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 $t=\frac{(T-T_c)}{T_c}$ and the ordering field, $h$, approach zero. However, the optimal scaling variables are the nonlinear scaling fields, $g_t=t+b_t{h}^2+c_t{t}^2+\cdots$ and $g_h=h(1+c_{h}t+\cdots )$. The nonlinearities yield correction factors to the leading power-law (and scaling) variation of thermodynamic quantities, $L$, of the form ($1+a_{L}t+b_L{t}^2+\cdots$), etc., where the correction amplitudes $a_L,b_L,\dots$, 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 $b_t{h}^2$ also generates nonanalytic contributions such as an additive, energylike term, varying as ${\left|t\right|}^{1-\alpha }$ in the zero-field susceptibility, and factors like ($1+c_L{\left|h\right|}^{2-\frac{1}{\Delta }}$) on the critical isotherm, $t=0\mathrm{}$. Irrelevant scaling fields yield further, in general distinct, nonanalytic corrections, and cause shifts in $T_c$ and various amplitudes although "universal" ratios remain constant.