  We analyze fermionic criticality in relativistic 2+1 dimensional fermion
systems using the functional renormalization group (FRG), concentrating on the
Gross-Neveu (chiral Ising) and the Thirring model. While a variety of methods,
including the FRG, appear to reach quantitative consensus for the critical
regime of the Gross-Neveu model, the situation seems more diverse for the
Thirring model with different methods yielding vastly different results. We
present a first exploratory FRG study of such fermion systems including
momentum-dependent couplings using pseudo-spectral methods. Our results
corroborate the stability of results in Gross-Neveu-type universality classes,
but indicate that momentum dependencies become more important in Thirring-type
models for small flavor numbers. For larger flavor numbers, we confirm the
existence of a non-Gaussian fixed point and thus a physical continuum limit. In
the large-$N$ limit, we obtain an analytic solution for the momentum dependence
of the fixed-point vertex.
