Addressing energy density functionals in the language of path-integrals I: Comparative study of diagrammatic techniques applied to the (0+0)-D O(N)O(N)-symmetric φ4\varphi^{4}-theory

The energy density functional (EDF) method is currently the only microscopic theoretical approach able to tackle the entire nuclear chart. Nevertheless, it suffers from limitations resulting from its empirical character and deteriorating its reliability. This paper is part of a larger program that aims at formulating the EDF approach as an effective field theory (EFT) in order to overcome these limitations. A relevant framework to achieve this is the path-integral (PI) formulation of quantum field theory (QFT). The latter indeed provides a wide variety of treatments of the many-body problem well suited to deal with non-perturbative interactions and to exploit a Lagrangian resulting from an EFT as a starting point. While developing the formalism in a general setting, we present a comparative study of such techniques applied to a toy model, i.e. the (0+0)-D $O(N)$-symmetric $\varphi^{4}$-theory. More specifically, our focus will be on the following diagrammatic techniques: loop expansion (LE), optimized perturbation theory (OPT) and self-consistent perturbation theory (SCPT). With these methods, we notably address the spontaneous breakdown of the $O(N)$ symmetry with care especially since spontaneous symmetry breakings (SSBs) play a paramount role in current implementations of the EDF approach.Comment: This article is part of a PhD project. The corresponding manuscript can be found at: arXiv:2210.1667

Single-boson exchange functional renormalization group application to the two-dimensional Hubbard model at weak coupling

We illustrate the algorithmic advantages of the recently introduced single-boson exchange (SBE) formulation for the one-loop functional renormalization group (fRG), by applying it to the two-dimensional Hubbard model on a square lattice. We present a detailed analysis of the fermion-boson Yukawa couplings and of the corresponding physical susceptibilities by studying their evolution with temperature and interaction strength, both at half filling and finite doping. The comparison with the conventional fermionic fRG decomposition shows that the rest functions of the SBE algorithm, which describe correlation effects beyond the SBE processes, play a negligible role in the weak-coupling regime above the pseudo-critical temperature, in contrast to the rest functions of the conventional fRG. Remarkably, they remain finite also at the pseudo-critical transition, whereas the corresponding rest functions of the conventional fRG implementation diverge. As a result, the SBE formulation of the fRG flow allows for a substantial reduction of the numerical effort in the treatment of the two-particle vertex function, paving a promising route for future multiboson and multiloop extensions

Approches de type intégrale de chemin pour l'étude de systèmes quantiques à N corps fortement corrélés

