Normal Form of the 2D Ising Renormalization Group Flows

We derive the scaling ansatz of the two-dimensional Ising model using Normal Form theory of dynamical systems, showing that no resonances with irrelevant variables can appear in the Renormalization Group flow equations. This clarifies the allowed asymptotic susceptibility and unifies a large literature of corrections to scaling. Our theory is general, describing a machinery for calculating corrections to scaling for any system by first classifying the flows into Universality Families, then adding analytic and singular corrections to parameterize the free energy and observables. In addition to describing currently known corrections to scaling we predict that the flow equations in the microcanonical ensemble will be non-analytic. This subtle and surprising fact indicates that the Renormalization Group may be more natural in some thermal ensembles than in others.