Tricritical crossover scaling ($f_{\mathrm{eq}}=|r{|}^{3/2}{s}^{-1/2}{\Phi }_{\pm }(X,Y)$)

For the stable sextic Landau free energy $f=r{m}^2/2+q{m}^4/4+s{m}^6/6-hm$, $s>0$, rescaling $m=(|r|/s{)}^{1/4}\psi$ gives the minimized order-parameter contribution $f_{\mathrm{eq}}=|r{|}^{3/2}{s}^{-1/2}{\Phi }_{\sigma }(X,Y)$, where $\sigma =\mathrm{sgn}r$, $X=q/(|r|s{)}^{1/2}$ and $Y=h{s}^{1/4}/|r{|}^{5/4}$. The two functions are ${\Phi }_{\sigma }={\min }_{\psi }(\sigma {\psi }^2/2+X{\psi }^4/4+{\psi }^6/6-Y\psi )$. In particular ${\Phi }_{-}(0,0)=-1/3$, but ${\Phi }_{+}(0,0)=0$. These separate branches and removal of smooth background terms are necessary when using the homogeneous form to infer mean-field critical exponents.

On the ordered branch of tricritical crossover scaling, at zero field and $X\to +\infty$, the minimizing square is $a={\psi }^2=(\sqrt{X^2+4}-X)/2\sim 1/X$. Its stationarity relation $Xa+a^2=1$ gives ${\Phi }_{-}(X,0)=-a/4-a^3/12\sim -1/(4X)$. Thus the minimized free-energy density becomes $-r^2/(4q)$ for fixed $q>0$ as $r\uparrow 0$, recovering the ordinary quartic continuous phase transition and its heat-capacity critical exponent $\alpha =0$. Replacing the scaling function by a nonzero constant in this limit would give an incorrect exponent.