The core of this thesis is the path-integral formulation of quantum field theory and its ability to describe strongly-coupled many-body systems of finite size. Collective behaviors can be efficiently described in such systems through the implementation of spontaneous symmetry breaking (SSB) in mean field approaches. However, as the thermodynamic limit does not make sense in finite-size systems, the latter can not exhibit any SSB and the symmetries which are broken down at the mean field level must therefore be restored. The efficiency of theoretical approaches in the treatment of finite-size quantum systems can therefore be studied via their ability to restore spontaneously broken symmetries. In this thesis, a zero-dimensional O(N) model is taken as a theoretical laboratory to perform such an investigation with many state-of-the-art path-integral techniques: perturbation theory combined with various resummation methods (Padé-Borel, Meijer-G, conformal mapping), enhanced versions of perturbation theory (transseries derived via Lefschetz thimbles, optimized perturbation theory), self-consistent perturbation theory based on effective actions (auxiliary field loop expansion (LOAF), Cornwall-Jackiw-Tomboulis (CJT) formalism, 4PPI effective action,...), functional renormalization group (FRG) techniques (FRG based on the Wetterich equation, DFT-FRG, 2PI-FRG). Connections between these different techniques are also emphasized. In addition, the path-integral formalism provides us with the possibility to introduce collective degrees of freedom in an exact fashion via Hubbard-Stratonovich transformations: the effect of such transformations on all the aforementioned methods is also examined in detail.Le cœur de ce travail de thèse est la formulation de la théorie quantique des champs basée sur les intégrales de chemin et sa capacité à décrire les systèmes quantiques à N corps fortement corrélés de taille finie. Les phénomènes collectifs gouvernant la phénoménologie de tels systèmes peuvent être efficacement décrits par l'implémentation de brisures de symétrie spontanées (SSB) dans le cadre d'approches de type champ moyen. Cependant, la limite thermodynamique n'étant pas pertinente pour des systèmes de taille finie, ces derniers ne peuvent manifester de SSB et les symétries brisées au niveau du champ moyen doivent donc être restaurées. L'efficacité d'approches théoriques à traiter les systèmes quantiques de taille finie peut donc être étudiée à travers leur capacité à restaurer les symétries brisées spontanément. Dans ce travail de thèse, nous prenons pour cadre théorique un modèle O(N) à zéro dimension pour réaliser une telle étude avec diverses méthodes de type intégrale de chemin : théorie des perturbations combinée avec différentes techniques de resommation (Padé-Borel, Meijer-G, conformal mapping), versions modifiées de la théorie des perturbations (transseries déterminées via le formalisme des Lefschetz thimbles, théorie des perturbations optimisée), théorie des perturbations auto-cohérente basée sur des actions effectives (auxiliary field loop expansion (LOAF), formalisme Cornwall-Jackiw-Tomboulis (CJT), action effective 4PPI,...), techniques de type groupe de renormalisation fonctionnel (FRG) (FRG basé sur l'équation de Wetterich, DFT-FRG, 2PI-FRG). Des connexions entre ces différentes méthodes sont aussi mises en exergue. De plus, le formalisme des intégrales de chemin nous offre la possibilité d'introduire des degrés de liberté collectifs de manière exacte à l'aide de transformations de Hubbard-Stratonovich : l'effet de telles transformations sur les méthodes susmentionnées est également étudié en détail

Approches de type intégrale de chemin pour l'étude de systèmes quantiques à N corps fortement corrélés

The core of this thesis is the path-integral formulation of quantum field theory and its ability to describe strongly-coupled many-body systems of finite size. Collective behaviors can be efficiently described in such systems through the implementation of spontaneous symmetry breaking (SSB) in mean field approaches. However, as the thermodynamic limit does not make sense in finite-size systems, the latter can not exhibit any SSB and the symmetries which are broken down at the mean field level must therefore be restored. The efficiency of theoretical approaches in the treatment of finite-size quantum systems can therefore be studied via their ability to restore spontaneously broken symmetries. In this thesis, a zero-dimensional O(N) model is taken as a theoretical laboratory to perform such an investigation with many state-of-the-art path-integral techniques: perturbation theory combined with various resummation methods (Padé-Borel, Meijer-G, conformal mapping), enhanced versions of perturbation theory (transseries derived via Lefschetz thimbles, optimized perturbation theory), self-consistent perturbation theory based on effective actions (auxiliary field loop expansion (LOAF), Cornwall-Jackiw-Tomboulis (CJT) formalism, 4PPI effective action,...), functional renormalization group (FRG) techniques (FRG based on the Wetterich equation, DFT-FRG, 2PI-FRG). Connections between these different techniques are also emphasized. In addition, the path-integral formalism provides us with the possibility to introduce collective degrees of freedom in an exact fashion via Hubbard-Stratonovich transformations: the effect of such transformations on all the aforementioned methods is also examined in detail.Le cœur de ce travail de thèse est la formulation de la théorie quantique des champs basée sur les intégrales de chemin et sa capacité à décrire les systèmes quantiques à N corps fortement corrélés de taille finie. Les phénomènes collectifs gouvernant la phénoménologie de tels systèmes peuvent être efficacement décrits par l'implémentation de brisures de symétrie spontanées (SSB) dans le cadre d'approches de type champ moyen. Cependant, la limite thermodynamique n'étant pas pertinente pour des systèmes de taille finie, ces derniers ne peuvent manifester de SSB et les symétries brisées au niveau du champ moyen doivent donc être restaurées. L'efficacité d'approches théoriques à traiter les systèmes quantiques de taille finie peut donc être étudiée à travers leur capacité à restaurer les symétries brisées spontanément. Dans ce travail de thèse, nous prenons pour cadre théorique un modèle O(N) à zéro dimension pour réaliser une telle étude avec diverses méthodes de type intégrale de chemin : théorie des perturbations combinée avec différentes techniques de resommation (Padé-Borel, Meijer-G, conformal mapping), versions modifiées de la théorie des perturbations (transseries déterminées via le formalisme des Lefschetz thimbles, théorie des perturbations optimisée), théorie des perturbations auto-cohérente basée sur des actions effectives (auxiliary field loop expansion (LOAF), formalisme Cornwall-Jackiw-Tomboulis (CJT), action effective 4PPI,...), techniques de type groupe de renormalisation fonctionnel (FRG) (FRG basé sur l'équation de Wetterich, DFT-FRG, 2PI-FRG). Des connexions entre ces différentes méthodes sont aussi mises en exergue. De plus, le formalisme des intégrales de chemin nous offre la possibilité d'introduire des degrés de liberté collectifs de manière exacte à l'aide de transformations de Hubbard-Stratonovich : l'effet de telles transformations sur les méthodes susmentionnées est également étudié en détail

