ϕ = 0, one obviously recovers the usual terminating form of the standard, i.e. diagonal, BCC energy and particle number kernels. The algebraic form of h(ϕ) and a(ϕ) can be explicitly obtained on the basis of standard Wick's theorem [START_REF] Wick | The Evaluation of the Collision Matrix[END_REF] as

h(ϕ) = H 00 (ϕ) + 1 2 k 1 k 2 H 02 k 1 k 2 (ϕ)W 20 k 1 k 2 (ϕ) + 1 8 k 1 k 2 k 3 k 4 H 04 k 1 k 2 k 3 k 4 (ϕ)W 20 k 1 k 2 (ϕ)W 20 k 3 k 4 (ϕ) + 1 24 k 1 k 2 k 3 k 4 H 04 k 1 k 2 k 3 k 4 (ϕ)W 40 k 1 k 2 k 3 k 4 (ϕ) , (2.14a) a(ϕ) = A 00 (ϕ) + 1 2 k 1 k 2 A 02 k 1 k 2 (ϕ)W 20 k 1 k 2 (ϕ) . (2.14b)
Eventually, this allows one to write the PBCC ground-state energy, resp. particle number, in terms of the connected Hamiltonian kernel h(ϕ), resp. connected particle-number kernel a(ϕ), and of the unexcited norm kernel N (ϕ) according to

E A 0 = 2π 0 dϕ e -iAϕ h(ϕ) N (ϕ) 2π 0 dϕ e -iAϕ N (ϕ) , (2.15a) A = 2π 0 dϕ e -iAϕ a(ϕ) N (ϕ) 2π 0 dϕ e -iAϕ N (ϕ)
.

(2.15b)

Gauge-rotated clusters amplitudes

In order to obtain gauge-rotated cluster amplitudes W n (ϕ), one first notices that R(0) = 1. Accordingly, the similarity-transformed amplitudes satisfy

U Z (0) = U , (2.16) such that W 0 (0) = 0 , (2.17a) W n (0) = U n ,
(2.17b) so that gauge-rotated cluster amplitudes are considered to be known at ϕ = 0 from a pre-processed BCC calculation. From these initial conditions, the W n (ϕ) are obtained for all values of the gauge angle through the resolution of a set of first order differential equations. In order to derive the set, let us rewrite Eq. (2.9) as where A 02 Z (ϕ) is the de-excitation part of A Z (ϕ). Introducing n-tuple excitations of the Bogoliubov vacuum through

e Z(ϕ) e U |Φ = e W (ϕ)
|Φ k 1 ...k 2n ≡ β † k 1 . . . β † k 2n |Φ , (2.22)
the set of coupled ODEs governing gaugle-rotated cluster amplitudes W n (ϕ) for n ≥ 0 is finally obtained by left-multiplying Eq. (2.20) with all possible n-tuple excitations of the vacuum Φ k 1 ...k 2n |. Up to the doubly-excited amplitudes W 2 (ϕ), this gives

d dϕ W 0 (ϕ) = i k 1 k 2 A 02 k 1 k 2 (ϕ) 1 2 W 20 k 1 k 2 (ϕ) (2.23a) d dϕ W 20 k 1 k 2 (ϕ) = i k 3 k 4 A 02 k 3 k 4 (ϕ) 1 2 W 40 k 1 k 2 k 3 k 4 (ϕ) -W 20 k 1 k 3 (ϕ)W 20 k 2 k 4 (ϕ) (2.23b) d dϕ W 40 k 1 k 2 k 3 k 4 (ϕ) = i k 5 k 6 A 02 k 5 k 6 (ϕ) 1 2 W 60 k 1 k 2 k 3 k 4 k 5 k 6 (ϕ) + W 20 k 1 k 5 (ϕ)W 40 k 6 k 2 k 3 k 4 (ϕ) + W 20 k 2 k 5 (ϕ)W 40 k 1 k 6 k 3 k 4 (ϕ) + W 20 k 3 k 5 (ϕ)W 40 k 1 k 2 k 6 k 4 (ϕ) + W 20 k 4 k 5 (ϕ)W 40 k 1 k 2 k 3 k 6 (ϕ) .
(2.23c)

The above ODEs stipulate that the evolution of gauge-rotated cluster amplitudes W n (ϕ) with the gauge angle ϕ is driven by A 02 Z (ϕ), knowing that A is the infinitesimal generator of the U (1) group, see Sec. 1.3. From Eq. (2.23), it appears that the derivative of W n (ϕ) systematically involves a contribution from W n+1 (ϕ), i.e.

d dϕ W k 1 ...k 2n (ϕ) = i k 2n+1 k 2n+2 A 02 k 2n+1 k 2n+2 (ϕ) 1 2 W k 1 ...k 2n+2 (ϕ) + . . . (2.24)
Thus truncating the gauge-rotated amplitudes to a certain excitation level W n (ϕ) effectively decouples the n + 1 first ODEs from the others. However this truncation implies a degree of approximation of the projection operator. The effect of this truncation will have to be gauged in numerical applications.

Restoration of particle number

In PBCC, the restoration of the average particle number A (Eq. (2.6b)) is ensured by the resolution of the first ODE (Eq. (2.23a)) that gives the derivative of W 0 (ϕ). Differentiating the off-diagonal norm kernel (Eq. (2.11a)) with respect to ϕ and inserting Eq. (2.23a) leads to

d dϕ N (ϕ) = d dϕ Φ|e iAϕ |Φ e W 0 (ϕ) = Φ|iAe iAϕ |Φ + d dϕ W 0 (ϕ) Φ|e iAϕ |Φ e W 0 (ϕ) = Φ|iA Z (ϕ)|Φ + Φ|iA Z (ϕ)W 1 (ϕ)|Φ Φ(ϕ)|Φ e W 0 (ϕ) = i Φ|A Z (ϕ)e W (ϕ) |Φ Φ(ϕ)|Φ e W 0 (ϕ) = ia(ϕ)N (ϕ) = iA(ϕ) , (2.25)
which means that the derivative of the off-diagonal norm kernel is equal to the off-diagonal particle-number kernel (times i) [START_REF] Duguet | Symmetry broken and restored coupled-cluster. Global gauge symmetry and particle number[END_REF]. With this at hand, one can demonstrate the restoration of the average particle number A (Eq. (2.6b)) using integration by part

2π 0 dϕ e -iAϕ A(ϕ) 2π 0 dϕ e -iAϕ N (ϕ) = -i 2π 0 dϕ e -iAϕ d dϕ N (ϕ) 2π 0 dϕ e -iAϕ N (ϕ) = -i e -iAϕ N (ϕ) 2π 0 -2π 0 dϕ d dϕ e -iAϕ N (ϕ) 2π 0 dϕ e -iAϕ N (ϕ) = -(-i) 2 A 2π 0 dϕ e -iAϕ N (ϕ) 2π 0 dϕ e -iAϕ N (ϕ) = A , ( 2.26)
where N (ϕ) is 2π-periodic4 , such that the average particle number is restored even when A(ϕ) is computed approximately.

Bogoliubov coupled cluster theory

One can recover BCC theory from PBCC by replacing the connected Hamiltonian kernel h(ϕ), resp. connected particle-number kernel a(ϕ), with its value at zero angle5 h(0), resp. a(0), in the PBCC ground-state energy, resp. particle number, (Eqs. (2.15a) and (2.15b)). As a result, the integral over the gauge angle acts trivially and one obtains

E A (BCC) 0 ≡ h(0) 2π 0 dϕ e -iAϕ N (n) (ϕ) 2π 0 dϕ e -iAϕ N (n) (ϕ) = h(0) , (2.27a) A (BCC) ≡ a(0) 2π 0 dϕ e -iAϕ N (n) (ϕ) 2π 0 dϕ e -iAϕ N (n) (ϕ) = a(0) , (2.27b)
such that algebraic forms of PBCC ground-state energy and particle number reads

E A (BCC) 0 = H 00 + 1 2 k 1 k 2 H 02 k 1 k 2 U 20 k 1 k 2 + 1 8 k 1 k 2 k 3 k 4 H 04 k 1 k 2 k 3 k 4 U 20 k 1 k 2 U 20 k 3 k 4 + 1 24 k 1 k 2 k 3 k 4 H 04 k 1 k 2 k 3 k 4 U 40 k 1 k 2 k 3 k 4 , (2.28a) A (BCC) = A 00 + 1 2 k 1 k 2 A 02 k 1 k 2 U 20 k 1 k 2 , (2.28b)
where the properties at zero angle of gauge-rotated cluster amplitudes and similaritytransformed operators

W(0) = U , (2.29a) H Z (0) = H , (2.29b) A Z (0) = A , (2.29c)
have been used.

Projected Hartree-Fock-Bogoliubov method

Another limit of PBCC is obtained by setting all gauge-rotated cluster amplitudes W n (ϕ) to zero for n > 0. Doing so, PBCC ground-state energy and particle number reduce to projected HFB (PHFB) ones

E A (PHFB) 0 ≡ 2π 0 dϕ e -iAϕ h (0) (ϕ) N (0) (ϕ) 2π 0 dϕ e -iAϕ N (0) (ϕ) , (2.30a) A (PHFB) ≡ 2π 0 dϕ e -iAϕ a (0) (ϕ) N (0) (ϕ) 2π 0 dϕ e -iAϕ N (0) (ϕ) , ( 2.30b)
where connected operator kernels are restricted to the 0th order in perturbation, which are nothing but the zero-body part of the similarity-transformed operators (Eqs. (B.1a) and (B.2a))

h (0) (ϕ) ≡ H 00 (ϕ) , (2.31a) a (0) (ϕ) ≡ A 00 (ϕ) , (2.31b)
and where the norm kernel N (0) (ϕ) reduces to its non-correlated part

N (0) (ϕ) = Φ(ϕ)|Φ . (2.32)
The latter is obtained by noticing that the correlated part of the norm kernel in Eq. (2.11a) is vanishing in this case

W (0) 0 (ϕ) = 0 , ( 2.33)
given that its value and first derivative are zero at ϕ = 0.

Projected Bogoliubov many-body perturbation theory

In this section, the projected Bogoliubov many-body perturbation theory (PBMBPT) is obtained from PBCC. Two variants of PBMBPT are actually introduced that differ in the way the gauge-rotated cluster operator W (ϕ) is evaluated.

Ground-state observables

The difference between PBMBPT and PBCC energies, resp. particle numbers, relates to the way h(ϕ), resp. a(ϕ), and N (ϕ) are evaluated. Expect for that, the ground-state energy, resp. particle number, in PBMBPT has the same formal structure as the PBCC one

E A (PBMBPT(n)) 0 ≡ 2π 0 dϕ e -iAϕ h (n) (ϕ) N (n) (ϕ) 2π 0 dϕ e -iAϕ N (n) (ϕ) , (2.34a) A (PBMBPT(n)) ≡ 2π 0 dϕ e -iAϕ a (n) (ϕ) N (n) (ϕ) 2π 0 dϕ e -iAϕ N (n) (ϕ) , ( 2.34b)
where the gauge-angle integration is made over n-order estimates of the connected operator kernel o (n) (ϕ) and norm kernel N (n) (ϕ). In the following, two ways of obtaining perturbative estimates are presented.

Perturbative off-diagonal amplitudes

It is possible to directly access p-order estimate of the gauge-rotated cluster amplitudes W (p) n (ϕ), see Ref. [START_REF] Duguet | Symmetry broken and restored coupled-cluster. Global gauge symmetry and particle number[END_REF]. If one were to resum all orders in perturbation, the exact PBCC amplitudes (Eq. (2.10)) would of course be recovered according to

W n (ϕ) = ∞ p=1 W (p) n (ϕ) . (2.35)
For example, only single and double gauge-rotated cluster amplitudes are non zero at first order 6 and reads as

W 20(1)
k 1 k 2 (ϕ) ≡ - Ω 20 k 1 k 2 E k 1 k 2 - 1 2 k 3 k 4 Ω 40 k 1 k 2 k 3 k 4 Z 02 k 3 k 4 (ϕ) E k 1 k 2 k 3 k 4 , (2.36a) W 40(1)
k 1 k 2 k 3 k 4 (ϕ) ≡ - Ω 40
k 1 k 2 k 3 k 4 E k 1 k 2 k 3 k 4 . (2.36b)
This procedure gives access to n-body gauge-rotated cluster amplitudes for n > 0 but does not provide W (p) 0 (ϕ) that encodes the correlated part of the norm kernel. This quantity is still obtained by solving the first ODE (Eq. (2.23a)) with W (p) 1 (ϕ) entering the right-hand side. Actually one can show that the resolution of Eq. (2.23a) is equivalent to the resolution of

d dϕ N (p) (ϕ) = ia (p) (ϕ)N (p) (ϕ) , (2.37)
which is the ODE on the full norm kernel used in Chap. 3, and where a (p) (ϕ) is the p-order estimate of the connected particle-number kernel

a (p) (ϕ) = A 00 (ϕ) + 1 2 k 1 k 2 A 02 k 1 k 2 (ϕ) p k=1 W 20(k) k 1 k 2 (ϕ) . (2.38)
This version of PBMBPT will be refered to as P P BMBPT where the "P" index stands for "perturbative".

Perturbative diagonal amplitudes and ODEs

Another possibility to design a PBMBPT is to start from perturbative estimates of unrotated cluster amplitudes U (p) n that, resummed to all orders, provide the exact BCC amplitudes (Eq. (2.2)) for n > 0 according to

U n = ∞ p=1 U (p)
n .

(2.39)

They can actually be recovered from the gauge-rotated ones introduced in Sec. 2.3.2 by taking the zero-angle limit, ϕ = 0. Starting from Eq. (2.36), this leads at first order to

U 20 (1)
k 1 k 2 ≡ - Ω 20 k 1 k 2 E k 1 k 2 , (2.40a) U 40 (1)
k 1 k 2 k 3 k 4 ≡ - Ω 40 k 1 k 2 k 3 k 4 E k 1 k 2 k 3 k 4 .
(2.40b)

The diagrammatic rules to systematically obtain p-order unrotated cluster amplitudes have been given in Ref. [START_REF] Arthuis | Bogoliubov Many-Body Perturbation Theory for Nuclei : Systematic Generation and Evaluation of Diagrams and First ab initio Calculations[END_REF] and processed up to second order in perturbation 7 . Gauge-rotated amplitudes W (p) n (ϕ) are then obtained by solving the set of ODEs (Eq. (2.23)), derived in the context of PBCC, with the initial conditions

W (p) 0 (0) = 0 , (2.41a) W (p) n (0) = U (p) n . (2.41b)
This version of PBMBPT will be refered to as P O BMBPT, where the O index stands for "ODE". While the associated cluster amplitudes W (p) n (ϕ) are still denoted as "perturbative", their content is not strictly perturbative in the residual interaction and are different from those obtained within the frame of P P BMBPT.

Connected operator kernel

Once perturbative estimates of gauge-rotated cluster amplitudes are obtained, either through P P BMBPT or P O BMBPT, one has to evaluate the connected operator kernel o (n) (ϕ). Because of the intrinsic non-linearity with respect to cluster amplitudes in the expression of o(ϕ) (Eqs. (2.13) and (2.14)), there is no unique way of defining the n-order connected operator kernel o (n) (ϕ).

A first possibility is to introduce perturbative estimates W (p) n (ϕ), with p ≤ n, in Eq. (2.14b) and keeps all terms

o (n) P µ BMBPT.a (ϕ) ≡ O 00 (ϕ) + 1 2 k 1 k 2 O 02 k 1 k 2 (ϕ) n p=1 W 20(p) k 1 k 2 (ϕ) + 1 8 k 1 k 2 k 3 k 4 O 04 k 1 k 2 k 3 k 4 (ϕ) n p=1 W 20(p) k 1 k 2 (ϕ) n q=1 W 20(q) k 3 k 4 (ϕ) + 1 24 k 1 k 2 k 3 k 4 O 04 k 1 k 2 k 3 k 4 (ϕ) n p=1 W 40(p) k 1 k 2 k 3 k 4 (ϕ) , ( 2.42)
where µ = P or O. Because of the quadratic term in W 1 (ϕ), o

PBMBPT.a (ϕ) actually receives contributions up to order 2n in perturbation. It is however possible to restrain the contribution up to order n in perturbation by removing appropriate parts of the product W 1 (ϕ)W 1 (ϕ). This provides another possibility to define the perturbative estimate of o(ϕ)

o (n) P µ BMBPT.b (ϕ) ≡ O 00 (ϕ) + 1 2 k 1 k 2 O 02 k 1 k 2 (ϕ) n p=1 W 20(p) k 1 k 2 (ϕ) + 1 8 k 1 k 2 k 3 k 4 O 04 k 1 k 2 k 3 k 4 (ϕ) p+q≤n p,q=1 W 20(p) k 1 k 2 (ϕ)W 20(q) k 3 k 4 (ϕ) + 1 24 k 1 k 2 k 3 k 4 O 04 k 1 k 2 k 3 k 4 (ϕ) n p=1 W 40(p) k 1 k 2 k 3 k 4 (ϕ) , (2.43)
which is trully perturbative. For the particle-number operator, which is a one-body operator, these two variants are the same as there is no non-linear contribution in cluster amplitudes such that the PBMBPT particle-number kernel is given by

a (n) P µ BMBPT (ϕ) ≡ A 00 (ϕ) + 1 2 k 1 k 2 A 02 k 1 k 2 (ϕ) n p=1 W 20(p) k 1 k 2 (ϕ) . (2.44)
In Chap. (3), a way to obtain o (n) P P BMBPT.b (ϕ) to arbitrary order will be discussed at length. The corresponding approach does not make use of gauge-rotated cluster amplitudes and is directly formulated via a perturbative expansion of gauge-rotated kernels themselves, which is the language in which BMBPT was historically derived [START_REF] Arthuis | Bogoliubov Many-Body Perturbation Theory for Nuclei : Systematic Generation and Evaluation of Diagrams and First ab initio Calculations[END_REF].

Bogoliubov many-body perturbation theory

Just as BCC can be recovered as a particular limit of PBCC, Bogoliubov many-body perturbation theory (BMBPT) can be recovered as a particular limit of PBMBPT. Stipulating that BMBPT connected operator kernels are independent of the gauge angle and noticing that they are indeed equal to PBMBPT ones at zero angle

o (n) BMBPT = o (n) PBMBPT (0) , ( 2.45)
gauge-angle integrals entering the expression of the PBMBPT ground-state energy and particle number (Eqs. (2.34a) and (2.34b)) trivially simplify according to

E BMBPT(n) 0 ≡ h (n) PBMBPT (0) 2π 0 dϕ e -iAϕ N (n) (ϕ) 2π 0 dϕ e -iAϕ N (n) (ϕ) = h (n) BMBPT , (2.46a) A BMBPT(n) ≡ a (n) PBMBPT (0) 2π 0 dϕ e -iAϕ N (n) (ϕ) 2π 0 dϕ e -iAϕ N (n) (ϕ) = a (n) BMBPT ,
(2.46b) such that BMBPT ground-state energy8 and particle number are directly given by the diagonal connected operator kernels. It does not matter which version of P µ BMBPT the reduction is operated from given that the zero angle reduction provides the same kernels in both cases. However, two possibilities in order to obtain the PBMBPT operator kernel have been given (Eqs. (2.42) and (2.43)) such that there are also two options for BMBPT.

In the first one, the BCC exponential ansatz for the wave function is used with perturbative amplitudes up to the order n in perturbation such that

o (n) BMBPT.a ≡ O 00 + 1 2 k 1 k 2 O 02 k 1 k 2 n p=1 U 20(p) k 1 k 2 + 1 8 k 1 k 2 k 3 k 4 O 04 k 1 k 2 k 3 k 4 n p=1 U 20(p) k 1 k 2 n q=1 U 20(q) k 3 k 4 + 1 24 k 1 k 2 k 3 k 4 O 04 k 1 k 2 k 3 k 4 n p=1 U 40(p) k 1 k 2 k 3 k 4 , (2.47)
which actually generates contributions up to order 2n in perturbation for the operator kernel. The other possibility is to evaluate the kernel at a strict given order in perturbation theory. Removing terms corresponding to orders higher than n in the operator kernel leads to

o (n) BMBPT.b ≡ O 00 + 1 2 k 1 k 2 O 02 k 1 k 2 n p=1 U 20(p) k 1 k 2 + 1 8 k 1 k 2 k 3 k 4 O 04 k 1 k 2 k 3 k 4 p+q≤n p,q=1 U 20(p) k 1 k 2 U 20(q) k 3 k 4 + 1 24 k 1 k 2 k 3 k 4 O 04 k 1 k 2 k 3 k 4 n p=1 U 40(p) k 1 k 2 k 3 k 4 . (2.48)
This last possibility is trully perturbative and corresponds to BMBPT as it was historically derived [START_REF] Arthuis | Bogoliubov Many-Body Perturbation Theory for Nuclei : Systematic Generation and Evaluation of Diagrams and First ab initio Calculations[END_REF]. For the particle-number operator, which is a one-body operator, these two variants are the same as there is no non-linear contribution in cluster amplitudes such that BMBPT particle-number kernel is given by

a (n) BMBPT ≡ A 00 + 1 2 k 1 k 2 A 02 k 1 k 2 n p=1 U 20(p) k 1 k 2 .
(2.49)

From Eq. (2.49), one can see that the particle number potentially receives corrections at each order in perturbation. Several ways to correct for this particle-number drift are presented in Ref. [START_REF] Demol | High-order Bogoliubov Many-Body Perturbation Theory[END_REF].

Left and right gauge-angle rotations

In this chapter, all quantities of interest were derived from a left-rotated Bogoliubov vacuum

Φ(ϕ)| L ≡ Φ|R(ϕ) , (2.50)
where the subscript L denotes that the rotated Bogoliubov vacuum is defined by the application of the gauge-angle operator to a bra state. However in the next chapter, off-diagonal quantities will be derived from a right-rotated Bogoliubov vacuum9

|Φ(ϕ) R ≡ R(ϕ)|Φ , ( 2.51)
where the subscript R denotes that the rotated Bogoliubov vacuum is defined by the application of the gauge-angle operator to a ket state. Even if the correspondence between the two definitions is trivial at the level of rotated states

|Φ(ϕ) L = Φ(ϕ)| † L = |Φ(-ϕ) R , (2.52)
it is less trivial at the level of Thouless transformations defined to the left, which lead to a pure de-excitation operator, or to the right, which lead to a pure excitation operator and at the level of kernels. The correspondence between quantities defined from a left-rotated state and from a right-rotated state is given in Tab. 2.1 for convenience.

Left definition Right definition Correspondence

L Φ(ϕ)| = Φ|R(ϕ) |Φ(ϕ) R = R(ϕ)|Φ |Φ(ϕ) L = |Φ(-ϕ) R [β R ] ( †) L (ϕ) = R -1 (ϕ)β ( †) R(ϕ) [β R ] ( †) R (ϕ) = R(ϕ)β ( †) R -1 (ϕ) [β R ] ( †) L (ϕ) = [β R ] ( †) R (-ϕ) U L (ϕ) = e -iϕ U U R (ϕ) = e +iϕ U U L (ϕ) = U R (-ϕ) V L (ϕ) = e +iϕ V V R (ϕ) = e -iϕ V V L (ϕ) = V R (-ϕ) A L (ϕ) = V † V L (ϕ) + U † U L (ϕ) A R (ϕ) = V † V R (ϕ) + U † U R (ϕ) A L (ϕ) = A R (-ϕ) B L (ϕ) = U T V L (ϕ) + V T U L (ϕ) B R (ϕ) = U T V R (ϕ) + V T U R (ϕ) B L (ϕ) = B R (-ϕ) Z 02 (ϕ) = B L (ϕ)A -1 L (ϕ) Z 20 (ϕ) = B * R (ϕ)A * -1 R (ϕ) Z 02 (ϕ) = Z 20 (ϕ) Z L (ϕ) = 1 2 k 1 k 2 Z 02 k 1 k 2 (ϕ)β k 2 β k 1 Z R (ϕ) = 1 2 k 1 k 2 Z 20 k 1 k 2 (ϕ)β † k 1 β † k 2 Z L (ϕ) = Z † R (-ϕ) [β Z ] L (ϕ) = e +Z L (ϕ) β e -Z L (ϕ) = β [ β † Z ] R (ϕ) = e -Z R (ϕ) β † e +Z R (ϕ) = β † [β Z ] L (ϕ) = [ βZ ] R (ϕ) = β [ β † Z ] L (ϕ) = e +Z L (ϕ) β † e -Z L (ϕ) = β † + Z 02 L (ϕ)β [β Z ] R (ϕ) = e -Z R (ϕ) β e +Z R (ϕ) = β + Z 20 R (ϕ)β † [ βZ ] L (ϕ) = [β Z ] R (-ϕ) [O Z ] L (ϕ) = e +Z L (ϕ) Oe -Z L (ϕ) [O Z ] R (ϕ) = e -Z R (ϕ) Oe +Z R (ϕ) [O Z ] L (ϕ) = [O Z ] † R (-ϕ) [O ij Z ] L (ϕ) = [O ji R ] † L (-ϕ) ([O ij Z ] L ) k 1 ...k i k i+1 ...k i+j (ϕ) = ([O ji R ] L ) * k i+1 ...k i+j k 1 ...k i (ϕ) O L (ϕ) = L Φ(ϕ)|Oe U |Φ O R (ϕ) = Φ|e U † O|Φ(ϕ) R O L (ϕ) = O * R (-ϕ) N L (ϕ) = L Φ(ϕ)|e U |Φ N R (ϕ) = Φ|e U † |Φ(ϕ) R N L (ϕ) = N * R (-ϕ) o L (ϕ) = O L (ϕ)/N L (ϕ) o R (ϕ) = O R (ϕ)/N R (ϕ) o L (ϕ) = o * R (-ϕ) R ++ k 1 k 2 (ϕ) = L Φ(ϕ)|β † k 1 β † k 2 |Φ L Φ(ϕ)|Φ R -- k 1 k 2 (ϕ) = Φ|β k 1 β k 2 |Φ(ϕ) R Φ|Φ(ϕ) R R ++ k 1 k 2 (ϕ) = R -- * k 2 k 1 (-ϕ)

Introduction

In chap. 2, PBMBPT was introduced as a limit of PBCC. This allowed to design two ways to obtain gauge-rotated amplitudes that ban be combined with two ways of computing observables. In all cases, the method was worked out to low orders. In this chapter, off-diagonal operator kernels themselves are expanded in perturbation. It thus corresponds to generate directly o (n) P P BMBPT.b (ϕ) as defined in Eq. (2.43). For a reason that will become clear later on, the diagrammatics at play in P P BMBPT is coined as the off-diagonal BMBPT diagrammatics from which the diagonal one encountered in straight BMBPT is recovered in a particular limit, i.e. diagonal BMBPT diagrams characterize the subset of off-diagonal BMBPT diagrams that are non-zero for ϕ = 0. In this context, the goal is to design a new version of the code ADG [START_REF] Arthuis | ADG: Automated generation and evaluation of many-body diagrams I. Bogoliubov many-body perturbation theory[END_REF], that automatically (1) generates all valid off-diagonal BMBPT diagrams 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). In this way, the capability of the ADG code is extended from BMBPT to P P BMBPT.

Many-body formalism

Projective eigenequations

Taking the hermitian conjugate of Eq. (1.5), for n = 0, and right-multiplying by an arbitrary auxiliary state |Θ (such that Ψ A 0 |Θ = 0), one obtains a projective equation of the form

O A 0 = Ψ A 0 |O|Θ Ψ A 0 |Θ . ( 3.1)
Choosing |Θ ≡ |Φ and expanding Ψ A 0 | around it leads to straight, i.e. symmetry-breaking, BMBPT [START_REF] Arthuis | ADG: Automated generation and evaluation of many-body diagrams I. Bogoliubov many-body perturbation theory[END_REF]. In the present work, the auxiliary state is taken as |Θ ≡ P A |Φ such that the symmetry is exactly restored by the presence of the projection operator P A whenever expanding (and eventually truncating) Ψ A 0 | around the Bogoliubov reference state. In this context, Eq. (3.1) becomes

O A 0 = Ψ A 0 |OP A |Φ Ψ A 0 |P A |Φ , ( 3.2)
such that introducing so-called off-diagonal norm and operator kernels1 In absence of the projection operator, one recovers BMBPT's master equation under the form

N (ϕ) ≡ Ψ A 0 |Φ(ϕ) , (3.3a) O(ϕ) ≡ Ψ A 0 |O|Φ(ϕ) , ( 3
O A 0 = O(0) , ( 3.5)
where intermediate normalization N (0) = Ψ A 0 |Φ = 1 with the unrotated Bogoliubov reference state has been used. Equations (3.4) and (3.5) are obviously equivalent in the exact limit but differ as soon as Ψ A 0 | is expanded around the U (1) breaking vacuum Φ| and truncated.

Imaginary-time formalism

Introducing the evolution operator in imaginary time 2

U(τ ) ≡ e -τ Ω , ( 3.6)
with τ real, allows one to write the ground state as 34

|Ψ A 0 = lim τ →∞ |Ψ(τ ) ≡ lim τ →∞ U(τ )|Φ Φ|U(τ )|Φ , ( 3.7)
where Φ|Ψ(τ ) = 1 for all τ . With this definition at hand, the off-diagonal kernels entering Eq. (3.4) read as

N (ϕ) ≡ N (ϕ) N (0) = lim τ →∞ Φ|U(τ )|Φ(ϕ) Φ|U(τ )|Φ , (3.8a) O(ϕ) ≡ O(ϕ) N (0) = lim τ →∞ Φ|U(τ )O|Φ(ϕ) Φ|U(τ )|Φ . (3.8b)
The off-diagonal kernels N (ϕ) and O(ϕ) are the many-body quantities to be approximated in a viable expansion method from which N (0) and O(0) at play in straight BMBPT can be obtained as a particular case [START_REF] Tichai | Bogoliubov many-body perturbation theory for open-shell nuclei[END_REF][START_REF] Arthuis | ADG: Automated generation and evaluation of many-body diagrams I. Bogoliubov many-body perturbation theory[END_REF][START_REF] Arthuis | Bogoliubov Many-Body Perturbation Theory formalism[END_REF].

Norm kernel

In PBMBPT, the off-diagonal norm kernel plays a particular role as its perturbative expansion is different from the expansion of an operator kernel [START_REF] Duguet | Symmetry broken and restored coupled-cluster. Global gauge symmetry and particle number[END_REF]. In fact, it can be trivially related to the particle-number operator kernel through

d dϕ N (ϕ) = i A(ϕ) , ( 3
.9)

2
The time is given in units of MeV -1 . 3 The result is obtained by inserting a complete set of energy eigenstates in both the numerator and the denominator. 4 The chemical potential λ is fixed such that Ω A 0 0 for the targeted particle number A 0 is the lowest value of all Ω A µ over Fock space, i.e. it penalizes systems with larger number of particles such that Ω

A 0 0 < Ω A µ
for all A > A 0 while maintaining at the same time that Ω A 0 0 < Ω A µ for all A < A 0 . This is practically achievable only if E A 0 is strictly convex in the neighborhood of A 0 , which is generally but not always true for atomic nuclei.

as it was already the case in Sec. 2.2.4. Accessing N (ϕ) via the integration of Eq. (3.9) ensures that Eq. (3.4), applied to O ≡ A, delivers the expected result A A 0 = A even when A(ϕ) is computed approximately through, e.g., perturbation theory, see Sec. 2.2.4. Further introducing the factorization of an arbitrary operator kernel

O(ϕ) ≡ o(ϕ) N (ϕ) (3.10)
where o(ϕ) denotes the connected/linked part of the operator kernel [START_REF] Duguet | Symmetry broken and restored coupled-cluster. Global gauge symmetry and particle number[END_REF], one arrives at the first-order ODE fulfilled by the norm kernel

d dϕ N (ϕ) = i a(ϕ) N (ϕ) , ( 3.11)
whose closed-form solution reads as

N (ϕ) = e i ϕ 0 dφ a(φ) . (3.12)
From the computation of a(ϕ), the off-diagonal norm kernel is consistently obtained. Eventually, the connected/linked part o(ϕ) of an operator kernel O(ϕ) is the sole quantity one needs to effectively focus on in order to implement the complete P P BMBPT formalism. This relates to the fact that o(ϕ) is size-extensive and properly scales with system size, which translates into the fact that it effectively displays a connected expansion.

Perturbation theory

Partitioning

The grand potential is split into an unperturbed part Ω 0 and a residual part Ω 1

Ω = Ω 0 + Ω 1 , (3.13)
such that the unperturbed part is given by

Ω 0 ≡ Ω 00 + Ω11 , ( 3.14)
where Ω11 , the one-body part of Ω 0 , is diagonal, i.e.,

Ω11 ≡ k E k β † k β k , ( 3.15)
with E k > 0 for all k, and the residual part is given by Ω 1 ≡ Ω 20 + Ω11 + Ω 02 + Ω [4] + Ω [6] , (3.16) where Ω11 ≡ Ω 11 -Ω11 . For a given number of interacting fermions, the key is to choose Ω 0 with a low-enough symmetry for its ground state |Φ to be non-degenerate with respect to elementary excitations. As already discussed, this leads for open-shell superfluid nuclei to choosing an operator Ω 0 that breaks particle-number conservation, i.e., while Ω commutes with U (1) transformations, one is interested in the case where Ω 0 , and thus Ω 1 , do not. The unperturbed grand potential Ω 0 is fully characterized by its complete set of orthonormal eigenstates in Fock space

Ω 0 |Φ = Ω 00 |Φ , (3.17a) Ω 0 |Φ k 1 k 2 ... = Ω 00 + E k 1 k 2 ... |Φ k 1 k 2 ... , (3.17b)
where the strict positivity of unperturbed excitations

E k 1 k 2 ... ≡ E k 1 +E k 2 +.
. . characterizes the lifting of the particle-hole degeneracy authorized by the spontaneous breaking of U [START_REF] Ishii | Nuclear Force from Lattice QCD[END_REF] symmetry in open-shell nuclei at the mean-field level.

In the particular case where |Φ solves the HFB variational problem, one has that Ω 20 = Ω11 = Ω 02 = 0 such that Ω 1 reduces to Ω [4] + Ω [6] . This choice defines the canonical version of BMBPT and reduces significantly the number of non-zero diagrams to be considered. However, this a priori hypothesis is not made such that the reference state |Φ and the corresponding unperturbed grand potential Ω 0 can be defined more generally, eventually leading to the appearance of non-canonical diagrams involving Ω 20 , Ω11 and Ω 02 vertices.

On the basis of the above splitting of Ω, one introduces the interaction representation of operators in the quasi-particle basis

O ij (τ ) ≡ e +τ Ω 0 O ij e -τ Ω 0 (3.18) = 1 i!j! k 1 ...k i+j O ij k 1 ...k i+j β † k 1 (τ ) . . . β † k i (τ )β k i+j (τ ) . . . β k i+1 (τ ) ,
where quasi-particle operators in the interaction representation are given by

β k (τ ) ≡ e +τ Ω 0 β k e -τ Ω 0 = e -τ E k β k , (3.19a) β † k (τ ) ≡ e +τ Ω 0 β † k e -τ Ω 0 = e +τ E k β † k . (3.19b)

Perturbative expansion

Expanding the evolution operator in powers of the perturbation Ω 1 [START_REF] Blaizot | Quantum theory of finite systems[END_REF] U(τ

) ≡ e -τ Ω = e -τ Ω 0 Te -τ 0 dτ Ω 1 (τ ) , (3.20)
where T denotes the time-ordering operator5 , one obtains [START_REF] Duguet | Symmetry broken and restored coupled-cluster. Global gauge symmetry and particle number[END_REF] the expansion of interest

6 o(ϕ) ≡ lim τ →∞ Φ|U(τ )O|Φ(ϕ) Φ|U(τ )|Φ(ϕ) = lim τ →∞ Φ|Te -τ 0 dtΩ 1 (t) O|Φ(ϕ) c = Φ|O|Φ(ϕ) - 1 1! +∞ 0 dτ 1 Φ|T [Ω 1 (τ 1 )O(0)] |Φ(ϕ) c + 1 2! +∞ 0 dτ 1 dτ 2 Φ|T [Ω 1 (τ 1 )Ω 1 (τ 2 )O(0)] |Φ(ϕ) c -. . . , (3.21)
where the lower index c refers to the restriction to connected diagrams. The timeindependent operator O could be inserted at no cost within the time-ordering by providing it with a fictitious and harmless time dependence t = 0. Indeed, all Ω 1 (τ k ) operators appear to the left of O and occur at a larger time given that their corresponding time variables are positive.

Invoking perturbation theory consists of truncating the Taylor expansion of the timeevolution operator in Eq. (3.21). Gathering all terms up to order p, o(ϕ) sums matrix elements of products of up to p + 1 time-dependent operators 7 . The running time variables are integrated over from 0 to τ → +∞ whereas the time label attributed to the operator O itself remains fixed at t = 0, i.e., contributions of order p contain a p-tuple time integral that needs to be performed to generate the end result under the required form.

Given the off-diagonal character of the kernels, each matrix element in Eq. (3.21) is computed via the application of off-diagonal Wick's theorem [START_REF] Balian | Nonunitary bogoliubov transformations and extension of Wick's theorem[END_REF], which is applicable to matrix elements of operators between any two (non-orthogonal) left and right product states. As a result, diagrams at play invoke a set of four off-diagonal unperturbed propagators defined in the quasi-particle basis

{β k , β † k } as G +-(0) k 1 k 2 (τ 1 , τ 2 ; ϕ) ≡ Φ|T[β † k 1 (τ 1 )β k 2 (τ 2 )]|Φ(ϕ) Φ|Φ(ϕ) , (3.22a) G --(0) k 1 k 2 (τ 1 , τ 2 ; ϕ) ≡ Φ|T[β k 1 (τ 1 )β k 2 (τ 2 )]|Φ(ϕ) Φ|Φ(ϕ) , ( 3.22b)
G ++(0) k 1 k 2 (τ 1 , τ 2 ; ϕ) ≡ Φ|T[β † k 1 (τ 1 )β † k 2 (τ 2 )]|Φ(ϕ) Φ|Φ(ϕ) , ( 3.22c)
G -+(0) k 1 k 2 (τ 1 , τ 2 ; ϕ) ≡ Φ|T[β k 1 (τ 1 )β † k 2 (τ 2 )]|Φ(ϕ) Φ|Φ(ϕ) . (3.22d)
By virtue of the off-diagonal elementary contractions worked out in Sec. 1.4.5 for the right definition of the gauge-rotated Bogoliubov vacuum8 , the four off-diagonal propagators are equal to

G +-(0) k 1 k 2 (τ 1 , τ 2 ; ϕ) = -e -(τ 2 -τ 1 )E k 1 θ(τ 2 -τ 1 )δ k 1 k 2 , (3.23a) G --(0) k 1 k 2 (τ 1 , τ 2 ; ϕ) = +e -τ 1 E k 1 e -τ 2 E k 2 R -- k 1 k 2 (ϕ) , (3.23b) G ++(0) k 1 k 2 (τ 1 , τ 2 ; ϕ) = 0 , (3.23c) G -+(0) k 1 k 2 (τ 1 , τ 2 ; ϕ) = +e -(τ 1 -τ 2 )E k 1 θ(τ 1 -τ 2 )δ k 1 k 2 , ( 3.23d)
where both normal propagators are actually related via antisymmetry under the exchange of time and quasi-particle labels. The higher generality and complexity of the off-diagonal BMBPT diagrammatics of present interest compared to the straight BMBPT diagrammatics discussed in Ref. [START_REF] Arthuis | ADG: Automated generation and evaluation of many-body diagrams I. Bogoliubov many-body perturbation theory[END_REF] is due to the presence of the anomalous propagator G --(0) (ϕ) that carries the full gauge-angle dependence. In particular, the possibility to form anomalous propagators significantly increases the combinatorics [START_REF] Duguet | Symmetry broken and restored coupled-cluster. Global gauge symmetry and particle number[END_REF]. Eventually, the two diagrammatics coincide in the limit ϕ = 0 given that G --(0) (0) = 0. All in all, the present extension of the ADG code amounts to dealing with this higher generality and complexity, which itself originates from the presence of different left and right vacua in the off-diagonal kernel o(ϕ) (see Eq. (3.21)). Equal-time propagators can solely arise from contracting two quasi-particle operators belonging to the same normal-ordered operator displaying creation operators to the left of annihilation ones. As a result, one finds that [START_REF] Duguet | Symmetry broken and restored coupled-cluster. Global gauge symmetry and particle number[END_REF]

G +-(0) k 1 k 2 (τ, τ ; ϕ) ≡ 0 , (3.24a) G --(0) k 1 k 2 (τ, τ ; ϕ) ≡ +e -τ (E k 1 +E k 2 ) R -- k 1 k 2 (ϕ) , (3.24b) G ++(0) k 1 k 2 (τ, τ ; ϕ) ≡ 0 , (3.24c) G -+(0) k 1 k 2 (τ, τ ; ϕ) ≡ 0 , (3.24d)
such that the sole non-zero equal-time contraction, and thus the sole contraction of an interaction vertex onto itself, is of anomalous character. Correspondingly, no contraction of an interaction vertex onto itself can occur in the diagonal case, i.e. for ϕ = 0.

Diagrammatic representation

The pedestrian application of the off-diagonal Wick's theorem becomes quickly cumbersome as the order p increases. Furthermore, it leads to computing independently many contributions that are in fact identical. By identifying the corresponding pattern, one can design a diagrammatic representation of the various contributions and evaluate their algebraic expressions such that a single diagram captures all identical contributions at once. In order to achieve this goal, one must first introduce the diagrammatic representation of the building blocks.

The operator O expressed in the quasi-particle basis is displayed in the Schrödinger representation in Fig. 3.1 as a sum of Hugenholtz vertices denoting its various normal-ordered contributions O ij . The antisymmetrized matrix element O ij k 1 ...k i k i+1 ...k i+j must be assigned to the corresponding square vertex, where i (j) denotes the number of lines traveling out of (into) the vertex and representing quasiparticle creation (annihilation) operators. The operator O(τ ) in the interaction representation possesses the same diagrammatic except that a time τ is attributed to each of the vertices, i.e., to each of the lines coming in or out of them.

In the canonical representation used in Fig. 3.1, all oriented lines go up, i.e., lines representing quasiparticle creation (annihilation) operators appear above (below) the vertex. Accordingly, indices k 1 . . . k i must be assigned consecutively from the leftmost to the rightmost line above the vertex, while k i+1 . . . k i+j must be similarly assigned consecutively for lines below the vertex. In the diagrammatic representation of the observable O A 0 , it is however possible for a line to propagate downwards. This can be obtained unambiguously starting from the canonical representation of Fig. 3.1 at the price of adding a specific rule. As illustrated in Fig. 3.2 for the diagram representing O 22 , lines must only be rotated through the right of the diagram, i.e., going through the dashed line, while it is forbidden to rotate them through the full line. Additionally, a minus sign must be added to the amplitude O ij k 1 ...k i k i+1 ...k i+j associated with the canonical diagram each time two lines cross as illustrated in Fig. 3.2.

Since the grand canonical potential Ω is involved in the evaluation of any observable O A 0 , its own diagrammatic representation is needed and displayed in Fig. 3.3. The only

O [0] =
O [2] = + + O [4] = + + + + O [6] = + + + + + + Rules to apply when departing from the canonical diagrammatic representation of a normal-ordered operator. Oriented lines can be rotated through the dashed line but not through the full line.

Ω [0] = Ω [2] = + + Ω [4] = + + + + Ω [6] = + + + + + + difference with Fig. 3.1 relates to the use of dots rather than square symbols to represent the vertices. The same is easily done for other operators of interest, i.e., H and A. It is to be noted that Ω 1 has the same diagrammatic representation as Ω except that Ω 00 must be omitted and Ω 11 replaced by Ω11 , which requires to use a different symbol for that particular vertex 9 .

As the off-diagonal Wick theorem contracts pairs of quasi-particle operators together, the lines entering the diagrammatic representation of operators are eventually connected in the computation of the kernel o(ϕ), thus, forming elementary contractions. Consequently, 9 There is no different symbol for Ω11 in the following although it must be clear that the vertex with one line coming in and one line coming out does represent Ω11 whenever it originates from the perturbative expansion of the evolution operator. This may be confusing whenever O = Ω since in this case there can also be a vertex Ω 11 at fixed time t = 0. the four unperturbed propagators at play also need to be represented diagrammatically, which is done in Fig. 3.4. Here, the convention is that the left-to-right reading of a matrix element corresponds to the up-down reading of the diagram.

Diagrams generation

With the building blocks at hand, off-diagonal BMBPT Feynman diagrams representing the contributions to o(ϕ) are generated by assembling them according to a set of topological rules [START_REF] Duguet | Symmetry broken and restored coupled-cluster. Global gauge symmetry and particle number[END_REF] 1. A Feynman diagram of order p consists of p vertices

Ω i k j k (τ k ), i k + j k ∈ {2, 4, 6}, along with one vertex O mn (0), m + n ∈ {0, 2, 4, 6}
, that are connected by fermionic quasi-particle lines, i.e., via non-zero propagators

G +-(0) , G -+(0) or G --(0) .
2. Each vertex is labeled by a time variable while each line is labeled by two time labels associated with the two vertices the line is attached to.

3. Generating all contributions to Eq. (3.21) requires to form all possible diagrams, i.e., contract quasi-particle lines attached to the vertices in all possible ways while fulfilling the following restrictions. a) Restrict equal-time propagators starting and ending at the same vertex to anomalous propagators. In the diagonal case, i.e. for ϕ = 0, no such selfcontraction may occur.

b) Restrict the set to connected diagrams, i.e., omit diagrams containing parts that are not connected to each other by either propagators or vertices. This implies in particular that the vertex O 00 with no line can only appear at order p = 0. c) Because of the time-ordering relations carried by the propagators (see Eq. (3.23)), normal lines linking a set of vertices must not form an oriented loop. For two given vertices

Ω i k j k (τ k ) and Ω i k j k (τ k )
, it means that normal lines must propagate between them in the same direction. Correspondingly, normal lines connected to the generic operator O at fixed time 0 must go out of it, i.e. upwards in time. Anomalous lines do not carry time-ordering relations and are thus not concerned by these restrictions. In the diagonal case, i.e. ϕ = 0, where no anomalous line may be formed, the above constrain imposes that contributing vertices O mn (0) can only have lines going out, i.e. one necessarily has n = 0. d) Restrict the set to vacuum-to-vacuum diagrams forming a set of closed loops with no external, i.e., unpaired, lines. This condition, together with the fact that G ++(0) (ϕ) is identically zero, strongly constrains which normal-ordered parts Ω i k j k (τ k ) and O mn (0) of the p + 1 involved operators can be combined, i.e., the condition

n a ≡ p k=1 (j k -i k ) + n -m ≥ 0 ,
must be fulfilled. The number n a corresponds to the number of anomalous propagators G --(0) (ϕ) in the diagram. In the diagonal case for which G --(0) (0) = 0, the set of combined operators are further reduced to n a = 0. e) Restrict the set to topologically distinct time-unlabelled diagrams, i.e., timeunlabelled diagrams that cannot be obtained from one another via a mere displacement, i.e., translation, of the vertices.

Diagram evaluation

Feynman expression

The way to translate off-diagonal BMBPT Feynman diagrams into their mathematical expressions follows a set of algebraic rules 1. Each of the p + 1 vertices contributes a factor, e.g., Ω ij k 1 ...k i k i+1 ...k i+j with the sign convention detailed in Sec. 3.3.3.

Each of the

n b ≡ p k=1 (j k + i k ) + n + m /2 , lines contributes a factor G gg (0) k 1 k 2 (τ k , τ k )
, where g and g characterize the type of elementary propagator the line corresponds to. According to Eq. (3.23), each of the n a anomalous propagators carries an exponential function of the two time labels and an anomalous contraction R -- k 1 k 2 (ϕ) while each of the n b -n a normal propagators carries an exponential function and a step function of the two time labels. convention based on the insertion of a fictitious semi-straight, e.g., horizontal line originating from the vertex that the self-contraction is forbidden to cross. Taking the semi-straight line as a reference point, the quasi-particle indices must be attributed to the equal-time propagator in the order the lines are crossed when going around the vertex in a clockwise fashion.

5. All quasi-particle labels must be summed over while all running time variables must be integrated over from 0 to τ → +∞.

6. A sign factor (-1) p+n c , where p denotes the order of the diagram and n c denotes the number of crossing lines in the diagram, must be considered 10 . The overall sign results from multiplying this factor with the sign associated with each matrix element.

7. Each diagram comes with a numerical prefactor obtained from the following combination a) A factor 1/(n e )! must be considered for each group of n e equivalent lines.

