Phase transitions beyond criticality: using analytic corrections to extend the validity of renormalization group scaling

Abstract

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 allows for 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 also produces an exponentially convergent approximation, accurately capturing both the phases and phase transition. Finally, we discuss preliminary work and challenges on extending our approach to the 3D Ising model and systems in an external field.