  Functional renormalization group (FRG) has become a diverse and powerful tool
to derive effective low-energy scattering vertices of interacting many-body
systems. Starting from a non-interacting expansion point of the action, the
flow of the RG parameter Lambda allows to trace the evolution of the effective
one-particle and two-particle vertices towards low energies by taking into
account the vertex corrections between all parquet channels in an unbiased
fashion. In this work, we generalize the expansion point at which the
diagrammatic resummation procedure is initiated from a free UV limit to a
cluster product state. We formulate a cluster FRG scheme where the
non-interacting building blocks (i.e., decoupled spin clusters) are treated
exactly, and the inter-cluster couplings are addressed via RG. As a benchmark
study, we apply our cluster FRG scheme to the spin-1/2 bilayer Heisenberg model
(BHM) on a square lattice where the neighboring sites in the two layers form
the individual 2-site clusters. Comparing with existing numerical evidence for
the BHM, we obtain reasonable findings for the spin susceptibility, magnon
dispersion, and magnon quasiparticle weight even in coupling regimes close to
antiferromagnetic order. The concept of cluster FRG promises applications to a
large class of interacting electron systems.
