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.

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 $\gamma =1$.