Phase transitions beyond criticality: normal form and analytic corrections for the 2D Ising Model
The renormalization group predicts universal scaling laws near critical phase transitions. But can we extend this understanding of the critical point to accurately capture the behavior throughout the surrounding phases? To this end, normal form theory provides a useful framework: analytic variable changes (in temperature or field, for example) extend the universal scaling function to the entire phase. We apply this idea to the 2D Ising model, where Onsager's exact solution enables quantitative tests of the accuracy of analytic corrections. By working in a special coordinate frame, in which the Fisher zeros lie on a straight line, we produce expansions of the free energy that converge for all temperatures. Even with minimal knowledge of the critical point, fitting the expansion to data (low- and high-temperature cluster expansions) deep within the phases produces similar results, accurately capturing both the phases and phase transition. Our approach may prove useful for mapping out phases in unsolvable statistical physics models or in experimental systems exhibiting critical transitions.