  The core of this thesis is the path-integral formulation of quantum field
theory and its ability to describe strongly-coupled quantum 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 (Pad\'e-Borel, Borel-hypergeometric, 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 the
aforementioned methods is also examined in detail.
