  The functional renormalization group (RG) in combination with Fermi surface
patching is a well-established method for studying Fermi liquid instabilities
of correlated electron systems. In this article, we further develop this method
and combine it with mean-field theory to approach multiband systems with
spin-orbit coupling, and we apply this to a tight-binding Rashba model with an
attractive, local interaction. The spin dependence of the interaction vertex is
fully implemented in a RG flow without SU(2) symmetry, and its momentum
dependence is approximated in a refined projection scheme. In particular, we
discuss the necessity of including in the RG flow contributions from both bands
of the model, even if they are not intersected by the Fermi level. As the
leading instability of the Rashba model, we find a superconducting phase with a
singlet-type interaction between electrons with opposite momenta. While the gap
function has a singlet spin structure, the order parameter indicates an
unconventional superconducting phase, with the ratio between singlet and
triplet amplitudes being plus or minus one on the Fermi lines of the upper or
lower band, respectively. We expect our combined functional RG and mean-field
approach to be useful for an unbiased theoretical description of the
low-temperature properties of spin-based materials.