Approches de type intégrale de chemin pour l'étude de systèmes quantiques à N corps fortement corrélés

The core of this thesis is the path-integral formulation of quantum field theory and its ability to describe strongly-coupled many-body systems of finite size. Collective behaviors can be efficiently described in such systems through the implementation of spontaneous symmetry breaking (SSB) in mean field approaches. However, as the thermodynamic limit does not make sense in finite-size systems, the latter can not exhibit any SSB and the symmetries which are broken down at the mean field level must therefore be restored. The efficiency of theoretical approaches in the treatment of finite-size quantum systems can therefore be studied via their ability to restore spontaneously broken symmetries. In this thesis, a zero-dimensional O(N) model is taken as a theoretical laboratory to perform such an investigation with many state-of-the-art path-integral techniques: perturbation theory combined with various resummation methods (Padé-Borel, Meijer-G, conformal mapping), enhanced versions of perturbation theory (transseries derived via Lefschetz thimbles, optimized perturbation theory), self-consistent perturbation theory based on effective actions (auxiliary field loop expansion (LOAF), Cornwall-Jackiw-Tomboulis (CJT) formalism, 4PPI effective action,...), functional renormalization group (FRG) techniques (FRG based on the Wetterich equation, DFT-FRG, 2PI-FRG). Connections between these different techniques are also emphasized. In addition, the path-integral formalism provides us with the possibility to introduce collective degrees of freedom in an exact fashion via Hubbard-Stratonovich transformations: the effect of such transformations on all the aforementioned methods is also examined in detail.Le cœur de ce travail de thèse est la formulation de la théorie quantique des champs basée sur les intégrales de chemin et sa capacité à décrire les systèmes quantiques à N corps fortement corrélés de taille finie. Les phénomènes collectifs gouvernant la phénoménologie de tels systèmes peuvent être efficacement décrits par l'implémentation de brisures de symétrie spontanées (SSB) dans le cadre d'approches de type champ moyen. Cependant, la limite thermodynamique n'étant pas pertinente pour des systèmes de taille finie, ces derniers ne peuvent manifester de SSB et les symétries brisées au niveau du champ moyen doivent donc être restaurées. L'efficacité d'approches théoriques à traiter les systèmes quantiques de taille finie peut donc être étudiée à travers leur capacité à restaurer les symétries brisées spontanément. Dans ce travail de thèse, nous prenons pour cadre théorique un modèle O(N) à zéro dimension pour réaliser une telle étude avec diverses méthodes de type intégrale de chemin : théorie des perturbations combinée avec différentes techniques de resommation (Padé-Borel, Meijer-G, conformal mapping), versions modifiées de la théorie des perturbations (transseries déterminées via le formalisme des Lefschetz thimbles, théorie des perturbations optimisée), théorie des perturbations auto-cohérente basée sur des actions effectives (auxiliary field loop expansion (LOAF), formalisme Cornwall-Jackiw-Tomboulis (CJT), action effective 4PPI,...), techniques de type groupe de renormalisation fonctionnel (FRG) (FRG basé sur l'équation de Wetterich, DFT-FRG, 2PI-FRG). Des connexions entre ces différentes méthodes sont aussi mises en exergue. De plus, le formalisme des intégrales de chemin nous offre la possibilité d'introduire des degrés de liberté collectifs de manière exacte à l'aide de transformations de Hubbard-Stratonovich : l'effet de telles transformations sur les méthodes susmentionnées est également étudié en détail

