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Much of our understanding of critical phenomena is based on the notion ofRenormalization Group (RG), but the actual determination of its fixed points isusually based on approximations and truncations, and predictions of physicalquantities are often of limited accuracy. The RG fixed points can be howevergiven a fully rigorous and non-perturbative characterization, and this is whatis presented here in a model of symplectic fermions with a nonlocal("long-range") kinetic term depending on a parameter $\varepsilon$ and aquartic interaction. We identify the Banach space of interactions, which thefixed point belongs to, and we determine it via a convergent approximationscheme. The Banach space is not limited to relevant interactions, but itcontains all possible irrelevant terms with short-ranged kernels, decaying likea stretched exponential at large distances. As the model shares a number offeatures in common with $\phi^4$ or Ising models, the result can be used as abenchmark to test the validity of truncations and approximations in RG studies.The analysis is based on results coming from Constructive RG to which weprovide a tutorial and self-contained introduction. In addition, we prove thatthe fixed point is analytic in $\varepsilon$, a somewhat surprising factrelying on the fermionic nature of the problem.

Gentle introduction to rigorous Renormalization Group: a worked fermionic example