Equivalent lines must all begin and end at the same vertices (or vertex, for anomalous propagators starting and ending at the same vertex), and must correspond to the same type of contractions, i.e. they must all correspond to propagators characterized by the same superscripts g and g in addition to having identical time labels.

b) Given the previous rule, an extra factor 1/2 must be considered for each anomalous propagator that starts and ends at the same vertex. c) A symmetry factor 1/n s must be considered in connection with exchanging the time labels of the vertices in all possible ways, counting the identity as one. The factor n s corresponds to the number of ways exchanging the time labels provides a time-labelled diagram that is topologically equivalent to the original one.

In order to illustrate the typical expression of off-diagonal Feynman BMBPT diagrams and to anticipate several key characteristics, let us compute the three second-order diagrams displayed in Fig. 3.6, i.e.,

PO2.2.1 = 1 4 k i Ω 02 k 3 k 4 Ω 22 k 3 k 4 k 1 k 2 O 20 k 1 k 2 × lim τ →∞ τ 0 dτ 1 dτ 2 θ(τ 2 -τ 1 )e -τ 2 E k 3 k 4 e -τ 1 E k 1 k 2 k 3 k 4 , (3.25a) PO2.2.2 = 1 2 k i Ω 02 k 3 k 5 Ω 13 k 3 k 1 k 2 k 4 O 20 k 1 k 2 R -- k 5 k 4 (ϕ) × lim τ →∞ τ 0 dτ 1 dτ 2 θ(τ 2 -τ 1 )e -τ 2 E k 3 k 5 e -τ 1 E k 1 k 2 k 4 k 3 , (3.25b) PO2.2.3 = 1 4 k i Ω 02 k 5 k 6 Ω 04 k 1 k 2 k 4 k 3 O 20 k 1 k 2 R -- k 6 k 4 (ϕ)R -- k 5 k 3 (ϕ) × lim τ →∞ τ 0 dτ 1 dτ 2 e -τ 2 E k 5 k 6 e -τ 1 E k 1 k 2 k 3 k 4 , ( 3.25c)
where the extended notation

E k a k b ... k i k j ... ≡ E k i + E k j + . . . -E k a -E k b -. . . , (3.26)
was introduced. In each case, the sign, the combinatorial factors and the three matrix elements directly reflect Feynman's algebraic rules listed above and are easy to interpret. Eventually, the final form of the integrand originates from expliciting the n b = 4 propagators via Eq. (3.23), which induces the presence of one off-diagonal elementary contractions per anomalous propagator. The three chosen diagrams display the same overall topology11 , i.e. while the vertex O 20 is at fixed time 0, the Ω 22 (Ω 13 ,Ω 04 ) vertex is at running time τ 1 and the Ω 02 vertex is at running time τ 2 . However, the three diagrams differ in their number of anomalous lines and, as such, clearly illustrate key consequences of going from diagonal to off-diagonal BMBPT. The first diagram, PO2.2.1, contains no anomalous line (n a = 0) and already occurs in straight, i.e. diagonal, BMBPT 12 . By turning the second vertex Ω 22 into Ω 13 (Ω 04 ), PO2.2.2 (PO2.2.3) contains n a = 1 (n a = 2) anomalous line(s) between the second and the third vertices. As a consequence, the integrands display typical structures that need to be scrutinized for the following.

• The fact that the two running variables τ 1 and τ 2 are positive is directly encoded into the boundary of the double integral.

• In PO2.2.1, the explicit step function characterizes the time ordering induced between Ω 22 and Ω 02 vertices by the two normal propagators connecting them. This step function, i.e. time ordering, remains at play in PO2.2.2 given than one normal line still connects the second and third vertices. Contrarily, the absence of step function in PO2.2.3 characterizes the fact that Ω 04 and Ω 02 are solely connected via anomalous propagators that do not induce any time-ordering relation between them. While in the first two cases the integral over τ 1 depends on the integral over τ 2 , both integrals are independent from each other in PO2.2.3.

• Grouping appropriately the exponential functions coming from the 4 propagators, the integrand displays one exponential factor per running time, i.e., per

Ω i k j k (τ k ) vertex. The relevant energy factor E k a k b ... k i k j .
.. multiplying the variable τ k in this exponential function denotes the sum/difference of quasi-particle energies associated with the lines entering/leaving the corresponding vertex.

The three diagrams exemplify the fact that the off-diagonal BMBPT diagrammatics differentiates itself by the presence of anomalous lines that, depending on the situation, may change the time-ordering structure between the vertices compared to the diagonal BMBPT diagram displaying the same overall topology.

Time-integrated expression

The expression obtained via the application of Feynman's algebraic rules does not yet constitute the form needed for the numerical implementation of the formalism. While the sign, the combinatorial factor and the matrix elements will remain untouched, the p-tuple time integral must be performed in order to obtain the needed expression.

A major part of Ref. [START_REF] Arthuis | ADG: Automated generation and evaluation of many-body diagrams I. Bogoliubov many-body perturbation theory[END_REF] was dedicated to the computation of the p-tuple time integrals via the introduction of the so-called time-structure diagram (TSD) underlying any given BMBPT diagram of arbitrary order and topology. The general theory of TSDs was given in Ref. [START_REF] Arthuis | ADG: Automated generation and evaluation of many-body diagrams I. Bogoliubov many-body perturbation theory[END_REF] and the way to implement it for more general off-diagonal BMBPT diagrams will be elaborated on later on. For now, it is sufficient to focus on the main consequence of the above analysis, i.e. while the presence of one anomalous line in PO2.2.2 does not change its time structure compared to PO2.2.1, turning the other propagator connecting the second and third vertices into an anomalous line does modify it. Consequently, while the TSD associated to diagrams PO2.2.1 and PO2.2.2 is T2.1 (see Fig. 3.11), it is replaced by T2.2 for PO2.2.3. Generically denoting as a k the energy factor multiplying the time label τ k in the integrand, the integrals associated with our examples are

T2.1 = lim τ →∞ τ 0 dτ 1 dτ 2 θ(τ 2 -τ 1 )e -a 1 τ 1 e -a 2 τ 2 = 1 a 2 (a 1 + a 2 ) , (3.27a) T2.2 = lim τ →∞ τ 0 dτ 1 dτ 2 e -a 1 τ 1 e -a 2 τ 2 = 1 a 1 a 2 , ( 3.27b)
the first (second) of which applies to PO2.2.1 and PO2.2.2 (PO2.2.3).

In order to obtain the final, i.e. time-integrated, expression, the factors a 1 and a 2 must be expressed back in terms of quasi-particle energies for each of the three diagrams. As discussed in Ref. [START_REF] Arthuis | ADG: Automated generation and evaluation of many-body diagrams I. Bogoliubov many-body perturbation theory[END_REF] for diagonal BMBPT diagrams, and as generalized to off-diagonal BMBPT diagrams below, the specific combinations of these factors emerging from the TSDs correspond necessarily to positive sums of quasi-particle energies that can be straightforwardly extracted from the diagram itself. Combining Eqs. (3.25) and (3.27) before inserting the appropriate combinations of quasi-particle energies, one obtains the desired expressions under the form

PO2.2.1 = 1 4 k i Ω 02 k 3 k 4 Ω 22 k 3 k 4 k 1 k 2 O 20 k 1 k 2 E k 1 k 2 E k 3 k 4 , (3.28a) PO2.2.2 = 1 2 k i Ω 02 k 3 k 5 Ω 13 k 3 k 1 k 2 k 4 O 20 k 1 k 2 E k 1 k 2 k 4 k 5 E k 3 k 5 R -- k 5 k 4 (ϕ) , (3.28b) PO2.2.3 = 1 4 k i Ω 02 k 5 k 6 Ω 04 k 1 k 2 k 4 k 3 O 20 k 1 k 2 E k 1 k 2 k 3 k 4 E k 5 k 6 R -- k 6 k 4 (ϕ)R -- k 5 k 3 (ϕ) . (3.28c)

Towards higher orders

Off-diagonal BMBPT diagrams of order p = 0 and 1 have been generated and evaluated manually within the NO2B approximation [START_REF] Duguet | Symmetry broken and restored coupled-cluster. Global gauge symmetry and particle number[END_REF], see chap. 5. The twenty corresponding diagrams are displayed in Fig. 3.7 for illustration. Among these twenty diagrams, only the three diagrams appearing in the first column (n a = 0) remain in straight, i.e. diagonal, BMBPT that has been dealt with in the first version of the ADG code [START_REF] Arthuis | ADG: Automated generation and evaluation of many-body diagrams I. Bogoliubov many-body perturbation theory[END_REF].

While it was already challenging to automatically generate and evaluate diagonal BMBPT diagrams of arbitrary orders and topologies, off-diagonal BMBPT reaches yet another level of complexity related to the proliferation of diagrams, itself increasing with the perturbative order, associated with the possibility to form off-diagonal propagators. Still, the step accomplished in Ref. [START_REF] Arthuis | ADG: Automated generation and evaluation of many-body diagrams I. Bogoliubov many-body perturbation theory[END_REF] happens to be of tremendous help to automatically generate and evaluate off-diagonal BMBPT diagrams as is explained below.

Generation of off-diagonal BMBPT diagrams

The automated generation of diagonal BMBPT Feynman diagrams via elements of graph theory was explained at length in Ref. [START_REF] Arthuis | ADG: Automated generation and evaluation of many-body diagrams I. Bogoliubov many-body perturbation theory[END_REF]. As a matter of fact, the strategy presently employed is not to follow a similar method to generate off-diagonal diagrams from scratch but rather to take advantage of having already done so for the diagonal ones, i.e. to start from the order p diagonal BMBPT diagrams to systematically produce their off-diagonal partners.

Basic analysis

Given that diagonal BMBPT diagrams constitute the base line for generating the offdiagonal ones, the eleven zero-, first-and second-order diagonal BMBPT diagrams generated from operator vertices containing four legs at most are displayed in Fig. 3.8 for reference. One recognizes in particular the three zero-and first-order diagrams PO0.1, PO1.1 and PO1.2 already appearing in Fig. 3.7 with a slightly different denomination whose aim is to group all diagrams originating from the same diagonal diagram.

Diagonal and off-diagonal diagrams of order p are actually derived from the same manybody matrix element in Eq. (3.21), except that the ket is gauge rotated in the off-diagonal case. The latter feature authorizes to contract pairs of quasi-particle annihilation operators in addition to only contracting one creation and one annihilation operators in the diagonal case. Starting from diagonal BMBPT diagrams of order p, the complete set of off-diagonal diagrams is obtained via two basic operations 1. adding self-contractions to each vertex, while changing the nature of the vertex accordingly, until the sum of lines entering and leaving the vertex is equal to the rank deg_max of the associated operator.

2. changing normal propagators linking two vertices into anomalous ones. This is achieved by turning the arrow associated with an outgoing line in the original propagator into an incoming line, thus changing the concerned vertex accordingly.

Let us now exemplified the two above operations that must eventually be applied systematically.

Considering the zero-order diagonal diagram PO0.1.1 (1) in Fig. 3.7 (i.e. PO0.1 in Fig. 3.8), and a two-body operator O (deg_max = 4), the vertex O 00 has no line entering or leaving it. Replacing it by O 02 and O 04 , one generates two valid off-diagonal diagrams containing one and two anomalous contractions denoted as PO0.1.1 (2) and PO0.1.1 (3) in Fig. 3.7. One can proceed similarly starting from the first-order diagram denoted as PO1.1.1 (1) in Fig. 3.7 (i.e. PO1.1 in Fig. 3.8). Adding a self-contraction to each of the two vertices provides three additional off-diagonal diagrams containing one or two anomalous lines and denoted as PO1.1.1 (4) , PO1.1.2 (1) and PO1.1.2 (4) in Fig. 3.7.

To illustrate the second operation, let us consider the first-order diagonal diagram PO1.2.1 (1) in Fig. 3.7 (i.e. PO1.2 in Fig. 3.8). This diagram contains four normal lines between the two vertices, each of which can be transformed into an anomalous line. Doing so generates four additional topologically-distinct off-diagonal diagrams denoted as PO1.2.1 (2) , PO1.2.1 (3) , PO1.2.1 (4) and PO1.2.1 (5) in Fig. 3.7.

Combining the transformation of normal lines into anomalous lines and the addition of self-contractions, one obtains the other topologically-distinct off-diagonal diagrams displayed in Fig. 3.7.

Similarity-transformed operator

An important feature is that the bottom vertex O m0 appearing in diagonal BMBPT diagrams is always at fixed time zero. Consequently, off-diagonal diagrams generated by adding self-contractions to it and/or by transforming a normal line leaving it into an anomalous line entering it possess the same time structure as the diagonal diagram it derives from. Indeed, a self-contraction carries no time dependence and thus cannot impact the time structure of the diagram. Furthermore, the fact that all Ω ij vertices are at higher times than O m0 remains true even if all the lines attached to the bottom vertex are changed into anomalous ones such that the time structure is invariant under this transformation.

A key consequence of the above observations is that all diagrams differing only by the number of self-contractions onto the bottom vertex and/or the number of anomalous propagators connected to it can be grouped into a single diagram in which the bottom vertex is replaced by its similarity transformed partner13 at gauge angle ϕ [START_REF] Duguet | Symmetry broken and restored coupled-cluster. Global gauge symmetry and particle number[END_REF] O Z (ϕ) ≡ e -Z(ϕ) Oe Z(ϕ) , (3.29) where Z(ϕ) is the Thouless operator, see Sec. 1.4.4 and Tab. 2.1. As explained at length in Sec. 1.5.3, the transformed operator O Z (ϕ) possesses the same formal structure as the initial operator O. As such, it is decomposed as a sum of terms O mn Z (ϕ) with the same overall rank as O, i.e. m + n ≤ deg_max. The only difference relates to the definition of the (gauge-dependent) matrix elements entering each term O mn Z (ϕ). The complete expression of these matrix elements in terms of the original ones14 are provided in App. B for an operator O characterized by deg_max = 6.

Exploiting this key property, one can reduce drastically the number of diagrams. Employing the transformed operator for the bottom vertex and forbiding any anomalous line to connect to it, the twenty off-diagonal diagrams displayed in Fig. 3.7 are recasted into the four (effective) off-diagonal diagrams displayed in Fig. 3.9. This feature being generic, the recasting procedure extends to any order p.

Systematic scheme

The analysis provided above puts us in position to state the systematic rules used to generate all order p off-diagonal (effective) BMBPT diagrams from the diagonal ones. While the method is straightforward, it is indeed important to discard topologically equivalent diagrams generated through this brute force procedure. Anticipating it, one can actually reduce the need to check for all of them, which is particularly beneficial given that the corresponding test scales factorially with the number of vertices in the diagrams. In practice, one can exploit the following features

• Diagonal BMBPT diagrams being generated with only one type of propagator and all lines going upward, the topology of the diagrams without self-contractions is fixed if one disregards the propagator types. As such, topologically equivalent off-diagonal diagrams cannot be produced by transforming normal propagators into anomalous ones. This step being done first, no check is required here.

• Topologically equivalent off-diagonal diagrams can only be generated by adding self-contractions on diagrams displaying a symmetry factor n s > 1. Consequently, the test to eliminate topologically equivalent off-diagonal diagrams can be limited to subsets of off-diagonal diagrams with self-contractions produced from a given diagram with such a symmetry factor.

Doing so one obtains many-body diagrams for which the number of at a given order is listed in Tab. 3.1. PBMBPT diagrams up to second order are given in App. D, together with their time-integrated expressions.

Evaluation of off-diagonal BMBPT diagrams

Having the capacity to generate all off-diagonal BMBPT Feynman diagrams of order p, the next challenge is to systematically derive their expression. Doing so on the basis of Feynman's algebraic rules is rather straightforward. However, it leaves the p-tuple time integral to perform in order to obtain the time-integrated expression of interest. In Ref. [START_REF] Arthuis | ADG: Automated generation and evaluation of many-body diagrams I. Bogoliubov many-body perturbation theory[END_REF], an algorithm was found to overcome this challenge without prior knowledge of the perturbative order or of the topology of the diagram. This eventually led to the identification of a novel diagrammatic rule. In the present section is explained how the method only needs to be slightly generalized in order to realize the same objective for off-diagonal BMBPT Feynman diagrams at play in P P BMBPT.

Time-structure diagrams

Obtaining the result of p-tuple time integrals in an automatic fashion was made possible via the introduction of the time-structure diagram underlying any given diagonal BMBPT diagram of arbitrary order and topology. While specificities encountered when dealing with more general off-diagonal BMBPT diagrams are presently detailed, refer to Ref. [START_REF] Arthuis | ADG: Automated generation and evaluation of many-body diagrams I. Bogoliubov many-body perturbation theory[END_REF] for the general theory of TSDs.

The key point was already alluded to in Sec. 3.3.5 and relates to the impact anomalous lines may have on the TSD attributed to a given off-diagonal BMBPT diagram. The main features are

• The running time labels (τ 1 , . . . , τ p ) are positive such that each Ω vertex entertains at least an ordering relation with the bottom vertex O Z (ϕ) independently of the network of lines running through the BMBPT diagram. Consequently, the TSD remains necessarily connected, independently of its topology.

• Contrarily to normal lines, anomalous lines do not induce any time ordering relation. This means that, while two Ω vertices connected by at least one normal line are time ordered, it is not the case if they are solely connected via anomalous propagators. Consequently, a link connecting two Ω vertices in the TSD associated to a diagonal BMBPT diagram will disappear when the two vertices become only connected via anomalous propagators in an off-diagonal partner diagram 15 . Whenever a Ω vertex ends up entertaining no time relation with any other due to the replacement of normal lines by anomalous ones, it becomes directly linked to the bottom vertex in the associated TSD.

• The addition of a self contraction to any given Ω vertex does not change the time structure of the diagram and thus the associated TSD.

In conclusion, the presence of anomalous lines may, depending on the situation, change the TSD associated to an off-diagonal BMBPT diagram compared to the diagonal diagram displaying the same topology. Eventually, the TSD associated to an off-diagonal BMBPT diagram can be obtained from the latter through the following steps

1. copy the off-diagonal BMBPT diagram, 2. remove all the anomalous propagators, 3. replace the normal propagators by links, 4. add a link between the bottom vertex at time 0 and every other vertex if such a link does not exist, 5. for each pair of vertices, consider all possible pathes linking them and only retain the longest one, 6. match the label a q associated to a given vertex in the TSD diagram to the sum/difference of quasi-particle energies associated with the lines entering/leaving the corresponding vertex in the BMBPT diagram.

The only difference with the procedure followed for diagonal BMBPT diagrams [START_REF] Arthuis | ADG: Automated generation and evaluation of many-body diagrams I. Bogoliubov many-body perturbation theory[END_REF] relates to step 2 that trivially stipulates to strip off anomalous propagators, if any. The procedure is illustrated in Fig. 3.10 for a third-order diagonal BMBPT diagram and for the particular off-diagonal diagram generated from it by turning the two normal lines connecting vertex Ω 40 to one of the two Ω 04 vertices into anomalous lines. Cleared of other informations, the TSDs tranparently characterize the time-ordering structure underlying the diagrams. In the first one, the three Ω ij vertices are at higher times than O 40 such that the two Ω 04 vertices are at higher times than Ω 40 without being ordered with respect to one another. From the graph theory viewpoint, the corresponding TSD is a tree, i.e., it contains no cycle, with two branches. In the second diagram, the fact that the two lines connecting Ω 40 to (one of the two) Ω 04 are anomalous relaxes the time-ordering between both vertices and, as a result, changes the nature of the associated TSD, i.e. a 3 is now directly linked to the bottom vertex. The corresponding TSD is also a tree with two branches.

Discussion

It is mandatory to generate the TSDs from the underlying (off-diagonal) BMBPT diagrams. Indeed, only in this case can the rank deg_max of the operators at play be employed to constrain the topology of the diagrams, eventually dictating the topology of allowed TSDs. Furthermore, going from diagonal to off-diagonal BMBPT diagrams may not only change the nature of the TSD associated to a particular diagram but also increase the list of allowed TSDs at a given order.

With this in mind and following the above rules, the 1/1/2/5 TSDs of order 0/1/2/3 associated to off-diagonal BMBPT diagrams generated from operators with deg_max = 4 or deg_max = 6 have been produced 16 and systematically displayed in Fig. 3.11. Interestingly, restricting one-self to diagonal BMBPT diagrams and deg_max = 4, T3.4 would have to be removed from the set of allowed TSDs, i.e. going from diagonal to off-diagonal BMBPT or going from deg_max = 4 to deg_max = 6 adds one allowed third-order TSD.

From the TSD back to the BMBPT diagram

In the end, different BMBPT diagrams of order p can have the same TSD, i.e., the same underlying time structure. At the same time, off-diagonal BMBPT diagrams originating from the same diagonal diagram may have different TSDs, i.e., different underlying time structures. Once the proper TSD is associated to the BMBPT diagram of interest, its computation follows the algorithm detailed in Ref. [START_REF] Arthuis | ADG: Automated generation and evaluation of many-body diagrams I. Bogoliubov many-body perturbation theory[END_REF]. In particular, the treatment of non-tree TSDs requires to turn it first into a sum of tree TSDs.

Once the expression of a tree TSD of order p has been obtained, the goal is to generate the actual time-integrated expression of the BMBPT diagrams associated to it. Rather than replacing the individual factors a q , q = 1, . . . , p, by their expressions for each BMBPT diagram, one introduces the notion of subdiagram, or subgraph, to directly obtain their combinations appearing in the denominator of the time-integrated expression of interest. In Ref. [START_REF] Arthuis | ADG: Automated generation and evaluation of many-body diagrams I. Bogoliubov many-body perturbation theory[END_REF], a subdiagram of a diagonal BMBPT diagram was defined as a diagram composed by a subset of vertices plus the propagators that are exchanged between them. As each vertex label a q in a TSD eventually stands for the sum/difference of quasi-particle energies associated with the lines entering/leaving the vertex in the associated BMBPT diagram, the combination of these labels denotes the sum/difference of quasi-particle energies associated with the lines entering/leaving the subdiagram grouping the corresponding vertices.

In the present context of off-diagonal BMBPT diagrams, one simply needs to slightly generalize the notion of subdiagrams such that the algorithm stipulated in Ref. [START_REF] Arthuis | ADG: Automated generation and evaluation of many-body diagrams I. Bogoliubov many-body perturbation theory[END_REF] to find the appropriate factors entering the time-integrated expression of the diagram applies. Thus a subdiagram of an off-diagonal BMBPT diagram is now defined as a diagram composed by a subset of vertices plus the normal propagators that are exchanged between them. This definition is obviously consistent with the one introduced earlier for strictly diagonal BMBPT diagrams given that the latter solely contain normal propagators.

With this definition at hand, the energy denominator of an off-diagonal BMBPT diagram associated with a tree TSD is obtained in the following way 1. Consider a vertex but the bottom one in the BMBPT diagram, a) determine all its descendants using the TSD, b) form a subdiagram using the vertex and its descendants, c) sum the quasi-particle energies corresponding to the lines entering the subdiagram, d) add the corresponding factor to the denominator expression, 2. Go back to 1. until all vertices have been exhausted.

Given that anomalous lines are excluded from the definition of a subdiagram, they systematically count as entering the subdiagram whenever they connect to a vertex belonging to it.

Let us illustrate the diagrammatic rule by focusing on the two third-order off-diagonal BMBPT diagrams17 displayed in Fig. 3.12.

1. The denominator in the time-integrated expression of the first diagram is obtained through the following steps a) The vertex at time τ 1 in the BMBPT diagram corresponds to vertex a 1 in the TSD. Its descendants are vertices a 2 and a 3 corresponding to BMBPT vertices at times τ 2 and τ 3 , respectively. The sum of quasi-particle energies associated to the lines entering the subgraph grouping the three vertices is

E k 1 k 2 k 3 k 4 ,
E k 1 k 2 k 3 k 4 E k 1 k 2 k 5 k 6 E k 3 k 4 k 7 k 8 ,
where each factor corresponds to a positive sum of quasi-particle energies.

2. The denominator of the second, off-diagonal, diagram containing two anomalous line and corresponding to a different TSD is obtained as a) The vertex at time τ 1 in the BMBPT diagram corresponds to vertex a 1 in the TSD. Contrarily to the previous case, vertex a 3 is not a descendant of a 1 anymore as is visible from the TSD such that the subgraph of interest solely groups a 1 and a 2 . The sum of quasi-particle energies associated to the lines entering the subgraph in the BMBPT diagram is E k 1 k 2 k 7 k 8 , thus, providing the first factor entering the denominator.

b) The situation of the vertex at time τ 2 in the off-diagonal BMBPT diagram is strictly the same as in the previous diagonal one. Consequently, the associated factor in the denominator is

E k 1 k 2 k 5 k 6 .
c) As in the diagonal BMBPT diagram, the vertex at time τ 3 has no descendant in the off-diagonal BMBPT diagram. Consequently, the subgraph corresponding to vertex a 3 reduces to itself. However, because the two anomalous lines carry two quasi-particle labels each, the sum of quasi-particle energies associated to the lines entering the subgraph has now become

E k 3 k 4 k 9 k 10 .
d) Eventually, the complete denominator reads as

E k 1 k 2 k 7 k 8 E k 1 k 2 k 5 k 6 E k 3 k 4 k 9 k 10 ,
where each factor corresponds to a positive sum of quasi-particle energies.

For completeness, let us work out another example highlighting additional features of interest, i.e the second-order off-diagonal diagram displayed in Fig. 3.13 together with its associated TSD 18 . Applying the diagrammatic rule, one obtains 1. The vertex at time τ 1 in the BMBPT diagram corresponds to vertex a 1 in the TSD.

Because it remains one normal line connecting it to the vertex at time τ 2 , a 2 is indeed its descendant. The subgraph of interest thus groups a 1 and a 2 . Due to the more general definition of subgraphs at play in the context of off-diagonal BMBPT, the anomalous line connecting the two vertices is excluded from it, together with the self contraction on the upper vertex. Consequently, the sum of quasi-particle energies associated to the lines entering the subgraph is

E k 1 k 2 k 4 k 5 k 6 k 7
, thus, providing the first factor entering the denominator.

2. The vertex at time τ 2 in the BMBPT diagram corresponds to vertex a 2 in the TSD. It has no descendant such that the corresponding subgraph reduces to itself, excluding the self contraction that the vertex exchanges with itself. The sum of quasi-particle energies associated to the lines entering the subgraph is E k 3 k 5 k 6 k 7 , thus providing the second factor entering the denominator.

3. Eventually, the complete denominator reads as

E k 1 k 2 k 4 k 5 k 6 k 7 E k 3 k 5 k 6 k 7 .

Output of the ADG program

A typical output for an off-diagonal BMBPT diagram is:

PO2.3.3 = (-1) 2 (2!)(3!) k i O 40 k 1 k 2 k 3 k 4 (ϕ)Ω 04 k 1 k 2 k 3 k 5 Ω 04 k 6 k 4 k 7 k 8 R -- k 6 k 5 (ϕ)R -- k 8 k 7 (ϕ) × lim τ →∞ τ 0 dτ 1 dτ 2 e -τ 1 E k 1 k 2 k 3 k 6 e -τ 2 E k 4 k 5 k 7 k 8 = (-1) 2 (2!)(3!) k i O 40 k 1 k 2 k 3 k 4 (ϕ)Ω 04 k 1 k 2 k 3 k 5 Ω 04 k 6 k 4 k 7 k 8 E k 1 k 2 k 3 k 6 E k 4 k 5 k 7 k 8 R -- k 6 k 5 (ϕ)R -- k 8 k 7 (ϕ) → T1: T1 = 1 a 1 a 2 a 1 = E k 1 k 2 k 3 k 6 a 2 = E k 4 k 5 k 7 k 8

Connection to time-ordered diagrammatics

In Ref. [START_REF] Arthuis | ADG: Automated generation and evaluation of many-body diagrams I. Bogoliubov many-body perturbation theory[END_REF], time-unordered and time-ordered diagrammatics emerging, respectively, from the time-dependent and the time-independent formulations of straight, i.e. diagonal, BMBPT were compared at length. The main outcome of the analysis related to the capacity of the time-unordered diagrammatics to resum at once large classes of timeordered diagrams. Correspondingly, it was shown that the new diagrammatic rule allowing for the direct obtention of the time-integrated results on the basis of time-unordered diagrams generalizes the resolvent rule at play for time-ordered diagrams.

As in Ref. [START_REF] Arthuis | ADG: Automated generation and evaluation of many-body diagrams I. Bogoliubov many-body perturbation theory[END_REF], the formal and numerical developments presented here rely on the timedependent formulation of P P BMBPT [START_REF] Duguet | Symmetry broken and restored coupled-cluster. Global gauge symmetry and particle number[END_REF]. While it is traditionally more customary to design many-body perturbation theories on the basis of a time-independent formalism [START_REF] Shavitt | Many-body methods in chemistry and physics : MBPT and coupled-cluster theory[END_REF], this task has so far not been attempted for P P BMBPT. While the end result has to be the same, the partitioning 19 of the complete order-p contribution to the observable O A 0 will differ in both approaches. In the absence of time-ordered diagrammatics associated to P P BMBPT, the same analysis as the one done in Ref. [START_REF] Arthuis | ADG: Automated generation and evaluation of many-body diagrams I. Bogoliubov many-body perturbation theory[END_REF] for straight BMBPT cannot be processed. Leaving this analysis to a future work, it can however be anticipated that timeunordered off-diagonal BMBPT diagrams will feature the same capacity to resum large classes of time-ordered diagrams at play in the, yet-to-be-formulated20 , time-independent version of P P BMBPT.

Conclusions

Bogoliubov MBPT perturbatively expands the exact solution of the Schrödinger equation around a so-called Bogoliubov reference state, i.e., a general product state breaking U (1) global-gauge symmetry associated with the conservation of good particle-number in the system. Given that the breaking of a symmetry cannot actually be realized in a finite quantum system, U (1) symmetry must eventually be restored exactly, which is made possible thanks to the recent formulation of the particle-number projected Bogoliubov many-body perturbation theory (P P BMBPT) [START_REF] Duguet | Symmetry broken and restored coupled-cluster. Global gauge symmetry and particle number[END_REF] that extends straight BMBPT on the basis of a more general diagrammatics.

In this context, the present chapter details the systematic generation and evaluation of diagrams at play in P P BMBPT operated by the second version (v2.0.0) of the code ADG. While the automated evaluation of the diagrams only requires a mere extension of the diagrammatic rule unrevealed in Ref. [START_REF] Arthuis | ADG: Automated generation and evaluation of many-body diagrams I. Bogoliubov many-body perturbation theory[END_REF], the method used to first generate all allowed diagrams is different from the one use in Ref. [START_REF] Arthuis | ADG: Automated generation and evaluation of many-body diagrams I. Bogoliubov many-body perturbation theory[END_REF]. Taking advantage of the capacity of the code ADG to already produce all valid BMBPT diagrams of order p, the set of rules to generate all those at play in P P BMBPT from those appearing in BMBPT was identified and implemented. While P P BMBPT is currently being numerically implemented up to order 1, the present work makes it possible to implement it to higher orders, e.g. 3 or 4, in the future. It allows to extend the work done in [START_REF] Demol | High-order Bogoliubov Many-Body Perturbation Theory[END_REF] for high-order BMBPT to realistic model spaces.

Eventually, the second version of the code ADG21 is kept flexible enough to be expanded throughout the years to tackle the diagrammatics at play in yet other many-body formalisms that either already exist, e.g. (un)rotated perturbative amplitudes (see Secs.2.3.2 and 2.3.3), or are yet to be formulated.

The full HTML documentation is available online under https://adg.readthedocs. io/, and help with eventual bugs of the program can be obtained by opening issues on the GitHub repository at https://github.com/adgproject/adg.

Chapter 4.

Expectation value approach to PBMBPT

Introduction

The goal of the present chapter is to design a way to compute other observables than the ground-state energy and particle number, i.e. observables that do not commute with H. As the eigenstates of H are not eigenstates of these operators, an expectation value method is required, i.e. the projective approach does not apply. The present work extends, in presence of symmetry breaking and possible restoration, the use of expectation value approach to observables already employed in standard closed-shell MBPT1 [START_REF] Strayer | Correlation Effects in Nuclear Densities[END_REF][START_REF] Roth | Hartree-Fock and many body perturbation theory with correlated realistic NN interactions[END_REF][START_REF] Tichai | Natural orbitals for ab initio no-core shell model calculations[END_REF].

Another goal is to go beyond low-order BMBPT by applying eigenvector continuation (EC) [START_REF] Frame | Eigenvector Continuation with Subspace Learning[END_REF]. In both cases, similar quantities need to be computed. A systematic method to access them is designed.

Trial many-body state

While the focus is on BMBPT, the presentation of the quantities to be calculated will be done in a more general context.

General form

Let us consider a symmetry-broken many-body state |Ψ c written as a linear combination of multi-quasiparticle configurations (e.g. up to six quasiparticle excitations)

|Ψ c ≡ C 00 |Φ + 1 2! k 1 k 2 C 20 k 1 k 2 |Φ k 1 k 2 + 1 4! k 1 k 2 k 3 k 4 C 40 k 1 k 2 k 3 k 4 |Φ k 1 k 2 k 3 k 4 + 1 6! k 1 k 2 k 3 k 4 k 5 k 6 C 60 k 1 k 2 k 3 k 4 k 5 k 6 |Φ k 1 k 2 k 3 k 4 k 5 k 6 + . . . , (4.1)
where the tensors

C 20 k 1 k 2 , C 40 k 1 k 2 k 3 k 4 and C 60 k 1 k 2 k 3 k 4 k 5 k
6 are fully antisymmetric with respect to permutation of their indices. For convenience, |Ψ c is supposed to be normalized 2 ( Ψ c |Ψ c = 1) such that coefficients of the linear combination verify

|C 00 | 2 + 1 2! k i |C 20 k 1 k 2 | 2 + 1 4! k i |C 40 k 1 k 2 k 3 k 4 | 2 + 1 6! k i |C 60 k 1 k 2 k 3 k 4 k 5 k 6 | 2 + . . . = 1 . (4.
2)

The state |Ψ c typically results from a symmetry-broken beyond mean-field calculation, e.g. BMBPT, BCC or BCI. It can be written has a many-body creation operator

C ≡ C 0 + C 1 + C 2 + C 3 + . . . , (4.3)
where the operator C n is given by

C n ≡ 1 (2n)! k 1 ...k 2n C 2n0 k 1 ...k 2n β † k 1 . . . β † k 2n , ( 4.4)
acting on the Bogoliubov vacuum, i.e.

|Ψ c = C|Φ = (C 0 + C 1 + C 2 + C 3 + . . .) |Φ . (4.5)
Given that the diagrams at play in the former scheme form a subset of those appearing in the latter, only the latter truncation scheme is presently discussed. Still, future numerical applications will aim at comparing the empirical merits of both truncation schemes to the expectation value approach. 2 When employing coefficients from BMBPT or BCC, C 00 = 1 as intermediate normalization Φ|Ψ c = 1 is used. This leads to the necessity to normalize the many-body state providing new coefficients, e.g.

C 00 = C 00 / Ψ c |Ψ c .
In the present work, the broken symmetry of the many-body state |Ψ c is possibly restored

|Ψ A c ≡ P A |Ψ c Ψ c |P A |Ψ c (4.6a) = P A C|Φ Φ|C † P A C|Φ (4.6b)
via the application of the projection operator P A (Eq. (1.20)). It is thus possible to recover the unprojected many-body state by removing the projection operator in Eq. (4.6). In this case, the denominator in Eq. (4.6) is one.

BCI coefficients

The Bogoliubov configuration interaction (BCI) method [START_REF] Ripoche | Combining symmetry breaking and restoration with configuration interaction: A highly accurate many-body scheme applied to the pairing Hamiltonian[END_REF][START_REF] Ripoche | Combining symmetry breaking and restoration with configuration interaction: Extension to z -signature symmetry in the case of the Lipkin model[END_REF] amounts to diagonalize the Hamiltonian H in the space of quasi-particle configuration

{|Φ k 1 k 2 ... }. The coefficients C BCI n directly result from this diagonalization 3 |Ψ BCI c = C BCI |Φ = C BCI 0 + C BCI 1 + C BCI 2 + C BCI 3 + . . . |Φ . ( 4.7)
In practice, the space of quasi-particle configurations is truncated. For example, considering configurations up to four quasi-particle excitation leads Bogoliubov configuration interaction with singles and doubles (BCISD). The expansion of the C n is thus restricted to double excitations

|Ψ BCISD c = C BCISD |Φ = C BCISD 0 + C BCISD 1 + C BCISD 2 |Φ . (4.8)

BCC coefficients

In order to write the BCC wave-function (Eq. (2.1)) in the langage of the C n coefficients

|Ψ BCC c = C BCC |Φ = C BCC 0 + C BCC 1 + C BCC 2 + C BCC 3 + . . . |Φ , ( 4.9)
one needs to expand the exponential of connected cluster amplitudes according to

|Ψ BCC c ≡ e U |Φ = e U 1 +U 2 +... |Φ = 1 + U 1 + U 2 + 1 2! U 2 1 + U 3 + U 1 U 2 + 1 3! U 3 1 + . . . |Φ . (4.10)
Identifying Eq. (4.9) and Eq. (4.10) allows one to express the C BCC n coefficients in terms of the U k coefficients according to

C BCC 0 = 1 , (4.11a) C BCC 1 = U 1 , (4.11b) C BCC 2 = U 2 + 1 2! U 2 1 , (4.11c) C BCC 3 = U 3 + U 1 U 2 + 1 3! U 3 1 , ( 4.11d)
. . .

i.e., the C n denote the disconnected cluster amplitudes. In practice, the connected cluster amplitudes are truncated at a given excitation level. For example, keeping U 1 and U 2 while setting all higher-rank connected amplitudes to zero leads Bogoliubov coupled cluster with single and doubles (BCCSD). The BCCSD wave-function is thus written as

|Ψ BCCSD c = C BCCSD |Φ = C BCCSD 0 + C BCCSD 1 + C BCCSD 2 + C BCCSD 3 + . . . |Φ , ( 4.12)
where the BCCSD coefficients are given by

C BCCSD 0 = 1 , (4.13a) C BCCSD 1 = U 1 , (4.13b) C BCCSD 2 = U 2 + 1 2! U 2 1 , (4.13c) C BCCSD 3 = U 1 U 2 + 1 3! U 3 1 . ( 4

BMBPT coefficients

In Sec. 2.3.3, cluster amplitudes U n were given as perturbative series. In the same way, the exact ground-state wave function can be expanded in perturbation according to

|Ψ c = ∞ n=0 |Ψ (n) c , ( 4.14)
where the n-order wave function is given by

|Ψ (n) c = C (n) |Φ = C (n) 0 + C (n) 1 + C (n) 2 + C (n) 3 + . . . |Φ . (4.15)
As the 0-order wave function is given by the reference vacuum (|Ψ (0) c = |Φ ), the 0-order coefficients are null except for the first one (C (0) 0 = 1). The n-order coefficient, n > 0, is obtained in the same way as the connected operator kernels o

(n) P µ BMBPT.b (ϕ) in Sec. 2.3.4.
In particular, they are given, up to triples, by

C (n) 0 = 0 , (4.16a) C (n) 1 = U (n) 1 , (4.16b) C (n) 2 = U (n) 2 + 1 2! n-1 k=1 U (k) 1 U (n-k) 1 , (4.16c) C (n) 3 = U (n) 3 + n-1 k=1 U (k) 1 U (n-k) 2 , ( 4.16d)
. . .

where the explicit expressions of U (k) n up to n = 3 and k = 2 are provided in App. C. In particular, the first-order coefficients are given by

C (1) 0 = 0 , (4.17a) C (1) 1 = U (1)
1 , (4.17b)

C (1) 2 = U (1)
2 , (

3 = 0 , (4.17d) . . . where U

(1) 3 = 0, as it is assumed that the interaction is of (effective) two-body character, whereas the second-order coefficients are given by

C (2) 0 = 0 , (4.18a) C (2) 1 = U (2)
1 , (4.18b)

C (2) 2 = U (2) 2 + 1 2! U (1) 1 U (1) 1 , (4.18c) C (2) 3 = U (2) 3 + U (1) 1 U (1)
2 .

(4.18d) . . . The excitation rank contained in the operator C increases with the perturbative order. At order k, C (k) n coefficients are potentially non zero up to n = 2k. The BMBPT wave function truncated at order P reads as

|Ψ [P ] c = P n=0 |Ψ (n) c , (4.19) = C [P ] 0 + C [P ] 1 + C [P ] 2 + C [P ] 3 + . . . |Φ , ( 4.20)
where the up to P -order coefficients C [P ] n are given by

C [P ] n = P m=0 C (m) n . (4.21)

Eigenvector continuation coefficients (from BMBPT)

The eigenvector continuation (EC) [START_REF] Frame | Eigenvector Continuation with Subspace Learning[END_REF] is a method that allows to solve a parametric problem, e.g. a system whose Hamiltonian H can be written as H = H [START_REF] Ishii | Nuclear Force from Lattice QCD[END_REF] where H(g) depends on a parameter g, under the hypothesis that H(g) is easily solvable for auxiliary values of the parameter, e.g. g < 1, and hardly solvable for the physical value g = 1.

Accessing several solutions S = {|Ψ(g 1 ) , |Ψ(g 2 ) , . . .} for g 1 , g 2 , . . . < 1, the eigenvector continuation ensures that diagonalizing H(1) in the non-orthogonal basis formed by the states of S, i.e. solving a generalized eigenvalue problem, provides an extremely good account of its exact eigenstates.

In the case of BMBPT, the parametric grand potential Ω(λ) is separated into an unperturbed part and a perturbed part according to

Ω(λ) ≡ Ω 0 + λΩ 1 , (4.22)
where one recovers the unperturbed grand potential for λ = 0 (Ω(0) = Ω 0 ) and the full grand potential for λ = 1 (Ω(1) = Ω). The parametric eigenstates of Ω(λ) are given by

|Ψ c (λ) = ∞ n=0 |Ψ (n) c (λ) = ∞ n=0 λ n |Ψ (n) c , ( 4.23)
such that the manifold generated by a set of these eigenstates S 1 = {|Ψ(λ 1 ) , |Ψ(λ 2 ) , . . .} is also generated by the set of perturbative corrections

S 2 = {|Ψ (0) c , |Ψ (1) c
, . . .} to the eigenstates of Ω.

Working at order P to generate S 2 , the dimensionality of the generalized eigenvalue problem is P + 1. In pratice, the problem is of extremely low dimension, e.g. at second order there are only three basis states. The challenge is thus not in the resolution of the generalized eigenvalue problem in itself, that reads as

ΩX = ENX , ( 4.24)
but rather in the computation of the matrix elements of the grand potential, and norm, in S 2

Ω pq ≡ Ψ (p) c |Ω|Ψ (q) c , ( 4.25a
)

N pq ≡ Ψ (p) c |Ψ (q) c . (4.25b)
It is to be noticed that the method designed in the following allows the computation of such matrix elements. The resolution of the generalized eigenvalue problem leads the up to the P -order EC ground-state wave function

|Ψ EC[P ] c = P n=0 c n |Ψ (n) c , (4.26) = C EC[P ] 0 + C EC[P ] 1 + C EC[P ] 2 + C EC[P ] 3 + . . . |Φ , ( 4.27)
where the coefficients c n originate from the diagonalization such that the up to P -order EC coefficients C EC [P ] n are given by

C EC[P ] n = P m=0 c m C (m) n .
(4.28)

Computation of observables

Let us consider a generic operator O that commutes with the particle-number operator 4A. We want to evaluate matrix elements of the operator O between, possibly different, many-body states generically defined through Eq. (4.1), i.e.

Ψ A c |O|Ψ A d = Ψ c |OP A |Ψ d Ψ c |P A |Ψ c Ψ d |P A |Ψ d , (4.29a) = Φ|C † OP A D|Φ Φ|C † P A C|Φ Φ|D † P A D|Φ . (4.29b)
We wish that the computation of the matrix element (4.29) is agnostic with respect to the origin of the coefficients defining the two states through the quasi-particle excitation operators C and D, i.e., it is applicable to any of the cases discussed above. Thus, our goal is to design a systematic method to compute Eq. (4.29) independently of the origin of C and D.

Diagrammatic method 4.4.1. Off-diagonal kernels and transformed operators

Expanding the projection operator P A according to Eq. ( 1.20) allows one to rewrite Eq. (4.29) via

Ψ c |OP A |Ψ d = 1 2π 2π 0 dϕ e -iAϕ Ψ c |OR(ϕ)|Ψ d ≡ 1 2π 2π 0 dϕ e -iAϕ O cd (ϕ) Φ|Φ(ϕ) , ( 4.30)
and

Ψ c |P A |Ψ c = 1 2π 2π 0 dϕ e -iAϕ Ψ c |R(ϕ)|Ψ c ≡ 1 2π 2π 0 dϕ e -iAϕ N cc (ϕ) Φ|Φ(ϕ) , ( 4.31)
where the generalized off-diagonal operator and norm kernels are defined according to

O cd (ϕ) ≡ Ψ c |OR(ϕ)|Ψ d Φ|Φ(ϕ) , (4.32a) N cc (ϕ) ≡ Ψ c |R(ϕ)|Ψ c Φ|Φ(ϕ) . (4.32b)
In O cd (ϕ), the rotation operator R(ϕ) can be brought to the right

O cd (ϕ) = Φ|C † OR(ϕ)D|Φ Φ|Φ(ϕ) = Φ|C † OD R (ϕ)|Φ(ϕ) Φ|Φ(ϕ) (4.33)
where the rotated operator D R (ϕ) is given by

D R (ϕ) ≡ R -1 (ϕ)DR(ϕ) . ( 4
O cd (ϕ) = Φ|C † OD R (ϕ)e Z(ϕ) |Φ = Φ|(C † ) Z (ϕ)O Z (ϕ)D RZ (ϕ)|Φ (4.35)
where Φ|e Z(ϕ) = Φ| was used 5 given that Z(ϕ) is a pure excitation operator. The similarity-transformed operators 6 entering the diagonal matrix element on the right-hand side of Eq. (4.35) take the generic form

O Z (ϕ) ≡ e -Z(ϕ) Oe Z(ϕ) . (4.36)
The same transformation can be applied to the norm kernel to give, e.g.

N cc (ϕ) = Φ|C † Z (ϕ)C RZ (ϕ)|Φ . (4.37)
Equation ( 4.35) has achieved the goal of commuting the off-diagonal matrix element of the initial product of operators into the diagonal matrix element of a product of operators depending on the gauge angle. In the present context, all operators at play are expressed in terms of quasi-particle creation and annihilation operators and are in normal-ordered form with respect to the Bogoliubov vacuum |Φ . Taking as an example a typical such operator of the form

O ij ≡ 1 i! 1 j! k 1 ...k i+j O ij k 1 ...k i+j β † k 1 . . . β † k i β k i+j . . . β k i+1 , ( 4.38)
its transformed partner reads as 7

O (ij) Z (ϕ) ≡ e -Z(ϕ) O ij e Z(ϕ) = 1 i! 1 j! k 1 ...k i+j O ij k 1 ...k i+j ( β † Z ) k 1 (ϕ) . . . ( β † Z ) k i (ϕ) (β Z ) k i+j (ϕ) . . . (β Z ) k i+1 (ϕ) , (4.39)
while its rotated and transformed partner reads as 5 Here the right convention for the rotated state is used, see Tab. 2.1 for correspondence. 6 At this point one has to notice that the hermitic character of an operator O is lost by the application of the thouless transformation. Furthermore the action of taking the adjoint of an operator does not commute with this transformation. In the following, the notation C † Z (ϕ) ≡ (C † ) Z (ϕ) always means that the thouless transformation is applied on the adjoint of the many-body creation operator C. 7 The notation O

O (ij) RZ (ϕ) ≡ e -Z(ϕ) O (ij) R (ϕ)e Z(ϕ) ≡ e -Z(ϕ) R -1 (ϕ)O ij R(ϕ)e Z(ϕ)
1 i! 1 j! k 1 ...k i+j O ij k 1 ...k i+j ( β † RZ ) k 1 (ϕ) . . . ( β † RZ ) k i (ϕ) (β RZ ) k i+j (ϕ) . . . (β RZ ) k i+1 (ϕ) , (4.40)
where transformed quasi-particle operators, resp. rotated and transformed quasi-particle operators, are expressed in terms of the original set of quasi-particle operators through a non-unitary, gauge-angle dependent, Bogoliubov transformation8 .

