Efficient vertex parametrization for the constrained functional renormalization group for effective low-energy interactions

We present an efficient approximation for the perturbative calculation of electron-electron interaction in multi-orbital lattice systems. By using ideas for channel decomposition, form factor expansion and the truncated unity functional renormalization technique we describe the interaction as arising from the non-local an orbital-dependent coupling of spin-diagonal and spin-changing particle-hole bilinears as well as particle-particle bilinears that each reside within the same orbital. This allows us to employ a rather fine momentum discretization which e.g. gives insights on the non-local screening of spin and charge interactions. Using this formalism we compute the effective low-energy interactions in a three-band model where the two bands away from the Fermi levels are integrated out with the constrained functional renormalization group (cfRG). We show that the cfRG adds important features to the effective model that cannot be found using the constrained random phase approximation (cRPA).