  Renormalization group methods are well-established tools for the (numerical)
investigation of the low-energy properties of correlated quantum many-body
systems, allowing to capture their scale-dependent nature. The functional
renormalization group (FRG) allows to continuously evolve a microscopic model
action to an effective low-energy action as a function of decreasing energy
scales via an exact functional flow equation, which is then approximated by
some truncation scheme to facilitate computation. Here, we report on our
transcription of a recently developed multiloop truncation approach for
electronic FRG calculations to the pseudo-fermion functional renormalization
group (pf-FRG) for interacting quantum spin systems. We discuss in detail the
conceptual intricacies of the flow equations generated by the multiloop
truncation, as well as essential refinements to the integration scheme for the
resulting integro-differential equations. To benchmark our approach we analyze
antiferromagnetic Heisenberg models on the pyrochlore, simple cubic and
face-centered cubic lattice, discussing the convergence of physical observables
for higher-loop calculations and comparing with existing results where
available. Combined, these methodological refinements systematically improve
the pf-FRG approach to one of the numerical tools of choice when exploring
frustrated quantum magnetism in higher spatial dimensions.