The next step consists in reexpressing all transformed operators in terms of the original set of quasi-particle operators and normal ordering them with respect to |Φ . In the end, one obtains

O (ij) Z (ϕ) ≡ i+j m=i j n=0 m+n≤i+j 1 m! 1 n! k 1 ...k m+n (O mn(ij) Z ) k 1 ...k m+n (ϕ) β † k 1 . . . β † k m β k m+n . . . β k m+1 , (4.41)
which defines a sum of normal-ordered terms. Each term has at least as many creation operators as the original operator (i) and possibly up to the total number of original quasiparticle operators (i + j). The number of annihilation operators ranges from 0 to the original number (j) such that the overall number of quasiparticle operators is bound to remain between i and i + j in each term. Once the normal ordering has been performed, we notice that the main conceptual difference between original and transformed operators relates to the fact that matrix elements of the latter depend on the gauge angle. Of course, the original operator is recovered in the unrotated limit, i.e. O Z (0) = O. This can also be done in the rotated and transformed case for which one obtains

O (ij) RZ (ϕ) ≡ i+j n=j i m=0 m+n≤i+j 1 m! 1 n! k 1 ...k m+n (O mn(ij) RZ ) k 1 ...k m+n (ϕ) β † k 1 . . . β † k m β k m+n . . . β k m+1 , (4.42)
defining a sum of normal-ordered terms. This time, each term has at least as many annihilation operators as the original operator (j) and possibly up to the total number of original quasiparticle operators (i + j). The number of creation operators ranges from 0 to the original number (i) such that the overall number of quasiparticle operators is bound to remain between j and i + j in each term. The original operator is recovered in the unrotated limit, i.e. O RZ (0) = O.

Applying the procedure to a generic, possibly non-hermitian, operator O provides

O Z (ϕ) ≡ O [0] Z (ϕ) + O [2] Z (ϕ) + O [4] Z (ϕ) + O [6] Z (ϕ) (4.43a) = (O 00 Z )(ϕ) + 1 1! k 1 k 2 (O 11 Z ) k 1 k 2 (ϕ)β † k 1 β k 2 + 1 2! k 1 k 2 (O 20 Z ) k 1 k 2 (ϕ)β † k 1 β † k 2 + (O 02 Z ) k 1 k 2 (ϕ)β k 2 β k 1 + 1 (2!) 2 k 1 k 2 k 3 k 4 (O 22 Z ) k 1 k 2 k 3 k 4 (ϕ)β † k 1 β † k 2 β k 4 β k 3 + 1 3! k 1 k 2 k 3 k 4 (O 31 Z ) k 1 k 2 k 3 k 4 (ϕ)β † k 1 β † k 2 β † k 3 β k 4 + (O 13 Z ) k 1 k 2 k 3 k 4 (ϕ)β † k 1 β k 4 β k 3 β k 2 + 1 4! k 1 k 2 k 3 k 4 (O 40 Z ) k 1 k 2 k 3 k 4 (ϕ)β † k 1 β † k 2 β † k 3 β † k 4 + (O 04 Z ) k 1 k 2 k 3 k 4 (ϕ)β k 4 β k 3 β k 2 β k 1 + . . . . (4.43b)
The expressions of the transformed matrix elements in terms of the original matrix elements of the operator are provided in App. B for the left definition of the rotated vacuum, see Tab. 2.1 for correspondence. It appears that each final normal-ordered term O mn Z (ϕ) potentially receives contributions from all the original terms O ij . While the expressions of the rotated and transformed matrix elements in terms of the original matrix elements of the operator have been derived, they are not provided in this document to limit its length. 1. The factor K ij k 1 ...k i k i+1 ...k i+j must be associated to a vertex, of a particular type, where i denotes the number of lines traveling out of the vertex and representing quasiparticle creation operators while j denotes the number of lines traveling into the vertex and representing quasiparticle annihilation operators.

Diagrammatic representation of operators

2. A factor 1/[i!j!] must multiply K ij k 1 ...k i k i+1 .
..k i+j given that the corresponding diagram contains j equivalent ingoing lines and i equivalent outgoing lines. i.e. lines representing quasiparticle creation (annihilation) operators appear above (below) the vertex. Accordingly, indices k 1 . . . k i must be assigned consecutively from the leftmost to the rightmost line above the vertex, while k i+1 . . . k i+j must be similarly assigned consecutively for lines below the vertex. As D RZ (ϕ) always corresponds to the bottom vertex we only consider lines going up, i.e. quasi-particle creation operators. In the same way C † Z (ϕ) always corresponds to the top vertex and so we only consider lines going down, i.e. quasi-particle annihilation operators. The rules to build and compute those diagrams are now detailed.

O [0] Z (ϕ) = O [2] Z (ϕ) = + + O [4] Z (ϕ) = + + + + O [6] Z (ϕ) = + + + + + +
1. Vacuum-to-vacuum, i.e. closed, diagrams consist of the three vertices

C †i 1 j 1 Z (ϕ), O i 2 j 2 Z (ϕ) and D i 3 j 3
RZ (ϕ), placed from top to bottom in that same order, connected, if possible, by fermionic quasi-particle lines, i.e. elementary propagators R -+ , see Eq. 1.64, forming a set of closed loops.

Each line is labeled by two quasi-particle indices and contributes a factor

R -+ k 1 k 2 = δ k 1 k 2 . 3. Each vertex contributes a factor K ij k 1 ...k i k i+1 .
..k i+j with the sign convention detailed in Sec. 4.4.2.

4. The contributions to O cd (ϕ) are generated by drawing all possible vacuum-to-vacuum diagrams. This is done by contracting the quasi-particle lines attached to the vertices in all possible ways.

5. All quasi-particle labels must be summed over.

6. A sign factor (-1) n c , where n c denotes the number of crossing lines in the diagram, must be considered. The overall sign results from multiplying this factor with the sign associated with each vertex factor.

7. Each diagram comes with a numerical prefactor 1/(n e )! for each group of n e equivalent lines. Equivalent lines must all begin and end at the same vertices.

Two "selection rules" limit drastically the number of non-zero diagrams.

1. Only trivial propagators R -+ need to be considered. Whenever a string of operators contains different numbers of creation and annihilation operators, the result is necessarily zero, i.e. for an arbitrary matrix element Φ|C †i

1 j 1 Z (ϕ)O i 2 j 2 Z (ϕ)D i 3 j 3 RZ (ϕ)|Φ to give non-zero contributions (diagrams), it is mandatory that n a ≡ 3 k=1 (j k -i k ) = 0.
2. No trivial contraction of a vertex onto itself is to be considered.

Low-rank examples

In the present section, C and D are limited to span up to doubles, i.e. four quasi-particle excitations. This is what defines "low-rank" in these examples. Furthermore, the operator O is limited to be, at most, of two-body character. Processing the diagrammatic rules for this example leads to 15 different contributions to O cd (ϕ). The first eight contributions are basically the same as the ones of second-order BMBPT, see Fig.

Discussion

Once off-diagonal kernels are calculated, observables with or without EC can be trivially accessed, see Sec. 4.4.1. Results without particle-number projection can be obtained by setting ϕ = 0 in all diagrams and equations. This allows one to bypass the steps to similarity transform and rotate the operator and corresponds to removing the projector from the outset in Eq. (4.6).

Higher-rank diagrams and their expressions can be obtained by extending the capabilities of the ADG program [START_REF] Arthuis | ADG: Automated generation and evaluation of many-body diagrams I. Bogoliubov many-body perturbation theory[END_REF]. This is envisioned in the future. As a matter of fact, the first numerical implementations will include up to triple, i.e. six quasi-particle, excitations in the definition of C and D.

The cost of this approach, for two-body observables and many-body states up to disconnected doubles, is driven by the diagram OCD2.8(ϕ). The evaluation of this diagram scales as N 6 , where N is the number of one-body states, which is the typical cost of BMBPT at second order.

Conclusion

A systematic method for the computation of matrix elements of operators between manybody states expressed in quasi-particle basis has been designed. This method is able to compute expectation value of observables, transition matrix elements and provides the necessary tools to design EC for BMBPT. The implementation of this scheme (without projection) to perform ab initio calculations of various observables in semi-magic nuclear ground-states is underway. Theses developments include up to triple, six quasi-particle, excitations in the definition of the operators C and D. Via EC, this gives the promise to

Introduction

The reach of ab initio many-body theories is rapidly extending over the nuclear chart. The computational load is particularly acute due to the relevance of three-nucleon interactions, knowing that including even more demanding four-body forces could eventually be mandatory [START_REF] Bazak | Four-Body Scale in Universal Few-Boson Systems[END_REF]. As a matter of fact, dealing with three-nucleon interactions in full makes the solving of the A-body Schrödinger equation rapidly too costly as the mass of the system grows. To circumvent the explicit treatment of three-body operators, ab initio calculations of mid-mass nuclei have been performed on the basis of the so-called normal-ordered 2-body (NO2B) approximation. This approximation captures the dominant effects of three-nucleon forces while effectively working with two-body operators. In large-scale no-core shell model (NCSM) calculations, the error induced by the NO2B approximation to the Hamiltonian was estimated to be of the order of 1-3% [START_REF] Roth | Medium-Mass Nuclei with Normal-Ordered Chiral N N + 3 N Interactions[END_REF][START_REF] Gebrerufael | Open-shell nuclei and excited states from multireference normal-ordered Hamiltonians[END_REF] up to the oxygen region.

The NO2B approximation consists of normal ordering the operator with respect to a many-body reference state and discarding the normal-ordered three-body component. While typically formulated with respect to an uncorrelated reference state, i.e. a Slater determinant, normal-ordering techniques and associated NO2B approximations can be also be devised with respect to a correlated reference state [START_REF] Kutzelnigg | Normal order and extended Wick theorem for a multiconfiguration reference wave function[END_REF], as is employed in the multireference IMSRG approach [START_REF] Hergert | The In-Medium Similarity Renormalization Group: A novel ab initio method for nuclei[END_REF][START_REF] Hergert | In-Medium Similarity Renormalization Group Approach to the Nuclear Many-Body Problem[END_REF], or even with respect to the fully correlated solution of the problem as is done in SCGF theory [START_REF] Carbone | Self-consistent Green's functions formalism with three-body interactions[END_REF]. In any case, the NO2B approximation has been employed so far on the basis of symmetry-conserving reference states. Only recently such an approximation has been employed in Gorkov SCGF [START_REF] Somà | Ab initio self-consistent Gorkov-Green's function calculations of semimagic nuclei: Formalism at second order with a twonucleon interaction[END_REF][START_REF] Somà | Ab initio Gorkov-Green's function calculations of open-shell nuclei[END_REF][START_REF] Somà | Ab initio self-consistent Gorkov-Green's function calculations of semi-magic nuclei: Numerical implementation at second order with a two-nucleon interaction[END_REF] and BMBPT [START_REF] Tichai | Bogoliubov many-body perturbation theory for open-shell nuclei[END_REF][START_REF] Tichai | Pre-processing the nuclear many-body problem[END_REF] in which U(1) symmetry associated with particle-number conservation is spontaneously broken by the (approximate) many-body state. In this context, proceeding to a naive truncation may lead to approximating a particle-number-conserving operator by a particlenumber-breaking one. A similar situation shall occur when using a SU(2)-breaking, i.e. deformed, reference state such that proceeding to a naive normal-ordered truncation of a rotationally invariant operator may lead to an angular-momentum non-conserving approximation.

The purpose of the present work is thus to design a general normal-ordering approximation of operators that is consistent with symmetries of the Hamiltonian while working with symmetry broken (and restored) reference state1 . Focusing on many-body formalisms in which U(1) symmetry associated with particle number conservation is broken (and potentially restored), a particle-number-conserving normal-ordered k-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 is designed and employed to check the particle-number conserving/violating character of the approximate operator.

Approximation

The standard NO2B approximation amounts to employing the full Hamiltonian to generate a mean-field Slater determinant reference state |SD and to truncating the normal-ordered form of H with respect to |SD at the effective two-body level for the beyond mean-field step. Thus, the NO2B approximation of a three-body operator O commuting with A reads as [6] = O [0] + O [2] + O [4] ,

O NO2B ≡ O -O
(

where all terms containing a normal-ordered product of six creation and annihilation operators have been discarded. A graphical representation of this approximation is given in Fig. 5 For O = H the naive extension of the NO2B approximation would imply working with a Hamiltonian H NO2B whose exact eigenstates are not eigenstates of A. While this can be considered as part of a systematic error induced by the approximation, it eventually leads, when restoring the symmetry of the approximate wave function, to energies that problematically depend on the particular way the symmetry restoration is formulated. This key feature is discussed and illustrated later on.

Naive extension

In this context, it is desirable to generalize the NOkB approximation of an arbitrary N-body operator in such a way that the truncated operator is particle-number conserving even if the reference state breaks U (1) symmetry.

Particle-number-conserving NOkB approximation

In the following, O denotes a normal-ordered N -body operator such that its naive NOkB (nNOkB) approximation, k < N , reads as

O NOkB ≡ k n=0 O [2n] .
(5.

3)

The nNOkB approximation suffers from the same pathology as the naive NO2B approximation regarding particle-number violation. One now wishes to design an approximation to O fulfilling the following three requirements 1. All normal-ordered terms of ranks higher than k must be discarded as the practical goal of normal-ordered approximations is to eventually work with an effective operator characterized by a maximum rank k < N .

2. The approximate operator must commute with A.

3. The error generated by the approximation must be minimal.

While the nNOkB approximation fulfills the first condition, it violates the second by fully retaining the normal-ordered contributions with ranks lower or equal than k. One can thus anticipate that fulfilling the second condition in addition to the first one requires to further approximate specific parts of the operator displayed in Eq. 5.3. In the following, a systematic procedure to achieve this goal is devised for any N and any k < N . It must be noted that there is no unique way to do so2 . Eventually, while the third condition may be anticipated based on reasonable arguments, it can solely be validated through benchmark calculations.

The particle-number-conserving normal-ordered k-body (PNOkB) approximation to O is given by

O PNOkB ≡ k n=0 õnn , ( 5.4)
where the n-body part is given in a single-particle basis by

õnn ≡ 1 n!n! l 1 ...l 2n õnn l 1 ...l 2n c † l 1 . . . c † l n c l 2n . . . c l n+1 , ( 5.5)
so that O PNOkB is manifestly particle-number conserving by construction. The n-body matrix elements õnn l 1 ...l 2n , n ≤ k, are recursively defined in decreasing order, from n = k down to n = 0, by

õkk l 1 ...l 2k ≡ Λ kk l 1 ...l 2k õnn l 1 ...l 2n ≡ Λ nn l 1 ...l 2n - k m=n+1
Λnn(mm) l 1 ...l 2n for n < k (5.6) where Λnn(mm) , m > n, is the m-body contribution to the n-body normal field associated with O PNOkB defined in the same way as the m-body contribution to the n-body normal field Λ nn(mm) associated with the full operator O introduced in Eq. (1.80).

It is clear that O PNOkB contains, through the fields Λ nn , n ≤ k, contributions from o nn up to n = N . It is the usual benefit of a normal-ordered approximation to capture the dominant part of all contributions to the original operator while working with effective operators of lower ranks.

To better appreciate the content of O PNOkB , let us consider its normal-ordered form in the single-particle basis. As proven in App. F.1, O PNOkB is obtained from O via a two-step process, i.e. by 1. removing all Λ ij fields with max(i, j) > k, 2. adding Λij defined below for max(i, j) ≤ k.

This leads to rewriting the approximate operator as

O PNOkB = max(i,j)≤k i,j=0 Λij , ( 5.7)
with

Λij ≡ Λ ij + Λij , ( 5.8)
where the extra term Λij is given by Λij

l 1 ...l i+j ≡ 1 i-j 2 ! 1 2 i-j 2 N n=i n-i 2 n κ =0   1 - 1 n κ + i-j 2 n κ    × l i+j+1 ...l 2i Λ ii(nn)(n κ ) l 1 ...l 2i - Λii(nn)(n κ ) l 1 ...l 2i κ l i+j+1 l i+j+2 . . . κ l 2i-1 l 2i (5.9) for i > j, by Λij l 1 ...l i+j ≡ 1 j-i 2 ! 1 2 j-i 2 N n=j n-j 2 n κ * =0   1 - 1 n κ + j-i 2 n κ    × l i+j+1 ...l 2j Λ jj(nn)(n κ * ) l 1 ...l i l i+j+1 ...l 2j l i+1 ...l i+j - Λjj(nn)(n κ * ) l 1 ...l i l i+j+1 ...l 2j l i+1 ...l i+j κ * l i+j+1 l i+j+2 . . . κ * l 2j-1 l 2j
(5.10)

for i < j and by Λij l 1 ...l i+j ≡ 0 (5.11)

for i = j. In the end, Λij is non-zero only if i -j ≥ 2 and n κ ≥ 1, or j -i ≥ 2 and n κ * ≥ 1. This is only possible for an initial N -body operator O with N ≥ 4, i.e. up to an initial three-body operator, no extra term is to be considered and the retained fields Λij are trivially given by the original fields Λ ij . The leading criterion behind the definition provided in Eqs. (5.4-5.6) is to ensure that O PNOkB shares the same normal fields as O for n ≤ k, i.e. Λnn = Λ nn , (5.12)

as those are believed to be the dominant contributions to the normal-ordered operator. This requirement possibly induces extra contributions Λij to the anomalous fields. This feature relates to the fact that Λij is obtained from the original operator through two successive normal orderings steps rather than one for Λ ij . When proceeding to anomalous contractions in the application of Wick's theorem, it possibly leads to the non-equivalent combinatorial prefactors

(n κ 1 + n κ 2 )! = n κ 1 !n κ 2 ! if n κ 1 ≥ 1 and n κ 2 ≥ 1 , (5.13)
in both procedures. Eventually, and as best illustrated through the examples worked out in App. F.1, the effect of the correction terms defined through Eqs. (5.9-5.11) is nothing but to modify the numerical prefactors of specific contributions to the normal-ordered anomalous fields expressed in the single-particle basis. Noticing that plainly omitting Λ ij with max(i, j) > k can also be viewed as the mere replacement of its original prefactor by an approximate one (0 in such cases), the function of the extra terms Λij is in fact not different, i.e. it corresponds to modifying the prefactors of a specific set of fields such that the initial objectives are fulfilled. The quantitative performance of the procedure can only be judged a posteriori by comparing many-body results obtained using the full operator and its PNOkB approximation.

PNO1B approximation of a two-body operator

Let us exemplify the PNOkB procedure by building the PNO1B approximation of the two-body operator

O = o 00 + o 11 + o 22 .
( where each contribution, in fact denoting matrix elements of the fields, is written in a compact form without indices such that the key informations, i.e. the original o nn term, the set of elementary contractions involved and the overall prefactor, can be easily extracted. The nNO1B approximation leads to just dropping O [2] = Λ 22 in Eq. (5.15). Using properties of elementary contractions

c † l 1 c † l 2 = : c † l 1 c † l 2 : + κ * l 1 l 2 , (5.17a) c † l 1 c l 2 = : c † l 1 c l 2 : + ρ l 2 l 1 , (5.17b) c l 2 c l 1 = : c l 2 c l 1 : + κ l 1 l 2 , ( 5.17c)
one can write

O nNO1B ≡ Λ 00 + Λ 20 + Λ 11 + Λ 02 = Λ 00 + 1 2!0! l 1 l 2 Λ 20 l 1 l 2 : c † l 1 c † l 2 : + 1 1!1! l 1 l 2 Λ 11 l 1 l 2 : c † l 1 c l 2 : + 1 0!2! l 1 l 2 Λ 02 l 1 l 2 : c l 2 c l 1 : ≡ õ00 + 1 2!0! l 1 l 2 õ20 l 1 l 2 c † l 1 c † l 2 + 1 1!1! l 1 l 2 õ11 l 1 l 2 c † l 1 c l 2 + 1 0!2! l 1 l 2 õ02 l 1 l 2 c l 2 c l 1 = õ00 + õ20 + õ11 + õ02 , ( 5.18)
where the matrix elements are given by õ00 ≡ Λ 00 - (5.19d) Equations (5.18-5. [START_REF] Shavitt | Many-body methods in chemistry and physics : MBPT and coupled-cluster theory[END_REF]) explicitly demonstrate that the nNO1B approximation leads to an operator that does not conserve particle number as it contains non-zero two-particleaddition õ20 and two-particle removal õ02 contributions. The application of the PNO1B approximation is more involved and leads to the construction of the corresponding operator

1 2 Tr[Λ 20 κ * ] -Tr[Λ 11 ρ] - 1 2 Tr[Λ 02 κ] = o 00 - 1 2 Tr[o 22 ρρ] - 1 4 Tr[o 22 κ * κ] , (5.19a
O PNO1B ≡ õ00 + õ11 , ( 5.20)
where the terms are to be obtained recursively on the basis of Eq. (5.6). The procedure starts with õkk = Λ kk , which is normal-ordered in the single-particle basis to generate õ(k-1)(k-1) etc. until õ00 is reached. In the present case, the procedure is trivial since it starts with k = 1 and leads directly to õ00 in the first step. Eventually, the result reads as

õ11 ≡ Λ 11 = o 11 + Tr[o 22 ρ] , ( 5

Other applications

The PNOkB approximation procedure is worked out in details for three other cases in App. F.1.2, i.e.

1. the PNO2B approximation of a 3-body operator, 2. the PNO2B approximation of a 4-body operator,

the PNO3B approximation of a 4-body operator,

The first case applies to state-of-the-art ab initio calculations [START_REF] Tichai | Bogoliubov many-body perturbation theory for open-shell nuclei[END_REF][START_REF] Tichai | Pre-processing the nuclear many-body problem[END_REF]. The second case illustrates how the extra terms Λij come into play for the first time when starting from a 4-body operator and may become of practical interest in the future in case four-nucleon interactions have to be accounted for at the effective two-body level. If one becomes capable of working at the effective three-body level, the third case provides the PNO3B approximation of a 4-body operator.

Finally, the main outcomes of the PNO3B and PNO4B approximations of a 5-body operator are briefly compared in App. F.1.2 to illustrate how the extra terms depend on the rank k used to approximate a given original N -body operator.

Testing [O (P)NOkB , A] = 0

A method is now introduced to test (formally and numerically) whether or not a given operator F is commuting with the particle-number operator A. Starting from a particlenumber-conserving operator O, the test is meant to be applied to both3 F ≡ O nNOkB and O PNOkB .

Particle-number projection

The test is based on the fact that, if F commutes with A, it must also commute with the particle-number projection operator P A (Eq. (1.20)). Given that P A is idempotent, i.e. (P A ) 2 = P A , and hermitian , i.e. P A † = P A , the commutation of F with P A can be re-expressed as

P A F P A † = P A F .
(5.25)

Based on the above, the ratio of singly over doubly projected mean-field matrix elements

Q A F ≡ Φ|P A F |Φ Φ|P A F P A † |Φ , ( 5.26)
is formed such that the particle-number conserving (violating) character of F corresponds to Q A F = 1( = 1).

Computation of Q A

F

The computation of Q A F relies on the representation of P A given in Eq. (1.20) such that

Q A F = 2π 0 dϕ 2π e -iϕA f (0) (ϕ) N (0) (ϕ) 2π 0 dϕ 2π 2π 0 dϕ 2π e -i(ϕ-ϕ )A f (0) (ϕ, ϕ ) N (0) (ϕ, ϕ ) , ( 5.27)
where singly-and doubly-rotated mean-field off-diagonal norm and connected operator kernels are given by

N (0) (ϕ) ≡ Φ(ϕ)|Φ , ( 5.28a
)

N (0) (ϕ, ϕ ) ≡ Φ(ϕ)|Φ(ϕ ) = N (0) (ϕ -ϕ ) , ( 5.28b)
and by

f (0) (ϕ) ≡ Φ(ϕ)|F |Φ Φ(ϕ)|Φ , ( 5.29a
)

f (0) (ϕ, ϕ ) ≡ Φ(ϕ)|F |Φ(ϕ ) Φ(ϕ)|Φ(ϕ ) , ( 5.29b)
where the gauge-rotated Bogoliubov state is defined through Φ(ϕ)| ≡ Φ|R(ϕ). Clearly, one has that

f (0) (ϕ) = f (0) (ϕ, 0) .
(5.30)

One-body operator

The numerical illustrations provided in the present chapter rely on the (P)NO1B of a two-body nuclear Hamiltonian, i.e. O ≡ H. Let us thus characterize Q A F for a generic, i.e. possibly particle-number violating, one-body operator

F ≡ f 00 + f 20 + f 11 + f 02 (5.31) = f 00 + 1 2 l 1 l 2 f 20 l 1 l 2 c † l 1 c † l 2 + l 1 l 2 f 11 l 1 l 2 c † l 1 c l 2 + 1 2 l 1 l 2 f 02 l 1 l 2 c l 2 c l 1 .
(5.32)

A graphical representation of the operator F is given in Tab. 5.1. Starting from a two-body operator O, the explicit expression of the matrix elements (f 00 , f 20 l 1 l 2 , f 11 l 1 l 2 , f 02 l 1 l 2 ) associated with O nNO1B were provided in Sec. 5.2.3, where it was formally proven that the particlenumber non-conserving terms f 20 and f 02 are indeed non zero in this case. For O PNO1B , the sole non-zero terms are f 00 and f 11 .

{c, c † }, |0 -6 -4 -2 0 +2 +4 +6 f [0] f 00
f [2] f 02 f 11 f 20 f [4] f [6] Table 5.1. Contributions to the one-body operator F in normal-ordered form with respect to the particle vacuum |0 and expressed in {c, c † }. The f ij contributions are sorted horizontally according to i -j and vertically according to i + j.

By virtue of the off-diagonal Wick's theorem [START_REF] Balian | Nonunitary bogoliubov transformations and extension of Wick's theorem[END_REF], singly-and doubly-rotated connected operator kernels write respectively as

f (0) (ϕ) = f 00 + 1 2 l 1 l 2 f 20 l 1 l 2 κ * l 1 l 2 (ϕ) + l 1 l 2 f 11 l 1 l 2 ρ l 2 l 1 (ϕ) + 1 2 l 1 l 2 f 02 l 1 l 2 κ l 1 l 2 (ϕ) , (5.33)
≡ f 00 (ϕ) + f 20(0) (ϕ) + f 11(0) (ϕ) + f 02(0) (ϕ) , and as (5.34) where singly and doubly gauge-rotated contractions are defined and calculated in App. F.2.1.

f (0) (ϕ, ϕ ) = f 00 + 1 2 l 1 l 2 f 20 l 1 l 2 κ * l 1 l 2 (ϕ, ϕ ) + l 1 l 2 f 11 l 1 l 2 ρ l 2 l 1 (ϕ, ϕ ) + 1 2 l 1 l 2 f 02 l 1 l 2 κ l 1 l 2 (ϕ, ϕ ) = f 00 +   l 1 l 2 f 11 l 1 l 2 ρ l 2 l 1 (ϕ -ϕ )   +   1 2 l 1 l 2 f 20 l 1 l 2 κ * l 1 l 2 (ϕ -ϕ )   e +2iϕ +   1 2 l 1 l 2 f 02 l 1 l 2 κ l 1 l 2 (ϕ -ϕ )   e -2iϕ ,
Using the change of variables φ ≡ ϕ -ϕ , f (0) (ϕ, ϕ ) can be expressed as

f (0) (ϕ, ϕ ) = f 00 + f 20(0) (φ) e +i2ϕ + f 11(0) (φ) + f 02(0) (φ) e -i2ϕ , (5.35)
i.e., it displays a Fourier decomposition in ϕ whose components are nothing but the singly gauge-rotated kernels associated with the various contributions to F . The appearance of non-trivial Fourier components, i.e. irreducible representation of U (1), constitutes a fingerprint of the particle-number non-conserving character of F . In the present case, two such non-trivial modes appear in connection with f 20 and f 02 . This result can obviously be extended to higher-body operators. Considering a general operator F containing arbitrary combinations of (an even number of) single-particle creation and annihilation operators, its doubly gauge-rotated mean-field connected kernel takes the form

f (0) (ϕ, ϕ ) ≡ f 00 + m-n ∈ 2Z f mn(0) (φ) e i(m-n)ϕ , ( 5
Q A F = 1 + Φ|P A f 20 |Φ Φ|P A f 11 |Φ + Φ|P A f 02 |Φ Φ|P A f 11 |Φ , ( 5.37)
which is (a priori not) equal to 1 when the particle non-conserving parts of F are (non) vanishing.

Results

Calculations set up

The nuclear Hamiltonian used in this work includes a chiral two-nucleon (2N) interaction at next-to-next-to-next-to leading order with a cutoff of Λ 2N = 500 MeV [START_REF] Entem | Accurate charge-dependent nucleon-nucleon potential at fourth order of chiral perturbation theory[END_REF]. The three-nucleon interaction is omitted to be able to compare results obtained with the full Hamiltonian to those obtained via its nNO1B and PNO1B approximations. The Hamiltonian is further softened using a similarity renormalization group (SRG) transformation with a flow parameter α = 0.08 fm4 [START_REF] Bogner | Similarity renormalization group for nucleon-nucleon interactions[END_REF][START_REF] Hergert | Unitary correlation operator method from a similarity renormalization group perspective[END_REF][START_REF] Roth | Unitary correlation operator method and similarity renormalization group: Connections and differences[END_REF][START_REF] Roth | Similarity-Transformed Chiral N N + 3 N Interactions for the Ab initio Description of C 12 and O 16[END_REF][START_REF] Jurgenson | Structure of p -shell nuclei using three-nucleon interactions evolved with the similarity renormalization group[END_REF] such that only up to transformed two-body operators are retained. Calculations are performed using the one-body eigenbasis of the spherical harmonic oscillator (SHO) Hamiltonian with frequency Ω = 20 MeV. One-and two-body operators are represented using all single-particle states up to e max = (2n + l) max = 10. Spherical HFB calculations are performed in J-coupled scheme while the (single and double) particlenumber projection (PHFB) is performed after the variation (PAV) on the basis of n int = 500 equally-spaced integration points over the interval 4 ϕ ∈ [0, π].

Energetics in O isotopes

Before coming to normal-ordered approximations, let us discuss the ground-state energetics obtained with the full Hamiltonian to set the orders of magnitude at play. Since three-body forces are presently discarded for the sake of the demonstration, computed energies are not meant to reproduce experimental data.

In Fig. 5.4, ground-state energetics along the Oxygen isotopic chain are displayed. Panel (a) provides HFB energies ranging from about -100 MeV in 14 O to about -220 MeV in 26 O. Panel (b1) displays static correlations associated with particle-number projection that are of the order5 of ±1MeV. This order of magnitude is to be compared with dynamical correlations displayed in panel (b2). Calculated via, e.g., BMBPT(2) [START_REF] Duguet | Symmetry broken and restored coupled-cluster. Global gauge symmetry and particle number[END_REF][START_REF] Tichai | Bogoliubov many-body perturbation theory for open-shell nuclei[END_REF] on the basis of the same Hamiltonian and model space, dynamical correlations range from -26 to -38 MeV in O isotopes. Eventually, both types of correlations can be captured consistently via PBMBPT [START_REF] Duguet | Symmetry broken and restored coupled-cluster. Global gauge symmetry and particle number[END_REF][START_REF] Ripoche | Projected Bogoliubov Many-Body Perturbation Theory[END_REF] or PBCC [START_REF] Duguet | Symmetry broken and restored coupled-cluster. Global gauge symmetry and particle number[END_REF][START_REF] Qiu | Particle-number projected Bogoliubov-coupled-cluster theory: Application to the pairing Hamiltonian[END_REF].

Results

(Q A H ) approx A O (a) (b1) (b2) (c1) (c2)
With results from the full Hamiltonian at hands, one can now analyze the effect of approximating it. Panel (c1) of Fig. 5.4 displays the difference of PHFB energies obtained with the two-body Hamiltonian and with its nNO1B or PNO1B approximations. While results are the same for single or double particle-number projections when using H PNO1B , it is not the case for H nNO1B . This observation is confirmed in Panel (c2) where the ratios

Q A H nNO1B and Q A H PNO1B are displayed. While Q A H PNO1B = 1 for all isotopes, Q A H nNO1B = 1
except in doubly closed-shell isotopes. This key result proves that H nNO1B is particlenumber violating, thus making PHFB energies dependent on the way the particle-number projection is defined. Contrarily, H PNO1B does behave as a particle-number conserving operator.

One further observes that the effect of the PNO1B approximation is significant, i.e., it is of the same order as the effect of the projection itself displayed in Panel (b1). This is not surprising given that ignoring the residual part of the two-body interaction has never been believed to be an appropriate approximation. In order to quantitatively gauge the quality of the PNOkB approximations, one must at least test the PNO2B approximation of a three-nucleon interaction, which is beyond the scope of the present chapter that is rather focusing on the symmetry violating/conserving character of a given approximation method.

Doubly gauge-rotated operator kernel

To further characterize the normal-ordered approximations, the doubly gauge-rotated mean-field connected Hamiltonian kernel h (0) (ϕ, ϕ ) is analyzed along with those associated with H nNO1B and H PNO1B . The real and imaginary parts of the kernels are shown in Fig. 5.5 for 18 O. They are displayed as contour plots with respect to the variables (φ = ϕ -ϕ , ϕ ).

Because H is particle-number conserving, its hamiltonian kernel solely depends on φ and is independent of ϕ , i.e., the Fourier expansion of h (0) (ϕ, ϕ ) with respect to ϕ in Eq. (5.36) only contains the trivial component, i.e., the irreducible representation of U (1), characterized by k = 0. While this feature is already manifest in the upper panels of Fig. 5.5, it is confirmed in Fig. 5.6 where the Fourier components are numerically extracted.

Performing the nNO1B approximation, the hamiltonian kernel displayed in the middle panels of Fig. 5.5 is obtained. It is clear that both the real and imaginary parts now vary with ϕ . This indicates that non-trivial components are present in the Fourier decomposition of Eq. (5.35) due to particle-number non-conserving contributions to H nNO1B . As visible from the middle panel of Fig. 5.6 non-zero Fourier components confirm the presence of f 20(0) (φ) and f 02(0) (φ), in agreement with the analytical derivation provided in Eqs. (5.33-5.36).

Moving from H nNO1B to H PNO1B , the hamiltonian kernel displayed in the bottom panels of Fig. 5.5 are obtained. The independence of the kernel with respect to ϕ is recovered, thus testifying of the particle-number conserving nature of the approximate Hamiltonian H PNO1B . This is confirmed in the lower panel of Fig. 5.6.

Results

f ij(0) (φ = π 2 ) [M ev] nNO1B -4 -2 0 2 4 k = i -j

Systematics

The analysis provided in Sec. 5.4.2 is now extended to Ca isotopes and to the variance operator. The energetics provided in Fig. 5.7 confirms the conclusions drawn earlier, i.e. while PHFB energies in O and Ca isotopes originating from H PNO1B are identical when using single or double projections, it is not the case for H nNO1B . It confirms that H nNO1B (H PNO1B ) is particle-number violating (conserving). HFB energy. Middle panel: static correlation energy brought about by the particle-number projection (after variation). Lower panel: difference between PHFB energies obtained with the full two-body Hamiltonian and with its nNO1B or PNO1B approximations. Energies are computed twice, i.e. using a single or a double particle-number projection.

As a curiosity, nNO1B and PNO1B approximations are further applied to the particlenumber variance operator. Corresponding results are displayed in Fig. 5.8. When using the full operator, PHFB calculations obviously deliver a null variance, independently of whether the single or the double projection is employed. Next, results obtained via the approximate nNO1B (PNO1B) one-body operator do (not) depend on the way the projection is performed, thus confirming that the approximate operator is particle-number violating (conserving).

Furthermore, results do depart significantly from the correct one in the PNO1B approximation, which is again not surprising given the expected crudeness of the one-body approximation to a two-body operator in general. Interestingly, nNO1B delivers essentially identical results to PNO1B when the single projection is employed. More surprisingly, nNO1B does provide essentially exact (i.e. null) results when the double projection is used. The reasons for this unexpected result are analyzed in App. F.2.3 where they are shown to

Conclusion

be specific to the one-body approximation of the particle-number variance operator and thus to be accidental.

Conclusion

In view of dealing efficiently with three-, possibly four-, nucleon interactions in ab initio calculations of open-shell nuclei, the present chapter addresses approximations based on normal-ordering techniques in the context of many-body methods in which the exact solution of the A-body Schrödinger equation is expanded around a symmetry-breaking reference state. Because a naive extension of the standard normal-ordered approximation, originally designed on the basis of symmetry-conserving states, may lead to symmetry-breaking approximate operators, a systematic approximation techniques delivering symmetry-conserving operators is necessary in this context. Focusing here on many-body formalisms in which U (1) global-gauge symmetry associated with particle number conservation is broken (and potentially restored), a particle-numberconserving normal-ordered k-body approximation (PNOkB) of an arbitrary N-body operator has been designed on the basis of Bogoliubov reference states. After laying down the general formalism, the explicit form of the approximate operator has been provided for various relevant combinations of N and k. Furthermore, numerical tests based on particle-number projected Hartree-Fock-Bogoliubov calculations have allowed to check (i) the particle-number violating character of a naive extension of the standard normal-ordered approximation and (ii) the particle-number conserving character of the newly designed PNOkB approximation. Using the PNOkB approximation, ab initio calculations based on formalisms exploiting the breaking and restoration of particle-number can thus be safely performed. The future formulation of an angular-momentum-conserving normal-ordered Chapter 6.

Symmetry reduction of tensor networks in many-body theory: Automated symbolic evaluation of SU (2) algebra

Introduction

While wave-function expansion methods are first formulated in terms of a generic singleparticle basis, their actual implementations typically exploit symmetry properties of the basis functions and of the targeted many-body state, e.g., with respect to angularmomentum, parity or even isospin projection quantum numbers. The adaptation of the generic formalism to a specific symmetry group defines a symmetry reduction of the manybody formalism. The goal is to use reduced many-body tensors associated to irreducible representations (IRREPs) of the symmetry group in order to pre-process a subset of the summations at play in the tensor networks defining the working equations.

In nuclear structure theory, this relates for example to the exploitation of rotational invariance associated with the conservation of total angular momentum and encoded in terms of the SU (2) non-abelian Lie group. In this particular case the reduction scheme will be referred to as the angular-momentum reduction.

Eventually, it turns out that the angular-momentum reduction poses a computationally non-trivial problem such that the derivation of the initial working equations requires the same amount of time as performing the angular-momentum reduction. However, there exists a highly systematic and elegant way to deal with this task through the use of the so-called Yutsis graphs [START_REF] Yutsis | Mathematical apparatus of the theory of angular momentum[END_REF], which is close in spirit to the use of Feynman's diagrams as a mnemonic device to represent contributions to physical observables.

Consequently, it is highly desirable to parallel the efforts done in automizing the generation of the original working equations [START_REF] Arthuis | ADG: Automated generation and evaluation of many-body diagrams I. Bogoliubov many-body perturbation theory[END_REF] by devising a framework that automatically performs the tedious symmetry reduction in an error-safe way. Currently, there is -to the best of our knowledge -no open-source library that can deal with the requirements imposed by nuclear structure many-body methods to perform symbolic manipulations of angular-momentum algebra. Typically, existing softwares are restricted to the numerical evaluation of coupling coefficients instead of performing symbolic manipulations including the simplification of complex tensor networks. There have been similar attempts for symbolic simplifications of angular-momentum expressions before without formally connecting to many-body theory [START_REF] Paul | Angular Momentum Diagrams[END_REF].

In this work, the first version of an automated tool performing graph-theory-based angular-momentum coupling is presented. Taking the symmetry-unrestricted expressions of a generic tensor network as an input, the code provides their angular-momentum-coupled form in an error-safe way in a matter of seconds. Several state-of-the-art many-body methods serve as examples to demonstrate the generality of the approach and to highlight the potential impact on the many-body community. Of course, this does not resolve the problem of writing an efficient, and most importantly error-free, implementation of the symmetry-restricted theory itself. While the generation of the source code itself is envisioned, it is, however, beyond the scope of the present thesis.

Symmetries and many-body theory 6.2.1. Symmetry group

Physical symmetries impact many-body formalisms at various stages of their elaboration. The existence of symmetries in finite systems is intimately connected to conservation laws, e.g., the existence of U (1) global gauge symmetry corresponds to particle-number conservation while SU (2) symmetry corresponds to angular momentum conservation. Mathematically, the invariance of a quantum system, characterized by its Hamiltonian H, is encoded in terms of transformation properties imposed by a symmetry group1 G whose action leaves the system invariant or, equivalently, the existence of a unitary linear representation U on the space of states such that

H = U (g)HU † (g) (∀g ∈ G) , ( 6.1)
which can be rewritten as

[H, U (g)] = 0 (∀g ∈ G) . (6.2)
Given eigenstates of the Hamiltonian

H|Ψ k = E k |Ψ k , ( 6.3)
Eqs. (6.1-6.2) stipulate that the transformed states

|Ψ k (g) ≡ U (g)|Ψ k (∀g ∈ G) , (6.4)
are also eigenstates with the same eigenvalue E k .

In the case of a discrete symmetry, such as parity or time reversal, the corresponding symmetry group is finite, e.g. Z 2 . Contrarily, continuous symmetries correspond to Lie groups allowing for a continuous parametrization of the (infinite number of) group elements in terms of a finite set of parameters. The present work focuses on the non-abelian SU (2) Lie group associated with the rotational invariance of nuclear systems. Relevant details about this symmetry group are provided in Sec. 6.3.3.

Symmetry groups enter the formulation of (nuclear) quantum many-body methods at three different levels (1) the symmetry group G ham of the Hamiltonian, specifying the invariance of the physical system under a given set of transformations along with the symmetry quantum numbers carried by its many-body eigenstates,

(2) the symmetry group G bas of the single-particle basis, specifying the symmetry properties of the computational basis,

(3) the symmetry group G ref of the reference state employed in the many-body expansion method of interest, specifying the symmetries of the auxiliary many-body problem that is solved to construct the reference vacuum.

While the symmetry group of the Hamiltonian is fixed by the target system under consideration, the symmetry properties of the single-particle basis and the reference state can be chosen freely such that various combinations of G bas and G ref can be employed.

Symmetries of the single-particle basis

Given H and its symmetry group G ham , there is infinitely many different single-particle basis spanning the one-body Hilbert space H 1 that can be used to represent the operator in second-quantized form. The single-particle basis functions are typically obtained as eigenstates of an auxiliary one-body Hamiltonian H bas whose symmetries are characterized by [H bas , U (g)] = 0 (∀g ∈ G bas ) . (6.5)

When choosing G bas = SU (2), one-body basis states are eigenstates of the squared total angular-momentum operator

J 2 ≡ J 2 x + J 2 y + J 2 z , ( 6.6)
where J x , J y and J z denote the Cartesian components of the total angular momentum vector. In most ab initio nuclear structure applications such a one-body basis is indeed employed, e.g., the eigenbasis of the three-dimensional spherical harmonic oscillator (SHO) Hamiltonian

H sHO = p 2 2m + 1 2 mω 2 r 2 , ( 6.7)
where m denotes the average nucleon mass and ω the HO frequency. It can be shown that

[H sHO , J 2 ] = 0 , (6.8a) [H sHO , J z ] = 0 , ( 6.8b)
such that the one-body eigenstates of H sHO are proportional to spherical Harmonics. In other frameworks, e.g. nuclear energy density functional calculations, the single-particle basis is possibly taken as eigenfunctions of the axially deformed HO hamiltonian that breaks rotational invariance and thus displays a smaller symmetry group G bas than H sHO .

Symmetries of the reference state

The rationale of expansion methods relies on the definition of a conveniently chosen A-body reference state |Φ that serves as starting point for the correlation expansion. Acting on the vacuum, the wave operator W yields the exact, e.g., ground state

|Ψ 0 = W |Φ . (6.9)
The wave operator is expanded and truncated according to a given many-body scheme, e.g., MBPT, SCGF or CC theory. The resulting equations are symmetry-unrestricted and therefore make no use of symmetry properties of many-body operators at play. In practice, the reference state is typically obtained as the, e.g., ground state of an 'unperturbed', i.e. zero-th order, Hamiltonian H ref capturing the average behavior of the system dynamics and charactetized by a symmetry group G ref

[H ref , U (g)] = 0 (∀g ∈ G ref ) , ( 6
E[|Φ G ref ] ≤ E[|Φ G ham ] . (6.12)
More importantly, this lowering is accompanied by a lifting of the degeneracy of |Φ G ham with respect to elementary excitations such that W can be expanded safely. In this case, however, the wave operator must not only capture dynamical correlations but also restore the symmetry G ham associated with the exact eigenstates of H. Because of the necessary truncation, a standard expansion of W is not capable of restoring the symmetry such that the symmetry contamination needs to be retrieve by the explicit inclusion of a symmetry projector in the definition of W , as it was already discussed in Chap. 2.

Reduction schemes and groups

A case of particular interest arises when the symmetry groups of the Hamiltonian, the single-particle basis and the reference state coincide, i.e.,

G sym ≡ G ham = G bas = G ref . (6.13)
In this setting, the common algebraic structure can be exploited to simplify the many-body formalism by expressing all working equations in terms of G sym -reduced tensors, thus potentially providing a tremendous gain in the required runtime and memory resources. In the present chapter, this situation is exploited relative to the SU (2) group (independently of the treatment of other symmetries such as U (1)).

Tensors and tensor networks

Due to the large variety of different expansion schemes used to represent the solution of the many-body problem, it is desirable to introduce a unifying language for the various frameworks at play. This common ground is provided by the language of tensors and tensor networks (TNs).

A mode-k symmetry-unrestricted tensor (SU-T), denoted as where the ellipses indicate indices that are not summed over2 . Consequently, tensors constitute the basic building blocks of various many-body formalisms 3 . A symmetry-unrestricted tensor network (SU-TN) denotes a set of SU-Ts combined according to a given contraction scheme specifying the way the tensors are contracted with each other. Furthermore, a SU-TN is said to be closed if all tensor indices are summed over and is said to be open otherwise.

T i 1 ...i k , ( 6
In many-body applications tensors typically appear in two broad classes (i) input tensors that are known prior to the actual solution of the Schrödinger equation in a given many-body framework, (ii) output tensors that are specific to a given many-body approach and are typically the objects being solved for.

Examples for input tensors are matrix elements of many-body operators, like the Hamiltonian, whereas examples of output tensors are CC amplitudes or dressed propagators in SCGF theory. In most non-perturbative many-body frameworks, like CC, IMSRG or SCGF, open TNs specify the working equations required to determine the unknown output tensors while the calculation of observables, e.g. the energy, relates to the evaluation of closed TNs.

Symmetry-reduced tensor networks

The goal of this work is to transform an initial SU-TN into a symmetry-reduced tensor network (SR-TN) incapsulating the symmetry reduction according to the associated symmetry group. To do so, the SU-Ts must be replaced by their symmetry-reduced counterparts. Given an initial SU-T, the corresponding symmetry-reduced tensor (SR-T) is obtained through a transformation f G sym

T k 1 ...k n f G sym -------→ Tk 1 ... kn , ( 6.16)
mitigating the symmetry reduction related to the group G sym . In the following, quantities with a tilde indicate symmetry-reduced objects 4 . The content of the indices themselves changes such that the set of quantum numbers labelling a SU-T and its SR-T counterpart are different. Thus, the SR-TN denotes the end product obtained via the replacement of the SU-Ts by their SR-T counterparts and via the adjustement of the contraction pattern

Angular-momentum algebra 6.3.1. Rationale

While the discussion on symmetry reduction and SR-TNs has been generic so far, the present work focuses on the SU (2) group. The goal is thus to obtain Angular-Momentum-Reduced Tensor Networks (AMR-TNs) from SU-TNs. The procedure requires to (i) replace all the SU-Ts by their AMR-T counterparts according to the transformation f SU (2) , (ii) constrain the contraction pattern to only be left with summations over the reduced set of quantum numbers.

In practice, step (i) involves a set of substitution rules for every many-body tensor at play that specify how the symmetry reduction is performed. The resulting AMR-TNand its computational complexity -may strongly depend on the choice made to perform this initial step 5 . From this point of view at least a minimum level of human input (and experience) is necessary to come up with the most convenient choice. This did not pose a severe limitation in any of the examples discussed below.

Other symmetries

While presently focusing on rotational symmetry, other symmetries can be exploited in the same way. A key example relates to intrinsic spin in quantum chemistry that is analogous to the total angular-momentum when using a ls-coupling scheme. The spin projection being only two-fold degenerate, i.e. m s = ± 1 2 , spin-restricted many-body theories benefit less from the symmetry reduction than in the j-coupling scheme that is at play in nuclear structure theory. Still, pre-processing the sums over spin projections is an important tool to reduce the computational cost and advance state-of-the-art expansion methods in strongly correlated electronic systems. Finite symmetry groups, e.g., the dihedral groups D n , may also arise in quantum molecules whereas cubic groups play an important role in the computation of homogeneous matter, e.g. the infinite electron gas or infinite nuclear matter, since periodic boundary conditions are employed to facilitate the calculation. In solid-state physics, symmetry properties of the many-body systems, e.g. helical symmetries in nano tubes, can also be exploited to reduce computational complexity.

