  The pseudofermion functional renormalization group (pf-FRG) has been put
forward as a semi-analytical scheme that, for a given microscopic spin model,
allows to discriminate whether the low-temperature states exhibit magnetic
ordering or a tendency towards the formation of quantum spin liquids. However,
the precise nature of the putative spin liquid ground state has remained hard
to infer from the original (single-site) pf-FRG scheme. Here we introduce a
cluster pf-FRG approach, which allows for a more stringent connection between a
microscopic spin model and its low-temperature spin liquid ground states. In
particular, it allows to calculate spatially structured fermion bilinear
expectation values on spatial clusters, which are formed by splitting the
original lattice into several sublattices, thereby allowing for the positive
identification of a family of bilinear spin liquid states. As an application of
this cluster pf-FRG approach, we consider the $J_1$-$J_2$ SU($N$)-Heisenberg
model on a square lattice, which is a paradigmatic example for a frustrated
quantum magnet exhibiting quantum spin liquid behavior for intermediate
coupling strengths. In the well-established large-$N$ limit of this model, we
show that our approach correctly captures the emergence of the $\pi$-flux spin
liquid state at low temperatures. For small $N$, where the precise nature of
the ground state remains controversial, we focus on the widely studied case of
$N=2$, for which we determine the low-temperature phase diagram near the
strongly-frustrated regime after implementing the fermion number constraint by
the flowing Popov-Fedotov method. Our results suggest that the
$J_1$-$J_2$-Heisenberg model does not support the formation of a fermion
bilinear spin liquid state.
