Shaping Lattice through irrelevant perturbation: Ising model

The leading irrelevant perturbation, which controls the deviation of critical square lattice Ising model with periodic boundary conditions from its continuous CFT analog is identified. An explicit expression for the coupling constant in terms of the anisotropy parameter is found. We calculate the next to leading $\sim 1/N^2$ corrections to the spectrum on both lattice theory and the perturbed CFT sides for several classes of states, always getting exact agreement. We discuss also how the perturbing operators and the higher integrals of motion are related.