All the aforementioned examples correspond to the algebraic reduction associated with exact symmetries of a many-body system. In recent years, exploiting emergent approximate symmetries has shown to be highly beneficial, in particular in the context of nuclear CIbased approaches. In this case, the symmetry group of the configuration basis G bas is larger than the actual symmetry group of the Hamiltonian, G ham ⊂ G bas , (6.18) thus exploiting algebraic properties that are not strictly realized in nature. A prime example is the symplectic symmetry group Sp(3) that is not an exact symmetry of the nuclear Hamiltonian but of the kinetic energy operator. In the symmetry-adapted no-core shell model (SA-NCSM), an A-body configuration basis is constructed from the Casimir operators of the approximate symmetry group SU (3) ⊂ Sp [START_REF] Van Kolck | Few-Nucleon Systems in a Quirky World[END_REF]. The use of symplectic algebra was shown to provide an efficient selection of many-body basis states, thus yielding computational savings in the diagonalization of the many-body Hamiltonian at the price of a more involved handling of many-body operators [START_REF] Dytrych | Ab initio symplectic no-core shell model[END_REF].

SU (2) group

In order to move closer to a concrete implementation of the above procedure, let us introduce details about the non-abelian compact SU (2) ≡ {R(Ω), Ω ∈ D SU (2) } Lie group associated with the rotation of a A-body fermion system characterized by an integer or a half-integer angular momentum. The group is parametrized by three Euler angles Ω ≡ (α, β, γ) whose domains of definition are

D SU (2) ≡ D α × D β × D γ = [0, 4π] × [0, π] × [0, 2π] . (6.19)
As SU (2) is considered to be a symmetry group of H, the commutation relations

[H, R(Ω)] = [T, R(Ω)] = [V, R(Ω)] = 0 , (6.20) hold for Ω ∈ D SU (2) .
Subsequently, the unitary representation of SU (2) on Fock space is utilized

R(Ω) = e -i αJ z e -i βJ y e -i γJ z . (6.21)
The components of the total angular-momentum vector make up the Lie algebra

[J i , J j ] = i ijk J k , ( 6.22)
where ijk denotes the Levi-Civita tensor. The Casimir operator J 2 of the group is built from the infinitesimal generators through a non-degenerate invariant bilinear form according to Eq. (6.6). Matrix elements of the IRREPs of SU (2) are given by the so-called Wigner D-functions [START_REF] Varshalovich | Quantum Theory of Angular Momentum[END_REF] ξJM

|R(Ω)|ξ J M ≡ δ ξξ δ JJ D J M M (Ω) , (6.23)
where |ξJM is an eigenstate of J 2 and J z

J 2 |ξJM = 2 J(J + 1)|ξJM , (6.24a) J z |ξJM = M |ξJM , (6.24b) with 2J ∈ N, 2M ∈ Z, J -M ∈ N and -J ≤ M ≤ +J.
The index ξ collects all quantum numbers of the many-body state but J and M . The (2J +1)-dimensional IRREPs are labeled by J and are spanned by the set of states {|ξJM , |M | ≤ J} for fixed J and ξ. An irreducible tensor operator T J of rank J is made of 2J + 1 operators T J K transforming under rotation as

R(Ω) T J K R(Ω) -1 = M T J M D J M K (Ω) , (6.25)
or, equivalently, fulfilling

[J z , T J K ] = K T J K , (6.26a) [J ± , T J K ] = (J ± K + 1)(K ∓ 1) T J K±1 , (6.26b)
where J ± = J x ± iJ y denote raising (lowering) operators. The nuclear Hamiltonian is an example of a spherical tensor operator of rank zero. Such operators are denoted as scalar or more rarely monopole. For higher spherical tensor ranks, J = 1, 2, 3, . . ., the notion of vector, quadrupole, octupole is established in correspondence to the terminology of multipolar expansion.

A powerful tool to treat spherical tensor operators is the celebrated Wigner-Eckart theorem

ξ 1 J 1 M 1 |T J M |ξ 2 J 2 M 2 = 1 Ĵ1 J 2 J J 1 M 2 M M 1 (ξ 1 J 1 |T J |ξ 2 J 2 ) , (6.27)
where the shorthand notation Ĵ ≡ √ 2J + 1 has been introduced. The theorem states that matrix elements of a spherical tensor can be written as a product of a geometric part, consisting in a Clebsch Gordan (CG) coefficient (see definition in Sec. 6.3.4) and corresponding to an orientation in Hilbert space, and of a reduced matrix element, containing the physical information of the system [START_REF] Suhonen | From Nucleons to Nucleus[END_REF] and being independent of magnetic quantum numbers. In the special case of scalar tensor operator one gets

ξ 1 JM |T 0 0 |ξ 2 JM = 1 Ĵ (ξ 1 J|T 0 |ξ 2 J) , (6.28)
and consequently, the initial and reduced matrix elements are independent of any projection quantum number. Therefore, in most cases reduced matrix elements are not used in the case of scalar operators.

Many-body states

In the following, the one-body basis B 1 is taken to be the eigenbasis of a SU (2)-invariant Hamiltonian H bas such that basis states are conveniently labeled as

|k = |n k l k j k m k t k , (6.29)
where n k denotes the radial quantum number, l k the orbital quantum number, j k the total angular-momentum quantum number, m k its projection and t k the isospin projection 6 . While the eigenstates of the aforementioned SHO Hamiltonian provide an example of practical interest, other bases characterized by the same symmetry are equally valid.

Even though the spin projection m s k does not appear explicitly it is fixed by l k and j k since l k and s k couple to j k . Later on, the AMR-Ts employed throughout the symmetry reduction will carry reduced labels k characterized by

| k ≡ |n k , l k , j k , t k , (6.30)
where the angular-momentum projection, i.e. the magnetic quantum number m k , is explicitly excluded. The tensor product of two one-body states defines a basis state of the two-body Hilbert space H 2 . These states are most easily obtained in the uncoupled representation as

|k 1 k 2 ≡ |k 1 ⊗ |k 2 .
(6.31)

Going to the coupled representation, the two total angular momenta j k 1 and j k 2 are coupled to a total two-body angular momentum J with projection7 M according to

| k1 k2 (J) ≡ m k 1 m k 2 j k 1 j k 2 J m k 1 m k 2 M |k 1 k 2 , (6.32)
where the vector space inner product

j k 1 j k 2 J m k 1 m k 2 M ≡ k 1 k 2 | k1 k2 (J) (6.33)
denotes the Clebsch Gordan coefficient mitigating the transformation from the uncoupled to the coupled basis. The inverse transformation of Eq. ( 6.32) is given by

|k 1 k 2 = JM j k 1 j k 2 J m k 1 m k 2 M | k1 k2 (J) . (6.34)
Note that the left-hand side of Eq. ( 6.32) defines a two-body eigenstate of J 2 . Along the same lines, the uncoupled three-body basis states of H 3 is obtained through the tensor product of three single-particle states

|k 1 k 2 k 3 ≡ |k 1 ⊗ |k 2 ⊗ |k 3 . (6.35)
Performing the angular-momentum coupling requires fixing the coupling order which is subsequently chosen to be

|[ k1 k2 (J 12 )] k3 (J) = m k 1 m k 2 m k 3 M 12 j k 1 j k 2 J 12 m k 1 m k 2 M 12 J 12 j k 3 J M 12 m k 3 M |k 1 k 2 k 3 , ( 6.36)
i.e., the first two particles are coupled to an intermediate two-body angular-momentum quantum number J 12 which is further coupled to the third particle to yield the overall (half-integer) three-body angular-momentum J. In analogy to the case of two particles, Eq. (6.36) defines a three-body eigenstate of J 2 . However, the initial choice of coupling order employed in Eq. (6.36) is arbitrary and an alternative coupling scheme is given by

|[ k1 [ k2 k3 (J 23 )](J) = m k 1 m k 2 m k 3 M 23 j k 2 j k 3 J 23 m k 2 m k 3 M 23 j k 1 J 23 J m k 1 M 23 M |k 1 k 2 k 3 , (6.37)
where the second and third single-particle states are coupled to an intermediate angular momentum J 23 that is subsequently coupled with j k 1 to an overall J. Both coupling schemes enable for the construction of a basis of H 3 . The transformation between the two representations is given by

|[ k1 k2 (J 12 )] k3 (J) = J 23 Ĵ12 Ĵ23 (-1) j k 1 +j k 2 +j k 3 +J j k 1 j k 2 J 12 j k 3 J J 23 | k1 [ k2 k3 (J 23 )](J) = J 23 Ĵ12 Ĵ23 j k 1 j k 2 J 12 j k 3 J J 23 |[ k3 k2 (J 23 )] k1 (J) , (6.38)
where the Wigner 6j-symbol was introduced 8 . Recursively, N -body states can be introduced for N ≥ 3, e.g. in the uncoupled representation

|k 1 . . . k N ≡ N i=1 |k i . (6.39)
Since in current ab initio implementations four-body operators play no dominant role yet, this extension is not discussed here.

Many-body matrix elements

The operator O λ µ being the component of a spherical tensor of rank λ, its uncoupled matrix elements are defined by

O λµ k 1 ...k 2n ≡ k 1 . . . k n |O λ µ |k n+1 . . . k 2n . (6.40)
By means of the transformation between uncoupled and coupled representations of the bra and ket states, coupled expressions for matrix elements can be derived 9 . Focusing on a two-body operator characterized by uncoupled matrix elements

O λµ k 1 k 2 k 3 k 4 , their angular-momentum-coupled counterparts are ÕJMλµJ M k1 k2 k3 k4 = m k 1 m k 2 m k 3 m k 4 O λµ k 1 k 2 k 3 k 4 j k 1 j k 2 J m k 1 m k 2 M j k 3 j k 4 J m k 3 m k 4 M , ( 6.41)
where the inverse relation is given by

O λµ k 1 k 2 k 3 k 4 = JM J M ÕJMλµJ M k1 k2 k3 k4 j k 1 j k 2 J m k 1 m k 2 M j k 3 j k 4 J m k 3 m k 4 M . (6.42)
Analogously, coupled three-body matrix elements are obtained as

ÕJMλµJ M k1 k2 J 12 k3 k4 k5 J 45 k6 = m k 1 m k 2 m k 3 M 12 m k 4 m k 5 m k 6 M 45 O λµ k 1 k 2 k 3 k 4 k 5 k 6 (6.43) × j k 1 j k 2 J 12 m k 1 m k 2 M 12 J 12 j k 3 J M 12 m k 3 M j k 4 j k 5 J 45 m k 4 m k 5 M 45 J 45 j k 6 J M 45 m k 6 M ,
where the inverse relation is given by

O λµ k 1 k 2 k 3 k 4 k 5 k 6 = J 12 M 12 JM J 45 M 45 J M ÕJMλµJ M k1 k2 J 12 k3 k4 k5 J 45 k6 × j k 1 j k 2 J 12 m k 1 m k 2 M 12 J 12 j k 3 J M 12 m k 3 M j k 4 j k 5 J 45 m k 4 m k 5 M 45 J 45 j k 6 J M 45 m k 6 M . (6.44)
Neither Eqs. (6.41-6.42) nor Eqs. (6.43-6.44) assume the underlying operator to be scalar. If it is indeed the case, the selection rules J = J and M = M hold such that the coupled matrix element is actually independent of M . For the three-body operator, the intermediate couplings do not however coincide, i.e. J 12 = J 45 and M 12 = M 45 .

Diagrammatic method

Even though all manipulations necessary to simplify angular-momentum expressions can be performed solely in terms of the expressions introduced in Sec. 6.3, it is at the heart of this work to introduce a more convenient representation of the involved algebraic steps that, additionally, allows for computer-aided derivations. As Feynman or Goldstone diagrams are used to efficiently capture the results of cumbersome derivations, diagrams can be introduced to restate complicated identities associated with angular momentum algebra [START_REF] Yutsis | Mathematical apparatus of the theory of angular momentum[END_REF]. A modern account of the underlying group-theoretic properties is provided in Ref. [START_REF] Paul | Angular Momentum Diagrams[END_REF].

Preliminaries

As seen in Sec. 6.3, CG coefficients constitute the basic building blocks of angularmomentum theory. However, CG coefficients are somewhat inconvenient due to the asymmetry with respect to the involved angular-momentum quantum numbers. A more symmetric representation can be obtained via the so-called Wigner coefficients, or 3jmsymbols10 ,

j 1 j 2 j 3 m 1 m 2 m 3 ≡ (-1) j 1 -j 2 -m 3 1 ĵ3 j 1 j 2 j 3 m 1 m 2 -m 3 , ( 6.45)
where the inverse relation reads

j 1 j 2 j 3 m 1 m 2 m 3 ≡ (-1) j 1 -j 2 +m 3 ĵ3 j 1 j 2 j 3 m 1 m 2 -m 3 . (6.46)
Wigner 3jm-symbols are invariant under an even number of column permutations,

j 1 j 2 j 3 m 1 m 2 m 3 = j 3 j 1 j 2 m 3 m 1 m 2 = j 2 j 3 j 1 m 2 m 3 m 1 , ( 6.47)
whereas an odd number of such permutations induces a phase factor

j 1 j 2 j 3 m 1 m 2 m 3 = (-1) j 1 +j 2 +j 3 j 2 j 1 j 3 m 2 m 1 m 3 . (6.48)
Wigner 3jm-symbols with opposite magnetic quantum numbers are related via the identity

j 1 j 2 j 3 m 1 m 2 m 3 = (-1) j 1 +j 2 +j 3 j 1 j 2 j 3 -m 1 -m 2 -m 3 . (6.49)
Furthermore, 3jm-symbols with one vanishing (j, m) couple simplify according to

j 1 j 2 0 m 1 -m 2 0 = (-1) j 1 -m 1 1 ĵ1 δ j 1 j 2 δ m 1 m 2 .
(6.50)

Nodes and edges

Wigner 3jm-symbols constitute the building blocks of the diagrammatic formalism. They represent the nodes entering the so-called Yutsis graphs. More specifically, a node carrying three outgoing edges, each labelled by a couple (j k , m k ), represents the 3jm-symbol

- j 1 j 2 j 3 m 1 m 2 m 3 = j 3 m 3 j 1 m 1 j 2 m 2
The node sign denotes a convention specifying the column order that must be used to write the corresponding 3jm-symbol, i.e. a minus (plus) sign stipulates that the edges and the associated angular-momentum labels must be read clockwise (counterclockwise). Furthermore, the node with one ingoing edge (j 3 ,m 3 ) represents the 3jm-symbol

- (-1) j 3 -m 3 j 1 j 2 j 3 m 1 m 2 -m 3 = j 3 m 3 j 1 m 1 j 2 m 2
with a negative projection -m 3 together with a phase (-1) j 3 -m 3 .

Starting from the two above definitions, the nodes with two and three in-going edges are obtained by applying the operation consisting of inverting the directions of all three edges at once (Eq. (6.49)). Starting for example from the node with three out-going edges, the node with three ingoing edges is produced

- (-1) j 1 -m 1 +j 2 -m 2 +j 3 -m 3 j 1 j 2 j 3 -m 1 -m 2 -m 3 = j 3 m 3 j 1 m 1 j 2 m 2
where the magnetic quantum numbers have been added to the phase at no cost given that m 1 + m 2 + m 3 = 0 holds. Additionally, performing the operation twice gives back the original node, thanks to the identity

(-1) 2(j 1 -m 1 +j 2 -m 2 +j 3 -m 3 ) = 1 . (6.51)
Through the previous operations, the node sign is not altered. Changing the sign carried by the node can be performed at the price of the phase factor Φ ns = (-1) j 1 +j 2 +j 3 , (6.52)

where the lower index ns stipulates the direct node sign change. Indeed, moving from a clockwise to a counterclockwise (or vice versa) reading of the node corresponds to performing one column inversion in the 3jm-symbol whose effect is characterized by Eq. (6.48). Notice that changing the node sign is equivalent to moving one edge across another one (indirect node sign change), such that doing this twice produce no additional phase.

Yutsis graphs

The network of 3jm-symbols generated via step (i) of the angular-momentum reduction of TNs (see Sec. 6.3.1) is represented via a Yutsis graph. Those graphs are thus obtained by contracting a set of nodes through their edges in a way that consistently represents the network of 3jm-symbols. Contracting the edges of two nodes is possible if both edges carry the same angular momentum quantum numbers (j, m) and go in the same direction, i.e., one must be going out of the first node while the other one must be going into the second node

+ j 3 m 3 j 1 m 1 j 2 m 2 - j 3 m 3 j 1 m 1 j 2 m 2
The contraction itself corresponds to summing over the common magnetic quantum number such that the internal edge does not carry it anymore

+ - j 3 j 1 m 1 j 2 m 2 j 1 m 1 j 2 m 2
Reading the nodes according to the definitions given previously, the algebraic expression resulting from the contraction reads as

m 3 (-1) j 3 -m 3 j 1 j 2 j 3 m 1 m 2 m 3 j 1 j 2 j 3 m 1 m 2 -m 3 . (6.53)
Given a Yutsis graph, the direction of an internal edge carrying angular momentum j can be reversed at the price of accounting for an additional phase factor Φ lr = (-1) 2j , (6.54) where the lr index stands for line reversal.

An example of practical interest relates to fully contracting the two nodes

+ j 3 m 3 j 1 m 1 j 2 m 2 - j 3 m 3 j 1 m 1 j 2 m 2
to generate a closed Yutsis graph

+ - j 3 j 1 j 2
actually corresponding to the so-called Wigner 3j-symbol j 1 j 2 j 3 , also called triangular delta or triangular inequality. The corresponding algebraic expression is given by

j 1 j 2 j 3 = m 1 m 2 m 3 (-1) j 1 -m 1 +j 2 -m 2 +j 3 -m 3 j 1 j 2 j 3 m 1 m 2 m 3 j 1 j 2 j 3 -m 1 -m 2 -m 3 =    1, if |j 1 -j 2 | ≤ j 3 ≤ j 1 + j 2 0, otherwise , (6.55)
which vanishes unless the inequalities are satisfied.

Unfactorizable graphs

Wigner 3nj-symbols provide relevant examples of Yutsis graphs that cannot be simplified via factorization rules, see Sec. 6.4.6. The first example is the Wigner 3j-symbol (n = 1) that has already been given in the previous section and whose value is either 1, or 0, depending on the satisfaction, or not, of a triangular inequality (Eq. ( 6.55)).

The second example is the Wigner 6j-symbol (n = 2) that can be graphically represented as a pyramidal structure Independently of which of the two diagrams is used, the corresponding algebraic expression for the Wigner 6j-symbol is

j 1 j 2 j 3 j 4 j 5 j 6 ≡ m 1 ...m 6 (-1) 6 k=1 (j k -m k ) × j 1 j 2 j 3 m 1 m 2 m 3 j 1 j 5 j 6 -m 1 -m 5 m 6 j 4 j 2 j 6 m 4 -m 2 -m 6 j 4 j 5 j 3 -m 4 m 5 -m 3 . (6.56)
The third case yields the Wigner 9j-symbol (n = 3), whose algebraic expression is given in terms of six 3jm-symbols by which can be rewritten as involving six nodes and nine edges. Higher-order 3nj-symbols only rarely arise in nuclear many-body theory. They do however naturally enter in the partial-wave decomposition of nuclear k-body Hamiltonians for k ≥ 4.

     j 1 j 2 j 3 j 4 j 5 j 6 j 7 j 8 j 9      ≡ m 1 ...m 9 j 1 j 2 j 3 m 1 m 2 m 3
     j 1 j 2 j 3 j 4 j 5 j 6 j 7 j 8 j 9      ≡ m 1 ...m 9 j 1 j 2 j 3 m 1 m 2 m 3 (-1) j 1 -m 1 +j 4 -m 4 +j 7 -m 7 j 1 j 4 j 7 -m 1 -m 4 -m 7 × j 4 j 5 j 6 m 4 m 5 m 6 (-1) j 2 -m 2 +j 5 -m 5 +j 8 -m 8 j 2 j 5 j 8 -m 2 -m 5 -m 8 × j 7 j 8 j 9 m 7 m 8 m 9 (-1) j 3 -m 3 +j 6 -m 6 +j 9 -m 9 j 3 j 6 j 9 -m 3 -m 6 -m 9 , (6.
In practice, Wigner 3nj-symbols play an important role since they can be pre-calculated and stored in cache in large-scale applications. This is typically done for 6j-symbols and if necessary for (a subset of) 9j-symbols. Since the number of 9j-symbols is very high for a selected model space it is often useful to re-express 9j-symbols as sums of products of 6j-symbols according to

     j 1 j 2 j 3 j 4 j 5 j 6 j 7 j 8 j 9      ≡ j x (-1) 2j x ĵ2
x j 1 j 4 j 7 j 8 j 9 j x j 2 j 5 j 8 j 4 j x j 6 j 3 j 6 j 9 j x j 1 j 2 (6.59) and resort to much smaller 6j-caches if the structure of the angular-momentum networks supports such strategies.

From tensor networks to Yutsis graphs

The crucial step is to extract the Yutsis graph associated to a SU-TN. As already discussed, this is done by expressing the original SU-Ts in terms of SR-Ts and a set of CG coefficients that are consecutively replaced by their 3jm-symbol equivalents. The next step consists of splitting each involved summation according to

k → n k l k j k m k t k → k m k . (6.60)
Given that the AMR-TNs do not depend on any m-quantum number, one can isolate the networks of 3jm-symbols along with the sums over the magnetic quantum numbers. Once this is done, the corresponding Yutsis graph can be extracted.

Factorization rules

Having the Yutsis graph at hands, the goal is to simplify it a much as possible. This corresponds to identifying specific subparts in the graph that can be reduced via the application of identities satisfied by the corresponding set of 3jm-symbols. Once the simplification process is completed, one is left with an expression involving irreducible Wigner 3nj-symbols (see Sec. 6.4.4). The benefit of using Yutsis graphs is that the search for reducible parts can be automated while their actual reductions can be realized by applying systematic factorization rules on the graph. The rules are characterized in terms of the length of the cycles involved in the process. Below, the factorization rules are introduced one after another with increasing degree of complexity, i.e. cycle length. For the proofs of the factorization formula, the reader is referred to Ref. [START_REF] Balcar | Introduction to the Graphical Theory of Angular Momentum[END_REF]. A more extensive list of angular-momentum-algebra identities that can be used to define factorization rules can be found in Ref. [START_REF] Varshalovich | Quantum Theory of Angular Momentum[END_REF].

Cycles of length two

The simplest factorization rule corresponds to the reduction of a 2-cycle or bubble. Algebraically, the corresponding identity is the orthogonality relation

m 1 m 2 j 1 j 2 j 3 m 1 m 2 m 3 j 1 j 2 j 3 m 1 m 2 m 3 = 1 ĵ2 3 δ j 3 j 3 δ m 3 m 3 j 1 j 2 j 3 , (6.61)
which can be rewritten as

m 1 m 2 (-1) j 1 -m 1 +j 2 -m 2 +j 3 -m 3 j 1 j 2 j 3 m 1 m 2 m 3 j 1 j 2 j 3 -m 1 -m 2 -m 3 = 1 ĵ2 3 δ j 3 j 3 δ m 3 m 3 j 1 j 2 j 3 . (6.62)
The diagrammatic representation of the factorization rule representing the identity stated in Eq. (6.62) is given in Fig. 6.1. The 2-cycle rule thus replaces two nodes connected through two edges by a single edge in a Yutsis graph.

-

+ = 1 ĵ2 3 j 1 j 2 j 3

Cycles of length three

The first factorization rule leading to a non-trivial Wigner 3nj-symbol corresponds to the factorization of a 3-cycle, or triangle, as displayed in Fig. 6.2. Algebraically, the factorization corresponds to the identity

m 4 m 5 m 6 (-1) j 4 -m 4 +j 5 -m 5 +j 6 -m 6 j 5 j 1 j 6 m 5 m 1 -m 6 j 6 j 2 j 4 m 6 m 2 -m 4 j 4 j 3 j 5 m 4 m 3 -m 5 = (-1) j 1 +j 2 +j 3 j 1 j 2 j 3 m 1 m 2 m 3 j 1 j 2 j 3 j 4 j 5 j 6 . ( 6

Cycles of length four

The most involved factorization rule employed in this work corresponds to a 4-cycle, or square, as displayed in Fig. 6.3. The underlying algebraic identity is given by

m 5 m 6 m 7 m 8
(-1) j 5 -m 5 +j 6 -m 6 +j 7 -m 7 +j 8 -m 8 j 5 j 1 j 6 m 5 m 1 -m 6 j 6 j 2 j 7 m 6 m 2 -m 7

j 7 j 3 j 8 m 7 m 3 -m 8 j 8 j 4 j 5 m 8 m 4 -m 5 = (-1) j 8 -j 1 -j 4 -j 6 j x m x ĵ2 x (-1) j x -m x j 1 j x j 4 m 1 -m x m 4 j 2 j x j 3 m 2 m
x m 3 j 1 j 4 j x j 8 j 6 j 5 j 2 j 3 j x j 8 j 6 j 7 .

(6.64) Equation ( 6.64) allows to factorize the topology involving four nodes into an irreducible part made of two 6j-symbols while leaving two nodes. Therefore, the resulting graph contains two nodes and three edges less than the initial one.

--

- - = j x
(-1) j x +j 8 -j 6 ĵ2

x j 1 j 4 j x j 8 j 6 j 5 j 2 j 3 j x j 8 j 6 j 7 + +

Cycles beyond length four

While the present version of the code (see Sec. 6.6) supports factorizations involving up to cycles of length four, there exist topologies, sketched in Fig. 6.4, which cannot be simplified through the above stated rules but require more involved identities. In principle, this restricts the range of applicability to topologies that do not contain only cycles of length five or higher. The smallest cubic graph that only involves cycles of length five is the so-called Peterson graph containing 10 nodes and 15 edges. Consequently, the simplest many-body diagram potentially leading to this topology must contain at least five two-body vertices, e.g., corresponding to a fifth-order MBPT diagram or a CC diagram with T 3 amplitudes. In the testing phase of the current version of the code, the factorization rules were applied to hundreds of many-body diagrams including topologies that are far beyond the current state-of-the-art applications. In none of the test cases a Yutsis graph involving only cycles of length five or higher appeared. Future versions of the program will be extended along these lines by including factorizations of more complex topologies, or including more elaborate techniques using for example the interchange rule [START_REF] Van Dyck | New heuristic approach to the calculation of general recoupling coefficients[END_REF].

Zero-line rule

Additionally, a 3jm-symbol with one vanishing (j, m) couple, called zero-line, is represented as follow

- (-1) j 2 -m 2 j 1 j 2 0 m 1 -m 2 0 = 0 j 1 m 1 j 2 m 2 = 1 ĵ1 j 1 m 1
where the zero-line does not carry a direction as the corresponding magnetic quantum number is both positive and negative. With resort to Eq. ( 6.50), one can reduce this node to finally lead one edge. The corresponding reduction rule is called zero-line rule.

Applications

In the following a number of different many-body formalisms are used to exemplify the steps involved in the process of symmetry reduction. The emphasis is on the angularmomentum reduction such that detailing the formalisms themselves is outside the scope of the present work. All examples considered here are used in state-of-the-art nuclear structure applications 11 .

The formulae derived below are not in the computationally most optimized form. Since the focus is on angular-momentum coupling, further reduction schemes are not discussed. For instance in all of the examples discussed below parity-and isospin-conservation can be exploited to yield more efficient implementations. However, processing SU (2)-symmetry yields by far the highest computational benefit due to the much smaller number of parity and isospin labels.

Many-body perturbation theory

Many-body perturbation theory can be formulated through an infinite power series for the exact ground-state energy and wave function [START_REF] Shavitt | Many-body methods in chemistry and physics : MBPT and coupled-cluster theory[END_REF]

E k (λ) = E (0) k + λE (1) k + λ 2 E (2) k + . . . (6.65a) |Ψ k (λ) = |Φ + λ|Ψ (1) k + λ 2 |Ψ (2) k + . . . , (6.65b)
where the lower index k enumerates excited states in the spectrum and |Φ ≡ |Ψ (0) k denotes the unperturbed reference state. The expansions from Eq. (6.65) are evaluated at λ = 1 to obtain the quantities corresponding to the original problem. Since in the following we are exclusively concerned with the description of nuclear ground states, i.e., k = 0, the subscript is dropped for simplicity. The starting point is given by the definition of a splitting of the full Hamiltonian

H = H 0 + λH 1 , (6.66)
into an unperturbed part H 0 and a perturbation H 1 such that the reference energy 12 is given by

E ref ≡ Φ|H|Φ = E (0) 0 + E (1) 0 . (6.67)
The first contribution to the correlation energy

∆E ≡ E 0 -E ref (6.68)
is obtained at second-order 13 . The simplest choice is to take |Φ as a Slater determinant.

In recent years, more sophisticated vacua have been used in order to account for static correlation effects in open-shell systems. Both multi-configurational reference states obtained from small CI diagonalizations [START_REF] Tichai | Open-shell nuclei from No-Core Shell Model with perturbative improvement[END_REF] and particle-number-broken HFB vacua [START_REF] Tichai | Bogoliubov many-body perturbation theory for open-shell nuclei[END_REF] have shown to provide computationally cheap benchmarks without loss in accuracy when employing softened chiral potentials.

In this work, a SU (2)-restricted HF slater determinant is used as a reference state, thus leading to the so-called canonical form of MBPT. The MBPT examples are worked out in detail to enable deeper understanding of each of the individual algorithmic steps.

Second-order energy correction

The second-order energy correction reads as 14

E (2) 0 = 1 4 abij H abij H ijab ab ij , ( 6.69)
where (a, b) and (i, j) denote particle and hole states, respectively, i.e., states that are unoccupied or occupied in the reference Slater determinant. Additionally, a short-hand notation for the energy denominator is used ab... ij... ≡ i + j + . . .ab -. . . , (6.70) where k denotes HF single-particle energies. According to the previous definitions, Eq. ( 6.69) provides a closed symmetry-unrestricted TN with the Hamiltonian tensor H pqrs as the central quantity.

Expressing SU-Ts appearing in Eq. (6.69) in terms of their AMR-T counterparts according to Eq. (6.42) yields

E (2) 0 = 1 4 ãbĩj 1 ãb ĩj J 1 J 2 J 1 H ãbĩj J 2 H ĩj ãb × m a m b M 1 m i m j M 2 j a j b J 1 m a m b M 1 j i j j J 1 m i m j M 1 j i j j J 2 m i m j M 2 j a j b J 2 m a m b M 2 , ( 6.71)
12 Contrary to chap. 3, where the k-order contribution to the ground-state energy refers to diagrams with k grand potential vertices and one Hamiltonian vertex, the k-order contribution in this case means k H 1 vertices, such that the first non-trivial correction does not appear at first order, see diagrams PO1.1 and PO1.2 in Fig. 3.8, but at second order in this convention. 13 Even though the object ∆E is usually referred to as correlation energy it does not mean that E ref does not contain correlation effect, e.g., when using a symmetry-broken or multi-configurational vacuum.

In the case of a HF vacuum, however, there is indeed no correlation contained in |Φ beyond those associated with Pauli's exclusion principle. 14 Using an arbitrary slater determinant as a reference state gives rise to one additional diagram at second order and eleven additional diagrams at third order. In any case the computational complexity is always driven by the canonical diagrams included in this discussion.

where it was used that the single-particle energies are m-independent, i.e. k = k and that the Hamiltonian is a scalar. The tensor network of Eq. (6.71) is split into an SU (2)invariant part, that does not depend on angular-momentum projection quantum numbers, and a part carrying the full dependence of magnetic quantum numbers that will be subsequently simplified. In a first step, CG coefficients are converted into 3jm-symbols, according to Eq. (6.46), yielding

E (2) 0 = 1 4 ãbĩj 1 ãb ĩj J 1 J 2 J 1 H ãbĩj J 2 H ĩj ãb Ĵ2 1 Ĵ2 2 × m a m b M 1 m i m j M 2 j a j b J 1 m a m b -M 1 j i j j J 1 m i m j -M 1 j i j j J 2 m i m j -M 2 j a j b J 2 m a m b -M 2 × (-1) 2j a -2j b +2j i -2j j +2M 1 +2M 2 , ( 6.72)
where the phase factor gives

(-1) 2j a -2j b +2j i -2j j +2M 1 +2M 2 = (-1) 4 (+1) 2 = 1 , (6.73)
given that j a , j b , j i and j j are half-integers and that M 1 and M 2 are integers. Focusing on the 3jm-symbols network, the second step consists in reversing all m quantum numbers in the second and fourth 3jm-symbols

m a m b M 1 m i m j M 2 j a j b J 1 m a m b -M 1 j i j j J 1 -m i -m j M 1 j i j j J 2 m i m j -M 2 j a j b J 2 -m a -m b M 2 × (-1) j a -m a +j b -m b +j i -m i +j j -m j +J 1 -M 1 +J 2 -M 2 (6.74)
at the price of an extra phase factor, where the magnetic quantum numbers have been added to the phase at no cost given that m i + m j -M 1 = 0 and m a + m b -M 2 = 0 hold and that M 1 and M 2 are integers. The analytical expression is now in the proper form to identify the associated Yutsis graph

+ - + - J 1 j a j b J 2 j j j i
Now that the working graph has been built, the next step consists in simplifying it via the application of appropriate factorization rules. The application of the 2-cycle rule, see Fig. 6.1, requires the direction of the edges carrying j a and j b to be reversed, thus bringing the phase Φ lr = (-1) 2j a (-1) 2j b = (-1) 2 = 1 and yielding the diagram

+ - + - J 1 j a j b J 2 j j j i
where the red box indicates the subpart of the diagram that will be factorizated in the next step. Factorizing the 2-cycle provides the intermediate factor

1 Ĵ2 1 j a j b J 1 δ J 1 J 2 (6.75)
and leaves the diagram

+ - J 1 j j j i
In the last step, the 3j-symbol is identified after reversing the orientation of the edges carrying j i and j j + -

J 1 j j j i
leading to the additional phase Φ lr = (-1) 2j i (-1) 2j j = (-1) 2 = 1 and providing the overall result 1 Ĵ2

1 j a j b J 1 j i j j J 1 δ J 1 J 2 . ( 6.76)
Replacing the m-dependent part of Eq. (6.72) by Eq. (6.76) finally provides the AMR form of the second-order energy correction

E (2) 0 = 1 4 J Ĵ2 ãbĩj J H ãbĩj J H ĩj ãb ãb ĩj , ( 6.77)
where triangular inequalities coming from 3j-symbols are assumed. While the initial SU-TN is of N 4 complexity, the AMR-TN is of Ñ 4 • (J max + 1) complexity, where Ñ is the number of reduced basis states k and J max corresponds to the maximum number of channels of two-body angular-momentum. In large model spaces the difference in runtime is improved by several orders of magnitude even for this very simple example.

Third-order energy correction

A more elaborate example is given by the third-order energy correction to the ground-state binding energy

E (3) 0 ≡ E (3) pp + E (3) hh + E (3)
ph , (

with [START_REF] Shavitt | Many-body methods in chemistry and physics : MBPT and coupled-cluster theory[END_REF]]

E (3) pp = 1 8 abcdij H ijab H abcd H cdij ab ij cd ij , ( 6.79a)
E (3) hh = 1 8 abijkl H ijab H klij H abkl ab ij ab kl , (6.79b) E (3) ph = - abcijk H ijab H kbic H ackj ab ij ac kj . (6.79c)
Following the same procedure as for the second-order energy correction, one obtains the Yutsis graph associated with the particle-particle contribution

+ - + - + - J 1 J 2 J 3 j a j b j c j d j i j j
and, similarly, the one associated with the hole-hole contribution

+ - + - + - J 1 J 2 J 3 j a j b j k j l j i j j
which are topologically equivalent. Due to the presence of one more Hamiltonian matrix element compared to the second-order energy correction, the number of 3jm-symbols, i.e. the number of nodes, is increased by two. In both cases, the red boxes indicate the subgraphs that are factorized by the application of the 2-cycle rule. Applying it twice and identifying the resulting Yutsis graph as a 3j-symbol leads to the result 1 Ĵ4

1 j a j b J 1 j c j d J 1 j i j j J 1 δ J 1 J 2 δ J 2 J 3 . (6.80) Considering the Ĵ2 1 Ĵ2 2 Ĵ2
3 factor coming from the prior conversion of CG coefficients into 3jm-symbols, the final AMR form of the two contributions is

E (3) pp = 1 8 J Ĵ2 ãb c dĩj J H ĩj ãb J H ãb c dJ H c dĩj ãb ĩj c d ĩj , (6.81a) E (3) hh = 1 8 J Ĵ2 ãbĩjkl J H ĩj ãb J H klĩj J H ãbkl ãb ĩj ãb kl , ( 6.81b)
which can be read as simple matrix-matrix products within each J channel. The symmetry reduction of the particle-hole term is more involved such that, following the same steps, the associated Yutsis graph is

+ + + - - - J 1 J 2 J 3 j a j b j c j k j i j j
where the edges associated to J 1 , J 2 and J 3 have been reversed at no cost, given that these indices are integers (Φ lr = (-1) 2J 1 (-1) 2J 2 (-1) 2J 3 = 1). The graph can be re-arranged in a more convenient way as

+ - - + + - j a j c J 3 j b J 2 j k J 1 j i j j
where the signs of the two nodes connected by the j a edge have been changed as the order of the indices associated to these nodes have been inverted. This Yutsis graph is nothing but a 9j-symbol, see Sec. 6.4.4. Consequently, the AMR form of the particle-hole term leads to the algebraic expression

E (3) ph = - J 1 J 2 J 3 Ĵ2 1 Ĵ2 2 Ĵ2 3 ãb cĩjk J 1 H ĩj ãb J 2 H kbĩ cJ 3 H ãc kj ãb ĩj ãc kj      j a j c J 3 j b J 2 j k J 1 j i j j      . ( 6.82)
In actual applications, this expression is conveniently re-written by expressing the 9jsymbol as a sum of products of three 6j-symbols

E (3) ph = - J 1 J 2 J 3 K Ĵ2 1 Ĵ2 2 Ĵ2 3 K2 × ãb cĩjk J 1 H ĩj ãb J 2 H kbĩ cJ 3 H ãc kj ãb ĩj ãc kj j a j b J 1 j i j j K j c J 2 j i j b K j k J 3 j k j j K j a j c , ( 6.83)
which can be obtained graphically by a successive application of the 4-cycle rule, the 3-cycle rule and finally the identification of a redundant 3j-symbol. The alternate form of the particle-hole term given in Eq. (6.83) allows to introduce Pandya-transformed matrix elements [95]

J 1 Ȏpqrs ≡ - J 2 Ĵ2 2 j p j q J 1 j r j s J 2 J 2 O psr q . (6.84)
Eventually, the particle-hole contribution can be written as

E (3) ph = J Ĵ2 ãb cĩjk J Hĩb ãj J Hk cĩb J Hã jk c ãb ĩj ãc kj , ( 6.85)
which reads as the trace of a two-fold matrix-matrix product of Pandya-transformed Hamiltonian matrix elements. Equation (6.85) clearly shows the computational benefit of an appropriate choice of coupling order, which in practice is not at all obvious.

Coupled-cluster theory

Contrary to a simple power series ansatz, coupled-cluster theory aims at a non-perturbative resummation of large classes of perturbation theory diagrams.

General formalism

The starting point of the CC framework is an exponential ansatz for the wave operator [START_REF] Shavitt | Many-body methods in chemistry and physics : MBPT and coupled-cluster theory[END_REF] such that the exact ground-state wave function reads as

|Ψ = e T |Φ , (6.86)
thus involving the cluster operator T defined as

T ≡ T 1 + T 2 + . . . + T A . ( 6.87)
The second-quantized form of the individual contributions to T is given by

T n ≡ 1 (n!) 2 a 1 ...a n i 1 ...i n t a 1 ...a n i 1 ...i n c † a 1 • • • c † a n c i n • • • c i 1 , (6.88)
where t a 1 ...a n i 1 ...i n denotes the connected cluster amplitudes. Thanks to the exponential form of the wave operator, the CC approach is manifestly size-extensize. In actual applications, T is truncated at a fixed truncation level defining a particular CC model, e.g.,

T CCSD ≡ T 1 + T 2 (6.89a) T CCSDT ≡ T 1 + T 2 + T 3 (6.89b) . . .
where the acronyms S,D,T,. . ., indicate the inclusion of single (S), double (D), triple (T), . . ., excitations. Working equations can be conveniently re-expressed in terms of the similarity-transformed Hamiltonian H ≡ e -T He T = (He T ) c , (

where the lower index c stipulates the connected character of the expansion. Since CC theory provides a non-Hermitian framework, the similarity-transformed operator H is non-Hermitian itself.

Energy equation

In the absence of three-body operators in the input Hamiltonian the CC correlation energy is given for arbitrary truncation by

∆E CC = Φ| H|Φ = ai f ia t ai + abij H ijab t ai t bj + 1 4 abij H ijab t abij . ( 6.91)
Equation ( 6.91) defines a closed SU-TN involving at most four internal contractions. Note that higher-order amplitudes affect the energy only implicitly by relaxing T 1 and T 2 without entering the energy equation explicitly.

Contrary to canonical MBPT, the CC energy equation involves one-body operators in the form of T 1 amplitudes and matrix elements of the Fock operator f pq . The Fock operator arises from a basis transformation of the one-body part of the Hamiltonian and is thus SU (2)-invariant. As the reference Slater determinant is characterized by J = 0, cluster amplitudes are irreducible SU (2) tensors of rank J = 0 such that a similarity-transformed operator Ō has the same irreducible SU (2) tensor rank as its non-transformed counterpart O. From there, Wigner-Eckart theorem (Eq. (6.27)) enables one to introduce the reduced matrix elements of one-body operator

ξ k 1 j k 1 m k 1 |T 1 |ξ k 2 j k 2 m k 2 = 1 ĵk 1 j k 2 0 j k 1 m k 2 0 m k 1 (ξ k 1 j k 1 |T 1 |ξ k 2 j k 2 ) , (6.92a)
ξ k 1 j k 1 m k 1 |F |ξ k 2 j k 2 m k 2 = 1 ĵk 1 j k 2 0 j k 1 m k 2 0 m k 1 (ξ k 1 j k 1 |F|ξ k 2 j k 2 ) , (6.92b)
where ξ k corresponds to all quantum numbers of the one-body state |k but j k and m k . In anticipation of the application of the diagrammatic approach, Eqs. (6.92) are re-expressed in terms of 3jm-symbols

ξ k 1 j k 1 m k 1 |T 1 |ξ k 2 j k 2 m k 2 = (-1) j k 2 +m k 1 j k 2 0 j k 1 m k 2 0 -m k 1 (ξ k 1 j k 1 |T 1 |ξ k 2 j k 2 ) , (6.93a)
ξ k 1 j k 1 m k 1 |F |ξ k 2 j k 2 m k 2 = (-1) j k 2 +m k 1 j k 2 0 j k 1 m k 2 0 -m k 1 (ξ k 1 j k 1 |F|ξ k 2 j k 2 ) . ( 6.93b)
Application to the first term of the energy equation yields ai

f ia t ai = ξ a ξ i j a j i (ξ i j i |F|ξ a j a )(ξ a j a |T 1 |ξ i j i ) × m a m i (-1) j i -m i +j a -m a j i 0 j a m i 0 -m a j a 0 j i m a 0 -m i , ( 6.94)
from which the m-dependent part can be extracted to yield the Yutsis graph

+ + j a j i 0 0
Reversing the direction of the j a edge (Φ lr = (-1) 2j a = -1) together with changing the sign of the leftest node (Φ ns = (-1) j a +j j ) allows to make use of the 2-cycle rule which leads to -(-1) j a +j j j a j j 0 = -(-1) j a +j j δ j a j j = δ j a j j , (

such that one obtains the final AMR form ai

f ia t ai = ξ a ξ i j a (ξ i j a |F|ξ a j a )(ξ a j a |T 1 |ξ i j a ) . (6.96)
For the second term of the energy equation, one has abij

H ijab t ai t bj = ξ a ξ b ξ i ξ j j a j b j i j j J Ĵ2 J H ĩj ãb (ξ a j a |T 1 |ξ i j i ) (ξ b j b |T 1 |ξ j j j ) × m a m b m i m j M (-1) j i -j j +M (-1) j a -j b +M (-1) j a +m i (-1) j b +m j × j i j j J m i m j -M j a j b J m a m b -M j a 0 j i m a 0 -m i j b 0 j j m b 0 -m j . ( 6.97)
Reversing the signs of m quantum numbers in the second 3jm-symbol, the m-dependent part of Eq. ( 6.97) can be brought into the canonical form

m a m b m i m j M (-1) j i -m i +j j -m j +j a -m a +j b -m b +J-M × j i j j J m i m j -M j a j b J -m a -m b M j a 0 j i m a 0 -m i j b 0 j j m b 0 -m j , ( 6.98)
delivering the Yutsis graph

+ - + - j a j b J j j j i 0 0
with two external edges carrying zero angular momentum. Applying twice the zero-line rule, one ends up with the graphical representation of a 3j-symbol, so that Eq. (6.98), that constitutes the m-dependent part of Eq. (6.97), is reduced according to 1 ĵa ĵb δ j a j i δ j b j j j a j b J , (

such that the final AMR expression is given by abij

H ijab t ai t bj = ξ a ξ b ξ i ξ j j a j b J Ĵ2 ĵa ĵb J H ξ i j a ξ j ξ b ãb (ξ a j a |T 1 |ξ i j a ) (ξ b j b |T 1 |ξ j j b ) . (6.100)
The detailed derivation of the last contribution to the energy equation is omitted given that it is formally identical to the derivation of the second-order MBPT correction, i.e. the appropriate Yutsis graph is the one displayed in Sec. 6.5.1. The final result reads as

1 4 abij H ijab t abij = 1 4 J Ĵ2 ãbĩj J H ĩj ãb J t ãbĩj . (6.101)

Amplitude equations

The unknown cluster amplitudes are obtained by solving a set of CC amplitude equations

0 = Φ a i | H|Φ , (6.102a) 0 = Φ ab ij | H|Φ , ( 6.102b)
. . . Equations (6.102) constitute a set of coupled non-linear equations which need to be solved iteratively for every external indices combination. It provides an example of open TN containing external indices that are not summed over.

In order to perform the symmetry reduction, one must sum over all magnetic quantum numbers, and in particular the external ones. This will lead to a closed Yutsis graph. To do so, an external coupling order has to be fixed. The coupling

1 ĵ2 a m a m i j a 0 j i m a 0 m i (6.103)
is used in the case of the T 1 amplitude equations and is such that

1 ĵ2 a m a m i j a 0 j i m a 0 m i t ai = 1 ĵa (ξ a j a |T 1 |ξ i j a ) , (6.104)
whereas the coupling

1 Ĵ2 m a m b m i m j M j a j b J m a m b M j i j j J m i m j M (6.105)
is used in the case of the T 2 amplitude equations and is such that

1 Ĵ2 m a m b m i m j M j a j b J m a m b M j i j j J m i m j M t abij = J t ãbĩj . ( 6.106)
The alternative couplings 151 Ĵ2

m a m b m i m j M j a j i J m a m i M j b j j J m b m j M (6.107) or even 1 Ĵ2 m a m b m i m j M j a j j J m a m j M j b j i J m b m i M (6.108)
can be used equally well. However it turns out that the resulting equation will look much simpler when the first option is employed since the coupling order is consistent with the one used for Hamiltonian matrix elements. This is an example where experience provides a strong guidance on the proper choice of the angular-momentum coupling scheme, even though ultimately all choices yield equivalent results.

