Cubic source coupling and third-order susceptibility ($f=r{m}^2+b{m}^4/2-B^{3}m⟹{\chi }^{(3)}=3/r$)

For $r,b>0$, minimization of the homogeneous Landau free energy gives $2rm+2b{m}^3=B^3$. The unique minimum near zero has $m=B^3/(2r)+O(B^9)$. Therefore its first two derivatives with respect to $B$ vanish at zero, whereas its third derivative is $3/r$. With $r=a(T-T_c)$ the first nonzero response diverges as $3/[a(T-T_c)]$. This is response to the variable $B$ through a cubic source, so it differs from the conventional third-order coefficient for a linearly coupled magnetic field. The missing factor $3!$ would give an incorrect susceptibility if one confused a series coefficient with a derivative.