terms in the parenthesis in scales like $\epsilon^4$. Thus, we can set $s=\lambda=1$ without loss of generality if we do not care about an overall factor of $\mathcal{F}$. Figure \[fig:Brazovskii\] illustrates the free energy $\mathcal{F}$ as a function of $m^2$. For $m^2\geq 0$, we have a graph with only a minimum at the origin. As $m^2$ decreases and crosses $-2^{-\frac 2 3} 3 \sim -1.89$, we see two inflection points away from the origin, representing the spatially modulated phase. At this point, the homogeneous phase at $a=0$ still has a lower energy. As we lower $m^2$ further, the energy of the spatially modulated phase becomes lower than that of the homogeneous phase, and the first order phase transition occurs.

![The relation between $m^2$ and $M^2$. Note that for any value of $m^2$, there is always a solution with $M^2>0$.[]{data-label="fig:m2M2"}](m2M2.pdf){width="7cm"}

The crucial point is that the origin $a=0$ remains semi-stable throughout the process since $d^2 \mathcal F / da^2 =2M^2 >0$ at $a=0$. This in turn is due to the fact that has a solution with $M^2 > 0$ for any value of $m^2$ including $m^2 < 0$. This is possible since the second term in the equation is proportional to 