Nonlinear scaling fields and corrections to scaling near criticality

Thermodynamic functions in the vicinity of an ordinary critical point are expected to obey asymptotic scaling laws as t=(T-Tc)Tc and the ordering field, h, approach zero. However, the optimal scaling variables are the nonlinear scaling fields, gₜ=t+bₜh²+cₜt²+⋯ and gₕ=h(1+cₕt+⋯). The nonlinearities yield correction factors to the leading power-law (and scaling) variation of thermodynamic quantities, L, of the form (1+aLt+bLt²+⋯), etc., where the correction amplitudes aL,bL,, are uniquely determined by the nonlinear scaling-field coefficients. It follows that "analytic" corrections to, e.g., the susceptibility, are directly related to those for the free energy and magnetization (in zero field). The term bₜh² also generates nonanalytic contributions such as an additive, energylike term, varying as |t|^1-α in the zero-field susceptibility, and factors like (1+cL|h|^2-1Δ) on the critical isotherm, $t=0$. Irrelevant scaling fields yield further, in general distinct, nonanalytic corrections, and cause shifts in Tc and various amplitudes although "universal" ratios remain constant.