 Mean-field scalar free-energy scaling

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 $\gamma =1$.