We present a functional renormalization group analysis of superconductivity
in the ground state of the attractive Hubbard model on a square lattice.
Spontaneous symmetry breaking is treated in a purely fermionic setting via
anomalous propagators and anomalous effective interactions. In addition to the
anomalous interactions arising already in the reduced BCS model, effective
interactions with three incoming legs and one outgoing leg (and vice versa)
occur. We accomplish their integration into the usual diagrammatic formalism by
introducing a Nambu matrix for the effective interactions. From a random-phase
approximation generalized through use of this matrix we conclude that the
impact of the 3+1 effective interactions is limited, especially considering the
effective interactions important for the determination of the order parameter.
The exact hierarchy of flow equations for one-particle irreducible vertex
functions is truncated on the two-particle level, with higher-order self-energy
corrections included in a scheme proposed by Katanin. Using a parametrization
of effective interactions by patches in momentum space, the flow equations can
be integrated numerically to the lowest scales without encountering
divergences. Momentum-shell as well as interaction-flow cutoff functions are
used, including a small external field or a large external field and a
counterterm, respectively. Both approaches produce momentum-resolved order
parameter values directly from the microscopic model. The size of the
superconducting gap is in reasonable agreement with expectations from other
studies.Comment: 22 pages, 16 figures, references added, some changes in the
introductio
