Multicritical even Landau potential ($r{\varphi }^2/2+{\lambda }_n{\varphi }^{2n}$)

For $n\ge 2$ and ${\lambda }_n>0$, this Landau free energy has mean-field critical exponents $\beta =1/[2(n-1)]$ and $\alpha =(n-2)/(n-1)$ when $r$ crosses zero linearly with temperature. For $n>2$, obtaining this behavior requires tuning the lower even interactions to zero.