= Solution
For a homogeneous <saddle-point approximation> define $r=a(T-T_c)>0$. With the cubic source coupling, minimizing the <Landau free energy> gives
$$
2rm+2bm^3-B^3=0.
$$
Its curvature $2r+6bm^2$ is positive, so there is a unique <local minimum> and <global minimum>. The solution near $B=0$ has
$$
m(B)=\frac{B^3}{2r}+O(B^9),
$$
because substituting this leading term makes the cubic self-interaction of order $B^9$. Therefore the <cubic source coupling and third-order susceptibility> give
$$
\boxed{\chi^{(1)}=\chi^{(2)}=0,\qquad\chi^{(3)}=\frac{3!}{2r}=\frac3{a(T-T_c)}.}
$$
Thus the first nonzero response diverges like $(T-T_c)^{-1}$ from above. The factor $3!$ is essential: the coefficient of $B^3$ is not itself the third <derivative>. This is susceptibility to the specified cubic source variable, rather than the usual third-order response to a linearly coupled field.
Back to article page
