Path-integral approaches to strongly-coupled quantum many-body systems

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 (Pade-Borel, Meijer-G, conformal mapping), enhanced versions of perturbation theory (trans-series 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. (author)