To exemplify the coupling of open TNs, a particular contribution to the CCSD doubles amplitude equation (Eq. (6.102b)) is chosen klcd H klcd t dj t ak t cbil , (

where (k, l, c, d) denote internal indices that are summed over while (a, b, i, j) characterize the external indices. The construction of the angular-momentum network originating from the application of the external coupling (6.105) to Eq. ( 6.109) requires to sum over a product of (i) two 3jm-symbols coming from the external coupling of a, b and i, j, Applying twice the zero-line rule to the leftmost and rightmost nodes ( 1 ĵa δ j a j k and 1 ĵj δ j j j d ) yields a Yutsis graph that is topologically equivalent to the one of the third-order particlehole contribution in MBPT, i.e., which corresponds to a 9j-symbol. The final expression reads as

J 1 J 2 K Ĵ2 1 Ĵ2 2 K2 ĵa ĵj ξ k ξ d lc J 1 H ξ k j a lcξ d j j (ξ d j j |T 1 |ξ j j j )(ξ a j a |T 1 |ξ k j a ) J 2 t cbĩl × j k i j k b K j k a j k j J j k c j k l K j k a j k j J 1 j k i j k l J 2 j k c j k b K . ( 6
.110)

In-medium similarity renormalization group

As a final example the in-medium similarity renormalization group (IMSRG) approach [START_REF] Tsukiyama | In-Medium Similarity Renormalization Group For Nuclei[END_REF][START_REF] Hergert | In-medium similarity renormalization group with chiral two-plus three-nucleon interactions[END_REF][START_REF] Morris | Magnus expansion and in-medium similarity renormalization group[END_REF][START_REF] Hergert | The In-Medium Similarity Renormalization Group: A novel ab initio method for nuclei[END_REF] is considered and provides a non-perturbative alternative to CC theory. In order to extend the diagrammatic angular-momentum coupling, the specific case of the treatment of non-scalar operators is taken as an example. Pioneering work has been performed in Ref. [START_REF] Parzuchowski | Ab initio electromagnetic observables with the in-medium similarity renormalization group[END_REF] where low-lying excited states and electromagnetic transitions were investigated for mid-mass closed-shell nuclei. Without the use of angular-momentum reduction, studies in the mid-mass regime would have been impossible from a computational point of view.

General formalism

The IMSRG is based on a unitary transformation of operators parametrized by a continuous parameter s ∈ [0, ∞)

O(s) = U (s) O U † (s) , (6.111)
which can be recast into a first-order ODE

d ds O(s) = [η, O(s)] , (6.112)
involving an anti-Hermitian generator η which can be chosen conveniently to obtain a desired decoupling pattern. A reasonable choice is given by the Wegner generator

η = [H od , H d ] , (6.113)
defined as the commutator of the suitably chosen 'diagonal' and 'off-diagonal' parts of H.

Even though the initial operator may contain up to two-body parts only, the evaluation of the commutator in Eq. ( 6.112) increases the particle rank of the operator, thus, inducing many-body operators up to the A-body level. In practice, the IMSRG(2) truncation is employed where one discards all higher-rank operators beyond the two-body level.

As discussed in Ref. [START_REF] Hergert | The In-Medium Similarity Renormalization Group: A novel ab initio method for nuclei[END_REF], the IMSRG(2) approximation is exact up to third order in perturbation theory for the evaluation of the ground-state energy while resumming large classes of higher-order diagrams.

Evolution of non-scalar operators

The form of Eq. (6.112) is completely generic and valid for an arbitrary Hermitian operator O independently of its transformation properties with respect to SU (2) symmetry. However, in practice the evaluation needs to take into account the specific tensorial properties of O.

The ground-state energy provides the simplest case since both O and the generator are scalar operators.

In the general case where O is a spherical tensor operator of rank λ, the evaluation of the AMR form of the commutator expansion is key

C λ µ ≡ [S λ 1 , T λ 2 ] λ µ , (6.114)
where S λ 1 µ 1 and T λ 2 µ 2 are spherical tensor operator of rank λ 1 and λ 2 , respectively, which are subsequently coupled to give a tensor of rank λ. Expanding the commutator gives

C λ µ = [S λ 1 T λ 2 ] λ µ -[T λ 2 S λ 1 ] λ µ . (6.115)
The proper definition of the spherical tensor product is given by

[S λ 1 T λ 2 ] λ µ ≡ µ 1 µ 2 λ 1 λ 2 λ µ 1 µ 2 µ S λ 1 µ 1 T λ 2 µ 2 , ( 6.116)
where the left-hand-side indeed defines a spherical tensor operator of rank λ.

As an example, a contribution to the two-body part of the evolved operator is considered. The m-scheme expression is given by

C λµ pqrs = 1 2 tu nt nu µ 1 µ 2 λ 1 λ 2 λ µ 1 µ 2 µ S λ 1 µ 1 pqtu T λ 2 µ 2 turs , (6.117)
where n p ∈ {0, 1} denotes the occupation number of the state |p and np ≡ 1 -n p . The occupation number is independent of the projection quantum number, i.e. n p = n p as well as np = np . The complete list of contributions is provided in Ref. [START_REF] Parzuchowski | Ab initio excited states from the in-medium similarity renormalization group[END_REF].

The application of Wigner-Eckart theorem to the left-hand-side of Eq. ( 6.117) gives

C λµ pqrs = 1 Ĵ1 J 1 J 2 M 1 M 2 j p j q J 1 m p m q M 1 j r j s J 2 m r m s M 2 J 2 λ J 1 M 2 µ M 1 (pqJ 1 |C λ |rsJ 2 ) , (6.118)
and similarly for tensors operators arising from the commutator expansion

S λ 1 µ 1 pqtu = 1 Ĵ3 J 3 J 4 M 3 M 4 j p j q J 3 m p m q M 3 j t j u J 4 m t m u M 4 J 4 λ 1 J 3 M 4 µ 1 M 3 (pqJ 3 |S λ 1 | tũJ 4 ) , (6.119a) T λ 2 µ 2 turs = 1 Ĵ5 J 5 J 6 M 5 M 6 j t j u J 5 m t m u M 5 j r j s J 6 m r m s M 6 J 6 λ 2 J 5 M 6 µ 2 M 5 ( tũJ 5 |T λ 2 |rsJ 6 ) . (6.119b)
In the following the standard external coupling of a tensor operator (see Eq. (6.105) for the scalar case) is employed

1 Ĵ1 m p m q m r m s M 1 M 2 µ j p j q J 1 m p m q M 1 j r j s J 2 m r m s M 2 J 2 λ J 1 M 2 µ M 1 . ( 6.120)
Applying Eq. (6.120) to Eq. (6.117) and inserting all the transformation displayed in Eqs. (6.119) yields

(pqJ 1 |C λ |rsJ 2 ) = 1 2 J 3 ...J 6 1 Ĵ1 Ĵ3 Ĵ5 tũ nt nũ (pqJ 3 |S λ 1 | tũJ 4 ) ( tũJ 5 |T λ 2 |rsJ 6 ) × µ 1 µ 2 µ M 1 ...M 6 {m i } λ 1 λ 2 λ µ 1 µ 2 µ × j p j q J 1 m p m q M 1 j r j s J 2 m r m s M 2 J 2 λ J 1 M 2 µ M 1 × j p j q J 3 m p m q M 3 j t j u J 4 m t m u M 4 J 4 λ 1 J 3 M 4 µ 1 M 3 × j t j u J 5 m t m u M 5 j r j s J 6 m r m s M 6 J 6 λ 2 J 5 M 6 µ 2 M 5 . ( 6.121)
The corresponding angular-momentum network is given by

- + + - - - - + - + j p j q J 3 j r j s J 6 J 1 J 2 λ λ 1 λ 2 j t j u J 4 J 5
The red boxes indicate subgraphs where the 2-cycle factorization rule is applied. The residual angular-momentum network corresponds to a 6j-symbol and a phase factor such that

(pqJ 1 |C λ |rsJ 2 ) = 1 2 λ (-1) J 1 +J 2 +λ J 3 λ 1 λ 2 λ J 2 J 1 J 3 × tũ nt nũ (pqJ 1 |S λ 1 | tũJ 3 ) ( tũJ 3 |T λ 2 |rsJ 2 ) . ( 6.122)
In the special case of scalar operators, i.e. λ = λ 1 = λ 2 = 0, the AMR form of the commutator simplifies to

(pqJ|C 0 |rsJ) = 1 2 Ĵ tũ nt nũ (pqJ|S 0 | tũJ) ( tũJ|T 0 |rsJ) , ( 6.123)
where the following property on 6j-symbol 0 0 0 j 1 j 2 j 3 = (-) 2j 1 δ j 1 j 2 δ j 1 j 3 ĵ1 (6.124) has been used. Equation (6.123) can be rewritten in terms of angular-momentum-coupled matrix elements (Eq. (6.41)) instead of reduced matrix elements giving

J C pqrs = 1 2 tũ nt nũ J S pq tũ J T tũrs . ( 6
.125)

The AMC program

The AMC program has been designed to work on any computer with a Python3 distribution. It requires no additional libraries. One however needs a L A T E X distribution installed in order to produce a pdf file associated to the output tex file. The AMC program consists in two parts

(1) a first part whose role is to translate strings of CG coefficients into a Yutsis graph and then proceed to its reduction,

(2) a second part whose role is to obtain the string of CG coefficients corresponding to a given SU-TN, i.e. a given many-body equation, feed it into (1) and retrieve the fully reduced expression in order to write the corresponding AMR-TN in a tex file.

This section is not meant to give a detailed presentation of the code but rather to explain its general operation.

Clebsch Gordan coefficients reduction

The first part of the AMC program is related to the reduction of strings of CG coefficients via a graphical representation. Starting from a given string of CG coefficients, with possibly some (-1) j-m phases and ĵ factors, the first step consists in translating each one of them into their 3jm-symbols counterparts according to Eq. (6.46). In the second step, the resulting string of 3jm-symbols is put into its canonical form, i.e., signs of projection quantum numbers m k in 3jm-symbols are inverted, with resort to Eq. (6.49), in order to obtain one positive (+m k ) and one negative (-m k ) occurences for each quantum numbers k. Furthermore, each negative occurence of a projection quantum number (-m k ) in a 3jm-symbol is accompanied with a phase (-1) j-m . This corresponds to the changes brought to the algebraic expression of the 9j-symbol going from Eq. (6.57) to Eq. (6.58). In this work it has been chosen, for convenience, that the treatment of 3jm-symbols with zero lines, i.e. with at least one zero index j = 0, is done prior to the Yutsis graph creation. As a 3jm-symbol with a zero line corresponds to a Kronecker delta between non-zero indices, along with some phase and factor, it can be easily processed without the knowledge of the entire Yutsis graph. In this way one is led to work with a closed Yutsis graph, which is somewhat more convenient.

Once the string of 3jm-symbols is brought into its canonical form, the corresponding closed Yutsis graph is created. The final step corresponds to the reduction of this Yutsis graph by successive applications of the reduction rules, see Sec. 6.4.6, together with potential applications of the line reversal ((-1) 2j ) and direct node sign change ((-1) j 1 +j 2 +j 3 ) if needed. In practice the reduction process is done according to the algorithm sketched in Fig. 6.5.

ReduceYutsisGraph(YutsisGraph Y) 1: while Y != 3j-symbol do 2:
if Y has bubble then 3:

Format and remove arbitrary bubble 4:

else if Y has triangle then 5:

Format and remove arbitrary triangle 6:

else if Y has square then 7:

Format and remove arbitrary square 8: else 9:

//Implement higher-order rules 10: end if 11: end while 12: return formula The final output is a string of 3j-and 6j-symbols together with some (-1) j phases and ĵ factors. Whereas the present work focuses on the reduction of SU-TNs into their AMR-TN counterparts, this part is interesting in itself such that the AMC program is able to provide the reduction of strings of CG coefficients independently of their origin.

Tensor networks reduction

The second part of the AMC program is related to the reduction of SU-TNs into their AMR-TN counterparts. The first step is to feed the code with expressions of the SU-TNs together with definitions of the associated SU-Ts along with their coupling schemes. This can be done in two ways. A first option is to input a python binary file (in a particular format not discussed here) which allows to easily interface the AMC program with ADG or any other symbolic python programs 16 . A second option, which is more user-friendly, consists in a text file that is written in a format sketched in Fig. 6.6.

declare C { mode=(2,2), latex="C" } declare S { mode=(2,2), latex="S" } declare T { mode=(2,2), latex="T" } declare nbar { mode=2, diagonal=true, latex="\bar{n}" } C_pqrs = 1/2 * sum_tu(nbar_t * nbar_u * S_pqtu * T_turs); The end product of this example is actually the one of Eq. ( 6.125). The first four lines correspond to the declaration of SU-Ts while the last one corresponds to the SU-TN that will be reduced. It is possible to include as many SU-TNs as necessary.

In a second step, the string of CG coefficients corresponding to the SU-TN is obtained by transforming each SU-T into its AMR-T counterpart, e.g. according to Eq. (6.42). The string of CG coefficients is then fed into the Yutsis graph part to be reduced. Additionally, the AMC program provides an option to automatically loop over equivalent indices of the SU-TNs, e.g. (t, u) indices for the SU-TN displayed in Fig. 6.6, in order to find the indices order that minimize the number of 6j-symbols and single-particle phases. In this way, the AMC program is able to find the most optimized form of the AMR-Ts.

Finally, the string of 3j-symbols, 6j-symbols, single-particle phases, ĵ factors and reduced matrix elements are gathered together. From this fully reduced expression, an output file is writen in L A T E X format.

Conclusions

In the present work, an automated tool to perform symbolic angular-momentum algebra has been designed. Imputing the working equations, i.e. symmetry-restricted tensor networks, at play in state-of-the-art nuclear many-body methods, the time-consuming and error-prone derivation of their angular-momentum-coupled form is performed in a matter of seconds. The design of the tool is based on the use of Yutsis graph representing networks of Wigner 3jm-symbols and fulfilling a set of factorization rules whose repeated applications eventually provide the angular-momentum-coupled form of the equations.

While examples of applications have been provided for many-body perturbation theory, coupled cluster theory and the in-medium similarity renormalization group method, the code can be interfaced with any many-body formalism of interest.

In view of obtaining the error prone, fast and numerically optimized implementation of involved many-body formalisms, the present code serves as the missing link between an automated tool used to generate the initial symmetry-unrestricted equations [START_REF] Arthuis | ADG: Automated generation and evaluation of many-body diagrams I. Bogoliubov many-body perturbation theory[END_REF] and an automated tool used to produce the efficient source code dedicated to numerical applications.

While the present work focuses on SU (2), extensions are envisioned for the future, e.g. to the subgroup of SU (2) at play in axially deformed nuclei, or to other symmetry groups.

Conclusion

In the past decades, ab initio many-body methods [20, 23-26, 28, 30-32, 34] have been established as a leading and rapidly-evolving domain of nuclear structure theory. The simultaneous development of chiral EFT Hamiltonians [START_REF] Weinberg | Nuclear forces from chiral lagrangians[END_REF][START_REF] Weinberg | Effective chiral lagrangians for nucleon-pion interactions and nuclear forces[END_REF][START_REF] Epelbaum | Modern theory of nuclear forces[END_REF][START_REF] Meißner | The long and winding road from chiral effective Lagrangians to nuclear structure[END_REF][START_REF] Epelbaum | Nuclear chiral EFT in the precision era[END_REF] rooted in QCD and of many-body methods, accompanied by a rapid growth in computing power, have extended the reach of ab initio calculations to the mid-mass region of the nuclear chart up to A ∼ 130 [START_REF] Binder | Ab initio path to heavy nuclei[END_REF].

First limited to mid-mass doubly-closed shell (or neighboring) nuclei, ab initio many-body methods have recently extended their reach beyond nuclear shell closures via the use of more general reference states. Some methods are based on multi-determinental reference state (MR), e.g. the MR-IMSRG [START_REF] Hergert | The In-Medium Similarity Renormalization Group: A novel ab initio method for nuclei[END_REF][START_REF] Hergert | Initio Calculations of Even Oxygen Isotopes with Chiral Two-Plus-Three-Nucleon Interactions[END_REF][START_REF] Hergert | In-medium similarity renormalization group for closed and open-shell nuclei[END_REF] or the NCSM with perturbative improvement [START_REF] Tichai | Open-shell nuclei from No-Core Shell Model with perturbative improvement[END_REF], whereas some others rely on the concept of spontaneous symmetry breaking in order to capture static correlations that are prototypical of open-shell nuclei. The broken symmetries in nuclear theory are usually (i) the U (1) gauge symmetry associated with particle-number conservation, in order to capture the superfluid character of singly-open-shell nuclei, and (ii) the SU (2) rotational symmetry associated with angular-momentum conservation, in order to capture the deformed character of doubly-open-shell nuclei. In the last ten years, novel ab initio many-body methods have been developped on top of symmetry-broken reference states, e.g. the Gorkov SCGF (GSCGF) framework [START_REF] Somà | Ab initio self-consistent Gorkov-Green's function calculations of semimagic nuclei: Formalism at second order with a twonucleon interaction[END_REF][START_REF] Somà | Ab initio Gorkov-Green's function calculations of open-shell nuclei[END_REF][START_REF] Somà | Chiral twoand three-nucleon forces along medium-mass isotope chains[END_REF], the Bogoliubov CC (BCC) formalism [START_REF] Duguet | Symmetry broken and restored coupled-cluster. Global gauge symmetry and particle number[END_REF][START_REF] Signoracci | Ab initio Bogoliubov coupled cluster theory for open-shell nuclei[END_REF], the Bogoliubov MBPT (BMBPT) [START_REF] Tichai | Bogoliubov many-body perturbation theory for open-shell nuclei[END_REF][START_REF] Arthuis | Bogoliubov Many-Body Perturbation Theory for Nuclei : Systematic Generation and Evaluation of Diagrams and First ab initio Calculations[END_REF] and even the Bogoliubov CI (BCI) [START_REF] Ripoche | Combining symmetry breaking and restoration with configuration interaction: A highly accurate many-body scheme applied to the pairing Hamiltonian[END_REF][START_REF] Ripoche | Combining symmetry breaking and restoration with configuration interaction: Extension to z -signature symmetry in the case of the Lipkin model[END_REF], the last one not being an expansion method. A difficulty encountered by these methods is that the symmetry breaking cannot actually occur in finite quantum systems such that results of many-body calculations carry a contamination associated with contributions originating from several irreducible representations of the broken symmetry group.

The present work has focused on the restoration of U (1) symmetry in ab initio many-body expansion methods, in particular in the particle-number-projected version of BMBPT that is coined as projected BMBPT (PBMBPT) [START_REF] Duguet | Symmetry broken and restored coupled-cluster. Global gauge symmetry and particle number[END_REF][START_REF] Duguet | Symmetry broken and restored coupled-cluster theory: I. Rotational symmetry and angular momentum[END_REF][START_REF] Qiu | Particle-number projected Bogoliubov-coupled-cluster theory: Application to the pairing Hamiltonian[END_REF] with the goal of performing ab initio calculations of singly-open-shell mid-and heavy-mass nuclei. Along these developments, several non-trivial formal and technical difficulties were standing on the way. It has been an integral part of this work to provide the most general solutions to these challenges that can also be used in other ab initio many-body methods.

Chapter 1 presented the formal ingredients needed all throughout the document, in particular to formulate PBMBPT. Chapter 2 introduced the fully non-perturbative projected BCC (PBCC) [START_REF] Duguet | Symmetry broken and restored coupled-cluster. Global gauge symmetry and particle number[END_REF][START_REF] Qiu | Particle-number projected Bogoliubov-coupled-cluster theory: Application to the pairing Hamiltonian[END_REF][START_REF] Qiu | Projected coupled cluster theory[END_REF] formalism and presented the extraction of its perturbative version, i.e. PBMBPT. This was done in two variants, based on gauge-rotated amplitudes and unrotated amplitudes. The introduction of PBMBPT provided the background of the four following chapters in which the original achievements of the thesis were discussed. In chapter 3, one version of PBMBPT was revisited such that off-diagonal kernels themselves were expanded in perturbation. Whereas low orders given in chap. 2 were worked out in Ref. [START_REF] Duguet | Symmetry broken and restored coupled-cluster. Global gauge symmetry and particle number[END_REF], the goal was to design a new version of the code ADG [START_REF] Arthuis | ADG: Automated generation and evaluation of many-body diagrams I. Bogoliubov many-body perturbation theory[END_REF] that could automatically generate all valid off-diagonal BMBPT diagrams to arbitrary order and evaluate their algebraic expression to be implemented for numerical applications. In chapter 4, a framework providing an efficient way to compute matrix elements between two (particle-number-projected) many-body states has been designed. This framework allows to compute other observables than the energy and particle number but also to implement the eigenvector continuation on top of (P)BMBPT.

The following chapters, while of direct interest to PBMBPT, have a broader reach. Chapter 5 presented an extension of the NO2B approximation, often used in nuclear manybody theory in order to capture the dominant effects of three-nucleon forces while effectively working with two-body operators, that is consistent with symmetries of the Hamiltonian while working with a symmetry broken reference state. This generalized approximation has already been employed in Gorkov SCGF [START_REF] Somà | Ab initio self-consistent Gorkov-Green's function calculations of semimagic nuclei: Formalism at second order with a twonucleon interaction[END_REF][START_REF] Somà | Ab initio Gorkov-Green's function calculations of open-shell nuclei[END_REF][START_REF] Somà | Ab initio self-consistent Gorkov-Green's function calculations of semi-magic nuclei: Numerical implementation at second order with a two-nucleon interaction[END_REF] and BMBPT [START_REF] Tichai | Bogoliubov many-body perturbation theory for open-shell nuclei[END_REF][START_REF] Tichai | Pre-processing the nuclear many-body problem[END_REF], it can be used in any many-body methods relying on the normal ordering of the Hamiltonian with respect to a U (1) symmetry-broken reference state. Last but not least, chapter 6 introduced a systematic method, based on the so-called Yutsis graph [START_REF] Yutsis | Mathematical apparatus of the theory of angular momentum[END_REF], used to perform the angular-momentum reduction of many-body equations in an automated and errorfree way. The first version of the AMC program that automatically performs this tedious symmetry reduction has been presented. Taking the symmetry-unrestricted expression of a generic tensor network as an input, the code provides its angular-momentum-coupled form in an error-safe way in a matter of seconds. This program is meant to be public and will be published in the coming months. It will provide a great tool to push various many-body methods to greater accuracy by implementing involved tensor networks associated with higher order truncations.

Whereas PBMBPT calculations of singly-open-shell nuclei are not appearing in this document, the implementation is close to completion, thanks to the presently designed formal tools, and first results should appear in the coming months. Our goal is to extend (P)BMBPT in several directions in the future. A first step along this line is the study of the order-by-order convergence of the BMBPT series. As BMBPT beyond first order presents particle-number perturbative corrections, a consistent adjustment protocol of these corrections at arbitrary order has to be designed. This has been achieved by the work of P. Demol and M. Frosini and will lead to a future publication [START_REF] Demol | High-order Bogoliubov Many-Body Perturbation Theory[END_REF]. A study of the importance of the remaining particle-number contaminations in BMBPT with respect to PBMBPT at a given order will be performed in the future. In the same way as particle-number-restored BMBPT, the implementation of U (1)-breaking BCC and its restored version are close to completion. On the longer term and in order to extend the reach of both BMBPT and BCC to doubly-open-shell nuclei, other extensions that consistently breaks (and restores) both SU (2) and U (1) symmetries are planned.

Adding n κ * , resp. n κ , anomalous κ * , resp. κ contractions, the prefactor becomes

Factor(n ρ , n κ * , n κ ) n ≡ 1 n ρ ! 1 n κ * ! n-n ρ 2 n-n ρ -2 2 . . . n-n ρ -2(n κ * -1) 2 (n -n ρ )! × 1 n κ ! n-n ρ 2 n-n ρ -2 2 . . . n-n ρ -2(n κ -1) 2 (n -n ρ )! = 1 n ρ !n κ * !n κ ! 1 2 n κ * 1 2 n κ 1 (n -n ρ -2n κ * )!(n -n ρ -2n κ )! . (A.3)
The term whose prefactor is given by Eq. (A.3) contributes to Λ ij with i = n -n ρ -2n κ * and j = n -n ρ -2n κ . This prefactor accounts for the factor 1/i!j! involved in the definition of the operator Λ ij in Eq. (1.78) and for the factor appearing in Eq. (1.80).

A.1.2. Example

Taking a particle-number conserving three-body operator O, the matrix elements Λ ij l 1 ...l i l i+1 ...l i+j are given by

Λ 00 ≡ 3 n=1 Λ 00(nn) = o 00 + l 1 l 2 o 11 l 1 l 2 ρ l 2 l 1 + 1 2 l 1 l 2 l 3 l 4 o 22 l 1 l 2 l 3 l 4 ρ l 3 l 1 ρ l 4 l 2 + 1 4 l 1 l 2 l 3 l 4 o 22 l 1 l 2 l 3 l 4 κ * l 1 l 2 κ l 3 l 4 + 1
6 l 1 l 2 l 3 l 4 l 5 l 6 o 33 l 1 l 2 l 3 l 4 l 5 l 6 ρ l 4 l 1 ρ l 5 l 2 ρ l 6 l 3 + 1 4 l 1 l 2 l 3 l 4 l 5 l 6 o 33 l 1 l 2 l 3 l 4 l 5 l 6 κ * l 1 l 2 κ l 4 l 5 ρ l 6 l 3 , (A.4a)

Λ 11 l 1 l 2 ≡ 3 n=1 Λ 11(nn) l 1 l 2 = o 11 l 1 l 2 + l 3 l 4 o 22 l 1 l 3 l 2 l 4 ρ l 4 l 3 + 1 2 l 3 l 4 l 5 l 6 o 33 l 1 l 3 l 4 l 2 l 5 l 6 ρ l 5 l 3 ρ l 6 l 4 + 1 4 l 3 l 4 l 5 l 6 o 33 l 1 l 3 l 4 l 2 l 5 l 6 κ * l 3 l 4 κ l 5 l 6 , (A.4b) Λ 20 l 1 l 2 ≡ 3 n=2 Λ 20(nn) l 1 l 2 = 1 2 l 3 l 4 o 22 l 1 l 2 l 3 l 4 κ l 3 l 4 + 1 2 l 3 l 4 l 5 l 6 o 33 l 1 l 2 l 3 l 4 l 5 l 6 κ l 4 l 5 ρ l 6 l 3 , (A.4c) Λ 02 l 1 l 2 ≡ 3 n=2 Λ 02(nn) l 1 l 2 = 1 2 l 3 l 4 o 22 l 3 l 4 l 1 l 2 κ * l 3 l 4 + 1 2 l 3 l 4 l 5 l 6 o 33 l 4 l 5 l 6 l 1 l 2 l 3 κ * l 4 l 5 ρ l 3 l 6 , (A.4d) Λ 22 l 1 l 2 l 3 l 4 ≡ 3 n=2 Λ 22(nn) l 1 l 2 l 3 l 4 = o 22 l 1 l 2 l 3 l 4 + l 5 l 6 o 33 l 1 l 2 l 5 l 3 l 4 l 6 ρ l 6 l 5 , (A.4e) Λ 31 l 1 l 2 l 3 l 4 ≡ Λ 31(33) l 1 l 2 l 3 l 4 = 1 2 l 5 l 6 o 33 l 1 l 2 l 3 l 4 l 5 l 6 κ l 5 l 6 , (A.4f) Λ 13 l 1 l 2 l 3 l 4 ≡ Λ 13(33) l 1 l 2 l 3 l 4 = 1 2 l 5 l 6 o 33 l 1 l 5 l 6 l 2 l 3 l 4 κ * l 5 l 6 , (A.4g) Λ 33 l 1 l 2 l 3 l 4 l 5 l 6 ≡ Λ 33(33)
l 1 l 2 l 3 l 4 l 5 l 6 = o 33 l 1 l 2 l 3 l 4 l 5 l 6 .

(A.4h)

In particular the quantities Λ 11 and Λ 20 , that are called normal and anomalous one-body fields, can be obtained as functional derivatives of the scalar part

Λ 11 l 1 l 2 = ∂Λ 00 ∂ρ l 2 l 1 , (A.5a) Λ 20 l 1 l 2 = ∂Λ 00 ∂κ * l 2 l 1 . (A.5b)
The graphical representation of O expressed in normal-ordered form with respect to |Φ is given in Tab. A.1. Contrarily to the case displayed in Tab. 1.1, non-diagonal contributions Λ 20 , Λ 02 , Λ 31 and Λ 13 appear.

{c, c † }, |Φ -6 -4 -2 0 +2 +4 +6 Λ [0] Λ 00 Λ [2] Λ 02 Λ 11 Λ 20 Λ [4] Λ 13 Λ 22 Λ 31 Λ [6] Λ 33
Table A.1. Contributions to the three-body operator O in normal-ordered form with respect to the Bogoliubov vacuum |Φ and expressed in the single-particle basis {c, c † }. The Λ ij contributions are sorted horizontally according to i -j and vertically according to i + j.

In the Slater determinant limit, for which ρ ip = δ ip , ρ ap = 0 and κ pq = 0

A.2. Quasi-particle basis

The derivation relates to the application of Wick's theorem with respect to |Φ , which is particularly straighforward using the set of quasi-particle operators of which |Φ is a

{c, c † }, |SD -6 -4 -2 0 +2 +4 +6 Λ [0] Λ 00 Λ [2] Λ 11 Λ [4] Λ 22
Λ [6] Λ 33

Table A.2. Contributions to the three-body operator O in normal-ordered form with respect to the slater determinant |SD and expressed in the single-particle basis {c, c † }.

Λ ij contributions are sorted horizontally according to i -j and vertically according to i + j.

vacuum. This procedure has already been discussed and examplified in Ref. [START_REF] Signoracci | Ab initio Bogoliubov coupled cluster theory for open-shell nuclei[END_REF] for a three-body operator O commuting with A. However, the expressions of the matrix elements O ij k 1 ...k i k i+1 ...k i+j were only provided for O [0] , O [2] and O [4] . Extending for completeness the results of Ref. [START_REF] Signoracci | Ab initio Bogoliubov coupled cluster theory for open-shell nuclei[END_REF], matrix elements up to O [6] are given in Eq. (A.7).

The graphical representation of the normal-ordered operator O with respect to |Φ , resp. |SD , and expressed in the quasi-particle basis, is given in Tab. A.3, resp. Tab A.4. One notices the presence of non-diagonal contributions in Tab. A.3 (as in Tab. A.1) and in Tab. A.4 (even if there is none in Tab. A.2). This is due to the use of the quasi-particle algebra that is agnostic regarding the symmetry-breaking or the symmetry-preserving character of the vacuum.

{β, β † }, |Φ -6 -4 -2 0 +2 +4 +6 O [0] O 00
O [2] O 11) 20) 11) 20) 40) 40)

O 20 k 1 k = l 1 l 2 Λ 11 l 1 l 2 U * l 1 k 1 V * l 2 k 2 -Λ 11 l 1 l 2 V * l 2 k 1 U * l 1 k 2 + Λ 20 l 1 l 2 U * l 1 k 1 U * l 2 k 2 -Λ 02 l 1 l 2 V * l 1 k 1 V * l 2 k 2 , (A.7b) O 11 k 1 k = l 1 l 2 Λ 11 l 1 l 2 U * l 1 k 1 U l 2 k 2 -Λ 11 l 1 l 2 V * l 2 k 1 V l 1 k 2 + Λ 20 l 1 l 2 U * l 1 k 1 V l 2 k 2 -Λ 02 l 1 l 2 V * l 1 k 1 U l 2 k 2 , (A.7c) O 02 k 1 k = l 1 l 2 Λ 11 l 1 l 2 U l 2 k 1 V l 1 k 2 -Λ 11 l 1 l 2 V l 1 k 1 U l 2 k 2 -Λ 20 l 1 l 2 V l 1 k 1 V l 2 k 2 + Λ 02 l 1 l 2 U l 1 k 1 U l 2 k 2 , (A.7d) O 40 k 1 k 2 k 3 k = l 1 l 2 l 3 l 4 Λ 22 l 1 l 2 l 3 l 4 -U * l 1 k 1 U * l 2 k 2 V * l 3 k 3 V * l 4 k 4 + U * l 1 k 1 V * l 3 k 2 U * l 2 k 3 V * l 4 k 4 -U * l 1 k 1 V * l 3 k 2 V * l 4 k 3 U * l 2 k 4 -V * l 3 k 1 U * l 1 k 2 U * l 2 k 3 V * l 4 k 4 + V * l 3 k 1 U * l 1 k 2 V * l 4 k 3 U * l 2 k 4 -V * l 3 k 1 V * l 4 k 2 U * l 1 k 3 U * l 2 k 4 + Λ 31 l 1 l 2 l 3 l 4 + U * l 1 k 1 U * l 2 k 2 U * l 3 k 3 V * l 4 k 4 -U * l 1 k 1 U * l 2 k 2 V * l 4 k 3 U * l 3 k 4 + U * l 1 k 1 V * l 4 k 2 U * l 2 k 3 U * l 3 k 4 -V * l 4 k 1 U * l 1 k 2 U * l 2 k 3 U * l 3 k 4 + Λ 13 l 1 l 2 l 3 l 4 -U * l 1 k 1 V * l 5 k 2 V * l 6 k 3 V * l 2 k 4 + V * l 5 k 1 U * l 1 k 2 V * l 6 k 3 V * l 2 k 4 -V * l 5 k 1 V * l 6 k 2 U * l 1 k 3 V * l 2 k 4 + V * l 5 k 1 V * l 6 k 2 V * l 2 k 3 U * l 1 k 4 , (A.7e) O 31 k 1 k 2 k 3 k = l 1 l 2 l 3 l 4 Λ 22 l 1 l 2 l 3 l 4 -U * l 1 k 1 V * l 3 k 2 V * l 4 k 3 V l 2 k 4 + V * l 3 k 1 U * l 1 k 2 V * l 4 k 3 V l 2 k 4 -V * l 3 k 1 V * l 4 k 2 U * l 1 k 3 V l 2 k 4 -U * l 1 k 1 U * l 2 k 2 V * l 3 k 3 U l 4 k 4 + U * l 1 k 1 V * l 3 k 2 U * l 2 k 3 U l 4 k 4 -V * l 3 k 1 U * l 1 k 2 U * l 2 k 3 U l 4 k 4 + Λ 31 l 1 l 2 l 3 l 4 -U * l 1 k 1 U * l 2 k 2 V * l 4 k 3 V l 3 k 4 + U * l 1 k 1 V * l 4 k 2 U * l 2 k 3 V l 3 k 4 -V * l 4 k 1 U * l 1 k 2 U * l 2 k 3 V l 3 k 4 + U * l 1 k 1 U * l 2 k 2 U * l 3 k 3 U l 4 k 4 + Λ 13 l 1 l 2 l 3 l 4 + V * l 5 k 1 V * l 6 k 2 V * l 2 k 3 V l 1 k 4 -U * l 1 k 1 V * l 5 k 2 V * l 6 k 3 U l 2 k 4 + V * l 5 k 1 U * l 1 k 2 V * l 6 k 3 U l 2 k 4 -V * l 5 k 1 V * l 6 k 2 U * l 1 k 3 U l 2 k 4 , (A.7f) O 22 k 1 k 2 k 3 k = l 1 l 2 l 3 l 4 Λ 22 l 1 l 2 l 3 l 4 + V * l 3 k 1 V * l 4 k 2 V l 1 k 3 V l 2 k 4 + U * l 1 k 1 V * l 3 k 2 U l 4 k 3 V l 2 k 4 -U * l 1 k 1 V * l 3 k 2 V l 2 k 3 U l 4 k 4 -V * l 3 k 1 U * l 1 k 2 U l 4 k 3 V l 2 k 4 + V * l 3 k 1 U * l 1 k 2 V l 2 k 3 U l 4 k 4 + U * l 1 k 1 U * l 2 k 2 U l 3 k 3 U l 4 k 4 + Λ 31 l 1 l 2 l 3 l 4 -U * l 1 k 1 V * l 4 k 2 V l 2 k 3 V l 3 k 4 + V * l 4 k 1 U * l 1 k 2 V l 2 k 3 V l 3 k 4 + U * l 1 k 1 U * l 2 k 2 U l 4 k 3 V l 3 k 4 -U * l 1 k 1 U * l 2 k 2 V l 3 k 3 U l 4 k 4 + Λ 13 l 1 l 2 l 3 l 4 -V * l 5 k 1 V * l 6 k 2 U l 2 k 3 V l 1 k 4 + V * l 5 k 1 V * l 6 k 2 V l 1 k 3 U l 2 k 4 + U * l 1 k 1 V * l 5 k 2 U l 6 k 3 U l 2 k 4 -V * l 5 k 1 U * l 1 k 2 U l 6 k 3 U l 2 k 4 , (A.7g) O 13 k 1 k 2 k 3 k = l 1 l 2 l 3 l 4 Λ 22 l 1 l 2 l 3 l 4 + V * l 3 k 1 U l 4 k 2 V l 1 k 3 V l 2 k 4 -V * l 3 k 1 V l 1 k 2 U l 4 k 3 V l 2 k 4 + V * l 3 k 1 V l 1 k 2 V l 2 k 3 U l 4 k 4 + U * l 1 k 1 U l 3 k 2 U l 4 k 3 V l 2 k 4 -U * l 1 k 1 U l 3 k 2 V l 2 k 3 U l 4 k 4 + U * l 1 k 1 V l 2 k 2 U l 3 k 3 U l 4 k 4 + Λ 31 l 1 l 2 l 3 l 4 + V * l 4 k 1 V l 1 k 2 V l 2 k 3 V l 3 k 4 -U * l 1 k 1 U l 4 k 2 V l 2 k 3 V l 3 k 4 + U * l 1 k 1 V l 2 k 2 U l 4 k 3 V l 3 k 4 -U * l 1 k 1 V l 2 k 2 V l 3 k 3 U l 4 k 4 + Λ 13 l 1 l 2 l 3 l 4 -V * l 5 k 1 U l 6 k 2 U l 2 k 3 V l 1 k 4 + V * l 5 k 1 U l 6 k 2 V l 1 k 3 U l 2 k 4 -V * l 5 k 1 V l 1 k 2 U l 6 k 3 U l 2 k 4 + U * l 1 k 1 U l 5 k 2 U l 6 k 3 U l 2 k 4 , (A.7h) O 04 k 1 k 2 k 3 k = l 1 l 2 l 3 l 4 Λ 22 l 1 l 2 l 3 l 4 -U l 3 k 1 U l 4 k 2 V l 1 k 3 V l 2 k 4 + U l 3 k 1 V l 1 k 2 U l 4 k 3 V l 2 k 4 -U l 3 k 1 V l 1 k 2 V l 2 k 3 U l 4 k 4 -V l 1 k 1 U l 3 k 2 U l 4 k 3 V l 2 k 4 + V l 1 k 1 U l 3 k 2 V l 2 k 3 U l 4 k 4 -V l 1 k 1 V l 2 k 2 U l 3 k 3 U l 4 k 4 + Λ 31 l 1 l 2 l 3 l 4 -U l 4 k 1 V l 1 k 2 V l 2 k 3 V l 3 k 4 + V l 1 k 1 U l 4 k 2 V l 2 k 3 V l 3 k 4 -V l 1 k 1 V l 2 k 2 U l 4 k 3 V l 3 k 4 + V l 1 k 1 V l 2 k 2 V l 3 k 3 U l 4 k 4 + Λ 13 l 1 l 2 l 3 l 4 + U l 5 k 1 U l 6 k 2 U l 2 k 3 V l 1 k 4 -U l 5 k 1 U l 6 k 2 V l 1 k 3 U l 2 k 4 + U l 5 k 1 V l 1 k 2 U l 6 k 3 U l 2 k 4 -V l 1 k 1 U l 5 k 2 U l 6 k 3 U l 2 k 4 , (A.7i) O 60 k 1 k 2 k 3 k 4 k 5 k = l 1 l 2 l 3 l 4 l 5 l 6 Λ 33 l 1 l 2 l 3 l 4 l 5 l 6 -U * l 1 k 1 U * l 2 k 2 U * l 3 k 3 V * l 4 k 4 V * l 5 k 5 V * l 6 k 6 + U * l 1 k 1 U * l 2 k 2 V * l 4 k 3 U * l 3 k 4 V * l 5 k 5 V * l 6 k 6 -U * l 1 k 1 U * l 2 k 2 V * l 4 k 3 V * l 5 k 4 U * l 3 k 5 V * l 6 k 6 + U * l 1 k 1 U * l 2 k 2 V * l 4 k 3 V * l 5 k 4 V * l 6 k 5 U * l 3 k 6 -U * l 1 k 1 V * l 4 k 2 U * l 2 k 3 U * l 3 k 4 V * l 5 k 5 V * l 6 k 6 + U * l 1 k 1 V * l 4 k 2 U * l 2 k 3 V * l 5 k 4 U * l 3 k 5 V * l 6 k 6 -U * l 1 k 1 V * l 4 k 2 U * l 2 k 3 V * l 5 k 4 V * l 6 k 5 U * l 3 k 6 -U * l 1 k 1 V * l 4 k 2 V * l 5 k 3 U * l 2 k 4 U * l 3 k 5 V * l 6 k 6 + U * l 1 k 1 V * l 4 k 2 V * l 5 k 3 U * l 2 k 4 V * l 6 k 5 U * l 3 k 6 -U * l 1 k 1 V * l 4 k 2 V * l 5 k 3 V * l 6 k 4 U * l 2 k 5 U * l 3 k 6 + V * l 4 k 1 U * l 1 k 2 U * l 2 k 3 U * l 3 k 4 V * l 5 k 5 V * l 6 k 6 -V * l 4 k 1 U * l 1 k 2 U * l 2 k 3 V * l 5 k 4 U * l 3 k 5 V * l 6 k 6 + V * l 4 k 1 U * l 1 k 2 U * l 2 k 3 V * l 5 k 4 V * l 6 k 5 U * l 3 k 6 + V * l 4 k 1 U * l 1 k 2 V * l 5 k 3 U * l 2 k 4 U * l 3 k 5 V * l 6 k 6 -V * l 4 k 1 U * l 1 k 2 V * l 5 k 3 U * l 2 k 4 V * l 6 k 5 U * l 3 k 6 + V * l 4 k 1 U * l 1 k 2 V * l 5 k 3 V * l 6 k 4 U * l 2 k 5 U * l 3 k 6 -V * l 4 k 1 V * l 5 k 2 U * l 1 k 3 U * l 2 k 4 U * l 3 k 5 V * l 6 k 6 + V * l 4 k 1 V * l 5 k 2 U * l 1 k 3 U * l 2 k 4 V * l 6 k 5 U * l 3 k 6 -V * l 4 k 1 V * l 5 k 2 U * l 1 k 3 V * l 6 k 4 U * l 2 k 5 U * l 3 k 6 + V * l 4 k 1 V * l 5 k 2 V * l 6 k 3 U * l 1 k 4 U * l 2 k 5 U * l 3 k 6 , (A.7j) O 51 k 1 k 2 k 3 k 4 k 5 k 6 = l 1 l 2 l 3 l 4 l 5 l 6 Λ 33 l 1 l 2 l 3 l 4 l 5 l 6 + U * l 1 k 1 U * l 2 k 2 V * l 4 k 3 V * l 5 k 4 V * l 6 k 5 V l 3 k 6 -U * l 1 k 1 V * l 4 k 2 U * l 2 k 3 V * l 5 k 4 V * l 6 k 5 V l 3 k 6 + U * l 1 k 1 V * l 4 k 2 V * l 5 k 3 U * l 2 k 4 V * l 6 k 5 V l 3 k 6 -U * l 1 k 1 V * l 4 k 2 V * l 5 k 3 V * l 6 k 4 U * l 2 k 5 V l 3 k 6 + V * l 4 k 1 U * l 1 k 2 U * l 2 k 3 V * l 5 k 4 V * l 6 k 5 V l 3 k 6 -V * l 4 k 1 U * l 1 k 2 V * l 5 k 3 U * l 2 k 4 V * l 6 k 5 V l 3 k 6 + V * l 4 k 1 U * l 1 k 2 V * l 5 k 3 V * l 6 k 4 U * l 2 k 5 V l 3 k 6 + V * l 4 k 1 V * l 5 k 2 U * l 1 k 3 U * l 2 k 4 V * l 6 k 5 V l 3 k 6 -V * l 4 k 1 V * l 5 k 2 U * l 1 k 3 V * l 6 k 4 U * l 2 k 5 V l 3 k 6 + V * l 4 k 1 V * l 5 k 2 V * l 6 k 3 U * l 1 k 4 U * l 2 k 5 V l 3 k 6 -U * l 1 k 1 U * l 2 k 2 U * l 3 k 3 V * l 4 k 4 V * l 5 k 5 U l 6 k 6 + U * l 1 k 1 U * l 2 k 2 V * l 4 k 3 U * l 3 k 4 V * l 5 k 5 U l 6 k 6 -U * l 1 k 1 U * l 2 k 2 V * l 4 k 3 V * l 5 k 4 U * l 3 k 5 U l 6 k 6 -U * l 1 k 1 V * l 4 k 2 U * l 2 k 3 U * l 3 k 4 V * l 5 k 5 U l 6 k 6 + U * l 1 k 1 V * l 4 k 2 U * l 2 k 3 V * l 5 k 4 U * l 3 k 5 U l 6 k 6 -U * l 1 k 1 V * l 4 k 2 V * l 5 k 3 U * l 2 k 4 U * l 3 k 5 U l 6 k 6 + V * l 4 k 1 U * l 1 k 2 U * l 2 k 3 U * l 3 k 4 V * l 5 k 5 U l 6 k 6 -V * l 4 k 1 U * l 1 k 2 U * l 2 k 3 V * l 5 k 4 U * l 3 k 5 U l 6 k 6 + V * l 4 k 1 U * l 1 k 2 V * l 5 k 3 U * l 2 k 4 U * l 3 k 5 U l 6 k 6 -V * l 4 k 1 V * l 5 k 2 U * l 1 k 3 U * l 2 k 4 U * l 3 k 5 U l 6 k 6 , (A.7k) O 42 k 1 k 2 k 3 k 4 k 5 k 6 = l 1 l 2 l 3 l 4 l 5 l 6 Λ 33 l 1 l 2 l 3 l 4 l 5 l 6 + U * l 1 k 1 V * l 4 k 2 V * l 5 k 3 V * l 6 k 4 V l 2 k 5 V l 3 k 6 -V * l 4 k 1 U * l 1 k 2 V * l 5 k 3 V * l 6 k 4 V l 2 k 5 V l 3 k 6 + V * l 4 k 1 V * l 5 k 2 U * l 1 k 3 V * l 6 k 4 V l 2 k 5 V l 3 k 6 -V * l 4 k 1 V * l 5 k 2 V * l 6 k 3 U * l 1 k 4 V l 2 k 5 V l 3 k 6 -U * l 1 k 1 U * l 2 k 2 V * l 4 k 3 V * l 5 k 4 U l 6 k 5 V l 3 k 6 + U * l 1 k 1 U * l 2 k 2 V * l 4 k 3 V * l 5 k 4 V l 3 k 5 U l 6 k 6 + U * l 1 k 1 V * l 4 k 2 U * l 2 k 3 V * l 5 k 4 U l 6 k 5 V l 3 k 6 -U * l 1 k 1 V * l 4 k 2 U * l 2 k 3 V * l 5 k 4 V l 3 k 5 U l 6 k 6 -U * l 1 k 1 V * l 4 k 2 V * l 5 k 3 U * l 2 k 4 U l 6 k 5 V l 3 k 6 + U * l 1 k 1 V * l 4 k 2 V * l 5 k 3 U * l 2 k 4 V l 3 k 5 U l 6 k 6 -V * l 4 k 1 U * l 1 k 2 U * l 2 k 3 V * l 5 k 4 U l 6 k 5 V l 3 k 6 + V * l 4 k 1 U * l 1 k 2 U * l 2 k 3 V * l 5 k 4 V l 3 k 5 U l 6 k 6 + V * l 4 k 1 U * l 1 k 2 V * l 5 k 3 U * l 2 k 4 U l 6 k 5 V l 3 k 6 -V * l 4 k 1 U * l 1 k 2 V * l 5 k 3 U * l 2 k 4 V l 3 k 5 U l 6 k 6 -V * l 4 k 1 V * l 5 k 2 U * l 1 k 3 U * l 2 k 4 U l 6 k 5 V l 3 k 6 + V * l 4 k 1 V * l 5 k 2 U * l 1 k 3 U * l 2 k 4 V l 3 k 5 U l 6 k 6 + U * l 1 k 1 U * l 2 k 2 U * l 3 k 3 V * l 4 k 4 U l 5 k 5 U l 6 k 6 -U * l 1 k 1 U * l 2 k 2 V * l 4 k 3 U * l 3 k 4 U l 5 k 5 U l 6 k 6 + U * l 1 k 1 V * l 4 k 2 U * l 2 k 3 U * l 3 k 4 U l 5 k 5 U l 6 k 6 -V * l 4 k 1 U * l 1 k 2 U * l 2 k 3 U * l 3 k 4 U l 5 k 5 U l 6 k 6 , (A.7l) O 33 k 1 k 2 k 3 k 4 k 5 k 6 = l 1 l 2 l 3 l 4 l 5 l 6 Λ 33 l 1 l 2 l 3 l 4 l 5 l 6 -V * l 4 k 1 V * l 5 k 2 V * l 6 k 3 V l 1 k 4 V l 2 k 5 V l 3 k 6 + U * l 1 k 1 V * l 4 k 2 V * l 5 k 3 U l 6 k 4 V l 2 k 5 V l 3 k 6 -U * l 1 k 1 V * l 4 k 2 V * l 5 k 3 V l 2 k 4 U l 6 k 5 V l 3 k 6 + U * l 1 k 1 V * l 4 k 2 V * l 5 k 3 V l 2 k 4 V l 3 k 5 U l 6 k 6 -V * l 4 k 1 U * l 1 k 2 V * l 5 k 3 U l 6 k 4 V l 2 k 5 V l 3 k 6 + V * l 4 k 1 U * l 1 k 2 V * l 5 k 3 V l 2 k 4 U l 6 k 5 V l 3 k 6 -V * l 4 k 1 U * l 1 k 2 V * l 5 k 3 V l 2 k 4 V l 3 k 5 U l 6 k 6 + V * l 4 k 1 V * l 5 k 2 U * l 1 k 3 U l 6 k 4 V l 2 k 5 V l 3 k 6 -V * l 4 k 1 V * l 5 k 2 U * l 1 k 3 V l 2 k 4 U l 6 k 5 V l 3 k 6 + V * l 4 k 1 V * l 5 k 2 U * l 1 k 3 V l 2 k 4 V l 3 k 5 U l 6 k 6 -U * l 1 k 1 U * l 2 k 2 V * l 4 k 3 U l 5 k 4 U l 6 k 5 V l 3 k 6 + U * l 1 k 1 U * l 2 k 2 V * l 4 k 3 U l 5 k 4 V l 3 k 5 U l 6 k 6 -U * l 1 k 1 U * l 2 k 2 V * l 4 k 3 V l 3 k 4 U l 5 k 5 U l 6 k 6 + U * l 1 k 1 V * l 4 k 2 U * l 2 k 3 U l 5 k 4 U l 6 k 5 V l 3 k 6 -U * l 1 k 1 V * l 4 k 2 U * l 2 k 3 U l 5 k 4 V l 3 k 5 U l 6 k 6 + U * l 1 k 1 V * l 4 k 2 U * l 2 k 3 V l 3 k 4 U l 5 k 5 U l 6 k 6 -V * l 4 k 1 U * l 1 k 2 U * l 2 k 3 U l 5 k 4 U l 6 k 5 V l 3 k 6 + V * l 4 k 1 U * l 1 k 2 U * l 2 k 3 U l 5 k 4 V l 3 k 5 U l 6 k 6 -V * l 4 k 1 U * l 1 k 2 U * l 2 k 3 V l 3 k 4 U l 5 k 5 U l 6 k 6 + U * l 1 k 1 U * l 2 k 2 U * l 3 k 3 U l 4 k 4 U l 5 k 5 U l 6 k 6 , (A.7m)

