The <Fourier transform> sends each spatial derivative to $iq_i$, so at Gaussian level
$$
\boxed{G(q)=a+\kappa q^2+\gamma q^4.}
$$
For $\kappa<0<\gamma$, minimizing over $s=q^2\geq0$ gives
$$
s_0=-\frac\kappa{2\gamma},
\qquad
\boxed{q_0=\sqrt{-\frac\kappa{2\gamma}}.}
$$
The minimum kernel is $G(q_0)=a-\kappa^2/(4\gamma)$. Gaussian fluctuations first diverge when this vanishes, so the <nonzero-wavevector soft-mode sphere> becomes unstable at
$$
\boxed{a_c=\frac{\kappa^2}{4\gamma}.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 344 / 2 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-344/2/b/solution