Brazovskii model

A Brazovskii model has a quadratic Fourier kernel minimized on a sphere of nonzero wavevector magnitude. The large phase space of soft modes can prevent the Gaussian mass from vanishing and turn a mean-field continuous transition into a fluctuation-induced first-order transition.

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 a self-consistent Gaussian treatment, fluctuations near the soft-mode sphere generate a positive mass shift that diverges as the renormalized mass approaches zero. The disordered phase therefore remains locally stable until its free energy crosses that of a finite-amplitude modulated phase, producing a first-order transition.