The Schwinger model is one of the simplest gauge theories, yet it is only solvable in the massless case. In order to obtain numerical results, the Kogut-Susskind lattice approach with staggered fermions is regularly used. I will show that, contrary to what it was believed, the lattice mass and the continuum mass are actually not the same, but they are related by a mass shift. This can be understood by considering the (anomalous) chiral symmetry in the massless case, and has the advantage of greatly improving convergence of the numerics. I will comment on the charge-q Schwinger model as well as the multiflavor Schwinger model.

I will discuss walking behavior in gauge theories and weakly first order phase transition in statistical models. Despite being phenomena appearing in very different physical systems, they both show a region of approximate scale invariance. They can be understood as a RG flow passing between two fixed points living at complex couplings, which we call complex CFTs. By using conformal perturbation theory, knowing the conformal data of the complex CFTs allows us to make predictions on the observables of the walking theory. As an example, I will discuss the two dimensional Q-state Potts model with Q>4.