B.3. Normal ordering in quasi-particle basis

O 00(00) (ϕ) = Λ 00 (ϕ) (B.2a) O 20(20) k 1 k 2 (ϕ) = Λ 20 k 1 k 2 (ϕ) (B.2b) O 11(
k 1 k 2 (ϕ) = Λ 11 k 1 k 2 (ϕ) (B.2c) O 11(
k 1 k 2 (ϕ) = l 1 Λ 20 k 1 l 1 (ϕ)R ++ l 1 k 2 (ϕ) (B.2d) O 02(02) k 1 k 2 (ϕ) = Λ 02 k 1 k 2 (ϕ) (B.2e) O 02(
k 1 k 2 (ϕ) = P (k 1 /k 2 ) l 1 Λ 11 l 1 k 1 (ϕ)R ++ l 1 k 2 (ϕ) (B.2f) O 02(
k 1 k 2 (ϕ) = - l 1 l 2 Λ 20 l 1 l 2 (ϕ)R ++ l 1 k 1 (ϕ)R ++ l 2 k 2 (ϕ) (B.2g) O 40(
k 1 k 2 k 3 k 4 (ϕ) = Λ 40 k 1 k 2 k 3 k 4 (ϕ) (B.2h) O 31(31) k 1 k 2 k 3 k 4 (ϕ) = Λ 31 k 1 k 2 k 3 k 4 (ϕ) (B.2i) O 31(
k 1 k 2 k 3 k 4 (ϕ) = l 1 Λ 40 k 1 k 2 k 3 l 1 (ϕ)R ++ l 1 k 4 (ϕ) (B.2j) O 22(22) k 1 k 2 k 3 k 4 (ϕ) = Λ 22 k 1 k 2 k 3 k 4 (ϕ) (B.2k) O 22(31) k 1 k 2 k 3 k 4 (ϕ) = P (k 3 /k 4 ) l 1 Λ 31 k 1 k 2 l 1 k 3 (ϕ)R ++ l 1 k 4 (ϕ) (B.2l) O 22 (40)
k 1 k 2 k 3 k 4 (ϕ) = - l 1 l 2 Λ 40 k 1 k 2 l 1 l 2 (ϕ)R ++ l 1 k 3 (ϕ)R ++ l 2 k 4 (ϕ) (B.2m) O 13(13) k 1 k 2 k 3 k 4 (ϕ) = Λ 13 k 1 k 2 k 3 k 4 (ϕ) (B.2n) O 13(22) k 1 k 2 k 3 k 4 (ϕ) = P (k 2 k 3 /k 4 ) l 1 Λ 22 k 1 l 1 k 2 k 3 (ϕ)R ++ l 1 k 4 (ϕ) (B.2o) O 13(31) k 1 k 2 k 3 k 4 (ϕ) = -P (k 2 /k 3 k 4 ) l 1 l 2 Λ 31 k 1 l 1 l 2 k 2 (ϕ)R ++ l 1 k 3 (ϕ)R ++ l 2 k 4 (ϕ) (B.2p) O 13(40) k 1 k 2 k 3 k 4 (ϕ) = - l 1 l 2 l 3 Λ 40 k 1 l 1 l 2 l 3 (ϕ)R ++ l 1 k 2 (ϕ)R ++ l 2 k 3 (ϕ)R ++ l 3 k 4 (ϕ) (B.2q) O 04(04) k 1 k 2 k 3 k 4 (ϕ) = Λ 04 k 1 k 2 k 3 k 4 (ϕ) (B.2r) O 04(13) k 1 k 2 k 3 k 4 (ϕ) = P (k 1 k 2 k 3 /k 4 ) l 1 Λ 13 l 1 k 1 k 2 k 3 (ϕ)R ++ l 1 k 4 (ϕ) (B.2s) O 04(22) k 1 k 2 k 3 k 4 (ϕ) = -P (k 1 k 2 /k 3 k 4 ) l 1 l 2 Λ 22 l 1 l 2 k 1 k 2 (ϕ)R ++ l 1 k 3 (ϕ)R ++ l 2 k 4 (ϕ) (B.2t) O 04(31) k 1 k 2 k 3 k 4 (ϕ) = P (k 4 /k 1 k 2 k 3 ) l 1 l 2 l 3 Λ 31 l 1 l 2 l 3 k 4 (ϕ)R ++ l 1 k 1 (ϕ)R ++ l 2 k 2 (ϕ)R ++ l 3 k 3 (ϕ) (B.2u) O 04(40) k 1 k 2 k 3 k 4 (ϕ) = l 1 l 2 l 3 l 4 Λ 40 l 1 l 2 l 3 l 4 (ϕ)R ++ l 1 k 1 (ϕ)R ++ l 2 k 2 (ϕ)R ++ l 3 k 3 (ϕ)R ++ l 4 k 4 (ϕ) (B.2v)
Appendix C.

Perturbative cluster amplitudes

First-order cluster amplitudes (Eq. (2.40)) are given by U 20( 1)

k 1 k 2 ≡ - Ω 20
k 1 k 2 E k 1 k 2 , (C.1a) U 40(1)
k 1 k 2 k 3 k 4 ≡ - Ω 40 k 1 k 2 k 3 k 4 E k 1 k 2 k 3 k 4 . (C.1b)
Second-order single amplitudes have been derived in Ref. [START_REF] Arthuis | Bogoliubov Many-Body Perturbation Theory for Nuclei : Systematic Generation and Evaluation of Diagrams and First ab initio Calculations[END_REF] and are given by U 20( 2)

k 1 k 2 = 1 6 P (k 1 /k 2 ) k 3 k 4 k 5 Ω 40 k 5 k 3 k 1 k 4 Ω 13 k 2 k 4 k 5 k 3 E k 1 k 3 k 4 k 5 E k 1 k 2 , (C.2a) + 1 2 k 3 k 4 Ω 40 k 1 k 2 k 3 k 4 Ω 02 k 3 k 4 E k 1 k 2 k 3 k 4 E k 1 k 2 , (C.2b) + 1 2 k 3 k 4 Ω 20 k 3 k 4 Ω 22 k 3 k 4 k 1 k 2 E k 3 k 4 E k 1 k 2 , (C.2c) + P (k 1 /k 2 ) k 3 Ω 20
k 1 k 3 Ω11 k 3 k 2 E k 1 k 3 E k 1 k 2 , (C.2d)
while second-order double amplitudes are given by U 40( 2)

k 1 k 2 k 3 k 4 = 1 2 P (k 1 k 2 /k 3 k 4 ) k 5 k 6 Ω 40 k 1 k 2 k 5 k 6 Ω 22 k 5 k 6 k 3 k 4 E k 1 k 2 k 5 k 6 E k 1 k 2 k 3 k 4 , (C.3a) + P (k 4 /k 1 k 2 k 3 ) k 5 Ω 40 k 1 k 2 k 3 k 5 Ω11 k 4 k 5 E k 1 k 2 k 3 k 5 E k 1 k 2 k 3 k 4 , (C.3b) + P (k 1 /k 2 k 3 k 4 ) k 5 Ω 20 k 1 k 5 Ω 31 k 2 k 3 k 4 k 5 E k 1 k 5 E k 1 k 2 k 3 k 4 , (C.3c)
where Ω11 has been specified in Sec. The algebraic (m-scheme) expressions corresponding to these three diagrams are given by

PO1.1.1 = - 1 (2!) k 1 k 2 O 20 k 1 k 2 (ϕ) Ω 02 k 1 k 2 E k 1 k 2 , (D.2a) PO1.1.2 = - 1 (2!) 2 k 1 k 2 k 3 k 4 O 20 k 1 k 2 (ϕ) Ω 04 k 1 k 2 k 3 k 4 R -- k 4 k 3 (ϕ) E k 1 k 2 k 4 k 3 , (D.2b) PO1.2.1 = - 1 (4!) k 1 k 2 k 3 k 4 O 40 k 1 k 2 k 3 k 4 (ϕ) Ω 04 k 1 k 2 k 3 k 4 E k 1 k 2 k 3 k 4 , (D.2c)
whereas their j-scheme counterparts are given by Let us now focus on the n-body contributions to anomalous fields associated with the full operator O and for which n ≥ max(i, j) and, e.g., i > j. One can write

PO1.1.1 = - 1 (2!) k1 n k 2 ĵ2 k 1 j k 1 O 20 n k 1 n k 2 (ϕ) j k 1 Ω 02 n k 1 n k 2 E k1 ξ k 1 n k 2 , (D.3a) PO1.1.2 = + 1 (2!) 2 k1 n k 2 k3 n k 4 j k 1 O 20 n k 1 n k 2 (ϕ) 0 Ω 04 k1 ξ k 1 n k 2 k3 ξ k 3 n k 4 j k 3 R -- n k 3 n k 4 (ϕ) E k1 ξ k 1 n k 2 k3 ξ k 3 n k 4 , (D.3b) PO1.2.1 = - 1 (4!) k1 k2 k3 k4 J Ĵ2 J O k1 k2 k3 k4 (ϕ) J Ω 04
k i O 20 k 1 k 2 (ϕ) Ω11 k 3 k 1 Ω 02 k 3 k 2 E k 1 k 2 E k 2 k 3 , (D.6a) PO2.1.2 = 1 2 k i O 20 k 1 k 2 (ϕ)Ω 02 k 1 k 3 Ω 02 k 4 k 2 R -- k 4 k 3 (ϕ) E k 1 k 4 E k 2 k 3 , (D.6b) PO2.1.3 = 1 2(2!) 2 k i O 20 k 1 k 2 (ϕ)Ω 04 k 1 k 3 k 4 k 5 Ω 04 k 6 k 2 k 7 k 8 R -- k 4 k 3 (ϕ)R -- k 6 k 5 (ϕ)R -- k 8 k 7 (ϕ) E k 1 k 4 k 3 k 6 E k 2 k 8 k 7 k 5 , (D.6c) PO2.1.4 = 1 (2!) k i O 20 k 1 k 2 (ϕ)Ω 02 k 1 k 3 Ω 04 k 4 k 2 k 5 k 6 R -- k 4 k 3 (ϕ)R -- k 6 k 5 (ϕ) E k 1 k 4 E k 2 k 6 k 5 k 3 , (D.6d) PO2.1.5 = 1 (2!) 2 k i O 20 k 1 k 2 (ϕ)Ω 13 k 5 k 1 k 3 k 4 Ω 04 k 5 k 2 k 6 k 7 R -- k 4 k 3 (ϕ)R -- k 7 k 6 (ϕ) E k 1 k 2 k 4 k 3 k 7 k 6 E k 2 k 5 k 7 k 6 , (D.6e) PO2.1.6 = 1 (2!) k i O 20 k 1 k 2 (ϕ) Ω11 k 3 k 1 Ω 04 k 3 k 2 k 4 k 5 R -- k 5 k 4 (ϕ) E k 1 k 2 k 5 k 4 E k 2 k 3 k 5 k 4 , (D.6f) PO2.1.7 = 1 (2!) k i O 20 k 1 k 2 (ϕ)Ω 13 k 5 k 1 k 3 k 4 Ω 02 k 5 k 2 R -- k 4 k 3 (ϕ) E k 1 k 2 k 4 k 3 E k 2 k 5 . (D.6g) PO2.2.1 = 1 (2!) k i O 20 k 1 k 2 (ϕ)Ω 22 k 3 k 4 k 1 k 2 Ω 02 k 3 k 4 E k 1 k 2 E k 3 k 4 , (D.7a) PO2.2.2 = 1 (2!) k i O 20 k 1 k 2 (ϕ)Ω 13 k 5 k 1 k 2 k 3 Ω 02 k 4 k 5 R -- k 4 k 3 (ϕ) E k 1 k 2 k 4 k 3 E k 5 k 3 , (D.7b) PO2.2.3 = 1 (2!) k i O 20 k 1 k 2 (ϕ)Ω 04 k 1 k 2 k 3 k 5 Ω 02 k 4 k 6 R -- k 4 k 3 (ϕ)R -- k 6 k 5 (ϕ) E k 1 k 2 k 4 k 6 E k 3 k 5 , (D.7c) PO2.2.4 = 1 (2!) k i O 20 k 1 k 2 (ϕ)Ω 04 k 1 k 2 k 3 k 5 Ω 04 k 4 k 6 k 7 k 8 R -- k 4 k 3 (ϕ)R -- k 6 k 5 (ϕ)R -- k 8 k 7 (ϕ) E k 1 k 2 k 4 k 6 E k 8 k 7 k 3 k 5 , (D.7d) PO2.2.5 = 1 (2!) k i O 20 k 1 k 2 (ϕ)Ω 13 k 5 k 1 k 2 k 3 Ω 04 k 4 k 5 k 6 k 7 R -- k 4 k 3 (ϕ)R -- k 7 k 6 (ϕ) E k 1 k 2 k 4 k 3 k 7 k 6 E k 5 k 7 k 6 k 3 , (D.7e) PO2.2.6 = 1 (2!) k i O 20 k 1 k 2 (ϕ)Ω 22 k 3 k 4 k 1 k 2 Ω 04 k 3 k 4 k 5 k 6 R -- k 6 k 5 (ϕ) E k 1 k 2 k 6 k 5 E k 3 k 4 k 6 k 5 . (D.7f) Off-diagonal BMBPT diagrams generated from the diagonal BMBPT diagram PO2.3 PO2.3.1 = 1 (2!) k i O 20 k 1 k 2 (ϕ)Ω 20 k 3 k 4 Ω 04 k 3 k 4 k 1 k 2 E k 1 k 2 E k 1 k 2 k 3 k 4 , (D.8a) PO2.3.2 = 1 (2!) k i O 20 k 1 k 2 (ϕ) Ω11 k 5 k 3 Ω 04 k 4 k 5 k 1 k 2 R -- k 4 k 3 (ϕ) E k 1 k 2 k 4 k 3 E k 1 k 2 k 5 k 3 , (D.8b) PO2.3.3 = 1 (2!) k i O 20 k 1 k 2 (ϕ)Ω 02 k 3 k 5 Ω 04 k 4 k 6 k 1 k 2 R -- k 4 k 3 (ϕ)R -- k 6 k 5 (ϕ) E k 4 k 6 E k 1 k 2 k 3 k 5 , (D.8c) PO2.3.4 = 1 (2!) k i O 20 k 1 k 2 (ϕ)Ω 04 k 3 k 4 k 5 k 7 Ω 04 k 6 k 8 k 1 k 2 R -- k 4 k 3 (ϕ)R -- k 6 k 5 (ϕ)R -- k 8 k 7 (ϕ) E k 4 k 3 k 6 k 8 E k 1 k 2 k 5 k 7 , (D.8d) PO2.3.5 = 1 (2!) k i O 20 k 1 k 2 (ϕ)Ω 13 k 7 k 3 k 4 k 5 Ω 04 k 6 k 7 k 1 k 2 R -- k 4 k 3 (ϕ)R -- k 6 k 5 (ϕ) E k 1 k 2 k 4 k 3 k 6 k 5 E k 1 k 2 k 7 k 5 , (D.8e) PO2.3.6 = 1 (2!) k i O 20 k 1 k 2 (ϕ)Ω 22 k 5 k 6 k 3 k 4 Ω 04 k 5 k 6 k 1 k 2 R -- k 4 k 3 (ϕ) E k 1 k 2 k 4 k 3 E k 1 k 2 k 5 k 6 . (D.8f) Off-diagonal BMBPT diagrams generated from the diagonal BMBPT diagram PO2.4 PO2.4.1 = 2(2!) 2 k i O 40 k 1 k 2 k 3 k 4 (ϕ)Ω 02 k 1 k 2 Ω 02 k 3 k 4 E k 1 k 2 E k 3 k 4 , (D.9a) PO2.4.2 = 2(2!) 4 k i O 40 k 1 k 2 k 3 k 4 (ϕ)Ω 04 k 1 k 2 k 5 k 6 Ω 04 k 3 k 4 k 7 k 8 R -- k 6 k 5 (ϕ)R -- k 8 k 7 (ϕ) E k 1 k 2 k 6 k 5 E k 3 k 4 k 8 k 7 , (D.9b) PO2.4.3 = 1 (2!) 3 k i O 40 k 1 k 2 k 3 k 4 (ϕ)Ω 02 k 1 k 2 Ω 04 k 3 k 4 k 5 k 6 R -- k 6 k 5 (ϕ) E k 1 k 2 E k 3 k 4 k 6 k 5 . (D.9c) Off-diagonal BMBPT diagrams generated from the diagonal BMBPT diagram PO2.5 PO2.5.1 = 1 (3!) k i O 40 k 1 k 2 k 3 k 4 (ϕ)Ω 13 k 5 k 1 k 2 k 3 Ω 02 k 5 k 4 E k 1 k 2 k 3 k 4 E k 4 k 5 , (D.10a) PO2.5.2 = 1 (3!) k i O 40 k 1 k 2 k 3 k 4 (ϕ)Ω 04 k 1 k 2 k 3 k 5 Ω 02 k 6 k 4 R -- k 6 k 5 (ϕ) E k 1 k 2 k 3 k 6 E k 4 k 5 , (D.10b) PO2.5.3 = 1 (2!)(3!) k i O 40 k 1 k 2 k 3 k 4 (ϕ)Ω 04 k 1 k 2 k 3 k 5 Ω 04 k 6 k 4 k 7 k 8 R -- k 6 k 5 (ϕ)R -- k 8 k 7 (ϕ) E k 1 k 2 k 3 k 6 E k 4 k 8 k 7 k 5 , (D.10c) PO2.5.4 = 1 (2!)(3!) k i O 40 k 1 k 2 k 3 k 4 (ϕ)Ω 13 k 5 k 1 k 2 k 3 Ω 04 k 5 k 4 k 6 k 7 R -- k 7 k 6 (ϕ) E k 1 k 2 k 3 k 4 k 7 k 6 E k 4 k 5 k 7 k 6 . (D.10d) Off-diagonal BMBPT diagrams generated from the diagonal BMBPT diagram PO2.6 PO2.6.1 = 1 (3!) k i O 40 k 1 k 2 k 3 k 4 (ϕ) Ω11 k 5 k 1 Ω 04 k 5 k 2 k 3 k 4 E k 1 k 2 k 3 k 4 E k 2 k 3 k 4 k 5 , (D.11a) PO2.6.2 = 1 (3!) k i O 40 k 1 k 2 k 3 k 4 (ϕ)Ω 02 k 1 k 5 Ω 04 k 6 k 2 k 3 k 4 R -- k 6 k 5 (ϕ) E k 1 k 6 E k 2 k 3 k 4 k 5 , (D.11b) PO2.6.3 = 1 (2!)(3!) k i O 40 k 1 k 2 k 3 k 4 (ϕ)Ω 04 k 1 k 5 k 6 k 7 Ω 04 k 8 k 2 k 3 k 4 R -- k 6 k 5 (ϕ)R -- k 8 k 7 (ϕ) E k 1 k 6 k 5 k 8 E k 2 k 3 k 4 k 7 , (D.11c) PO2.6.4 = 1 (2!)(3!) k i O 40 k 1 k 2 k 3 k 4 (ϕ)Ω 13 k 7 k 1 k 5 k 6 Ω 04 k 7 k 2 k 3 k 4 R -- k 6 k 5 (ϕ) E k 1 k 2 k 3 k 4 k 6 k 5 E k 2 k 3 k 4 k 7 . (D.11d) Off-diagonal BMBPT diagrams generated from the diagonal BMBPT diagram PO2.7 PO2.7.1 = 1 (3!) k i O 20 k 1 k 2 (ϕ)Ω 31 k 3 k 4 k 5 k 1 Ω 04 k 3 k 4 k 5 k 2 E k 1 k 2 E k 2 k 3 k 4 k 5 , (D.12a) PO2.7.2 = 1 (2!) k i O 20 k 1 k 2 (ϕ)Ω 22 k 5 k 6 k 1 k 3 Ω 04 k 4 k 5 k 6 k 2 R -- k 4 k 3 (ϕ) E k 1 k 2 k 4 k 3 E k 2 k 5 k 6 k 3 , (D.12b) PO2.7.3 = 1 (2!) k i O 20 k 1 k 2 (ϕ)Ω 13 k 7 k 1 k 3 k 5 Ω 04 k 4 k 6 k 7 k 2 R -- k 4 k 3 (ϕ)R -- k 6 k 5 (ϕ) E k 1 k 2 k 4 k 3 k 6 k 5 E k 2 k 7 k 3 k 5 , (D.12c) PO2.7.4 = 1 2(3!) k i O 20 k 1 k 2 (ϕ)Ω 04 k 1 k 3 k 5 k 7 Ω 04 k 4 k 6 k 8 k 2 R -- k 4 k 3 (ϕ)R -- k 6 k 5 (ϕ)R -- k 8 k 7 (ϕ) E k 1 k 4 k 6 k 8 E k 2 k 3 k 5 k 7 . (D.12d) Off-diagonal BMBPT diagrams generated from the diagonal BMBPT diagram PO2.8 PO2.8.1 = 1 (2!) 3 k i O 40 k 1 k 2 k 3 k 4 (ϕ)Ω 22 k 5 k 6 k 1 k 2 Ω 04 k 5 k 6 k 3 k 4 E k 1 k 2 k 3 k 4 E k 3 k 4 k 5 k 6 , (D.13a) PO2.8.2 = 1 (2!) 2 k i O 40 k 1 k 2 k 3 k 4 (ϕ)Ω 13 k 7 k 1 k 2 k 5 Ω 04 k 6 k 7 k 3 k 4 R -- k 6 k 5 (ϕ) E k 1 k 2 k 3 k 4 k 6 k 5 E k 3 k 4 k 7 k 5 , (D.13b) PO2.8.3 = 1 2(2!) 3 k i O 40 k 1 k 2 k 3 k 4 (ϕ)Ω 04 k 1 k 2 k 5 k 7 Ω 04 k 6 k 8 k 3 k 4 R -- k 6 k 5 (ϕ)R -- k 8 k 7 (ϕ) E k 1 k 2 k 6 k 8 E k 3 k 4 k 5 k 7 . (D.13c)
Λ ij(nn) l 1 ...l i+j = n-j 2 n κ = i-j 2 Λ ij(nn)(n κ ) l 1 ...l i+j = n-i 2 n κ =0 n κ ! n κ + i-j 2 ! 1 2 i-j 2 l i+j+1 ...l 2i Λ ii(nn)(n κ ) l 1 ...l 2i κ l i+j+1 l i+j+2 . . . κ l 2i-1 l 2i = 1 i-j 2 ! 1 2 i-j 2 l i+j+1 ...l 2i   Λ ii(nn) l 1 ...l 2i - n-i 2 n κ =0   1 - 1 n κ + i-j 2 n κ    Λ ii(nn)(n κ ) l 1 ...l 2i    × κ l i+j+1 l i+j+2 . . . κ l 2i-1 l 2i , (F.3)
where Λ ij(nn)(n κ ) denotes the contribution from o nn to Λ ij containing n κ anomalous contractions κ. Indeed, several contractions patterns associated with different numbers of normal and anomalous contractions can lead from o nn to Λ ij , knowing that the minimal number of κ contractions is (i -j)/2. The goal of the above rewriting is to factorize (i -j)/2 anomalous contractions κ in order to express each contribution Λ ij(nn)(n κ ) in terms of its diagonal (i.e. i = j) partner Λ ii(nn)(n κ ) . Summing Eq. (F.3) over n-body contributions with n = i, . . . , N , one obtains

Λ ij l 1 ...l i+j = 1 i-j 2 ! 1 2 i-j 2 l i+j+1 ...l 2i   Λ ii l 1 ...l 2i - N n=i n-i 2 n κ =0   1 - 1 n κ + i-j 2 n κ    Λ ii(nn)(n κ ) l 1 ...l 2i    × κ l i+j+1 l i+j+2 . . . κ l 2i-1 l 2i . (F.4) A similar relation holds for Λij associated with O PNOkB Λij l 1 ...l i+j = 1 i-j 2 ! 1 2 i-j 2 l i+j+1 ...l 2i    Λii l 1 ...l 2i - k n=i n-i 2 n κ =0   1 - 1 n κ + i-j 2 n κ    Λii(nn)(n κ ) l 1 ...l 2i    × κ l i+j+1 l i+j+2 . . . κ l 2i-1 l 2i , (F.5)
the difference being that the sum over n only extends up to k instead of N for Λ ij given that õnn is null for n > k to begin with. Using Eq. (F.2), the combination of the two above identities allows one to relate both sets of fields through Λij

l 1 ...l i+j = Λ ij l 1 ...l i+j + Λij l 1 ...l i+j , (F.6) with Λij l 1 ...l i+j ≡ 1 i-j 2 ! 1 2 i-j 2 N n=i n-i 2 n κ =0   1 - 1 n κ + i-j 2 n κ    l i+j+1 ...l 2i Λ ii(nn)(n κ ) l 1 ...l 2i - Λii(nn)(n κ ) l 1 ...l 2i × κ l i+j+1 l i+j+2 . . . κ l 2i-1 l 2i , (F.7)
where Λii(nn)(n κ ) is in fact zero for n > k. The case of anomalous field with i < j is obtained through the same procedure by factorizing κ * contractions instead of κ ones. Of course, the standard NOkB approximation is recovered from the PNOkB one whenever the reference state is particle-number conserving, i.e. whenever |Φ reduces to a slater determinant |SD . The nNO2B approximation leads to just dropping O [3] = Λ 33 . The application of the PNO2B approximation is more involved and we now proceed to the construction of the corresponding operator

F

O PNO2B ≡ õ00 + õ11 + õ22 , (F.11)
where the different terms are to be obtained recursively on the basis of Eq. (5.6). In the present case, it leads to õ22 = Λ

PNO3B approximation of a four-body operator

Let us now build the PNO3B approximation of the four-body operator where the extra term is nothing but Λ20 . Four-body operators are thus the first for which such an extra term appears.

In order to check the consistency of the derivation, the extra term can also be obtained via the application of Eq. (F.7) to the present case of interest. This gives Λ20 = 1

2-0 2 ! 1 2 2-0 2 4 n=2 n-2 2 n κ =0   1 - 1 n κ + 2-0 2 n κ    l 3 l 4 Λ 22(nn)(n κ ) l 1 ...l 4 - Λ22(nn)(n κ ) l 1 ...l 4 κ l 3 l 4 = 1 4 Tr[Λ 22(44)(n κ =1) κ] = 1 16 Tr[o 44 κ * κκ] , (F.25)
which is indeed consistent with Eq. (F.24). Inspecting Eq. (F.24), one realizes that the difference between Λ20 and Λ 20 eventually boils down to using a prefactor

F.1.3. Alternative approximation

The procedure to define the PNOkB approximation is not unique. Given the three objectives stated in Sec. 5.2.2, two independent approaches cannot differ drastically in their philosophy and cannot produce very different approximate operators. In the present section, an alternative approximation procedure based on the quasi-normal ordering of Ref. [START_REF] Kong | An algebraic proof of generalized Wick theorem[END_REF] is briefly investigated to highlight the similarities and differences.

Quasi normal-ordering

The quasi normal-ordering introduced in Ref. [START_REF] Kong | An algebraic proof of generalized Wick theorem[END_REF] is stipulated through the set of equations

c † l 1 c l 2 ≡ N[c † l 1 c l 2 ] + γ l 2 l 1 , (F.35a) c † l 1 c † l 2 c l 4 c l 3 ≡ N[c † l 1 c † l 2 c l 4 c l 3 ] + A(N[c † l 1 c l 3 ]γ l 4 l 2 ) + A(γ l 3 l 1 γ l 4 l 2 ) , (F.35b) . . . ,
where N[. . .] denotes the quasi normal-ordering defined by the given of the elementary contraction γ. The antisymmetrization operator A(. . .) is the one defined in Ref. [START_REF] Shamasundar | Cumulant decomposition of reduced density matrices, multireference normal ordering, and Wicks theorem: A spin-free approach[END_REF]. At this point, the contraction is defined as any two-index tensor such that the above procedure is very general. It happens that such a definition is sufficient to have a generalized Wick's theorem governing the way the product of two quasi normal-ordered operators can be systematically decomposed into a sum of quasi normal-ordered operators weighted by a set of contractions [START_REF] Kong | An algebraic proof of generalized Wick theorem[END_REF].

In the present context, the quasi-normal ordering is only used as a systematic way to reshuffle as a large part as possible of an initial string of single-particle operators into operators of lower ranks in view of neglecting the quasi normal-ordered term having the same rank as the original string. Compared to the normal ordering with respect to the Bogoliubov vacuum associated with standard Wick's theorem, it is presently possible to only invoke a symmetry-conserving elementary contraction such that each individual term in the quasi normal-ordered form is a scalar under U (1) transformations. For the rest, the expectation value of a quasi-normal-ordered operator defined in such a general way does not vanish a priori, i.e.

Φ|N[c

† l 1 c l 2 ]|Φ = 0 , (F.36a) Φ|N[c † l 1 c † l 2 c l 4 c l 3 ]|Φ = 0 , (F.36b) . . . .
As a result, the quasi normal ordering does not present the practicalities of more traditional Wick's theorems.

Thus, the approach proceeds in two steps. First, the original operator expressed in quasi normal-ordered form is truncated to produce the particle-number-conserving quasi-normalordered k-body approximation (PQNOkB). Second, the resulting operator is brought into a normal-ordered form with respect to the Bogoliubov vacuum according to standard Wick's theorem in view of using it in the many-body formalism of interest.

PQNO1B of a two-body operator

Applying quasi normal-ordering to a particle-number-conserving two-body operator O leads to

O ≡ l 1 l 2 o 11 l 1 l 2 c † l 1 c l 2 + 1 4 l 1 l 2 l 3 l 4 o 22 l 1 l 2 l 3 l 4 c † l 1 c † l 2 c l 4 c l 3 , = Λ 00(qno) + l 1 l 2 Λ 11(qno) l 1 l 2 N[c † l 1 c l 2 ] + 1 4 l 1 l 2 l 3 l 4 Λ 22(qno) l 1 l 2 l 3 l 4 N[c † l 1 c † l 2 c l 4 c l 3 ] , (F.37)
where the n-body fields Λ nn(qno) are given by Λ 00(qno) ≡

l 1 l 2 o 11 l 1 l 2 γ l 2 l 1 + 1 2 l 1 l 2 l 3 l 4 o 22 l 1 l 2 l 3 l 4 γ l 3 l 1 γ l 4 l 2 , (F.38a) Λ 11(qno) l 1 l 2 ≡ o 11 l 1 l 2 + l 1 l 2 l 3 l 4 o 22 l 1 l 3 l 2 l 4 γ l 4 l 3 , (F.38b) Λ 22(qno) l 1 l 2 l 3 l 4 ≡ o 22 l 1 l 2 l 3 l 4 . (F.38c)
The PQNO1B approximation of O is obtained by neglecting the effective two-body part Λ 22(qno)

O PQNO1B ≡ Λ 00(qno) + l 1 l 2 Λ 11(qno) l 1 l 2 N[c † l 1 c l 2 ] ≡ o 00(qno) + l 1 l 2 o 11(qno) l 1 l 2 c † l 1 c l 2 , (F.39)
where o 00(qno) = Λ 00(qno) -

l 1 l 2 Λ 11(qno) l 1 l 2 γ l 2 l 1 (F.40a) = - 1 2 l 1 l 2 l 3 l 4 o 22 l 1 l 2 l 3 l 4 γ l 3 l 1 γ l 4 l 2 , o 11(qno) l 1 l 2 = Λ 11(qno) l 1 l 2 (F.40b) = o 11 l 1 l 2 + l 1 l 2 l 3 l 4 o 22 l 1 l 3 l 2 l 4 γ l 4 l 3 .
The above quantities are independent of the specific choice made for the elementary contraction γ. In the present case, γ is taken as the normal density ρ of the Bogoliubov vacuum. With this definition at hand, the PQNO1B operator is brought into a normalordered form with respect of the Bogoliubov state on the basis of standard Wick's theorem

O PQNO1B = Λ 00 + l 1 l 2 Λ 11 l 1 l 2 : c † l 1 c l 2 : , (F.41)
where

Λ 00 ≡ o 00(qno) + l 1 l 2 o 11(qno) l 1 l 2 ρ l 2 l 1 (F.42a) = l 1 l 2 o 11 l 1 l 2 ρ l 2 l 1 + 1 2 l 1 l 2 l 3 l 4 o 22 l 1 l 2 l 3 l 4 ρ l 3 l 1 ρ l 4 l 2 , Λ 11 l 1 l 2 ≡ o 11(qno) l 1 l 2 (F.42b) = o 11 l 1 l 2 + l 1 l 2 l 3 l 4 o 22 l 1 l 3 l 2 l 4 ρ l 4 l 3 .
Comparing Eq. (F.42) to Eq. (5.16), one observes that the contribution to Λ 00 originating from the two-body operator via two anomalous contractions is not present in the PQNO1B operator. The latter thus corresponds to a more drastic appoximation of the original operator O than the PNO1B operator. Performing the PQNO2B approximation of a three-body operator, the PQNO2B operator similarly misses a three-body contribution to Λ 11 associated with two anomalous contractions that is present in the PNO2B operator.

F.2. Double particle-number projection F.2.1. Gauge-rotated contractions

Matrix elements of the doubly gauge-rotated contractions are defined through

ρ l 1 l 2 (ϕ, ϕ ) ≡ Φ(ϕ)|c † l 2 c l 1 |Φ(ϕ ) Φ(ϕ)|Φ(ϕ ) , (F.43a) κ l 1 l 2 (ϕ, ϕ ) ≡ Φ(ϕ)|c l 2 c l 1 |Φ(ϕ ) Φ(ϕ)|Φ(ϕ ) , (F.43b) κ * l 1 l 2 (ϕ, ϕ ) ≡ Φ(ϕ)|c † l 1 c † l 2 |Φ(ϕ ) Φ(ϕ)|Φ(ϕ ) , (F.43c)
such that singly-rotated ones are nothing but ρ(ϕ) ≡ ρ(ϕ, 0) , (F.44a) κ(ϕ) ≡ κ(ϕ, 0) , (F.44b) κ * (ϕ) ≡ κ * (ϕ, 0) .

(F.44c) Doubly gauge-rotated contractions can be expressed in terms of the Bogoliubov transformation defining |Φ and of the gauge angles via [START_REF] Ring | The nuclear many-body problem[END_REF] ρ(ϕ, ϕ

) = ρ + V * Z 20 * (ϕ -ϕ )U † , (F.45a) κ(ϕ, ϕ ) = e -2iϕ κ -V * Z 20 * (ϕ -ϕ )V † , (F.45b) κ * (ϕ, ϕ ) = e +2iϕ κ * -U * Z 20 * (ϕ -ϕ )U † . (F.45c)
Setting ϕ = 0 in Eq. (F.45), singly and doubly gauge-rotated contractions appear to be trivially related through

ρ(ϕ, ϕ ) = ρ(ϕ -ϕ ) , (F.46a) κ(ϕ, ϕ ) = e -2iϕ κ(ϕ -ϕ ) , (F.46b) κ * (ϕ, ϕ ) = e +2iϕ κ * (ϕ -ϕ ) . (F.46c)

F.2.2. Projection

With Eq. (F.46) at hand, the singly-(left-) projected mean-field matrix element on the value A of the one-body operator F reads as

Φ|P A F |Φ = 1 2π 2π 0 dϕe -iϕA f (0) (ϕ)N (0) (ϕ) = l 1 l 2 f 11 l 1 l 2 1 2π 2π 0 dϕe -iϕA ρ l 2 l 1 (ϕ)N (0) (ϕ) + 1 2 l 1 l 2 f 20 l 1 l 2 1 2π 2π 0 dϕe -iϕA κ * l 1 l 2 (ϕ)N (0) (ϕ) + 1 2 l 1 l 2 f 02 l 1 l 2 1 2π 2π 0 dϕe -iϕA κ l 1 l 2 (ϕ)N (0) (ϕ) = Φ|P A f 11 |Φ + Φ|P A f 20 |Φ + Φ|P A f 02 |Φ , (F.47)
while the doubly-(left-and right-) projected, on possibly two different values A and A , is

Φ|P A F P A |Φ = 1 2π 2π 0 dϕ e +iϕ A 1 2π 2π 0 dϕe -iϕA f (0) (ϕ, ϕ )N (0) (ϕ -ϕ ) = 1 2π 2π 0 dϕ e +iϕ (A -A) 1 2π 2π 0 dϕe -i(ϕ-ϕ )A f (0) (ϕ, ϕ )N (0) (ϕ -ϕ ) = 1 2π 2π 0 dϕ e +iϕ (A -A) 1 2π 2π-ϕ -ϕ dφe -iφA N (0) (φ) × l 1 l 2 f 11 l 1 l 2 ρ l 2 l 1 (φ) + 1 2 f 20 l 1 l 2 e +2iϕ κ * l 1 l 2 (φ) + 1 2 f 02 l 1 l 2 e -2iϕ κ l 1 l 2 (φ)
and the difference between PHFB and HFB normal density matrices ∆ρ proj . The latter being expected to be small, the doubly-projected nNO1B particle-number variance is expected to be small as well.

F.3. Effect of particle-number projection

As seen in panel (b) of Fig. 5.4, the particle-number projection provides further/no/lesser binding in 14,22 O/ 16,24 O/ 18,20,26 O, knowing that the effect is essentially negligeable in 22,26 O.

In order to analyze this trend, the HFB state and energy can be decomposed into their particle-number projected components according to the sum rules

1 1 = A>0 c 2 A , (F.55a) H 00 = A>0 c 2 A E A , (F.55b)
where the projected weights and energies are defined as 16 O and 24 O display a single component corresponding to their number of particles, i.e. the HFB vacuum reduces to a Slater determinant in these doubly closed-shell nuclei such that the subsequent particle-number projection does not provide any static correlations. While in 14 O the distribution is slightly skewed towards smaller particle numbers than the average value 14, it is the opposite in 18,20 O. As seen in Fig. F.4, the skewness of the distribution eventually impacts the energy associated with each component obtained under the constraint that the energy sum rule (Eq. (F.55b)) must be fulfilled. In particular, the projected energy associated with the largest component, i.e. with the targeted particle number, ends up being more/less negative than the average, i.e. the HFB energy, in 14 O/ 18,20 O.

c 2 A ≡ 2π 0 dϕ 2π e -iϕA N (0) (ϕ) , (F.56a) E A ≡ 2π 0 dϕ 2πc 2 A e -iϕA h (0) (ϕ)N (0) (ϕ) . (F.56b)

G.2.1. Normal ordering in single-particle basis

Λ 00 (ϕ) = O 00 + 1 2 k1 n k 2 ĵ2 k 1 j k 1 O 20 n k 1 n k 2 j k 1 R ++ n k 1 n k 2 (ϕ) (G.1a) + 1 8 k1 n k 2 k3 n k 4 ĵk 1 ĵk 3 0 O 40 k1 n k 2 (ljt) k 1 k3 n k 4 (ljt) k 3 j k 1 R ++ n k 1 n k 2 (ϕ) j k 3 R ++ n k 3 n k 4 (ϕ) j k 1 Λ 20 n k 1 n k 2 (ϕ) = j k 1 O 20 n k 1 n k 2 + 1 2 k3 n k 4 ĵ-1 k 1 ĵk 3 0 O 40 k1 n k 2 (ljt) k 1 k3 n k 4 (ljt) k 3 j k 3 R ++ n k 3 n k 4 (ϕ) (G.1b) j k 1 Λ 11 n k 1 n k 2 (ϕ) = j k 1 O 11 n k 1 n k 2 + 1 2 k3 n k 4 ĵ-1 k 1 ĵk 3 0 O 31 k3 n k 4 (ljt) k 3 k1 n k 2 (ljt) k 1 j k 3 R ++ n k 3 n k 4 (ϕ) (G.1c) j k 1 Λ 02 n k 1 n k 2 (ϕ) = j k 1 O 02 n k 1 n k 2 + 1 2 k3 n k 4 ĵ-1 k 1 ĵk 3 0 O 22 k3 n k 4 (ljt) k 3 k1 n k 2 (ljt) k 1 j k 3 R ++ n k 3 n k 4 (ϕ) (G.1d) J Λ 40 k1 k2 k3 k4 (ϕ) = J O 40 k1 k2 k3 k4 (G.1e) J Λ 31 k1 k2 k3 k4 (ϕ) = J O 31 k1 k2 k3 k4 (G.1f) J Λ 22 k1 k2 k3 k4 (ϕ) = J O 22 k1 k2 k3 k4 (G.1g) J Λ 13 k1 k2 k3 k4 (ϕ) = J O 13 k1 k2 k3 k4 (G.1h) J Λ 04 k1 k2 k3 k4 (ϕ) = J O 04 k1 k2 k3 k4 (G.1i)

G.2.2. Normal ordering in quasi-particle basis

Zero-and one-body matrix elements (ljt) k 1 k2 (ϕ)

O 00(00) (ϕ) = Λ 00 (ϕ) (G.2a) j k 1 O 20(20) n k 1 n k 2 (ϕ) = j k 1 Λ 20 n k 1 n k 2 (ϕ) (G.2b) j k 1 O 11(11) n k 1 n k 2 (ϕ) = j k 1 Λ 11 n k 1 n k 2 (ϕ) (G.2c) j k 1 O 11(20) n k 1 n k 2 (ϕ) = - n k 3 j k 1 Λ 20 n k 1 n k 3 (ϕ) j k 1 R ++ n k 3 n k 2 (ϕ) (G.2d) j k 1 O 02(02) n k 1 n k 2 (ϕ) = j k 1 Λ 02 n k 1 n k 2 (ϕ) (G.2e) j k 1 O 02(11) n k 1 n k 2 (ϕ) = n k 3 j k 1 Λ 11 n k 3 n k 1 (ϕ) j k 1 R ++ n k 3 n k 2 (ϕ) + n k 3 j k 1 Λ 11 n k 3 n k 2 (ϕ) j k 1 R ++ n k 3 n k 1 (ϕ) (G.2f) j k 1 O 02(20) n k 1 n k 2 (ϕ) = - n k 3 n k 4 j k 1 Λ 20 n k 3 n k 4 (ϕ) j k 1 R ++ n k 3 n k 1 (ϕ) j k 1 R ++ n k 4 n k 2 (ϕ) . (G.2g) Two-body matrix elements J O 40(40) k1 k2 k3 k4 (ϕ) = J Λ 40 k1 k2 k3 k4 (ϕ) (G.3a) J O 31(31) k1 k2 k3 k4 (ϕ) = J Λ 31 k1 k2 k3 k4 (ϕ) (G.3b) J O 31(40) k1 k2 k3 k4 (ϕ) = - n k 5 J Λ 40 k1 k2 k3 n k 5 (ljt) k 4 (ϕ) j k 4 R ++ n k 5 n k 4 (ϕ) (G.3c) G.2. Similarity-transformed matrix elements J O 22(22) k1 k2 k3 k4 (ϕ) = J Λ 22 k1 k2 k3 k4 (ϕ) (G.3d) J O 22(31) k1 k2 k3 k4 (ϕ) = n k 5 J Λ 31 k1 k2 n k 5 (ljt) k 3 k4 (ϕ) j k 3 R ++ n k 5 n k 3 (ϕ) -(-1) j k 3 +j k 4 +J J Λ 31 k1 k2 n k 5 (ljt) k 4 k3 (ϕ) j k 4 R ++ n k 5 n k 4 (ϕ) (G.3e) J O 22(40) k1 k2 k3 k4 (ϕ) = - n k 5 n k 6 J Λ 40 k1 k2 n k 5 (ljt) k 3 n k 6 (ljt) k 4 (ϕ) j k 3 R ++ n k 5 n k 3 (ϕ) j k 4 R ++ n k 6 n k 4 (ϕ) (G.3f) J O 13(13) k1 k2 k3 k4 (ϕ) = J Λ 13 k1 k2 k3 k4 (ϕ) (G.3g) J O 13(22) k1 k2 k3 k4 (ϕ) = - n k 5 J Λ 22 k1 n k 5 (ljt) k 2 k3 k4 (ϕ) j k 2 R ++ n k 5 n k 2 (ϕ) - n k 5 J Ĵ 2 J Λ 22 k1 n k 5 (ljt) k 4 k3 k2 (ϕ) j k 4 R ++ n k 5 n k 4 (ϕ) j k 1 j k 2 J j k 3 j k 4 J -(-1) j k 3 +j k 4 +J J Λ 22 k1 n k 5 (ljt) k 3 k4 k2 (ϕ) j k 3 R ++ n k 5 n k 3 (ϕ) j k 1 j k 2 J j k 4 j k 3 J (G.3h) J O 13(31) k1 k2 k3 k4 (ϕ) = - n k 5 n k 6 J Λ 31 n k 5 (ljt) k 3 n k 6 (ljt) k 4 k1 k2 (ϕ) j k 3 R ++ n k 5 n k 3 (ϕ) j k 4 R ++ n k 6 n k 4 (ϕ) + J Λ 31 k1 n k 5 (ljt) k 2 n k 6 (ljt) k 3 k4 (ϕ) j k 2 R ++ n k 5 n k 2 (ϕ) j k 3 R ++ n k 6 n k 3 (ϕ) -(-1) j k 3 +j k 4 +J J Λ 31 k1 n k 5 (ljt) k 2 n k 6 (ljt) k 4 k3 (ϕ) j k 2 R ++ n k 5 n k 2 (ϕ) j k 4 R ++ n k 6 n k 4 (ϕ) (G.3i) J O 13(40) k1 k2 k3 k4 (ϕ) = n k 5 n k 6 n k 7 J Λ 40 k1 n k 5 (ljt) k 2 n k 6 (ljt) k 3 n k 7 (ljt) k 4 (ϕ) j k 2 R ++ n k 5 n k 2 (ϕ) j k 3 R ++ n k 6 n k 3 (ϕ) j k 4 R ++ n k 7 n k 4 (ϕ) (G.3j) J O 04(04) k1 k2 k3 k4 (ϕ) = J Λ 04 k1 k2 k3 k4 (ϕ) (G.3k) J O 04(13) k1 k2 k3 k4 (ϕ) = n k 5 J Λ 13 n k 5 (ljt) k 1 k2 k3 k4 (ϕ) j k 1 R ++ n k 5 n k 1 (ϕ) + J Λ 13 n k 5 (ljt) k 3 k4 k1 k2 (ϕ) j k 3 R ++ n k 5 n k 3 (ϕ) -(-1) j k 1 +j k 2 +J J Λ 13 n k 5 (ljt) k 2 k1 k3 k4 (ϕ) j k 2 R ++ n k 5 n k 2 (ϕ) -(-1) j k 3 +j k 4 +J J Λ 13 n k 5 (ljt) k 4 k3 k1 k2 (ϕ) j k 4 R ++ n k 5 n k 4 (ϕ) (G.3l) J O 04(22) k1 k2 k3 k4 (ϕ) = - n k 5 n k 6 J Λ 22 n k 5 (ljt) k 3 n k 6 (ljt) k 4 k1 k2 (ϕ) j k 3 R ++ n k 5 n k 3 (ϕ) j k 4 R ++ n k 6 n k 4 (ϕ) + J Λ 22 n k 5 (ljt) k 1 n k 6 (ljt) k 2 k3 k4 (ϕ) j k 1 R ++ n k 5 n k 1 (ϕ) j k 2 R ++ n k 6 n k 2 (ϕ) - n k 5 n k 6 J Ĵ 2 J Λ 22 n k 6 (ljt) k 3 n k 5 (ljt) k 2 k1 k4 (ϕ) j k 2 R ++ n k 5 n k 2 (ϕ) j k 3 R ++ n k 6 n k 3 (ϕ) + J Λ 22 n k 5 (ljt) k 1 n k 6 (ljt) k 4 k3 k2 (ϕ) j k 1 R ++ n k 5 n k 1 (ϕ) j k 4 R ++ n k 6 n k 4 (ϕ) × j k 1 j k 2 J j k 3 j k 4 J n k 5 n k 6 J (-1) j k 3 +j k 4 +J Ĵ 2 J Λ 22 n k 6 (ljt) k 4 n k 5 (ljt) k 2 k1 k3 (ϕ) j k 2 R ++ n k 5 n k 2 (ϕ) j k 4 R ++ n k 6 n k 4 (ϕ) + J Λ 22 n k 5 (ljt) k 1 n k 6 (ljt) k 3 k4 k2 (ϕ) j k 1 R ++ n k 5 n k 1 (ϕ) j k 3 R ++ n k 6 n k 3 (ϕ) × j k 1 j k 2 J j k 4 j k 3 J (G.
j k 1 R ++ n k 5 n k 1 (ϕ) j k 3 R ++ n k 6 n k 3 (ϕ) j k 4 R ++ n k 7 n k 4 (ϕ) + J Λ 31 n k 5 (ljt) k 1 n k 6 (ljt) k 2 n k 7 (ljt) k 3 k4 (ϕ) j k 1 R ++ n k 5 n k 1 (ϕ) j k 2 R ++ n k 6 n k 2 (ϕ) j k 3 R ++ n k 7 n k 3 (ϕ) -(-1) j k 1 +j k 2 +J J Λ 31 n k 6 (ljt) k 3 n k 7 (ljt) k 4 n k 5
(ljt) k 2 k1 (ϕ)

