Deformations of infrared-conformal theories in two dimensions

We study two exactly solvable two-dimensional conformal models, the critical Ising model and the Sommerfield model, on the lattice. We show that finite-size effects are important and depend on the aspect ratio of the lattice. In particular, we demonstrate how to obtain the correct massless behavior from an infinite tower of finite-size-induced masses and show that it is necessary to first take the cylindrical geometry limit in order to get correct results. In the Sommerfield model we also introduce a mass deformation to measure the mass anomalous dimension, $\gamma_m$. We find that the explicit scale breaking of the lattice setup induces corrections which must be taken into account in order to reproduce $\gamma_m$ at the infrared fixed point. These results can be used to improve the methodology in the search for the conformal window in QCD-like theories with many flavors.