  This thesis is concerned with ground state properties of two-dimensional
fermionic superfluids, in which fluctuation effects like the renormalization of
the order parameter or infrared singularities are important. In the superfluid
state, the fermionic two-particle vertex develops rich and singular dependences
on momentum and frequency, for which an efficient parametrization in terms of
boson-exchange interactions in the particle-hole and particle-particle channels
is formulated. Based on this decomposition of the vertex, flow equations for
the effective interactions are derived, extending existing
channel-decomposition schemes to i) the description of symmetry breaking in the
Cooper channel and ii) the inclusion of those two-loop renormalization
contributions to the vertex that are neglected in the Katanin scheme. In the
second part of this thesis, the superfluid ground state of various model
systems is studied. For the attractive Hubbard model at weak coupling, the
momentum and frequency dependence of the two-particle vertex and the frequency
dependence of the self-energy are determined on one- and two-loop level.
Results for the suppression of the superfluid gap by fluctuations are in good
agreement with the literature. The two-loop approximation captures the singular
infrared behaviour that is expected in a fermionic superfluid at zero
temperature. For the repulsive Hubbard model at weak coupling, the momentum
dependence of the two-particle vertex and the d-wave superfluid gap are
determined on one-loop level as a function of the interaction, the next-nearest
neighbour hopping and the fermionic density. The results for the critical
scales and superfluid gaps are in qualitative agreement with the literature and
suggest the existence of an optimal value of the next-nearest neighbour hopping
for pairing.
