Phase transitions beyond criticality: extending Ising universal scaling functions to describe entire phases

Universal scaling laws only apply asymptotically near critical phase transitions. We propose a general scheme, based on normal form theory of renormalization group flows, for incorporating corrections to scaling that quantitatively describe the entire neighboring phases. Expanding Onsager's exact solution of the 2D Ising model about the critical point, we identify a special coordinate with radius of convergence covering the entire physical temperature range, 0<T<. Without an exact solution, we demonstrate that using solely the critical singularity with low- and high-temperature expansions leads to exponentially converging approximations across all temperatures for both the 2D and 3D Ising free energies and the 3D magnetization. We discuss challenges and opportunities for future work.