Abstract
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.
Highlights
Emergent scale invariance is a key to many of our current scientific and engineering challenges, including cell membranes [1], turbulence [2], fracture, and plasticity [3,4], and the more traditional continuous thermodynamic phase transitions
We show in the Supplemental Material [62] that the true singularity of the magnetization at the critical point is M ∼ t1=2Wðxt−27=25Þ1=3, where W is the Lambert-W function defined by WðzÞeWðzÞ 1⁄4 z, and xðuÞ is a complicated but explicit function of the irrelevant variable u. [The traditional log and log-log terms follow from the asymptotic behaviors of WðxÞ at large and small x
We have shown how normal form theory leads to a systematic procedure for handling the singularity in renormalization group (RG) flows
Summary
Emergent scale invariance is a key to many of our current scientific and engineering challenges, including cell membranes [1], turbulence [2], fracture, and plasticity [3,4], and the more traditional continuous thermodynamic phase transitions. This allows us to apply a branch of dynamical systems theory, normal form theory [20,21], to provide a unified description applicable to all of these systems.
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