Normal Form for Renormalization Groups

• Published in: Physical review. X Vol. 9; no. 2
• Main Authors: Raju, Archishman, Clement, Colin B., Hayden, Lorien X., Kent-Dobias, Jaron P., Liarte, Danilo B., Rocklin, D. Zeb, Sethna, James P.
• Format: Journal Article
• Language: English
• Published: College Park American Physical Society 23-04-2019
• Subjects: Bifurcation theory; Canonical forms; Critical phenomena; Critical point; Fixed points (mathematics); Magnets; Mathematical analysis; Nonlinearity; Phase diagrams; Phase transitions; Physicists; Polynomials; Scaling; Self-similarity; Singularities
• ISSN: 2160-3308; 2160-3308
• DOI: 10.1103/PhysRevX.9.021014

The results of the renormalization group are commonly advertised as the existence of power-law singularities near critical points. The classic predictions are often violated and logarithmic and exponential corrections are treated on a case-by-case basis. We use the mathematics of normal form theory to systematically group these into universality families of seemingly unrelated systems united by common scaling variables. We recover and explain the existing literature and predict the nonlinear generalization for the universal homogeneous scaling functions. We show that this procedure leads to a better handling of the singularity even in classic cases and elaborate our framework using several examples.