= Tricritical crossover scaling
{title2=$f_{\rm eq}=|r|^{3/2}s^{-1/2}\Phi_\pm(X,Y)$}
For the stable sextic <Landau free energy> $f=rm^2/2+qm^4/4+sm^6/6-hm$, $s>0$, rescaling $m=(|r|/s)^{1/4}\psi$ gives the minimized order-parameter contribution $f_{\rm eq}=|r|^{3/2}s^{-1/2}\Phi_\sigma(X,Y)$, where $\sigma=\operatorname{sgn}r$, $X=q/(|r|s)^{1/2}$ and $Y=hs^{1/4}/|r|^{5/4}$. 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>.
Back to article page