Addressing energy density functionals in the language of path-integrals I: comparative study of diagrammatic techniques applied to the (0 + 0)-D

The energy density functional (EDF) method is currently the only microscopic theoretical approach able to tackle the entire nuclear chart. Nevertheless, it suffers from limitations resulting from its empirical character and deteriorating its reliability. This paper is part of a larger program that aims at formulating the EDF approach as an effective field theory (EFT) in order to overcome these limitations. A relevant framework to achieve this is the path-integral formulation of quantum field theory. The latter indeed provides a wide variety of treatments of the many-body problem well suited to deal with non-perturbative interactions and to exploit a Lagrangian resulting from an EFT as a starting point. While developing the formalism in a general setting, we present below a comparative study of such techniques applied to a toy model, i.e. the (0 + 0)-D O(N)-symmetric $$\varphi ^{4}$$-theory. More specifically, our focus will be on the following diagrammatic techniques: loop expansion, optimized perturbation theory and self-consistent perturbation theory. With these methods, we notably address the spontaneous breakdown of the O(N) symmetry with care especially since spontaneous symmetry breakings play a paramount role in current implementations of the EDF approach

Addressing energy density functionals in the language of path-integrals II: Comparative study of functional renormalization group techniques applied to the (0+0)-D O(N)O(N)-symmetric φ4\varphi^{4}-theory

International audienceThe present paper is the second of a series of publications that aim at investigating relevant directions to turn the nuclear energy density functional (EDF) method as an effective field theory (EFT). The EDF approach has known numerous successes in nuclear theory over the past decades and is currently the only microscopic technique that can be applied to all atomic nuclei. However, the phenomenological character of the EDF method also comes with important limitations, such as the lack of an explicit connection with quantum chromodynamics (QCD). As was argued in the first paper of this series, reformulating the EDF framework as an EFT would enable us to overcome these limitations. In particular, path-integral (PI) techniques are suited to achieve such a purpose as they allow to design numerous non-perturbative approximations and can take Lagrangians possibly derived from EFTs of QCD as inputs. In our previous paper, we have illustrated such technical features for diagrammatic PI techniques in a study of the (0+0)-D $O(N)$-symmetric $\varphi^{4}$-theory. In the present work, we consider another class of PI techniques, i.e. functional renormalization group (FRG) approaches, that we apply on the same toy model. Despite our explicit interest for the nuclear many-body problem, the presented study is also directed towards FRG practitioners from various fields: technical details are provided for FRG techniques based on 1-particle-irreducible (1PI), 2-particle-irreducible (2PI) and 2-particle-point-irreducible (2PPI) effective actions (EAs), coined respectively as 1PI-, 2PI- and 2PPI-FRGs, and the treatment of the $O(N)$ symmetry is also addressed thoroughly. Connections between these various FRG methods are identified as well