  In the derivation of low-energy effective models for solids targeting the
bands near the Fermi level, the constrained random phase approximation (cRPA)
has become an appreciated tool to compute the effective interactions. The
Wick-ordered constrained functional renormalization group (cfRG) generalizes
the cRPA approach by including all interaction channels in an unbiased way.
Here we present applications of the cfRG to two simple multi-band systems and
compare the resulting effective interactions to the cRPA. First we consider a
multiband model for monolayer graphene, where we integrate out the
$\sigma$-bands to get an effective theory for $\pi$-bands. It turns out that
terms beyond cRPA are strongly suppressed by the different $xy$-plane
reflection symmetry of the bands. In our model the cfRG-corrections to cRPA
become visible when one disturbs this symmetry difference slightly, however
without qualitative changes. This study shows that the embedding or layering of
two-dimensional electronic systems can alter the effective interaction
parameters beyond what is expected from screening considerations. The second
example is a one-dimensional model for a diatomic system reminiscent of a CuO
chain, where we consider an effective theory for Cu 3d-like orbitals. Here the
fRG data shows relevant and qualitative corrections compared to the cRPA
results. We argue that the new interaction terms affect the magnetic properties
of the low-energy model.
