Scaling hypothesis for critical phenomena ($F_s(t,h)={\ell }^{-D}{F}_s({\ell }^{y_t}t,{\ell }^{y_h}h)$)

The scaling hypothesis treats the singular equilibrium free-energy density as a generalized homogeneous function of thermal and field-like scaling variables. Choosing the blocking factor to make the thermal variable order one gives $F_s=|t{|}^{D/y_t}{f}_{\pm }(h/|t{|}^{y_h/y_t})$. It concerns the singular contribution after subtracting analytic backgrounds. Dangerously irrelevant couplings or marginal logarithms can qualify this simple two-variable form. Differentiation yields scaling relations for critical exponents.

Differentiate $f_s=|t{|}^{2-\alpha }{g}_{f,\pm }(h/|t{|}^{\Delta })$ with respect to its conjugate field. A finite nonzero one-sided ordered-branch amplitude gives $\beta =2-\alpha -\Delta$. To approach a finite critical isotherm as $t\to 0$ at fixed small $h$, the large-argument function must behave as $g_m(x)\sim C\mathrm{sgn}(x)|x{|}^{1/\delta }$. Cancelling the remaining thermal power yields $\beta =\Delta /\delta$. An analytic background, branch selection and thermodynamic limit must be distinguished from the singular free energy being differentiated.

For a scalar quartic Landau free energy $r_{t}t{m}^2/2+u{m}^4/4-hm$ with $r_t,u>0$, rescaling $m=(r_t/u{)}^{1/2}|t{|}^{1/2}\psi$ gives this minimized scaling form, with $a=r_t^2/u$ and $b=\sqrt{u}/r_t^{3/2}$. The two functions minimize $\pm {\psi }^2/2+{\psi }^4/4-H\psi$. In particular $f_{+}(0)=0$ and $f_{-}(0)=-1/4$. Below the transition the equilibrium field dependence has a cusp at zero; pure-phase derivatives are one-sided. The form gives order-parameter critical exponent $\beta =1/2$ and magnetic-susceptibility critical exponent $\gamma =1$.