× j k 2 R ++ n k 5 n k 2 (ϕ) j k 3 R ++ n k 6 n k 3 (ϕ) j k 4 R ++ n k 7 n k 4 (ϕ)
-(-1) j k 3 +j k 4 +J J Λ 31

n k 5 (ljt) k 1 n k 6 (ljt) k 2 n k 7
(ljt) k 4 k3 (ϕ) (ϕ)

× j k 1 R ++ n k 5 n k 1 (ϕ) j k 2 R ++ n k 6 n k 2 (ϕ)
× j k 1 R ++ n k 5 n k 1 (ϕ) j k 2 R ++ n k 6 n k 2 (ϕ) j k 3 R ++ n k 7 n k 3 (ϕ) j k 4 R ++ n k 8 n k 4 (ϕ) . (G.3o)

G.3. Set of ODEs for cluster amplitudes

The j-coupled version of the set of ODEs (Eq. 2.23) is given by

d dϕ W 0 (ϕ) = i 2 k1 n k 2 ĵ2 k 1 j k 1 Ã02 n k 1 n k 2 (ϕ) j k 1 Wn k 1 n k 2 (ϕ) (G.4) d dϕ j k 1 Wn k 1 n k 2 (ϕ) = i 2 k3 n k 4 ĵ-1 k 1 ĵk 3 j k 3 Ã02 n k 3 n k 4 (ϕ) 0 Wk 1 n k 2 (ljt) k 1 k3 n k 4 (ljt) k 3 (ϕ) -i n k 3 n k 4 j k 1 Ã02 n k 3 n k 4 (ϕ) j k 1 Wn k 1 n k 3 (ϕ) j k 1 Wn k 2 n k 4 (ϕ) (G.5) d dϕ J Wk 1 k2 k3 k4 (ϕ) = i 2 k5 n k 6 J Ĵ-2 Ĵ 2 j k 5 Ã02 n k 5 n k 6
(ϕ) J Wk 1 k2 J k5 ; k3 k4 Jn k 6 (ljt) k 5 (ϕ)

-i n k 5 n k 6 j k 1 Ã02 n k 5 n k 6
(ϕ) j k 1 Wn k 1 n k 5 (ϕ) J Wk 3 k4 n k 6 (ljt) k 1 k2 (ϕ)

+ j k 2 Ã02 n k 5 n k 6
(ϕ) j k 2 Wn k 2 n k 5 (ϕ) J Wk 3 k4 k1 n k 6 (ljt) k 2 (ϕ)

+ j k 3 Ã02 n k 5 n k 6
(ϕ) j k 3 Wn k 3 n k 5 (ϕ) J Wk 1 k2 n k 6 (ljt) k 3 k4 (ϕ)

+ j k 4 Ã02 n k 5 n k 6 (ϕ) j k 4 Wn k 4 n k 5 (ϕ) J Wk 1 k2 k3 n k 6 (ljt) k 4 (ϕ) . (G.6)

G.4. Perturbative diagonal cluster amplitudes

First-order single amplitudes Second-order single amplitudes (G.9a)

j k 1 Ũ (1) n k 1 n k 2 = - j k 1 Ω20 n k 1 n k 2
j k 1 Ũ (2) n k 1 n k 2 1 = 1 6 k3 k4 k5 J ĵ-2 k 1 Ĵ 2    J Ω40 k5
j k 1 Ũ (2) n k 1 n k 2 2 = 1 2 k3 n k 4 ĵ-1 k 1 ĵk 3 0 Ω40 k1 n k 2 (ljt) k 1 k3 n k 4 (ljt) k 3 j k 3 Ω02 n k 3 n k 4
k1 k2 k3 k4 k1 k2 (G.9b)

j k 1 Ũ (2) n k 1 n k 2 3 = 1 2 k3 n k 4 ĵ-1 k 1 ĵk 3 j k 3 Ω20 n k 3 n k 4 0 Ω22 k3 n k 4 (ljt) k 3 k1 n k 2 (ljt) k 1 k3 k4 k1 k2
(G.9c) ĵ2

j k 1 Ũ (2) n k 1 n k 2 4 = n k 3    j k 1 Ω20 n k 1 n k 3 j k 1 Ω11 n k 3 n k 2 k1 k3 k1 k2 + j k 1 Ω20 n k 2 n k 3 j k 1 Ω11
k 1 n k 1 n k 2 n k 3 D(ljt) k 1 n k 1 n k 2 (ϕ) C †(ljt) k 1 n k 2 n k 3 (ϕ) Õ11 (ljt) k 1 n k 3 n k 1 (ϕ) (G.11a) OCD2.2(ϕ) = 1 4 (ljt) k 1 (ljt) k 3 ĵk 1 ĵk 3 n k 1 n k 2 n k 3 n k 4 C †(ljt) k 1 n k 1 n k 2 (ϕ) 0 Õ22 k1 (ljt) k 1 n k 2 k3 (ljt) k 3 n k 4

Résumé

La physique nucléaire est le domaine de la physique qui vise à la description des noyaux atomiques. Contrairement à ce que l'on pense généralement de la physique nucléaire, les propriétés des noyaux ne sont pas encore pleinement comprises et le domaine a récemment accompli des progrès majeurs tant d'un point de vue théorique qu'expérimental. L'une des spécificités des systèmes nucléaires est que la force internucléonique qui les rend liés ne peut pas être directement dérivée de la chromodynamique quantique (QCD). Alors que des tentatives de connexion de l'interaction nucléaire à la QCD sous-jacente par des calculs sur réseau ont été effectuées [START_REF] Ishii | Nuclear Force from Lattice QCD[END_REF][2][START_REF] Van Kolck | Few-Nucleon Systems in a Quirky World[END_REF], des résultats physiquement pertinent ne sont pas encore disponibles. Compte tenu de ce fait, des méthodes plus phénoménologiques ou effectives ont été conçues pour effectuer des calculs réalistes. Une autre difficulté réside dans le contenu de l'interaction nucléaire elle-même, e.g. spin-orbite, tenseur, termes spin-orbite quadratiques, ce qui rend sa manipulation particulièrement complexe. De plus, il existe deux sources de caractère non perturbatif. La première correspond à la forte répulsion à courte portée associée à l'interpénétration des nucléons, qui est une source non perturbative de type ultraviolet. La seconde correspond aux longueurs de diffusion associées à l'existence d'un état proton-neutron faiblement lié, le deutéron, et d'un état di-neutron virtuel, c'est-à-dire une source non perturbative de type infrarouge. De plus, le traitement des nucléons en tant que particules ponctuelles, alors qu'ils sont en fait composites, nécessite l'utilisation de forces à trois, quatre, . . . , jusqu'à A nucléons en principe. En pratique, la limitation aux forces à trois nucléons s'est avérée fournir des résultats assez précis.

Étant donné un modèle de l'hamiltonien nucléaire, l'objectif est maintenant de relever le défi de résoudre l'équation de Schrödinger à A corps. Les noyaux atomiques contiennent jusqu'à environ 300 nucléons, de sorte que la plupart des noyaux ne peuvent pas être considérés comme de très petits ou très gros systèmes. Par conséquent, les méthodes exactes disponibles pour les systèmes à quelques nucléons trouvent rapidement leurs limites théoriques et de computationnelles, à mesure que le nombre de nucléons augmente, tandis que les effets physiques provenant de la taille finie des noyaux empêchent l'utilisation d'une approche statistique. La théorie nucléaire à basse énergie vise à décrire les propriétés de l'état fondamental (masse, rayon, déformation, moments multipolaires...) et des états excités ainsi que leurs différents modes de désintégration (forte, électromagnétique et électrofaible). En outre, une description unifiée des systèmes nucléaires nécessite une description des systèmes à couches fermées et des systèmes à couches ouvertes, des mouvements collectifs de petite et de grande amplitude, de l'interface entre la structure et la réaction pour accéder à la fission spontanée et induite, de la fusion et de l'émission de nucléons au niveau de la drip-line... Ce défi doit être relevé sur l'ensemble de la carte nucléaire, c'est-à-dire pour environ 3400 noyaux déjà observés [START_REF] Audi | The NUBASE2016 evaluation of nuclear properties[END_REF] et les milliers restant à découvrir.

Alors que les propriétés de volume des noyaux peuvent être principalement décrites en utilisant des approches macroscopiques comme le modèle de la goutte liquide (LDM) [5], des méthodes microscopiques, telle que l'approche fonctionnelle de la densité pour l'énergie (EDF), sont nécessaires pour une description cohérente des propriétés statiques et dynamiques sur la carte nucléaire. Dans ce qui suit, l'accent est cependant mis sur les méthodes dites ab initio qui peuvent être caractérisées par un ensemble commun de propriétés.

1. Les nucléons sont considérés comme les degrés de liberté élémentaires, c'est-à-dire que les quarks et gluons ne sont pas explicitement pris en compte, de sorte que les nucléons sont traités comme des objets ponctuels. Dans ce contexte, les degrés de liberté collectifs, comme tout autre phénomène nucléaire, sont censés émerger de la description des nucléons en interaction.

2. Les interactions internucléoniques sont basées sur la QCD sous-jacente pour préserver le lien avec la physique des hautes énergies. Le paradigme actuel de la physique nucléaire ab initio consiste à utiliser des interactions dérivées de la théorie effective des champs chirale (χEFT) qui sont ajustées dans le secteur à deux corps pour les forces à deux nucléons, dans le secteur à trois corps pour les forces à trois nucléons, etc.

3. La solution de l'équation de Schrödinger à A corps est développée de manière systématique, permettant ainsi de contrôler la troncation sur le résultat et d'évaluer les incertitudes associées.

4. Les erreurs sur les résultats, provenant de l'hamiltonien d'entrée, de la troncation analytique et du traitement numérique, sont finalement estimées.

Ces caractéristiques distinguent les méthodes ab initio des autres approches du problème nucléaire à A corps qui reposent sur des interactions phénoménologiques et pour lesquelles une évaluation approfondie des erreurs est souvent compliquée, voire impossible. Les méthodes ab initio fournissant une solution essentiellement exacte de l'équation de Schrödinger à A corps, à savoir Fadeev-Yakubovski [6-8], Green's function monte carlo [START_REF] Pudliner | Quantum Monte Carlo calculations of nuclei with A <˜7[END_REF][START_REF] Wiringa | Quantum Monte Carlo calculations for light nuclei[END_REF][START_REF] Wiringa | Quantum Monte Carlo calculations of A = 8 nuclei[END_REF] et le no-core shell model [START_REF] Navrátil | Shell-model calculations for the three-nucleon system[END_REF][START_REF] Navrátil | Large-basis shell-model calculations for p -shell nuclei[END_REF][START_REF] Navrátil | Few-nucleon systems in a translationally invariant harmonic oscillator basis[END_REF][START_REF] Quaglioni | Ab initio no-core shell model and microscopic reactions: Recent achievements[END_REF][START_REF] Navrátil | Recent developments in no-core shell-model calculations[END_REF], ont été limités aux noyaux légers jusqu'à A ∼ 12 en raison de leur complexité exponentielle en A. Au cours des quinze dernières années, et avec le développement de méthodes à complexité polynomiale, une extension significative des méthodes ab initio par rapport aux nombres de masse accessibles a été possible. Les approches ab initio applicables aux systèmes à couches fermées partent généralement d'un déterminant de Slater unique, par exemple Hartree-Fock (HF), comme état de référence et prennent en compte les corrélations dynamiques via l'inclusion d'excitations de type particule-trou. La méthode la plus simple à cet égard est la théorie des perturbations à N corps (MBPT) [START_REF] Goldstone | Derivation of the Brueckner many-body theory[END_REF][START_REF] Hugenholtz | Perturbation theory of large quantum systems[END_REF][START_REF] Shavitt | Many-body methods in chemistry and physics : MBPT and coupled-cluster theory[END_REF]. Cette méthode a été abandonnée dans les années 1960 par la communauté nucléaire en raison du noyau dur (que l'on croyait) inhérent à l'interaction nucléaire. Ce n'est que récemment avec le développement d'hamiltoniens plus souples, générés par des transformations du groupe de renormalisation (SRG) qui ont dompté la source ultraviolette de non perturbativité, que MBPT a été revisité avec grand succès [START_REF] Tichai | Hartree-Fock many-body perturbation theory for nuclear ground-states[END_REF][START_REF] Hu | Ab initio nuclear many-body perturbation calculations in the Hartree-Fock basis[END_REF][START_REF] Tichai | Open-shell nuclei from No-Core Shell Model with perturbative improvement[END_REF]. Plusieurs méthodes à N corps resommant les contributions perturbatives à tout ordre ont été développés pour décrire les systèmes de masse moyenne, par exemple coupled cluster theory (CC) [START_REF] Kowalski | Coupled Cluster Calculations of Ground and Excited States of Nuclei[END_REF][START_REF] Bartlett | Coupled-cluster theory in quantum chemistry[END_REF][START_REF] Hagen | initio coupled-cluster approach to nuclear structure with modern nucleon-nucleon interactions[END_REF][START_REF] Piecuch | Left-eigenstate completely renormalized equation-of-motion coupled-cluster methods: Review of key concepts, extension to excited states of open-shell systems, and comparison with electronattached and ionized approaches[END_REF][START_REF] Binder | Extension of coupled-cluster theory with a noniterative treatment of connected triply excited clusters to three-body Hamiltonians[END_REF], self-consistent Green function (SCGF) [START_REF] Dickhoff | Self-consistent Green's function method for nuclei and nuclear matter[END_REF][START_REF] Cipollone | Chiral three-nucleon forces and the evolution of correlations along the oxygen isotopic chain[END_REF][START_REF] Carbone | Self-consistent Green's functions formalism with three-body interactions[END_REF] ou encore in-medium similarity renormalization group (IMSRG) [START_REF] Tsukiyama | In-Medium Similarity Renormalization Group For Nuclei[END_REF][START_REF] Hergert | In-medium similarity renormalization group with chiral two-plus three-nucleon interactions[END_REF][START_REF] Morris | Magnus expansion and in-medium similarity renormalization group[END_REF][START_REF] Hergert | The In-Medium Similarity Renormalization Group: A novel ab initio method for nuclei[END_REF]. Pour les noyaux à double couches fermées, toutes ces méthodes non perturbatives concordent bien avec les calculs NCSM quasi exacts pour les énergies de l'état fondamental des noyaux dans le régime A ∼ 20, et sont considérées comme constituant les méthodes de référence pour les noyaux de moyenne masse.

Ces méthodes d'expansion ont pu, sur la base d'hamiltoniens chiraux réalistes, étendre leur portée de la carte nucléaire jusqu'à A ∼ 130 au cours de la dernière décennie [START_REF] Binder | Ab initio path to heavy nuclei[END_REF], mais sont restées longtemps limitées aux noyaux à deux couches fermées (ou voisin). En s'éloignant des fermetures de couches nucléaires, la description à déterminant de Slater unique devient qualitativement erronée en raison des dégénérescences présentes dans le spectre des particules uniques, nécessitant un traitement approprié des corrélations statiques. Afin de surmonter cet inconvénient, des états de référence plus généraux sont nécessaires. Une première option pour surmonter cette difficulté est de s'appuyer sur des méthodes multi-références (MR), tenant compte des différents états produits qui contribuent sensiblement à la fonction d'onde. Cette idée a été suivie pour développer MR-IMSRG [START_REF] Hergert | The In-Medium Similarity Renormalization Group: A novel ab initio method for nuclei[END_REF][START_REF] Hergert | Initio Calculations of Even Oxygen Isotopes with Chiral Two-Plus-Three-Nucleon Interactions[END_REF][START_REF] Hergert | In-medium similarity renormalization group for closed and open-shell nuclei[END_REF] en physique nucléaire ou MR-CC en chimie quantique [START_REF] Mukherjee | Use of Cluster Expansion Methods in the Open-Shell Correlation Problem[END_REF][START_REF] Jeziorski | Spin-adapted multireference coupled-cluster approach: Linear approximation for two closed-shell-type reference configurations[END_REF][START_REF] Musiał | Multireference coupled-cluster theory: The easy way[END_REF]. Plus récemment, cette idée a été employée dans le contexte de MBPT en utilisant des états de référence NCSM [START_REF] Tichai | Open-shell nuclei from No-Core Shell Model with perturbative improvement[END_REF]. Une deuxième option consiste à exploiter le concept de brisure spontanée de symétrie, brisant par exemple la symétrie U (1) associée à la conservation du nombre de particules afin de capturer le caractère superfluide des noyaux à une couche ouverte. Les noyaux à double couches ouvertes peuvent également être traités via la brisure de la symétrie SU (2) associée à la conservation du moment angulaire, permettant aux noyaux de se déformer. Briser la symétrie U (1) permet de faire face à l'instabilité des paires de Cooper et de capturer l'effet dominant de la source infrarouge de non perturbativité au niveau de l'état de référence. Via l'utilisation d'un vide plus général de Bogoliubov, la dégénérescence d'un déterminant de Slater à une couche ouverte par rapport aux excitations particules-trous est levée et transformée en une dégénérescence par rapport aux transformations de symétrie du groupe de symétrie brisée (U (1) dans ce cas). En conséquence, l'expansion mal définie des quantités exactes autour d'un déterminant de Slater est remplacée par une expansion bien définie autour d'un état de Bogoliubov. La brisure de symétrie est utilisée depuis des décennies par la communauté EDF [START_REF] Ring | The nuclear many-body problem[END_REF][START_REF] Bender | Self-consistent mean-field models for nuclear structure[END_REF][START_REF] Duguet | Breaking and restoring symmetries within the nuclear energy density functional method[END_REF][START_REF] Duguet | The Nuclear Energy Density Functional Formalism[END_REF], c'est-à-dire au niveau du champ moyen. Au cours des dix dernières années, de nouvelles méthodes ab initio à N corps ont été développées à partir d'états de référence à symétrie brisée, par ex. Gorkov SCGF (GSCGF) [START_REF] Somà | Ab initio self-consistent Gorkov-Green's function calculations of semimagic nuclei: Formalism at second order with a twonucleon interaction[END_REF][START_REF] Somà | Ab initio Gorkov-Green's function calculations of open-shell nuclei[END_REF][START_REF] Somà | Chiral twoand three-nucleon forces along medium-mass isotope chains[END_REF], Bogoliubov CC (BCC) [START_REF] Duguet | Symmetry broken and restored coupled-cluster. Global gauge symmetry and particle number[END_REF][START_REF] Signoracci | Ab initio Bogoliubov coupled cluster theory for open-shell nuclei[END_REF], Bogoliubov MBPT (BMBPT) [START_REF] Tichai | Bogoliubov many-body perturbation theory for open-shell nuclei[END_REF][START_REF] Arthuis | Bogoliubov Many-Body Perturbation Theory for Nuclei : Systematic Generation and Evaluation of Diagrams and First ab initio Calculations[END_REF] et même Bogoliubov CI (BCI) [START_REF] Ripoche | Combining symmetry breaking and restoration with configuration interaction: A highly accurate many-body scheme applied to the pairing Hamiltonian[END_REF][START_REF] Ripoche | Combining symmetry breaking and restoration with configuration interaction: Extension to z -signature symmetry in the case of the Lipkin model[END_REF], le dernier n'étant pas une méthode d'expansion. Une difficulté rencontrée par ces méthodes est que la brisure de symétrie ne peut pas réellement se produire dans les systèmes quantiques finis et les résultats des calculs à N corps comportent une contamination associée à des contributions de nombres de particules non ciblés, c'est-à-dire de plusieurs représentations irréductibles du groupe de symétrie brisé.

Afin de surmonter cette difficulté, la symétrie explicitement brisée doit finalement être restaurée, ce qui a constitué un défi de longue date au niveau formel. La dégénérescence par rapport aux transformations du groupe U (1) doit donc être levée en restaurant la symétrie. Alors qu'un protocole de restauration de symétrie approprié reste encore à être formulé dans le cadre de GSCGF, des extensions des formalismes MBPT et CC incluant la brisure et la restauration de symétrie ont été récemment conçus pour restaurer systématiquement la symétrie à n'importe quel ordre de troncation [START_REF] Duguet | Symmetry broken and restored coupled-cluster. Global gauge symmetry and particle number[END_REF][START_REF] Duguet | Symmetry broken and restored coupled-cluster theory: I. Rotational symmetry and angular momentum[END_REF][START_REF] Qiu | Particle-number projected Bogoliubov-coupled-cluster theory: Application to the pairing Hamiltonian[END_REF]. Le présent travail se concentre sur la restauration de la symétrie U (1) dans les méthodes d'expansion à N corps ab initio, en particulier dans la version projetée sur le nombre de particules de BMBPT qui est dénommé BMBPT projetée (PBMBPT). L'objectif actuel est de mettre en oeuvre le formalisme PBMBPT en vue d'effectuer des calculs ab initio des noyaux à une couche ouverte de masses moyenne et grande. Parallèlement à ces évolutions, plusieurs problèmes formels et techniques non triviaux se dressaient. Le présent travail a consisté à L'objectif du chapitre 4 est de concevoir un moyen de calculer d'autres observables que l'énergie de l'état fondamental et le nombre de particules, c'est-à-dire des observables qui ne commutent pas avec H. Étant donné que les états propres de H ne sont pas des états propres de ces opérateurs, une méthode de la valeur moyenne est requise, c'est-àdire que l'approche projective ne s'applique pas. Le présent travail étend, en présence d'une brisure de symétrie et d'une restauration possible, l'utilisation de l'approche de la valeur moyenne aux observables déjà utilisés dans la version standard de MBPT pour les couches fermées [START_REF] Strayer | Correlation Effects in Nuclear Densities[END_REF][START_REF] Roth | Hartree-Fock and many body perturbation theory with correlated realistic NN interactions[END_REF][START_REF] Tichai | Natural orbitals for ab initio no-core shell model calculations[END_REF]. Un autre objectif est d'aller au-delà de BMBPT aux bas ordres en appliquant la méthode eigenvector continuation (EC) [START_REF] Frame | Eigenvector Continuation with Subspace Learning[END_REF]. Dans les deux cas, des quantités similaires doivent être calculées. Une méthode systématique pour y accéder est conçue. La mise en oeuvre de ce schéma (sans projection) pour effectuer des calculs ab initio de divers observables dans des états fondamentaux nucléaires semi-magiques est en cours. Ces développements incluent jusqu'aux excitations triples, six quasi-particules, dans la définition des opérateurs C et D. Via EC, cela donne la promesse de capturer des excitations triples complètes au prix de leurs calculs perturbatifs dans la construction de C et D.

En pratique, la gestion complète des interactions à trois nucléons rend la résolution de l'équation de Schrödinger à A corps rapidement trop coûteuse à mesure que la masse du système augmente. Pour contourner le traitement explicite des opérateurs à trois corps, les calculs ab initio des noyaux de masse moyenne sont généralement effectués sur la base de l'approximation normal-ordered 2-body (NO2B). Cette approximation capture les effets dominants des forces à trois nucléons tout en travaillant avec des opérateurs à deux corps effectifs. Elle consiste à ordonner normalement l'opérateur par rapport à un état de référence à N corps et à éliminer la composante à trois corps effective. Elle a été utilisée jusqu'à présent sur la base des états de référence qui conserve la symétrie. Dans ce contexte, le but du chapitre 5 est donc de concevoir une approximation de l'ordre normal des opérateurs qui soit cohérente avec les symétries de l'hamiltonien tout en travaillant avec un état de référence à symétrie brisée (et restaurée). En se concentrant sur les formalismes à N corps dans lesquels la symétrie U (1) associée à la conservation du nombre de particules est brisée (et potentiellement restaurée), l'approximation particlenumber-conserving normal-ordered k-body (PNOkB) d'un opérateur arbitraire est conçu sur la base d'états de référence de Bogoliubov. Un test numérique basé sur des calculs de Hartree-Fock-Bogoliubov projetés est conçu et utilisé pour vérifier le caractère de conservatif ou non du nombre de particules de l'opérateur approximé.

Finalement le chapitre 6 traite de la réduction du moment angulaire permettant d'utiliser des tenseurs à N corps réduits associés à des représentations irréductibles (IRREPs) du groupe de symétrie SU (2) afin de prétraiter un sous-ensemble des sommations en jeu dans les réseaux de tenseurs définissant les équations de travail. Il s'avère que cette réduction pose un problème formel non trivial, de sorte qu'elle nécessite autant de temps à être dérivée à la main que les équations de travail initiales. Cependant, il existe une manière très systématique et élégante de faire face à cette tâche en utilisant les graphes de Yutsis [START_REF] Yutsis | Mathematical apparatus of the theory of angular momentum[END_REF]. Par conséquent, il est hautement souhaitable de mettre en parallèle les efforts déployés pour automatiser la génération des équations de travail originales [START_REF] Arthuis | ADG: Automated generation and evaluation of many-body diagrams I. Bogoliubov many-body perturbation theory[END_REF] en concevant un cadre qui effectue automatiquement la réduction de symétrie fastidieuse sans erreur. Dans ce travail, la première version d'un outil automatisé (le programme AMC) effectuant la réduction du moment angulaire basé sur la théorie des graphes est présentée. Prenant en entrée les expressions sans restriction de symétrie d'un réseau de tenseurs génériques, le code fournit leur forme couplée en moment angulaire en quelques secondes.

List of Figures

  .6c) One is typically concerned with the treatment of a N -body operator 2 O commuting with the particle-number operator A O ≡ o 00 + o 11 + o 22 + o 33 + . . .

  e +iA ϕ = δ AA ,(1.14) whereas the IRREPs fulfill a first-order ordinary differential equation (ODE)-i d dϕ e iAϕ = Ae iAϕ . (1.15)A tensor operator O of rank k 3 and a many-body state |Ψ A n of A particles transform under global gauge rotation according to R(ϕ)OR -1 (ϕ) = e ikϕ O ,(1.16)

F. 3 .

 3 Considering such a particle-number wave packet

  (ij) Z (ϕ) denotes the transformed operator of O ij such that the upper label (ij) is a sole reminder of the normal-ordered nature of the original operator but does not characterize the normalordered nature of the transformed operator. Contrarily, O mn Z (ϕ) does denote the normal-ordered part of the transformed operator O Z (ϕ) of O containing m (n) quasiparticle creation (annihilation) operators.expansion many-body methods are built. The HFB theory relies on the Ritz variational principle find the best mean-field approximation to the ground state of H in the manifold of Bogoliubov vacua |Φ . As |Φ violates particle-number symmetry A|Φ = A|Φ ,(1.93)

. 100 )

 100 Left-multiplying and right-multiplying equation (1.100) by R leads to [H, R] = 0 , (1.101)

  .114) which, with the resort to relation (1.35a) -Tr[κκ * ] + Tr[ρ 2 ] -Tr[ρ] = 0 , (1.115)

  .3b) where |Φ(ϕ) ≡ R(ϕ)|Φ , leads to the working form O A 0 = 2π 0 dϕ e -iAϕ O(ϕ) 2π 0 dϕ e -iAϕ N (ϕ) . (3.4) Equation (3.4) constitutes the master equation on which the PBMBPT formalism built.

Figure 3 . 1 .Figure 3 . 2 .

 3132 Figure 3.1. Canonical diagrammatic representation of normal-ordered contributions to the operator O in the Schrödinger representation.

Figure 3 . 3 .

 33 Figure 3.3. Canonical diagrammatic representation of normal-ordered contributions to the grand potential operator Ω in the Schrödinger representation.

Figure 3 . 4 .

 34 Figure 3.4. Diagrammatic representation of the four unperturbed elementary one-body propagators G gg (0) (ϕ). The convention is that the left-to-right reading of a matrix element corresponds to the up-down reading of the diagram.

Figure 3 . 5 .

 35 Figure 3.5. Convention to draw and read anomalous self-contractions. The example is given for a Ω 13 displaying a self-contraction.

Figure 3 . 6 .

 36 Figure 3.6. Selected second-order off-diagonal Feynman BMBPT diagrams.

Figure 3 . 7 .

 37 Figure 3.7. Zero-and first-order off-diagonal Feynman BMBPT diagrams. Diagrams are grouped vertically according to the number n a of anomalous lines they contain.

Figure 3 . 8 .

 38 Figure 3.8. Zero-, first-and second-order diagonal Feynman BMBPT diagrams generated from operator vertices containing four legs at most, i.e., with deg_max = 4.

Figure 3 . 9 .

 39 Figure 3.9. Zero-and first-order effective off-diagonal BMBPT diagrams recasting the twenty displayed in Fig. 3.7.

Figure 3 . 10 .

 310 Figure 3.10. Production of the TSDs associated with a third-order diagonal BMBPT diagram and with an off-diagonal diagram obtained from it by turning two among the eight normal lines into anomalous ones.

Figure 3 . 11 .

 311 Figure 3.11. Zero-, first-, second-and third-order TSDs corresponding to off-diagonal BMBPT diagrams generated from operators containing four or six legs at most, i.e., with deg_max = 4 or deg_max = 6.

Figure 3 . 12 .

 312 Figure 3.12. Fully-labelled third-order diagonal and off-diagonal BMBPT diagrams displayed in Fig. 3.10 and their associated TSDs. The bottom vertex corresponds to O 40 Z (ϕ).

Figure 3 . 13 .

 313 Figure 3.13. Fully-labelled second-order off-diagonal BMBPT diagram and its associated TSD.

3. 7 .

 7 Connection to time-ordered diagrammatics Diagram 3.3:

Contents 4 . 1 . 73 4. 4 .

 41734 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.2. Trial many-body state . . . . . . . . . . . . . . . . . . . . . . . 68 4.2.1. General form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.2.2. BCI coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.2.3. BCC coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.2.4. BMBPT coefficients . . . . . . . . . . . . . . . . . . . . . . . . 70 4.2.5. Eigenvector continuation coefficients (from BMBPT) . . . . . . 72 4.3. Computation of observables . . . . . . . . . . . . . . . . . . . . Diagrammatic method . . . . . . . . . . . . . . . . . . . . . . . 73 4.4.1. Off-diagonal kernels and transformed operators . . . . . . . . . 73 4.4.2. Diagrammatic representation of operators . . . . . . . . . . . . 76 4.4.3. Diagrammatic rules . . . . . . . . . . . . . . . . . . . . . . . . 78 4.4.4. Low-rank examples . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.4.5. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Z

  (ϕ) denotes the transformed operator of O ij such that the upper label (ij) is a sole reminder of the normal-ordered nature of the original operator but does not characterize the normalordered nature of the transformed operator. Contrarily, O mn Z (ϕ) does denote the normal-ordered part of the transformed operator O Z (ϕ) of O containing m (n) quasiparticle creation (annihilation) operators. Consistently, the notation O mn(ij) Z (ϕ) is used to further represent the part of O mn Z (ϕ) originating from the O ij part of O.

=

  Normal-ordered operators in the Schroedinger picture can be displayed diagrammatically. Considering the transformed operators O Z (ϕ), C † Z (ϕ) and D RZ (ϕ), canonical diagrams representing their normal-ordered contributions O ij Z (ϕ), C †ij Z (ϕ) and D ij RZ (ϕ) are shown in Figs. 4.2, 4.3 and 4.4, respectively. Such a representation based on fully-antisymmetrized vertices compacted into a point characterizes so-called Hugenholtz diagrams. Focusing on a generic operator K as an example, the various diagrams contributing to it must be understood in the following way 9 .

3 .

 3 In the canonical representation used in Figs. 4.2, 4.3 and 4.4, all oriented lines go up,

4 .Figure 4 . 1 .

 441 Figure 4.1. Rules to apply when departing from the canonical diagrammatic representation of a normal-ordered operator. Oriented lines can be rotated through the dashed line but not through the full line.

Figure 4 . 2 .

 42 Figure 4.2. Canonical diagrammatic representation of normal-ordered contributions to the operator O Z (ϕ) in the Schroedinger picture.

Figure 4 . 3 .

 43 Figure 4.3. Canonical diagrammatic representation of normal-ordered contributions to the operator C † Z (ϕ) in the Schroedinger representation.

Figure 4 . 4 . 4 . 4 . 3 .

 44443 Figure 4.4. Canonical diagrammatic representation of normal-ordered contributions to the operator D RZ (ϕ) in the Schroedinger representation.

4 . 5 .

 45 The seven additional contributions correspond to non-connected diagrams, see Fig.4.6, and arise because the numbers C †00 Z (ϕ), O 00 Z (ϕ) and D 00 RZ (ϕ) are potentially non-zero. Let us notice that the sum of the diagrams OCD0.1(ϕ), OCD1.1.1(ϕ) and OCD1.2.1(ϕ) is null for ϕ = 0 if the two unprojected states are orthogonals 10 , i.e. Ψ c |Ψ d = 0.

Figure 4 . 5 .

 45 Figure 4.5. Connected diagrams contributing to O cd (ϕ)

Figure 4 . 6 .

 46 Figure 4.6. Non-connected diagrams contributing to O cd (ϕ)These diagrams generate fifteen contributions to O cd (ϕ) that are given in App. E. The angular-momentum coupled form of these diagrams is given in App. G.5.Equation (4.37) for N cc (ϕ) can also be translated into a set of vacuum-to-vacuum diagrams. Is is trivial to understand that the corresponding diagrams can be obtained from

Figure 4 . 7 .

 47 Figure 4.7. Diagrams contributing to N cc (ϕ) and N dd (ϕ)

91 5. 3 . 91 5. 3 . 1 . 92 5. 3 . 2 . 92 5. 4 . 98 5. 5 .

 91391319232924985 k-body approximation in particle-number-breaking theories Contents 5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.2. Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.2.1. Naive extension . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.2.2. Particle-number-conserving NOkB approximation . . . . . . . . 86 5.2.3. PNO1B approximation of a two-body operator . . . . . . . . . 88 5.2.4. Other applications . . . . . . . . . . . . . . . . . . . . . . . . . Testing [O (P)NOkB , A] = 0 . . . . . . . . . . . . . . . . . . . . . . Particle-number projection . . . . . . . . . . . . . . . . . . . . Computation of Q A F . . . . . . . . . . . . . . . . . . . . . . . . 92 5.3.3. One-body operator . . . . . . . . . . . . . . . . . . . . . . . . . Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.4.1. Calculations set up . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.4.2. Energetics in O isotopes . . . . . . . . . . . . . . . . . . . . . . 94 5.4.3. Doubly gauge-rotated operator kernel . . . . . . . . . . . . . . 96 5.4.4. Systematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Figure 5 . 2 .

 52 Figure 5.2. Representation of the naive extension of NO2B approximation for a three-body operator O normal-ordered with respect to the quasi-particle vacuum |Φ . Left column: normal-ordered form with respect to |0 expressed in {c, c † }. Middle column: normalordered form with respect to |Φ expressed in {c, c † }. Right column: normal-ordered form with respect to |Φ expressed in {β, β † }. Red crosses indicate terms that are suppressed in the naive extension of the NO2B approximation such that red dashed lines separate suppressed terms from retained ones.

11 = o 11 +

 1111 Tr[o 22 ρ] , 22 κ * ] .

Λ22

  (5.20-5.21) fully define the PNO1B approximation of the two-body operator. It is particle-number conserving by construction. It is useful to further characterize the operator by closely inspecting its normal-ordered contributions. Focusing for example on the normal field Λ11 , one obtainsΛ11 = õ11 = o 11 + Tr[o 22 ρ] = Λ 11 , (5.23)which indeed satisfies the systematic property Λii = Λ ii . Eventually, the complete set of normal-ordered contributions to O PNO1B expressed in the single-particle basis relates to those of the original two-body operator through thus observes that the non-zero fields are strictly equal to those associated with the original operator, i.e., there is no extra term Λ11 when approximating a two-body operator.A graphical representation of the PNO1B approximation of a two-body operator O is given in Fig.5.3.

20 O

 20 04 O 13 O 22 O 31 O 40

Figure 5 . 3 .

 53 Figure 5.3. Representation of the PNO1B approximation of a two-body operator O. Left column: Normal-ordered form with respect to the particle vacuum |0 expressed in {c, c † }. Middle column: Normal-ordered form with respect to the Bogoliubov vacuum |Φ expressed in {c, c † }. Right column: Normal-ordered form with respect to |Φ expressed in {β, β † }. The red crosses, dashed lines and tildes embody the effect of the PNO1B approximation, i.e. (1) red crosses indicate the suppressed terms, (2) red dashed lines separate suppressed terms from retained ones and (3) red tildes represent the retained terms that are modified.

Figure 5 . 4 .

 54 Figure 5.4. Energetics along oxygen isotopes. Panel (a): HFB energy. Panel (b1): static correlation energy brought about by the particle-number projection (after variation). Panel (b2): dynamical correlation energy computed via BMBPT(2). Panel (c1): difference between PHFB energies obtained with the full two-body Hamiltonian and with its nNO1B or PNO1B approximations. Energies are computed twice, i.e., using a single or a double particle-number projection. Panel (c2): Ratios Q A H nNO1B and Q A H PNO1B of singly-over doubly-projected energies obtained on the basis of H nNO1B and H PNO1B .

Figure 5 . 5 .

 55 Figure 5.5. Real (left) and imaginary (right) parts of the doubly gauge-rotated mean-field connected Hamiltonian kernel h (0) (ϕ, ϕ ) in 18 O. Upper panel: full Hamiltonian. Middle panel: nNO1B. Lower panel: PNO1B.

Figure 5 . 6 .

 56 Figure 5.6. Fourier decomposition of the doubly gauge-rotated mean-field connected Hamiltonian kernel h (0) (ϕ, ϕ ) with respect to ϕ for fixed φ = π/2 in 18 O. Upper panel: full Hamiltonian. Middle panel: nNO1B. Lower panel: PNO1B.

Figure 5 . 7 .

 57 Figure 5.7. Energetics along oxygen (left) and calcium (right) isotopes. Upper panel:HFB energy. Middle panel: static correlation energy brought about by the particle-number projection (after variation). Lower panel: difference between PHFB energies obtained with the full two-body Hamiltonian and with its nNO1B or PNO1B approximations. Energies are computed twice, i.e. using a single or a double particle-number projection.

Figure 5 . 8 .

 58 Figure 5.8. Particle-number variance obtained via PHFB calculations along oxygen (left) and calcium (right) isotopes. Upper panel: results obtained with the full two-body operator. Lower panel: results obtained via the nNO1B and PNO1B approximation to the full operator. Results are computed twice, i.e. using a single or a double particle-number projection.

Contents 6 . 1 . 6 . 4 .

 6164 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.2. Symmetries and many-body theory . . . . . . . . . . . . . . . 103 6.2.1. Symmetry group . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.2.2. Symmetries of the single-particle basis . . . . . . . . . . . . . . 104 6.2.3. Symmetries of the reference state . . . . . . . . . . . . . . . . . 104 6.2.4. Reduction schemes and groups . . . . . . . . . . . . . . . . . . 105 6.2.5. Tensors and tensor networks . . . . . . . . . . . . . . . . . . . 105 6.2.6. Symmetry-reduced tensor networks . . . . . . . . . . . . . . . . 106 6.3. Angular-momentum algebra . . . . . . . . . . . . . . . . . . . . 107 6.3.1. Rationale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.3.2. Other symmetries . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.3.3. SU (2) group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6.3.4. Many-body states . . . . . . . . . . . . . . . . . . . . . . . . . 109 6.3.5. Many-body matrix elements . . . . . . . . . . . . . . . . . . . . 111 Diagrammatic method . . . . . . . . . . . . . . . . . . . . . . . 112 6.4.1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 6.4.2. Nodes and edges . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.4.3. Yutsis graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 6.4.4. Unfactorizable graphs . . . . . . . . . . . . . . . . . . . . . . . 115 6.4.5. From tensor networks to Yutsis graphs . . . . . . . . . . . . . . 117 6.4.6. Factorization rules . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.5. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6.5.1. Many-body perturbation theory . . . . . . . . . . . . . . . . . 120 6.5.2. Coupled-cluster theory . . . . . . . . . . . . . . . . . . . . . . . 126 6.5.3. In-medium similarity renormalization group . . . . . . . . . . . 130 6.6. The AMC program . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.6.1. Clebsch Gordan coefficients reduction . . . . . . . . . . . . . . 134 6.6.2. Tensor networks reduction . . . . . . . . . . . . . . . . . . . . . 135 6.7. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

kA

  ...k... B ...k... C ...k... f G sym -------→ k Ã... k... B... k... C... k... . (6.17)

  Translating the central node through the line carrying j 5 to the upper right corner and accounting for the change in the ordering of the edges attached to the upper-left and the lower-right nodes by chanding their sign, the diagram can be equivalently represented as a square with two diagonal edges

j 4 j 5 j 6 m 4 m 5 m 6 j 7 j 8 j 9 m 7 m 8 m 9 × j 1 j 4 j 7 m 1 m 4 m 7 j 2 j 5 j 8 m 2 m 5 m 8 j 3 j 6 j 9 m 3 m 6 m 9 ,

 69789

  [START_REF] Qiu | Projected coupled cluster theory[END_REF] and relates to a diagrammatic representation given by the following hexagon

3 Figure 6 . 1 .

 361 Figure 6.1. Factorization rule for 2-cycles, or bubbles, giving rise to a 3j-symbol.

. 63 )j 1 j 2 j 3 j 4 j 5 j 6 j 3 m 3 j 2 m 2 j 1 m 1 Figure 6 . 2 .

 6362162 Figure 6.2. Factorization rule for 3-cycles, or triangles, giving rise to a 6j-symbol.

Figure 6 . 3 .

 63 Figure 6.3. Factorization rule for 4-cycles, or squares, giving rise to two 6j-symbols.

  ... ...

Figure 6 . 4 .

 64 Figure 6.4. Schematic picture of a generic higher-order topology that cannot be simplified in terms of the triangle or square rules. Node signs are left out for simplicity.

(

  ii) two 3jm-symbols with zero-edges coming from the application of Wigner-Eckart theorem to the T 1 amplitudes, (iii) four 3jm-symbols coming from the coupling of H and T 2 matrix elements, yielding eight 3jm-symbols and eleven summations over magnetic quantum numbers, eight corresponding to one-body indices (m a , m b , m c , m d , m i , m j , m k , m l ), two originating from the decoupling of H and T 2 (M 1 ,M 2 ) and one (M ) coming from the external coupling of double amplitude equation. The corresponding Yutsis graph is given by

Figure 6 . 5 .

 65 Figure 6.5. Algorithm used to reduce Yutsis graphs in the AMC program.

Figure 6 . 6 .

 66 Figure 6.6. Illustration of the AMC program input format.

Figure D. 1 .

 1 Figure D.1. The three off-diagonal diagrams at first order.

D. 2 .D. 2 . 1 .D. 2 . 2 .

 22122 Diagrams and algebraic expressions at order 2 in PBMBPT Time-structure diagrams At this order, there is two TSDs with one being a tree TSD. Many-body diagrams and expressions At second order in PBMBPT, there are 37 off-diagonal diagrams. These diagrams are displayed in Fig. D.2. The algebraic (m-scheme) expressions corresponding to these 37 diagrams are given in the following 1 . Off-diagonal BMBPT diagrams generated from the diagonal BMBPT diagram PO2.1 PO2.1.1 =

Figure D. 2 .

 2 Figure D.2. The 37 off-diagonal diagrams at second order.

Figure F. 1 .

 1 Figure F.1. Representation of the PNO2B approximation of a three-body operator O. Left column: Normal-ordered form with respect to the particle vacuum |0 expressed in {c, c † }. Middle column: Normal-ordered form with respect to the Bogoliubov vacuum |Φ expressed in {c, c † }. Right column: Normal-ordered form with respect to |Φ expressed in {β, β † }. The red crosses, dashed lines and tildes embody the effect of the PNO2B approximation, i.e. (1) red crosses indicate the suppressed terms, (2) red dashed lines separate suppressed terms from retained ones and (3) red tildes represent the retained terms that are modified.

Figure F. 2 .

 2 Figure F.2. Representation of the PNO3B approximation of a five-body operator O. Left column: Normal-ordered form with respect to the particle vacuum |0 expressed in {c, c † }. Middle column: Normal-ordered form with respect to the Bogoliubov vacuum |Φ expressed in {c, c † }. Right column: Normal-ordered form with respect to |Φ expressed in {β, β † }. The red crosses, dashed lines and tildes embody the effect of the PNO3B approximation, i.e. (1) red crosses indicate the suppressed terms, (2) red dashed lines separate suppressed terms from retained ones and (3) red tildes represent the retained terms that are modified.

F. 3 . 26 Figure F. 3 .

 3263 Figure F.3. Distribution of good-particle components c 2A in each oxygen isotope. The dashed line denotes the average particle number in the underlying HFB vacuum.

k3 n k 2 (

 2

n k 3 n k 1 k2 j k 1 j k 2 J j k 3 j k 4 J - 1 2 j k 1 j k 2 J j k 4 j k 3 Jk3 k4 n k 5 (ljt) k 1 k2 j k 1 Ω11 n k 1 n k 5 k2 2 j k 2 Ω11 n k 2 n k 5 k1k1 k2 n k 5 (ljt) k 3 k4 j k 3 Ω11 n k 3 n k 5 k1 k2 k4 k5 k1 k2 k3 k4 + J Ω40 k1 k2 k3 n k 5 (ljt) k 4 j k 4 Ω11 n k 4 n k 5 k1j k 2 Ω20 n k 2 n k 5 Jk3 k4 k1 n k 5 (ljt) k 2 k2 k5 k1 k2 k3 k4 -(- 1 ) j k 1 +j k 2 +J j k 1 Ω20 n k 1 n k 5 Jk3 k4 k2 n k 5 (ljt) k 1 k1j k 4 Ω20 n k 4 n k 5 Jk1 k2 k3 n k 5 (ljt) k 4 k4 k5 k1 k2 k3 k4 -(- 1 ) j k 3 +j k 4 +J j k 3 Ω20 n k 3 n k 5 Jk1 k2 k4 n k 5 (ljt) k 3 k3G. 5 .

