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If we expand the argument of $\ln$ in $q_1, q_2$ around $\left(q_1,q_2\right)=\left(0,0\right)$ and set $q_1,q_2$ to zero, we find $$(1+x^2)^2-2x(1-x^2)2=(1+x^2)^2-4x(1-x^2)=(x^2+2x-1)^2$$ which is obviously minimized at the root of $x^2+2x-1$, giving $x_c = \sqrt{2}-1$. Because the model is taken in the $N\rightarrow\infty$ limit, the integral for $F/N$ ...



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