$$\begin{split}
\mathcal{F}= &\frac 1 2 \int \frac{d^d q}{(2\pi)^d} \phi(\vec q)\phi(-\vec q) \left[ m^2+ (|\vec{q}|-q_0)^2\right]\\
&+ \frac{\lambda}{4!}\int \frac{d^d q_1}{(2\pi)^d}\frac{d^d q_2}{(2\pi)^d}\frac{d^d q_3}{(2\pi)^d} \phi(\vec q_1)\phi(\vec q_2)\phi(\vec q_3) \phi(-\vec q_1-\vec q_2 - \vec q_3)\;.
\end{split}$$ Note the unconventional dispersion relation $\omega=m^2+ (|\vec q|-q_0)^2$. It has the Van Hove singularity $d\omega/dq=0$ at $|\vec{q}|=q_0$, which will play an important role in the following.

In the mean field approximation, the system undergoes a phase transition at $m^2 = 0$, and there is a spatially modulated phase for $m^2 < 0$. The phase is characterized by the position dependent expectation value of $\phi(\vec x)$, $$\label{E:phivev}
\left< \phi(\vec x)\right> = 2 a \cos \vec q \cdot \vec x\;,$$ with $|\vec q | = q_0$. The phase transition is of the second order in this approximation.

Let us examine if this picture is modified by thermal fluctuations. The inverse susceptibility is defined by $$M^2 := G^{-1}(|\vec q|=q_0)=m^2 - \Sigma(|\vec q|=q_0)\;,$$ where $\Sigma$ represents thermal loop contributions. In the Hartree approximation, $$\label{E:rcorrected}
M^2 = m^2 + \frac{s \lambda}{M} + \lambda a^2\;,$$ where $s= \pi S_d q_0^{d-1}/(2\pi)^d$ with $S_d$ the area of the $(d-1)$-sphere. There are higher order corrections, but they will not affect the behavior of $M^2$ near $a=0$. The first 