 1212351522553554452552k41215514554k41435535 k5 k6 J (-1) j k 3 +j k 4 +J Ĵ 2 k3 k4 k5 k1 k2 k3 k4+ J Ω40 k3 k4 k1 n k 5 (ljt) k Ω31 Ω31Ω31 Ω31 Diagrammatic contributions to O cd (ϕ) G.5.1. Connected contributions Angular-momentum coupled form of the second-order-type contributions to O cd (ϕ) (Eq. (E.1)) are given by OCD2.1(ϕ) = (ljt) k 1

k 1 (ljt) k 3 ĵk 1 ĵk 3 n k 1 n k 2 n k 3 n k 4 D(ljt) k 1 n k 1 n k 2 3 n k 3 n k 4 k 1 (ljt) k 3 ĵk 1 ĵk 3 n k 1 n k 2 n k 3 n k 4 Õ02 (ljt) k 1 n k 1 n k 2 n k 4 (n k 5 J 1 n k 1 n k 5 (ljt) k 1 n k 5 n k 5 JĴ2 Õ11 (ljt) k 1 n k 1 n k 5 (ljt) k 1 n k 5 n k 1 n k 2 D(ljt) k 1 n k 1 n k 2 1 n k 2 n k 1 n k 1 n k 2 D(ljt) k 1 n k 1 n k 2 1 n k 2 n k 1 n k 1 n k 2 Õ20 (ljt) k 1 n k 1 n k 2 1 n k 2 n k 1 G. 5 . 3 . 1 ĵ2 k 1 n k 1 n k 2 C(ljt) k 1 n k 1 n k 2 1 n k 2 n k 1

 1334112334133411245115155115152112121211212121121215312112121 (ϕ) 0 C †04 k1 (ljt) k 1 n k 2 k3 (ljt) k 3 n k 4 (ϕ) Õ20 (ljt) k (ϕ) 0 D40 k1 (ljt) k 1 n k 2 k3 (ljt) k 3 Ĵ2 C †(ljt) k (ϕ) J Õ13 k2 k3 k4 (ϕ) J D40 k3 k4 k1 k2 (ϕ(ϕ) J D40 k2 k3 k4 (ϕ) k3 k4 k5 k6 J Ĵ2 J C †04 k1 k2 k3 k4 (ϕ) J Õ22 k3 k4 k5 k6 (ϕ) J D40 k5 k6 k1 k2 (ϕ) (G.11h)G.5.2. Non-connected contributionsAngular-momentum coupled form of the non-connected diagrams contributing to O cd (ϕ) (Eq. (E.2)) are given byOCD0.1(ϕ) = D00 (ϕ) Õ00 (ϕ) C †00 (ϕ(ϕ) C †(ljt) k (ϕ) Õ02 (ljt) k (ϕ) C †(ljt) k Diagrammatic contributions to N cc (ϕ)Angular-momentum coupled form of the diagrams contributing to N cc (ϕ) (Eq. (E.3)) are given by NCC0.1(ϕ) = C00 (ϕ) C †00 (ϕ) , (G.13a) NCC1.1.1(ϕ) = 1 2 (ljt) k (ϕ) C †(ljt) k (ϕ) , (G.13b)

3. 1 . 4 . 1 .

 141 Canonical diagrammatic representation of normal-ordered contributions to the operator O in the Schrödinger representation. . . . . . . . . . . . . . . 3.2. Rules to apply when departing from the canonical diagrammatic representation of a normal-ordered operator. . . . . . . . . . . . . . . . . . . . . . . 3.3. Canonical diagrammatic representation of normal-ordered contributions to the grand potential operator Ω in the Schrödinger representation. . . . . . 3.4. Diagrammatic representation of the four unperturbed elementary one-body propagators G gg (0) (ϕ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Convention to draw and read anomalous self-contractions. . . . . . . . . . 3.6. Selected second-order off-diagonal Feynman BMBPT diagrams. . . . . . . 3.7. Zero-and first-order off-diagonal Feynman BMBPT diagrams. . . . . . . . 3.8. Zero-, first-and second-order diagonal Feynman BMBPT diagrams generated from operator vertices containing four legs at most. . . . . . . . . . . 3.9. Zero-and first-order effective off-diagonal BMBPT diagrams recasting the twenty displayed in Fig. 3.7. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10. Production of the TSDs associated with a third-order diagonal BMBPT diagram and with an off-diagonal diagram obtained from it by turning two among the eight normal lines into anomalous ones. . . . . . . . . . . . . . 3.11. Zero-, first-, second-and third-order TSDs corresponding to off-diagonal BMBPT diagrams generated from operators containing four or six legs at most. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12. Fully-labelled third-order diagonal and off-diagonal BMBPT diagrams displayed in Fig. 3.10 and their associated TSDs. . . . . . . . . . . . . . . . . 3.13. Fully-labelled second-order off-diagonal BMBPT diagram and its associated TSD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rules to apply when departing from the canonical diagrammatic representation of a normal-ordered operator. . . . . . . . . . . . . . . . . . . . . . . 4.2. Canonical diagrammatic representation of normal-ordered contributions to the operator O Z (ϕ) in the Schroedinger picture. . . . . . . . . . . . . . . . 4.3. Canonical diagrammatic representation of normal-ordered contributions to the operator C † Z (ϕ) in the Schroedinger representation. . . . . . . . . . . . 4.4. Canonical diagrammatic representation of normal-ordered contributions to the operator D RZ (ϕ) in the Schroedinger representation. . . . . . . . . . . 4.5. Connected diagrams contributing to O cd (ϕ) . . . . . . . . . . . . . . . . . 4.6. Non-connected diagrams contributing to O cd (ϕ) . . . . . . . . . . . . . . . 4.7. Diagrams contributing to N cc (ϕ) and N dd (ϕ) . . . . . . . . . . . . . . . . . 5.1. Representation of the NO2B approximation for a three-body operator O normal-ordered with respect to a Slater Determinant |SD . . . . . . . . . . Titre : Théorie de perturbation à N corps de Bogolioubov projeté: Solutions aux difficultés formelles et techniques Mots-clés : ab initio, théorie à N corps, théorie de perturbation, brisure de symétrie, restauration de symmétrie, outils automatisés Résumé : Étant donnée l'interaction entre les nucléons, la résolution de l'équation

Title:

  Projected Bogoliubov Many-Body Perturbation Theory: Overcoming formal and technical challenges Keywords: ab initio, many-body theory, perturbation theory, symmetry breaking, symmetry restoration, automated tools Abstract: 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 mean field 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 the atomic nucleus. Thus, the restoration of U(1) symmetry beyond the mean field is necessary for a correct 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. Université Paris-Saclay Espace Technologique / Immeuble Discovery Route de l'Orme aux Merisiers RD 128 / 91190 Saint-Aubin, France

Table 2 .1. Left and right gauge-angle rotations correspondence Chapter 3. Automated generation and evaluation of many-body diagrams in PBMBPT Contents 3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Many-body formalism . . . . . . . . . . . . . . . . . . . . . . . .

 2 Norm kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	3.2.1. Projective eigenequations . . . . . . . . . . . . . . . . . . . . .
	3.2.2. Imaginary-time formalism . . . . . . . . . . . . . . . . . . . . .
	3.2.3.

3.3. Perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . .

	3.3.1. Partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	3.3.2. Perturbative expansion . . . . . . . . . . . . . . . . . . . . . .
	3.3.3. Diagrammatic representation . . . . . . . . . . . . . . . . . . .
	3.3.4. Diagrams generation . . . . . . . . . . . . . . . . . . . . . . . .
	3.3.5. Diagram evaluation . . . . . . . . . . . . . . . . . . . . . . . .
	3.3.6. Towards higher orders . . . . . . . . . . . . . . . . . . . . . . .

3.4. Generation of off-diagonal BMBPT diagrams . . . . . . . . .

	3.4.1. Basic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	3.4.2. Similarity-transformed operator . . . . . . . . . . . . . . . . . .
	3.4.3. Systematic scheme . . . . . . . . . . . . . . . . . . . . . . . . .
	3.5

. Evaluation of off-diagonal BMBPT diagrams . . . . . . . . . .

	3.5.1. Time-structure diagrams . . . . . . . . . . . . . . . . . . . . . .
	3.5.2. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	3.5.3. From the TSD back to the BMBPT diagram . . . . . . . . . .
	3.6

. Output of the ADG program . . . . . . . . . . . . . . . . . . . . . 3.7. Connection to time-ordered diagrammatics . . . . . . . . . . . 3.8. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Table 3 . 1 .

 31 Number of (P)BMBPT diagrams generated from operators containing at most four (deg_max = 4) or six (deg_max = 6) legs.

	Method	Order	0 1	2	3	4
	BMBPT	deg_max = 4 1 2	8	59	568
		deg_max = 6 1 3	23	396	10716
	PBMBPT deg_max = 4 1 3	37	919	31968
		deg_max = 6 1 6 213 19530	. . .

  thus, providing the first factor entering the denominator. b) The vertex at time τ 2 in the BMBPT diagram corresponds to vertex a 2 in the TSD. It has no descendant such that the corresponding subgraph reduces to itself. The sum of quasi-particle energies associated to the lines entering the subgraph is E k 1 k 2 k 5 k 6 , thus providing the second factor entering the denominator. c) The vertex at time τ 3 in the BMBPT diagram correspond to vertex a 3 in the TSD. It has no descendant such that the corresponding subgraph reduces to itself. The sum of quasi-particle energies associated to the lines entering the subgraph is E k 3 k 4 k 7 k 8 , thus, providing the last factor entering the denominator.

d) Eventually, the complete denominator reads as

  .34) Expressing the rotated vacuum |Φ(ϕ) as a Thouless transformation applied to |Φ (see Sec. 1.4.4 and Tab. 2.1), one can rewrite O cd (ϕ) according to

  .1. 02 O 11 O 20 O 04 O 13 O 22 O 31 O 40 O 06 O 15 O 24 O 33 O 42 O 51 O 60 Representation of the NO2B approximation for a three-body operator O normal-ordered with respect to a Slater Determinant |SD . Left column: normal-ordered form with respect to |0 expressed in {c, c † }. Middle column: normal-ordered form with respect to |SD expressed in {c, c † }. Right column: normal-ordered form with respect to |SD expressed in {β, β † }. The contributions o ij /Λ ij /O ij are sorted horizontally according to i -j and vertically according to i + j. Red crosses indicate terms that are suppressed in the NO2B approximation such that red dashed lines separate suppressed terms from retained ones.

	o 00		Λ 00	O 00
	o 11 o 22	-→	Λ 11 Λ 22	-→
	o 33		Λ 33
	Figure 5.1.

O

  04 O 13 O 22 O 31 O 40 O 06 O 15 O 24 O 33 O 42 O 51 O 60

				O NO2B , A = 0 .	(5.2)
	A graphical representation of this naive extension of the NO2B approximation is given in
	Fig. 5.2.
	o 00		Λ 00		O 00
	o 11 o 22	-→	Λ 02 Λ 11 Λ 20 Λ 13 Λ 22 Λ 31	-→	O 02 O 11 O 20
	o 33		Λ 33

Whereas this truncation is transparent and appropriate in the symmetry-conserving context, it has not been discussed in connection with symmetry-breaking reference states. It happens that, even if O commutes with A, the approximate operator O NO2B obtained via a naive generalization of Eq. (5.1) on the basis of a Bogoliubov reference state |Φ , does not commute with A, i.e. O

  .14) This constitutes the simplest possible case. While it is not of interest for realistic ab initio calculations, it is used in the present chapter to illustrate the difference between a nNOkB approximation and the newly designed PNOkB one.

	and involves the following normal and anomalous fields
	Λ 22 = o 22 ,				(5.16a)
	Λ 20 =	1 2	Tr[o 22 κ] ,				(5.16b)
	Λ 11 = o 11 + Tr[o 22 ρ] ,				(5.16c)
	Λ 00 = o 00 + Tr[o 11 ρ] +	1 2	Tr[o 22 ρρ] +	1 4	Tr[o 22 κ * κ] .	(5.16d)

The normal ordering of O with respect to |Φ reads in the single-particle basis as O = Λ 00 + Λ 20 + Λ 11 + Λ 02 + Λ

22

(5.15)

  One first observes that, even if O has no constant term o 00 to begin with, O PNO1B does acquire one. The corresponding contributions originate from the two-body operator whose normal-ordered two-body part is omitted. When normal ordering O PNO1B in the singleparticle basis, this term combines with the fully contracted part obtained from õ11 to generate the complete fully contracted part of the original two-body operator. One further notes that õ00 entering O PNO1B is different from the one appearing in Eq.(5.19), which underlines the fact that the PNO1B approximation does not correspond to performing the nNO1B approximation before dropping particle-number non-conserving terms.Equations

						.21a)
	õ00 ≡ Λ 00 -Λ00(11)
	= o 00 -	1 2	Tr[o 22 ρρ] +	1 4	Tr[o 22 κ * κ] ,	(5.21b)
	where the only needed intermediate quantity is
	Λ00(11) = Tr[õ 11 ρ]
	= Tr[o 11 ρ] + Tr[o 22 ρρ] .	(5.22a)

  In the simplest case, the vacuum is chosen to be a Slater determinant |Φ G ham obtained from a SU (2)-restricted Hartree-Fock mean-field calculation, i.e. |Φ G ham thus typically belongs to the trivial IRREP of SU (2) and U (1), i.e. it carries good angular momentum J = 0 and a fixed number of particles. Dynamic correlations are introduced via the action of the wave operator that generates summations over elementary particle-hole excitations.In open-shell systems, the above reference state is improper due to the partial filling of the last occupied shell. This leads to a degeneracy with respect to particle-hole excitations, thus, signalling the existence of a Goldstone mode and the ill-definition of the previously performed expansion of W . This problem can be circumvented by lowering the symmetry group of H ref , i.e. by taking a well-chosen subgroup G ref ⊂ G ham . This typically leads to breaking U (1) symmetry in singly open-shell nuclei and/or SU (2) in doubly open-shell nuclei. The lower symmetry of H ref induces a lower reference energy due to the enlarged variational space

	G ref = G ham .	(6.11)
	In nuclear systems,

.10)

such that |Φ typically belongs to the trivial IRREP of G ref . In the following, the reference state |Φ G ref thus carries a subscript specifying the symmetry group of the Hamiltonian it is the ground-state of.

  .14) is a multi-variate data array carrying k indices with (possibly different) index ranges I 1 , . . . , I k . Given a set of SU-Ts, A, B, C, . . ., a contraction is defined as the summation over a common index, e.g.,

k A ...k... B ...k... C ...k... , (6.15)

Table A . 3 .

 A3 02 O 11 O 20O[4] O14 O 13 O 22 O 31 O 40 O [6] O 06 O 15 O 24 O 33 O 42 O 51 O 60 Contributions to the three-body operator O in normal-ordered form with respect to the Bogoliubov vacuum |Φ and expressed in {β, β † }. The O ij contributions are sorted horizontally according to i -j and vertically according to i + j. O 14 O 13 O 22 O 31 O 40 O [6] O 06 O 15 O 24 O 33 O 42 O 51 O 60

	{β, β † }, |SD	-6	-4	-2	0	+2 +4 +6
	O [0]				O 00
	O [2]			O 02 O 11 O 20
	O [4]

Table A . 4 .

 A4 Contributions to the three-body operator O in normal-ordered form with respect to the slater determinant |SD and expressed in {β, β † }. The O ij contributions are sorted horizontally according to the value of i -j and vertically according to the value of i + j.

	O = Λ 00 ,	(A.7a)

  ≡ 1 -P αβ -P αγ -P αδ , (C.4b) P (αβ/γδ) ≡ 1 -P αγ -P αδ -P βγ -P βδ + P αγ P βδ , (C.4c)

	3.3.1. In Eqs. (C.2) and (C.3), P (. . . / . . . ) denotes a
	general permutation operator [49] that is used in order to antisymmetrize unequivalent
	indices and is given by
	P (α/β) ≡ 1 -P αβ ,	(C.4a)
	P (α/βγδ)

.1.2. Examples PNO2B approximation of a three-body operator

  Let us exemplify the case of largest interest in current nuclear structure ab initio calculations, i.e. the PNO2B approximation of the three-body operatorO = o 00 + o 11 + o 22 + o 33 .

										(F.8)
	The normal-ordered form of O with respect to |Φ reads in the single-particle basis as
							O =		Λ 00
							+ Λ 20 + Λ 11 + Λ 02
							+ Λ 31 + Λ 22 + Λ 13
									+ Λ 33 ,	(F.9)
	and involves the set of normal and anomalous fields
	Λ 33 = o 33 ,							(F.10a)
	Λ 31 =	1 2	Tr[o 33 κ] ,							(F.10b)
	Λ 22 = o 22 + Tr[o 33 ρ] ,					(F.10c)
	Λ 20 =	1 2	Tr[o 22 κ] +	1 2	Tr[o 33 ρκ] ,			(F.10d)
	Λ 11 = o 11 + Tr[o 22 ρ] +	1 2	Tr[o 33 ρρ] +	1 4	Tr[o 33 κ * κ] ,	(F.10e)
	Λ 00 = o 00 + Tr[o 11 ρ] +	1 2	Tr[o 22 ρρ] +	1 4	Tr[o 22 κ * κ] +	1 6	Tr[o 33 ρρρ] +	1 4	Tr[o 33 ρκ

* κ] . (F.10f)

  One thus observes that the non-zero fields are strictly equal to those associated with the original operator, i.e. there is no so-called extra term when approximating a three-body operator.A graphical representation of the PNO2B approximation of a three-body operator O is given in Fig.F.1. 04 O 13 O 22 O 31 O 40 O 06 O 15 O 24 O 33 O 42 O 51 O 60

	o 00		Λ 00				O 00
	o 11 o 22	-→	Λ 02 Λ 11 Λ 20 Λ 13 Λ 22 Λ 31	-→		O 02 O 11 O 20
	o 33		Λ 33
							(F.12a)
			õ11 = Λ 11 -Λ11(22)
			= o 11 -	1 2	Tr[o 33 ρρ] +	1 4	Tr[o 33 κ * κ] ,	(F.12b)

22

= o

22

+ Tr[o

33

ρ] , O

  O = o 00 + o 11 + o 22 + o 33 + o 44 .

	Λ00(33) =	1 6	Tr[õ 33 ρρρ] +	1 4	Tr[õ 33 ρκ * κ]
					=	1 6	Tr[o 33 ρρρ] +	1 4	Tr[o 33 ρκ * κ] +	1 6	Tr[o 44 ρρρρ] +	1 4	Tr[o 44 ρρκ * κ] .	(F.22f)
	The two above equations fully define the PNO3B approximation of the four-body operator.
	It is particle-number conserving by construction. It is useful to further characterize the
	operator by closely inspecting its normal-ordered contributions. Focusing for example on
	the normal field Λ22 , one obtains
										Λ22 = õ22 + Tr[õ 33 ρ]
												= o 22 + Tr[o 33 ρ] +	1 2	Tr[o 44 ρρ] +	1 4	Tr[o 44 κ * κ]
												= Λ 22 ,	(F.23)
	which indeed satisfies the systematic property Λii = Λ ii . Focusing now on the anomalous
	field Λ20 , one obtains
	Λ20 =	1 2	Tr[õ 22 κ] +	1 2	Tr[õ 33 ρκ]
		=	1 2	Tr[o 22 κ] +	1 2	Tr[o 33 ρκ] +	1 4	Tr[o 44 ρρκ] +	1 8	Tr[o 44 κ * κκ]
		=		1 2	Tr[o 22 κ] +	1 2	Tr[o 33 ρκ] +	1 4	Tr[o 44 ρρκ] +	1 16	Tr[o 44 κ * κκ] +	1 16	Tr[o 44 κ * κκ]
		= Λ 20 +	1 16	Tr[o 44 κ * κκ] ,	(F.24)
												O =	Λ 00
												+ Λ 20 + Λ 11 + Λ 02
												+ Λ 40 + Λ 31 + Λ 22 + Λ 13 + Λ 04
												+ Λ 42 + Λ 33 + Λ 24
												+ Λ 44 ,	(F.18)
	and involves the following normal and anomalous fields
	Λ 44 = o 44 ,								(F.19a)
	Λ 42 =	1 2	Tr[o 44 κ] ,				(F.19b)
	Λ 40 =	1 8	Tr[o 44 κκ] ,			(F.19c)
	Λ 33 = o 33 + Tr[o 44 ρ] ,	(F.19d)
	Λ 31 =	1 2	Tr[o 33 κ] +	1 2	Tr[o 44 ρκ] ,	(F.19e)

(F.17)

The normal ordering of O with respect to |Φ reads in the single-particle basis as

  1/8 instead of 1/16 in the term Tr[o 44 κ * κκ]. This originates from the non-equivalent combinatorial prefactor at play when generating contributions of the type Tr[o 44 κ * κκ] with three anomalous contractions to Λ20 and Λ 20 via the application of Wick's theorem to O PNO3B and O, respectively. The difference between the normal-ordered fields of the PNOkB operator and of the original operator, if any, can always be rephrased in terms of modified prefactors of certain contributions involving strings of anomalous contractions. Eventually, the complete set of normal-ordered contributions to O PNO3B expressed in For illustration, a graphical representation of the PNO3B approximation of the five-body operator O is given in Fig. F.2. 11 O 20 O 04 O 13 O 22 O 31 O 40 O 06 O 15 O 24 O 33 O 42 O 51 O 60 O 08 O 17 O 26 O 35 O 44 O 53 O 62 O 71 O 80 O 010 O 19 O 28 O 37 O 46 O 55 O 64 O 73 O 82 O 91 O 100

	o 00		Λ 00	O 00
	o 11		Λ 02 Λ 11 Λ 20 Λ 11 Λ 11	O 02 O
	o 22 o 33	-→	Λ 04 Λ 13 Λ 15 Λ 24 Λ 33 Λ 42 Λ 51 Λ 22 Λ 31 Λ 40	-→
	o 44		Λ 35 Λ 44 Λ 53
	o 55		Λ 55

  3m)

	J O	04(31) k1 k2 k3 k4	(ϕ) = -	n k 5	n k 6	n k 7	J Λ 31 n k 6	(ljt) k 3	n k 7	(ljt) k 4	n k 5

  j k 4 R ++

															n k 7	n k 4	(ϕ)	(G.3n)
	J O	04(40) k1 k2 k3 k4	(ϕ) =	n k 5	n k 6	n k 7	n k 8	J Λ 40 n k 5	(ljt) k 1	n k 6	(ljt) k 2	n k 7	(ljt) k 3	n k 8	(ljt) k 4

  de Schrödinger à A corps permet d'accéder aux propriétés des états quantiques des noyaux. La théorie des perturbations à N corps (MBPT) basée sur un état de champ moyen d'Hartree-Fock permet de traiter les corrélations dynamiques mises en jeu dans les noyaux à doubles couches fermées. Au-delà des fermetures de couches, la brisure spontanée de la symétrie U(1) associée à la conservation du nombre de protons et de neutrons permet en sus l'inclusion des corrélations statiques au niveau du champ moyen et la formulation de la théorie des perturbations à N corps de Bogoliubov (BMBPT). Néanmoins, les résultats des calculs BMBPT présentent des contaminations du fait que la brisure de symétrie n'est qu'émergente dans les systèmes quantiques finis tels que le noyau atomique. Ainsi, la restauration de la symétrie U(1) au-delà du champ moyen est nécessaire pour une description correcte et donne lieu à la formulation de la méthode BMBPT projetée (PBMBPT). Le but est d'implémenter PBMBPT afin d'effectuer des calculs ab initio pour les noyaux à couche ouverte et de masse intermédiaire. Le présent travail apporte des solutions systématiques aux problèmes formels et techniques qui se posent lors de l'implémentation numérique de PBMBPT: une méthode automatique et sûre permettant de générer les diagrammes PBMBPT ainsi que les expressions correspondantes, un formalisme permettant de calculer d'autres observables que l'énergie, une extension de l'approximation en produit normal aux méthodes avec brisures de symétries ainsi qu'un outil automatique et sûr de réduction sphérique de réseaux de tenseurs.

In this document operators are given in math style, e.g. the particle-number operator A, while eigenvalues are given in roman style, e.g. the particle-number A.

In the following, the notation deg_max will be used as the maximum rank of an operator. It gives 2N for a N -body operator.

Even though both Eqs. (1.27a) and (1.27b) gives four relations on U and V matrices, the last four are redundants.

To say "The" Bogoliubov vacuum is an abuse because a state in quantum mechanics is always defined up to a phase.

Also called pairing tensor.

An alternative definition |Φ(ϕ) ≡ R(ϕ)|Φ is used in Chap. 3. Consequences of this choice are summarized in Tab. 2.1.

The quantities Λ ij are called fields as in Hartree-Fock-Bogoliubov theory. This will become clear later on.

It is usually called the chemical potential or the Fermi energy.

Λ 11 and Λ 20 correspond to matrix elements of Ω in the basis {c l , c † l } normal-ordered with respect to |Φ , see Sec. 1.5.1 and App. A.1.

Up to a phase.

The expansion is defined on top of any Bogoliubov vacuum, not necessarily the HFB one. In the context of PBCC, the vacuum can be obtained through a mean-field calculation that is informed of the symmetry restoration.

When U (1) gauge symmetry is not broken the particle-number projection acts as the identity such that |Ψ PBCC = |Ψ BCC .

Inserting the BCC wave-function ansatz instead of the PBCC one would lead to the BCC ground-state energy E A BCC = Φ|He U |Φ , where intermediate normalization Φ|Ψ BCC = 1 is used.

In the case where the resolution of the set of ODEs leads to non 2π-periodic W 0 (ϕ), there would be a non-zero surface term [e -iAϕ N (ϕ)] 2π 0 contribution to the particle number.

It is to be noted that in the exact limit, the connected operator kernel is indeed independent of ϕ, which makes the projection and thus the integral over the gauge angle trivial[START_REF] Duguet | Symmetry broken and restored coupled-cluster. Global gauge symmetry and particle number[END_REF].

It is the case for a Hamiltonian containing two-body forces only or approximated via the PNO2B approximation, see App. F.1.2.

Complementing Eq. (2.40), second-order unrotated cluster amplitudes are given in App. C.

Here the superscript A is not present because there is no symmetry restoration and thus the BMBPT ground-state is not an eigenstate of A.

This change of definitions, that do not impact observables in the end, is due to historical reasons. While the definitions used in the present chapter are considered with Ref.[START_REF] Qiu | Particle-number projected Bogoliubov-coupled-cluster theory: Application to the pairing Hamiltonian[END_REF] published very recently, the definitions used in the next chapter follow the original publication on PBCC and PBMBPT[START_REF] Duguet | Symmetry broken and restored coupled-cluster. Global gauge symmetry and particle number[END_REF].

Off-diagonal kernels given in Eq. (3.3) relates to those introduced in Eq. (2.5) via a complex conjugation and the replacement ϕ → -ϕ, see Sec.

2.3.6 for a detailed correspondence.

The time-ordering operator orders a product of operators in decreasing order according to their time labels (i.e., larger times to the left) and multiplies the result with the signature of the permutation used to achieve the corresponding reordering.

In agreement with Eq. (3.5), straight BMBPT is recovered from Eq. (3.21) for ϕ = 0 given that O A 0 = O(0) = o(0) in this formalism.

The expansion starts at order p = 0 that corresponds to the term containing no Ω 1 operator and no time integral in Eq.(3.21).

For the left definition of rotated vacuum and thouless transformation, the non-zero anomalous contraction is R ++ (Eq. (1.65)) whereas it is R --for the right definition, see Tab. 2.1 for correspondence.

A normal line can be interpreted as G -+(0) or G +-(0) depending on the ascendant or descendant reading of the diagram. Similarly, the ordering of quasi-particle and time labels of a propagator depends on the ascendant or descendant reading of the diagram. All the lines involved in a given diagram must be interpreted in the same way, i.e., sticking to an ascendant or descendant way of reading the diagram all throughout. In the present work, the chosen convention corresponds to reading diagrams from top to bottom, which further relates to reading the matrix element it originates from in a left-right fashion. This is the convention employed to represent the four propagators in Fig.3.4.

The reading of an anomalous line linking two different vertices is unambiguous as long as one stick to the up-down convention displayed in Fig.3.4. However, the up-down reading of a self-contraction is potentially ambiguous depending on the way the line is actually drawn. As illustrated in Fig.3.5, one must further fix a

In case a line is drawn such that it crosses itself, the crossing(s) must be omitted when evaluating p.

The number of quasi-particle indices on which summation is performed increases by one per anomalous propagator due to the fact that the matrix R --(ϕ) is not diagonal in quasi-particle space.

This diagram is the one denoted as PO2.2 in Fig.6of Ref.[START_REF] Arthuis | ADG: Automated generation and evaluation of many-body diagrams I. Bogoliubov many-body perturbation theory[END_REF].

The application of a similarity transformation to an operator has been discussed in Sec. 1.5.3 for the left definition of the rotated vacuum. Here the right definition is used, see Tab. 2.1 for correspondence.

Matrix elements of similarity-transformed operator are given for the left definition of the rotated vacuum.Here the right definition is used, see Tab. 2.1 for correspondence.

As a TSD stores only the minimal information required for time-ordering but not all, such a disappearance may be associated with the appearance of one or several new links with respect to the TSD of the diagonal BMBPT diagram. Hence TSDs must always be produced starting from a given off-diagonal BMBPT diagram and not from the TSD of its parent BMBPT diagram.

The two TSDs appearing in Fig.3.10 are denoted, respectively, as T3.3 and T3.2 (with a cyclic permutation of (a 1 , a 2 , a 3 )) in Fig.3.11.

The results obtained in Eq. (3.28) for the three second-order diagrams displayed in Fig.3.6 are straightforwardly recovered by the application of the diagrammatic rule.

It is worth noting that the TSD of the off-diagonal diagram of interest is unchanged compared to the diagonal diagram it is generated from. However, the companion diagram with one more anomalous line joining the first and second Ω vertices relates to a different TSD.

A valid partitioning relates to splitting the complete order p in a sum of terms that are individually proportional to a fraction of the form 1/(E k i ...k j . . . E k u ...k v ) with p energy factors in the denominator. Any other form does not constitute a valid partioning in the present context.

The fact that anomalous propagators/contractions are not diagonal in their quasi-particle indices should lead to a rather unconventional time-ordered diagrammatics that shall itself lead to an interesting variant of the resolvent rule.

While the formal work presented in this chapter, that led to the new version of ADG, is an original part of this thesis work, the extension of the ADG code itself has been performed by P. Arthuis and is thus not reported on in the present chapter.

In Refs.[START_REF] Strayer | Correlation Effects in Nuclear Densities[END_REF][START_REF] Roth | Hartree-Fock and many body perturbation theory with correlated realistic NN interactions[END_REF][START_REF] Tichai | Natural orbitals for ab initio no-core shell model calculations[END_REF], the expectation value is rewritten in terms of a connected expression, without explicit normalization left, prior to truncating the expansion. A consequence is that the (size-extensive) expression appearing at order P does not take the form of the expectation value of an operator in a truncated wave function. Contrarily, the present scheme consists of truncating the wave functions perturbatively before computing the matrix element of the operator, which leads to the appearance of an explicit expansion of the norm in the denominator and of disconnected terms in the numerator.

The BCI method allows one to access both the ground-state and low-lying excited states. The notation |Ψ BCI c corresponds to one of these states.

In principle we could deal with operators that are not scalar with respect to gauge symmetry such that [A, O] = kO and k = 0, e.g. beta decay process, but we stick to scalar ones here for simplicity.

Transformed quasi-particle operators are given in Sec. 1.4.6 for the left definition of the rotated vacuum, see Tab. 2.1 for correspondence. Rotated and transformed quasi-particle operators are given in App. E.5.

While several of the considerations below are identical to those provided in chap. 3, they are repeated here such that the present chapter is self-contained.

This sum of diagrams actually reduces to the evaluation of the overlap between |Ψ c and |Ψ d .

See chap. 6 for the method used to produce the angular-momentum coupled form of the equations in a systematic way.

The symmetry-breaking nature of the many-body methods of present interest concerns the approximate many-body state while making use of a symmetry-conserving Hamiltonian. Approximating the Hamiltonian in a symmetry-violating way is conceptually different and more problematic as one wishes to eventually employ a symmetry-restored approximate many-body state[START_REF] Duguet | Symmetry broken and restored coupled-cluster. Global gauge symmetry and particle number[END_REF][START_REF] Duguet | Symmetry broken and restored coupled-cluster theory: I. Rotational symmetry and angular momentum[END_REF][START_REF] Qiu | Particle-number projected Bogoliubov-coupled-cluster theory: Application to the pairing Hamiltonian[END_REF].

In App. F.1.3, an alternative particle-number conserving approximation based on the quasi-normal ordering proposed in Ref.[START_REF] Kong | An algebraic proof of generalized Wick theorem[END_REF] is investigated.

The test constitutes a necessary but not sufficient condition to prove that F commutes with A, i.e., the test can thus be used to prove that O nNOkB does not commute with A but can only indicate that O PNOkB probably commutes with A. This limitation is not problematic in the present case given that O PNOkB does commute with A by construction.

Working with a Bogoliubov state carrying good, e.g. even, number-parity quantum number, the integration over the gauge angles can indeed be reduced to the interval [0, π] in Eq.(5.27).

In EDF calculations, particle-number projection typically lowers the energy, which is not the case with the presently used chiral 2N interaction. Because PHFB calculations with realistic nuclear interactions are rather novel, this feature is briefly analyzed in App. F.3 by looking at the decomposition of the HFB vacuum into its particle-number projected components.

Chapter 5. Normal-ordered k-body approximation in particle-number-breaking theories k-body approximation based on deformed Slater determinant or Bogoliubov reference states is envisioned.

See Sec. 1.3 for a detailed discussion of the U (1) group.

Indices may appear more than twice, a feature uncommon for traditional contractions as in the theory of general relativity.

This statement assumes that one is working with a basis expansion method. Complementary methods such as lattice effective field theory or real-time methods like Green's function Monte Carlo are not based on tensors.

The 'tilde' in Tk 1 ... kn , although useful to distinguish SR-Ts from SU-Ts, will actually be omitted in the application section.

This step is not uniquely defined as several choices for the same group can be envisioned.

Notice that both j k and m k are half-integers, such that they verify (-1) 2j k = (-1) 2m k = -1 and (-1) 4j k = (-1) 4m k = 1. In the following, some total angular-momentum indices (J, M ) will be integers, such that they verify (-1) 2J = (-1) 2M = 1.

While the two-body state does indeed depend on M , the label is omitted for brevity given that the reduced tensors eventually built in that basis are diagonal in M and independent of it.

An explicit definition of Wigner 6j-symbol is provided in Eq. (6.56).

This connection between both sets of matrix elements (tensors) provides the transformation f SU (2) alluded to in Sec.6.3.1.

Although the 3jm-symbols are usually referred to as 3j-symbols in the litterature, this terminology is used here in order to distinguish them from the 3j-, 6j-and 9j-symbols etc. that appear in the theory.

Whereas the focus of this work is on the development of PBMBPT, applications presented in this section are based on other many-body theories in order to illustrate the versatility of this approach. However, the AMC program has been interfaced with ADG in order to provide coupled formulae for (P)BMBPT at any order. Coupled formulae for the three off-diagonal diagrams appearing in PBMBPT at first order are provided in App. D.

Such a choice is sometimes referred to as cross coupling since it involves angular-momentum coupling of bra and ket single-particle states.

The AMC program has also been interfaced with an homemade code that automatically derives expressions for the normal-ordered components of an operator, see App. A.

Labels i, j, . . . denotes hole indices, i.e. occupied single-particle states in the Slater determinant, whereas a, b, . . . denote particle indices, i.e. occupied single-particle states in the Slater determinant. Indices p, q, . . . denotes either of those.

J-scheme counterparts have been derived, with resort to the AMC program, but are not reported here not to overload the document.

These sum rules are obtained from the Fourier expansion of uncorrelated singly-rotated off-diagonal kernels N (0) (ϕ) = Φ(ϕ)|Φ and O (0) (ϕ) ≡ o (0) (ϕ)N (0) (ϕ) = Φ(ϕ)|O|Φ computed at ϕ = 0.

Acknowledgements

capture full triples at the price of their perturbative calculations in the construction of C and D. The n-body matrix elements o nn l 1 ...l n l n+1 ...l 2n are fully antisymmetric with respect to the permutation of the n first, resp. n last, indices, so that all terms coming from n ρ normal ρ contractions can be recasted into a single term with an appropriate prefactor, and similarly for all terms coming from n κ * anomalous κ * contractions or n κ anomalous κ contractions. For the term with n ρ normal ρ contractions, there are n ρ creation, resp. annihilitation, operators among n to be considered. Furthermore there are n ρ ! ways to pair n ρ creation and n ρ annihilation operators. Thus, the prefactor resulting from performing n ρ normal ρ contractions and no anomalous contraction of an initial n-body operator is Factor(n ρ , 0, 0

Appendix A. Normal ordering

B.2. Normal ordering in single-particle basis

where P αβ denotes the permutation operator that exchanges α and β indices in a given expression. Furthermore, second-order triple amplitudes have been given in Ref. [START_REF] Tichai | Pre-processing the nuclear many-body problem[END_REF] and read as

E.1. Connected contributions

Applying the diagrammatic rules explained in Sec. 4.4.3 to the first eight linked/connected diagrams contributing to O cd (ϕ) and displayed in Fig. 4.5 gives

E.2. Non-connected contributions

Applying the diagrammatic rules explained in Sec. 4.4.3 to the seven unlinked/disconnected diagrams contributing to O cd (ϕ) and displayed in Fig. 4.6 gives

E.3. Diagrammatic contributions to N cc (ϕ)

As pointed out in Sec.

E.4. Example of derivation -OCD2.2(ϕ)

We

E.5. Rotated and transformed Bogoliubov transformation

The rotated and transformed quasi-particle operators β RZ (ϕ), β † RZ (ϕ) are defined by the application of the similarity transformation associated to the Thouless operator Z(ϕ) to rotated quasi-particle operators β R (ϕ), β † R (ϕ)

and are related to quasi-particle operators β, β † through a non-unitary Bogoliubov transformation

where the non-unitary Bogoliubov matrix O † RZ (ϕ) is equal to

Proof. Starting from the definition of rotated and transformed quasi-particle operators (E.5) and inserting the rotated Bogoliubov transformation, given in Eq. (1.49) for the left definition of the rotated vacuum, one gets

where the relations

which follow from the definition of Z 20 (ϕ)

and from the transposition of Eq. (1.28a) applied for the gauge-rotated Bogoliubov transformation, have been used. For each (i, j) such that max(i, j) ≤ k one has to prove that

Appendix F.

Normal

The n-body normal fields, for which i = j = n, are given by Λnn l 1 ...l 2n = Λnn(nn)

Λnn(mm)

Λnn(mm)

where the property Λnn(nn) = õnn and the definition of õnn in Eq. (5. The two above equations fully define the PNO2B approximation of the three-body operator.

It is particle-number conserving by construction. Recent ab initio BMBPT calculations [START_REF] Tichai | Bogoliubov many-body perturbation theory for open-shell nuclei[END_REF] have been performed on the basis of this approximation although this was not explicited at the time.

It is useful to further characterize the operator by closely inspecting its normal-ordered contributions. Focusing for example on the normal field Λ11 , one obtains where again no extra term arises. Eventually, the complete set of normal-ordered contributions to O PNO2B expressed in the single-particle basis relates to those of the original three-body operator through The nNO3B approximation leads to just dropping O [4] = Λ 44 . The application of the PNO3B approximation is more involved and we now proceed to the construction of the corresponding operator

where the different terms are to be obtained recursively on the basis of Eq. ( 5 (F.26i)

Among the non-zero fields, one (plus its hermitian conjugate) contains an extra contribution, i.e. a term with a modified prefactor compared to the original operator O.

PNO2B approximation of a four-body operator

Let us now further approximate a four-body operator by building its PNO2B approximation.

Starting from the operator defined in Eq. (F.17), the nNO2B approximation now leads to dropping O [3] + O [4] = Λ 42 + Λ 33 + Λ 24 + Λ 44 . The application of the PNO2B approximation is more involved and we now proceed to the construction of the corresponding operator

where the different terms are to be obtained recursively on the basis of Eq. (5.6). In the present case, it leads to õ22 = Λ where the extra term is nothing but Λ20 . One notices that it is the same as in the PNO3B approximation of the four-body operator. Eventually, the complete set of normal-ordered contributions to O PNO2B expressed in the single-particle basis relates to those of the original four-body operator through

(F.32i)

Among the non-zero fields, one (plus its hermitian conjugate) contains an extra contribution, i.e. a term with a modified prefactor compared to the original operator O.

PNO3B and PNO4B approximations of a five-body operator

We have also worked out the PNO3B and the PNO4B approximations to a five-body operator in full details. As the expressions become lengthy, they are not reported here. Still, it is interesting to note that the particular form of the extra terms depends on which PNOkB, e.g. PNO3B or PNO4B, approximation is performed starting from the same original five-body operator. This point can be illustrated by only reporting how the normal-ordered contributions to both O PNO3B and O PNO4B relate to those of the original five-body operator. One has

for the PNO3B approximation and

for the PNO4B approximation. While in both cases the extra term entering Λ31 , and coming from the five-body part of O, is the same, one observes that Λ20 does acquire an extra term in the PNO3B approximation, coming both from the four-and five-body part of O, but it does not in the PNO4B approximation.

=

where N (0) (φ), ρ(φ), κ * (φ) and κ(φ) are 2π-periodic functions.

F.2.3. Particle-number variance

In Fig. 5.8, a small (non-zero) value was obtained from the nNO1B approximation of the particle-number variance operation the basis of the double particle-number projection. While a zero variance must be obtained for the exact particle-number variance operator, the fact that its nNO1B approximation systematically delivers a (almost) zero value is not immediately obvious. This feature, for which the double projection is essential, is now briefly analyzed.

Doubly-projected naive NO1B expectation value

The doubly-projected expectation value of the nNO1B approximation of an operator O can be systematically written as

where

and where the explicit form of õ00 and õ11 are given in Eq. (5.19).

Praticular case of particle-number variance

If A 2 is approximated at the nNO1B level, Eq. (F.49) specifies to

where the expressions

were used along with the identity (Eq. (1.35a))

originating from the unitarity of the Bogoliubov transformation. Having constrained the average particle number in the Bogoliubov vacuum to equate the targeted particle number A, i.e.

Tr[ρ] = A , (F.54) the doubly-projected nNO1B approximation to the particle-number variance is eventually given by

G.1. Introduction

In this appendix are gathered several j-coupled expressions, derived with ressort to the AMC program, and corresponding to different parts of the manuscript.

G.2. Similarity-transformed matrix elements

The j-coupled version of similarity-transformed matrix elements, see App. B, is given in the following subsections. NCC1.2.1(ϕ) = 1 24 k1 k2 k3 k4 J Ĵ2 J C40 k1 k2 k3 k4 (ϕ) J C †04 Dans le premier chapitre, les ingrédients formels nécessaires tout au long du document sont présentés, en particulier pour formuler PBMBPT. On y trouve une présentation des équations fondamentales, de la représentation utilisée pour les opérateurs, des groupes de symétrie de jauge, de l'algèbre de Bogoliubov, de la mise en ordre normal, de la théorie de Hartree-Fock-Bogoliubov (HFB) et de sa version contrainte.

Dans le second chapitre, le formalisme projected Bogoliubov coupled cluster (PBCC) [START_REF] Duguet | Symmetry broken and restored coupled-cluster. Global gauge symmetry and particle number[END_REF][START_REF] Qiu | Particle-number projected Bogoliubov-coupled-cluster theory: Application to the pairing Hamiltonian[END_REF][START_REF] Qiu | Projected coupled cluster theory[END_REF] est introduit. Cela est fait en deux variantes, basées sur (i) les amplitudes dépendantes de l'angle de jauge et (ii) les amplitudes indépendantes de l'angle de jauge. Cette méthode d'expansion à N corps est une extension du formalisme Bogoliubov coupled cluster (BCC) [START_REF] Signoracci | Ab initio Bogoliubov coupled cluster theory for open-shell nuclei[END_REF][START_REF] Henderson | Quasiparticle coupled cluster theory for pairing interactions[END_REF]. Alors que les deux formalismes peuvent modéliser des systèmes quantiques fortement corrélés en permettant à l'état de référence de champ moyen de briser la symétrie U (1), PBCC hérite et améliore encore la précision énergétique de la théorie BCC en conservant la symétrie U (1). La théorie de perturbation à N corps de Bogoliubov projetée (PBMBPT) est ensuite obtenue à partir de PBCC. Deux variantes de PBMBPT sont en fait introduites et diffèrent dans la manière dont l'opérateur de cluster dépendant de l'angle de jauge W (ϕ) est évalué. Alors que la présentation de PBCC est reproduite à partir de Ref. [START_REF] Qiu | Particle-number projected Bogoliubov-coupled-cluster theory: Application to the pairing Hamiltonian[END_REF], l'extraction des deux variantes de PBMBPT constitue une oeuvre originale. L'implémentation numérique associée à ce travail est presque terminée mais les résultats n'apparaissent pas dans ce document.

Dans le chapitre 3, les noyaux d'opérateurs hors diagonaux eux-mêmes sont développés en perturbation. Cela correspond donc à générer directement o (n) P P BMBPT.b (ϕ) comme défini dans l'Eq. (2.43). À partir des diagrammes en jeu dans P P BMBPT, dénommés hors diagonaux, les diagrammes diagonaux sont récupérés dans une limite particulière, c'est-àdire que les diagrammes BMBPT diagonaux caractérisent le sous-ensemble de diagrammes BMBPT hors diagonaux qui sont non nuls pour ϕ = 0. Dans ce contexte, l'objectif est de concevoir une nouvelle version du code ADG [START_REF] Arthuis | ADG: Automated generation and evaluation of many-body diagrams I. Bogoliubov many-body perturbation theory[END_REF], qui (1) génère automatiquement tous les diagrammes BMBPT hors diagonaux valides et (2) évalue automatiquement leur expression algébrique à mettre en oeuvre pour des applications numériques. Ceci est réalisé à n'importe quel ordre de perturbation p pour un hamiltonien contenant à la fois des interactions (sommets) à deux corps (quatre pattes) et à trois corps (six pattes). De cette façon, la capacité du code ADG est étendue de BMBPT à P P BMBPT.

Plusieurs méthodes à N corps modernes servent d'exemples pour démontrer la généralité de l'approche et mettre en évidence l'impact potentiel sur la communauté à N corps. Bien sûr, cela ne résout pas le problème de l'écriture d'une implémentation efficace, et surtout sans erreur, de la théorie à symétrie restreinte elle-même.

Alors que les calculs PBMBPT des noyaux à une couche ouverte n'apparaissent pas dans ce document, la mise en oeuvre est presque terminée, grâce aux outils formels actuellement conçus, et les premiers résultats devraient apparaître dans les prochains mois. Notre objectif est d'étendre (P)BMBPT dans plusieurs directions à l'avenir. Une première étape dans cette direction est l'étude de la convergence ordre par ordre de la série BMBPT. Comme BMBPT au-delà du premier ordre présente des corrections perturbatives au nombre de particules, un protocole d'ajustement cohérent de ces corrections à un ordre arbitraire doit être conçu. Ceci a été réalisé par les travaux de P. Demol et M. Frosini et conduira à une future publication [START_REF] Demol | High-order Bogoliubov Many-Body Perturbation Theory[END_REF]. Une étude de l'importance des contaminations restantes du nombre de particules dans BMBPT par rapport à PBMBPT à un ordre donné sera effectuée à l'avenir. De la même manière que PBMBPT, l'implémentation de BCC et sa version restaurée sont presque terminées. À plus long terme et afin d'étendre la portée de BMBPT et BCC à des noyaux à double couches ouvertes, d'autres extensions qui brisent (et restaurent) à la fois les symétries SU (2) et U (1) sont prévues.

List of Tables