  We present a multiloop pseudofermion functional renormalization group (pffRG)
approach to quantum spin systems. As a test case, we study the
spin-$\tfrac{1}{2}$ Heisenberg model on the kagome lattice, a prime example of
a geometrically frustrated magnet believed to host a quantum spin liquid. Our
main physical result is that, at pure nearest-neighbor coupling, the system
shows indications for an algebraic spin liquid through slower-than-exponential
decay with distance for the static spin susceptibility, while the pseudofermion
self-energy develops intriguing low-energy features. Methodologically, the
pseudofermion representation of spin models inherently yields a strongly
interacting system, and the quantitative reliability of a truncated fRG flow is
\textit{a priori} unclear. Our main technical result is the demonstration of
convergence in loop number within multiloop pffRG. Through correspondence with
the self-consistent parquet equations, this provides further evidence for the
internal consistency of the approach. The loop order required for convergence
of the pseudofermion vertices is rather large, but the spin susceptibility is
more benign, appearing almost fully converged for loop orders $\ell \geq 5$.
The multiloop flow remains stable as the infrared cutoff $\Lambda$ is reduced
relative to the microscopic exchange interaction $J$, allowing us to reach
values of $\Lambda/J$ on the subpercent level in the spin-liquid phase. By
contrast, solving the parquet equations via direct fixed-point iteration
becomes increasingly difficult for low $\Lambda/J$. We also scrutinize the
pseudofermion constraint of single occupation per site, which is only fulfilled
on average in pffRG, by explicitly computing fermion-number fluctuations.
Although the latter are not entirely suppressed, we find that they do not
affect the qualitative conclusions drawn from the spin susceptibility.
