Nonzero-wavevector soft-mode sphere

For $G(q)=a+\kappa q^2+\gamma q^4$ with $\kappa <0<\gamma$, the minimum occurs on the sphere $q=q_0=\sqrt{-\kappa /(2\gamma )}$. The minimum value is $a-{\kappa }^2/(4\gamma )$.

In three dimensions, let $J(q)=r+\gamma (q^2-q_{\ast }^2{)}^2$, with $q_{\ast }>0$ and small $r>0$. Near the soft shell, $J(q)\simeq r+4\gamma q_{\ast }^2(q-q_{\ast }{)}^2$, soThe radial integral uses ${\int }_{-\infty }^{\infty }ds/(r+A{s}^2)=\pi /\sqrt{Ar}$. The $r^{-1/2}$ divergence obstructs the zero-mass limit of a self-consistent Gaussian variational approximation at a fixed nonzero ordering wavevector. This is the mechanism behind the Brazovskii fluctuation-induced first-order transition; Gross, Ignatiev and Chakraborty's study of fluctuation-driven ordering analyzes the corresponding standard Brazovskii model.